592 Parameter Estimation in Siructurai and Reduced Form Models

mators for Functions of Parameters and Structural Coefficients of EconometricModels,” Journal of the American Statistical Association, 74:365, March, (85-193,

Zellner, Arnold and Henri Theil [1962].Estimation of Simultaneous Equations,

hree-Stage Least Squares: Simultancous‘onometrice, Wz), January, 54-78.

FURTHER READINGS

Addison, John T. and W,Stanley Siebert [1979]. The Marketfor Labor: An AnalyticalTreatment, Santa Monica, Calif: Goodyear. See especially Chapter 12, “WageInflation,”pp, 414-464.

Barro, Robert J. [1987], Macroeconomics, Second Edition, NewYork: John Wiley andSons. See especially Chapter 16. “The Interplay between Nominal and Real Vari-

ait Is the Evidence?,” pp. 455-461.de Marchi, Neil and ChristopherL. Gilbert, eds. [1989]. “History and Methodology

of Econometrics,” Special Issue of the Oxfard Economic Papers, 41:1, January. Alively discussion on the historical development of econometrics with specialemphasis on simultaneous equations procedures.

Desai, Meghnad (1976], applied Econometrics, New York: McGraw-Hill, See espe-cially Chapter 7, “Dynamic Simultaneous Equation Models: Wages and Prices.pp. 205-232, and Chapter 8. “Macroeconomic Models: Simulation and PolicyApplications,” pp. 233-268.

Ehrenberg, Ronald G. and Robert S. Smith [1985]. Modern Labor Ecomomies, SecondEdition, Glenview, INL: Scott, Foresman. Sce especially Chapter16, “Inflation andUnemployment,” pp. 524-564.

Feisher, Belton and Thomas Kniesner[1984], Labor Econontics: Theory, Evidenceand Policy, Third Edition, Englewood Cliffs, NJ.: Prentice-Hall. See especiallyChapter 12, “Unemploymentand Wage Inflation,” pp. 462-527

. Robert E., Jr. and Thomus J. Sargent [1981}, eds., Rational Expectations undEconametric Practice, Minneapolis: University of Minnesota Press. A collectionof oft-cited papers dealing with rational expectations.

McCallum, Bennett T, [1989], Monetary Economics: Theoryand Policy, New York:Macmillan. See especially Chapter 9, “Inflation and Unemployment: AlternativeTheories,” pp. 174-200,

Sargent, Thomas J. [1987], Macroeconomic Theory, Second Editivn, Bostun: Aca-demic Press, See especially Chapter 16, “The Phillips Curve.” pp. 438-446.

Tauchen, George, ed. [1990}, Special Issue on “Sabving Nontinear Rational Expecta-tions Models,” Journal of Business and EconomicStatistics, 8:1, January. 1-52.Acollection of eleven articles introducing and comparing alternative computa-tional algorithmsfor solving nonlinear rational expectations models.

Wallis, Kenneth F. [1973], Tupics itt Applied Econometrics, London: Gilishing, Ltd. See especially Chapter 4, “Simultaneous Equation §127.

Lue:

Chapter I

Whether and How MuchWomen Work for Pay:Applications of LimitedDependent VariableProcedures

most ofthe available evidence suggests that female laborsupply, measured either as laborforce participation or as hours of

work, is considerably more wageand propertyincomeelasticthanmatelabor supply.

MARK R KILLINGSWORTH (1983. p 432]

“Oneunexpectedfinding is that working wives in Canada tendtowork fewer hours per year whenpaid moreper hour... Ourresulting uncompensated wageelasticities ofhours of work areshownto be very similar to those reported byother researchers formen.”

ALICE NAKAMURA, MASAO NAKAMURA, and DALLAS CULLEN (1979. p. 787}

“High taxes, sometimes bydiminishing the consumptionaf thetaxed commodities, and sometimes by encouraging smuggling.frequently afford a smaller revenueto government than what mightbe drawnfrom more moderate taxes.”

ADAM SMITH (4776, BookV. p. 835)

_ reasonableestimates ofan aggregatelabor supplyelasticity andofan overall marginal labor tax rateare both low enough to suggestthat broad-based cuts in labor tax rates would not increase[government] reventa

DION FULLERTON(4982. p. 20)

593

$94 Whether and How Much Women Wark for Pay

Manycontemporary public policy debates involve issues concerning thesupply oflabor. For example, controversies surrounding welfare payments,socialsecurity, and the income tax systemoften focus on incentives for work,Even macroeconomic analyses of unemployment and wage rigidity oftentaise microeconomic issues concerning choices between leisure and laborsupply.

Thenotion oflabor supply, however, encompasses several dimensions:the size and demographic composition of the population, the proportion ofthe working-age population that is working for pay or seeking such work

(called the laborforce parti ion rate, LFPR), the numberofhours workedper week or per year, and the quality of labor. In this chapter we focus onfactors that affect the decisions of household members, particularly women,on whetherto work for pay and,if so, for how many hours. The econometrictools that we employincludevariouslimited and censored dependentvariableprocedures.

In the United States since World WarI] the agggregate LFPR was fairlystable to 1968 and then trended upward. In (948, for example, the aggregateLFPR was 58.8%, in 1968 it was 59.6%, but in 1988 it stood at 65.9%. Thistrending pattern in the aggregate LFPR, however, masks two dramatic and

almostoffsetting drifts, those of males and those offemales, For example, in1948, 1968, and 1988 the LFPR rate for females increased from 32.7%to41.6% to 56.1%, while that for males decreased from 86.6% to 80.1% to76.2%."

Labormarket analysts have offered numerous exptanations for growthin the female LFPR.’ Although increases in male real wages and male realincomes since World WarII have provided opportunities for married women10 reduce their commitmentto the laborforce (an incomeeffectJ, this nega-tive effect has apparently been more than offset by improvements in the realwages offered women (both incomeandsubstitution effects) and by increasedavailability of daycare services for children, as well as by the enhancedemploymentpossibilities that are available to women. Moreover, the greater

educational attainment of females not only has resulted in improved real‘wages (see Chapter 5), but may also have changed women's preferencestoward obtaining meaningful work outside the home. These factors, pluswidespread growth in the availability ofbirth control, have undoubtedly con-\ibuted to the substantial decline in birth rates since World War ll, a phe-nomenon that has greatly facilitated increased femate labor force participa-tion. Finally, rising productivity in the household due to labor-savingtechnological progress (such as freezers, microwave ovens, self-defrostingrefrigerators, and automatic dishwashers)hasalso contributed to encouragingwomen to engage in part- and full-time employmentin the labor market,

Theexplanations underlying the gradual decline in the LFPR of malesare somewhatdifferent from those explaining the increased female LFPR andare highlighted when onedisaggregates the male population by age. For maleteenagers there has been little if any change in the LFPR from World WarIlto the present, and for prime-aged males (those in the 25~54 age group) the

Whether and How Much Women Work for Pay $9

aggregate LFPR has beenrelatively stable, decliningslightly from araund 96%or 97% in 1949 to about 94% in 1984,

Thedecline in the overall LFPR of males since World War I is dueprimarily to decreases in the LFPR ofthose in the 55-64 and 65 and over agegroups. While the sharp drop in the LFPR ofmales who are 65 years ofageand over from 47% in 1949 to 16.1% in 1984 might not be surprising, givengrowth in earned benefits available from private pension and U.S.social secu-city programs,there has also been a substantial decrease in the LFPR of malesin thetraditional preretirement years ofages 55-64;in 1949, for example, thisLFPR was 87.5%, while by 1984 i1 had fallen 18.9 percentage points to68.6%?

Econometric research on labor supply has 2 tong history. Much ofthiscesearch has been policy oriented, attempting to measure the eflects on laborsupply ofchangesin transfer programsortheeffecis on tax revenues andlaborsupply of changes in incometax rates. Although the ordinaryJeast squaresestimatorwasused in much early econometric research,several factors causedeconometricians to worry about biases resulting from the use of OLS. Forexample, the domain of the dependentvariable in many regressions—hoursworked—is censored; observed hours worked are never negative, and in manysamples the numberofhours worked is zero for a large numberofindividuals.Further, labor supply data are often truncated in that data on wages are typ-ically unobserved for individuals who are not working.’ Finally, for thoseworking individuals for whom the wageis observed,issues emerge concerningthe simultaneity of wages and hours worked.

To deal with limited dependentvariable, sample selectivity, censoredand truncated data, and otherstatisticalissues that arise in the empirical anat-ysis of labor supply, within thelast two decades, econometricians have devel-oped and implemented a hostofsophisticated econometric techniques.In thischapter we examineseveral of the more commonlimited dependent variableestimators, focusing on the analysis of labor supply, particularly the laborforce decisions of women.

Webegin in Section 11.1 with definitions of variables and 2 briefdis-cussion of common data sources. Then in Section 11.2 we review the ectnomic theory oflabor force participation and labor supply. Here we examineindividual and household labor supply decision making, summarize theeffects on labor supply of changes in aggregate demand,briefly discuss theallocation of time over thelife cycle, and note implications for labor supplyofthe fixed costs of employment.

In Section 11.3 we consider econometric issues and representativeempiricalfindings, using the paradigm of Mark R.Killingsworth [1983] thatdistinguishes first- and second-generation studies of labor supply. We thenassess the sensitivity of empirical results to alternative econometric proce-duresthatrely on differingstatistical and economic assumptions; this discus-sion is drawn in large part from Thomas A, Mroz [1987]. We conclude thissection with an overview ofissues in dynamic labor supply.

Finally, in the hands on portion ofthis chapter we enable you to obtain

596 Whether and How Much Women Work for Pay

first-hand experience with implementing empirically 4 variety of estimationtechniques,including the OLS, instrumentalvariable, logit, probit, Tobit, andgeneralized Tobit (Heckit) procedures. This is accomplished by engaging youin a numberofexercises, each of which uses a portion of Mroz'soriginal 1976Panel Swdy of Income Dynamicsdata set.

DEFINITIONS AND COMMONDATA SOURCES

The supply oflaboris typically defined as the amountofeffort offered by apopulation ofa givensize. In turn, this amount is conveniently decomposedinto four factors: (1) the percentage of the population engaged in or seekinggainful employment,usually called the laborforce participation rate; (2) thenumber of hours people are willing to work per day, per week, or per yearwhile they are in the laborforce;(3) the amountofeffort that people put forthper hour or day while at work; and (4) the level of training and skills thatworkers bring to their jobs. In this chapter we focus attention primarily oncomponents 1 and 2 of labor supply and emphasize the lahor supply ofwomen.’ Not much is known yet concerning component3;° someaspects ofcomponent4 are discussed in Chapter5 of this book.

In practice, researchers have used a wide variety of measures of laborsupply; a survey by Glen Cain and Harold Watts {1973a, Table 9.3], forexample,identified [8 distinct measures of labor supply. liis therefore imper-ative that we devote some attention to measurementissues.’

Statistics on labor force participation rates (LFPR) and numberofhourssupplied are gathered by a number ofprivate and governmental organiza-tions. In terms of LFPR in the United States, official data on mortthly laborforce, employment, and unemploymentstatistics are typically announced bythe Bureau of LaborStatistics on thefirst Friday ofeach month and are drawnfrom the Current Population Survey.” This survey is based on interviews con-ducted by an employee of the U.S. Bureau of the Census during the weekfollowing the twelfth day of the previous month—called the reference week.The sampleconsists of people in about 60,000 households in over 700 sepirate geographic areas. Households are chosen to reflect the principal characteristics of the population (e.g., age, gender, race, ethnicity, and marital sta-

tus), as taken from the most recent decennial census. The same household isincluded in the survey for four consecutive months, is removed from the sur-vey for eight months, and thenis included again for a final four consecutivemonths,

As currently defined in the United States, the /abor forceis limited tothose individuals 16 years of age and older whoare notinstitutionalized inprisons, mentalinstitutions, and the like and who either are employed, aresearchingfor gainful employment,or are unemployed. Anylabor supplied byindividuals under age 16 is ignored, presumably because such an amountissmall. Note that students aged 16 or over are counted as being in the labor

11.1 Definitions and Common Data Sources 597

force, and manyin fact participate on a part-time or part-year basis. Separatedata are kept on the civilian and total laborforces; the latter adds residentmilitary personnelto thecivilian labor force. For our purposesin this chapterthe focus of the economic theory and econometric analyses of labor supplywill be confinedto the civilian portion of the laborforce.”

The employed are defined as those individuals who worked one hour ormore for wages orsalary during the reference week or did 15 hours or moreof unpaid work in a family business or farm. Individuals who were absentfrom work because ofvacation,illness, inclement weather,or strikes and lock-outs are also counted as employed but are included in the separate subcate-gory “with a job but not at work.” The wnemployed are those who are onlayoff from a job, who have nojob but have looked for work duringthe pre-ceding four weeks and were available for work during the reference week, orwho are waiting to report to 4 new job within the next 30 days.

Onthebasis of questions asked by the interviewer the laborforce status

of eachcivilian aged 16 or overis determined in an ati-or-nothing manner. Apersonis either in or outofthe laborforce; bydefinition, one cannotbe partlyin the labor force and partly out. Thustheentire civilian noninstitutionalizedpopulation P is decomposed into employed, £, unemployed, L/N, and out ofthe labor force. O, where

Pok+ UN+0 CLL)

The laborforce LFis limited to those who are employed or unemployed,

LFoE + UN (1.2)

while the laborforce participation rate, in percentages, is measuredas

LFPR = 100 (LF/P) (1.3)

The unemploymentrate UR, also measured in percentages,is

UR & 100 (UN/LF) (1.4)

An importantfocusofthis chapteris the LFPR of women.Asis seen inTable 11.1, this LFPR varies considerably by age and has increased dramat-ically over timeforall age groups. In 1988, for example, the LFPR for womenaged 25-34 was 72.7%, more than twice the 34.9% of 33 years earlier. Sharpincreases in the LFPR of women aged 20-24 and 35~44also occurred atter1955 (especially after 1965), while the LFPR of womenaged 45~54 increasedmore than 50% from 43.8% in 1955 10 69.0% in 1988.

In terms of hours oflabor supplied, in practice most empirical laboreconomists use hours of employmentas a measure of hours supplied. How-ever,this employment measure might notreflect the numberofhours an indi-vidualis willing to supply at a given wagerate. Thisraises severalissues. Manyemployers specify standard weekly hours of work equa) to, say, 35, 37.5, or40 hours per weck, but to someextent, employces can choose among employ-

598 Whether and How Much Women Work for Pay

Table 11.4 U.S. Female LaborForceParticipation, 1955-4985,Women Aged 20-54

Year 20-24 25-34 35-44 45-541955 45.9 34.9 4t6 43.8

1960 46.1 36.0 43.4 49.8

1965 49.9 38.5 46.1 50.9

1970 «$7.7 45.0 Sul 54.4

1975 64.1 54.6 55.8 54.6

1980 69.0 65.4 65,8 59.9

1985 1B 9 TB 64.4

i983 7: 72.7 75.2 69.0

Susaver Reproduced from Duniel $. Hamermesh and AlbertRees, The Ecurumas of Work und fay. Fourth Edition, NewYork: Harper & Raw, 1988, Table 1,2, (948 dats are from theJanuary 1989 issue of US. Hureau of Labor Statistics, Zenploy=‘mend anut Earnings

ers with differing standard workweeks. Employers also vary in terms of possibilities for part-time and overtime work. Furthermore,firms with the same40-hourstandard workweek often offer different yearly hours of work becauseofvarying layoff, vacation, and holiday policies. Finally, occupational choicehas a working-hours dimension. The weekly or yearly hours of work maydiffer if one chooses to becomean elementary school teacher rather than a real-tor, an assembly line worker rather than a funeral director, or a New YorkCity lawyer rather than a small-town one. Hence, although there are con-straints, employees canstill exercise a substantial choice over hourg at work.

ft is also useful to distinguish hours at work from hours paid. SinceWorld War il in the United States, the wedge between hours at work andhours paid has increased, owing to the growing importance of paid holidays,paid vacations, and other paid leaves. The U.S. Chamber of Commerce hasestimated, for example, that paid time notat work as a percentoftotal wageand salary costs increased from 5.9% in 1955 to 7.3%, 9.4% and 10.3% in1965, 1975, and 1985, respectively." In this chapter, unless otherwise speci-fied, the measure oflabor supply used refers to hours at work, not hours puid.

One important measurementissue concerns the source of the hours atwork data. The monthiy Current Population Survey asks the respondent tolist not only his or her hours at work during the reference week, but also thehours ofworkfor all other adult household members. Inability to recall one’sown hours the previous week or those of other household members canresultin serious measurement error. Moreover, the self-employed and many sala-ried workers might not accurately recoltect when they were working and whenthey were not. Measures ofhours at work are thereforelikely to includeerrors,

Analternative governmental source of data on hours at work is based‘on surveysoffirms and establishments rather than of households. Data pro-

11.2 Economic Theory of Labor Force Participation $99

vided by business firms and establishments for all manufacturing firms arecollected by the U.S. Census Bureau once every five or six years (CensusofManufactures), while a smalier sample of manufacturing firmsis surveyedeach year (Annual Survey ofManufactures)."' Data from a samipteofall firmsand establishments, including those in nonmanufacturing sectors, are gath-ered on a monthly basis and are published in summary form by the Bureauof LaborStatistics (the Employment and Earnings Surveys). Although laboreconomists have long advocated that the government match data from itshousehold surveys with those from establishments, such data have not yetbeen developed, in part because ofissues of confidentiality and budgetaryconsiderations.”

A numberof nongovernmentalinstitutions produce data on individualand household labor supply. Two that are ofparticular importance to empir-ical labor economists are the Center for Human Resource Research at OhioState University, which conducts National Longitudina! Surveys (NLS), oftencalled the Parnes data, and the Survey Research Center at the University ofMichigan, which provides data from its Survey of Working Conditions. Qual-ity of Employment Survey, and the Panel Study of Income Dynamics{PSID).” The NLS andthe PSID have becomeincreasingly useful to empir-ical researchers, owing in part to the fact that both generate panet data, thatis, they interview the same households yearafter year. Moreover, in the PSID,attempts are madeto distinguish desired hours oflabor supply from hours atwork.

Although labor force participation and hours of labor supply are the

principalfocus ofthis chapter, other variables are also typically used in empir-ical studies explaining labor supply. Such variables include the wage rate,measures of experience, the educational attainment of the individual, andnonlabor (often called property) income. Measurement ofthese variablesintroduces additional problems of accuracy andinterpretation. For a discus-sion of measurementissues for the wage and education variables, see Section5.2.A of Chapter 5 in this book; the measurement of property income andother variables in the context oflabor supply is discussed by, amongothers,MarkKillingsworth (1983, pp. 87-100}.

{t is important that difficulties in the measurementoflabor supply andits determinants be borne in mind when working through the economic the-

ory and econometric analyses surveyed in this chapter. As we shall sec, mea-surementissues turn out to be very important.

THE ECONOMIC THEORY OF LABOR FORCEPARTICIPATION AND LABOR SUPPLY

We now briefly review the economic theory underlying labor supply. Webegin by examining the individuat’s participation decision and the hours sup-plied by the individual, given that participation. This framework is then

600 Whether and How Much Women Work for Pay

extended to encompass the household rather than the individual as a decisionunit. Hypotheses concerningthe effects ofdemandshifts, knownas the addedand discouraged worker hypotheses, are addressed. A more general treatmentofthe allocation of time, both static and overthelife cycle, is presented, fol-lowed bya discussion of implicationsoffixed costs of employment, By neces-sity this review of the economic theory underlying labor supplyis brief; morethorough anddetailed analyses are found in the labor economics textbookslisted underthe heading Further Readings at the end ofthis chapter, and espe-cially in Mark Killingsworth [1983, Chapters 2, 5 and 6).

41.2.A Individual Labor Supply

‘The neoclassical model of the supply oflaboris in essence an application ofthe theory of consumer behavior. The individual is assumed to allocate timeto market work and to nonmarketable activities; the latterare typically called,erroneously, “leisure.” Utility is maximized by choosing combinations ofgoods andleisure hours subject to time, price, and income constraints.’

More specifically, let the individual's preferences be represented by atwice differentiable utility function / = U(G. L), indicating the utility (UV)obtained from consuming alternative quantities ofgoods (G) andleisure (L).{Note: Hereafter, U refers toutility, not to the unemployed; further, L isleisure, not labor.) The marginalutilities of G and of L (AfU, = 8U/aG andMU, = aUsaL) are both assumed to be positive, and the utility function isconcave in G and L, implying that 2U/8G", PU/IL? < 0, and eUs8GAL >

0.

Along anindividual's indifference curve, alternative combinations of Land G generate the samelevelof satisfaction. In Fig. 11.1, four indifferencecurves are drawn, Jy, /,, 42, and J;, each corresponding with successivelygreaterlevels of U.

‘Two important properties ofthe indifference curve are its slope anditsshape. Theslope is derived as follows. The total differential oftheutility func-tion UG, £)is

(@U/AG)dG + (@U/AL)dL = dU (15)

Along given indifference curve, dU’ = 0. Substituting dU= 0 into Eq.(11.5)and rearrangingyield the slope ofindifference curves in Fig. 11,1, dG/dL, alsoknownasthe negative of the marginalrate of substitution ofleisure for con-sumergoods and denoted here as — MRSz6.

Oers, (116)

Since by assumption, A/U, and MU,are both positive, indifference curvesslope downward. Further, concavity implies that indifference curves are con-vex to the origin. Hence, although it is possible to substitute £ for G and holdutility fixed, the greater the ratio of L to G, the greater the marginal amountof L required to compensate for giving up a marginal amountof G.

11.2 Economic Theory of Labor Force Pat

Goods6c

hy

Slope = ~ P/F,

HoursofLeisure L—

Figure 14.4 indiflerence Curves and the Budget Constraint

Since indifference curves that are farther away from the origin representsuccessively higher levels of utility, the utility-maximizing individual willchoose the highest possible indifference curve. given his or her budget

constraint.Three factors affect the budget constraint—prices. nonlabor income,

and time. First, Ict the unit price of goods be P,, and let the exogenous andconstant wage rate be P,; thus the individual's real wage is P,/P,. Second, tet

the individual's nonlabor or property income be denoted as ¥: in terms ofconsumption goods the real amount of nonlabor income is V//P,. Third,because only a finite number of hours 7 are available in any time period,leisure (nonmarket) hours £ plus hours devoted to market work H mustexhaust 7. that is, L + H = T. Further, labor income equals the productPH, and thereal labor income forgone by choosing one more unitof leisureinstead of working equals P,/P..

If it is further assumed thatthe individual spends all of his or her avail-able income, the above three factors imply the budget constraint

YoPH+V = P(T-L)+¥ = PG (1.7)

where ¥, total money income,is the sum oflabor and nonlabor income. —

Equation (11.7) is often rewritten in two ways. To emphasize the notion

602) Whether and How Much Women Work for Pay

that an individual's “full income” F is spent on goods and leisure, GaryBecker [1965] has added P,L to both sides of Eq. (11.7), obtaining

FoY¥t+PbsPiG+PLorhT+ ¥ (11.8)

In this formulation the full income budget constraint consists ofthe totalamountoftime available, T. evaluated at the constant wagerate P,. plus non-

labor income F. This full incomeF is thentotally expended onleisure (PL)and on goods (PG).

Alternatively, to facilitate graphical analysis, one can rewrite Eq. (11.7)in termsofreal income,

G -T+—|- Tek (19)Py Py Py

When Eg.(11.9) is graphed as in Fig. #1.1, the budget line embodying theincomeconstraintis line 8’B, with intercept equal to ((P,/P.) - T + ¥/Pe)and slope cqual to —(P,/P,). Note that even if L = T (whenall time isdevoted to feisure and none to market work), the budget line 8°8 docs not

cross the horizontal axis unless nonlabor income¥is zero.Maximization of utility subject to the budget constraint Eq. (11.9)

involves choosing the set of G and L thatis feasible (on the budget linc) andon the highest indifference curve that touches the budget line 2’8. In

Fig. 11.1, such a point is at P, where the slope of the indifference curve4, (—MRS,,) equals the slope of the budget line — P,/P,. At this point theindividual purchases goods OG’. chooses leisure OH’, and supplies hours oflabor #7’T to the market.

More formally, the individual solves his or her maximizatioa problemby maximizing U = U(G, L) subject to the budget constraint 2,G = P,(f — L) + ¥. This is done by setting up the Lagrangian function,

W = UG. L)— MPG — PAT -— Ly - 14 (1.10)

takingfirst partial derivatives ofW with respect to G and L,setting them equal

to zero, and then solving. This yields

aval _ MU:aujaG MU,

Notice that according to Eq. (11.11), at the point at whichutility is maxi-mized, the AfRS,,; (the negative of which. by Eq. (11.6). equals the slope oflhe indifference curve) is equal to the real wage rate P,/P,; (the negative ofwhich, by Eq, (11.9), equals the slope of the budget line 8’B).

The way in which Fig. 11.1 has been drawn allows the individual tomaximizehis or herutility at an interior solution point P where L << T and#H > 0, that is, where the individual participates in the laborforce with non-zero H. This need notbe thecase and in fact is critical to understanding thelaborforce participation decision. To see this, assume that the wagerate the

tee *| ra

PB

= MRSiy = R aL

11.2 Economic Theoryof Labor Foree Participation 603

individual would earn were he or she to work in the markel was ?;, which is

tower than P,. Further, let the individual have the same aonlabor income 7

and preferencesas before. In such a case the budget constraintfacing the indi-

vidual would be PuG = Pi(T — L) + V, which is drawnin Fig. 11.1 as the

Hatter budget line 8” 8. Now the highest indifference curve the individual

could attain, given his or her preferences and the budget constraint 8” B. is

1,. Al point # the indifference curve /, touches the budget line B’#, where L

= Tand H = 0, that is, at the point at which the individualis « nonpartici-

in thelaborforce, spendingall of his or hertimein leisure (nonmarket

ies). Anyhigherindifference curveis simply unattainable toindividuals

with such preferences and budget constraints.

Points such as B represent a cornersolution to the individual’s utility

maximization problem, rather than an interior solution. Note in particular

that at the corner solution 8, instead of Eq. (11.11) holding, whereMRS,, =

p./P,, the slope of the indifference curveis steeper than that of the budget

line, implying that MRS, > P,/P.. This suggests thatthe laborforce partic.

ipation decision can be simply envisaged as corresponding to whether the

individual'sutility maximization problem,given budget constraints, yields a

corner solution or an interior solution. Specifically, if at the solution point,

MRS, P,{P,, then H > Qand L < T—aninteriorsolution occurs; but

if instead at the solution point, AARS,., > P./P,, then H 2 Qand L = T—acornersolution obtains.

An important concept underlying the labor force participation decision

is the notion ofthe reservation wage. At point # the slope ofthe indifference

curve /,, ~MRS;,; indicates how much exira earnings the individual would

require to be induced to give up one unit ofleisure, when he or she is not

working at all, This amountof extra earnings is called the reser lion wage.

which we denote as »*.' Note that as drawn in Fig. 11.1 budget line

B’B,thereservation wage ¥* is greater than the market wage P,. that is, the

extra satisfaction from an hourofleisureis greater than the wage rate. How-

ever, if the wage rate rose so that the budgetfine rotated upward from B°B

to BB, then at some point the wage rate would exceed the reservation wage,

resulting in a positive labor force supply. Hence the condition for positive

labor force participation is simply that P, > w*.

Several important implications of this economic theory of labor force

participation are worth noting. First, for individuals with identical reservation

wages those with higher (potential) wage rates are more likely to be in the

laborfarce. Second,for individuals with identical potential wage rates those

with lower reservation wages are more likely to be

in

the labor force. Since

“workaholics” have lower reservation wages than, say, avid hobbyists, given

identical potential wage rates, workaholics are morclikely to be in the labor

force. Similarly, womenwith very young children at home and withonlyvery

expensive daycare possibilities are likely to have higher reservation wages than

single, career-oriented women with no children; other things being equal, we

would expect the fatter to have higher laborforce participation rates. Note

604° Whether and How Much Women Work for Pay

that such differences in preferences among individuals are reflected in theshape andslope oftheir indiflerence curves. Moreover,for a given individualthe shape ofthe indifference curve may change at variouspoints during hisorherlife cycle.

Thetheoretical framework presented above can be generalized lo casesin which only several options for hours at work are available to employees,for example, one could only work zero, 20, or 40 hours per week. In suchsituations the optimal cornersolution is the largest amountof hours of workfor which the market wageis greater than or equalto the reservation wage.

We now consider the effects on labor supply of changes in nonlaborincomeandin the wage rate. Suppose that the nonlabor incomeofthe indi-vidual increased from TB to TC in Fig. 11.1 but that the wage rate remained.constantat P,. This results in a new budgetline C’C parallelto the old budgetline B’B. Relative to the initial equilibrium at P, the increase in nonlaborincome now permits the utility-maximizing individual to move to a higherindifference curve, 7,, tangent to the new budget line C’C at point P’, wherethe hours ofleisure consumed increases to OH”, the amountoflabor supplieddecreases 10 H” 7, and the amount ofgoods consumed increases to OG". Thisoutward movement from P to P’reflects a pure income effect in which Z andG both increase in response to the increase in nonlabor income. Note, how-ever, that the pure incomeeffect on hours oflabor supplied is negative,reflect-ing the implicit assumption here thatleisure is a normal good.

Matters are not quite as simple when the individual experiences achange in the wage rate, since the response reflects both income and substi-

ttion effects. To see this, consider Fig. 11.2. Let the initial equilibrium be atpoint P, where the /y indifference curve is tangent 10 the origina! budgetfineB’B, sesulting in OG units of goods and O# hours ofleisure cortsumed andHThours oflabor supplied. NowIct the wage rate increase while goodsprices,preferences, and nonlabor income remain unchanged. The increase in thewage rate results in an upwardly rotated new budget line 8”8, tangent to the

higher indifference curve /, at point P’. At this new equilibrium the ntility-maxitaizing individual increases goods consumed from OG to OG’, decreasesthe amountof leisure chosen from OH to OH", and increases hours oflaborsupplied to the market from HT to H‘T.

It is useful to decompose the movement from P to /” into pure income

and compensated substitution effects. The compensated substitution effect isdefined as the response ofthe individual to a change in the wagerate holdingutilityfixed. Hence the compensated substitution effect involves a movementalong the origina! indifference curve Jy. To show this graphically, let usnotionally decrease the individual's nonlabor income by imposing a tax onitsuch that the new notional budget line D’D has the same slope as the newbudget line 8”B (reflecting the higher wage rate) but is just tangent to theoriginal indifference curve /, at point P”. Note that in termsofutility thisnotional budget line D’D just offsets or compensates for the improvement inearning powerthat results from the increased wagerate.

11.2 Economic Theoryof Labor Force Participation 605

HoursofLeisure £ ~ —~

Figure 44.2 Income andSubstitution Effects in Response toa Changein the Wage Rate

Asis shownin Fig. | !.2, the movementfrom Pto P” is the compensatedsubstitution effect: it results in reduced leisure OH” and increased labor sup-ply "7, reflecting the fact that leisure has become more expensive as thewage rate has increased. Once this compensated substitution effect has beenisolated, the pure incomeeffect(with no prices changing)is the residual move-ment from P" to P’, with leisure increasing from OH” to OH" andlabor sup-plied to the market falling from H’T to H’T. Incidentally, although the

nomenclature can be confusing,it is worth noting that in muchofthelitera-ture the total movement from P to P’ in response to an increase in the wagetate is called the uncompensated or gross substitution effect. Hence the

uncompensated or gross substitution effect is the sum of the compensatedsubstitution effect and the pure incomeeffect.

An important implication of the above graphical analysis is that inresponse to an increase in the wage rate, the utility-maximizing individualrespondsin two different ways: The uncompensated substitution effect resultsin more labor supply andless leisure, while the pure incomeeffect results inless labor supply and more leisure. Figure 11.2 has been drawn in such a waythat the positive compensated substitution effect on labor supply dominates

606 Whether and How Much Women Work for Pay

the negative pure incomeeffect, but this need not be the case. Had we drawnthe indifference curves with different shapes (but still downward sloping andconvex to the origin}, we could have obtained a new equitibrium in which thepure incomeeffect dominated the compensated substitution effect. [t is there-fore important to note that even whenleisure is a normal good, on the basisof economictheory, one cannot determinea priori whether an increase in thewage rate, ceteris paribus, results in increased or reduced hours oflabor sup-

Dlied; this depends on the relative magnitudes of the compensated substitu-tion and pure income effects, which in turn are empirical, not theoreticalissues. In termsoftraditionallabor supply curves, therefore, one needs empir-ical evidence to determine whether the labor supply curve is upward slopingor backward bending,

Two further remarks are worth noting. First, the above analysis ofincomeandsubstitution effects has assumed thatinterior solutions occur bothbefore and after the wage rate increase, thatis, it is conditional on positivelabor force participation. Although it is not demonstrated here, it can beshownthat if the original solution was a corner one with no participation,then a sufficiently large wageincrease, ceteriy paribus, could result in positivelabor force participation, as could a reduction in nenlabor income, ceterisparibus, Hence participation decisions can also be assessed in terms of income

and substitutioneffects.Second, following the traditional theory of consumer demand as, for

example, in R. G. D. Alien [1938] and Angus Deaton and John M. Muell-bauer [1980], one can write the gross, compensated substitution and pure

incomeeffects of a wage rate change in termsof the calculus and in elasticityform. In particular, this results in the well-known Slutsky equation,

aH aH auoa) ae of LBPlam” aPleew * ay (142)

where Yis money income. Thefirst term on theright-handside of Eq. (1 1.12)is the compensated substitution effect with utitity held constant, and the sec-ond term is the pure incomeeffect. If one multiplies the entire expression inEq. (11.1) by P,/f/ and then multiplies and divides the incomeeffect by Y,

oneobtains a Slutsky relation in elasticity form.

oH P, oH P, Hoon ¥So aS + ses 11.13;OP, Hi yon OP, Hlynt Y or a »

which in turn can be rewritten as

in, = Cue + 5° tee (14)

where 5S;is the share of labor in total money income and where the g and ¢superscripts on the substitution elasticities refer to gross and compensatedchanges, respectively.’*

11.2 Economic Theoryof Labor Force Participation 607

14.2.8 Household Labor Supply

Thetheoretical framework just presented envisages an individual as makinglabor supply decisionsin isolation. in fact, however, the labor supply decisionis typically made in the context of decisions taken by other members of thehousehold or family within which the individualresides. Kittingsworth (1983,Chapter 2] distinguishes three approaches that link family membership tolabor supply.

In the “male chauvinist” modelit is assumed that, when making herlabor supply decisions, the wife views her husband's earnings as a form ofProperty or nonlabor income, whereas the husband decides on his labor sup-ply solety on the basis of his own wage and the family's actual nonlaborincome,withoutreference to his wife's labor supply decisions."’ The onlydif:ference in this model compared to that considered in Section 11.2.4 is thatin the male chauvinist framework the property income V relevant to thewife's labor supply decision is assumed to include the husband’slabor incomeas well as all nonlabor income,"

A second approach assumes the existence of a family or householdaggregate utility function U = U(G, £,, L,,..., £,), where £,is the leisureconsumed bythe ith individual in the family. This family utility function ismaximized subject to a family budget constraint."

The family utility approach has becomea familiar one in labor supplyanalyses, owing in part to the fact that manyof the well-known comparativestatics results of the individual utility maximization framework carry overwith lite adaptation. An important generalization that is afforded by thefamily utility function approachis that there now are four substitutioneffects.In particular, not only do both family members have own-substitutioneffects(the response of the ith family member's labor supply to a change in his orher own wagerate), but two cross-substitutioneffects also occur, involving theith family member's labor supply response to a change in the wagerate of thejth family memberand vice versa.

Because ofthe assumed existence of an aggregate familyutility function,the compensated cross-substitution effect of a change in the ith individual's‘wagerate on the jth member's iabor supply must equal the effect of the jthmember's wage rate on the ith member's labor supply. Further, although

these compensated cross-substitution effects must be equal in magnitude,their signs can beeither positive (indicating substitutability) or negative {com-plementarity). However, because the pure incomeeffects on the two familymembers need not be identical, the gross (uncompensated) cross-substitutioneffects are not necessarily equal. Hence the gross effect of a change in thewife's wage rate on the husband's labor supply need not equalthegrosseffectof a changein the husband’s wagerate on the wife’s labor supply.

A special case of the family utility function framework, considered indetail by Malcolm S. Cohen, Samuel A. Rea, and Robert I. Lerman [1970]and by Orley Ashenfelter and James J. Heckman (1974), occurs when the

608 Whether and How Much Women Work for Pay

compensated cross-substitution effects equa! zero forall family members. Inthis instance a change in the ith member's wage rate imparts only a pureincome effect on the labor supply of the jth individual. This implies that thelabor supply function ofthe jth family member depends only on his or berwage rate and on the sum ofall other family members’ labor incomes plusfamily nonlabor income.

Becauseofits analytical simplicity, the family utility function approachis commonin empirical labor supply studies. It is worth noting, however, thatthis framework has a number of drawbacks. Because the aggregate utilityfunction implies that the family derives utility from consumption as a whole,the distribution ofgoods consumption does not matter; while this might makesense for family “public goods” such as heat andlight, it is obviously ques-tionable for “private goods” such as food, clothing, and entertainment. More-over, the familyutility function approachis silent concerning the process thatactually generates an aggregate utility function that is mutually agreeable toall family members.

Owing in part to such considerations, a variety of alternative frame-works have been developed thattreat labor supply within a householdsetting,yet emphasize the individual behavior of family members. Jane Leuthold[1968], for example, has presented a model in which individuals’ reactions toother family members’ labor supplies and wage rates are analogous to theCournot or Stackelberg reaction curves of duopolists or oligopolists. As inmodern game theory, such a framework requires stringent assumptions toensure the existence and stability of a unique equilibrium.”

Bargaining models of family members’ behavi fe another example ofrecent developments in “family economics” emph: 2 the individual util-ities approach. As formulated by, among others, Marilyn E. Manger and Mur-ray Brown (1979, 1980] and Marjorie B. McElroy and Mary Jean Horney[1981], family members arrive at decisions about labor supply and consump-lion spending through a complex process of bargaining. Without stringentassumptions, such models seldom generate closed form solutions that areamenable to empirical implementation, As with the aggregate utility functionapproach,in these bargaining models, the total effects of changes in, say, thehusband's wage on the wife’s labor supply need not equalthe total effects ofchangesin the wife’s wage on the husband's labor supply.” However, unlike

y utility function framework,in the bargaining models, both theleveln of consumption among household members can affect the

labor supply of each member. Unfortunately, these bargaining models havenot, as yet, generated any discriminating testable restrictions.

Manitalstatus is clearly important in the modeling of houschold laborsupply. In this context, recent work by Gary Becker (1974, 5981, 1988] secksto endogenizejointly the marital status, consumption, and labor supply deci-sions of individuals, but at this point, no fully empirically implementablemodelhas been developed.

11.2 Economic Theory of Labor Furee Pas

41.2.C Effects on Labor Supply of ChangesinAggregate Demand

‘Therelative importanceof incomeandsubstitution effects hasalso played an

importantrole in assessing the effects on labor supply ofcyclical changes inaggregate demand. Here the important issue concerns the net impact onLFPR ofchanges in aggregate demand,for example, of recessions or booms.In the 1930s an hypothesis emerged that when the usual breadwinner became

unemployed, additional members of the household enter the labor force inorder to maintain the family’s income,the implication being that labor forceParticipation rates would rise along with unemployment. This wascalled theadded-worker hypothesis.” A counter view, known as the discouraged-workerAypothesis, holds that when unemploymentincreases, searching for employ-ment becomesso disheartening that some of the unemployed give up andwithdraw from the labor force, while others who would ordinarily enter thelaborforce choose not to do so.

These two hypotheses have very different implications for the measure-ment and interpretation of the aggregate unemployment rate. According tothe added-worker hypothesis, the labor force is above the long-term trend dur-

ing recessions. implying that the unemploymentrate is overestimated in thesense that creating one new job might reduce the number of unemployed bytwo people. Onthe other hand, accordingto the discouraged-worker hypoth-esis, the size of the aggregate laborforce varies in the samedirection as aggre-gate demandrather than in the opposite direction. Since unemploymentismeasured as the number of people who report themselves as secking work.

the discouraged-worker hypothesis suggests that unemploymentis underesti-mated, since one should add to those enumerated as unemployed another

group, the hidden unemployed, who would be searching for employmentwere the search notas likely to befruitless. Which of these two hypothesesismost valid has been viewed as having importantpoticy implications, since thetheories can differ sharply in implying whether increases in the measuredunemployment rate might warrant expansionary or contractionary mucro-

economic policy responses.In terms of economic theory the added-worker hypothesis essentially

invoives a houschold incomeeffect. since it implies that the reduction inaggregate demand lowers the income of those family members whoare in thelabor force (those who lose their jobs or are put on shorter workweeks),

thereby reducing the nonlabor incomeavailable to family members who arenot currently in the labor force. Given this reduction in nonlabor income,reservation wages drop, the demandfor leisure (a normal good)falls, andlabor supply increases,

In contrast, the discouraged-worker hypothesis involves a substitutioneffect for wo reasons. First, when aggregate demandforlaborfalls, real wagerates will often also tend to fall, making leisure hours less expensive. Second,

610 Whether and How Much Women Work for Pay

the probability of successfully finding a job with a given wagefalls during a

recession.Ifone defines the expected wage from job search as the product of

the wage rate of people who have the job and the probability of finding the

job, then both components of the expected wage from job search may fall

‘during recessionaryperiods, thereby towering theattractiveness ofjob search

and reducing the opportunity costofleisure hours.

It is of course possible that both the added-worker (income)effect and

the discouraged-worker (substitution) effect coexist; the added and discour-

aged workers might well be different groups of people. Which group tends to

dominate is again an empirical issue. Although rescarchers such as Jacob

Mincer[1962], Shelly Lundberg, [1985] and Olivia S. Mitchell [1980] have

obtained results suggesting a significant number of added workers, many of

them married women, the added-worker impact emerges only from families

whose normal breadwinnerloses a job, and this incomeeffect tends to occur

for only a relatively smalt portion ofthe labor force (unemploymentrates in

the United States seldom go above 10%). By contrast,the fall in expected real

wages that occurs in recessionsaffects nearly every household. and so itis not

surprising that empirical researchers have generally concluded that the dis-

couraged-worker(substitution) effect is considerably larger than the added-

worker effect2> Hence, other things being equal, the labor force tends to

shrink during recessions and grow during periods of economicrecovery.”

44.2.D The Allocation of Time: A More GeneralTreatment

The theoretical framework that we havediscussed so far assumes that all time

is allocatedeither to market workorte leisure. This is clearly unrdalistic, since

much nonmarket time is spent in “producing” consumable services at home,

using both time and goods. For example, the preparation of meals at home

consumesboth time and food; this timeis neither leisure nor market work.fn an important paper, Gary Becker [1965] called such activities nonmarket

work, We now briefly examinefactors that affect market labor supply when

the choiceset ofalternative activities is expanded to also include nonmarket

work.Becker’sessential insight was that neither time nor consumer goodsare

consumed by themselves, since timeis usually not enjoyed without goods,

and virtually all goods require time in order to be consumed. This implies

that consumption decisions of individuals and households are made from a

menuofalternative activities with varying amounts of both goods and time

inputs. The total price of any consumption activity is the sum ofits market

goods price plus the value of time required to produce or consume it, Other

things being equal, increase in the wage rate raises the relative price of

time-intensive activities, induces substitution against them, and thereby

mightresult in an increase in hours worked. Note that in this broader frame-

41.2 Economic Theory of Labor Force Participation 611

work the individual's supply oflabor hours is determined jointly with con-sumption;it no longerinvolves just a tabor-leisure tradeoff.

More formally, a model with such time consumption costs begins witha utility function U = U(Z,, Z;,.... Zn). where the Z, are household activ.ities depending on both goods and time inputs.” Households are assumed tocombine time and market goods to produce the “basic commodities” Z,, sub-Ject to household production functions Z, ZAG,.4).i= 1 2a), wherethe G, and ¢, are goods and time inputs. Beckerspecifies that the household

production function is of the fixed coefficient type,

Z=Gfa and Z = t/b, Lt)

where the 4, and 6, are units of goods and units of time per unit ofactivity,respectively, Since the time and money income constraints imply that

T=D,+H and TPG =P+ (LI6): 7

one can rewrite the full income constraint (11.8}—the income that would berealized if T = H—as

Dd PaG, + Phe =H PP+ hak (IL ET}

When Eq. (11.15) is substituted in Eq. (11.17), one obtains the

expression

a+ PY Zb = xy Z,

(11.18)

where

mos Poa, + Pb, is... mt 11.19)

Notethat 2,is the full cost of consuming or producing activity Z,, includingboth the goods cosi and forgone-carnings components.

If one maximizes the utility function subject to the budget constraint(11.19), one obtains from thefirst-order conditions the equilibrium relation

MU,— igedc.ym as :MU, j miei (11.20) According to Eq. (11,20), the utility-maximizing individual consumes/pro-duces activities andj such that the marginalrate ofsubstitution between anytwo activitics just equals the ratio oftheirfull costs, including timecosts.

To see the uscfulness ofthis more general (ramework,considertheeffectofa wage increase on an individual’s marketlabor supply. The incomeeffectwill bring about an increase in the consumption ofall normal commodities;

612) Whether and How Much Women Work for Pay

if the time-intensive nonmarket commodities are normal, the income effectwill cause less time to be devoted 10 market work and more to be allocated tononmarketactivities. In termsof substitution effects, however, note that thewage increase will raise the cost of all commodities (assuming that eachrequires a nonzero amountoftime) butwill raise the relative cost of the moretime-intensive ones. This results in a shift away from more time-intensivetoward less time-intensive nonmarket activities and toward market laborsupply.

Within the household, for example, the increased wage mightresult inpurchases of time-saving devices such as a microwave oven and a computer-activated water-sprinkling system. Note that although the income and substi-tution effects on an individual’s labor supply are therefore essentially similarin sign to those derived in the earlier, simplercase, in this more general frame-

work, insights can be gained into factors aflecting the division of nonmarkettime into more andless time-intensive activities. Further, to the extent thattechnological advances result in increasing the labor productivity of homeactivities, such advances could be viewed as facilitating increased marketlabor supply. Hence, although this more general treatment oftime probablyhas principal implications for “home economics,” it also has a nontrivial

impact on market labor supply because of interactions among market andnonmarket activities,

The most important contributionofthe more general treatmentoftimeto understanding family labor supply behavior, however, concerns householddecisions regarding family size and child-rearing. Since child-rearingis clearlya time-intensive process, the full costs of having children depend in part onthe opportunity costs of the parents’ time. Although beyondthescope ofthischapter, it is worth noting that a substantialliterature exists relating fertilityrates to, among other things, wage rates of married women and incomes ofmarried men; not surprisingly, empirical evidence suggests that other thingsbeing equal, married women with higher earnings potentials appear to havefewerchildren and participate to a greater extent in the laborforce.» A num-berofstudies have also been made ofhowone might valuate the contributionof the nonmarket working time of married women,including that portionallocated to child-rearing; on this, see, for example, Reuben Gronau [1973a.b, c. 1977, 1986].

Muchrecentresearch in the labor supply literature has involved an evenmore general treatment oftime, analyzing the allocation oftime overthelifecycle.” Significantly, once one considers the longerlife cycle horizon, deci-sions involving education, training, hours of labor supplied at different ages,and retirementail become interdependent and 10 a certain extent endoge-nous, What may have been exogenous or predetermined in a static contextRow becomes in part endogenous. Someofthese issues are discussed brieflyin Chapter5, particularly in Section §.1. For our present purposes, however,it is worth noting that not only do married women currently appear to behaving smaller families than in previous decades, but the time duration

(2.2 Economic Theory of Labor Force Pai

pation 613

between children has also declined, implying that the time devoted to child-rearing has been compressed, while that allocated to market laborsupply hasexpanded.

14.2.E traplications of Costs of Employment

In general,it is costly to hold a job. Afeneycosts mustbe incurred by workers,to pay for transportation to work, purchase appropriate work clothing, buymeals away from home, and, in manycases, pay for babysitting, daycare, orotherchild care services. Timecosts are also typically incurred by workers,since most people take timein traveling from home 10 job; further, childrenmust often be transported to child care locations before the Parentcan travelto work. There are also nonpecuniary costs to job holding, such as dealingwithtraffic hassles or traveling through arcas where personal safety may be atisk, A substantial portion of these job costs are fixed in the sense that theyare incurred regardless of whether one works ten utes or eight hours perday: a portion ofchild care costs, however, is variable, depending on theamountoftime for which child care is required. We now briefly consider theeffects on labor force participation and labor supply of changes in fixed andvariable costs of employment.

John F. Cogan (1977, 1980b, 1981] has modeled the effects ofjob costson labor force participation and labor supply.** When job-holding costs areonly lime costs and are fixed, the budget constraintline of Fig. 11.1 becomeskinked, with a flat, horizontal segment emanating from H = 0 to, say,H = h’, indicating that thefixed time costof/’ contributes nothing to earn-ings. Asa result, eitherindividuals tend to perform no market work at all (acorner solution)or they enter the labor force ata significant numberofhoursper week (a “very interior” solution); hencefixed job time costs imply thatinterior solutionsare Jess likely to occur near the corner point where H = 0,One implicationis that because ofthe fixed time costs ofholding a job, muchof the upward-sloping portion oflabor supply curves will not be observed indata on hours of work.

Coganalso showsthat ifan exogenousincrease in fixed timecosts occurs(say, an increase in traffic congestion, implying a ionger commuting time),the individual's reservation wage 1is raised, and owing to the reduced fullincomethatis available as a result ofthe increased fixed timecost {a down-ward parallel shift in the budget constraint, total hours availablestilt fixed atT), hours of laborsupplied H will decrease; provided that L is a normal good,hours in £ will also decrease, compensating in part for the increased fixedtime costs of employment.| Costs of employmentare in practice very significant and are of primeimportance in determining the labor supply behavior of married women. Forworking women, Cogan [1981] estimates that in the United States in 1967,average annualfixed costs as a percentage of the average annual carnings ofworking women were 28.3%; furthermore,this figure increases considerably

614

11.3

Whether and How Much Women Work for Pay

with the numberofchildren in the household under age six and with addi-tional years of education for the mother. Cogan also reports that for an aver-age woman thediscontinuity in the labor supply function from H = 0 toH > Ooccurs at about 1300 hours annually (more than half-time work), indi-cating that when the equilibrium solution changes from a cornerto aninteriorone, the result is a very substantial increase in the numberof hours worked.

This concludes our brief overview of the theoretical issues underlyingthe lshor force participation and labor supply decisions of individuals andhouseholds. We now move on to a moredetailed discussion of econometricissues and empiricalresults, focusing particular attention on factors thataffectfemale labor supply.

ECONOMETRIC ISSUES AND REPRESENTATIVEEMPIRICAL FINDINGS

Thevast proportion of empirical studies of labor supply, beginning with thepioneering effort of Erika Schoenberg and Paul H, Douglas [1937], has beenbased on the neoclassica? analysis of choice, as outlined in the previous sec-tion. This empirical literature has been surveyed by, among others, Mark R.Killingsworth [1983] and Killingsworth and James J. Heckman [1986]. Forour purposesit is useful to employ the paradigm developed by Killingsworth,in which studies are classified as first or second generation depending on theattention paid to econometric issues and the underlying economic theory.”

11.3.A_ First Generation Studies.

First-generation studies of labor supply are characterized by Killingsworth asrelying on ordinary least squares estimation of parameters in equations whosefunctional form is chosen either arbitrarily or on the basis of specific ad hocconsiderations; in particular, the functional forms are not derived explicitlyfrom utility or indirect utility functions. Although dates are not precise, first-generation research began in the 1930s and continued into the early 1970s; awell-known first-generation study using aggregate data is by Glen Cain[1966], while representative studies using microdata are found in the volumeedited by Glen Cain and Harold Watts (1973b].

Someexamples oflinear specifications that are commonly used in first-generation research include

H=a+ bW+ Vite (11.21a)

!H,= 4,4 2 bW, + o¥ +6 (11.21)

=

4

a+ bW,+e(V+ 30 WH) +6 (11.21¢)pier

A,

11.3 Econometric Issues 615

where H is hours work time, JVis the real wage rate, V is per period realproperty income, ¢ is a stochastic term, and the i, j subscripts refer to differinghouschold members. Equation (11.21) refers to the labor supply of a givenindividual, Eq. (11.21b) to the labor supply of a given household memberallowing for nonzero intrahouschold (cross-substitution) effects on /’s laborsupply, and Eq.(11.2ic) constrains these intrahousehold substitution effectsto zero, Intercept termsin these equations are often made a function ofvar-ious variables such as age, gender, and race,reflecting tastes for work. Further,in many studies, transformations of the variables in Eqs. (11.21) are used,including iogarithmsand polynomials. Finally. as Killingsworth notes, ques-tions concerning the sources underlying the error term « are typicallyignoredin first-generation studies. Most ofthis research proceeds on the assumptionthatthe errorterm is randomlydistributed and makes nodistinction betweenthe measurement error or omitted regressors that might have generated «. As

weshall see,this last issue receives much more explicit treatment in second-generation research.

in terms of empirical findings, most first-generation studies concludethat female labor supply is considerably more sensitive to changes in wagerates and property incomethan is male labor supply. Most also find thatJei-

sure is a normalgood for both men and women. Another commonfindingisthat an income-compensated increase in one’s own wage increases one’s labor

supply.

Although these qualitative conclusions are roughly consistent acrossstudies, the quantitative estimates of the labor supplyelasticities obtained

from first-generation studies of labor supply differ very dramatically. AsHeckman, Killingsworth, and MaCurdy(1981, Table 1.1, p. 80] note, esti-mates ofthe compensated own-wage rate laborsupplyelasticities range widelyfrom ~0.05 to +0.96 for males and from —0.05 to +2.00 for females; grossown-wage rate labor supply elasticities vary from —0.45 to +0.55 for malesand from ~0.!0 to + 1.60 for females, while labor supply elasticities withfespect to property income extendfrom 0 to —0.16 for males and from —-0.1to —0.75 for females. Moreover. in almost ali cases the symmetry of com-pensated wage effects among household members is rejected byfirst-genera-tion empirical results, and at times the estimated compensated own-wageeffect on laborsupply has been positive, contrary to theory. These disappoint-ing results (and some well-timed financial support from research fundingagencies) have fed labor researchers to examine more ctosely the theoryunderlying labor supply models,as well as the econometric andstatistical pro-cedures that have been used to obtain parameterestimates.

Three issues have received particularly close attention because of theunsatisfactory first-gencration results, First, althoughfirst-generation empiri-cal studies appear to have used interchangably alternative measures of laborsupply H such as labor force participation, hours worked per week, weeksworked per year, lifetime participation, and the fraction ofa lifetime spentworking, in second-generation research,distinct theoretical models have been

616 Whether and How Much Women Workfor Pay

constructed to consider separately the participation, hours of work per week,lifetime participation, and lifetime hours of work decisions. These distincttheoretical models have helped in part to reconcile persistent differences inestimates of labor supply responses obtained by fitting the same function todata on temporally different perspectives of labor supply.

Second, empirical labor economists began focusing more attention onfunctional form and econometric technique. In turn, this entailed a more

explicit treatment of unobservable variables and introduced theissue ofsam-ple selection bias. Equations such as (11.21) were now viewed more formallyas representing the outcomeofutility maximization and in that context werefound to be woefully inadequate. Specifically, with even the simplest type ofutility maximization model, the labor supply function for a single individualcan be stated as

H=at b+ ch +¢ ifandonlyif Ww > W, (11.228)

H=0 otherwise (11.226)

where H/, is the individual's reservation wage, derived by setting // in Eq.(11.22a) equalto zero and solving for

First-generation studies ignored this distinction,typically either by fit-ting equationssuch as Eq. {11.21a) to a random sample taken from the pop-

ulation with the labor supply of nonworkingindividuals set at zero”—here-after called Procedure I—orby fitting the equation by OLSusing populationsubsamples that consist only of working individuals—called Procedure II."Procedure I (setting H for all nonworkers equal to zero) implicitly assumesthat Egs. (11-21) or Eq. (11.22a) holds for all values of W”, not justgor values

in excess of the reservation wage, and therefore involves a model misspecifi-cation,yielding inconsistent estimates of the parameters. Use of Procedure I(limiting the sample to workers) results in a nonrandom selection ofc, sinceby Eq.(11.22a) an individual will be included in the estimation subsampleifand only if H > 0, thatis, ifand only ife > —(a + 6W + cV) —J. Hencethe data on H is censored, and « must necessarily be correlated with HW and

V(unless & = c 0,or if everyone in the population works, as was virtuallythecasein certain studies of the labor supply of prime-aged males).

Since observations under Procedure If are systematically selected into

theestimation subsample according to the criterion « > —J, OLS parameterestimates based on such subsamples do not provide consistent estimaies ofthe total labor supply response parameters. This « > —J condition thereforedraws attention to corner solutions in which labor force participation {abinary variable} is determined and interior solutions in which hours of workare then determined, given positive LFP. As Heckmanetal. [1981] empha-size, while generated from the same preference function, the hours of workequation and the participation equation are fundamentally very different.

Thethird issue that drew the attention of researchers who were unsat-isfied with first-generation empirical studies wasthat of nonlinear budget con-

11.3 Econometric Issues 617

straints. Once oneintroduces into the model common incometax structures,varioustypes ofincometransfer programs, fixed costs of work, or employers"preferences for minimum hours of work,the simplestraight-line budget con-straints drawn in Figs. 11.1 and 11.2 disappear and give way to complex formswith kinks, discontinuities, gaps, and nonconvexities. Moreover, when taxrates depend on thelevel of income,the after-tax wage rate becomes endog-enous, and simultaneous equations problems can emerge. Mostfirst-genera-tion work ignored these phenomenaaltogether,in spite ofthe fact that in thelate 1960s and early 1970s a great deal of policy-oriented econometricrescarch focused on the labor supplyeffectsofalternative social benefit pro-grams{such as the negative income tax), programsthat clearly implied com-plex budget constraints.

These shortcomings of first-generation studies suggested numerousopportunities for improvement. As Heckman, Killingsworth, and MaCurdynoted,

analysis and technique go together, and first-gencration work, whichglossed over some ofthe structural aspects of labor supply decisions andused empirical techniques that did not adequately address some ofthecomplexities of that structure, suffered from 2 numberofserious prob-lems. The results ofsecond-generation work . . . suggest that solving theseproblems makes a considerable difference for estimates oflabor supplyparameters, with correspondingly considerable implications for analysisund policy.”

11.3.8 Second-Generation Studies

We now turn to a dicussion of second-generation studies, Wefirst considerissues of specification and estimation, then we introducetaxes into the laborsupply model, andfinally, we briefly summarize empirical findings from sec-ond generation research.

11.3.8.1 SPECIFICATION AND £STIMATIONISSUES A distinguishing featureof:second-generation modelsoflabor supplyis the explicit treatmentofutilityfunctions and unobservable variables. As an example, consider the utilityfunction

Us (WUE + 0) + PL Ur + ep (1.23)where HW’ and

V

are real wage and real property income,total time 7" is nor-malized to | so that // is the proportion oftime spent at market work, | ~ df=

L

is the proportion oftimein leisure (nonmarket aclivities), ¢ is an unob-servable error term varying from one person to another, and the term in thefirst set of square brackets is real goods consumption G, The error term ¢isinterpreted as representing interpersonaldifferences in tastes for leisure andfor goods consumption.” Note that Eq.(11.23) implies that even though twopeople have identical W and V, they are permitted to derive differing utilityU from the same amountofobservable H.

618 Whether and How Much Women Work for Pay

The marginal rate of substitution Af implied by the utility function

(11.23), Af=

(@U/OL)/(8U/AG), can be shown to equal

M = [BAL ~ by (WUE + 0) + MAL —+ ey) (0124)

where 6 & B/(« + ). To computethe reservation wage W’, evaluate Af in

Eq.(11.24)at the point where H = Oand L = 1. Atsucha point,

W, = [BL — by] De + VIAL - e] (11.25)

Anindividual with a given valueof ¢will provide positive labor supply if and

only if(hereafter abbreviated as iff) 1’ > HW. Setting H’> Wand rearranging

implies that H > Qiffe, > —J, where

gee and J=[(t— 6)— bY/W] (11.26)

This implies that

H>o iffe>-J (27)

H=0 iff < -J (11,28)

Further, if H > 0, then H is determined by the condition that theJ = W

in Eq. (11.24). Therefore if one sets Mf = W’ in Eq. (11.24) and solves for

H, one obtains the empirical labor supply function (given H > 0) as

He(l-b)-byW+a H>o (11.29)

More generally, one can specify a utility function U = U(G. L, e), in

which case the marginal rate of substitution function Af is M(G, L. ¢) =

MOVH + ¥. 1 — #, e)andthe reservation wage function at Hf = Ois HW, =

M(V, I, e). The individual will work iff 17 > HW, and for these individuals,

W’equals M, implying thatthe labor supply function can be derived by solv-

ing W = M(WH + V1 — H, e) for H > 0.

Although the labor supply function (11.29) looks quite similar to first-

generation equations such as Eq.(11.21), in fact the equation system (1 1.27),

(11.28), and (11.29) provides much more information and does so in an inte-

grated and coherent manner. To see this, first note that Eqs. (11.27) and

(11.28) emphasize a crucial threshold condition summarizing the labor force

participation (LFP) decision (for given V and W’, an individual will work iff

& > —4J). Second, the equation system (11.27)-(11.29) highlights the fact

that the labor supply function is in fact two functions—Eq. (11.29), which

holds if W > W, and Eq. (11.28), which holds iff #’ < 3. Third, notice

that the same observable variables, unobservable variables, and parametcrs—

namely, W, V, ¢, and b—affect both the LFP decision and the amount of

labor force supply conditionalon positive LFP. While first-generation models

overlooked these aspects of labor supply, second-generation models empha-

size Them.Let us now briefly examine someofthe principles of estimation

implied by second-generation models of jabor supply.

11.3 Economerric Issues 619

Initially, assume that measures ofthe real wage rate }Vare available forall individuals in the population, including nonworkers. For the ah individ-ual, from Eq. (11.26), denote values concerning tastes for work as en, 2 —¢,andthe value ofJ given 5, W,, V,, and e, as J,. A commonassumptionis that¢#, has a mean ofzero, has a standard deviation of oy, and is normally dis-tributed in the population as a whole, implying that the standardized normalvariable ¢z/¢ has a meanofzero and a variance of |.

Giventhis assumption concerning the population distribution of¢,,, theprobability that a given individual/ will workis given by

Pi works) = Pien/on) > (~Jfen)

=f Aodt = 1 ~ F(-Ffou) 11.30)iow

wherefand are the standard normal and cumulative normaldensity func-tions, respectively. Using Eqs. (11-27) and (11.28), write thelikelihood func-tion for a sample ofindividuals who are either working or not working as

eT]- Fl-c ~ 8)" + busty -“TL A1-a - 4)+ (131)<r

where 2 is the set of people who are working, @is the set of people who arenot working, b* = b/oy, and (1 — 6)" 3 (1 — boy.

Equation (11.31) is the standard probit equation, andits parameters &*and (1 — 5)* can be estimated by maximizingthe samplelikelihood functioné (orits logarithm) with respect to 5* and (1 — b)*.* Given these estimates,one can then uniquely compute the maximum likelihood estimates of a, as/fo* + (1 — 5)*] and of b as b*/[5* + (1 — &)*}. Hypothesistests can beimplemented usingthelikelihood ratio test procedure.

An interesting feature of this probit modelis that although it providesestimates of # and «,, and therefore provides information on parameters gov-erning labor supply,its computation in Eg.(11.31) uses information only onV,, W,, and whether the individual works, but does notutilize any informa-tion on hours worked, H,, Further, the conditional expectation of F[—J/oy]in Eq. (11.26) simply equals the probability that individual / is not working,thatis, the probability that H; = 0. A numberofalternative procedures canbe used to incorporate the additional information provided by nonzero dataon H,,

Forthe moment, assumeagain that data on wagesare available for bothworkers and nonworkers. The probability that individual { works 4, hoursand that this amount #,is positive can be written as

Pli works H, hours and H,; > 0] =Pli works H] - P[H, > Oli works H,] = P[i works HJ + 1 = fl)

620) Whether and How Much Women Work for Pay

where the probability density of observing individual i working exactly H,hours is fley/o)/on, Hence the likelihood function for the entire sample of2 workers and 2’ nonworkers is

€ = T] Menfoudton) « TT Ft-sion 1132)eu ne

where ey, © H, — J,and J, [(1 — 6) — B(V,/1%,)]. Equation(11.32) is calleda Tobitlikelihood function.” Nonlinear methods can be employed to findestimates of 5 and o, that maximize the samplelikelihood function (1 1.32),and hypotheses can be tested by using thelikelihood ratio test procedure.”

Thefirst part of the right-hand side of Eq. (11.32) refers to workers,while the secondrefers to nonworkers. Moreover,the first part is identical tothe likelihood function that is implicit in a traditional OLS regression,whereas the secondpart resembles the nonworking probability portion of theprobit likelihood function (11.31). In a sense, therefore, Tobit combinesfegression and probit frameworks. Notealso that if everyone worked, then

thelast part of Eq. (11.32) would disappear, and estimation of Ea.(I 1.29) vyOLS would yield unbiased estimates of 6 and s,. However, if some indiuals do not work, Tobit is more appropriate than OLS, since Tobit not onlypredicts the number of hours worked for each worker (see Eq. (11.29)), but

also produces the estimated probability that an individual will not work and

thereby helps explain why a number of observationsare clustered at the pointH = 0 (see Eq. (11.28).

The probit and Tobit models that we have discussed so far are based ona simple form ofJ derived from the specific utility function(1 1.23), namely,J.= [( — 8) — b(V,/5¥)]" In a more generalcase, one might write a laborsupply model

Ha XG t+ uy, ifX6 + ty > 0 (11.33)H.=0 ifXP + tn SO, P= N.S N

where X,is a vector ofregressors, 8 is the corresponding unknowncoefficientvector, Nis the total numberofindividuals, and ty, is an independently nor-mailydistributed “tastes for work” random variable with mean zero and vaance o*, The stochastic index X,8 + uy, is observed only when it is positiveand thus results in censored samples. This index can be substituted into theprobit likelihood function (11.31) or the Tobit likelihood function (11.32) inplace of J, and maximum lihood estimates of # and 6 can then beobtained by using nonlinear optimization procedures. |

It is tempting but incorrect to interpret the Tobit ¢ coefficients as mea-suring the effect on E(#,) given a change in X for individuals who are work-ing, that is, QE(H,)/@X, = 8 for those H, > 0, where EC) refers to theexpected value operator. In the above mode!(11.33), and ignoring individualsubscripts i, the E(H) = XBF(z) + af{z}, where z= XBjo, fiz) is the unitnormaldensity, and F(z) is the cumulative normal distribution function. As

(1.3 Econometric issues 621

has been emphasized by Amemiya [1973], the expected value ofH for obser-vations that are above the threshold (with positive labor supply}, now calledH*. is simply X¢ plus the expected value ofthe truncated normal error term,

EUP) = EWU > 0) = EU[ty > —Xp) = X68 + of(zyF(z) (1134)This implies that the celationship amongexpected values ofall observations,E(H), the expected value conditional upon being above the limit, EU"), andthe probability of being above the limit, F(z), is EH) = FIZJEUP),

John F. McDonald and Roben A. Moffitt [1980] have noted thatifonedifferentiates this last expression with fespect to X,, where Y, is the kth regres-sor, one obtains

SEU) yay SEUPY eppey GED)ax, A) ae + BUY Se (11.38)

In the labor supply context the total effect of a change in XY, on E(H) cantherefore be decomposed into two intuitively appealing components: ( 1) thechange in hours worked for individuals who are already working, weighted bythe probability of working, plus (2) the change in the probability of working,weighted by the expected value of hours for those who work.McDonald and Moffitt go on to show thatit is relatively simple to com-pute these two componentsofthe totaleffect, as well as their relative impor-lance. Specifically, one can use as an estimate of F(z) the fraction of the sam-ple abovethelimit (thatis, the samplelaborforce participationrate), and onecan estimate E(#*) using Eq. (1 1.34), along with entries from standard nor-maltables evaluating F(z), parameter estimates, and data evaluated at samplemeans.Also, the derivative 8E(H*)/A.X, equals 8, - 4, where A = {l — [xft2yF(z)) — [(=)'/F(z)']}, a fraction between zero and 1° Finally, since d#(2z)/

8X, = f\z)Bi/o, one can estimate the Proportion ofthe total effect of a changein X, on labor supply dueto the effect from those already working simply asA”

Thefraction 4therefore not only provides the appropriate adjustmenton 4, to obtain consistent estimates ofthe effects of changes in Y; on H forthose already abovethe threshold index, but also indicates the proportion ofthe total effect due to induced changes in the behavior of those above thethreshold. Thisdistinction can be important; for example, not onlyis it oftenuseful for policy purposes to know the effect of a Policyvariable such as as aspecific incometax rate change on total labor supply, but iis also pertinentto be ableto distinguish what portion ofthe total effect fepresents a changein LFPR (and hence a change in unemploymentrates) irom that portion ofthe effect on people who are already working.

Although the Tobit procedure utilizes information on both LFP andhours supplied given positive Participation,it is but one ofseveral proceduresthat are available for estimating labor supply models. Two variants on oneProminentalternative,called the selection bias-corrected regression method,

622 Whether and How Much Women Work for Pay

use both regression analysis and probit techniques, but as we now see, they

do so in a mannerthatis very different from Tobit.To understandsclection bias more clearly, considera labor supply func-

tion generated from a utility-optimizing framework and having the form H,

= V6 + up, where the normally distributed t¢, have a population mean of

zero and standard deviation of¢,as in Eq. (11.33). (Often, //, or someof the

X, are logarithmic transformations.) Suppose that one estimated this equation

by OLS using data only from working individuals. As was noted in the pre-vious section, such a procedure results in inconsistent estimates of the param-

eters in Eg. (11.33), since with such sampies, the conditional expectation of

My, is HONZErO and ity, is correlated with X.The nature of the OLS bias has been considered by, among others,

Arthur S. Goldberger (1981] and William H. Greene [1981]. Under the

assumption that all independent variables and the dependent variable are

multivariate normalitydistributed in the population (this therefore excludes

dummyvariables as regressors), Goldberger obtains the strong result that the

OLS regression coefficients are biased downward in the sense that the OLS

coefficient vector is a scalar multiple of the “true” labor supply coefficient

vector, where thescalarlies within the 0-1 interval. Moreover, Greene shows

that to obtain consistent estimates of the true labor supply parameters in Eq.

(11.33), all one need dois divide each clementof the OLS coefficient vector

by the proportion of observations for which, in our context, // > 0.

Although Goldberger shows analytically that this remarkable result does

not hold when the multivariate normality assumptionis relaxed, Greenefinds

that with a variety of nonnormalsituations (including dummyvariable regres-

softs), this simple adjustment of OLS estimates providess surprisingly robust

approximation to maximum likelihood estimates ofthe labor supply param-

eters in Eq. (11.33). The OLS-based estimates of standard errors, however,

are inconsistent, and as Greene shows, they cannotbe easily adjusted. Hence

one possible wayofdealing with sampleselectivity is to do probit estimation

of the LFP decision based on all observations, then do OLS estimation ofthe

hours worked equation, limiting the sample to workers, and finally, obtain

consistent estimates of the hours worked parameters by multiplying the OLS

coefficient vector by the sample proportion of observations for which H > 0.

Note thattestsofsignificance cannot be done with this procedure, since OLS-

based estimated standard errors are unreliable.A second variant on the sampleselectivity bias procedure involves OLS

estimation ofan expanded regression equation. Recall that, using Eq. (11.34),

one can write the conditional expectation of hours worked for workers as

E(H*)= E(H|H > 0) = X68 + K (11.36)

where K,, the conditional mean of u, given H > 0,is

Kao: AARZ)) = 0 (11.37)

}, is oftencalled the inverse of Mill's ratio, andin reliability theory it is known

as the hazard rate.Since KX, is essentially an omitted variable in the linear

11.3 Econometric Issues 623

regression equation H, = X,8 + ty, James Heckman [1976, 1979, 1980] hassuggested adding an estimateof4, as a regressorto such an equation and thenestimating the expanded regression equation by OLS while limiting the sam-ple to individuals for which H > 0.

Heckmansuggests estimatingA, initially on the basis ofa probit regres-sion using data from all workers and nonworkers and shows that when thisestimateofA, is appended asa regressorto the equation #/, = X,8 + ity. OLSestimates of the labor supply parameters are consistent, However, since the

disturbance term from this expanded regression equation turns out to be het-eroskedastic, OLS estimates of 6 are inefficient. and estimates of standarderrors are biased and inconsistent.| Yet another alternative procedure for handling this sample selectivity

ue has been suggested by Randall J. Olsen [1980]. Olsen proposes estimat-

ing by OLS 2 linear probability model in which the dependent variable is a0-1 dummyvariable (equaling1 if Hf > 0, otherwise zero) and the regressorsare the sameas jn Eq.(11.33), then retrieving the fitted values from this OLSTegression (calling them /), and finally, adding (P — ) as a regressor in thehours worked equation confined to the sample for which // > 0. Olsen showsthat under the uniform distribution this procedure provides consistent esti-matesof the labor supply parameters in (11.33)."'

The Goldberger-Greene, Heckman, and Olsen two-step procedure:in a sense generalizations of the Tobit technique, since the undertyingutilityfunction and labor supply parameters in thefirst-stage probit estimation arenot constrained to equal numerically the values that sre obtained trom theOLS oradjusted OLS estimates in the second stage, even if the underlyingeconomic theory suggests that they should be identical, By contrast, the Tobitprocedure constrains these parameters to be equal in both the labur force par-ficipation and labor supply equations. Note, however, that if one initially

specifies a more general model with discontinuities in the labor supply sched-ule ducto, say, fixed costs of work, then the Iwo-stage estimator might wellbe preferable.

Unfortunately, the Goldberger-Greene, Heckman, and Olsen two-stepstructural procedures require wage data for workers and nonworkers, as doesTobit. A major problem confronting second-generation researchers, however,is the fact that typically wage data are available only for workers. This dataproblem creates other econometric issues, to which we now turn ourattention.

A numberofalternative procedures have been employed by second-gen-eration researchers to obtain estimates oflabor supply parameters when wagesare unobserved for nonworkers. Many ofthese approaches use an additionalpopulation wage equation of the form

W,= ZT + en, (11,38)

where Z, is a vector ofvariables observed forall workers, including, for exam-ple, age, education, gender, region, and experience, and «,, is a mean zeronormally distributed random error term representing the effects of unob-

624 Whether and How Much Women Work for Pay

served factors such as motivation and abilities.” Killingsworth [1983] has dis-tinguished eight estimation procedures that attempt in alternative ways todeal with missing wage data for nonworkers.” We now summarize hisdetailed discussion.

In Procedure1, wo steps are taken. First, the wage equation (11.38)isestimated by OLS using data on workers only, and then the resulting OLS

parameters and data on Z, are used to create predicted wages H’, for bothworkers and nonworkers, Second, the predicted wages are used as a repressorin the hours worked equation H, = X,8 + ty, which is estimated by OLSusing data on all individuals and setting hours worked of nonworkers equalto zero. As was noted carlier, this procedure produces inconsistent parameterestimates, since it assumes that the hours worked equation applies to all pco-pie, rather than only to those for whom H > 0.

ProcedureHf is simply a one-step procedure in which the hours workedequation

H, = X68 + tn (1139)

is estimated by OLS using data only on workers. A variant of Procedure [1uses predicted rather than actual wages as 3 regressor butstill limits the sam-

ple for which Eq. (11.39) is estimated to individuals who work. Recalt thatsince people for whom H = 0 are excluded from estimation, Procedure [Isuffers from sample selectivity and yields biased parameter estimates.

Procedure If is the Tobit estimator discussed above bul uses as a mca-sure of the unobserved wage for nonworkers the predicted wage from Proce-dure I. Unfortunately, this procedure results in inconsistent parameteresti-

mates, since it is highly likely that sample selectivity problems also bedevilestimation of the wage equation (11.38), which by necessity is estimated byusing data only on workers.“ Specifically, the disturbance term ey, in Eq.(11,38)reflecting unobserved factors thataffect wagerates (such as productiv-ity, abilities, and motivation) is very likely correlated with the disturbanceterm #,, tepresenting unobservedfactors that influence labor supply (such astastes for work). This implies that aw», the covariance between en, and uy,issonzero; alternatively, it implies that the wage variable in the hours workedequation (11.39) is endogenous, correlated with the u,,, disturbance term. As

a result, neither use of imputed wage duta derived from OLSestimates nor

use of actual wage data (if they were somehow available for al} workers) wouldyicid consistent estimates ofthe labor supply model under any of Procedures1, II, or Ut. To be valid, each of these procedures would require that 1’, beexogenousor, equivalently, that oy, be zero.

Procedure {V is one of several different procedures discussed by Heck-man (1974b, 1976a}. It attempts to deal with this endogeneity issue by usinga reduced form procedure. Let the marginal rate of substitution functionderived from a utility function have the linear form

M, = X3& + tn (11.40)

Ty

11.3 Econometric issues 625

where the X* matrix is the Y matrix of Eq. (11.39) plus an additional columnvector, hours worked H,, and where the mean zero normallydistributed ¢s,random error term represents unobservable taste factors. When expression

(11.40)is solved for that wage rate at which Hf = 0, one obtains the reserva-

tion wage equation

W, = XF0+ «, unary

where the X** matrix is the X* matrix of Eq. (11.40) with the W, and #,vectors deleted.

Heckman begins by adopting an analytically convenient proportionalityhypothesis (consistent with linear labor supply) where hours of work //, areproportionalto the difference between W, and Hi, whenever HY > 1, andotherwise are zero:

H, = 6(W, — BY) = DW, — XO — beg, iP, > W, U1.42a)

H,=0 iis W, (1142b)

Theparticipation equation implied by this equation system is

Pli works) = PEW, > WW) = PIZT + en > XO + ea]

= Pl> ~4) (11.43)

where ¢p, 3 eu, — €2, and J, = Z,P ~ XT6. Since eu, and ¢g, are zero meannormally distributed random variables, so toois tn; the varianceoftp, iS ob= of, + oh + 2owg. The likelihood function for LFP observations from asample consisting of workers 2 and nonworkers 2 can then be written as

e= TP - F-sool TFadeo)l (11.44)nr

whose estimation involves standard probit procedures.The hours of work equation that is consistent with this framework is

most interesting. Heckman first estimates the wage determination equation(11.38) by OLS or GLS, using data for workers only, and then directly sub-stitutes this into Eqs.(11.424) and (11.42b), resulting in

H, = UZ) - XFe+ y, iT) >0H,=0 W[-}<a

(1145a)

i1.45b)

Hence by substituting out the endogenous wage in Eqs. (11.42) using Eq.(11.38), Heckman obtains a seduced form equation (11.45), whose right-handvariables are all exogenous. Three issues merit brief discussion.

First, note that certain variables in the Z matrix from the wage deter-mination equation (11.38) might be identicalto those in the X** matrix fromthe reservation wage equation (11.41), for example, age, education, gender,race, and years ofwork experience. This raises issues of identification, a prob-lem that arises whenever wages are endogenous. Heckman [1974b, 1979]

notes that a condition sufficient foridentification is that at least onc variable

626 Whether and How Much Women Warkfor Pay

included in Z must be excluded from ¥**." Second, the data in the Z andX** matrices of the reduced form equations(11.45)are typically observed forboth workers and nonworkers, in contrast with traditional labor supply equa-

tions such as Eq, (11.29), for which data on H’ are available only for workers.Hence it appears possible to utilize data on the entire population, not justworkers. Third, note that one could estimate Eq. (11.454) by nonlinearleast

squares (or by OLS with the parameterestimates from Eq. (11.38) directly

substituted in), using data on the entire population and setting J2 0 fornonworkers. Killingsworth [1983] calls such a reduced form estimationmethod Procedure IV.

A majorproblem with Procedure IV, unfortunately,is that if it is basedon data for all individuals with nonworkers #f set to zero,it involves a mis-specification similar to Procedure I, since in fact Eq. (11.45a) refers only toworkers. If one instead estimates Eq. (11.45a) only for workers, sample selec-tivity problems emerge, similar to those in Procedure {1. Hence while it isapparently promising, Procedure IV does not solve the hours worked-wagerate endogencity problem.

There is, however, a related two-stage estimation procedure that effec.tively deals with this problem. In Procedure V, one takes advantageofthe factthatép, in Eqs. (11.45a) and (11.45b) is normally distributed and that H, cannever be negative. One then specifies the Tobit likelihood function (11.32)adjusted to accountfor the fact that the disturbance term de,is the productof an unknown parameterto be estimated and the mean zero normallydis-tributed random variable ¢p,. This yields

€ = TT RoeatbooVben) - [] Fi—SJon) (11.46)

where be, = H, — bJ,and J, is defined underneath Eq.(11.43).In the second stage the parameter estimates that are obtained by maxi-

mizing the Tobit samplelikelikood function (11.46) with respect to &. I’, #,and op, along with data on X** and Z,, are used to calculate the inverse Millsratio A, = fl—J/op[1 — A—J/op)], that is, one uses parameters and datato estimate the probability that an individual with given XYand Z,will work.This estimateof 4,is then inserted as an additionalvariable in the wage equa-tion (11.38), whose parameters are estimated by OLS using data on workersonly. Note that with Procedure V, estimation of the panticipation-hoursworked equationsis based on a reduced form Tobit specification, while esti-tation ofthe wagestructural equation for workers allows for sample selectiv-ity. However, since the underlying parameters in the reduced form equationare not constrained to equality with structural parameters of the wage equa-tion, parameter estimates are notefficient.

In Procedure VI, also discussed in Heckman [1974b], parameters oftheentire equation system consisting of Eqs. (11.38) and (11.41) and the Tobitequations(11.42a) and (11.42b) are estimated by usingfull information max-imum likelihood (FIML)simultaneous equation techniques, with cross-equa-

11.3 Econometric Issues 627

tion parameter constraints fully imposed. This FIML specification builds onthe assumptionofjoint normally distributed random disturbance terms ew, inthe wage equation (11.38) and ¢g, in the reservation equation(11.41), havingcovariance gue and implying that the random variable ep, = ey, — ¢4, in thehours equation (11.42) is also joint normally distributed with ey, havingcovariance ope © oly — dwg. AS a result, the likelihood function for the wagetate, hours of work, and participation decisions of working individuals in set{and nonworking individuals in set 1” is

@ = [Tf Aben/bom enfow) «TE Fl-Jyo) (1.47)ren ver

wherej is thejoint probability density functionfor ¢,, and ¢,, and other termsare as defined above.

The important distinction between the FIML (11.47) and Tobit (11.32)likelihood functionsis therefore that while FIML treats W’ and H us simul-tancously determined becauseofthe possible correlation between ev and ¢,,Tobittreats H’ us strictly exogenous. Further, unlike the Tobit specification(11.45), the FIML estimation of Eq.(11.47) utilizesall available informationon Wand H for workers.

A common feature ofall the above estimation proceduresis that theyassumethat labor supply falls continuously to zero as wage rates or propertyincomedec! ; See, for example, Eqs. (11.33) and (11.42a). If there are fixedcosts of employment, however, one might want to develop a procedure thatallows for discontinuous labor supply schedules, that is, one that permits thelowest nonzero numberof hours supplied by a worker to be substantially inexcess of zero. Two estimation procedures, hoth known as generalized Tobitestimators and often called Heckit procedures, allow for such discontinuities.

In Procedure VI/ a three-stage method is implemented. Stage | is thesameas in Procedure V, except that one now estimates parametersofa probit{not a Tobit) likelihood function for participation, such as Eq. (11.44). Byusing a probit rather than a Tobit specification, continuity ofthe labor supplyscheduleis not imposed. In the second stage of Procedure VII, one uses the

probit parameter estimates to computeprobit estimates of the inverse Millsratio X, = A—Jfon)/{1 ~— F(-J/ep)] and then appends A,as an additionalregressorto the wage equation (11,38) to obtain OLS selection bias-correctedparameterestimates using data on workers only; Killingsworth [1983, p. 159}notes that the parameter estimate on A, in this second stage should be inter-preted as an estimate of oy», the covariance between ey, and en.

{n the third and final stage of Procedure VII, one estimates by OLS thereduced form hours equation (1 {.45a) with the probit inverse Mills ratio vari-able added, based on data for workers only, which, along with the siructuratwage parameter estimates from Stage 2, yields consistent estimates of thelabor supply structural parameters. In this case the parameterestimate on A,is interpreted as an estimate of [(ouo/e) + blewo/op)], where oyp is thecovariance between u,,, and ¢,,. Formulaefor the asymptotic variance-covari-

628 Whether and How Much Women Workfor Pay

JAMES HECKMAN

Labor Econometrician

ore than anyother economist in the last

two decades, James Heck-man has affected theeconometric modeling of

the labor force participa-tion and labor supply de-cisions. Heckman’s con-tributions began with his197} Ph.D. thesis atPrinceton, “Three Essayson Houschold Labor Sup-ply and the Demand for MarketGoods.” Since then, as detailed in

this chapter, Heckman’sresearch has

essentially built the core of what are

now called the “second-generationmodels” of labor supply. Two distin-guishing features of this research arethat the various estimating equationsare derived explicitly from neoclassi-cal utility theory and that the sto-chastic disturbance termsare an in-tegral part of the model, rather thanbeing “tacked on™ at the end ofthederivation. Heckman's analysis of

self-selection in labor supply has pro-duced a new methodologyforsolvingself-selection and sample selection

problems, often called the “Heckit”

procedure;it is now widely used in avanity ofcontexts. Although they are

not discussed in this chapter, Heck-man has also developed and imple-mented empirically a numberof dy-namicdiscrete choice models, as wellas models for the analysis of du-

ration data. Someofthisresearch has involved newuses of semiparametricand nonparametric meth-ods.

Born in Chicago in

1944, Heckman pursued

undergraduate studies atColorado College, major-ing in mathematics, earn-ing a Phi Beta Kappa key,and graduating summacum laude. He then went

on to study economics at Princeton,and after spending a summer as Jun-ior Economist at the Council of Eco-nomic Advisors in Washington,D.C., Heckman was awarded thePh.D.degree in 1971, studying underStanley Black, E. Phillip Howrey,Harry Kelejian, Riehard Quandt,and Albert Rees.

Heckman’s first academic ap-

poiniment was at Columbia Univer-sity. In 1973 he accepted a positionat the University of Chicago, andin 1985 he was named its Henry

Schultz Professor of Economics. In1988 he moved to Yale University,where he nowis the A. Whitney Gris-wold Professor of Economics. Heck-man has been elected a Fellow of theEconometric Society and of theAmerican Academy of Arts and Sci-

ences. In 1983 he was awarded thedistinguished Jobn Bates ClarkMedal by the American EconomicsAssociation.

11.3 Econumetric Issues 629

ance matrix of estimates obtained from this three-stage procedure have beenderived by Lung-Fei Lee (1982], but consistent estimates of Standard errorsfor the third-stage parameters can be obtained in a computationallyless bur-densome mannerby using the Halbert White (1980} robust estimator,

Finally, in Provedure VIIT thefirst two Stages are identical to those inProcedure VII, butin the third stage, instead of using the reduced form spec-ification (1 1.45a), one employs instrumentat variable and selectivity bias-cor-rected techniques to estimate parameters of the structural hours equation(11,33) for workers, More specifically, A, is added as a regressor to thefirstequation in (11.33), and instrumental variable estimation is undertaken withthe actual wage rates of workers replaced by fitted values derived from thesecond-stage selectivity bias-corrected wage equation estimates: hence theprobit A, appears as a regressor in both the wage and hours equations.

{t is worth noting thatif in fact the labor supply schedule is continuous,parameter estimates from the Heckit multistage Procedures VII and VIIIshould be identicalto those from the FIML Procedure Vi. Onthe other hand,if the labor supply schedule is discontinuous owing to, say, fixed costs ofemployment, then the Heckit multistage generatized Tobit estimators of Pro-cedures VII and VIII shoulddiffer from FIML. However, unless one specifiesexplicitly the nature of the optimization problem facing individuals—includ-ing, for example, the budget constraint implicationsoffixed casts of employ-ment, the Heckit parameterestimates are “black box”in the sense that theyarc unable to provide insights on what factors have generated the disconti-nuities. For further discussion of such importantspecificationissues, see Kil-lingsworth [1983, pp. 161-168], who considers theoretical and econometricaspects of several explicit models of discontinuous labor supply.

14.3.8.2 INCORPORATING TAXES INTO LABOR SUPPLY To this point, ourdiscussion of labor supply has ignored the presence oftaxes, Since consump-tion expenditures derive from after-tax income,the relevant budget constraintsel for the analysis of individual labor supplyis the after-tax constraint, notthe before-tax one. Taxes on wage and Property income can significantlyaffect labor supply, in part because taxes ofien introduce a wedge betweenaverage and marginalafter-tax wage rates. Moreover, since much public pol-icy discussion has focused on the labor supply effects of changing statutoryprovisionsofthe incometax code,it is important that we focus attention onthe effects of taxes on labor supply. Second-generation research on this topichas been substantial, with noteworthy contributions made by Harvey S.Rosen [1976], Gary Burtless and Jerry A. Hausman [1978], Hausman [1979;1980; [981a, b; 1985], Terence J. Wales and Alan D. Woodland [1979],James J. Heckman and Thomas E. MaCurdy (1981), and Alice Nakamuraand Masao Nakamura (1981a].

Tosee the impactoftaxcs, let us assumethattheindividual receivesrealproperty income V and carns a gross (before-tax) real wage of Wper hourworked,Asis shownin Fig. 11.3,in the absence oftaxes the budget constraint

630 Whether and How Much Women Work for Pay

Goods, G|

Slope =WU = 2)

Leisure Proportion —

Figure 11.3. The Effects of Varying Marginal Tax Rates onthe Budget Constraint

is 1F, with the nonvertical portion having a slope of ~ #’. Suppose, how-ever, that the individualis taxed on the sum of property and wage income.Ifthis total incomeis C, or less, let the tax rate be 100 - +, percent;if totalincomeis greater than C,, let the marginaltax rate on total income above C,be 100- 7, percent, where +; > 7,. In Fig. 11.3 this implies thatif the individ-ual works hours between zero and | — L, and chooses leisure amountbetween L, and 1, the marginal after-tax wage rate is W(1 - 7,), while if theindividual works more than | — L, hours (chooses less £ than £,), the mar-ginal after-tax wage rate is (1 — 7;); thus the kinked after-tax budget con-straint is 1/,XF;. Note also that in Fig. 11.3 the vertical distance between thetwo budget constraints represents the total amount of taxes paid at variousproportions of L. .

The budget constraint set in Fig. 11.3 is rather simple. It is relativelystraightforward to extend the analysis to a progressive tax structure with mdistinct and successively flatter linear segments, correspondingto increasingmarginal tax rates 7, > T,-) > * + + >11; in such a case the budget constraintset would be quasi-concaveto the origin. Note that with such a quasi-concavebudget constraint set, given strictly convex indifference curves, the equilib-

H1.3 Econometric Issues 631

rium point of tangency between the indifference curve and the budgetsetisunique andoccurseither at oneofthe kinksor at an interior location betweenkinks.

However, if one introduces other real-world statutory tax provisionsinto the analysis such as exemptions, deductibles, and transfer paymentstiedto hours worked, then the budget constraint set may include nonconcaveregions, and as a resuli, multiple tangenciesofindifference curves to the bud-get set might occur. As Wales and Woodland [£979] and Hausman [198 1a]emphasize,in such cases, to determinetheutility-maximizing hours of work,one must know the form oftheutility function and compare utility at thevarioustangencypoints. This introduces complications into the estimation ofthe effects oftaxes on labor supply.

One way of simplifying the potentially complex budget constraint setinvolves linearization; this has been implemented empirically by, among oth-ers, Hall [1973] and Wales [1973]. Specifically, for an individual who workssomewhere along segment VX, when confronted with the budgetset | /.XF,,that individual could be assumed to act as if he or she faced a simplestraight-line budget constraint 1 /,XF, with reat property incomeequalto V,and a real marginal wage rate of H’(1 — 1,). The effects of small changes inproperty incomeor before-tax wagerates could then be analyzed by assuming

that the individual does not shift onto a different segment of the budget con-straint set. Incidentally, in this casc the after-tax property income I, is theamountofdisposable incomethatis available to the individual assuming thatH = Oand L = }. Following W. Erwin Diewert [1971], Burtless and Haus-man [1978] call this the individual'svirtual income: hence virtual income cor-responds with the before-tax property income V and the tax rate 7).

Similarly, for an individual who is working somewhere along segmentXF,, when confronted with the same kinked budget constraintset, that indi-vidual might be envisaged as acting as ifhe or she faced the simple straight-line budget constraint 1 V,XF; with a virtual incomeof V; and a real marginalwagerate of W(t ~ 7;). Hence complex budget constraints can be made con-siderablyless complicated simply by linearizing them atthe individual’s equi-librium point. Note that in econometric implementation,at each observationthe measures ofafter-tax W and virtual income V used must be changed toreflect the after-tax wage and property income corresponding with the indi-vidual’s marginaltax rate at the equilibrium point,

Althoughthis linearized budget constraint approachis attractive, it alsointroduces new econometric problems, Several issues are of particular importance in empirical implementation, First, because the after-tax JV and ¥ mea-sures depend on # (the marginal tax rate + varies with A = | — ZL) thedisturbanceterm in the hours worked equation is by construction correlatedwith the after-tax measures of Wand V; hence estimation by OLSprovidesinconsistent estimates of the parameters, and instrumental variable (IV)esti-mation maybe preferable, 1V estimates have been reported by, amongothers,Jerry A. Hausman and David A, Wise [1976] and Harvey S. Roscn [1976].Rosen's studyis of special interest, since in it he examines tax perception.

632 Whether and How Much Women Work for Pay

Rosenfindsthat he cannotreject the null hypothesis that individuals perceive

theeffects of taxes andreactto the net after-tax wage rather than to the before-

lax gross wage,

Second, and closely related to the previous point, it might be that in

response to a change in before-tax V or W, the individual would shift to an

entirely different segment on the budget constraintset. Estimation and model

predictions based on predetermined marginaltax rates would therefore be

unreliable. As might be expected, such estimation problemsincrease when the

concave budgetset is kinked rather than continuous, that is, when marginal

tax rate changes are discrete rather than continuous. Problems of estimation

become even more severe if the budget set has nonconcave and kinked

portions.

Such problems with the linearization or local approximation approach

have given rise to a variety ofalternative procedures in which the complete

‘budget set is evaluated.” In one procedure the previous no-tax estimation

framework in which the probability ofH > 0 is estimated along with an hours

worked equation conditional on H > O—a single-threshold framework—is

generalized to include multiple marginal tax rates by permitting multiple

thresholds, each corresponding with a different segmentof the budget set in

which the hours worked decision entails a different marginal tax rate. This

type of modelcan be estimated by using the ordered probit function; see Ta-

keshi Amemiya [1975, 1981] for a discussion ofthis estimator and Antoni

Zabalza [1983] and John C. Ham and Cheng Hsiao (1984) for empirical

implementations.Yet another procedure assumes that random disturbances arise either

from individuals’ errors in optimization or from measurementerrorof and

involves evaluating utility at each point ofthe individual's budget constraint

sel (including the kink points), given values ofthe relevant structural param-

eters of an assumed direct or indirect utility function. This procedure, which

is oftencalled the complete budget constraint approach (CBC).literally solves

the individual's utility maximization problem foreach set of parameter values

andis obviously very computationally intensive. Notice in particular thatfor

each set of parameter values, the individual's entire maximization problem is

solved: estimation procceds, as has been noted by Wales and Woodland

[1980]. by maximizing the likelihood of observed hours with respect to the

entire set ofstructural parameters ofthe utility or indirectutility function.

The Wales-\WWoodland approach has been generalized by Burtless and

Hausman [1978] and by Hausman [1980]to allow the disturbance term in

the hours worked equationto originate not only from errors in optimization

orerrors in the measurement of H. but also from omitted variables that are

known by the individual but are not observable by the econometrician,

Remarkably, this more general conceptual framework can be implemented

withoutresort to the laborious Woodland-Wales maximizing procedure, pro-

vided that one makes very strong assumptions aboutthe functional form of

the hours worked equation.

11.3 Econometric Issues 633

_The above discussion therefore suggests that while the presence of taxesconsiderably complicates the analysis of labor supply, empirical analysis canproceed, provided that oneis willing to make tradeoffs concerning computa-tional complexity and flexibility of the functional form. This type oftradeoffis, of course, a familiar quandary to the practicing econometrician.

11.3.8.3 EMPIRICAL FINDINGS We now conclude ourdiscussion of second-generation researchin labor supply by providing a brief summary ofempiricat

findings. More detailed surveys can be found in, among others, Heckman,Killingsworth, and MaCurdy[1981, pp. 106-112}. Kiltingsworth [1983, pp.179-206}; and Killingsworth and Heckman [1986, pp. 185-197].

Mostofthe second-generation studies avoid the pitfalls associated withfirst-generation research. In particular, second-generation analyses explicitlydistinguish the LFP function from the hours supptied fisnction, yet they alsonote that both derive from a commonutility-maximizing framework. Withinthis framework a substantial proportion of second-generation studies treathours worked and the wage rate as endogenous. Finally, most of the second-generationstudies take accountoftheselectivity bias in estimating tabor sup-ply functions, using one or more of the Procedures V through VII summa-

sized above,__ A vast majority of second-generation work has been concerned either

with fixed costs and discontinuities in the labor supply function (Cogan[5980b, 1981}, Giora Hanoch [1976, 1980], Heckman [1980], and T. PaulSchultz (1980]) or with taxes and nonlinear budget constraints (Burtless andHausman [1978], Rosen [1976], and Wales and Woodland [1979]). With thenotable exception of Hausman [1980; 198ta, b; 1983] and Alice Nakamuraand Masao Nakamura [1981a}, most studies have focused on either fixedcosts or taxes but have not considered them jointly. As might be expected,the numerous second-generation studiesdiffer not only in estimation proce-dures. employed, but also in terms of the underlying data samples, This het-*erogeneity makesit somewhatdifficult to provide a concise comparison andsummary of research results,

Two criteria might be used in assessing this research. First, do second-generation results suggest importantstytized facts of labor supply that ure notapparent from first-generation studies? Second, do estimates oflabor supplyresponses based on second-generation researchdiffer appreciably from thosebasedon OLS estimation of first-generation equations? The answertothefirstquestion appears to be “perhaps,” while the answer to the second is anemphatic “yes.”

In termsofthefirst criterion,like thefirst-generation studies, most (butnotafi, as we shall soon see) second-generationresults generally conclude thatfemale labor supply is considerably more responsive to changes in properyincome and wagerates than is malelabor supply. Killingsworth (1983, p, 206}goes even further, concluding that “female labor supply responses are some-whatlarger than was apparentin first-generation research”(sce also his quote

634 Whether and How Much Women Work for Pay

at the beginningofthis chapter). He notes as well that second-generation stud-ies that are based on continuous labor supply schedules (using Procedures It,V, and VI) tend to produce greaterestimatesoffemale labor supplyelasticitiesthan do second-generation studies that allow for a discontinuouslabor supply

schedule (Procedures VII and VIII), Moreover, most second-generation

research explicitly considering discontinuities in the labor supply schedulefinds that such discontinuities are substantial. Estimates of the “pointofdis-continuity”—the minimum amountofannual hours at which wages are suf-ficiently high to overcomefixed costs—are reasonably consistent over diversestudies. For example, Heckman [1980] and Cogan [1980b, 1981) report esti-mates that range between only 1126 and (383 hours, while Hanoch’s lowestestimate is 820 hours. |

Unfortunately, however, like their first-generation counterparts, interms of wage and incomeelasticity estimates, the second-generation esti-mates oflabor supply responsesare verydiverse. In fact, noting that estimatesof the female uncompensated wageelasticity of annual hours vary from

—0.30 10 + 14.00, Killingsworth and Heckman (1986, p. 185] reach the sad-

dening conclusion that

it is not uncommonfor authors of empirical papers on labor supply topointto results in otherstudiessimilar to the ones they have obtained. but

...such comparisons may not always be informati ts all too casy to

find at least one otherset of results similar to almost anyset of estimates‘one may have obtained.*

In spite ofthis diversity, and in termsofthe secondcriterion for evalu-

ating second-generation studies, on the basis of research to 1980, Heckman,

Killingsworth, and MaCurdy [1981] have argued that clasticity ‘estimates

obtained from sccond-generation studies tend to be greater—sometimes con-

siderably greater—in absolute value than those based on first-generation tech-

niques. Theycite two types of evidence to support theirtentative conclusion.

First, an important Monte Carlo study by Wales and Woodland[1980] indi-

cated that, given joint normality, OLS estimates are badly biased toward zero

(though perhaps not as much in the wage equation asin the hours equation),

whereas techniques that correct for sample selection bias are much closer to

the “true” parameter values. Second, studies of female labor supply by,

amongothers, Cogan [1980a], Heckman (1976a}, and Schultz [1980], found

that labor supply elasticity estimates that were based on second-generationestimation procedures were consistently larger than those based on first-gen-

eration OLS methods.Notall labor econometricians agree with Heckman, Killingsworth, and

MaCurdy's assessment. In particular, already in the late 1970s, Alice Naka-

mura and Masao Nakamura [1979] reported results of a second-generation

study that found female labor supply to be basically unresponsive to changes

in wage rates, similar to the findings reported by others for males (sce their

quote at the beginning ofthis chapter); additional supporting findings were

11.3 Econometric Issues 635

presented later in Nakamura and Nakamura [1981a: 1985a,b]. This contro-versy about whether males and females respond differentially to wage rate

changes is of considerable interest, and therefore in Section 11.3.C we willdiscuss it further when we consider the Mroz [1987] anticle.

Finally, in terms of analyzing the effects of taxes on labor supply, notmuch is known yet concerning the consequences andrelative merits of usinglinearized (LBC) or compicte budget constraint (CBC)estimation procedures.Kilingsworth (1983, pp. 204-205] notes that the LBC and CBC estimatesdiffer considerably, but since most CBC studies employed restrictive utilityfunction specifications or limited samples,it is not clear how muchof thedifference is due to LBC versus CBC,versus the choice ofa restrictive func-tional form.

The empirical results ontheeffects oftaxes on labor supply reported byHausman[1981a] are of particularinterest. however. Hausman incorporateseffects of a progressive incometax schedule,as well as provisions of numerousUS. transfer programsresulting in budget constraint sets with both concaveand nonconcave regions. Following the previous tradition, Hausmantreatsthe husbandas the primary workerin the family and specifies that in makingher labor supply decisions, the wife treats the husband's incomeas predeter-mined.” Significantly, the functional form and stochastic specificationemployed by Hausman constrain leisure to be a normal good. Fixed costs ofworking are incorporated into the model. Hausman computes virtualincomes correspondingto cach individual's equilibrium marginaltax rate andemploysa specification in which increases in income neverraise desired hoursof work. However, heterogeneity in preferences among individuals can resultin income effects that vary across the population. A CBC estimation proce-dure is employed, and estimation results are compared on the basis of a com-bined concave, nonconcave budget constraintset with results from a globallyconcave approximation to a complex budget constraint set.

For husbands, Hausman finds that the uncompensated wageelasticityis essentially zero but that the incomeeffect is negative and substantial.Results from the concave and nonconcare estimation procedures are similar.Given these estimated incomeeffects, Hausmanconcludes that for husbandswith averagetastes and in the $8000~$ 12.000 federaltax bracket in 1980, theProgressive incometax structure reduces labor supply by 8%compared 10 aNo-tax situation and by 7% compared to a proportionaltax system generatingequaltax revenue.

For wives, Hausman usesresults from his carlier research that suggestedthat the wives’ wage rates before taxes are not a function of hours worked;Mmorcover, he also concludes that sampleselectivity is not a significant prob-teminthe female wage rate equation. Hausmanestimates the uncompensatedwageclasticity for the average woman workingfull time as being equal to arather substantial 0.995 and,iffixed costs are taken into account, at 0.906.Theincomeelasticity is negative and substantial, reflecting in part the leisureRormality assumption. These results imply that progressive income taxes

JERRY A. HAUSMAN

Modeling the Effects of Taxes on Labor Supply

erry Hausman’s con-Jtributions to the the-

ory and practice ofeconometrics have beennumerous and wide rang-

ing. While students mightrecognizehim bestfor theoft-used specification testprocedure that he devel-oped, Hausman has writ-

ten extensively in econo-metric theory on simul-taneous equations estima-tion,the interpretation ofinstrumen-talvariable estimators, estimation ofmodels with errors in variables, com-putational algorithms for nonlinearstructural models, and panel data es-

timators.Although Hausman has made

major contributions in econometrictheory, he considers himself essen-tially an applied econometrician. Hehas long been interested in public f-nance (he currently serves as a Mem-ber of the Massachusetts Governor'sAdvisory Council for Revenue andTaxation), and this interest led himto research the effects of taxes onjJabor supply. One of Hausman’s im-portant econometric findings, dis-cussed in this chapter,is that the U.S.income tax system leads to greater

deadweight loss than had previouslybeen believed and that an equal rev-enue-generating tax structure with

personal exemptions and then a pro-portional tax would involve muchsmallerefficiency losses.

In addition to his work on the ef-fects of taxes on labor supply, Haus-

manhas published empir-

ical studies on the effectsof alternative rate struc-tures on the demand forelectricity, on factors af-fecting the purchase andutilization ofenergy-usingdurable goods, on the re-

lationship between pat-ents and research and de-velopment, and most

recently, on the effects ofcompetition and technical

progress in telecommunications.Born in West Virginia in 1946,

Jerry Hausmanobtained his A.B. de-gree (summa cum laude) from Brown.

University in 1968. Aftcr spending

twoyears with the U.S. Army Corpsof Engineers in Anchorage, Alaska,Hausman resumed studies at OxfordUniversity, where he was a MarshallScholar. At Oxford, Hausman earneda B.Phil. degree in 1972, and aD.Phil. in 1973; his‘dissertation wasentited “A Theoretical and Empiri-cal Study of Vintage Investment andProduction in Great Britain.”

Hausman’s first academic ap-pointment was at MIT in 1972, andaside from occasional visiting ap-pointments, he has been there cversince, teaching courses in economet-tic theory, microeconomics, public

finance and a course in the industrialorganization of tclecommunications.Hausman has beenelected a Fellowof the Econometric Society, and in(985 he was awarded the John BatesClark Medal by the American Eco-nomic Association.

636

11.3 Econometric Issues 637

reduce wives’ labor supply in large part by decreasing the net after-tax wage.Specifically, to the extent that the wife is a “secondary” worker, Hausmanhotes that she faces a “marriage tax,” since thefirst dollar of her earnings istaxed at the rate that is applicable to the last dollar of her husband's earn-ings.** Hausmanalso examinestheeffects oftaxes on labor supplyof femaleheadsoffamilies and reports significant sensitivity to taxes.

Overall, Hausmanestimates that the progressive federal U.S.incometaxstructure existing in 1980 results in an 8.6% declinein labor supply comparedto a no-tax situation and estimates that the deadweightloss as a percent ofincometax revenuesraised is 28.7%. Owing to progressivity, deadweight lossincreases sharply with market wage rates and associated marginal taxes. Bycontrast, 2 proportional incometax system that generated identical tax reve-Nues would reduce labor supply by onlyslightly more than 1%, and the aver-age deadweight loss as a proportion of tax revenues raised would be only7.19, Arguing that a proportional tax system with aninitial exemption pro-vides an alternative way of implementing progressivity without entailing aslarge a deadweightloss, Hausmanconcludes that “the progressive income taxsysiem maybe a costly way to seek incomeredistribution. Other, more cost-effective meansof incomeredistribution maywell exist.”

Hausman’s findings are not universally accepted, however. For exam-ple, in commenting on Hausman [1983]. Heckman (1983) conjectures thatHausman’s ratherlarge wage responsiveness estimates reflect at least in partthe effects of his assumption, imposed on the data, that leisure is a normalgood.

11.3.C Onthe Sensitivity of Results to AlternativeStatistical and Economic Assumptions: TheMroz Study

In reviewing second generation empirical results in the previous section weemphasized thatthe substantial heterogeneity that is observed in the estimatesofwage and incomeelasticities could be ducin part to the fact that the variousstudies differin terms of estimation method,functional form, and data base.This suggests that an informative research project would involve undertal ing.detailed sensitivity analyses of existing alternative second-generation estim:tion methods but using a common dala base and functional form. That isPrecisely the approach taken by Thomas Mroz [1987], who employs datafrom the 1976 Panel Study of Income Dynamics (PSID) sampleconsisting of753 white married women aged 30-60, of whom 428 worked al some timeduring 1975, Using these data, Mroz compares elasticity estimates based ona variety ofalternative statistical and economic assumptions that were madeby authors ofprevious studies,

Mroz's study is of considerable interest because, although it does notfocus on new methodological developments, it yields important results con-cerning sensitivity, and it follows up on Nakamura and Nakamura (1979;

638 Whether and How Much Women Work for Pay

198 1a; 1985a, b} in challenging the conventional wisdom oflabor economiststhat the responsivencss of female labor supply to changes in wage rates andincomeis larger than that of men.” We now examine the Mroz study and theassociated sccond-generation empiricalliterature in more detail.

Mroz begins by noting the very large diversity af reported estimates offemale labor supply responses to variations in wage rates and income.” Tomakethe results of various studies comparable, Mroz evaluates the impliedeffects of these studics on annual hours worked in 1975, assuming that thewife's wage is $4.50 and her hours worked are 1500, the husband’s wage is$7.00 and his hours are 2000, the household nonlabor incomeis $1000, andthe labor and nonlabor marginal tax rates are 0.339 and 0.280, respectively.Mcoz’s evidence on the wide range oflabor supply response estimates is pre-

sented in Table 11.2, which reproduces Table | of Mroz[!987, p. 766]. Thereit is scen that while the vast majority of uncompensated wage effect estimates

Table 11.2 MROZ's Survey of Estimates of Marrled Women's Labor Supply

Responses

Estimation Method Wage Effect incomeEffect

[1973] Instrumentalvariables 29 ~16.9‘Cogun[1980a] Tobit 865 —32,3Cogan (1980a] Instrumental variables 349Cogan [1980b} Tobit 632Cogan [1980b) Fixed costs 196Cogan[1981] Fixed costs 269Greenhalgh [1980) Instrumentalvariables 213Hausman [!981a] Convex budgetsets 328 OfHausman[19813] Nonconvex budget sets 335Hausman [1981a] Fixed costs 305

Heekman [1976a] Tobit 1462Heckman [19763] Generalized Tobit -499

Heckman[1980] Generatized Tobit i401

Layard etal. [1980] Tobit 128Layard et al. [1980] Instrumental variables 22

Leuthold [1978] 1967estimates 14Leuthold [1978] 1969estimates 45Leuthold [1978] 1971 estimates 33Nakamura and Generalized Tobit -16

Nakamura (198ta)Schultz (1980] Tobit 123 -67.0

Schultz [1980] Instrumentalvariables —26 19 ‘eter Ties effects are evaluated ot 3 wife's wage of$4.30, wife hours at £500, buaband's wage of $7.00, ustand?s hour at2000, huosehold nantshut income of 51000, and Tabor and rontabar raarginal tax rite of €.339and 0.280especie, Theiacomne effect is per $1000 in 1975 dtiare‘Suurce Data from Thamas A. Mroz “The Serutvtyofan Empirical Model of Married Women’s Hours uf Wark to Economica SiatitenAtsumptionn” Economica. Vol. $8, No.4, 1987, TAB 1,9. 766

11.3 Econometric Issues 639

ate positive, the range is substantial, varying from —499 to +1462 for a $1change in the wagerate; estimated incomeeffects per $1000 in 1975 dollarsare primarily negative but range from — 125 to +51. Further, there does notappear to be any obviousrelationship between the estimation method thatisemployed and the magnitudeorsign ofthe estimated labor supply response.

Mroz proceeds by choosing a simple functional form for the labor sup-ply equation, one that Nicholas Stern [1986] derived explicitly from an indi-rectutility function. This labor supply equation is

H, = dy + a, In(W) + a ¥, + aZ, + (11.48)

where H,is the ith woman's hours of work in 1975, HV, is a measure of herwage rate, V, is a measure of other incomereceived by the household inthousands of 1975 dollars, Z, is a vector of additional control variables(including the wife’s age, her years of schooling, the numberofchildren lessthan six years old in the household, and the numberofchildren between theages ofsix and 18), and¢; is a stochastic disturbance term. Al a given wagefate, Mroz computes the uncompensated wageeffect on hours worked for a$1 wage rate change as simply dO’, = a,/W’, and the income effect asof/aV’, = a;. The corresponding uncompensated wage and incomeelastici-ties are then computed as 3 in H./d in W, = a,/H,and dln Hfain Vi = aVsH,, respectively.

The 1976 PSID data sample (for year 1975) is employed by Mroz in thisstudy because only in 1976 did the PSID survey directly interview wives inhouseholds. Duringall other years the household head supplied informationaboutthe wife’s labor market experiences during the previous year. As Mroznotes, “one suspects that the own reporting is more accurate,andit is for thisfeason that manyrecentstudies of married women’s labor supply have usedthese data.”* Annual hours worked H, is measured as the product of thenumberofweeks the wife worked for pay in (975 and the average number ofhours of work per week during the weeks she worked. The wage rate 1’, ismeasured as average hourly earnings, calculated as total labor income of thewife in 1975 divided by H,. Note that if there is a random measurement errorin annual hours worked, a spurious negative correlation will emerge between.hours worked and this wage rate measure. Finally, the Property income mea-sure V, is defined as the houschold’s total money income minus the wife'slabor income,in thousands of 1975 dollars.

For a “base case estimation,” Mroz reports results when parameters inEq.(11.48) are estimated by OLS and the sampleis restricted to those whowork(recall from Section 11.3.B.1 thatthis is Procedure It)

Hy = a — 17 n( WW) — 4.2 V, — 342 KL6, ~ 115 K618, + others(81) 3.2) (131) (29)

(11,49)

Here KL6,is the numberofchildren underage six in the household, K618,is the numberofchildren between ages six and 18, numbers in parentheses

od Whether and How Much Women Work for Pay

are estimated standard errors (adjusted for arbitrary formsofheteroskedastic-

ity, using the Halbert White [1980] procedure), and “others” refers to the

additional variables (wife's age and wile’s education), whose paramcleresti-

mates, along with the intercept, are nol reported by Mroz. Note that these

OLS estimates imply that both the uncompensated wageeffect and income

effect are negative (the elasticities are — 17/1500 = —09l 13 and —4,2/1500

= —0,0028, respectively) and that the presence ofchildren in the household.

(particularly those under age six) reduces labor supply substantially, other

things being equal. The negative OLS estimated wage effect is not surprising.

since as was noted above, if there is a random measurementerrorin //,, then

a spurious negativerelationship occurs between #/, and 1, ;

With Eq. (11.49) as a basis of comparison, Mroz then examines the

effects of three variations: (1) alternative exogeneity assumptions on the

fegressors. issues that are raised in particular by Nakamura and Nakamura

[1981a]; (2) statistical controlforself-selection into thelabor force—a classic

issue; and (3) the impacts of controlling for taxes. We begin by summarizing

results that occur once Mroz controls for possible correlation between regres-

sors and the disturbance term in Eq. (11.48) through the use of an instru-

mentalvariable procedure such as two-stageleast squares QSLS).

Treating In (3¥,) as an endogenous variable that might be correlated

with the disturbance term ¢, in Eq. (11.48). Mroz uses as instruments the

wife’s reported labor market experience (the number of years the woman has

worked for pay since her eighteenth birthday), this variable squared,and seve

erat different polynomialsin the wife's and husband's age and education. This

variant on Procedure II {sce Section 11.3.B.1) results in an estimated 2SLS

equation whose wage and income parameters are much different from the

OLSestimates of Eq.(11.49):

H,

=

aa + 672 In (IK) — 64, — 283 KL6,— 85 K618, + ---(217) (3.6) (47) (36F

(11.50)

Note that unlike the OLS estimates, here the 2SLS estimate ofthe uncom-

pensated wage response is positive and highly significant; in fact. this large

2SLS estimate on the wage coefficientis similar to some ofthe largest wage

effects reponed in Tabie 11.2. The 2SLS incomeeffect, however, is more neg-

ative than the OLS estimate, and the 2SLS estimates of theeffects of children

onlabor supply, while also negative, are smaller than OLS estimates. Further,

using the clasticity formulae underneath Eq. (11.48) and evaluating deriva-

tives at H, = 1500 and V, = I (recall ¥, is measured in thousands of 1975

dollars), we see that the implied 2SLS. uncompensated wage clasticily estimate

is 672/1500 = 0.448, while the income elasticity estimate is very small,

—6.41/1500 = —0.0043.

Fellowing 3 conjecture by Nakamura and Nakamura [1981a, p. 480],

Mroz then examines whetherthe disturbance term in Eq. (11.48) reflects in

part the unobservable preferences of the individual, inctuding her attitude

IL3 Econometric issues 64

toward work. If this were true,it is likely that a variable such as the wile’labor market experience would alse be correlated with the disturbance ternand thatas a result this variable andits square would not be legitimate instruments. Mroz therefore recstimates Eq. (11.48) by 2SLS but excludesthe experience variable andits square as instruments. Thisyields an equation with ;much smaller andstatistically insignificant estimated wageeffect:

H, = dy + 46 In (WV) — 4.4, - 337 KL6, - 112 K618, + ~~(220) (3.3) «31 (30)

(151With the experience variables excluded as instruments the uncompensatecwageelasticityfalls to 46/1500

=

0.031, and the incomeelasticity to —4.4#1,1500 = —0.0029. Such low estimates match someof the smallest estimate:reported in Table 11.2. The estimated effects of children on labor supplyhowever, are not affected as muchandin factare quite similar in the variousOLS and 2SLS specifications. These results therefore demonstrate that esti-mates of the wage and incomeresponses are very sensitive to assumptionsconcerning the endogeneity ofregressors and the choice of instruments.

To makethe choice of instrumentsless arbitrary, Mroz proceeds with 3variant of the Durbin-Wu-Hausman-Whitespecification test.* In this contextthe null hypothesisis that the experience instruments are uncorrelated with«, and the alternative hypothesis is that they are correlated. Mroz showsthata test ofthis hypothesis amounts to testing whetherthe estimated wage coel-ficient in the 2SLS equation (11.50) with the experience variables included asinstruments is equal to the estimated wage cocfiicient in the 2SLS equation(11.51) with experience variables excluded as instruments. He derives the for-mulae for the asymptotic variances and covariances and obtains an asymp-totic normaltest statistic of 3.0, impiying that the null hypothesisis rejectedat any reasonable level of significance. Mroz therefore concludes that thewife's previous labor market experienceis not exogenousin the labor supplyequation and that as a result the experience variables are invalid instruments,consistent with the Nakamura and Nakamura {19813} conjecture. Interest-ingly, Mroz undertakes similar exogeneitytests for the KL6,. K618,, and ¥,variables andfinds that the exogeneityofthese children and nonwite incomevariables cannotbe rejected at reasonable levels of significance.”

Second,as was discussed in Section !1.3.B.1.a major problem with eachof the above Procedure II estimatesis that they are based on a subsample ofworking women only and therefore do not control for self-selection into thelaborforce. To examine the impactofignoringself-selection Mrozesti-mates the labor supply equation using the Heckit multistage Procedure VINEIn particular, using the full sample of 753 married women, Mrozfirst esti-mates a probit model of labor force participation in which the explanatoryvariables include KL6, K618, J” polynomials in the wife’s age and education,the education of the wife’s mother and father. the county's unemploymentfate, a dummyvariable for large city. and the wife's previous labor market

642) Whether and How Much Women Work for Pay

experience and its square (note that the wife’s wage is not included). Of the18 regressors, only the coefficients on KL6, V, experience, and its square aremore than twice their estimated standard error. Mroz then uses this probitmodelto estimate the inverse Mills ratio \ (see the discussion in Section11.3.B.1, especially that on Procedures VII and VIII) and appends this vari-

able as a regressor in a wage rate determination equation in which the othertight-hand variables are the samc as in the estimated probit equation butwhere the sampleis limited to workers (428 of the 753 observations). In thethird andfinal stage, Mroz uses thefitted value from the wage determinationequation as an instrument in a 2SLS estimation ofthe hours worked equation

(11.48) but with the inverse Mills ratio 4, added as a regressor. It is worthnoting that since the wife’s previous labor market experience andits squareare regressors in the first two stages of this Heckit procedure, this particularestimation assumes that these experience variables are exogenous.

The resulting estimated labor supply equation has a small positiveuncompensated wage effect and a smail positive incomeeffect, neither ofwhichisstatistically different from zero:

H, = ay + 122 In(W} + 3.9 V, + 53 KL6;(225) (45) (173)

— 87 K618, — 7584, + ---G4) a7 152)

In this case, when evaluated at the same wage and incomeas earlier, the esti-

mated uncompensated wage elasticity is but 0.081 (122/1500), and theincomeelasticity is 0.0026 (3.9/1500). Surprisingly, the estimated coefficienton young children KL6 is now positive (but insignificantly different fromzero), while that on K618 is still negative but considerably smailer in absolutevalue than in Eq. (11.50). Since the ratio of the estimated coefficient on 4, toits standard erroris 753/171 = 4.43, a numberthalis greaterthanthecritical

value for the normal distribution at reasonablelevels of significance, Mrozrejects the null hypothesis that there is noself-selection bias.

However, when the same multistage Heckit estimation Procedure VII

is employed butthe wife’s labor market experience andits square are omittedas regressors in the probit and wage determination equations, a rather differ-ent labor supply equation is obtained:

H, = a + 641n(W) ~ 1.0 V, — 183 KL6,(227) (9.2) (408)

— 106 K618, — 294a,+ ---(34) (467) (11.53)

Not only is the uncompensated wage effect coefficient small and insignifi-cantly different from zero, but the incomeeffect is negative, small, and alsoinsignificant. Further, the estimated coefficientonA, is insignificantly differentfrom zero, implying that one cannotreject the null hypothesis that there is

noself-selection bias.*

11,3 Econometric tssues 643

These results suggest that whether the experience variables are assumedto be exogenous has a decisive impact on whether theself-selection bias isstatistically significantin the labor supply equation. Totest for exogeneity ofthe experience variables, Mroz uses the samespecification test procedure aswas described earlier, but with the asymptotic variance formulae adjusted toinclude sample selectivity. The nui! hypothesis of exogeneity reduces again to

@ very simple test—whether the estimated wage coelticient in Eq. (11.53) isequalto that in Eq.(11.52); since the difference in estimated coefficientsis 58(122 — 64) and the estimated asymptotic standard error is 163, the nullhypothesis of exogeneity is not rejected.

‘On thebasis ofresults from these fourestimated equations, Mroz drawsa umberofconclusions, First, the estimated uncompensated wageeffect ispositive but typically very smail, provided that one treats the wife's experiencevariables as “endogenous” and/or oneallows for sampleselectivity bias in thewage determination andlabor supply equations. Further, under the sameesti-mation procedures the incomeeffect is usually negative butalso rather small,

Mroz[1987, p. 794] therefore concurs with Nakamura and Nakamura(1979; 1981a; 1985a, b] and rejects the conventional wisdom based on sec-ond-generation studies, concluding that

the small income and wage effects found in this study provide a muchmore accurate picture of the behavioral responses of working women to

variations in nontabor income and wages than those found in most pre-viousstudies.

Suchfindings suggest as well that the modest responsiveness of married wom-en's labor supply to changes in wage rates and incomeis not muchdifferentfrom that foundin studies of the labor supply of prime-aged matried males.Second, the sampleselectivity bias is significant only if the wife’s experiencevariables are treated as exogenous. Third, the wife’s labor market experiencevariables are exogenousprovided that oneallows for sampleselectivity bias,

Fourth,theresults are consistent with the view that a woman'stastes for workare unobserved omitted variables that affect both her previous and currentlabor marketparticipation and hours of work, over and abovetheir influenceon her wagerate.”

In addition 10 estimating the labor supply equation using the Heckitmultistage Procedure VII, Mroz employs the full Tobit simultaneous equa-tion estimation Procedure VI. Recall that unlike Procedures VII and VIU,which allow for discontinuities in the labor supply schedule dueto,say,fixedcosts of employment, the Tobit Procedure VI assumes continuity. In fact, theProcedure V1 estimates will be equal to those from Procedures VI] and VIIwhenever the labor supply schedule is continuous in wages. Mroz reports thathis Procedure VI Tobitestimates of the labor supply equation are

H, = a, + 261 In(W,) — 22.9 V, — 1035 KL6, ~ 97 K618, + - + -(357) (4) (140) (48)

(11.54)

$44 Whether and How Much Women Work for Pay

Hence the Tobit uncompensated wage elasticity estimate is also quite small(261/1500 = 0,174) and insignificantly different from zero, while the incomeeffect is significant and larger than those estimated above {—22,9/1500 =0.015)but is stil! rather small, Further,with the Tobit specification the nullhypothesis of exogeneityof the experience variables is rejected. Mrozthentests for this Tobit specification (11.54) as a special case of the muitistageHeckit Procedure VIII and in all cases rejects the null hypothesis, This leadsMroz to conclude that the restrictions implied by the Tobit Procedure VImust be rejected, a conclusion thatis consistent with the view that fixed costsof employment are substantial. . | |

In the third phase of his study, Mroz assesses the effectsof including ameasure of the marginal tax rate into the labor supply equation. FollowingHall [1973]. Mroz uses a linearization ofthe budgetset in which themarginalafler-tax wage rate replaces the previous wage measure and virtual income is

measured as the intercept of the linearized budget set at zero hours of work;untike Hall, however, Mroz treats both the after-tax wage rate and virtualincome as endogenous variables. Using similar estimation Procedures asbefore,” Mroz finds that the estimated wage coefficient falls and in factbecomes negative but that the magnitude of the change is at most 33 hoursandis always within 20% of one standard deviation oftheestimates obtainedwithout taxes. This leads Mroz [1987,p. 786] to conclude, “{nthe light of theother possible sources of bias examined in the previous sections, the influ-encesof taxes on the estimates of the labor supply parameters appears to beat most a second order effect.” . |

Finally. following Rosen [1976], Mroz rewrites Eq. (11.48) in after-taxform as the expanded equation

H, = a) + a,in[Wl — 7)} + bn — 1)

+ aV,+ mYV, + aiZ,+ 6 . (11.58)

where z, is the marginal tax rate and YV, is virtual income. The null hypoth-esis that married women optimally take taxes into considerationis that b =ad, = 0, while the alternative hypothesis is that 6, # 0, a; # 0. One mightalso consider testing the hypothesis that married women completely ignoretaxes when making labor supply decisions. This corresponds lo the nullhypothesis that 6, = —a, and b, = 0, while the alternative hypothesis is b,# ~ay, b; # 0. Unlike Rosen, Mrozfindsthat neither of these two distinctnull hypotheses can be rejected, perhapsreflecting the relativelylarge standarderrors on the estimated wage coefficients and income terms. |

As can be seen, therefore, the significance of Mroz’s study is thatit highlights the empirical consequences of employing alternative assump-tions concerning exogeneity, sample selectivity, and taxes on estimates ofthe labor supply responsiveness of married women. These assumptions domatter.

11,3 Econometric Iss

eS 645

44.3.0 Issues in Dynamic Labor Supply

Oneinterpretation of Mroz's results on the importance ofthe Sabor forceexperience variablesis that labor supply might better be viewed in a dynamiclife cycle context, Before concluding this chapter, therefore, we briefly over-view the rapidly growing theoreticalliterature an dynamic labor supply.

Within thelast decade, considerable theoretical research has focused ondynamic labor supply models. In dynamic models oflabor supply, economicagents act knowing that today’s decisions will have consequences in thefuture. Further, in dynamic models, the accumulation of nonhuman and/orhuman capital is modeled explicitly. Our overview ofthis literature is bynecessity brief, more detailed discussions can be found in Killingsworth[1983, Chapter 5], Killingsworth and Heckman (1986, pp. 144-179], and,inparticular, Pencavel [1986]. We begin by discussing models in which wagesat each point in time are exogenous and then examine models in which wagesare determined endogenously through human capital accumulation,

Although discussions ofthe timing of labor supply and human capitalaccumulation duringthelife cycle uf women have a longhistory in labor eco-nomics, one ofthe mostinfluential early contributions on dynamic issues isthat by Jacob Mincer [1962]. Mincer suggested that one way of thinkingthrough dynamicissues is simply to reinterpret the static analyses of laborsupplyin lifetime terms, where variables such as consumption,leisure, workat home, wages, budget constraints, and time are envisaged aslifetime vari-ables. Note that in this view, the woman's real asset holdings should be rein-terpreted as those occurring at the beginningofthe (lifetime)time period andnot as her actuallifetime or current exogenous income,

Following Killingsworth and Heckman [1986], tet us assume that thelife cycle consists of T periods, and let us then sort single-period reat wagerates in descending order so that w(1) denotes the highest real wage and w(7")denotes the lowes. For the moment, ussume that consumptionandleisure atdifferent points in time are pertect substitutes. Under these conditions anindividual will work sometime duringherlife if #(1) exceeds her lifetime res-ervation or shadow wage rate A{RS(O), where the latter is the mutrginal rate ofsubstitution evaluatedat zero lifetime haurs of work. To determine the totalnumberof periods worked, compare discounted real wage rates to (he MRS;the individual will work exactly & time periodsif

w(k) = MRS(K) > wik + 1) (11.56)

where at least oneofthe inequalitiesis strict." Thus the total number ofperi-ods worked & is a function of an unobserved taste or houschold productionvariable ethat aflectslifetimeutility, real initial wealth A, and the “marginalwage"w(K), that is,

k= KDWAT, e] (1.87)

646 Whether and How Much Women Work for Pay

In this case the proportion ofall periods in the woman’s lifetime that is

devoted to work, #. is simply A = &/T. which together with Eq. (11,57)

implies that

h = Alntk), A. T, e] (10.58)

Onepractical problem with implementing empirically equations such

as Eq.(11.57) or Eq. (11.58)is that it would appearthat each requires & or /t

data on labor supply over the entirelife cycle——a truly formidable obstacle.

As Mincer[1962] and Heckman [1978] have noted. however,ifone abstracts

from “transitory”factors, one might assumethatthe timing of work over the

life cycle is random.If this were true, andif all individuals worked at some

point in their lives, then one could estimate parameters of Eq. (11.58) by

replacing /t (whichrefers to lifetime participation) with, say, Q, where @ now

refers to participation as of a given date; the resulting parameter estimates.

could then be used to obtain estimates ofelasticities of labor supply with

respect to (permanent) wage changes and with respect to income(here,initial

wealth).While apparently promising, however, this approach suffers from the

practical ambiguity of determining the appropriate measure of the “marginal

wage” w(k) that could clearlydiffer from the wage prevailing as of the date

referenced by the Q variable; specifically. while data onlifetime labor supply

might not be necessary, onestill needs to be able to determine which partic-

ular wage rate—of al! the wages the individual will carn during her lifetime—

happensto be the appropriate marginal wage rate in Eq. (11.56).

Another problem with this approach, emphasized by Killingsworth and

Heckman[1986, pp. 148-149], is that using estimates of Eq.(11.58) with @

replacing / to obtain measures of substitution and incomeeffects is appro-

priate only when

A

is strictly positive, that is, only when individuals have an

interior solution 10 their lifetime labor supply optintization problem. As

Heckman (1978) notes, although appropriate data are rather scarce, there is

reasonto believe that a nonnegligible portion ofthe female population never

undertakes market work for pay. Therefore this approach might not provide

usefulestimates of income and substitution effects.

Muchrecent work has focused on modeling more explicitly the factors

that affect an individual's equilibrium sequence oflifetime labor supply, lei-

sure, and consumption;the solution to thelifetime optimization problem is

often called the individual's equilibrium dynamics. A common approach is

to assumeperfect certainty and specify lifetime utility as an additively sepa-

rable utility function

rv= Vast syei(cw, LO) (11.59)

=

where Tis fixed, L(f) + (4) exhausts all available time during period ¢, 5 is

the individual's subjective rate of time preference, and t[-] is the utility func-

fat

11.3 Economettic Issues 647

tion for period 1, depending on consumption C(¢) andleisure L(2). Inciden-tally, in this additively separable specification, C(s) and C(z’), as well as L(0)and L 1), are not assumed to be perfect substitutes for ¢ # £5?

Lifetimeutility is maximized subject to a lifetime budget constraint

rAQ) + SO + yt FO) — PCI) = 60)

=o

where A(0)is the individual's initial asset holdings, r is the market rate ofinterest, and (2), H(¢), and P(1) are the wagerate, hours of work, and pricelevels in period 1, respectively.

More formally, using Lagrangian procedures, maximize Eq. (11.59)over C(t) and H(z) subjectto the lifetime budget constraint(11.60), and inter-pretthe Lagrangian multiplier in front of Eq. (11.60), say, A, as the marginalIifetime utility of initial asset holdings. Thefirst-order conditions yield anintertemporal equilibrium plan, given a set ofwage rates andprice levels P(r)and Pt), 2 = 0,..., T. for C(t), £0), and A. Onecan ofcourse assess howthe equilibrium paths of differing individuals would vary as a result of theirholding different -1(0) or facing diverse wage rates #¥(1); such mental experi-ments are called comparative dynamics and should be distinguished {rom theindividual's initial intertemporal equilibrium sequence,called her equilil-riwn dynamics.

| In carrying through comparative dynamics, one mustofcourse be care-ful in specifying what is being held fixed. One commonapproach,called theFrisch 4-fixed or Frisch demand notion, examines the equilibrium timepatheffects of a changein, say, (1), treating the marginallifetimeutility ofassetsa as fixed; an alternative procedure, dubbed A-variable, allows this marginalutility ofassets to vary in response to a change in #(2)."In either case it canbe shown thatvariation in a wage at any given date may have consequencesnotonly at that peint in time, but also at other dates.

; It is worth noting here that within the labor economics literature, muchink has been spilt concerningthe effects ofchanges in “permanent”or “tran-sitory” incomeonlabor supply overthelife cycle. Controversy has occurred,

for example, concerning whether one should include only measures of per-manentincomein labor supply equations or whether one should also includea measure oftransitory income. Although muchofthisliterature is informalrather than rigorous, it can be assessed within a comparative dynamicsframework.

Specifically, define permanent income W’, as the present value of thestream of anindividual's future rates from ¢ = 0 tor = T, discounted at rater, and transitory income w,as the difference between the actual wage (1)and the permanent wage W,, that is, w, = 4() — W, AsKillingsworth andHeckman [ 1986} emphasize, empirical measures of W, have been constructedin practice by using essentially ad hoc procedures, dependingin large part onthe nature ofthe available data.

648 Whether and Row Much Women Work for Pay

‘One bodyofliterature adopts whatKillingsworth and Heckman [1986]

call the PO position, according to which hours of work and labor foree pat-

ticipation in any period ¢ depend on the permanent wage only, analogous to

Mikon Friedman’s [1957] permanent incometheory of consumption. Under

the PO hypothesis, one need never include the transitory wage measure w, in

ati equation such as Eq.(11.50), since its coefficient should be zero.

An alternative to the PO position,called PT, hypothesizes that both MH,

and w(1) should appearin labor supplyequations; moreover. accordingto this

literature, the coefficient on w, should be positive and algebraically greater

than the coefficient on H,, since the latter represents the sum of a positive

substitution effect and a negative incomeeffect, whereas the formerreflects

only a positive substitution effect.”

Using comparative dynamic procedures with both A-constant and d-

variable, Killingsworth and Heckman [1986, pp. 156-158] show that con-

trary to PO,leisure and labor supply arc responsive to transitory wage differ-

ences, owing to thefact thatlife cycle asset accumulationis affected. Further,

contrary 10 PT. it is not necessarily the case that labor supply at time 7 is

positively correlated with ,. However, if one examines instead the compar-

ative dynamics of changes in exogenous income(such as the incomeofother

famity members) on an individual’s equilibrium dynamics (rather than

changes in her 4’, or w,), it turns out that labor supply depends only on per-

manent exogenous income and not on transitory exogenous income.**

Not surprisingly, Killingsworth and Heckman conclude that much of

the ambiguity in the transitory-permanent income controversyarises because

of a lack of proper theoretical foundations. As an alternative, they suggest

employing the Frisch demand framework, in which the marginal utility of

assets X constitutes a type of “permanent wage” and variations in the

observed wage #1/(4) with d constant correspondtoa type of “transitory” wage

variation. since with A constant it can be shownthat diffegences in f’(r) must

always be negatively correlated with L(1}.Ourdiscussion on dynamic labor supply to this point has assumed that

wages are exogenous. By contrast, and as discussed morefully in Chapter 5

ofthis book {see especially Section 5.1},there is a great deal of literature sug-

gesting that wage rates for women are endogenous.affected in large past by

imtermittent labor supply and expectations concerning child-rearing. More

specifically, since current wagerates dependin part on humancapital endow-

ments, and since human capital formation is affected by educational attain-

ment and on-the-job training, the choices that women make concerning mar-riage, child-rearing, and work affect their time paths of wagerates. As a result,

in the more complete and necessarily more complex dynamic models of

female labor supply, educational attainment, marriage, the desired number

and spacing ofchildren, the time path of market labor supply, and the timepaths of wages and consumptionare all endogenous.

In large part because of their mathematical complexity, formal theoret-

ical models ofjoint labor supply and wage determination have not yet yielded

U4 Additional Remarks on Econometric Analyses 649

much empirical insightinto the dynamics of women’s work and wages. How-ever, this topicis currently attracting a great dealofattention both from the-orists and from econometricians.”

ADDITIONAL REMARKS ON ECONOMETRICANALYSES OF LABOR SUPPLY

As we haveseen,the literature onfactors that affect female labor supplyvides a useful example of how economic theory, measurement issues, andeconometric technique caninteract 10 enhance our understanding of impor-tant public policy issues, Butthis literature has also amplyillustrated that inSpite of enormousprogress in theory, technique, and, to some extent, mea-surement, it is still frustratinglydifficult to “nail down”precise estimates ofimponantclasticities. While critics of recent econometric practice mightinterpret this difficulty as further evidence that the payofis of state-of-the arteconometric research are often meager, many economelricians who work onlabor supply remain strongly optimi: Killingsworth [1983, p. 432], forexample, concludes his summary of what we do and don't know about laborsupply by stating, “At least temporarily, then, uncertainty about. and varia-tion in, actual magnitudes oflabor supply estimates seems larger thanit usedto be.” Whether Killingsworth’s 1983 assessmentthatthis large rangeofelas-licity estimates is but a temporary condition in the history of the analysis oflabor supply remainsto be seen, .

Finally,it is worth noting that somestudents oflabor supply argue thatnereason for the large variability in labor supply elasticityestimates, whichis evident in Table 11.2, is that existing theoretical models of labor supply aresimply too crude. Although we have attempted in this chapter to discuss thecurrentlabor supplyliterature in a relatively thorough mannerand have dem-onstrated some ofits complexity, in fact the labor supply literature isextremely large, muchofitstill needs to be synthesized, and by necessity wehave ignored potentially important strandsofresearch. For example, we haveonly skimmed theliterature on family or household d ion-making pro-cesses, as well as the significance of hours worked quantity restrictions onlabor supply.” Further, we have entirely overlooked labor supply ues Concerning the quality of work, the amountofleisure time spent at work, and thetisk and uncertainty ofjob search. In terms of econometric matters we haveconsidered only the relatively simple limited dependent variable models.*and among other omissions we have not dealt with issues concerning the pos-sibility of obtaining better econometric estimates of labor supplyelasticitiesby employing data from well-designed experiments,”” nor have we examinedissues ofspecification,estimation, and inference whenthe labor Supply obse:vations are taken from a pane! datasel.” Fortunately, readers whoare inter-ested in pursuingthese related issues can begin by working through the tefer-ences given here and those provided in leading labor economics textbooks,such as thosereferenced al the end ofthis chapter.

650 Whether and How Much Women Work for Pay Exercises 65)

44.5 HANDS ON WITH LIMITED DEPENDENT VARIABLE y Manyofthe data sets that are used to estimate models oflabor supplyTECHNIQUES TO ASSESS THE LABOR SUPPLY OFMARRIED WOMEN

We have now cometo the hands-on portion ofthis chapter. The goal of the

following exercises is to engage you in obtaining firsthand experience inimplementing and interpreting alternative procedures that attempt to mea-sure factors affecting female tabor force participation and hours worked; inparticular. you will have the opportunity to carry out on your own a number

of estimation methods, including most of the second generation proceduresdiscussed in Section 11.3.B.1.

Note: Althoughthe exercisesofthis chapter have been carefully designedto introduce you to someofthe most important computational and interpre-tive issues in the econometric analysis of labor supply, they are but an intro-duction, and yourinstructor may use the data set in this chapter for otheruseful and more elaborate exercises. For example, although data on the hus-band’s hours of work and wages are provided in the MROZdata file on yourdiskette, we make no use of those data in the exercises. Such data could beuseful in a model offamily labor supply. Alternatively, your instructor mightwantto provide you with a different set of data. Hence we urge you to en-visage these exercises as a beginning step to understanding better the applic-ability of limited dependent variable procedures to the analysis of laborsupply.

Theexercises of this chapter can be summarized as follows. Because itis extremely important to examine data sets before becoming involved ineconometric estimation, we highly recommendthatall readers work throughExercise 1, “Inspecting Mroz’s 1975 Panel Study of Income Dynamics Data,”Moreover,this exercise is essential because in it you generate two dataseriesthat are necessary for completion of subsequent exerciseg. In Exercise 2 weintcoduce you to a commonbutinappropriate estimation procedure in whichan hours worked equationis estimated by OLS with hours of nonworkers setto zero (Procedure I). Then in Exercise 3 we invite you to explore the femalelaborforce participation decision using OLS,probit, and logit estimation pro-

cedures and to examine numericalrelationships among them. In Exercise 4we ask you to compare the conditiona! OLS and Tobit estimators ofthe hoursworked equation (Procedures 1] and [1f) and to implement the Goldberger-Greeneprocedure for assessing the OLS bias. Next, in Exercise 5 we have youconsideridentification issues when both the wage rate and labor supply equa-tions are estimated by using Procedure IV. In Exercise 6 we lead you throughestimation of the hours worked equation based on theincreasingly commonHeckit multistage procedure (generalized Tobit), which Killingsworth calledProcedure VIII. In Exercise 7 we have you estimate a Heckit model in whichtaxes are taken into account, and finally, in Exercise 8 we work with you inspecifying and estimating an extended Tobit model (Procedure V1).

Cercecocce

are large,often involving over 16,000 observations, and it is simply infeasibleto provide such a large data base on your 360K data diskette. However, the1976 PSID data set employed by Mroz [1987]in his sensitivity study is usefuland convenient,since it is rather small(it contains 753 observations on 19variables and occupies only about 60K ofspace on yourdata diskette), anditpermits replication of some of Mroz’s most important findings.”

In the subdirectory called CHAPILDATofyourdata diskette you willfind a data file called MROZ.This data file contains 753 observations on mar-ried white women aged 30-60 in 1975 for i9 variables. The first 428 obser-vationsare those for women whose hours ofworkin 1975 were positive, whilethe final 325 observations are those for women who did not work for pay in1975, Each 80-column row ofthis data set corresponds to one observation.

Thefirst variable, LFP. is a labor force participation dummyvariablethat equals 1 if the woman's hours of work in 1975 were positive; olherwise,it equals zero. WHRSis the wife’s hours of work in 1975, while KL6 andK618 indicate the numberofchildren in the houschold under age six andbetween ages six and 18, respectively. WA is the wife’s age in years, WEis thewife’s educational attainment in ycars of schooling, WWis the wife's 1975average hourly earnings in 1975 dollars, and RPWG is the wife's wagereported at the time ofthe 1976 interview,in dollars. The HHRSvariable isthe husband's hours worked in 1975, HA is his age, HE is his educationalattainment in years of schooling, and HW is his 1975 wage in 1975 dollars.FAMINCis the family incomein 1975dollars; henceto calculate the wife'sproperty income, one must subtract the product of WW and WHRSfromFAMINC. MTRis the wife’s marginal tax rate evaluated if her hours ofworkwere zero. MTRis taken from published federal tax tables (it excludes stateandlocal incometaxes but includes any applicable social security benefits).WMEDisthe wife’s mother’s years of schooling, and WFEDis the wite’sfather’s years of schooling. UN is the unemploymentrate in the county ofresidence,in percentage points, while CIT is a dummyvariable that equals |if the family lives in a large city (a Standard Metropolitan Statistical Area,SMSA);otherwise,it equals zero. Finally, AX is the wile’s previous labor mar-ket experience, in years. Further details on these variables are given in theheoe file in the subdirectory CHAPII.DAT and in Mroz

EXERCISE 1: Inspecting Mroz’s 1975 Panel Study of IncomeDynamics Data

The purpose of this exercise is to help you become familiar with salient fes-tures of the data series in the MROZdata file. While exploring this data set,you will compute arithmetic means, standard deviations, and minimum and

652 Whether and How Much Women Work for Pay

maximum values for variables in the entire sample and for the data sortedinto various subgroups. You will also construct and save two variables thatwill be used in the subsequent exercises ofthis chapter.

(a)

(b)

(c)

To check whether the data are the same as those employed by Mroz

{1987, Table 111, p. 769], compute and print outthe arithmetic meansand standard deviations ofeach of the 19 variables in the MROZ datafile, using the entire sample of 753 observations. The results that youobtain from sucha calculation should equal (for each variable named,

followed with parentheses enclosing first the mean and then the stan-dard deviation): LFP (0.56839, 0.49563), WHRS (740.57636,871.31422), KL6 (0.23772, 0.52396), K618 (1.35325, 1.31987), WA(42,53785, 8.07257), WE (12.28685, 2.28025), WW (2.37457, 3.24183),RPWG (1.84973, 2.41989), HHRS (2267.27092, 595.56665), HA(45.12085, 8.05879), HE (12.49137, 3.02080), HW (7.48218, 4.23056),FAMINC (23080.59495, 12190.20203), MTR (0.67886, 0.08350),WMED(9.25100, 3.36747), WFED (8.80876, 3.57229}, UN (8.62351,3.11493), CIT (0.64276, 0.47950), and AX (10.63081, 8.06913). Doyourresults match those of Mroz? (Because ofdiffering rounding con-ventions, your computer software program might generateslightly different valucs. But the means and standard deviations should be veryclose to those reported here.) Also compute minimum and maximum

values for each of these variables; this is a particularly useful practice,since by doing this, one can often spot data coding errors. Are any ofyour min-maxvalues “suspicious? Why or why not?Now comparethe sample ofworking women(thefirst 428 observationsin the MROZ data file) with those not working for pay in [975 (thefinal325 observations). Compute arithmetic means and standard deviationsfor cach of the 19 variables in these two subsamples, then print andcompare them. Mroz notes that arithmetic means in the working andnonworking samples are quite similac for variables such as WA, WE,K618, HA, HE, and HHRS.Doyoualso find this to be the case? How-ever, meansin the working and nonworking samples tendto differ more

for variables such as KL6 and HW. Howdotheydiffer, and what mightthis imply concerning the reservation wages of womenin these two sam-

ples? Why? To the extent that the previous labor market experience ofwomen (AX)reflects their preferences or tastes for market work, onemight expect means ofAX to differ between the working and nonwork-ing samples. Is this the case? Interpret this difference. Are there anyotherdifferences in meansofvariablesin the two subsamples that mightaffect labor force participation or hours worked? Ef so, comment on

them.In the model of labor supply estimated by Mroz,it is assumed that inmaking her labor supply decisions the wife takes as given the house-hold’s entire nonlabor income plus her husband’s labor income. Mroz

d

Exercises 653

calls this sum the wife’s property income and computes it as total familyincome minusthe labor income earned by the wife. For the entire sam-ple of 753 observations, computethis property incomevariable (named,say, PRIN) as PRIN = FAMINC — (WHRS - WW).Also calculate andprint its mean and standard deviation; these should equal 20129 and11635, respectively. Save PRIN for use in subsequent exercises ofthischapter.One ofthe variables that is often employed in empirical analyses oflaborforce participation is the wage rate. As was emphasizedearlier inthis chapter, however, the wage rateis typically not observed for womenwhoare not working. Some analysts have attempted to deal with thisproblem (albeit in an unsatisfactory manner—see Section 1 (.3.B.3) byestimating a wage determination equation using data on workers onlyand then using the resulting parameterestimates and characteristics ofthe nonworking sample to construct fitted or predicted wages for eachofthe nonworkers.

Restricting your sample to workers (the first 428 observations},take the naturallogarithm ofthe wife’s wagerate variable WW andcallthis log-transformed variable LWW. Compute and print out the meanand standard deviation of LWW for this sample. Then, for the entiresample of 753 observations, construct the square ofthe wile’s experiencevariable and call it AX2, thatis, generate AX2 = AX*AX< (and, for lateruse, the squareofthe wife’s age, WA2 = WA*WA), Next, following thehumancapitalliterature on wage determination summarized in Chapter5 ofthis book and using only the 428 observations from the workingsample, estimate by OLS a typical wage determination equation inwhich LWWis regressed on a constant term, WA, WE,CIT, AX, andAX2. Does this equation make sense? Why or why not? Then use theparameterestimates from this equation and values of the WA, WE, CIT,AX, and AX2 variables for the 325 women in the nonworking sample1o generate the predicted orfitted log-wage for the nonworkers. Call thisfitted log-wage variable for the nonworkers FLWW, Computethe arith-metic mean and standard deviation of FLWW for the nonworkers, andcompare them to those for LWW from the working sample. Is the dif=ference in means substantial? How do youinterpret this result? Finally,fortheentire sample of753 observationsandfor use in subsequent exer-cisesof this chapter, generate a variable called LWW

1

for whichthefirst428 observations(the working sample) LWWI = LWW andfor whichthe last 325 observations (the nonworking sample) LWWI = FLWWfrom above. Note that your constructed LWW

1

variable should includeeither the actual or a predicted wagefor each individual in the sample.To ensure that you have computed the data series LWW! correctly,compute and print out its mean and standard deviation; they should.equal 1.10432 and 0.58268,respectively. Save the LWW

|

data series foruse in subsequent exercises ofthis chapter,

654 Whether and How Much Women Workfor Pay

EXERCISE 2: Estimating the Hours Worked Equation Using Procedure |

The purpose ofthis exercise is to introduce you to a common,but unfortu-nately inappropriate, procedure for estimating the hours worked equation.Specifically, in this exercise you implement Procedure 1, in which youesti-mate by OLS an hours worked equation using the entire sample of 753 obser-valions and predicted wages for nonworkers and you set hours worked fornonworkers equal to zero. Such an equationis often called a truncated nermalregression equation. You also compute the implied responses to changes in

the wage rate and in property income, both in level and in elasticity form.

{a) Inspect the WHRSvariable (wife's hours worked) and verify that when-ever the LFP(labor force participation) variable equals zero, the valueof WHRSis also zero. Then,using the OLS estimation procedure andtheentire set of 753 observations in the MROZ data file, regress WHRSon a constant term and on the variables KL6, K618, WA, WE, LWW!(constructed in part (d) of Exercise 1), and PRIN (constructed in part(c) of Exercise 1). Do thesigns of the estimated parameters agree withyour intuition? Why or why not? Whatis the value of R*? Why mightthis value be so low when Procedure I is employed?

(b) Using the above OLS parameter estimates and theelasticity formulaeunderneath Eq. (11.48) evaluated at the same points as noted at the bot-

tom of Table 11.2, computetheelasticity of hours worked with respectto wages and with respect to property income. Is the wageelasticity acompensated or an uncompensated one? Why? Then,following Mroz’sprocedures, which are also discussed underneath Eq. (11.48), computethe implied response of hours worked to a $1 change in the wage rate,evaluated at the same point. How doesthis estimate compare with thosepresented in Table 11.2? Finally, compute the implied response of hours

worked te a $1000 increase in property income. Camment on how thisestimate compares with those presented in Table 11.2.

(c} Although this Procedure | is simple and easy to implement, it has anumberofserious flaws. Whatare the principal shortcomings?

EXERCISE 3: Comparing OLS,Problit, and Logit Estimates of theLaborForce Participation Decision

Thegoal of this exercise is to help you gain experience with simple limiteddependentvariable estimation techniques by computing and then comparingOLS,probit, and logit estimates of a typical labor force participation equa-tion, The numerical comparison of these various estimators is based in largepart on the work of Takeshi Amemiya [1981], who has derived relationshipsamong them.”

fa)

(b)

Exercises 655

Econometrics textbooks typically point out that if the dependent vari-able in an equation is a dichotomous dummyvariable, and if an equa-tionis estimated by OLS in which this dependentvariableis related lin-carly to an intercept term, a numberof regressors, and a stochastic crrorterm, then the resulting equation (often called a linear probability

model) suffers from at least two defects: (1) thefitted values are not con-fined to the 0-1 interval, and so their interpretation as probabilities isinappropriate; and (2) the residuals from such an equation are hetero-skedastic. Note that in our context, LFP is such a dichotomous depen-dent variable.

Using OLS estimation procedures and the MROZ data file for all753 observations, estimate parameters ofa linear probability model inwhich LFP is related linearly to an intercept term, the LWW1 variableconstructed in part (d) of Exercise 1, KL6. K618, WA. WE, UN,CIT,

thewife's property income PRIN (constructed in part (c) of Exercise 1),

and a stochastic error term. Do signsofthese OLS estimated parametersmake sense? Why or why not? Comment on the appropriateness of

using the OLS estimated standard errors to conducttests of sta 1significance. Next retcieve and print outthe fitted values from thisesti-mated linear probability model. For how manyobservationsare the fit-ted values negative? For how many are they greater than 1? Why doesthis complicate the interpretation of this model? What is &? in this

model? Does it have any useful interpretation? Why or why not?Onepossible estimation procedure that is more appropriate when thedependent variable is dichotomousis based on the assumption that thecumulative distribution of the stochastic disturbances is the logistic;the resulting maximum likelihood estimatoris usually called logit.”

With LFP as the dependent variable and with a constant term,LWWI (from part (d) of Exercise 1). KL6. K618, WA, WE, UN,CIT,and PRIN (from part (c} of Exercise 1) as eaplanatoryvariables, use theentire sample of 753 observations in the MROZdatafile and estimateParameters based on a logit maximum likelihood procedure. Do thesigns of these estimated logit parameters make sense? Why or why not?Whichof the estimated logit parameters is significantly different fromzero? interpret. Does the nonlinear logit computational algorithm inyour computer software program reach convergence quickly, after only

a few iterations(say, fewer than five)? Some computer programsprovide“goodness-of-fit” output for the estimated logit model, such as a pseudo-R? measure or a measure indicating what percentofthe predictions are

“correct.” Check your computer output and manual!for the interpreta-tion of any such goodness-of-fit measures.” Finally, compare yourlogitestimates to the OLS orlinear probability mode! estimates from part (a).In particular,following Takeshi Amemiya [1981], each of the OLS slope

Parameterestimates should approximately equal 0.25 times the corre-sponding logit slope parameter estimate. How well does Amemiya’s

656 Whether and How Much Women Work for Pay

«c)

{d)

(e)

approximation work in this sample? Further, Amemiya shows that eachof the OLS intercept and OLS dtunmyvariable imtercept terms shouldapproximately equal 0.25 times the correspondinglogit parameteresti-mate, plus 0.5. Do your OLSandlogit intercept and dummyvariableintercept terms correspond well with Amemiya’s approximation?

Another commonestimation procedure thatis used in the estimation ofdichotomous dependent variable models is based on the assumptionthat the cumulative distribution of the disturbances is normal; this isusually called the probit model. Likelihood functions based on the

probit model have been discussed in Section 11.3.B.1 of this chapter.With LFP as the dependent variable and with a constant term,

LWW1(from part (d) of Exercise 1), KL6, K618, WA, WE, UN,CIT,and PRIN(from part (c) of Exercise 1) as explanatory variables, use theentire sample of 753 observations in the MROZdata file and estimateparameters based on a probit maximum likelihood procedure. Do thesigns ofthese estimated probit parameters make sense? Why? Which ofthe estimated probit parameters is significantly different (rom zero?Why? Does the probit computational algorithm in your computersoft-ware program reach convergence quickly,after onlya fewiterations? Isconvergence more orless rapid than with the logit model? Asin part (b),interpret any goodness-of-fit measures that are provided as output byyour computer software program.Because the cumulative normaldistribution and thelogistic distributionare very close to each other, in most cases thelogit and probit estimatedmodels will be quite similar.” Are the sample maximized log-likelinoodsin your estimated logit and probit models similar? Which is larger?Whataboutthe signs and statisticalsignificance ofthe estimated param-eters—are they similar? The estimatedeffect of a change in a regressoron the probability of participating in the labor force. aP/AX,, is equal toPe( — P)*B;, the logit model and fUP)*B», in the probit model, wherePis the probability of LFP, 8,; and @,, are the estimated logit and probitcoefficients, respectively, on the fh explanatory variable, and /( P) is thecumulative normal function corresponding to P.”* Evaluate these esti-mated derivatives for the logit and probit models, using the sampleLEPRof 425/753 = 0.568 as an estimate of P and notingthat f(?) ={(0.568) = 0.393. At this sample mean,are the estimated effects similarfor the logit and probit models? What happens if you evaluate theseeffects at thetail of the distribution, such as at P = 0.9 where f(P) =0.175?Sincethelogistic distribution has a variation of «7/3, whereas the vari-ation from the probit modelis usually normalized to unity, one way tocompare the logit and probit estimates is to multiply cach ofthelogitestimates by Vile & 1.73205/3.14159 = 0.5513, and then comparethese transformed logit parameters to the actual probit estimates. Ame-miya [1981] argues, however, that a better approximation emerges if

Exercises 657

one mullplies the logit parameter estimates by 0.625 and then com-pares these transformed logit parameters to the probit estimates. Whichof these two approximations best transforms your logit estimates tocomparable probit estimates? Why?

(f) In this exercise we have compared OLS linear probability model, logit,and probit estimates of a labor force participation equation and havefocused on numericalrelationships amongthe estimated parameters. Aswas pointed out in Section 11.3.B.1, however, there are seriousstatisti-cal problems with each of the particular OLS,togit, and probit proce-

dures that were employed in this exercise. What are these problems?

EXERCISE 4: Relating the Tobit and Conditional OLS Estimates

The purposeofthis exercise is to engage you in the estimation and interpre-tation ofa labor supply model based on the Tobit Procedure {Lf and to enrichyour understanding of how this Tobit model relates to the conditional OLS.frameworkofProcedure II, You will also have the opportunityto implementempirically the analytical results of Goldberger [1981], Greene [1981], and

McDonald and Moffitt [1980], which were discussed in Section 11.3.B.1.

(a) First, estimate a conditional OLS model of labor supply (Procedure [1).In particular,restricting your sample to the women who worked for payin 1975 (thefirst 428 observations in the MROZdatafile), run a regres-sion of WHRSon constant term, KL6, K618,. WA, WE, PRIN, andLWW(sec part (d) of Exercise 1 for a discussion of LWW and PRIN).Compare yourresults to those obtained by Mroz, reproduced in Eq.(11.49). Your standard error estimates might differ from those reportedby Mroz,since his estimates are adjusted for heteroskedasticity using theHalbert White [1980] robust estimation procedure. If your computer

software program permits, also compute the White robust standarderrors; yourresults should be very close to those reported by Mroz.

(b) In Section 11.3.B.1 it was pointed out that these conditional OLS esti-mates are biased estimates of the parameters in Eq. (11.33). Whyaretheybiased? Following Goldberger and Greene, computeconsistent esti-mates ofcach ofthe labor supply parameters using the LFP adjustment.In particular, catculate the sample proportion of observations for whichWHRSis positive—in the MROZdatafile this is 428/753 = 0.568.Then divide each ofthe conditional OLS parameterestimates from part{a) by this proportion. According to Goldberger and Greene, these trans-formed conditional OLS estimates are consistent estimatesof thelaborsupply parameters in Eq. (11.33). Are these transformed,consistentesti-mates plausible? What can yousayregarding thestatistical significanceofthese transformed parameters? Why?

(c} Now obtain consistent estimates using the Tobit estimation method

658 Whether and How Much Women Work for Pay

(d)

(e)

(Procedure II). In particular, using the Tobit maximum likelihood pro-cedure {the sample likelihood function corresponding to Eq. (11.33) isEq. (11.32) with J, now equal to X,8 + iy,) and the same functionalform as in part (a), compute the Tobit parameterestimates and asymp-totic standard errors, How do these estimates compare to the Goldbergerand Greene approximations from part (b)? What can you sayregarding,

thestatistical significance of these Tobit parameters?As was discussed in Section 11.3.B.1, McDonald and Moffitt haveshown (see Eq.(11.35)) that the total effect of the change in a regressoron expected hours worked in the Tobit model can be decomposed intotwo parts: the change in hours worked for those already workingweighted by the probability ofworkingplus the changein the probabilityof working weighted by the expected value of hours worked for thosewho work. Using the sample proportion of those who work (428/753 =0.568) as an estimate of F(z), the Tobit parameter estimates from part{c), and data evaluated at sample means,calculate 4 as outlined under-neath Eq.(11.35). Note: For F(z) = 0.568, z = 0.175 andf(z) = 0.393.Ofthetotal change in hours worked due to a $1 change in the wagerate,what amountresults from changesin hours worked from those who arealready working? What amount comesfrom new entrants into the labor

force? Whatis the proportion ofthetotal effect on hours worked of achangein any ofthe variables that derives from those women who arcalready working?Thelikelihood function (11.32) for the Tobit model indicates that theTobit can be envisaged as combining a probit modelof laborforce par-ticipation with a standard regression model of hours worked for thosewho work {see Section 11.3.B.1). One might therefore concludethatifthe sample were limited to those who work, Tobit estimates would benumerically equivalent to OLS estimates. Would such a conclusion becorrect? Why or why not? Verify your intuition numerically, using datain the MROZdata file, by estimating a Tobit model,restricting the sam-ple to the first 428 observations(the workers) and-comparing these Tobitestimates with the OLS estimates ofpart (a).fn this exercise we have numerically related the Tobit and conditionalOLS estimates in a model oflabor supply. As was pointed outin Section11.3.B.1, however,there are scriousstatistical problems with the Tobitand conditional OLS procedures that were employed in this exercise.Whatare these problems?

EXERCISE 5: Identifying Parameters In a Reduced Form Estimation

The purpose ofthis exercise is to explore issues of identification of the struc-tural parameters when the reduced form Procedure [V method is imple-mented empirically. In essence, this procedure involves first estimating a wage

ia

Exercises 659

equation using a sample of workers only, then using the entire sample ofobservations with hours of nonworkers set to zero to estimate a reduced formhours worked equation derived from a reservation wage relationship, andfinally, identifying the structural parameters ofthe reservation wage equationby using the reduced form and the wage equation parameterestimates. As weshall see, the reservation wage equation is underidentified,just identified, oroveridentified depending on whetherzero, one, or more than oneregressorinthe wage equation is excluded from the reservation wage equation.

(a) Retrieve from Exercise 1, part (d), the parameterestimates from the

(b)

OLS wage determination equation in which LWW wasregressed on aconstant, WA, WE, AX, WA2, and CIT and the sample was confinedto those who worked (the first 428 observations). This corresponds tothe equation, with / subscripts deleted for simplicity,

LWW = g. + WA + g, WA2 + g, WE+ g CIT + g, AX + ew (11.61)

where the g's are OLS parameterestimates and ey. is the OLS residual.Next specify three alternativestructuralreservation wage equations, andthen using a transform of Heckman’s proportionalityrelation (11.42),write out their corresponding reduced form representations. Specifically,three possible structural representations for the log of the reservationwage (L.WR) equation, analogousto Eq. (11.41). are

Just identified:

LWR = ay + a, WA + a WA2 + a, WE + a, CIT+ ds KL6 + a, K618 + a, PRIN + a, UN + (11.62)

Overidentified:

LWR = by + b, WA + b, WA2 + b, WE + b, KL6+ b, K618 + 6, PRIN + 4,UN + ¢ (11.63)

Underidentified:

LWR = @ + ¢, WA + ¢, WA2 +c WE + CIT + 6, AX+ o KL6 + ¢, K618 + & PRIN + 6 UN + ¢ (11.64)

where PRIN is the property incomevariable construcied in part {c) ofExercise { and « is a random disturbance. Note that in Eqs. {11.62},a 1.63), and (11.64) there are one (AX), two (AX and CIT), and zerovariables, respectively, included in the LWW equation (11.61) butexcluded from the LWR equation. For each ofthese three reservationwage cquations, first employ a log-transformation of Heckman’'s pro-portionality relation (£1.42),

H, = d(LWW, — LWR,)

H,=0

iff LWW, > LWR, (11.65)

iff LWW, < LWR, (11.65b)

60 Whether and How Much Women Work for Pay

(ec)

(a)

cS

(fy

then substitute into Eqs. (11.65) for LWW the OLS estimated wage

determination equation (11.61) and for LWRone of the LWR specifi-

cations in Eqs. (11.62)-(11.64), Having done this, you should bave

obtained three alternative reduced form equations for hours worked,

each in terms of the underlying structural parameters, exogenous vari-

ables, and disturbances.Using the entire sample of 753 observations in the MROZ data file

andsetting hours of nonworkers to zero, estimate by OLS the reduced

form equation corresponding to the just identified equation (11.62)

derived in part {b), using the g estimates from Eq. (11.61) to solve

for the @ parameters in Eq. (11.62). This procedure is often called

indirect least squares, Do these structural estimates of the a param-

eters make sense? Why or why not? What about your estimate of

d? Next, employ a more direct estimation procedure. Specifically.

using your reduced form equation corresponding to Eq. (1 1.62),

construct transformed regressors variables as the product of the

original regressors in Eq. (11.61) and their estimated g coetticients,

and then estimate by OLS the resulting equation with these trans-

formed variables as regressors. Verify that you obtain the same

structural estimates of the @ and d parameters as you did with the

indirect least squares procedure. Are the standard error estimates

from this direct procedure appropriate for doing inference? Why orwhy not? .To understand complicationsthat arise when the reservation wage equa-

tion is over-identified,first show thatin the reduced form equation cor-

responding to Eq, (11.63) derived in part (b) 2 parameter restriction

must be imposed so that a unique estimateof d is obtained. Using the

entire sample of 753 observations in the MROZ data file and setting

hours of nonworkers to zero, impose this parameterrestriction, estimate

by OLS the constrained, overidentified reduced form equation. and

solvefor the b parameters in Eq. (11.63) and for d. Do these structuralestimates of the # and ¢ parameters make sense? Why or why not?

Derive, interpret, and then implement empirically a test for the over-

identified model (11.63) as a special case of the just identified model

11.62).

Show at if you were to estimate by OLS the reduced form equation

corresponding to the underidentified reservation wage equation a 1.64)

derived in part (b), there is no way that you can obtain unique estimates

of the ¢ and ¢ structural parameters. If you nonetheless estimated this

feduced form equation by OLS, what woutd the R? be relative to that

from the just identified reduced form estimation ofpart (c)? Why does

this occur?There is a serious drawback with the Procedure IV estimation you have

donein this exercise. Whatis this problem?

Exercises 661

EXERCISE 46: Implementing the Heckit Generalized Tobit Estimator

The purpose of Exercise 6 is io have you implement and interpret the Heckit

multistage Procedure VIIL. This is accomplished by having you replicateresults reported by Mroz (1987]. Youalso will compare the Heckit sampleselectivity procedure to an OLS-based method due to Olsen [1980].

(a) In the first stage of the Heckit procedure, one estimates a probit LFP

equation and retrieves from this estimated equation the inverse Mills

ratio A. Mrozfirst generates a numberofpolynomial transformationsofthe wife's age, education and experience variables to be used as explan-atory variables in the LFP equation. Forthe entice sample of753 obser-vations in the MROZdata file, following Mroz, generate AX2 oAX#AX, WA? & WAsWA, WE2 & WE*WE, WA3 = WA2*WA, WE3= WE2+WE, WAWE = WA*WE, WA2WE = WA2«WE, and WAWE2

= WA*WE2. With this sample, estimate by maximum likelihood aprobit model in which LFP is the dependent variable and the explana-tory variables include 2 constant term, Ki.6, K618, WA, WE, WA2,

WE2, WAWE, WA3, WE3, WA2WE, WAWE2, WFED, WMED,UN.CIT, and PRIN (this last variable was calculated in part (c) of Exercise1). From this estimated probit model, calculate the inverse Mills ratiofor each observation,save this variable, and call it INVRI. (Some com-puter software programsofferthis calculation as an optional command:for others, it must be computed bybrute force, using Eq. (11.37) andvalues from the normal distribution.) Now redo the probit estimation,

this time adding the experience variables AX and AX2 and calling thecorresponding inverse Mills ratio values INVR2, Commenton the sta-listical significance of parameters estimated in these two probit equa-tions. Note that the LWW! wagevariable is excluded as an explanatoryvariable in this probit model. Since econumic theory supgests that LFPis affected by the wage rate, why is this LWWvarisble excluded? Inwhat sense, however, might it be included indirectly?

{b) Next, restricting your sample to those who work for pay(the first 428observations in MROZ). estimate by OLS a wage determination eqttion allowing for sample selectivity and compare results to an equation

that does not take into account the sampleselectivity. In particular, fol-

lowing Mroz, let LWW be a linear function of a constant term, KL6,

K618, WA, WE, WA2, WE2, WAWE, WA3, WE3, WA2WE, WAWE2,WMED,WFED,UN,CIT, and PRIN.Callthis set of explanatoryvari-ables "Set A,” Estimate this equation by OLS (if your software permits,employ the White [1980] robust standard error procedure), and com-mentonthe signs andstatistical significance of the estimated parame-ters. Redo this OLS estimation, addingto Set A the experience variablesAX and AX2.Interpret any changes in results. Then, with the same 42%

162 Whether and How Much Women Workfor Pay

()

observations, estimate by OLS two wage determination equations thatallow for sampleselectivity:first, an LWW wage determination equationwith the Set A variables and the INVR1 inverse Mills ratio variable(from part (a)) included as regressors and second, an LWW wage deter-mination equation with the Set A variables included, but with the AX,AX2 experience measures and the corresponding INVR2 variable alsoincluded. Commenton the sensitivity of the estimated parameters toinclusion ofthe experience variables and to the sample sclectivily adjust-ment.Is sample selectivity significant? (You may use large sample dis-tribution theory and the White {1980] robust standard error method toconduct statistical inference.)Finally, restricting your sample to those who workfor pay {the first 428observations in MROZ), use the fitted values from the wage determi-nation equation in part (b) as instruments in a sample selectivity-adjusted two-stage least squares estimation of the hours worked equa-tion. Specifically, following Mroz, for a base case comparison, firstemploy instrumental variable (IV) estimation (with the White robuststandard error procedure ifit is available in your software} of an hoursworked equation in which WHRSis the dependent variable and theexplanatory variables include a constant term, KL6, K618, WA, WE,LWW,and PRIN;call this set of explanatory variables “Set B.” fn thisIV or 2SLS estimation, treat LWW as an endogenousvariable, and usethe Set A variables defined in part (b} to form instruments. How do yourresults compare with those reported by Mroz, reproduced in Eq.(IL51)? (Note: Mroz’s standard error estimates employ the Whitetobust standard error procedure. If your sofiware does not permitthis,your OLS standard error estimates will differ slightly from those in Eq.(11.51). Next allow for sample selectivity, but exclude the experiencevariables. Specifically, using the same Set A variables plus the INVRIinverse Mills ratio variable to form the instrument for LWW,estimateby IV or 2SLS an hours worked equation with the Set B variables asfegressors (using robust standard error methods if possible), but withINVR!added as a regressor. How do your results compare with thosereported by Mroz, given in Eq. (11,53)? Is sampleselectivitysignificant?Whyor why not? (Vote: Mroz’s standard error estimates are based on a

formula derived in his Appendix—his estimates will difler slightly fromthose of the White robust standard error procedure.) Then estimate amodel by 2SLS in which the experience variables AX and AX2 areincluded along with the Set A variables in thefirst-stage wage determi-nation equation, but in which sample selectivity is not taken intoaccount and only the Set B variables are regressors. Compare yourtesults with those of Mroz, reproduced in Eq. (11.50). Finally, includethe experience variables AX and AX2, the Set A variables, and theINVR2 inverse Mills ratio as variables in forming the instrument forLWW,andthenestimate by 2SLS the hours worked equation includingas repressors the Set B variables and INVR2. Yourresults should con-

{d)

{e)

s 663

form closely with these reported by Mroz, reproduced in Eq.(11.52). Issample selectivity important when the experience variables are treatedas exogenous? Why or why not?

Interpret yourfindings in part (c), commenting in particular on theimportance of allowing for sampleselectivity and how this sensiti isaffected by the assumption of exogeneity of the experience variables.Mroz concludes that with his preferred specifications the uncompen-

sated wage effect for his sample is smail, as is the estimated incomeeffect. Do you agree? Why or why not? (You might want to compareresults here with those from otherstudies, reported in Table 11,2.)Now compare the Heckit inverse Mills ratio procedure to the OLS-basedlinear probability model method proposed by Olsen [1980]. In particu-lar, using a mode! either with or without the AX and AX2 experiencevariables, follow the Olsen procedure outlined in Section 11.3.B.1undemeath Eq. (11,37) and use as a regressor in the wage rate deter-mination and hours worked equations,instead ofthe inverse Mills ratio,thefitted probability minus | from an OLS-estimated linear probabilitymodel. Compare these results with those obtained from the Heckit pro-cedure. Do they differ substantially?Tn whatsense is the estimation procedure ofthis exercise a generalized

Tobit estimator?

EXERCISE 7: Incorporating Income Faxesinto a Modelof LaborSupply

The purpose ofthis exercise is to enable you to assess the impacts of incor-Pporating income taxes on your estimates of the wage and income responsive-hess of labor supply. This will involve creating virtual income and after-taxwagerate variables that are consistent with a linearized budget constraint(LBC) budget constraint specification, estimating by using the Heckit gener-wen procedure, and comparing results with those reported by Mroz

1 .

@

(b)

Ourfirst task is to create several tax-related variables. For the entiresample of 753 observations in the MROZdaia file,first create and savea virtual property income variable defined as VPRIN = (1 —MTR)*PRIN, where PRIN was created in part (c) of Exercise { andMTRis the marginaltax rate variable provided in the MROZdatafile.Next, generate and save LTAX = LOG(I— MTR)andthelogarithm ofan after-tax wage variable as LTWW = LTAX + LWW1, where LWWIis the wagevariable created in Exercise J.

To implementthe Heckit multistage procedure thatallows for sampleselectivity, we must initially estimate a probit LFP equation for theentire sample of 753 observations and compute from this estimation theHeckman inverse Mills ratio. Following Mroz,first generate polynomial

364 Whether and How Much Women Work for Pay

transformations ofthe wife's age, education, and experience variables,

10 be used as explanatory variables in the LFP cquation. In particular,

for the entire sample of 753 observations in the MROZdata file, gen-

erate and save WA2 = WA*WA, WE2 = WE*WE, WA3 = WA2=WA,

WE3 = WE2*WE, WAWE = WA*WE, WA2WE = WA2*WE, and

WAWE2 = WA*WE2.With this sample, estimate by maximum likeli-

hood a probit model in which LFP is the dependent variable and the

explanatory variables include a constant term, KL6, K618, WA, WE,

WA2, WE2, WAWE, WA3, WE3, WA2WE, WAWE2. WFED,

WMED,UN,CIT, and PRIN (thislast variable was calculated in part

(c) of Exercise 1), From this estimated probit model, calculate the

inverse Mills ratio for each observation,save this variable, and call it

INVR. (Some computer software programs offer this calculation as an

optional command;forothers it must be computed bybrute force using

Eq. (11.37) and values from the normal distribution.) .

(c} Nowestimate a model withae included, similarto walPresentedin

. (11.55), by 2SLS, Specifically, using as exogenous variables a -

aot cm KL6, K618, WA, WE, WA2, WE2, WAWE, wa3, WE3,

WA2WE, WAWE2, WMED, WFED,UN, CIT, PRIN and the inverse

Mills ratio variable INVR from part (b), form an instrument for the log

of the after-tax wage rate variable LTWW (created in part (a). With

these instruments for LTWW,estimate by 2SLS a model in which

WHRSisa linear function of a constant term, LTWW, LTAX, KL6,

K618, WA, WE, PRIN, VPRIN, INVR, and a random disturbance

term. If your software permits, obtain the White robust standard ertor

estimates. Comment on the sign and magnitude of the estimated wage,

property income, and tax coefficients. Using the information given

underneath Eq. (11.35) and your estimated variance-covariance matrixof the estimated coefficients, formulate and test the hypothesis that

women optimally take income taxcs into account when making labor

supply decisions. Then formulate and test the bypothesis tha women

entirely ignore incometaxes when making labor supply decisions, Inter-

pret yourtest results. Do your conclusions agree with those reported by

Mroz and by Rosen [1976]? Why or why not? |

{d) Experiment with any two other plausible specifications for the probit,

wage determination, and/or hours worked equations with laxes

included, and commentonthesensitivity of your results to changes in

specification. Do Mroz’s results on the unimportance of income taxes

appear to be robust?

EXERCISE 8: Specifying and Estimating an ExtendedTobit Modei

In thisexercise you will specify and estimate an extended Tobitsimultancous

equations model in which parameters of the wage fate, reservation wage {and

ly, hours of work), and labor force participation equations are estl-

Exercises 665

mated simultaneously by using the method of full information maximumlikelihood (FIML), with cross-equation parameter constraints imposed. Aswasnoted in Section 11.3.B.1, Killingsworth [1983] hascalled this ProcedureVi.

(a) We begin by specifying the wage equation for workers only, whereLWWisspecified to be a linear function ofa constant term, WA, WA2,WE, CIT, AX,and a random disturbance term ex, as in Eqs. (11.38)and (11,61), Next we specify the functional formofthe reservation wageequation derived from the marginalrate ofsubstitution function.In particular, let the logarithm ofthe reservation wage equation be LWR andtet it have the functional form as in the overidentified equation (11.63)but denote the disturbance term in Eq. (11.63) by ex, as in Eq. (141),Finally, following Heckman [1974b], specify that hours worked,WHRS,is a proportion ¢ofthe difference between LWWand LWRiifLWw, > LWR,andis zero iff LWW, < LWR, analogous to Eqs.(11,65a) and (1).65b). Finally, analytically substitute into the hours

worked equations (11.65a) and (11.65b) your specifications for theLWW and LWR equations. and note that in Eq. (11.65a) the distur-bance term. now denoted dep,is equal to dep, = dew, — em). which issimilar to Eq.(11.45a) but with dreplacing b.

(by) Now assume that ey, and eg, are joint normally distributed with vari-ances oj. and o} and covariance ows. This implies that in the hoursworked equation that you constructed in part (a) the disturbance term€p, is normally distributed with variance oj, = 6% + o) — 2owe and isjoint normallydistributed with e», having covariance guy = ah ~ one.Given these distributional assumptions, write out thelikelihood func-tion for the entire sample of 753 observations, analogous to Eq.(11.47).where d now replaces and othertermsare as defined by Eq. (11.47).

(c) Using the appropriate computersoftware and the 753 observations fromthe MROZdatafile aswell as thelikelihood function constructed in part(b), estimate by maximum likelihood the parameters appearing in theextended Tobit model consisting of the wage and hours worked equa-tions. Mroz reports that whenhe estimated a model similar10 this, heobtained a large positive and statisticallysignificant coefficient on LWWbut a negative and marginally significant coefficient on PRIN. Are yourresults qualitatively similar to those of Mroz?

{d) Finally, following Mroz,specify an alternative modelin which the expe-rience variable AX is excluded from the wagerate determination equa-tion, Asin parts (a) and (b). write out the correspondinglikelihood func-tion, and then estimate parameters using the FIML method. Compareyourresults to those obtained in pact (c). Mroz found that when the AXexperience variable was omitted from the wage determination equation,the estimated FIML coefficient on LWWin the hours worked equationwas positive but much smaller than when AX was included (and statis-tically insignificantly different from zero). However, the estimated coef

666 Whether and How Much Women Work for Pay

ficient on PRIN was negative andstatistically significant, but smaller inabsolute value than when the AX variable was included in Eq. (11.61).Do yourresults conform with those of Mroz? Why or why not?The mode!specified and estimated in this exercise is based implicitly on

an important assumption concerning the continuity of the hours

worked relationship. Whatis this assumption, and under whatpractical

considerations might it be violated? What estimation procedures areavailable that do not require this strong assumption?

CHAPTER NOTES

. These data are taken from the 1989 Economic Report ofthe President, Washing-

ton, D.C.: U.S. Government Printing Office, Table B-36,p. 349.

. For a detailed discussionoffactors affecting the timeseries growth ofthe female

jabor force supply, see James P. Smith and Michael Ward [1985] and the refer-

ences cited therein.

. Reasonsfor this decline are discussed and surveyed by, amongothers, John Pen-cavel (1986).

|. To a first approximation the effect of truncation occurs when sample data are

drawn from a subset of'a larger populationof interest (e.g., those in the laborforceare selected from those in the population as a whole), whereas censoring occurs

whenvalues over a range of a variable are reported simply as beingless than (orgreater than) some given value(e.g., the unobserved reservation wage is less thanthe market wage facing an individual). Unlike truncation, censoringis essentially

a defect in the sample data.Fora survey ofissues concerning the labor supply of maies, see John Pencavel

1986}.Veeever, see theclassic study by P. Sargant Florence (1924).For a more detailed discussion of measurementissues, see, for example, Harold

‘Watts et al. [1977] and Shirley J. Smith [1983].Detailed data on U.S, government household surveys are found in two U.S.

Bureau of LaborStatistics publications, an occasional one called Handbook of

Labor Statistics, Technical Notes, and the monthlypublication called Employment and Earnings. International comparisonsof laborforce participation andinternational data sources are discussed in, among others, Constance Sorrentino11983]. The discussion that follows is based in farge part on Danie) Hamermesh

and Albert Rees [1988], Chapter 1.Issues involved in assessing factors that affect the supply of laborto the U.S. mil-itary are discussed in, among others, Cotin Ash, Bernard Udis, and Robert F.

McNown [1983].. These figures are taken from Hamermesh and Rees {1988, Table 2.5}.. Another,less commonly used data set is one that focuses particular attention on

the variety ofsources of income.Iis the U.S. Bureau ofCensus Survey ofIncome

and Program Participation, undertaken as a joint project with the U.S. Depart-ment of Health and Human Services. For a discussion ofthis data set, see U.S.

Department of Commerce, Bureau of the Census [1987].. For additionaldiscussion, see Daniel S. Hamermesh [1990].

”as

2 28,

Notes 667

. Descriptionsofthe data gathered at Ohio State are found in Parnes et al. [1970],white those ofdata collected at the University of Michigan are presented in Mor.gan et al, [1966, 1974].

}. Fora classic presentationofthe individual labor supply model, sce H. Gregg Lewis(1957),

}. This reservation wage notionis apparently ducto Jacob Mincer[1963]. Importantextensions and applications include those of Reuben Gronau{1973b] and JamesHeckman (1974b};also sce Gronau { 1986).

. The expressions in Eqs. (11.12}-(11.14) implicitly assumethat interior solutionsoccurboth before and after the small change in P,.

- This choice of nomenclature is somewhat unfortunate, since while chauvinism is

clearly undesirable, this particular model might or might not be a reasonabledescription of houschold decision making.

. For examples ofthe use of the male chauvinist model, see, amongothers, Bowenand Finegan [1965, 1969] and Hausman (1981a).

. Applicationsofthis model include Marvin Kosters [1966]: Malcolm Cohen, SamRea, and Robert Lerman [1970}; Robert Hail [1973]: Orley Ashenfelter andJames Heckman [1974]; and Jerry Hausman and Paul Ruud (1984).This duopoly type of model of labor supply has been implemented by J. S. Ash-worth and David T. Utph (1981).

. Fora discussionofsuch indirect incomeeffects, see Killingsworth (1983, pp. 36-38}.

|. The added worker hypothesis appears to have been first pointed out by WladimirS. Woytinsky[1940].‘On this, see, for example, Mincer [1962, 1966] and Belton M. Fleisher and GeorgeP. Rhodes, Jr. [1976].

}. Ronald G. Ehrenberg and Robert S. Smith [1985. p. 204] note that if one incor-

porated estimates of discouraged workers into official unemploymentrale statis-tics, thea the U.S. unemployment rate in 1973, for example. would have increasedfrom 4.9 to 5.6%. There is some evidence, however,that for a substantial aumberofdiscouraged workers the attachmentto the laborforce is loose, and soit is not

clear how one should interpret the welfare consequences of the discouragedworkereffect; see, for example, Pau! Flaim [1984]. For this and other reasons.considerable controversy exists concerning whether published unemploymentrates should include estimates ofdiscouraged workers. A usefuldiscussionofthisset of issues is found in the final report of the National Commission on Employ-ment and UnemploymentStatistics (1979. pp. 44-49].

. The discussion that follows is based in large part on Addison and Sieben {1979pp. 79-85).

. For a discussion of such family economics issues and additional references, seeGary S. Becker (1981, 1988).

. A partial list of studies includes the theoretical and econometric analyses by Maretin Browninget al. (1985), Kim B. Clark and Lawrence H. Summers [1982], Gil-bert R. Ghez and Gary S. Becker (1975), James J, Heckman [1978], James J.

Heckman and Thomas E. MaCurdy(1980, 1986], Richard Layard and JacobMincer [1985], Thomas E, MaCurdy [1981], Robert A. Moffitt [1984}, Alice Nak-

amura and Masao Nakamura [1985a,b], and T. Paul Schultz [1980].Other earlystudies cstimating the effectsoffixed costs on labor supply includethose by Giora Hanoch (1976, 1980] andJerry A. Hausman (1980).

368 Whether and How Much Women Work for Pay

29.

30.B.

32.33.

<3

35.

36.

43,

44,

45.

46.

. In practi

Thisclassification scheme was actually developed earlier by James J. Heckman,Mark R.Killingsworth, and Thomas &, MaCurdy [1981].Sce, for example, Robert E. Hall (1973).

This procedure is foliowed by cach of the authors in Cain and Watts [1973b],except for Hall (1973].Heckman,Killingsworth, and MaCurdy [198t, pp. 83-84),

Whenthis type of modelis implemented using paneldata, it is often assumed thatalthough ¢varies across people at anygiven point in time, for any oneperson itis fixed overtime. A discussion ofeconometric techniques that exploit panel datais beyondthe scope ofthis chapter: see, however, Zvi Griliches, Bronwyn H.Hall,and Jerry A. Hausman [1978]. Griliches and Hausman [1986}; and Cheng Hsiao[1986].Recall thatthe first-order conditionsforut maximization at an interior solu-tion implythat the marginal rate of substitution Af equals the real wage rate 1’,that is, that the slope of the indifference curve equals the slope ofthe budgetline.Probit analysis has a long history in biometrics; sce, for example, David J. Finney[1947] and Jerome Corfield and Nathan Mantel [1950}. For early discussions inthe context of econometrics, see James Tobin {1955] and Arthur S. Goldberger[1964, pp. 248-251].

Forfurtherdiscussion,see, itter afia, Tobin (1958), Amemiya {1973}, and Mad-dala [1983, Chapter 6]. The etymology of Tobitis unclear, although Goldberger[1964, p. 253] refers to it as Tobin's probit. Maddala (1983, p. 151] suggests thatthere is presumably no connection with the prophet Tobit or the book ofTobit inthe Apocrypha. It is known that James Tobin achieved a distinguished record inthe U.S. Navy during World Waril, including anti-submarineduty, Further, Kil-lingsworth (1983, fo. 7, p. 142) notesthat the novelist Herman Wouk wasa class-

mate of Tobin's in naval officer training school and that in his best-seller TheWoukcreated a midshipman character named “Tobit.”

. Computationalissues concerning estimation of the Tobit model have been con-sidered by. amongothers, Richard N. Rosett and Forrest D. Nelson [1975] andRayFair [1977].

. For a discussion of more general utitity functions and their implied consumerdemandand labor supply equations, see Arthur S. Goldberger[1987].

. McDonald and Moffitt emphasize that 4 will equal | only if X is infinity, in whichcase F(z) = 1 and f(z) = 0. ‘

. this is typically done byevaluating the expressions at sample meansofthe variables and using estimated parameters.

- Forfurtherdiscussion, see Olsen [ 1980). and for a generalization to accommodateother distributions, see Lung-Fei Lec [1982].

. For further discussion ofissues concerning the functional form ofthe wage determination equation (11.38), see Chapter 5 ofthis book.

See Killingsworth [1983, Table 4.1, p. 15t] and Killingsworth and Heckman.(1986, Table 2.26,p, 192]for a very brieftabular summary ofthese cight methods.Oneofthe first statements of this sample selectivity problem in the context oflaborsupply is found in Reuben Gronau [1973a).

In practice, this factor ofproportionality often applics to the differencein the nat-ural logarithms of W, and ¥,,Forfurtherdiscussion of identification issues, see Killingsworth[1983, pp. 154~156, especially footnotes 14 and 17) and Olsen [1980].

dus

tin &

Notes 669

47, There are other important problemsin analyzing thelabor supplyeffects oftaxes.To determine the appropriate marginal after-tax wage rate, researchers typicallymust employpublished statutory tax cate information. However, actual marginalafter-tax wage rates may differ substantially from statutoryrates, owing to com-plex exemptionsand deductible expense provisions, underreporting or cunceal-mentofincome by individuals, and/orto discretionary administrative behaviorbytax-collecting agencies. With the exception of William Gould [1979], hardlyany empirical work has been doneto date on the labor supply implicationsoftaxavoidance and tax evasion; however, for theoretical discussiuns ofthis Topi, seeJonathan C. Baldry [1979]. Ame J. Isachsen and Steiner Strom [1980], AgoarSandmo {1981}, and James J. Heckman (1983). Further,it

is

widely believed thatthe available data on hours worked measure actualhours worked witherror, whichimplies that the budget consiraint segmentfacing the individual might be misspec-ified. This issue has been addressed by, among others. Burtless and Hausman(1978), Hausman [1980], and Wates and Woodland [1980].

48. This is slightly beter than the cynical summaryoffered

by

George Johnson (1976,P. 107, fa. 9] concerning estimates of input substitutability in production: “InFecent years, economists have narrowedits value duwn to a ringe between zeroandinfinity.”

49. Wales and Woodkaind [1979] and Zabalza [1983] employ the CES direct utilityfunction, while Burttess and Hausman[1978] and Hausman (1980. 19819, 1983]adopt a form that places a priori inequality (negative) restrictions on the totulincome elasticity. Ashworth and Ulph { 1981] restrict their sample to people whowork at least cight hours per week bul donot corrcct for the potent

50. Orley Ashenfelter and James J. Heckman [1974] consider a joint labor supplymode! but do not incorporate the effects of taxation. For another attempt atincluding tases in the family labor supply context, see Jerry A. Hausmanand PaulRuud {1984).

SI. Note that in countries such as Canada where spouses file separate income tax

returns, Unis marriage taais less of an issue. For further discussion, see Alice Nak-amura and Masao Nakamura [198 la}.

$2. Hausman(19814, p. 63]. Also see Hausman [1986].53. This conventional wisdom has also beenpersistently attacked by Alice Nakamurs.

and Masao Nakamura [1981b; 1985a, bj.34. A more complete survey offemale labor supply is presented in Table 2.26ofKil-

lingsworth and Heckman [1986, pp. 189-192].55. Mroz [1987, p. 769].56. See James Durbin [1954], De-Min Wu (1973), Jerry Hausman (1978). and Hale

bert White (198:$7, That KL6,the number of children under age sin, and perhaps KG18, are exoge-

Rous is 4 somewhat surprising finding. One possibilityis that the power of theDurbin-Wu-Hausman-White specification test may be low, Particularly when thefit from the first-stage regressionsis low, For a discussion of such powerissues farthis test, see Nakamura and Nakamura { [985c}.

58. It is worth noting Uvat when the two experience variables are excluded fromthefirst-stage probit equation, the sample likelihood function falls from — 39% to— 450, and ofthe 16 regressors, only KL6, and V haveestimated coefficients thatare more than twice their estimated standard errors.

a

70 Whether and How Much Women Work for Pay

59.

6h.

62.

63.

65.66.

67.

68.

69.

70.

Mm.

72.

Mrozalso reports thatthese findings are essentially unchanged whenthefirst-stageprobit equationis altered to reflect alternativedistributions such as thelogit or lognormal. Furthermore,tests for the exogencity of KL6, K618. and 1 uniformlyindicate nonrejectionof the null hypothesis of exogeneity.

. It is worth noting, however, thal use of the simple conditional meanor inverseMills ratio adjustment to account for sample selectivity is at best a “first-orderapproximation” when taxes afe incorporated into the model; unless strong

assumptions are made similar to those specified by Hausman [1981a], the appro-priate procedure is very complex and computationally cumbersome. Mroz’sresults concerningthe effects of taxes shouid therefore be viewed as lentative.

Note the similarity ofthis frameworkto thatof progressive taxationin static laborsupply discussed in Section 11.3.B.2; in both cases the budget constraint set con-sists of numerous segments havingdiffering slopes.

ion ofthis additive separability is that C(z) or L£(2) does notaffect themarginal utility of Cir’) or £(°) for ¢ # 2". Further, if leisure is a normal good ineachtimeperiod, then additive separability implies thatleisure times at differentpoints overthelife cycle must be net substitutes (holdinglifetimeutility constant).Forfurtherdiscussion, sce Angus Deaton [1974].Forfurtherdiscussion, see Martin Browning, Angus Deaton, and Margaret Irish[1985] andthereferences cited therein.

}. See Harold Waits, Dale Poirier, and Chris Mallar [1977], among others, for 2study implementing PO and Edward Kalachek, Wesley Mellow, and FredricRaines [1978]for PT.Fordetails, see Killingsworth and Heckman [1986. pp. 159-161].Fora generaltheoretical discussion on models ofendogenous dynamicwage deter-mination, see Yoram Weiss [1986]. Recent econometric studies on various aspectsofdynamic labor supply include Joseph Altonji [1986], Richard Blundell and IanWalker [1982]. James Heckman and Thomas MaCurdy[1980}, Richard Layardand Jacob Mincer (1985), Thomas MaCurdy[1981], Robert Moffitt [1984], AliceNakamura and Masao Nakamura [1985a,b], and James P. Smith [1977, 1980].The labor supply effects of such quantity restrictions have been considered by,

amongothers, Angus Deaton and John M. Muelibauer {1981}, Robert A. Moff[1982], and John C. Ham [1982,1986].For a survey of econometric problems with more general qualitative dependentvariable specifications, see, for example, Takeshi Amemiya [1981. 1984], RichardBlundell [1987], James Heckman [1976b], G. S. Maddala [1983], and Daniel

McFadden [1984]. In the context offa laborforce participation studies, seeJules Thecuwes [1981] for an application of the multinomiallogit specification.Fordiscussions on inference from experimental data,sce, for example, Dennis J.Aigner (1979}; Michael C. Keetcy [1981]; Michael C. Keeley, Philip K. Robins,Roben G.Spiegelman, and Richard W. West [1978]; and Charles E. Metcalf[1973}.Paneldata issues are discussed by. amongothers, Zvi Griliches, Bronwyn H. Hall,

and Jerry A. Hausman [1978]; Gritiches and Hausman[1986]; and Cheng Hsiao[1986).Thanksare due to Thomas Mroz, who madethis data set available for ouruse.

Foran empirical comparison ofthe OLS,logit, and probit estimators in the con-text of the laborforce participation decision, see Morley K. Gunderson [1980].Gunderson [1980,p. 217] finds that, “Whenthe probability oflabor force partic-

References 671

ipationis .50, .. . , the results ofall three statistical techniques are similar. How-ever, when the actual probability of panicipation is nearer to zero or onc. then the

probit orlogit results differ substantially from the finear probabilityfunction.”73, Historicalaspects ofthelogit specification in bioassay studies are discussed bricily

in Joseph Berkson [1951].74. A discussion ofgoodness-of-fit measures in the context oflimited dependent vari«

able models is given in, for example, G. S. Maddala [1983. pp. 37-41, 76-77].75, However, amongothers, Joseph Berkson [1951] has argued thatthelogitis pref

crable to the probit.76. For a discussion andderivation, see G. S, Maddala (1983, pp. 23-24).

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and the Esti-2, June,

FURTHER READINGS (All are textbooksin labor economics)

Addison, John T. and W. Stanley Siebert [1979], TheMarketfor Labor: An anabyticalTreatment, Santa Monica, Calif: Goodyear.

Ehrenberg, Ronald G. and Robert S. Smith [1985], Modern Labor Eeunumics: Theoryund Pablic Policy, Second Edition, Glenview, IIL: Scott, Foresman, ‘

Fleisher, Belton and Thomas Kniesner [1984], Labur Economics: Theory, EvidenceandPolicy, Third Edition, Englewood Ctiils, N.J.: Prentice Hall.

Gunderson, Mosley and W. Craig Riddell {1988}, Labor Market Economics: Theory,Evidence ‘and Policyin Canada, Second Edition, Toronto: McGraw-Hill Ryerson,Ad.

Hamermesh, Daniel S. and Albert Rees [1988], The Economics of Work und Pay.Fourth Edition, NewYork: Harper & Row.

MeConnell. Campbell R. and Stantey L. Brac [1986], Contemporary Labor Econom.ies, New York: McGraw-Hill.

Reynolds, Lloyd G., Stanley H. Masters, and Colletta H. Moser {1987], Economics ofLabor. Englewood Cliffs, N.J.: Prentice Hall.

its continued from copyright

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¢ Research, 1961, p. 35. Reprinted withissiun. Griliches: Technology, Edudand Productivity, NY: Basil Blackwell,1988, p. 120. Reprinted with per-

on. Forester, oll, The Informationnology Review, Cambridge, MA.MITPress, 1985, Reprinted withission. Machlup, Journut of Politicalomy, 82:4, July/August, 1974, p. 892,inted with permission. Halll and Tay-dtacrocconomics: Theory, Petformance.

. 237, Reprinted withliches, “Commems ontor/Kendrick, eds... Fron

ofQuantitative Econumics, Vol. Il, Am-im; North-Holland, 1974, p. 335. Re>xd with permission, Koopmans:ited from “Report ofthe Modeling Re-¢ Group, 1978, with permission fromativnal Academy ofSciences, Washing-DC. Reprinted by permission ofRICAN SALESMAN, © TheNationalich Bureau, Inc., 424 North Third St,ngton, lowa 52601-5224. Clarke,nometric Measurement ofthe Duration

Svertising Effects on Sales.” Journal of”eting Research, 13:4, Novem!

. p. 355. Reprinted with permission.talensee, The Econumics ofAdvertising,

jorth-Holland, 1972, p. 38.

mied. with permission. Fuss, Mcen and Mundlak, “A Survey of Fune-1 Formsin the Economics Analysis ofaction,” in Fuss/MeFadden, eds. Prom-mEconomics: A Ducal Approach to The-ind Applications, Amsterdam: North-ind, 1978, p. 266. Reprinted withission, Berndt and Wood, “Engi-

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neering and Econometric Interpretations ofEnergy-Capitat Complementarity.” Ameri-can Economic Review, 69:3, June 1979, p.351. Reprinted with permission, Fried-man, “The Role ofMonetary Policy.” Anter.icun EconomieReview, 5B:1, Macch 1968, p.UL Reprinted with permission. Okun,“Poswar Macroeconomic Performance,” inMartin S. Feldstein, ed., The American Econ.omy in Transition, Chicago: The Universityof Chicago Press. 1980, p. 166. Reprintedwith permission. Solow, “What We Knowand Don't KnowAbout Inflation.”” Technol-ogy Review, 81:3, December 1978/Jan-wary 1979, p. 31. Reprinted with permissionfrom Technology Review, copyright 1978,Lucas, Jr. and Sargent, “After KenyesianMacroeconomics,” in After the PhillipsCurve: Persistence of High Inflation andHigh Unemployment, BOSTON: FederalReserve Bank ofBoston, ConferenceSeries,No, 19, p. 69. Gordon, “The Theory ofDomestic Inflation.” american EconomicReview, 67:1, March 1977, p. 132, Reprintedwith permission. Killingsworth, LaborSupply, NY: Cambridge University Press,1983, p. 432. Reprinted with permission.Nakamura, Nakamura, and Cullen, “JobOpportunities, the Offered Wage, and theLabor Supply of Married Women,” stmert-can Econumics Review, 69:5, Devernber1979, p. 787, Reprinted with permission.Fullerton, “On the Possibility of an InverseRelationship between Tax Rates and Gav-ernment Revenues,” Journal uf Public Evo-numnies, 19:1, October 1982, p. 20. Reprintedwith permission. Jack Johnston, Ecouv-metic Methods, Ye, © 1984, p. 335. Re»printed with permission of McGraw-Hifl,Inc. Rudiger Dornbusch and StanleyFischer, Macrovcunomics, 4Je, © 1987, p.315. Reprinted with permistion of McGraw.Hill, Ine.

4: courtesy of Guy Orcutt. Page 5:ice V. Wilkes, David J. Wheeler, and+y Gill, Preparation ofProgramsjor antcpnie Digital Computer. © V9S1, ce1 1985; Adgison-Wesley Publishingany. Reprinted with permission ofthecher. Page 24: courtesy of Harry M.owitz. Page 38: courtesy ofSteven A.

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tesy of Robert Hall Page 258: counesy ofJames Tobin. Page 329: courtesy of Hea-drik Houthakker. Page 334: photo ofDaniel McFadden: courtesy ofTito Simboli.Page 395: courtesy of Richurd Schmnalensee.Page 406: courtesy of Harry V. Roberts.Page 459: courtesy of W. Erwin Diewen,Page 440: courtesy of Dale Jongenson.Page 537: courtesy of Rubert E. Lucas, Je,Page 546: courtesy of Ray C. Fair. Page628: courtesy of James Heekman. Page636: courtesyof Jerry A. Hausman.

Author Index

Aaker, David, 407, 441,4a7

Aaron, Henry J., 674Abel, Andrew, 260,

296f, 297f, 298,299

Abernathy, William J.,676, 96f, 98

Abowd,John M., 210f,214, 522, 5810, 584

Abraham, Katherine,176, 214, 218, 219

Ackley, Gardner, 363,437f, 441

Adelman,Irma, 144f,146

Addison, John T., 211f,

214, 222, 592,667£, 671, 679

Ager, MargaretE., 505Ahl, David H., 186

Aigner. Dennis J., 182,214, 310, 354f,356, 357, 439f,441, 447, 4996,503, 670f, 671

Aitchison, John, 144f,

146Akerlof, George A.,

219, 212f 214

Alberts, William, 96f,97f, 98

Albion, Mark S., 440£,443

Alchian, Armen, 96f,98

Aliber, Robert Z., 586

Allen, R.G. D., 606,67

Almen,Shirley. 237,288, 299

Altonji, Joseph G., 539,584, 670f, 671

Alworth, Julian, 302Amel, Dean F., 146f,

146Amemiya, Takeshi,

621, 632, 654-656,670f, 671

Anderson, Richard G..475, 501

Anderson, Theodore

W.. 186Ando,Albert, 269,

297f, 299

Andress, Frank J., 966.98

Andrews. Emily S., 676Andrisani, Paul, 189,

216Anton,James J., 96f, 98Arbeiter, Larry. 407Archibald, G.

Christopher, $81f,585

Arndt, Johan, 414, 446

Arnold, Stephen J., 407,

437, 441Arrow, Kenneth J.. 157.

214, 365, 444, 452,50%

Ash, Colin, 666f, 671Ashenfelter, Orley, 179,

210f, 2436, 214,215, 216, 218, 221,222, $39, 585, 607,667f, 6698, 671,674, 676, 678, 679

Ashley, Richard, 356f,357, 374, 383, 398,427, 438f, 4408,441

Ashworth, 5. S., 667f,6698

Assmus, Gert, 392Atkinson,Scott E., 484,

SolAtrostic, B. K., 211f.

214Auerbach,Alan J.,

297f, 299, 675Angarten,Stan, 196,

145f, 146Ayanian, Robert, 4401,

44t

Bailey, Elizabeth E.,500f, 501

Bailey, Martin J., 504Baily, Martin N., 587Bain, Joseph, 415, 441Baldry, Jonathan C.,

669f, 671

Barnett, Wiliam A.,299, 501f, 501

Barquin-Stolleman.Joan A., 146

Barro, Robert J., 582f.585, 592

Bartik, Timothy, £30,146Barton, Margaret, 676Basmann, Robert L.,

371, 441, 583f, 585

Bass, Frank M., 380,391, 43Bf, 441. 442

Beach, Charles M...356f, 357

681

Author Index

Jeauvais, Edward, 358

Becker, Gary, 152, 155,156, 169, 170, £80,181, 211f, 2126,214, 216, 602, 610,667f, 671, 673

Begg, David K. H., 549,585

Behrman,Jere R., 189,215

Ben-Porath, Yoram,210f, 215

Bentzel, Ragnar, 577,585

Ben-Ur, Joseph, 438f,446

Bergman,Barbara R..223

Bergson, Abraham, 453,301

Berkson,Joseph, 671f,672

Berndt, Ernst R., 146f,146, 217, 297f,299, 301, 311f,355f, 357, 449,450, 454, 456, 472,479, 484, 486, 487,SOOF, SIF, 501,$02, 504, 505, 06,674

Betancourt, RogerR.,298f, 299, 354f,356f, 357

Bischoff, Charles W.,250, 253. 286, 293,2981, 299

Bitros, George C., 296f,299

Black, Fisher, 38, 55f,56

Blanchard, Olivier J..260, 296f, 2978,298, 299, S81f,S836, 585

Blank, David M., 393,398, 4376, 439f,442

Blattenberges, Gail R.,315, 354f, 355f,359

Blaug, Mark, 212f, 215

Blinder, Alan, 156,182-184, 188, 191,203-208, 211f, 215

Bloch, Fasrelt, 178, 202,215

Bloch, Harry, 441f, 442

Bloom,David, 182, 215Blum,James, 676Blume, Marshall E.,

55f, 56Blundell, Richard, 670f,

672, 678

Bohi, Douglas R.. 328,S31, 344, 357

Bojanek, Robert, 415,445

Borden, Neil H., 363,

380, 424, 442Boskin, MichaelJ., 638,

672Boston Consulting

Group, 60, 96f, 98

Bowen, Howard R., 216Bowen,William G.,

6670, 672

Bowles, Samuel, 213f,215

Box, George E. P., 145f,146, 299, 320, 357

Boyer, Kenneth D.,4411, 442

Bradford, David F.,297f, 299

Brealey, Richard, 54f,558, 56f. 57

Brechling, Frank P. R.,S80f, 585

Breusch, TrevorS., 568,S84f, 585

Brinner, Roger E., 5810,585

Brown, Charles, 210f,218

Brown,Charles V., 67!

Brown, James A. C.,

198, 1440, 146,356f, 357

Brown, Murray, 505,676

Brown, Philip, 56f, $7

Brown, Randall S.. 484.502

Browning, Martin, 667f.670f, 672

Brue, Stanley L.. 180f,218, 222, 679

Brunner. Karl, 303,447, 588. 590

Buchanan, Norman,365, 442

Bultez, Alain V.. 437f,43RE, 442, 444

Burdett, Kenneth, 212f,215

Burmeister, Edwin, 55f,57, 58

Burridge,Peter, 580f,585

Burstein. Meyer L.,1446, 146

Burtless, Gary, 355f,357, 631-633.669F, 672

Butler, John £., 951, 98

Butler, Richard, 186,215

Cable, John, 439f, 442

Cain, Glen G., 182,214, 215, $96, 614,8686, 672, 674

Card, David, 539, 585Carliner, Geoffrey. 187,

25

Carman, David, 407,

441

Cartwright, David W.,125, 126f, 144f,

146Caves, Douglas W.,

354f, 357, 482, 502Ceef, Christopher, 18F

Chamberiain, Gary,167, 215

Chambertin, EdwardH., 368, 442

Chambers, Robert G.,S00f, 502, 506

Chapman, L. Duane,3558, 358

Chen,Carl R., 560, 57

296f. 299, 452, S01Chiang, Judy S. Wang.

482, 502

Chirinko, Robert S.,298f, 299

Chiswick, Barry. 187,

2110, 2138, 214,ans

Chow, Gregory C., 56f,57, 97f, 98, 104,118, 119, 136-137,139-142, 1456,146, $49, 585.

Chow, Simeon,440f,442

Christ, Carl F., 54f, 57,99, 505, 677

Christensen, Laurits R.,81-84, 90-92, 98,247, 299, 354f,

357, 453, 480, 482,484, 499f, 502

Chu, Shih-Fan, 499f,503

Ciccolo, John H.. Jr.2971, 300

Clark, J. Maurice, 233,296f, 300

Clark, Kim B., $22,585, 6671, 672

Clark, Peter K., 298f,300

Clarke, Darral G., 361,

387, 389-391, 398,415, 4398, 440f,442

Cobb, Charles, 68, 451,4986, 503

Cochrane, Donald, 5Coen, Robert M., 296f,

297f, 300. 583f,585

Cogan, John F., 613,633, 634, 638, 672

Cohen,Malcoim S.,607,6674, 672

Cole, Rosanne, 124,126f, 138~139,143, 1448, t45f,146

Comanor,William S..4408, 4411, 442

Comins, KathyA.,228f, 296f, 301

Constuner Reports,4408, 442

Conway, Delores, 191,215

Cooper, Guy M., 55f,58

Cooper,J. Phillip, 288,300

Corcoran, Mary, 184,188, 215

Corio, Marie, 3556, 357Cornfield, Jerome, 668f,

672Court, Andrew T., 110,

144f 147

Cowing, Thomas G..81, 98, 482, 503

Cowles, Alfred W. II,20. $4f, 57

Cowling, Keith, 439fCox, David R., 1456,

146

Cragg, John G., 59Culten, Dallas, $93,

678

Dasgupta, Partha, 96f,98

Davies, Nicholas, 439f,

Author Index 683

Day. George S., 96f,97f, 98, 99

Daymont, Thomas,189, 216

Deaton, Angus, 606,670f, 672, 673

de Leeuw, Frank, 236,300

Deloitte, Haskins andSells, 263, 300

de Marchi, Neil, 550,585, 592

Desai, Meghnad, 295f,300, S80f, 585, 592

Dhalta, Norman K.,439F, 442

Dhrymes, Phoebus,145f, 147, 296f,300, 3558, 357

Diamond, Peter A..5811, 585

Diewen, W. Erwin,100, 304, 453,459-460, 4991,

S00f, 501, 503,631, 673

Dillon,John, 453, 503Dolan, Robert J. 96f,

98Donner, Arthur, $16,

586Dorfman, Robert, 365,

442Dornbusch, Rudiger,

224, 300, 5831, 86Douglas, Paul H., 68,

450, 451, 4981,503, 614, 678

Douglass, George K.,2121, 216

Dréze, Jacques, 499f,506

Dubin, Jeff, 334Dubrul, Stephen M.,

144f, 147

Duesenberry, James,239, 296f, 298f,300, 586

684 Author Index

Dutberger, Ellen R.,145f, 146f, 146,

147Duncan, Greg, 184,

188, 210f, 215, 216Durbin, James M., 282,

296f, 300, 347,357, 438f, 442,568, 582f, 584f,586, 669f, 673

Dutton, John M., 95f,98

Eastlack, Joseph O.,4401, 442

Eatwell, John, 445Eckstcin, Otto, 586, 588Ehrenberg, Ronald G.,

213f, 216, 220,

222, 592, 672, 675,677, 679tein, Albert, 20, 57

Eisner, Robert, 252,269, 296f, 2971,300

Ekelund, Robert B., Jr.,393, 443

England. Paula, 213f,216

Epple, Dennis, 130, 147Epstein, Roy J., 19f,

550, 583f, 586Estes, William K., 95f,

98Evans, MichaelK., 236,

301

Fair, Ray C., 266, 292,301, 304, 4411,

443, 535, 545, 546,

549, $52, 583F,586, 668f, 672

Fama, Eugene F., $50.56f, 57, 59

Farber, Henry S.. 176,179, 2146, 218,216

Farley, John U., 392,439f, 441,443

Farris, Paul W., 440f,443

Feldstein, Martin S.,269, 296f, 301,590, 674, 675

Ferber, Marianne A.,192, 216

Ferguson, James M.,4a0f, 443

Field, Barry C., 502,504

Filer, Randall K., 189,

216

Finegan, T. Aldrich,6676, 672

Finney, David J., 668f,

Fischer, Stanley, 224,300, 582f, 586

Fisher, Franklin M.,56f, 57, 1436, 145f,146f, 147, 312,354f, 355f, 357,371, 442, 577,

SB2f, 586Fisher, Irving, 516,

S80f, 586Flaim,Paul, 667f, 673Fleisher, Belton M.,

222, 592, 6676,673, 679

Flemming, John S.,269, 301

Flood, Robert P., 549Florence, P. Sargant.

666f, 673Fomby, Thomas B..

97f, 98Foot, David K., 296f,

301Forester, Tom, 102, 147Frankel, Marvin, 4991,

503Franklin, Benjamin,

150Fraumeni, Barbara,

486, 500f, 504, 506Freeman, Richard B..

171, 181, 216

French, Kenacth R..56f, 57

Friedlacnder, Ann F..

481, 482, SO0E,50t, 502, 505

Friedman, James W.,A37f, 443

Friedman, Milton, 210f,216, 314, 357, 507,521, 586, 648, 673

Frisch, Ragnar, 452,503

Fromm, Gary, 297,

300Fudenberg, Drew, 96f,

9Fullerton, Don, 297f,

299, 301, 302, 593,

673Fuss, Melvyn A., 2976,

299, 449, 501f,502, $03, 504, 506

Gabor, Andre, 354f,

Galbraith, John

Kenneth, 363,437f£, 443

Gallant, A. Ronald, 96f,99, 500f, 503

Garber, Peter M., 549,586

Gardner, David, 99

Geisel, Martin S., 355f,359

Ghemawat, Pankaj, 79,80f, 80, 97f, 99

Ghez,Gilbert R.. 156,216, 667f, 673

Gilbert, Christopher L.,515, 550, 580f,5B5, $86, 592

Gilroy, Curtis L.. 676Gintis, Herbert, 213fGloricux, G., 269, 302Goddeeris, John H.,

2108, 216

Godfrey, Leslie G., 568,S84F, $86

Goett, Andrew A., 3566,357

Goldberger, Arthur S.,97f, 99, 1440, 147,191, 192, 216,298f, 301, 3551,356f, 358, 4381,443, 499f, $03,550, 5531, 535.574, $80f, 5831,587, 588, 622, 657,668f, 673

Gollop, Frank M., 506Gordon, Margaret S.,

677Gordon, Robert A., 677Gordon, Rober J.,

143f, 1458, 147,149, 507, 540,SBIE, S820 S83,587

Gorman,John A., 228f,295f, 301

Gorman, William M..144f, 147

Gould, John P.. 298f,301

Gould, William, 669f,673

Gramm,William P.,393, 443

Granger, Clive W.3.,289, 301, 359, 374,383, 398, 427,439f, 441, 443

Green, Carole A., 192,216

Greene, William H..81-84, 90-92, 98,622, 657, 673

Greenhalgh, Christine,638, 673

Greenwood, Joen, 1456,1468, 147

Gregory, Allan W.,S00f, 503

Grenier, Gilles, 187,216

Grice, Joseph, 675

Griffiths, William E..57, 99, 148, 218

Griliches, Zvi, 97f, 99,102, 115-116,144f, 1456, 1468,146, 147, 148, 149,166, 167, 217, 224,236, 237, 264, 301.304, 305, 3551,358, 387, 396, 443,504, 587, 668f,670f, 673, 675, 677

Gronau, Reuben, 2136,217, 612, 667f,668F, 673, 674

Grunfeld, Yehuda, 239,301

Guilkey, David, 482,503

Gujarati, DamodarN..56f, 57

Gunderson, Morley K..213f, 217, 222,6701, 674, 679

Haavelmo, Trygve, 243,297f, 301. 584f,587

Hall, Bronwyn H., 6681,670f, 673

Hall, Robert E., 146f,148, 153, 217, 224,245, 246, 247,2966, 297f, 301,S81f, 582f, $87,631, 644,6671.668f, 674

Hallett, A. J. Hughes.S83f, 587

Halvorsen, Robert, 325.358, 484, 501

Ham,Joho C., $39,587, 632, 6708,674

Hamermesh, Daniel,172, 214f, 217,222, 5798, $87,SORE, 6668, 674.679

Author Index 685

Hamitton, David. 97,99, 135, 148

Hanoch, Giora, 170,ITF, 217, 476,500F, 503, 633,667f, 674

Hansen, Alvin, 363,437f

Hansen, Bent, S80f, 587Hansen, Lars P., 298f,

301, $01f, $03,564, 5830, 587

Hansen, W. Lee, 169,207

Hanser, Philip, 359Hanssens, Dominique

M,, 4408, 443Hanushek, Eric A..2 217

Harberger, Arnold C.,308, 504

Harman,Harry H.,1456, 148

Harper, Michael! J.,2978, 301

Hart, Peter, $90Hartman, RaymondS.,

328, 3556, 358Harvey, Andrew C.,

289, 298f, 302,304, 359

Haugh, Larry D., 382.445

Haugh, William D.,212¢

Hausmun, Jerry A.,355f, 357, 4388,443, 499f, 503,550, 565, SOF.587, $88, 629,631-638,6671,668F, 6691, 670f,672, 673, 674, 675

Hausman, Leonard J.,215

Hax, Amoldo.60, 96f,9

Hayashi, Fumio, 260,297f, 302

Author Index

Haydel, Juanita M..56f, 57

Heady, Earl, 453, 303Heck, Jean Louis, 57Heckman, JamesJ.,

157, 186, 191,2116, 212f, 215,217, 607, 614-617,623-629,633, 634,637, 638, 645-648,665, 667f, 668.6696. 6708, 671.675, 676

Heien, Dale M., 144f,148

Helliwell, John F., 269.302, 305

Helpman, Elhanan, 96f,9

Helvacian, Nurhan,146-147

Henderson, J. Stephen,384f, 358

Hendry, David F.. 305,583f, 588

Herriges, Joseph A.,354f, 357

Hesse, Dieter M.. 484,50!

Hickman,Bert G.,296f, 302, 5836,585

Hicks, Sir John R., 227,2959, 302

Hill, M. N., 216, 223Hill, R. Carter, 57, 976,

98, 99, 148, 218Hirsch, Wemer, 96f, 99.

Hirschberg, Joseph G.,310, 356, 357

Hirschey, Mark, 441f,443

Hirschleifer, Jack, 212f,217

Hirschmann,WilfredB., 96f, 99

Hodge, James H.. 147Hogben, Lancelot, 329Holland, Daniel M.,

297, 302

Holmiund, Bertil, 2108,216

Holt, Charles C., 581%,588

Holtmann, Alphonse,482, 503

Holzer, Harry, 2136,217

Hood, William C., 584f,588

Horney, Mary Jean,677

Hornstein, Zmira, 675Horsky, Dan, 440f, 443Hough, Douglas E..

2126 218Houston, Franklin S..

4396, 443, 447

Houthakker, Hendrik

S., 2-3, 5-6, 196,144f, 148, 212f,217, 308, 310, 328.329-330, 356f,

387, 358, 439f,443, 501

Hsiao, Cheng, 668f,670f, 674, 675

Huang. Chi-fu, 59Hubbard, Elbert

(Green), |Hudson, Edward A.,

SOOF, 504Hulten, Charles R.,

231, 296f, 299, 302Hunter, Jerald, 2976,

298f, 302Huss, William R., 349,

354f, 358

Ibbotson, Roger, 55f, 57Intriligator, Michael D.,

301, 304, 305, 503,504, 587, 675, 677

Irish, Margaret, 670f,

Isachsen, Arne 3., 669f,675

James, S. F., 19FJefferson, Thomas, 361

Jenkins, Gwilym M.,289, 299, 320, 357

Jensen, Michaet, 38,55f, 56

Jeuland, Abel P., 96f,

98Jhally, Sut, 4370, 444Johnson, George E.,

179, 2116, 2138,214, 217, 6698,676

Johnson, Samuel, 362Johnson,Stanley R..

978, 98Johnson, Thomas, 182,

217

Johnston, J., 100, 449,504

Jones, Deborah J., 505Jones, F, L., 2136 217Jorgenson, Dale W.,

124f, 147, 149,

237, 242, 247, 250,

296f, 297F, 2986,299, 301, 302, 453,479-481, 486,499f, SO0f, SOI,502, 503, 504, 506,582f, 588

Joskow, Paul L., 96f, 99Jovanovic, Boyan, 176,

2426, 217, 219Judge, George G.. 56,

57, 97f, 99, 145f,148, 214f, 218

Kalachek, Edward,670f, 676

Kaldor, Nicholas, 365,444

Kalish, Shlomo, 440f,

Kamalich, Richard,175, 191, 218

Kaysen, Carl, 146f, 147,312, 3546, 355f,357

Keeley, Michael C.,670f, 676

Kehver, Ken, 677

Keim, Donald B., 56f,37

Kelejian, Harry H.,296f, 2986, 299,3560, 357

Kelly. Michael, 439f,442

Kemmis, Robyn, 581f,585

Kendrick, David A.,301, 503

Keynes, John M., 305Khaled, Mohammed S.,

5008, 501Kiefer, Nichotas M.,

2126, 218

Kiker, Billy F., 2108, 218Killingsworth, Mark .,

157, 182, 213%,215, 216, 218, 223,593, 595, 599, 600,607, 614, 615, 617,

624, 626-627, 629,

633-635, 645-649,665, 6674, 668F,6698, 670f, 675,676

King, Mervyn A., 297f,302

Kirshner, Daniel. 3556,358

Klaassen, Leo H.. 148,591

Kleidon,Allan W., 56f.37

Kein, Benjamin, 410,432, 446

Klein, Lawrence R.,VBF, 296f, 303, 510,550, 575. 576,S79F, SBOE, 583f.S8df, 588

Kline, Stephen, 437f,444

Kmenta, Jan, 437, 440f,444, 4991, 504,305, 5836, 589

Kaiesner, Thomas J.204f, 218, 222,592, 679

Knox, A. D., 296f, 303Koopmans, Tjalling C..

18f, 306, 358, S84f,588

Kopcke, Richard W.,

226, 237, 240, 241,254, 261. 270, 279.284-287, 291,

2958, 2971, 303Kosters, Marvin, 667f,

676Kotler, Philip, 437f,

444, 487Koyck, Leendert M..

148, 303, 355f,358, 4391, 444, 591

Krinsky, Itzhak, SOOf,504

Krishnamurthi,Lakshman, 440f.444

Krueger, Alan B, 2116,212f, 218

Krugman, Pau! R., 96f,99

Kuh, Edwin E., 239,303

Kulatilaka, Nalin, 475,

504Kuskin, Mark, 178,

202, 215Kuznets, Simon, 2108,

216

Labrune, Francis, 358Lafontaine, Francine,

500f, 504Laidler, David E. W.,

580f, 585, 588

Lakonishok, Josef, 56f,57

Lambin, Jean-Jacques,438f, 444, 447

Lancaster, Kelvin, 149Lancaster, Kent M.,

44, 442Lancaster. Tony, 2126,

218Landau, Ralph, 124f.

147, 149, 302

Author Index 687

Langford, David, 18f

Laslavic, Thomas J..S6f. 57

Lau,Lawrence J., 453,480, 4991, SO0f,502, 564

Lau, Sheila, 32, 59Layard, Richard G.,

215, 216, 218, 221,222, 505, 638,6671, 6708, 674,676, 678, 679

Lazear, Edward, 156,176, 186, 218

Lec, Lung-Fei, 629,668f, 676

Lee, Tsoung-Chao, 57,99, £48, 218

Lehmann,Donald R..392, 439, 441, 443

Leibowitz, Arlecn, 211f,28

Leiss, William. 437f.444

Lenin, ViadimirI., 306Leone, Robert P., 391,

439f, 441, 444

Lerman,RobertI., 607,667f. 672

Lerner, Abba P.. 437f,444

Leuthold.Jane H., 638,

676Levy, Haim, 58Levy, Robert A.. 558,

58Lewis, H. Gregg, 176,

178, 179, 202,2U3Ef, 218, 6671,676, 677

Lieberman, Marvin B.,96f, 99

Lilien, Gary, 447Lindsay, Robert, 302Lintner, John, 550, 58

Lipsey, Richard G..518, 524, $26.S8If, 588

Little, John D. C., 414,415, 448

8 AuthorIndex

Litzenberger, RobertHL, 59

Liviatan, Nissan, 395,444

Lloyd, Cynthia B., 2134,258, 223, 676

Lorenz, Wilhelm. 2118.221

Lorie, James, 406Lovell, C. A. Knox,

482, 503

Lucas, Robert E., Jr.,270, 298f, 303,

455, 504, 507, 509,527, 530, 531, 534,536, $37, 549,

582f, 588, $89, 592Lundberg, Shelly, 610,

676Lutkepohl, Helmut, 57,

99, 148, 218Lydall, Harold, 210f,

218

MacBeth, James D..556, 57

Machlup, Fritz, 150, 218MacKinnon, James G..

356f, 357MaCurdy, Thomas,

179, 218, 615, 617,629, 633, 634,6678, 6686, 6708,675, 616

Madansky, Albert, 584f,589

Maddala, G. S., 549,553K, SB4P, 589,668F, 6708, 6711.676

Mahajan, Vijay, 414,444

Majd, Saman, 966, 99Majluf, Nicholas, 60,

96f, 99Makridakis, Spyros G.,

386f, 358, 359Malinvaud, Edmond,

345, 589

Malkiel, Burton G., 20,550, 566, 58, 59

Mailar, Chris, 670f, 679Mankiw, N. Gregory,

$39, 589Mann, Henry B., 58éf,

589

Manser, Marilyn E.,676

Manser, Martin H., 18fMantel, Nathan, 668f,

672Marden, William D.,

2126 218Markowitz, Harry M.,

23, 24, 55f, 58, 59Marsh, Terry A... 56f,

STMarshall,Alfred, 450

Martin, Stephen, 441f,444

Mason, William, 166,217

Massy, William F., 371,44a

Masters, Stanley H.,222, 679

McCallum, Bennett T..345, $49, 561, 562.563, 5B1f, S82,589, 592

McCann, John M., 387,442

McCollum, James F.,$16, 586

McConnell, CampbellR.. 180, 218, 393,446, 679

MeDonald, John F.,96F, 99, 621, 657,668f, 676

McElroy, Marjorie B.,55f, $8, 471, 504,677

McFadden, Daniel, 334,355F, 3568, 357,358, 449, 499F,SOOF, 503, 504,306, 670f, 677

McGowan,John J.,145£, 1461, 147

McGuinness, Tony,439f, 442

McNown,Robert F.,666f, 671

Medoff, James, 176,219

Mellow, Wesley, 670f,676

Melrose, Kendrick B.,380, 444

Meltzer, Aian N., 303,447, 588, 590

Menill, Lynch,Pierce,

Fenner and Smith,39

Messer. Karen, 208,219

Metcalf, Charles B..670, 677

Meyer, John R., 239,303

Meyer, Lawrence H.,674

Middleton, Kenneth A.,96F, 99

Milgate, Murray. 445Miller, Merton H.,

2976, 303

Miller, P. B., 414, 445

Mills, Gordon, 590Milne, A. A.. 116

Mincer, Jacob, 152,* 155, 163, 168, 169,

172, 174, 176, 188,189, 196, 208,206, 24208 2138,219, 610, 645, 646,667f, 670F. 676,677

Minhas, Bagichs, 452,501

Mishkin, Frederic, 549,589

Mitchell, Bridger M.,

328, 358Mitchell, Olivia S., 610,

677

waist

Modigiliani, Franco,S550, 58, 269, 2978.

299, 303, $39,S82f, 589

Moff, Robert A., 621,657, 6676, 668f,6701, 676, 677

Montgomery, David B..96f, 97f, 98, 99,439f, 440f, 444

Montmarquette,Claude, 391, 501

Morgan, Chris, 18fMorgan, James N..

210f, 219, 667f, 677Moriguchi, C., 391, 444Morimune, Kimio,

A38f, 444

Morley,J. Kentfield, 20Moroney, John R.. 357,

359, 502Morison, Catherine J..

484, 485, SOIT,502, 504, 505, 506

Moser, Colletta H., 222,679

Mount, TimothyD.,3550, 358

Mroz, Thomas A., 595,635, 637, 638-644,650, 652, 661, 663,6691, 677

Muellbauer, John M..606, 670f. 673

Mueller, M. G., 300Muller, Eitan, 414, 444Mundlak,Yair. 391,

444, 449, 503Murphy, Kevin M..

172, 187. 219, 410,432, 446

Murray, Michael P.,354f, 358

Musgrave,John C..228f, 296, 301

Mussa, Michael, 298f, 303

Muth, John F.. 268,298f, 303. 319,358, 536, 589

Myers, John G., 447Myers, Kenneth H..

4371, 444Myers, Stewart C., $4f,

358, 566, 57, 297F,302

Nadisi, M. ishag, 252,297F, 298f, 300,302, 485, 505

Naert, Philippe A..37, 438F, 442,444

Nakamura, Alice, 593,629, 633-635, 637,638, 640, 641, 643,6671, 669f. 670f.677, 678

Nakamura, Masao, 593,629, 633-635, 637,638, 640, 641, 643,6671, 6691. 670f.677, 678

Nathan,Felicia, 211f.219

National Bureau ofEconomicResearch. 102, 148

Navasky,Victor, 18fNaylor, Thomas H.,

96f, 9Nefici, Salih, 582f, 589Nelson,Charles R.. 56f,

58, 289, 303, 308,310, 3141, 350,352, 353, 354f.359, 360

Nelson,Forrest 0..668f, 678

Nelson, Randy A., SOIL.505

Nerlove, Mare, 81-84,87-94, 99, 100,237, 303, 365, 401,417, a37f, 444,454, 457, 505,583f, 589

Neumark, David, 213f,219

Author Index 689

Newbold, Paul, 439f,442

Newman,Peter, 445Nicholls, William H..

424, 445Nicholson, Walter, 96f,

100Nickell, Stephen J.,

296f, 303

Niemi, Beth T.. 213f,218, 223

Norsworthy,Randolph, 500f,505

Norton,Peter, 19f

Oaxaca. Ronald, 182,188, 191, 203-208,20

Oberhofer, Walter,499f, 505, 5831,

589O'Connor, Charles J.,

2101, 224Ofek, Haim, 174, 189,

219Ohta, Makoto, 130.

146F, 148

Oi, Walter, 159, 210f,210 219

Okun, Arthur M., 507,

540, 541, 583f.589, 590

Olsen, Randall J.. 623,661, 663, 6680, 678

Olson, Kenneth, tOrcutt, Guy H., 3-5.

18f, 19f

Oswald, Andrew J.,179, 219

Otuteye. Eben, 440f,445

Our,Tae H., 407, 436,441

Packard, Vance, 437f,445

Padilla, Arthur H..2146, 218

Author Index

Pagan, Adrian R., 305Palda, Kristian S., 380,

390, 402, 438f,440, 445

Palm,Franz, 264, 297f,304

Papademos, Lucas,$82f, 589

Park, Joon Y., 50Of,505

Park, Rolla E., 358Park, Soo-Bin, 584f,

391Parkin, J. Michael, 524,

526, S80f, 581f,588

Parnes, Herbert S.,

2108, 219, 667f,678

Parsons, Leonard J.,380, 438f, 440f,442, 443, 448

Patterson, J. Wayne,96f, 100

Pazderka, Bohumir,407, 436, 441

Pechman, Joscph A.,674

Peck, Stephen C., 308,310, 311f, 350,

352, 353, 354f,359, 360

Pencavel, John, 79,

218, 220, 645,666f, 678

Perkins, J. W., S84f,585

Perloff, Jeffrey M., 540,590

Perry, George L., 581,5820, 583f, 590

Petly, Sir William, 152Phelps, Edmund S.,

521, $89, 590

Phillips. A. William,

512, 513, 515,579f, 580f, 590

Phillips, Peter C. B.,

SO0F, 505, 545, 590

Pierce, David A., 382,4398, 445

Pindyck, Robert S., 96f,

98F, 99, 100, 356f,

359, 484, 505,583f, 590

Plourde, Andre, 356f,359

Pogue, Gerald A., 55f,38

Poirier, Dale, 670f, 679Polachek, Solomon,

174, 175, 188, 191,

211, 213f, 214f,207, 218, 219

Pollak, Robert A., 143f,148, 189, 215

Pollay, Richard W..404, 445, 448

Porter, Michael E., 96f,100

Poterba, James M.,2971, 303, 581f, 590

Prais, S. J., 196. 356f,357

Prucha,IngmarR., 485,505, 578, 590

Psacharopoulos.George, 171, I71f,212f, 220

Puig, Carlos, 355f, 358Pulley, Lawrence, 358Purdy, David L., 385

Quandt, Richard, 380,A38f, 445

Quigley, John M., 211f,217

Raines, Fredric, 670f,676

Raj, S. P., 440%, 444Rao, Ambar G., 414,

440f, 442, 445

Rao, Ram C., 440f, 445Rapping, Leonard J.,

96f, 100, 309, 527,530, 531, 534,582f, 588

Rasche, Robert, 269,299

Rasmussen, Arne, 365,445

Rea, Samuel A., 607,667f, 672

Reder, Melvin W..

214f, 220

Rees, Albert, 172, 214,217, 222, $79,587, 598L, 666f,674, 679

Rehn, Gésta, 580, 587Reimers, Cordelia, 187,

220Reinganum, Marc R.,

58Renahack, Robert, 315,

3548, 3558, 359Reynolds, Lloyd G.,

679Rhodes, George F., Jr.,

667f, 673Riddell, W. Craig, 222,

679Riley, John G.. 210fRizzuto, Ronald, 211.

220Robb, A.Leslie, 500f,

504Roberts, Harry V., 191,

215, 405-407,440f, 445

Robins, Philip K., 670f,67%Roester. Theodore W.,

393, 446Roll, Richard, 55f, 566,

57, 58Rose, Nancy L.,. 96f, 99Rosen, Harvey S., 629,

633, 644, 66d, 678Rosen, Sherwin, 130,

148, 2108. 211f,2U2F, 214, 216,220, 372, 445

Rosenberg. Nathan, 101Rosett, Richard N.,

668F, 678

Ross, Stephen A., 37,38, 52, 55f, $7, 58

Rotemberg, Julio R.,484, 505, 539,583f, 589, 590

Rothenberg. ThomasJ.,467, 505

Rothschild, Michael,

2968. 298f. 301,403

Rowe, Robert D., 391,

445Rowley, John C. R.,

295f, 303Roy, A. D., 165, 220Ruback, Richard S.,

S6f, 58Rubinfeld, Daniel L..

986, 100, 356f, 359Russell, R. Robert, 96f,

100Ruud,Paul, 667£, 6691,

675Ryan, David, 356f, 359

Salinger, Michacl A.,297f, 303

Samuelson,Paul A.,

97f, 100, 498f, 505,517, 518, $80f, $90

Sandmo, Agnar, 669,678

Sargan, J. Denis, 305,4998, 505, 590

Sargent, Thomas J.,2986, 303, SOIf,503, 507, 549,

S8If, 5821, 589,590, 592

Sarnat, Marshall, 58Sasieni, Maurice W..

414, 445

Savin, N. Eugene, 479,SO0f, 502

Sawa, Takamitsu, 584f,591

Scarf, Herbert E., 504Scherer, F. Michael,

956, 100

Schmalensec, Richard,361, 365, 368-372,374, 376, 380, 383,

393-396, 398, 410,415, 427, 429, 430,A37E, 438F, 4391,440F, 445

Schmidt, Peter, 179,213f, 220

Schneider, Lynne, 410,AIL, 432, 446

Schoenberg, E. H.. 380,410, 446

Schoenberg, Erika, 614,678

Scholes, Myron, 38,55f, 56

Schudson, Michael,437f, 446

Schultz, George P., 586

Schulz,Henry, 405,446

Schultz, Randall L.,4391, 440f, 443,444, 448

Schultz, T. Paul, 633,634, 638. 667f, 678

Schultz. Theodore W..#52, 220, 671

Scarle, Allan D., 96f,100

Selickacrts, Willy, 302

Shapiro, Carl, 222, 220

Sharpe, William F., 40,S46, 55f, 561, SB

Sheffrin, Steven M.,549, 591

Shell, Karl, 143f, 147,371, 443

Shoven, John B., 504

Sicktes, Robin C., 482,503

Siebert, Calvin D., 297f,

Siebert, W. Stanley,

2Ibf, 214, 222,592, 667f, 671, 679

Silk, Alvin J., 415, 439f,AAOf, 442, 444, 445

Author Index 691

Silverstein, Gerald,228f, 296f, 301

Simon, Hermann, 414,446

Simon,Julian L., 370,414, 4381, 4398,AAIE, 446

Simon, Leonard S.,SOF, 443

Sims, Christopher A..2976, 304, 4391,4401, 446, 582,390

Singer, Burton. 212f,217

Singleton, Kenneth J.,2981, 301, 564,$831, 587

Sinquefield, Rex, S5f.37

Smeeding, Timothy,241f, 220

|. Seymour, 56f. 57, Adam, 150, 152.

210F, 220, 593, 678‘Smith, James P.. 186,

187, 2136,666. 670,674, 675, 678

Smith, Robert S., 210£.221, 222, $92,6671, 673, 679

Smith,Shirley J., 666f.

Smith, V. Kerry, 81, 98Snetsinger, Douglas W..

407, 436, 441Solnik, Bruno, 55f, $8Solow, Rober M., 363.

446, 452, 301. 507,

517, 518, S80f,S81f, 590, 591

Sorrentino, Constance,666f, 678

Spady, Richard H., 481.505

Spann, Robert, 358Spence, A. Michael, 60,

96f, 100, 157, 221

92 Author index

Spiegelman, Robert G.,670f, 676

Spitzer, John 3. 145f,146f, 148

Stafford, Frank P,, 211f,204f, 217, 221

Steiner, Peter O.. 365.442

Stern, Nicholas, 639,678

Stevens, Joseph C,,4381, 446

Stigler, George J., 440f,446

Stigler, Stephen M., 149Stiglitz, Joseph,96f, 98,

2108, 2L1f, 2126,220, 221

Stoker, Thomas M.,S796, 591

Stone, Richard, 116,144f, 148

Strauss, Lewis L., 306,359

Strauss, Robert, 179,220

Stromm,Steiner, 669f,675

Strowz, Robert H.. 270,300

Summers, Anita A..2016 221

Summers, Lawrence H.,2U1f, 2N2E 218,297f, 304, 522,539, 580f, SBE585, 589, 590, 591,6671, 672

Swan, Craig, 2951, 304Swinton, David, 2134,

221

Taubman, Paul, 166,

189, 2108, 211f,215, 221

Tauchen, George,592

Taylor, Jim, 580f,591

Taylor, John B., 224,301, 359, 510, $41,$49, 582F, 583f,586, $87, 591

Taylor, Lester D., 315,321, 323, 328, 343,354f, 355f, 359,397, 439f, 443, 446

Telser, Lester G., 365,380, 384, 410, 411.438f, 4408, 446

Teng, Sinn-Tsair, 440f,446

Thaler, Richard H.. 56f,58, 59

‘Theeuwes,Jules, 670f.678

Theil, Henri, 97f, 100,298f, 304, 355,359, 391, 446,5530, SB4f, 591,392

Thomas, Annie, 95f, 98Thomas, Howard, 99Thomas, R. L.. 295f,

304Thompson, Gerald L.,

440f, 446Thurow, Lester, 170,

182, 2128, 2138,2148, 221

‘Thursby, Jerry G., 475,501

Thurstone, L. L.. 95f,100

Tidwell, Paul W., 1456,146

Tinbergen, Jan, 144f,148, 2981, 304,$50, 579f, 583f,50

Tirole, Jean. 96f. 99,100

Tobin, James M., 55f,59, 256, 258-259,304, 582f, 591,668F, 678

Toevs, Alden L., 500f,505

Toyama, Nate. 359Treadway, ArthurB.,

270, 298f, 304Tretheway. Michael W.,

482, 502Treynor, Jack L., 55f.

59Triggs, C. M., 439%, 442Triptett, Jack E., 120,

123, 124f, 128,1446, 1456, 148,149, 217, 219, 220,614

Trivedi, Pravin K..295f, 303

Turnovsky, Steven J.,269, 299, 582f, 391

Twain, Mark, 20, 59‘Tyrrell, Timothy J.,

3550, 358

Udis, Bernard, 666f, 671

Uhler, Robert G., 311f,353, 354f, 359

Ulph, David T., 667f,669f, 671

Usher, Dan, 302, 304Uzawa, Hirofumi, 499f,

506

Vandaele, Walter, 356f,359

Varian, Hal R.. 96f,100

Veall, Michael R., SOOf,

Verdon, Walter A., 393,446,

Verleger, Philip K., Jr.,355f, 359

Verma, Vinod K., 372.447

Vernon, John M., 96f,99

Viscusi, W. Kip. 216f,221

von Furstenberg,

George M., 297f,304

Wachtel, Paul, 2118,

220Wachter, Michacl L..

540, 582f, 590, 591‘Wagner, Joachim, 21 If,

221‘Wagner, Wayne,32. 59Wakeman,Frederic,

437%, 447Wald, Abraham,584fWales, Terence J., 210f,

211f, 221, 355f,359, 500f, 503,629, 631-634,669F, 679

Walker, lan, 670f, 672,

678Wail, Kent D., 55f. 57Wallis, Kenneth F.,

295f, 298f, 304,4400, 447, 525,527, 549, 580f,S81f, 585, 591,592

Walsh, John R., 210f,221

Walters, Alan A., 101,4998, 506

Walwick, Nancy J.,515, 591

Ward, Michacl, 666f,

678Watkins, G. Campbell,

501f, 502Watson, Mark W., 580f

Watson, Thomas J.. 1,18f

‘Wats, Harold W.. 135,149, 596, 614,

666F, 6636, 6701,672, 674, 679

‘Waugh,Frederick V..104, 106-110,132-135, 142,144f, 145£, 1466,

149, 401, 417,4378, 444

Waverman, Leonard,SOIL, $02

Wayne, Kenneth, 671,96f, 98

Webb, Aifted, 675Weil, Robman, 406Weinberg, Charles B.,

391, 392, 447Weisbrod, Burton, 240f,

Weiserbs, Daniel, 397,439%, 446

Weiss, Doyle L., 391,392, 439f, 443, 447

Weiss, Yoram, 156,157, 215, 222,6701, 679

Welch,Finis, 171f, 172,186, 187, 2121,213f, 219, 221,

222Wenz, Kenneth L.. 96f,

99West, Richard W., 670f,

676‘Weyant, John P., 360Wheelwright, Steven C..

356f, 358, 359Whitaker, John K.,

590White, Halbert, 53, 59,

208, 209, 219, 222,299, 356f, 359.

629, 640, 657. 661,662. 6691, 679

White, Kenneth J.,Svor, 504

Wildasin, David, 297f,304

Wilkes, Maurice V..it

Wilkinson, Maurice,96f, 100

Willis, Robert, 162,

167, 171f, 172,208, 210f, 212f,222

Wilson, Thomas A..4401, 441f, 442

Windal, Pierre, 391,

392, 439, 447

Author Index 693

Winer, Russel S., 439,447

Winfrey, Robert, 227,296f, 304

Wise, David A., 210F.222, 631, 675

Witteveen, Hendrik J.148, 591

Wold, Herman, 577,585

Wolf, Barbara L., 211,221

Wolfe, JN. 443

Wolpin. Kenneth, 210f,22

Wormer. N. Keith, 96f,100

Woo, Chi-Keung. 354f,359

Wuod. David O.. 297F,301, 449, 430, 472,486, 487, SOIL.502

Woodland, Alan D.,355K, 359, S0OF.506, 629, 631-634,669f. 679

Wouk, Herman,est

Woytinsky, Wladimir.6671, 679

Wu, De-Min, 4381, 447,584f, $91, 669F.679

Wykoff, Frank, 231.296f, 302

Wyplosz, Charles, 297f,299

Yao, Dennis A.. 96f,98

Yelle, Louis E.. 95f.100

Yellen, Janet L., 211f,212f, 214, 222

Yoshikawa, Hiroshi,260, 297f, 304

Youmans, Kenwood C.,213f, 217

1 Author Index

Zabalza, Antoni, 632,669F, 676,679

Zeides, Stephen P., 539.591

Zeliner, Arnold, 214f,222, 264, 297f,

304, 355f, 359,

391, 438f, 4391,441, 447, 4998,

506, 522, S8tf,S84f, 584, 591, ]592 q

Zimmenman,MartinL., 96f, 100

Subject Index

ies:

distribution of, 161as omitted variables, 166-

167

Accelerator model ofinvestment:

alternative distributed lagsfor, 234-237

empirical implementations,237-239, 281-283,291-292

flexible, 234naive, 233theory, 233-234

Adaptive expectations, SeeExpectations

Added worker hypothesis, See

Labor supplyAdding-upconditions ofshare

‘equations, 470-474,478-479, 491-492

Adjusiment cost models, 269—

270, 484-485

Advertising:as barrierto entry, 415,

brand loyalty model of, 388,392, 423-424

consistent market share

specifications, 383-385to counteract cyclical

TMacrocconomicfluctuations, 363

cumulative effect on sales,387-390, 401-404

current elfects models of,387-389, 423

distributed lag models of,385-389, 423

duration levels, 389-393effects on aggregate

consumption, 393-400,427-429

effects ofban on cigaretteconsumption, 409-414,432-436

expenditures, 362, 373frequency. 374Granger causality with sates,

380-383, 427-429importance ofcopy quality,

407-409, 436-437as information, 372Fingering effects models of,

386-393, 423,measurementof, 373-375optimal advertising-sales

ratios, 366-371

role ia changing consumers’tastes, 371-372

rote ofregression analysis inanalyzing, 416

scale economics in, 415-416

shape of advertisingresponse function, $14~als

simultancity with sales,375-380, 417-420,429-432

stock of, 435temporal aggregation and

data interval bias, 389~393

theoretical determinants ofadvertising outlay, 365-at

theoretical effects on sales,31-373

Arbitrage pricing model, 37-38, 52-53

AREMOS/PC, 16, 17ASCII code, 7

Autocorrelation:Breusch-Godireytest for,

$67-570Cochrane-Oreutt estimation

procedure, 5, 54, 92,93, 283, 345, 348, 356f,388, 558

Durbin-Watsontest for, 54,92, 347, 497

Durbin’s m- and h-testprocedure, 235, 347,534, 567-568

estimation with laggeddependentvariable.281-283, 294-295,520, 423- 424, $52

Hitdeeth-Lu estimationprocedure, 54, 92, 93,253, 282-283, 345,347, 388, 424, 558

local-giobat problems, 348maximum likelihood

estimation procedure,345, 348

Prais-Winsten estimationprocedure, 356f

with simmultaneity, 291-292,432, 534-535, $52,586, $58, 561-565. 569

in systemsof equations,476-479, 497

testing with laggeddependentvariable,235, 281-283, 294~295, 347, 404, $34-$35, 556, 567-570

vector, 477-479Autoregressive integrated

moving average(ARIMA)models, 268,289-291, 353, 382,427-829

695

396 Subject Index

Auniliary regression equation,16-77, 322

BASSSTAT.16, 17Bayesian procedures, 39Bertrandoligopoly model,

69Beta value:

definition, 27, 33different in January, 50-51estimation of, 34-41

exemstudies, 48-50for gold, 45stability, 35-36, 48

Blinder, Atan, [85-186BMDP/PC,16, 17Boston Consulting Group, 60,

960Box-Coxtransformation, 128,

142-143, 211f, 231Box-Jenkins ARIMA models,

268, 289-291, 319-320, 352~353, 382-383, 390-400, 564

Box-Pierce Q statistic, 290,

Sox-Tidwell transformation,128

Brand loyalty model, SeeAdvertising

‘CADET, 246

Capital asset pricing model:beta value, 27, 33-34estimation of, 34~41sisk-return linear

telation: F=32as special case of APM, 52-

33test of, 37-39use in constructing

portfolios, 47Capital stock:

constant exponential decay,228-232

detetioration and

depreciation, 230-232gr055, 228mortalitydistributions, 227net, 228one-hoss shay deterioration,

perpetual inventorymethod, 228

Cash flow model ofinvestment, 239-242,284-286, 5St

Causality, See Grangercausality

Censored data, 595, 616, 6666‘Center for Research on

Securities’ Prices, 37CHANCE,12, 19fChow test, 48, $1, 93-94,

139-141, 201-208,525-526

Cigarette advertising ban,effects of, 409-414,432-436

Cobb-Dougias productionfunction, 68-75, 248-256, 266, 286-287,450-452, 488-489

Cochrane-Orcutt procedure,See Autocorretation

Coefficientofdetermination:Partial, 108relationship to R?, 108-110separate, 109

Compensated substitutioneffect, 604, 65-008

Computer commands andoperations:

BASIC,10CD(change directory), 9COPY, 8-9DISKCOPY,8DOS(disk operating

system), 8EDLIN,10FORMAT,6MD(inake directory), 9operating system, &pathnames, 8-9RENAME. 9texteditor, 10TYPE,10word processors, 10XCOPY, &

‘Computers in econometrics:briefhistory, 2-6,Pioneering studies, 340

Conoco takeaver, 48-50Constant elasticity of

substitution (CES)production function,252-254, 292-294,452-457, 488-489, $30

Consumption function, $51~555, 566, 370-572, 576

Cost function:briefhistory of functional

formsfor, 457-458choice between production

and cost function, 457-460

general, 63generalized Leontief, 460-

465multiproduct, 481-482Iransiog. 81, 470-475, 483variable cost function, 482-

434Cournotoligopoly model, 369Cowles Commission, 2, 21Cross-correlograms, 399, 428-

anyCurrent effects model, See

Advertising

DATA-FIT, 16, 17Data interval bias, 389-393Dirichlet distribution, 500fDiscrimination,effects on

wages:by gender, [8B-190, 205

by race, 181-187, 203-205theory and measurement

issues, 180-184Discouraged worker

hypothesis. See Laborsupply

Distributed lags:accelerator model, 234-237,

281-283advertising-sales models,

385-389, 401-408Almonlag, 236, 284-289with autocorrelation, 292-

294, 316

cash flow model, 241-242,284-286

specification, 236

.

models, 316-320

geometric, 235, 317, 386inverted V-Ing, 236neoclassical model of

invesiment, 249-256polynomial distributed lag,

236-237, 284-289

rational tag, 237, 582fwith simuhtancity, 291-292Tobin's ¢ model, 287-289

Double-in-ten formuis, 307,339, 354f

Duality theory, 65Dummyvariables, 48-50, 51,

TEI-114, 136-14:151, 161-166,19:

Durationintervals, 389-393,421-422

Durbin-Watson test statistic:test for first order

autocorrelation, 54

use with lagged dependentvariables, 235, 281~

283, 386,

Economies ofscale:definition, 63empirical evidence, 81~83general, 61imponance of, 61and shape of average cost

‘curve, 64six-tenths rute, 61

ECONOMISTWORKSTATION,16,7

EDSAC Computer, 5-6, 356Education,rate of return to:empirical evidence, 169-

172stability over time, (71-172

Elasticity ofsubstitution:for generalized Leantiet

function, 464465,492-494

Hicks-Alten partial inmultiple input case,463

production function withconstant, 482-457

short: and long-run, 484for anslog function, 475,

492-494two-input case, 452

Electricity demand:autocorrefationin models

of, 346-348data guality for analysis of,

34-315

demand charges, 310double-in-ten formula, 307,

339empiricat evidence on, 328

335energy charges. 310with equipmentstocks

included explicitly,312-315

with equipment stocksincluded indirectly,315-320

essential facts, 308-312forecasting, 310, 344-346,

349-353functional form for models.

of, 326-328heteroskedastic errors in

modeis of, 34)marginal and intramarginal

prices, 321-324multicblock rate schedules,

309, 321-326short-run model, 312-313simuhtaneity ofprice and

quantity, 324-326,38

Typicai Etectric Bills data,325-326

Enteringdata, 13-15Error correction process, 529ET, 16, 17Event studies, 48-30Expectations:

adaptive, 267-269, 317~320, 529-530, $38-539, 544, 564

rational, 268-269, 319-320,484-485, 509, $38-539, 541, $44, $49,556, 561-568, $79

Experience curves, SeeLearningcurves

Feedback, 382Final form, 399, 428-429Forecast

advertising-salesrelationships, 398-400,433

double exponentialsmoothing procedure,350

dynamic, 271, 274-275

Subject Index 697

electricity demand, 307~308, 344-346, 349-352

exponential growth,Procedure, 349

‘ex post dynamic, 274-276,

Holt’s two-parameterexponential smoothingprovedure, 351

investment in fixed business

capital, 271-276, 294-295

single exponentialsmoothing procedure,380

static, 271timeseries procedures,

unit costs with learningcurve effects, 94-95

Fourier functionall form, 482Ful income, 602Full information maximum

likelihood, 463, 485,545, 552-554, 556,573-875,578, 626-627, 665

Functional form:Bor-Cox procedure, 128,

142-143, 2116

choice of, 127-128, 161~162, 326-328

flexible production and costmodels, 482

GAUSS,16, 17Genera? equilibrium models,

479-481Generatized Box-Cox

functional form, 482Generalized Leontief

functional form, 327,460-455, 478, 489-498

Generalized Tobit model,Limited dependentvariable models

Geometry mean,computation of, 194

Gord, 45Government spendingimpact

muhtiplier, $24Granger causality, 380-383,

427-429Gross output, 488

98 Subject Index

iross substitutioneffect, 605-606

Jausman specificationtest,379-380, 409, 419,430-432,565-567, 641

Jazard rate, 622leckit model, See Limited

dependent variablemodels

Yedonic prices:for advertising copy quality,

409

wy

applied to asparagus, 105-110, 132-135, 142143

applied to automobiles,110-116

applied to computerindustry, 118-126,136-143

briefhistory, 105-116chained, 141-142

incorporated by U.S,Bureau of EconomicAnalysis, 125

make effects, 129in models of cost and

production, 481—482as outcome ofsupply-

demand relationships,117118, 129-131

parameter stability, 12131

test for constantrate ofprice change, 139

with unobserved attributes,128-129

leteroskedasticity:in electricity demand

equations, 328, 34in labor force particip:

equations, 655obtained when using

averaged data, 127robust standard errors with,

53, 208, 209, 341, 629,640, 657, 661, 662

functions, 208-209

transformation of data toensurehomoskedasticity, 137,328, 341-52

,, White testfor, 53, 208-209

Houthakker-Taylor stateadjustment model,397-398

Humancapital theory:equalizing differences. 152-

153

‘measurementissues inempiricalimplementation, 159-160

on-the-job training asinvestment, 155-157

general training, 155specific training, 155-

156, 178role ofabilities, 155schooling as investment,

ESI-158serecning as an alternative

hypothesis, 157-158Hypothesis testing:

individual tests, 36, 78, 86~87, 113, 137

joint tests, 78-79, 86-87,T13-EE4, 139-141

Lagrange multiplier test,465-467, 494-496

likelihood ratio test, 465-467, 494-496, 554,$75, 579

nonlinear, 97f, 475, 534overidentifying restrictions,

$54, $57, 873-575, $79relation to A? vatue, 468-

469, 519, SBIEWald test, 465-867, 494-

496

Identification, 376-378, 419-420, 479, $24-525,552, 556, $70-573,625-626, 658-660

Indirect least squares, 377~379, S71, 660

Info World, 12, 19Information set, $43, 561-565

Instrumental variableimation:

consistency with rationaleapectationshypothesis, 267-268,561-565

general, 316, 319, 378-379,395~396, 484-985,556, 569, 664

Interaction terms, 163, 165,197-199, 200

Intertemporal utility function,527-528, 539-540,646-648

Inverse Mills ratio, Seesampleselectivity

Investment in fixed businesscapital:

accelerator model, 234-239,281-283, 295-292

adjustment cost models,269-270

autocorrelation, 293-294cash flow model, 239-24

284-286, 551changes in business

ventaries, 226‘empirical comparison of

models, 270-277, 294-295

forecasting, 294-295gross, net and replacement,

definitions of, 229moving average errors in,

267-268neoclassical model, 242~

256, 286-287non-static expectations,© 268-269putty-clay model, 250-254,

286-287eesidential construction,

226simuttancity issues, 266—

267, 291-292time series/autoregressive

model, 263-265, 283-284, 289-291

Tobin's q model, 256-263,287-289

Iterative minimum distanceestimator, $45, $47-549, 579

erative three-stage feastsquares (13SLS)estimator, 463, 474,484, 485, 492, $82,$53, $56, 573

Iterative Zellner-fficient(IZEF)estimator, 463,474, 488-496, 556

Jacobian, 576-578

Klein's Modet 1 of the US.economy, 550-558,$65~567, 572-578

Kolmogorov-Smimovtest fornormality, 54, 195

Koyck transformation, 235,267, 318, 386, 529

Laborforee, measurementof,

See Unemployment,labor supply

Laborforce participation rate,1. 5594-595,

added worker hypothesis,609-610

allocation oftimeframework, 610-613

backward bending function,606

comparative dynamics, 647complete budget constraint

approach to, 632, 635discouraged worker

hypothesis, 609-610,667F

effects from cyclicalaggregate demand,609-610

equilibrium dynamics.637

first generation studies of,614-617

Frisch demand notionimplications ofcosts of

employment, 613-614,627-629, 666

linearized budget constraint,631, 63:

male chauvinist model, 607,667t

measurementissues, $96-599

second generation studiesof, 617-637

and taxes, 617, 629-633,644, 663-664, 6697

theory, dynamic taborsupply, 645-649

theory, household laborsupply. 607-609

theory, individual laborsupply, 600-606

See also UnemploymentLagrange multiplier test

procedure, 465-467,474-475, 494-496

Lagrangianfunction, 65, 602,

Learning curves:autocorrelation, 92‘empirical evidence, 79-80estimation of, 85and experience curves, 66forecasting units costs with,

94-95general, 13, 61-62integration with Cobb-

Douglas function, 71-18

measurementissues, 75-76‘omitted variable bias, 76-78overview, 66-68slope, 67, 80, 85

Lemer index of monopoly

powerLikelihood test

procedure, 465-467.474-475, 494-496,$54, 575, 579, 619, 620

LIMDEP, 16, 17Limited dependent variable

models:general, $9Sgeneralized Tobit, 596,

627-629, 650, 661-664, 665

Heckit, $96, 627-629,651,642-644, 650, 661-664

logit, 333, 596, 650, 654-657

Brobit, 596, 619-620, 623,625, 627, 629, 641,642, 650, 654-658, 668

Subject Index 699

Tobit, 596, 620-624, 626~627, 629, 643, 644,650, 657-658

Limited informationmaximum likelihootestimator,552

in prices,

Linear probability model, 655Lingering effects models, Sev

AdvertisingLogit model, See Limited

dependent variablemodels

Lydiametries, 364, 404

MacUser, 12, 19fMacWeek, 12, 19fMacWorld, 12, 19fMarginal rate of substitution,

600-605, 618, 624,645

estimation, 463, 473-475, 494-497, $73-579

Mcanvalue function, 453Meta-analysis, 392-393MicroTSP, 15, 17Minflex Laurentfunctional

form, 482Minimum chi-square

estimator, 462-463Minimumdistance estimator,

545,547Minimum expected loss

estimator, 584fMINITAB,16, 17Mode! T Fords, 66-65Money illusion, 536, 538Monte Carlo techniques, 475,

573Movingaverage error models,

20, 413, 427

64,Multicollinearity, 136, 319,

339Multifactor productivity

growth, 485-486, 497-498

MYSTAT,16

00 Subject Index

datural rate of

unemployment, 522Acoctassical model of

investment:‘empirical implementation,

248-256, 286-287theory, 243-245user cost ofcapital, 245—

248

Neo-Keynesian models, $39,S41

Nested logit model, 333Net marginat revenue

product, 366New classical school, 538Non-aceelerating inflation rate

of unemployment, 540Nonhomothetic functions, 469Nonlinearleast squares, 377,

418, 434, 626Normality:

chi-square goodness of fitteat, 54, 195

Kolomogorov-Smirnov testprocedure, 54, 195

‘Okun's Law, $40Omitted variable bias:

auxiliary regression, 76-77,322, 344

in clectricity demandcontext, 321-324, 343-a

Jearning curves, 76-78, 86retums to schooling, 166-

167On-the-job usining. rate of

relum to:affected by intermittent

labor forceparticipation, 174

empiricat evidence, 172-196

‘Overidentifying restrictions.test for, $54, 337, 573~575, 579

Panel data, 167, 649, 668fPartial adjustment hypothesis,

315, 320, 346-348, 408PC-GIVE, 16, 17PC Magazine, 12, 190PC-TSP, 15, 17, 18, 348PC Week, 12, 196

PDQ, 16. 17Permanent income measure,

314, 647-648

Phiblips curve:carly history, 312-517expectations augmented,

precursors to, 516-517stability, 517-527underlying theory, 512-513,

518-520Pinkham,Lydia E. Medicine

Company, 402, 420-424

Pooled crass-section, timeseries estimation, 436—4ayT

Ponfolio:aggressive, 14defensive, 34diversification, 23-27, 32efficient, 31optimal, 27

Potentiat output, 522, $41Price elasticities, 464-165,Price indexes:chained priceindexes, 141-

142quality change via matched

model procedure, 101—105

See also Hedonic priveindexes

Probit models, Sec Limiteddependent variablemodels

Production function:brief historicat review of

nal forms for,

choice between productionand cost function, 457--

460‘Cobb-Douglas, 68-75, 248-

256, 450-452constantelasticity of

substitution (CES)function, 452-457

definition, 63integration with learning

curves, 71-75,

PSTAT, 16, 17

R? measure:changes in to conduct

hyportiesis tests, 519.Seif

high value docs not implyvalid model, 499

interpretation in CAPMframework, 40, 47

interpretation in linearprobability model, 655

relation to correlationcoefficient, 131-135

relation to partial andseparate coefficients ofdetermination, 108-409, 131-135

in reverse regressions, 456,489

in simple and mukipteregressions, 86-87,131-135

in systems of equations,468-469, 496

value in regression ofactualon fitted y. 135

Rate of retum:definition, 21-22marginal return, 26marginal variance, 26risk-free, 22risk premium, 22to schooling, 161-163

Rational expectations, SeeExpectations

Rational predictor, 319RATS, 15, 17Reduced form model:

to compute indirect feastsquares estimates, 571

equivalence to structuralestimates, 573-575

gencral, 376-380, 533, 535,$84, $45, 555-555, 556

tase in forecasting orsimulation, 526-527

use in fabor supply models,624-626, 658-660

Reservation wage, 603, 613,616, G18, 625, 645

Residential End-Use EnergyPlanning System(REEPS), 332-335

Returns to scale:for Cobb-Douglas function,69

definition, 63empirical evidence, 80-83estimation of, 87-94and shape ofaverage cost

for translog function, 476Reverse regression, 45, 134,

191-192,455-456Risk:

definition, 22linearrelationship with

return, 27-32market, 32and portfolio diversification,

2-27premium, 22tiskefree return, 22specif, 32systematic, 32unique, 32unsystematic, 32

Root mean squared error:

to evaluate forecasts, 272-277, 291, 294-298

Sales:effects from advertising,

371-373, 393-400Granger causality with

advertising, 380-383See also Advertising

Sample selectivity:general, SBIf, 595inverse Milts ratio, 191,

622, 627, 642, 661~664, 670F

in labor supply equations,616-629, 642-644,661-666

in wage rate equations,

190SAS/STAT,16, 17Saturation model, 313SCA SYSTEM, 16, 18Screening hypothesis of

education, 157-158Scemingly unrelated

estimator, 462-263,S71, S74

Setection biss-correctedtegression, 621-623

Shadow values, 484

Share models:adding-up conditions for,

470-471, 491-492

autocorrelationin, 477-479consistent specification in

advertising context,

383-385, 424-426in cost functions, 470-474,

491-492,

SHAZAM, 15, 18Shephard's Lemma, 461, 370,

983, 485Simultaneous equations

estimation:in advertising-sales models,

378-380, 429-432alternative estimators, $50-

558, $70-$73with autocorrelation, 291-

292, 569in cost and production

models, 474-475in electricity demand

in investment models, 291~292

Singular equation systems,345, 472-474, 478-479, 491-492

Slutsky equation, 606SORITEC,15, 18Sorting data. 195, 652-653SPSS/PC +, 16, 18SST,16, 18STATA,16, 18STATGRAHICS,16, (8

Statistical software programs:data, VLdata terminator, U1header, 11overview, 15-17reviews of, 12vendors for, 17-18

STATPRO,16, 18STATVIEW 512+, 16, 18‘Structural equation modet,

375, 380, $45, 558,570, $73-875

Structurally recursive models,556, 575-578

Successiveleast squares, 556,317

Subject Index 701

Symmetric generalized‘McFadden functions!form, 482

SYSTAT,16, 17

Taylor's series expansions, $53Technical progress:

biased, 498disembodied, 485, 397embodied, 486and the measurement of

multifactor

productivity growth,485486, 497198

and shifts in average costcurve, 64-65

Temporal aggregation, 389~393, 420-422

Theil’s incquality coefficient,295

‘Three Mite Island, 48-50Three-stage least squares, 463,

474-475, 485, 492,$45, 552, 553, 556,564, 570, 573

‘Time SeriesAutoregressiveinvestment models:

empitical implementation,

24-265, 283-284,249-291

theoretical foundations,263-264

Tobin's g model ofinvestment:

average vs. marginal g, 259-261

empirical implementation,259-263, 280, 287-289

theory, 257-259Tobit model, See Limited

dependent variablemodels

Transfer function, 428-829Transitory income, 647-648Translogfunction, 81, 327.

469-476, 480, 489-398

Truncated data, $95, 666FTruncated normal repression

equation, 654‘TSP, 15, $48, $53, 555f‘Two-stage least squares:

in advertising-sales models,379-340, 398, 417-420, 432-433

702 Subject Index

Two-stage least squares (cont)with autgcorrelation, 432,

534-535consistency with rational

‘expectations

hypothesis, 267-268,561-565

in cost of productionfunction content, 463,474

difference from two-stepordinary east squaresestimation, 420, 5$8—561

general, $52in investment models, 291-

2

in labor supply models,640

in macro-cconomenicmodels, 545, 552-554,556-567, 570-573

in Phillips curve context,525, $33-535

Fefation to suycessiye leastSquares estimator, 577

Uncompensated substitutioneffect, 605-608, 639-G44, 654, 662

Unemployment, measurementof, 530-512, $31-532,oo7t

User cost of capitdefinition, 245expected capital gains term,

248measurement of, 245-248

Value-added output, 188Virtual income, 631, 663

Wage rates, determinants offdifferences forsingle and

married women, 197

198discrimination as source,

180-190

effects of cyclical variations,192

efficiency wages, 175, 192empirical evidence on, 167~

179estimating experience-

carnings profiles, 199--200

human capital theory, 152+

sample selectivity issues, 191sercening hypothesis, 157-

188

union-nonunion relativewages, 176-179, 196—197, 201-203

Wald test procedure. 465-467,474, 194-196

WeberFechner law ofpsychophysics, 372

Zelingr-cfficient estimator,462-463, 474, 488—492, 545, S71, 578

eo