SYMBOLS USED IN CHAPTER 2 - Home - Springer978-94-011-2972-5/1.pdf · SYMBOLS USED IN CHAPTER 2 c e K n p q( Q t ... SYMBOLS USED IN CHAPTER 3 ... Layard & Walters (1978, p89)

A P PEN D I X: S Y M B 0 L S USE D

SYMBOLS USED IN CHAPTER 2

c

e

K

n

p

q(

Q

t

individual's available capabilities, supplied in job j.

working conditions, job j

labor cost

environmental quality of a firm (job)

return to time share A, activity i

short run production function

learning curve parameter

investment outlay

labor input index, job j

number of operations, learning curve

number of workers, job j

output price

production function

output

job requirements, job j

experience

wage, job j

learning curve parameter

elasticity of x to Y

300 CAPABILITIES, ALLOCATION AND EARNINGS

share of worker time devoted to activity i

Langrangean multiplier

SYMBOLS USED IN CHAPTER 3 (IF NEW OR DIFFERENT)

a o initial capability level

e education type

g(r) constraint function for job choice

h( ) ,h() constraint functions for job choice (upper bounds and lower bounds)

s

u(

u

5,A

p

rate of return to s years of schooling

upper and lower bounds on available jobs, capability j.

length of schooling

utility function

utility

shadow prices (Kuhn-Tucker multipliers)

marginal rate of substitution, job requirement to wage

individual discount rate

APPENDIX: SYMBOLS USED 301

SYMBOLS USED IN CHAPTER 4 (IF NEW OR DIFFERENT)

y

~i

~ij

~i

personal characteristics individual i

output individual i, activity j

earnings

job characteristics, job j

relative dispersion (available to required) capability i

unobserved tastes, individual i

index for individual i to select job j (0,1)

coefficient of variation, required capability i

share of time devoted to activity i (section 4.4)

coefficients earnings function (section 4.3)

mean of distribution of capability j (i=a: available, i=r: required)

unobserved skills, individual i

profit from hiring worker i

standard deviation of distribution capability j (i=a:available, i=r: required)

index for individual i to want job j

coefficients utility function

of

NOTES

1 INTRODUCTION

1 This section draws on Hartog(198la, Chapter 7), where

much more detail can be found, as well as additional

references.

2 For a critical evaluation of the DOT-measurements, see

Ann Miller, et al (1980).

3 The discussion draws heavily from Peterson and Bownas

(1982) .

4 It is interesting to note in passing, that psychol

ogical research has identified a limited number of

stable types of preferences. To quote from Peterson and

Bownas (1982, p.83):"Evidence collected by many

investigators ... indicates that vocational preferences

are highly stable (after age 21) for individuals over

long periods of time (up to 25-30 years). They are

also reasonably effective predictors of future occupa

tional classification for persons- -especially within

broad occupational families as opposed to highly

specific occupations". Perhaps not surprisingly, they


have poor predictive value for performance within

occupations.

5 Hunter and Schmidt (1982) give some empirical evidence

that the dispersion of the dollar value of performance

across individuals differs among occupations.

6 A concept often used is skill. It may refer to specific

job skills and then reflects proficiency in performing

a particular activity, e.g., typing. The distinction

between unskilled, semi-skilled and skilled workers

can be based on required training time for proficiency

in a particular job, ranging from at most a few days

for unskilled to a few years for skilled work. Skills

are activity-specific, but they can be measured

uniformly along the time-scale. The concept of skill

will not be used here. See also the discussion in

Rumberger (1983).

7 Note that this suggests an optimal allocation of

individuals to jobs (assigning the top ten percent of

the labor force to jobs requiring that level), but

since these scales are defined separately for each

type of ability, this can never be real ized (unless

the correlation between the abilities is perfect).

NOTES 305

2 LABOR DEMAND

1 For simplicity, the dependence of wages on number

hired is suppressed; the results of this section are

not affected, as they also hold for a monopsonistic

firm.

2 If (2.50) is correct, it implies (using (2.49), and

noting that O<~<l):

[l+At(l-~)/hl }-~/l-~ > A(l+t(l-~)/hl }-~/l-~

l+H(l-~) /hl < {l+t (l-~) /hl } A -(l-~) h

which is indeed correct, since O<A<l.

3 Note that the tax system could subsidize the provision

of working conditions as consumption goods. If tw is

the marginal wage tax rate (including social security

premiums, etc), the worker receives (l-tw)w from his

wage and the marginal valuation of the worker in terms

of net wages is transformed into savings of w(l-tw)-l

for the firm. Similarly if the outlays are deductible,

the cost reduces to e(l-tc)' where tc is the marginal

corporate tax rate. Then (2.52) becomes

aC(l_t ) ae c

306

With wage

under the

CAPABILITIES, ALLOCATION AND EARNINGS

savings increasing and cost decreasing,

usual conditions this means an increased

consumption of working conditions. In fact, firms have

an incentive to stimulate working conditions over

raising wages, as it is a cheaper way of increasing

worker utility.

4 In a newspaper ad, a firm selling air conditioning

systems refers to research claiming that every addi

tional degree Celsius above 20° reduces labor produc-

tivity by 4%. NRC, May 5, 1987, p.16.

3 SCHOOLING AND SUPPLY

1 The result applies that for a function

N J: f (z) dz,

aN/ax

2 Just to illustrate the tremendous size of the choice

set: Kodde and Theunissen (1984), in an analysis of

educational careers in the Netherlands, note that

after secondary education, individuals can choose

among at least 130 types of education in higher

education (p.118).

NOTES 307

3 It is assumed that there is only one intersection that

also satisfies the second-order condition for an

earnings maximum.

4 The decomposition of (3.21) into two separate condi

tions only makes sense if one takes the scale of

measurement of capability levels as a useful reference.

Test scores should neither be standardized for age nor

for education. Note that the scaling problem does not

apply to (3.19).

5 It will be assumed that for each individual both

educations yield a sufficient rate of return.

6 The analogy of the present model to that of Rosen is

quite close and the results obtained above for the

case of individuals ranked by aOA/aOB are very similar

to his results of assigning different workers to

identical tasks (section III).

7 See Varian (1978), p.259. It is assumed that 'con

straint qualification' holds for the binding con

straints.

8 Varian (1978), p.268.

9 In particular for lengthy education, it is better to

measure the increases relative to some reference

education, to make the evaluation with first-order

derivatives more adequate: it would not be apt to

measure these derivatives at zero-years of education.

10 This result is found from setting OJ=O in the general

school effect equation

(aw/aae-ojahj/aae-I~magm/aae) ,


and bringing in the other conditions mentioned in the

text (M=l, aw/aae = 0).

11 A model with initial allocation based on schooling,

but without transfer cost, is developed in Hartog

(198lb). Empirical support for the assumptions made

here is presented in Hartog (1983). Taubman (1975)

shows that the effect of ability on earnings is

stronger at more advanced stages of the career. Note

that the present analysis focuses on labor supply

behavior, and is not an equilibrium model of the labor

market. For an elaborate theory, see MacDonald (1982).

12

13

A simple specification of dgm (e) might occur with

~ (r, e) = &;.(r) - am (e) :s 0 and e only affecting the

intercept, i. e. , dgm (e) = -dam (e) .

If s works as a screen by a separate effect of school-

ing length on the constraint(s), this will still be

irrelevant for condition (3.37); any such effect will

be included in the wage function w(s), but will not be

affected by initial (unobservable) capability levels.

14 Note that the two conditions now have a joint interval

boundary because drB/drA is not differentiated by

individuals.

15 Since Am > 0, and since it is supposed that constraint

relaxing has a positive effect, -dgme is taken to be

positive. For an illustration, see note 12.

16 Note 2 cites the existence of 130 types of higher

education curriculums. From the survey by Peterson and

NOTES 309

Bownas(1982), one may conclude that the relevant number

of capability types is far less.

17 In case of utility maximization, the discussion should

be cast in terms of the distribution of welfare rather

than of earnings.

18 In secondary school, students choose Latin and Greek

significantly more often if in elementary school they

scored high at language; they choose sciences sig

nificantly more often if in elementary school they

scored high at mathematics. Similar results hold on

type of secondary school chosen.

4 EQUILIBRIUM AND OPTIMUM

1 Lucas(1977) specifies only Zj and refers to McFadden's

axiom on the Independence of the Irrelevant Alterna

tive. This seems an unnecessary strong condition to

impose, and is not copied here.

2 For continuity, Tinbergen's symbols have been replaced

by those of the present book. Details on deriving the

solution are given in Tinbergen(1956); some generaliza

tion and error correction is available in Van Batenburg

and Tinbergen (1984).

3 Both available degrees a 1 and az and demanded degrees

r 1 and r z are assumed to be independently distributed.

The solution is obtained by calculating first an

individual's optimum supply of attributes given his


available levels and the utility and wage function.

The transformation of ' available' levels into ' sup

plied' levels then transforms an 'available' distribu

tion into a 'supplied' distribution, i.e., transforms

the moments of the bivariate normal 'available'

distribution, into moments of a normal 'supplied'

distribution. These moments are then equated to those

of the required distribution, which allows to solve

for the parameters Ai j .

4 The results reported here, in Tables 4.1, 4.2, A4.l

and A4.2 were calculated by Peimin Zhang whose work as

a research assistant was financed by the Department of

Economics of Queen's University in Canada.

5 A generalization of Tinbergen's theory to an arbitrary

number of attributes while maintaining the basic

structure has been given by Epple(1987); see also

Epple(l984) .

6 Such a result is already contained in Roy(1951) .

7 Note that with increasing returns to time share (as

discussed in Chapter 2), the output frontier for a

given worker (say, of capability r3 in the terminology

of section 2.3.2) would be convex to the origin:

I

9 I

". ~

1\ I~

NOTES 311

If a represents the output for one man devoting a time

share a to producing ql and (1-a) to qz' then qlO >

ql(a)/a and qzo > qz(l-a)/(l-a), where qi (x) means qi

produced at time share x. Clearly specialization is

now advantageous. This can even be reinforced if

workers specializing in ql or qz can be chosen at

optimal capability level r 1 and r z .

8 In equilibrium marginal rates of substitution for the

activity outputs should be equal among consumers A and

B, since they face the same price ratio.

9 This statement depends on the crossing of individual

output frontiers; see below in main text.

10 See e.g., Layard & Walters (1978, p89). For related

literature on

Dreze(1976) ,

(1979) .

the problem, see Dreze and Hagen(1975),

Duncan and Stafford(1980), Weddepohl

11 Atkinson and Stiglitz(1980), Lecture 17.

12 The thesis would seem to appear quite generally to any

bargaining when represented members differ in preferen

ces.

13 Benarot, Kamens and Meyer (1989). The analysis is

still very preliminary and appears to hide more varia

tion than the authors conclude to (cf the final table

in their paper).


5 IMPLICATIONS FOR EMPIRICAL WORK

1 The same result occurs with any other selec tion rule

and correlated errors between the selection function

and the earnings function.

2 Since marital status of males was not known, this was

predicted from a random drawing of a number between 0

and 1, assigning the individual to 'married' if the

number is not above p, with p the known proportion of

married males in the relevant age group. The procedure

was suggested and implemented by Gerard Pfann.

3 Work by John Ham (1982) indicates that estimation

from midpoints may produce only very mild bias.

APPENDIX 5.1.

1 A detailed account of the data collection is given

(in Dutch) in Hartog & Pfann (1985).

2 It was K. Molenaar who discovered that the 1952

questionnaires were still available in a university

archive in Nijmegen.

3 Some schools had school years beginning in April

rather than in September. For these schools, half the

pupils of half the schools were included in the sample;

this yielded 369 answers (among a total of 5823).

4 The probability of non-response for a standard male

(all dummies at zero, IQ = 100, scholastic achievement

NOTES 313

score = 7) in the probit-only model is the area under

the standard normal distribution up to z .279-

(.005) 100 + (.032)7 = + .052. The other calculations

are similar.

6 ALLOCATION

1 The program was written by Geert Ridder at the Univer

siteit van Amsterdam (GRMAX).

2 Note that

where the subscript n refers to the n- th explanatory

variable in the vector Xi .

3 To avoid complicated statements, the expresssion

'relative to the effect on the probability of obtaining

the highest job level', is usually suppressed.

4 Cases of misreporting (i. e., the first year of a new

school is before the last year of a preceding school)

were eliminated.

5 Measuring education in years, the coefficients are

estimated on 2000 observations and predictions are

made for 245 cases. The specifications with education

dummies and childhood variables use 1300 observations

to estimate and 110 to predict.


6 In view of the high computer cos t and limits on the

number of explanatory variables, various alternative

specifications have been tried in turn to allow some

selection of variables.

7 Checks have indicated that the effect of experience is

quite independent of that of years educated.

8 The effect of family background through schooling has

been ignored.

9 Since job level is measured at an ordinal scale,

comparison of expected job levels, strictly speaking,

is an arbitrary standard. However, Table 6.8 indicates

that only at two job levels (4 and 7) the upward

effect of education is not substantially smaller for

the group with the higher actual education. This

points in the same direction.

10 The comparison is made between two predicted job

levels, and not with observed job level as reference.

Otherwise, the resul t would be confounded wi th the

prediction error from the model. The predicted job

level is not necessarily equal to the observed job

level; this explains why downward shifts may occur at

the lowest observed level and upward shifts at the

highest observed level.

11 See Maddala (1983, p. 46) for an exposition of the

model.

12 These conditions are spelled out in Madda1a (1983,

p.48). The computer routine used was GRMAX (See note 1)

NOTES 315

13 Extensive work on the effect of market situation on

allocation is reported in Teu1ings (1990).


7 EARNINGS

1 The estimates are only consistent if the error term in

the price equation is uncorrelated both with the

errors in the supplier's valuation function and those

in the demanders' valuation function, see section 5.4.

2 They are available in: Hartog, Van Ophem and Pfann

(1985), Hartog (1986a,b)

3 Lang and Dickens (1988) argue that such a test as

applied here has limited value. Adding job characteris

tics after allowing for worker characteristics would

also produce significant coefficients if only worker

characteristics were relevant, because of correlation

between unobserved worker characteristics and observed

job characteristics (such correlation follows from the

hedonic model, see section 4.2). At the least, this

calls for datasets that contain as many worker charac

teristics as possible. It also calls for structural

models specifying the relations between worker and job

characteristics.

4 This derivation of the multinomial logit model is due

to McFadden; see Maddala (1983).

5 Using the theory of hedonic prices, the predicted sign

is ambiguous. The offered wage-hours relation may be

parabolic, due to fixed cost at low hours and declining

marginal productivity at high hours. Moffitt (1984),

in the context of a labor supply model simultaneously

estimates an hours and a wage equation and finds a

NOTES 317

quadratic wage relation with negative slope for weekly

hours above 34.

6 If three outlyers are removed from the observations at

job level 7, the intercept increases to 11.38 (1.62),

the effect of test score is reduced to .825 (2.02) and

the coefficient of >. increases to -18.50 (1.90). At

job level 5 IQ and>. are highly collinear. If >. is

deleted, the intercept is reduced to 19.54 (5.12), the

coefficient of IQ is reduced to -.044 (1.34) and of

test score to .629 (1.86). If IQ is deleted, the

coefficient of >. becomes - .349 (1.41), with other

coeffcients not much affected.

7 The routine was provided by Centrum voor Wiskunde en

Informatica (C.W.I.), Universiteit van Amsterdam.

8 The earnings coefficients are the coefficients on

dummies for each education job level combination

(with some job levels combined, as indicated) in a

regression equation that also includes age, age

squared, experience with present employer, sex, and an

intercept, all highly significant.

9 The dataset is known as the NPAO-Mobility survey. The

original sample consists of 2677 observations. Earnings

are reported as after - tax earnings in Dutch guilders

per period, turned into hourly earnings by dividing by

reported usual hours of work. Education is recorded

according to the Standard Classification of Education

(used by the Dutch National Statistical Office, CBS),

classifying schools in 5 levels that work out to


durations in multiples of 3 years (from 6 to 18). The

number of observations used for the regressions in

Table 7.14 is 540 for total, 394 for males and 140 for

females.

10 The following translation of schooling levels into

years is made: some grade school =4,

school =8, some high school =10, finish

=12, some college =14, finish college

college =17.5.

8 APPLICATIONS, CONCLUSIONS, EXTENSIONS

1 See section 6.2.5.

2 See section 8.3.

3 See section 7.4.2.

finish grade

high school

=16, beyond

4 Such cost would occur for transition from one alloca

tion to another. It is not sure what would happen if

there were a transition to a different system of

allocating individuals to jobs and job levels.

NAME INDEX

Atkinson, A.B. 311 Bartik, T.J. 149 Becker, G.S. 3 Benarot, A. 311 Bierens, H.J. 230 Bownas, D.A. 6,303,309 Brasse, P. 280 Braverman, H. 19,20,142 Brown, B.W. 86 Brown, J.N. 149 Conen, G.J.M. 145 Davies, J.B. 89,135 Dickens, W.T. 316 Dreze, J.H. 311 Dronkers, J. 144 Duncan, G.J. 263, 311 Edwards, R.C. 238 Epple, D. 149,151,242,310 Hagen, K.P. 311 Ham, J. 312 Hanushek, E.A. 48,49 Hartog,J. 1,4,5,34,47,105,133,135,

195,230,236,267,278,283, 303,308,312,316

Hay, J.W. 233,273,275 Heckman, J.J. 147 Hirschman, A.O. 130


Hoffman, S.D. 263 Holmlund, B. 40 Huijgen, F. 145 Hunter, T.E. 295,304 Jovanic, B. 296 Kamens, D. 311 Kodde, D.A. 89,306 Lancaster, K. 135 Lang, K. 316 Lau, L.J. 48 Layard, P.R.G. 311 Levin, H.M. 48,249 Lindsay, C.M. 135 Lucas, R.E.B. 91,92,309 Lutz, B. 144 MacDonald, G.M.T. 89,119,133,308 Maddala, G.S. 146,181,233,270,314 Mangione, T. 154 Markusen, T.R. 119 Marshall, A. 19 McCormick, E.J. 4 McKelvey, R.D. 212,252 Meyer, J. 311 Miller, A. 303 Miller, R.A. 296 Moffitt, R. 316 Molenaar, K. 312 Oosterbeek, H. 265 Peterson, N.G. 6,303,308 Pfann, G.J. 237,283,312,316 Quinn, R. 154 Ridder, G. 283 Riesewijk, B. 145 Ritzen, J.M.M. 132 Rosen, S. 68,87,89,99,119,148,151 Rosen, H.S. 149 Roy, A.D. 310 Rumberger, R.W. 145,304 Saks, D.H. 86 sattinger, M.A. 118 Schmidt, F.L. 295,304 Seashore, S. 154 Sedlacek, G. 147 Sengenberger, W. 144 Sicherman, N. 268

NAME INDEX

Sikking, E. 280 smith, A. 18 Spence, A.M. 79 Stafford, F.P. 311 staines, G. 154 stiglitz, J.E. 311 Taubman, P. 308 Terza, J.V. 233 Teulings, C. 315 Theunissen, M.A.M. 89,306 Thurow, L.C. 226,264 Tinbergen, J. 91,98,105,259,309 Tsang, M.C. 249 Van Batenburg, P. 105,309 Van Hoof, J.J. 144 Van Ophem, J.C. 237,316 Varian, H. 307 vriend, N.J. 277 Walters, A.A. 311 Watson, C. 89 Weddepohl, C. 311 Weiss, A. 36 Willis, R.J. 87,89 Zanders, H.L.G. 153 zavoina, W. 212,252

321

SUBJECT INDEX

ability index 113 achievement test score 48 aptitudes 98 Babbage's Principle 19,30 bid function 96 cognitive abilities 6 comparative advantage 61,107 compensating differential 97 constraint lifting 73 corporate tax 305 curriculum (educational) 55,132 diploma (school) 90 discount rate 56 earnings distribution 88 economies of scale 133 education length 55 educational production function 48 efficiency units 121 effort 83 envelope-property 94,136 Envelope Theorem 73 equity 254 excess demand 210 excess supply 210 Gumbel distribution 270 hiring standards 44 hours (of work) 123

CAPABILITIES, ALLOCATION AND EARNINGS 323

Information 133 integration 22 intelligence 6 IQ 165 job complexity 4,8 job design 122 job difficulty 108 job enrichment 36 job matching 295 job requirements 5,7,41 labor market segment 26,226 labor union 131 learning curve 34 leisure 123 Lindahl solution 128 minorities (ethnic) 277 mismatch 254 multiple job holding 124 non-cognitive abilities 48 non-response 164 overeducation 223 overqualification 211 overutilization 98 Pareto-efficiency 240 personality variables 6 polarization 20,28,31 product differentiation 135 psychomotor abilities 6 public goods 126 quality of work 18 returns to education (rate of) 56 safety regulations 130 scholastic achievement 167 scholastic test score (deciles) 202 screening 49 selectivity bias 146 self-selection 146 separatibility 106 skill bumping 142 social efficiency 240 sorting: vertical 57,201

horizontal 58 specialization 22,109 sUbstitution: quality - quantity 16

324

tax rate: wage tax 305 corporate tax 305

Taylor system 36 Tiebout hypothesis 128 time share sensitivity 23 trickling down 104,142 undereducation 223 underqualification 211 underutilization 98 vocational preference 6 worker alienation 18 working conditions 124

SUBJECT INDEX

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SYMBOLS USED IN CHAPTER 2 - Home - Springer978-94-011-2972-5/1.pdf · SYMBOLS USED IN CHAPTER 2 c e K n p q( Q t ... SYMBOLS USED IN CHAPTER 3 ... Layard & Walters (1978, p89)