LABOR MARKET EFFICIENCY AND THE DYNAMIC BEHAVIOR OF REGIONAL UNEMPLOYMENT DIFFERENTIALS

Richard B. Tiller*

Abstract

In dynamic complex economies where unemployment is never zero, relative economic opportunities can be greatly effected by how evenly unemployment is distributed throughout the economy. Accordingly, a study of the temporal behavior of geographic differentials in unemployment rates will yield useful information about how well the labor market performs its resource allocation function. Using spectral methods, this paper tests the competitive theory predictions concerning the dynamic behavior of regional unemployment differentials. While there is evidence that unemployment rates tend to equalize, the results of this study indicate that it is a slow process in which competitive forces are relatively weak. As a result, the secular shift in employment growth from the North to the South and West has been the source of long run regional unemployment differences.

I. Introduction

In dynamic complex economies where unemployment is never zero, relative economic opportunities can be greatly effected by how smoothly unemployment is distributed throughout the economy. Compared to the wage differential literature, however, surprisingly little attention has been given to how well unemployment rate differentials function as a labor allocation device. The little empirical evidence that does exist is mixed (see Gallaway [5], Peterson and Muller [ 13], and Wheaton [ 17] ).

This paper tests the competitive theory predictions concerning regional unemployment rate differentials in a dynamic setting using spectral methods. The empirical analysis is based on the premise that the strength of competitive forces is reflected in the dynamic relationship between regional differences in unemployment rates and job growth. Over the long run these two variables should be independent of each other, if, allowing for structural differences, competitive forces equalize jobless rates across markets.

This hypothesis is tested on State time series data over a period in which there were dramatic shifts in regional growth rates. While there is evidence that unemployment rates tend to equalize, the results of this study indicate that it is

*Economist, Bureau of Labor Statistics, U.S. Department of Labor, USA.

Date Received: March 1984.

21

RICHARD B. T I L L E R

a slow process in which competitive forces are relatively weak. As a result, the long run shift in employment growth from the North to the South and West has been the source of long run regional unemployment differentials. However, there are important differences in the way State labor markets adapted to growth or decline. The simple frostbelt-sunbelt dichotomy is somewhat misleading.

If. The Competitive Hypothesis in a Dynamic Setting

In a generalized Borts-Stein neoclassical multi-market model regional growth rates need not equalize. Given differences in income and price elasticities of regional product demands, shifts in the composition of national demand are sufficient to generate permanent interregional differences in output and employment growth (Richardson [14]). However, the theory does predict that competitive market adjustments will eliminate labor shortages in rapidly growing regional economies and prevent chronic labor surpluses in declining areas. The two classical market mechanisms critical to this efficient allocation of labor are, I) interarea relative price-wage flexibility which tends to divert commodity trade to excess supply areas and, 2) quantity adjustments in the form of labor moving to those areas where returns are highest. Of the two, labor mobility is generally believed to be the most important. This follows from the Hecksher-Ohlin model which shows that commodity trade alone can equalize factor payments only under very restrictive conditions. In contrast, perfectly elastic regional labor supply functions are sufficient to equalize earnings under very general conditions.

If the classical model is expanded to include unemployment, then the labor supply response depends not only on wages but also on the probability of obtaining employment (one minus the unemployment rate). A worker's propensity to change location is, thus, a function of expected wages--the real wage market differential discounted by the unemployment rate (Todaro [16]). Adding unemployment does not change the basic nature of the classical market clearing mechanism. Both wage adjustments and labor mobility act to equilibrate expected earnings. The difference is that unemployment rates may substitute for wages as an allocative mechanism (Hall [7]). In general, impediments to wage adjustments will stimulate labor mobility and vice versa with the result that unemployment rates tend to equalize across markets (Gallaway [5] and Behman Ill. Traditional theory, however, has little to say about the dynamics of this process, particularly the length of the adjustment period.

Recent applications of neoclassical aggregate equilibrium theory of stochastic environments to multisector labor models provides a useful dynamic framework for evaluating the competitive hypothesis. Although this theory was developed to explain national business cycles, it also has direct applications to the issues raised in this paper. Lilien [I0], for example, has adapted the Lueas- Prescott unemployment equilibrium model to the problem of interseetoral labor allocation. While Lilien's sectors are industrial, his model easily applies to geographic sectors.

There are two important characteristics of the new neoclassical models. First, regional labor markets are part of a complex dynamic economy in which technology and relative product demands are continually subjected to stochastic change. Secondly, the degree to which the geographic structure of unemployment is influenced by these shocks depends on labor supply

22

LABOR MARKET EFFICIENCY AND THE DYNAMIC BEHAVIOR

adjustments. Stochastic equilibrium models assume some degree of friction is inherent in this process. Many of the new neoclassical models are based on a search theory view of the labor market that treats these frictions as arising solely from information lags resulting from the constantly changing demand conditions (Phelps [12]). Since workers are efficient collectors and processors of information, the relationship between relative unemployment rates and demand shifts is transitory. Other models, such as those developed by Lilien [10] and Black [2], stress technological constraints to shifting specialized human capital between sectors as a primary source of friction. This sluggish adjustment generates unemployment of considerable persistanee that contributes to the development of business cycles. However, as time passes, constraints disappear, competitive forces become stronger, and the economy approaches its long run equilibrium state. In all of these models, long run labor supply functions are highly elastic in terms of sectoral differences in expected wages. Hence normal unemployment rate differentials are independent of demand shifts, a conclusion that corresponds to the conventional neoclassical regional growth model.

From this discussion it follows that it is the nature of the dynamic behavior of geographic unemployment differentials that is the key to testing the competitive hypothesis. Some degree of short run sensitivity of relative unemployment to growth must be observable to stimulate wage and/or migration response to restore equilibrium. In fact, invariance of the unemployment rate structure to relative growth fluctuations of business cycle duration is not necessary to the new neoclassical position. Business cycle related unemployment is treated simply as a transitory deviation from normal levels (Lilien [10]). The essential theoretical condition is that unemployment differentials arising from unbalanced growth be liquidated in the long run.

In reality, the important issue is not one of strict invariance of relative unemployment to growth. At any given moment that is a matter of degree. The issue is whether differences in regional growth have a substantial impact on the normal spatial distribution of unemployment. Thus, if the long run invariance property fails, the consequences for the competitive hypothesis depend upon the quantitative nature of this failure. First, consider weaker forms of the competitive hypothesis. Provided that relative labor supply elasticities increase as the period of adjustment lengthens, then unemployment differentials at least will tend to converge over time. As long run supplies become less elastic, this weak form of the competitive hypothesis approaches the Keynesian case in which the classical long run result is a limiting property of a prolonged adjustment process where there are lengthy, if not permanent, departures from long run equilibrium.

If geographic unemployment differentials are more sensitive to long than short run changes in relative demands, then market forces may operate in fundamentally different ways from the predictions of the competitive hypothesis. Theories that do not view regional labor market imbalances as resulting from adjustment lags in an otherwise convergent process may be more appropriate. For example, poIarization theory predicts that labor market imbalances arising from differential growth will be perpetuated by the combined effects of migration selectivity and external economies (Myrdal [I l]).

III. Testing the Competitive Hypothesis

A dynamic equation relating regional relative unemployment rates to relative employment growth is used to test the competitive hypothesis. Let

23

RICHARD B. TILLER

relative unemployment be defined as the ratio of the region's unemployment rate to the national rate. Similarly, define the region's employment growth differential as the proportional deviation from the national average. The relative unemployment rate for the ath area at time t, U(a, t), is specified as determined by, l) relative demand and supply disturbances represented by a distributed lag in the deviations of relative employment growth from its normal level, E(a, t), 2) a slow moving component, N(a, t), which reflects the many institutional and structural determinants of labor market conditions besides employment growth, and 3) a random disturbance term, V(a, t), as follows:

U(a, t) - b(j)E(a,t-j) + N(a,t) + V(a,t) a = l,...,r (I)

where lim b(j) -- 0

j-~oo

{X)

jEb(j).=O = k< co .

These equations may be viewed as the final forms of a large dynamic multi-equation system where U, E, and V can be represented as ARMA processes. Setting V(t)=0, the b(j)'s describe the internal dynamics governing inter-area differences in unemployment rates. This behavior can be succinctly characterized by the cumulative interium multipliers,

~E(a,t)bU(a't) Ii I-1 = j~0 b(j), I=1,2 . . . . . (2)

Setting l=0 yields the impact multiplier which shows the initial effect on relative unemployment of a change in the area's growth rate. The permanent effect on U(a,t) of a (one unit) change in E(a,t), that is maintained indefinitely, is given by the total multiplier as 1 goes to infinity.

The competitive hypothesis imposes certain restrictions on the behavior of these multipliers as the period of adjustment increases. In the short run U(a,t) will be sensitive to E(a,t). Thus low order multipliers will have negative (positive) values if demand (supply) rather than supply (demand) does most of the initial shifting. The total multiplier will be zero. Under these conditions equation (I) describes a natural rate model in which the N(a,t)'s represent the normal regional unemployment rate structure which may vary over time but are independent of demand disturbances.

Weaker forms of the competitive hypothesis imply that successive multi- pliers decline in magnitude with the total multiplier approaching a non-zero value. In other words, the system tends to return to equilibrium after an initial disturbance, but this could be a slow and incomplete process. A polarization process would also imply a non-zero total multiplier, but with higher-order mul- tipliers greater in magnitude than lower-order multipliers. That is, regional un- employment differentials resulting from growth disturbances increase over time.

These questions are more easily dealt with empirically by transforming equation (1) into the frequency domain. This approach provides a natural meas- ure of the short and long run relationships between U(a,t) and E(a,t) with the important advantage of not having to specify and estimate specific dynamic models for each area. Of course, the specific distributed lag function cannot be directly inferred from the frequency domain but this is not necessary, as will be shown.

24

LABOR M A R K E T E F F I C I E N C Y AND THE DYNAMIC BEHAVIOR

The spectral representation of equation (1) is obtained by taking fourier transforms of both sides of the equation which yields (omitting the regional subscript)

where U(f) =b(f)E(f) + R(f) (3)

f = the fraction of a cycle completed in a unit time period by the fth frequency component

b(f) = ~b( j )exp( - i2Yf j ) J

E(f) = E E(t) exp(-i2Yft)

R(f) = V(f) + Y(f) = ER(t)exp(-i2TTft) t

i 2 = - 1 .

Thus, the spectral representation decomposes the relationship between U and E into a large number of frequency components each associated with a particular cycle length. Cycle frequency, f, varies continuously from very high values corresponding to short transitory movements such as white noise and seasonality to very low values corresponding to the permanent or secular component which consists of nonperiodic movements approaching an infinite duration. The obvious advantage of this decomposition is that the short and long run may be linked in a nonparametric way to various ranges within this frequency continuum (Granger and Hatanaka [6]).

Of particular interest is b(f) which is the frequency response function of the unspecified distributed lag of equation (I). It may be expressed in a more useful form as,

b(f) = G(f)exp[-iPh(f)] (4)

where G(f) is the gain function which assumes only non-negative values and Ph(f) is the phase shift.

The transfer function may be interpreted as a linear regression of U on E at each frequency. The gain, which is similar to a slope coefficient, measures the sensitivity of regional unemployment differentials to growth differences of varying duration. The relationship between U(f) and Eft) is not exact because of the presence of R(f). The coherence function provides a measure of the strength of association between U(f) and G(f) analagous to the traditional R-squared,

G 2 (f)

coh = U2(f)E2(f) (5)

Clearly, a zero coherence implies a zero gain at a given frequency. Another useful statistic is the spectral densityof U(f) which divides its variance into that due to a corresponding oscillation in E and that due to the residual,

var[U(f)] =G 2 (f)var[E(f)] + var [R(f)]. (6)

The total variance of U(t) may be divided into the frequency sums of these variances,

25

RICHARD B. TILLER

TT s var [U(t)] = _I~G2(f)var[E(f)] ~y var[R(f)]. (7)

These spectral statistics provide a precise means for testing the competitive hypothesis. If the normal structure of geographic unemployment rates is invariant to growth differences, the coherence at and near the zero frequency will be zero. If this is not the case, the behavior of the gain over frequency provides information about the nature of the departure from the long run competitive result. A gain function that declines as cycle frequency falls is consistent with a converging adjustment process while an inverse relationship between the gain and frequency suggests a polarization process. Finally, the variance decomposition measure is useful in quantifying the strength of competitive forces in the overall variance of the unemployment rate structure.

III. Empirical Results

Data and Method of Estimation

This study uses monthly State employment and unemployment data from the Current Population Survey, widely accepted as the most reliable source of labor force estimates. The sample period covers 1967 through 1982. Data for each of the fifty States are not available for the entire sample period. The 15 most populous States are individually identified and the remaining States are grouped by census divisions. The sample period is of particular interest because it covers a time in which there was a dramatic acceleration in regional shifts in growth (Jackson and Masnick [9]). It also is a period that contained the most severe business contractions of the post-WWII era.

The relative unemployment rate for a State, RUR, is defined as the ratio of its rate to the national average. Relative employment growth, REG, is computed as the State's growth relative to its employment level in January 1967 divided by the corresponding national growth rate. These two series were prefiltered using a quasi-difference of the form (I-.95L), where L is the one period lag operator. This filter was selected to attenuate potential non- stationary elements in the two series. The gain and coherence statistics were derived from a smoothed cross-periodogram of the two series. To form the cross-periodogram, the finite fourier transform of RUR was multiplied by the complex conjugate of the fourier transform of REG. To obtain consistent esti- mators, the real and complex parts were smoothed using a symmetric triangular seven-point weight function. The weights reflect at the origin (zero frequency). Before smoothing at the origin, the first periodogram ordinate was replaced with the second. For a discussion of the properties of these estimators see Fuller [4].

Findings

Table I presents the eoherences between relative unemployment rates and employment growth for periodicities greater than one year. Assuming that the prefiltered RUR and REG are bivariate stat ionary normal time series, Snedecor's F stat is t ic is used to test the null hypothesis that the coherenees are zero (Fuller [4]). Only eoherences significant at the .10 level are shown in Table I. For analytical convenience the frequency range, reported in terms of periodicities, may roughly be divided into two hands. The long run or secular period is defined as consisting of those components with periods of eight or more years. Those fluctuations of greater than one but less than eight years are referred to as

26

LABOR MARKET EFFICIENCY AND THE DYNAMIC BEHAVIOR

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27

RICHARD B. TILLER

cyclical since this band is dominated by cycles of national business cycle length. Within the sample period the duration of national business cycles, as measured by NBER, ranged from about 1.5 to 6 years.

Out of a total of 22 States and divisional groupings presented in Table I, 15 have significant coherences in the secular range. Eleven of these States have coherences of .50 or higher in that range. Over the business cycle range there are significant coherences for all States. Thus, the strong form of the competitive hypothesis is rejected; the geographic unemployment rate structure is sensitive to long run growth differentials. In addition, the unemployment structure is also altered by business cycle fluctuations, a result in agreement with previous studies (Hyclak and Lynch [8], Tiller and Bednarzik [15]).

Part A of Table II presents the average values of the gains over the secular (S) and cyclical (C) ranges, as previously defined, as well as for the higher frequency range (H) consisting of transitory influences of no more than one year duration. These averages were computed by weighting the gain at each frequency by its coherence. Gains associated with nonsignificant coherences were given zero weight.

The weighted gains generally decline in value as the duration of growth fluctuations increases. The median value of the gain over all States declines from 5.29 for the high frequency range to 1.06 for the secular range. This is consistent with a converging but prolonged adjustment process. However, this average performance masks some major differences among States.

As a background for discussing these differences, Table IIl gives the mean values for RUR and REG and their average yearly change over the sample period. The change in relative growth by State reflects the well known long run interregional shift in employment from the North East and North Central to the South and West, a phenomena that can be traced back to at least the end of WW II. Since the mid-1960s there has been an acceleration in this trend and a rising public concern that it has come at the expense of the North. Table II, however, indicates that there has been a fundamental difference in the way the North East and North Central regions have responded to decline. In the North East, particularly in New England, declining secular growth has not caused unemployment rates to rise relative to the nation as a whole, as evidenced by the zero or low gain value for all States, but Pennsylvania, in this region.* Part B of Table II reinforces this point by showing that the proportion of the total variance in relative ]~nemployment due to secular growth is less than 9 percent in the North East. ~ This insensitivity in relative unemployment in New Jersey, New York and the New England States occurred even though these States experienced not only below average growth but also declining relative growth. In sharp

1The gain statistics contain no information about the direction of correlation between RUR and REG. The phase angle is sometimes used to infer sign but can be difficult to interpret. Band limited regressions (Engle [3]) were estimated by the frequency interval S, C, and H, as defined above. All significant slope coefficients had negative signs. Thus the net relationship between RUR and REG over the short and long run is negative, a result consistent with this paper's emphasis on the importance of demand disturbances.

2To compute the variances, the spectral densities of the prefiltered RUR and REG series were adjusted for filter bias by multiplying by the inverse of the square of the filter's transfer function.

28

LABOR MARKET EFFICIENCY AND THE DYNAMIC BEHAVIOR

TABLE II

GAIN ESTIMATES AND PROPORTION OF TOTAL VARIANCE IN RELATIVE UNEMPLOYMENT RATES (RUR) DUE TO I~MPLOYMENT

GROWTH (REG) BY LENGTH OF CYCLE*

A. Weighted Gains B. Propor t ion of RUR due to REG

S C H S C H

Northeast New England Massachusetts 0.00 5.32 6.22 0.03 0.02 0.02 Other 0.00 1.26 2.93 0.03 0.04 0.04

Middle Atlantic New Jersey 0.00 0.50 5.80 0.08 0.01 0.04 New York 0.73 4.32 6.77 0.06 0.03 0.02 Pennsylvania 1.81 2.07 5.84 0.56 0.04 0.07

North Central East North Central

Illinois 3.21 3.26 5.24 0.65 0.05 0.02 Indiana 3.47 3.01 5.62 0.50 0.06 0.03 Michigan 3.49 6.13 7.34 0.48 0.11 0 .08 Ohio 2.45 2.41 6.71 0.42 0.05 0.05 Wisconsin 4.60 2.61 5.86 0.31 0.03 0.09

West North Central Missouri 2.98 1.52 5.98 0.39 0.04 0.09 Other 0.81 0.61 2.78 0.10 0.03 0.08

South South Atlantic

Florida 0.00 2.53 3.13 0.00 0.05 0.07 North Carolina 2.66 4.18 5.43 0.15 0.11 0.10 Virginia 1.25 1.65 4.13 0.12 0.03 0.12 Other 0~ 1.08 2.76 0.02 0.03 0.07

East South Central 3.28 1.06 0.48 0.52 0.03 0.01 West South Central

Texas 0.59 2.38 4.27 0.20 0.09 0.11 Other 0.47 0.15 3.23 0.04 0.02 0.08

West Mountain 0.48 1.02 2.97 0.34 0.06 0.06

Pacific California 2.94 1.48 5.33 0.73 0.03 0.03 Other 0.88 1.29 3.17 0.08 0.03 0.07

Median 1.06 1.86 5.29

IDuration of cycles in years: S = 8 to infinity C = greater than 1 but less than 8 H = l or less

29

RICHARD B. TILLER

TABLE III

AVERAGE VALUES OF RELATIVE UNEMPLOYMENT RATES (RUR) AND EMPLOYMENT GROWTH (REG), 1967-1982

Mean Yearly Change RUR REG RUR REG

Northeast New England

Massachusetts 1. 025 0.937 0.002 -0.005 Other 0. 987 1.005 0.006 -0. 003

Middle Atlantic New Jersey 1.074 0. 944 -0.003 -0. 006 New York 1.091 0. 840 -0.005 -0. 015 Pennsylvania 1.017 0.915 0.011 -0.012

North Central East North Central

Illinois 0. 908 0. 952 0. 023 -0. 009 Indiana 0.976 0. 937 0.020 -0. 010 Michigan 1.312 0. 946 0. 036 -0. 010 Ohio 1.026 0. 956 0. 021 -0. 010 Wisconsin 0. 814 1.012 0. 030 -0. 002

West North Central Missouri 0. 830 0. 940 0. 007 -0.009 Other 0.659 1.064 0.009 0. 001

South South Atlantic

Florida 1.003 1.320 -0. 018 0. 037 North Carolina 0. 872 1.034 -0.004 0.002 Virginia 0. 774 1.122 -0.007 0. 009 Other 0.939 0. 898 -0.000 -0. 003

East South Central 1.005 0. 965 0.011 -0. 005 West South Central

Texas 0. 789 i . 174 -0.009 0. 022 Other 0. 999 1. 012 -0. 015 0.010

West Mountain 0.994 1. 275 -0.023 0. 033 Pacific

California 1.262 1.105 -0. 032 0. 012 Other i . 274 0. 895 -0. 013 0. 001

contrast, States in the North Central region have gains over the secular range that are 3 to 4 times the median which is also reflected in the large proportion of variance in RUR due to secular fluctuations in REG. In addition, some of these mid-west States exhibit weak convergence of RUR towards the national average, as evidenced by gains over the secular range that exceed those over the cyclical range.

States in the South and West generally have tighter labor markets as a result of being the beneficiaries of the interregional shift in employment. Here too, there are major differences within regions. Although Florida has experienced the most rapid growth of any of the States identified in this study,

30

LABOR MARKET EFFICIENCY AND THE DYNAMIC BEHAVIOR

its relative unemployment rate has not been effected, as evidenced by a zero gain over the secular range. Other growth States in the South--North Carolina, Texas, and Virginia--have nonzero secular gain values but the total variance in their RUR series due to growth is relatively low. In the East South Central, declining growth results in a comparatively large increase in unemployment. Unemployment in other States in the South Atlantic is not much effected by declining relative growth. In the West, California's unemployment has been very sensitive to its higher than average growth. As a result, its relative unemployment has fallen an average 3 percent age points per year over the sample period.

IV. Conclusions

The findings of this study support a weak form of the competitive hypothesis somewhat closer to the traditional Keynesian than neoclassical view of the regional growth process. There is evidence that unemployment differentials tend to narrow over time, but it is a slow process. Over a much longer sample period than the one used in this study (16 years), one might observe that all States have zero coherences between unemployment and growth in the very low frequency range. Nevertheless, the adjustment to long run growth differentials entails more than just temporary unemployment if the latter is defined (liberally) as unemployment that persists no longer than business cycle duration.

It is possible that the non-zero coherences observed over the secular range resulted not because competitive forces were weak but because of the presence of unique and powerful exogenous forces, such as the drastic rise in energy prices, increases in international competitiveness, and poorly managed monetary and fiscal policies. Clearly, these forces were the source of considerable trauma during the 1970s. Under the strong version of the competitive hypothesis much of the adverse interregional impact should have occurred within the business cycle range. In addition there are differences between regions that are difficult to explain. For example, New England, which clearly was susceptible to these disruptive forces, behaved in a manner consistent with the strong form of the competitive hypothesis but the North Central did not. While this may suggest major differences in regional labor market efficiency, it may simply reflect the fact that New England, which has been in decline for decades, had a longer time to prepare. Ih any case, the findings of this study support the view that long run shifts in growth do generate persistent labor market imbalances. Only over the very long run (several decades) are these imbalances likely to be liquidated by market forces.

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RICHARD B. TILLER

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