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Tests of labor market rigidities and the Roy Model
Economics Letters 61 (1998) 243–250
Tests of labor market rigidities and the Roy Model Edward Funkhouser* Department of Economics, University of California, Santa Barbara CA 93106 -9210, USA Received 9 September 1997; accepted 16 April 1998
Abstract Previous studies examining labor market rigidities using the Roy Model have not incorporated the demand side. In this paper, I show that when the demand side is included, selection models of labor market rigidities based on the Roy Model in general lead to the prediction of mean wage differentials across sectors. This result is not surprising since the allocation of workers between sectors is determined by the marginal worker, not the mean or median worker, and an inducement differential may be necessary for demand to equal supply in each sector of the labor market. Previous tests of labor market rigidities are tests of the joint null hypothesis that under competition wages would be equalized across sectors and that wages are in fact equalized across sectors. They may, therefore, attribute differences in wage structure to rigidities that are, in fact, the result of equalization of supply and demand within the inducement model developed. An example is presented from the paper of [Dickens, W., Lang K., 1985. A test of the dual labor market theory. American Economic Review, 75(4), 792–805] using the Panel Study of Income Dynamics. 1998 Elsevier Science S.A. All rights reserved. Keywords: Wage differentials; Self-selection; Roy Model JEL classification: J23; J31
1. Introduction Tests of labor market rigidities have been based on the prediction that with full employment and competition, the allocation of workers will be that which equalizes wages across sectors of the labor market. Evidence in favor of unions exerting monopoly control, labor market segmentation between the formal and informal sectors of the labor market, public–private wage differentials, the payment of efficiency wages, and barriers to migration has utilized the idea that, in the absence of a rigidity, wages of otherwise similar workers in similar jobs should be equalized across sectors of the labor market. The finding of either a significantly different intercept (union–non-union and public–private differentials) or the joint significance in the test of differences in coefficients between sectors (segmentation, barriers to mobility, and industry differentials) has been viewed as support for the rigidities hypothesis. In this paper I show that, within the two-sector version of the Roy Model used in most tests of labor market rigidities, differences in unconditional mean wages across sectors – even after corrections for *Tel.: 11 805 8933490; fax: 11 805 8938830; e-mail: [email protected] 0165-1765 / 98 / $ – see front matter 1998 Elsevier Science S.A. All rights reserved. PII: S0165-1765( 98 )00133-5
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unobserved heterogeneity – are consistent with competition between sectors. It is the wage of the marginal worker, not the wage of the mean or median worker, that must equalize between sectors. Since most tests of labor market rigidities in the literature examine the equality of mean, rather than marginal, wages, these tests provide little evidence on the existence of labor market rigidities. The selection model proposed in most studies of labor market rigidities is based on the Roy model (Roy, 1951; Heckman, 1979; Heckman and Honore, 1990; Vijverberg, 1993). In the Roy Model, the (logarithm of the) marginal product of each worker is symmetrically distributed around the unconditional mean marginal product in each sector. The basic insight of the traditional two-sector model – the marginal worker must be indifferent between employment in either sector – remains in the Roy Model. But because the condition that demand equal supply in each sector rarely implies that the mean (or median) worker is the marginal worker, both actual mean wages and the wages of the mean (or median) worker in the unconditional distribution of wages (including all workers whether or not they are employed in the sector) will usually differ across sectors. The existence of such an equilibrium in a competitive labor market runs counter to the intuition of previous tests of labor market rigidities. Since previous studies have attributed such differences to rigidities or compensating differentials, the finding that a competitive equilibrium is associated with wage differentials between sectors should force reexamination of some of the earlier findings.
2. Inducement model A feature of most versions of the Roy Model is the exogenous determination of the distribution of wages in each sector and, as a result, perfectly elastic demand for labor at a given wage differential between sectors. In this section, I model the wage difference in the Roy framework by including the demand for and the supply of workers in each sector. On the supply side, workers choose to enter each sector according the wage differential as in current versions of the Roy Model.1 On the demand side, the marginal product of labor is a declining function of employment in each sector. Consider a labor market with two sectors, Y and Z. In each sector, the logarithm of marginal product includes a non-stochastic component, fk (k equals Y,Z), common across workers and an individual component, eki , that is distributed with mean zero. Wages offered and received in each sector are equal to the marginal product in that sector: w Yi 5 fY 1 eYi
(1a)
w Zi 5 fZ 1 eZi
(1b)
where eYi and eZi are jointly distributed f(eYi ,eZi ) and eki and fk are independent for k5Y,Z. By construction, the non-stochastic part of the marginal product, fk , is the mean marginal product over all workers.2 1
With exogenously determined mean marginal product in each sector, the fraction of workers entering each sector depends on the variance of earnings in each sector, the correlation of earnings across sectors, and the mean earnings difference between sectors. 2 I refer to the distribution over all workers including those not working in the sector as the unconditional distribution.
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2.1. Supply Each worker knows the stochastic component to own marginal product in each sector and will supply labor to Sector Y if the wage in that sector is greater than the wage in Sector Z. With no barriers to mobility, the probability that a worker chooses to be in sector Y is: S S Pr(Y) 5 Pr(w Yi 2 w Zi . 0) 5 Pr(eYi 2 eZi . 2 [w Ym 2 w Zm ])
(2)
S
where w km refers to the wage received by the unconditioanl mean worker in sector k. A similar probability exists for the choice of sector Z. Define V5 eYi 2 eZi with unconditional mean 0 and finite variance s 2V 5 s 2e Y 1 s 2e Z 22se Y, e Z . The supply of workers to sector Y, LY , can be written as a function of the difference in the wage offered the mean workers in the unconditional distributions, w SYm 2w SZm , necessary to induce workers into the sector: S S LY 5 F([w Ym 2 w Zm ] /sV )N
(3)
or, as an inverse supply function: w SYm 2 w SZm 5 F 21 (LY /N)sV
(4)
in which F is the standardized cumulative distribution of V.
2.2. Demand In this section, I incorporate demand into the Roy Model. In the model presented, the marginal product is assumed to decline by an identical amount for each worker as the number of workers in the sector increases. The non-stochastic component of the marginal product, fk , must therefore be expressed in terms of the number of workers employed, Dk , and the firm offers: D w Ym 5 fk [Dk ] 1 eki f k9 , 0 k 5 Y,Z
(5)
in which Dk is the number of workers employed in sector k. Because eki is constant for all values of fk [Dk ], the wage paid to other workers is shifted by the stochastic component around the unconditional mean value. Moreover, because eYi 2 eZi is constant for all values of fY [DY ]2 fZ [DZ ], the mean wage differential between sectors can be expressed in terms of the difference in the wages offered to the mean worker in the unconditional distribution in each sector. Noting that the mean worker may not be the same individual in both sectors, the difference in the wage offered to the mean workers across sectors at any level of employment in Sector Y is: w DYm 2 w DZm 5 fY [DY ] 2 fZ [N 2 DY ] where N is the total labor force.
(6)
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2.3. Equilibrium Equilibrium occurs when the difference in the marginal product of the mean workers across sectors equals the difference in the wage offered to the mean workers across sectors. At this point, the marginal worker is indifferent between the two sectors. Equilibrium in Sector Y therefore occurs when:
fY [DY ] 2 fZ [N 2 DY ] 5 F 21 (LY /N)sV
(7)
Equilibrium in Sector Y implies equilibrium in Sector Z. The equilibrium condition is based on the difference in the unconditional mean wage offered across sectors and is shown in Fig. 1.3 The figure includes the demand and supply curves to both sectors. Along the horizontal axis is the proportion employed in Sector Y. Because full employment has been assumed, the proportion employed in Sector Z is increasing from right to left. The left vertical axis includes the wage offered by employers and demanded by workers (given the wage in Sector Z) in
Fig. 1. Demand and supply for each sector.
3
Fig. 1 is based on:
fY 51502 pY ; fZ 57520.5pZ ; eYi |N(0,900); eZi |N(0,400); rYZ 51 where pk is the proportion of employment in sector k.
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Sector Y. The right vertical axis includes the wage offered by employers and demanded by workers (given the wage in Sector Y) in Sector Z. S S Equilibrium occurs when w DYm 2w DZm equals w Ym 2w Zm , shown at point A in sector Y and at Point B in sector Z. With this differential in the unconditional mean wage, the number of workers supplied to each sector exactly equal demand.4 The intuition behind the existence of the inducement differential is straightforward. Employment in each sector is that at which the marginal worker is indifferent between sectors. In equilibrium, demand must equal supply in each sector and the marginal worker will typically not be the mean or median worker. It is clear from Eq. (7) that the wage differential depends on the proportion of workers willing to work in each sector, not the ordering of those workers.
3. Implications The main implication of the preceding model is that labor market outcomes that are consistent with imperfect markets may also be consistent with a competitive model within the Roy model framework used by most researchers. This differential will vary across labor markets for different labor inputs, including those by skill group, demographic group, and geographic area. Similar to other competitive models, the inducement model allows for predictions whether changes in demand and supply determine observed changes in prices and quantities. Because non-competitive models do not yield similar predictions, the consistency of the predictions with actual changes in demand and supply for each type of labor can be used to distinguish between classes of models. Given the equilibrium within the inducement model, changes in demand and supply are identical to those in any other model of supply and demand in the labor market. Changes in the valuation of the non-wage attributes of each sector, or differences across types of labor, are represented by a shift of the supply curve to each sector. Changes in the demand for a given type of labor, or differences across types of labor, are represented by a shift in the demand curve in that sector. The inducement model predicts that increases in demand in one sector are associated with both an increase in the proportion of workers in that sector and an increase in the equilibrium wage differential. Similarly, an increase in supply to a sector (perhaps because of change in the non-wage rewards) is associated with an increase in the proportion of workers in that sector and a decrease in the equilibrium wage differential. To examine these predictions, it is necessary to examine different types of labor that may be considered separate inputs in the labor market. These types of labor may be across geographic region, across skill group, or across race or gender. Changes in the correlation between the equilibrium wage differential and the proportion of the group working in each sector provides evidence on demand and supply differences between the markets for each factor. 4
The mean wage necessary to induce each proportion of the labor force into Sector Y given the unconditional mean wage in Sector Z is given by w SYm 5F 21 (LY /N) sV 1 fZ . Similarly, the mean wage necessary to induce each percentage of the labor S force into Sector Z given the unconditional mean wage in Sector Y is given by w Zm 5F 21 (LZ /N)sV 1 fY . These relations are shown by the supply to Sector Y and supply to Sector Z curves in Fig. 1 and are identically symmetric around the mean wage in the other sector.
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3.1. An example I examine the correlation between the inducement differential and the proportion of workers utilizing the same data employed by Dickens and Lang (1985) in their study of dual labor markets in the United States. The sample includes 2812 male heads of households in the 1980 Wave of the Panel Study of Income Dynamics (PSID) between the ages of 18 and 65 that worked more than 1000 hours in the reference year and had non-missing values for education, metropolitan area (SMSA) residence, whether they were ever married, race, and potential labor market experience.5 Dickens and Lang estimated a two-sector (primary, secondary) switching model in which the choice of sector was not known. The determinants of the logarithm of the hourly wage include SMSA status, never married status, years of education, white race status, and potential years of labor market experience. The determinants of choice of sector include the same independent variables except potential labor market experience. They find evidence in favor of the dual labor market theory because the structure of wages is statistically different in the two sectors and the coefficients in the equation determining choice of sector are not proportional to the difference in the coefficients determining wages in the two sectors. They then examine whether non-economic barriers may be the causes for differences across sectors. In this section, I show that the inducement model provides another competitive explanation for these findings given separate marginal product generation in two sectors. To be comparable with the Dickens and Lang paper, the probability of working in each sector and the predicted wage in each sector can be calculated from the coefficients of the switching regressions model reported in Table 1 of their paper. 82.6% of the full sample is predicted to work in the primary sector with a predicted mean hourly wage differential of 0.833 log points. In Table 1, I calculate the proportion in each sector and the mean wage differential for several categories that might be considered different labor inputs: education level, race, location, and marital status. For each case, predicted values were calculated on the entire sample by imposing values on the studied variable while holding all other variables at their actual values. In the top rows, the proportion in the primary sector and the predicted primary–secondary wage differential are shown for education equal to 10 years, 12 years, and 16 years, controlling for all other variables. 81.2% of those with 10 years of education are predicted to work in the primary sector with a predicted mean log wage differential between sectors of 0.7. In contrast, 86% of those with 16 years of education are predicted to be in the primary sector with a mean log wage differential between sectors of 1.1. The proportion in the primary sector is positively related to the primary–secondary equilibrium wage differential. In the next rows, the values calculated for whites and other races are shown. A higher proportion of other races are predicted to be in the primary sector (89.8% to 66.3%) with a higher mean log wage differential (0.877 to 0.732) For race as well, the proportion in the primary sector is positively related to the primary–secondary equilibrium wage differential. Within the inducement model, these findings for education and race suggest that demand factors for these different types of labor are important. In the next rows of the table, the predicted values for marital status (ever married, never married) are shown. In contrast to the results for education and race, there is a negative relationship between the proportion in the primary sector and the primary–secondary wage differential. Within the 5
Experience is calculated to be age minus education minus six. Two persons with negative experience were eliminated. The resulting sample has characteristics very close, but not identical to those reported by Dickens and Lang.
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Table 1 Percent primary sector and primary / secondary wage differential Percent primary a
Primary / secondary wage differential b
Education 10 years 12 years 16 years
0.812 0.829 0.860
0.696 0.822 1.074
Race White Other race
0.898 0.663
0.877 0.732
Marital status Ever married Never married
0.822 0.871
0.835 0.817
Number of observations52812 a
, Sample includes all male heads of household in 1980 PSID with over 1000 hours of work and non-missing values for hourly wage, education, age, SMSA status, never married status, and potential labor market experience. b , Entries are averaged over entire sample assigning indicated values to variable of interest for all cases.
inducement model, this suggests that supply factors are important in explaining different sectoral attachment by marital status. The inducement model indicates that even when the ‘‘correct’’ sectoral wage equations are calculated, there may exist differences in the wage structure and these differences may be correlated with observable variables that are separate factors of production. The calculations in this section indicate whether the observed differentials are consistent with demand or supply-driven inducement differentials. But it is important to emphasize that the calculations do not distinguish between the inducement differential and other non-competitive explanations for the existence of sectoral wage differences. But they do suggest that studies that conclude that because wage structures are not equalized, there must be rigidities may be premature. For example, in the finding that the profile of returns to education is steeper in the primary sector than in the secondary sector may be due to rigidities or it may be due to inducement differentials that vary by skill level. In the data of Dickens and Lang, variation in the inducement differential is consistent with different demand for each education level after all other observables have been controlled for. Similarly, differences in the black–white differential across sectors observed may be due to rigidities or may also be due to differences in demand for black and white labor within the inducement model.
4. Concluding remarks Previous studies examining labor market rigidities using the Roy Model have not incorporated the demand side. In this paper, I show that when the demand side is included, selection models of labor market rigidities based on the Roy Model in general lead to the prediction of mean wage differentials
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across sectors. This result is not surprising since the allocation of workers between sectors in the Roy Model is determined by the marginal worker, not the mean or median worker. Since they are tests of the joint null hypothesis that under competition expected wages would be equalized across sectors and that wages are in fact equalized across sectors, previous tests of labor market rigidities may attribute differences in expected wages to rigidities that are, in fact, the result of equalization of supply and demand within the inducement model. Corrections for self-selection control for the truncation in the bivariate distribution [eY ,eZ ], but do not control for the inducement differential. Since the inducement differential is likely to be correlated with observable characteristics, the estimated coefficients used in most tests of labor market rigidities may also be biased.
Acknowledgements I have benefitted from the comments of Rodney Garratt and Stephen Trejo.
References Roy, A.D., 1951. Some thoughts on the distribution of earnings. Oxford Economic Papers 3, 135–146. Dickens, W., Lang, K., 1985. A test of the dual labor market theory. American Economic Review 75 (4), 792–805. Heckman, J., 1979. Sample selection as specification error. Econometrica 47, 153–161. Heckman, J., Honore, B., 1990. The empirical content of the Roy model. Econometrica 58 (5), 1121–1149. Vijverberg, W., 1993. Measuring the unidentified parameter of the extended Roy model of selectivity. Journal of Econometrics 57 (1-3), 69–89.
Tests of labor market rigidities and the Roy Model Edward Funkhouser* Department of Economics, University of California, Santa Barbara CA 93106 -9210, USA Received 9 September 1997; accepted 16 April 1998
Abstract Previous studies examining labor market rigidities using the Roy Model have not incorporated the demand side. In this paper, I show that when the demand side is included, selection models of labor market rigidities based on the Roy Model in general lead to the prediction of mean wage differentials across sectors. This result is not surprising since the allocation of workers between sectors is determined by the marginal worker, not the mean or median worker, and an inducement differential may be necessary for demand to equal supply in each sector of the labor market. Previous tests of labor market rigidities are tests of the joint null hypothesis that under competition wages would be equalized across sectors and that wages are in fact equalized across sectors. They may, therefore, attribute differences in wage structure to rigidities that are, in fact, the result of equalization of supply and demand within the inducement model developed. An example is presented from the paper of [Dickens, W., Lang K., 1985. A test of the dual labor market theory. American Economic Review, 75(4), 792–805] using the Panel Study of Income Dynamics. 1998 Elsevier Science S.A. All rights reserved. Keywords: Wage differentials; Self-selection; Roy Model JEL classification: J23; J31
1. Introduction Tests of labor market rigidities have been based on the prediction that with full employment and competition, the allocation of workers will be that which equalizes wages across sectors of the labor market. Evidence in favor of unions exerting monopoly control, labor market segmentation between the formal and informal sectors of the labor market, public–private wage differentials, the payment of efficiency wages, and barriers to migration has utilized the idea that, in the absence of a rigidity, wages of otherwise similar workers in similar jobs should be equalized across sectors of the labor market. The finding of either a significantly different intercept (union–non-union and public–private differentials) or the joint significance in the test of differences in coefficients between sectors (segmentation, barriers to mobility, and industry differentials) has been viewed as support for the rigidities hypothesis. In this paper I show that, within the two-sector version of the Roy Model used in most tests of labor market rigidities, differences in unconditional mean wages across sectors – even after corrections for *Tel.: 11 805 8933490; fax: 11 805 8938830; e-mail: [email protected] 0165-1765 / 98 / $ – see front matter 1998 Elsevier Science S.A. All rights reserved. PII: S0165-1765( 98 )00133-5
E. Funkhouser / Economics Letters 61 (1998) 243 – 250
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unobserved heterogeneity – are consistent with competition between sectors. It is the wage of the marginal worker, not the wage of the mean or median worker, that must equalize between sectors. Since most tests of labor market rigidities in the literature examine the equality of mean, rather than marginal, wages, these tests provide little evidence on the existence of labor market rigidities. The selection model proposed in most studies of labor market rigidities is based on the Roy model (Roy, 1951; Heckman, 1979; Heckman and Honore, 1990; Vijverberg, 1993). In the Roy Model, the (logarithm of the) marginal product of each worker is symmetrically distributed around the unconditional mean marginal product in each sector. The basic insight of the traditional two-sector model – the marginal worker must be indifferent between employment in either sector – remains in the Roy Model. But because the condition that demand equal supply in each sector rarely implies that the mean (or median) worker is the marginal worker, both actual mean wages and the wages of the mean (or median) worker in the unconditional distribution of wages (including all workers whether or not they are employed in the sector) will usually differ across sectors. The existence of such an equilibrium in a competitive labor market runs counter to the intuition of previous tests of labor market rigidities. Since previous studies have attributed such differences to rigidities or compensating differentials, the finding that a competitive equilibrium is associated with wage differentials between sectors should force reexamination of some of the earlier findings.
2. Inducement model A feature of most versions of the Roy Model is the exogenous determination of the distribution of wages in each sector and, as a result, perfectly elastic demand for labor at a given wage differential between sectors. In this section, I model the wage difference in the Roy framework by including the demand for and the supply of workers in each sector. On the supply side, workers choose to enter each sector according the wage differential as in current versions of the Roy Model.1 On the demand side, the marginal product of labor is a declining function of employment in each sector. Consider a labor market with two sectors, Y and Z. In each sector, the logarithm of marginal product includes a non-stochastic component, fk (k equals Y,Z), common across workers and an individual component, eki , that is distributed with mean zero. Wages offered and received in each sector are equal to the marginal product in that sector: w Yi 5 fY 1 eYi
(1a)
w Zi 5 fZ 1 eZi
(1b)
where eYi and eZi are jointly distributed f(eYi ,eZi ) and eki and fk are independent for k5Y,Z. By construction, the non-stochastic part of the marginal product, fk , is the mean marginal product over all workers.2 1
With exogenously determined mean marginal product in each sector, the fraction of workers entering each sector depends on the variance of earnings in each sector, the correlation of earnings across sectors, and the mean earnings difference between sectors. 2 I refer to the distribution over all workers including those not working in the sector as the unconditional distribution.
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2.1. Supply Each worker knows the stochastic component to own marginal product in each sector and will supply labor to Sector Y if the wage in that sector is greater than the wage in Sector Z. With no barriers to mobility, the probability that a worker chooses to be in sector Y is: S S Pr(Y) 5 Pr(w Yi 2 w Zi . 0) 5 Pr(eYi 2 eZi . 2 [w Ym 2 w Zm ])
(2)
S
where w km refers to the wage received by the unconditioanl mean worker in sector k. A similar probability exists for the choice of sector Z. Define V5 eYi 2 eZi with unconditional mean 0 and finite variance s 2V 5 s 2e Y 1 s 2e Z 22se Y, e Z . The supply of workers to sector Y, LY , can be written as a function of the difference in the wage offered the mean workers in the unconditional distributions, w SYm 2w SZm , necessary to induce workers into the sector: S S LY 5 F([w Ym 2 w Zm ] /sV )N
(3)
or, as an inverse supply function: w SYm 2 w SZm 5 F 21 (LY /N)sV
(4)
in which F is the standardized cumulative distribution of V.
2.2. Demand In this section, I incorporate demand into the Roy Model. In the model presented, the marginal product is assumed to decline by an identical amount for each worker as the number of workers in the sector increases. The non-stochastic component of the marginal product, fk , must therefore be expressed in terms of the number of workers employed, Dk , and the firm offers: D w Ym 5 fk [Dk ] 1 eki f k9 , 0 k 5 Y,Z
(5)
in which Dk is the number of workers employed in sector k. Because eki is constant for all values of fk [Dk ], the wage paid to other workers is shifted by the stochastic component around the unconditional mean value. Moreover, because eYi 2 eZi is constant for all values of fY [DY ]2 fZ [DZ ], the mean wage differential between sectors can be expressed in terms of the difference in the wages offered to the mean worker in the unconditional distribution in each sector. Noting that the mean worker may not be the same individual in both sectors, the difference in the wage offered to the mean workers across sectors at any level of employment in Sector Y is: w DYm 2 w DZm 5 fY [DY ] 2 fZ [N 2 DY ] where N is the total labor force.
(6)
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2.3. Equilibrium Equilibrium occurs when the difference in the marginal product of the mean workers across sectors equals the difference in the wage offered to the mean workers across sectors. At this point, the marginal worker is indifferent between the two sectors. Equilibrium in Sector Y therefore occurs when:
fY [DY ] 2 fZ [N 2 DY ] 5 F 21 (LY /N)sV
(7)
Equilibrium in Sector Y implies equilibrium in Sector Z. The equilibrium condition is based on the difference in the unconditional mean wage offered across sectors and is shown in Fig. 1.3 The figure includes the demand and supply curves to both sectors. Along the horizontal axis is the proportion employed in Sector Y. Because full employment has been assumed, the proportion employed in Sector Z is increasing from right to left. The left vertical axis includes the wage offered by employers and demanded by workers (given the wage in Sector Z) in
Fig. 1. Demand and supply for each sector.
3
Fig. 1 is based on:
fY 51502 pY ; fZ 57520.5pZ ; eYi |N(0,900); eZi |N(0,400); rYZ 51 where pk is the proportion of employment in sector k.
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Sector Y. The right vertical axis includes the wage offered by employers and demanded by workers (given the wage in Sector Y) in Sector Z. S S Equilibrium occurs when w DYm 2w DZm equals w Ym 2w Zm , shown at point A in sector Y and at Point B in sector Z. With this differential in the unconditional mean wage, the number of workers supplied to each sector exactly equal demand.4 The intuition behind the existence of the inducement differential is straightforward. Employment in each sector is that at which the marginal worker is indifferent between sectors. In equilibrium, demand must equal supply in each sector and the marginal worker will typically not be the mean or median worker. It is clear from Eq. (7) that the wage differential depends on the proportion of workers willing to work in each sector, not the ordering of those workers.
3. Implications The main implication of the preceding model is that labor market outcomes that are consistent with imperfect markets may also be consistent with a competitive model within the Roy model framework used by most researchers. This differential will vary across labor markets for different labor inputs, including those by skill group, demographic group, and geographic area. Similar to other competitive models, the inducement model allows for predictions whether changes in demand and supply determine observed changes in prices and quantities. Because non-competitive models do not yield similar predictions, the consistency of the predictions with actual changes in demand and supply for each type of labor can be used to distinguish between classes of models. Given the equilibrium within the inducement model, changes in demand and supply are identical to those in any other model of supply and demand in the labor market. Changes in the valuation of the non-wage attributes of each sector, or differences across types of labor, are represented by a shift of the supply curve to each sector. Changes in the demand for a given type of labor, or differences across types of labor, are represented by a shift in the demand curve in that sector. The inducement model predicts that increases in demand in one sector are associated with both an increase in the proportion of workers in that sector and an increase in the equilibrium wage differential. Similarly, an increase in supply to a sector (perhaps because of change in the non-wage rewards) is associated with an increase in the proportion of workers in that sector and a decrease in the equilibrium wage differential. To examine these predictions, it is necessary to examine different types of labor that may be considered separate inputs in the labor market. These types of labor may be across geographic region, across skill group, or across race or gender. Changes in the correlation between the equilibrium wage differential and the proportion of the group working in each sector provides evidence on demand and supply differences between the markets for each factor. 4
The mean wage necessary to induce each proportion of the labor force into Sector Y given the unconditional mean wage in Sector Z is given by w SYm 5F 21 (LY /N) sV 1 fZ . Similarly, the mean wage necessary to induce each percentage of the labor S force into Sector Z given the unconditional mean wage in Sector Y is given by w Zm 5F 21 (LZ /N)sV 1 fY . These relations are shown by the supply to Sector Y and supply to Sector Z curves in Fig. 1 and are identically symmetric around the mean wage in the other sector.
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3.1. An example I examine the correlation between the inducement differential and the proportion of workers utilizing the same data employed by Dickens and Lang (1985) in their study of dual labor markets in the United States. The sample includes 2812 male heads of households in the 1980 Wave of the Panel Study of Income Dynamics (PSID) between the ages of 18 and 65 that worked more than 1000 hours in the reference year and had non-missing values for education, metropolitan area (SMSA) residence, whether they were ever married, race, and potential labor market experience.5 Dickens and Lang estimated a two-sector (primary, secondary) switching model in which the choice of sector was not known. The determinants of the logarithm of the hourly wage include SMSA status, never married status, years of education, white race status, and potential years of labor market experience. The determinants of choice of sector include the same independent variables except potential labor market experience. They find evidence in favor of the dual labor market theory because the structure of wages is statistically different in the two sectors and the coefficients in the equation determining choice of sector are not proportional to the difference in the coefficients determining wages in the two sectors. They then examine whether non-economic barriers may be the causes for differences across sectors. In this section, I show that the inducement model provides another competitive explanation for these findings given separate marginal product generation in two sectors. To be comparable with the Dickens and Lang paper, the probability of working in each sector and the predicted wage in each sector can be calculated from the coefficients of the switching regressions model reported in Table 1 of their paper. 82.6% of the full sample is predicted to work in the primary sector with a predicted mean hourly wage differential of 0.833 log points. In Table 1, I calculate the proportion in each sector and the mean wage differential for several categories that might be considered different labor inputs: education level, race, location, and marital status. For each case, predicted values were calculated on the entire sample by imposing values on the studied variable while holding all other variables at their actual values. In the top rows, the proportion in the primary sector and the predicted primary–secondary wage differential are shown for education equal to 10 years, 12 years, and 16 years, controlling for all other variables. 81.2% of those with 10 years of education are predicted to work in the primary sector with a predicted mean log wage differential between sectors of 0.7. In contrast, 86% of those with 16 years of education are predicted to be in the primary sector with a mean log wage differential between sectors of 1.1. The proportion in the primary sector is positively related to the primary–secondary equilibrium wage differential. In the next rows, the values calculated for whites and other races are shown. A higher proportion of other races are predicted to be in the primary sector (89.8% to 66.3%) with a higher mean log wage differential (0.877 to 0.732) For race as well, the proportion in the primary sector is positively related to the primary–secondary equilibrium wage differential. Within the inducement model, these findings for education and race suggest that demand factors for these different types of labor are important. In the next rows of the table, the predicted values for marital status (ever married, never married) are shown. In contrast to the results for education and race, there is a negative relationship between the proportion in the primary sector and the primary–secondary wage differential. Within the 5
Experience is calculated to be age minus education minus six. Two persons with negative experience were eliminated. The resulting sample has characteristics very close, but not identical to those reported by Dickens and Lang.
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Table 1 Percent primary sector and primary / secondary wage differential Percent primary a
Primary / secondary wage differential b
Education 10 years 12 years 16 years
0.812 0.829 0.860
0.696 0.822 1.074
Race White Other race
0.898 0.663
0.877 0.732
Marital status Ever married Never married
0.822 0.871
0.835 0.817
Number of observations52812 a
, Sample includes all male heads of household in 1980 PSID with over 1000 hours of work and non-missing values for hourly wage, education, age, SMSA status, never married status, and potential labor market experience. b , Entries are averaged over entire sample assigning indicated values to variable of interest for all cases.
inducement model, this suggests that supply factors are important in explaining different sectoral attachment by marital status. The inducement model indicates that even when the ‘‘correct’’ sectoral wage equations are calculated, there may exist differences in the wage structure and these differences may be correlated with observable variables that are separate factors of production. The calculations in this section indicate whether the observed differentials are consistent with demand or supply-driven inducement differentials. But it is important to emphasize that the calculations do not distinguish between the inducement differential and other non-competitive explanations for the existence of sectoral wage differences. But they do suggest that studies that conclude that because wage structures are not equalized, there must be rigidities may be premature. For example, in the finding that the profile of returns to education is steeper in the primary sector than in the secondary sector may be due to rigidities or it may be due to inducement differentials that vary by skill level. In the data of Dickens and Lang, variation in the inducement differential is consistent with different demand for each education level after all other observables have been controlled for. Similarly, differences in the black–white differential across sectors observed may be due to rigidities or may also be due to differences in demand for black and white labor within the inducement model.
4. Concluding remarks Previous studies examining labor market rigidities using the Roy Model have not incorporated the demand side. In this paper, I show that when the demand side is included, selection models of labor market rigidities based on the Roy Model in general lead to the prediction of mean wage differentials
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across sectors. This result is not surprising since the allocation of workers between sectors in the Roy Model is determined by the marginal worker, not the mean or median worker. Since they are tests of the joint null hypothesis that under competition expected wages would be equalized across sectors and that wages are in fact equalized across sectors, previous tests of labor market rigidities may attribute differences in expected wages to rigidities that are, in fact, the result of equalization of supply and demand within the inducement model. Corrections for self-selection control for the truncation in the bivariate distribution [eY ,eZ ], but do not control for the inducement differential. Since the inducement differential is likely to be correlated with observable characteristics, the estimated coefficients used in most tests of labor market rigidities may also be biased.
Acknowledgements I have benefitted from the comments of Rodney Garratt and Stephen Trejo.
References Roy, A.D., 1951. Some thoughts on the distribution of earnings. Oxford Economic Papers 3, 135–146. Dickens, W., Lang, K., 1985. A test of the dual labor market theory. American Economic Review 75 (4), 792–805. Heckman, J., 1979. Sample selection as specification error. Econometrica 47, 153–161. Heckman, J., Honore, B., 1990. The empirical content of the Roy model. Econometrica 58 (5), 1121–1149. Vijverberg, W., 1993. Measuring the unidentified parameter of the extended Roy model of selectivity. Journal of Econometrics 57 (1-3), 69–89.