The Inter-industry Wage Differential, Educational Choice, and Directed Search

Transcription

1 The Inter-industry Wage Differential, Educational Choice, and Directed Search Mark R. Kurt November 23, 2010 Abstract The inter-industry wage differential (IWD), i.e. the differences in wages of homogeneous workers attributable to the industry in which those workers are employed, persists across time and across country, but lacks a compelling theory regarding its existence. A candidate theory, directed search, offers a competitive theory of wage dispersion. Workers direct their searches across industries taking into account both the wages offered and the probability of hire. In equilibrium, wages equalize in expectation but not in realization. I extend and calibrate a directed search model (Shimer, 2005a) to reconcile several facts seen in the labor market: the existence of the IWD, the increase in the size of the IWD early in the earnings profile, and the existence of highly educated workers in low wage industries. The model is extended to include multiple periods and educational choice. Embedded within the educational choice is an information friction in which workers and firms are unable to observe the human capital of the educated workers in the initial period. In the second period, workers and firms directly observe human capital causing sorting among workers. This information friction increases the dispersion of workers within a period, in addition to an increase in the IWD across the earnings profile. The IWD is sensitive to both the cost of education and the probability of becoming skilled, since these factors affect the proportions of educated and skilled in the economy. A sensitivity analysis for the cost of education and probability of skill is performed. Results from the calibration exercise demonstrates the model s ability to qualitatively replicate all three facts.

2 Fall 2007 Kurt, Mark R. 2 1 Introduction The inter-industry wage differential (IWD) is a pervasive feature of the labor market, persisting across time and countries. The IWD is the residual wage component attributed to industry after controlling for observable characteristics of the worker. This feature of the labor market is puzzling, especially within the context of a competitive labor market in which homogeneous workers receive the same wage. The existence of the IWD implies an arbitrage opportunity for workers to switch to the high wage industries, eliminating the IWD over time. One explanation for the IWD is unobserved heterogeneity. However, many studies find a persistent differential across industries after controlling for observable heterogeneity. Krueger and Summers (1987, 1988) document the differential s existence over time, across countries and in various planned economies. Krueger and Summers argue the IWD cannot be reconciled by heterogeneity of human capital and reason non-competitive theories underpinned the differential. Murphy and Topel (1986, 1987) offer an alternative competitive hypothesis, measurement error. The econometrician cannot observe the true amount of human capital an individual possesses. They develop an innovative way of measuring human capital by viewing industry switchers. However, it is difficult to accurately identify industry switchers in data (Brown and Light, 1992). The inability to isolate switchers has lead to inconsistent conclusions from empirical work (Gibbons and Katz, 1992). More complete data was not available until Abowd, Kramarz, and Margolis (1999). They utilize the switching approach with French panel data where isolating switchers is possible. They decompose the IWD into three distinct parts: unobserved individual heterogeneity, firm effects and industry effects. Their contribution highlights the dominant role of unobserved heterogeneity in human capital. Abowd, Kramarz, and Margolis s result raises questions regarding the nature of the unobserved component. Kurt (2007a) explores the unobservable human capital hypothesis using the Baccalaureate and Beyond data set (B&B). The data couple detailed variables on new college graduates educational experiences, including college transcripts, and employment outcomes early in the earnings profile. Even with this wealth of heterogeneity, the IWD persists. The differential is positively correlated with post-degree work experience. The IWD increases early in the earnings profile of college graduates in the sample versus a small decrease in the overall IWD in the CPS (Table 1). Why individuals sort into industries that persistently have differences in wages is puzzling and still largely unanswered. There are two candidate theories: non-competitive theories and market frictions. In this paper, I explore the role of search frictions in generating wage dispersion across industries. I extend a directed search model from Shimer (2005a). In directed search, individuals

3 Fall 2007 Kurt, Mark R. 3 make employment application choices based not only on the wage offers but also on the probability of hire. Expected wages equalize across industries in expectation but not in realization. Shimer s model features heterogeneous workers and firms which yields wage dispersion. This is achieved through a coordination friction. Shimer s directed search model generates many features seen in data: wage dispersion among similar workers, the correlation of wages with capital intensity, the existence of wage dispersion early in workers careers, and the non-monotonicity of wages in skill type within a competitive framework. There are several features that Shimer s model cannot generate. In data, highly educated workers are dispersed across industries (Table 2). In Shimer s model, high productivity workers do not match with low productivity firms (industries). The value of some candidate matches is too low to attract high productivity workers. The model, also, has no implications for the evolution of wages. From the B&B, I see an increase in the IWD early in workers careers. My model extends the directed search model in two ways. I add a second period and an informational friction regarding worker type. This friction is formalized as an educational choice with a possibility that education will lead to increased human capital. This educational choice allows workers to equalize wages along a second margin, skill type. Workers take into account which skill type is optimal given the cost of education and the probability of employment conditional on skill. With these two extensions to the model, I generate a persistent IWD which increases early in the earnings profile, and match the education distribution of workers across industries. Directed search provides a reasonable framework for understanding the IWD. It has been proven as an effective means of providing a theory for wage dispersion which is the result of a competitive equilibrium (Moen 1997). Directed search does not rely on wage bargaining to determine wages outside of the model as in random search models. The assumption of wage bargaining lacks a theoretical foundation to discipline the manner in which wages are determined. Another advantage is that the matching friction is endogenous to the model. In random search models, agents have no control regarding from whom they receive job offers. In directed search, workers only apply to types of firms they would be willing to accept an offer and vice versa. 2 A Model of Educational Choice While the Shimer model produces wage dispersion, it is less successful when replicating the distribution of skills across industries (Kurt, 2007b). This is due to the fact that the value of the match between high productivity workers and low productivity industries is too low to attract any applications from those workers. Furthermore, the model does not have

4 Fall 2007 Kurt, Mark R. 4 any implications for workers across the earnings profile. Two extensions of the model are proposed to address both issues. First, an informational friction is embedded in a directed search model. This friction is modeled through an educational choice. A second period is then added where the informational friction is relaxed. 2.1 Environment Workers are risk neutral agents and are indexed by m {1, 2}. In period one, workers are uneducated, m = 1, and educated, m = 2. In period two, workers are either low skilled, m = 1 or high skilled, m = 2. Firms also are risk neutral and indexed by n {1, 2} but are the same across periods. Firms are grouped into industries determined by their type, a type n firm is in industry n. Each firm has a single job vacancy. The mass of each industry type, n, is denoted by ν n. 2.2 Educational Choice At the beginning of period one, workers are identical in their endowment of human capital, h 1, and have a mass equal to 1. Some workers choose to enroll in school and receive a degree for a cost, c. Some educated workers will become skilled, acquiring human capital where h 2 > h 1. This will happen with probability s. With probability (1 s), the worker remains with the initial endowment of human capital, h 1. Neither workers nor firms know which workers out of the educated group have become skilled. Therefore, workers are only known as educated and uneducated in period one, even though some workers are skilled within the educated group. The mass of educated workers is ɛ and 1 ɛ for the uneducated workers. At the beginning of period two, matches are dissolved and skilled workers are revealed to all workers and firms. The mass of skilled workers is ɛs and (1 ɛs) for unskilled. Firms within industries post wages and workers apply for positions. 2.3 Production In order for any production to occur, a worker and a firm must meet and form a match. If a worker does not form a match with another firm and vice versa, nothing is produced. In other words, a job vacancy without a worker cannot produce anything and vice versa. Workers and firms are matched in single pairs, i.e. a worker cannot match with multiple workers and vice versa. The output that results from a match in period t is x t m,n of a homogeneous consumption good. Firms within industry n have k n units of capital. I assume a Cobb-Douglas production function governing output:

5 Fall 2007 Kurt, Mark R. 5 x t m,n = (h t m) α (k t n) (1 α) 2.4 Wage Posting Wages are determined through a three stage wage posting game. In stage one, wages, wm,n, t are determined in which firms within each industry post a single wage for each type of worker. In period one, wages are posted for uneducated and educated workers only. In period two, wages are only posted for skilled and unskilled workers. In the second stage, workers view the posted wages and make employment decisions to apply to one industry based not only on the wages but the probability of being employed. In stage three, firms receive applications. Conditional on receiving applications, firms hire the most profitable worker, produce x t m,n, and pay wages wm,n. t 2.5 Anonymity Restriction Following Shimer, I impose an anonymity restriction. I assume workers are unable to distinguish among firms within an industry. Similarly, firms are unable to distinguish a particular worker within a type. This inability to parcel individual workers and firms within types creates a coordination friction. Workers cannot coordinate their actions with each other to apply strategically to firms. This leads to the possibility of multiple workers applying for the same vacancy while other vacancies receive no applications within the same industry. Moreover, I require identical workers and firms to have identical strategies. This restriction will also serve to ensure that the competitive search equilibrium is unique. 3 The Social Planner s Problem Before examining the decentralized problem, I will set up the planner s problem. The planner seeks to maximize output subject to the resource constraints. In order to achieve this objective, the planner will instruct workers to apply for jobs and for firms within industries to hire certain workers. 3.1 Application Process The planner faces the same anonymity restriction faced by workers and firms. As in the decentralized case, only symmetric strategies are available to the planner. The planner gives identical instructions to all workers (firms) of the same type. If the planner instructs one type m worker to apply for a position in industry n, she must tell all type m workers to

6 Fall 2007 Kurt, Mark R. 6 apply to industry n. The planner has the option to tell a worker m to apply to a position in industry n with probability p t m,n. For each type m worker, application probabilities will sum to one across industries. The probability of applying to an industry, in part, determine the expected number of applications a firm within an industry receives. This is referred to as expected queue length, qm,n. t The expected queue length is equal to the probability a type m worker applies to a firm in type n industry times the ratio of mass of workers of that type and the size of the industry. qm,n t = p t µ t m m,n ν n The number of applications a firm receives for a position is a Poisson distributed random variable. The probability a firm in industry n gets exactly z applications from worker of type m is 1 z (qt m,n) z e qt m,n. The probability a firm in industry n will receive at least one application is e qt m+1,n (1 e q t m,n ). The first term is the probability no applications from workers with a higher productivity than type m are received by a firm in industry n. 1 second term is the probability that at least one type m application is received. In order to calculate aggregate output in the economy, the planner multiplies the mass of the industry n and the probability of hire by the match specific output, x t m,n. The 3.2 Objective function The planner solves the following problem: [ 2 ] V (ɛ) = max E t β t 1 Y t (q t, ɛ) q t,ɛ,q t+1 t=1 [ 2 { 2 2 } ] = max E t β t 1 ν n e qt m+1,n (1 e qm,n t )x t m,n ɛc q t,ɛ,q t+1 t=1 n=1 m=1 2 s.t. ν n qm,n t = µ t m(ɛ, s), for all m and t. n=1 where µ t m is the mass of type m worker in time t. The planner chooses expected queue lengths in order to maximize output subject to the constraints. In period one, the planner cannot observe who is skilled if at least some workers become educated. Note that expected output is equal to: 1 Note that if q m+1 = 0 then e qt m+1,n = 1

7 Fall 2007 Kurt, Mark R. 7 E[x 1 2,n] = (1 s)(h α 1 kn 1 α ) + s(h α 2 kn 1 α ) The planner not only chooses queue length, but also educational choice of the workers. The planner takes into account the cost of education, c, the probability of becoming skilled, s, and the probability of employment when choosing ɛ. I formulate the Lagrangian and take the first order conditions with λ m as the multiplier on the mt constraints. It will be instructive to go through the first order conditions (FOCs) determining expected queue lengths and the educational choice. dl : dɛ (λ1 2 + βsλ 2 2) (λ βsλ 2 1) c, and = c if ɛ > 0. (1) dl dq t m,n : λ t m e (qt m+1,n +qt m,n) x t m,n e qt m,n (1 e q t m 1,n ) x t m 1,n and q t m,n 0 (2) These two FOCs are at the heart of the planner s problem. They determine the mass of workers choosing education and the expected queue lengths across workers and industries that maximize output in the economy. Interpreting for λ t m as the expected income of a worker m in period t will add an understanding of the FOCs. The interpretation of equation (1) is the more straightforward of the two conditions. This FOC determines the proportion of workers who become educated in period one. This choice directly determines the mass of skilled agents in the economy in period two. The first term on the left hand side is the value from becoming educated over the worker s lifetime. The second term on the left hand side is the opportunity cost of education. If a worker chooses to pursue an education, an individual gives up the lifetime stream of income associated with an uneducated worker. The difference between the first and second term may not be less than the cost of education, c, for a worker to become educated. From the equation (2), we see that the planner will only instruct a type m worker to apply for a position in industry n if the additional output of the application is greatest versus another industry. The first term on the right hand side is the benefit of the additional application. The second term is the benefit if the additional application does not arrive. The difference is the marginal benefit of the application. If the right hand side of equation 2 is less than the left, it implies that the marginal product of the application is not high enough relative to an application in other industries. As a result, the planner will decrease q t m,n until either the FOC holds with equality or q t m,n = 0 2. If the right hand side is higher, the planner will increase q t m,n until value or an additional application is equal to the expected income of 2 If q t m,n = 0, then the productivity of the match is too low to attract any type m applicants to industry n in time t.

8 Fall 2007 Kurt, Mark R. 8 worker m. For any qm,n t > 0, expected wages equalize across industries. Notice that λ t m appears in both equations. This implies the planner simultaneously adjusts q and ɛ such that expected incomes are equalized across the industries and education groups, provided that the productivities from the matches are high enough. Proposition 1. Any queue lengths, qm,n, t probability of education, ɛ, and expected incomes, λ t m, satisfying the resource constraints and complementary slackness condition are socially optimal. 3 4 Competitive Search Equilibrium 4.1 Worker s Problem A worker makes educated choices, as well as, decisions regarding which industry to submit a job application. Typically, in a directed search model, workers only equalize expected wages across firm types. The addition of educational choice allows workers to equalize expected wages across skill types as well. Workers adjust their educational and application strategies so that they are indifferent between going to school and not going to school as well as which industry to apply for a position. It will be instructive to go through the worker s problem. Two general first order conditions determine the worker s decisions. The first general condition necessary for optimality from the worker s problem over application strategies is as follows. λ t m = e q (1 m+1,n t e qt m,n ) w qm,n t m,n if qm,n t > 0. (3) The term involving the queue lengths is the probability that a worker is hired conditional on applying for the position. This is multiplied by the wage offered by a firm in industry n. Workers take queue lengths as given. Each individual worker believes that her application strategy will not affect the expected queue length due to her infinitesimal size relative to the total mass of workers. Workers adjust their application strategies such that they are indifferent among jobs in either industry as long as the productivity of the match is high enough, i.e. q t m,n > 0. The second condition is taken with respect to the educational choice and can be simplified using equation 3 to: (λ t 2 + βsλ t+1 2 ) (λ t 1 + βsλ t+1 1 ) c 0 and = if ɛ > 0 (4) 3 The planner s problem is concave in q and ɛ with linear resource constraints. A detailed proof is in a forthcoming appendix.

9 Fall 2007 Kurt, Mark R. 9 Workers adjust their educational strategies such that the benefit from going to school is equal to the tuition cost plus the opportunity cost of education. This equation is exactly the same as the planner s decision rule for education. For a given {λ}, queue lengths and proportion of educated are determined as a result of the application and education strategies which are governed by equations (3) and (4). 4.2 Firm s Problem Now I turn to the firm s problem in period t in industry n. Firms understand the wages they set will determine the expected queue lengths. If a firm would like to increase the probability of hiring a worker m, they increase the wage posted. An increase in the wage offer increases the expected queue length. As the expected queue length increases, the number of applications increases on average. This causes the probability that a type m worker becomes employed at that firm in industry n to decrease. Workers are willing to accept the decrease in employment probability in exchange for a higher wage in the event they are hired. Firms maximize their expected profit by taking into account the productivity of a match and the probability of hiring a worker of type m: 2 πn t = e q m+1,n (1 e qt m,n )(x t m,n wm,n) t (5) m=1 The first term in equation (5) is the probability of hiring a worker m. The second term is the revenue generated from the match less the wages paid for each worker type. The firm s problem is globally concave in q; similar to the planner s problem. A Competitive Search Equilibrium is a set of wages, wm,n, t expected queue lengths, qm,n, t probability of education, ɛ, and expected incomes, λ t m where worker s maximize utility, firm s maximize expected profits, and the labor market clears. 4.3 Wage determination From the competitive search equilibrium, we will derive the equilibrium wage equation using the optimal allocations from the planner problem. Firms know the exact relationship between wages and expected queue length determined by the expected income of the worker. This relationship can be used to substitute wages out of the firm s problem, simplifying it. 4 I rewrite the worker s decision rule for application strategies, equation (3), by multiplying both sides by q m,n t. The right hand side of equation (3) is equal to the expected wage bill of the firm for a type m worker. Substituting the left hand side of equation (3) into the firm s problem, equation (5), yields: 4 See Shimer 2005 for a more detailed exposition

10 Fall 2007 Kurt, Mark R πn t = e q m+1,n (1 e qt m,n )x t m,n λ t m qm,n t (6) m=1 Firm s maximize expected profit by choosing expected queue lengths given λ. Taking the derivative with respect to qm,n t yields the same condition as for the planner, equation (2), less summing across industries. λ t m e (qt m+1,n +qt m,n) x t m,n e qt m,n (1 e q t m 1,n ) x t m 1,n and q t m,n 0 (7) Equation (7) holds with equality if q t m,n > 0. Wages are only defined where the productivity of the match is high enough to attract a positive mass of applicants. Substituting λ t m from equation (7) into equation (3) yields the following wage equation: wm,n t = qt m,ne qt m,n } {x tm,n e (qtm,n qm 1,n) (1 e qtm 1,n )x tm 1,n (1 e qt m,n ) (8) The wage is equal to the marginal value of an application. The first term in equation (8) is the probability that exactly one type m applicant arrives for a position in industry n conditional on applying. If more than one type m application is received then the additional output from that application is zero. The bracketed term is the total output of the match of a worker m employed in industry n less the expected output of the next lower productivity type worker. For m = 1, the second term in the brackets is eliminated as x t 0,1 = 0. In this case the marginal value of an m = 1 applicant is equal to the hiring probability multiplied by output, x t 1,n. 5 Calibration The model predicts wages across both industries and across the earnings profile. Implications regarding employment probabilities, unemployment and vacancy rates, and skill premia are also products of the model. In this section, I calibrate the model to match the behavior of wages in the U.S. economy in the mid 1990 s. I choose this time frame to coincide with three interview dates in the B&B between 1992 and Several parameters must be chosen. 5.1 Capital Stock Given the production technology, the model implies that industries differ by their capital stock per vacancy, k n. The question is what counterpart in data most closely matches k n. The capital to labor ratio lends itself as an obvious candidate. One benefit of using the

11 Fall 2007 Kurt, Mark R. 11 capital ratio, is data to calculate the ratio is readily available. I use the quality adjusted equipment capital to labor ratio. I choose to use the equipment capital series as it has a higher correlation with wages than structures and equipment. 5 Gordon (1985) argues that the quality change bias is large for equipment and is not properly accounted for in data. The result is large distortions in the capital stock. Gordon includes a quality index from 1947 to 1982 which must be extended. In order to extend the Gordon index, we adopt a procedure similar to Greenwood et. al. (2000). Each of the four major components from the Personal Consumption Expenditures price index table (BEA table 2.3.4) is weighted by the actual expenditure of the corresponding component for each year from From this point the Gordon index is regressed on the four weighted components of the PCE index from ln(gordon t ) = A t γ 1 + B t γ 2 + C t γ 3 + D t γ 4 + ɛ t ln(gordon t ) is the Gordon series for capital equipment. A t is the Personal Consumption Expenditures for durable goods price index. B t corresponds to non-durable goods. Where as C t is the covariate for services and D t for transportation. The coefficients are multiplied by their respective component for each year. The result is a quality adjusted price index over our sample period. With an appropriate price index, we can deflate the capital equipment series. Before k n can be calculated we must aggregate the total labor hours for each industry. This is achieved using aggregate hours in the CPS and applying population weights. The total capital stock is divided by aggregate labor hours which yields, k. The average capital to labor ratio equals 60.1 units of capital per labor hour. I divide the economy into two equally sized industries, low and high productivity. Next, I calculate the capital to labor ratio for the low and high industry, and 97.8, respectively. The concern is matching the relative capital stocks not the levels. Therefore, I normalize the capital stock, k 1, in the low productivity industry to be 1. The resulting capital stock, k 2, for the high productivity industry is approximately 4.4. The production function is constant returns to scale with α set to match the labor share of income in the U.S. economy, Vacancies and Unemployment The mass of the labor force and total mass of vacancies is not determined by the model. The relative size of the labor force to the total number of positions available must be identified 5 Various capital to labor ratios were calculated with minimal impact on the results.

12 Fall 2007 Kurt, Mark R. 12 from the data. Identifying the size of the labor force is straightforward. I use the seasonally adjusted employment and unemployment numbers in the CPS. The mass of filled vacancies is matched to data from the BLS. Questions arise regarding how to identify the number of unfilled vacancies. Before 2000, no data source existed which focused on the number of vacancies in the economy. Earlier work (Shimer, 2005b) uses an index based on column square inches of employment classified advertisements. This measure is not the appropriate measure for use in a directed search model for a couple of reasons. First, the level of vacancies is important for the model, not only the relative quantity of vacancies. In order to calculate the proportion of jobs to workers, I require the number of job openings. Second, new job postings have migrated away from newspaper ads. This creates a progressively increasing downward bias in actual number of vacancies late in the series. In 2000, the BLS created the JOLTS, Job Openings and Labor Turnover Survey, to measure the demand side of the labor market. The JOLTS uses unfilled vacancies as an economic indicator of unmet demand for labor. The survey provides monthly data regarding the number of current openings in the U.S. economy. 6 I take the seasonally adjusted average number of job openings as the number of vacancies in the model. For consistency, I calculate the average number of employed (130,760,984), unemployed (7,483,714) and positions open (3,506,878) in the CPS and JOLTS data from The resulting mass of jobs is normalized to one and the mass of workers is Human Capital In a standard neo-classical growth model with human capital, agents invest in human capital in order to increase their wages. In directed search, additional human capital does not guarantee higher wages, rather an increase in the amount of human capital raises expected wages. Expected wages can rise in one of three ways: wage offers increase, the probability of employment increases, or both. Mincerian regressions result in biased estimates of the amounts of human capital when viewed through the lens of a directed search framework. The reason is simple; the dependent variable is actual wages in a Mincerian regression, not expected wages. Thus, differences in employment probabilities by skill type are not captured. This value is used as a lower bound for the amount of human capital. To determine an upper bound, I use the values obtained from an estimation of the static version of Shimer s model using Generalized Method of Moments. The model is estimated to match wages in the U.S. economy in the mid 1990s. The estimates obtained from the model are the upper bound on the value of human capital for several reasons. First, the 6 The JOLTS measures available positions that could start within thirty days of posting and where applicants are being actively recruited.

13 Fall 2007 Kurt, Mark R. 13 duel role of human capital is accounted for in the directed search framework. Second, the model does not include experience; biasing the human capital estimates upward. I index high school graduates human capital, h 1, to one and 2.75 for college workers, h 2. 7 College educated agents are made up or both skilled and unskilled. The actual amount of human capital of the skilled can be solved from the following expression: H s = H e (1 s)h u s Using the calibrated value for s, the upper bound for H s is roughly 3.2. The model is calibrated to match the college premium from the CPS, which is approximately 40%. The value for H s is 2.75 which accords with the average college premium implied by the model. This value clearly falls within the bounds. 5.4 Cost of Education The cost of education is calibrated to the 4 year tuition expense in addition to the opportunity cost of an education. In 2002, the Nation Center of Education and Statistics (NCES) issued a report detailing costs of postsecondary education in the U.S. from 1992 to The main conclusion is that once non-loan financial support was taken into consideration, costs remained fairly constant. I use the median four year tuition cost of public and private institutions less non-loan financial aid, $44, The opportunity cost of going to school is 4 years of lost wages of a high school graduate. I use 4 years of the median income of a high school graduate aged from the CPS, $64, 300. Total costs associated with education is $108, 500. The level of the cost is not the important feature but rather the relative cost of education to earnings is what needs to be captured. One method would be to take the ratio of educational cost to the present discounted value of a college educated worker. This approach might understate the actual cost of education. The initial investment in education is large early in a worker s life and the benefits are spread out incrementally over a worker s lifetime. High educational costs are often cited as a major reason people choose not to earn a college degree. Dynarski (2003) finds that reductions in student aid in the 1980s caused a decrease in the enrollment rates of those affected by one-third. I incorporate this financial constraint by matching c to the ratio of educational cost to the present discounted value of a college educated worker for first 10 years of wages. Student loans can be taken out for financially constrained students in most cases. The repayment term is typically 10 years. The cost of education in the model works out to 32.8% of per period wages. The cost of education 7 See Kurt 2007b for complete details regarding the estimation procedure. 8 All monetary amounts are in 2007 USD.

14 Fall 2007 Kurt, Mark R. 14 is adjusted in the model until it is 32.8% of per period wages. For the two period model, c = Probability of Skill The probability of becoming skilled if a worker chooses to go to school is the most difficult to identify. I propose two methods of approximating its value. The first involves using the B&B data set. The second utilizes a measure of skill developed by Ingram and Neumann (2005). The model implies that wages increase for workers that are revealed to be skilled. Not only does a skilled worker s productivity increase but the supply of skilled workers in the economy decreases in the second period 9. The combination of increased scarcity and increased productivity makes skilled workers more valuable to a prospective employer. As a result, upward pressure is placed on both employment probability and wages. A way of measuring the percentage of workers who become skilled is to look at the percentage of workers who experienced an increased real wage in the B&B. Almost 75% of workers have higher wages in the second period. If only the skilled workers from the educated pool have increased wages, then s =.75. However, skilled workers are not the only educated workers who see increased wages. For some parameterizations, Unskilled but educated workers may become employed in the high productivity industry. These workers also see wage increases as well. Hence, the upper bound of the probability of acquiring skill in college is.75. Another method of obtaining the probability of skill comes from Ingram and Neumann (2005). They employ a factor analysis to decompose average skill requirements for occupations into four distinct factors. I use the measure of skill referred to as F1 or intelligence. The intelligence factor is distributed normally, N (0, 1). Merging F1 with the CPS by occupation codes yields a means of decomposing s. If a college worker ends up in a high F1 position then I will think of him as skilled. In 1992, approximately 70% of all college graduates possess an F1 greater than 1. At the same time, only 13.5% of high school graduates have an F1 above 1. The two separately identified measures of skill seem to coincide. The probability of skill will be chosen to be below the upper bound estimates with s = Results Table 3 lists the values used from the calibration exercise. A constrained maximization routine is used to calculate the optimal allocations of educated workers and expected queue lengths from the planner s problem. The optimal allocations are used to determine wages and 9 In period one, the high type of workers is the educated group made up of skilled and unskilled workers.

15 Fall 2007 Kurt, Mark R. 15 employment probabilities derived from the worker s and firm s problem. From the calibration there will be two parameters of interest: the cost of education and the probability of skill. A sensitivity analysis is performed to better understand the impact of these parameters on the predictions of the model. Table 4 displays the proportion of workers that receive an education in period one and those revealed as skilled in period two. The model predicts that more workers choose to educate than in the data. This is a common prediction. There are many explanations as to why more people choose not enter college. One theory is that the costs from the calibration understate the true cost of education. Even though I partially account for this by incorporating the loan restriction, there may be other costs, such as geographical constraints (Henderson, Shalizi, Venables, 2001) or inefficiencies in the financial aid system (Edlin, 1993). The expected queue lengths are listed in Table 5. The expected queue length is the average number of applicants that will arrive for a position. There is a positive portion of each worker type employed in every industry in both periods. In order for this to be the case, the productivity of the match between all worker types and industries must be high enough to induce dispersion of educated and skilled across the economy. This feature accords well with the data. Note that low productivity firm will receive fewer applications in expectation than the capital intensive industry. Even the amount of applications received by the capital intensive industry seems small. However, a firm receives applicants of both types of workers within a period. For example in period one, 30% of firms in the capital intensive industry receive an uneducated applicant and 97% receive an educated applicant for the same position. 29% of firms in the high type industry receive both applicants. When this occurs, the educated, higher productivity worker is hired over the lower productivity worker. Expected queue lengths are used to determine the wages in Table 6 and employment probabilities in Table 7. Recall the probability of employment is the first term from the equilibrium wage equation (8). Wages from the model are lower for those without an education in the capital poor industry and higher in the capital intensive industry. The uneducated workers are offered the highest wage in the first period of any worker type. The explanation can be found in Table 7. It is unlikely that an uneducated worker will be offered a job; only 32.6% of uneducated workers are hired in the high productivity industry. The high wage is offered as compensation to offset the low employment probability. The planner is willing to offer a high wage to entice an uneducated worker to apply for a position in the capital intensive industry as insurance against the vacancy being left unfilled. Wages are higher for both education types in the high productivity industry. Higher wages, relative to the low productivity industry, are coupled with lower employment probabilities. This argument holds for each wage offered by the high productivity industry.

16 Fall 2007 Kurt, Mark R. 16 Overall, wages in the second period for the skilled worker are higher relative to the educated in the first period. There are two forces acting on the wages: the mass of the skilled workers and productivity of the skilled workers versus the educated workers. The increased scarcity of skilled workers translates into more competition for the high productivity workers in both industries, driving expected wages up in the second period. Increased productivity or the skilled relative to the educated also increases wages. Employment probabilities increase for all types of workers across periods. The logic for the skilled workers is straightforward. The expected productivity of the match is higher, relative to the expected productivity of the educated. There are also fewer skilled workers relative to educated workers. This increases the skilled workers value, driving up wages and employment probabilities. Thus, skilled workers experience a higher wage offer and employment probability. Both the uneducated unskilled workers and educated unskilled workers see wages decrease in the second period. This is a result of the increase in the proportion of unskilled workers and increase in the difference of relative productivities. The total mass of unskilled workers increases in the second period as the unskilled yet educated are revealed. This causes the expected wages of unskilled workers to decrease through lower actual wages, lower employment probabilities, or both. In this case, wages decrease but employment probability increases. The reason behind the increasing employment probabilities lies in the decreasing mass of high productivity workers in period two. High productivity workers crowd out unskilled workers for jobs. As the mass of high productivity workers decreases from period one to period two, it becomes more likely that a low productivity worker is hired. Wages decrease for low type workers between the two periods. Although unskilled workers productivity remains unchanged, the total mass of unskilled increases in period two. This increase in the realized quantity of unskilled workers puts downward pressure on their wages. Table 8 displays the average wages workers receive across time by educational attainment. Wages actually decrease for the uneducated workers in the economy over the early portion of the earnings profile. This is due to the influx of educated but unskilled workers in period two. The increase in the size of the unskilled group drives down wage offers. The model abstracts from growth. This feature would not be present if I incorporated technology progress or on-the-job human capital accumulation. The college premium is calculated by taking the ratio of the wages in Table The college premium increases between the two periods. The increase is driven by the large increase in wages of the revealed skilled workers. Table 9 lists the employment rates by education type across both periods. Employment rates are higher for educated workers 10 Commonly referred to at the skill premium.

17 Fall 2007 Kurt, Mark R. 17 across both periods as seen in the data. The IWD statistic is the weighted adjusted standard deviation in wages across industries. I calculate the IWD first by squaring the differences of the average wage in each industry less the average wage for the entire economy. The average industry wage is simply the weighted average of the actual wage offered to each type of worker with in the industry. Then, the squared terms are summed together and the square root is taken. This measure is analogous to the measure developed by Krueger and Summers. IW D t = 2 ( w n t w t ) 2 (9) n=1 where w n t is the average wage in industry n at time t and w t is the average wage over both industries at time t. Table 10 displays the IWD for each period and the ratio of the IWD in period two over period one. The IWD is large and positive in both periods and it increases in size across industries. Average wages decrease in the low industry and increase in the high industry across periods. The match productivity of the skilled workers in high type industries drives almost all of the increase in the IWD in period two. Skilled workers are found in both industries. However, the match productivity is too low in the capital poor industry to attract a large quantity of skilled workers. Low type industries offer a higher probability of employment as compensation for the lower wage, but it is not enough to entice large quantities of skilled workers in equilibrium. Unskilled workers play a secondary role in determining the magnitude of the IWD. The quantity of unskilled workers increases and drives down the unskilled wage. More unskilled workers are employed in the capital poor industry resulting in a decreasing average industry wage in period two. The end result is increased wages dispersion across industries in period two. The model qualitatively captures the existence of and the increase in the IWD while generating matches of educated workers in both industries. However, the IWD is more prominent in the model than what is observed in the CPS and B&B. Further investigation is required to determine if these two features of the IWD are robust to changes in some key parameters, cost and skill. These parameters may also affect the magnitude of the IWD. 6.1 Sensitivity Analysis The parameter values chosen for the cost of education and probability of skill acquisition may be open to debate. The probability of becoming skilled, s, is difficult to identify in the data. Even defining what is meant by skilled is troublesome. This parameter affects

18 Fall 2007 Kurt, Mark R. 18 wages in two distinct ways. First, as the probability of skill increases, it is more likely that an educated worker will become skilled resulting in a higher expected wage in the second period. Second, the expected productivity of an educated worker increases in period one. This occurs because a greater proportion of educated workers become skilled as the value of s increases. The increase in expected productivity raises the expected wage in period one. Both effects raise the value of an education causing greater numbers of workers to choose an education. The cost of education, while not as elusive as the probability of skill, is also difficult to identify in data. Unlike the probability of skill, the cost of education does not affect the expected productivity of educated workers. The cost does not directly affect the benefit of an education. As the cost of education increases, the value of an education decreases. Fewer people choose to educate. When the cost increases, the quantity of educated decreases forcing expected wages up. The rising cost of education is partially offset by increased expected wages, since fewer workers become educated. 6.2 Changes in the Probability of Skill The probability of skill is varied around the calibrated value,.65. I consider values of s {.304,.845}. Table 11 displays the proportions of worker types for each period. One can see that as s decreases from the maximal value less workers become skilled (column 4). Moreover, fewer workers choose to receive an education. The value from an education decreases as s decreases. Tables 12 and 13 list the wages of worker types across industries and across periods for different values of s. A decrease in skill affects wages in a uniform fashion for the low skilled workers across industries. There are relatively more unskilled in the economy and firms can offer lower wages and still secure a worker. Wages for the educated workers also decrease. This is a result of the lower expected productivity of the match, recall s directly affects the expected productivity, x t m,n. Once skilled workers are revealed, wages increase by a larger margin as s decreases. A skilled worker s productivity remains constant as s decreases. However, the mass of skilled workers decreases. The quantity effect on wages dominates the increase in the hiring probability. Therefore, wages increase. When s decreases to 50%, skilled workers become so scarce that capital poor firms are unable to attract any workers. Even if capital poor industries offer a nearly guaranteed job, the productivity of the match is simply not high enough to attract any skilled applicants. Tables 14 and 15 display employment probabilities for each worker type across periods for each industry. Employment rates increase as s decreases across industries and across all types of workers. Uneducated workers and unskilled workers face less competition from

19 Fall 2007 Kurt, Mark R. 19 educated and skilled workers. As a result of increased competition for jobs, firms can reduce compensation and still attract workers. This results in lower wages and higher employment probabilities. Average wages by education type are displayed in Table 17. At first glance, average wages of educated workers in period two seems at odds with the earlier exposition of wages. Average wages for all education types are increasing as s increases. At the same time, wage of skilled workers are declining. However, the wage of the skilled worker is still higher than the unskilled. In addition, there are more skilled workers in the economy. Both of these factors drive up the average wage of the educated in period two. The IWD is influenced by changes in s as well. As s decreases, the IWD within a period tends to increase. The increase in the IWD across periods is robust in values of s. Only when all workers choose not to educate does the IWD remain constant across the earnings profile. This occurs because without the educational choice, there is no information friction. This friction is at the heart of the increasing IWD. In the first period, educated workers are compensated based on the expected productivity. This expected productivity is lower than the skilled workers productivity. The inability to sort out skilled workers in period one depresses their wages. In period one, the informational friction increases the mass of unskilled workers in the capital rich industry. Unskilled workers slip in under the guise of being educated. This tends to cause a large increase in wages in the capital intensive industry and a slight decrease in the capital poor industry across periods. Only when s becomes very small, does one see the IWD ratio decrease. This is caused by a lack of skilled workers in the economy to perpetuate a large differential in the second period despite their increasing wage. The probability of becoming skilled has wide ranging effects on employment decisions, educational choices, wages, and hiring probabilities both within a period and across the earnings profile. Even with moderate changes in s the model still produces a large IWD that increases across periods and education dispersion across industries. 6.3 Changes in the Cost of Education The aforementioned methodology utilized to parameterize the cost of education may not capture the actual cost. Therefore, a sensitivity analysis is performed. Table 19 displays the proportions of workers by education and skill type across periods as vales of c vary. As the cost of education increases, fewer people choose to educate. The increased cost of education must be offset for workers to be indifferent between an education or not. Less workers receiving a degree creates fewer educated workers driving up their expected wage. Education costs must more than double for all workers to choose to forgo an education. Conversely, when a 30% reduction in cost occurs, c =.65, everyone in the economy chooses

20 Fall 2007 Kurt, Mark R. 20 an education. As in the case of s, educated and skilled workers are dispersed across the economy for wide ranging values of c. Tables 20 and 21 list wages by worker type across industries and across periods. Wages react differently to a change in the cost of education than they do to changes in the probability of skill. Wages in each period do not react uniformly. Wages of uneducated in period one and unskilled in period two are decreasing as c rises. The substitution effect is dominating where the mass of uneducated is increasing causing downward pressure on wages. Wages for the educated workers are fairly constant for both industries in period one. Unlike changes in the probability of skill, the cost of education does not affect the expected productivity of the worker. The result is that the bulk of the movement in the expected wage is accounted for in the employment probabilities which are increasing quickly in c. Employment probabilities are found in Tables 22 and 23. Although, skilled workers do see an increase in wages as c increases, this is a quantity effect. Fewer skilled workers in the economy drive up skilled wages and hiring probabilities. The cost of education also affects the IWD. The IWD is non-monotonic in costs for each period. Changes in the cost of education cause changes in the quantity of educated workers. The productivity of the match is affected due to these quantity changes. When costs are very high, educated workers in period one and skilled in period two do not apply to capital poor industries. This reduces the amount of wage dispersion across industries. Once costs decrease to around 1.10, the value of the match in the capital poor industry increases enough to attract some educated/skilled applicants. Once both types are employed in each industry, wage dispersion rises rapidly across industries rapidly. After this initial increase in the IWD, it declines as costs continue to decline. At the same time, the ratio of the differential increases smoothly. As cost decreases, the quantity of skilled workers increases driving the IWD. The informational friction acts to mute the IWD in the first period causing the IWD to be larger in the second period when skill is revealed. 7 Conclusion This paper extends a directed search model by adding a multi-period component as well as educational choice. The model successfully generates a persistent and increasing IWD over the earnings profile and the dispersion of educated workers across industries. Educational choice directly affects the IWD within a period and across time. When workers choose to educate with a positive probability there is a larger differential over time. This is due to the fact that workers and firms are unable to distinguish who is skilled within the education group early in the agent s life. The revelation of skill in the second period changes the