Human Capital and Market Size

Transcription

1 Human Capital and Market Size Cecilia Vives University of the Basque Country UPV/EHU May 2016 Abstract This paper studies how the size of the labour market aects workers' decision to invest in human capital. We consider a model of mismatch where rms rank workers according to their level of skills. In this framework, the job nding probability of a worker depends on market size, market tightness and his ranking in the distribution of skills. When market tightness is below one, bigger markets provide more incentives to acquire human capital to the workers with higher rank but lower incentives to the workers with lower rank. Therefore, bigger labour markets have a larger dispersion of skills. We also nd that returns to skills increase with the size of the market. Keywords: Human Capital, Skill, City Size JEL Classication Numbers: J24, R23 I would like to thank Salvador Ortigueira, Javier Fernández Blanco, Diego Puga, Belén Jerez and Alfredo Salgado for their helpful comments and suggestions. Departamento de Fundamentos del Análisis Económico I, University of the Basque Country UPV/EHU. address: cecilia.vives@ehu.eus 1

2 1 Introduction The distribution of skills diers across cities. Big cities have a higher proportion of high skilled workers. Glaeser and Resseger (2010) nd a 2.8 increase in the percentage of college graduates in a city for an increase of 1% in population size for US in However, in big cities the distribution of skills is also more unequal than in small ones, as it has been found in Eeckhout et al. (2014) and Combes et al. (2012). These dierences in the skill composition are an important factor in order to understand spatial wage inequality. Combes et al. (2008) estimate that dierences in the skill composition of the labour force account from 40 to 50% of aggregate spatial wage disparities in France. The uneven distribution of skills across locations may be the result of either workers migrating to those locations that favour more their level of skills or workers accumulating more human capital in those locations. Combes et al. (2012) nd a small role for migration in accounting for the dierence in skills between denser and less dense areas. Therefore, in this paper we consider how the size of cities aect the incentives to acquire human capital. This issue is of interest because it implies that the size of cities may be a determinant of the distribution of skills of the entire economy. We develop a static model of mismatch with endogenous human capital. Under this setting, the number of workers and vacancies in the market are random. If there is an excess of workers, the ones with the lowest level of skills will be unemployed. However, workers can invest in human capital before entering the market in order to improve their ranking. In this environment, we dene market tightness as the ratio of the expected number of vacancies to the expected number of workers. Therefore, the size of the market is large if the expected number of workers is large, for a given level of market tightness. We nd that the eect of market size on the probability of nding a job for a worker with a given rank depends on market tightness. If market tightness is below 1, the job nding probability of the workers with higher rank increases with market size whereas this probability decreases for the workers with lower rank. The eect of market size on the job nding probability determines the eect on the distribution of human capital in equilibrium. The workers with lower rank invest less in human capital when the size of the market is large but the workers with higher rank invest more. Therefore, the model generates a more unequal distribution of skills in bigger markets. Wheeler (2001), Bacolod et al. (2009) and Eeckhout et al. (2014) have documented 2

3 the allocation of skills across locations. Wheeler (2001), using county-level data from the 1990 Census of Population and Housing, estimates a negative relationship between the percentage of population with low levels of education and population and a positive relationship for the case of population with high levels of schooling, thus implying that workers have higher levels of education in bigger cities. Bacolod et al. (2009) deal with the eect of city size on the distribution of education and other kind of skills, like intelligence or social skills, with data from the % Census sample and the National Longitudinal Survey of Youth They nd that the eect of city size on the mean education of workers is larger than on the other measures of skills. Their work also highlights that simplifying the level of skills to two levels might be too restrictive. Although they do not nd a strong eect of city size on the mean level of intelligence and social skills, they do nd that it aects the distribution, as larger cities accumulate more people with skills at the top and at the bottom of the distribution. Eeckhout et al. (2014) ndings also suggest that the distribution of skills in big cities have fatter tails. Using the information from the Current Population Survey of 2009, when they measure skills with years of education, they nd a small positive correlation between city size and average skills but a positive relation between city size and the standard deviation of the skill measure. With respect to wages, the model generates higher returns to skill in bigger cities. This is in line with the evidence in Wheeler (2001) and Bacolod et al. (2009). The model implies that the distribution of human capital in equilibrium is such that the job nding probability of a worker with a given level of human capital is independent of market size. That is, workers with the same rank have dierent probabilities depending on the size of the market, but workers with the same level of human capital do not. There have been proposed dierent theories that can explain why big cities attract skilled workers. Most of them are based on dierent kinds of agglomeration economies. Glaeser and Resseger (2010) consider that cities foster the learning of skilled workers. In Andersson et al. (2007), bigger markets provide with better assortative matching. Hendricks (2011) relates the skill composition to the size of the business services sector. At the country level, Redding and Schott (2003), propose that skilled-intensive production has increasing returns to scale. Other theories are based on the idea that only the most skilled can aord the tougher competition of bigger markets, as in Nocke (2006) and Behrens et al. (2014). On the other hand, Eeckhout et al. (2014) explain the higher inequality in skills in big cities through complementarities between the skills 3

4 of workers in the production function. In contrast with the previous literature, in our model workers choose their level of skills. 1 Instead of an exogenous distribution of skills and workers sorting across locations, in our model workers remain in their location but can choose how much human capital to accumulate. Our model is also related with the literature that deals with the sources of agglomeration economies. One of the explanations for the existence of agglomeration economies is that the matching process is more ecient in bigger cities (Duranton and Puga (2004)). This is consistent with the ndings of Gan and Li (2016) for the market of PhD. economists and the evidence in Bleakley and Lin (2012) on occupational switching. Most of the empirical research nd that the number of matches at the aggregate level is a constant-returns-to-scale function of unemployment and vacancies (Petrongolo and Pissarides (2001)). However, Shimer (2007) shows that a mismatch model with a matching function that features increasing returns to scale is also consistent with the data. The human capital investment decision in our model is closely related to Moen (1999). His formulation does not require heterogeneity of the workers or rms in order to nd a non-degenerate distribution of skills. However, unlike us, in his model the size of the market plays no role. The rest of the paper is organized as follows. Section 2 describes the setting. Section 3 considers the problem of the workers. In Section 4 we calculate the equilibrium. In Section 5 we use the model to analyse the relationship between the size of the labour market, the distribution of human capital and wages, and, nally, Section 6 concludes. 2 Setting We analyse an economy that lasts for one period in which we can identify 3 stages: In the rst stage a continuum of unemployed workers are born and vacancies are created. In the second stage, each worker decides how much human capital to accumulate. Finally, workers and vacancies go to the labour market in order to get matched and receive their payos. We assume that the number of workers, U, and vacancies, V, is random and independently distributed with mean U > 0 and V > 0 and variance σu 2 and σ2 V. Let the 1 The exception is Redding and Schott (2003), where workers can choose to be either skilled or unskilled. 4

5 associated distribution functions be denoted by F U (u) and F V (v). We assume that these distributions are continuous. In particular, we will consider the case in which U and V are normally distributed. The assumption of a normal distribution is motivated by the fact that when σu 2 = U and σ2 V = V for U and V large enough, the normal distribution is a good approximation to the Poisson distribution, previously used in Shimer (2007). In addition we will also consider the case in which F U (u) and F V (v) are continuous with support (0, ). In the rst stage, a realization of U and V, which we will denote u, and v, is drawn. We assume that each rm has one vacancy, so referring to a vacancy or a rm is equivalent. At this stage workers are identical and do not know which realization has taken place. In the second stage, the decision problem of each worker consists in choosing a costly level of human capital (or skill), h. Workers are risk neutral. The key assumption in this model is that the workers decide how much human capital to accumulate, they do not know u and v. However, we assume that they know the distribution of these two variables. If we think of education as one of the main factors to acquire human capital, this assumption is reasonable. Individuals usually decide their level of human capital at early stages in life, without knowing the exact conditions of the labour market. Each worker can accumulate a level of human capital h 0 at a cost C (h). The cost function is twice continuous dierentiable and satises C(0) = 0, C (0) = 0, C (h) > 0 for h > 0, and C (h) > 0. We also assume that for some h, C (h) > 1. After the investment decision has taken place, all workers and rms observe u and v and enter the labour market in order to get matched. We assume perfect competition in the labour market. The chances of getting matched for a worker will depend on his level of human capital and on the realizations of U and V. If a vacancy and a worker with a level of human capital equal to h match in the market, they together produce h units of an homogeneous output. In this case, the payo of the worker will be w 0 and the payo of the rm will be π = h w 0. If a worker does not match any rm, he does not produce and receives 0. Similarly, if a rm does not match any worker, prots are zero. 5

6 3 Workers' problem In this section, we will deal with the workers' problem. First, we will set the problem they face at the third stage, when they enter the labour market. We will derive which workers will be employed and at what wages. In the second part of this section, we will describe how workers choose their level of human capital. 3.1 The labour market When the agents enter the labour market, the level of human capital of all workers is already determined. Let G(h) denote the cumulative distribution function over human capital that arises in equilibrium. G = G (h) can also be considered as the ranking of the worker, with higher G meaning a higher ranking. Initially, we will assume that this distribution is continuous with a bounded support. We will denote the minimum level of h in equilibrium h min and the maximum h max. There is perfect competition in the labour market. Firms seek to maximize their prots and workers seek to maximize their wage. In this setting, the number of matches will equal min {u, v} and rms employ the most productive workers 2, that is, rms rank the workers according to their level of human capital. This last result and continuity of the distribution of human capital imply that a worker with a level of skills h 0 will nd a job if there are more vacancies than workers with a higher level of skills than him, which can be expressed as (1 G(h 0 ))u v. The payos of the workers will depend on the lowest level of human capital among those employed. We will refer to this worker as the marginal worker and denote his level of skills as h m (u, v). The payos in equilibrium of a worker, when there are u workers and v vacancies in the market, is w (h, u, v) and is given by: h when u < v w (h, u, v) = h h m (u, v) when (1 G (h)) u v u (1) 0 when v < (1 G (h)) u This function is derived in Appendix A. The payo function states that if there is an 2 If v < u, the workers with higher human capital are employed. If this is not true, then there is some h and h with h > h and h is employed but h is not. This implies that π (h, u, v) π (h, u, v), thus w (h, u, v) > w (h, u, v) 0 but in this case h strictly prefers to be employed while it is not. This cannot be an equilibrium. 6

7 excess of jobs, v > u, workers' wage is their productivity. However, if there is an excess of workers, v u, the worker is employed only if (1 G (h)) u v; then the wage of an employed worker is the dierence between his productivity and the productivity of the marginal worker. 3.2 Workers' expected earnings At the second stage of the period, workers choose the level of human capital that maximizes their expected earnings. The worker only knows the distributions from where u and v will be drawn. Notice that the lack of information at the second stage of the process is the only source of friction in this economy. Once the agents go to the market, the matching process is ecient, both in generating the maximum number of matches possible and in selecting the most productive workers. The expected earnings, E (h), is a function of h, dened as: E (h) = E [w (h, U, V )] C(h) where E [ ] is the expectation operator. As it can be seen from (1), the expected payo depends on the job nding probability of the worker. For a worker with rank G = G (h), this is the probability that (1 G) u v. Let denote it as P (G). In Lemma 1 we show that the job nding probability is strictly increasing in G when F U (u) and F V (v) are continuous with support (0, ). In Lemma 2 we nd under what conditions the same result is also true when U and V are normally distributed. These results are important to ensure that G (h) is continuous in equilibrium. Lemma 1. If F U (u) and F V (v) are continuous with support (0, ), P (G) is a continuous function, strictly increasing and dierentiable in G [0, 1). Furthermore, P (0) > 0. Proof. See Appendix B. Lemma 2. If U and V are normally distributed with 2σV 4 U > V, P (G) is a continuous function, strictly increasing and dierentiable in G [0, 1]. Furthermore, P (0) > 0. Proof. See Appendix B. Since we are interested in the normal distribution as an approximation of the Poisson distribution, the condition imposed in Lemma 2, 2σV 4 U > V, is not restrictive. For the 7

8 normal distribution to approximate the Poisson distribution, it must be that σ 2 V = V. Therefore, the condition is clearly satised. We will next decompose the job nding probability in such a way that we can calculate the expected earnings. First of all, since the problem of the worker is to choose h, we must make explicit again the dependence of G on h. Lemma 2 imply that P (G (h)) can be expressed as: P (G (h)) = P (0) + ˆ h h min P ( )) ) G ( h G ( h d h To simplify notation, let p h (h) = P (G (h)) G (h). Then, P (G (h)) = P (0) + ˆ h Lemma 1 and h min p h (h) d h (2) According to this equation, P (G (h)), can be calculated as the sum of two terms. The rst term is the probability that (1 G (h)) u v for G (h) = 0, which is the probability that there is an excess of vacants. The second term is the integral of p h (h), which is the density of (1 F (h)) u = v, that is, of h being the marginal worker. This denition of the job nding probability allows us to calculate the expected earnings of a worker as: ˆ h ( h) ( E (h) = P (0) h + p h h h ) d h C(h) (3) h min The rst term of the equation accounts for the wage when there is an excess of vacancies. And the second term accounts for the case when there is an excess of workers and h nds a job for all possible values of the marginal worker. Notice that equation (3) also denes the expected earnings for h < h min and h > h max. The decision problem of the worker is: The worker takes G(h) and w (h, u, v) as given. max {E (h)} (4) {h 0} 8

9 4 Equilibrium An equilibrium solution to the problem outlined above can be described by a tuple (G (h), w (h, u, v)) such that: 1. w (h, u, v) satises (1). 2. Workers solve (4). 3. G (h) satises E (h 1 ) = E (h 2 ) for h 1, h 2 on the support of G (h) and E (h) < E (h 1 ) for h that does not belong to the support. The third point is due to the fact that, ex ante, all the individuals are equal. Thus, in equilibrium they all must obtain the same expected earnings. For any h not in the support, the expected earnings must be lower. Given that the support of G (h) is connected, this third condition implies that the derivative of the expected earnings with respect to human capital is zero for any h in the support of G (h). This derivative is: E (h) h = P (0) + ˆ h h min p h ( h) d h + hp h (h) p h (h) h C (h) = 0 Notice that terms 3 and 4 account for the increment in the job nding probability, what is referred to as the rat race. However, at this point the worker is the marginal worker and his wage is 0. Therefore, the gain from increasing the probability of nding a job is 0. This means that, in contrast with Moen (1999), in this setup the ranking of workers does not produce rat race. On the contrary, each worker chooses the optimal level of h given his job nding probability. Substituting (2), it simplies to: Therefore, in equilibrium: E (h) h = P (G (h)) C (h) = 0 P (G (h)) = C (h) (5) This condition means that the marginal cost of acquiring an extra unit of human capital must equal the marginal benet, which is simply P (G (h)). The individuals with a higher ranking will enjoy a higher probability of nding a job, which increases expected earnings. However, in equilibrium, all individuals must obtain the same expected earnings. Compare two workers with dierent levels of human capital. One 9

10 worker has a higher position in the ranking than the other and a higher probability of nding a job. How much more human capital than the other he has does not aect the probability, since it only depends on the relative position in the ranking of each of them. In order for the low skilled worker to be willing to remain low skilled, the high skilled worker must accumulate a quantity of human capital such that the cost associated to it makes the expected earnings of both workers equal. Therefore, expected earnings are equalised through the cost of obtaining human capital. The following proposition shows that there exists a function G (h) that satises equation (5) and that when the distribution of human capital is given by G (h), the economy is in equilibrium. Proposition 1. An equilibrium of the model exists with the associated distribution of human capital satisfying P (G (h)) = C (h). G (h) is a continuous distribution. See Appendix B. Proposition 1 shows that the distribution of human capital only depends on the job nding probability and the marginal cost. The marginal cost is exogenous but the job nding probability depends on the ranking of the workers and the distribution of U and V. Since we are interested in the eect of the size of the labour market on G (h), we need to relate it to the distributions of U and V. We had denoted the mean of U by U and the mean of V by V. Now, let V = θu with θ > 0. We can interpret U as market size and θ as market tightness. 5 The eect of market size on the distribution of human capital and wages We have seen that the size of the labour market can aect the distribution of human capital if it aects the job nding probability. Therefore, we begin by studying this latter relationship. Consider the case in which U and V are normally distributed. We allow the standard deviation of U and V to depend on their means. Therefore, let σ U = s ( U ) and σ V = s ( θu ), with s () a dierentiable function. This assumption implies that the coecients of variation of U and V are also functions of U and θu. We denote the coecient of variation of U as cv ( U ) = s(u). The following result shows that the eect U of market size on the job nding probability depends on this coecient. 10

11 Proposition 2. Consider U N ( U, s ( U )) and V N ( θu, s ( θu )). If cv ( U ) is strictly increasing in U, the job nding probability of a worker with ranking G > (<) 1 θ, is strictly decreasing (increasing) with market size and the job nding probability of a worker with ranking G = 1 θ is constant. If cv ( U ) is strictly decreasing in U, the job nding probability of a worker with ranking G > (<) 1 θ, is strictly increasing (decreasing) with market size and the job nding probability of a worker with ranking G = 1 θ is constant. If cv ( U ) is constant in U, the job nding probability is unaected by market size. Proof. See Appendix B. If the coecient of variation is strictly increasing, bigger markets imply a higher job nding probability for the workers with lower rank and a lower job nding probability for the workers with higher rank. If the coecient of variation is strictly decreasing, the opposite occurs. On the other hand, if the coecient of variation is constant, the size of the labour market has no eect on the job nding probability. (a) Inow of labour supply (b) Inow of vacancies Figure 1: Coecient of variation for the Spanish provinces In order to look at this relationship in the data, we use time series at the regional level. Given that we are interested in the variability of the series rather than on their levels, we nd more appropriate to use data from administrative sources rather than surveys. Specically, we use the monthly data for Spain from the Servicio Público de Empleo Estatal (SEPE) for the 52 provinces and autonomous cities for the period SEPE provides data on labour supply, unemployment and the inow of labour supply for each month. We use these three variables as measures of u. For the 11

12 number of vacancies, we only have information of the inow of vacancies created each month. We compute the mean and the coecient of variation of these variables for each region. Figure 1 shows a negative relationship between the coecient of variation and the mean. The same occurs for the variables labour supply and unemployment, represented in Figure 5 in Appendix C. From now on, we will focus in this case. Notice that, if we use the normal distribution as an approximation of the Poisson distribution, the coecient of variation is decreasing. In Figure 2, we represent the job nding probability in this case (in which U and V are normally distributed with σu 2 = U and σ2 V = V ) as a function of G for θ = 0.6 and dierent market sizes. Consistent with Proposition 2, the job nding probability is increasing with market size for G > 1 θ = 0.4 and decreasing with market size for G < 0.4. The three functions cross at 1 θ = 0.4, thus, the probability of nding a job for the worker with G = 0.4 is the same for all market sizes. Figure 2: Job nding probability, P ( G, U, θ ) The level of market tightness plays an important role in the relationship between market size and the job nding probability. The larger is θ the higher the proportion of workers who will have a higher probability in a bigger market. In particular, if θ > 1, all workers have G > 1 θ. Thus, all of them will have a higher probability when the market is bigger. The result in Proposition 2 does not rely on the assumption of normality. In Appendix C, we show that the same pattern arises when U and V follow a Gamma distribution. 12

13 5.1 The distribution of human capital The eect of market size on the distribution of human capital is driven by the eect on the job nding probability. If market size increases, the probability for the workers with rank below 1 θ decreases. This means that their marginal benet of investing in human capital is lower. In order for the marginal cost to be also lower, their level of human capital must decrease. On the contrary, for the workers with rank above 1 θ, the job nding probability increases with market size. Therefore, they choose a higher level of h. An important aspect is whether market tightness is greater or lower than one. If it is lower than one, bigger markets imply a more unequal distribution, since the workers with lower rank have a lower level of h and the workers with higher rank have a higher level of h. If it is higher than one, all workers have rank above 1 θ. Therefore, all of them will invest more in human capital in bigger labour markets. Proposition 3 formalises this result. Proposition 3. Consider U N ( U, s ( U )) and V N ( θu, s ( θu )) with cv ( U ) decreasing. Let h 0 be such that G (h 0 ) = 1 θ. Then, a. G (h) is strictly increasing in U for h < h 0, strictly decreasing in U for h 0 < h and constant for h 0. b. h min is strictly increasing (decreasing) in U if θ > (<) 1. h max is strictly increasing in U. c. As U tends to innity the distribution converges to a discrete distribution with θ workers with h = h max and max {0, 1 θ} workers with h = 0. Proof. See Appendix B. In addition, Proposition 3 deals with the case that market size tends to innity. Again, the result depends on the level of market tightness. If it is higher than 1, all workers invest h max ; if it is lower than one, only a proportion of workers invest h max while the rest choose h = 0. Figure 3 represents the cumulative distribution and the density function for the same numerical examples as in Figure 2. As market tightness is below one, we nd that the density becomes more dispersed and the range of the support expands, as market size increases. Proposition 3 also implies that when market tightness is higher than 1, the mean level of human capital is higher in bigger markets. However, for market tightness below 13

14 (a) Cumulative (b) Density Figure 3: Distribution of human capital Figure 4: Mean level of h 1, we must rely on numerical examples. Let C (h) = 1 2 h2. Then, Equation (5) simplies to P (G) = h. We have represented in Figure 4 the mean level of human capital as a function of market size for dierent levels of market tightness. The minimum level of market size chosen is 20 since the normal distribution is only a good approximation of the Poisson distribution when U is big enough. With this cost function, we know that the mean level of human capital when market size tends to innity is θ. The Figure shows that when market tightness is below 0.7, the mean level is decreasing with market size but for market tightness above this level the mean is increasing. 5.2 The expected wage We can analyse the eect of market size on the returns to skill. Let the expected wage of a worker with human capital h be w e (h). The expected wage is the expected payo conditional on nding a job. w e (h) = E [w (h, U, V )] P (G (h)) The expected payo, E [w (h, U, V )], can be derived from the equilibrium condition 14

15 that equalises the expected earnings of all workers. Since the expected earnings of the worker at the bottom are easier to compute, we will use E (h) = E (h min ). Substituting E (h) = E [w (h, U, V )] C(h) into this equilibrium condition, we obtain E [w (h, U, V )] = E (h min ) + C(h). Thus, the expected wage becomes: w e (h) = E (h min) + C(h) P (G (h)) Since we are interested in the eect of market size on w e (h), we need to know how it aects P (G (h)) for a worker with a given level of human capital. Equilibrium condition (5) implies that market size does not aect the job nding probability. Since P (G (h)) = C (h), when market size increases, the ranking of the worker will change so that the job nding probability always equals the marginal cost. Proposition 4. If θ < 1, returns to skill are incresing with market size. Proof. See Appendix B. This is an implication of the equilibrium condition. The expected earnings must be the same for all workers. As market size increases, the expected earnings of the worker at the bottom decreases. Therefore, the expected earnings of all workers must decrease in the same quantity. Given h, the cost of acquiring human capital is constant. This implies that all workers have a decrease in their expected payos equal to their decrease in expected earnings. As the job nding probability of a worker with low rank is small, the expected wage must decrease more in order to obtain the same reduction in the expected payo. This implies that we can explain higher returns to skill in bigger cities through gains in matching eciency when market tightness is lower than 1. 6 Conclusions In this paper, we have explored the implications of city size on the incentives to invest on human capital. To this end, we have developed a model in which workers' chances of getting a job depend on the size of the market and their relative position in the distribution of skills. The results of this model are consistent with the hypothesis that investment in human capital increases with market size. 15

16 References Andersson, F., S. Burgess, and J. I. Lane (2007). Cities, matching and the productivity gains of agglomeration. Journal of Urban Economics 61 (1), Bacolod, M., B. S. Blum, and W. C. Strange (2009). Skills in the city. Journal of Urban Economics 65 (2), Behrens, K., G. Duranton, and F. Robert-Nicoud (2014). Productive cities: Sorting, selection, and agglomeration. Journal of Political Economy 122 (3), Bleakley, H. and J. Lin (2012). Thick-market eects and churning in the labor market: Evidence from us cities. Journal of Urban Economics 72 (2-3), Combes, P. P., G. Duranton, and L. Gobillon (2008). Spatial wage disparities: Sorting matters! Journal of Urban Economics 63 (2), Combes, P.-P., G. Duranton, L. Gobillon, and S. Roux (2012). Sorting and local wage and skill distributions in france. Regional Science and Urban Economics 42 (6), Duranton, G. and D. Puga (2004). Micro-foundations of urban agglomeration economies. Handbook of regional and urban economics 4, Eeckhout, J., R. Pinheiro, and K. Schmidheiny (2014). Spatial sorting. Journal of Political Economy 122 (3), Gan, L. and Q. Li (2016). Eciency of thin and thick markets. Journal of Econometrics 192 (1), Glaeser, E. L. and M. G. Resseger (2010). The complementarity between cities and skills. Journal of Regional Science 50 (1), Hendricks, L. (2011). The skill composition of u.s. cities*. International Economic Review 52 (1), 132. Moen, E. R. (1999). Education, ranking, and competition for jobs. Journal of Labor Economics 17 (4), Nocke, V. (2006). A gap for me: Entrepreneurs and entry. Journal of the European Economic Association 4 (5),

17 Petrongolo, B. and C. A. Pissarides (2001). Looking into the black box: A survey of the matching function. Journal of Economic Literature 39 (2), Redding, S. and P. K. Schott (2003). Distance, skill deepening and development: Will peripheral countries ever get rich? Journal of Development Economics 72 (2), Shimer, R. (2007). Mismatch. American Economic Review 97 (4), Wheeler, C. H. (2001). Search, sorting, and urban agglomeration. Economics 19 (4), Journal of Labor 17

18 A Payos If the payos of the workers at the third stage are given by (1), then prots are given by π (h, u, v) = h w (h, u, v). a) When u < v: In this case there will be unmatched vacancies. For rms to remain indierent about their vacancies being unmatched, prots must be zero for all rms. Therefore, w (h, u, v) = h. b) When v u. (a) For the workers who remain unemployed their payo will be zero. (b) For the workers who get employed: i. We derive the wage of the marginal worker. To simplify notation, let h m (u, v) = h m. According to (1), w (h m, u, v) = h m h m = 0. In order to prove it, suppose it is not, then w (h m, u, v) = ɛ > 0. Since the distribution of human capital has a connected support, there is an unemployed worker with h = h m ɛ and 0 ɛ < ɛ. In equilibrium this worker must be indierent about being unemployed, so his wage is 0. But this implies, π (h, u, v) > π (h m, u, v), which is not possible in equilibrium. ii. To derive the wage of all the workers with a level of human capital above the marginal worker, note that rms must be indierent about whom to hire, so for any h,h h m, π (h, u, v) = π (h, u, v). In particular, if h = h m, π (h, u, v) = π (h m, u, v) = h m. Therefore, w (h, u, v) = h h m. (c) For the case of h (h min, h max ): when v < (1 F (h)) u, the worker is unemployed and obtains the payo w (h, u, v) = 0 and when (1 F (h)) u v u the worker is employed and obtains the payo w (h, u, v) = h h m. (d) For the case of h = h min : when v < (1 f 0 ) u, the worker is unemployed and obtains the payo w (h, u, v) = 0, and when (1 f 0 ) u v u the worker has some positive probability of being employed (derived in footnote 5). Since h min = h m when (1 f 0 ) u v u. Both the workers employed and unemployed obtain payo w (h min, u, v) = h min h m = 0. 18

19 B Proofs Proof to Lemma 1: Let U G = (1 G) U. Then, the job nding probability is the probability that u G v, where the density function of U G is f UG (u G ) = 1 f ( ug 1 G U 1 G). Therefore, P (G) = = = P (G) = ( f 0 V (v) f v U f V (v) > 0 and v > 0. (1 G) 2 v 1 G Proof to Lemma 2: 1 G ) ˆ ˆ v 0 ˆ 0 ˆ 0 ˆ > 0, thus f U G f U v 1 G 0 ( f V (v) F U ( ) ug f V (v) du G dv 1 G f U (u) f V (v) dudv ) dv v 1 G v dv. For any v on the interval (0, ) and G < 1, (1 G) 2 ( v 1 G) > 0. On this interval, it is also true that Let X = (1 G) U V. Then, X is normally distributed with mean µ = (1 G) U V and variance σ 2 = (1 G) 2 σu 2 + σ2 V. Denote the associated distribution function by F X (0). The job nding probability is given by ( ) ( P (G) = F X (0) = Φ µ ) V (1 G) U = Φ σ ( (1 G) 2 σu 2 + ) 1/2 σ2 V The derivative is: Therefore, P (G) > 0 if P (G) = φ ( ) Uσ ( V (1 G) U ) 1 2 σ 3 σ 2 Uσ ( V (1 G) U ) 1 2 σ 3 > 0 This can be rewritten as U ( 2σ G ) V > 0 19

20 substituting σ 2 = (1 G) 2 σ 2 U + σ2 V U we obtain ( 2 ( (1 G) 2 ) σu 2 + σv 2 2 ) + 1 G V > 0 The most restrictive case is G = 1. Therefore, P (G) > 0 if U2σ 4 V V > 0. Proof to Proposition 1: 1. G (h) that satises equation (5) exists and is a continuous distribution: In Lemma 1 it is shown that P (G) is strictly increasing for G [0, 1]. Therefore, in equation (5), we can invert P (G) and obtain G (h) = P 1 (C (h)) for h [h min, h max ], with h min dened by P (0) = C (h min ) and h max by P (1) = C (h max ). Since the range of P (G) lies in [0, 1]. The assumption that C (h) continuous and ranging from 0 to a value greater than 1 guarantees the existence of G (h). SinceP (G) and C (h) are continuous and strictly increasing G (h) is a continuous distribution. 2. The equilibrium conditions are satised: In section 4, we have shown that if G (h) satises equation (5), then E (h 1 ) = E (h 2 ) for any h 1, h 2 [h min, h max ]. But, given h 1 [h min, h max ], we have to show that E (h) < E (h 1 ) h that does not belong to the support, that is for h < h min and for h > h max. (a) E (h) < E (h min ) for any h < h min. The expected earnings of a worker with h h min are E h hmin (h) = P (0) h C (h). Since E h h min (h) = C (h) < 0, the maximum of E h hmin (h) is achieved when E h h min (h) = P (0) C (h) = 0, that is, at h min. (b) E (h) < E (h max ) for any h > h max. The expected earnings of a worker with h h max are ˆ h E h hmax (h) = P (0) h + = P (0) h + p h h min ˆ hmax p h h min ( h) ( h h ) d h C(h) ( h) ( h h ) d h C(h) The second equality is due to p h (h) = 0 for h > h max since G (h) = 0 in this 20

21 range. Then, ˆ hmax ( h) E h h max (h) = P (0) + p h d h C (h) = P (1) C (h) h min Since E h h max (h) = C (h) < 0, the maximum of E h hmax (h) is achieved when E h h max (h) = 0, that is, at h max. Proof to Proposition 2: Following the same procedure as in the proof of Lemma 1, the job nding probability can be expressed as: P (G (h)) = Φ ( ) (θ (1 G (h))) U σ ( U ) with σ ( U ) = ((1 G (h)) 2 ( s ( U )) 2 ( ( )) 2 ) 1/2 + s θu Therefore, the derivative of the job nding probability with respect to market size is given by: P (G (h)) = Φ ( ) (θ (1 G (h))) σ ( U ) Uσ ( U ) ( σ ( U )) 2 We need to nd the sign of σ ( U ) Uσ ( U ). Computing this term gives: σ ( U ) Uσ ( U ) = (1 G (h)) 2 s ( U ) ( s ( U ) Us ( U )) +s ( θu ) ( s ( θu ) θus ( θu )) Since s () is the standard deviation, it is never negative. If s ( U ) Us ( U ) > (<) 0 for all U, s ( θu ) θus ( θu ) is also positive (negative). Then, the sign of σ ( U ) Uσ ( U ) equals the sign of s ( U ) Us ( U ). In turn, this is positive (negative) if cv ( U ) is strictly increasing (decreasing) in U. Proof to Proposition 3: a. Given a worker with human capital h, Equation (5) must be satised. Taking derivatives on both sides, we obtain: G (h) = P(G(h),U,θ) P(G(h),U,θ) G(h) The denominator is always positive. The numerator is positive (negative) when G > 21

22 (<) 1 θ. Therefore G(h) is positive (negative) when G < (>) 1 θ. If h 0 satises G (h 0 ) = 1 θ, then h < (>) h 0 when G < (>) 1 θ. b. h min satises P ( 0, U, θ ) = C (h min ). Therefore, h min = This is positive (negative) if θ > (<) 1. P(0,U,θ) C (h min ) h max satises P ( 1, U, θ ) = C (h max ). Therefore, h max = P(1,U,θ) C (h max ) This is positive if θ > 0, which is always satised. c. As U tends to innity P ( G, U, θ ) tends to 1 for G > 1 θ, that is, for G (1 θ, 1]. Therefore, h tends to h max = C 1 (1) for 1 (1 θ) = θ workers. As U tends to innity P ( G, U, θ ) tends to 0 for G < 1 θ, that is, for G [0, 1 θ). Therefore, h tends to h min = C 1 (0) = 0 for (1 θ) 0 = θ workers. Proof to Proposition 4: Let h > h 0. The skill premium is: w e (h ) w e (h) h h = E(h min )+C(h ) P (G(h )) E(h min)+c(h) P (G(h)) h h Taking the derivative with respect to market size, we obtain: ((w e (h ) w e (h)) / (h 1 h)) = 1 P (G(h )) P (G(h)) E (h min ) h h This is the product of two terms. The rst term is negative. We can compute the second term, E(h min). We have that E (h min ) = P (0) h min C (h min ). Therefore, E (h min ) = P (0) h min When θ < 1, E(h min) < 0. Therefore, ((we (h ) w e (h))/(h h)) > 0. 22

23 C Additional Figures (a) Labour supply (b) Unemployment Figure 5: Coecient of variation Figure 6: Job nding probability, P ( G, U, θ ), when U follows a Gamma distribution with mean and variance U and V follows a Gamma distribution with mean and variance V. This implies that the shape parameter is U for the case of U and V for the case of V and that the scale parameter is 1 in both cases. 23