Entrepreneurs, Jobs, and Trade Elias Dinopoulos University of Florida Bulent Unel Louisiana State University March 6, 2013 Abstract We propose a simple theory of personal income distribution, equilibrium unemployment, and interindustry trade, in which product markets are perfectly competitive and labor markets exhibit search related frictions. Individuals, based on their managerial talent, choose to become self-employed entrepreneurs and acquire more managerial capital, or they become workers and face the prospect of unemployment. We analyze the effects of trade on a small-open jobless economy and a two-country global economy. In the case of a small-open economy, improvements in international competitiveness raise the possibility of immiserizing recessions with higher unemployment, lower GDP, and lower aggregate welfare. Reductions in the costs of acquiring managerial capital or appropriate job-vacancy subsidies generally lead to lower unemployment rate, higher aggregate welfare, and higher income inequality. In a two-country global economy, a country exports the good with lower labor-market frictions or lower costs of managerial capital acquisition. Unilateral job-creating policies have asymmetric effects on income inequality and unemployment across countries, and ambiguous effects on welfare in each country. JEL Classification: F1, J2, J3, J6, L1 Keywords: Inequality, Managerial Capital, Search and Matching, Trade, Unemployment Elias Dinopoulos: Department of Economics, University of Florida, Gainesville, FL 32611. E-mail: elias.dinopoulos@cba.ufl.edu; Tel: (352) 392-8150. Bulent Unel: Department of Economics, Louisiana State University, Baton Rouge, LA 70803. E-mail: bunel@lsu.edu; Tel: (225)578-3790.
1 Introduction The last decade has witnessed substantial changes in the U.S. distribution of earnings. According to Haskel et al. (2012), since 2000 real earnings of most U.S. workers have declined independently of educational status, and corporate profits increased substantially reaching 12.4 percent of GDP in 2010 (the highest percentage in the pase 60 years). In addition, the share of U.S. income accounted by the top 1 percent earners rose steadily from 13.5 percent in 1995 to 16.5 in 2000, reaching 18.3 percent in 2007. During the same period the process of globalization steadily intensified caused by further improvements in communication and transportation technology leading to reductions in trade barriers and a surge in trade. The 2007 financial crises added two features to this process: persistent high unemployment in advanced countries, and significant changes in volume and terms of trade among many countries. U.S. aggregate unemployment rate has stayed above 8 percent in the past five years, and Europe has experienced double digit unemployment rates with unemployment reaching over 25 percent in countries such as Greece and Spain. In addition, Gopinath et al. (2012) documented that the global trade collapse of 2008-2009 lead to significant changes in trade prices and volume across countries. These developments have generated renewed interest among economists in the nexus among unemployment, personal income distribution, and globalization. This interest is guided by some of the following questions. Is more trade job-creating or job-destroying? What are the effects of international competitiveness captured by better terms of trade on personal income distribution, unemployment, and welfare? What is the relationship between comparative advantage and labor market frictions in a global economy with persistent unemployment? Can one identify the nature and effects of economic recovery policies? Do unilateral job-creating policies have any adverse effects on other countries? In this paper, we propose a simple theory of personal income distribution, equilibrium unemployment, and trade addressing these important questions. Labor is the only factor of production and consists of a fixed measure of individuals differing in managerial talent 1
(ability). The economy produces two goods, a traditional and a modern one, under perfect competition. The traditional sector consists of a measure of single-worker firms as in Pissarides (2004, Chapter 1), whereas the modern sector consists of multiple worker firms managed by self-employed entrepreneurs with high managerial talent as in Lucas (1978). Individuals with low managerial talent choose to become workers. An entrepreneur may enhance the productivity of her firm by acquiring managerial capital which depends positively on managerial talent. In contrast, worker productivity is independent of ability. The choice between becoming an entrepreneur or a worker depends on expected earning considerations. We assume, for tractability purposes, that entrepreneurs do not face the prospect of unemployment and their earnings equal firm profits, whereas workers have to search for jobs. Labor markets for workers exhibit search frictions leading to equilibrium unemployment, which equals the weighted sum of sector-specific unemployment rates and changes with shifts of labor between the traditional and modern sectors. 1 Having developed a tractable model, we first study a small open jobless economy trading with the rest of the world at fixed terms of trade. An increase in the relative price of modern good leads to a movement of resources from the traditional to the modern good. The effects on aggregate unemployment depend on sector specific differences in labor market distortions. Where the derived parameter measuring the job-finding rate (the probability that a worker finds a job) is higher in the modern sector than in the traditional sector, then an increase in the price of the modern good reduces aggregate unemployment. The opposite occurs if the job-finding rate in the traditional sector is higher. An increase in the the relative price of the modern sector raises the economy s aggregate expenditure, increases income inequality between employed workers and entrepreneurs, and has an ambiguous effect on aggregate welfare. Under the assumption that the distribution of entrepreneurial talent is Pareto, aggregate welfare is a U shape function of the relative price of modern good. 1 Botero et al. (2004) among others assert that labor market rigidities stemming primarily from national labor regulation policies different across countries, affect adversely labor force participation, and lead to higher unemployment especially among young workers. 2
Next we illustrate the usefulness of the small open economy framework by highlighting a few novel comparative statics properties. We employ the assumptions that the job-finding rate in the modern sector is higher than the corresponding one in the traditional sector, and that the distribution of managerial talent is Pareto. We analyze the effects of international competitiveness measured by the absolute difference between the economy s terms of trade (relative price of its exportable) and the autarky price. Within the context of our static model, greater international competitiveness can occur in two ways: through an exogenous improvement in the terms of trade caused by lower trade costs or the global financial crises; or product market policies that change the domestic price such as tariffs and/or export subsidies. Under parameter assumptions implying lower labor-market distortions in the modern sector, the effects of international competitiveness depend on the initial trade pattern. Where the economy starts with exporting the traditional good and exhibiting low trade volume, improved international competitiveness may lead to an immiserizing recession with lower GDP, higher unemployment, and lower welfare. Where the economy exports the modern good, improved international competitiveness caused by better terms of trade reduces unemployment, increases welfare, and leads to higher GDP. We then discuss the effects of two job-creating labor-market policies within the context of a small open jobless economy. A policy leading to lower costs of managerial capital or one generating lower costs of posting job-vacancies in the modern sector have identical effects. They lead to lower unemployment, higher GDP, and higher welfare. As such, they are preferable to product market policies that might lead to a recession. These results are consistent with the generalized theory of distortions and welfare. Finally, we analyze the relationship between labor-market rigidities and comparative advantage by employing a two-country framework. Comparative advantage depends on a derived parameter based on the following factors: the cost of acquiring managerial capital and market tightness of modern sector relative to that in the traditional sector. A country is more likely to export the modern good if it has lower costs of managerial capital and 3
a relatively lower labor-market frictions in modern sector. A move from autarky to free trade has asymmetric effects on income inequality (between entrepreneurs and workers) and the rate of unemployment across two countries. Global free trade, however, has ambiguous national welfare effects. Furthermore, our analysis show that a unilateral job-creating policy also has asymmetric effects on income inequality and the rate of unemployment across countries, while having ambiguous effect on welfare in each country. This paper is related to a growing literature that investigates the nexus between trade and unemployment. Researchers in this literature have identified different sources of labor market frictions that lead to equilibrium unemployment. In Brecher s (1974) model, for example, unemployment stems from minimum wages, whereas in Copeland s (1989) and Davis and Harrigan s (2011) models efficiency wages play key role in generating equilibrium unemployment. Kreickemeier and Nelson (2006), Amiti and Davis (2011), and Egger and Kreickemeier (2009) use models of fair wages to generate unemployment. Our model is more closely related to the trade literature that uses the standard Diamond Mortensen Pissarides search and matching frictions in the labor markets as a source of unemployment. 2 In an important contribution, Davidson et al. (1999) develop a general equilibrium search model of trade between two countries to investigate the extent to which results obtained in traditional, full-employment trade models must be modified. 3 In their model, both sectors share the same production technology which is also common across countries. However, job turnover (break-up) rates vary across sectors and countries, and these differences determine the pattern of trade. Although there are workers and entrepreneurs in their model, their supplies are exogenously given. In addition, their model assumes away managerial capital acquisition, and thus trade does not affect firm 2 Helpman et al. (2011) provides a comprehensive review of this literature by emphasizing the role of firm heterogeneity. 3 Davidson et al. (1999) build on Davidson et al. (1998) that study two-sector, closed-economy, general equilibrium models with frictional unemployment. Davidson and Matusz (2009) provides several extensions of the Davidson et al. (1999) model to study how international trade affects labor markets and how differences in labor markets affect international trade. 4
productivity. 4 Helpman and Itskhoki (2010) introduce search and matching frictions leading to equilibrium unemployment in a trade model with product differentiation, monopolistic competition, and scale economies. 5 Helpman et al. (2010) extends Helpman and Itskhoki (2010) by introducing match-specific heterogeneity in ability across workers to study the impact of trade on wage distribution. Imperfectly competitive product markets complicate their analysis, and necessitate the use of a two-country framework because firms in these models are not price takers. In addition, they do not consider the effects of self-employed entrepreneurs by assuming that the supply of workers is exogenous. Our theory makes possible the study labor-market frictions and trade without additional complications and distortions stemming from product market imperfections, and also allows us to analyze the effects of international trade and policies in the context of a small open jobless economy. Moreover, our theory complements Helpman et al. (2010) by highlighting the nature of income inequality fueled by firm profits stemming from heterogeneous entrepreneurial talent (as opposed to income inequality based on differences in wages generated by heterogeneous worker ability). The rest of the paper is organized as follows. Section 2 describes the model. Section 3 introduces international trade considerations by analyzing the properties of a small open jobless economy and a two-country global economy. Section 4 concludes. 2 The Model Consider an economy producing two homogeneous goods indexed by i = 1, 2. The economy is populated by a continuum of identical families with measure equal to one. Each family has a supply of labor consisting of all family members. Without loss of generality, we assume that the size (mass) of each family is equal to one as well. Family members possess 4 Dutt et al. (2009) develop a model of trade with unemployment to study the impact of trade on unemployment. Although search and matching frictions lead to unemployment in their model, trade results from either Heckscher-Ohlin or Ricardian comparative advantage. 5 See also Felbermayr et al. (2011) who use a similar set-up to study the relationship between trade and unemployment in a context of symmetric countries. 5
different ability levels indexed by a. In the present context, ability is a broad generic term measuring managerial talent, propensity to learn, level of training or education and any other feature helping an individual to acquire managerial (entrepreneurial) capital. The distribution of abilities within each family is given by an exogenous cumulative distribution G(a) with density g(a) and support [1, ). Decisions are made sequentially. An individual first decides whether to become an entrepreneur or a worker. An entrepreneur with a given entrepreneurial talent chooses the optimal level of managerial capital, followed by how many workers to employ. The latter choice is conditioned by the existence of hiring costs stemming from search and frictions in the labor market. The entrepreneur engages in bargaining with the firm s workers to determine the negotiated wage while treating the level of managerial capital and hiring costs as sunk. Entrepreneurial income equals firm profits. Each worker decides first in which sector to search for a job based on wage and unemployment considerations. Once committed to a sector, a worker cannot switch sectors. Finally, we assume that each family engages in appropriate income transfers among its members equalizing their income and utility levels independently of their employment statues and ability. Accordingly, aggregate welfare equals the sum of individual utilities across all family members. 2.1 Preferences Family members have identical preferences over the two homogeneous goods as described by the following utility function u = ( ) 1 β ( ) β q1 q2, (1) 1 β β where q i is the consumption of good i = 1, 2, and 0 < β < 1 is an exogenous, constant parameter. Denoting with e income of an individual, one can write the demand for good i as q i = β i e/p i, (2) 6
where β 1 = 1 β, β 2 = β, and p i is the price of good i. We choose good 1 as the numeraire by setting its price equal to one (i.e., p 1 = 1). To further simplify notation, we rewrite the relative price of good 2 as p. Substituting q i from (2) into (1) yields the following indirect utility function v(e, p) = ep β, (3) which increases with income e and decreases with the relative price of good 2, p. The linear dependence of indirect utility on income indicates that individuals are risk neutral. 6 2.2 Production and Job Creation We refer to good 1 as the traditional good and to good 2 as the modern good. Production of traditional good 1 is carried out by small, identical, and single-worker firms. The concept of single-worker firms has been used extensively in the Diamond-Mortensen-Pissarides (DMP) literature of equilibrium search unemployment (see, e.g., Chapter 1 in Pissarides, 2004). Each firm is free to post a job vacancy and hire one worker. Vacancies are not instantaneously filled because of matching and search frictions. When a firm hires a worker, the worker produces one unit of good 1 independently of ability. Upon matching, the firm and worker bargain over revenue earned by selling one unit of output at p 1 = 1. Assuming equal bargaining power and zero value of outside options (e.g., no unemployment insurance), each worker receives w 1 = 1/2. Good 2 in the modern sector is produced by a continuum of heterogeneous firms, each owned and managed by an entrepreneur. The production technology of sector 2 incorporates two features: heterogeneous managerial skill (or organization capital) denoted by z; and diminishing returns to scale with respect to labor capturing span of control stemming from a concave production function, as in Lucas (1978). The production function of a firm 6 Risk aversion considerations can be readily introduced into the analysis by assuming that utility takes the form u = (q 1 /β 1 ) β 1 (q 2 /β 2 ) β 2, with β 1 + β 2 < 1 such that the indirect utility is concave in income. However, we confine ourselves to the risk-neutral case to simplify the subsequent exposition. 7
managed by an entrepreneur with organization capital z is given by y 2 (z) = ( ) z 1 η ( ) l η, (4) 1 η η where l is the number of workers employed and η (0, 1) is an exogenous parameter. According to (4), organizational capital z also represents firm productivity and exhibits diminishing returns for any given level of labor l. Thus, we hereafter use the terms organization capital and firm productivity interchangeably. The above formulation delivers a market structure with heterogeneous firms producing a homogeneous product under perfect competition. Heterogeneous firm productivity stems from endogenous formation of organizational capital which, in turn, is based on exogenous distribution of managerial talent. As a result, this paper complements seminal studies of firm heterogeneity, such as Hopenhayn (1992) and Melitz (2003), which postulate an exogenous distribution of firm productivity and internal scale economies. Following the insights of human capital theory, we assume that individuals choose to become entrepreneurs or workers depending on earning considerations and costs of acquiring managerial capital. In particular, we postulate that an individual with ability (managerial talent) a faces λz 2 /2a costs of acquiring z units of organizational capital, where λ > 0 is an exogenous parameter. The costs of acquiring managerial capital are measured in units of good 1, decline with the level of ability, and increase with the level of managerial capital. The proposed cost function captures, albeit in a reduced form, the idea that human capital formation is costly involving various inputs, experience, schooling, on-the-job training, etc. These dynamic elements are not explicitly modeled but captured by shift parameter λ. This quadratic cost function delivers linear marginal costs and leads to an interior, closedfrom solution. Higher values of λ are associated with higher total and marginal costs of managerial-capital acquisition. In what follows, we will refer to an increase in λ as a rise in costs of acquiring managerial capital. 7 7 The production-function based approach to human capital formation has long tradition going back to seminal studies by Becker (1994) and Ben-Porath (1967) among many others. 8
An entrepreneur with managerial talent a maximizes his earnings e 2 (a), equal to firm profits, by choosing the level of managerial capital z and the number of employees l. As in the traditional sector 1, hiring in the modern sector is costly due to search and matching frictions. A firm with productivity z may hire l workers instantaneously by incurring hiring cost c 2 l, measured in terms of good 1. In addition, the firm incurs a wage bill denoted by w 2 l, where w 2 is the negotiated wage between the entrepreneur and employees as will be shown below. As a result, an entrepreneur incurs managerial capital costs and two types of labor costs: hiring costs and the wage bill. As will be discussed in the next section, hiring-cost parameter c 2 depends on sector-specific labor market conditions, and therefore is common across all firms producing good 2. This discussion leads to the following expression for entrepreneurial income where y 2 (z) is given by (4). e 2 (a) max } {py 2 (z) w 2 l c 2 l λz2, (5) 2a To obtain a solution for e 2 (a), we first determine the negotiated wage rate w 2 and labor input l recognizing that bargaining occurs after an entrepreneur completes the process of hiring and acquiring managerial capital. Upon a match an employee cannot be replaced without cost, and thus a hired worker is not interchangeable with an outside worker. Consequently, hired workers have bargaining power. The wage bargaining process is modeled as in Stole and Zwiebel (1996a): the firm engages in bilateral bargaining with each worker and internalizes the effect of a worker s departure on wages of remaining workers. In our model all workers are identical in regards to output productivity, since the latter is independent of ability by assumption. 8 As a result, a firm treats each worker as marginal and firm surplus from a worker departure equals the marginal change in firm value (profits) with respect to labor. We assume that the value of outside options for each worker is zero (i.e., workers 8 Helpman et al. (2010) develop a model of unemployment and inequality where worker productivity depends on ability and firms engage in ex-ante screening for worker ability. Screening leads to more productive firms offering higher wages. Our model complements their analysis by focusing on endogenous firm productivity (based on managerial capital acquisition) and equally productive workers. 9
do not receive unemployment benefits); and therefore worker surplus equals the negotiated wage. Assuming equal bargaining power, the Stole and Zwiebel (1996a) solution requires that total surplus be equally split between the entrepreneur and a worker according to l [py 2(z) w 2 l] = w 2, where w 2 is the negotiated wage rate and p is the relative price of good 2. 9 Because hiring costs c 2 l and managerial-capital acquisition costs λz 2 /2a are sunk (i.e., they are paid before the bargaining process), these terms do not appear in the right hand side. The solution to this ordinary differential equation provides the following expression for the negotiated wage w 2 = p [ ] ηz 1 η. (6) 1 + η (1 η)l According to (6), the negotiated wage increases with price p and firm productivity z, and decreases with firm size measured by the number of employees l. 10 Faced with this wage rate, an entrepreneur chooses the number of employees l by taking into account hiring costs to maximize (5). This maximization yields w 2 = p [ ] ηz 1 η = c 2. (7) 1 + η (1 η)l Thus, all entrepreneurs (irrespective of managerial capital and firm size) pay the same wage to workers by setting it equal to hiring cost per worker, that is w 2 = c 2. The above equation can be solved for the number of workers hired by a firm with productivity z l = η [ ] 1 p 1 η z. (8) 1 η (1 + η)c 2 Equation (8) is the firm demand for labor in the presence of labor market frictions. This demand for hired labor (l) declines with the level of hiring costs per worker (c 2 ), and increases with the relative price of good 2 (p) and firm productivity (z). Thus, firms with higher productivity and lower hiring costs per employee employ more workers. 9 Incorporating asymmetric bargaining power between an entrepreneur and workers as in Stole and Zwiebel (1996b) does not alter the main results. 10 It can easily be verified that (6) is the solution to the differential equation by substituting (6) in the former and performing the calculations. 10
2.3 Talent Allocation An entrepreneur with managerial talent a maximizes earnings (economic profits) by choosing the level of managerial capital z after taking into account labor costs. Substituting w 2 = c 2 from (7) and (8) into equation (5), and maximizing the resulting function with respect to z yields z = a λ [ p (1 + η)c η 2 ] 1 1 η. (9) Equation (9) indicates that the optimal level of acquired organizational capital increases with managerial talent a and price p; and decreases with hiring cost per worker c 2, and the cost of acquiring managerial capital λ. We are now in the position to determine how the distribution of managerial talent determines the choice between becoming an entrepreneur or a worker. This choice is affected by expected income considerations. Substituting w 2 = c 2 from (7), l from (8), and z from (9) into equation (5) provides a closed-form expression for maximum income e 2 (a) earned by an active entrepreneur e 2 (a) = a [ 2λ p (1 + η)c η 2 ] 2 1 η. (10) According to (10), the return on entrepreneurship e 2 (a) is an increasing linear function of managerial talent a and behaves similarly to the optimal level of organizational capital z: it increases with relative price p, and declines with the costs of managerial capital acquisition λ and hiring c 2. Active entrepreneurs do not face the prospect of unemployment and receive e 2 (a) with certainty. Observe also that indirect utility (3) is linear in income implying that individuals are risk neutral. The next section will formally establish that the ex-ante income of each worker is equalized across the two sectors, that is all workers receive the same expected wage. Worker income is independent of managerial talent a by assumption, and entrepreneurial income increases linearly with the level of managerial talent, as indicated by (10). As a result, there exists a unique cutoff level of managerial talent a > 0 such that all individuals 11
with talent a < a choose to become workers, whereas all individuals with talent a a choose to become self-employed entrepreneurs. Deriving an explicit expression for the cutoff level of managerial talent requires determination of expected worker income in one of the two sectors. It is simpler and more convenient to focus on the traditional sector, where worker income is e 1 = w 1 = 1/2 and, of course, does not depend on the level of managerial talent. Let ζ 1 denote the probability of a worker to find a job in sector 1 (to be determined in the next section). Then expected income of a worker then is simply E[e 1 ] = ζ 1 w 1 = ζ 1 /2. An individual chooses to become an entrepreneur if and only if e 2 (a) E[e 1 ]. The cutoff level of managerial talent a, at which an individual is indifferent between being an entrepreneur and a worker, is given by e 2 (a ) = ζ 1 /2, and using (10) yields [ (1 + η)c η ] 2 a 1 η 2 = λζ 1. (11) p As in Lucas (1978), only the most talented individuals become entrepreneurs. However, unlike Lucas (1978), here entrepreneurs acquire managerial capital and workers face the threat of unemployment. The cutoff level of managerial talent a is endogenous and increases with per-worker hiring cost c 2, the probability of finding a job as a worker ζ 1 (capturing part of the opportunity costs of entrepreneurship), and the cost of managerial capital acquisition λ. The cutoff level of managerial talent decreases with relative price p as expected. Finally, combining equations (9) and (11) yields z = ( ) 1 ζ1 2 a, (12) λa which states that the equilibrium level of managerial capital increases with the probability of finding a job in the traditional sector ζ 1, and decreases with the talent cutoff level a and the cost of acquiring managerial capital λ. 12
2.4 Taking Stock For a given level of managerial capital and hiring costs, the firm equilibrium is identical to the textbook short-run equilibrium of a perfectly competitive firm producing a homogeneous good under a U shaped average cost curve and increasing marginal costs of production. Equation (5) can be employed to express total costs as a function of output. Solving for the number of employees from production function (4) and substituting the resulting expression into C(l, z) = (w 2 + c 2 )l + λz 2 /2a yields C(y 2, z) = Φ(z)y 1/η 2 + λz 2 /2a, where Φ(z) = (w 2 + c 2 )η[(1 η)/z] (1 η)/η. The first component of C(y 2, z) is an increasing and convex function of output and corresponds to variable production costs; the second component corresponds to fixed costs. Accordingly, for a given level of managerial capital z and per-unit labor costs w 2 + c 2, each firm faces a U shaped average cost curve and increasing marginal costs. As a result, each entrepreneur chooses the output level that maximizes e 2 (y 2, z) = py 2 C(y 2, z) by setting price equal to marginal costs p = [Φ(z)y 2 /η] (1 η)η = (w 2 + c 2 )[(1 η)y 2 /z] (1 η)/η as in Lucas (1978). The present model differs from the standard textbook model of output choice by a competitive firm in several important aspects. First, each entrepreneur differs in managerial talent leading to different levels of managerial capital, firm productivity, and firm size. Second, each firm engages in bargaining with its employees and uses its monopsony power to drive the negotiated wage down to the level of hiring costs per worker, that is w 2 = c 2. Third, economic profits are not driven down to zero: a fraction of economic profits covers short-run fixed costs stemming from acquisition of managerial capital; and the remaining fraction of profits covers the return to entrepreneurial talent which serves as a fixed primary factor of production. In the long-run, the return to entrepreneurial talent determines the allocation of resources among individuals who decide to become self-employed entrepreneurs and individuals who decide to become workers according to (11). Thus entry costs into the modern sector equal expected worker income. Fourth, hiring costs and the negotiated wage are endogenously determined through interactions between labor market frictions and 13
general equilibrium forces, as the next subsection demonstrates. The present model also differs from models of monopolistic competition with heterogeneous firms such as those descending from the seminal work of Melitz (2003). In Melitz-type models each firm produces a horizontally differentiated product under internal increasing returns to scale, and draws its fixed productivity from a random distribution after incurring fixed entry costs. In the present model, each good is homogeneous and produced under diminishing returns and increasing marginal production costs. Firm productivity is endogenous and reflects the underlying distribution of managerial talent. Entry into the modern sector does not depend on fixed costs, as in Melitz, and reflects the opportunity cost of entrepreneurship; and aggregate profits are distributed among active self-employed entrepreneurs. In summary, the present model complements existing models of firm heterogeneity that emphasize the role of R&D and productivity uncertainty and abstract from the determinants of personal income distribution. In contrast, our model follows the tradition of human-capital formation and talent allocation as sources of firm heterogeneity and personal income distribution. One notable advantage of our approach, as we will later discuss, is the analysis of trade and personal income distribution within the context of a traditional small open economy facing fixed terms of trade and perfectly competitive product markets (as opposed to symmetric countries with monopolistically competitive product markets). Another notable feature is incorporating the option of becoming a self-employed entrepreneur which has profound implications for the behavior of aggregate equilibrium unemployment. 2.5 Equilibrium Unemployment Workers are risk neutral and decide whether to search for a job in the traditional or modern sectors. The labor market exhibit search frictions as in the standard DMP theory of unemployment. As we indicated above, we assume that once committed to a sector a worker cannot move to another sector, i.e. there is no ex-post inter-sectoral worker mobility. 14
Let N i and V i denote the number of unemployed workers searching for jobs and total number of posted job vacancies in sector i = 1, 2. We assume that the number of successful matches (e.g., hired workers) in sector i is given by Cobb-Douglas matching function M i = m i V ξ i N 1 ξ i, where parameter m i measures matching efficiency, and ξ (0, 1) is a constant parameter. Assuming that M i < N i, the job finding rate ζ i M i /N i, interpreted as the probability that an unemployed worker matches with a firm is given by ζ i = m i (V i /N i ) ξ < 1, where V i /N i is the number or vacancies per unemployed worker. The term ζ i is an index of labor market tightness and captures the notion that a higher job-finding rate implies a greater number of vacant jobs per unemployed worker, that is more firms are posting vacancies and therefore the demand for labor increases leading to a tighter labor market. In what follows we use the terms job-finding rate and labor market tightness interchangeably. Following Blanchard and Gali (2010), we assume that a firm can hire workers instantaneously by incurring a cost of hiring a worker expressed in units of the traditional good. Hiring cost per worker in sector i is given by c i (ζ i ) = τ i ζ γ i, (13) where the exogenous parameter τ i is an index of labor market frictions; ζ i is the job-finding rate or the labor market tightness index; and γ > 0 is a constant parameter. According to equation (13), higher market tightness ζ i implies greater instantaneous hiring costs per worker c i, for any given degree of labor market frictions τ i. As shown by Blanchard and Gali (2010, footnote 6), the above hiring cost function can be derived from a standard constant returns to scale Cobb-Douglas matching function. Accordingly, if the matching function is given by M i = m i V ξ i N 1 ξ i, then the two parameters in (13) are given by γ = (1 ξ)/ξ > 0 and τ i = m 1/ξ i. As a result, τ i declines with matching efficiency m i, and therefore corresponds to an index of labor market frictions (rigidities) in sector i. Consider now sector 1 where each firm employs one worker. Entry to the market is unrestricted, but each firm faces entry costs equal to the cost of posting a vacancy (denoted by ν 1 and measured in terms of good 1). It then follows that χ 1 = ν 1 /c 1 represents the 15
probability that a firm fills a vacancy, that is χ 1 is the hiring rate in the traditional sector. Because each firm receives half of generated revenue, expected profit is χ 1 /2. As a result, free-entry condition χ 1 /2 = ν 1 implies c 1 = 1/2. Substituting c 1 = 1/2 into (13) yields the equilibrium values of the job-finding rate and hiring costs ζ 1 = (2τ 1 ) 1 γ, c 1 = 1 2. (14) We assume that τ 1 < 1/2 to ensure that the job-finding rate ζ 1 is smaller than one. Note that the equilibrium probability of finding a job in this sector decreases with the degree of labor market frictions τ 1 as expected. Accordingly, greater labor-market frictions lead to lower labor market tightness in sector 1. In the absence of heterogeneous organizational capital, hiring cost per employee c 1 and labor market tightness ζ 1 are exogenously determined by labor-market (as opposed to product-market) parameters. To determine the equilibrium values of job-finding rate ζ 2 and hiring-cost per employee c 2, note that ex-ante labor mobility implies that a worker must be indifferent between assigned to the traditional or modern sectors. Because workers are risk-neutral, equilibrium in the labor market requires equalization of expected worker earnings between sectors. Expected earnings of a worker searching for a job in sector 1 is simply ζ 1 /2, and expected earnings of a worker searching for a job in sector 2 is ζ 2 w 2 = ζ 2 c 2. Using equations (13) and (14), ζ 1 /2 = ζ 2 c 2 implies ζ 2 = 2 1 γ τ 1 γ(1+γ) 1 τ 1 1+γ 2, c 2 = 1 2 ( τ2 τ 1 ) 1 1+γ. (15) To determine the unemployment rate among workers, first note that the measure of individuals who decide to enter the labor force as workers is G(a ) since the size of population is normalized to one. Observe also that ζ 2 N 2 = a lg(a)da, where l is given by (8). Using (9) together with (11) in the integration yields N 2 = 2η E[a a ] (1 η) a, (16) 16
where E[a a ] = a ag(a)da is the expected level of entrepreneurial talent in sector 2. The aggregate unemployment rate among workers is defined as the total number of unemployed workers in both sectors divided by the total number of workers searching for jobs. The number of unemployed workers in sector i is U i = (1 ζ i )N i, and the aggregate number of workers searching for a job is equal to the supply of workers G(a ) = N 1 + N 2 because entrepreneurs are fully employed. As a result, the economy-wide unemployment rate is given by U = (U 1 + U 2 )/G(a ) and can be written as U = (1 ζ 1 ) + (ζ 1 ζ 2 ) N 2 G(a ) = (1 ζ 2η 1) + (ζ 1 ζ 2 ) (1 η) E[a a ] a G(a ), (17) where parameters ζ 1 and ζ 2 are given by (14) and (15), and N 2 /G(a ) < 1 is the fraction of workers assigned to the modern sector. In Helpman and Itskhoki (2010), the supply of workers is exogenous, and as a result, aggregate unemployment is affected only through changes in N 2. In contrast, in the present model self-employed entrepreneurs influence worker supply G(a ), which is endogenous and affects the level and behavior of aggregate unemployment rate. 2.6 Aggregation Armed with these variables, we can determine total revenue in each sector [ Y 1 = ζ 1 N 1 = ζ 1 G(a ) 2η E[a a ] ] (1 η) a, (18a) py 2 = where ζ 1 is given by (14). 11 py 2 (a)g(a)da = (1 + η)ζ 1 a (1 η) Aggregate hiring cost in each sector is given by C 1h = c 1 ζ 1 N 1 = ζ [ 1 G(a ) 2η E[a a ] ] 2 (1 η) a, C 2h = a c 2 l(a)g(a)da = ηζ 1 1 η E[a a ] a, (18b) E[a a ] a. 11 Equation (7) implies that py 2 (z) = (1 + η)c 2 l/η and equations (8), (9), and (11) imply that c 2 l = ηζ 1 a/[(1 η)a ]. Thus, py 2 (a) = (1 + η)ζ 1 a/[(1 η)a ], and integrating this last expression across all entrepreneurs yields (18b). 17
Traditional good 1 is also used in formation of managerial capital. Because each entrepreneur uses λz 2 /2a units of good 1, equation (12) implies that z 2 /2a = ζ 1 a/(2a ). Integrating this expression across all entrepreneurs yields the total amount of good 1 used in the formation of organizational capital, C z, C z = ζ 1 E[a a ] 2 a. (19) Thus total amount of resources used in hiring and the formation of organizational capital (measured in units of good 1) is given by C = C 1h + C 2h + C z = ζ 1 2 [ G(a ) + E[a ] a ] a. (20) Equations (18a), (18b), and (20) imply that the value of total output, measured in units of good 1, used in hiring and in the formation of organizational capital equals half of the economy s GDP, that is C = (Y 1 + py 2 )/2. 2.7 Income Distribution and Welfare Equations (10) and (11) imply that the income of an entrepreneur with ability a is given by e 2 (a) = ζ 1 a/(2a ). Worker income in sector 1 is w 1 = 1/2 and worker income in sector 2 is w 2 = c 2. As a result, ex-post personal income distribution is given by e(a) = 0 if a < a and worker is unemployed 1/2 if a < a and worker is employed in sector 1 c 2 if a < a and worker is employed in sector 2 ζ 1 a/(2a ) if a a and individual is self-employed entrepreneur (21) where ζ 1 and c 2 are given by (14) and (15), respectively. Because ζ i is the probability of finding a job in sector i, it follows from the law of large numbers that ζ i N i is the number of workers and w i ζ i N i is the total wage paid the workers in sector i. Using the above income distribution, aggregate income is E = ζ 1 2 [ G(a ) + E[a ] a ] a. (22) 18
Comparing (22) to (20) reveals that E = Y 1 +py 2 C and E = C = (Y 1 +py 2 )/2. In words, aggregate earnings equal aggregate revenue minus the value of good 1 covering organization capital and hiring costs; it also reveals that half of the economy s GDP covers hiring and organization costs and half of GDP covers earnings of workers and entrepreneurs. Using (3) and (21), one can readily determine the welfare of each individual. However, to address the impact of trade on welfare, we need to aggregate the individuals welfare. The assumption of identical families allows us to treat aggregate welfare as each family s welfare which equals the sum of the individual utilities. Integrating the resulting indirect utility functions across all individuals within a family yields V = Ep β, (23) where E is the aggregate expenditure given by (22). 2.8 Closed-Economy Equilibrium In the closed-economy equilibrium, Q 1 = Y 1 C and Q 2 = Y 2, where Q i is aggregate quantity consumed of good i. Substituting these into Q 1 /pq 2 = (1 β)/β from equation (2) implies E[a a ] a = BG(a ), B β(1 η) 2(1 β)(1 + η) + β(3η + 1). (24) The left hand side (LHS) is a decreasing function of managerial ability cutoff a, 12 whereas the right hand side (RHS) is an increasing function of a. As a result, equation (24) determines the unique closed-economy equilibrium cutoff level of managerial talent a. Because 0 < B < 1, the equilibrium cutoff level ensures that both goods are produced, i.e. N i > 0 for i = 1, 2. 13 Observe that the equilibrium cutoff level of managerial talent a does not 12 Using Leibniz rule, de[a a ]/da = d [ a ag(a)da ] /da = a g(a ) < 0. 13 Equation (24) can be written as N 2 /G(a ) = 2ηE[a a ]/[(1 η)a G(a )] = 2ηB/(1 η) < 1 implying that the equilibrium share of workers assigned in sector 2 is strictly positive and strictly less than one. This result combined with the supply of workers G(a ) = N 1 + N 2 implies N i > 0, i.e. the closed-economy equilibrium is characterized by incomplete specialization of production. 19
depend on parameters capturing labor market rigidities such as hiring costs per employee, costs of posting and maintaining job vacancies, and costs of acquiring organizational capital. This property facilitates the analysis of comparative advantage as will become obvious in the next section. Lemma 1. The closed-economy equilibrium cutoff level of managerial talent a satisfying equilibrium condition (24) exists and is unique. Once the closed-economy cutoff level of managerial talent is determined one can easily determine other endogenous variables in the model. First, substituting the cutoff talent level a into (21) and (22) determines the distribution of personal income and aggregate income, respectively. Second, substituting a into (12) yields the equilibrium cutoff level of managerial capital ζ = (a ζ 1 /λ) 1/2, where ζ 1 is exogenous and given by (14). Third, equations in (15) determine the job-finding rate ζ 2 and hiring cost per worker c 2, respectively. Fourth, substituting c 2 into (11) yields the relative price of good 2, p; and substituting p into (23) determines aggregate welfare. Using equilibrium condition (24) in (17) provides the following expression for aggregate unemployment rate U = (1 ζ 1 ) + (ζ 1 ζ 2 ) 2ηB 1 η, where parameter B is defined in (24) and expression 2ηB/(1 η) is the equilibrium share of workers assigned to sector 2. Finally, note that the autarkic relative price of modern good p A can be written as a function of a from (11) p A = (1 + η)c η 2 [ ] 1 η λζ1 2 a, (25) where ζ 1 and c 2 are given by (14) and (15). The autarkic price depends on virtually all parameters: it increases with labor-market tightness in the traditional sector and with the cost of managerial capital acquisition; and declines with parameters increasing the 20
cutoff level of managerial talent a (which is determined by (24)). This completes the characterization of closed-economy equilibrium. 3 Open Economy In this section, we extend our framework to analyze the nexus of trade, unemployment, inequality, and welfare. We discuss two cases: first, we employ the traditional framework of a small, open, jobless economy taking its terms of trade (relative price of good 2) as given; second, we employ a two-country framework to analyze the role of labor market rigidities and costs of managerial capital acquisition as sources of comparative advantage. 3.1 A Small Open Jobless Economy Consider now a small open economy trading with the rest of the word at a fixed international relative price of good 2 denoted by p T. Following the standard practice of trade theory, we assume balanced trade: Y 1 Q 1 C + p T (Y 2 Q 2 ) = 0, where Y i is quantity supplied and Q i is quantity demanded for good i. The balanced trade condition can be written in an equivalent form as E = Y 1 + p T Y 2 C = Q 1 + p T Q 2, where aggregate income E is given by (22). In other words, the value of aggregate income (expenditure) equals the value of production net of hiring and organization capital costs (treated as intermediate inputs). Substituting the international price p T into equation (11) provides the equilibrium level of managerial talent [ (1 + η)c η ] 2 a 1 η 2 T = λζ 1. (26) p T where ζ 1 and c 2 are given by (14) and (15). Once a T is determined, we can readily solve for all endogenous variables as in the closed-economy equilibrium. 21
If the economy produces both goods, then there must be an upper bound, denoted by p max T, on the relative price of good 2. 14 The trade pattern depends on several factors that determine the ranking between the closed-economy equilibrium price p A, which is given by (25), and the (exogenous) international price p T : where p A = p T, there is no trade; where 0 < p T < p A, the economy exports the traditional good; and where p A < p T < p max T, the economy exports the modern good. Proposition 1. Consider a small open economy producing both goods and trading with the rest of world at a fixed international price of good 2 denoted by p T. Let p A denote the autarky price determined by (24) and (25). The economy does not engage in trade if p A = p T ; exports the traditional good 1 iff 0 < p T < p A ; and exports the modern good 2 iff p A < p T < p max T. As the international price p T increases, the economy switches from importing the modern good to exporting it. This pattern of trade reflects general-equilibrium forces and the law of comparative advantage. Where the economy exports the modern good, firm heterogeneity is irrelevant to whether small or large firms export. This feature stems from the assumptions of perfect competition and the absence of fixed foreign-market access costs. 15 We continue the analysis of a small-open economy by investigating the effects of trade on unemployment, income distribution, and welfare. Without loss of generality we consider the case where the economy is incompletely specialized and under free trade exports the 14 Note that equation (26) states that a T is monotonically decreasing in p T from infinity to zero. As the relative price of good 2 increases and the cutoff level of managerial talent declines, the modern sector expands because more individuals want to become self-employed entrepreneurs and more workers would like to find employment in the modern sector. This process is reflected in the corresponding increase in the fraction of workers assigned in the modern sector N 2 /G(a T ) 1. Note also that N 2 /G(a T ) = 2ηE[a a T ]/[(1 η)a T G(a T )] declines monotonically in a T, and approaches unity for a sufficiently small value a T = a min. For example, if the distribution of managerial talent is Pareto G(a) = 1 a k, where k > 1 is the shape parameter, then the share of workers in sector 2 is given by N 2/G(a ) = 2ηk/[(1 η)(k 1)(a k 1)]. As a result, a min = [1 + 2ηk/(1 η)(k 1)] 1/k > 1. Replacing a T in (26) with a min yields p max T. 15 The introduction of fixed costs would presumably resolve this indeterminacy in the usual fashion with larger firms exporting and smaller firms serving the domestic market. Incorporating fixed costs will derail the focus of the present paper, increase the algebraic complexity and cloud the intuition of the main results. We differ the analysis of firm heterogeneity and export status to future research. 22
modern good 2, that is p A < p T < p max T. The rate of aggregate unemployment is given by 2η U = (1 ζ 1 ) + (ζ 1 ζ 2 ) (1 η) E[a a T ] a T G(a T ), (27) where cutoff level a T is determined by (26). Because the share of workers allocated to the modern sector declines (increases) monotonically with a T (p T ), the effect of trade on unemployment depends on the ranking of sector-specific job-finding rates ζ 1 and ζ 2, which are determined by (14) and ( 15). It is obvious from (27) that the unemployment rate is independent of the international price if ζ 1 = ζ 2 = ζ. In this special case (which holds when τ 1 = τ 2, i.e. both sectors have identical matching efficiency), the rate of unemployment is U = (1 ζ). Where the job-finding rate in the modern sector exceeds the one in the traditional sector (i.e., ζ 1 ζ 2 < 0), then more exposure to trade reduces aggregate unemployment. Where the job-finding rate in the traditional sector exceeds the job-finding rate of the modern sector (i.e., ζ 1 ζ 2 > 0), more trade exposure increases aggregate unemployment. Proposition 2. Consider a small open economy producing both goods and exporting modern good 2 under free trade (p A < p T < p max T ). A move from autarky to free trade has no impact on the unemployment if sector-specific job-finding rates are equal (i.e., ζ 1 = ζ 2 = ζ). More trade reduces (increases) the unemployment rate if the job-finding rate in the modern sector is higher (lower) than the job-finding rate in the traditional sector, i.e. ζ 2 > ζ 1 (ζ 2 < ζ 1 ). Since an increase in the relative price of good 2 from p A to p T lowers the managerial talent cutoff from a to a T, equation (21) implies that such an increase only raises entrepreneurs income. This leads to rise in income inequality between entrepreneurs and employed workers. In addition, equation (22) indicates that aggregate expenditure decreases with a T. Accordingly more trade increases aggregate expenditure E. 23