Melitz meets Pissarides: Firm heterogeneity, search unemployment and trade liberalization

Transcription

1 Melitz meets Pissarides: Firm heterogeneity, search unemployment and trade liberalization Gabriel Felbermayr, Julien Prat, and Hans-Jörg Schmerer PRELIMINARY VERSION April 2007 Abstract Motivated by strong evidence on labor market turnover and on the selection effect of trade liberalization, we introduce search unemployment à la Pissarides (2000) into the Melitz (2003) trade model of heterogeneous firms. The equilibrium is highly recursive and allows general analytical solutions. We discuss two trade liberalization scenarios: a reduction of variable trade costs, and the entry of new trading countries. Trade liberalization weeds out inefficient firms while offering new opportunities to more productive firms. Consequently, real search costs are lower relative to productivity for the average firm, inducing it to create more jobs. Hence, In both scenarios, the long-run equilibrium rate of unemployment falls. Additionally, we study the effects of increasing fixed costs for export. Rising fixed costs also force less productive firms out of the market and therefore decrease unemployment on the long run. Calibrating the model towards the U.S. economy, we find that trade liberalization leads to modest but robust gains in unemployment. JEL Classification: F12, F15, F16 Keywords: Trade liberalization, unemployment, search model, firm heterogeneity. We are very grateful Hartmut Egger, Marc Melitz, and Gianmarco Ottaviano as well as seminar and conference participants at IZA Bonn, Konstanz, Vienna, and Tübingen for comments and discussion. Part of this paper was drafted when Felbermayr was visiting the university of Zürich Corresponding author. University of Tübingen, Economics Department, Nauklerstrasse 47, Tübingen, Germany. gabriel.felbermayr@uni-tuebingen.de. University of Vienna and IZA, Bonn. University of Tübingen. 1

2 1 Introduction Does globalization lead to unemployment? While of enormous interest to policy-makers, the academic economist s answer to this question is bound to be ambiguous. The theory of the second best teaches that dismantling barriers to trade in an environment featuring additional pre-existing distortions in the labor market that give rise to unemployment can either exacerbate or attenuate the welfare damage of those distortions (Lipsey and Lancaster, 1956). Hence, the answer to the above question will almost necessarily depend on modeling details. 1 It therefore comes with some relief that merging the leading macroeconomic explanation of unemployment the Pissarides (2000) search and matching model with the most recent version of the Krugman (1980) trade model the Melitz (2003) model with heterogeneous firms yields an unambiguous conclusion: globalization modeled as various types of trade liberalization lowers the equilibrium long-run rate of unemployment. The Melitz (2003) trade model is a natural starting point to study unemployment in an open economy. It combines increasing returns to scale in production and product differentiation with heterogeneous firms. It has been successfully applied to a number of questions and has done very well in empirical studies (see the survey of Helpman, 2006). In that model, trade liberalization puts inefficient firms out of business while offering new opportunities to efficient ones. That selection effect leads to an aggregate productivity gain. There is ample evidence for this effect, both from microeconometric and from aggregate data (Bernard et al., 2003) Similarly, the search and matching model of Mortensen and Pissarides (1994) has been successfully applied to a large number of environments, both in theoretical and empirical studies. Its main advantage over other models of unemployment is that it is based on a robust, empirically verifiable presumption, namely the existence of labor market turnover. Any conceivable labor market is likely to feature turnover, regardless of a country s institutional details or workers and trade unions preferences. 2 If job creation is not instantaneous due to search frictions, turnover yields equilibrium unemployment. Davidson and Matusz (2004) argue that the minimalist perspective on labor markets embodied in search-matching models is particularly useful in general equilibrium trade models. Bringing the two models together is complicated by the fact that the basic search-matching 1 Moreover, as Davidson and Matusz (2006) forcefully argue, short- and run-long effects of trade liberalization on unemployment may have opposite signs. 2 The empirical relevance of labor market turnover has been widely documented over the past decade (Davis et al., 1996). 2

3 model relies on perfect competition and constant returns to scale. This assumption is not satisfied in Krugman (1980) type trade models. Bertola (1994) and Stole and Zwiebel (1996a,b) show that individual wage bargaining in multiple worker firms with market power leads to an over-hiring effect as employers exploit local monopoly power. We follow Felbermayr and Prat (2007) and Ebell and Haefke (2006) to solve this issue. Importantly, firms with different productivities turn out to pay the same wage rate. This is an important feature, since it implies that the general equilibrium of the merged Melitz-Pissarides model is recursive. Product market equilibrium (average productivity) is essentially independent from labor market outcomes (labor market tightness), but the inverse is not true. Trade liberalization leads to a long-run improvement in average productivity, which in turn reduces the average firm s real cost of posting vacancies relative to its marginal revenue. The new steady state reached after trade liberalization displays a lower equilibrium rate of unemployment. Hence, trade liberalization offers an additional channel for welfare gains by reducing the severity of search frictions. 3 This result stands in contrast to Egger and Kreickemeier (2006), who integrate the fair wage hypothesis into the Melitz model and find that globalization increases unemployment. This result, however, hinges on an ad-hoc wage formulation of a reference wage. 4 Davidson et al. (1988) is the first paper to integrate search unemployment into a trade model. In a stream of subsequent papers (e.g., Davidson et al., 1999), the authors have explored different product market configurations. 5 However, they generally keep the twin assumptions of perfect competition and constant returns to scale and their focus is on the effect of search frictions on patterns of comparative advantage. The paper that is closest to our work is Davidson et al. (2007), which analyzes search unemployment in the heterogeneous firms environment proposed by Albrecht and Vroman (2002). That paper studies how the existence of search frictions affects the endogenous determination of firm productivities. In our paper, the direction of causality is exactly the opposite, as we stick with the Melitz (2003) assumption of an exogenous sampling distribution (though, of course, the ex post distribution of productivities is endogenous). 3 Note, however, that welfare statements are difficult in the search-matching framework. It is possible that trade liberalization actually lowers welfare (defined as intertemporal utility) as the rate of unemployment drops below its efficient level. 4 Janiak (2006) also integrates search unemployment into the Melitz (2003) model and counter-intuitively finds opposite results. His model differs from ours in several respects. First, in the largest part of the paper, he takes wages as exogenous. Second, he allows for external scale effects, which imply a negative counter-factual correlation between unemployment and country size. 5 The authors show that the existence of labor market turnover implies modifications of many standard theorems in international economics (Davidson et al., 1988, 1999). 3

4 The empirical relation between unemployment and trade liberalization is unclear. Blanchard (2005) writes that there is much intuitive appeal to the idea that globalization causes turmoil on labor markets,...but the data just do not show it... (p. 27). Our model is consistent with this finding. The rate of labor market turnover (the equilibrium flow into unemployment) does increase, but so does labor market tightness. As a result, the rate of unemployment actually falls. Davis et al. (1996) show for the U.S., that sectors in which productivity increases, feature higher turnover rates and net job creation, a result that is in tune with our theoretical results. The remainder of this paper is organized as follows. In chapter 2, we present the closed economy version of our model and show that the equilibrium of the Melitz-cum-Pissarides model displays a surprisingly strong degree of recursivity and hence tractability. In chapter 3 we derive our key results on the effects of trade liberalization on equilibrium unemployment. We distinguish two globalization scenarios: (i) a decrease in variable trade costs, and (ii) an increase in the number of trading partners (e.g., due to entry of emerging economies. Chapter 4 shows that our model can be readily calibrated and presents simulation results for the U.S. economy. Finally, chapter 5 provides the conclusions and an outlook on further research. 2 The model under autarky 2.1 Setup The economy is populated by an infinitely lived representative household whose size is normalized to unity. The household has a logarithmic instantaneous felicity function and a subjective discount rate γ (0, 1). Time is discrete and all payments are made at the end of a time period. C is the quantity of a single final output good Y which can be consumed or invested. The household s utility evolves according to U = ln C + γu, (1) where a prime denotes a realization in next period. Under the assumption of perfect financial markets, the household faces the usual intertemporal budget constraint, A = (1 + r) A + (1 u ) w C, where (1 + r) A denotes dividend payments, r is the interest rate taken as exogenous by the household, u is the rate of unemployment, (1 u) w denotes wage income, again taken as given by the household, and valuation gains are disregarded. Dynamic optimization yields the usual Euler equation C /C = γ (1 + r). We focus on stationary equilibria. Hence, the interest rate is just a function of the exogenous and constant time preference rate r = (1 γ) /γ. 4

5 Y can be either consumed or used as a fixed input in the entry process, as fixed input requirements in the production process, or resources spent in the search for workers. Labor is the unique factor of production. It is inelastically supplied by the household whose mass we normalize to unity. Using a continuum of intermediate inputs, a large number of firms produce the final consumption good under conditions of perfect competition. Denoting the quantity of such an input q (ω), we posit the same production function as Blanchard and Giavazzi (2003) Y = [ ] σ M 1 σ q (ω) σ 1 σ 1 σ dω, σ > 1, (2) ω Ω where the measure of the set Ω is the mass M of available intermediate inputs and σ denotes the elasticity of substitution between any two varieties of inputs. The normalization by M 1/σ rules out external scale effects which would generate a counter-factual negative correlation between unemployment and country size. 6 The price index dual to (2) is P = [ 1 1/(1 σ) p (ω) dω] σ, (3) M ω Ω where p (ω) is the price of input ω. We choose the final output good as the numéraire. Then, the profit maximizing quantities of intermediate inputs are q (ω) = Y M p (ω) σ. (4) At the intermediate inputs level, a continuum of monopolistically competitive firms produce each a unique variety. Thus we may also index firms by ω. Inputs producers operate under increasing returns to scale. Denoting w (ω) the wage rate of firm ω and f fixed operating costs, both in terms of the numéraire, total production costs are c(ω) = w (ω) q (ω) + f. (5) ϕ (ω) The labor market is characterized by search frictions. Marginal recruitment costs are increasing at the aggregate level because of congestion externalities while they are exogenous from the point of view of an individual firm. The aggregate matching function exhibits constant 6 The normalization allows to focus on the new selection effect of trade introduced by Melitz (2003) and rules out the well-understood love for variety channel. We reintroduce external scale effects in Appendix; however, this amendment only strengthens our findings. (2) deviates from Blanchard and Giavazzi (2003) in that we take σ as independent from the mass of intermediate input producers M. 5

6 returns to scale so that the contact rates between firms and workers solely depend on the ratio θ of vacancies, v, to job seekers, u. Firms post vacancies which are filled at the rate m(θ). The vacancy filling rate m(θ) is a decreasing function of the vacancy-unemployment ratio θ. The cost of posting vacancies is proportional to the parameter c, so that increasing employment by entails spending (c/m(θ)) in recruitment costs. 2.2 Pricing behavior on product markets Denote by l the labor demand of an intermediate input producer. Further, assume that each period a measure δ of intermediate producers exits the market and worker-firm matches are destroyed with probability χ. Let δ and χ be stochastically independent. The market value of an intermediate producer with productivity ϕ can be written as 1 { ( J (l, ϕ) = max p (q) q w (l, ϕ) l cv + (1 δ)j l, ϕ )} v 1 + r ( ) 1 Y σ s.t. (i) p (q) = q σ 1 M (ii) q = ϕl (iii) l = (1 χ)l + m (θ) v where l is the level of employment next period. Constraint (i) takes into account the fact that intermediate producers face isoelastic demand curves, (ii) is the firm production function and (iii) gives the law of motion of employment at the firm level. vacancy posting reads The first order condition for c m (θ) = (1 δ)δj (l, ϕ) δl. (6) Quite intuitively, profit maximization implies that the firm sets the shadow value of labor equals to the recruitment cost. Substituting the constraints into the asset value and differentiating with respect to l yields the shadow value of labor [( ) J (l, ϕ) 1 σ 1 = ϕp (l, ϕ) w (l, ϕ) l 1 + r σ [( ) 1 σ 1 = ϕp (l, ϕ) w (l, ϕ) 1 + r σ w (l, ϕ) l w (l, ϕ) l + l The second equality follows from the optimality condition (6). l + (1 δ) J ] (l, ϕ) (1 χ) l ]. (7) c (1 χ) m (θ) The first term in the square brackets corresponds to marginal revenues, the second and third terms give the marginal costs of expanding the labor force while the fourth term is the steady-state mobility cost. The cost 6

7 of the marginal worker differs from the wage since the firm takes into account the effect of additional employment on the wage of previously employed (inframarginal) workers. In other words, firms act as monopsonists when they set their optimal labor demand schedules. explained in Bertola and Caballero (1994) and Stole and Zwiebel (1996a; 1996b), this leads to over-employment relative to the benchmark case where the firm takes the wage as given. Replacing the first order condition (6) on the left-hand side of (7), we obtain a tractable pricing rule p (l, ϕ) = 1 ϕ ( ) [ σ w (l, ϕ) + σ 1 w (l, ϕ) l + l c m (θ) As ( )] r + s, (8) 1 δ where s χ + δ + χδ is the actual rate of job destruction. Without search frictions and local monopsony power of the firm, marginal costs would just be w/ϕ. In the present case, this term is augmented by the over-employment effect and recruitment costs. 2.3 Wage bargaining Let E (ϕ) and U denote the asset values of a worker employed at a firm with productivity ϕ and of an unemployed worker, respectively. The advantage of holding a job over unemployment is equal to the difference between the wage and the opportunity cost of unemployment ru. The surplus from being employed by a firm with productivity ϕ is therefore given by E (ϕ) U = w (l, ϕ) ru r + s. (9) Before production takes place, employees and the firm can engage in an arbitrary number of pairwise negotiations. Wage contracts are unenforceable: the firm may fire any employee and symmetrically any employee may decide to quit. In other words, prior to production, individual negotiations over wage contracts can be re-open by any worker or the firm. Stole and Zwiebel (1996a) formally characterize the stable division of production into wages and profits such that neither any employee nor the firm can improve upon his gains by renegotiating. They show that the stable profile can be derived as the unique subgame perfect equilibrium of an extensive form game where the firm and workers play the alternating-offer bargaining game of Binamore et al. (1986) within each bargaining session. 7 We consider the continuous limit of the stable profile by letting the worker size goes to zero. It can be derived considering the following Nash-bargaining 7 Stole and Zwiebel (1996a) also establish that the stable outcome can be characterized by the Shapley value of a corresponding cooperative game. This interpretation is the one favored by Acemoglu and Hawkins (2006). 7

8 condition ( ) J (l, ϕ) (1 β) (E (ϕ) U) = β V l, (10) where V is the value of an unfilled vacancy and β [0, 1] is the bargaining power of the worker. 8 In equilibrium, there are no arbitrage opportunities and so firms post vacancies until V = 0. Individual bargaining implies that each employee is treated as the marginal worker. This is why the value of the job for the firm is equal to the shadow value of labor. To solve the Nashbargaining problem (10), notice that the optimality condition (6) does not vary with the level of the control variable v. This implies that the size of continuing firms remains constant through time, so that l = l in (7). This condition and the envelope theorem enable us to rewrite the first line of (7) as follows J (l, ϕ) l = ( 1 r + s ) [( σ 1 σ ) ϕp (l, ϕ) w (l, ϕ) ] w (l, ϕ) l l Reinserting this expression together with (9) into (10) yields ( ) σ 1 w (l, ϕ) w (l, ϕ) = β ϕp (l, ϕ) + (1 β)ru β l σ l ( ) ( ) 1 σ 1 Y σ = β ϕ (ϕl) 1 σ w (l, ϕ) + (1 β)ru β l. (11) σ M l The first and second term of the above expression weight, with the bargaining share, the worker s outside option and the average revenue per worker. The third term reflects that the marginal revenue being bargained upon declines as more and more workers are employed. Equation (11) is a linear differential equation in l. One can verify by direct substitution that its solution reads 9 w (l, ϕ) = (1 β)ru + βϕp (l, ϕ) ( ) σ 1 σ β.. (12) This equation is the counterpart of the Wage Curve in the standard search-matching model. In order to derive the counterpart of the Job Creation condition, we reinsert the demand function (4) into (12) and differentiate the resulting equation with respect to l, which yields w (l, ϕ) l = 1 [ ( )] σ 1 βϕp (l, ϕ). l σ σ β 8 One can endogenize the bargaining power by explicitly modeling the bargaining game. For instance, in Binmore et al. (1986), β is equal to the ratio of the firm s and worker s discount factors. 9 See Bertola and Gariabldi (2001) or Ebell and Haefke (2006) for a detailed solution of this ODE by the method of variation of parameters. Note also the similarity of expression 12 to equation (17) in Bertola and Caballero (1994). 8

9 This expression allows us to substitute ( w (l, ϕ) / l) l in (8) and, after a few simplifications, to obtain w (l, ϕ) = ϕp (l, ϕ) ( ) σ 1 σ β ( ) r + s c 1 δ m (θ). (13) As in the standard search-matching model, the Job Creation condition states that wages are decreasing in the severity of labor market frictions. Finally, we express the Wage Curve as a function of θ by replacing (12) into (13), so that ( ) ( ) β r + s c w (l, ϕ) = ru + 1 β 1 δ m(θ). (14) The Wage Curve highlights that wages are constant across firms. This somewhat surprising result concurs with the findings in Stole and Zwiebel (1996a) and Wolinsky (2000) that multipleworker firms exploit their monopsony power until employees are paid their outside option. In our enlarged set-up, the outside option is augmented by the recruitment costs that the firm would have to pay if it were to replace the worker. As intuition might suggest, the surplus extracted by employees is increasing in the severity of labor market frictions. In the remainder of the paper, we make use of (14) to drop the dependence of w on l and ϕ. Let us also mention for future reference that equations (13) and (14) imply that prices are a linear function of ϕ, so that p(ϕ 1 )ϕ 1 = p(ϕ 2 )ϕ Firm entry and exit As in Melitz (2003), the timing of the entry process is in two stages. First, prospective entrants have to develop a new intermediate good variety. By sinking f E units of the final output good, they acquire a new blueprint with certainty. Then, they discover their productivity levels and may or may not start operation. Hence, the entry fee f E gives access to a blueprint and a productivity draw. As in Melitz (2003), the firm-specific value of ϕ is constant through time and uncorrelated to the destruction rate δ, which is identical across firms. Firms draw their productivity from a sampling distribution with c.d.f. G (ϕ) and p.d.f. g (ϕ). This distribution is known to prospective entrants. Free entry therefore requires that expected profits be zero. Since f E is sunk, firms which draw a realization of ϕ too low to cover the fixed costs f do not find it optimal to start production at all. This gives a second relation, the zero cutoff profit (ZCP) condition, which ensures that the marginal entrant makes zero profits. 9

10 Before turning to an analysis of these conditions, we need to define the average productivity level. Let µ (ϕ) denote the ex-post distribution of productivity among active firms, i.e., conditional on a productivity draw that makes entry into the market worthwhile. Let ϕ denote the productivity of the marginal entrant. Following Melitz (2003), we define an average productivity level ϕ, which has the property that the quantity q ( ϕ) is equal to average output per firm Y/M. Given the demand function (4), this choice implies p ( ϕ) = P = 1. Using the proportionality of optimal prices to simplify the aggregate price index given in (3), we obtain an explicit expression for the average productivity level ϕ (ϕ ) [ + 0 ] 1 ϕ σ 1 σ 1 µ (ϕ) dϕ = [ + ϕ ϕ σ 1 g (ϕ) dϕ 1 G (ϕ ) ] 1 σ 1. (15) where the second equality follows from the definition of µ (ϕ). The above expression gives a mechanical link between the average productivity ϕ and the cutoff productivity ϕ. Notice that d ϕ (ϕ ) /dϕ > 0 iff ϕ > ϕ, which is a regularity assumption that always holds in our set-up. We may now use the definition of ϕ to analyze the zero cutoff profit and free entry conditions. 10. Let π (ϕ) denote the optimal profit per period as a function of the firm s productivity ( ) ( ( ) ) c c π (ϕ) = p(ϕ)ϕl(ϕ) wl(ϕ) χl(ϕ) f = l(ϕ) ϕ w χ f. (16) m (θ) m (θ) where the second equality follows from p(ϕ)ϕ = p( ϕ) ϕ = ϕ. By definition, the market value of firms with a productivity above ϕ is positive. At the margin, the cutoff productivity ϕ is such that π (ϕ ) r + δ cl(ϕ ) m(θ) = 0, (17) since π (ϕ ) is the stream of operating profits and l(ϕ )c/m (θ) is the firm s recruitment costs at the time of entry into the market. Notice that, due to the linearity of the adjustment cost function, entering firms jump to their optimal levels of employment. Gradual convergence can be restored either by considering that recruitment costs are convex in the number of posted vacancies, as in Bertola and Caballero (1994), or by assuming that firms can post only one vacancy, as in Acemoglu and Hawkins (2007). In both cases, firms slowly converge to the steady-state employment levels characterized in this paper. Given the concavity of the marginal revenues functions, the convergence process also implies that the shadow values of labor and thus 10 Note that the model features quasi rents in equilibrium as firms with ϕ > ϕ will make strictly positive profits. These profits are completely absorbed in expectations by the costs of entry paid by firms that end up with productivity levels ϕ < ϕ 10

11 wages decrease with the vintage of the firm. This greatly complicates the aggregation procedure since then the workers outside option depends on the cross-sectional distribution of wages. 11 Given that this mechanism is not material to our analysis, we adopt a more parsimonious specification wherein convergence to optimal employment is achieved within the first period of activity, as in Melitz (2003). The proportionality of prices enables us to relate the operating profits of the cutoff firm ϕ and of the average firm ϕ π ( ϕ) + f π (ϕ ) + f ( ) l( ϕ) ϕ σ 1 = l(ϕ ) = ϕ. (18) Reinserting this expression into (17) yields the zero cutoff profit (ZCP ) condition ( ) ( ( ) c ϕ σ 1 π ( ϕ) = (r + δ) l( ϕ) + f m(θ) ϕ 1). (ZCP) The free entry condition (F E) ensures that the entry costs f E match the expected discounted stream of profits of firms participating in the entry stage. Given that the cost of entry is paid upfront, free entry is satisfied when + ( π (ϕ) f E = r + δ cl(ϕ) m(θ) ϕ ) g (ϕ) dϕ = (1 G (ϕ )) Combining (ZCP ) and (F E) we find that in equilibrium ( π ( ϕ) ( c )) l( ϕ) r + δ m(θ). (FE) ( ) ϕ σ 1 f E /f ϕ = (r + δ) 1 G(ϕ ) + 1. (19) This condition is similar to equation (12) in Melitz (2003). It implies that, given the average productivity ϕ, there exists a unique cutoff productivity ϕ such that (F E) and (ZCP ) are simultaneously satisfied. One can now use the definition of the average productivity given in (15) and combine it with (19) to jointly determine ϕ and ϕ. 2.5 Autarky Equilibrium We have the following set of equilibrium conditions. First, the definition of the average productivity given in (15) together with (19) determines ϕ and ϕ as a function of exogenous parameters only. Given ϕ, the wage and job creation curves displayed in (13) and (14) can be solved jointly 11 See Koeniger and Prat (2007) for a numerical analysis of a model with firm entry and convex adjustment costs. 11

12 Π Zero Cutoff Profit Free Entry f f E Π( ϕ) ϕ ϕ Figure 1: Equilibrium threshold productivity to yield equilibrium values of the wage rate w and labor market tightness θ. In order to express the equilibrium tightness as a function of the fundamental parameters, we decompose the asset equations for E and U ru = b + θm(θ) + ϕ (E (ϕ) U) µ (ϕ) dϕ re (ϕ) = w + s (U E (ϕ)), where b is the flow value of non market activity. Using the Nash-bargaining condition (10) together with (1 δ) ( J (l, ϕ) / l) = c/m(θ), yields the following expression for the opportunity cost of employment ( ) β cθ ru = b + 1 β 1 δ. Reinserting that latter expression into (13), (14) and using the normalization p(ϕ)ϕ = ϕ we obtain ( ) ( W : w = b + β cθ 1 β 1 δ + β JC : w = ϕ ( σ 1 σ β ) 1 β ( r+s 1 δ ) ( ) r+s c ) 1 δ m(θ) c m(θ). (20) The Job Creation (JC) condition is a decreasing function of θ whereas the Wage curve (W ) ( ) is increasing. Thus, if b < σ 1 σ β 0 ϕdg(ϕ), the model has a unique equilibrium. 12 Figure 12 Given that ϕ > 0 ϕdg(ϕ), the existence condition proposed in the main text is sufficient but not necessary. 12

13 2 shows how those two curves pin down the equilibrium value of θ. Once θ is known, the unemployment rate can be solved for via the standard Beveridge curve u = s s + θm (θ). (21) The JC curve provides a relationship between average productivity and the wage rate, thereby allowing changes in the composition of firms to affect the unemployment rate. A higher average productivity ϕ shifts the JC curve up and leaves the W curve unchanged. 13 It follows that ϕ raises θ and so lowers the rate of unemployment. The economics behind this finding is intuitive: as the average productivity increases, active firms post more vacancies 14 and the equilibrium labor market tightness increases. Wage ϕ( σ 1 σ β ) f W f E b JC θ Figure 2: Equilibrium labor market tightness. On the other hand, the labor share of aggregate output does not directly depend on the 13 This discussion and Figure 2 make it clear that the results are not driven by our modelization of the bargaining process. One could derive a qualitatively similar impact of PMR by using the large-firm model of Pissarides (2000) or even more directly by taking wages as exogenous. The crucial feature is that the Wage Curve does not shift upward with ϕ. This would not be true if wages were directly indexed to the firm s idiosyncratic productivity, as in the fair wage model analyzed by Egger and Kreickemeier (2006). 14 Notice that this mechanism is similar to the effect of an increase in the match productivity in the standard search-matching model (see Pissarides, 2000, page 20). Yet, it does not lead to the same undesirable negative correlation between growth and unemployment. 13

14 average productivity. To see this, multiply both sides of (W ) by (1 u). Recognizing that ϕ (1 u) = Y, we find that the sum of wage payments equals a constant share, (σ β) / (σ 1), of total revenues net of recruitment costs. When workers have no bargaining power, the labor share is equal to (σ 1) /σ = ρ as in the model without search frictions. We close the characterization of the equilibrium by deriving the equilibrium mass of operating firms. First, we use the (ZCP ) condition to solve for the average value of employment. Reinserting the expression of firm s profits (16) into (17), we obtain ( ) l (ϕ c ) ϕ w (r + δ + χ) = f. m (θ) Inserting the (JC) condition and recognizing that l (ϕ) = (ϕ/ϕ ) σ 1 l (ϕ ), we find l (ϕ) = ( ) ϕ σ 1 ( ) ( ) f σ β ϕ ϕ 1 β. (22) The mass of operating firms follows from the aggregate identity Ml ( ϕ) = 1 u, so that M = = ( ) ( θm (θ) ϕ s + θm (θ) ϕ ( θm (θ) s + θm (θ) ) ( (r + δ) ) 1 σ ( ϕ f ) ( ) 1 β σ β f E 1 G(ϕ ) + f ) 1 ( ) 1 β ϕ. (23) σ β where the second equality follows from the equilibrium condition (19). Equation (23) makes clear that, for a given level of employment, both setup and fixed costs have a negative effect on the mass of operating firms. Since fixed costs also lower the equilibrium rate of unemployment, their overall effect on M is a priori ambiguous. But, as shown in the next section where we simulate the model, this countervailing effect is of second-order for reasonable parameter values. 3 The effect of trade liberalization Given the recursivity of the model, the effects of trade liberalization on equilibrium unemployment are straightforward to study. It suffices to know how aggregate productivity changes. For the sake of simplicity we consider trade between n + 1 perfectly identical countries. We analyze three globalization scenarios. We first consider a marginal drop in variable trade costs (iceberg costs). This scenario captures the effect of improvements in transportation technology or of a round of multilateral trade liberalization, for example through the WTO. Second, we study the effects of an increase in the number n of countries that undertake international trade amongst 14

15 them. This scenario captures the entry of new countries, such as China or India, into the global marketplace. Third, we study the effects of increasing fixed costs for export. In all these scenarios we focus on the long-run and neglect the complex dynamic adjustment process in the two state variables of the model, θ and ϕ. 15 We follow Melitz (2003) and model trade liberalization as the incomplete merger of all countries. Incomplete, because international trade still is affected by variable trade costs. Firms who consider entering a foreign market bear additional fixed costs (beachhead costs) f x, which are associated to the costs of distribution, marketing, or of legal regulations. However, since foreign trade involves iceberg trade costs τ 1, for given ϕ, prices faced by consumers are higher, and (since σ > 1) foreign revenues are lower than domestic ones. This is so because Iceberg transport costs means that for each unit of the exported good to arrive at the foreign market, a firm has to ship an amount of τ > 1 goods. The rest melts away during the process of transportation. This implies that only relatively high productive producers achieve sufficient foreign sales to cover up the beachhead costs. This gives rise to an additional cut-off productivity level ϕ x. In equilibrium, producers sort into three regimes. If ϕ < ϕ, the producer exits immediately upon paying the entry fee f E. If ϕ ϕ < ϕ x, the producer sells on the domestic market only. If ϕ ϕ x, the producer sells on the domestic as well as on the foreign market. Due to perfect symmetry, if a producer finds it optimal to sell on one foreign market, she finds it optimal to sell on all n foreign market. Export prices are given by p x (ϕ) = τp (ϕ), with p (ϕ) being determined by (8). Export sales to any partner country and the respective revenues at the firm level are given by q x (ϕ) = τ σ q (ϕ) and π x (ϕ) = τ 1 σ The zero profit conditions that determines the marginal operating producer (the one that sells exclusively on the domestic market) is identical to the one discussed under autarky (17). Entry into a foreign market involves a similar zero profit condition for the marginal exporter, which can be stated as π(ϕ x) r + δ cl(ϕ x) = 0, (24) m(θ) where π (ϕ x) relates to operating profits (net of f x ) realized on the export market. For given θ, equation (24) pins the productivity level ϕ x beyond which producers find it optimal to enter the foreign market. 15 For this reason, we refrain from welfare considerations. 15

16 We can now rewrite the aggregate productivity level as a function of the domestic and foreign cut-offs ϕ and ϕ x : [ ϕ ϕ σ 1 g(ϕ)dϕ ϕ t = 1 G(ϕ + ) + ϕ x ϕ σ 1 g(ϕ)dϕ 1 G(ϕ x) ] 1 σ 1 (25) That expression consists of two parts, the first for firms that are active on the domestic market only, and the second for firms that are active on the domestic and all foreign markets. The first term is equal to the average productivity condition for all active firms under autarky 16, given by equation (15). The condition that pins down average productivity of exporting firms is set up in exactly the same way and given by [ ϕ x (ϕ x) = 0 ] 1 ϕ σ 1 σ 1 µ x (ϕ)dϕ [ ] 1 ϕ = ϕσ 1 g(ϕ)dϕ σ 1 x 1 G(ϕ x) (26) The aggregated average productivity must equal the weighted average productivity for the same reasons as already discussed under autarky 17 ϕ t = [ 1 ( M ϕ σ 1 + nm x (τ 1 ϕ x ) σ 1) ] 1 σ 1. (27) M t M denotes the number of firms selling on the domestic market, M x denotes the number of firms additionally exporting and M t represents the sum of M and M x. 18 Let ϱ = (1 G(ϕ x))/(1 G(ϕ )) denote the ex ante probability 19 of drawing a productivity level high enough for entry into the export activity. Then, since M x is the share of active domestic firms that also engage in exports, M x = ϱm, must hold. Under free trade the zero profit condition not only depends on domestic firms exporting to foreign markets, it also depends on the foreign exporting firms supplying their own varieties on the regarded home market. Hence, the zero cutoff profit condition depends on a fraction of nϱ exporting firms.as under autarky, the marginal and the average exporting firm have proportional sizes. Equation (18) becomes π( ϕ x ) + f x π(ϕ = l( ϕ x) x) + f x l(ϕ x) = ( ) σ 1 ϕx (28) 16 Notice, only the condition remains unchanged when going from autarky to free trade, which does not imply that the values for ϕ and ϕ remain constant. 17 In equilibrium, all parameters of the model are pinned down only on the average productivity and the average productivity itself is pinned down by the cutoff productivity 18 Each firm that serves the foreign market also serves the domestic market, and therefore counts twice for M t. 19 Conditional on successful market entry. ϕ x 16

17 for exporting firms. Combining equation (24) with equation (28) allows us to rewrite the Zero Cutoff Profit Condition for firms that additionally participate in international trade as ( ) [ ( ) c σ 1 ϕx π( ϕ x ) = (r + δ) l( ϕ x ) + f x m(θ) ϕ 1] (ZCP ) x The free entry condition also needs some modification and becomes { ( ) [ ( )]} π( ϕ) c π( f E = ([1 G(ϕ ϕx ) c )] l( ϕ) + nϱ r + δ m(θ) r + δ l( ϕ x). (29) m(θ) This is so because a firm must take into consideration the fact that going from autarky to free trade opens up a new channel for potential profits. In setting up the conditions that pin down the free trade equilibrium, we follow Bernard, Redding and Schott (2007), who set up the free entry condition in the open economy according to (1 G(ϕ ))[ π d +nϱ π x ] = f E, where π d denotes average profit of domestic producers generated on the domestic market and π x denotes average profit of domestic and foreign exporters generated through export. Since firms do not know their productivity in advance, the free entry condition is fulfilled if expected profits on foreign and domestic markets equal sunk entry costs. Combining the free entry- and both zero profit conditions yields f E (r + δ) = ([1 G(ϕ )] { f [ ( ) [ ϕ σ 1 ( ) σ 1 ϕx ϕ 1] + nϱf x ϕ 1]}, (30) x which unambiguously pins down both cutoff productivity levels. As shown in Melitz (2003), there exists a direct link between ϕ and ϕ x, which allows us to relate both variables in form of ϕ x = ϕ τ ( ) 1 fx σ 1. (31) f Hence, it is possible to express equation (30) so that it only depends on ϕ. This is so because the average productivity ϕ and ϕ x are also implicitly pined down by ϕ and ϕ x. Due to this implicit dependency on ϕ, comparative statics are implemented quite easily which will be done in chapter (3) when we analyze the effects of a variation in some of the variables connected to free trade. Further readjustments concerning unemployment, Job-Creation curve and number of active firms have to be done in order to fully characterize the free trade equilibrium. discussed earlier, the Job-Creation curve shifts up- or downward depending on the change of aggregate average productivity, followed by a new intersection point with the Wage curve Which as mentioned before remains unchanged since it doesn t depend on aggregated average productivity. As 17

18 Once, the economy is in equilibrium again wages also adjust to an new equilibrium level due to an higher market tightness. This is so because of higher average profits due to additional export sales and a higher average productivity of the employed labor force. As under autarky, wage bargaining leads to an equalization of wages throughout the different firms. The new equilibrium level of unemployment is derived via the Beveridge curve, which directly depends on the market tightness θ. The new number of active firms decreases since a great number of the least productive firms were forced out of the market and a relatively small number of moderately productive firms increase their sales since they start with an additional participation in international trade. Formally speaking, the number of active firms is now pinned down by M t l( ϕ t ) = (1 u) = Ml( ϕ) + M x l( ϕ x ) = M(l( ϕ) + nϱl( ϕ x )), where l( ϕ) is implicitly given by equation (22) and l( ϕ x ) is given by ( ) σ 1 ( ) ( ) ϕx fx σ β l( ϕ x ) = ϕ t 1 β Which, due to the same reasoning as under autarky, gives rise to M = = ϕ x ( ) ( ) [ ( ) θm(θ) 1 β ϕ σ 1 ( ) ϕt + 1 ( ) ϕ σ 1 ( ) ] x ϕt s + θm(θ) σ β ϕ f ϱ ϕ x f x (32) ( ) ( ) ] θm(θ) 1 β f 1 E ϕ t [(r + δ) s + θm(θ) σ β 1 G(ϕ ) + f + nϱf x. (33) We can t say for sure if the number of active firms rises or falls in going from autarky to free trade. This ambiguity is due to the fact, that although M is increasing in ϕ t and θ, it is decreasing in the term in squared brackets, due to the rising cutoff productivity for domestic supply and additional fixed costs for export. As we will see later on, calibrating the model will clarify this ambiguity. The number of export firms is a fraction of ϱm firms. 3.1 Comparative Statics of the Open Economy In this section we want to do some comparative statics in order to theoretically show how trade liberalization affects unemployment. Our variables of choice are i) iceberg transport costs, ii) the number of trade relations and iii) fixed costs for export. Thereafter we want to further illustrate these findings by analyzing our calibration results, which will be done in the next chapter. 18

19 Decreasing Iceberg Transport Costs. As observed over the last decades, transportation costs have decreased due to an improvement of transportation technology. Hence, studying the effects of varying transportation costs in our model appears to be quite important for a better understanding of the globalization process. Formally speaking, this can be done by differentiating the equation that pins down both cutoff productivity variables with respect to τ. This is so because, as already said, the model solely depends on the cutoff productivity ϕ due to the matter of fact that all endogenous variables of interest are implicitly pinned down by ϕ. We prove in the Appendix that decreasing transportation costs in form of a falling τ increase the cutoff productivity for solely domestic suppliers and has decreasing effects on the cutoff productivity for the export sector of the economy ϕ τ = nf x j (ϕ x) ϕ x τ < 0, (34) fj (ϕ ) + nf x j (ϕ x) ϕ x ϕ [ ( ) ] σ 1 where k(ϕ) = ϕ(ϕ) ϕ 1 and j(ϕ) = (1 G(ϕ))k(ϕ) were used to simplify the equation (30) in order to solve for equation (34) by total differentiation 21. The negative sign follows from j (ϕ) < 0 for all ϕ. In order to proof that ϕ x τ equation (31) for ϕ x τ = ϕ x τ + ϕ x ϕ ϕ τ > 0, one has to solve the total differential of which is simplified by inserting equation (34) and yields ϕ x τ = f x τ fϕ j (ϕ ) fϕ j (ϕ ) + nf x ϕ xj (ϕ x) > 0. (35) The strict positive sign is also due to j (ϕ) < 0. An decrease in ϕ x leads to an increasing number of moderately high productivity firms that can afford additional export, analogously an increasing ϕ leads to an exit of the least productive firms. The sector of domestic supply shrinks, whereas the export sector expands. Following from the dependency of average productivity on the cutoff productivity in both sectors, average productivity in the first one must fall and in the latter must rise due to the change in cutoff productivity. It follows immediately that the exit of a huge number of very low productive firms accompanied by an market entry of moderately high productivity firms in the export sector of the economy increases average productivity and therefore decreases unemployment. Thus, the Darwinian selection effect described by Melitz (2003) lowers relative recruitment costs and therefore triggers the decrease in unemployment. Increasing Number of Trade Relations. We now turn to the effects of an increasing number of trade relations. Total Differentiation of equation (31) yields ϕ x n 21 Notice the similarity of this equation to equation (E.2) in Melitz (2003). = ϕ x ϕ ϕ n. Additionally 19

20 differentiating equation (30) with respect to n 22 allows us to derive ϕ n = f x ϕ j(ϕ x) fϕ j (ϕ ) + nf x ϕ xj (ϕ x) (36) Hence, ϕ n > 0 and ϕ x n > 0 since j(ϕ) > 0 for all ϕ and j (ϕ) < 0 for all ϕ holds. Thus, a great number of both sector s least productive firms are forced to exit the market. This gives rise to an increase in the aggregated average productivity, followed by an increase in the equilibrium theta and an decrease in the rate of unemployment. Since aggregated revenue is fixed for our economy and export sales rise due to the higher number of foreign firms that engage in the regarded home market, all firms will incur a profit loss on the domestic market. Hence, we can determine the losers and winners in our economy. The former are those firms that are not productive enough for successful participation in free trade. The latter ones are those firms, that additionally export to foreign markets. Since the number of markets where export firms are active increased, those additional revenues generated on the foreign markets are high enough to compensate the export firms for their losses on domestic markets. Increasing Fix Costs for Export. As proven in the appendix, comparative statics with respect to fixed costs for export yields the following theoretical results for both zero cutoff productivity levels. ϕ f x = ϕ x f x = n [1 G(ϕ x)] fj (ϕ ) + nf x j (ϕ )(ϕ x/ϕ ) < 0 (37) 1 (nj(ϕ x) ) nf x j (ϕ + fj (ϕ ) ϕ > 0 (38) x) f x Hence, rising fixed costs forces the moderately productive firms to retreat from taking part in international trade. They won t be forced out of the market, but they will stick to the domestic supply only. Average productivity in this sector is rising due to the exit of the least productive firms. 23 The cutoff productivity for domestic supply will fall below its erstwhile equilibrium level, followed by an influx of new solely domestic supporting firms. Since a relatively large number of firms with the lowest productivity enter the market, the average productivity in the domestic sector falls. The effect on aggregate average productivity is therefore not that clear-cut as in the scenarios discussed above. Melitz (2003) pointed out, that the aggregate average productivity is rising in decreasing fixed costs for export if the new exporters are more productive than the 22 Due to the same reasoning as discussed when we derived the comparative statics for an increasing τ 23 In this sector, not in the whole economy. 20

21 average productivity level. Hence, the effect of f x on the aggregate average productivity and thus unemployment is somehow ambiguous. We therefore address this issue to the calibration chapter. 4 Quantitative analysis In this section, we calibrate the model to match the statistics of interest for the U.S. economy. Then we simulate the model in order to perform comparative statics to analyze how transport costs, the number of trading relations and fixed costs affect the rate of unemployment in the open economy equilibrium. 24 The purpose of this section is to gain a sense of the magnitudes involved and to sort out ambiguities. To solve the extended model, we can proceed as before. First of all, we fix the elasticity σ and compute the equilibrium mass of producers M. We first run a baseline exercise in which only the selection effect is present, and then generalize the analysis to simultaneously allow for the Blanchard-Giavazzi competition effect. Our simulation results confirm out theoretical results discussed earlier in this paper since unemployment rates decrease due to the competition effect. 4.1 Calibration Productivity distribution. distribution, so that We assume that firms sample their productivity from a Pareto G(ϕ) = 1 ( ) ϕ γ. ϕ The shape parameter γ > 0 measures the rate of decay of the sampling distribution and ϕ > 0 is the minimum possible value of ϕ. This parametric assumption is standard in the literature on heterogeneous firms (see Axtell (2001), and Helpman et al. (2004) for a recent application). It is justified by the observation that the log-density of firms log-sizes is well approximated by an affine function. Reinserting the expression of G(ϕ) into (15) yields ( ϕ = γ γ + 1 σ ) 1 σ 1 ϕ. (39) As expected, the average productivity ϕ is an increasing function of the cut-off productivity ϕ. Notice that this expression implies that, for the the average productivity to be bounded, the 24 Note that we confine ourselves to analysis of the steady state. 21

22 rate of decay γ has to be higher than σ 1. Using (39) to simplify the equilibrium condition (19), we obtain ϕ = (( ) ( ) ) 1 σ 1 1 f γ ϕ. γ + 1 σ r + δ f E By definition, ϕ has to be greater than ϕ, so that the term on the right-hand side must be greater than one. For reasons explained before, ϕ is increasing in f and decreasing in f E. The effect of f/f E on average productivity is larger the higher the elasticity of substitution σ. The reason for this is that a higher σ corresponds to a lower markup, which makes it more difficult for low productivity firms to survive. In order to parameterize g(ϕ), we notice that the density of firm size S(l) is given by 25 S(l) = µ(ϕ) dϕ dl = γ ϕ ( ) ϕ γ ( ϕ ϕ (σ 1) l ) = ( γ σ 1 ) ( l l ) γ σ 1 ( 1 l Thus employment levels are also Pareto distributed with a rate of decay equal to: γ/ (σ 1). Empirical evidence suggests that the Zipf distribution accurately approximates the dispersion of firm sizes. This implies that the rate of decay should be close to one. We target the value estimated by Axtell (2001) using the 1997 data from the U.S. Census Bureau, so that γ = (σ 1). In order to pin down the value of the lower bound of the distribution, we notice that the absolute value of ϕ is intrinsically meaningless. Hence, without loss of generality, we can set ϕ so as to normalize to one the mean of the sampling distribution. Since + ( ) γ E [ϕ] = ϕg(ϕ)dϕ = ϕ, γ 1 it follows that ϕ = ( ) γ 1 γ. ϕ ). Matching function. The matching function is assumed to be Cobb-Douglas 25 The expression of l (ϕ) follows from reinserting (39) into (22) to obtain ( ) ( ) ( ) l (ϕ) = ϕ σ 1 σ β γ. f ϕ σ 1 β γ + 1 σ m (θ) = m 0 θ α. (40) 22

23 We follow the standard practice in the search-matching literature and set the elasticity parameter α to 0.5. In the absence of well-established estimates, we set the bargaining power β = α. 26 To calibrate the scale parameter m 0, we use empirical estimates of the job finding rate and labor market tightness. Given the CRS property of the matching function, the equilibrium tightness must be equal to the ratio of these two rates. Shimer (2005) estimates the monthly rate at which workers find a job to be equal to Hall (2005) finds an average ratio of vacancies to unemployed workers of over the period going from 2000 to Accordingly, we match an equilibrium tightness of 0.5 by setting the monthly job filling rate to 0.9. Reinserting these values into (40), we find that m 0 = Separation shocks. Job separations occur either because the firm leaves the market or because the match itself is destroyed. We consider that the first type of shock arrives at a Poisson rate of 0.11 per year. This implies that the annual gross rate of firm turnover is equal to 22%, as suggested by the estimates in Bartelsman et al. (2004). The match-specific shocks account for the job separations which are left unexplained by the firm-specific shock. Given that Shimer (2005) estimates the monthly rate of job separation to be 0.034, it follows that the rate of arrival of match-specific shocks χ should be equal to per year. 4.2 Calibrating the free trade equilibrium For sake of simplicity, we reintroduce the assumption of f = f x. Following Melitz (2003), we know that r(ϕ x)/r(ϕ ) = (ϕ x/ϕ ) σ 1 and that both cutoff productivity levels are interrelated to each other according to equation (31), which allows us to simplify the probability of successfully export, conditional on successful market entry, as follows: ϱ = (1 ( ) G(ϕ x)) ϕ γ (1 G(ϕ )) = = τ γ (41) Notice that the above simplification is possible, since the cumulative distribution function is known. The average productivity of domestic firms that sell their varieties on domestic markets basically remains the same as under autarky and is given by equation (15), whereas the average productivity for the export sector is determined by equation (26). According to Melitz (2003), 26 The equality of the bargaining power and matching function elasticity is known as the Hosios condition in the search-matching literature. Note, however, that in our case this condition is not sufficient to ensure an efficient allocation due the over-hiring externality. ϕ x 23