On the coexistence of different personnel policies: The role of unions

Transcription

1 On the coexistence of different personnel policies: The role of unions Christian Holzner July 23, 2014 Abstract This paper explains the coexistence of unionized and non-unionized personnel policies in an on-the-job search model where workers can decide on their search intensity. A unionized personnel policy is one where unionized firms have to offer at least a minimum value of employment and to commit to a certain seniority salary scales. A non-unionized personnel policy is unconstrained and allows firms to increases the wage of a worker at will. Non-unionized firms can thus counter outside offers, while unionized firms cannot. The differences between the personnel policies manifest themselves in a lower search intensity but a higher quitting rate for workers employed at unionized firms, in higher wages for unemployed workers starting at unionized firms and lower wage gains for employed workers moving from unionized firms to other firms. JEL Classification: J31, J41, J64 Keywords: personnel policies, unionization, on-the-job search. I like to thank Carlos Carrillo-Tudela, Melvyn Coles and Philipp Kircher for their comments. CESifo and University of Munich; holzner@ifo.de. The usual disclaimer applies. 1

2 1 Introduction The fact that unionized and non-unionized firms coexist in many developed economies suggests that there is a benefit from being unionized that makes it worthwhile to accept the restrictions associated with union membership. Machin (1999) summarizes the role of unions in the following way: Trade unions have traditionally been seen as defenders of egalitarian pay structures. This is reflected in equal pay for equal work notions and the standardization of pay setting mechanisms, often in the form of rigid pay scales attached to jobs rather than individuals. In this paper I analyze under which conditions unionized and non-unionized personnel policies coexist. A unionized personnel policy is one where unionized firms have to offer at least a minimum value of employment implied by a certain wage contract and have to commit to a certain seniority salary scale. 1 A nonunionized personnel policy is unconstrained. Thus, non-unionized firms can freely choose the wage contract and can grant wage increases to workers whenever it is in their interest. I analyze the question of coexistence of both types of personnel policies in an on-the-job search model. I use an on-the-job search model, because search frictions give firms monopsony power (see Manning, 2001), which enables them to use different personnel policies, and because workers can reduce firms monopsony power by searching on-the-job. I combine the two most commonly used on-the-job search models in the search literature, the model by Burdett and Mortensen (1998) and by Postel-Vinay and Robin (2002a,b), since they assume different firm strategies if their workers search on-the-job. The Burdett-Mortensen model assumes that firms commit not to counter outside offers, i.e., commit to a certain wage contract and do no deviate from the implied salary scale like unionized firms, while the Postel-Vinay and Robin model assumes that firms counter offers that their workers receive from other employers, i.e., flexibly increase wages whenever it is in their interest like non-unionized firms. 2 I depart from most on-the-job search models by endogenizing 1 See Freemann and Medoff (1984), Gosling and Machin (1995), Booth and Frank (1996), Machin (1997), Metcalf et al. (2001) and more recently Zangelidis (2008) for the importance of formal seniority scales in unionized firms in the UK and the fact that unions are associated with a lower level of earnings inequality. See Dustmann and Schönberg (2009) for the importance of fixed wage schedules in unionized firms in Germany. 2 The retention policy of firms in the directed on-the-job search literature based on Delacroix and Shi (2006), Shi (2009) and Menzio and Shi (2010a, 2010b) does not differ from the one in Burdett and Mortensen (1998). 2

3 the search intensity of employed workers, because the different personnel policies provide different incentives for workers to search on the job. The decision to counter outside offers increases competition between the incumbent and the outside firm and results in a higher payoff for the worker. The prospect of such a pay rise increases workers incentive to search for an outside offer. This moral hazard problem of rent seeking on-the-job search lowers firms profits and makes it attractive for firms to commit not to match outside offers, i.e., to commit to a fixed salary scale, as argued by Postel-Vinay and Robin (2004). Postel-Vinay and Robin (2004) is also the model closest related to the model presented here. They compare a matching with a non-matching personnel policy. The first difference to this paper is that they do not allow firms to use optimal wage-tenure contracts. This is restrictive as the work by Stevens (2004), Burdett and Coles (2003, 2007) and Carrillo-Tudela (2009a) indicates, who show that rising wage-tenure contracts successfully reduce the quitting rate in a Burdett-Mortensen on-the-job search environment. The second difference is that I do not normalized the search cost of workers to zero. This is important, since it implies that wage-tenure contracts (more specifically step contracts) are also optimal for non-unionized firms that follow a non-unionized personnel policy. Step contracts are optimal, because they increase a worker s value of employment at the fastest possible rate and therefore reduce the rent seeking search effort to a minimum. The reduced search effort increases the joint match surplus and thus non-unionized firms profits. The third difference is that I introduce a monopoly union that maximizes the wage sum of its members in order to endogenize the degree of unionization in an economy. It chooses the minimum value of employment and the salary scale that unionized firms have to offer. The minimum value of employment and salary scale are chosen such that they optimally trade off the benefits of higher average wages for its members with the lower degree of unionization that a higher value of employment brings about. The novelty of paper is to provide rich set of testable predictions regarding the difference of personnel policies between unionized and non-unionized firms that can be tested. The first testable prediction follows from the incentives to search actively for another job implied by the different personnel policies. Since outside firms know that unionized firms do not counter outside offers, they will offer the contacted worker no more than necessary in order to induce the worker to change jobs. Firms contacting workers at non-unionized firms know, however, that the incumbent firm will compete in order to keep their workers. Thus, the model predicts that workers at unionized firms search less for another job than workers at non-unionized firms. The higher search intensity does not necessarily trans- 3

4 late into a higher job-to-job transition rate, because workers at non-unionized firms will be promoted in order to stay with the firm. While workers at unionized firms will once they are contacted by another firm change their employer, since their own employer does not counter the outside offer. The model therefore predicts that the job-to-job transition rate of workers at unionized firms is higher than at non-unionized firms (if both firms are equally productive). The model also predicts differences in payments between workers that are employed at unionized and non-unionized firms. The first prediction follows from the fact that workers employed at unionized firms are only offered slightly more than their current value of employment in order to induce them to change jobs, while workers employed at non-unionized firms are offered a significantly higher value of employment, because their employer competes with the outside firm. The model hence predicts that workers employed at unionized firms experience lower wage gains after employer changes than workers employed at non-unionized firms. The theory also predicts that unionized firms will offer a higher discounted sum of wages than non-unionized firms, if the exogenous job finding rate, i.e., if the exogenous degree of competition for employed workers, is relatively low. The counteroffer strategy of non-unionized firms also implies that the wage increases at non-unionized firms is more random than the wage increases at unionized firms. The randomness of wage gains can be measured with the variance of the return to tenure at unionized versus non-unionized firms. The model thus predicts that the variance of the return to tenure of workers employed at unionized firms is smaller than the variance of the return to tenure of workers employed at non-unionized firms. Finally, the model predicts that the degree of unionization decreases with a higher exogenous job finding rate, i.e., with a higher exogenous degree of competition for employed workers. The level of the exogenous job finding rate can only be measured by the job-to-job transitions at unionized firms, since the non-unionized personnel policy of matching outside offers leads to a lower job-to-job transition rate at non-unionized firms, which implies that a higher total job-to-job transition rate can be associated with a higher or lower degree of unionization. One can therefore only condition the degree of unionization on the job-to-job transition rate at unionized firms. Thus, the model predicts that industries, where workers at unionized firms are less likely to leave their employer, are more unionized. The paper is structured as follows. Section 2 presents the theoretical model, analyzes workers payoffs and search decisions as well as the optimal wage-tenure contracts 4

5 offered by unionized and non-unionized firms. Section 3 characterizes the equilibrium and presents the conditions under which unionized and non-unionized personnel policies coexist. Section 4 presents the testable predictions and section 5 concludes. 2 The model 2.1 Framework The model has an infinite horizon is set in continuous time and concentrates on steady states. Workers and firms are infinitely lived and risk neutral. A worker s stay in the labor market is exponentially distributed at rate δ. The exit rate also serves as discount rate for workers and firms. The economy is populated by a unit mass of homogenous workers that face a unit mass of equally productive firms. Firms have to decide whether or not to be unionized. Unionized firms are indexed as type-u firms, non-unionized firms as type-n firms. γ denotes the fraction of unionized firms. I assume a monopoly union, which chooses the salary scale of wages w u (.) paid by unionized firms as well as the minimum value of being employed E u, i.e., the starting point on the wage tenure profile, that has to be offered by unionized firms in order to maximize the sum of wages paid to all workers employed at unionized firms. The monopoly union has to take into account the participation constraint of firms, i.e., unionized firms have to be at least indifferent between being unionized and not being unionized. Unionized firms can decide to be more generous and offer a higher value of being employed than the minimum value E u. However, unionized firms are committed to the salary scale and are not allowed to deviate. This implies that unionized firms are not able to react to outside offers of competing firms by promoting a worker. Non-unionized firms, however, can counter outside offers from firms that want to poach their workers. They can also condition their offers on the applicant s current value of being employed and the competing firm s type in order to ensure that the offer is high enough to attract the worker. Firms and workers are brought together through a sequential and undirected search process. Unemployed and employed workers meet a firm at rate λ + s, where s is the search intensity chosen by a worker given the convex search cost function c (s) (with c (0) = 0 and c s (0) = 0). Workers that search with positive intensity s > 0 are said to search actively. Unemployed workers receive unemployment benefits b, and all firms are assumed to have the same productivity p > b. I also assume that all firms have to pay a 5

6 wage that is at least as high as the minimum subsistence level w, with w b. 3 An unemployed worker accepts an offer if it is at least as high as the value of being unemployed. An employed worker will quit into unemployment, if the value of being employed drops below the value of being unemployed. Furthermore, an employed worker will only change jobs if the outside firm offers a higher value than the value of employment at the incumbent firm. 2.2 Workers A worker encounters a unionized firm at rate [λ + s] γ and a non-unionized firm at rate [λ + s] (1 γ). Unemployed workers will only accept a value of employment E, if it is at least as high as the value of being unemployed U. Restricted by the minimum value of employment required by the monopoly union E u U unionized firms offer unemployed workers E u. I assume that non-unionized firms offer workers a value of employment equal to the value of unemployment U, i.e., that the minimum subsistence level is low enough to ensure that a non-unionized firm is able to offer E n (0 w n (.)) = U. The respective value of being unemployed is given by, δu = max {b + (λ + s) γ [E u U] c (s)}, (1) s where the optimal search intensity chosen by unemployed workers satisfies, c (s (U)) = γ [E u U]. (2) The optimal search intensity chosen by unemployed workers increases with the fraction of unionized firms γ and the monopoly union s minimum value of employment E u. A worker employed at a unionized firm will not gain from meeting another firm, since the monopoly union s commitment to the salary scale prohibits that the incumbent firm counters an outside offer. Thus, the contacted unionized and non-unionized firms offer the worker only epsilon more than the worker s current value of employment in order to induce the worker to quit. Workers employed at a unionized firms therefore have no incentive to search actively. The respective value of employment at tenure t is therefore 3 If I assumed no minimum subsistence level w, firms would offer fee-contracts to all workers such that workers pay up-front for their job and are paid their marginal product while being employed (see Stevens, 2004). Since workers would have no incentive to search actively, fee-contracts solve the moral hazard problem. 6

7 given by, { } δe u (t w u (.)) = max w u (t) + s Ėu (t w u (.)) c (s), (3) with s (E u (t w u (.))) = 0. Note, that the value of being employed can change with tenure t if the salary scale set by the monopoly union is not flat. The change in the value function with tenure t is denoted by Ėu (t w u (.)). If a worker at a non-unionized firm meets another firm, competition between the incumbent and the outside firms ensures that the worker will receive a wage equal to the marginal product p. The value of being employed at a wage equal to the marginal product is equal to E (p) = p/δ. The value of being employed at a non-unionized firm at tenure t is therefore given by, δe n (t w n (.)) = max s { } w n (t) + Ėn (t w n (.)) + [λ + s] [E (p) E n (t w n (.))] c (s), where the optimal search intensity of workers at non-unionized firms satisfies, c (s (E n (t w n (.)))) = [E (p) E n (t w n (.))]. (5) Note, that the optimal search intensity is independent of the fraction of unionized firms, since all firms that meet a worker employed at a non-unionized firm will enter Bertrand competition. (4) 2.3 Non-unionized firms The expected profit of a firm is given by the number of workers hired h n (E) and the expected discounted profit per hired worker J n (E) for a given value of employment E offered to the newly hired worker. The hiring rate for unemployed workers is given by, h n (U) = [λ + s (U)] u, and the hiring rate of workers employed at unionized firms at a value E u is given by, h n (E u ) = λl u (E u ), where l u (E u ) stands for the number of workers employed at unionized firms at the value of employment E u. Note, that no workers are hired from other non-unionized firms, since these workers are paid their marginal product at the incumbent firm due to Bertrand competition between the outside and the incumbent firm. The expected profit of a non-unionized firm is therefore given by, Π n = [λ + s (U)] uj n (U) + λ E(p) E u J n (E) l u (E) de. (6) Non-unionized firms take the worker s current value of being employed into account and offer the lowest value E necessary to attract the worker. Conditional on the offered value 7

8 E [U, E (p)) non-unionized firms choose the optimal wage tenure contract to maximize the profit from employing a worker J n (E) J n (0 w n (.)). The profit from employing a worker at a wage tenure contract w n (.) at tenure t depends on the flow profit p w n (t) and on the discount rate, which is given by the worker s exogenous exit rate δ and the rate (λ + s (E n )) at which workers contact other firms. Note that workers that contact other firms are paid the marginal product. The profit from employing a worker at tenure t is therefore given by, [δ + λ + s (E n (t w n (.)))] J n (t w n (.)) = p w n (t) + J n (t w n (.)). (7) The associated optimal control problem of a non-unionized firm that maximizes the flow profit for a fixed value of being employed is given by, max w n w subject to the following differential equations, 0 (p w n (t)) ψ (t) dt, ψ (t) = [δ + λ + s (E n (t))] ψ (t), (8) Ė n (t) = δe n (t) w n (t) (λ + s (E n (t))) [E (p) E n (t)] + c (s (E n (t))), (9) with starting values E = E n (0) and ψ (0) = 1 and E n (t) [U, E (p)] for all t. differential equation (8) determines the discount rate at any tenure t and the differential equation (9) the worker s value of employment at any tenure t as implied by equation (4). The solution for the optimal control problem can be found in the appendix and the result is stated in the following Lemma. Lemma 1 It is optimal for non-unionized firms that contact a worker with a value of being employed equal to E [U, E (p)) to offer a step contract {w, T n (E)}. Proof. See Appendix A.1. The optimal wage tenure contract w n (.) for a given value of employment E [U, E (p)) is defined such that workers are paid w until the time to promotion T n (E) or until they meet another firm. Thereafter, they are paid the marginal product p. Step contracts are optimal for non-unionized firms, because they decrease the search cost that reduces the match surplus. To see this consider the example that the non-unionized firm pays a constant wage. This implies that employed workers search until they meet another firm. The associated search cost reduces the match surplus. With the step contract workers 8 The

9 initially search with the same intensity as with a constant wage contract (since they are hired at the same value of being employed). However, a worker employed at a step contract decreases his search intensity the closer he gets to the time of promotion. Thus, the overall search cost associated with a step contract is lower than the overall search cost associated with a constant wage contract. By offering a step contract non-unionized firms are thus able to lower overall search costs and to increase the match surplus. The higher match surplus then leads to higher firm profits. 2.4 Monopoly union and unionized firms The objective function of the monopoly union is equal to the sum of wages paid to all workers at unionized firms, i.e., W = 0 w u (t) l u (E u (t w u (.))) dt. (10) The monopoly union chooses the salary scale of wages w u (.) paid by unionized firms as well as the minimum value of being employed E u. The monopoly union is constraint by the fact that unionized firms must be indifferent between being unionized and not being unionized, i.e., Π n = Π u. The expected profit of a unionized firm is similarly to that of a non-unionized firm given by the hiring rate times the profit from employing a worker, i.e., E(p) Π u = [λ + s (U)] uj u (0 w u (.), E u ) + λ J u (0 w u (.), E) l u (E) de, (11) E u where the profit from employing a worker J u (0 w n (.), E) at tenure t = 0 depends on the salary scale w u (.) and the offered value of employment E, i.e., [δ + λ] J u (t w u (.), E) = p w u (t) + J n (t w u (.), E). (12) Unionized firms will offer a value of employment E E (p) above the minimum value of employment E u, if it is necessary to ensure that a contacted worker is willing to accept the offer. Given that unionized firms must be indifferent between being unionized and not being unionized, the monopoly union s optimization problem can be rewritten as maximizing the expected profit of unionized firms subject to a certain level of the wage sum W. Thus, the monopoly union s problem can be decomposed into two parts. First, the optimal salary scale w u (.) is chosen such that it maximizes the unionized firms profit from employing 9

10 a worker J u (0 w u (.), E) subject to a fixed value of employment E [E u, E (p)). In a second step the monopoly union chooses the minimum value of employment E u, i.e., the lowest possible starting point on the salary scale. Similar to Stevens (2004) the optimal salary scale has a step profile where workers are paid w until tenure T u (E). After tenure T u (E) they are paid the marginal product p. Tenure T u (E) is chosen such that the value of employment equals E, i.e. E u (0 w, T u (E)) = E. This is formally stated in the following Lemma. Lemma 2 The profit of unionized firms from employing a worker at a value of employment E [U, E (p)) is maximized by a step salary scale {w, T u (E)}. Proof. See Appendix A.2. The proof of Lemma 2 relies on showing that for any E [U, E (p)) a step salary scale maximizes the expected profit from employing a newly hired worker J u (0 w u (.), E) by minimizing the probability that the employed worker quits. The quitting probability is minimized by taking the worker s quitting incentives away at the fastest possible rate. Since the unionized firm is indifferent if a worker that earns the marginal product leaves or stays, the profit from employing a worker is maximized by increasing the worker s value of employment at the fastest possible rate up to the value E (p). This is ensured by paying the minimum wage w in the beginning and the marginal product p from tenure T u (E) onward, where tenure T u (E) is chosen in such a way that the implied value of employment at the point of hiring equals the intended value E. The monopoly union maximizes the sum of wages of all workers at unionized firms by choosing the minimum value of employment E u or equivalently the maximum time to promotion T u (E u ) for a given step salary scale {w, T u (E)}, i.e., W = max T u (E u ) T u (U) [wlu (w) + pl u (p)], (13) where l u (w) denotes the number of workers earning a wage w and l u (p) the number of workers earning a wage equal to the marginal product at unionized firms. The monopoly union takes the participation constraint of unionized firms, i.e., Π u = Π n, into account by considering the impact that a change in the maximum time to promotion T u (E u ) has on the fraction of unionized firms, i.e., the function γ = f (T u (E u )) is implicitly defined by the participation constraint of unionized firms Π u = Π n. The monopoly union is also constrained by the reservation time to promotion T u (U) that makes unemployed workers 10

11 indifferent between accepting or not accepting a job offer, since a lower offer will result in zero hirings. The monopoly union s first order condition trades of the positive effect of a shorter time to promotion on the sum of wages and the negative effect on the fraction of unionized firms, i.e., [ ] w lu (w) T u (E u ) + p lu (p) T u (E u ) = w lu (w) + p lu (p) dγ γ γ dt u (E u ). (14) 2.5 Baseline salary scale For notational simplicity I will use the following re-normalization of the time to promotion. The lowest value of being employed that the monopoly union can set is equal to the value of being unemployed. Given the definition of the offered time to promotion at unionized firms all workers hired by unionized firms are offered a time to promotion T u (E) T u (U). Following Burdett and Coles (2003) we denote the wage profile (w, T u (U)) as the baseline salary scale. Given the definition of the baseline salary scale, let τ be the position of a worker in a unionized firm on the baseline salary scale with τ [ T u (U), 0]. Note, that τ < 0 and that firms that offer a position τ > τ offer a value of employment E u (τ ) > E u (τ). Similarly, when the monopoly union chooses the minimum value of being employed E u that unionized firms have to offer, it chooses a certain point [ T u (U), 0] on the baseline salary scale. The baseline salary scale can also be used to re-normalize the time to promotion at non-unionized firms. Non-unionized firms offer step contracts with a finite time to promotion. Workers employed at unionized firms at a position τ on the baseline salary scale are offered a step contract (w, ˆτ (τ)) by non-unionized firms that gives the same value of employment. Thus, the function ˆτ (τ) is implicitly defined by E n (ˆτ (τ)) = E u (τ). The following Lemma compares the time to promotion offered by unionized and non-unionized firms. Lemma 3 Given a value of employment E [U, E (p)) the time to promotion at a nonunionized firm is longer than at a unionized firm, i.e., dˆτ (τ) dτ = Ėu (τ ) Ė n (ˆτ (τ )) > 1, for all τ [ T u (U), 0]. 11

12 Proof. See Appendix A.3. The intuition is as follows. A worker employed at a non-unionized firm with a certain time to promotion values his employment higher than a worker employed at a unionized firm with the same time to promotion. The reason is that workers employed at nonunionized firms are in addition to workers that are employed at unionized firms paid the marginal product if they meet another firm. It follows that the value of employment E can only be identical across unionized and non-unionized firms, if the time to promotion is longer at non-unionized firms. 2.6 Steady state flows All unemployed workers that meet a unionized or a non-unionized firm will be offered a value of employment high enough to ensure that they chose to become employed. The measure of unemployed workers in the economy is therefore given by equating in- and outflows, i.e., u = δ δ + λ + s (U). (15) The number of workers employed at unionized firms at a value E u (τ) is denoted by l u (E u (τ)). Since unemployed workers hired by unionized firms start by definition at the position τ = on the baseline salary scale and flow out immediately as the value of employment increases with tenure. It follows that the measure of workers employed at E u () equals the inflow from unemployment, i.e., l u (E u ()) = l u (E u ) = [λ + s (U)] γu. The number of workers with a value of employment E u (τ) is equal to the number of workers that were hired at a value of E u and did not leave the firm over the period T u (E u ) + τ > 0 (note, that τ is negative). Workers exit this cohort either because they exit the labor market (at rate δ) or because they meet a non-unionized firm (at rate λ (1 γ)). If workers meet another unionized firm they are offered E u (τ) (plus epsilon) and stay in the pool l u (E u (τ)). The number of workers employed at τ on the baseline salary scale is therefore given by, l u (E u (τ)) = l u (E u ) e (δ+λ(1 γ))(t u (E u )+τ). (16) The number of workers earning the minimum subsistence level w at unionized firms is 12

13 then obtained by integrating over [, 0), i.e., l u (w) = l u (E u ) 0 = l u (E u ) 1 e (δ+λ(1 γ))t u (E u ). δ + λ (1 γ) e (δ+λ(1 γ))(t u (E u )+τ) dτ, (17) After workers are promoted and paid their marginal product, i.e., at τ 0, workers only leave the firm, if they have to exit the labor market. The number of workers employed at the value E u (p) is therefore given by equating the outflow of workers, δl u (p), to the inflow, l u (E u (0)), i.e., l u (p) = l u (E u ) e (δ+λ(1 γ))t u (E u ), (18) δ The total number of workers employed at unionized firms is therefore given by, l u (w) + l u (p) = [λ + s (U)] γu δ + λ (1 γ) e (δ+λ(1 γ))t u (E u ). (19) δ [δ + λ (1 γ)] Equivalently, the total number of workers employed at unionized firms can also be obtained by equating the inflow from unemployment,l u (E u ) = [λ + s (U)] γu, to the outflow, [δ + λ (1 γ)] l u (w) + δl u (p). The number of worker at unionized firms at a value E n (ˆτ) is denoted by l n (E n (ˆτ)). It consists of worker that were hired from unemployment and did not exit nor meet another firm over the period T n (U) + ˆτ > 0 and of all workers that were hired from unionized firms over the same period and did not leave the firm, i.e., l n (E n (ˆτ)) = l n (U) e ˆτ +λ (1 γ) T n (U) [δ+λ+s(en (ˆτ ))]dˆτ (20) ˆτ T n (U) l u (ˆτ (τ)) e ˆτ ˆτ(τ) [δ+λ+s(en (ˆτ ))]dˆτ dˆτ (τ), where the number of workers employed at unionized firms, i.e., l u (ˆτ (τ)) = l u (E u (τ)), is defined in equation (16) and the inflow of workers from unemployment by l n (U) = [λ + s (U)] (1 γ) u. The number of workers at non-unionized firms earning the minimum subsistence level w is then given by, l n (w) = 0 T n (U) l n (E n (ˆτ)) dˆτ. (21) The number of workers that earn the marginal product at non-unionized firms can be obtained by equating the outflow δl n (p), to the inflow of workers that were either promoted because they reached the time to promotion, l n (E n (0)), or because they contacted 13

14 another firm, i.e., δl n (p) = l n (E n (0)) + 0 T n (U) (λ + s (E n (ˆτ))) l n (E n (ˆτ)) dˆτ. (22) The total number of workers employed at non-unionized firms, l n (w) + l n (p), can also be obtained by equating the inflow from unemployment, l n (U) = [λ + s (U)] (1 γ) u, and from unionized firms, λ (1 γ) l u (w), to the outflow δ [l n (w) + l n (p)]. Rearranging and substituting l u (w) implies, l n (w) + l n (p) = [λ + s (U)] (1 γ) u δ + λ (1 γ) + λγ [ 1 e ] (δ+λ(1 γ))t u (Eu ) δ [δ + λ (1 γ)]. (23) In addition, the labor market resource constraint implies, 1 u = l u (w) + l u (p) + l n (w) + l n (p). 3 Equilibrium 3.1 Equilibrium definition In equilibrium unemployed workers accept only wage contracts that offer them a value of employment at least as high as the value of unemployment. Employed workers change employers if they are offered a value of being employed that is higher than their current value. Unemployed and employed workers chose their search intensity optimally according to equations (2) and (5). In equilibrium the number of unemployed workers u and the number of workers employed at unionized and non-unionized firms need to be consistent with steady state turnover as characterized by equations (15) to (22). Unionized and non-unionized firms offer step salary scales or step contracts as characterized in Lemmas 1 to 3. Non-unionized firms offer a value of employment that ensures that the worker is just willing to accept the offer. Unionized firms behave similarly except that they have to offer a value of employment that is at least as high as the minimum value E u set by the monopoly firm. Since non-unionized firms always make positive profits (e.g. if they offer the level of unemployment benefits for ever), all firms have to make positive profits in equilibrium. Unionized and non-unionized firms only coexist in equilibrium, if they make the same profit. In this case the equilibrium fraction of unionized firms γ > 0 has to satisfy Π n = Π u, where Π n is defined in equation (6) and Π u in equation (11). In case unionized firms exist in equilibrium the monopoly union will take the participation constraint of unionized firms, Π u = Π n, the search and acceptance strategies of 14

15 workers and the steady state flow equations (15) to (19) into account when maximizing the sum of wages of all workers at unionized firms. The optimal level of the minimum value of employment E u is set by the monopoly union in such a way that it has to satisfy equation (14). 3.2 Firms values from employing a worker I will first compare the profits from employing a worker J u (τ) and J n (ˆτ) of both types of firms before analyzing the coexistence of unionized and non-unionized firms. The profit of a unionized firm from employing a worker at the position τ on the baseline salary scale can be derived by integrating (δ + λ) J u (τ) = [p w] + J u (τ) with terminal condition J u (0) = 0, i.e., J u (τ) = [p w] 0 τ e [δ+λ]τ dτ. (24) Similarly, the profit of a non-unionized firm from employing a worker J n (ˆτ) is given by, J n (ˆτ) = [p w] 0 ˆτ e 0 ˆτ [δ+λ+s(e n (ˆτ ))]dˆτ dˆτ. Since the time to promotion at non-unionized firm is scaled differently compared to unionized firms I change the variable of integration at non-unionized firms from ˆτ to τ in order to make the profits from employing a worker comparable at the same position τ of the baseline salary scale. Since the function ˆτ (τ) is implicitly defined by E n (ˆτ (τ)) = E u (τ) I get, dˆτ (τ ) dτ = Ėu (τ ) Ė n (ˆτ (τ )) = [p w] e δτ [p w + c (s (E n (ˆτ )))] e 0 ˆτ [δ+λ+s(e n (ˆτ ))]dˆτ, where the respective derivatives of the values of employment are obtained by integrating equations (3) and (4) with the terminal conditions E u (0) = E n (0) = p/δ and differentiating them accordingly, i.e., E u (τ) = w δ + p w e δτ, and (25) δ E n (ˆτ) = p 0 δ [p w + c (s (E n (ˆτ )))] e 0 ˆτ [δ+λ+s(e n (ˆτ ))]dˆτ dˆτ. (26) ˆτ Changing the variable of integration allows one to write the profit of a non-unionized firm as a function of the position τ on the baseline salary scale, i.e., J n (τ) = [p w] 0 τ [p w] [p w + c (s (E n (ˆτ (τ ))))] eδτ dτ. (27) 15

16 The following Lemma compares profits from employing a worker for unionized and nonunionized firms over the whole baseline salary scale. Lemma 4 J n (τ) J u (τ) for τ [τ, 0], where τ satisfies J n (τ ) = J u (τ ) at τ < 0, and J n (τ) < J u (τ) for τ < τ. Proof. See Appendix A.4. The following Figure 1 illustrates the single crossing property of the profit from employing a worker stated in Lemma 4. J u (τ) J n (τ) τ -T u (U) τ* Figure 1: Single crossing property The intuition for the single crossing property is as follows. Close to the time to promotion, i.e., τ 0, workers at non-unionized firms search very little and are thus only marginally more likely to contact an outside firm compared to workers at unionized firms. The remaining time to promotion is according to Lemma A.3 still longer at nonunionized firms. This implies that close to the time to promotion the value of employing a worker is higher for a non-unionized firm than for a unionized firm, i.e., J n (τ) J u (τ) for τ [τ, 0]. However, the further away a worker is from the time to promotion the higher is workers search intensity at non-unionized firms. Since the associated aggregate search costs increase according to the convex cost function overproportionally the further away the worker is from the time to promotion the profit of a non-unionized firm from employing a worker increases by less than the profit of a unionized firm. 16

17 3.3 Coexistence of unionized and non-unionized firms Unionized and non-unionized firms only coexist, if the expected profit for both types of firms is the same, i.e., Π u = Π n. Since the hiring rate is the same for both types of firms, i.e., 0 Π n = [λ + s (U)] uj n ( T u (U)) + λ Π u = [λ + s (U)] uj u () + λ 0 J n (τ) l u (E u (τ)) dτ, J u (τ) l u (E u (τ)) dτ, the equal profit condition reduces to comparing the profits from employment over the whole baseline salary scale, i.e., J u () J n ( T u (U)) = λγ 0 [J n (τ) J u (τ)] e (δ+λ(1 γ))(t u (E u )+τ) dτ, where equation (16) and l u (E u ) = [λ + s (U)] γu is used to substitute l u (E u (τ)). The single crossing property of J u (τ) and J n (τ) stated in Lemma 4 implies that the equal profit condition (28) can only be satisfied, if unionized firms make higher profits from hiring unemployed workers than non-unionized firms, i.e., if J u () J n ( T u (U)) > 0, because for τ close to 0 the profit from employing a worker at a nonunionized firm is higher than the profit at a unionized firm (compare Figure 1). Note that J u () J n ( T u (U)) > 0 has to hold even though the maximum time to promotion T u (E u ) set by the monopoly union is potentially smaller than the time to promotion T u (U) offered to unemployed workers. The exogenous rate λ at which workers employed at unionized firms change to other firms is a key parameter in determining whether both types of firms coexist. (28) At low values of λ a match at a unionized firm is rarely destroyed. This implies that the match surplus is close to its maximum value at any point of the baseline salary scale, i.e., lim λ 0 J u (τ)+e u (τ) = p/δ. Furthermore, a low job finding rate λ implies less poachings from unionized firms. This decreases profits of non-unionized firms more than of unionized firms. Thus, there exists a parameter range of the exogenous job finding rate [0, λ ] in which only unionized firms exist, if the monopoly union sets the minimum value of employment equal to E u = U. The threshold job finding rate λ is defined such that the equal profit condition holds at γ = 1 and E u = U, i.e., J u ( T u (U)) J n ( T u (U)) = λ 0 T u (U) 17 [J n (τ) J u (τ)] e δ(t u (U)+τ) dτ,

18 where J u (τ) and J n (τ) are given by equations (24) and (27). At higher values of the job finding rate λ profits of unionized firms decrease more than profits of non-unionized firms, since the personnel policy of unionized firms to not match outside offers in order to avoid that workers search actively is counteracted by the exogenously high job finding rate of workers. At very high values of the job finding rate λ is the profit decreasing effect of active search of workers employed at non-unionized firms so small compared to the profit decreasing effect caused by the exogenous job finding rate that all firms find it optimal to become non-unionized in order to avoid paying the higher life-time value that unionized firms have to pay to unemployed workers. This happens in the parameter range [λ, ). The threshold job finding rate λ is hence defined such that the equal profit condition holds at γ = 0, i.e., J u () = J n ( T u (U)). In order to see why unionized and non-unionized firms coexist in the job finding range [0, λ ) and not only in the range (λ, λ ) one needs to consider the monopoly union s optimization problem. If the minimum value E u offered would be equal to the value of being unemployed, i.e., E u = U, then unionized and non-unionized firms would only coexist, if the job finding rate is in the parameter range (λ, λ ). The monopoly union, however, maximizes the sum of wages of all workers at unionized firms and will demand a higher minimum value E u and thereby decrease the profit of unionized firms. In fact, in the parameter range of the exogenous job finding rate [0, λ ] the monopoly union has always an incentive to increase the minimum value E u above the value of being unemployed U, as long as only unionized firms exist. Formally, the union s objective function (13) at γ = 0 simplifies to W = λu [w 1 ] e δt u (E u ) + p e δt u (E u ), δ δ and the first derivative with respect to T u (E u ), W T u (E u ) = λu [p w] e δt u (Eu) < 0, is negative, which implies that unions decrease the time to promotion T u (E u ) as long as only unionized firms exist. As soon as some firms become non-unionized, i.e., γ > 0, the monopoly union chooses the optimal minimum value E such that the reduction in the fraction of unionized firms (implied by the increase in the minimum value) just offsets the increase in average wages for those workers that are still employed at unionized firms, i.e., [ ] w lu (E u (w)) + p lu (E u (p)) dγ γ γ dt u (E ) = w lu (E u (w)) T u (E + p lu (E u (p)) ) T u (E ). 18

19 Thus, by setting the minium value E high enough the monopoly union ensures that unionized and non-unionized firms coexist also in the parameter range [0, λ ]. The following Proposition characterizes the coexistence of unionized and non-unionized firms. Proposition 1 (i) If λ [0, λ ) unionized and non-unionized firms coexist, i.e., γ (0, 1). If λ [λ, ) all firms will chose not to be unionized, i.e., γ = 0. The fraction of unionized firms γ is unique and decreases with λ on [0, λ ). (ii) If λ [0, λ ] unionized firms offer a higher value of employment to unemployed workers than non-unionized firms, i.e., E > U (T u (E ) < T u (U)), if λ (λ, λ ), unionized firms offer E U (T u (E ) T u (U)). Proof. See Appendix A.5. Part (i) establishes the coexistence of unionized and non-unionized firms for λ [0, λ ). It also shows that the fraction of unionized firms decreases with the exogenous job finding rate λ. Part (ii) shows that the monopoly union forces unionized firms to make a more generous offers to unemployed workers than non-unionized firms if the exogenous job finding rate is sufficiently small, i.e., λ [0, λ ]. For λ (λ, λ ) the minimum value chosen by the monopoly union can be higher or equal to the value of being unemployed. The reason is that a higher minimum value of employment increases the search intensity of unemployed workers and thereby the hiring rate and thus employment at unionized firms. Higher employment at unionized firms increases the sum of wage and thus the monopoly union s objective function. On the other hand decreases a higher minimum value the profit and thus the fraction of unionized firms and the respective number of employed workers. The later effect always dominates for λ close enough to the second threshold λ and therefore induces the monopoly union to set E = U. 4 Empirical predictions The theoretical model explains the coexistence of unionized and non-unionized firms based on the assumption that unionized firms have to offer at least the minimum value of employment E that the monopoly union sets and that the monopoly union demands that wages increase according to a fixed step salary scale. This implies that unionized firms do not counter outside offers of workers that are contacted by other firms. Since outside firms know this, they will offer the contacted worker no more than necessary in 19

20 order to induce him to change jobs. The incentives to search actively for another job are therefore small for workers employed at unionized firms. Firms contacting workers at non-unionized firms know, however, that the incumbent firm will compete in order to keep their workers. Workers employed at non-unionized firms will therefore gain from meeting another firm and are hence willing to search actively for other jobs. This gives the first testable prediction. Hypothesis 1: Workers at unionized firms search less for other jobs than workers at non-unionized firms. The higher search intensity does not translate into a higher job-to-job transition rate, because workers at non-unionized firms will be promoted and thus stay with the firm. While workers at unionized firms will once they are contacted by another firm change their employer, since their own employer does not counter the outside offer. In a more general model, one would need to take the productivity of the incumbent firm into account before testing job-to-job transition rates. If unionized firms are more productive than nonunionized firms, non-unionized firms are less likely to be successful when bidding for a worker. This might turn the result around and imply that workers at unionized firms are less likely to change their job. The effect therefore only holds for equally productive firms. Hypothesis 2: The job-to-job transition rate of workers at unionized firms is higher than at non-unionized firms, if both firms are equally productive. The model also predicts differences in payments between workers that are employed at unionized and non-unionized firms. The first prediction follows from the fact that workers employed at unionized firms are only offered slightly more than their current value of employment in order to induce them to change jobs, while workers employed at non-unionized firms are offered a significantly higher value of employment, because their employer competes with the outside firm. In the model presented above, where all firms are equally productive, workers at non-unionized firms are paid the marginal product and will not change jobs. However, if firms differ in productivity, more productive firms will be able to hire from less productive non-unionized firms. And in these cases one should observe higher wage gains for workers that come from a non-unionized firm than for workers that come from a unionized firm. Hypothesis 3: Workers employed at unionized firms that change jobs experience lower wage gains (discounted sum of wages) than workers employed at non-unionized firms. 20

21 Furthermore, unionized firms will according to Proposition 1 offer unemployed workers a higher discounted sum of wages than non-unionized firms, if the job finding rate λ is in the range [0, λ ]. In the range (λ, λ ) the discounted sum of wages is weakly higher. The same relationship does not hold for employed workers, since all firms offer employed workers just the value that is needed in order to ensure that the contacted worker changes employer. This leads to the following hypothesis. Hypothesis 4: Unemployed workers that are hired by unionized firms receive in expectation a higher discounted sum of wage than unemployed workers hired by non-unionized firms. The counteroffer strategy of non-unionized firms also implies that wage growth at nonunionized firms is more random than wage growth at unionized firms. The randomness of wage growth can be measured with the variance of the return to tenure at unionized versus non-unionized firms. Hypothesis 5: The variance of the return to tenure of workers employed at unionized firms is smaller than the variance of the return to tenure of workers employed at nonunionized firms. Finally, Proposition 1 implies that unionization, i.e., the level of γ, decreases with the exogenous job finding rate λ. The level of the exogenous job finding rate can only be measured by the job-to-job transitions at unionized firms, since the non-unionized personnel policy of matching outside offers leads to a lower job-to-job transition rate at non-unionized firms, which implies that a higher total job-to-job transition rate can be associated with a higher or lower degree of unionization. Still, since job-to-job transitions at unionized firms are according to theory exogenous, one can test the following hypothesis. Hypothesis 6: Industries, where workers at unionized firms are less likely to leave their employer, are more unionized. 5 Conclusions This paper compares two types of personnel policies, a unionized personnel policy where firms have to pay at least a minimum value of employment and commit to a certain salary scale, i.e., they are not allowed to increase the wage of a worker because he receives a wage offer from another firm, with a non-unionized personnel policy where firms can 21

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