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This paper studies wage bargaining in a simple economy in which both employed and unemployed workers search for better jobs. The axiomatic Nash bargaining solution and standard strategic bargaining solutions are inapplicable because the set of feasible payoffs is nonconvex. I instead develop a strategic model of wage bargaining between a single worker and firm that is applicable to such an environment. I show that if workers and firms are homogeneous, there are market equilibria with a continuous wage distribution in which identical firms bargain to different wages, each of which is a subgame perfect equilibrium of the bargaining game. If firms are heterogeneous, I characterize market equilibria in which more productive firms pay higher wages. I compare the quantitative predictions of this model with Burdett and Mortensen's [1998. Wage differentials, employer size and unemployment. International Economic Review 39, 257–273.] wage posting model and argue that the bargaining model is theoretically more appealing along important dimensions.

This paper modifies the Burdett and Mortensen (1998) model of on-the-job search by allowing wages to be determined via strategic bargaining rather than posted unilaterally by firms. There are both empirical and theoretical motivations for studying such a model. Empirically, Mortensen (2003, Section 4.3.4) argues that a bargaining model provides a better description of the data than does the wage posting model, although his model of bargaining is not totally explicit. Theoretically, bargaining is pervasive in search models without on-the-job search and so it is intellectually interesting to understand how bargaining affects the equilibrium of a model of on-the-job search. Moreover, there are significant theoretical shortcomings of the wage posting model. Coles (2001) explains that the equilibrium of the Burdett and Mortensen (1998) model is not time consistent (or renegotiation-proof) because once a firm lures a worker away from her old employer, it has an incentive to cut its wage. By construction, an equilibrium wage in the bargaining model is renegotiation-proof. Finally, the wage posting model is not readily amenable to a study of out-of-steady state dynamics. Doing so is much easier in the bargaining model I develop here. Search theorists are increasingly aware of the need to incorporate on-the-job search into their models. In part this is because job-to-job transitions are pervasive in the United States economy. According to conservative estimates, job-to-job transitions are about half as common as unemployment-to-employment transitions (Blanchard and Diamond, 1989). Using evidence from a newer data set, Fallick and Fleischman (2004) argue that half of all new employment relationships result from a job-to-job transition rather than a movement from unemployment or out of the labor force into employment. But the interest in on-the-job search models is also a consequence of the novel theoretical results that they generate. Burdett and Mortensen (1998) develop a wage-posting model in which firms offer high wages to attract workers from other firms and to reduce worker turnover. They show that the unique equilibrium of the labor market is characterized by a continuous wage distribution, even if all workers and firms are identical. If firms are heterogeneous, higher productivity firms pay higher wages. This paper has spawned a number of extensions. Stevens (2004) and Burdett and Coles (2003) allow firms to post wage contracts rather than just a single wage. The latter paper shows that if workers are risk averse, equilibrium involves a distribution of contracts, each with an upward-sloping wage profile. Postel-Vinay and Robin (2002) allow firms to match outside offers and show that workers may voluntarily take a wage cut in order to move to a firm that is likely to be more aggressive in matching outside offers in the future. Cahuc et al. (2006) explicitly model the bargaining game between a worker and one or more potential employers. Moreover, many of these models have been tested using matched worker–firm data sets; Mortensen (2003) is a prominent example. At the same time, there is a substantial gap between this model and the ‘standard’ labor market model of search, summarized in Pissarides's (2000) textbook. In the simplest version of that model, only unemployed workers search for jobs. When a worker and firm meet, the wage is set in accordance with the axiomatic Nash (1953) bargaining solution. Pissarides shows that this results in the worker and firm splitting the gains from trade, with the worker's share determined by her (exogenous) bargaining power. There have been some attempts to introduce on-the-job search into the bargaining model. Pissarides (1994) assumes that a worker and firm split the surplus from matching. The equilibrium of the resulting model is qualitatively different from the equilibrium of the Burdett and Mortensen (1998) model: If workers and firms are homogeneous, then all workers earn the same wage at all jobs, so there is no wage dispersion. The natural conclusion is that whether there is wage dispersion in a homogeneous agent economy with on-the-job search depends critically on whether firms post wages or wages are bargained. This paper revisits this conclusion. The first finding is that the axiomatic Nash bargaining solution is inapplicable in this environment. Nash, 1953, p. 129 writes “The only important thing is the set of those pairs (u1,u2)(u1,u2) of utilities which can be realized by the players if they cooperate. We call this set B and it should be a compact convex set in the (u1,u2)(u1,u2) plane.” Unfortunately, in the model with on-the-job search, the set of feasible payoffs is typically nonconvex because an increase in the wage raises the duration of an employment relationship. This possibility is absent from models without on-the-job search, but is central to wage setting in the environment of interest to this paper. Since Nash and most of the subsequent literature impose convexity, it is unclear how to extend his results to a more general environment.1 Instead, I focus on a strategic bargaining game. I assume that when a worker and firm first meet, they bargain over the wage for the duration of the employment relationship, taking as given the wage bargained by other workers and firms, the “wage distribution.” I model bargaining as an infinite horizon alternating offers game with a small risk that bargaining breaks down between offers. I require that any wage ww that is paid in a market equilibrium be a subgame perfect equilibrium of the strategic bargaining game when the risk of breakdown is sufficiently small. The existing literature on such games, including Rubinstein (1982), Shaked and Sutton (1984), and Binmore et al. (1986) shows that under some conditions there is a unique subgame perfect equilibrium in this strategic bargaining game. Unfortunately, these results are also inapplicable to my environment because all of these papers also assume that the set of feasible payoffs is convex. When I extend their approach to handle models with nonconvex payoffs, I find that the subgame perfect equilibrium of the bargaining game with a given wage distribution is no longer unique. Instead I get a precise characterization of the set of subgame perfect equilibria. In a market equilibrium, each wage in the support of the wage distribution corresponds to one of these subgame perfect equilibria. In an environment with homogeneous firms and on-the-job search, I find there are many market equilibria. There is a continuum of market equilibria each characterized by a different continuous wage distribution. In each market equilibrium every wage in the support of the distribution is a subgame perfect equilibrium of the bargaining game. Depending on how employed workers behave when they encounter a firm paying their current wage, there may also be a continuum of market equilibria with a degenerate wage distribution and more generally a continuum of market equilibria with an n-point wage distribution for arbitrary n. I then extend the model to have heterogeneous firms, with a continuous distribution of productivity x across firms. I provide a simple characterization of market equilibria in which more productive firms pay strictly higher wages: There is a function φx(y)φx(y) such that for each firm type x , φx(x)⩾φx(y)φx(x)⩾φx(y) for all y in a neighborhood of x . This is a generalization of a naïve application of the Nash bargaining solution to this model (see Mortensen, 2003, Section 4.3.4), which imposes the stronger condition that φx(x)⩾φx(y)φx(x)⩾φx(y) for all firm types x and y. This paper proceeds as follows. Section 2 lays out the basic model with homogeneous workers and firms and discusses convexity of the set of feasible payoffs. Section 3 characterizes the set of market equilibria with a continuous wage distribution, while Section 4 shows that, if workers never switch employers when they are indifferent, the model has many market equilibria characterized by a mass of firms paying the same wage. I argue that such market equilibria seem contrived compared to the ones with a continuous wage distribution, since they are broken if firms are concerned that workers might sometimes accept equal outside offers. Section 5 explores the model with heterogeneous firms. I provide a concise definition of a market equilibrium when more productive firms pay higher wages. I then show that, like the Burdett and Mortensen (1998) model, the strategic bargaining model of on-the-job search predicts the productivity of each worker conditional on her wage and the entire wage distribution. Moreover, the model implies that some wage distributions cannot be produced by this model regardless of the distribution of productivity. Section 6 discusses the connection between this paper and existing attempts to use the Nash (1953) bargaining solution to set wages in models with on-the-job search. Finally, the paper concludes in Section 7 by evaluating the advantages and disadvantages of bargaining and wage posting models of on-the-job search.

The Burdett and Mortensen (1998) model has become an important workhorse of theoretically motivated empirical labor economics. This paper introduces a related model of bargaining and on-the-job search that delivers results that are qualitatively, if not quantitatively, similar to the wage posting model. Why might an economist prefer one model to the other? The wage posting model has one undeniable appeal: It has a unique market equilibrium. Even in the simplest model with homogeneous workers and homogeneous firms, and even if one is willing to ignore the less robust market equilibria with mass points in the wage distribution, the bargaining model admits a multiplicity of market equilibria, each characterized by a continuous wage distribution. Future research should explore which of these market equilibria is most plausible. For example, one can prove that there is only one wage distribution, F(x+z)/2F(x+z)/2, such that all View the MathML sourcew∈[w̲,w¯] are local maxima of (E(w)-U)J(w)(E(w)-U)J(w). With any other wage distribution View the MathML sourceFw̲ and View the MathML sourcew̲>(x+z)/2, it is easy to show that View the MathML source(E(w̲)-U)J(w̲) is a local minimum. The characterization of market equilibrium with heterogeneous firms, condition (23), therefore suggests that only the wage distribution F(x+z)/2F(x+z)/2 is the limit of market equilibria of heterogeneous agent economies, with wages monotonic in productivity, as heterogeneity grows less important. Along other dimensions, the bargaining model seems more attractive than the posting model. Consider the out-of-steady state dynamics of the two models. In the wage posting model, the payoff-relevant state of the economy is described by the unemployment rate u and the distribution of wages paid to employed workers G. Burdett and Mortensen prove that if these are at their steady state values, then there is a market equilibrium in which the wage offer distribution F is constant over time. But suppose instead the economy starts off out of steady state. Does it converge to steady state? What do the nonstationary dynamics look like? Although it is possible to answer these questions under special conditions, a general characterization of the nonstationary dynamics remains elusive ( Shimer, 2003). In the bargaining model, the characterization of market equilibrium when the economy is away from steady state is trivial—in fact, it was not necessary to mention the unemployment rate uu or the distribution of wages paid G anywhere in the paper. Whether a wage distribution F is a market equilibrium is independent of whether uu and G are in steady state. Allowing for aggregate shocks, e.g. changes in the arrival rate of offers λλ, further complicates the posting model. First is the question of whether firms should be able to post offers that are contingent on the aggregate shock. If they can, one can show that firms will use the shock in order to artificially create an upward-sloping wage profile, much as in Stevens's (2004) and Burdett and Coles's (2003) deterministic wage contracting models. This conclusion seems unappealing, and so one is led to assume that the firms cannot make wage offers contingent on the aggregate state. But in such a model, the payoff relevant state of the economy is the aggregate shock, the unemployment rate, and the wage distribution across workers. Solving for a market equilibrium is complex at best. In this environment, the bargaining model is appealing along two dimensions. First, it is natural to assume that workers and firms continually re-bargain in the face of shocks. Second, the payoff relevant state is again only the aggregate shock, and so it is possible, at least in principle, to find a solution to the model in which the wage offer distribution depends on current and expected future values of the shock. Finally, the bargaining model addresses an important theoretical concern with the wage posting model. In the latter model, wages are time-inconsistent, since a firm would like to cut the wage as soon as the worker agrees to take a job. Although reputation concerns might keep firms paying high wages, reputations are complicated to model and usually ignored; a notable exception is Coles (2001). In the wage bargaining model, a worker and firm can re-bargain at any time and the old wage would remain a subgame perfect equilibrium.