The Immobile Incumbent Problem in a Model of Short-Term Wage-Posting

3.1 The Distribution of Employment and Employment Flows in a RPE

First we will show that in any RPE the equilibrium distribution of employment across firm can be solved for independent of the wage distribution because workers will always want to move from a lower to a higher wage firm. We now derive the steady-state distribution of employment in an RPE.

In steady-state, the unemployment rate will be given by:

The fraction of employed workers employed in firms at position f in the wage distribution or lower, G f will, equating employment flows into and out of the mass of firms below f, in steady-state solve:

that can be solved to yield:

A firm at position f in the wage offer distribution will, in equilibrium, have quits equal to recruits so will have the following level of employment:

that can be solved to yield:

where the quit rate of the firm is given by:

and the flow of recruits is:

Note that we have been able to solve for these distributions without reference to the equilibrium wage distribution; this is a consequence of the assumption in the Burdett-Mortensen model that workers will always move for a wage increase, however small, and that there can be no mass points in the equilibrium wage distribution. The implication is that all possible RPEs must have the same distribution of employment across firms. Additional useful results are:

and:

which implies that the elasticity of recruits with respect to the position in the wage distribution is equal to minus the elasticity of quits with respect to the same variable.^[2] What follows below will make extensive use of the equations derived in this section. These equations are identical to those obtained in the canonical Burdett-Mortensen model which is set in continuous time. This is useful as it makes comparison easy.

3.2 The Value Functions of Workers

Denote the value function for an unemployed worker by V ^u , and for a worker employed at the start of the period in a firm at position f in the wage distribution by V f . Define the value functions to be the value of that state before any job offers or unemployment shocks arrive i.e. at the beginning of a period. Assume the discount factor for workers is β _w . Given this, we will have in a steady-state:

and:

Differentiating (11) we have that:

that solves to:

This is useful in what follows. Note that the reservation wage of workers will be equal to b because of the assumption that job offers arrive at the same rate whether employed or unemployed.

3.3 The Employer’s Decision

The state variable for an employer is its inherited level of employment^[3] and the assumption that firms follow Markov strategies means that this is a sufficient statistic for the strategy followed by the firm. As we are interested in the steady-state and not transitional dynamics one can – without loss of generality – use as the state variable the position of a firm in the steady-state employment distribution, f₀ – this makes the derivation of the equilibrium a lot easier. Denote by Π f 0 the value function of a firm at position f₀. This value function will obviously depend on the strategies used by other firms but as these are constant in a steady-state we do not make this dependence explicit in the interests of notational simplicity. Each period the firm will choose the wage it pays, w. Workers will then quit from and be recruited to this firm according to a rule that will be derived shortly. Given these decisions, next period, the firm will end up with a level of employment that must be in the range N 0 , N 1 – so that one can think of the firm as ending up at some, possibly different point, point in the steady-state employment distribution. The reason is that N 0 is the level of employment in a firm that was at that level last year, only ever recruits from unemployment and always loses workers to other firms when they have alternative job offers – as long as w ≥ b (and all firms will do this) one can never do worse than this. Similarly one can never do better than N 1 as the highest wage firm can do no better than only having quits to unemployment (which are exogenous) and recruiting all workers, employed and unemployed, whenever they have a job offer.

We will assume that employers set wages to maximize the present discounted value of profits, treating the wages offered by all other firms as exogenous. The position in the employment distribution next period will be a function of the initial position and the wage offered this period – denote this function by ψ f 0 , w – we derive this function below. So under the assumption that firms follow a stationary Markov strategy i.e. that workers expect a firm that deviates from the equilibrium strategy this period to return next period to the equilibrium strategy (which will not be the same wage if employment has changed), the value function for the employer can be written as:

where β is the discount factor of the employer (we retain the possibility that this differs from the workers’ discount factor). In a steady-state rank-preserving equilibrium we will have:

Substituting (15) into (14) we have, after some re-arrangement, that:

so one can think of the firm as maximizing a level of current profits where the employment it will have is the current employment level and the ‘wage’ is a weighted average of the current wage and the future wage with the weight on the future wage being the discount factor.

To make any progress we need to derive an expression for the function ψ f 0 , w – this comes from the mobility rule followed by workers.

3.4 Workers’ Mobility Decision

Now let us consider how we can derive ψ f 0 , w . Suppose all workers think that a firm currently at position f₀ and offering w will be at position ψ f 0 , w next period. Suppose that workers can observe the current employment level in the firm and the current wage offered – we return to this below. If workers are assumed to believe that employers are following a Markov strategy and they expect reversion to equilibrium behaviour, the state variable of the firm next period will be ψ f 0 , w and the value of a job at such a firm is V ψ f 0 , w . Now define f as the solution to the following equation:

The right-hand side is the value one obtains from a job in the deviating firm – the current offered wage and then the future value of a job at the position the firm ends up with. The left-hand side is the value from accepting a job at a firm that is following the equilibrium strategy and is at position f. The implication of (17) is that a worker in the deviating firm will accept a wage offer from a firm at a position above f and, symmetrically, a worker in a firm at position below f will accept a wage offer from the deviating firm. Equation (17) assumes a solution in the unit interval but there is no point in a firm making setting a wage w so high that (17) implies a solution in which even f = 1 has the left-hand side less than the right-hand side as the current wage can be lowered without affecting quit and recruitment rates. Similarly there is no point in a firm making setting a wage w so low that (17) implies a solution in which even f = 0 has the left-hand side greater than the right-hand side as the lowest equilibrium offer will be the utility from unemployment and offering lower than this will make all workers quit and an inability to recruit anyone which cannot be profit-maximizing.

Equation (17) has three notions of the position of a firm – its initial position f₀, the position it is viewed as being equivalent to by workers when making job decisions and ψ, the position that the firm will end up at the end of the period. If the firm pays a wage w = w(f₀), then we will have f = ψ = f₀. If w > w(f₀), then we will have f > ψ > f₀ while w < w(f₀) implies f < ψ < f₀. Although (17) has been written to derive f given (f₀, w, ψ) it is more convenient to think of the firm having f as the choice variable and (17) then tells us the wage required to have this level of attractiveness to workers. Using this idea, change notation to let us write ψ f 0 , f . From the mobility decision for workers one can derive what the relationship between f, ψ, and f₀ will be. From (17) we know that the offer from the deviating firm will be regarded as equivalent to a firm at position f so that quit and recruitment decisions will be based on this and hence employment will satisfy:

And with the change in the decision variable to f, and using (17), (16) becomes:

where ψ = ψ f 0 , f . If employers do not discount the future so that β = 1, the maximand in (19) reduces to {p − w(ψ)} N ψ i.e. it is only the steady-state level of profits that is relevant as one would expect. But if there is some discounting of the future this is not the case. And if workers discount the future then if a firm is currently at a position f₀ but wants to be at position ψ next period it has to make sure workers evaluate its current job offer at a position f > ψ > f₀ so that the term [ V ( f ) − V ψ ] is positive and this adds to the employer costs. Intuitively, the labour supply curve facing an employer is less elastic in the wage when there is discounting and short-term wage contracts – this intuition will be important later.

Figure 1 shows how the labour supply curve facing an employer is affected by the assumption of no-commitment in wage-setting – this is helpful for understanding the intuition of the results. Figure 1 first shows the steady-state relationship between wages and employer size – this can be thought of as the long-run labour supply curve facing the firm as if it paid a wage w for ever it would end up with a level of employment on this curve. Now consider a firm starting on this long-run labour supply curve and considering paying a wage that deviates from it. There are two reasons for why the short-run labour supply curve will be less elastic than the long-run labour supply curve. First, because it takes time for employment to adjust. Suppose that workers expect the new wage to be paid for ever (or that workers are myopic) so that the quit rate and recruitment flow faced by a firm doing this deviation is the same as that faced by a firm offering the new wage on a permanent basis. But employment in the deviating firm this period will be 1 − q N 0 + R and this will be lower than the employment level in a firm that has always paid the new wage because the initial level of employment is lower – call this the naive short-run labour supply curve. But there is a second effect at work if workers are forward-looking and not myopic. Because employment is lower at the end of the period in a firm that has only paid w this period compared to one that has always paid it, workers do not expect the new higher wage to be sustained – they expect the wage to fall back in the subsequent period. This means that the quit rate will be higher and flow of recruits lower giving a second reason why employment will be less sensitive to current wages than the long-run labour supply curve would suggest. The size of this second effect depends on how forward-looking are workers – if they do not care about the future at all they will only pay attention to the current wage in making mobility decisions. This is represented by the ’sophisticated’ labour supply curve in Figure 1 This way of understanding the model will be of use in giving an intuition for some comparative statics results later.

Figure 1:

Long- and short-run labour supply curves.

In deriving the above, we assumed that workers can observe both the current level of the offered wage and the current level of employment in the firm. They can then deduce whether the firm is deviating from the steady-state and then use this information to derive what they expect future wages to be. But, suppose that workers only observe the current wage offer and not the current level of employment or the history of wages. Workers would then be unable to detect deviation from the steady-state as long as the offered wage is an equilibrium offer for some firm. As no firm will deviate from the equilibrium, workers would simply assume that the offered wage is the equilibrium offer for the firm. Decisions would then be based on the current wage alone as workers would assume this wage will be maintained in the future. From the employer perspective, this case is then isomorphic to the case β _w  = 0 where workers are myopic. This makes the myopic worker case of more interest than one might have thought.

3.5 Solving for the Rank-Preserving Equilibrium

We are now in a position to derive the RPE, assuming that it exists (an issue we return to later). We are looking for an RPE equilibrium in which each firm chooses its current wage to maximize the present discounted value of profits treating the decisions of all other firms as given and these decisions have the rank-preserving property. We will use the first-order conditions that are necessary for profit maximization and show that these have a unique solution.

We can use the differential equation (20) to obtain some comparative static results.

Let us provide some intuition for these results. First, consider part (a). If employers have a higher discount factor, they put greater weight on future relative to current profits. This reduces the temptation to exploit the short-run immobility of workers by cutting current wages so tends to make employers choose higher wages. The higher wages of one employer then spills over to raise the wages of other employers. There is a certain paradox here – one can interpret (somewhat loosely) a higher β as a longer period of committed wages. It would seem in the interest of an individual employer to commit to wages for a longer period but this is to the collective disadvantage of employers.

Now, consider part (b). If workers have a higher discount factor, they put greater weight on future wages relative to current wages. This makes their mobility decision less responsive to current wage offers from employers as was explained earlier. This has the effect of making the labour supply curves to employers less elastic and this leads to lower wages. Again there is a certain paradox here – workers want to make employers think they are more responsive to current wages than they really are and being myopic is one way to do this.

Part (c) shows that when one puts the two effects together it is that in part (b) that dominates. Wages are higher the lower is the discount factor.

It is perhaps useful to compare the RPE with the equilibrium in the canonical B-M model in which employers make a one-off decision about wages to maximize steady-state profits. The equilibrium in this case is well-known - all firms must make the same level of profits that is given by:

Using (5) and (6) this can be written as:

which, differentiating, can be written as:

Comparison of (20) and (23) shows that the equilibria are identical when β = 1 (though note that no assumption is needed on β _w ). This is what we would expect – if firms do not discount future profits they only care about steady-state profits and this is essentially the canonical model. More generally, if we compare the two equilibria we have the following result:

These results have been derived assuming that job offer arrival rates are the same whether employed or unemployed – what would we find if they were different? As is well-known the lowest wage would be the reservation wage and that this is itself a function of the wage offer distribution. If (as most estimates suggest) the offer arrival rate is higher for the unemployed than the employed then a result like Proposition 2 would probably be true as the reservation wage is higher the higher are wages generally.

For what it is worth,^[4] the closed-form solution for the equilibrium wage distribution is given by:

where:

All of this has assumed that an RPE exists, and used necessary conditions to show that there is a unique RPE if one exists. But as the next section shows this is problematic.

7.1 The Commitment Assumption

It has been assumed that firms can commit to wages for a period of finite length – so the model can be thought of as one of limited commitment. Given that Moscarini and Postel-Vinay (2013) assume full commitment is possible and Coles and Mortensen (2011) assume no commitment is possible (their model being set in continuous time) and both have existence of an RPE one might conclude there is something unusual about the limited commitment case. However, in this section, we show that the guaranteed existence of an RPE in these models derives from features other than the different level of commitment that is assumed feasible for firms.

To see this, first consider the case where all other features of the model of this paper are retained but firms are assumed to be able to commit to the future path of wages but constrained to pay the same wage to all workers in each period – this is the assumption made by Moscarini and Postel-Vinay (2013). Commitment on future wages is only of value to employers if workers are not myopic. From this we can deduce that the commitment and no-commitment equilibria coincide if workers are myopic. This in turn implies that there can be non-existence in a model with full commitment of wages. If workers are not myopic the commitment and no-commitment equilibrium differ but for β _w close enough to zero there will, by continuity, be non-existence of an RPE if there is non-existence in the myopic worker case.

Given that Moscarini and Postel-Vinay (2013) – henceforth MPV – prove existence of an RPE in a much more general economic environment (e.g. they allow for aggregate productivity shocks and transitional dynamics), this raises the question of the source of the difference in conclusions. The difference stems from one seemingly innocuous difference in assumption about the timing of wage payments. In this paper it is assumed that production occurs and wage payments are made at the end of each period. In contrast, in MPV it is assumed that wages are paid at the beginning of the next period. This might be thought to be a trivial difference but it has the consequence in MPV of introducing into their model an initial wage (which they call the time 0 wage), which transfers value between workers and firms before the firm chooses an initial contract (quite what this time 0 wage corresponds to in the real world is unclear). This time 0 wage has no role in influencing mobility decisions of workers and, hence, no allocational role. MPV discuss the role played by this initial wage and emphasize correctly that the precise mechanism by which it is determined does not influence the nature of the equilibrium in later periods. The RPE result does not necessarily apply to this initial wage – MPV discuss a case where “if the firm has some employees at time 0 and all the bargaining power, it optimally pays the lowest possible wage….by making them just indifferent between staying or quitting into unemployment” (MPV 2013, p. 1557). Because larger firms will pay higher wages from time 1 onwards from the RPE results – this implies that the time 0 wage must be lower in larger firms in order to make sure the value of a job in all firms is equal to the value of unemployment.^[5]

Although the mechanism by which the time 0 wage is determined is unimportant for the nature of the continuation equilibrium and so might be thought a mere technicality, the existence of the pre-contractual time 0 wage is vital for their result on the existence of an RPE – if there is no pre-contractual wage then the proof of the existence of a rank-preserving equilibrium fails. This can be seen most clearly in the two-period example considered by MPV where the removal of what they term the period 0 wage causes the main result to fail. What happens in the MPV model is that there is an immobile incumbent problem but it shows up solely in the initial wage as it is efficient to exploit incumbent workers using this initial wage as it has no allocational consequences. But if one assumes that no such initial wage exists then there is no guarantee of the existence of an RPE in their model because firms can then only use current wages to exploit the relatively immobile incumbents. To summarize, the existence of an RPE in Moscarini and Postel-Vinay (2013) rests on the availability, at the beginning of time before any contracts have been signed, of a transfer between workers and employers that has no allocational role.

Now consider Coles and Mortensen (2011) who assume that wages are set in continuous time. This can be thought of as the limit of the model of this paper as the length of the period goes to zero. If we restrict attention to Markov strategies then, as discussed at the end of the previous section, the equlibrium converges to one in which all employers pay b. The intutition is that in continuous time firms are unable to commit to paying wages higher than b for more than an instant, so workers’ mobility decisions do not respond to wage changes and the labour supply to employers becomes completely inelastic inducing employers to pay the minimum possible wage. Coles and Mortensen point out that if there is Markov behaviour wage deviation will have (almost) no effect on the expected value of the worker’s future earnings and thus there will be no turnover response. But if turnover does not respond to the wage deviation, then cutting wages is a profitable strategy. This can be thought of as a version of the Diamond paradox (Diamond 1971) – if workers have no threat to leave, wages get driven down to reservation level. Coles and Mortensen want to avoid this prediction but one could argue that it is sensible. Somewhat loosely (because transitions are certainly happening in continuous time so more than one transition might be made per ‘period’) one can interpret the length of a period as the length of time for which employers are able to commit to paying a certain level of wages. Then the limiting result is what we might expect to happen if firms cannot commit to wages for more than an instant (a second, a millisecond?) – one goes to work in the morning and the employer changes the wage in the course of the working day. If this equilibrium does not correspond to what we observe this is because in the real world firms are able to commit to wages for a certain period of time so that the finite time period model is the correct one to use and the Diamond paradox problem does not arise in that model. The continuous time model is not simply a convenient way to represent a finite time model – there is a substantive difference in the nature of the equilibrium. We should not be surprised by this conclusion that the extent of possible commitment affects the real equilibrium.

But Coles and Mortensen (2011) do not assume that agents pursue Markov strategies and this allows them to prove the existence of an RPE. In particular they assume that an employer who deviates from the steady-state equilibrium is assumed to pay the reservation wage thereafter. Given these expectations there is no point in an employer doing anything different so this is an equilibrium.^[6] However even allowing such punishment strategies does not overcome the potential non-existence problem when firms can commit to wages for a finite period. The simplest way to see this is to consider the case where workers are myopic – in this case it is the current wage alone that influences workers’ decisions and beliefs about what might follow deviations from the equilibrium strategy are irrelevant. So, models with non-Markov strategies cannot be guaranteed to have an RPE if employers are able to commit to wages for any finite period.

This section has argued that the problem of potential non-existence of an RPE is likely to occur whenever current wages have some allocational role (which seems plausible) and the restriction to non-Markov strategies is not crucial.

7.2 Directed Search

The model here has assumed that search is undirected and one might wonder whether similar problems arise if search is directed (Delacroix and Shi 2006; Kircher 2009; Menzio and Shi 2010, 2011; Moen 1997) which can be thought of as the case where it is costless to sample all vacancies in the market. This section gives an example of a very stylized directed search model in which the non-existence of a RPE occurs and for similar reasons as in the undirected search model. Assume the number of firms and workers to be fixed and equal. The timing of events and actions in each period is assumed to be the following:

1. Each firm has only one job which can be filled or vacant at the start of each period. The state variable for the firm is whether the job was filled the previous period.

2. Employed workers lose their job with probability δ and become unemployed.

3. Employers then set wages for that period which they must pay if their job is filled at the end of the period. Employers are assumed myopic so choose the wage to maximize expected profits in the current period.

4. All workers, both employed and unemployed can, with probability λ, see all the wages offered. Workers are assumed myopic so apply for the vacant job that offers the highest expected utility in the current period.

5. The probability of filling a vacant job, h a depends on the number of applicants, a according to the usual formula in directed search models h a = 1 − e − a . A worker who applies to a job with a applicants will get the job with probability μ a = h a / a . Workers with job offers accept or reject job offers, there is production and wage payments.

In this model firms with filled jobs are trying to set wages to deter their workers from applying to other jobs and quitting while firms with vacant jobs are trying to set wages to attract recruits. Our aim is not to provide a complete characterization of equilibrium but to give an example of an equilibrium where firms with filled jobs choose to pay lower wages than firms with vacant jobs (within the set-up this is the only dimension of employer size). That this is possible is summarized in the following result.

This Propostion shows that something akin to the lack of an RPE can occur in directed as well as undirected search models and for the same reasons – firms have more monopsony power over existing workers than over new recruits so there is a temptation to pay lower wages to existing workers.

7.3 Wage Discrimination

This section considers what happens when we relax the assumption that employers have to pay the same wage to all workers in each period and allow the employer to pay different wages to different workers although we maintain the assumption that wages must be set before any other wage offers have been received. In this situation the only wage differentiation of interest to employers will be that the employer will choose different wages for new recruits and existing workers. Allowing employers to vary wages with seniority has been considered by Burdett and Coles (2003, 2010) in the context of undirected search and Shi (2009) in the context of directed search – both papers assume that the employer can commit to wage-tenure contracts. We assume workers are myopic so recruitment and quits depends only on the current wage. Suppose that the employer offers a wage w _R to new recruits, that this leads to a flow of recruits R w R (which will depend on what other firms are offering), and a wage w _S to senior existing workers that leads to a quit rate q w S . If the firm has an initial level of employment N₀ (which will be its state-variable) the value function for the firm in a steady-state will be given by:

A very useful result (proved in the Appendix) is that Π N 0 is linear in N₀ i.e. can be written as Π N 0 = π 0 + π 1 N 0 . Equation (40) can then be written as:

Secondly it means that all offered wages to senior workers must yield the same equilibrium level of expected profits per worker – denote this by π S * i.e. we must have:

Thirdly the separability in the profit function between the part of profits from new recruits and the part from senior workers that can be noted in (41) means there is no reason why a firm that pays high wages to new recruits should also pay high wages to existing workers. But the smallest amount of heterogeneity in productivity across firms will resolve this indeterminacy – high productivity firms will want to pay high wages both to new recruits and to exisitng workers. We will assume that those firms which pay high wages to new recruits also pay high wages to senior workers and the equilibrium we derive can then be thought of as the limit as productivity differentials between firms go to zero. If this is the case we would then appear to have something that resembles a RPE – large firms pay higher wages than small firms to both new recruits and senior workers and these differentials will persist through time. But though the problem of the lack of an RPE has been solved by allowing contracts of this type, another potential problem arises, namely that it may be optimal for an employer to pay a lower wage to senior workers than to new recruits something that does not seem plausible given that senior workers are generally paid more than new recruits.

The following Proposition provides a sufficient condition for this to be the case.

The root of this result is the same as the cause of the non-existence of an RPE in the canonical model – if turnover rates are low, employers have more monopsony power over existing workers and they will pay them lower wages. The immobile incumbent problem takes a different form here, but it remains a problem.