Employment Dynamics in a Signaling Model with Workers' Incentives
Abstract:
Many firms adjust employment in a "lumpy" manner - infrequently and in large bursts. This paper shows how workers' incentives can function as an endogenous cost that leads to lumpy firm-level employment adjustments such as mass layoffs. The signaling model features a firm that has private information about idiosyncratic demand shocks and relies on workers' effort for production. When workers observe the firm's employment choice but not the underlying shock, the equilibrium is characterized by partial pooling in which the firm hoards labor to maintain workers' incentives. In response to small negative shocks, the firm initially underadjusts in order to conceal information from workers; it conducts a mass layoff after business conditions deteriorate past a certain point.
1 Introduction
Neoclassical models suggest that a profit-maximizing firm should adjust its workforce frequently in response to idiosyncratic demand or productivity shocks. If wages are not fully flexible, a firm might want to increase or reduce its workforce each time its marginal product of labor rises or
falls. In reality though, many U.S. firms conduct layoff episodes infrequently and in large bursts.1 While fixed costs of adjustment can produce lumpy
patterns, in a flexible labor market such as the U.S. such explicit fixed costs are not necessarily large enough to explain observed layoff patterns. The theoretical model presented in this paper explores an additional reason that a firm might make lumpy adjustments to employment instead of more
gradual changes: the negative effect of layoff episodes on the incentives and effort provision of the firm's workers.
In the signaling model of this paper, the firm has private information about idiosyncratic shocks to its business condition (i.e. productivity or demand for its product). The firm's revenue depends on its business condition and on the non-contractible effort exerted by its homogeneous workers.
When each worker's compensation is based on a combination of the worker's own productivity and the firm's privately observed business condition, workers use their beliefs about the firm's profitability to inform their effort decisions. Since the firm's optimal choice of the size of its labor force
depends on its business condition, workers view the firm's decision of employment (which is observable to every worker) as a signal. Therefore, the firm chooses its employment level strategically considering its impact on workers' beliefs and effort provision.
Using this framework I show that there generally exists an equilibrium in which firm types with intermediate levels of business conditions pool their actions, leading to labor hoarding in response to small shocks. These firm types have incentives to delay layoffs and bear the costs of being
overstaffed for an extended period of time in order to discourage workers from shirking or looking for other jobs. This layoff avoidance strategy leads to "pent-up" adjustments: a layoff becomes optimal only if business conditions deteriorate beyond some threshold. This employment adjustment
consolidates all previous periods of falling labor demand into a single layoff episode, and thus the adjustment is much larger than the magnitude of the underlying shock. The setup of this model suggests that lumpy employment patterns are generated by factors other than the frequency of large
shocks; they may be more common in industries in which monitoring is costly or infeasible but workers' effort is important.
A principal-agent problem underlies the contracting environment between the firm and its workers, with the realistic implication that workers' compensation is increasing in their employer's productivity.2 Although this microfoundation leads to workers sharing directly in the firm's production, workers may also derive indirect benefits from working at better firms and thus the type of profit-sharing arrangement that arises in the
model can encompass any situation in which workers have a stake in the business prospects of their employer. For example, the firm's condition could determine whether the worker will have future promotion opportunities with salary increases or it may impact the worker's future layoff probability.
Compensation that is increasing in effort and business conditions is meant to capture workers' incentives to invest in their inside option when they are employed by a better firm. Instead, if workers believe that their current employer is not doing well, they may shirk, start looking for other
jobs, and/or invest less in firm-specific human capital.
Although my signaling model is mostly concerned with the effort response to a firm's choice of employment, workers' wage and bonus structures play an important role. Therefore, in an extension to the model I consider the case in which the firm offers workers a wage contract before the signaling
game takes place. For a meaningful signaling game, I only consider parameters values for which the firm finds it too costly to propose a contract that implements high effort for all possible business conditions. Instead, some states of the world result in low effort provision from workers.3
The literature about labor adjustments at firms spans many decades. A closely related paper is Bentolila & Bertola (1990), which looks at the optimal labor adjustments of a firm that faces shocks to demand and explicit (per-worker) costs of hiring or
firing.4 In the equilibrium of their model, a firm makes no adjustment to employment in response to very small shocks; layoffs are optimal only when negative
shocks are sufficiently large. However, the explicit per-worker adjustment costs in their model introduce a wedge between a firm's genuine employment choice and its optimal adjustments. Consequently, the magnitude of any layoff episode in their model is smaller than
required by a full adjustment, which is the key difference between their result and mine. Another classic paper, Azariadis (1975), looks at the contracting decision of a firm with its workers when the firm has private information about business conditions. Like much of the
implicit contracting literature, in his paper labor contracts play an insurance role and the model hinges on the assumption that workers are risk-averse.5 In
contrast, workers in my model are risk-neutral and labor contracts are designed to elicit workers' effort.
Several recent papers look at the interaction between workforce decisions at firms and workers' incentives to provide effort. Golosov & Menzio (2015) take a macro approach, incorporating effort provision into a general equilibrium search model. Their model
shows how state-contingent contracts to induce effort can provide an amplification mechanism for aggregate separations. However, since their model has only single-worker firms, it does not speak to the lumpiness of firm-level layoffs. Jeon & Shapiro (2007) look at
a contracting game in which workers do not observe whether the firm has experienced a negative shock; subsequently the firm chooses its employment level and then workers decide whether to provide effort. Their paper has a rich characterization of workers' value functions and outside options but,
differently from my model, theirs has only two periods and, more importantly, only two types of firms. Levin (2002) has a setting similar to mine with a key difference that he assumes multilateral relational employment contracts can be implemented between a firm and its
employees. The form of such contracts implies a stark result: labor-hoarding can be sustained as an equilibrium because any layoff would invalidate the firm's promise to pay bonuses and trigger low effort from workers. In contrast, the bilateral and verifiable nature of contracts in my model leads
to some very different conclusions. For example, Levin shows that multilateral contracts (and thus labor hoarding) are more likely when when workers' bargaining power is low whereas I show that labor hoarding and mass layoffs are less likely when workers share less in
the profits of their employers.
Methodologically, some features of my model are closely related to Spence (1973). The signal senders in his paper are students who undertake education, where education is less costly for high types. In my model, firms are the signal senders and a large workforce is less
costly for good firms. In the equilibrium of the Spence model, high types choose more education to separate from low types. The equilibrium of my model instead has some intermediate types of firms pooling on employment choices. This partial pooling result is driven by two mechanisms: the binary
nature of workers' incentives and the non-monotonicity of firm types' preferences over workforce size. Regarding the latter mechanism, since it is costly for a firm to depart up or down from its optimal workforce size, in equilibrium some firm types choose their optimal workforce rather than incur
the costs of pooling.6
In a survey paper, Hamermesh & Pfann (1996) draw attention to a gap in the current literature on labor and capital adjustment patterns at firms: the source of adjustment costs are not well understood. They say, "Without knowing the source of the costs we have little hope of using estimates of parameters describing the lag structures that we specify, no matter how sophisticated they are, to extrapolate to the likely effects of policies that impose gross or net costs of adjusting inputs". And, "Discovering the size of adjustment costs and how these too vary by industries' and workers' characteristics should be high on anyone's research agenda in the study of factor demand." The signaling model in my paper shows that workers' effort can act as an endogenous cost that resembles a fixed cost of labor adjustment at firms. By building a micro-founded model, it sheds light on the size of the effect and how it may vary with industry and worker characteristics.
2 Evidence of and explanations for lumpy layoffs
This section highlights the lumpy nature of firm-level layoffs in the U.S. and discusses possible explanations. First I present evidence from previous literature that illustrates the costs of downward employment adjustments. Then I use microdata from large U.S. companies to discuss a few stylized facts about the size, frequency, and reasons for mass layoffs. Finally, I evaluate explanations such as indivisibilities, stock price reactions, and legal considerations and conclude that none of these reasons provide a full explanation for the prevalence of lumpy adjustments.
2.1 Related literature
A number of papers in the literature show that infrequent and large employment changes appear to be a general feature of U.S. microdata. Using plant-level output and employment measures for a single firm, Hamermesh (1989) is one of the first to conclude that employment appears to be lumpy due to fixed costs of adjustment. Caballero, et al. (1997) also find evidence of infrequent and large adjustments, based on differences between intensive and extensive labor margin adjustments in a large number of establishments. Bloom (2009) looks at the joint behavior of firm-level capital and labor adjustments in response to uncertainty shocks. With regard to labor adjustment, he finds "high" fixed costs of labor adjustments in the U.S. (around 2.1% of a firm's annual revenue) and only "limited" per capita costs of hiring and firing. Cooper, et al. (2007), using a search model calibration, also reach a clear conclusion that the lumpy pattern of employment adjustments in the U.S. appears to be due to large fixed costs of hiring and/or firing. However, analysis in Davis, et al. (2013) shows that hiring activities have very minimal returns to scale, which suggests that fixed costs of hiring are not the primary driver of lumpy employment patterns. Davis & von Wachter (2011) emphasize that not only are mass layoffs common in the U.S., they are a particularly important policy issue because of their substantial negative effects on workers.
2.2 Mass layoffs at large U.S. companies
This section uses microdata from the 2007-2012 Challenger Job Cut Reports to show the prevalence of large, lumpy mass layoffs. The reported reason for many of these mass layoffs is weak financial conditions; this section discusses some evidence about the relationship between firm-level financial
conditions and mass layoff decisions.
The Challenger Job Cut dataset captures all layoffs by U.S. employers announced in the media, company financial reports, and those reported as part of the Worker Adjustment and Retraining Act (WARN) requirement. The chart and table below limit the sample to companies listed on U.S. stock
exchanges, mainly because listed companies are generally larger and more likely to have public reporting of their layoffs.7 At these companies, an average
reported layoff affects more than 900 workers (or 7% of the workforce) at a given firm in a given quarter. In fact, nearly 85% of these episodes are above the 50 worker threshold that is generally used to define a "mass layoff".8
Even during the years of the Great Recession, mass layoffs appear to be relatively infrequent at a company level. As shown in Figure 1, 78% of publicly-listed companies had no layoff episode reported during the six year sample period. Indeed, more than 10% had only one large layoff and a further 4% had only two large layoffs. A histogram of layoff frequency at nearly four thousand publicly-listed companies is shown below.
Figure 1: Distribution of Mass Layoff Episodes per Company, 2007-2012
Source: Challenger layoff data. Mass layoff episodes are defined as the number of quarters in which a company reports layoffs at at least one establishment.
To understand the reasons that companies conduct mass layoffs, the next table summarizes the explanations recorded in the Challenger dataset.9
Table 1: Reported Reason for Mass Layoff Episode
Source: Challenger layoff data.| Reason | % of episodes |
|---|---|
| Weak demand | 30% |
| Reorganization | 26% |
| Bankruptcy/closing | 21% |
| Cost reduction | 10% |
| Outsourcing/relocation | 3% |
| Acquisition/merger | 3% |
| Loss of contract/funding | 2% |
| Other reasons | 1% |
Table 1 above shows that "weak demand" is the primary reason reported for 30% of mass layoff episodes. This number could actually be a lower bound if companies under-report "weak demand" because of the negative signal it sends to investors.
In a companion paper, Weingarden (2016), I use the Challenger dataset to show that a firm's propensity to conduct layoffs increases in response to declines in firm-level revenue and profits. Moreover, I find that the response to firm-level financial conditions happens over one year or more. While a fall in financial conditions at a firm triggers an immediate rise in its layoff propensity, past quarterly financial declines also have significant marginal effects. Consistent with the empirical evidence, in this paper layoffs are a pent-up response to the accumulation of negative firm-level shocks.
2.3 Reasons for lumpy layoffs
While it has been shown that generic fixed costs of firing can match the patterns of employment to generate lumpy changes, are the sources of these costs observable? As shown in Table 1, some mass layoffs are motivated by indivisibilities such as
bankruptcy/closings or reorganization (perhaps eliminating divisions within a company or replacing a large number of workers with new workers or new technologies). However, at large U.S. companies a third of mass layoffs appear to be conducted in response to deteriorating business conditions; a
similar result emerges for the reasons reported by a broader set of companies compiled by BLS Mass Layoff Statistics.
Anecdotally, workers' incentives are cited as a reason that firms hoard labor and conduct lumpy layoffs. In Bewley (1999), interviews with corporate managers reveal some of the interaction between worker effort and mass layoff decisions by firms. In particular, one
human resources manager says, "Continued layoffs are demoralizing. It is better to take your hit all at once". Another echoes this sentiment, saying "Layoffs should be done quickly. You should pay those left more and say that you are done". This highlights a common corporate strategy of
consolidating layoffs into large, infrequent episodes rather than gradual layoffs that could damage the attachment of remaining workers to the firm.
Although two other reasons are often mentioned as explanations for the mass layoff phenomenon, neither seems to be adequate. First, the anticipated reaction of stock prices to mass layoff announcements probably cannot explain why U.S. firms choose mass layoffs over smaller, more frequent layoff
episodes. Farber & Hallock (2009) have found that a company's stock price generally declines in response to a layoff announcement but they found that the effect on stock prices is closer to zero in their more recent data. Accordingly, anticipated stock market
reactions may have very little impact on layoff decisions at companies- if the reaction to layoff announcements is negative then companies might even prefer small dismissal episodes to mass layoffs. Second, legal considerations may dissuade companies from laying off workers one-at-a-time, but these
considerations are probably not large enough to explain the size and extent of mass layoffs. For example, Oyer & Schaefer (2000) show that companies shifted away from individual dismissals after 1991, when penalties for employment discrimination increased.
Nonetheless, this shift was only evident for workers in protected categories, suggesting that companies do consider the legal implications of dismissals, but the magnitude of the effect is limited.
Not only are there few clear benefits to mass layoffs, but such lumpy adjustments may actually have diseconomies of scale. For example, mass layoffs can trigger advance notification provisions, increase the administrative burden on human resources departments, increase the number of workers that might initiate lawsuits, and lead to a disconnect between a company's optimal and actual labor force size. This reveals an important discrepancy: in a flexible labor market such as the U.S., there are not many costs to conducting frequent layoffs that affect a small number of workers. However, firm-level employment dynamics suggest that U.S. firms behave as if there are large fixed (or non-convex) costs to such layoffs. This paper shows that workers' incentives can generate costs that are large and non-convex, and may therefore help explain the extent of lumpy layoff patterns seen in the data.
Brief outline
This paper presents a model in which the firm chooses its employment strategically in order to elicit effort from workers. Section 3 presents the timing and payoffs of the signaling game; the firm's production and workers' compensation both depend on worker effort and the firm's exogenous business condition realization. Section 4 focuses on the strategic interaction between the firm and its workers in this one period model. The strategies of the firm and workers are outlined in the context of a perfect Bayesian equilibrium, where the firm's choice of employment is a signal to workers about its most recent business condition realization. Section 5 presents a numerical example and provides a more rigorous discussion of the equilibrium properties of the model, including conditions under which partial pooling occurs in equilibrium. A multiperiod model is presented in Section 6 and its dynamics are illustrated with a numerical example. Finally, Section 7 concludes. Proofs and lengthy discussions about assumptions are relegated to the Appendix.
Overview
In the signaling model of this paper, a single firm draws a random "business condition" parameter, which could be a random realization of the firm's productivity or of the demand for its final output. The firm's revenues are determined both by its business condition parameter and the effort
exerted by its workers; the firm's costs are determined by the number of workers and the wage paid to these workers. Workers are homogeneous and their expected contributions to production are increasing in effort and the firm's business condition, with the marginal impact of effort being higher in
better business conditions. Effort is non-contractible and therefore the firm offers each of its workers a wage that is based on the worker's individual contribution to the firm.
The signaling model features asymmetric information between the firm and its workers. After the firm has received private information about its business condition, it decides its employment level, which is an observable signal that workers use to update their beliefs. Workers decide effort based
on their beliefs about the firm's business condition and provide more effort when they expect business conditions to be good.
In the remainder of this section I describe each of these aspects in more detail. In particular, in Section 3.1 I describe the firm's production technology and workers' effort provision. In Section 3.2 I outline the timing of the game, including the wage contracting stage, while in Section 3.3 I define both players' strategies and workers' beliefs in the signaling game, and finally in Section 3.4 I present a solution to the full information benchmark case.
3.1 Production technology
Workers each contribute y to production, where y is a random variable. Its distribution, and thus ultimately its expected value, depends on the effort exerted by the worker and on the business condition of the firm. I denote worker effort
by e and I assume that it can take only two values
$$ e \in \left\{ e_L, e_H \right\}$$, with $$ e_L<e_H$$.10 Workers have a disutility of effort, $$ \Psi(e)$$ that is strictly increasing in
e. The business condition of the firm is denoted by z, such that higher values of z correspond to better business conditions. The firm type z is drawn from density $$ g(z)$$ with full support on
$$ [z_{min},z_{max}]$$, where
$$ z_{min} \geq 0$$.
Business condition parameter z can be interpreted as a realization of the firm's (1) productivity and/or (2) external demand. If z is a productivity shock then it affects workers' expected contributions to output, whereas if it is a demand
shock then it affects the firm's revenues. To allow for either interpretation of z, I refer to it more generally as the firm's exogenous business condition.
As anticipated above, the expected value of y depends on e and z (it is weakly positive and strictly increasing in both variables). In particular,
| $$\displaystyle q(e,z)=E[y\vert e,z]$$ | (1) |
Thus $$ q(e,z)$$ represents the expected individual contribution of any worker given effort choice e and business condition parameter z. This function is characterized by increasing differences in z and e,
The assumption that $$ q(e,z)$$ has increasing differences in its two arguments means that the firm's business conditions and workers' effort are productive complements. Therefore, a firm with
better business conditions is more inclined to encourage high effort from workers. The increasing differences property also leads workers to prefer providing high effort for an employer with a higher value of z, since (as it will be shown later) each worker earns a share
of his individual contribution to output.
The firm's production technology aggregates the individual worker contributions of its n workers using a fixed factor of production. This fixed factor of production is also subject to a congestion effect, $$ X(n)$$, leading to higher costs as employment increases. In this setting $$ X(n)$$ is increasing and strictly convex: $$ X'>0$$, $$$ X">0$$$, and also satisfies $$ X(0)=0$$. The firm's expected production with n workers, business condition realization z, and workers' effort choice e is therefore,
The main role of $$ X(n)$$ in this model is to produce decreasing returns to scale, which ensures that the firm's profit function is concave in n. The separability
of $$ q(e,z)$$ from $$ X(n)$$ has the additional useful property that the effort of
individual workers can be separated from the production technology of the firm. Since effort only enters the first term of the expression, each worker contributes $$ q(e,z)$$ to production and has
a constant marginal return to effort that does not depend on the total number of workers at the firm. When $$ Q(e,z,n)$$ is determined as above and y is observable (and
contractible) for each worker, team production is not a concern and there are no externalities between workers. Thus the firm can contract with individual workers in a standard principal-agent framework.
The next section discusses the relevant principal-agent contracting problem and the sequence of events in each period of the model.
3.2 Model timing
At the beginning of the period, before receiving private information about z,11 the firm offers workers a state-contingent
wage contract that will depend on the worker's contribution to production. Although effort is non-contractible, workers' individual contributions are observable and verifiable ex-post in Stage 4.
Stage 0 (Wage contracts): The firm offers a contract that specifies the wage payment $$ w(y)$$ as a linear share $$ \omega ^*$$ of the worker's individual contribution y,
| $$\displaystyle w(y)= \omega^* y\\ $$ | (3) |
Workers are risk-neutral and have a limited (zero) liability. For the main body of the paper $$ \omega ^*$$ is treated as an exogenous parameter, however, the firm's optimal endogenous contract
offer is derived in Appendix section 10.2 in terms of the model primitives. The principal-agent problem shows that any optimal contract must have
$$ \omega^* \in [0,1)$$ but I only look at the interesting setting in which $$ \omega ^*$$ is
non-zero (implemented with Assumption 2 in Section 3.4.2).
Stage 1 (Private information): The firm observes private information about its business condition, z. Workers know the prior distribution of business conditions, $$ g(z)$$ but they do not observe the firm's realization of z.
Stage 2 (Employment signal): Based on its realization of z, the firm chooses its number of employees, n according to its optimal strategy laid out in Section 4.1. Workers observe
n and know that higher levels of n are more profitable for firms with higher levels of z. Therefore, as detailed in Section 4.2, workers use the signal of n to update their beliefs about the firm's business condition.
Stage 3 (Effort decision): Workers decide effort e based on their posterior beliefs about z.
Workers are risk-neutral with additively separable utility (and no income effects). Their utility for a given level of effort e is,
Since the expected value of y, $$ q(e,z)$$, has increasing differences in e and z, workers obtain relatively
greater benefits from providing high effort when the firm's business condition is good. The workers' optimal strategy is given in Section 4.3.
Stage 4 (Production and payments): Workers' contributions are realized. The firm earns revenue from production (as described in Section 3.1) and pays workers as agreed in Stage 0. The firm's realized profits are,
In each period of the model described above, the signaling game takes place after wage contracting in Stage 0. For expositional ease, the wage parameter $$ \omega ^*$$ is taken as exogenous in the following section, which focuses on the strategic interaction between the firm and its workers as given by the signaling game in Stages 1-4.
3.3 Perfect Bayesian equilibrium
The notation for strategies and beliefs is introduced here and is used again in Sections 4 and 5 to characterize the equilibrium solution.
A perfect Bayesian equilibrium consists of (i) a strategy for the firm, which is a choice of employment n for each (privately observed) z; (ii) a strategy for workers, which is a choice of effort e given the signal n; and (iii) workers' beliefs given n such that:
- The firm's strategy $$ a:Z\rightarrow N$$ maximizes expected profits, given its private information about business condition z and given the workers' strategy for choosing effort;
- Workers' beliefs $$ \mu:N\rightarrow \Delta_Z$$ assign a probability distribution of Z based on the signal n. The probability that the firm is type z conditional on the worker observing action n is denoted $$ \mu(z\vert n)$$ such that $$ \int_{Z}\mu(z\vert n)dz=1$$ for each $$ n \in N$$. Beliefs are determined by Bayes' rule where possible and otherwise, if workers observe an n off the equilibrium path, beliefs are based on the equilibrium dominance refinement;
- The workers' strategy $$ b:N\rightarrow E$$ selects an effort level that maximizes each worker's expected utility, given the firm's strategy and the signal n that is used to inform beliefs;
- The firm's strategy $$ a(z)=n$$ and workers' strategy $$ b(n)=e$$ are sequentially rational best responses and workers' beliefs $$ \mu(z\vert n)$$ are consistent given the strategy profiles.
The optimal strategies and the off-path refinement for workers' beliefs are detailed in Section 4. First, however, it is helpful to define the full information benchmark.
3.4 Full information benchmark
This section characterizes the solution to a counterfactual model in which there is no private information (i.e. if both the firm and its workers observe z).
3.4.1 Firm's profit maximization problem
In this benchmark case in which workers have full information about z, the firm's choice of n does not serve as a signal. The firm's expected profits are denoted $$ \pi(e,z,n)$$, where e can be any anticipated effort level by workers.
The firm's optimal choice of n in the full information case is based on
$$ \pi(e,z,n)$$ as defined above. Although this represents the firm's choice of employment under full information, the notation introduced below is used in proofs and propositions in later
sections.
Notation: $$ N^*(e,z)$$ is the firm's "genuine" labor force choice that would be optimal in each period of the model, under full information (if the level of employment had no signaling value to workers), where e can be any anticipated level of effort from workers.
| $$\displaystyle N^*(e,z)=$$arg$$\displaystyle \underset{n \geq 0}{\max} \ \ \pi(e,z,n)$$ | (7) |
Since $$ X(0)=0$$ and
$$ \omega^*<1$$, there always exists some employment level $$ n \geq 0$$ for which
$$ \pi \geq 0$$. Based on the profit function, $$ N^*(e,z)$$ is increasing in z (for a given level of effort e), which means that firms with better business conditions prefer to employ more workers, all else equal.
Next, in Section 3.4.2, I discuss the workers' utility maximization problem in the full information case and I introduce an additional assumption for the exogenous value of $$ \omega ^*$$. In Section 3.4.3 I conclude the full information benchmark model by characterizing the equilibrium strategies of the firm and its workers.
3.4.2 Workers' utility maximization problem
When there is full information and workers observe z directly, they choose effort to maximize expected utility as given in Stage 3 of the signaling game. Workers have a trade-off when choosing e: higher effort leads to higher expected wages (based on expected output $$ q(e,z)$$) but it also imposes a utility cost, $$ \Psi(e)$$. Workers' expected utility is described by,
when the wage contract is treated as exogenous or pre-determined with parameter $$ \omega ^*$$. In this full information case, workers choose e depending on the observed level of z.
| $$\displaystyle e_F^*(z)=$$arg$$\displaystyle \underset{e}{\max} \ U(e,z)$$ | (9) |
Since $$ q(e,z)$$ has increasing differences, workers generally prefer $$ e=e_L$$ for
lower levels of z and $$ e=e_H$$ for higher levels of z. The workers' indifference point in terms of z is used
frequently in later analysis and thus is defined formally below.
Notation: The worker's indifference point between providing low and high effort is denoted by $$ \tilde{z}$$ in the full information case in which z is observed by both the firm and the workers. It is assumed, for convenience, that each worker chooses high effort at
$$ z=\tilde{z}$$.
Before proceeding, I make an assumption about $$ \tilde{z}$$ to ensure that a worker prefers to provide low effort for some values of z and high effort for other values of z: the worker's indifference point $$ \tilde{z}$$ should lie in the domain of z.
This assumption is essentially a restriction on the magnitude of $$ \Psi(e_L)$$ and $$ \Psi(e_H)$$ relative to share $$ \omega ^*$$ that is specified by the wage contract. The left-hand side of the inequality establishes that the expected wage payment must be large enough that it outweighs the marginal cost of providing high effort, at least for the best possible state (zmax). The right-hand side ensures that the share is not so high that the worker finds it optimal to provide high effort even in the worst state (zmin). This rules out the uninteresting cases in which the worker chooses only eL for all possible z or only eH for all possible z. As mentioned earlier, the optimal value of $$ \omega ^*$$ is given in Appendix section 10.2 and a numerical example satisfying Assumption 2 is provided in Section 11. Regardless of whether $$ \omega$$ is endogenous or exogenous, $$ \tilde{z}$$ is inversely related to it. With a higher sharing parameter, workers are willing to provide high effort for lower levels of z.
3.4.3 Equilibrium of full information benchmark
Using the notation introduced above, I conclude the discussion of the full information case by stating the equilibrium strategies of the firm and its workers. Since workers observe z directly in this case, their beliefs about z are
obviously consistent with its true value and their strategy depends only on the realized value of z.
When workers observe z directly, their strategy is simple: they choose eL for values of z below $$ \tilde{z}$$ and eH for higher z realizations. The workers' strategy in the counterfactual full information case is,
| $$\displaystyle \ b_F(z)=\begin{cases}e_L \ \text{for} \ z \in [z_{min},\tilde{z}) \\ e_H \ \text{for} \ z \in [\tilde{z}, z_{max}]\\ \end{cases}$$ | (10) |
The firm's strategy when workers observe z is to choose its genuine employment level based on eL where z is low, and based on eH where z is high. In particular, the firm's strategy in the counterfactual full information case is,
| $$\displaystyle \ \ a_F(z)=\begin{cases}N^*(e_L,z) \ \ & \text{for } z \in [z_{min},\tilde{z}) \\ N^*(e_H,z) \ \ & \text{for } z \in [\tilde{z}, z_{max}]\\ \end{cases}$$ | (11) |
Therefore, the firm's strategy is to choose the genuine employment level (based on the realization of z) that is a best response to the workers' effort provision strategy.
The following property of $$ a_F(z)$$ is invoked in Section 4, where it is used to show that even with asymmetric information, the worker can generally determine z when the firm chooses its genuine level of n.
In other words, the strategy of the firm entails a unique choice of $$ n=a_F(z)$$ for each possible realization of z.
Now that I have characterized an equilibrium of the full information benchmark case, I will return to the signaling model in which the firm has private information about z and the firm's choice of n is a signal to workers about the true underlying value of z. The next section presents a candidate equilibrium of the signaling model that includes pooling for a range of z values.
4 Strategies and Beliefs
In the following discussion, the notation from Section 3.3 is used to expand upon the optimal strategies and equilibrium beliefs in the signaling model. Here I lay out the structure and notation for the candidate equilibrium in which firms pool for intermediate values of z. Throughout, I restrict attention to pure (non-mixed) strategies.
4.1 Firm's optimal strategy with private information
In the full model, the firm faces a profit maximization problem that includes signaling considerations. The firm privately observes the realized level of z in Stage 1 and then chooses its labor force size n in
Stage 2. Under asymmetric information, instead of choosing its genuine labor force size for each value of z, the firm's optimal strategy anticipates the workers' subsequent effort strategy in Stage 3.
The firm's profit maximization problem with signaling considerations is to choose the optimal n for each z subject to the workers' strategy for effort provision.
| $$$\begin{split}\text{}\underset{n \geq 0}{\max} \ \ \pi(e^*,z,n) \\ \text{s.t. } e^*= b(n) \end{split}$$$ | (12) |
In general, the firm may want to pursue a strategy in which it chooses its genuine level of n for every realization of z or, alternatively, some firm types may benefit from a pooling strategy in which they choose the same level of
n for multiple realizations of z. Although pooling is costly since it generally requires a firm to deviate from its optimal genuine level of employment, it may be attractive to some firm types. For example, when the firm draws z just below $$ \tilde{z}$$ (i.e. close to the workers' effort indifference point), pooling with a firm type above $$ \tilde{z}$$ provides an increase in worker effort from eL to eH and the costs of pooling are relatively small.12 However, for firms with very low z realizations, pooling may be prohibitively costly because it would require significant distortions to n.
With asymmetric information, an equilibrium may be characterized by partial or complete pooling. In the equilibrium proposed below, the firm's pooling actions are confined to an unique, compact interval $$ z \in [\underline{z}, \overline{z}]$$ around $$ \tilde{z}$$.13 Firm types in this pooling interval choose the same pooling level of employment, NP, and outside of the pooling interval firm types choose their genuine full information level of employment. In the signaling model with asymmetric information between the firm and its workers, the firm's partial pooling strategy is,
| $$\displaystyle \ \ a(z)=\begin{cases}N^*(e_L,z) \ \ & \text{for } z \in [z_{min},\underline{z}) \\ N_{P} & \text{for } z \in [\underline{z},\overline{z}] \\ N^*(e_H,z) \ \ & \text{for } z \in (\overline{z}, z_{max}] \\ \end{cases}$$ | (13) |
When z is either lower or higher than the pooling interval, the optimal strategy in the candidate equilibrium corresponds to the firm's strategy $$ a_F(z)$$ in the
full information benchmark. In the signaling game, the non-pooling component of the firm's strategy can be defined as $$ N_S(z)$$, where
$$ N_S(z)\equiv a(z) \equiv a_F(z)$$ for all
$$ z \notin [\underline{z},\overline{z}]$$.
Although here the pooling interval location, employment level, and resulting beliefs were taken as given, they will be established formally in Section 5.
4.2 Workers' beliefs about z
Based on the standard common knowledge and rationality assumptions in strategic games, workers know the firm's strategy even though they do not observe the true value of z. Therefore, when workers observe n they form beliefs about
z. Workers' beliefs about z, denoted as $$ \mu(z\vert n)$$, are probability distributions that depend on whether the firm's action was a
separating or pooling action.
Given the firm's strategy in the signaling model, there are two different situations that can arise with regard to workers' beliefs: (i) workers observe a value of n that is part of the firm's strategy (either pooling or non-pooling); or (ii) workers observe a value of n that is off-path (i.e. not part of the firm's strategy). In Section 4.2.1 I discuss workers' beliefs in case (i) in which they observe an on-path value of n and then in Section 4.2.2 I discuss case (ii) in which they apply an equilibrium refinement to update their beliefs after observing an off-path value of n.
4.2.1 On-path beliefs
Pooling action observed: A pooling level of employment is one type of on-path action that comes from the firm's optimal strategy, $$ a(z)$$, presented above. If the
firm chooses a pooling action, workers cannot work out the true value of z from the firm's strategy because any
$$ z \in [\underline{z}, \overline{z}]$$ would lead to the same choice of employment. Therefore, the workers' belief function places positive probability on all possible z realizations for which the firm optimally chooses the pooling action ($$ n=N_P$$).
When pooling action NP is observed, workers' beliefs are:
| $$\displaystyle \ \mu(z\vert n=N_P)=\begin{cases}\frac{g(z)}{G(\overline{z})-G(\underline{z})} & \text{for } z \in [\underline{z},\overline{z}]\\ \ 0 & \text{for all other } z\\ \end{cases}$$ | (14) |
After workers observe $$ n=N_P$$, Bayesian updating of their prior $$ g(z)$$ normalizes the
distribution over the possible z realizations that could have led to the pooling action.
Non-pooling action observed:
Instead, the firm might choose a value of n that is consistent with a value of $$ N_S(z)$$, the non-pooling component of the firm's equilibrium strategy. This choice
of n would also constitute an on-path action for the firm. In this case (unlike with pooling), the worker can determine the exact value of z given the firm's choice of n and thus the belief function is degenerate
with probability mass of one at the true z. More formally,
When any action $$ N_S(z)$$ is observed, workers' beliefs are:
| $$\displaystyle \ \mu(z\vert n=N_S(z))=\begin{cases}1 \ \ & \text{for } z=N_S^{-1}(z) \\ 0 & \text{for all other } z\\ \end{cases}$$ | (15) |
Since $$ a_F(z)$$ is invertible in z (as established by Claim 1), $$ N_S(z)$$ is also invertible. Thus when workers see a particular
$$ n \in N_S(z)$$ they are able to tell exactly which z was realized.
Finally, workers' beliefs assign a zero probability to off-path actions by the firm. Consistent beliefs contain no information about the firm type when workers observe a value of n that does not correspond to the firm's equilibrium strategy (i.e. an off-path action). However, certain firm types might prefer the equilibrium outcome to the potential deviation; a belief refinement is included in the equilibrium definition to reflect the fact that certain deviations are attractive to certain firm types and not to others. In the next section I present a general introduction to the belief refinement used for off-path actions in this model.
4.2.2 Refinement for off-path beliefs
The refinement invoked here for workers' beliefs about a firm's off-path behavior is "equilibrium dominance", as discussed in Cho & Kreps (1987). Equilibrium dominance requires that any firm choosing a particular off-path action has done so because the action could have improved its payoff for at least one configuration of workers' beliefs, relative to the payoff that the firm expects to receive in equilibrium. If workers were to observe an action that is not part of the firm's equilibrium strategy (in other words, any off-path employment level $$ N_O \notin a(z)$$), they use the equilibrium dominance refinement to update beliefs. As I show in Section 5, this refinement limits the types of firms from which a deviation could come. This has important implications for the existence of a pooling interval in the equilibrium of the model.
4.3 Workers' optimal strategy with asymmetric information
This section outlines the worker's optimal strategy for effort provision in the signaling model in which workers observe n and use it to inform their posterior beliefs about z. To simplify the model exposition, from now on I will consider
the case in which workers' contributions are linear in z such that $$ y=zf(e)$$, where
$$ 0 \leq f(e_L)$$ and
$$ E[f(e_L)]<E[f(e_H)]$$ and therefore the expected value of y exhibits increasing differences in e and z as
required by Assumption 1.
With a slight abuse of notation, let $$ E_{\mu}[z\vert n]=\int_{z_{min}}^{z_{max}}z \mu(z\vert n)dz$$. This expression represents a worker's posterior expectation of z after observing the firm's choice of n. When each worker's expected contribution (and therefore also expected compensation) is linear in z, his or her strategy in the signaling model depends only on his or her posterior beliefs about z. The strategy of workers is therefore,
| $$\displaystyle \ \ b(n)=\begin{cases}e_{L} \ \ & \text{for }n \text{ such that } E_{\mu}[z\vert n] \in [z_{min},\tilde{z}) \\ e_{H} & \text{for }n \text{ such that } E_{\mu}[z\vert n] \in [\tilde{z},z_{max}]\\ \end{cases}$$ | (16) |
where
$$ E_{\mu}[z\vert n]$$ is the expected value of z based on workers' posterior beliefs. The workers' optimal strategy is to provide low effort when they believe
z is low and provide high effort when they believe z is at or above their indifference point $$ \tilde{z}$$.
This formulation, however, does not fully satisfy the perfect Bayesian equilibrium criterion that beliefs be consistent and strategies be (sequentially rational) mutual best responses. The analysis presented here instead lays the groundwork for the next section, in which the equilibrium properties are discussed more rigorously.
5 Equilibrium Characterization
This section shows that a perfect Bayesian equilibrium of the signaling model can be characterized by partial pooling in the firm's employment decision across type-space z in each period. I first present an example for illustrative purposes and then derive a series of lemmas about the properties of any pooling interval in the signaling model. At the end of Section 5.2 I look at conditions for the existence of a pooling interval in equilibrium.
5.1 Numerical example with partial pooling equilibrium
Figure 2 shows an example of the firm's optimal employment strategy in each period of the signaling model. In particular, $$ g(z)$$ is normal and
$$ X(n)$$ has constant elasticity - see Appendix section 11 for specific parameter values.
The dashed line is the firm's counterfactual genuine choice of n; this is the level of employment that a firm would choose under full information (if workers could observe z directly). The solid line is the firm's optimal strategy in the
signaling model (when workers cannot observe z). This line has a flat region in an interval around $$ \tilde{z}$$, which is the indifference point where workers switch
from eL to eH. This flat portion is the model's predicted pooling equilibrium. Note that, when a pooling interval exists, the interval bounds are always around $$ \tilde{z}$$, the only discontinuity in the graph is at
$$ \underline{z}$$, and firm types within the pooling interval always have higher employment in the signaling model than in the full information benchmark. Although the binary nature of workers'
effort leads to a discontinuity in the firm's labor demand even in the counterfactual case, the model of asymmetric information shows both pooling and a larger discontinuity than the counterfactual model. In the multiperiod model equilibrium, pooling will produce underadjustment; the large
discontinuity will produce a lumpy overadjustment that was discussed in the introduction.
This graph also highlights the implications of asymmetric information. In the signaling model, firm types below
$$ \underline{z}$$ (and those above
$$ \overline{z}$$) choose their genuine employment levels and get low (high) effort from workers, just as in the full information case. In the pooling interval however, firms choose a higher level
of employment in the signaling model equilibrium compared with their genuine choices under full information. For the lower-type firms in the interval, those below $$ \tilde{z}$$, asymmetric
information allows them to pool with better type firms to get high effort from workers. However, the incentives for lower-type firms to pool imposes an externality on the higher-type firms in the pooling interval. In particular, the firm types above $$ \tilde{z}$$ would have been better-off in the full information equilibrium in which they chose genuine levels of employment and received high effort from workers. Asymmetric information leads to an inefficient outcome: some firm types below
$$ \tilde{z}$$ have an incentive to mimic those above $$ \tilde{z}$$, which leads to labor
hoarding by all firms in the pooling interval (since the higher types do not want to be perceived as low types).
The next section discusses the general properties of any pooling interval in this model. In particular, I show the existence of an equilibrium with partial pooling on a central interval by first describing the interval, characterized by a pooling employment level (Section 5.2.1) and lower/upper interval bounds (Section 5.2.2). Next, I show that this proposed equilibrium has no profitable deviation. Finally, in Section 5.2.3 I state sufficient conditions for the existence of an equilibrium with partial pooling.
5.2 Pooling in equilibrium
As a first step in characterizing the equilibrium with partial pooling in Section 5.2.1, I assume that this equilibrium exists and I discuss the equilibrium beliefs that workers hold when pooling is observed (Lemma 1), the pooling level of employment in relation to firms' genuine preferences (Lemma 2), and workers' beliefs regarding possible off-path deviations (Lemma 3).
5.2.1 Pooling level of employment and workers' beliefs
In order for a pooling interval to exist in equilibrium, it must be profitable for at least some firm type relative to the alternative of choosing a genuine level of employment. The first lemma asserts that pooling can only be sustained if it induces workers to choose high effort.
Proof: If workers' posterior beliefs lead to
$$ E_{\mu}[z\vert n=N'_P]<\tilde{z}$$ upon observing employment level $$ N'_P$$, then
from the workers' strategy defined in eq.(16), firms that pool on $$ N'_P$$ would get low effort from workers. However, holding worker effort constant at
eL,
$$ N^*(z,e_L)$$ is the genuine choice of employment for these firm types. The genuine choice of employment is at least weakly more profitable than the pooling action $$ N'_P$$ when
$$ E_{\mu}[z\vert n=N'_P]<\tilde{z}$$, so firms would have a profitable deviation from $$ N'_P$$. Therefore, the pooling interval must be characterized by
$$ E_{\mu}[z\vert n=N_P] \geq \tilde{z}$$.
The lemma above shows that the pooling interval has no profitable deviation only if workers provide high effort after observing the pooling level of employment. This is the case only if their expectation of z for the pooling interval is at least $$ \tilde{z}$$, where $$ \tilde{z}$$ denotes their indifference point between providing low or high
effort.
In the proposed equilibrium with partial pooling, a firm with type $$ \overline{z}$$ is the best-type firm that pools. As the next lemma shows, this firm type has an incentive (and the capacity) to deviate from any pooling level of employment, unless the pooling level coincides with its genuine choice of employment.
Proof: See Appendix section 9.
Therefore, eq.(18) defines the level of pooling employment in terms of the interval upper bound. The firm type
$$ \overline{z}$$ is included in the pooling interval by virtue of choosing its genuine level of employment and it has no reason to try to move out of the pooling interval, since as established by
Lemma 1, pooling induces high effort provision from workers. Aside from the firm at
$$ \overline{z}$$, which is choosing its genuine level of employment even with pooling, the pooling action involves over-employment for other firm types in the interval. Their genuine choices for
n are lower than the pooling level (in Figure 1, note that firms' genuine employment choices when workers observe z lie below the pooling level of employment). Although these firms prefer lower employment levels, their ability to deviate from
the pooling equilibrium and still receive high effort from workers depends crucially on workers' beliefs about off-path actions.
Based on Lemma 2 above, the pooling level of employment NP entails over-employment for all firm types in the pooling interval (except the firm type at upper bound
$$ \overline{z}$$). Therefore, the potentially attractive deviations for firms in the pooling interval would be off-path actions with lower levels of employment than the pooling level,
$$ N_O<N_P$$.14
As discussed in Section 4.2.2, when workers see a deviation to an off-path action NO, their posterior beliefs are updated with the equilibrium dominance refinement. With this refinement, workers' beliefs about z when observing off-path actions NO depend upon the types of firms that might have reason to deviate from the equilibrium strategy. The lemma below characterizes workers' beliefs for the most plausible deviation to a lower level of employment,
that $$ E_{\mu}[z\vert n=N_O]<E_{\mu}[z\vert n=N_P]$$ when they observe off-path action $$ N_O<N_P$$.
Proof: See Appendix section 9.
Since workers use their posterior beliefs about z to decide effort, workers' beliefs about off-path actions help to pin down the firm's on-path equilibrium strategy. As shown in the next section, although NO might appear to be an
attractive deviation for some firm types in the pooling interval, the resulting impact on workers' posterior beliefs imposes a constraint on the deviations that would actually be profitable.
In the next section, I first show that the only equilibrium pooling level without a profitable deviation is located such that workers provide high effort for $$ n=N_P$$ (Lemma 4) but low effort for any lower n, including any off-path action (Corollary 4a). Therefore, firms face a trade-off: pool to get high effort or choose a genuine level of employment but get low effort from workers. I show in Lemma 5 that the incentives for lower-type firms to pool is monotonically increasing in z. This result is used to derive a cutoff for the pooling interval (Corollary 5a); firms below this cutoff value find pooling too costly relative to their genuine choice of employment even though their genuine employment choice leads to low effort from workers. The section concludes with Proposition 1 in which I provide a complete definition of the pooling interval, including its bounds and the relevant firm and workers' strategies for pooling.
5.2.2 Pooling interval bounds
This section introduces two equations, both in terms of interval lower bound $$ \underline{z}$$ and upper bound $$ \overline{z}$$, that determine the pooling interval location. The first equation is summarized in the lemma below.
| $$\displaystyle \int_{\underline{z}}^{ \overline{z}}z\frac{g(z)}{G(\overline{z})-G(\underline{z})}dz =\tilde{z}$$ | (19) |
Proof: From Lemma 1, the pooling action must have
$$ E_{\mu}[z\vert n=N_P] \geq \tilde{z}$$. However, if pooling action $$ N'_P$$ resulted in
$$ E_{\mu}[z\vert n=N'_P] >> \tilde{z}$$, then some firm types in the pooling interval would have a profitable deviation to a lower level of employment. A deviation to
$$ N'_P-\epsilon$$ would still lead to beliefs
$$ E_{\mu}[z\vert n=N'_P-\epsilon]>\tilde{z}$$ and worker effort choice eH. Since this logic can be repeated for any $$ N'_P$$ with
$$ E_{\mu}[z\vert n=N_P']>\tilde{z}$$, in equilibrium the only pooling action without a profitable deviation must satisfy
$$ E_{\mu}[z\vert n=N_P] = \tilde{z}$$.
This lemma shows that the equilibrium pooling action is one that minimizes the over-employment associated with pooling but still satisfies the belief requirement of Lemma 1. The corollary below discusses the implications of workers' beliefs about on-path pooling
actions for their beliefs about off-path actions. Specifically, the pooling interval is located such that any level of employment lower than the pooling level induces workers to provide low effort.
Corollary 4a (Off-path beliefs) Off-path employment level $$ N_O<N_P$$ induces workers' beliefs
$$ E_{\mu}[z\vert n=N_O]<\tilde{z}$$ and workers' effort choice
$$ e(n=N_O)=e_{L}$$.
Proof: Lemma 3 concludes that
$$ E_{\mu}[z\vert n=N_O]<E_{\mu}[z\vert n=N_P]$$. Since Lemma 4 established that NP results in beliefs
$$ E_{\mu}[z\vert n=N_P] = \tilde{z}$$, it follows that
$$ E_{\mu}[z\vert n=N_O]<\tilde{z}$$ when $$ N_O<N_P$$.
In other words, firms cannot choose a lower employment level than the pooling level and still get high effort from workers. Therefore, any firm considering pooling faces the choice between its genuine employment choice or the pooling level of employment, which is higher than the genuine level
for low-type firms.
The following notation is used to denote the change in profits that a firm of type z could expect in a deviation from the pooling level of employment, where workers' on-path beliefs lead to eH for the pooling level of employment
(Lemma 1) and off-path beliefs lead to eL for any deviation to a lower level of employment (Corollary 4a).
Notation: When the pooling level upper bound is $$ \hat{z}$$ and the pooling employment level is
$$ N_P(\hat{z})=N^*(e_H, \hat{z})$$ based on the result of Lemma 2,
$$ \Delta \Pi_P(z, \hat{z})$$ is the difference in profits between pooling and not pooling for firm type z,
$$ \Delta \Pi_P(z, \hat{z})= \pi(e_H,z,N_P(\hat{z}))-\pi(e_L,z,N^*(e_L,z))$$ where $$ z<\hat{z}$$
$$ \Delta \Pi_P(z, \hat{z})$$ therefore represents the difference between the firm's profits with the pooling level of employment and high effort from workers, compared to the firm's profits with
its genuine level of employment but low effort from workers.
As established below, this difference is increasing in firm type; firms with lower levels of z are more sensitive to the costs of pooling. Firms with intermediate levels of z (those close to $$ \tilde{z}$$) benefit more from pooling in order to get high effort. Of course, firms with very high levels of z can choose $$ N^*$$ and still get eH. The next lemma establishes that, for firms with low to intermediate values of z, the incremental profits from pooling to get high effort are increasing in firm type.
If pooling is weakly profitable for a certain firm type $$ \underline{z}$$ then it is also profitable for all firm types $$ z \in [\underline{z},\hat{z}]$$ compared to the alternative of $$ N^*$$ with eL. Furthermore, $$ \forall \hat{z} \in [\tilde{z},z_{max}]$$, there exists a unique $$ \underline{z} \in (0, \hat{z})$$ where $$ \Delta \Pi_P(\underline{z}, \hat{z})=0$$.
Proof: See Appendix section 9.
This result is used in the next corollary to define the lower bound of the pooling interval in terms of its upper bound (and later, in Proposition 2, the sorting condition is used to show that an equilibrium with partial pooling has no profitable deviation).
Although the sorting condition is satisfied for any
$$ \hat{z} \geq \tilde{z}$$ (all plausible values of
$$ \overline{z}$$), the upper bound of pooling is also pinned down by the beliefs condition of eq.(19), which is the first equation that defines the pooling interval
bounds.
The second such condition, eq.(20) below, defines
$$ \underline{z}$$ as the type of firm that is indifferent between choosing its genuine employment level and getting low effort from workers or choosing the pooling action, which entails
over-employment but results in high effort.
.
Corollary 5a (Zero profit from pooling at lower bound)
In an equilibrium with partial pooling, if
| $$\displaystyle \Delta \Pi_P(\underline{z},\overline{z})=\pi(e_H,\underline{z},N^*(e_H, \overline{z}))-\pi(e_L,\underline{z},N^*(e_L,\underline{z}))= 0 \ $$ | (20) |
then firms have no profitable deviation from pooling action $$ a(z)=N_P$$ when $$ z \in [\underline{z}, \overline{z}]$$.
Proof: Lemma 2 established that a deviation to a higher level of employment is not attractive to these firm types
$$ z \in [\underline{z}, \overline{z}]$$. Of all deviations to a lower level of employment, $$ N^*$$ is the most attractive, however, Corollary 4a established that any employment level less than NP results in low effort from workers. When
$$ \Delta \Pi_P(\underline{z}, \overline{z})=0$$, Lemma 5 established that for all
$$ z \in [\underline{z}, \overline{z}]$$, pooling on
$$ N_P=N^*(e_H, \overline{z})$$ is more profitable than a genuine level of employment with low effort from workers. Therefore, firms in the pooling interval defined above have no profitable
deviation from NP.
The lemma above uses the sorting condition to show that no firm type in the pooling interval would have a profitable deviation away from pooling when
$$ \Delta \Pi_P(\underline{z}, \overline{z})=0$$. However, the reverse is also true: when Corollary 5a holds, no firm type outside of the pooling interval would have
an incentive to deviate by choosing the pooling level of employment (this is shown formally in Proposition 2). Since pooling in this model requires over-employment for low firm types, types below
$$ \underline{z}$$ find pooling for high effort too costly relative to their alternative genuine employment choice with low effort.
To conclude this section, I use Lemmas 1-5 and their corollaries to state Proposition 1. This proposition focuses on the pooling interval; it characterizes the bounds of the interval, the firm's strategy when $$ z \in [\underline{z}, \overline{z}]$$, and the workers' strategy when NP is observed.
Iff
$$ \ \exists \ \left\{\underline{z},\overline{z} \right\}$$, with
$$ z_{min} < \underline{z}<\overline{z} < z_{max}$$ s.t. eq.(19) and eq.(20);
then in equilibrium, the firm's optimal action on the pooling interval is
$$ a(z)=N_P$$
$$ \forall z \in[\underline{z},\overline{z}]$$ where NP is given by eq.(18); workers' beliefs are consistent,
and their optimal action when pooling occurs is given by eq.(17).
The proposition above characterizes a pooling interval in which firms pool on an employment level that is at least weakly higher than their genuine choice. Once pooling arises, because of the beliefs that would result from an off-path deviation, higher-type firms in the interval (those above
$$ \tilde{z}$$) cannot "opt-out" lest they be perceived as a lower-type firm. Lemma 5 tells us that all of the firms in the pooling interval earn higher profits by
pooling than they would earn from choosing a genuine employment level when workers perceive them as a lower-type firm. Therefore, in this model, pooling necessarily occurs on a compact interval.
In the next section, I show that the equilibrium with partial pooling that was characterized here is robust to potential deviations (Proposition 2). In Proposition 3, I present a sufficient condition for existence. I address the issue of uniqueness by establishing that the most natural candidate, a separating equilibrium, does not exist in the signaling model (Proposition 4).
5.2.3 Equilibrium existence
As the next proposition establishes, when the pooling interval bounds can be defined based on Proposition 1, a perfect Bayesian equilibrium of the signaling model includes a pooling interval. The firm's strategy is to pool for levels of z in the pooling interval but choose its genuine level of employment for other values of z. In equilibrium, workers provide high effort to firms that pool or choose even higher employment levels than NP; workers provide low effort to firms with lower levels of employment.
| $$\displaystyle \ \ b(n)=\begin{cases}e_{L} \ \ & \text{for }n<N_P \\ e_{H} & \text{for }n \geq N_P\\ \end{cases}$$ | (21) |
and there is no profitable deviation from this equilibrium.
Proof: See Appendix section 9.
When the conditions in Proposition 1 are satisfied, Proposition 2 shows that a partial pooling equilibrium has no profitable deviation. Firm types below
$$ \underline{z}$$ find pooling too costly; firm types above
$$ \overline{z}$$ can get high effort from workers by choosing their genuine employment level. For reasons discussed earlier, firm types within the pooling interval have no profitable deviation
from the pooling level of employment.
In fact, the partial pooling equilibrium stated above exists under fairly general conditions. To anticipate the results of Proposition 3, both $$ \underline{z}$$ and $$ \overline{z}$$ are well-defined values in the domain of z whenever $$ z_{min}=0$$ and $$ z_{max}=\infty$$.
Proof: See Appendix section 9.
The proof uses Lemma 5 to show that a solution for
$$ \underline{z}$$ in terms of
$$ \overline{z}$$ based on eq.(19) yields
$$ 0<\underline{z}<\overline{z}$$ and therefore
$$ \underline{z}$$ is well-defined whenever $$ z_{min}=0$$. For
$$ \overline{z}$$ to satisfy eq.(20), it is enough that
$$ z_{max}=\infty$$ and $$ \tilde{z}$$ is not excessively large. When these conditions are
met, there exists an equilibrium of the signaling model with partial pooling on an interval that lies within the domain of z.
The next proposition establishes that the signaling model has no equilibrium in which all firm types choose their genuine levels of employment, which argues for the uniqueness of the proposed equilibrium with partial pooling.
Proof: Let
$$ \tilde{z}^-=\tilde{z}- \epsilon$$. In a fully separating equilibrium, a firm of type
$$ \tilde{z}^-$$ would get low effort from its genuine employment choice
$$ N^*(e_L,\tilde{z}^-)$$ and a firm of type $$ \tilde{z}$$ would get high effort from its
genuine employment choice
$$ N^*(e_H,\tilde{z})$$. However, if a firm of
$$ \tilde{z}^-$$ type were to get high effort, its optimal genuine labor choice would be
$$ N^*(\tilde{z}^-,e_H)$$. By an envelope theorem argument, the firm at
$$ \tilde{z}^-$$ has a profitable deviation to
$$ N^*(e_H,\tilde{z})$$, since workers observing
$$ n=N^*(e_H,\tilde{z})$$ provide eH in the fully separating candidate equilibrium. The higher employment choice imposes a second-order loss in profits,
while the benefit of workers switching from low to high effort provision provides a first-order gain. Therefore, the proposed fully separating equilibrium always has a profitable deviation for some firm type just below $$ \tilde{z}$$.
The proposition above shows that the signaling model does not have an equilibrium that corresponds to the full information benchmark in which every firm chooses its genuine employment level.
6 Employment Dynamics
Having presented the main mechanisms at work in the one period signaling model above, this section sheds some light on the main question of this paper, namely the implications of signaling for the path of employment adjustments.
Throughout this discussion I focus on the negative shocks that a firm experiences over time.15 In particular, I focus on shocks for which zt-1 and/or zt are in the pooling interval. For other values of z below
$$ \underline{z}$$ and above
$$ \overline{z}$$, the signaling model is uninteresting because the pooling equilibrium looks the same as the full information benchmark.
The intuition of the model dynamics can be grasped by looking at the graph below. In this stylized example, the firm has been doing well for a long period of time. Assume it starts from point A, where it is choosing a genuine level of employment and workers can infer the exact value of
z. At some point in time, the firm experiences a negative shock to point B (that lies in the pooling interval). The firm reduces employment, but by less than prescribed by the full information benchmark. If the firm gets another shock to point C, it makes almost no
adjustment to employment.16 Only a further shock to a level of z below the pooling interval will prompt a layoff (for example,
the shock from Point C to Point D). At this point, the firm conducts a lumpy, mass layoff. Any negative shock that takes the firm below Point D will lead it to make the same employment adjustments as in the case of full information.
While this graph shows an example of the model dynamics, the next section develops a fully fledged multiperiod model to study the magnitude of employment adjustments as a function of the previous level of z and the size of the most recent shock.
6.1 Set-up of a dynamic model
This section illustrates a firm's optimal employment choices over multiple periods where the firm and its workers maximize expected profits/utility and employment choices reflect signaling considerations. In the following analysis, zt is privately observed
by the firm and evolves according to a first-order Markov process. Although the firm experiences an idiosyncratic shock at the beginning of every period, this model will show that firm types in the pooling interval may hoard labor across multiple periods. Layoffs, when they happen in the model, are
often pent-up and disproportionate relative to the most recent change in business conditions.
The dynamic model consists of multiple periods of the signaling game, given by Stages 1-4 from the one period model. Wage contracts are treated as state-dependent but predetermined and it is assumed that workers observe the firm's business condition when they receive
their wages in Stage 4.17
The level of the firm's business conditions evolves as a first-order Markov process (i.e. the realization of zt depends on zt-1 in the previous period but does not depend explicitly on any earlier history of z realizations). The process evolves as $$ g: \Delta_{Z_{t-1}} \rightarrow \Delta_{Z_t}$$ and workers have rational expectations about the firm's business condition from one period to the next. Therefore, $$ g(z_t\vert z_{t-1})$$ summarizes their prior at the beginning of period t.
6.1.1 Value functions
In this section I present the multiperiod profit maximization problem of the firm and multiperiod utility maximization problem of the worker, both of which are additively separable across periods with constant discount rates $$ \beta$$. Although these are formulated as recursive problems, Claim 2 will show that equilibrium beliefs and strategies can depend on the state variable zt-1 but are otherwise
stationary.
The firm's dynamic profit maximization problem is,
| $$\displaystyle V_t^\pi(z_t)=\underset{n_t}{\max} \ \pi_t(e^*_t,z_t,n_t)+ \beta E_t[V_{t+1}^\pi(z_{t+1})\vert z_t]$$ | (22) |
where
$$ V_t^\pi(z_t)$$ is the firm's value function with current business condition zt and $$ \pi(.)$$ denotes the firm's one period expected profit function from eq.(6). These functions are indexed by t for the case of asymmetric information in which workers' beliefs depend on zt-1 and their beliefs determine the firm's pooling strategy.
The workers' dynamic utility maximization problem is,
| $$\displaystyle V_t^E(z_t)=U(e^*_t,z_t) + \beta E_t[(1-s_{t+1}(z_{t+1}))V_t^E(z_{t+1})\vert z_t]$$ | (23) |
s.t.$$ \ \ e^*_t=$$ arg$$ \underset{e_t}{\max} \ E_{\mu}[V_t^E(z_t)\vert n_t, z_{t-1}]$$
where
$$ V_t^E(z_t)$$ is the value to a worker of being employed by a firm of type zt and $$ U(.)$$ is the workers' one period expected utility. A worker's probability of layoff from a firm of type $$ z_{t+1}$$ is given by
$$ s_{t+1}(z_{t+1})$$, which is indexed by time to reflect its dependence on the firm's previous employment choice.
To summarize eq.(23) above, a worker's value of employment at firm of type zt is the worker's expected utility at time t plus the expected future value of employment at the firm. That future
value is discounted by the probability of being laid off between time t and time t+1; the value of being unemployed is set to zero (without loss of generality).
In Claim 2 below I show that, for both the firm and the workers, each period of the dynamic problem can be reduced to the same decision as in the one period model.
Proof (Signaling model): The claim above holds even when there is asymmetric information between the firm and its workers. In this case, nt has signaling value and workers use their beliefs to choose effort. However, despite the
differences from the full information benchmark, the timing of the multiperiod model dictates that workers observe the true value of z in Stage 4 when their wages are paid. Since the worker knows the exact value of zt-1 at time t, the firm would not be able to deceive the worker about the longer history of z through its choice of nt. For the same reasons as stated in the proof above, the
firm's optimal strategy in the dynamic signaling model is stationary and analogous to eq.(13) in the static model: any deviation from its optimal static strategy in a given period of the dynamic model would entail lower profits in that period and no increase in
profits in the following periods. The workers' optimal strategy in the dynamic signaling model is analogous to eq.(16) in the one period model. Of course, unlike the one period model, at each time t of the multiperiod model all
equilibrium strategies (and workers' beliefs) depend on zt-1.
To provide a more formal definition based on the discussion above, a Markov perfect Bayesian equilibrium for the multiperiod signaling game consists of (i) the firm's optimal strategy,
$$ a(z_t\vert z_{t-1})=n_t$$; (ii) the workers' optimal strategy,
$$ b(z_t\vert n_t, z_{t-1})=e_t$$; and (iii) workers' beliefs,
$$ \mu(z_t\vert n_t,z_{t-1})$$. In equilibrium, workers' beliefs must be consistent and strategies must be sequentially rational best responses.
Obviously there are infinitely many possible sequences of shocks and adjustment paths that could arise in the equilibrium of a dynamic model. Although the equilibrium of the multiperiod model is well-defined, an additional complication arises: the pooling bounds and pooling level of employment change as zt-1 changes.18 Therefore, in order to characterize the changing employment levels, the next section uses three dimensional graphs to show how adjustments to employment depend on past business conditions zt-1 and realized shocks $$ \Delta z=z_{t}-z_{t-1}$$.
6.1.2 Illustration of model dynamics
The signaling model in this paper is not a steady state model but rather one of adjustments over the short- to medium-term. In line with the main research question, I focus primarily on negative shocks for which the signaling model dynamics differ from the full information benchmark. These
involve either $$ z_{t+1}$$ and/or zt in the pooling interval.
When z is initially high and is then affected by a negative shock that brings it into the pooling region, this leads to inaction and labor hoarding - an initial underadjustment. When another negative shock (or series of shocks) arrives and is large enough to result in
z below the pooling interval, this leads to a lumpy downward change in employment - an overadjustment relative to the counterfactual model. The next two sections highlight the signaling model's underadjustment and overadjustment dynamics, respectively.
Initial underadjustment
This section shows the change in employment in the signaling model when the firm experiences a negative shock into the pooling interval. This can be understood as either a shock from Point A to B or from Point B to C in Graph 3.
The graph below compares adjustments in the signaling model to the adjustments in a model with full information. This comparison is reported as the ratio of adjustments; the ratio is equal to one when the signaling model adjusts by the same number of workers as the counterfactual model. A ratio
of less than one indicates that the signaling model shows labor hoarding relative to the counterfactual model.
The counterfactual model always has negative adjustments in response to negative shocks, whereas the signaling model has smaller adjustments in many instances and sometimes even shows small positive adjustments to negative shocks. In all of the cases graphed above, once the level of z falls into the pooling interval, the signaling model shows underadjustment relative to the full information benchmark (i.e. a ratio of <1 in the graph).
Subsequent overadjustment
Here I look at the next possible type of shock, from an intermediate value of z to a value below the pooling interval, such as the change from Point C to Point D in Graph 4.
The three dimensional graph below shows that changes in employment in the signaling model can be larger, in some cases much larger, than employment changes in the counterfactual full information setting. While the adjustment ratio is around 2 for many combinations of initial levels and shocks
(indicating that these adjustments are twice as large as those in the benchmark model), in this example the ratio is well above 7 when z starts near the pooling lower bound. In the signaling model, these lumpy patterns come from one-time overadjustments that occur when
the firm switches from pooling to choosing its genuine level of employment in response to a negative shock. This shock does not need to be large for an overadjustment to take place: with a small shock, the counterfactual model predicts a small adjustment whereas the signaling model shows a very
large adjustment. Therefore, the ratio in the graph is often highest for the smaller-sized shocks.
Summary- The simulations presented in this section illustrate the role of private information and workers' incentives. In the signaling model, negative shocks that lead to pooling result in underadjustment. Firms experiencing shocks that are not too large hoard labor while in the pooling interval and can appear inactive. However, a subsequent negative shock can lead to a layoff that is considerably larger than the adjustment predicted by the counterfactual full information model. The inaction and lumpy layoff pattern in the multiperiod signaling model matches many empirical findings from other papers that were mentioned in the introduction.
6.2 Aggregate shocks: the case of recessions
As emphasized by Lazear, et al. (2013), lower outside options might have positive effects on workers' effort provision.
The most straightforward extension is one in which workers must shirk in order to search for a new job.19 When a recession lowers the value of workers' outside options, it could therefore decrease their indifference point between providing low/high effort from $$ \tilde{z}$$ to $$ \tilde{z}'<\tilde{z}$$. When their outside option is weak, workers are willing to stay at a firm with values of $$ z \in [\tilde{z}',\tilde{z}]$$, whereas before the recession began they would have wanted to leave these firm types. The next section provides a summary of this and other findings from the signaling model with workers' effort decisions.
6.3 Summary of equilibrium dynamics
The equilibrium strategies described in the multiperiod model above have the following implications for the dynamics of employment adjustments:
- When both zt-1 and $$ z_t \in [\underline{z}_t,\overline{z}_t]$$, the model shows labor hoarding (i.e. over-employment) relative to the full information genuine $$ N^*$$ benchmark and the firm underadjusts in response to negative shocks.
- When zt-1 and zt are both either very low or very high, the magnitude of adjustments are the same as in the model with full information.
- When $$ z_{t-1} \in [\underline{z}_t,\overline{z}_t]$$, a negative shock that pushes zt below $$ \underline{z}_t$$ leads to a mass layoff episode. These adjustments move the firm from labor hoarding to a genuine choice of employment. The mass layoff is a pent-up "overadjustment" that reflects changes in labor demand from all previous labor hoarding periods.
- Workers' incentives for effort may be affected by the value of their outside option. If workers must shirk in order to search on the job, $$ \tilde{z}$$ defines the point at which workers are indifferent between staying at their employer or not. In the signaling model, if the expected value of workers' outside options falls during a recession then workers would be willing to provide high effort for firm types that would have otherwise inspired shirking. In this manner, the signaling model may be consistent with evidence in Lazear, et al. (2013) that worker effort increased during the 2008 Recession.
Although it is difficult to summarize the full set of possible scenarios, some of the dynamics described above are consistent with the stylized facts discussed in Section 2. Many firms adjust their employment infrequently, yet there are firms that make disproportionate changes to employment in a short period of time. The mechanisms presented here might help to explain why U.S. employment adjustments might be lumpy even in the absence of large non-convex statutory or technological firing costs.
7 Discussion
Mass layoffs are common in practice, but why do firms choose lumpy patterns over more gradual employment adjustments? This model suggests that the signaling mechanism may contribute to the lumpiness of firm-level employment adjustments when the firm has private information about its business condition, workers' incentives matter for production, workers' contracts have a profit-sharing structure, and the terms of these contracts are predetermined.20 While information asymmetry is a key ingredient for the signaling model, the gap between the firm and the workers' information does not need to be large. One may think that executives have better information than production workers; early departures by key executives could reveal negative information to less-informed workers. However, even if the firm's only informational advantage is a more accurate one-period ahead forecast- perhaps because it has better knowledge of its production technology- this still produces enough information asymmetry to generate lumpy employment dynamics in the model. Several features of workers' contracts also play a role in generating these employment dynamics. As in a standard principal-agent problem, workers' incentives matter for production and effort is non-contractible. Although workers' remuneration cannot depend directly on effort, the optimal agreement is essentially a piece rate-style contract in which workers are paid according to what they produce. The model also relies on the assumptions that workers produce more when working for a better employer and that contracts specify earnings as a share of output. The latter feature is akin to profit-sharing, which could also represent some of the non-pecuniary benefits that workers derive from providing high effort when working for a better employer.21 A final assumption is that wages are not negotiated within each period; contracts are agreed upon ex-ante. Although there is some contractual rigidity in wages, workers' earnings are actually state-contingent because they depend on worker output (which is determined by effort and, importantly, by the firm's business condition). The features described above can be summarized as: (1) non-contractible effort, (2) interaction between worker effort and firm quality, and (3) quasi-rigid wages. Although it is difficult to develop a complete and realistic microfoundation, all three features are quite common in practice. In fact, they are probably most common when workers have long-term relationships with their employers (so employers care about worker morale) and effort matters but there is no effective monitoring technology. As I show in this paper, bonus contracts can be optimal under these conditions. In reality, workers often earn bonuses that are a function of their own effort and the employer's business condition (where a worker's contribution may be inferred from metrics of output such as sales, publications, or billable hours). Consistent with the model, there is some evidence that industries that with performanced-based compensation are more exposed to mass layoffs.22 A model of endogenous labor adjustment costs has important implications for both positive and normative policy conclusions. First, looking at the model's key assumptions could help policy makers better predict the industries that are more prone to mass layoffs. This could be used to allocate unemployment insurance budgets or to create targeted job-finding resources for the workers most at-risk. Second, labor hoarding may be an important driver of aggregate productivity dynamics- Burnside, et al. (1993) find, "...a significant proportion of movements in the Solow residual are artifacts of labor hoarding type behavior". A better understanding of labor hoarding dynamics could potentially lead to new conclusions about the propagation of aggregate and sectoral shocks. Third, Davis & von Wachter (2011) emphasize that mass layoff episodes are an important policy issue because of their sizable adverse effects on workers. Since mass layoffs appear to be particularly detrimental to workers' health and careers, policies encouraging more gradual adjustments may improve workers' welfare.23 The signaling model presented in this paper suggests that the prevalence of large mass layoffs could potentially be reduced by decoupling workers' incentives from their employer's profitability. The role of workers' effort in the model also shows that endogenous adjustment costs can have large effects on employment dynamics at a micro level. This suggests that exogenous adjustment costs (for example, statutory firing costs, mandated severance payments, and layoff prenotification policies) may have less of an impact on employment frictions than previously thought.
8 Conclusion
This paper shows that a signaling model in which workers reduce their effort in response to layoffs can generate lumpy employment adjustments that seem prevalent in U.S. microdata. In this model, workers' incentives depend on the quality of their employer but workers only observe the firm's
employment decisions. Although labor hoarding is costly, some firm types will postpone layoffs in the face of small shocks to conceal negative information and continue receiving high effort from their workers. The equilibrium employment strategy of firms leads to pent-up adjustments: when a firm's
business conditions get substantially worse it makes a full adjustment. This adjustment entails laying off all workers that were not needed in the previous periods of decline, which brings the firm to its genuinely optimal (lower) level of employment. The microfoundation in this paper provides some
predictions about the relationship between labor contracts, firms' financial conditions, and employment dynamics and therefore can inform policies that try to anticipate and mitigate the negative effects of mass layoffs.
Proof of Claim 1:
Since the profit function is strictly concave in n, for each e the firm's profit maximization problem has a unique maximum for each z. When the set of N is convex, the Maximum Theorem leads to the further conclusion that the optimal employment correspondence for each level of effort e, $$ N^*(e,z)$$, is a single-valued (continuous) function in z. Therefore, for each e, $$ N^*(e,z)$$ is invertible in z. The composite function $$ a_F(z)$$ is given by $$ N^*(e_L,z)$$ for $$ z<\tilde{z}$$ and $$ N^*(e_H,z)$$ for $$ z \geq \tilde{z}$$. Since $$ N^*(e,z)$$ is increasing in both z and e and $$ e_L<e_H$$, $$ N^*(e_L,\tilde{z})<N^*(e_H,\tilde{z})$$ and the function $$ a_F(z)$$ is also invertible.
Proof of Lemma 2 (Pooling level):
The first step, based on eq.(13), is to notice that the pooling level of employment can only lie in the interval defined by $$ N_P \in [N^*(e_L, \underline{z}), N^*(e_H,\overline{z})]$$. If NP were lower or higher than that range, it would encroach on the level of employment that another firm type chooses in the equilibrium strategy. This would contradict the definition of the pooling interval bounds as $$ [\underline{z},\overline{z}]$$. The next step is based on Lemma 1, which concluded that workers' beliefs about NP are $$ E_{\mu}[z\vert n=N_P] \geq \tilde{z}$$ and workers' equilibrium effort choice is $$ b(n=N_P)=e_H$$. Upon observing an employment level greater than the pooling level, workers' beliefs would be slightly higher ( $$ E_{\mu}[z\vert n>N_P]>\tilde{z}$$) and thus workers would still provide high effort; see Lemma 3 for a related proof based on the equilibrium dominance refinement. For a firm of type $$ \overline{z}$$, if the pooling employment level were lower than its genuine choice such that $$ N_P<N^*(e_H, \overline{z})$$, it would have a profitable deviation to $$ n=N^*(e_H, \overline{z})$$. Combining these results shows that $$ N_P=N^*(e_H, \overline{z})$$ is the unique pooling employment level.
Proof of Lemma 3 (Off-path refinement):
Equilibrium dominance leads workers to refine their beliefs; only firm types that might stand to improve their equilibrium payoff about the firm type would have chosen a given action. Therefore, when off-path action $$ N_O<N_P$$ is observed and $$ N_P=N^*(e_H, \overline{z})$$, workers look at the types of firms that would want to choose an employment level lower than NP. Workers' beliefs are refined to:
| $$\displaystyle \ \mu(z\vert n=N_O)=\begin{cases}\frac{g(z)}{G(\overline{z})-G(\underline{z})} & \text{for} \ z \in [\underline{z},\overline{z})\\ \ 0 & \text{for all other } z\\ \end{cases}$$ | (24) |
Thus, when an off-path action is observed, workers' beliefs put no weight on the possibility that $$ z=\overline{z}$$ and put more weight on the lower realizations of z relative to when they observe NP. This leads to the result that $$ E_{\mu}[z\vert n=N_O]<E_{\mu}[z\vert n=N_P]$$ when $$ N_O<N_P$$.
Proof of Lemma 5 (Partial sorting condition):
When getting low effort from workers, the firm can do no better than choosing its optimal genuine employment level $$ N^*$$. However, the profits that it gets may be smaller than the profits it
would get from pooling on employment to get high effort from workers. To illustrate the firm's trade-off between choosing $$ N^*$$ with eL or the pooling
level NP with eH, Figure 6 shows the relevant profit calculations for the numerical example. Although the graph shows one example of the partial sorting condition, the main
properties (derived below) hold more generally for any parameter configurations of the model.
Changes in profits with respect to z $$ (\Delta \Pi_P(z, \hat{z}))$$:
- At z=0, the firm's profits from pooling with $$ N_P=N^*(e_H, \hat{z})$$ are strictly less than its profits from choosing its genuine level of employment, $$ N^*$$. The firm's profits at z=0 are: $$ \pi(e_H,0,N_P)=-X(N_P)$$ and $$ \pi(e_L,0,N^*(e_L,z))=0$$ from pooling and not pooling, respectively. Thus, $$ \Delta \Pi_P(z=0,\hat{z})<0$$. Note that $$ \Delta \Pi_P(z, \hat{z})$$ is defined as the difference between the profits that low-type firms would get from pooling, $$ \pi(e_H,z,N_P)$$, and the profits that they would get from not pooling, $$ \pi(e_L,z,N^*(e_L,z))$$.
- At $$ z=\hat{z}$$, the pooling employment level with high effort is strictly more profitable than the firm's genuine employment level with low effort,
$$\displaystyle \pi(e_L,\hat{z},N^*(\hat{z},e_L))<\pi(e_H,\hat{z},N^*(e_H, \hat{z})) \ \Rightarrow \Delta \Pi_P(\hat{z},\hat{z})>0.$$ (25)
| $$\displaystyle \frac{\partial \pi(e_L,z,N^*(e_L,z))}{\partial z}=(1- \omega^*) q_z(e_L,z)N^*(e_L,z)$$ | (26) |
The derivative (with respect to a change in z) of the firm's profit with NP and eH is,
| $$\displaystyle \frac{\partial \pi(e_H,z,N_P)}{\partial z}=(1- \omega^*) q_z(e_H,z)N_P$$ | (27) |
Since $$ 0<q_z(e_L,z)<q_z(e_H,z)$$ and $$ 0 \leq N^*(e_L,z) < N_P$$ for all $$ z \in [0, \hat{z}]$$,
| $$\displaystyle \frac{\partial \pi(e_L,z,N^*(e_L,z))}{\partial z}<\frac{\partial \pi(e_H,z,N_P)}{\partial z} \ \ \ \forall \ z \in [0, \hat{z}]$$ | (28) |
- From the properties detailed above, I can conclude that $$ \Delta \Pi_P(z, \hat{z})$$ is increasing in z for all $$ z \in [0, \hat{z}]$$. The value of the firm's profits from pooling with high effort versus not pooling with low effort cross only once and thus for any $$ \hat{z}$$ there is a unique value of $$ \underline{z} \in (0, \hat{z})$$ for which $$ \Delta \Pi_P(\underline{z}, \hat{z})=0$$.
Proof of Proposition 2 (Equilibrium with partial pooling):
The workers' equilibrium strategy indicates that workers provide high effort iff they observe $$ n \geq N_P$$. Otherwise, if they observe a lower level of employment they provide low effort. This means that the relevant comparison for a profitable deviation to an employment level below NP is $$ \Delta \Pi_P(z, \overline{z})$$, which calculates the difference in profits from pooling with high effort compared to separating with low effort.
- When $$ \Delta \Pi_P(\underline{z}, \overline{z})=0$$, a firm of type $$ \underline{z}$$ is indifferent between NP and $$ N^*(e_L,\underline{z})$$. Based on the Partial sorting condition of Lemma 5, for $$ z \in [z_{min},\underline{z})$$ (i.e. firm types below the pooling interval) the firm's profits would be lower from pooling; firms with $$ z \in [z_{min},\underline{z})$$ have no incentive to deviate from their optimal non-pooling (genuine) employment level, with the resulting low effort.
- As established by the Partial sorting condition above, the difference in profits between pooling and not pooling is greater than zero for all firm types in the pooling interval,
$$\displaystyle \pi(e_H,z,n)-\pi(e_L,z,N^*(e_L,z)) > 0 \ \ \forall z \in (\underline{z},\overline{z})$$ (29)
A firm with type $$ z \in (\underline{z}, \overline{z})$$ within the pooling interval would have no incentive to deviate from the equilibrium because of the impact on workers' beliefs (and their resulting effort decision). - A firm with type $$ z=\overline{z}$$ is choosing its genuine optimal employment level in the pooling equilibrium and getting high effort from workers, so it has no incentive to deviate (either up or down) from $$ N_P=N^*(e_H, \overline{z})$$.
- Finally, there is no incentive for a firm with $$ z \in (\overline{z},z_{max}]$$ to deviate from its genuine choice of employment in equilibrium. Since the firm is already getting high effort from workers, its choice of $$ N^*(e_H,z)$$ is better than any profit it could earn from choosing a different n from its optimal genuine employment level.
Proof of Proposition 3 (Sufficient condition for existence):
The bounds of the pooling interval are pinned down by two equations: eq.(19) defines "Beliefs" of workers and eq.(20) defines a "Zero profit"condition for firms. Figure 7 shows the three other conditions described
below and illustrates the intuition of the solution.
Existence is established by the fact that both equations define continuous functions. The implicit function for $$ \underline{z}(\overline{z})$$ from eq.(19) is continuous and decreasing in $$ \overline{z}$$. The implicit function for $$ \underline{z}(\overline{z})$$ from eq.(20) is continuous and increasing in $$ \overline{z}$$. For eq.(19), $$ \underline{z}=\tilde{z}$$ when $$ \overline{z} = \tilde{z}$$. Instead, at $$ \overline{z} = \tilde{z}$$ eq.(20) has a lower $$ \underline{z}$$, specifically $$ \underline{z} \in (0, \tilde{z})$$. At the other extreme, $$ \underline{z}=\tilde{z}$$ corresponds to $$ \overline{z}>\tilde{z}$$ in eq.(20).
Given the bounds described above, the functions must intersect at some point. Since both functions are monotonic they will cross only once: the solution is unique. Finally, to conclude the proof, it should also be clear from the graph that since $$ \left\{\underline{z}=\tilde{z},\overline{z} = \tilde{z} \right\}$$ is not a solution to eq.(20), the pooling interval bounds are given by $$ z_{min} < \underline{z}<\overline{z} < z_{max}$$. As stated in Proposition 1, if $$ \tilde{z}<<z_{max}$$ then the system of equations has a solution whenever the domain of z is given by $$ z_{min}=0$$ and $$ z_{max}=\infty$$. This additional assumption, $$ \tilde{z}<<z_{max}$$, comes from the requirement that $$ 2 G(\tilde{z})<G(z_{max})$$, which means that there exists a point $$ \underline{z}=0$$ and $$ \overline{z}>\tilde{z}$$ satisfying eq.(19). The last statement establishes that a solution for eq.(19) exists even when $$ \underline{z}=0$$, whereas $$ \underline{z}$$ is always greater than zero in eq.(20). This range of values for eq.(19) is depicted as the leftmost vertical line in Figure 7.
10 Appendix: Microfoundations of Assumptions
In the baseline model, two important assumptions were introduced to make the model more tractable. The first important assumption in this model is that effort can take only two values, eH or eL. This assumption can be micro-founded from a variety of utility configurations, as discussed in Section 10.1. The second assumption is that contracts between workers and firms consist of a wage with expected value strictly increasing in z. This feature is microfounded in the context of a principal-agent problem in which the firm makes a take-it-or-leave-it contract offer to the worker in Stage 0; the details of such a contract are discussed Section 10.2 below. Alternative models of workers' increasing compensation could involve either worker-firm bargaining or career concerns in which workers' expected tenure and success with a firm depends on the firm's business condition. The principal-agent framework is the most quantifiable of the three options mentioned above, since a bargaining model in which the firm negotiates with each worker over the expected surplus is sensitive to assumptions about the firm's outside options and a career concerns model necessarily spans multiple periods.
10.1 Effort provision
Throughout the model I assume that effort is a binary choice that leads to discontinuous outcomes. A binary effort decision rule by workers could actually stem from a utility maximization problem with corner solutions: once the compensation for effort becomes sufficiently high, the worker may find it optimal to switch from the minimum effort to the maximum effort level. This would typically only arise if the disutility from effort is not convex. The restriction to binary effort provision by workers is common in many models, such as Shapiro & Stiglitz (1984); it is also common in the principal-agent mechanism design literature. In this signaling model, the effort choice can stem from the workers' decision problem of whether or not to shirk in order to search on the job for a new employer. Notwithstanding the idea that the binary decision of a worker stems from deciding whether to search for a new job, the effect of search on effort (i.e. shirking) is crucial in this model. If the only disciplining force on the firm's layoffs were the threat of an immediate quit, then the model's partial pooling equilibrium would not result.
10.2 Derivation of $$ \omega ^*$$ from an ex-ante principal agent contract
In an extension to the model I consider the case in which the firm offers workers a wage contract before the signaling game takes place. For a meaningful signaling game, I only look at scenarios in which the firm does not always implement high effort. Since high effort is costly, the firm is willing to accept low effort provision from workers in some states of the world (this implies a mild restriction on parameter values).24 The firm offers each worker a wage contract ex-ante before either the firm or the worker has any knowledge of z beyond the prior $$ g(z)$$. Effort and business conditions are not verifiable, but contracts can be specified in terms of the worker's individual contribution y. It is assumed that contracts can only be written in terms of output share $$ \omega$$, which affects the firm's expected profits ($$ \pi$$), the workers' expected wages, and their indifference point between providing low and high effort ($$ \tilde{z}$$). Furthermore, the worker is subject to limited liability (normalized to zero). Therefore, a feasible contract requires weakly positive payments to the worker for any realized value of y. A general formulation of the principal-agent problem is presented here and a numerical example is solved for in the table of Appendix section 11.
The assumption above restricts contracts to those linear in the worker's contribution. Without this restriction, an optimal contract would still be weakly increasing in the expected value of y but it would generally contain non-linearities based on threshold
quantities of y.25 However, a contract that is linear in $$ q(e,z)$$ could arise naturally if output is restricted to two levels
$$ \left\{0, q_{H}\right\}$$ and effort and business conditions determine the probability that qH is realized. In that context,
a contract to elicit high effort for higher values of z would offer payment $$ \omega>0$$ when qH results, which would mean
that the worker's expected payment is linear in $$ q(e,z)$$, the expected level of individual output. In this setting, the firm offers a contract to each worker that maximizes its own expected
profit (as given in Stage 4). The firm can anticipate the workers' effort provision strategy and thus it designs the ex-ante contract taking the workers' ex-post effort decision as given. For each worker, the firm chooses an optimal sharing parameter to solve its
uninformed principal problem,
$$ \omega^*=$$arg$$ \underset{\omega} {\max} \ \ \ \ \pi(e^*,z,n=a(z),\omega)$$
s.t. $$ e^*=b(n)$$ Incentive compatibility
$$ 0 \leq \int_{z_{min}}^{z_{max}}\omega q(e^*,z)dG(z)$$ Participation constraint with
$$ \overline{U}=0$$
$$ 0 \leq \omega q(e,z)$$
$$ \forall \ e$$ and z Limited liability; minimum of $$ -l=0$$
In this case, the limited liability constraint is more stringent than the participation constraint and, since
$$ q(e,z) \geq 0$$, both can be reduced to the condition
$$ \omega \geq 0$$. As shown by Proposition 4, when the firm has private information about z an equilibrium with full separation
does not exist. Here, in addition to determining the bounds of the pooling interval (analogous to those in Proposition 1), the value of $$ \omega$$ is an
endogenous ex-ante choice for the firm. In choosing $$ \omega ^*$$, the firm anticipates the pooling actions that will arise in the equilibrium of the signaling game in Stages
1-4 of the model. For non-pooling values of z,
$$ a(z)=N^*(e^*,z, \omega)$$. For pooling values of z, however, the profit function depends on NP and the bounds of the pooling
interval described by
$$ [\underline{z},\overline{z}]$$.
$$ \pi(.)=\int_{z_{min}}^{\underline{z}}\pi(e_L,z,N^*(e_L,z,\omega),\omega)dz+\int_{\underline{z}}^{ \overline{z}}\pi(e_L,z,N_P(\omega),\omega)dz+\int_{\overline{z}}^{z_{max}}\pi(e_H,z,N^*(e_H,z,\omega),\omega)dz$$
The bounds of the interval and the level of pooling employment are defined in the proposition below.
$$ \omega^*=$$argmax$$ \underset{\omega \geq 0} \ \ \pi(e^*,z,n=a(z),\omega)$$ s.t.
$$ \int_{\underline{z}}^{ \overline{z}}z\frac{g(z)}{G(\overline{z})-G(\underline{z})}dz =\tilde{z}(\omega)$$
$$ \pi(e_H,\underline{z},N^*(\overline{z},e_H,\omega),\omega)-\pi(e_L,\underline{z},N^*(e_L,\underline{z},\omega),\omega) = 0$$ $$ e^*=b(n)$$ then in equilibrium, the firm's optimal strategy on the pooling interval is:
$$ a(z)=N_P$$ $$ \forall z \in[\underline{z},\overline{z}]$$, where NP is given by $$ N_P=N^*(\overline{z},e_H,\omega)$$
workers' beliefs are consistent, and their optimal strategy when pooling occurs is:
$$ b(n=N_P)=e_H$$
The problem above is most interesting when the endogenous value of $$ \omega ^*$$ is such that the workers' indifference point $$ \tilde{z}(\omega^*)$$, implicitly defined in terms of $$ \omega ^*$$, lies in the interior of the domain of z. This is equivalent to Assumption 2 that was discussed in Section 3.4.2 when $$ \omega ^*$$ was treated as exogenous. When that condition is satisfied, the firm's equilibrium strategy is characterized by partial pooling as established in Propositions 2 and 3. For the ex-ante principal-agent contract, one such example in which Assumption 2 is satisfied endogenously is provided in the second column of the table in Appendix section 11 below. The optimal wage contract implements low effort for values of z below $$ \underline{z}$$ (i.e. below the pooling interval lower bound) and implements high effort for all other z.
11 Appendix: Example Parameters
The numerical examples are based on the following parameters and functional forms:
Example Parameters
| Graphs | Principal-Agent Problem |
|---|---|
| $$ X(n)=\frac{1}{4}n^4$$ | $$ X(n)=\frac{1}{2}n^2$$ |
| $$ \Psi_L=0$$, $$ \Psi_H=0.5$$ | $$ \Psi_L=0$$, $$ \Psi_H=0.01$$ |
| $$ f(e_L)=0.6$$, $$ f(e_H)=0.7$$ (deterministic) | $$ f(e_L)=0.3$$ $$ ,f(e_H)=0.5$$ (deterministic) |
| $$ \omega^*=0.33$$ (exogenous) | $$ \omega^*=0.089$$ (endogenous) |
| $$ \tilde{z}=15$$ | $$ \tilde{z}(\omega^*)=0.56$$ |
| $$ \underline{z}=8.5$$ and $$ \overline{z}=17.8$$ | $$ \underline{z}=0.50$$ and $$ \overline{z}=0.62$$ |
|
$$ g(z_t\vert z_{t-1}) \sim N(z_{t-1},5)$$ (with $$ z_0=16$$) on domain $$ [0, 35]$$ |
$$ g(z)$$ uniform on $$ [0,1]$$ |
Bibliography
Review of Economic Studies 47(2):321-338.
Journal of Political Economy 83(6):1183-1202.
International Economic Review 35(4):821-32.
Review of Economic Studies 57:381-402.
Journal of Political Economy 102(5):841-877.
Harvard University Press.
Econometrica 77(3):623-685.
NBER Working Papers 18208, National Bureau of Economic Research.
Journal of Political Economy 101(2):245-73.
American Economic Review 87(1):115-37.
Quarterly Journal of Economics 102(2):179-221.
Journal of Monetary Economics 54 (Supplement):56-78.
Journal of Economic Perspectives 20(3):3-26.
Quarterly Journal of Economics 128(2):581-622.
NBER Working Papers 17638, National Bureau of Economic Research, Inc.
Labour Economics 16(1):1-11.
Unpublished manuscript.
American Economic Review 79(4):674-89.
Journal of Economic Literature 34(3):1264-1292.
Journal of the European Economic Association 5(5):1043-1063.
Journal of Economics & Management Strategy 6(4):899-913.
Unpublished manuscript.
The Quarterly Journal of Economics 117(3):1075-1103.
Journal of Labor Economics 18(3):405-26.
RAND Journal of Economics 31(2):345-358.
American Economic Review 74(3):433-44.
Quarterly Journal of Economics 87(3):355-74.
Unpublished manuscript.