Lecture note on moral hazard explanations of efficiency wages

Transcription

1 Lecture note on moral hazard explanations of efficiency wages (Background for this lecture is the article by Shapiro and Stiglitz, in the reading list) The value function approach. This approach is used in the Shapiro-Stiglitz model, to be discussed in this note below. It will later be used in several other models, such as the Mortensen-Pissarides and Pissarides search and matching models; and some of the models on turnover costs. It is an important technical tool for theoretical labor market analysis. The value function approach is closely related to Bellman s principle of dynamic programming. This principle implies that complex dynamic problems can be simplified tremendously, by essentially considering one step at the time. Usually, practical applications of this principle imply that individual decision makers, starting from a particular state, may in the future transfer to one or more other states. The rates of transition to new states may depend on both stochastic factors, and on the individual s own actions.

2 The key to using the value function approach is to view the value of being in a given current state as an asset, with a particular return. This return has two main parts: First, a net current return. Secondly, an expected change in asset value due to transitions to new states. This principle is exemplified below when using the Shapiro- Stiglitz model below. Typically for a worker, one such state is unemployment, and another state is employment. The worker may transfer from the current state of unemployment, to a possible future state of employment. The rate at which this transfer occurs may depend on the individual s actions (e.g. on search effort, or his/her wage demand), or on worker characteristics (e.g. the worker s productivity). Relevant states may however also directly depend on ones action. Thus, in the Shapiro-Stiglitz model below, one state is being employed and working hard, and another state is being employed and taking it easy.

3 Moral-hazard based efficiency wage theory: The Shapiro- Stiglitz (SS) model Basic idea: Workers must be motivated to be productive. High wage serves as motivator. The standard (canonical) model in this literature is that of Shapiro-Stiglitz (SS), who make the following assumptions: - the wage paid by firms is constant, - firms do not monitor workers perfectly, - workers are identical, and firms fire workers when they are caught shirking. These assumptions together are shown to lead to efficiency wages. The efficiency wage is here the same as a non-shirking wage, required for the worker not to shirk on the job. SS consider the following asset values: V u = the (present discounted lifetime) value (or simply value ) of currently being unemployed. V n = the value of currently being employed and working hard, V s = the value of currently being employed and shirking (meaning that one takes it easy on the job). When working hard, a worker puts up effort e and has productivity 1. When shirking, there is no effort, and no productivity from this worker.

4 Firms hire workers and monitor them (check that they are productive). This checking does not occur continuously, only at particular points of time, with stochastic intervals of expected length 1/q, which are assumed to be exponentially distributed. This implies that regardless of how long time has elapsed since the last check, the expected amount of time to the next check is 1/q. If a worker is caught shirking, he is fired immediately, and replaced by another worker. If the worker is checked and found to work hard, he is retained in the firm. He is also retained whenever there is no checking. Individual jobs are assumed to disappear at a rate b. This implies that the expected lifetime of a job is 1/b, for a worker who does not shirk. These assumptions permit the following asset equations to be derived, for the value of shirking (V s ) and the value of nonshirking (V n ) respectively : (1) rv s = w + (b+q)(v u V s ) (2) rv n = w - e + b(v u V n ) In (1), rv s is the return to the asset V s, assuming a common interest rate r. The first term on the right-hand side is the current return, w (the wage), for a shirking worker, who does not put up any effort. The second term expresses the expected value of the change in the asset, (b+q)(v u V s ), where b+q is the total rate at which this change occurs for a shirker (remember that a shirker also suffers a probability q of being caught and then fired). V u V s expresses the asset change once it occurs, i.e., once the worker loses his job. This change is generally negative, i.e. there is a utility loss associated with losing ones job.

5 In (2) there is a lower current return, w-e, since the worker puts up effort e when working hard, and thus the current utility is lower. This is however rewarded in the future, since the capital loss V u V n is suffered with smaller probability (since q is missing in (2)). Intuitively, when the worker now is checked, he keeps his job and continues receiving the return w-e. An essential part of the solution to the SS model is derivation of a non-shirking condition (NSC), i.e., a condition on the wage which makes it attractive for a worker to work hard instead of shirking. The NSC will hold if the constraint V n V s is fulfilled. Note that whenever this constraint is binding, the solution chosen by the firm will imply that V n = V s. (This must be the case at equilibrium, since in the opposite case, V n > V s, the nonshirking constraint has no influence on the firm s wage decisions. We will later see that this is contradictory, because the firm here always wants to set the wage as low as it can.) The NSC condition V n V s, together with equations (1)-(2), can then be used to solve for the two endogenous variables V n and V s, and in addition find a constraint on w, the socalled NSC constraint. This is found to be as follows: (5) w rv u + (r+b+q)e/q w(ns), The wage level given by equality in (5), called w(ns), is what we call the non-shirking wage. We can show that firms will here never want to set the wage higher than w(ns).

6 We next need to find an expression for V u, which can be found in a way similar to the other value functions (1) and (2): (6) rv u = v + a(v n V u ). Here v is the current return from being unemployed (unemployment benefits or leisure), and a is the rate of job finding among unemployed workers, i.e., the rate at which the capital gain V n V u is realized. Combining (6) with (5) yields the following non-shirking constraint: (9) w v + e + (a+b+r)e/q At a long-run equilibrium we have the following steady-state relationship between total labor force N, employment L and unemployment U = N-L: (10) a(n-l) = bl, where N is total labor force, and L is total employment. au = a(n-l) is the number of workers who find jobs at any given time, and bl similarly the number of workers who lose their jobs at any given time (given no shirking). These two numbers must be equal in a market equilibrium. Define now u = (N-L)/N = U/N as the rate of unemployment, and 1-u = L/N as the rate of employment. Thus from (10), a = bl/(n-l) = b(l/n)(n/(n-l)) = b(1-u)/u = b/u b. This implies that a+b = b/u.

7 Using (10) in (9) we now find a new non-shirking constraint, which can be expressed as follows: e bn (11) w e + v + + r = e + v + ( e / q)( b / u + r) w( ns). q N L (11) shows that the non-shirking wage requires there to be some unemployment at equilibrium. We see namely that when the unemployment rate u goes to zero, w(ns) tends to infinity. We however see that as q tends to infinity (the case of perfect monitoring), u will tend to zero and we get a (nearly) standard competitive solution. In this case there is continuous checking of workers, and the moral hazard problem does not have any implications for the solution. Firms choose employment by maximizing profits, leading to the standard condition (12) F (L) = w(ns). This can be illustrated in a simple diagram, in the SS paper and in class.

8 Extension 1 A restriction on the problem posed in the SS model is that firms are required to pay fixed wages over time to their workers. We now consider an extension of the basic SS model, to the case where the firm uses a two-step wage schedule, and may pay a lower initial wage. We will see that this in principle may alleviate, or even eliminate, the problem of unemployment in their model. Assume now the following: - When the worker first arrives, he is initially paid a low wage. - After the worker is monitored for the first time, the wage is increased to a higher level. Assume that the wage is kept constant after that time. We can consider the initial period (up to the first monitoring of the worker) as a test period, during which the wage is kept low. If performance is observed to be satisfactory when the worker is controlled, the worker will get a raise after the control. In important point is that the worker does not know exactly at what time the next control comes. This is stochastic, and as before, the expected period up to first control is 1/q (technically, controls arrive with constant rate q). We must now extend the value function concept introduced in the SS model, to consider two different values for a worker, namely 1) the values (of working and shirking) of just starting to work, V n (1) and V s (1), and the values of continuing in the firm working and shirking after the first control and possible raise in wages, V n (2) and V s (2).

9 We have the following asset equations for newly hired workers: (13) rv n (1) = w 1 e + q[v n (2) V n (1)] + b[v u - V n (1)] (14) rv s (1) = w 1 + (q+b)[v u V s (1)]. We are now interested in the non-shirking constraint for newly hired workers, which can be found setting V n (1) V s (1), yielding (15) q[v n (2) V u ] e. The most important point to notice from (15) is that the nonshirking constraint on newly hired workers only depends on the value of the contract after the possible future raise, V n (2), and not at all on the initial contract value, V n (1). This implies, importantly, that the initial wage w 1 can be set as low as we want, without incentives for newly hired workers being affected. We find that w 2 (ns) has the same form as before, namely (16) w 2 (ns) = rv u + [(r+b+q)/q]e. The value of unemployment can be found as (17) rv u = b + a[v n (1) - V u ].

10 V n (1) cannot be set lower than V n, since the worker otherwise would not have any incentive to join the firm. Assume however that it can be set as low, i.e., V n (1) = V u. (remember that, from (15), V n (1) does not affect incentives of workers to shirk or work). From (17) this implies (18) V u = b/r. This implies that the value of being unemployed in this case only depends on the current benefits from unemployment. Assuming that (15) holds with equality, (19) V n (2) = V u + e/q. From (13) we now find: (20) (r+b+q)v n (1) = w 1 -e+qv u +e+bv u. Since V n (1) = V u, this implies (21) w 1 = rv u = b.

11 The main implications of there results are as follows: 1. The worker works for free in the initial period (up to the first control). The worker then gets compensation for the foregone unemployment benefit b, but not for effort e. 2. The labor market clears in this case. This follows from V n (1) = V u in this case. V n (1) is here just the expected contract value of a new worker, which at equilibrium in this case is the expected value of unemployment.

12 Extension 2 Consider briefly an extension of the above framework to the case of regular cyclical fluctuations for the firms in question. Assume that the output price facing the firm is either p H or p L < p H. Assume that when the firm is in state i, the rate of transition to the other state equals b(i), i = H,L. This in turn implies that on average the firm is in state H for a period of 1/b(H), and in state L for a period of 1/b(L). The fraction of the time the firm is in state H is then b(l)/[b(h)+b(l)]. This corresponds to regular business cycles which are perfectly foreseen (in a statistical sense). Consider now a firm with an efficiency wage structure otherwise as in the SS model. One can then show that the firstorder conditions for the firm with respect to employment, in the high-demand and the low-demand state, are as follows: (22) (23) p ( L) f '( L) = w( L) = ( q + r + s + a( L)) e p ( H ) f '( H ) ( q + r + s + a( H ) + b( H )). q e q Here a(l) and a(h) are worker rehiring rates in the two states (if states are uncorrelated across firms, i.e. all cycles are fully specific to individual firms, a(l) = a(h) = a; if cycles are correlated, a(l) a(h) in general).

13 The implication is that we have a domain for relative prices defined by p( L) q + r + s + a (24), 1 p( H ) a + r + s + a + b( H ) where employment is fixed. The intuitive reason for this result is that when the firm lowers employment going into a recession, and workers are aware of this, such a lowering of employment leads to a risk of layoffs for workers. It turns out that this risk of layoff must be compensated in the high-demand state, in the form of a higher efficiency wage, since this risk makes it more difficult to enforce worker effort. An alternative for the firm is to keep employment constant over the cycle, in which case no increase in the wage is necessary going into the high-demand period. The domain for relative prices above gives the condition under which employment is kept constant. A main implication of this result is that being employed is more risky in high-demand periods than in low-demand periods, whenever firms change employment over the business cycle. Under low demand, firms employment may be lower, but those who have a job, have a safe job, since the situation cannot get worse, only better. Under high demand, all jobs are less safe because there is a possibility of going into a recession, leading to layoffs. (A basic assumption is here that all workers have the same probability of employment when going into a recession.)