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Endogenous employment rate in the efficiency wage shirking approach.

Abstract

The aim of New Keynesian theorists is to obtain Keynesian results on the basis of maximizing behavior. Accordingly, the New Keynesian shirking models depict a world of fully rational maximizing agents where equilibrium unemployment is the main consequence of the payment of efficiency wages. The problem is that oversimplified nature of most shirking models has until now prevented a full investigation of the interdependence of unemployment, the effort supplied by workers and labor demand. This article shows that the existence of this interdependence weakens the whole approach. In particular, when the unemployment rate is considered a truly endogenous variable, the stability of the macroeconomic equilibrium is generally incompatible with the existence of unemployment ascribed to the fact that firms pay efficiency wages. (JEL J41)

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Introduction

The shirking version of the efficiency wage approach is one of the most popular explanations of unemployment proposed by New Keynesian Economics. The general aim of New Keynesian theorists is to obtain Keynesian results, such as equilibrium involuntary unemployment, on the basis of maximizing behavior. Accordingly, the shirking models [Calvo, 1979; Calvo and Wellisz, 1979; Shapiro and Stiglitz, 1984; Yellen, 1984; Bowles, 1985; Akerlof and Yellen, 1985 and 1986; Katz, 1986; Stiglitz, 1987; Weiss, 1991; Pisauro, 1991; Simmons, 1991; Phelps, 1994; Albrecht and Vroman, 1996; and Laszlo, 2000] depict a world of fully rational maximizing agents where a functional relationship between workers' effort and other variables generates a moral hazard problem (firms may not know the effort supplied by each worker perfectly) that disrupts the competitive framework. In particular, these models assume that (identical) workers will shirk more as their wages fall, thus forcing maximizing firms to pay an equilibrium wage above the Walrasian one in order to obtain an adequate (profit maximizing) level of effort from workers. The aggregate result is less-than-full employment with any re-equilibrium path based on underbidding by the unemployed destined to be ineffective.

Although the existence of equilibrium unemployment is the main consequence of efficiency wages, the oversimplified nature of most models has until now prevented a full investigation of the interdependence existing between unemployment, the effort supplied by the workers, and labor demand. In particular, many authors admit that the effort should be considered as a continuous function both in the unemployment rate and the real wage. (1) Nevertheless, they consider the unemployment rate as given and the effort as a function of the real wage alone, arguing that this assumption does not alter the results [Shapiro and Stiglitz, 1984, p. 435, note 4; Stiglitz 1987, p. 8, note 11]. On the contrary, this article aims to show that modeling the unemployment rate as an endogenous variable bears three important consequences. First, building of micro and macroeconomic labor demand functions becomes non-trivial and badly behaved (upward sloping) segments of the demand schedules are a usual result, so that multiple equilibria may occur. Second, even if well-behaved labor demand schedules are assumed, the stability requirements of the macroeconomic equilibrium are demanding. Third, the stable efficiency wage macroeconomic equilibria are meaningless for the theory since they all correspond to particular cases in which unemployment would exist even without efficiency wages and to utterly unrealistic values for the unemployment rate (always above 50 percent). The existence of these problems can jeopardize the New Keynesians' ambition of using the shirking models in order to obtain equilibrium involuntary unemployment results on the basis of maximizing behavior.

This article is organized as follows. In the following section, the microeconomic equilibrium for the efficiency wage shirking model is obtained on the basis of a continuous effort function in both the wage and the unemployment rate. Next, the same model is used to discuss the characteristics of the microeconomic labor demand. Then, macroeconomic equilibrium is obtained and its stability requirements are discussed. Finally, the last section sums up the main results.

The Microeconomic Equilibrium

The following analysis examines the efficiency wage as a cause of unemployment (and not of wage differentials), (2) accepting the hypotheses on which the New Keynesian shirking models rest. (3) In particular, it is assumed that there are many profit maximizing, identical firms in the economy. Each firm considers other firms' decisions (and hence, also the unemployment rate) as given in choosing its profit maximization programme. The product market is perfectly competitive and the price of the good produced is equal to 1. There are many identical workers in the economy. At a given unemployment rate, workers increase their shirking (and run a higher risk of being caught and fired) (4) when the wage rate declines, whereas, they work harder when the wage rises (mainly because the utility loss connected with unemployment is now larger). Firms cannot perfectly monitor the effort supplied by the workers [Shapiro and Stiglitz, 1984, pp. 435-37; Yellen, 1984, section 2.1; Stiglitz, 1987, pp. 4 ff.; Pisauro, 1991, pp. 329-31; Romer, 1996, p. 446; and Laszlo, 2000, p. 248].

On the basis of these assumptions and considering effort as a continuous function in both the wage and the unemployment rate, (5) the microeconomic equilibrium can be obtained as follows. For any given level of the unemployment rate, u, a definite relationship between effort and the wage rate exists. (6) The effort function, e = e(w, [bar.u]), referred to in this article (and in the main contributions, such as in Stiglitz, 1987, p. 5 and Laszlo, 2000, p. 247), which has a minimum value equal to zero and a maximum value higher than one, is represented by the curve drawn in Figure 1 and possesses the following properties: (7)

e(w, [bar.u]) = 0 if w [less than or equal to] [w.sub.1]

[de(w, [bar.u])]/[dw] > 0

[[[d.sup.2]e(w, [bar.u])]/[d[w.sup.2]]]{[> 0 if w < [w.sub.2]]/[< 0 if w > [w.sub.2]]}. (1)

[FIGURE 1 OMITTED]

Each firm's output depends on the number of workers hired ([L.sub.i]) and the effort supplied by these workers. Since efficiency wage models assume that labor input and effort enter the production function multiplicatively [Pisauro 1991, p. 330; Romer 1996, p. 443], the microeconomic production function is:

[y.sub.i] = [f.sub.i] ([L.sub.i]e(w, [bar.u])). (2)

and the single firm's profit is:

[[pi].sub.i] = [f.sub.i]([L.sub.i]e(w, [bar.u])) - w[L.sub.i]. (3)

The first order conditions for profit maximization are therefore:

[[[partial derivative][f.sub.i]([L.sub.i]e(w, [bar.u]))]/[[partial derivative][L.sub.i]]] = w, (4)

[[[partial derivative][f.sub.i]([L.sub.i]e(w, [bar.u]))]/[[partial derivative][L.sub.i]e(w, [bar.u])]][[de(w, [bar.u])]/[dw]] = 1, (5)

and the second order conditions (omitted here) are satisfied if [[[d.sup.2]e(w, [bar.u])]/[d[w.sup.2]]] < 0.

The ratio between (4) and (5) gives condition

[[de(w, [bar.u])]/[dw]] = [e(w, [bar.u])]/w, (6)

which determines the efficiency wage, w* (with w* > [w.sub.2], so that [[[d.sup.2]e(w*, [bar.u])]/[d[w*.sup.2]]] < 0), that each profit maximizing firm has to pay and the resulting level of effort, e*, that it obtains from workers (again figure 1). On the other hand, condition (4) determines the number of workers, [L*.sub.i], that the firm has to hire. Therefore, each firm should pay a wage high enough to minimize the cost of a unit of efficient labor, that is, minimize the ratio w/e and hire workers up to the point at which the marginal product of a unit of efficient labor is equal to its cost. The fundamental difference with respect to the competitive approach is that each firm simultaneously determines both the wage, w*, that it pays and the number of workers, [L*.sub.i], that it hires.

The Microeconomic Labor Demand Function

Since w* and [L*.sub.i] can vary only if u varies, the microeconomic labor demand function is an implicit relationship between w* and [L*.sub.i] obtained by considering all the possible values of the unemployment rate. Consequently, the labor demand function, [L*.sub.i] (w*), plays a minor role in the efficiency wage approach, while the function [L*.sub.i](u), which shows the relationship between the unemployment rate and the single firm's labor demand, plays a major role.

In any event, since the single firm's labor demand must fulfill condition (4), the behavior of [L*.sub.i] as a function of u (and, implicitly, as a function of w*) depends on the way in which both the marginal productivity function [d[f.sub.i]([L.sub.i]e(w*, [bar.u]))]/[d[L.sub.i]] and the efficiency wage, w*, vary when u varies. Hence, the behavior of [L*.sub.i] depends ultimately on the characteristics of the production function and those of the e(w*, u) function.

In general, the behavior of e* when u varies is not easily ascertained because of the inability of the efficiency wage approach to obtain a univocal formal specification of the effort function (either by intertemporal utility maximization or by empirical evidence). (8) A similar problem also arises for the effort function described previously. In the latter case, it is uncontroversial to assume that given the wage rate, workers consider it worse to be fired when unemployment is high since they presumably will have a harder time finding another job, so that

[de([bar.w], u)]/[du] > 0. (7)

However, this assumption and the related condition

[dw*]/[du] < 0 (w* tends to zero as u [right arrow] 1), (8)

are not enough to establish a priori whether the equilibrium level of effort increases or decreases when the unemployment rate rises, that is, to obtain a univocal sign for the comparative static derivative ([de(w*, u)]/[du]). In fact, workers will expend a higher level of effort if u rises and their wages remain unchanged (due to relation 7). However, if u rises, the equilibrium wage decreases (due to relation 8) and, ceteris paribus, the effort decreases too (due to relations 1) and vice versa. Therefore, a direct assumption on the sign of the comparative static derivative ([de(w*, u)]/[du]) must be made. Together with conditions (7) and (8), it is convenient to assume:

([de(w*, u)]/[du]) > 0 [for all] u, with 0 [less than or equal to] u [less than or equal to] 1. (9)

Relation (9) is justified by the circumstance that assuming the opposite (a negative value of the derivative) would be incompatible with the main characteristics of the efficiency wage approach. In fact, an important characteristic of efficiency wage models is that e* tends to equal zero as full employment approaches. Since workers consider 1 - u as an approximation of the probability of being hired by another firm after having been fired for shirking, they will shirk in full employment (they will furnish the lowest possible level of effort) even in the presence of a wage rate that tends to be infinite. (9) This implies that at least in the vicinity of u = 0, the derivative cannot show a negative value. However, relation (9) goes one step further and rules out the possibility (fully legitimate from a theoretical viewpoint) that the derivative shows a negative value for any value of the unemployment rate, and thus, bears important implications for the uniqueness of the equilibrium. (10)

Less problems arise with the production function. In fact, it is generally true that, under whatever assumption regarding the effort function e(w*, u), a different marginal productivity function [d[f.sub.i]([L.sub.i]e(w*, [bar.u]))]/[d[L.sub.i]] exists for each level of the unemployment rate. This circumstance implies that [[partial derivative][f.sub.i]([L.sub.i]e(w*, u))]/[[partial derivative][L.sub.i]] may show lower or higher values for higher values of the unemployment rate and hence, for lower values of the wage rate. Hence, the single firm's labor demand can either rise or fall when the unemployment rate rises (and the wage rate falls) as shown in figure 2 (where the wage decreases from [w*.sub.1] to [w*.sub.2] and [w*.sub.3], while the labor demand first increases from [L*.sub.1] to [L*.sub.2], and then decreases from [L*.sub.2] to [L*.sub.3]); and vice versa. However, the most common production functions easily ensure that:

d/[du] ([[partial derivative][f.sub.i]([L.sub.i]e(w*, u))]/[[partial derivative][L.sub.i])] > 0, (10)

so that condition (9) is enough to get well-behaved (i.e. monotonically upward sloping) [L*.sub.i](u) schedules. (11)

[FIGURE 2 OMITTED]

On the contrary, if condition (9) is rejected, ([d[L*.sub.i]]/[du]) < 0 (and therefore ([d[L*.sub.i]]/[dw*]) > 0) cannot be prevented. This circumstance would not constitute a major problem for a theory in which the traditional re-equilibrium path based on underbidding by unemployed workers does not operate. However, the possible non-monotonicity of the function [L*.sub.i](u) can cause multiple equilibria and, in general, difficulties in dealing with the micro (and, aggregating, the macro) economic characteristics of the efficiency wage models. Since this paper only deals with a unique equilibrium case, condition (9) is assumed to hold.

The Macroeconomic Equilibrium

In aggregate, the (short-run) labor demand function ALD(u) can be obtained as the sum of single firms' labor demands [L*.sub.i](u). Well-behaved microeconomic functions give rise to monotonically upward sloping ALD(u) schedules with a positive horizontal intercept, since as the economy approaches full employment, w* tends to infinity whereas e* and the marginal productivity of labor tend to be nil, like the labor demand. (12) The macroeconomic equilibrium can be obtained as the solution of the system made of the ALD(u) and the relation:

u = [LS - L]/[LS], (11)

which defines the unemployment rate as a function of the number of employees, L, and of the labor supply, LS. In fact, the condition for achieving the equilibrium is that the aggregate labor demand corresponding to the equilibrium level (u*) of the unemployment rate generates an unemployment rate equal to u*, so that ALD(u*) = LS - LSu*. A well-behaved ALD(u) schedule gives rise to a unique equilibrium, as shown in figure 3, whereas, multiple equilibria may be the outcome of non-monotonicity in the [L*.sub.i](u) functions. In this scheme, if ALD(u*) < LS, the underbidding offers of unemployed workers are refused by maximizing firms and the equilibrium position features involuntary unemployment.

[FIGURE 3 OMITTED]

The Stability Requirements of the Macroeconomic Equilibrium

In the case of a well-behaved ALD(u) and a unique macroeconomic equilibrium, the conditions that must be fulfilled to obtain stability results are demanding and bear important implications. To achieve these conditions, the dynamics of the labor demand and the unemployment rate can be represented by the following model:

AL[D.sub.t] = g([u.sub.t-1]), (12)

[u.sub.t] = [LS - AL[D.sub.t]]/[LS]. (13)

In these relations, the single firm's labor demand at time t (and thus the aggregate labor demand AL[D.sub.t]), is a (generic, at this stage) function g of the unemployment rate of the previous period [u.sub.t-1] (which in turn depends on the firms' employment choices of that period). Whereas the current unemployment rate depends upon the current firms' labor demand. (13) Linearizing relation (12) around u*, the following relation is obtained:

AL[D.sub.t] = [eta][u.sub.t-1] + [thetav], (14)

so that:

[u.sub.t] = 1 - [[thetav]/[LS]] - [[eta]/[LS]] [u.sub.t-1]. (15)

Therefore, the equilibrium is stable if |[eta]/[LS]| < 1 and since no economic reason justifying this condition exists, stability is not the general outcome of models founding equilibrium unemployment only on efficiency considerations.

If the AL[D.sub.t] function is thoroughly linear, the fact that labor demand tends to be nil close to full employment implies that in relation (14), [thetav] < 0 and the fulfillment of the stability condition |[eta]/[LS]| < 1 implies that the equilibrium unemployment rate exhibits absolutely unrealistic values (always above 50 percent) and that the existence of equilibrium unemployment cannot be ascribed solely to the circumstance that firms pay efficiency wages. (14) In fact, if [eta] < LS, even for small values of the parameter [thetav], aggregate labor demand would never reach the level necessary to employ the whole labor supply: and this would be true also for u = 1. Now, since when u = 1 effort is at its highest level, in the case of well-behaved [L*.sub.i](u) functions the firms' labor demand for a positive wage is at least of the same amount as in a Walrasian, competitive system (in which effort is equal to one). In these circumstances, if labor demand was lower than labor supply, it would also be lower in a Walrasian context (for a wage rate above zero), implying that the existence of unemployment cannot be ascribed to the fact that firms pay efficiency wages. (15)

The conclusion is that all the equilibria in which unemployment depends on efficiency considerations (cases in which the traditional approach would determine full employment) are unstable, whereas all the stable equilibria correspond to cases in which there would be unemployment even without efficiency wages.

The graphic analysis of the stability of the efficiency wage equilibria is proposed in figure 4 where relation (13) and two different ALD schedules are depicted: the AL[D.sup.I] (obtained from relation (14) for [eta] > LS) that generates an unstable equilibrium and the AL[D.sup.II] (obtained for [eta] < LS that generates a stable equilibrium. When the latter is obtained, it is evident that the highest possible level of labor demand (with u = 1 and therefore, with the highest level of the effort) is below the full employment level LS.

[FIGURE 4 OMITTED]

For the general case of a non-linear AL[D.sub.t] function, the results appear to be strengthened rather than weakened given the difficulty of providing economic reasons justifying an ALD in which the requirements of stability (a slope less than LS approaching the equilibrium), plausibility (an equilibrium unemployment rate below 50 percent), and theoretical coherence (unemployment caused by efficiency considerations) are simultaneously met. Moreover, there seems to be no chance of obtaining such a function from any microeconomic history.

Concluding remarks

On the basis of the above discussion, it is possible to conclude that the efficiency wage shirking approach, which was born with the scope of obtaining Keynesian equilibrium unemployment results on the basis of maximizing behavior, faces major difficulties when the unemployment rate is considered as a truly endogenous variable. In particular, this article has reached three main conclusions. First, building micro and macroeconomic labor demand functions becomes a non-trivial problem if the effort is considered as a continuous function in both the unemployment rate and the real wage and multiple equilibria may occur. Second, even assuming well-behaved labor demand functions, the stability requirements of macroeconomic equilibrium are demanding. Finally, in general, all the stable unemployment equilibria appear to be meaningless for the theory since they correspond to particular cases in which unemployment would exist even without efficiency wages and to unrealistic values of the unemployment rate (always above 50 percent).

Given this scenario, it is not surprising that many recent New Keynesian theoretical contributions on the theme [Bulkley and Miles, 1996; Marti, 1997; and Altenburg and Straub, 1998] have ceased to identify efficiency wages as the sole cause for equilibrium unemployment and have proposed models that combine efficiency wages with the existence of other market imperfections. Thus, though aware of the weakness of the efficiency wage approach, New Keynesians have not yet given up the attempt to find equilibrium unemployment on maximizing behavior. It is too soon to judge the success of these attempts since until now these models combining different causes of equilibrium unemployment have been neither fully specified nor adequately developed. However, a judgment based only on the soundness of the efficiency wage models would have to be negative, so that Keynesian involuntary unemployment equilibrium results and maximizing behavior still remain incompatible.

Footnotes

(1) In general, effort can be considered as a function of a number of variables: wage, amount of spare time, probability of being fired after being caught shirking, unemployment rate (on the basis of which it is possible to build an approximation of the probability of being hired by another firm after having been fired for shirking), the wage paid by other firms, and unemployment benefits.

(2) This circumstance explains why hereafter, the wages paid by other firms do not enter the effort function.

(3) For a discussion of these hypotheses and of New Keynesian paradigm in general, see Nistico and D'Orlando, 1998.

(4) The firing leaves no trace in each worker's history so that being fired does not constitute a signal of low efficiency--all the workers are equal and equally efficient. This assumption is highly unrealistic but crucial for the consistency of the moral hazard version of the efficiency wages literature. Models that remove this assumption belong to the adverse selection version of the efficiency wages literature [Weiss, 1991].

(5) In the simplified version of the efficiency wage shirking model originally proposed by Shapiro and Stiglitz [1984], the effort function was discrete (that is, either workers shirk furnishing a zero level of effort, or they work expending a given positive level of effort). Also, the unemployment rate did not explicitly enter the effort function, the single firm's labor demand function, and the aggregate labor demand function. It has been shown elsewhere that these simplifications prevent the model from obtaining involuntary unemployment results [D'Orlando 1998, pp. 76-83].

(6) It should be possible to obtain the formal properties of the effort function from workers' intertemporal utility maximization. However, the conclusions of several attempts in this direction are not univocal, since they depend strictly upon the utility functions and the time horizon referred to by the analysis [Simmons, 1991]. Moreover, the soundness of the whole utility maximizing procedure has been disputed [Nistico and D'Orlando, 1998, Appendix A5]. These circumstances may explain why most models simply assume the properties of the effort function, also admitting that these properties can conflict with empirical evidence [Stiglitz, 1987, p. 36, note 69].

(7) It is worth noting that the minimum level of effort furnished by each worker has to be nil and cannot be positive. Otherwise, firms may maximize their profits paying a wage below the efficiency wage and accepting the resulting (minimum) level of effort [see also Akerlof and Yellen, 1986; contra see Shapiro and Stiglitz, 1984, p. 436, p. 438, note 15].

(8) See note six above.

(9) According to Shapiro and Stiglitz [1984, p. 438], "it is immediately evident that no shirking is inconsistent with full employment."

(10) These implications are discussed in the last paragraph of this section and passim in the next section.

(11) For example, the function [y.sub.i] = [L.sub.i.sup.y][e.sup.[delta]] yields upward sloping [L*.sub.i](u) schedules (if [gamma] + [delta] = 1).

(12) It is important to notice that in order to get ALD(u) = 0 as u [right arrow] 0, it is not necessary that the productive contribution of a shirker be nil. Instead, only the labor marginal productivity of shirking workers must be lower than w* (with w* [right arrow] [infinity] as u [right arrow] 0). However, the conclusion that ALD(u) = 0 as u [right arrow] 0 is strengthened if e* becomes nil as u = 0. For example, this result is reached in the model by Shapiro and Stiglitz [1984], where the No Shirking Constraint function (which depicts the minimum level of wage that workers accept not to shirk) never crosses the (vertical) labor supply curve, even in the presence of a wage rate tending to infinity [See also Phelps, 1994, p. 32; contra see Calvo 1979, p. 105].

(13) Since the current unemployment rate is generated by the current firms' employment decisions, in a discrete-time context, firms do not know the rate before they make these decisions, and hence, current unemployment rate cannot determine these decisions.

(14) It is true that the existence of equilibrium unemployment cannot be ascribed solely to the fact that firms pay efficiency wages. However, the conclusion is that within the efficiency wage model, stable efficiency wage unemployment equilibrium exists only if causes of unemployment other than the payment of efficiency wages exist.

(15) Since the particular effort function considered here prevents workers from furnishing a positive level of effort when the wage is nil, the highest competitive (Walrasian) labor demand would correspond to w = 0 and will be higher than the highest efficiency wage labor demand, since for u = 1 it is w* > 0. For this reason, it is still possible to have a stable unemployment equilibrium due to efficiency wages for the particular case in which the competitive full employment equilibrium is realized for a level of wage equal to zero (or not perceivably different from zero).

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FABIO D'ORLANDO*

*Universita di Roma "La Sapienza"--Italy. The author wishes to thank Sergio Nistico, Giorgio Rodano, Francesco Saraceno, Benedetto Scoppola, Domenico Tosato, an anonymous referee and numerous seminar participants for their helpful comments and suggestions. The usual caveat applies.
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