Efficiency wages, partial wage rigidity, and money nonneutrality.

Lai, Ching-Chong

1. Introduction

Since the 1970s, the persistently high unemployment rates in many industrial economies have made more and more economists believe that involuntary unemployment is one of the major stylized facts of modern economies. Therefore, a satisfactory macroeconomic labor model should explain well such a stylized fact. The efficiency wage theory has in recent years generally been regarded as a powerful vehicle for explaining why involuntary unemployment has persisted in the labor market. In constructing a business cycle model, "a potential problem of the efficiency-wage hypothesis is the absence of a link between aggregate demand and economic activity" (Yellen 1984, p. 204). Hence, until Akerlof and Yellen (1985) presented the near-rational model, efficiency wage theories still left unanswered the question of how changes in the money supply can affect real output.(1) By utilizing the idea of partially rigid wages, this paper interprets why changes in money supply and other demand management policies can cause changes in aggregate employment and output.

In macroeconomic theory, the wage is simply regarded as the amount of money that employees receive and is assumed to be exactly equal to the average cost of labor to employers.(2) In practice, the components of wages are more complicated than the simple economic setting would suggest. There exist some gaps between the amounts that trading partners pay and receive. For example, the actual average cost of labor to employers is equal to the wage that employees receive after the addition of hiring and training costs, firing (severance pay) and retirement (pension) costs, various taxes and insurance fees, sometimes traffic and housing outlays, and so on. Some of these costs, especially taxes, insurance, and traffic fees, are set by the process of political negotiations. The resetting processes relating to these costs are always time-consuming and controversial in modern democratic societies, and these costs are not as flexible as other components of wages determined by competitive markets or monopsonists. Since some components of wages are always inflexible, partial rigidity of wages is thus a realistic specification for economic modeling. When we recognize that wages have the property of partial rigidity, it is logical to expect that money nonneutrality will hence result.

The basic tenet of the efficiency wage theory is that the effort or productivity of a worker is positively related to his real wage and firms have the market power to set the wage. Therefore, in order to maintain high productivity, it may be profitable for firms not to lower their wages in the presence of involuntary unemployment. The main reasons that are provided for the positive relationship between worker productivity and wage levels include nutritional concerns (Leibenstein 1957), morale effects (Akerlof 1982), adverse selection (Weiss 1980), and the shirking problem (Shapiro and Stiglitz 1984).(3) The shirking viewpoint proposed by Shapiro and Stiglitz (1984) is the most popular version of the theory.(4) Its essential feature is that firms cannot precisely observe the efforts of workers due to incomplete information and costly monitoring; equilibrium unemployment is therefore necessary as a worker discipline device. We thus adopt a shirking model as the analytical framework of this paper to examine the effects of partial rigidity of wages.

The rest of this paper is organized as follows. Section 2 derives the effort function of a representative worker. Section 3 examines a typical firm's labor demand and wage-setting behavior. Section 4 studies the labor market equilibrium. Finally, some concluding remarks are presented in section 5.

2. The Worker's Optimization Problem

Our analysis starts by considering a simple economy where each of many identical firms hires a number of ex ante homogeneous workers to perform some task.(5) The worker enjoys on-the-job leisure and dislikes working hard. Owing to team production or some unobservable disturbances, the firm cannot determine the actual effort of an individual worker. Thus, it is impossible to reward individual workers according to their particular productivity, suggesting that there would be a probabilistic penalty to induce work effort. Such a probabilistic penalty in the shirking model is typically represented by the threat of firing. Employers will therefore monitor the performance of employees and lay off workers who are found not to be working hard.

The worker's probability of being fired [Rho] is assumed to be negatively related to his work effort e. For simplicity, [Rho] is specified as

[Rho] = 1 - e; 0 [less than or equal to] e [less than or equal to] 1. (1)

Effort e is assumed to be the fraction of the standard paid-for hours that the worker actually works, while the number of standard hours is assumed to be fixed and normalized to unity.

When a worker is fired, he will try to find another job. The probability of finding another job is assumed to be the employment rate (1 - u). The unemployment rate u here is defined to be the ratio of the number of unemployed to the total number of workers.

Following the static analytical framework of Pisauro (1991), there are three states of nature that an employed worker may face. First, he is not fired and receives a real wage [w.sub.i] with the probability of (1 - [Rho]). Second, he is dismissed but finds another job at a real wage w; the associated probability is [Rho](1 - u). Third, he is fired and cannot find another job; hence, he becomes unemployed (and enjoys all-day leisure, e = 0) and receives unemployment benefits b from the government. The probability in this case is [Rho]u. Moreover, firms are assumed to be identical and to pay the same real wage (w = [w.sub.i]). The three states are thus reduced to two.(6) Accordingly, the expected income of an employed worker, namely [y.sup.e], is

[y.sup.e] = (1 - [Rho]u)w + [Rho]ub. (2)

The worker enjoys consumption of goods by spending income y and dislikes putting forth any effort e. For ease of analysis, his utility is specified as

U(y, e) = v(y) - e; v[prime] [greater than] 0, v [double prime] [less than] 0, v(0) = 0.(7) (3)

From Equations 1, 2, and 3, the expected utility of a typical worker E[U(y, e)] is

V [equivalent to] E[U (y, e)] = [1 - (1 - e)u][v(w) - e] + (1 - e)uv(b). (4)

Since unemployment benefits are not the focus of this paper, in what follows, we will set b = 0.

A utility-maximizing worker will choose his effort level at which the expected marginal gain from effort equals the expected marginal cost of effort. Taking the first differentials of Equation 4 with respect to e, we have the first-order condition as

[V.sub.e] = u[v(w) - e] - [1 - (1 - e)u] = 0. (5)

The second-order condition for an interior maximum is satisfied because [V.sub.ee] = -2u [less than] 0.

The effort function can be solved from Equation 5 as

e = e(w, u) = 1/2 [v(w) - 1/u + 1], (6)

with

[e.sub.w] = v[prime](w)/2 [greater than] 0, [e.sub.u] = 1/2[u.sup.2] [greater than] 0, [e.sub.ww] = v[double prime](w)/2 [less than] 0, and [e.sub.wu] = 0.

Intuitively, the result of [e.sub.w] [greater than] 0 indicates that an increase in the employed worker's real income w raises his expected cost of being fired. The worker will thus provide a greater effort level to prevent his being dismissed. The positive relationship between a worker's effort and the real wage is the basic tenet of efficiency wage theories.

The outcome of [e.sub.u] [greater than] 0 states that, when the unemployment rate rises, it becomes more difficult for the employed to find an alternative job if dismissed so that the worker will exert more effort. This result confirms the Shapiro and Stiglitz (1984) contribution that unemployment is a worker discipline device. In addition, the results of [e.sub.ww] [less than] 0 and [e.sub.wu] = 0 are relevant for the discussions that follow.

When there is no unemployment (u = 0), from Equation 5 it can be seen that [V.sub.e] = -1. This result means that full employment is inconsistent with any positive effort level. This outcome is similar to that of the no-shirking condition reported by Shapiro and Stiglitz (1984). Clearly, for a lay-off to be perceived as a real penalty, it must be the case that the probability of a fired worker finding a new job is smaller than one, which implies that the labor market equilibrium in this model must be characterized by unemployment.

3. The Firm's Optimization Problem

Let us assume there are many identical firms with the same production technology f(en), which has the usual properties of a standard production function with f[prime] [greater than] 0 amd f[double prime] [less than] 0. In this production function, n is the number of workers and en represents the actual working hours or the effective labor force of the firm. The output of the firm is also affected by an unobservable disturbance [Mathematical Expression Omitted], which is a random variable with a probability density function q([Epsilon]) and an expected value equal unity, E([Epsilon]) = 1. The unobservable disturbance presumption indicates that the firm cannot determine the actual effort level of an individual worker.

Given the effort function in Equation 6, the expected profits [[Pi].sup.e] of a risk-neutral firm are

[Mathematical Expression Omitted], (7)

where p is the price of the good, t is the lump-sum or the specific tax per worker, and the term W (W = pw) is the nominal wage that the firm pays. As is typically the case in efficiency wage models, the wage affects the labor quality. Furthermore, it is the effective labor units en that enter the production function instead of the crude number of employed workers n.

As mentioned in section 1, there exist some gaps between the wage that the employer pays and the wage bill that the employees actually receive. The imposition of the lump-sum tax is an example of inserting a tax wedge (t) between the cost to the firm (W + t) and the wage of the worker (W). Here we specify, for simplicity, that the nominal value of the lump-sum tax is fixed. This indicates that the government does not adjust its taxation even if the economy experiences a change in the price level.(8)

The firm's goal is to maximize Equation 7 by choosing its employment and wage. Since there are many small firms in the economy, each firm makes its employment decision based on the belief that it cannot influence the unemployment rate. The first-order conditions with respect to n and W are

[Mathematical Expression Omitted], (8)

[Mathematical Expression Omitted]. (9)

Equation 8 states that, given the quality of labor, the quantity of labor employed is that where the marginal revenue from labor (pef[prime]) equals the marginal cost of labor (W + t). Equation 9 explains that, given the quantity of labor, the quality of labor hired is that where the marginal revenue of wage ([e.sub.w]nf[prime]) equals the marginal cost of wage (n). The unobservable disturbance has no impact on the risk-neutral firm's optimal decisions. The second-order conditions are fulfilled due to [e.sup.2]f[double prime] [less than] 0 and [e.sup.2][e.sub.ww]nf[prime]f[double prime] [greater than] 0.

Before proceeding, it should be noted that, when there is no taxation (t = 0) as with the standard efficiency wage model of Yellen (1984), from Equations 8 and 9, we can obtain

w [e.sub.w]/e = 1. (10)

This result indicates that a profit-maximizing firm will set its wage at the level at which the effort-wage elasticity is unity. This result is dubbed the Solow condition by Akerlof and Yellen (1986).

However, in their review of efficiency wage models, Akerlof and Yellen (1986) argue that an effort-wage elasticity of unity is too high and propose a simple model with external costs to illustrate an effort-wage elasticity lower than unity. When the lump-sum tax is imposed, the effort-wage elasticity is less than one; that is,

w[e.sub.w]/e = W/W + t [less than] 1. (11)

This result has been reported in Schmidt-Sorensen (1990) and Pisauro (1991).

The firm's labor demand and wage-setting functions can be solved from Equations 8 and 9 as(9)

n = n(u, p), (12)

W = W(u, p), (13)

with

[n.sub.u] = [e.sub.u]/[e.sup.2][e.sub.ww]f[double prime] [[e.sub.ww](f[prime] + enf[double prime]) + [([e.sub.w]).sup.2]nf[double prime]] [greater than] 0,(10) (12a)

[n.sub.p] = - 1 - w[e.sub.w]/e/pe[e.sub.ww]f[double prime] [[e.sub.ww]f[prime] + [([e.sub.w]).sup.2]nf[double prime]] [greater than] 0, (12b)

[W.sub.u] = [e.sub.u][e.sub.w]/e[e.sub.ww] [less than] 0, (13a)

[W.sub.p] = W/p + [e.sub.w](1 - w[e.sub.w]/e)/[e.sub.ww] [less than] W/p. (13b)

Intuitively, an increase in the unemployment rate raises the quality of workers ([e.sub.u] [greater than] 0) and motivates the firm to hire more workers and pay lower wages ([n.sub.u] [greater than] 0, [W.sub.u] [less than] 0), as shown in Equations 12a and 13a. It is worth noting from the wage-setting function in Equation 13 that, given the unemployment rate of the economy and the price level, the firm will set a corresponding optimal wage for its employees. This outcome explains why the wage does not continuously fall to eliminate the excess supply of labor in the presence of involuntary unemployment.

Moreover, when the lump-sum tax is absent, as in the specification for the standard efficiency wage model, the Solow condition holds and the results of [n.sub.p] and [W.sub.p] in Equations 12b and 13b become [n.sub.p] = 0 and [W.sub.p] = W/p. These results show that the firm's nominal wage increases proportionately with an increase in the price level, whereas its employment remains intact. Consequently, both real variables, namely, the real wage and employment, are invariant to changes in the nominal variable, i.e., the price, and hence money is neutral. This result is the same as in the case of the standard efficiency wage model in Yellen (1984). In fact, from the Solow condition in Equation 10, we can see that the level of the real wage is solely determined by the effort function. Since the worker's utility and the firm's profits are dependent only on real rather than nominal variables,(11) any variation in price is supposed to be neutralized by the firm's nominal wage policy, resulting in no impact on real variables. This is the fundamental reason why "[a]ny efficiency wage model based on pure maximization must, of necessity, be a real model" (Akerlof and Yellen 1986, pp. 18-19).

When the nonindexation lump-sum tax is imposed, the effort-wage elasticity is less than unity, The results in Equations 12b and 13b thus become [n.sub.p] [greater than] 0 and [W.sub.p] [less than] W/p. In other words, nominal wages increase less than proportionally as the price level increases and the firm's employment increases. As a consequence, money is not neutral (a price shock has a positive effect on the firm's employment decision). Intuitively, even though the firm has an incentive to neutralize the impacts of price changes by adjusting its nominal wage proportionally, a rise in the price level still lowers the real average labor cost of the firm (W + t)/p by way of lowering the real lump-sum tax burden tip due to the fixed nominal tax t. The lower labor cost alters the perceived employment-wage tradeoff faced by firms and thus motivates the firm to provide more jobs at a lower real wage.(12) Consequently, money is not neutral because of the rigidity of the cost of labor tax. As the lump-sum tax is a part of the firm's total labor cost, we can thus conclude that the partial rigidity of wages will result in money nonneutrality.

4. The Labor Market Equilibrium

Suppose that the number of identical firms is m so that the market labor demand function is

[N.sup.d] = mn(u, p). (14)

Owing to firms having market power to determine their employment and wages, the market labor demand is not a function of wages. In other words, the labor market equilibrium is prominently determined by the demand-side rather than by the Walrasian Auctioneer in this shirking model.

Under the assumption that all workers are homogeneous, we must exclude the situation in which the expected utility of a worker who participates in the labor market is less than the utility of an individual who does not.(13) This implies that every worker is willing to look for a job. However, not all workers can be hired when there is less than full employment. Hence, given the number of total workers, the higher the unemployment rate is the fewer the workers who are employed. Let N be the total number of workers and [N.sup.s] the number of workers who can get a job. The relationship between [N.sup.s], u, and N is

[N.sup.s] = (1 - u)N. (15)

To avoid workers' exertion to a minimum (zero effort), we have shown that the labor market equilibrium must result in unemployment. This implies that [N.sup.d] = [N.sup.s] holds at a positive unemployment rate. Therefore, the labor market equilibrium condition is

mn(u, p) = (1 - u)N. (16)

The equilibrium unemployment rate [u.sup.*] can be solved from Equation 16 as

[Mathematical Expression Omitted]. (17)

The impacts of changes in p on the firm's wage-setting will further vary by way of changing the unemployment rate. By substituting Equation 17 into Equation 13, the market equilibrium wage [W.sup.*]is

[W.sup.*] = [W.sup.*](p) = W[[u.sup.*](p), p], (18)

with

[Mathematical Expression Omitted]. (14)

As stated in section 3, money is neutral in the standard efficiency wage model with t = 0. This result is demonstrated by Equation 17 and 18 under the constraint of the Solow condition w[e.sub.w]/e = 1, that is, [Mathematical Expression Omitted] and [Mathematical Expression Omitted]. When the nonindexation lump-sum tax is introduced, the results in Equations 17 and 18 show that the market equilibrium wage level will rise less than proportionally with the price increase, whereby the equilibrium unemployment rate will fall in response. In other words, the aggregate employment and thus aggregate output will rise as the price level increases.(15)

Graphically, we name the loci of the combinations of N and u, which respectively satisfy Equations 14 and 15, as the LD curve and the LS curve. Their slopes are du/d[N.sup.d] = 1/m[n.sub.u] [greater than] 0 and du/d[N.sup.s] = -1/N [less than] 0. The positively sloping LD curve reflects the fact that an increase in the unemployment rate motivates the worker to furnish more work effort and thus increases each firm's labor demand. The negatively sloping LS curve reveals that a rise in the unemployment rate reduces the number of workers who can get a job. Both curves are drawn in the right-hand panel of Figure 1, and the intersection of both curves determines the market equilibrium employment [N.sup.*] and the equilibrium unemployment rate [u.sup.*].

A curve WS depicting the firm's wage-setting behavior in Equation 13 is drawn in the left-hand panel of Figure 1. The negatively sloping WS curve states the result of [W.sub.u] [less than] 0 in Equation 13a; that is, a higher unemployment rate raises the work effort and thus enables the firm to lower its wage offer without hurting labor productivity. By substituting the equilibrium unemployment rate [u.sup.*] into the WS curve, one can obtain the market equilibrium wage [W.sup.*]. There are N ex ante homogeneous workers who are willing to work at the market wage [W.sup.*] and to provide the effort level according to the effort function in Equation 6. However, the number of total available vacancies that all firms in the economy want to afford is only [N.sup.*]. There are (N - [N.sup.*]) workers who want to work at the prevailing wage but cannot find a job and become unemployed involuntarily.

When t = 0, a rise in the price level makes each firm raise its nominal wage proportionally, and hence employment remains intact. This change shifts the WS curve upward to [WS.sub.1] and leaves the LD curve unchanged. As a result, the market equilibrium wage increases from [W.sup.*] to [Mathematical Expression Omitted], whereas the market equilibrium employment [N.sup.*] and the unemployment rate [u.sup.*] remain unaltered. When t [greater than] 0, a rise in p motivates each firm to raise its nominal wage less than proportionally and increase its employment simultaneously. The WS curve thus shifts upward to [WS.sub.2], and the LD curve shifts downward to [LD.sub.1]. The result is that the market equilibrium nominal wage increases to [Mathematical Expression Omitted], and the equilibrium unemployment rate falls to [Mathematical Expression Omitted].

The result of money nonneutrality is crucially based on the assumption of the rigidity of the lump-sum tax. The tax we define is a representative specification of a lot of costs. Since these costs are some parts of the total wage or the average cost of labor that employers pay, we name such channels as the partial rigidity of the wage.

Why are these costs rigid or, in a more plausible setting, not under full indexation? It is because these costs are always determined by tedious political processes. The outcomes of political negotiations (laws) are public goods. These public goods, once produced, are available for use by all citizens without cost. The free-rider problem then naturally appears. The producers of public goods create a positive externality so that such goods tend to be underproduced. This means that these costs cannot be adjusted quickly or precisely enough to be under full indexation. In sum, the notion of the externalities of public goods is the essential reason for justifying why some parts of wages are not fully flexible and why changes in aggregate demand policies are the causes of fluctuations in aggregate supply.

For the same reasons, the argument can be applied to the idea of partial rigidity of prices. As we know, the price is always defined as the amount of money that buyers pay and is assumed to be the actual average revenue of sellers. From a broader point of view, the actual average revenue of sellers is equal to the price that buyers pay after the deduction of transportation and storage costs, advertising and wastage costs, various taxes and insurance fees, and so on. Some of these costs are also determined by tedious political negotiations. As a consequence, partial rigidity of prices exists and is one of the reasons why changes in money supply and other demand management policies can cause fluctuations in aggregate employment and output.(16)

5. Concluding Remarks

The efficiency wage theory is generally regarded as a plausible explanation as to why wages do not fall to clear labor markets in the presence of involuntary unemployment. In a paper, Blinder (1988, p. 290) claims that "the simplest, and to me the most appealing, of these [theories addressing the involuntary unemployment question] is the efficiency wage model. It also seems to accord best with common sense." In their popular macroeconomic textbook, Blanchard and Fischer (1989, p. 489) claim that "in labor markets, notions of efficiency wages have a definite ring of truth."

However, any efficiency wage model based on pure maximization must, of necessity, be a real model. "They have nothing to say about nominal magnitudes, and hence allow no role for nominal money, until they are altered to include fixed costs of changing nominal wages or prices. Nor, in their current state of development, do they have much to say about fluctuations in employment" (Blinder 1988, p. 290). In this paper, we use the idea of partial rigidity of wages to point out a linkage between aggregate demand policies and aggregate supply in a pure maximization shirking-type model of efficiency wages. We believe that the partial rigidity of wages is a very common phenomenon in our modern democratic societies, and surely such a linkage is also suitable in other macroeconomic models.

We would like to thank one anonymous referee for helpful comments and suggestions. Any remaining errors are entirely our responsibility.

1 Mankiw (1985) proposed a similar idea of the menu cost at more or less the same time.

2 Hence, the only possibility for establishing a link between aggregate demand and economic activities is by means of the (full) rigidities of wages (or prices) at least in the short run.

3 Another famous version of efficiency wages without an effort function is the labor turnover model of Salop (1979).

4 Excellent surveys of the efficiency wage literature have been provided by Yellen (1984), Akerlof and Yellen (1986), and Katz (1986).

5 The analytical framework we use is a static framework, which closely follows Pisauro (1991). A dynamic framework originating from Shapiro and Stiglitz (1984) can be found in Chatterji and Sparks (1991).

6 For details, see Pisauro (1991).

7 This is a simple utility function form that keeps the following analysis as simple as possible. A more general form of the utility function U(y, e) with [U.sub.y] [greater than] 0, [U.sub.yy] [less than] 0, [U.sub.e] [less than] 0, [U.sub.ee] [less than] 0, U(0, 0) = 0 does not change the basic results regarding money nonneutrality in this paper. A detailed mathematical appendix concerning the results reported in this paper can be obtained from the authors upon request.

8 Even if this taxation could be discretely reset or automatically adjusted according to some degree of indexation of the price level, it is very hard to believe that full indexation can exist in the real world. It is accordingly very reasonable to specify that the labor tax cannot be adjusted to full indexation due to incomplete information and the complicated process of political negotiations. Without loss of generality, we thus assume that the lump-sum tax is complete nonindexation. However, it is easy to understand that the qualitative comparative statics of the partial-indexation setting are the same as those of the nonindexation setting.

9 Since the effects of changes in the lump-sum tax are not the focus of this paper, we do not report the comparative static results in relation to this tax in the discussion that follows. An analysis of the lump-sum tax can be found in Pisauro (1991).

10 It is assumed that (f[prime] + enf[double prime]) [greater than] 0. This assumption holds under the usual production function form f(en) = [(en).sup.[Alpha]] with 0 [less than] [Alpha] [less than] 1. Akerlof and Yellen (1985) and Pisauro (1991) made the same assumption in their efficiency wage models.

11 Since the price level is exogenous to the firm, the firm is indifferent when it comes to maximizing nominal or real profits.

12 The money wage increase is less proportional than the price increase, and thus the real wage decreases.

13 The assumption of utility differences between employment and unemployment underlies models of trade unions (see, e.g., Oswald 1985) and models of job search (e.g., Devine and Kiefer 1991). The literature on the relationship between employment, unemployment, and subjective well-being has been reviewed by Warr (1987), Banks and Ullah (1988), and Feather (1990). These reviews show that unemployment is generally found to be negatively correlated with well-being. Recently, Korpi (1997) has provided some empirical evidence showing that, relative to employment, unemployment has an ambiguously negative effect on well-being.

14 Here, we suppose that d[e(1 - u)N]/du = [e.sub.u](1 - u)N - eN [less than] 0. This assumption excludes the situation in which a rise in the economy's unemployment rate raises the effective labor force of the total employed workers of the economy. A similar assumption is made in Pisauro (1991, p. 337).

15 This indicates that the aggregate supply curve for the economy will be positive in the output-price quadrant. As a result, changes in money supply and other demand management policies can cause fluctuations in aggregate employment and output.

16 The idea of externality has been successfully used to construct such business cycle models as the menu costs model of Mankiw (1985) and the profit sharing model of Weitzman (1985). The sources of externality they propose arise from the market structures of imperfect competition and fixed-share contract. In welfare economics, we learn that imperfect competition and public goods may explain why resource allocation in free markets cannot be Pareto efficient. Here, it is interesting and important to emphasize that imperfect competition and public goods are the causes of business cycles.

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