Exploring the interaction between efficiency wages and labor market frictions

The Quarterly Review of Economics and Finance 40 (2000) 121–137

Exploring the interaction between efficiency wages and labor market frictions Miles B. Cahilla* a

Department of Economics, College of the Holy Cross, One College Street, Worcester, MA 01610, USA

Abstract This paper explores the combined effects of efficiency wages and labor market matching frictions. A combined efficiency wage-frictional model is developed in which separate efficiency wage, frictional, and undistorted models are nested. It is found that the inclusion of efficiency wages puts upward pressure on wages and raises unemployment, while the friction puts downward pressure on wages and raises unemployment. Thus, it appears that unemployment generated by the frictional model cannot completely fulfill the role of unemployment as a discipline device, and vice versa. Other results show that the combined model has significantly different characteristics than its components. © 2000 Bureau of Economic and Business Research, University of Illinois. All rights reserved. Keywords: Efficiency wages; Frictional unemployment

1. Introduction In recent years, there has been intense interest in explaining the natural rate of unemployment. Two empirically supported models used to explain this phenomenon are efficiency wages and frictional matching.1 Because efficiency wages affect the productivity of workers and frictional matching affects the cost of labor, it is a natural exercise to explore the interaction between the two models. This paper builds a model in which the individual efficiency wage and frictional elements can be isolated, yet analysis reveals that the combined model exhibits characteristics that are fundamentally different from either model. For

* Corresponding author. Tel: ⫹1-508-793-2682; fax: ⫹1-508-793-3708. E-mail address: [email protected] (M.B. Cahill) 1062-9769/00/$ – see front matter © 2000 Bureau of Economic and Business Research, University of Illinois. All rights reserved. PII: S 1 0 6 2 - 9 7 6 9 ( 9 9 ) 0 0 0 4 8 - 4

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example a numerical simulation exhibits a situation in which unemployment and wages in the combined model exhibit more variability than either of the component models. It is also found that the efficiency wage and frictional distortions each increase the unemployment rate when introduced into a labor market in which the other distortion is present. This implies that the unemployment generated by the matching friction can never completely fulfill the role of unemployment as a discipline device for efficiency wages. Similarly, unemployment generated by efficiency wages can never provide enough frictional unemployment to eliminate firms’ search costs. This apparently counterintuitive result is explained through examining the interaction between the models. Not surprisingly, unemployment in the combined model is not necessarily the sum of the unemployment that would be generated under efficiency wage and frictional models alone. The paper provides examples for which the combined unemployment is higher or lower than the sum of the individual models. Finally, this paper suggests some implications of these results for theoretical and empirical research. 1.1. Efficiency wages and matching models Although there are various types of efficiency wage models, this paper concerns itself with the efficiency wage incentive scheme used to overcome monitoring problems.2 In this context, formalized by Shapiro and Stiglitz (1984), firms which can only sporadically monitor employees pay a wage above the market clearing rate. The high wages, in combination with resulting unemployment, provide a sufficient incentive to eliminate shirking. A key result of this literature is that involuntary unemployment is necessary to discipline workers — unemployment results in queues for jobs, placing a cost on any workers fired for shirking. Because this involuntary unemployment exists without the Keynesian assumption of sticky wages, efficiency wages have been used in many macroeconomic models.3 While these models have relied exclusively on efficiency wages to generate unemployment, the combined model below shows that efficiency wages raises the unemployment rate even when frictional unemployment is present. The many different types of frictional models have a longer history in the literature explaining unemployment.4 In general, a friction is any factor that prevents job-seeking workers from finding a desired job when such a job is available, or employee-seeking firms from finding desired employees when such employees are available. The cause of a friction is usually either (a) an information imperfection or geographical barrier that imposes costs on unemployed workers who are searching for firms with vacancies (or vice versa), or (b) asymmetric information problems that make it difficult for heterogeneous workers to find an optimal job, when jobs are also heterogeneous. The former type of model has been used to model unemployment, vacancies and the relationship between the two (called the “Beveridge Curve”). The latter type of model has been used to model unemployment in the context of optimal search behavior.5 In either case, a “friction” caused by asymmetric information prevents the labor market from clearing. These frictions essentially create adjustment costs.6 The matching model below, based on the Romer (1996) formulation, follows the adjustment cost literature by assuming that it is costly for firms to recruit new employees. Since recruiting costs become prohibitively high without a significant pool of unemployed workers,

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long run unemployment results. While the fact that labor market frictions alone cause unemployment is well known, the model below shows that the friction raises the unemployment rate even when unemployment has already been generated by the efficiency wage environment. It should be noted that the model used in this paper is not of the matchingbargaining type of model that is also common in the literature. When heterogenous workers and firms seek matches and bargain over wages, the results are likely to differ considerably. The following section details the combined efficiency wage-matching model. The standard Shapiro and Stiglitz (1984) model will be used to derive the minimum wage for which workers will work, and the standard Romer (1996) framework will be used to derive the condition that defines the maximum wage firms will be willing to pay. Next, the presentation of the equilibria for the combined model and each of its component models are detailed. Before the conclusion, results are presented. These results include a comparison of the equilibria in each of the models and a comparative static analysis.

2. The model 2.1. Jobs and job matching As noted earlier, the job market in this model is based on the Romer (1996) framework.7 Firms are organized around an endogenously determined number of identical jobs. Jobs may be filled or vacant, and a filled job employs one worker. Denote F as the number of filled jobs, and V as the number of vacancies. The total number of workers in the labor force is N; denote L (L⫽F) as the number of employed workers. A worker in a job is capable of producing output at rate A. It costs rate c (exogenously determined) to keep a job in existence, whether it is filled or not.8 The parameter c can be considered the cost of capital when the job is filled or vacant. Jobs earn a wage rate w, which will be endogenously determined. Workers vacate jobs at exogenous rate b, where b is the probability of a job separation for any filled position. The value of filled and unfilled jobs can be represented by asset value equations. The interest rate is exogenously determined, and is denoted by r. The value of a filled job (VF) is implicitly defined by the following asset value equation. rV F ⫽ A ⫺ w ⫺ c ⫹ b共V V ⫺ V F兲

(1)

This equation specifies that the rate of return on a filled job must be the productivity less costs (A ⫺ w ⫺ c) plus the return associated with the job becoming vacant (VV⫺VF) weighted by the probability of it becoming vacant (b). Another way to conceptualize the asset value equation is to divide both sides by r. In this case, the value of a filled position is the present discounted value of the output of the job (A) less costs (⫺w⫺c), plus the probability of the job becoming vacant (b) times the present discounted value of the job becoming vacant (VV) and the lost present discounted value of the filled position (⫺ VF). The value of a vacant job (VV) is implicitly defined in Eq. 2. rV V ⫽ ⫺c ⫹ ␣ 共V F ⫺ V V兲

(2)

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The parameter ␣ denotes the rate vacant jobs are filled; it is a probability to any single firm. The expected length of time a job remains vacant is therefore (1/␣). The rate vacancies are filled is determined by a matching function. Denote ␮ ⫽␮ (V, (N⫺L)) as the matching function which determines the flow of workers to jobs (␮) for a given rate of vacancies advertised (V), the (exogenous) efficiency of the matching technology (␺), the number of unemployed workers searching for jobs (N⫺L), and (exogenous) elasticity parameters (␾,␥). Following Blanchard and Diamond (1989) and Romer (1996), the following explicit form for the matching function is used.

␮ ⫽ ␺ 共N ⫺ L兲 ␾V ␥

(3)

The efficiency of matching parameter (␺) captures factors related to the availability of employment agencies, newspapers, etc. The parameter ␾ is the elasticity of matches with respect to the unemployment level. This elasticity is constrained to 0 ⱕ ␾ ⱕ 1, reflecting the idea that high levels of unemployment impede the search process for individual workers. The parameter ␥ is the elasticity of vacancies. The assumption 0 ⱕ ␥ ⱕ 1 captures diminishing returns to vacancies, which can also be interpreted as congestion effects.9 Note that the stream of matches a firm receives at a given level of unemployment (N⫺L) and vacancy rate (V) is increasing in ␺, ␾, and ␥. The matching function defines the rate ␣, as the rate vacancies are filled (␣V) must equal matches (␮).

␣ ⫽ ␮ 共V, 共N ⫺ L兲兲/V

(4)

As noted in the introduction, the frictional model will be compared to a frictionless model. The matching friction is eliminated by assuming perfect efficiency of matching (␺3⬁). That is, as (␺3⬁), (␣3⬁) and the matching friction is eliminated; firms can draw any number of unemployed workers and instantly fill any vacancies. 2.2. Workers Employed workers (receiving wage w) are productive (that is produce output at rate A) if they expend effort at predetermined rate e⬎0 on the job. Workers lose utility at rate e from expending effort, so they will shirk unless given sufficient incentives. If unemployed, the rate of unemployment benefits (or returns from home production) is s⬎0. Every dollar of income results in one unit of utility. Shirking workers (expending less effort than e) are caught at exogenous rate q⬎0; the average time before a shirking worker is caught is therefore (1/q). When (1/q) ⫽ 0, shirking workers are caught immediately, and no special incentives are needed to ensure productivity; the minimum wage firms must pay is equal to w⫽s⫹e. When (1/q)⬎0, firms rely on efficiency wages to obtain effort from employees. Below it will be shown that in equilibrium, there will be no shirking, and thus no firings. As noted above, workers (whether shirking or not) leave their jobs with exogenous probability b, so the expected tenure of a non-shirking worker is (1/b). Unemployed workers find new jobs with endogenously determined probability a; the expected duration of unemployment is accordingly (1/a).

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Using the assumptions above, the value of a job to a non-shirking worker (VN E ) is implicitly defined in the asset value equation below, where VU is the value of being unemployed and VE is the value of being employed. rV EN ⫽ w ⫺ e ⫹ b共V U ⫺ V E兲

(5)

The value of a job to a shirking worker (VSE) is implicitly defined in Eq. 6. rV ES ⫽ w ⫹ 共b ⫹ q兲共V U ⫺ V E兲

(6)

The value of unemployment (VU) is implicitly defined in Eq. 7. rV U ⫽ s ⫹ a共V E ⫺ V U兲

(7)

The accession rate, a, is determined by the matching function. Specifically, a is defined in Eq. 8. a ⫽ ␮ 共V, 共N ⫺ L兲兲/共N ⫺ L兲

(8)

2.3. Maximum and minimum wage conditions Firms will keep vacancies open—that is, recruit labor—as long as the value of a vacancy is not negative (VV⬎0). It is assumed that there is free entry and exit to the market. This competitive assumption restricts (VV ⫽ 0). Thus, solving the equation VV ⫽ 0 for the wage as a function of employment (and parameters) defines the maximum wage condition, a function of the quantity of labor.10 Solving Eq. 1 for VF and substituting the resulting expression into Eq. 2 reveals the following, after rearranging. V V ⫽ 关 ␣ 共 A ⫺ w ⫺ c兲 ⫺ c共r ⫹ b兲兴/共 ␣ r ⫹ r 2 ⫹ rb兲

(9)

From Eq. 9, it is clear that VV ⫽ 0 if the numerator is equal to zero. Setting the numerator to zero and rearranging reveals the maximum wage that firms are willing to pay workers: w ⫽ A ⫺ c ⫺ 共r ⫹ b兲c/ ␣

(10)

In the absence of the matching friction (that is, when (1/␣)⫽0), the wage is simply set equal to the marginal product of labor (A), less non-wage costs (c). When the matching friction is present, an employee also represents future expected costs associated with finding a replacement, and this cost is deducted from the wage. Specifically, c is the cost of holding open the vacancy and (1/␣) is the expected duration of the vacancy. This cost is discounted with the interest rate (r) and the probability of departure (b), leaving (r⫹b)c/␣ as the discounted expected turnover cost of an employed worker. Below, it will be shown that this maximum wage condition is related to the level of employment in the steady state. Where Eq. 10 showed the maximum wage firms will pay workers, we must derive an expression defining the minimum wage for which workers will be productive. That is, an incentive compatibility condition must be derived. Using Equations 5, 6, and 7, Shapiro and Stiglitz (1984) show that the “no shirking condition” ensuring workers work productively

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depends on both the wage and the duration of unemployment (1/a). Specifically, by solving S for the condition V N E ⱖ V E , the boundary of the no shirking condition is: w ⫽ s ⫹ e ⫹ 共e/q兲共a ⫹ b ⫹ r兲

(11)

Below, it is shown that the no shirking condition relates the wage to the quantity of employed labor in the steady state. When monitoring is perfect ((1/q) ⫽ 0), Eq. 11 reduces to w ⫽ s ⫹ e. That is, the wage is equal to the reservation wage, regardless of employment. This is an individual rationality condition. 2.4. Steady state conditions For a steady state to exist, flows into jobs (␮) must equal flows out of jobs (bL). Thus, the following steady state condition must hold:

␮ 共V, 共N ⫺ L兲兲 ⫽ bL

(12)

From Eq. 12, steady state expressions for ␣ and a are determined. First consider ␣. Rearranging the matching function Eq. 3 reveals that V ⫽ ( ␮ /(N ⫺ L) ␾ ) 1/ ␥ . Substituting this expression into Eq. 4 and applying steady state condition Eq. 12 to substitute bL for ␮ provides the steady state expression for the rate vacancies are filled (␣) as a function of employment. This is given in Eq. 13.

␣ ⫽ ␺ 1/␥共N ⫺ L兲 ␾/␥/共bL兲 共1⫺␥兲/␥

(13)

The steady state expression for the accession rate (a) requires applying Eq. 12 to Eq. 8. a ⫽ bL/共N ⫺ L兲

(14)

The steady state maximum wage condition is obtained by applying Eq. 13 to Eq. 10. After rearranging, this becomes: w ⫽ A ⫺ c ⫺ c共r ⫹ b兲共bL兲 共1⫺␥兲/␥/ ␺ 1/␥共N ⫺ L兲 ␾/␥

(15)

This condition has several important properties. First note that in the absence of the friction (␺3⬁), firms are willing to pay workers a maximum of the marginal product of labor (w ⫽ A⫺c).11 However, in the frictional model, the wage is decreasing in L and asymptotic to full employment; for any positive employment level, recruiting costs drive the maximum wage below the marginal product. These properties are proven as lemmas 1 and 2 in the appendix. Not surprisingly, the wage is increasing in the matching elasticity of unemployment (␾), and the efficiency of matching (␺). The minimum wage condition (that is, the individual rationality and incentive compatibility condition) is obtained by applying condition (Eq. 14) to the no shirking condition (Eq. 11). The following, steady state no shirking condition is obtained: w ⫽ s ⫹ e ⫹ 共e/q兲关bN/共N ⫺ L兲 ⫹ r兴

(16)

Eq. 16 defines the minimum wage firms must pay to elicit effort. It has very well-known properties, as detailed in Shapiro and Stiglitz (1984). Specifically, when monitoring is

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imperfect ((1/q)⬎0), w is increasing in L, asymptotic to full employment, and above the reservation wage, s⫹e. As noted above, when monitoring is perfect ((1/q) ⫽ 0), Eq. 16 simply becomes the individual rationality condition, w ⫽ s⫹e. However, it must be noted that L is constrained to the domain [0, N]. When L ⫽ N, competition for workers will bid the wage up to the maximum wage firms will be willing to pay. Eq. 17 formalizes this assumption. 兵w ⫽ s ⫹ e if L 僆 关0, N兲, w ⱖ s ⫹ e if L ⫽ N 其

(17)

From the pairs of maximum and minimum wage conditions, four models have now been specified. Denote model “C” as the combined efficiency wage-frictional model, which is defined by the frictional maximum wage condition (Eq. 15, ␺⬍⬁) and the no shirking condition (Eq. 16). Denote model “EW” as the standard Shapiro and Stiglitz (1984) efficiency wage model, defined by the frictionless maximum wage condition (w ⫽ A ⫺ c) and the no shirking condition. Denote model “F” as the standard (Romer, 1996) frictional matching model, defined by the frictional maximum wage condition and the individual rationality constraint (Eq. 17). Finally, denote model “U” as the undistorted model, defined by the undistorted minimum and maximum wage constraints. 2.5. Equilibrium The non-linear nature of the constraints prevents an analytical solution for equilibrium values of L.12 However, by analyzing the properties of each model, it is possible to show that an equilibrium exists for each of the models, and some inferences about the equilibria can be made. Further results will be illustrated by numerically solving the equilibrium conditions for sets of parameter values. Proposition 1, in the appendix, establishes that a single equilibrium {w, L} exists for models C, EW, F, and U as long as the intercept of the maximum wage condition curve is greater than that of the minimum wage condition.13 Denote the equilibrium for each of the models C, EW, F, and U as {w *C, L *C }, {w *EW, L *EW}, {w *F, L *F}, and {w *U, L *U} respectively. Also denote the unemployment rate for each model as u *C, EW, F or U. Fig. 1 below displays the equilibria. On the graph, frictional and undistorted maximum wage conditions (Eq. 15) are labeled MPF (for marginal product) and MP, respectively; efficiency wage and undistorted minimum wage conditions (Eq. 16 and 17) are labeled NSC (for no shirking condition) and IRC (for individual rationality condition) respectively. The equilibria of models C, EW, F and U are labeled EC, EEW, EF, and EU respectively.

3. Results These results are presented using the assumption that the marginal product of labor (A) is constant. It is straightforward to show that the substantive results also hold—and are sometimes extended—when there are diminishing returns to labor. When the results differ below, a note is made.

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Fig. 1. Equilibrium in models C, EW, F, and U.

3.1. Ranking the equilibria From our knowledge of the shapes of the minimum and maximum wage conditions, it is possible to partially rank the equilibrium wage and employment level of each model. Proposition 2 establishes that, ceteris paribus, adding an efficiency wage distortion raises the equilibrium wage in an economy with a matching friction, while adding a frictional distortion lowers the equilibrium wage in an efficiency wage environment.14 Formally,15 Proposition 2: The equilibrium wage in each of the models C, EW, F and U can be ordered from highest wage to lowest wage as follows: w*U ⫽ w*EW ⬎ w*C ⬎ w*F, ceteris paribus. This result is to be expected, because firms must raise wages to overcome monitoring problems, and lower wages to offset turnover costs.16

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Proposition 3 shows that the undistorted model U has the highest equilibrium level of employment, while the combined model C has the lowest level of employment. Proposition 3: The equilibrium level of employment for models C, EW, F and U is ordered as follows: L*U ⬎ L*EW ⬎ L*C, L*U ⬎ L*F ⬎ L*C, ceteris paribus. Without further assumptions, nothing can be said about the relative levels of equilibrium employment in models EW and F. However, it can be shown that if q and ␺ are sufficiently small, employment in the pure efficiency wage model (EW) is less than employment in the pure frictional model (F). This is because as q falls, the minimum wage condition becomes less steep, and the intercept gets larger. As ␺ decreases, matching is more costly, and maximum wage condition becomes steeper. Proposition 3 implies an important result. It establishes that adding either a matching friction or efficiency wage considerations to a model raises the steady-state unemployment rate. That is, it appears that the unemployment generated by a labor market friction can never completely fulfill the role of unemployment as a discipline device for efficiency wages, and vice versa. Neither type of unemployment is sufficient to eliminate the other.17 Closer analysis explains this apparently counterintuitive result. First consider why a frictional element always raises the unemployment rate. Recall that competition ensures that firms always pay workers a wage equal to the marginal product of labor (less non-wage costs) in each model. When a labor market friction is added to an efficiency wage environment, turnover costs rise, lowering the effective marginal productivity of labor and thus the wage. These lower wages will only meet the no shirking condition if the unemployment rate rises; the magnitude of the change in wages and employment depends on the sensitivity of both the minimum and maximum wage conditions to changes in employment. Now consider why efficiency wages always raise the unemployment rate. In the pure frictional environment, firms pay workers the reservation wage, but the no shirking condition mandates that firms pay a wage above the workers’ reservation wage. Thus, when imperfect monitoring is introduced, firms must raise the wage. The rise in wages moves the economy back along the maximum wage condition for labor to a lower level of employment. Again, the extent to which wages and employment changes depends on the sensitivity of the wage conditions to employment. 3.2. Superadditivity and subadditivity of employment From the discussion above, it should be clear that the unemployment generated by the models is not simply additive. The unemployment effects of the individual efficiency wage and matching frictions may be “super” or “sub” additive in the combined model. Super and subadditivity is shown on Fig. 2 below. On each diagram, the equilibrium levels of unemployment for models C, EW and F are labeled uC, uEW, and uF respectively. Model U has no steady state unemployment in either diagram. By comparing the unemployment rates in the different models, it is clear that on the left hand diagram, the labor market exhibits superadditivity (as uC⬎uEW⫹uF). On the right hand diagram, the matching and efficiency wage distortions are magnified, and the combined model in fact exhibits less unemployment than the sum of the component models do—there is “subadditivity” of unemployment.18 An implication of the existence of super and subadditivity is that it is not straightforward

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Fig. 2. Graphical examples of superadditivity and non-superadditivity of unemployment.

to determine the relative importance of efficiency wages and matching frictions in determining unemployment in a combined environment. In the case of superadditivity, some unemployment is created by the interaction of the two distortions; in the case of subadditivity, some unemployment is eliminated by the interaction. This fact has obvious implications for the growing body of empirical research that is concerned with measuring the extent of frictional unemployment. 3.3. Comparative static results It is not surprising, but nonetheless important, that the combined model C reacts differently to shocks than do models EW, F or U. For example, the equilibrium wage in model F changes only if there is a change in the reservation wage (s⫹e), and the wage in model EW changes only if there is a change in the net marginal product (A⫺c), while the equilibrium wage in model C changes if there is a change in any parameter value. Further, if the marginal product of labor increases, model F predicts that the level of employment will rise but the wage will stay the same; model C predicts that both employment and wages will rise. The nonlinear nature of the frictional maximum wage condition does not make it possible to analytically compare many comparative static results. Though clearly not general results, numerical examples demonstrate that the models will act differently from one another when subjected to shocks. Table 1 shows how the equilibrium unemployment rate (u), a measure of unemployment additivity (␬) and the wage (w) in a numerical example reacts to a 10%

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Table 1 Effect of a 10% increase in each parameter value on the unemployment rate (u), unemployment additivity ratio (␬), and wage rate (w) on the combined (C), efficiency wage (EW), frictional (F) and undistorted (U) models. Initial specification Model

Unemployment rate (u) Additivity ratio (␬) Wage rate (w)

C

EW

F

U

4.97% 1.16 6.36

3.53%

0.74%

0.00%

8.00

2.00

8.00

10% change in each parameter value Parameter

A c s e r b N ␺ ␥ ␾ q

Initial value

12.000 4.000 1.000 1.000 0.025 0.015 1.000 0.150 0.600 0.400 0.075

% change in u Model C

EW

F

U

% change in ␬

⫺18.9% 11.9% 2.0% 10.6% 2.7% 11.2% 0.0% ⫺4.6% 6.3% 7.0% ⫺7.6%

⫺12.4% 5.4% 1.3% 9.0% 0.4% 7.1% 0.0% 0.0% 0.0% 0.0% ⫺6.8%

⫺3.6% 4.1% 0.4% 0.4% 1.4% 2.4% 0.0% ⫺3.2% 1.4% 8.3% 0.0%

0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%

⫺0.5% 0.7% 0.0% ⫺0.3% 0.5% 0.1% 0.0% ⫺1.0% 4.7% ⫺2.5% 0.3%

% change in w Model C

EW

F

U

14.8% ⫺6.7% 0.4% 1.8% ⫺1.1% ⫺0.7% 0.0% 3.1% ⫺3.8% ⫺4.1% ⫺1.5%

15.0% ⫺5.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%

0.0% 0.0% 5.0% 5.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%

15.0% ⫺5.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%

increase in value of each of the parameters in turn.19 While the numerical model is hypothetical, parameter values were chosen to replicate some basic attributes of the U.S. economy. In particular, Blanchard and Diamond (1989) estimates were used for the matching elasticity of unemployment (␾) and vacancies (␥) using the assumption of constant returns to scale (␾⫹␥ ⫽ 1). Capital costs (c) are set at 1/3 of total productivity (A). The probability of an employed worker becoming unemployed (b) was chosen to match the appropriate monthly transition probability computed from the Current Population Survey (0.015, Ehrenberg and Smith 1994, 587) from the same period. Other parameters were chosen so the combined model approximates the current U.S. unemployment rate (5%), and the monthly probability of an unemployed worker finding a job (a) (the examples have a ⫽ 0.29, while Ehrenberg and Smith 1994, 587, report a ⫽ 0.31 for white males). In many cases, the combined model exhibits more wage and unemployment variability than either of the component models.20 Specifically, unemployment varies more in the combined model than the component models when there are shocks to any parameter except N or ␾, and wages vary more in the combined model except for shocks to A, s, or e. It is also apparent that many variables affect super and subadditivity of unemployment. The percent change in ␬ measures the direction of the change in additivity, where a positive change is for greater additivity; increases in A, e, ␺ and ␾ reduce additivity while increases in the other parameters increase additivity.

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4. Conclusions It is clear that a combined model exhibits markedly different characteristics than either a standard efficiency wage or frictional model. Not only do comparative static results differ, but the interaction of efficiency wages and costly matching itself can produce or eliminate unemployment. If, as evidence suggests, the real economy suffers from both efficiency wage and frictional distortions, there are important implications for theoretical, empirical, and, by extension, policy research. For example, a researcher using a frictional model to estimate the impact of the Internet reducing recruiting costs may overestimate the unemployment effects, while underestimating the effects on wages. A researcher attempting to determine the overall impact of efficiency wages on unemployment may over or underestimate the effects, depending on whether super or subadditivity is present. Similarly, a researcher attempting to isolate the level of frictional unemployment in an labor market may under or overestimate the true impact of frictional factors. The unique properties of the model suggests that a dynamic combined model is worth testing. In recent years, efficiency wage and frictional models have been increasingly used to explain the labor market in a dynamic context. The motivation for their use comes from results that show that classical models of the labor market cannot explain important data regularities.21 Although a variety of assumptions have been introduced to address these problems, efficiency wage and frictional models separately show some promise.22 While this study does not give any indication of the behavior of a dynamic combined model, it suggests the possibility of interactions that are worth exploring.

Acknowledgments The author is grateful to Jack Barron, John Carlson, John Pomery, John F. O’Connell, George Kosicki, and two anonymous referees for guidance and assistance. The author also acknowledges financial assistance from a Purdue University Research Foundation grant.

Notes 1. See Krueger and Summers (1986), Cappelli and Chauvin (1991), and Blanchflower and Oswald (1994) for different approaches that find support for efficiency wages. See Blanchard and Diamond (1989) for an estimation of a matching model, and Koning, Ridder, and van den Berg (1995), Holzer (1993), Warren (1991), McCallum (1987) and Kaloski (1987) for empirical studies of frictional unemployment. 2. Other types of efficiency wage models address employee turnover, health, crime, and other factors. Efficiency wages have also been considered as “gifts”, and not discipline devices (Akerlof, 1982). See Weiss (1990), Katz (1986) for a review of the efficiency wage literature. Several alternative solutions to the monitoring problem have been proposed, including peer pressure (Barron and Gjerde, 1997), performance

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3. 4. 5.

6. 7.

8. 9. 10.

11. 12.

13.

14.

15. 16.

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bonds (Becker and Stigler, 1974), employment fees (Carmichael, 1985), deferred compensation (Akerlof and Katz, 1989) and reputation (Shapiro and Stiglitz, 1984; Strand, 1987, and Akerlof and Katz, 1989). See Shapiro and Stiglitz (1984), Bulow and Summers (1986), Summers (1988), and Drago (1989), for examples of this literature. See Mortensen (1986) for a survey. See Hosios (1990) for a discussion of the literature on matching functions. See Romer (1996) and Hosios (1990) for generalized matching models. They contain elements of McCall (1970), Jovanovic (1979), and Lippman and McCall (1981), among others. See also Pissarides (1990) for a study of search models and unemployment. The seminal work in this area is that of Stigler (1961, 1962). See Hamermesh and Pfann (1996) for a general review of the adjustment cost literature, including the literature that relates labor market search to adjustment costs. However, the equilibrium in the model is different from Romer’s formulation. To define the minimum wage condition, Romer assumes that workers and firms split the total surplus between them, while the model below uses efficiency wages. Further, Romer does not explicitly derive a maximum wage condition in which the maximum wage firms are willing to pay depends on level of employment. If there is an additional search cost for filling a position, the results of the model are not changed significantly. This matching function was estimated by Blanchard and Diamond (1989), and empirical support was found for the indicated parameter restrictions. This an alternative approach to dynamic programming; constructing a standard maximization problem will obtain the same result. Romer (1996) uses this approach to characterizes a search and matching model and Shapiro and Stiglitz (1984) use this approach to derive the no shirking condition. Later, diminishing marginal product of labor is considered. Note that in Eq. 15, the variable L is raised to the (1⫺␥)/␥ power, and (N⫺L) is raised to (␾/␥). Equating Eq. 15 and 16 implicitly defines L, which can be used in conjunction with Eq. 15 or 16 to define w. Note that Romer (1996) uses a splitting the surplus rule to determine the wage for which workers accept jobs, and thus the minimum wage condition. The model above differs in that when the market is below full employment, the equilibrium wage is equal to the reservation or efficiency wage (depending on the model). If all firms offer wage w ⱖ s ⫹ e , workers will accept such a wage offer rather than remain unemployed. If the market is at full employment, firms bid the wage up to the (net) marginal product of labor, which may be significantly above the reservation wage. When the marginal product of labor is decreasing in employment, the efficiency wage would also raise the wage if added to an undistorted model, and the matching friction would lower the wage (except if the undistorted equilibrium wage is s ⫹ e). All proofs are provided in the appendix. Without further assumptions, it is not possible to compare wage levels between models U and C when there are diminishing returns. The relative sizes of the equilibrium wage depends on the slope of the frictional and undistorted maximum

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17.

18.

19.

20.

21.

22.

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wage conditions. For relatively steeply sloped maximum wage conditions, or low search costs, w*C⬎w*U; for relatively flat conditions, or high search costs, w*U⬎w*C. It should also be noted that the results depend on the structure of bargaining power; see note 17, below. Note that this result would not necessarily exist in other types of frictional models. For example, in a model with heterogenous workers and firms, a worker finding a good match would be able to extract a rent from her employer if she has bargaining power. If sufficiently large, this rent, combined with the frictional unemployment, could fulfill the role of efficiency wages. Furthermore, Proposition 3 would not necessarily apply to such a model; workers could bargain for a wage above the minimum in the frictional framework. Because it is not possible to analytically solve for equilibrium values of unemployment, necessary and sufficient conditions for superadditivity cannot be shown analytically. A large array of parameter combinations are likely sufficient for generating (or eliminating) it. It is necessary for the undistorted model to have zero unemployment, and for the unemployment levels in the individual efficiency wage and frictional models to be low (for example see Fig. 2). Thus, superadditivity is facilitated by large labor productivity, and low unemployment benefits. Note that these numerical examples are based on a particular parameter specification and should not be considered general results. Further, it should be noted that the model is not dynamic—the examples only show how the labor market reacts to one-time shocks. Dynamic efficiency wage and frictional models have been shown to have interesting properties not captured in this static model including labor hoarding (see Orphanides, 1990; and Hamermesh and Pfann, 1996). Note that wages do not change in reaction to many shocks because the assumption of constant marginal product. However, the combined model does have more wage variance even under diminishing marginal labor productivity. Further, it should be noted that under certain parameter specifications, the combined model exhibits dampened changes in unemployment when compared to the other individual models. Although there are a variety of different models that attempt to simulate the business cycle, Real Business Cycle (RBC) models, pioneered by Kydland and Prescott (1982) and Long and Plosser (1983) are the benchmark. As reported in Andolfatto (1996), when compared to actual U.S. data, the standard RBC model does not have as much variation in output; total hours and employment do not vary enough relative to output, are too highly correlated with output, and do not lag output; and productivity and the real wage are too highly correlated with output and do not lead the cycle. See Stadler (1994) for a review of RBC models and their variants. Dynamic efficiency wage models include Gomme (1998), MacLeod, Malcomson and Gomme (1994), Orphanides (1993), Danthine and Donaldson (1991), and Strand (1991). Dynamic frictional models include Andolfatto (1996), who uses a wellspecified search model, and MacLeod et al. (1994), who assumed that workers changing jobs pay a cost. Non-efficiency wage and non-frictional models include Hansen (1985), who assume workers must work a fixed number of hours (or none at all); Benhabib, Rogerson, and Wright (1991), who assume shocks to home produc-

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tion; and Burnside, Eichenbaum, and Rebelo (1990), who assume labor must be hired before true productivity is known.

Appendix Denote the frictional labor maximum wage condition as w ⫽ w D F (L; 䡠), the undistorted D labor maximum wage condition as w ⫽ w U (L; 䡠), the efficiency wage minimum wage S S condition as w ⫽ wEW (L; 䡠), and the undistorted minimum wage condition as w ⫽ wU (L; 䡠). Lemma 1:

d关w FD共L; 䡠 兲兴 ⬍ 0. dL

Proof: Trivial by differentiating Eq. 15. Q.E.D. Lemma 2: w FD共L; 䡠 兲 is negatively asymptotic to full employment. Proof: This is clear by taking the limit of Equation 15: limL3N 共w兲 ⫽ ⫺⬁ Q.E.D. Proposition 1: A single equilibrium {w, L} exists for models C, Ew, F, and U if (A ⫺ c) ⱖ s ⫹ e ⫹ (e/q)(b ⫹ r). D S Proof: If ( A ⫺ c) ⱖ s ⫹ e ⫹ (e/q)(b ⫹ r), w D F (L⫽0, 䡠), w U (L⫽0, 䡠) ⱕ w EW(L⫽0, 䡠), S D D w U(L⫽0, 䡠). Lemma 1 shows that (d[w F (L, 䡠)]/dL) ⬍ 0 and (d[w U (L, 䡠)]/dL) ⫽ 0 @ L僆 S S [0, N]. Shapiro and Stiglitz (1984) shows that w EW (L, 䡠), w U (L, 䡠) are not decreasing in L the domain L僆[0, N]. Thus, there exists a unique equilibrium {w C僆 [s ⫹ e ⫹ (e/q)(b ⫹ S r). ⬁], L C僆 [0, N]} s. t. w FD (L, 䡠) ⫽ w EW (L, 䡠), {w EW僆 [s ⫹ e ⫹ (e/q)(b ⫹ r), ⬁], U S L EW僆 [0, N]} s. t. w D (L, 䡠) ⫽ w EW(L, 䡠), {w F僆 [s ⫹ e, ⬁], L F僆 [0, N]} s.t. S D S w FD (L⫽0, 䡠) ⫽ w U (L, 䡠) and {w U僆 [s ⫹ e, ⬁], LU僆 [0, N]} s. t. w U (L⫽0, 䡠) ⫽ w U (L, 䡠). Q.E.D. Proposition 2: The equilibrium wage in each of the models C, EW, F and U can be ordered from highest wage to lowest wage as follows: w*U ⫽ w*EW ⬎ w*C ⬎ w*F, ceteris paribus. S Proof: By d[w FD (L, 䡠)]/dL ⫽ 0, and competitive assumptions, w *U ⫽ w *EW . By d[w EW (L; S S 䡠)]/dL ⬎ 0 and Lemma 1 w *EW ⬎ w *C. By Lemma 1 and w EW(L⫽L i ; 䡠) ⬎ w U(L⫽L i , 䡠) @ L i 僆[0, N], w *C ⬎ w *F. Q.E.D. Proposition 3: The equilibrium level of employment for models C, EW, F and U is ordered as follows: L*U ⬎ L*EW ⬎ L*C, L*U ⬎ L*F ⬎ L*C, ceteris paribus. D Proof: L *U ⬎ L * EW as d[w U (L, 䡠)]/dL ⫽ 0 and L *U(w⫽w i ; 䡠) ⬎ L *EW(w⫽w i ; 䡠) S @ w i 僆[s ⫹ e, ⬁). L *EW ⬎ L *C by d[w EW (L; 䡠)]/dL ⬎ 0 and Lemma 1. L *U ⬎ L *F by S D Lemma 1 d[w U(L; 䡠)]/dL ⫽ 0兩L僆 (0, N), and L ⱕ N. L *F ⬎ L *C as d[w U (L; 䡠)]/dL ⬍ 0 and L *U(w⫽w i ; 䡠) ⬎ L *EW(w⫽w i ; 䡠) @ w i 僆[(s ⫹ e), ⬁). Q.E.D. References Akerlof, G.A. (1982). Labor contracts as partial gift exchange. Quarterly Journal of Economics 97(4) (November), 543–569.

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