I argue that in contrast to the literature to date efficiency wage and bargaining solutions will typically be independent. If the bargained wage satisfies the efficiency wage constraint efficiency wages are irrelevant. If it does not, typically we have the efficiency wage solution and bargaining is irrelevant.

Frank Walsh

English

Research Papers in Economics

 
 
 
 
 
 
UCD CENTRE  FOR  ECONOMIC  RESEARCH 
 
 
 
WORKING  PAPER  SERIES  
 
 2005 
 
 
             Efficiency Wages and Bargaining 
 
    
             Frank Walsh, University College Dublin  
 
 
 
WP05/23 
 
November 2005 
 
 
 
 
 
 
 
UCD SCHOOL OF ECONOMICS 
UNIVERSITY COLLEGE DUBLIN 
BELFIELD  DUBLIN  4 
 
 
 
 
 
 
 
 
  
 
 
 
 
Efficiency Wages and Bargaining 
Frank Walsh 
Economics Department 
University College Dublin 
Belfield, Dublin 4 
Frank.walsh@ucd.ie 
Phone: 353-1-7168697 
 
 
 
 
 
 
 
Abstract 
 
I argue that in contrast to the literature to date efficiency wage and bargaining 
solutions will typically be independent.  If the bargained wage satisfies the efficiency 
wage constraint efficiency wages are irrelevant.  If it does not, typically we have the 
efficiency wage solution and bargaining is irrelevant.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   1 
Introduction 
 
A number of papers that have modelled the interaction between the efficiency 
wage and bargaining models and find that efficiency wages weaken [Lindbeck and 
Snower (1991)] or strengthen [Sanfey (1993), Toulemonde (2003) or Strand (2003)]
1 
the workers payoff from the bargaining game In this paper I show that the models will 
be independent in most cases.  A bargained wage that satisfies the efficiency wage 
constraint is unaffected by the constraint.  If the constraint is not satisfied, typically 
we have the efficiency wage solution and bargaining is irrelevant
2. 
 
I. The efficiency wage 
A worker has the following utility at any point in time t : ) (x g w u - =  where  
w is the wage, x is effort, and g(x) is a convex function of effort. The flow value of a 
job at any point in time (t) in a stationary equilibrium is: 
)] ( ) ( [ ) ( ) ( t V t V b x g w t rV
u - + - =      (1) 
The Poisson arrival rate of exogenous firm shutdowns isb and the discount rate is r.  
Workers get the wage stream plus the expected capital gain/loss from moving to 
unemployment (V
u) times the exogenous arrival rate of separations. We focus on a 
stationary equilibrium.  From (1) the value of a job or rent is: 
r b
rV x g w
V V
u
u
+
- -
= -
) (
  (2) 
If F(.) is a well behaved production function and n is employment, the value of the 
firms expected profit stream is: 
                                                 
1 Hoel (1988) looks at the impact of efficiency wages on local versus centralised wage bargaining. 
2 There is also an empirical literature examining whether worker rents are due to efficiency wages or 
rent sharing.  See Konings and Walsh (1994) or Machin and Manning (1992).    2 
r b
n w n x F
+
+ -
=
) , (
p   (3) 
We assume that firms must pay a wage that satisfies the following inequality to satisfy 
the concave efficiency wage constraint illustrated in Figure 1. 
) (x w w
e ‡   (4) 
From (3), taking the number of workers as given, we draw isoprofit curves for the 
firm in x,w space.  The firms preferred wage effort bundle is where the isoprofit curve 
Q* is tangent to the efficiency wage constraint in Figure 1.   
We can solve explicitly for the Shapiro and Stiglitz (1984) efficiency wage 
model.  Equation (1) is the value of not shirking and q is the arrival rate of 
supervisors.  When we equate (1) with the value of 
shirking )] ( ) ( )[ ( ) ( t V t V q b w t rV
s u s - + + = the efficiency wage and rents for 
effort x are
3:  
q
x g
V V
rV A x g rV
q
q r b
x g w
u
u u e
) (
) ( ) )( (
= -
+ = +
+ +
=
  (5)
 
Yellen (1984), Katz (1986) or Weiss (1991) discuss the different efficiency 
wage models.  Apart from the shirking model these include the turnover cost model 
[Salop (1979)], sociological models such as the fair wage hypothesis [Akerlof and 
Yellen (1990)] or adverse selection models [Weiss (1991)].  These models all predict 
that firms will offer a wage effort combination where rents [equation (2)] are positive, 
but only in the shirking model would we expect rents to increase with effort as in (5).  
For example the  fair wage hypothesis  is that firms pay higher wages to placate 
                                                 
3 Malcolmson (1999) shows that even when the underlying assumptions of  Shapiro and Stiglitz do not 
hold workers earn rents when  monitoring is difficult as long as reputation effects are not too strong 
and there is some unemployment.  Akerlof and Katz (1989) show workers will earn rents even if firms 
use upward sloping wage profiles.   3 
disgruntled workers who perceive they are being treated unfairly or in  the turnover 
cost model wage premiums prevent costly turnover.  In these two models firms satisfy 
the efficiency wage constraint by putting workers on a higher indifference curve in 
wage effort space.  In Figure 1 if the efficiency wage model is the shirking model 
worker utility is increasing with effort along the efficiency wage constraint.  In other 
efficiency wage models the efficiency wage constraint in Figure 1. is an indifference 
curve.  
 
II.The Nash Bargaining Solution 
  Holding employment fixed
4 and where (2) and (3) are the worker’s and firm’s 
objective functions we solve for the Nash bargain over x and w by maximising: 
m m m mp Q P
r b
w x V w x V w x S
u - -
+
= - =
1 1 1
) , ( ] ) , ( [ ) , (    (6) 
A crucial difference between this paper and the previous literature is that the 
efficiency wage constraint does not enter the workers objective  function.  M ost 
previous studies imposed a binding parametric relationship between wages and effort 
whereas we see from Figure 1., bargaining solutions to the right of w
e(x) are feasible.  
Strand (2003) used the efficiency wage constraint as the workers fallback utility in the 
bargaining game on the basis that firms must hire workers before bargaining with 
them in a matching framework.  The assumption that workers would work at the 
efficiency wage while bargaining seems unreasonable. This will lead to a bargaining 
outcome that benefits either workers or firms and makes the other party worse off.  
Since working at the efficiency wage during bargaining must be by mutual agreement 
                                                 
4 See Oswald (1993) for a model where insiders will not bargain over employment, while the norm in 
the matching literature is that bargaining is between the worker and firm.   4 
this will not take place.  We set the threat point equal to the exogenously given value 
of unemployment for the worker and to zero for the firm
5. 
w
x
w
x
Q
Q
P
P
=           (7) 
This is the condition that the solution must be on the contract curve in w and x 
space where workers indifference curves are tangent to firms isoprofit curves.  
Differentiating P and Q the slopes of the indifference and isoprofit curves respectively 
are: 
) (
1
|
x g dw
dx
x
P=   
) , (
|
n x F
n
dw
dx
x
Q=     (8) 
From ( 8) we see that the indifference/isoprofit curves will have positive and 
diminishing/increasing slopes respectively as in Figure 2.  The curves P0..P1 
represent indifference curves for the workers while Q0..Q1 are isoprofit curves while 
Q* is the isoprofit curve where the firm chooses its efficiency wage optimally. The 
contract curve CC shows the various combinations that satisfy (7).
6:  
x x ng n x F = ) , (    (9) 
This is the condition that the level of effort will be set such that its marginal 
product will equal the disutility of effort to workers
7.  A lower value of µ moves the 
bargaining solution to the right along the CC curve in Figure 2.  
As we noted earlier in many models the efficiency wage constraint will be an 
indifference curve.  It follows that the efficiency wage solution   will be on the 
contract curve as in Figure 2a.  High values of µ lead to bargaining solutions on the 
CC curve to the left of w
e that are not incentive compatible. Both parties are better off 
                                                 
5 Following Binmore et al. (1986) we use the outside option as the threat point assuming the motivation 
for bargaining is the fear of a breakdown.  If impatience was the motivation the threat point would be 
zero for both parties.  This would not affect the qualitative results. 
6 This is horizontal because utility is separable in wages and effort. 
7 Strand (2003) notes that bargaining yields the efficient level of effort but this is conditional on 
employment which is possibly set inefficiently given the wage premium.   5 
at the efficiency wage solution.  Bargaining solutions to the right of w
e are incentive 
compatible and unaffected by the efficiency wage constraint.  
The situation is more complicated for the shirking model.  From (5) the slope 
of the efficiency wage constraint is 
A x g dw
dx
x ) (
1
= .  Since A>1 this is flatter than the 
isoprofit curve [equation (8)] at the point where the CC curve and efficiency wage 
constraint intersect (wb,x*) in Figure 2b.  It follows that the isoprofit curve will be 
tangent to the efficiency wage constraint to the left of this point at (we,xe).  The 
following discussion will look at three regions of the CC curve in Figure 2b.   
 
(1)  From Figure 2b any solution on the CC curve where w>wb is incentive 
compatible and the efficiency wage constraint is irrelevant.  Solutions 
where w<wb are not incentive compatible. 
(2)   Figure 3a draws an indifference curve labelled P** through  the firm’s 
preferred efficiency wage solution (we,xe).  This intersects the CC curve at 
w**.  If bargaining yields a wage less than w** on the CC curve both 
workers and firms prefer (we,xe) to any such solution, (we,xe) is the 
solution and bargaining is irrelevant. 
(3)    A  bargained wage between w** and w b is not incentive compatible.  
Neither is (we,xe) because workers  prefer  the bargaining solution to 
(we,xe).  The best outcome for firms that is incentive compatible and which 
workers will accept is illustrated in Figure 3b.  Draw an indifference curve 
through the bargained wage labelled P***.  The point where this intersects 
the efficiency wage constraint gives the profit maximising contract that is 
incentive compatible and which workers will accept over the bargaining   6 
solution.  In this region between w** and w
e the efficiency wage constraint 
does affect the bargaining outcome. 
 
In summary, a bargained wage  greater than wb  means  the efficiency wage 
constraint  is irrelevant.  A bargained wage less than w**  means  bargaining is 
irrelevant.  A  bargained wage between w** and w b  means the wage effort 
combination will lie on the efficiency wage constraint between (we,xe) and (wb,x*).  
In this last case the efficiency wage constraint does not change worker utility relative 
to the bargaining solution but leads to a lower wage effort combination which is 
incentive compatible. 
 
IV. Conclusion 
Efficiency wage and bargaining models are used widely and are not mutually 
exclusive.  If wages are determined by bargaining it may be difficult to monitor 
workers or notions of how fairly they are being treated may affect worker’s 
performance.  This paper suggests that from the point of view of a researcher 
modelling the labour market in this situation life is simpler than the earlier literature 
suggests.  This paper does not show which  model  should be used but suggests that 
once we make this choice we need not worry about interactions with the other model. 
There may also be implications for modelling endogenous union membership.  
An i ncrease in the importance of firms where efficiency wages are paid  may 
undermine the role of unions other things equal. 
 
 
   7 
References 
 
Akerlof, G.A. and Katz, L.F. (1989), "Worker's Trust Funds and the Logic of Wage 
Profiles. " Quarterly Journal of Economics,  August. 523-537 
 
Akerlof, George and Janet L. Yellen (1990) “The Fair Wage Hypothesis and 
Unemployment” Quarterly Journal of Economics, May, issue 2.  pp 255-283. 
 
Binmore, Ken, Ariel Rubenstein and Asher Wolinsky (1986) “The Nash Bargaining 
Solution in Economic Modelling” Rand Journal of Economics, Vol. 17, No. 2 
 
Carmichael, H. Lorne, (1985) “Can Efficiency Wage Unemployment be 
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Hoel, Michael (1989) “Efficiency Wages and Local Versus Central Wage 
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Katz, Lawrence F. (1986) “Efficiency Wage Theories: A Partial Evaluation” In 
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   8 
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Figure 1: The Efficiency wage and level of effort 
 
 
 
 
 
Wage 
u rV  
Effort 
) (x w w
e =  
we 
xe 
Q*   9 
Figure 2: The Contract curve and the efficiency wage constraint. 
(a) Other efficiency wage models. 
 
(b) The Shapiro and Stiglitz model 
 
 
 
 
 
 
Wage 
Effort 
) (x w w
e =  
x* 
Q0 
Q1  P0 
P1 
CC 
we 
Q* 
Wage 
Effort 
u rV A x g w + = ) (
we 
x* 
Q0 
Q1  P0 
P1 
CC 
wb 
xe 
Q*   10 
Figure 3 a: Wage/effort combinations when  the bargained wage violates the no-
shirking condition 
 
 
Figure 3b: Wage/effort combinations when the bargained wage violates the no-
shirking condition 
 
 
 
 
Wage 
Effort 
u rV A x g w + = ) (  
we 
x* 
P*** 
CC 
wb 
xe 
w** 
Wage 
Effort 
u rV A x g w + = ) (
we 
x* 
P** 
CC 
wb 
xe 
Q* 
w** 

C.(1979) “A model of the natural rate of unemployment” American Economic Review Shapiro Carl

Efficiency Wage Models of Unemployment”

Efficiency Wage Theories: A Partial Evaluation” In

Efficiency Wages: Models of Unemployment, Layoffs and Wage Dispersion:

Efficient Contracts are on the Labour Demand Curve”

Evidence of Efficiency Wage Payments in UK Firm Level Panel Data”,

Individual Employment Contracts.

On the Interaction Between Efficiency Wages and UnionFirm Bargaining Models”,

Relative wages, Efficiency wages, And Keynesian Unemployment, American Economic Review, Papers and

Efficiency Wages and Bargaining

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