It is well known that profit sharing arrangements Pareto-dominate fixed wage contracts. Share agreements are (far) less than ubiquitous, however. This paper offers a solution of this „fixed wage puzzle“ by adopting a perspective of bounded rationality. We show that share arrangements that fulfill „plausible“ constraints are not generally acceptable to both firms and unions.

Jochen Michaelis

Jürgen Jerger

English

Research Papers in Economics

 
Arbeiten aus dem 
 
OSTEUROPA-INSTITUT 
REGENSBURG 
Wirtschaftswissenschaftliche Abteilung 
 
Working Papers 
 
 
 
Nr. 283 April 2010 
The Fixed Wage Puzzle: 
Why Profit Sharing Is So Hard to Implement 
 
Jürgen JERGER* and Jochen MICHAELIS
†  
 
 
 
 
 
 
 
* Osteuropa-Institut Regensburg and University of Regensburg, Universit¨atsstr. 31, 
93053 Regensburg, Germany, Email: juergen.jerger@wiwi.uni-regensburg.de.  
Phone: ++49-(0)941-9432697. 
†
 University of Kassel, Nora-Platiel-Str. 4, 34127 Kassel, Germany,  
Email: michaelis@wirtschaft.uni-kassel.de, Phone: (0) 561-8043562 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
OSTEUROPA-INSTITUT  
REGENSBURG 
Landshuter Str. 4 
93047 Regensburg 
Telefon:  0941 943 5410 
Telefax:  0941 943 5427 
E-Mail: oei@osteuropa-institut.de 
Internet: www.osteuropa-institut.de 
 
ISBN 978-3-938980-32-3 
 Contents
Abstract..................................................................................... v
1 Introduction.............................................................................. 1
2 Basic Model and Fixed Wage Solution ................................................... 2
3 Changeover Scenarios................................................................... 3
3.1 Three Plausible Suggestions ...................................................... 3
3.2 Total Wage Equivalence on Impact ................................................ 3
3.3 Total Wage Equivalence after Employment Adjustment ............................. 4
3.4 Share Offer of Firms............................................................... 4
4 Conclusions ............................................................................. 6Abstract
It is well known that proﬁt sharing arrangements Pareto-dominate ﬁxed wage contracts. Share
agreements are (far) less than ubiquitous, however. This paper offers a solution of this „ﬁxed
wage puzzle“ by adopting a perspective of bounded rationality. We show that share arrangements
that fulﬁll „plausible“ constraints are not generally acceptable to both ﬁrms and unions.The Fixed Wage Puzzle: Why Proﬁt Sharing Is So Hard to Implement
1 Introduction
Itis wellknown thatproﬁt sharing arrangementsPareto-dominate ﬁxedwage contracts. In
particular, proﬁt sharing schemes may give rise to higher productivity of workers (Bental
and Demougin 2006); allow for an efﬁcient risk distribution (Brouwer 2005); and foster
employmentbyreducingthemarginalcostoflabor(Weitzman1985; JergerandMichaelis
1999). Despite these arguments, share agreements are more the exception than the rule
(Pendleton et al. 2001).
The inconsistency between the desirability of proﬁt-sharing schemes in theory and their
rare use was already noted in Weitzman (1985) and termed the „ﬁxed wage puzzle“ by
Kirstein and Kirstein (2007). Weitzman’s explanation of this puzzle invokes an external-
ity. Whereas the excess labor demand created by share contracts is beneﬁcial for a society,
individual ﬁrms have an incentive to revert back to a ﬁxed-wage contract if everyone else
maintains proﬁt sharing. This externality, however, vanishes when there is a negotiation
over the parameters of a share contract (Holmlund 1990).
In this paper, we show that the ﬁxed wage puzzle can be understood by invoking a mild
form of bounded rationality. Speciﬁcally, we assume that instead of thinking through the
whole bargaining and adjustment processes of a changeover from ﬁxed wages to share
contracts, unions and ﬁrms consider to implement a share solution with some plausible
properties to be detailed below. A share contract is reached if and only if both parties
prefer it relative to the ﬁxed wage solution.
In the next section, we present the basic model and the ﬁxed wage solution that serves
as a benchmark. Section 3 presents three changeover scenarios, section 4 concludes.
1OSTEUROPA-INSTITUT REGENSBURG Working Paper Nr. 283
2 Basic Model and Fixed Wage Solution
We consider an economy populated by a large number of monopolistically competitive
and symmetric ﬁrms producing revenue R by means of labor N according to R = N,
where 0 <  < 1.   (1;2
) denotes a revenue shock. Since we conﬁne ourselves to
the sectoral level and the analysis of symmetric ﬁrms, we can do away with ﬁrm indices.1
Each ﬁrm faces a monopoly union whose behavior is driven by the maximization of the
utilitarian Stone-Geary function
U = N 

W 1 

1   

 
B1 

1   


; (1)
where W denotes the real wage, B the exogenously given alternative income, and 
 > 0
the Arrow-Pratt measure of relative risk aversion.
The sequence of events is as follows. First, the union sets the wage rate W after which
the revenue shock  is revealed. Then the ﬁrm decides about the level of employment.
This sequence ensures that ﬁxed wage contracts do not completely isolate workers from
uncertainty since employment will depend on the realization of .
Solving the dynamic game, we have ﬁrst to consider the ﬁrms’ decision, which is sum-
marized by the labor demand schedule
N =


W
 1
1 
: (2)
Expected employment is thus given by
E(N) =
 
W
 1
1 
 ; (3)
where  = 1+
2

2(1 )2. Note that E(N) depends positively on 2
 due to the convexity of
the labor demand schedule (2).
Maximizing the expected value of (1), observing (3), yields W =   B, where  
1=(
 1) and   +(1 )
. The optimal wage does not depend on 2
, i.e. the revenue
shock shows up in employment only.2 For further reference, expected utility and expected
proﬁts in the ﬁxed wage solution are given as follows:
E(U) =
1   



B
 1
1 
 (4)
E() = (1   )


B
 
1 
 (5)
1Our focus on the incentives to arrive at a share agreement also allows us to ignore the
macroeconomic consequences.
2This property, however, is due to the fact that the labor-demand schedule is iso-elastic
(MacDonald and Solow 1981).
2The Fixed Wage Puzzle: Why Proﬁt Sharing Is So Hard to Implement
3 Changeover Scenarios
3.1 Three Plausible Suggestions
The „straightforward“ comparison between ﬁxed and ﬂexible wage arrangements looks
at the differential, both in union utility and proﬁts, between the two bargaining outcomes
(Jerger and Michaelis 1999). However, since the theoretical possibility of a Pareto-
improvement via a bargaining process might be difﬁcult to evaluate, ﬁrms and unions
might consider „plausible“ instead of optimal share arrangements. Clearly, the difﬁculty
is the speciﬁcation of plausibility. In this paper, we offer three different scenarios. First,
we look at a share contract that ensures that the total wage is equal to the ﬁxed wage
„on impact“ , i.e. when employment is still at the level that prevails under a ﬁxed wage
contract. Second, we analyze a contract that leaves the wages of incumbent workers un-
changed after the adjustment of employment under the share contract took place. Finally,
we consider whether it makes sense for a ﬁrm to offer a share contract unilaterally.
A share contract speciﬁes workers’ remuneration as some base wage ! that is comple-
mented by a share  of the difference between revenue per worker and the base wage.3
Formally: W = !+(R(N) !N)=N. Clearly, a given level of total remuneration can
be achieved via an inﬁnite number of combinations of ! and . To pin things down, we
assume ! = B.
3.2 Total Wage Equivalence on Impact
A plausible minimum requirement from the viewpoint of workers for a change from ﬁxed
wages to a share contract is that incumbent workers’ wages remain the same as long as
employment is unaltered. This requirement implies  =
( 1)
  . Expected employment
is given by E(N) = (=B)1=(1 ). Clearly, total remuneration is lower than under a
ﬁxed wage contract after the adjustment to this employment level. Expected utility and
expected proﬁts are readily computed as
E(U) =
1
1   

 
 
1 
   1
 
B
 1
1 
 (6)
E() = (1   )
2 
   

B
 
1 
; (7)
where    ((2   )   1)=(   ). Comparing (4)-(5) to (6)-(7) leads to the following
result:
Proposition 1 Unions will agree to a share contract (!;) =

B;
( 1)
 

if and
only if the degree of relative risk aversion 
 is less than unity. Firms will always
accept this share contract since proﬁts are higher than under ﬁxed wages.
3This revenue sharing is economically equivalent to proﬁt sharing (Michaelis 1997).
3OSTEUROPA-INSTITUT REGENSBURG Working Paper Nr. 283
We are only aware of two empirical studies that directly estimate the crucial parameter,
i.e. the unions’ degree of relative risk aversion. Farber (1978) in a study of the US United
Mine Workers over the period 1948-1973, ﬁnds a 
 of about 3. Carruth and Oswald
(1985) derive an estimate of 0.8 – slightly below the benchmark of proposition 1 – from
the behavior of unions in the coal and steel industries in the United Kingdom over the
period 1950-1980. The literature on the degree of relative risk aversion of households
arrives at values of 
 between 1 and 5 (Watt 2002, Dohmen et al. 2010). Hence, risk
aversion might well explain the reluctant use of share contracts.
3.3 Total Wage Equivalence after Employment Adjustment
Incumbent workers may object to the changeover scenario of the last subsection for the
simplereasonthatinequilibrium, i.e. aftertheadjustmentofemployment, theirtotalwage
will be lower than the ﬁxed wage. Thus, the next plausible suggestion is a share contract
that leaves wages unchanged after employment is adjusted to the proﬁt-maximizing level.
With ! = B, this requires  =
( 1)
1  . The algebraic solution can be omitted because the
intuition is rather straightforward. It is summarized in the following
Proposition 2 Unions will always agree to a share contract (!;) =

B;
( 1)
1 

.
Firms, however, will not accept this share contract since proﬁts are lower than
under ﬁxed wages.
Unions beneﬁt unambiguously since employment rises, whereas total remuneration per
worker remains the same (by assumption). This clearly implies declining proﬁts.
3.4 Share Offer of Firms
The two previous scenarios looked at speciﬁc share contracts. Now we ask whether it
may beneﬁt a ﬁrm to unilaterally offer some positive share  to the union. The rationale
for that may be that the union is willing to reduce the base wage component in order to
foster employment. This in turn might lead to higher proﬁts. The sequence of events thus
is    !      N, where the monopoly union sets !. Looking at the ﬁrms’ problem, we
start from the expected proﬁt function E() = (1   )(1   )(=!)

1 . Maximizing
with respect to  yields the ﬁrst-order condition
@E()
@
= 0 )
@!
@
=  
!(1   )
(1   )
: (8)
If the ! requested by the monopoly union declines in  according to (8) or faster, ﬁrms
have an incentive to offer such a share component to their workers. Writing union utility
as a function of both ! and  and maximizing with respect to !, treating  as given, leads
to the optimal base wage demand ! =
B
(1 )+. From this, it is clear that unions reduce
4The Fixed Wage Puzzle: Why Proﬁt Sharing Is So Hard to Implement
their base wage claim according to
@!
@
=  
!(1   )
(1   ) + 
: (9)
By comparing this result to (8), it is immediately clear that for any positive , the reduc-
tion of the base wage claim is lower than what ﬁrms would need in order to offer share
contracts. This establishes
Proposition 3 Firms have no incentive to offer a positive share of proﬁts to work-
ers, although unions would react by lowering the base wage.
Only if the moderation of the base wage is complemented by an increase in labor pro-
ductivity, a unilateral offer of a share contract might be possible (see Koskela and Sten-
backa 2006).
5OSTEUROPA-INSTITUT REGENSBURG Working Paper Nr. 283
4 Conclusions
In most ﬁrms, ﬁxed wage contracts are in effect and serve as a natural benchmark for
a share contract. From this benchmark, however, plausible changeover suggestions are
likely to be refused by one party involved. Fixed wage contracts thus exert quite some
„normative power of the factual“ – and this power may well prevent more efﬁcient share
agreements. We show that plausible share arrangements are unlikely or impossible to be
accepted. Only if workers display a relatively moderate degree of risk aversion and agree
to a share contract under which total remuneration declines relative to the ﬁxed wage
contract, there is a possibility for a mutually acceptable changeover from ﬁxed wages to
a share arrangement.
One should note that this analysis is an application of the general and recently very
fruitful analysis of bounded rationality. Under unbounded rationality (and the assump-
tions concerning technology and union utility in this paper), a share agreement clearly
Pareto-dominates a ﬁxed wage agreement. We thus offer a new perspective on the well-
known ﬁxed wage puzzle.
6The Fixed Wage Puzzle: Why Proﬁt Sharing Is So Hard to Implement
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7

Defending Expected Utility Theory&quot;,

Employee share ownership and proﬁt-sharing in the European Union, Luxembourg: Ofﬁce for Ofﬁcial Publications of the European Communities.

Flexible and Committed Proﬁt Sharing with Wage Bargaining: Implications for Equilibrium Unemployment&quot;,

Incentive Contracts and Total Factor Producitivty&quot;,

Individual Preferences and Union Wage Determination: The Case of the United Mine Workers&quot;,

Individual Risk Attitudes: New Evidence from a Large, Representative, Experimentally-Validated Survey&quot;, IZA Discussion Paper 1730, forthcoming in:

Inefﬁcient Intra-Firm Incentives Can Stabilize Cartels in Cournot Oligopolies, mimeo,

Managing uncertainty through proﬁt sharing contracts from medieval Italy to Silicon Valley&quot;,

Miners’ Wages in Post-war Britain: An Application of a Model of Trade Union Behaviour&quot;,

The Fixed Wage Puzzle : Why Profit Sharing Is So Hard to Implement

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