Journal of Economics
Vol. 53 (1991), No. 2, pp. 161-184
Zeitschrift
f~ir National6konomie
@ Springer-Verlag 1991
Printed in Austria
Long-term Union-Firm Contracts
By
Geir Asheim, Bergen-Sandviken, Norway,
and Jon Strand, Oslo, Norway*
(Received March 12, 1990; revised version received December 6, 1990)
We explore the possibility for self-enforcing long-term contracts between
a risk averse union and a risk neutral firm, when these have the option
to strike an efficient bargain at every stage, and the state of the world is
variable. It is shown that any long-term efficient wage agreement satisfying
individual rationality constraints involves a more even income stream to the
workers (except for the case when the discount rate is high) and can be
implemented by a Subgame-perfect equilibrium (by the threat of returning
to short-term bargaining). Moreover, any such constrained efficient agreement
can be supported by the threat of triggering agreements which themselves
are constrained efficient, i.e., it can be implemented by a Renegotiation-proof
equilibrium.
1. Introduction
In recent years we have witnessed several interesting developments
in the theory of labor markets. Two of the most important are the
theory of "implicit labor contracts," and the theories of labor unions
* This paper is a revised version of Strand (1988) and is part of the research
project "Wage Formation and Unemployment" at the SAF Center for Applied
Research at the Department of Economics, University of Oslo. We would like
to thank seminar participants at Cambridge University for pointing out an error
of an earlier version. Thanks also to Terje Lensberg, Kjell Erik Lommerud,
Lars Thorlund-Petersen, and two referees, as well as seminar participants at
the 1989 EEA Conference in Augsburg and at the Universities of British
Columbia, Haifa, and Maryland for helpful comments.
162
G~ Asheim and J. :Strand:
and union-firm behavior, The first of these emphasizes the possibilities
for intertemporal risk shifting between risk averse workers and risk
neutral firms, given that markets for contingent claims outside of the
firm-worker relationship are imperfect or nonexistent. A long-term
relationship between workers and firms may then work to smooth the
stream of wage payments, relative to that resulting from a series of
short-term congaqts, while employment.often,is,set at an efficient level. 1
In the existing union-~rm literature, dn the 0therhand, the emphasis
is instead placed on wage and wage-employment bargaining 2 in static
models where the time dimension plays a small role, at least in the
formal execution of the Uni0n-firm gamei3.... ....... ....
In the present paper w e will attempt ~8~:bUild a bridge between
the theory .of efficient union,finn bargaining on the one side and
implicit contract theory on the other. We will take as our point of
departure a situation where the state of the world facing a firm is
stochastically variable, where al! workers in :the finn (who are fixed in
number and infinitely long lived) are organized in a single union and
at arty given,time have the opportunity to b a r g ~ o v e r their wage. (and
employment). For a given state of the ~world we' can then identify an
efficient short-term solution with a particular outcome .of the bargaining
game played at that stage..' Assuming (as we will throughout this paper)
that~this game always resuffs in efficient employment, the short-term
wage agreement wiif :in generaldepend on the state of the world, with
a higher wage in ihigh~ than low demand periods.
While each short'term agreement is efficient Within the corresponding state of the world, a sequence of such agreements is n o t e x a n t e
efficient in an intertemporal sense, given that the finn is risk neutral and
1 See the original contributions by Azariadis (,1975)and Baily (197.4) and
i~ier surveys .by :Azariadis and Stiglitz (1983)and Rosen (i985)i some of
tills literature, e. g. the Azariadis paper, in fact does not predict perfectly
efficient employment'but then generally in the' form of e'0veremployment"in
low-demand states.
2 The so-called monopoly union model of Oswald (1982) and others can
be viewed as a special case of wage bargaining with all bargaining power to
the union.
3 ExceptiOns to this are Espinosa and Rbee (i989) and Strand (i989) Who
study 'an infinitely repeated game between a monopoly Unionand a firm. in
these c0ntributions the derived sOlut{ons are as a ~mie %emi'-effieient'~ in the
sense that employmefit lies be:tweehits levds in the standard monopoly union
and efficient:bargaining models. See a!s0 the literature on firms, investment
decisions inthe ~face :of labor ufiiohs, elg. Grout (1984), Strand (1987), and
Hoel and Moene (1988).
Long-term Union-Firm Contracts
163
the union (whose members are assumed identical) risk averse, workers
can neither borrow nor lend to or from an external capital market nor the
firm, and both parties discount the future equally heavily. The natural
question which then arises, and which we will attempt to answer, is
whether there exist long-term contracts which are self-enforcing, i.e.
they do not rely on binding contracts, and which may even so improve
upon the sequence of short-term agreements, in the sense of making at
least one of the parties strictly better off. Hence, the problem has to be
analyzed through the application of non-cooperative game theory.
Our approach to this question is to derive a set of efficient long-term
agreements, constrained by the requirement that they can be implemented by Subgame-perfect equilibria of the associated infinite horizon
game between the firm and the union. In this game the enforcement
mechanism is at the outset assumed to consist of the threat of reverting
to the sequence of short-term bargaining solutions forever, should either
of the players decide not to abide by the wage agreement prescribed
by the equilibrium in any given subgame; this is in fact an optimal
penal code as defined by Abreu (1988). In section 3 we derive the
set of such efficient long-term wage agreements, which turn out to
minimize the wage variation between states: whenever the rate of
discount ~- (used by both players) is sufficiently close to zero, there
always exist implementable long-term agreements with the same wage
in both states. When r is higher, solutions with a constant wage across
states cannot be implemented by a Subgame-perfect equilibrium, while
solutions with less wage variation than under short-term bargaining
are still admitted. For r sufficiently high, there is no implementable
long-term wage agreement better than that resulting from short-term
bargaining.
Above the firm and the union discipline each other by the threat
of reverting to short-term bargaining forever should a deviation occur.
However, since there are mutually advantageous implementable longterm agreements, this implicitly assumes that the parties can commit
themselves not to renegotiate. Hence, allowing for renegotiation, the
threat supporting the long-term agreements is not credible if these
agreements are indeed viable. Section 4 resolves this dilemma by
demonstrating that in the present context any constrained efficient
long-term agreement can be supported by the threat of initiating agreements that themselves are constrained efficient. We call such equilibria
rene gotiation-proof .
These equilibria are interesting from a technical point of view since
the punishments are only triggered if a deviation occurs just prior to
the time at which the state changes. Hence, although deviations are
perfectly observable, the execution of the threat is dependent upon
164
G. Asheim and J. Strand:
a random event. 4 All constrained efficient long-term agreements can
be supported in this way since such stochastic punishments are as
deterring to a deviating player as an infinite sequence of short-term
bargaining solutions would be. The intuition behind this result is as
follows: assume that the union deviates by pushing the wage up in the
high-demand state. If then a change from high-demand to low-demand
occurs, there exist constrained efficient long-term agreements initiated
in this new low-demand state which can be used as a harsh punishment
against the union in such an eventuality. A converse argument holds
for the firm deviating in the low-demand state.
The analysis demonstrates that the risk averse labor union, that in
principle could obtain a very high wage in the good state of the world,
by bargaining aggressively, often will have incentives to refrain from
doing so, if it thereby can obtain some income smoothing in the form
of a higher wage in the bad state. This is more relevant the less heavily
the parties discount the future (or alternatively, the shorter lasting each
state is on the average), and the greater would be the wage variation
given a sequence of short-term bargaining solutions.
We conclude our analysis in Section 5 by requiring short-term and
long-term bargaining to be governed by the same principle, e.g. the
Nash bargaining solution. We show that such consistency is in fact
obtainable.
The paper yields a theory of implicit firm-union wage (and wageemployment) contracts which can be confronted with other models in
the literature attempting to do the same. One such model is Horn
and Svensson (1986), where a dominant (monopolistic) union writes
state-contingent risk sharing contracts with a firm, facing the firm
with a given expected level of utility, and where workers' wage in
low-demand periods is higher than that resulting from the respective
spot solutions. Malcolmson (1983) considers the set of state contingent
contracts between unions and firms that may be viable in the absence
of outside enforcement. The union here serves the dual role of enforcer
(e,g. by carrying out a strike if the firm breaks its part of the contract)
and as a verifier of the state of the world that actually occurs. While
the former paper may seem to be well in the spirit of standard implicit
contract theory, no mechanism for ex p o s t enforcement of the solution
is specified. Enforcement is a major issue in the latter paper, but only
4 Compare this to Green and Porter (1984) where individual deviations are
not observable, and where a symmetric punishment is triggered stochastically,
even if all players follow equilibrium behavior, in order to deter the players
from deviating. In contrast, in our equilibrium, punishments are only triggered
if a deviation in fact occurs.
Long-term Union-Firm Contracts
165
in a somewhat ad hoc way: there is no formal model showing that it
will actually be optimal for a union to carry out a particular threat (e.g.
a strike) ex post.
Our work is perhaps most closely related to Thomas and Worrall
(1988) who consider a long-term contract between a risk averse worker
and a risk neutral firm. The contract covers an infinite sequence of
periods, in each of which the productivity of the firm is random and
independently and identically distributed. They show, like we do, that
income smoothing will be the result of an optimal incentive constrained
worker-firm contract. Moreover, like ours their solution is subgame-perfect. They, however, focus on quits as the mechanism of reneging for
workers (which is never relevant within our union-firm formulation)
and disregard the issue of renegotiation-proofness (which is the main
focus of our paper). By imposing renegotiation-proofness our analysis
yields a more coherent analysis of the concept of self-enforcement in
such models.
We will also argue that the present theory is relevant for the explaining of "sticky wages" in unionized economies, over business cycles
where firms' productivities vary considerably, in particular when these
cycles are reasonably short. The model can easily be shown to yield
this prediction regardless of whether employment is fixed or variable.
While similar results are provided by Thomas and Worrall (1988), we
extend their results by considering renegotiation-proof equilibria, and
furthermore, by treating the case where a particular firm faces business
cycles of various length (with the states in successive periods being
correlated). Our analysis should then help to provide a more general
paradigm for understanding the phenomenon of wage inflexibility with
or without employment variability, in unionized economies.
2. Preliminaries
Consider a firm whose work force consists of a given number n of
infinitely living workers, all belonging to a union. Time is divided into
an infinite number of discrete stages. At every stage, the firm faces a
stochastic output price p, which can take two values Pl and P2, with
pl > p2, and which is observable by firm and workers but possibly not
by outsiders. The transitions between the states 1 and 2 are governed
by geometric distributions with parameters ql and q2. This implies that,
given state i, this state remains for an expected number 1/q~ of stages.
The two states and the transitions between them are meant to capture
business cycles in a rudimentary, but tractable manner. More states
166
G. Asheim and J. Strand:
would augment realism as well as analytical complication, and will not
be considered here.
The firm discounts the future at the rate r > 0, is risk neutral,
and is endowed with the production function f(nl), with fr(n~) > O,
f ' ( n ~) < 0, where n ~ < n is the number of workers employed. Workers
are risk averse with utilities u(y) only depending on income y, i.e.
u~(y) > O, u"(y) < 0 everywhere, and they can neither borrow nor
lend. 5 When workers are employed in the firm, y is synonymous with
wage income w. When workers are unemployed in state i, y equals
some alternative income bi (e.g. unemployment benefits or alternative
wage) plus a possible supplementary income from the firm. In order to
highlight the firm as an insurance provider, we assume that the workers
also discount the future by the rate r.
We assume that the firm and the union at every stage strike an
efficient bargain. Such efficiency implies that employment in state i,
n,,:, is independent of the wage rate w and is given by p~ff(ni) - bi >_ 0
and ni <_ n, with pif~(ni) - b~ = 0 if n~ < n. It is reasonable that the
number n - n~ of unemployed workers can be positive only in state 2
(the low demand state). Furthermore, with strictly concave worker utilities and no worker moral hazard, efficiency implies full unemployment
insurance. Therefore, the firm pays a supplementary income w - bi to
laid-off workers. These assumptions imply (a) that employment is not
influenced by the (short- or long-run) wage bargaining and will, hence,
not be discussed in the remainder of our paper, and (b) that given
state i, the utilities and profits depend solely on the wage rate w and
are given by u(w) and 7ri(w) = p~f(ni)+ b~(n- n i ) - wn, respectively.
We postulate as a benchmark a particular standard short-term wage
agreement in each state i, yielding a specific split of the net surplus
pif(n~) - bini by the implementation of the wage rate w~. This pair
of standard short-term agreements reflects the bargaining power of the
parties in short-term negotiation and is not affected by the history (e.g.
earlier wage rates) of the firm-union relationship. Since it is common
knowledge that short-term bargaining will, depending on the state,
lead to one of these standard agreements, negotiation is not called for,
and either party can at any stage unilaterally enforce the wage level
corresponding to this agreement. We will return to the details of this
short-term bargaining outcome in Section 5; here we will only assume
that w~ > w~.
The essential aspect of this solution in the present context is that
even though it is efficient within any given state, it is intertemporally
5 This is of course a strong assumption, but it is meant to capture the
notion that credit and capital markets are generally imperfect.
Long-term Union-Firm Contracts
167
inefficient since it exposes workers to income risk that in principle
could be borne by the risk neutral firm. The question arises whether
the firm and the union could credibly sustain an "implicit contract"
which both parties are willing to accept and honor, and which implies
less, preferably no, wage variability.
In order to characterize such contracts, we need some game theoretic
formalism. We will be concerned with two infinite horizon multi-stage
games, G~, i E {1, 2}, where i corresponds to the state at stage 0. Given
that the state at stage 0 is equal to i, the players k (firm) and 1 (union)
each take an action by announcing the wage rate they offer/demand at
stage 0. The action pair (w0k, W~o) results in the wage rate w0 = u,~
if w~ r Wlo, w0 is equal to their common value otherwise. Hence,
either side can force the standard short-term agreement, although it is
feasible for the parties to agree on other wage rates. Furthermore, at
stage 0 nature draws the state to arrive at stage 1, denoted i0. We have
that i0 equals i with probability (1 - qi), i0 equals the other state with
probability q~. In general, the state at stage t is given by it-i, and the
action pair (w tk ~' wl) results in the wage rate wt = wSz t - t if w~ r w t1,
wt is equal to their common value otherwise. When taking their actions
at stage t, the players are informed of the history at stage t, consisting
of the previous wage offers/demands as well as the previous and current
states. Hence, for any t _> 0, the game G,: is characterized by a set of
t-histories, H.it, given by
{(
l.
~t-1,
~ wx)
t
(w x,
IR 2, ix
O<x<t-1}
{1, 2};
,
(Hio = {i}) and a pair of strategies at stage t, hitJ :
Hit
]R 2
with j = k, l, being a function from the set of t-histories to the set
of actions. A strategy for player j (firm or union), hi,J consists of a
sequence of stage t strategies, (Tit,
J t = 0, 1, 2, . . . . A strategy profile
(hi,k hi)l determines the payoffs (expected discounted profits/utilities)
of the players through the wage path it generates. A wage path of Gi
determines at any stage t _> 0 the wage rate as a function of the history
of the states. If a strategy profile ( 5rki, ~ I i) generates a particular wage
path wi, then (a~, al) is said to implement wi.
What expected profits/utilities can the firm/union secure by insisting
on the short-term agreement at every stage? To answer this question,
let Hi~ denote the expected discounted profits of the finn in state i,
given that the short-term agreement is implemented at every stage. For
168
G. Asheim and J. Strand:
i = 1, 2, expected profits are determined by
H~ = 7rl(w~) + [(1 - ql)" H~ + ql" H~]/(1 + r) ,
(1)
H~ = 7r2(w~) + [(1 - q2)" II~ + q2' II~]/(1 + r) ,
(2)
yielding
H~ = [(r + q2)" ~r~(w~) + ql" ~r2(w~)]/D,
(3)
[I~ ~---[q2" 7rl(w~) ~- (T ~- ql)' 7r2(w~)]/D ,
(4)
where D = r 9(r + ql + q2)/(1 + r). It is natural to expect 7rl(W~) >
7r2(w~), such that II~ > II~, since with Pl > P2 the firm should be
able through short-term bargaining to secure for itself a higher profit
in the good than in the bad state. However, whether zrl(w~) > 7r2(w~)
or not is of no consequence for what follows. Note that in any case,
the relative difference between II~ and II~ tends to zero as r ~ 0.
In an equivalent way, let U[ denote the expected lifetime discounted
utilities of a representative worker (who is infinitely living) in state i,
given that the short-term agreement is implemented at every stage. For
i = 1, 2, expected utilities are determined by
U~ = u(w~) + [(1 - q~). U~ + ql 9 U~]/(1 + r),
(5)
U~ = u(w~) + [(1 - q2)" U89~ + q2" U~]/(1 + r),
(6)
yielding
U~ = [(r + q2) " u(w~) + ql " u(w~)]/D ,
(7)
g~ = [q2" u(w~) q- (r -'k ql)" u(w~)]/ D .
(8)
We see that since u(w{) > u(w~) and r > 0, U~ > U~, but again the
relative difference between U~ and U~ tends to zero as r --~ 0.
Let a~ denote the strategy of Gi that for any t and hit E Hit satisfies
trOt(hit) = w~~-1" Since the short-term agreement can be enforced
by either side by playing a~, it follows that FI~ (U~) is the lowest
individually rational expected profits (utilities). That is, at any stage of
the game the firm cannot be forced to a lower payoff than H~, and the
workers can never be forced to a lower payoff than U,/~. We can then
state:
Lemma 1: In the game Gi, II~ (U~) is the minmax payoff for the firm
(union).
It is also a rather trivial observation that the strategy profile (a~, a~)
is a Subgame-perfect equilibrium (SPE) since it holds for either player
Long-term Union-Firm Contracts
that for any t and hit E Hit,
subgame defined by hit.
169
is a best reply to a'~lt,.~, in the
Lemma 2." The strategy profile (a~, a~) is a Subgame-perfect equilibrium of Gi with equilibrium payoffs II~ and Ui~.
Note that the SPE (a~,a:}) would be less robust (in the sense of
involving weakly dominated strategies) if wt = zt,~ only if demand
exceeds offer (i.e. 'w~ > w~), but wt C [w~, w~:] otherwise. Then the
firm could gain by offer less than w~ in state 1 (if the wage demand
of the union turned out to be even lower) without any risk of a loss;
similarly for the union in state 2. Since we have wt = w~,_~ also when
w I < u.,tk, this argument is less appealing, and the SPE (a.~, ai ~) is less
open to criticism.
Lemmas 1 and 2 imply that threatening each player with the SPE
( ~ , ~ ) in state i, after a deviation has occurred, constitutes an optimal
penal code (Abreu, 1988). Hence, ( ~ , a~) can be used to support the
most collusive equilibria. This result will prove very useful in determining what payoffs (combinations of expected profits/utilities) can be
realized as a Subgame-perfect equilibrium, and thereby, what long-term
agreements the two parties will honor without binding contracts/third
party enforcement.
3. Individually Rational Long-term Contracts
We will be particularly concerned with efficient long-term agreements. It will turn out to be useful first to investigate the set of efficient
long-term agreements given that third party enforcement is available.
Note again that due to the workers' risk aversion and the firm's risk
neutrality, full intertemporal efficiency in the union-firm game entails a
constant wage rate such that workers face no income risk. However, in
order to be individually rational for the firm to accept a constant wage
rate w, its expected discounted profits, [(r + q2)" 7rl(W) + ql" 7r2(w)]/D
if initiated in state 1 and [q2 ' 7rl(W) -[- ( r + q l ) " 7 r 2 ( w ) ] / D if initiated
in state 2, must weakly exceed II~ and II~, respectively. This implies
that an individually rational and efficient binding contract initiated in
state i must satisfy w _< if,;, where Wl and Y'2 are determined by [using
equations (3) and (4) as well as the definition of 7r.i(-)]
('I" -~- q2)" (Wis -- W l ) ~- q l " (W~ -- ~ 1 ) = 0
q 2 ' (W~ -- W2) -+- ('r --~-q l ) " (W~ -- ~tl2) = 0 .
(9)
(10)
170
G. Asheim and J. Strand:
Similarly, in order to be individually rational for the union to accept
a constant wage rate w, its discounted utilities, (1 § r) 9 u ( w ) / r =
[(r + q2)" u ( w ) + ql " u ( w ) ] / D = [q2 " u ( w ) + (r + ql)" u ( w ) ] / D , must
weakly exceed U,[ if initiated in state i. This implies that an individually
rational and efficient binding contract initiated in state i must satisfy
w >_ _wi, where ~_uI and w__2 are determined by [using equations (7)
and (8)]
(r § q2)" [u(w~) - ~(Wl)] § ql ' [U(W~) -- ~(~U1) ] = 0 ,
(1 1)
q2' [u(w~) -- u(w2) ] § (r + ql)" [u(w~) -- u(w2) ] = 0 .
(12)
Since w~ > w~, we have that w~ > "t~1 > ~tU2 > W~ and w~ > _w1 >
~'2 > w~. Furthermore, by the concavity of u(.) it follows that [using
(9) and (11), and (10) and (12)]
~,~
-
~,~ -
~(~_,~)
~(~_,~)- ~ ( ~ )
w~
~(w~)
~
~ - ~_,~
~_~ - ~
-
'
hence, wi > w_~ for i = 1, 2. Let l=I~ ((;i) denote the m a x i m u m expected
profits (utilities) realizable by a binding contract initiated in state i
through a constant wage equal to _wi (~i). Figure 1 shows the efficient
payoffs in either state as well as the fixed wage rates corresponding to
each of the m a x i m u m levels of II,i and Ui.6
Without binding contracts not all of these efficient long-term agreements are viable. In particular, without third party enforcement, an
agreement without wage variability needs to be individually rational in
both states, i.e. such an agreement exists only if _w1 <_ ~2. The problem
is that an agreement for a constant wage w initiated in state 1 needs to
be individually rational for the firm in state 2, i.e.,
q2.
(w~
-
~)
+ (~- + q l ) - ( w g
-
~)
>_ 0 ,
(13)
and an agreement for a constant wage w initiated in state 2 needs to
be individually rational for the union in state 1, i.e.,
(r § q2)" [u(w~) -- u(w)] -I- ql" [u(w~) -- u(w)] < 0 .
(14)
6 Since u(.) is strictly concave, it would have been more correct to draw
the frontiers in Figure 1 (as well as those in Figures 2 and 3) strictly concave,
not (piecewise) linear as in the figure(s).
L o n g - t e r m U n i o n - F i r m Contracts
171
wl
9~'
U1 i / " "
-/if1
~'~
Initiated in the
high demand state
t
s
.,...~" ~!:1
]~!
[I 1
~2
./.o"
/t~
Initiated in the
low demand state
t
tl'2
"11'2I
tx,~
t
II
Fig. 1
From conditions (13)-(14) it follows that a long-term agreement without wage variability can be individually rational in both states only if
r is sufficiently small.
Let us refer to wage paths which are efficient among the individually
rational ones in both states for both players as constrained efficient
long-term agreements. If w a < if'2, then clearly a constant wage equal
to w 1 maximizes expected profits when the agreement is initiated
in state 1, while a constant wage equal to z~2 maximizes expected
utilities when the agreement is initiated in state 2 (since these wage
172
G. Asheim and J. Strand:
paths maximize the respective payoffs even with binding contracts). If
~ > z~2, let wl,~b2 be the pair of wage rates with least variability
satisfying
q2" (w~ - Lffl ) -4- 0" q-
qx)" ( w ~
--/~'2) = 0 ,
(r + q2)" [u(w~) -- U(Wl)] + ql" [U(W~) -- U(W2)] = 0 .
(15)
(16)
Note that if r is sufficiently large, such that (r + ql)(r + q~) >_
qlq2 9 u'(w~)/u'(w~), then only Wl = 'w~ and E'2 = w~ satisfy (15)
and (16). Furthermore, let Wl denote the wage path initiated in state 1,
while ~b2 denotes the wage path initiated in state 2, each consisting
of ~'1 in state 1 and z~2 in state 2. Then it can be argued 7 that w l
maximizes expected profits when the agreement is initiated in state 1,
while w2 maximizes expected utilities when the agreement is initiated
in state 2. Refer to the corresponding payoffs as 17]' and U~.
When initiating an agreement in state 2, a firm seeking to maximize
its expected profits given the individual rationality constraints would
always want to force the short-term wage w~ for as long as the current
state 2 lasts. The reason is as follows: the firm gains by waiting during
the remaining stages of state 2, in which w~ is relatively favorable to the
finn, and then initiating the profit maximizing agreement when the state
changes to 1. Call this wage path w2, whereby the initial w~ as long
as state 2 lasts is followed by a wage equal to w__1 in subsequent stages
if '~-'1 <- 'g'2, and by w~ otherwise. It follows that w2 is a constrained
efficient long-term agreement which, given state 2, maximizes expected
profits subject to the individual rationality constraints, yielding the
7 Let w be an agreement initiated in state 2. Suppose w satisfies individual
rationality constraints at all future stages, as well as providing the union with
higher expected utilities initially than ~2. Due to positive discounting, we may
w.l.o.g, assume that w differs from ~b2 only for a finite number of stages.
Consider a period of time for which state 2 lasts followed by a period of
time for which state 1 lasts. Assume that this is the last such pair of periods
for which w differs from ~b2. Due to equal discounting and the firm's risk
neutrality, the individual rationality constraints imply that w is a "discounted
mean-preserving spread" of @2 for these two periods. Since the union is
risk-averse, it follows that the expected utilities for the union at the beginning
of this pair of periods is lower for w than for @2. We may therefore construct
an agreement which compared to w is even better for the union in expected
ex ante terms, by not deviating from ~2 during this pair of periods. By an
inductive argument, a contradiction is obtained, implying that no w with the
above properties exists. Similarly for an agreement initiated in state 1.
Long-term Union-Firm Contracts
173
payoff II,~. Note that along w2 the union is held down to its minmax
payoff Ui~ in the current state 2 as well as in all future states 1, but
not (unless a high r precludes wage smoothing) in future states 2, as
this would introduce inefficient wage variability. Likewise, a union
initiating a contract in state 1 has the same kind of incentives to
wait during the remaining stages of state 1, in which w~ is relatively
favorable to the union, and then to initiate the utility maximizing
agreement when the state changes to 2. Call this path @~, whereby the
initial w~ as long as state 1 lasts is followed by w2 in all subsequent
stages if ~Jl -< w2, and by @2 otherwise. As above, @~ is a constrained
efficient long-term agreement which given state 1 maximizes expected
utilities subject to the individual rationality constraints, yielding the
payoff U~'. Note that along @1 the firm is held down to its minmax
payoff 1-I;~ in the current state 1 as well as all future states 2, but
not (unless r is too high) in future states 1, as this would introduce
inefficient wage variability.
The constrained efficient long-term agreements can now be characterized as follows.
Case 1: ILl I ~ IU 2.
Either (a) the wage path has constant wage in the open interval
('~'1, w2) (if this exists), or (b) if initiated in state 1, the wage path
consists of a wage in the closed interval [~2, w~] in the current state
followed by a constant wage equal to ~2, or (c) if initiated in state 2,
the wage path consists of a wage in the closed interval [w~, ~-'1] in the
current state followed by a constant wage equal to u_h.
Case 2." "~"1 > IL'2"
Either (a) if initiated in state 1, the wage path consists of a wage
in the closed interval ['u_,l,w~] in the current state followed by @2, or
(b) if initiated in state 2, the wage path consists of a wage in the closed
interval [w.~,~b2] in the current state followed by _wl.
Cases 1 and 2 are illustrated in Figures 2 and 3, respectively.
It is clear that any wage path that at some stage is not individually
rational for one of the players cannot be implemented by a Subgameperfect equilibrium, since this player would be better off by enforcing
the standard short-term agreement. However, by Lemmas 1 and 2, all
the constrained efficient long-term agreements described above can be
implemented by SPEa.
Proposition 1: Any constrained efficient long-term agreement can be
implemented by a Subgame-perfect equilibrium by the threat of triggering (oi~, ~ri~) at stage t + 1 if a deviation occurs at stage t.
174
G. Asheim and J. Strand:
'tk~2E
(71 ....
U{'
w!
Initiated in the
high demand state
t
l
.."...""88%.
".......... ""...
"%".~"'-.
"-...
W s
k
1
'~t/7>2
,
-'1L
7.U2
'//!2 [-
..-'
1I
U2
1)i
t
N~
)
<[-
.....iii..................
!ZI
if'2
..""
~2
11
Initiated in the
low demand state
11;
""%"',
~1~ *h-'l
112
Fig. 2. iv__1 ~ ~-~2.
Proof." Consider a deviation by the firm at a stage t at which the state
is 1. Since ( a ~ , a~,) is triggered, by (1) the firm at stage t receives the
expected profits
7rl(W~) -}- [(1 - q l ) " ]7I~ -Jr- q i " I151/(1 -~ r ) = 1-[~ ;
i.e., a deviation by the firm from a constrained efficient long-term
agreement is not profitable. Similarly for a deviation in state 2 and
for a deviation by the union.
Q.E.D.
Long-term Union-Firm Contracts
i/.12
I!'"~
If I
'""""%"-..
175
Initiated in the
high demand state
I--2_
""%.. )
/
.-
...
...........
...........""
U~
,i'2
]
[--
tt~
[I' 1"
t
H1
--L__I
Initiated in the
low demand state
t
U;
~,2
u'~ F
[-t
F[
Fig. 3. ~_v1 > if;2.
Proposition 1 is based on the parties having an implicit contract
which prescribes that if either party deviates from the negotiated
long-term agreement, the standard short-term agreement will be used
for ever after. This implicitly assumes that the parties are able to
commit themselves not to renegotiate. However, if individually rational
long-term agreements yielding a strict Pareto-improvement over the
standard short-term agreement can be negotiated initially, it seems hard
to argue that the parties without third party enforcement would choose
not to renegotiate such a long-term agreement after a deviation has
occurred. Hence, even though the implicit contract of Proposition 1
176
G. Asheim and J. Strand:
is individually rational, the threat of triggering the standard short-term
agreement for ever after is not credible in view of collective rationality.
This again questions the viability of long-term agreements supported
by such a threat.
4. Collectively Rational Long-term Contracts
The problem posed at the conclusion of the previous section can
be resolved if any constrained efficient long-term agreement can be
implemented by a Subgame-perfect equilibrium by the threat of triggering other constrained efficient long-term agreements. In such cases
there can be no renegotiation because the agreement specified by the
contract to be implemented in case the original agreement is broken is
itself constrained efficient: we may then say that the implicit contract
is renegotiation-proof. 8
In Proposition 2 below we state and prove that the problem can
be resolved in this way, i.e., a deviation from any constrained efficient
long-term agreement can be sufficiently punished by initiating another
constrained efficient long-term agreement. However, as the following
discussion will motivate, there are deviations that ex post cannot be punished this way. What instead supports the renegotiation-proof equilibria
is that deviations ex ante are discouraged by rather severe consequences
whenever a constrained efficient long-term agreement can be used as a
punishment.
To analyze this issue, assume w_1 < ~'2, and consider the constrained efficient long-term agreement with constant wage equal to ~b2.
In state 2, the agreement holds the firm down to its lowest individually
rational payoff I-[3 . Still, by forcing the standard short-term agreement
8 The literature on renegotiation is divided into two lines of research.
One seeks a game-theoretic refinement of the concept of a Subgame-perfect
equilibrium, the other is based on contract theory. The former includes Farrell
and Maskin (1989; defining a Strongly renegotiation-proofequilibrium), Bernheim and Ray (1989; Consistent equilibrium), Asheim (1991; Pareto-perfect
equilibrium), as well as somewhat different concepts by Pearce (1987) and
Abreu et al. (1989). The latter includes Hart and Moore (1988) in the context
of incomplete contracts and Hart and Tirole (1988) and Laffont and Tirole
(1990) in the context of asymmetric information. The present contribution
studies complete and non-binding contracts in a game of complete information
and is hence more closely related to the former branch of the literature.
Our requirement for renegotiation-proofness is a strong one, implying Strong
renegotiation-proofness, Consistence, and Pareto-perfectness.
Long-term Union-Firm Contracts
177
unilaterally it can lower the wage rate and increase its profit at the
current stage. However, if it does so and the state is still 2 at the next
stage, it cannot be punished for this deviation, since an individually
rational long-term agreement with lower expected profits is not available. Therefore, this deviation has ex post increased the profits to the
firm.
On the other hand, the firm can be punished if it deviates under
these circumstances and the state changes from 2 to 1. Because given
that state 1 has arrived, the wage path @1 (with u:~ as long as state 1
lasts, followed by a constant wage equal to w2) is constrained efficient.
This increases the wage rate and lowers the flow of profits to the firm as
long as state 1 lasts and therefore does punish the firm for its deviation.
It is sufficient to discipline the firm by punishing only if it deviates at a stage after which the state changes from 2 to 1. The
punishment consists of starting the wage path @1. Correspondingly,
it turns out to be sufficient in order to discipline the union to punish
only if it deviates at a stage after which the state changes from 1
to 2. The punishment then consists of starting the wage path w2. In
order to induce the parties to adhere to the punishment paths, the same
punishments are used, i.e. (re)start @1 (w2) if the firm (union) deviates
from @1 or we.
This is similar to what Abreu (1988) calls a simple penal code;
still it is different in the sense that whether the punishment is being
executed depends on a random event.
Note that this penal code can be used only because the state at stage
t + 1 (i.e. it) is not known to the players when announcing their wage
offer/demand at stage t. The chance of a transition from one state to
another after any stage induces the parties to refrain from deviations.
Proposition 2: Consider a constrained efficient long-term agreement
~b.i (initiated in state i) as well as the wage paths @1 and _W2. Let the
strategy profile ( e^l~', 6-I) of Gi be defined by the property that "&~ is
being implemented as long as neither of the following occurs: (a) The
firm deviates from the implementation of ~bi at a stage after which
the state changes from 2 to 1, or (b) the union deviates from the
implementation of @i at a stage after which the state changes from 1
to 2. If (a) occurs, then @1 is started at the following stage, while if
(b) occurs, then we is started at the following stage. Deviations from @1
and _W2 are in the same way punished by (re)starting @~ or _W2. Then
^k ^/
(a i o-i) is a Subgame-perfect equilibrium.
Proof" First note that the firm has no incentive to deviate in state 1
while the union has no incentive to deviate in state 2.
178
G. Asheim and J. Strand:
The gain for the firm of deviating at stage t, when the state is 2,
and the wage path wi specifies the play of "u?~, is
Since the punishment wage path li/1 yields expected profits II~, the
expected loss is
q 2 (I]] - II[)/(1 + r) ,
where l:I'~ is the expected profits from @i at stage t + 1 given that the
state changes to 1 after stage t. Now recall
1]~ = 7r2(w,~) + [(1 - q J . II~ + q2" II~]/(1 + r) ,
(2)
and note that
1]; -- 7r2(E,;) + [(1 - q2)" I]; + q2" l~I~]/(1 + ~9,
^
(17)
.
where II~ is the expected profits from @i as long as the current state 2
lasts. Since @~; is a constrained efficient agreement, it follows that
II~ >_ II~. This combined with (2) and (17) implies that the gain of a
deviation does not exceed the expected loss.
The gain for the union of deviating at stage t, when the state is 1,
and the wage path @i specifies the play of 's is
^
,
~('w~) - ~(~/
-
Since the punishment wage path w2 yields expected utilities U~, the
expected loss is
ql" (0~ - U~)/(1 + r ) ,
where U-~ is the expected utilities from @i at stage t + 1 given that the
state changes to 2 after stage t. Now recall
U~ = ~(m~) + [(1 - q J . Ui~ + q~. ~ ] / ( 1 + r),
(5)
and note that
O~ = u(~b~) + [(I - ql)" O[ + q1" 0~]/(I + r),
(18)
where U{ is the expected utilities from @i as long as the current state l
lasts. Since w.i is a constrained efficient agreement, it follows that
Long-term Union-Firm Contracts
^
179
.
U~ _> Ui~. This combined with (5) and (18) implies that the gain of a
deviation does not exceed the expected loss.
The proposition follows since ~bl and w2 are themselves constrained efficient long-term agreements.
Q.E.D.
The implications of this result are striking: all constrained efficient
long-term agreements can be implemented not only by Subgame-perfect
equilibria, but also by renegotiation-proof equilibria. To understand this
result, note that a deviation by the firm in state 2 from an established
constrained efficient wage path, i.e. setting w~, needs to be punished
only if this deviation occurs at a stage after which the state changes
from 2 to 1. Given that the state has changed, a punishment consisting
of w~ as long as state 1 lasts, followed by w2 in all subsequent stages
if w 1 _< z02, by ~/2 otherwise, is constrained efficient (in fact, it maximizes expected utilities for the union given the individual rationality
constraints). By the proof of Proposition 2, this threat, executed only if
the deviation occurs at a stage after which the state changes, is sufficient
to deter the firm from defecting. Furthermore, since the punishment is
constrained efficient, it cannot be made better for both parties through
renegotiation. Likewise, a deviation by the union in state 1 needs to
be punished only if this occurs at a stage after which the state changes
from 1 to 2, and the punishment is in this case analogous. Finally,
deviations from the punishment paths are punished in the same way.
This equilibrium is in principle quite similar to how van Damme
(1989) constructs efficient and renegotiation-proof equilibria in a repeated prisoners' dilemma, one difference being that the triggering of
the punishment paths here depends on a random event.
5. Short-term and Long-term Bargaining
When determining the set of constrained efficient long-term contracts, we have only characterized the efficient frontier; we have not
indicated what particular point on this frontier will be the outcome of
efficient bargaining between the firm and the union. This contrasts our
assumption that there be a unique "standard" outcome of the short-term
bargaining problem. Instead, it would be desirable that the outcome
of long-term bargaining be governed by the same principle as the
short-term bargaining.
One standard assumption is that the outcome of short-term bargaining be determined by the Nash bargaining solution. As demonstrated
180
G. Asheim and J. Strand:
by Rubinstein (1982) and Binmore et al. (1986), this solution can be
defended in a strategic setting if delays to negotiation are costly to both
parties.
However, if the Nash bargaining solution is assumed to be applied
for short-term bargaining, it should be used even when the parties
bargain over the gains from striking a constrained efficient long-term
agreement. In order to satisfy such a consistency requirement, the
following will be maintained here:
(1) The outcome of long-term bargaining should prescribe two ("standard") long-term agreements, one initiated in each state.
(2) The pair of standard long-term agreements should be such that if
the disagreement point is the standard short-term agreement of the
current state followed by the standard long-term agreement of the
other state, then the Nash bargaining solution chooses the standard
long-term agreement of the current state and vice versa.
Since the Nash bargaining solution satisfies individual rationality, condition (2) implies that in either state it does not pay for any of the parties
to delay negotiations in order for the state to change and thereby obtain
the standard long-term agreement of the other state.
These conditions can be formalized as follows: let Ri denote the
set of pairs of payoffs, (H, U), for the firm and the union respectively,
associated with individually rational long-term agreements initiated in
state i, while C i is the efficient frontier of Ri (i.e., the set of payoff pairs
associated with constrained efficient long-term agreements initiated in
state i). Let dl(H, U) denote the payoff pair realized in state i given
that the firm and the union follow the standard short-term agreement as
long as the current state 1 lasts, and that they then realize the payoffs
(II, U) when the new state 2 arrives. We have
dl(II, U ) = [(1 + r ) .
(Trl(w~),u(w~))+ ql. (II, U)]/(7" + q l ) -
By (1) and (5), this expression can be rewritten as
dl(H, U) = (I]~, U~) + ql" [(H, U) - (II~, g 89
+ ql) 9
(19)
Similarly, by defining the analogous function d2(II, U), we have
d~(H, u ) = (FI~, U~) + q2. [(II, u ) - (II~, Ui~)]/(r + q2) 9
(20)
It follows from (19) and (20) that dl maps any pair in R2 into R1 and
vice versa for d2.
Long-term Union-Finn Contracts
181
Let furthermore NI(I1, U) be the set of Nash bargaining solutions
in state 1, given that the disagreement point is (II, U) (which is a
payoff pair in state 1), and given that the solution must correspond
to an individually rational long-term agreement. Define N.2(II, U) similarly. Since Nash bargaining is efficient, N1 maps any pair in RI
into C1 _C R1, and analogously for N.).
Conditions (1) and (2) then entail that the outcome of long-term
bargaining is a pair of long-term agreements, one initiated in state 1
with payoffs (I]~, U~') E /~1, and the other initiated in state 2 with
payoffs (H~, U.~) E R2, satisfying
(rI;, u ; ) z
(21)
(22)
Proposition 3: A pair of standard long-term agreements fulfilling the
conditions (1) and (2) exists.
Proof. It is evident that Hi and/~2 are nonempty (note that (II~, Ui~) E
/~i, i = 1, 2) and compact. Furthermore, we have that R1 and /~2 are
convex since
d2U
till
-- u11(w)/'n, < 0
i = 1, 2
where w is the wage agreed upon for as long as the initial state lasts.
By (19) and (20), di, i = 1, 2, are clearly single-valued and continuous,
while Ni, i = 1, 2, due to the properties of the Nash bargaining solution,
are single-valued and continuous since/L:, i = 1, 2, are convex. Hence,
Nl(dl(N,2(d2(.)))) is single-valued and continuous.
Then (21) and (22) imply that we need to show the existence of a
payoff pair (Ill', U~') C R~ satisfying
This follows from Brouwer's fixed-point theorem, since (a) H1 is nonempty, compact, and convex and (b) N1 (dl (J~v;2(d2(')))) is single-valued
and continuous, and maps any pair in/~1 into C1 _C R1.
Q.E.D.
182
G. Asheim and J. Strand:
Remark: Since NI maps any pair in R1 into C1 C_/~1, and analogously
for N2, it follows that a standard long-term agreement is constrained
efficient.
One problem remains. The continuation of a standard long-term
agreement is, given that the state has changed at least once, in general
not itself a standard long-term agreement of the current state. Likewise,
the wage paths (@1 and w2) supporting a standard long-term agreement
in general are not standard long-term agreements, either. Given that a
standard long-term agreement exists in either state, it may seem hard to
argue that the parties instead would want to carry out the continuation
of a standard long-term agreement (or if called for the wage paths @1
and w2). However, the following defense can be made:
The long-term bargaining takes the existing (implicit) contract as its
point of departure. If no long-term bargaining has taken place earlier,
then by default the existing contract is that which in any contingency
implements the standard short-term agreement. On the other hand, if
there exists a contract specifying the implementation of a particular
long-term agreement if there are no deviations and if there are wage
paths to follow if there are deviations from the original agreement, then,
in any contingency, the wage path specified by the existing contract is
the fall-back for new long-term bargaining. 9 In particular, if the wage
path is constrained efficient, then there is nothing to bargain over.
Since the contracts discussed in Section 4 have the property that
in any contingency the wage path specified is constrained efficient, no
more bargaining will occur after the parties have entered into such a
contract.
References
Abreu, D. (1988): "On the Theory of Infinitely Repeated Games with Discounting." Econometrica 56: 383-396.
Abreu, D., Pearce, D., and Stacchetti, E. (1989): "Renegotiation and Symmetry
in Repeated Games." ST/ICERD Discussion Paper TE/89/198, London
School of Economics.
Asheim, G. B. (1991): "Extending Renegotiation-Proofness to Infinite-Horizon
Games." Games and Economic Behavior 3, forthcoming.
9 If not even this very weak form of commitment is admitted, then attempts
by the firm and the union to attain mutually beneficial wage smoothing would
seem futile.
Long-term Union-Firm Contracts
183
Azariadis, C. (1975): "Implicit Contracts and Underemployment Equilibria."
Journal of Political Economy 83:1183-1202.
Azariadis, C., and Stiglitz, J. E. (1983): "Implicit Contracts and Fixed-Price
Equilibria." Quarterly Journal of Economics 98: 1-2.
Baily, M. N. (1974): "Wage and Unemployment under Uncertain Demand."
Review of Economic Studies 41: 37-50.
Bemheim, B.D., and Ray, D. (1989): "Collective Dynamic Consistency in
Repeated Games." Games and Economic Behavior 1: 295-326.
Binmore, K., Rubinstein, A., and Wolinsky, A. (1986): "The Nash Bargaining Solution in Economic Modelling." Rand Journal of Economics 17:
176-188.
Espinosa, M.P., and Rhee, C. (1989): "Efficient Wage Bargaining as a
Repeated Game." Quarterly Journal of Economics 104: 565-588.
Farrell, J., and Maskin, E. (1989): "Renegotiation in Repeated Games." Games
and Economic Behavior 1: 327-360.
Green, E. J., and Porter, R. H. (1984): "Noncooperative Collusion under Imperfect Price Information." Econometrica 52: 87-100.
Grout, P. (1984): "Investment and Wages in Absence of Binding Contracts:
A Nash Bargaining Approach." Econometrica 52: 449-460.
Hart, O., and Moore, J. (1988): "Incomplete Contracts and Renegotiation."
Econometrica 56: 755-785.
Hart, O., and Tirole, J. (1988): "Contract Renegotiation and Coasian Dynamics." Review of Economic Studies 55: 509-540.
Hoel, M., and Moene, K. O. (1988): "Profit Sharing, Unions and Investments."
Scandinavian Journal of Economics 90: 493-505.
Horn, H., and Svensson, L. E. O. (1986): "Trade Unions and Optimal Labour
Contracts." Economic' Journal 96: 323-341.
Laffont, J.-J., and Tirole, J. (1990): "Adverse Selection and Renegotiation in
Procurement." Review of Economic Studies 57: 597-625.
McDonald, I. M., and Solow, R. M. (1981): "Wage Bargaining and Employment." American Economic Review 71: 896-908.
Malcolmson, J. M. (1983): "Trade Unions and Economic Efficiency." Economic Journal (Conference Papers) 94:51~55.
Oswald, A.J. (1982): "The Microeconomic Theory of the Trade Union."
Economic' Journal 92: 576-595.
Pearce, D. (1987): "Renegotiation-Proof Equilibria: Collective Rationality
and Intertemporal Cooperation." Cowles Foundation Discussion Paper
No. 855.
Rosen, S. (1985): "Implicit Contracts: A Survey." Journal of Economic Literature 23:1144-1175.
Rubinstein, A. (1982): "Perfect Equilibrium in a Bargaining Model." Econometrica 50: 97-109.
Strand, J. (1987): "Bargaining, Limited Liability and Putty-Clay: The Inefficiency of Investment and Employment." Memorandum No. 16, Department of Economics, University of Oslo.
(1988): "Long-term Union-Firm Contracts." Memorandum No. 12, Department of Economics, University of Oslo.
-
-
184
Asheim and Strand: Union-Firm Contracts
(1989): "Monopoly Unions versus Efficient Bargaining: A Repeated
Game Approach." European Journal of Political Economy 5: 473-486.
Thomas, J., and Worrall, T. (1988): "Self-Enforcing Wage Contracts." Review
of Economic Studies 55: 541-554.
van Damme, E. (1989): "Renegotiation-Proof Equilibria in Repeated Prisoners'
Dilemma." Journal of Economic Theory 47: 206-217.
-
-
Addresses of authors: Professor Geir B. Asheim, Institute of Economics,
Norwegian School of Economics and Business Administration, Hellevn. 30,
N-5035 Bergen-Sandviken, Norway; Jon Strand, Department of Economics,
University of Oslo, Box 1095 Blindern, N-0317 Oslo 3, Norway.