The RAND Corporation
The Strategic Choice of Managerial Incentives
Author(s): Steven D. Sklivas
Source: The RAND Journal of Economics, Vol. 18, No. 3 (Autumn, 1987), pp. 452-458
Published by: Blackwell Publishing on behalf of The RAND Corporation
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RAND Journal of Economics
Vol. 18, No. 3, Autumn 1987
The strategic choice of managerialincentives
Steven D. Sklivas*
Do firms with separate owners and managers maximize profits? We address this question
for an oligopoly wheremanagers compete in quantitiesor prices, as in the Cournotor Bertrand
models, and owners choose their managers' incentives. Wefind that there is a strategic aspect
in the problem of selecting incentives and that profit-maximizing behavior does not result.
In particular, in the oligopoly we study, the behavior offirms competing in quantity (price)
more closely resembles perfectly competitive (collusive) behavior than Cournot (Bertrand)
behavior.
1. Introduction
* Economists have long debated the objective function of large corporations. Some have
suggested that large firms are more concerned with maximizing revenues or market share
rather than profits, and that market power is their ultimate goal. Simon (1957), Cyert and
March (1963), Marris (1964), Williamson (1964), Galbraith (1967), and Baumol (1977)
have all questioned the validity of the profit-maximization hypothesis. The complexity of
managerial decision processes and the separation of ownership and management have been
stressed as reasons for the deviation from profit maximization. Here we examine the implications of the separation of ownership from management for an oligopoly competing in
prices or quantities, as in the Bertrand or Cournot models.
We consider a two-stage, owner-managergame. In the first stage owners simultaneously
choose their managers' incentives. In the second stage each manager chooses the firm's
price or quantity. Owners receive the resulting profits, and each manager is rewarded according to the incentives chosen by his owner.
A manager's incentives induce him to adopt a particular type of behavior that will
affect his own as well as his rival manager's action. This game is similar to other two-stage
games (e.g., Dixit, 1980; Fudenberg and Tirole, 1984) where a player's decision in the first
stage affectsthe equilibrium actions taken by all playersin the second stage. Since an owner's
profits are determined by the actions of all managers, his optimal choice of managerial
incentives takes into account their effect on rival managers' actions as well as on those of
his own manager. In addition, an owner's optimal choice of managerial incentives will
depend on rival owners' choices. This introduces a strategic element into his decision.'
* Columbia University.
I am indebted to Rich McLean, Rafael Rob, Xavier Vives, Andrew Postlewaite, Avinash Dixit, Al Phillips,
Gerry Faulhaber, and Steve Matthews for their helpful suggestions and comments. I am especially grateful to
Richard Kihlstrom, my advisor, for valuable discussions and encouragement. I also thank the Center for the Study
of Organizational Innovation for secretarialsupport.
1 This is in contrast to many standard incentives problems (e.g., Holmstrom, 1979) where the principal is
concerned only with the action of his agent and where the optimal contract is independent of other contracts.
452
SKLIVAS /
453
The separation of ownership and management gives the owner the opportunity to
commit his firm to not maximizing its profit. We show that owners will take advantage of
this opportunity because of the effects on rival managers' equilibrium actions. We also show
that when managers compete in quantities, the result more closely resembles competition
than Cournot behavior; conversely, when they compete in prices, the result more closely
resembles collusion than Bertrand behavior.
In Section 2 we describe the model for the two-stage, owner-manager game. Section 3
sets out the results for quantity competition, and Section 4 gives the results for price competition. In Section 5 we brieflydiscuss relatedempiricalwork, and in Section 6 we summarize
our results.2
2. The model
* We examine a duopoly in which the firms, each having one owner and one manager,
play a two-stage game. In the first stage the owners simultaneously write and publicly announce contracts with their managers that specify how they will be rewarded. In the second
stage the managers simultaneously choose their firms' output.3 Owners receive the resulting
profits, and managers are rewarded according to their contracts. By applying the Nash
equilibrium to both stages of the game, we obtain a subgame-perfect equilibrium as our
solution.
Owner i measures his manager's performance according to some function of his firm's
profits (lli) and revenues (Ri), which are readily observed indicators of performance. We
call this measure gi, i = 1, 2. The higher is gi, the higher is manager i's bonus or the lower
is the likelihood that he will be fired. Because firm i's output (xi) does not enter manager
i's utility directly, he chooses xi to maximize gi. Let gi( *) represent manager i's incentives.
For simplicity, we make gi a linear combination of profits and revenues:
gi = Xi IIi(xI, X2) + (I1-
=Ri(x1,x2)-X C(xi),
i)Ri(xl,
X2)
i= 1,2.
(1)
Owner i simply chooses the parameter Xito determine his manager's incentives.4
Definition 1. (x4, X2) is a Nash equilibrium in the managers' subgame if and only if
x4* = argmax gi(xi, x,4), ij= 1, 2, i *j.
Since gi depends on Xi, we can write x4 and x4 as functions of (XI, A2).
We assume that the owner knows the functional form of demand and costs.5
Definition 2. (X1?,X*) is a Nash equilibrium in the owner's subgame if and only if
XI'= argmax 11i(x'(Xi, Xj7),x4(Xi, Xj*)), i, j = 1, 2, i # j.
To solve the owner-manager game we express x* and x4 as functions of (XI, 2) and
then find (X1*,Xl').The market outcome is x?*(X*, X2*),X*(X1 , Xl'),i.e., the Nash equilibrium
2
Vickers (1985) presents the separation of ownership and management as a problem of truthful revelation
of preferences and reaches similar conclusions. Fershtman and Judd (1985) independently and simultaneously
obtain results similar to those in this article.
3Later we extend this model to price competition.
4 We exclude output as an argument of gi for two reasons. First, the owner may not be able to observe it
directly. Second, when the firm produces many products, "output" may be difficult to characterize.
One may also interpretRi(*)- X-C(x,) as accounting profits. In this case the owner decides how heavily costs
are to be weighted in determining accounting profits, which he then uses to evaluate his manager's performance. I
owe this interpretation to Richard Kihlstrom.
5 Even if owners are uncertain about costs, the same qualitative results are obtained.
454
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THE RAND JOURNAL
OF ECONOMICS
quantities in the managers' subgame that result from the Nash equilibrium incentives of
the owners' subgame.6
3. Quantity competition
* We examine the previous model with linear demand, homogeneous products, and constant marginal cost, c. We normalize by setting c = 1. We then have P = a - bx, where P
is price (a > 1) and x = xl + x2. We find manager i's best-response function, qi(xj; Xi),by
maximizing gi( *) over xi. As Xiis decreased, costs are weighted less, and 0i( - ) shifts out.
Hence, decreasing Xi commits manager i to more "aggressive"behavior, i.e., for every x;
manager i responds with a greater xi:
(2)
xi= (a - i- bxj)/2b = 4i(xj; Xi).
The Nash equilibrium quantities as a function of (X1, X2)are
(3)
x* = (a - 2Xi+ Xj)/3b.
Notice that as the owner i makes his manager more aggressive, by decreasing Xi, his
own firm's output increases, while his rival'sdecreasesin equilibrium. We have the following
profit function for the owners:
lIj(Xj, Xj)= [M+ Xj(6- a - Xj)- 2X?]/9b,
(4)
where M = a 2- 3a- 3Xj+ 2aXj + Xj2.
The owner's best-response function and Nash equilibrium incentives are given in equations (5) and (6), respectively:
Xi= (6 - a - Xj)/4
XM= (6-a)/5,
(5)
i= 1,2.
(6)
Proposition 1. In the owner-manager game managers behave more aggressively than profit
maximizers, i.e., MI< 1, i = 1, 2. This results in outputs that are higher than in the Cournot
model, yet still below the social optimum, i.e., (a/2b) > x?(Xic, XI) > x*(1, 1), i = 1, 2.
The fact that MI < 1 follows from a > 1, and X? = XI'< 1 implies that
xi*(X*, XI) > x* (1, 1), i= 1, 2.
1)/Sb is less than a/b, the social
One can see that xl(XtI, XI) + xj(X* , XA) = 4(aoptimum.
In Proposition 1 we see firms in the owner-manager game act as profit maximizers
with less than true cost. They produce outputs greater than the Cournot output, yet still
below the socially optimal level. This results in firms' having lower profits than the profitmaximizing firms in the Cournot model. These results are generalized to n-firms in the
Appendix, where we show that MIincreases monotonically with n and asymptotically approaches 1.
The intuition for the seemingly paradoxical result of profit-maximizing owners' committing their managers to non-profit-maximizing behavior can be seen in Figure 1. Figure 1 illustrates the Cournot outcome, C, and the outcome of the owner-manager game, S.
In choosing Xian owner implicitly chooses his manager's best-response function, so we can
think of owners as playing a game in best-response functions. Starting from point C, the
Nash equilibrium in the managers' subgame resulting from XI = 1 = X2, owner 1 can
increase his profits by decreasing XI, which shifts + 1(x2;XI)out. By committing his manager
to more aggressive behavior, owner 1 moves the equilibrium quantities down along
02(XI; X2 = 1), and thereby increases x1 and decreases x2*. This increases owner l's profits
6 We only consider situations where demand and cost ensure a unique Nash equilibrium in the managers'
subgame. These are the conditions for a unique solution to the Cournot model (Friedman, 1977).
SKLIVAS /
455
FIGURE 1
THE SOLUTIONS TO THE OWNER-MANAGER AND COURNOT GAMES
x22
(X2;XA = A;)
4\1
1=
4\1(x2;
|
b
2's
IFIRM
ISOPROFIT
CURVES
1's ISOPROFIT
~~~~~FIRM
IC
k2xl
CURVES
2(x= X2
=
1
lop~~~~~~~~~~~~~~~~~x
because +02(X2; X2 = 1) is negatively sloped and firm l's isoprofit curve is flat at any
point along k1(x2; XI = 1). So XI = 1 cannot be a best response to X2 = 1, and both owners'
committing their managers to profit-maximizing behavior cannot be an equilibrium. One
can see that X? is a best response to X* by the fact that firm l's isoprofit curve is tangent
to
+02(X1;
X2
X*) at S, and similarly
that XI is a best response
to Xl.
Firm 1's unilateral deviation from profit maximization increases its profits and lowers
firm 2's. By placing some positive weight on revenue, owner 1 wishes his firm to deviate,
while owner 2 prefers that firm 1 continue to maximize profits. Although the opportunity
to commit to not maximizing profits introduces additional conflict into the duopoly by
causing both firms to have lower profits in equilibrium, the resulting profits are still positive.
4. Price competition
In this section we examine the owner-manager game when managers compete in prices.
We analyze this for the case of symmetric product differentiation, linear demand, and
constant marginal cost, c. We write linear demand as
U
0<#<
xi=a-Pi+(3Pj,
1,
i,j= 1,2,
i*j,
0<c<a/(
-@f),
(7)
where Pi is firm i's price.
We solve the game as before. We find manager i's best-response function, 4ti(Pj; Xi),
by maximizing gi over Pi. By increasing Xi owner i makes his manager less aggressive, i.e.,
he responds with a greater Pi for any Pj:
Pi = (a + Xic+ OPj)/2= 4i(Pj; Xi).
(8)
The Nash equilibrium prices as a function of (XI, X2) are
Pi
= (2a
+ 2Xic + a3 + 3Xjc)/(4
-
f2).
(9)
456
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OF ECONOMICS
Notice that as Xivaries, both prices move in the same direction. This yields the following
profit function for the owners, where K = (2a + at: + Xjflc)(2a + ad + #Xjc - 4c + 32C)
is a constant:
lli(Xi, Xj)= [K+ Xj(2afl2c + a(33c + 3 c2Xj-
62c2
34c2 + 8c2)
+
+ X?(2#2C2
- 4c2)]/(4
-
fl2)2.
(10)
The owner's best-response functions and Nash equilibrium incentives are given in
equations (1 1) and (12), respectively:
Xi=(2a2
MI=
+a3
(2a32 + a#3
+3XjC
-
+
-6f2C
6f2C + f4C
4C
+ 8c)/c(8 -4f2)
+ 8c)/c(8 - 432
(11)
- 33).
(12)
Proposition 2. In the owner-manager game firms that compete in prices behave less aggressively than profit maximizers, i.e., A* > 1. This results in higher prices than in the Bertrand
model, i.e., PRI(X*,Xl') > PRI(1, 1).
The fact that MI > 1 results from a/(1 - fi) >
P
(X*
, X*)
>
P*(1,
C,7
and XA = Xt' > 1 implies that
I)
The consequences of the separation of ownership and management reverse under price
competition; firms act as profit maximizers with greater than true cost, with the result that
prices are higher. Firms also receive higher profits than in the Bertrand model.
Figure 2 illustrates the outcomes of the Bertrand model, B, and the owner-manager
game, 0. Again we see that XI = 1 cannot be a best response to X2 = 1, because owner 1
FIGURE 2
THE SOLUTIONS TO THE OWNER-MANAGER
AND BERTRAND GAMES
P2A
FIRM 1's ISOPROFIT CURVES
1,(p2;X
=
1 (P2; X =Xi)
1)
FIRM 2's ISOPROFIT CURVES
' P1
Notice that A* is increasing in a and that A* = I for a
=
(1 -
13)c,but we assume that a > (1 - #)c.
SKLIVAS /
457
can increase his profits by committing his manager to less aggressive behavior (i.e., he can
increase XI).P* and Pt both increase as prices move up along 4/2(P1;X2= 1) from the point
B, and thus firms 1 and 2 earn higher profits. Hence, profit-maximizing behavior is not an
equilibrium. In the equilibrium to the owner-manager game each owner's isoprofit curve
is tangent to the opposing manager's best response at the intersection, 0, so that neither
owner can gain higher profits by changing his manager's incentives.
Because firm l's unilateraldeviation from profit maximization raisesboth firms' profits,
commitment has a cooperative effect on the price-competing duopoly: both firms earn
higher profits.8
5. Some empirical evidence
* Empirical studies of executive compensation that test the validity of the profit-maximization hypotheses appear in Roberts (1959), McGuire, Chiu, and Elbring (1962), Williamson (1963), and Yarrow (1972), who surveys this work. Although the motivation given
for the lack of profit-maximizing behavior was quite different from that provided here,
Roberts did find that executive compensation was strongly related to sales, as did McGuire
et al. But Lewellan and Huntsman (1970) and Yarrow (1972) cast doubt on these conclusions
about sales maximization. While the empirical studies do not provide the most appropriate
test for the hypotheses presented in this article, they offer some support for the argument
in Section 3 that managerial compensation is often based on revenues.
Lackman and Craycraft(1974) provide a more appropriate test of the hypotheses presented here. They estimate the demand and cost relationships in the corrugated specialties
industry and test various theories of oligopoly pricing. They find that the Cournot model
is inadequate to predict prices and quantities and that incorporating revenue maximization
into the firms' reaction function increases the model's predictive capacity. Their work supports the hypotheses of Section 3, but none of the tested models in their article corresponds
exactly to the one presented here. We would like to see a test of the model of oligopolistic
behavior proposed here.
6. Summary
* The separation of ownership and management gives owners the opportunity to commit
their managers to non-profit-maximizing behavior. We found that owners in an oligopoly
will take advantage of this opportunity. If duopolists compete in quantity, both firms earn
lower profits. Conversely, if duopolists compete in price, both firms earn higher profits.
Appendix
*
For the n-firmcase equations (2)-(6) become equations (Al)-(A5). Let:
n
A-i= I Xi
n
and
x-i=>
j=1
j=1
joi
joi
X,
- i- bx)
xi = (1/2b)(a
(Al)
xi*= ( l b(n+ l1))(a- n~i- X-,
+ Xi(n(n+ 1) -(n
Hi(xi, Xj)= (l/b(n + 1)2)[A
(A2)
-
l)a -(n -1)X,)
-
nX?]
Xi= (1/2n)[(n(n+ 1)- (n - l)X-,)- (n- l)a]
8
One can confirm that profits are less than the joint-profit-maximizing level.
(A3)
(A4)
458
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THE RAND JOURNAL OF ECONOMICS
Il = [n(n + 1)-(n-
1)a]/[(n2
1)],
+
i= 1
n,
(A5)
lim Xi = 1
n bPoo
(8aX/8n)= [(a- 1)(n2- 2n- 1)]/[(n2+
A=a2+aXS-a(n+
1)-(n+
1)2] > 0
for
n>2
(by a> 1)
1)X-i+aX-+(X_i)2.
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