Competition, Risk and Managerial Incentives

Michael Raith

University of Chicago and CEPR �

February 2001

Abstract

This paper examines how the degree of competition among �rms in an industry a�ects

the optimal incentives that �rms provide to their managers. A central assumption is that

there is free entry and exit in the industry, which implies that changes in the nature

of competition lead to changes in the equilibrium market structure. The main result is

that as the intensity of product market competition increases, principals unambiguously

provide stronger incentives to their agents to reduce costs, and hence agents work harder.

At the same time, more intense competition also leads to a higher volatility of both

�rm-level pro�ts and managers' compensation. Consequently, managers' incentives are

positively correlated with �rm-level risk, consistent with empirical evidence.

JEL codes: D43, L13, L22

Keywords: Competition, incentives, risk, X-eÆciency

�Correspondence: Michael Raith, Graduate School of Business, University of Chicago, 1101 E. 58th

Street, Chicago, IL 60637, USA; email: michael.raith@gsb.uchicago.edu. I would like to thank Jean De

Bettignies, Guido Friebel, Luis Garicano, Allison Garrett, Canice Prendergast, and Lars Stole for very

helpful comments.

1 Introduction

This paper presents a theory of how competition between �rms in an industry interacts

with the design of managerial incentives within �rms. It predicts that an increase in

competition unambiguously induces �rms to provide stronger incentives to their managers.

At the same time, greater competition leads to an increase in the volatility of pro�ts,

thereby creating a positive correlation between the strength of incentives and �rm-level

risk. These results help to explain discrepancies between theory and evidence regarding

how managerial incentives are related to competition, and how they are related to risk.

I analyze managerial incentives within an explicit oligopoly framework, and demon-

strate how they are determined by the fundamentals that de�ne a market: costs, con-

sumers' tastes, and market size. The model presented here has two salient features. First,

the intensity of price competition among �rms is measured by the substitutability of prod-

ucts for consumers.1 Second, market structure is determined endogenously: �rms enter

the market until anticipated post-entry pro�ts equal the cost of entering.

Each �rm is run by a principal who hires an agent. By exerting e�ort, the agent can

reduce his �rm's expected marginal cost, but realized cost is stochastic. Each principal

o�ers her agent variable pay contingent on realized cost (linear contracts are optimal in

this model). The optimal bonus depends on the agent's risk aversion, the volatility of

costs, and the value of a cost reduction.

An increase in product substitutability a�ects managerial incentives in two ways. First,

there is a \business stealing e�ect": since �rm-level demand functions are now more

elastic, a �rm with a cost advantage will �nd it easier to attract business from its rivals.

Hence, for given prices set by its rivals, competition increases a �rm's marginal bene�t

of reducing its costs. Second, there is a \wealth e�ect": for a given number of �rms,

equilibrium prices and pro�ts fall. A �rm whose rivals charge lower prices loses market

share, and this decreases the marginal bene�t of reducing its costs.

With endogenous entry and exit, however, �rms' pro�ts are zero in equilibrium. For a

given number of �rms in the market, an increase in the substitutability of products leads to

lower pro�ts, but this will induce some �rms to exit, until the remaining �rms' pro�ts are

1 The speci�c model I use is a circular-road model, in which product heterogeneity is measured by the

\transport cost" that consumers incur when traveling along the circle.

zero again. Since equilibrium pro�ts are una�ected by changes in competition, the wealth

e�ect vanishes, and what remains is the business stealing e�ect. Thus, when competition

increases, principals unambiguously provide stronger incentives to their managers.

By a�ecting the elasticity of demand, greater competition also increases the variance

of each �rm's pro�ts, even when the variance of costs remains unchanged. Consequently,

the model predicts a positive relationship between the volatility of �rms' pro�ts and

the optimal strength of incentives for managers. The variance of pro�ts and optimal

incentives are a�ected by competition independently: a positive correlation between risk

and incentives results without any direct causal link between them.

An increase in market size has a similar e�ect on incentives as an increase in product

substitutability. When market size grows, each �rm stands to gain more from reducing its

costs, and therefore increases the incentives it provides to its agent. At the same time, the

volatility of �rms' pro�ts increases, and again the implication is a positive relationship

between incentives and risk.

These results help to explain two separate con icts between principal-agent theory and

available evidence. The �rst is the relationship between product market competition and

managerial incentives. It seems fair to say that most economists believe that competition

not only improves allocative eÆciency but also leads to greater eÆciency within �rms

and helps to reduce managerial slack.2 Empirical evidence for this view is weak, but

generally supportive.3 E�orts to pin down this idea theoretically, however, have produced

mixed results. Hart (1983) demonstrated that competition can mitigate agency problems

by providing a principal with better information about her agent's actions. Scharfstein

(1988), however, showed that Hart's result is sensitive to how managerial preferences are

2 This view is most prominently associated with Leibenstein (1966) and Machlup (1967), but has

been expressed by many others, including Adam Smith and John Hicks. See also Nickell (1996) for a

discussion.

3 See Nickell (1996) again. Graham, Kaplan and Sibley (1983), for example, �nd that signi�cant

productivity gains resulted from deregulation in the U.S. airline industry. Coughlan (1985) and Slade

(1998) �nd that �rms are more likely to delegate decisions to agents, and provide them with greater

incentives, the higher is the degree of product substitutability in a market. On a more \macro" level,

Porter (1990) shows that the global success of an industry is strongly associated with the extent of

competition it faces domestically.

2

speci�ed. Schmidt (1997) has argued that by reducing pro�ts, competition can lead to

an increased e�ort by a manager to prevent bankruptcy. However, both Hermalin (1992)

and Schmidt (1997) have pointed out that if more intense competition leads to lower

pro�ts, the marginal bene�t of reducing costs or raising demand may decrease as well,

leaving the e�ect of competition on incentives ambiguous.4 Other papers model a �rm's

market environment more explicitly, but nevertheless take market structure as given, such

as Martin (1993) and Boone (2000).5

In contrast to the previous literature, I obtain unambiguous predictions. First, by

studying �rms' contracting decisions in an explicit oligopoly framework, I identify a link

between competition and incentives that is largely absent in more abstract models: with

greater competition, �rms demand functions are more elastic, and a \business stealing

e�ect" induces �rms to provide stronger incentives to managers.6 Second, the e�ect that

lower pro�ts may reduce the bene�t of managerial e�ort (which I called the \wealth e�ect"

above) is the main reason for the ambiguity of previous results. Here, this e�ect vanishes

because market structure is endogenous and hence �rms' pro�ts are zero.7

A second unresolved issue in principal-agent theory is the relationship between risk

and managerial incentives. A central prediction of the theory is that a risk-neutral prin-

cipal will provide weaker incentives to a risk-averse agent the more the evaluation of the

agent's performance is subject to errors. As Prendergast (1999, 2000a, 2000b) points

out, however, empirical evidence of an inverse relationship between risk and incentives is

scarce. Evaluating two dozen empirical studies of the relationship between performance-

4 For more detailed reviews of the literature, see Schmidt (1997) and Nickell (1996).

5 In contrast, Stennek (2000) treats the number of �rms as endogenous, but does not model competition

among �rms explicitly.

6 The role of product substitutability is also emphasized in recent papers by De Bettignies (2000) and

Grossman and Helpman (2000), who study how the degree of vertical integration of �rms in an industry

depends on competition. Grossman and Helpman assume that because of specialization, nonintegrated

�rms can produce at lower cost, which is more valuable when the market is more competitive (as here). In

De Bettignies' model the (downstream) entrepreneur is better informed about market conditions than the

(upstream) investor, which renders his input in raising product quality more important when competition

is more intense. In both models, nonintegration is costly because of the holdup problem.

7 Put di�erently, a wealth e�ect can arise only if market structure does not change when a market

becomes more or less competitive. I will discuss this point in the Conclusion.

3

pay sensitivity and measures of risk in various occupations, Prendergast (2000b) concludes

that studies �nding a positive or insigni�cant relationship greatly outnumber those that

�nd the negative relationship predicted by theory. Similarly, in a survey of research on

franchising, Lafontaine and Slade (2000) conclude:\Theory predicts that more risky units

should tend to be operated by the parent company. The evidence, however, strongly

rejects this predicted tendency."

Considering a principal-agent relationship within an oligopoly context sheds light on

the relationship between agent performance, �rm performance, and optimal incentives.

On one hand, my model predicts the standard negative relationship between incentives

and the riskiness of the agent's measured performance. On the other hand, competition

simultaneously increases both the value of cost reductions and the variance of �rms' prof-

its, without a�ecting the riskiness of the agent's measured performance. Hence, because

of the greater value of e�ort, �rms provide stronger incentives to their managers. This

leads to a positive correlation between managerial incentives and �rm risk even though

in this model there is no direct relationship between the two.

2 Model

1. Entry: n �rms enter the market and position themselves symmetrically around a

circle of circumference 1. To enter, a �rm must incur a setup cost of F . Each �rm

consists of a risk-neutral principal and a risk-averse agent. The principal makes all entry,

personnel and pricing decisions, while the agent has in uence over the �rm's marginal

costs. The number of �rms is endogenously determined by the assumption of free entry

and exit, which implies that in equilibrium, each �rm's pro�t net of the setup cost must

be nonnegative. For simplicity, I treat n as a continuous variable.

2. Contracts: By default, each �rm's expected marginal cost is �c. By exerting e�ort

ei, �rm i's agent can reduce expected marginal cost by ei. Moreover, cost is a�ected by a

random in uence ui, where ui is normal with zero mean and variance �2, and independent

of the other �rms' cost shocks. Thus, �rm i's (constant) marginal cost ci is given by

ci = �c� ei � ui. I assume that realized cost is contractible, which allows the principal to

reward the agent for his e�orts to reduce costs. Each principal o�ers her agent a linear

4

contract consisting of a salary si and variable pay bi(�c � ci), where bi is a \piece rate".

The agent's total compensation hence is wi = si + bi(�c� ci).

3. E�ort choice: All agents simultaneously choose e�ort levels, which are unobserv-

able. Each agent's utility is given by � expf�r[wi�ke2i =2]g, where r is the agent's degree

of (constant absolute) risk aversion and ke2i =2 is the agent's disutility of exerting e�ort.

Under these assumptions, it is optimal for the principal to o�er a linear contract, cf.

Holmstr�om and Milgrom (1987).8 The agent's expected wage upon choosing e�ort ei is

si + biei, and the variance of his wage is b2i�2. The certainty equivalent of his utility is

hence given by

si + biei �1

2rb2i�

2 �k

2e2i : (1)

The agent accepts any contract (si; bi) that gives him an expected utility of at least U ,

which we normalize to zero.

4. Prices: After agents have exerted e�ort, each �rm learns its realized cost ci, which is

its private information. The �rms then simultaneously choose prices, and �rm i's realized

gross pro�t is given by �i = (pi � ci)qi(p), where pi is i's price and qi(p) is its demand

(to be determined below), which is a function of the vector of prices p.

5. Demand: The circle is populated by a continuum of consumers with a uniform

density of d. Each consumer buys exactly one unit of the good. If a consumer located at

x purchases from �rm i located at zi, his resulting utility is

Vi(x) = y + a� pi � t(x� zi)2:

Here, y is income, a is the utility of consuming the most preferred variety (namely x), and

t(x� zi)2 is the disutility associated with consuming variety i instead, which is quadratic

in the distance between the consumer and the �rm.9

8 Holmstr�om and Milgrom illustrate this point in a dynamic environment in which both the agent's

e�ort choice and the measurement of his performance are continuous in time.

9 Linear transport costs, in contrast, would lead to discontinuous pro�t functions for the same reason

as in the standard Hotelling model with linear transport costs: A �rm i whose price is low enough to

attract the consumer located at �rm i+ 1 immediately captures all of i+ 1's consumers. This does not

cause any problems in the circle model when �rms have identical costs, but would substantially complicate

the analysis here where �rms' realized costs are di�erent.

5

The unit transport cost t is the measure of the \toughness of competition" in this

model.10 The main results of this paper establish how the equilibrium changes as the

market becomes more competitive as re ected by a decrease in t. In addition, I will study

how a change in market size via the density of consumers d a�ects the equilibrium market

structure and optimal incentives.11

To keep the analysis tractable, I restrict the parameters of the model such that in

equilibrium, every �rm competes only with its immediate neighbors. What I am ruling

out, therefore, is that one �rm's cost is so much lower than a neighboring �rm's cost that

the �rst �rm can capture all of the second �rm's consumers.

To make these restrictions explicit, I �rst determine an upper bound to the number

of �rms that are viable in the industry. No �rm will ever charge a price exceeding a,

the consumers' utility of consuming their most preferred product. Given a market size

of 1, a is therefore the maximal joint pro�t of the �rms in the industry. In a symmetric

equilibrium, each �rm's expected pro�t therefore cannot exceed a=n� F , which leads to

�n = a=F as an upper bound for n.

To rule out large random cost di�erences I assume that the variance of cost shocks

�2 is suÆciently small such that realized costs far above or far below the mean can be

ignored:12

10 Sutton (1991, p.9) de�nes the \toughness of price competition" as a function linking market structure

to prices or unit margins, not to the level of prices or unit margins observed in equilibrium. An alternative

measure of the toughness of competition is considered by Symeonidis (2000b): if �rms are assumed to

maximize the sum of their own and a fraction � of their rivals' pro�ts, then variations in � describe a

continuum of di�erent intensities of price competition ranging from Bertrand (� = 0) to perfect collusion

(� = 1). Such a measure seems less appropriate for our purposes because it is unclear where � comes

from. Instead, it seems preferable to determine the \competitiveness" of a market from fundamentals

such as consumer preferences and transport costs.

11 It is straightforward to paramaterize the circumference of the circle as well. Not surprisingly, in

this extended model the equilibrium number of �rms is always proportional to the circumference, which

eliminates any interesting interaction between the circumference and agents' incentives. Hence, I con�ne

the formal analysis here to the case where the circumference is unity.

12 This assumption is commonly made in both �nancial economics and the industrial organization

literature, see e.g. Vives (1999, Chapter 8). Whenever random stock prices, demand intercepts or costs

are described by normally distributed variables, possible negative realizations are ignored in the analysis.

6

Assumption 1 �2 < t2

3�n4.

Moreover, the agents' disutility of e�ort must be suÆciently convex so that an agent

will never want to cut costs to a level that allows his �rm to attract all of a neighbor's

customers. That is, the optimal piece rate o�ered by the principal must not be too high.

This case is ruled out by the assumption that the disutility parameter k is suÆciently

large:

Assumption 2 2kt(1 + kr�2) > �nd.

These conditions, which are suÆcient for the existence of a symmetric interior equilibrium,

are eventually violated as the toughness of competition t approaches zero, and Assumption

2 is violated if the density of consumers d exceeds some upper bound. This is a consequence

of the assumption that the cost reduction is linear in e�ort and that disutility is quadratic.

The bene�t of this assumption is that it allows me to calculate closed-form expressions

for the resulting equilibrium.

3 Equilibrium

I solve for a symmetric equilibrium of the game (i.e. an equilibrium in which all �rms

choose the same contracts) by backward induction.

5. Demand: A consumer located between �rms i and i + 1 at distance x from i will

purchase from i if

pi + tx2 � pi+1 + t�1

n� x

�2; (2)

where 1=n is the distance between the two �rms. Rearranging (2), the marginal consumer

between i and i+ 1 is given by

�x =1

2n+

n

2t(pi+1 � pi);

which lies strictly between 0 and 1=n as long as jpi+1�pij does not exceed t=n2. A similar

expression describes the marginal consumer between i�1 and i. Firm i's total demand is

qi = d�1

n+

n

2t[(pi+1 � pi) + (pi�1 � pi)]

�; (3)

7

and expected demand is

E(qi) = d�1

n+

n

t(E(p)� pi)

�; (4)

where E(p) is the price �rm i expects its rivals to set.

4. Prices: Each �rm sets its price without knowing the other �rms' costs, and hence

maximizes its gross pro�t

�i = (pi � ci)E(qi) = (pi � ci)d�1

n+

n

t[E(p)� pi]

�(5)

with respect to pi. Solving the f.o.c. for pi leads to

pi =t

2n2+

E(p) + ci2

: (6)

In a symmetric equilibrium, the expected price E(p) must equal the r.h.s. of (6) if the

�rm's cost is equal to its expected value E(c). Solving for E(p) we obtain

E(p) = E(c) +t

n2: (7)

This is the standard equation one obtains for the equilibrium price if all �rms have the

same marginal cost, cf. Tirole (1988, p.283). Substitute (7) into (6) to obtain i's equilib-

rium price as a function of its cost ci:

pi =t

n2+

E(c) + ci2

= ci +t

n2+

E(c)� ci2

: (8)

To obtain expected demand, substitute (7) and (8) into (4):

E(qi) = d�1

n+

n

2t(E(c)� ci)

�: (9)

The resulting expected gross pro�t in equilibrium is

�i = (pi � ci)E(qi) =dt

n

�1

n+

n

2t(E(c)� ci)

�2: (10)

For this expression to be valid requires that E(qi), given by (9), be nonnegative, or

ci � E(c) �2t

n2: (11)

The upper bound on �2 imposed by Assumption 1 ensures that realizations of ci that

violate (11) can be safely ignored.13

13 More precisely, Assumption 1 can be rephrased as 2p3� < 2t

�n2: Thus, since n � �n, (11) can be

violated only if ci deviates from its mean by more than 2p3� = 3:46�, the probability of which is

well below 0.1 percent. By making Assumption 1 more restrictive, any desired con�dence level can be

achieved. On the other hand, for all results below that rely on Assumption 1, only the much weaker

condition �2 < 4t2=n4 is needed.

8

3. E�ort choice: Firm i's agent maximizes (1) with respect to ei and hence chooses

the e�ort ei = bi=k. His expected utility then is

si +b2ik�

1

2rb2i�

2 �1

2kb2i :

For the agent to obtain an expected utility of at least zero, the principal must pay him a

(negative) salary of

si = �1

2k(1� kr�2)b2i : (12)

2. Contracts: After realization of ci, �rm i's expected pro�t, net of the agent's total

compensation wi, is

�i = �i � wi =dt

n

�1

n+

n

2t(E(c)� ci)

�2� si � b(�c� ci): (13)

Given that ci = �c� ei � ui, we have �c� ci = ei + ui, and hence (13) equals

dt

n

"1

n+ n

E(c)� (�c� ei � ui)

2t

#2� si � bi(ei + ui): (14)

Substitute si from (12) and ei = bi=k into (14) to obtain �rm i's expected net pro�t after

realization of ci:

�i =dt

n

"1

n+ n

E(c)� �c+ bik+ ui

2t

#2+

1

2k(1� kr�2)b2i � bi

bik+ ui

!

=dt

n

24 1

n+ n

E(c)� �c+ bik

2t

!2

+n2u2i4t2

+nui

t

1

n+

E(c)� �c+ bik

2t

!35

+1

2k(1� kr�2)b2i � bi

bik+ ui

!: (15)

The �rm's expected pro�t before realization of ci is then obtained by taking the expected

value of (15) over ui:

dt

n

24 1

n+ n

E(c)� �c+ bik

2t

!2

+n2�2

4t2

35� 1

2k(1 + kr�2)b2i : (16)

Under Assumption 2, this pro�t function is strictly concave in bi. For given expectations

E(c) about other �rms' average costs, �rm i maximizes (16) with respect to bi, which

leads to

bi = kd2tn � n(�c� E(c))

2kt(1 + kr�2)� nd: (17)

9

In a symmetric equilibrium, all �rms choose the same contract, and each agent chooses the

same e�ort. It follows that E(c) = �c� e = �c� b=k with b given by (17). The equilibrium

value of b chosen by all �rms hence is the solution to

b = kd2tn � n(�c� (�c� b=k))

2kt(1 + kr�2)� nd;

or

b =d

n(1 + kr�2): (18)

1. Entry: Substitute b from (18) into (16) to obtain each �rm's expected net pro�t

upon entry. Firms enter the market as long as the expected net post-entry pro�t minus

the setup cost,dt

n3+

nd�2

4t�

d2

2kn2(1 + kr�2)� F; (19)

is nonnegative. The derivative of (19) with respect to n can be written as

d2

n3

d

k(1 + kr�2)�

2t

n

!+ d

�2

4t�

t

n4

!; (20)

where the �rst term in (20) is negative because of Assumption 2, and the second term is

negative because of Assumption 1. Hence pro�t is decreasing in n, and �rms enter until

(19) equals zero.

Notice that the second term in (19) is increasing in �. This is a quite general feature

of models with stochastic costs or demand (see e.g. Raith 1996): since pro�ts are convex

functions of equilibrium prices, expected pro�ts are increasing in the variance of �rm-

speci�c random in uences. Intuitively, this holds because, in many models, the gain in

pro�t from a given cost advantage (relative to a �rm's rivals) exceeds the loss in pro�t

from a cost disadvantage of the same magnitude.

4 Determinants of Market Structure

According to (18), managerial incentives depend on the number of �rms, which is endoge-

nous. Therefore, to see how managerial incentives vary with the toughness of competition

and with market size, we �rst need to know how market structure itself varies with these

fundamentals:

10

Proposition 1 (a) The equilibrium number of �rms is increasing in t. In other words,

when products become more substitutable, �rms exit or merge.

(b) The equilibrium number of �rms increases, but less than proportionally, with mar-

ket size.

Proof: (a) Given Assumption 1, (19) is increasing in t. The result then follows because

(19) is decreasing in n. (b) First, n must increase with s because (19) is increasing in

s for any given n. To see that the increase is less than proportional, it suÆces to show

that (19) falls when s and n both increase by the same factor � > 1. Substitute �s for s

and �n for n in (19). Then the derivative of the resulting expression with respect to � at

� = 1 is

�d

2n3t(4t2 � n4�2) < 0;

which establishes the result.

While both parts of Proposition 1 may look familiar from other models, they are

not as obvious here, because of two distinct features of this model, stochastic costs and

endogenous cost reductions.

First, it is a standard property of any spatial model of competition that, for a given

number of �rms, price competition becomes more intense and equilibrium pro�ts fall

as the transport cost decreases. The model studied here also has this general feature.

However, because of the second term in (19) discussed above, given uctuations in costs

translate into larger uctuations in prices and demand when competition increases. Other

things equal, this raises expected pro�ts. The upper bound on �2 imposed by Assumption

1 ensures that the �rst, \normal" e�ect always dominates the second.

Second, the result that n increases less than proportionally with market size is a

standard feature of exogenous-sunk-cost models, cf. Sutton (1991, ch. 2). Here, however,

�rms' costs are endogenously determined by the incentives that �rms provide to their

managers. As Sutton has shown, when investments in quality improvements or cost

reductions are endogenous, then increases in market size may lead to an escalation of

�rms' strategic investments. If �rms' initial pro�t gains are dissipated by escalating

investments in R&D or advertising, then entry of new �rms may not occur, and the

market may remain concentrated irrespective of market size.14 This escalation e�ect is

14 A model with cost reductions that exhibits this property is analyzed in Dasgupta and Stiglitz (1980).

11

ruled out by Assumption 2. For the interpretation of concentration measures, however,

it is important to keep in mind that the negative correlation between market size and

concentration established by Proposition 1 (b) is sensitive to the speci�cations of the

model, and need not hold generally.

5 Competition and Managerial Incentives

I now turn to how, in a free-entry equilibrium, managerial incentives vary with the tough-

ness of competition and with market size. The �rst main result follows immediately from

Proposition 1 (a):

Proposition 2 The equilibrium level of b is decreasing in t, and hence �rms' expected

marginal costs are increasing in t. That is, as the market becomes more competitive, �rms

provide stronger incentives to their managers, and expected marginal costs fall.

Proof: According to (18), b does not directly depend on t but is decreasing in n. It then

follows from Proposition 1 (a) that b is decreasing in t in the free-entry equilibrium. Since

stronger incentives induce greater cost reductions, expected marginal cost are increasing

in t.

To better understand Proposition 2, substitute (6) into (5), which gives �rm i's pro�t

as a function of the price E(p) it expects its rivals to set:

d

4

�t

n2+ E(p)� ci

��1

n+

n

t(E(p)� ci)

�=

nd

4t

�E(p)� ci +

t

n2

�2(21)

Di�erentiate this pro�t with respect to ci to obtain the marginal gain of reducing cost:

d(ci � E(p))n2 � t

2nt(22)

Condition (11) implies that (22) is negative, i.e. that, as expected, a cost reduction in-

creases �rm i's expected pro�t.

For a given number of �rms, an increase in product substitutability has two e�ects.

1. The marginal bene�t of a cost reduction (22) is increasing in t. Since (22) is nega-

tive, this means that a decrease in t increases a �rm's gain from reducing its cost.

Intuitively, when t decreases, demand becomes more elastic, making it easier for

12

a �rm to increase its demand by cutting its price. This \business stealing e�ect"

implies that for given prices of the rivals, a �rm has an increased incentive to lower

its cost, and hence to raise bi.

2. On the other hand, an increase in t also leads to lower equilibrium prices. This

price change has a negative e�ect on a �rm's incentives to lower costs, which I call

the \wealth e�ect": Since (22) is decreasing in E(p) and (22) itself is negative, a

decrease in t decreases a �rm's gain from reducing its cost. Intuitively, when �rm

i's rivals reduce their price, �rm i's quantity will fall, since �rm i's optimal price

response does not fully match a fall in E(p), cf. (6). Since the value of reducing cost

is proportional to market share, this value decreases when the rivals' prices fall.15

With endogenous entry, however, the wealth e�ect is eliminated: rather than reducing

the incentives to invest, a decrease in pro�ts for any given n induces �rms to exit, such

that in equilibrium �rms always make zero pro�t. What remains is the business stealing

e�ect. Each �rm remaining in the market has a larger share of total demand and hence

has a greater incentive to make �xed investments (here, ei) in cost reduction. To this end,

principals unambiguously provide stronger incentives to their agents.

As a side remark, notice that b according to (18) does not depend on t. This means

that for given n, the business stealing and the wealth e�ects exactly cancel each other.

This is a feature of the spatial framework used here and should not be expected to hold

generally. What is general, however, is that free entry eliminates the wealth e�ect, leaving

only the business stealing e�ect as a link between competition and managerial incentives.

For given t, managerial incentives also depend on market size:

Proposition 3 The long-run equilibrium level of b is increasing in d, and hence expected

marginal costs are decreasing in b. That is, as the market grows, �rms provide stronger

15 The wealth e�ect described here corresponds to Hermalin's (1992) change-in-the-relative-value-of-

actions e�ect and Schmidt's (1997) value-of-a-cost-reduction e�ect. It is distinct from Hermalin's income

e�ect according to which agency costs are falling in gross pro�ts when the agent's participation constraint

is not binding. Notice that Schmidt's value-of-a-cost-reduction e�ect, too, arises only when the agent's

participation constraint is not binding. But is an implication of Schmidt's assumption that the agent is

risk-neutral but wealth constrained. Here, where the agent is risk-averse, the wealth e�ect is present even

though the agent's participation constraint is always binding.

13

incentives to their managers, and expected marginal costs fall.

Proof: This follows immediately from the fact that b according to (18) is proportional to

d=n, and that n grows less than proportionally with d.

For given n, an increase in the density of consumers increases each �rm's demand

proportionally. This implies a proportional increase in the bene�t of reducing costs, and

hence in the optimal incentives that principals provide to their agents. While the agents'

compensation rises, the initial gain in pro�ts because of higher demand outweighs this

cost increase, and the net e�ect is an increase in pro�ts. Additional �rms then enter the

market, which partially o�sets the e�ect of market size on incentives.

Propositions 2 and 3 state what would be many economists' intuition: competition

exerts a downward pressure on costs. Importantly, this is not a selection e�ect whereby

more eÆcient �rms drive out less eÆcient ones. Instead, all �rms are driven to reduce

their costs by providing their managers with the necessary incentives.

6 Managerial Incentives and Risk

When competition becomes more intense, �rms' demand functions become more elastic.

As a consequence, given uctuations in costs have a greater impact on realized pro�ts:

Proposition 4 The variance of each �rm's gross pro�t (pi � ci)qi and net pro�t is de-

creasing in t. That is, as the market becomes more competitive, �rm-level risk increases.

Proof: Using (3), �rm i's realized gross pro�t is

�i = (pi � ci)d�1

n+

n

2t(pi+1 � pi + pi�1 � pi)

�;

which using (8) simpli�es to

�i = d

t

n2+

E(c)� ci2

!�1

n+ n

ci+1 + ci�14t

�nci2t

�: (23)

In equilibrium, we have E(c) = �c� b=k and ci = �c� b=k � ui, and then (23) reduces to

�i = d�t

n2+

ui

2

��1

n� n

ui+1 + ui�1

4t+

nui

2t

�: (24)

14

Given that E(ui) = E(uiuj) = 0 and E(u2i ) = �2, the expected value of (24) is dt=n3 +

nd�2=(4t), and so we have

�i � E(�) = d

ui

n+

nu2i4t

�n�2

4t�

ui+1

4n�

ui�1

4n�

nuiui+1

8t�

nuiui�1

8t

!: (25)

The variance of �rm i's gross pro�t is Ef[pii � E(�)]2g. With E(ui) = E(uiuj) = 0 and

E(u2i ) = �2 as well as E(u3i ) = 0, E(u4i ) = 3�4 and E(u2iu2j) = �4, we obtain from (25):

Var(�i) = d2 9

8

�2

n2+

5

32

n2�4

t2

!: (26)

According to (26), the variance of pro�t is decreasing in t. The derivative of (26) with

respect to n has the same sign as 5n4�2 � 36t2, which under Assumption 1 is negative.

Since according to Proposition 1, n is increasing in t, it follows that the variance of pro�ts

is decreasing in t both for given n, and when changes in n are taken into account.

Moreover, since the variance of an agent's wage is b2i�2 (see no.3 in Section 2), it

follows that the variance of each agent's compensation is decreasing in t. Therefore, the

variance of a �rm's pro�t net of the agent's compensation is decreasing in t as well.

According to Proposition 4, a change in competition endogenously leads to a change

in �rm-level pro�t risk, even though the source of uncertainty, namely the stochastic

nature of production costs, is held constant. The reason is again the increase in the price

elasticity of demand that results from a decrease in t: for given prices, (3) implies that

each �rm's realized demand uctuates more strongly when t is lower. While equilibrium

prices fall, their variance does not, as inspection of (8) shows. The overall result is an

increase in the variance of pro�ts.

Propositions 2 and 4 immediately imply

Corollary 1 Variations in the degree of competition induce a positive correlation between

agents' incentives (b) and the variance of �rms' pro�ts.

This result holds even though the principal-agent model used here is entirely standard.

The reason is that agents are paid for observed cost reductions, which I assumed are

contractible. According to (18), the optimal piece rate b is inversely related to both

actual cost risk �2 as well as the agent's risk aversion r, just as in the standard principal-

agent model. The variance of the �rm's pro�t, in contrast, has no direct consequences for

the agent, and hence plays no role in the design of the optimal contract.

15

Suppose instead �rms' pro�ts were contractible but costs not. In this case, optimal

relative performance evaluation would lead to a very similar result: from the realized

pro�ts of every �rm one can unambiguously determine the di�erence between each �rm's

price and the average industry price. Since this average price is close to E(p) with high

probability, one can then quite accurately estimate any �rm's cost.

Only if an agent's compensation is based on his own �rm's realized pro�ts but not on

that of other �rms as well, one would expect a more ambiguous relationship between �rm

risk and incentives because of a tradeo� of two forces: as emphasized in this paper, an

increase in competition makes cost reductions more valuable for each �rm individually,

which would lead to stronger incentives. On the other hand, a higher variance of pro�ts

also implies a higher variance of agents' measured performance, which because of their

risk aversion would be a reason to reduce incentives.

The signi�cance of Corollary 1 lies in its potential to explain the apparent contradiction

between theory and evidence discussed in the Introduction. Any direct test of the standard

principal-agent model rests on the premise that variations in the riskiness of performance

measures are correlated with variations in \measurement errors". As the theory presented

here demonstrates, however, this premise is problematic. A high level of measured �rm risk

(such as the variance of �rm-level pro�ts) might be the consequence of intense competition

in a market. But with intense competition, the marginal value of an agent's e�ort is

also high, while the precision of the agent's measured performance may not depend on

competition. The result then is a positive correlation between risk and incentives.

Prendergast (2000b) o�ers an alternative explanation for this positive relationship.

In his model a principal chooses between instructing an agent to carry out a speci�c

project, and delegating the choice of project to the agent, who is assumed to have superior

information about the payo�s of di�erent projects. An increase in the riskiness of these

payo�s increases the value of the agent's information by increasing the di�erence between

expected payo�s of the projects that the agent and the principal, respectively, believe are

optimal. This leads to the prediction that when delegation is costly (because of higher

monitoring costs or risk aversion), delegation and pay for performance are optimal if the

level of risk is high, whereas non-delegation and a �xed wage are optimal if the level of

risk is low.

16

In Prendergast's model, a change in risk directly a�ects the bene�t of delegation by

changing the option value of the agent's information. Here, in contrast, an increase in

competition leads to both more volatile pro�ts and a higher value of each agent's e�ort.

Hence, both risk and incentives are endogenously determined as functions of the degree

of competition. It is important to notice that the higher value of cost reductions results

from more elastic demand functions alone and not from a higher volatility of demand.

Thus, I obtain a positive correlation between risk and incentives without any causal link

between them.

Lafontaine and Slade (2000) have suggested that an observed positive correlation be-

tween risk and incentives may be the result of reverse causation: when for other reasons

�rms provide agents with greater incentives, the agents' actions are likely to lead to greater

sales variability. Such an e�ect does not arise, however, in the (standard) principal-agent

model used here. Since ci = �c� ei � ui, the piece rate bi a�ects only the expected level of

�rm i's cost and hence pro�t, but not its variance.

The results of the model imply that empirical studies of the factors determining man-

agerial incentives should control for the toughness of competition, measured by product

substitutability, for two reasons: �rst, competition directly a�ects the value of managerial

e�orts. Second, �rm risk is itself a function of both competition and underlying demand or

cost shocks. In particular, competition can a�ect measures of �rm risk without a�ecting

the diÆculty of measuring a manager's performance.

A corollary is that testing the predictions of standard principal-agent theory requires

precise estimates of the measures on which agents' compensation is actually based. When

compensation is not entirely based on �rm performance but also on accounting measures

or subjective criteria, then measures of �rm risk give a distorted picture: �rm risk may be

high because of intense competition, while errors in measuring agents' performance may

be small.

Finally, an increase in market size has a similar e�ect as an increase in the toughness

of competition:

Proposition 5 The variance of each �rm's gross pro�t is increasing in d. That is, as

the market grows, �rm-level risk increases.

Proof: by inspection of (26), given that n increases less than proportionally with d.

17

Corollary 2 Variations in market size induce a positive correlation between agents' in-

centives and the variance of �rms' pro�ts.

The underlying logic of these results is similar as for a change in t: when the density

of consumers increases, new �rms enter and the circle becomes more densely populated

with �rms. The result is a higher elasticity of demand functions, which leads to a higher

volatility of pro�ts, while inducing �rms to provide stronger incentives. Thus, once again

we obtain a positive correlation between �rm risk and incentives without any direct causal

link between them.

7 Conclusion

I have argued in this paper that when product market competition becomes more intense,

�rms will provide stronger incentives to their managers to reduce costs, even though pro�ts

become more volatile. This occurs because greater competition leads to more elastic

demand functions, implying that given price cuts lead to greater increases in demand. As

as a consequence, both the value of managers' e�orts and the volatility of pro�ts increase.

This prediction is unambiguous because market structure is endogenous: in equilibrium,

�rms always make zero pro�ts, and hence there is no negative e�ect of competition on

incentives via the level of �rms' pro�ts.

These results help to simultaneously reconcile two unresolved issues in principal-agent

theory. First, while most economists appear to believe that competition positively a�ects

incentives, theoretical research suggests a much more ambiguous relationship. Second,

most empirical work �nds a positive relationship between risk and incentives, which is at

odds with a basic result of principal-agent theory.

The IO approach used here resolves these con icts and o�ers new perspectives on the

relationships between incentives, competition and risk. These seem important from both

a theoretical and an empirical point of view:

� In oligopoly, the substitutability of products for consumers is a very natural mea-

sure of the \degree of competition" between �rms. Most importantly, more intense

competition does not simply reduce pro�ts but also increases the marginal value of

investments (such as cost reductions) that allow a �rm to steal business from others.

18

� As emphasized by Sutton (1991) and others, changes in the nature of competition

between �rms in a market typically lead to changes in market structure. Treating

market structure as endogenous eliminates any e�ects of competition on incentives

that may arise because of changes in the level of pro�ts. This does not mean that

\wealth" e�ects do not exist in reality. On the contrary, they can arise when-

ever market structure fails to adjust to changes in competition. This may be the

consequence of regulatory restrictions, or simply of integer constraints: when �rm

numbers are discrete and entry costs large, small changes in competition may not

induce entry or exit. In practice, however, entry, exit and mergers are pervasive

features of most markets, which suggests that market structure should be treated

as endogenous.

� Testing for the negative relationship between risk and the strength of incentives

predicted by principal-agent theory is problematic unless competition is taken into

account: greater competition increases both the value of agents' e�orts and �rm-

level risk, without necessarily a�ecting the quality of performance evaluations.

� The common interpretation of a high degree of concentration in an industry as

evidence of lack of competition is inconsistent with the assumption of free entry

and exit: in the model presented here, and in fact any model with free entry, an

increase in the substitutability of products leads to the exit of �rms and hence

an increase in market concentration. Thus, if markets vary in the toughness of

competition, then high levels of concentration are indicative of intense competition,

not a lack of it. Sutton (1991) and Symeonidis (2000a) provide empirical evidence

of this positive relationship between competition and concentration. On the other

hand, when markets vary in size, markets that are less concentrated will tend to be

larger markets and hence will also be more competitive. Thus, since increases in

the toughness of competition and increases in market size have opposite e�ects on

market structure, concentration measures alone are likely to be poor indicators of

the degree of competition.

The results obtained in this paper seem quite robust. They rely on the central assumption

that an increase in the substitutability of products raises the marginal value to each �rm

19

of gaining a cost or quality advantage over its rivals. This assumption is neither speci�c

to the circle model chosen here, nor to the assumption that e�ort is directed at reducing

costs. Symeonidis (2000b), for example, shows within a linear-demand model that with

free entry, an increase in product substitutability leads �rms to increase their investments

in quality.

In practice, when market conditions are uncertain, managers must be induced to make

the right business decisions (as in Prendergast 2000b and De Bettignies 2000), rather than

simply to \work hard". This can be achieved through pay for performance, which helps to

align managers' goals with those of the shareholders, but also entails agency costs. Via the

elasticity of demand, greater competition increases the value of making good decisions,

and then �rms will provide stronger incentives in response. Hence, the logic of the theory

presented here also applies to more general contexts.

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22