WORKING 

 PAPERS

Competition, Contracts, and Innovation

Christopher J. Metcalf

John D. Simpson

WORKING PAPER NO. 298 

December 2009

___________________________________________________________________________

FTC Bureau of Economics working papers are preliminary materials circulated to stimulate discussion

and critical comment. The analyses and conclusions set forth are those of the authors and do not

necessarily reflect the views of other members of the Bureau of Economics, other Commission staff, or

the Commission itself. Upon request, single copies of the paper will be provided. References in

publications to FTC Bureau of Economics working papers by FTC economists (other than

acknowledgment by a writer that he has access to such unpublished materials) should be cleared with the

author to protect the tentative character of these papers.

______________________________________________________________________________

BUREAU OF ECONOMICS

FEDERAL TRADE COMMISSION

WASHINGTON, DC 20580
Competition, Contracts, and Innovation

Christopher J. Metcalf and John D. Simpsony

December 15, 2009

Abstract

Our paper contributes to the literature on the relationship between innovation

and market power by considering how changes in the intensity of product market

competition a§ect innovation when managerial compensation is a linear function

of Örm proÖts. Changes in the intensity of product market competition a§ect

both the return from innovation and the cost of inducing managers to innovate.

Several recent papers account for both the returns-to-investment e§ect and the

agency-cost e§ect in analyzing the e§ect of additional product market competition

on incentives to innovate (see e.g., Schmidt (1997), Raith (2003), and Piccolo,

DíAmato, and Martina (2008)). Our model di§ers from these papers in the type

of contract that we assume Örms can use to induce innovation. With linear proÖtsharing contracts, the cost of a non-drastic innovation declines as product market

competition increases because the increment gained from innovation becomes a

larger fraction of the total proÖt. We argue that this decline in the cost of

attaining innovation as competition increases means that competition will often

lead to more innovation even in models where the returns to innovation otherwise

would fall as competition increases.

1 Introduction and Literature Review

Does market power facilitate innovation? If so, under what circumstances should competition policy tolerate the short-term allocative ine¢ ciency associated with market

power in order to obtain higher levels of innovation? These two questions have been a

We thank two anonymous referees and the seminar audience at Oberlin College. Any errors are

our own. The views expressed are those of the authors and do not necessarily reáect those of the

Federal Trade Commission.

yBureau of Economics, Federal Trade Commission, 600 Pennsylvania Ave NW, Washington, DC

20580. E-mail: cmetcalf@ftc.gov and jsimpson@ftc.gov. Contact author: Metcalf.

1
central issue in Industrial Organization for 50 years. While simple answers have proved

elusive, numerous papers have made signiÖcant progress toward addressing these questions. Our paper seeks to add to this stock of knowledge by considering how product

market competition a§ects innovation when managerial compensation is a linear function of Örm proÖts.

Changes in the intensity of product market competition a§ect both the return from

innovation and the cost of inducing managers to innovate. An extensive literature

examines the e§ect of product market competition on the return from innovation and

Önds that greater product market innovation can lead to either increased innovation

or decreased innovation depending on the assumptions made about factors such as the

nature of competition before and after the innovation, whether the innovation can be

readily copied, and whether rivals can also innovate. (see e.g., Arrow (1962); Tirole

(1988); Schumpeter (1947); Qiu (1997); Vives (2008)). Another set of paper examines

the e§ect of product market competition on the cost of inducing managerial e§ort and

Önds that increased competition, measured as the number of entrepreneurial Örms (i.e.,

without agency problems), either reduces agency problems or increases agency problems

depending on the agentís utility function (e.g., Oliver Hart (1983) and Scharfstein

(1988)). Finally, several recent papers account for both the returns-to-investment e§ect

and agency-cost e§ect in analyzing the e§ect of additional product market competition

on incentives to innovate (see e.g., Schmidt (1997), Raith (2003), and Piccolo, DíAmato,

and Martina (2008)).

Similar to this last set of papers (especially Piccolo et al.), we examine the overall

e§ect that product market competition has on innovation in the presence of managerial

contracts. However, our model di§ers from these papers in the type of contract that

we assume Örms can use to induce innovation. These other models focus on the case

where a Örm can tailor a contract to induce managers to attain a proÖt or cost target

2
associated with some particular level of innovation. In contrast, we assume that Örms

are restricted to using linear proÖt-sharing contracts with their managers. While

linear contracts are clearly a special case, representing one extreme of how contracts

can be written, economic theory1 and empirical studies2suggest that this special case

oftentimes does not di§er far from reality.

With linear proÖt-sharing contracts, the cost of a non-drastic innovation declines as

product market competition increases because the increment gained from innovation

becomes a larger fraction of the total proÖt.3 To see this, suppose a Örm can only get a

manager to undertake a project by o§ering this manager a share of proÖts. Then, the

managerís share of overall proÖts must be su¢ ciently high that the managerís share

of the incremental proÖts generated by the innovation exceeds the managerís cost of

innovating. Now consider the case where the cost of a minor innovation equals half

of the incremental proÖts from innovation. A monopoly Örm is unlikely to undertake

such an innovation because it does not want to give the manager half of its pre-existing

1Holmstrom and Milgrom (1987) identify some circumstances where optimal contracts are a linear

function of proÖts. Prendergast (1999) and Milgrom (1991) show that contracts that link compensation

to a speciÖc goal (in this case some level of innovation) can divert an agentís e§ort from other important

goals (in this case, other tasks necessary for proÖt maximization). Finally, Hall and Murphy (2003)

note that shareholders may be reluctant to o§er highly non-linear contracts because such contracts

o§er managers no incentive to perform in some possible states of the world.

2Empirical studies suggest that managerial compensation has a large linear component. Managers

tend to own a large share of the stock of the Örms they manage: Holderness et al. (1999) found

that the mean percentage of common stock held by a Örmís o¢ cers and directors was 21 percent in

1995. Returns on this stock comprise a large component of managersíincentive compensation: Hall

and Liebman (1998) pp. 674-5 found that returns on stock and stock options account for nearly all

of the relationship between managerial compensation and Örm performance and that the return on

stock accounts for over half of such incentive compensation. While the returns on stock options could

be highly non-linear in theory, Hall and Murphy (2003) (pp. 50,59) found that ìmost employee stock

options expire in ten years and are granted with an exercise price equal to the market price on the date

of the grant.î The value of such an option would be captured by an upward sloping curve positioned

between the value of the share price and the value of the call option if exercised immediately. (See

Brealey and Myers (1988) p. 480). Thus, the relationship between the value of these options and the

Örmís overall proÖt is closer to linear than it may initially seem. Hall and Murphy (2003) also note that

stock options indexed to a competitorís stock price are ìnonexistentî as an empirical phenomenon.

3Piccolo et al. also Önd that the cost of inducing e§ort using proÖt-based compensation falls as

competition increases. However, a di§erent mechanism leads to this result in their paper. SpeciÖcally,

in their model, an increase in competition (as measured by product substitutability) makes Örms more

similar thereby making it easier for owners to determine if their manager exerted e§ort.

3
monopoly proÖts to get half the beneÖt from a minor innovation. In contrast, a Örm

in a more competitive market might innovate because its pre-existing proÖts are much

smaller relative to the incremental proÖt from innovation.

In this paper, we argue that this decline in the cost of attaining innovation as competition increases means that competition will often lead to more innovation even in

models where the returns to innovation otherwise would fall as competition increases.

To show this, we use a model in which two symmetric Örms, both of which can innovate, compete as di§erentiated Bertrand competitors and face a linear-quadratic

demand function. While the additional proÖt that a Örm gains by innovating declines

as competition increases in this model (e.g., see Qiu (1997) and Piccolo et al. (2008)),

once we account for the cost of attaining innovation (assuming linear proÖt sharing contracts), the overall return on innovation increases with competition over a wide range

of parameter values.

It is sometimes argued that antitrust policy should tolerate the short-term allocative

ine¢ ciency associated with monopolies in order to gain the beneÖts associated with

higher levels of innovation. However, our results suggest that a Örm with market power

may be less able to exploit such opportunities for innovation because, with market

power, it is more costly to incentivize managers to innovate. This suggests that antitrust

policy should be less willing to tolerate monopolies to increase the incentives to innovate

than would otherwise be the case. That said, there may still be instances where greater

market power is needed to encourage innovation. For instance, Qiu (among others)

has noted that spillover e§ects of innovation can rapidly decrease a Örmís incentive to

innovate.

4
2 Model

We consider a three stage model: In stage one, each of two symmetric Örms simultaneously can contract with its manager to make an investment that would lower its

marginal cost; in stage two, these managers simultaneously decide whether to undertake costly investments to lower marginal cost; and, in stage three, these managers

simultaneously select prices so as to maximize Örm proÖts.

Focusing initially on stage 1, we assume that Örm iís proÖt, i, is divided between

its risk-neutral owner (principal), who is the residual claimant, and its risk-neutral

manager (agent), who is compensated by a linear proÖt-sharing contract of the form of

the form wi = bii where bi 2 R+. We assume for simplicity that the agentís reservation

wage is zero since we only wish to examine the principalís incentive to induce the agent

to undertake an action.4 We assume that managers know the model and the previous

actions of themselves and their rivals. We assume that owners know the model but

cannot observe the actions of managers.5 Given these assumptions, we then assume

owners simultaneously select biin order to maximize expected proÖt.

In stage 2, managers simultaneously decide whether to undertake costly investments

to reduce their constant marginal costs from their initial level c, where  > 0 is a

scaling parameter and 0 < c < 1. Firm iís manager can either expend e§ort at utility

cost g to reduce iís marginal cost by x to (c  x) (where 0 < x < c) or expend no

e§ort and have marginal cost remain c. We denote the choice variable for i as xi 2

f0; xg.

In stage 3, we assume that the Örms compete as di§erentiated Bertrand competitors.

4This justiÖes the zero intercept of the contract since the principle will set the assured wage to zero.

5These assumption are made to construct an agency problem, but we could justify them by modeling

costs directly by assuming that Örms can observe total costs but cannot allocate these costs between

Öxed and variable costs. Suppose Örm i has constant marginal costs ci and Öxed cost fi, where fi

is stochastic and distributed according to some distribution. Thus, the Örm cannot infer the actions

of the manager from marginal costs levels, which to it are unobservable, or from total costs, which

include the stochastic Öxed cost term.

5
SpeciÖcally, we assume that each Örm i with competitor j faces a linear-quadratic

demand function given by:6

Di(pi; pj ) = 1

1  


2

( (1  
)  pi + 
pj ) with 0 < 
 < 1 (1)

Where piis the price set by Örm i, pjis the price set by Örm j, and 
 is a parameter

measuring the degree of product di§erentiation. The degree of product di§erentiation

decreases with 
 and thus competition between the Örms increases with 
.

We limit our analysis to non-drastic innovation of the following form: we assume

that x is small enough that both Örms always produce a positive quantity. This

assumption is equivalent to the more analytically tractable assumption that 
 < 
,

where 
 is the highest 
 such that q

  0 for a Örm that is at a cost disadvantage from

not investing when its rival did.7

Assumption 1 
 < 
 

1

2

qx(22c+x)

(1c)

2 + 9 

1c+x

1c



The proÖt function of a Örm is: i = Di(pi; pj ) (pi  MCi). Thus, the Örmís proÖt

is i = Di(pi; pj ) (pi  (c  x)) if the manager innovates and i = Di(pi; pj ) (pi  c)

if the manager does not innovate

The equilibrium concept is subgame perfect Nash equilibrium in pure strategies,

which we solve using backward induction. Since investment has a discrete size as in

Bester and Petrakis (1993), we investigate how the maximum cost at which a Örm would

innovate changes as 
 changes. In other words, we examine how the range of g under

which there is equilibrium investment changes with 
 rather than how the optimal size

6This demand function can also be rewritten as: Di(pi

; pj ) = (1
)

1
2



  pi +




(1
)

(pj  pi)



.

7SpeciÖcally, we assume x < M in[x; c] where x = 
1



2  2c  
 + c
  


2 + c
2



. The nondrastic constraint is not binding from x > c  if 
 < 1

2

q

8 + (1  c)

2  (1  c)1



.

6
of the investment changes with 
.

In particular, we focus the analysis on the region of gís where it is the unique

pure strategy equilibrium for both owners to induce both managers to invest (given a

pure strategy equilibrium in stage 2). Although we do make note of some behavior

outside this region, we focus on this case for simplicity, since there are not the multiple

equilibria that exist in other regions, and because when the game leaves this region, it is

certain that innovation has decreased since innovation was at a maximum. In addition,

the region may be of interest because it represents the region where innovation lowers

production cost for all of the industry and thus the region where consumers beneÖt

most from such innovation.

3 Innovation E§ort without an Agency Problem

We begin by considering the case where an owner directly manages the Örm. This case

serves as a benchmark for comparison of the case with proÖt-sharing contracts.

Looking forward in stage 2, the Bertrand-Nash equilibrium price in stage 3 given

by e§orts xi and xjis:

pi (xi; xj ) = 

4  


2



2(c  xi) + (c  xj ) + 2  
  


2

 (2)

The equilibrium proÖt given the e§orts xi and xjis:

i (xi; xj ) = 

2

(4  


2

)

2

(1  


2

)



(2 + 
) (c  xi) + 
 (c  xj ) + 2  
  


2

2

(3)

Denote asym = i (x; 0)  i (0; 0); as the proÖtability of Örm i innovating when

Örm j does not innovate, and sym = i (x; x)  i (0; x), as the proÖtability of Örm i

innovating when Örm j innovates. Then:

7
asym (
; ; x; c) = x (2  


2

)

(4  


2

)

2

(1  


2

)



x



2  


2



+ 2 (1  c)



2  
  


2

 (4)

sym (
; ; x; c) = asym (
; ; x; c) 

2

2x2
 (2  
2

)

(4  


2

)

2

(1  


2

)

(5)

Note from equations 4 and 5 that asym > sym since the extra term in sym

is negative for 
 > 0. Since asym is the incentive to innovate when a Örmís rival

does not innovate and sym is the incentive to innovate when a Örmís rival innovates,

asym > sym means that innovation can be thought of as strategic substitutes.

The types of equilibria are determined by several parameter regions. If and only

if sym > g, investing is the dominant strategy for both Örms. If asym > g but

sym < g, then there are two pure-strategy asymmetric Nash equilibria. In these

equilibria, investment reduces the cost of only the investing Örmís output.8 Finally if

and only if asym < g, then declining to invest is a dominant strategy for both Örms.

The e§ect of increasing the intensity of competition (increasing 
) is that the equilibrium weakly moves in the direction of less innovation investment when considering

the extent of the symmetric both-invest region, but the equilibrium may move in either

direction when considering the extent of the asymmetric one-invest region.

Proposition 1 sym > 0, and as 
 increases, sym decreases; while, asym >

0, and as 
 increases, asym decreases then increases. Furthermore, sym (
) <

sym (0) = asym (0) < asym (
)

Proof. See Appendix.

8There is also one mixed strategy equilibrium (Örm i and j invest with some probability). In the

mixed strategy equilibrium, investment reduces the cost of all output in the market if both Örms invest,

reduces the cost of only the output of the investing Örm if one Örm invests, and does not reduce cost

if neither Örm invests.

8
Let us Örst focus only on the case where it is the unique pure strategy equilibrium

for both owners to invest. In this case, proposition 1 shows that an ownerís additional

proÖt from having a lower marginal cost declines as 
 (competition) increases and

thus the amount that an owner would invest to achieve this lower marginal cost also

declines as 
 increases. (This is also shown in Ögure 1). This result, variants of which

are noted by Qui (1997) and Piccolo et al. (2008), supports the Schumpeterian view

that market power leads to greater innovation. This result, however, only describes

how the beneÖt from innovating (the additional proÖt from having a lower marginal

cost) changes as competition increases. To understand how a Örmís overall proÖt from

innovating changes as competition increases, we must also consider how the cost of

attaining such innovation changes as competition increases. The next section does this

using this result as a benchmark. As noted earlier, we focus on the case where both

Örms innovate because it represents the region where innovation lowers production cost

for all of the industry and thus the region where consumers beneÖt most from such

innovation.

For example, assuming  = 1, c = 1=4, and x = 1=20, then the return to investing

when the rival invests is shown by the Ögure 1.

9
asym

sym

0

g

Figure 1: Baseline Example: asym and sym for

(; c; x) = 1;

1

4

;

1

20 

As an aside, let us now brieáy consider the parameter region where it is a pure

strategy equilibrium for only one owner to invest. Proposition 1 shows that an ownerís

additional proÖt from having a lower marginal cost Örst decreases and then increases

as competition (
) increases. Combining this result with the result for the case where

both owners invest, and recalling that asym > sym, shows that the form of the

equilibrium can change in a number of ways as 
 increases. SpeciÖcally, depending

on the level of g (the cost of innovating), the relationship between competition and

investment may result in the following: (1) no innovation for any level of competition (g > asym (
)), (2) no innovation followed by innovation by only one Örm

(asym (0) < g < asym (
)), (3) innovation by both Örms, followed by innovation by one Örm, followed by no innovation, followed by innovation by only one Örm

(min asym < g < asym (0)), (4) innovation by both Örms followed by innovation

by only one Örm (sym (
) < g < min asym), (5) innovation by both Örms for all

levels of competition (g < sym (
)). Note that if innovation is costless, g = 0, then

10
it is the dominant strategy for both Örms to invest.

These results arise in large part because of the lumpiness of innovation investment.

This lumpiness gives rise to asymmetric outcomes in a symmetric situation, and this

asymmetry in turn leads to the increase in investment after the initial decrease. If

investment was not lumpy, then as the return to investment falls with increased competition (given that the opponent is also investing) both Örms would reduce investment

leading potentially to a symmetric equilibrium. Since investment is lumpy, as the return to investment decreases one or both of the Örms will stop investment, which creates

the opportunity for an asymmetric outcome where one Örm invests and the other does

not. Investing when the opponent does not has a di§erent relationship to the intensity

of competition than investing when the opponent also invests because investing entails

getting away from the opponent instead of matching the opponent. As the level of

competition increases, the beneÖt of distancing from the opponent eventually increases

because the baseline proÖt of not investing su§ers greatly from the increase in competition. While we believe this last set of results are interesting, we do not use them in

the remainder of the paper.

4 Principal-Agent Model: Managerial Contracts

In this section, we consider the case where the owner and the manager are separate

and thus where the owner must contract with the owner to obtain innovation. Recall

that the owner of Örm i is constrained to use contracts of the form: wi = bii.

Thus, for innovating to be the dominant strategy for manager i, the following incentive

compatibility conditions must hold.

biasym > g (6)

11
bisym > g (7)

Since asym > sym, equation 7 is the binding condition in terms of deÖning the

region of 
 such that innovating is a dominant strategy for managers of both Örms.

DeÖne 




(g; ; c; x) as the implicit function deÖned by: sym (
; ; c; x) = g. DeÖne




 (b; g; ; c; x) such that for all 
 that 0 < 
 < 
 (b; g), the equilibrium of the stage

2 has both managers investing as a dominant strategy. Since the manager receives

only a share of the additional proÖt from innovating, an immediate result is that for

any b, fewer situations induce both Örms to invest than in the model without an agency

problem:

Proposition 2 For any b 2 (0; 1), 


 (b; g) < 


(g).

Proof. From proposition 1, sym > 0,

@sym

@
 < 0, and 0 < b < 1.

@sym

@
 < 0

implies that this implicit function is well-deÖned. If 
 is such that bsym = g, it must

be that sym > g.

A second result is that an owner must pay a manager a larger share of Örm proÖt

in order to obtain innovation as competition increases. However, because the Örmís

overall proÖt decreases as competition increases, the total amount paid to managers

decreases as competition increases. Denote bsym as the minimum b that will induce the

agent to innovate given g and the other Örm investing, i.e., bsym =

g

sym

.

9

Proposition 3 For the case where both rivals invest, when g < sym, the fraction

of proÖts needed to induce the manager to invest increases as competition increases,

@bsym

@
 > 0, while both the payment to the manager and the return to the owner from

innovating decrease, @bsymi(x;x)

@
 < 0 and @(1bsym)i(x;x)@
 < 0.

9

bsym =

g(


24)

2

(1


2

)

x(2
2)(2(c+x)(2
2
)x(2
2))

12
Proof. See Appendix.

Denote basym as the minimum b that will induce the agent to innovate given g and

the other Örm not investing, i.e., basym =

g

asym

.

10 Since asym > sym, it is the

case that bsym > basym.

Since the principal will not pay an excess sum to induce an action, the only pure

strategy equilibria of stage 1 are (bsym; bsym), (basym; 0), (0; basym), and (0; 0) (See Table

2.) In stage 2, the choices of b from the stage 1 determine the equilibrium investments.

If the principals choose (bsym; bsym) then investing is a dominant strategy for both

agents, and thus (x; x) is the unique stage 2 Nash Equilibrium. If the principals

choose (bsym; basym) then investing remains the dominant strategy for the agent of Örm

1. However, since the agent for Örm 2 will only invest when the agent for Örm 1

declines to invest, the unique stage 2 Nash Equilibrium is now (x; 0). Therefore,

an equilibrium is not formed by (bsym; basym) because principal 2 would unilaterally

deviate to (bsym; 0) since 2 (0; x) > (1  basym)2 (0; x), similarly for (basym; bsym). If

the principals choose (bsym; 0) then investing is a dominant strategy for the agent of Örm

1 while not investing is a dominant strategy for the agent of Örm 2. Thus, the unique

stage 2 Nash Equilibrium is (x; 0). However, an equilibrium is not formed by (bsym; 0)

because principal 1 would unilaterally deviate to (basym; 0) since (1  basym)1 (0; x) >

(1  bsym)1 (0; x), similarly for (0; bsym). If the principals choose (basym; basym) then,

there are two pure strategy Nash equilibria: (x; 0) and (0; x). Therefore, an equilibrium

is not formed by (basym; basym) (with a pure strategy equilibrium in stage 2) because the

principal whose agent does not invest in stage 2 would unilaterally deviate from basym

to 0 since i (0; x) > (1  basym)i (0; x): If the principals choose (0; 0) then for both

agents not investing is a dominant strategy, thus the unique Nash equilibrium is (0; 0).

Neither bsym nor basym is ever a dominant strategy in stage 1 because in response to 0

10 basym =

g(4


2

)

2

(1


2

)

x(2
2)((
+2)(2(c+x)(1
)+x
)2x)

13
and bsym (respectively) another strategy may give a higher payo§.

Stage 1 has (bsym; bsym) as an equilibrium if incentive condition 8 holds.

(1  bsym) i (x; x) > i (0; x) (8)

Stage 1 has (basym; 0) or (0; basym) as an equilibrium, if incentive conditions 9 and

10 hold.

(1  basym) i (x; 0) > i (0; 0) (9)

j (0; x) > (1  bsym) j (x; 0) (10)

Stage 1 has (0; 0) as an equilibrium if the negation of 9 holds.

There is a unique (bsym; bsym) equilibrium if condition 8 holds as well as condition 9

and the negation of 10.11

We Önd that for any set of parameter values (; c; x; 
), there is a level of investment

cost, gsym, such that for g < gsym, the unique pure strategy equilibrium is for both

owners to induce their managers to invest. Furthermore, we Önd that gsym, which

deÖnes a parametric threshold beyond which investment decreases, has a non-monotonic

relationship to the intensity of competition (
). Generally as competition increases,

gsym declines at Örst when competition is low, then increases as competition intensiÖes,

and Önally declines as competition further intensiÖes. (See Ögures 2, 3, and 4). We

Önd that the region where increased competition leads to greater innovation increases

in extent as c and x fall but does not depend on . Thus, the e§ect in which greater

11There is a unique (basym; 0) or (0; basym) equilibrium if conditions 9 and 10 hold as well as the

negation of condition 8. There is a unique (0; 0) equilibrium if the negation of condition 9 holds as

well as the negation of condition 8. With the negation of conditions 9 and 8, stage 1 has bi = 0 as a

dominant strategy, and therefore (0; 0) as the unique equilibrium, if condition 10 holds.

14
competition leads to greater innovation is most extensive where cost is relatively low

and where innovation leads to relatively small cost reductions.

Proposition 4 There is gsym (; c; x; 
) such that if g < gsym then the unique (pure

strategy) equilibrium is for both principals to induce the agents to invest by setting

b

 = bsym and if g > gsym then both principals inducing investment is not an equilibrium.

There is an x and a c(x) such that if x < x and c < c(x), then there are 
L (c; x) and


U (c; x) such that @gsym

@
 > 0 if 
L < 
 < 
U ; otherwise @gsym@
  0.

12 Finally, @
L

@ = 0,

@
L

@c > 0,

@
L

@x > 0,

@
U

@ = 0,

@
U

@c < 0, and @
U@x < 0.

Proof. See Appendix.

12For instance at 
 =

1

2

, then @gsym

@
 > 0 if x < 190511 and c < 1 

321

190x.

15
Figure 2: The Investment Increases with Competition Intensity Region:

For each (c; x), the upper bound and lower bound are shown for the region of 
ís for which

as 
 increases, the extent of the investment region increases.

16
The results from proposition 4 can also be visualized through two examples in which

gsym declines slightly over a narrow range when competition is low, increases signiÖcantly over a broad range as competition intensiÖes, and Önally declines sharply as

competition further intensiÖes. In example 1,  = 1; c =

1

3

; x =

1

6

, thus an innovation

would reduce marginal cost by 50 percent. In this example, the highest level of cost

for which a Örm with an agent would invest when its rival also invests (gsym) initially

falls almost imperceptibly, then rises over almost the entire range of the substitution

parameter 
, then falls sharply as competition becomes very intense. In example 2,

 = 1; c =

1

2

; x =

1

5

, thus, an innovation would reduce marginal cost by 40 percent.

In this example, gsym declines modestly when the substitution parameter is low, increases substantially when the substitution parameter has moderate values, and then

falls sharply for very high levels of substitution. As we noted earlier, the e§ect in which

greater competition leads to greater innovation is strongest when cost is low relative

to demand and where innovation leads to relatively small cost reductions. These two

examples in particular show that the the competition e§ect is substantial and extensive

for reasonable parameter values.

0.0 0.2 0.4 0.6 0.8 0.000

0.005

0.010

0.015

0.020

0.025

0.030

gsy m

Figure 3: Example 1: (; c; x) = 1;

1

3

;

1

6



17
0.2 0.4 0.6 0.8 0.024

0.025

0.026

0.027

0.028

0.029

0.030

0.031

gsy m

Figure 4: Example 2: (; c; x) = 1;

1

2

;

1

5



In our model, the relationship between product market competition and innovation

changes dramatically when we add the assumption that Örms can only use proÖt-sharing

contracts to motivate managers to innovate: Absent agency problems, the level of

innovation declines monotonically with greater product market competition; when Örms

must use proÖt-sharing contracts to solve agency problems, the level of innovation

initially declines slightly, then rises over a broad region, and then falls sharply.

The intuition for this result can be seen by combining the equality formed by condition 8 and bsym =

g

sym

to get equation 11, where R =

sym

(x;x)

(the ratio of the

incremental proÖt to the total proÖt).

gsym = sym  R (11)

Equation 11 divides gsym into a factor that represents the beneÖt of inducing innovation, sym, and a factor that represents the inverse of the cost to the owner of

inducing innovation, R. Intuitively, the owner must o§er the agent a su¢ ciently high

share of total proÖts that the agentís share of incremental proÖts exceeds the cost of

innovation. Therefore, the cost of inducing innovation depends on the ratio of the

18
incremental proÖt to the total proÖt because the owner will pay a fraction of the total

but the size of the fraction depends on the size of the increment relative to g.

13

The change of the boundary of the investment region, @gsym

@
 , can then be written as

in equation 12 (since @sym

@
 < 0 and @R@
 > 0 from Proposition 3).14

@gsym

@
 = 









@sym

@










R +









@R

@










sym (12)

The maximum cost where both Örms innovate decreases with competition if and

only if







@sym

@








R is greater than







@R

@








sym. In other words, if the e§ect of the

decrease in the incremental proÖt is greater than the e§ect of the decrease in the cost of

investing multiplier (relative to the levels) then investment decreases with competition.

This can be understood as the decrease in the return to innovation dominating the

decrease in the cost of inducing innovation. The maximum cost where both invest

increases with competition if







@R

@








sym >







@sym

@








R. This can be understood as

the decrease in the cost of inducing innovation dominating the decrease in the return

to innovation. Therefore, the non-monotonic relationship between product market

competition and innovation results from one e§ect being dominant over the other for

some ranges of competitive intensity but not all.

13Formally, the cost can be divided into two parts, a fraction of the incremental proÖt and a fraction

of the baseline proÖt. The fraction of the incremental proÖt must equal g to induce innovation, which

determines bsym. This same fraction of the baseline proÖt also becomes part of the cost. Therefore

for a given level of incremental proÖt, if the baseline proÖt decreases then the cost decreases because

the necessary fraction stayed the same but the total payment decreased. Alternatively, for a given

level of total proÖt, as more of the proÖt is shifted into the increment from the baseline, the necessary

fraction of the total decreases since g stayed the same but sym increased and therefore the cost

decreases (because the total proÖt stayed the same).

14 In proposition 3, we Önd @bsymi(x;x)

@
 < 0, which is equivalent to g

@(1=R)

@
 < 0.

19
5 Generalization for ProÖt Functions

This section considers the generality of our results by deriving conditions for general

(stage 2) proÖt functions under which the incentive e§ects driving our results hold.

First, the cost of innovation, bsym (x; x), can be divided into two parts: (1) the satisfaction of the incentive compatibility constraint: bsymsym and (2) a rent to the

manager: bsym ( (x; x)  sym) = bsym (0; x).

15 The incentive compatibility constraint payment is constant with respect to 
 since bsym (
) sym (
) = g. Only the

rent to the manager, bsym (
)  (0; x) (
), changes with intensity of competition. From

the incentive compatibility constraint, bsym =

g

sym

, so the managerís rent is equal to

g

sym(
)

 (0; x) (
).

Proposition 4 follows from the cost of inducing innovation decreasing with competition due to proÖt sharing contracts, which occurs if and only if Condition 13 holds.16

@ (x; x) =@


 (x; x)

>

@ (0; x) =@


 (0; x)

(13)

Condition 13 requires that an increase in the intensity of competition has a more

negative proportional e§ect on the proÖt of not matching the innovation of the rival

than the proÖt of matching it so that the incremental proÖt decreases less than the

baseline proÖtñincreasing the ratio of the incremental proÖt to the total proÖt. With

constant marginal costs, Condition 13 can be reÖned. Using the envelope theorem, the

cost of innovation decreases with the intensity of competition if and only if Condition

14 holds (abusing some notation).17

15 Inducing innovation when the opponent innovates is worthwhile only if (1  bsym)  (x; x) >

 (0; x), which can be rewritten as sym > bsym (x; x)ñthe return being greater than the cost.

16Since @

g

sym(
)

 (0; x) (
) =@
 = g

((x;x)(0;x))(@(0;x)=@
)(0;x)(@(x;x)=@
@(0;x)=@
)

((x;x)(0;x))2

which simpliÖes to g

(x;x)(@(0;x)=@
)(0;x)(@(x;x)=@
)

((x;x)(0;x))2 , the overall e§ect on cost is negative if and

only if  (x; x) (@ (0; x) =@
) <  (0; x) (@ (x; x) =@
).

17Since from the hypothesis xi and xj are Öxed:

@i=@
 =

 @p

i

@
  Di + (p



i  c + x



i

)

@Di

@p

i



+ (p



i  c + x



i

)



@Di

@p

j

@p

j

@
 +

@Di

@
 . From the FOCís:

20


@Di(x;x)

@p

j

@p

j

(x;x)

@
 +

@Di(x;x)

@
 

=Di (x; x)

>



@Di(0;x)

@p

j

@p

j

(x;0)

@
 +

@Di(0;x)

@
 

=Di (0; x)

(14)

If the cost of innovation is decreasing, then overall innovation increases with competition if either the return to innovation increases with competition or if the cost of

innovation falls more quickly than the return. In a model such as Singh-Vives (1984),

where the return to innovation increases with competition, the cost reducing e§ect of

proÖt sharing contracts simply ampliÖes the increasing return to innovation. In a

Schumpeterian model where the return to innovation falls with competition (such as

the one we consider), the rates of change of the return and the cost must be compared.

The cost decreases more quickly than the return to innovation as 
 increases if and

only if 15 holds at a given g.

g

 (x; x) (@ (0; x) =@
)   (0; x) (@ (x; x) =@
)

( (x; x)   (0; x))2 <



@ (x; x)

@
 

@ (0; x)

@
 

(15)

The extent of the both invest region expands with 
 if there is an increase in the

highest g such that sym  bsym (x; x). The highest such g is gsym deÖned earlier,

and then condition 15 yields condition 16. More stringent than Condition 13, Condition

16 requires that the proportional e§ect on proÖt when both invest be less negative than

a proportional e§ect on the laggard, where the laggard e§ect is less negative than in 13

since it is proportional to the average of the two proÖts.18 Intuitively, this condition

requires that the cost of inducing innovation not only decreases as in Condition 13 but



Di + (p



i  c + x



i

)

@Di

@p

i



= 0, then @i (xi; xj ) =@
 = (p



i  c + x



i

)



@Di

@p

j

@p

j

@
 +

@Di

@
 

. Finally, the

price-cost margins cancel.

18Condition 16 is can be derived alternatively by taking the derivative of gsym directly, yielding

@gsym

@
 =

((x;x)(0;x))(((x;x)+(0;x))@(x;x)=@
2(x;x)@(0;x)=@
)

(x;x)

2 > 0.

21
with a high enough rate.19

@ (x; x) =@


 (x; x)

>

@ (0; x) =@


( (x; x) +  (0; x)) =2

(16)

6 Conclusion

This paper analyzes the e§ect of greater product market competition on innovation

using a model that makes two key assumptions. First, the model assumes that two

symmetric Örms, both of which can innovate, compete as di§erentiated Bertrand competitors and face a linear quadratic demand function. With this assumption, a Örmís

beneÖt from making a non-drastic innovation declines as product market competition

increases. Second, the model assumes that a Örm can only incentivize managers to

innovate by o§ering them a Öxed share of proÖts. With this assumption, a Örmís cost of

investing in a non-drastic innovation declines as product market competition increases.

The overall e§ect of increasing product market competition in this model depends

on whether the reduction-in-beneÖt e§ect or the reduction-in-cost-of-innovating e§ect

dominates. In our paper, we Önd that the reduction-in-beneÖt e§ect dominates where

product market competition is either very low or very high, however, the reduction-incost e§ect dominates for intermediate levels of product market competition. Based on

these results and our general analysis, the argument that Örms with market power are

more innovative is weaker once one accounts for one plausible cost of innovating.

Of course, our Önding that Örms with substantial market power have a high cost of

innovating is premised on the assumption that Örms can only incentivize managers by

o§ering them linear proÖt sharing contracts. Thus, at this point, it is useful to revisit

this assumption. As noted earlier, economic theory suggests that a linear proÖt-sharing

19A version of Condition 14 is available but is much more cumbersome since the price-cost margins

do not cancel.

22
contract may perform well in some circumstances, and empirical work suggests that

the incentive contracts between Örms and top management often have a signiÖcant linear proÖt-sharing component. On the other hand, if innovation is very important, a

monopoly Örm would have an incentive to use some other type of contract because it

does not want to o§er managers a signiÖcant share of pre-existing proÖts to gain a comparatively small incremental proÖt. However, these other types of incentive contracts

are likely to be costly because contracts that target one particular goal can harm a Örm

by diverting attention from other important goals. Hence, irrespective of the incentive

contract a monopoly Örm uses, such a Örm may Önd it costlier to induce innovation

than would a competitive Örm.

Put di§erently, several treatments of agency problems within Örms note that agency

problems are eliminated if the manager can own the Örm. Compared to a monopoly

Örm, it is much less expensive for a competitive Örm to get a manager part of the way

toward this goal because the competitive Örmís pre-existing proÖts are smaller relative

to the incremental proÖts from innovating.

Finally, our results illustrate a more general relationship between agency problems

and competition and may o§er an explanation regarding the relationship of Örm size

and innovation productivity. If an agency problem increases the costliness of inducing

innovation, then an increase in the intensity of competition may increase innovation if

it reduces the size of these frictions. The type of problem we consider is that contracts

are constrained to be particularly coarse, but a similar mechanism may well hold for

less extreme sets of contracts or other frictions that occur from the stochastic nature of

proÖts and multi-task nature of management. In addition, it is thought that small Örms

are important innovators (Gans and Stern, 2003), which is perhaps because they have

an advantage in the production of research since they can more properly incentivize

their research employees. Our paper shows that one reason for this could be that the

23
frictions related to the costliness of incentivizing employees are directly related to the

ratio of the incremental proÖt from an innovation to the total proÖtña friction that

would be smaller for a smaller Örm or an up-and-coming Örm.

24
Firm 1 ProÖt Firm 2 proÖt

(I; I) I;I =



2

(1
)(1c+x)

2

(1+
)(2
)

2 I;I =



2

(1
)(1c+x)

2

(1+
)(2
)

2

(I; D) I;D =



2

(x
+(2



2

)(1c+x))

2

(1


2)(4
2)

2 I;D =



2

(x
(2



2

)(1c))

2

(1


2)(4
2)

2

(D; I) D;I =



2

(x
(2



2

)(1c))

2

(1


2)(4
2)

2 D;I =



2

(x
+(2



2

)(1c+x))

2

(1


2)(4
2)

2

(D; D) D;D =



2

(1
)(1c)

2

(1+
)(2
)

2 D;D =



2

(1
)(1c)

2

(1+
)(2
)

2

Table 1: Stage 2 Payo§s to Investment Decision

b2 = 0 b2 = basym b2 = bsym

b1 = 0 1 (0; 0);

2 (0; 0)

1 (0; x);

(1  basym) 2 (x; 0)

1 (0; x);

(1  bsym) 2 (x; 0)

b1 = basym

(1  basym) 1 (x; 0);

2 (0; x)

(1  basym) 1 (x; 0);

(1  basym) 2 (0; x)

(1  basym) 1 (0; x);

(1  bsym) 2 (x; 0)

b1 = bsym

(1  bsym) 1 (x; 0);

2 (0; x)

(1  bsym) 1 (x; 0);

(1  basym) 2 (0; x)

(1  bsym) 1 (x; x);

(1  bsym) 2 (x; x)

Table 2: Stage 1 Payo§s, Assuming Pure Strategy (Asymmetric) Stage 3 Equilibrium

in which Manager 1 Invests.

7 Appendix

Proof. DeÖne: d = (1  c), f = (2d + x), and h = (d + x).

sym: The expression for sym =



2x(2
2

)

(1


2)(4
2)

(2d  2h
  2h
2) is the product of two factors. The Örst factor is always positive. The second factor has

one root in the range 0 < 
 < 1: 


 =

(1c+x)

p

9(1c)

2+10x(1c)+3x2

22c+x

, such that

if 0 < 
 < 
then sym > 0 and if 


 < 
 < 1 then sym < 0. Since


 < 
 



, if 
 < 
 then sym > 0. The expression for @sym

@
 is the product of

two factors. The Örst factor, 2x2

(4


2)

3

(1


2)

2 , is always negative. The second factor

is (8h  8f 
 + 2h
2 + 8f 
3  7h
4  4f 
5 + 3h
6 + f 
7). This second factor has no

roots in the range in the range 0 < 
 < 1 and is positive in this range. Therefore

@sym

@
 < 0.

asym: Since asym > sym, it is also therefore always positive. The expression asym is continuous at 
 = 0 and 
 = 
. The di§erence, asym (
)asym (0),

is positive being a product of two factors: x2


4(4


2)

2

(1


2)

> 0 and a Örst-degree polynomial in (; c) which is always positive. The expression for @asym

@
 is the product of

two factors. The Örst factor, x(2


2

)

(


24)2

(1


2)

, is always positive. The second factor is:

(8d  8f 
 + 2d
2 + 8f 
3  7d
4  4f 
5 + 3d
6 + f 
7). This polynomial has a single

25
root in (0; 
), 


, such that if 0 < 
 < 
 then @asym

@
 > 0 and if 


 < 
 < 
 then

@asym

@
 < 0 . Also sym(0) = asym(0) = (1=4)

2

(2  2c + x).

Proof. Proposition 3

1. Since @bsym(
)sym(
)

@
 =

@bsym(
)

@
 sym (
) + bsym (
)

@sym(
)

@
 =

@g

@
 = 0 and

@sym(
)

@
 < 0, therefore it must be the case that @bsym(
)@
 > 0.

2. @bsym(
)i(x;x)(
)

@
 = g



i(x;x)i(0;x)

0

(
)i(0;x)(
)i(x;x)

0

(
)

(i(x;x)(
)i(0;x)(
))2



is negative if and only if

i(0;x)

0

i(0;x) <

i(x;x)

0

i(x;x)

since (i (x; x) (
)  i (0; x) (
))26= 0 for 
 < 
. This inequality

holds for 
 < 
.

3. @(1bsym)i(x;x)

@
 =

((i(x;x)(
)i(0;x)(
))2+gi(0;x)(
))i(x;x)

0

(
)gi(x;x)(
)i(0;x)

0

(
)

(i(x;x)(
)i(0;x)(
))2

is negative if:

i(0;x)

0

i(0;x) <

i(x;x)

0

i(x;x)

(which is true from part 2) and if g < g~ where

g~ 

(i(x;x)(
)i(0;x)(
))2i(x;x)0(
)

i(x;x)(
)i(0;x)

0

(
)i(0;x)(
)i(x;x)

0

(
)

, which is true since g >~ sym > g.

Proof. Proposition 4: DeÖne RHS 

(i(x;x)+i(0;x))

2i(x;x)

and LHS 

@i(0;x)

@
 =

@i(x;x)

@


Since for 
 < 
, asym > 0 and sym > 0, there is a g low enough that the

principal can induce innovation in stage 2.

Condition 8 holds if g < gsym where gsym =

(i(x;x)i(0;x))2

i(x;x)

.

Condition 9 holds if g < g0 where g0 =

(i(x;0)i(0;0))2

i(x;0) .

The negation of 10 holds if g < gasym where gasym =

(j (x;0)j (0;x))(i(x;x)i(0;x))

j (x;0) .

Since gsym < g0, gsym < gasym, and gsym < sym, if g < gsym then the unique

equilibrium is both owners inducing both managers to invest; furthermore, it is an

equilibrium for both Örms to invest only if g < gsym.

@gsym

@
 =

i(x;x)i(0;x)

i(x;x)

2



(i (x; x) + i (0; x)) @i(x;x)

@
  2i (x; x)

@i(0;x)

@
 

which is T 0

if: RHS S LHS.

RHS is always positive and always decreasing. LHS is always positive but is

decreasing if and only if (x < 1

2

and c < 1  x and 
 < 
g(c; x) for some 0 < 
g(c; x) <


). As 
 ! 0

+,

@gsym

@
 < 0 since: lim
!0+ RHS = RHSj
=0 =

1

2



1 + (1c)

2

(1c+x)

2



and lim
!0+ LHS = LHSj
=0 =

1c

1c+x

so that (RHS  LHS)


=0 =

x

2

2(1c+x)

2 > 0.

As 
 ! 


,

@gsym

@
 < 0 since: lim
!
 RHS = RHSj
=
 =

1

2

and lim
!
 LHS =

LHSj
=
 = 0. Therefore, @gsym

@
 will begin and end negative but may become positive

if the LHS decreases fast enough.

Using Mathematicaís Cylindrical Algebraic Decomposition Algorithm, we Önd that

there is a c(x) and x < 1=2 such that if and only if c < c(x) and x < x, then there

are 
L and 
U such for 
L < 
 < 
U ,

@gsym

@
 > 0; otherwise @gsym@
 < 0, where 
L, 
U , c,

and x are deÖned as the roots of polynomial equations. Taking the derivatives of 
L

26
and 
U results in further implicitly deÖned functions, which are similarly signed by the

Cylindrical Algebraic Decomposition Algorithm.

References

[1] Arrow, Kenneth, J. 1962. ìEconomic Welfare and the Allocation of Resources for

Invention.îIn The Rate and Direction of Inventive Activity: Economic and Social

Factors (ed. R. Nelson). (NBER) Princeton University Press.

[2] Bester, Helmut, and Emmanuel Petrakis. 1993. "The Incentives for Cost Reduction in a Di§erentiated Industry." International Journal of Industrial Organization

11(4): 519-534.

[3] Brealey, Richard A., and Stewart C. Myers. 1988. Principles of Corporate Finance

(3rd edn.) New York: McGraw Hill Book Company.

[4] Gans, Joshua and Scott Stern. 2003. The Product Market and the Market for

íIdeasí: Commercialization Strategies for Technology Entrepreneurs. Research Policy. 32(2): 333-350.

[5] Hall, Brian, and Je§rey Liebman. 1998. "Are CEOs Really Paid Like Bureaucrats?"

Quarterly Journal of Economics, 113(3): 653-691.

[6] Hall, Brian J., and Kevin J. Murphy. 2003. "The Trouble with Stock Options."

Journal of Economic Perspectives 17(3): 49-70.

[7] Hart, Oliver. 1983. "The Market Mechanism as an Incentive Scheme." Bell Journal

of Economics, 14(2): 366-82.

[8] Holderness, Cli§ord G., Randall S. Kroszner, and Dennis P. Sheehan. 1999. "Were

the Good Old Days That Good? Changes in Managerial Stock Ownership since

the Great Depression." Journal of Finance 54(2): 435-469.

[9] Holmstrom, Bengt, and Paul Milgrom. 1987. "Aggregation and linearity in the

provision of intertemporal incentives." Econometrica 55: 303-28.

[10] Holmstrom, Bengt, and Paul Milgrom. 1991. "Multitask Principal-Agent Analyses: Incentive Contracts, Asset Ownership, and Job Design." Journal of Law,

Economics, and Organization 7: 24-52.

[11] Piccolo, Salvatore, Marcello DíAmato, and Riccardo Martina. 2008. "Product Market Competition and Organizational Slack under ProÖt-Target Contracts." International Journal of Industrial Organization 26(6): 1389-1406.

[12] Prendergast, Canice. 1999. ìThe Provision of Incentives in Firms.î Journal of

Economic Literature, 37:7-63.

27
[13] Qiu, Larry. 1997. "On the Dynamic E¢ ciency of Bertrand and Cournot Equilibria."

Journal of Economic Theory, 75: 213-229.

[14] Raith, Michael. 2003. "Competition, Risk, and Managerial Incentives." American

Economic Review, 93(1):1425-1436.

[15] Scharftstein, David. 1988. "Product-market Competition and Managerial Slack."

RAND Journal of Economics, 19(1): 147-155.

[16] Schmidt, Klaus. 1997. "Managerial Incentives and Product Market Competition."

Review of Economic Studies, 64: 191-213.

[17] Schumpeter, Joseph. 1947. Capitalism, Socialism and Democracy (2nd edn.) London: Allen and Unwin.

[18] Tirole. Jean. 1988. The Theory of Industrial Organization. Cambridge, MA, The

MIT Press.

[19] Vives, Xavier. 2008. "Innovation and Competitive Pressure." Journal of Industrial

Economics 56(3): 419-469.

28