Innovation, Market Structure and the Holdup Problem:

Investment Incentives and Coordination

Abraham L. Wickelgren

Federal Trade Commission*

awickelgren@ftc.gov

Abstract: I analyze the innovation incentives under monopoly and duopoly provision of 

horizontally differentiated products purchased via bilateral negotiations, integrating the market 

structure and innovation literature with the holdup literature. I show that competition can 

improve local incentives for non-contractible investment. Because innovation levels are 

generally strategic substitutes, however, there can be multiple duopoly equilibria. In some 

circumstances, monopoly can provide a coordination device that can lead to greater expected 

welfare despite inferior local innovation incentives. The conditions for this to be the case, 

however, are quite restrictive.

 

*

 I thank Cindy Alexander, Jeremy Bulow, Joe Farrell, Ezra Friedman, Stephen Holland, Dave Meyer, Dan 

O’Brien, Charissa Wellford, and seminar participants at the Bureau of Economics of the Federal Trade 

Commission and the Economic Analysis Group of the Antitrust Division of the Department of Justice for 

comments. This paper does not reflect the views of the Federal Trade Commission or any individual 

Commissioner. All errors are my own.
1

I. Introduction

The question of what type of market structure provides the best innovation incentives 

dates back to Kenneth Arrow (1962). While many papers have analyzed this issue in a variety of 

settings (Arrow, 1962; Partha Dasgupta and Joseph Stiglitz, 1980; Jeremy Bulow, 1982; and 

many others), there has been no work on the effect of market structure on non-contractible 

investment (such as innovation) incentives in markets where trade occurs in bilateral contracts 

rather than in a spot market. In addition, most prior papers dealing with bilateral contracts and 

non-contractible investments have assumed the existence of bilateral monopoly (Oliver 

Williamson, 1985; Jean Tirole, 1986; Sanford Grossman and Oliver Hart, 1986; Oliver Hart and 

John Moore, 1988 among others), bypassing the question of how market structure affects 

investment incentives. Bilateral monopoly, however, is not the only situation where trade occurs 

via individually negotiated contracts rather than in spot markets. When customers are not final 

consumers, but rather firms purchasing inputs, these firms will often negotiate with multiple 

suppliers. 1 Even in cases where there is a bilateral monopoly, that monopoly will be often 

created by a choice between alternative suppliers in a prior period. In fact, this paper grows out 

of analysis of a proposed merger between the two dominant suppliers of accounting software for 

large law firms where the issues analyzed in this paper had direct policy relevance.

In markets where trade is governed by bilateral contracts, the issue of output distortion 

does not arise because trade is negotiated individually.2 Thus, the effect of market structure on 

welfare will only be through its effect on non-contractible investments (as is standard in the 

holdup literature). Moreover, the effect of market structure on product innovation incentives is 

 

1

 Consider the market for various types of business planning software. Companies such as Oracle or 

PeopleSoft offer large, customizable software packages for business that perform a myriad of essential 

tasks such as billing and accounting, human resources management, supply chain management and many 

others. Large companies do not buy these products "off the shelf." Rather, they send out a request for 

proposal to several firms and negotiate the best deal with the company they prefer.

2

 Of course, there can be bargaining failures. In the model below, I assume that bargaining always results 

in the efficient transaction taking place. Even when bargaining failures can occur, however, market 
2

substantially different when there are no set prices. For example, the “replacement effect” and 

the “product inertia effect” that greatly influence innovation incentives in standard models (Shane 

Greenstein and Garey Ramey, 1998 is one such example) are not present in this paper. I show 

that competition does alleviate the holdup problem by providing superior incentives for noncontractible investment. Since only the sellers are making non-contractible investments, if they 

had all the bargaining power, investment incentives would be optimal under duopoly and 

monopoly. Because the split of the surplus is determined by a bargaining game, however, I show 

that a seller only gets the entire marginal surplus from product improvements (though not the 

entire total surplus) when the buyer has a binding outside option (an option that gives the buyer at 

least half the total surplus available from trade with its preferred seller). Otherwise, the seller 

only receives half the marginal surplus from product improvements. Competition alleviates the 

holdup problem, then, because competition increases the options available to buyers, making it 

more likely that a buyer will have a binding outside option when negotiating with its preferred 

supplier. When more buyers have binding outside options, the marginal return (to the seller) 

from product improvement is closer to the social optimum.

Competition does not always increase total welfare, however, because it can create a 

coordination problem. There can be multiple equilibria because the two firm’s innovation levels 

are (in most circumstances) strategic substitutes. This is due to a market share effect; the more 

my rival innovates the larger is her market share and the smaller is mine, reducing my incentive 

to innovate. 

When there are two competing suppliers there is always one duopoly equilibrium that 

generates at least as much welfare as the monopoly outcome. When there are multiple duopoly 

equilibria, however, there will sometimes be one or more equilibria that generate less welfare 

than the monopoly outcome. This is more likely when non-contractible investment (innovation) 

 

structure still won’t affect the degree of output distortion unless the probability of a bargaining failure is 

correlated with market structure.
3

costs are small, but not too small. Since it turns out that more asymmetric duopoly equilibria 

generate more welfare, when multiple equilibria exist, small innovation costs induce the 

monopolist to develop the two products asymmetrically. If innovation costs are too small, 

however, then the only possible duopoly equilibrium will be asymmetric as well. Of course, for 

monopoly to be superior even in these cases, buyers’ value from trading outside the market must 

also not be too small or the inferior development incentives will overwhelm any potential 

coordination advantage. In addition, the best duopoly equilibrium (from a social welfare 

standpoint) must not occur with probability one when there are multiple equilibria.

This is not the first paper to study the relationship between competition and noncontractible investment incentives. Leonardo Felli and Kevin Roberts (2000) have also shown 

that competition can alleviate the holdup problem when only sellers invest. Their model, 

however, assumes that sellers can make take it or leave it offers to buyers. This guarantees that 

sellers get the entire marginal surplus from the transaction (whether there is competition or not). 

Thus, it does not address the central question of this paper: what is the effect of competition on 

ex ante investment incentives when trade is negotiated? Like Felli and Roberts (2000), Harold 

Cole et. al. (2001) model the effect of competition in a model with match specific investments 

(not product innovation investments as in this paper). An additional difference between their 

paper and this one is that their analysis of competition considers the effect of adding closer 

substitutes, whereas in this paper, the number of available products is fixed, but there is more 

competition when the products are under separate ownership. Neither of the above papers 

considers the effect of ownership on the holdup problem. Che and Ian Gale (2000) also show that 

competition in the form of contests can help improve sellers’ incentives to make non-contractible 

cooperative investments. Because they focus on contests, however, their paper also does not 

consider the effect of competition when price is negotiated.
4

The plan of the rest of the paper is as follows. Section II develops the duopoly model, 

while Section III develops the monopoly model. Section IV discusses the welfare comparisons 

between the two. Section V concludes. All proofs are in the appendix.

II. The Duopoly Model

There are two producers, A and B, each producing their own, differentiated, product, at 

zero marginal cost (I will also refer to the products as A and B). Analyzing differentiated 

products is natural in this setting since negotiated trade is far less likely in a market for 

homogenous products. I consider horizontal differentiation, since vertical differentiation does not 

make sense in the context of bilaterally negotiated trade.

Consider a linear city model where product A is located at point zero and B is at point 

one. There is a unit mass of customers who are uniformly distributed over the interval from zero 

to one. Customers only have use for one unit of one of the two products (a business only needs 

one accounting software program or one supply chain management software program, for 

example).

There are two periods in the model. In period 0, products A and B start with an equal 

general value to customers, V. The products are differentiated, however, by their location in 

product space. A customer at point e between 0 and 1 gets a value from product A in period 0 of 

V –ke, where k is the parameter measuring the cost of purchasing a product whose specifications 

are one unit away from one’s optimal specifications. Similarly, this customer’s value from 

product B is V –k(1-e). I assume that each customer’s value of ε is common knowledge to the 

customer and both suppliers.3 During period 0, the suppliers of product A and B choose an 

 

3

 This assumption allows me to use a bargaining solution without uncertainty to determine the price that 

each customer pays for the product and eliminates the possibility of bargaining failures. While this will not 

exactly reflect reality, for many markets it is not that far off. In the business planning software market, for 

example, the customers send out detailed requests for proposals, host demonstrations where they inquire 

about the capabilities that they are most interested in, and ask for specific customization of the software to 

meet their needs. All of this provides the supplier with detailed information about how important different 

capabilities are to that customer, giving it a very good idea of how the customer will value its product 

relative to its competitors.
5

amount of, non-stochastic, product development innovation. If supplier i develops its product by 

an amount di then it increases the value of its product by that amount. This costs the supplier 

2

2

( )

i i

d

c

C d = . Customers do not maker purchase decisions until period 1. At this time, a 

customer that purchases product A receives a value of V+dA-ke, whereas, if it purchases product 

B it receives a value of V+dB-k(1-e).

Thus, I am following Yeon-Koo Che and Donald Hausch (1999) in considering the case

of cooperative investments rather than selfish investments, i.e., investments that benefit one’s 

trading partner rather than oneself. In the markets I have in mind (such as software markets), the 

dominant form of innovation is product improvement innovation, which is fully cooperative. 

Notice that I do not allow contracts in period 0 between customers and firms. There are 

several reasons for this. First, Che and Hausch (1999) prove, for cooperative investments in a 

bilateral setting, that there is no ex ante, renegotiation proof, contract that can improve upon ex 

post negotiation. To the extent this result is different when there are two suppliers, this only 

strengthens the result that competition often alleviates the holdup problem. Second, firms may

not know the identity of all their potential customers in period 0, making ex ante contracts 

infeasible. Third, the type of contract suggested by W. Bentley MacLeod and James M. 

Malcomson (1993) for inducing efficient investments in a similar situation requires ensuring that 

a customer’s outside option always binds. This necessitates payments from seller to buyer if the 

buyer does not trade with the seller. Unless the seller gets the entire surplus from their 

interaction, up front payments from the buyer to the seller will be strictly less than this payment. 

Thus, potential buyers have a strong incentive to misrepresent their desire to purchase this 

product so as to enter into this type of contract that gives them positive surplus even if they do not 

intend to trade. These losses could easily exceed the seller’s share of the added surplus from 

more efficient development incentives. Finally, in many cases, such as the market for accounting 
6

software for law firms that motivated the paper, we do not observe such contracts, thus we need a 

theory to explain the effect of market structure on welfare in these situations. 

Since ε is common knowledge, and I assume that purchase decisions are made by 

bargaining without transactions costs, each customer will choose the product that gives it the 

greatest value. Thus, a customer will choose product A if and only if its *

2

e ≡ e

− +

<

k

d d k A B

.

The price each consumer pays is determined by the outcome of a bargaining game 

between the consumer and the two potential suppliers.4 I posit an alternating offer bargaining 

game of the type used in Patrick Bolton and Michael D. Whinston (1993). The only difference 

between that bargaining game and this one is that in their game there are two buyers and one 

seller rather than two sellers and one buyer. Thus, the equilibrium of this bargaining game is a 

straightforward application of their equilibrium. They show that the equilibrium of the three 

player bargaining game is identical to the equilibrium of an outside option bargaining game 

between the two parties with the largest joint surplus where the party with the alternative trading 

partner has an outside option of trading with its less preferred partner and obtaining the entire 

surplus from that trade. It is well known that the solution to this outside option bargaining game 

(in the current application) gives the buyer the larger of half the surplus from the transaction and 

the surplus it could get from its outside option (getting the other product for free) (Ariel 

Rubenstein, 1982; Avner Shaked and John Sutton, 1984). As a result, a buyer located at e ≤e *

will pay the following price for product A:

(1a) ( ),( ) ( (1 ) )}

2

1

p (e) Min{ V d ke V d ke V d k e A A A B = + − + − − + − − .

Similarly, a buyer located at e > e * will pay the following price for product B:

(1b) ( (1 ) ),( (1 ) ) ( )}

2

1

p (e) Min{ V d k e V d k e V d ke B B B A = + − − + − − − + − .

 

4

 Bargaining over price is common for intermediate goods. It is certainly the dominant method of trade 

when large firms purchase major software systems.
7

Given these prices, A and B in period 0 choose dA and dB respectively to maximize their 

profits. The profit functions are given below, where f is the probability density function for the 

uniform distribution ( f(e) = 1iff e ∈ [ 0,1],0 otherwise ).

(2a) ( ) ( ) ( )

*

0

A A

d C dA = ∫ p f −

e

p e e e

(2b) ( ) ( ) ( )

1

*

B B

d C dB = ∫ p f −

e

p e e e .

Differentiating (2), with these pricing functions, gives the following first order 

conditions:

(3a) )

3

2 2

(

2

1

)

2

(

k

d d V k

F

k

d d k

cd F

A B A B

A

− − +

−

− +

= and

(3b) ))

3

2

(1 (

2

1

)

2

1 (

k

d d V k

F

k

d d k

cd F

A B A B

B

− + +

− −

− +

= − .

Notice that the producer obtains the full marginal increase in value from product 

development from its customers with binding outside options and half the increase in value from 

its customers whose outside options are not binding.

Customers that prefer A have B as their outside option. For those A customers with 

smaller relative preferences for A over B, 

k

d d k

k

d d V k A B A B

3 2

2 − +

< ≤

− − +

e , the outside 

option of B will be binding in their bargaining game with A. These customers will get the surplus 

that they would receive if they bought B for free, and any increase in the value of A from more 

development accrues entirely to the producer of A. However, for those customers with larger 

relative preferences for A, for whom B is not a binding outside option, increases in the value of 

the product from added development are split evenly between the customer and the producer. 

Similarly, customers who prefer product B, but not by too much 

(

k

d d V k

k

d d k A B A B

3

2

2

− − +

< <

− +

e ), have a binding outside option of purchasing product A. 
8

But for those customers with very strong preferences for B over A, purchasing product A is not a 

binding outside option.

While in models of innovation with non-negotiated prices, innovative investments are 

strategic substitutes, this is not always the case here. Typically, innovative investments are 

strategic substitutes because increased innovation by the rival decreases one’s market share, 

reducing one’s incentive to invest. With negotiated prices, however, there is an additional effect. 

When one’s rival innovates this improves the outside option for one’s customers. This increases 

the fraction of one’s customers whose outside option is binding. And for those customers, the 

firm gets all, rather than just half, the added surplus from innovation. Because I assume that 

customers are uniformly distributed between zero and one, the market share effect will always 

dominate (innovative investments will be strategic substitutes) so long as at least some consumers 

buy each product. If, however, no consumers buy one product, then increased investment in that 

product will only affect the outside option for the other firm’s customers, but will not reduce its 

market share, making investment by the non-selling firm a strategic complement for investment 

by the selling firm. If the density of consumers were greater near the middle of the unit interval 

than at the extremes, then investment by the firm selling to a positive (but very small) fraction of 

consumers could be a strategic complement for investment by the firm selling to the vast majority 

of the customers.

Differentiating the marginal benefit functions (the right hand sides of (3a) and (3b)) with 

respect to rival’s development gives the precise conditions for when innovation is a strategic 

substitute or complement. For dA, dB will be a strategic substitute for dA if and only if:

(4a) )

3

2 2

(

3

2

)

2

(

k

d d V k

f

k

d d k

f

A B A B

− − +

>

− +

Similarly, dA will be a strategic substitute for dB if and only if:

(4b) )

3

2

(

3

2

)

2

(

k

d d V k

f

k

d d k

f

A B A B

− + +

>

− +
9

Figure 1 depicts the reaction functions for dA and dB for one particular set of parameter 

values.

Figure 1

0.5 1 1.5 2

dA

0.5

1

1.5

2

dB

V = 2.5, c = .75, k =1

The dashed line is B’s reaction function. The solid line is A’s reaction function.

Notice that there are three possible equilibria in the figure (the reaction functions 

intersect three times). There is a symmetric equilibrium where both products are developed 

equally and two equilibria where only one product is developed and sold. Of course, the number 

of equilibria is dependent on the values of the parameters. If product development is very costly 

(c is large), then there will be a symmetric equilibrium. In Figure 1, however, product 

development is cheap enough that if one firm (say A) is not going to develop its product then B’s 

level of development will be large enough that no customers will buy product A, giving A no 

incentive to develop its product. 

One can also see in Figure 1 that when dB, for example, is very small, it becomes a 

strategic complement for dA. A similar effect occurs for dB when dA is very small (the reaction 

functions have a positive slope in these regions). 
10

Figure 2 depicts the reaction functions when product development is relatively more 

important (its is cheaper and the initial value of the product is smaller). In this case, there five, 

rather than three, equilibria. In addition to the symmetric equilibria and the equilibria at the 

corners, there are two interior asymmetric equilibria.

Figure 2

0.5 1 1.5 2

dA

0.5

1

1.5

2

dB

V = 1,c = .6,k = 1

The interior asymmetric equilibria occur when one product (say B) is developed enough 

more than A that the outside option of purchasing product B is binding for all customers that 

purchase product A. The reverse, however, is not true. Thus, even though A has a smaller 

market share, its development incentive is only slightly smaller than B’s since A receives the 

entire surplus from added development from all its customers. This ensures that its market share 

is large enough to justify its development level. 

Because I assume that customers are uniformly distributed, explicitly solving the duopoly 

first order conditions (3) requires considering six possible cases. There is the fully interior case, 

where 1

3

2

3 2

2 2

0 <

− + +

<

− +

<

− − +

<

k

d d V k

k

d d k

k

d d V k A B A B A B

. In this case both A and B 

have positive sales, and, moreover, both A and B sell to some customers with binding outside 
11

options and to some customers without binding outside options. I call this case (2I) since two 

firms sell and the outside option cutoff points are interior. There is the case where all of one 

firm’s customers have a binding outside option, but the other firm has customers of both types 

and both firms make sales. I call this case (2B); two firms sell and one has all its customers with 

a binding outside option. This case requires that the following condition hold: 

1

3

2

2

0

3

2 2

<

− + +

<

− +

< <

− − +

k

d d V k

k

d d k

k

d d V k A B A B A B

. Notice that this condition 

indicates that it is firm A whose customers all have a binding outside option. Of course, there is 

an analogous equilibrium where it is firm B whose customers have the binding outside option. 

Since these two equilibria are identical, I consider them as one equilibrium type. The third case 

where both firms make positive sales is where all the customers of both firms have binding 

outside options. I call this case (2BB) (for two binding outside options). It requires that: 

k

d d V k

k

d d k

k

d d V k A B A B A B

3

2

1

2

0

3

2 2 − + +

< <

− +

< <

− − +

. Notice, that when both firms 

make sales, it is not possible that either firm will have all its customers have a non-binding 

outside option since the marginal customer is necessarily indifferent between the two goods.

The three remaining cases all involve only one firm making sales. For each case, there is 

an equilibrium where the firm that does not make sales is A and where that firm is B. I will 

describe the case where A makes no sales. When A makes no sales, its value can be such that for 

some customers purchasing A for free gives them a binding outside option while for some 

customers it does not. I call this case (1I) (one firm sells, the outside option cutoff is interior). 

This requires that: 1

3

2

0

2

<

− + +

< <

− +

k

d d V k

k

d d k A B A B

. Alternatively, the value of A could 

be such that all consumers have a binding outside option: 

k

d d V k

k

d d k A B A B

3

2

0 1

2

− + +

< < <

− +

. This is case (1B). There is also case (1NB), where A 
12

is not a binding outside option for any customers. This requires that: 

0 1

3

2

2

< <

− + +

<

− +

k

d d V k

k

d d k A B A B

.

While there are six distinct types of equilibria that can obtain, only some of subset of 

these equilibria will be feasible for any given set of parameter values. To determine when any of 

these equilibria are feasible, one must solve the first order conditions for dA and dB under the 

assumption that a given equilibrium obtains and then determine for what parameter values the 

development levels are consistent with the conditions for that equilibrium.

For example, if equilibrium (2I) exists then the first order conditions are:

(5a)

k

d d V k

k

d d k

cd A B A B

A

3

2 2

2

1

2

− − +

−

− +

= and

(5b) ))

3

2

(1 (

2

1

)

2

1 (

k

d d V k

k

d d k

cd A B A B

B

− + +

− −

− +

= − .

Solving these two equations for equilibrium development levels gives:

(6)

6 1

2 2

−

+

= =

ck

V k

d d

I

B

I

A

.

It is easy to see that the second order conditions in this case require that ck>1/3, and that these 

development levels are consistent with the conditions for (2I) if and only if 

c

ck V

c

ck

2

4 1

4

2 1 −

≤ ≤

−

. If V is larger than this upper bound then every customer’s outside option 

binds, while if V is too small the customer at 1⁄2 will not get positive value from either product 

(when development levels are given by (6)).

Table 1 gives the development levels and the conditions for each of the six possible 

equilibria. The entries in the table are obtained exactly as they were obtained for (2I) in the 

above paragraph.
13

Table 1

Equilibria Development Levels Conditions on ck Conditions on V

(2I)

6 1

2 2

−

+

= =

ck

V k

d d

I

B

I

A

3

1

ck ≥

c

ck V

c

ck

2

4 1

4

2 1 −

≤ ≤

−

(2B)

2

2

2

2

1 10 12( )

(1 2 ) 2 (1 )

;

1 10 12( )

3 (1 2 )

ck ck

V ck k ck d

ck ck

V k ck d

B

B

B

A

− + −

− + −

=

− + −

+ −

=

12

5 13

2

1 +

≤ck ≤ or

1

12

5 13

< ≤

+

ck

c

ck k ck V

2

4 1

3 (2 1)

−

− ≤ ≤ or

3 (2 1)

2

4 1

≤ ≤ −

−

V k ck

c

ck

(2BB)

c

d d

BB

B

BB

A

2

2 2 1

= =

2

1

ck ≥

c

ck V

2

4 − 1

≥

(1I)

0

1

=

I

A

d ; 

6 1

1 4

+

+

=

ck

V k

d

I

B

2

1

0 ≤ck ≤ or

1

2

1

< ck ≤

c

ck V

c

ck 2 1

2

1 2 +

≤ ≤

−

or

c

ck k ck V2 1

3 (2 1)

+

− ≤ ≤

(1B)

0

1

=

B

A

d ; 

c

d

B

B

1 1

=

0 ≤ck ≤1

c

ck V2 + 1

≥

(1NB)

0

1

=

I

A

d ; 

c

d

NB

B

2

1 1

=

2

1

0 ≤ck ≤

c

ck V

2

1− 2

≤

Note that for equilibria (2B) and (1I) there are two different conditions for ck and for V. 

The first condition for V are applicable only when the first condition for ck holds and similarly for 

the second conditions. Thus, in (2B), for example, the equilibria will exist if and only if 

12

5 13

2

1 +

≤ck ≤ and 

c

ck k ck V

2

4 1

3 (2 1)

−

− ≤ ≤ or 1

12

5 13

< ≤

+

ck and 

3 (2 1)

2

4 1

≤ ≤ −

−

V k ck

c

ck . Also, note that the lower bounds on ck in the first three equilibria 

ensure that the second order conditions are satisfied. The second order conditions for the last 

three equilibria hold for any ck>0.
14

Inspection of Table 1 reveals the above figures do not represent special cases; for many 

parameter values there are multiple equilibria. In Section IV, I will discuss how the equilibria 

compare in terms of social welfare.

III. The Monopoly Model

Because trade is negotiated individually under perfect information, market structure does 

not affect the purchase decisions of any customers (though, it will affect the price). When one 

firm owns both A and B, it doesn’t allow them to compete against each other for any customer, 

eliminating this outside option for every customer. I assume that there exists an inferior third 

product that provides an exogenously given value v to all customers regardless of location. This 

product would not affect the equilibrium in the duopoly model because I assume that v ≤V − k : 

even if A or B did not develop their product at all one of these products would be at least as good 

an outside option for all customers as the third product. Now that customers are only offered A 

or B, but not both, however, this third product serves as every customer’s outside option. As a 

result, a customer at e ≤e * now pays the following price for product A:

(7a) ( ),( ) }

2

1

p ( ) Min{ V d k V d k v A A Ae = + − e + − e − .

Similarly, a buyer located at e > e * will pay the following price for product B:

(7b) ( (1 ) ),( (1 ) ) }

2

1

p ( ) Min{ V d k V d k v B B Be = + − − e + − − e − .

With these pricing functions, the monopolist’s profit function is the following:

(8) ( ) ( ) ( ) ( ) ( ) ( )

1

*

*

0

M A A B

d C dB = ∫ p f d − C d + ∫ p f −

e

e

p e e e e e e

Differentiating (8) gives the following first order conditions in the monopoly case:

(9a) )

2

(

2

1

)

2

(

k

d v V

F

k

d d k

cd F

A B A

A

− +

−

− +

= and
15

(9b) )) 2(1 (

2

1

)

2

1 (

k

d k v V

F

k

d d k

cd F

A B B

B

− − +

− −

− +

= − .

Because v ≤V − k , the right hand sides of (9) are (at least weakly) smaller than the right 

hand sides of (3), the first order conditions in the duopoly case. The reason the marginal benefit 

to innovating is smaller with monopoly than duopoly is that, since the value of every consumer’s 

outside option smaller, a smaller fraction of consumers will have a binding outside option. Thus, 

the seller will more often split the gross social surplus from the transaction rather than get the 

residual surplus over the consumer’s outside option. So, even though the total price the producer 

receives for the product is (weakly) greater for every consumer, the marginal effect of innovation 

on price is (weakly) smaller with monopoly than with duopoly.

Since there can be multiple solutions to the monopolist’s first order conditions, (9), to 

accurately compare the monopoly case to the duopoly one, I must determine which solution 

maximizes profits. To do so, I first explicitly solve the first order conditions to determine the 

candidate profit maximizing development levels for any given set of parameter values. This is 

done in the same way as I determined the possible solutions to the duopoly first order conditions 

above. Of course, since every customer’s outside option is now to purchase the third product and 

obtain surplus v, the conditions for when each possible solution is valid are different. (I give each 

of these possible solutions the same abbreviated label as in the duopoly case except that I add the 

letter M in front in the monopoly case.) In addition, there are also two other possible solutions 

when both products are sold: the outside option can now be not binding for all customers of 

either one or both products. I call these two cases (M-2NB) and (M-2NBNB) respectively. 

(There is also the possibility of a case where one outside option is always binding and the other is 

not, however, one can show that there is no solution to the monopolist’s first order conditions that 

is consistent with this case. It also turns out there is no solution in case (M-2B) either.)
16

For both products to have positive sales and an interior outside option cutoff, (M-2I) 

requires that 1

2

2

2

0 <

− − +

<

− +

<

− +

<

k

d v k V

k

d d k

k

dAv V A B B

. Notice that this is identical 

to the duopoly conditions for (2I) except that 

k

dA− 2v + V

replaces 

k

d d V k A B

3

− 2 − + 2

and 

k

dB− k − 2v + V

 replaces 

k

d d V k A B

3

2 − + +

. The conditions for the other cases are also identical 

to the duopoly conditions with the same replacements. The conditions for the two new cases are:

(10) 1

2

2

and 2

2

0 <

− − +

<

− + − +

<

− +

<

k

d v k V

k

d d k

k

d v V

k

d d k A B A A B B

 (M-2NB) and 

(11) ,1

2

2

2

0,

k

d v V

k

d d k

k

dBv k V A B A− +

<

− +

<

− − +

 (M-2NBNB).

After obtaining the solutions to the monopolist’s first order conditions (and the conditions 

for when those solutions are valid) for every case, I compare the profit among solutions that exist 

for overlapping parameter values. For example, if equilibrium (M-2I) exists then the first order 

conditions are:

(12a)

k

d v V

k

d d k

cd A B A

A

− +

−

− +

=

2

2

1

2

 and

(12b) )) 2

(1 (

2

1

)

2

1 (

k

d d V k

k

d d k

cd A B A B

B

− + +

− −

− +

= − .

Solving these two equations for equilibrium development levels gives:

(13)

2 1

2 2 2

+

+ −

= =

− −

ck

v k V

d d

M I

B

M I

A

.

It is easy to show that these development levels are consistent with the conditions for (M-2I) if 

and only if 

c

V v

c

ck

2

1

2

4

2 1

≥ − ≥ −

−

. If V-2v is larger than this upper bound then the outside 

options are never binding, while if V-2v is too small they are always binding when development 

levels are given by (13).

When equilibrium (M-1I) exists, on the other hand, the first order conditions are:
17

(14a) = 0 Acd and

(14b) )) 2(1 (

2

1

1

k

d d V k

cd A B

B

− + +

= − − .

Solving these two equations for equilibrium development levels gives:

(15)

2 1

2 2

0;

1 1

+

+ −

= =

− −

ck

v k V

d d

M I

B

M I

A

.

One can show that these development levels are consistent with the conditions for (M-1I) if and 

only if 

c

V v

c

ck 1

2

2

2 1

≥ − ≥ −

−

 and ck<1/2 or 

c

k ck V v

1

(1− 2 ) ≥ − 2 ≥ − and 1/ 2 ≤ck ≤1 . If V2v is larger than this upper bound then the outside options are never binding, while if V-2v is too 

small they are always binding.

In comparing (M-2I) and (M-1I), one can see that both solutions are valid when 

c

V v

c

ck

2

1

2

2

2 1

≥ − ≥ −

−

 and ck<1/2 or when 

c

V v

c

ck

2

1

2

4

2 1

≥ − ≥ −

−

 and 1/ 2 ≤ck ≤1 . Since 

the monopolist will never choose the one that yields lower profit, I must compare the profit in the 

two solutions under these conditions. Using the monopolist’s profit function, (8), I can write the 

profit difference between (M-1I) and (M-2I) as follows:

(16)

4(1 2 )

(1 2 ) 2 ( 2 )

2

ck

k ck c V v

+

− + −

When ck<1/2, this is obviously negative, so the monopolist will never choose (M-2I). If 

1/ 2 ≤ck ≤1 , then the monopolist will only choose (M-2I) if:

(17) ]

2

(2 1)

,

2

(2 1)

2 [

c

k ck

c

k ck V v− −

− ∈ − .
18

 Since (1 2 )

2

(2 1)

k ck

c

k ck

> −

−

 whenever ck<1, (M-2I) will be more profitable than (M-1I) if 

and only if 

c

k ck V v

2

(2 1)

2

−

− > − . This condition only limits the possible parameters for when 

(M-2I) or (M-1I) can be chosen if 

4

1+ 5

ck < , since otherwise 

c

k ck

c 2

(2 1)

2

1 −

− > − .

By conducting comparisons of this type between all possible solutions of the 

monopolist’s first order conditions, one can determine the profit maximizing development levels 

for all possible parameter values. The results are summarized in Table 2. (I do not need to worry 

about second order conditions in Table 2 since the development levels have been compared to all 

other possible local maxima.)
19

Table 2

Case Development Levels Conditions on ck Conditions on V

(2I)

2 1

2

2 2

+

+ −

=

=

− −

ck

v k V

d d

M I

B

M I

A

4

1 5

2

1 +

≤ck < or

≤ck +

4

1 5

c

ck V v

c

k ck

4

2 1

2

2

(2 1) −

≤ − ≤

−

− or

c

ck V v

c 4

2 1

2

2

1 −

− ≤ − ≤

(2BB)

c

d

c

d

M BB

B

M BB

A

2

1

2

1

2

2

=

=

−

−

1

4

1 5

≤ <

+

ck or

ck ≥ 1

c

V v

c

ck ck

2

1

2

2

2 2(1 2( )

2

≤ − ≤−

− + + −

 or 

c

V v

2

1

− 2 ≤−

(2NB) DNE DNE DNE

(2NBNB)

c

d

c

d

M NBNB

B

M NBNB

A

4

1

4

1

2

2

=

=

−

−

ck>1/2

V v

c

ck 2

4

2 1

≤ −

−

(1I) 0

1

=

M − I

A

d ; 

2 1

1 2 2

+

+ −

=

−

ck

k v V

d

M I

B

2

1

0 ≤ck ≤ or

4

1 5

2

1 +

≤ck < or

1

4

1 5

≤ <

+

ck

c

ck V v

c 2

2 1

2

1 −

− ≤ − ≤ or

c

ck ck

V v

c 2

(2 1)

2

1 −

− ≤ − ≤− or

c

ck ck

V v

c 2

2 2(1 2( ) )

2

1

2

− + + −

− ≤ − ≤

(1B) 0

1

=

M − B

A

d ; 

c

d

M B

B

1 1

=

−

ck ≤1

c

V v

1

− 2 < −

(1NB) 0

1

=

M − NB

A

d ;

c

d

M NB

B

2

1 1

=

−

ck ≤1/ 2

V v

c

ck 2

2

2 1

< −

−
20

IV. Welfare Comparisons

I define social welfare as total surplus, the total value purchased by customers less the 

costs of innovation. Given that the social planner will want to assign each customer to the 

product it values most, the social welfare function can be written as follows:

(18) ( ) ( ) ( (1 ) ) ( ) ( ) ( )

*

0

1

*

A B d C dA C dB SW = ∫ V + d − k f d + ∫ V + d − − k f − −

e

e

e e e e e e .

The first order conditions for social welfare maximization are the following:

(19a) )

2

(

k

d d k

cd F

A B

A

− +

= and

(19b) ))

2

(1 (

k

d d k

cd F

A B

B

− +

= − .

By comparing these first order conditions with those of the duopoly and monopoly 

models ((3) and (9) respectively) one can see that the social planner wants larger local innovation 

incentives than exist in either duopoly or monopoly, but that local innovation incentives are

closer to the social optimum under duopoly than monopoly.

Of course, socially inferior local innovation incentives do not necessarily imply socially 

inferior innovation levels. To do precise welfare comparisons, one must compare all of the 

possible equilibria in the duopoly model for various parameter values to the outcome for those 

parameter values in the monopoly case. When there are multiple equilibria with duopoly, welfare 

comparisons may also depend on the probability of any of particular duopoly equilibria obtaining. 

This is particularly true because, as the first proposition demonstrates, welfare varies among 

duopoly equilibria in a particularly straightforward fashion.

Proposition 1. Whenever there are multiple duopoly equilibria, social welfare is always largest 

when development levels are most asymmetric.
21

The advantage of asymmetric equilibria is that they have less duplicative investment and 

more customers are taking advantage of the product that invests more. The disadvantage is that 

customers are, on the average, purchasing products that are farther away from their ideal location 

in product space. It turns out, however, that when this effect dominates for a particular 

equilibrium, more asymmetric equilibria do not exist.

Because, when multiple equilibria are possible, more asymmetric equilibria generate 

more social welfare than the less asymmetric ones, the level of social welfare in a duopoly is 

uncertain. It will depend on the probability that any given equilibria, among those that are 

possible, will obtain. When one firm owns both products, however, this uncertainty is absent. 

While there still will be several different solutions to the monopolist’s first order conditions, (9), 

Table 2 identifies exactly which solution maximizes profits for any particular set of parameter 

values. If the relative profit levels are closely enough aligned with the relative welfare levels 

among these solutions, then having a monopoly could provide a socially beneficial coordination 

device. Of course, whether or not the (potential) coordination benefits are ever sufficient to 

overcome the inferior local development incentives in monopoly will depend on just how inferior 

the development incentives are. This will depend on v, the value of the third product that 

provides the outside option for buyers once A and B are under common ownership. The larger v

is the larger the fraction of customers that will have a binding outside option for any given set of 

parameter values, thus reducing the difference in product development incentives between 

monopoly and duopoly. The second proposition compares the total welfare in duopoly and 

monopoly market structures for the largest v the model allows (the best case for monopoly).

Proposition 2. Assume v=V-k.
22

(a) If 

2

1

3

1

≤ck < and 

c

ck V

2

1− 2

≤ or 1

2

1

≤ck < and V

c

ck ≤2 + 1

, then expected 

welfare is strictly greater with monopoly whenever there is a positive probability 

of all feasible duopoly equilibria occurring.

(b) If 

2

1

3

1

≤ck < and 

c

ck V

c

ck

2

4 1

2

1 2 +

≤ ≤

−

or

4

1 5

2

1 +

≤ck < and 

c

ck k V

c

ck ck 2 1

2

2

(2 1) +

+ ≤ <

−

 or 1

4

1 5

≤ <

+

ck and 

c

ck k V

c

ck ck 2 1

2

2

2 2(1 2( ) )

2

+

+ ≤ <

− + −

, then there exists some probability 

distribution for what duopoly equilibria will obtain over feasible duopoly 

equilibria where monopoly generates greater expected welfare than duopoly.

(c) For any other parameter values, duopoly generates as much or more welfare 

than monopoly for any probability distribution over feasible duopoly equilibria.

What Proposition 2 says is that while it is possible for the coordination benefits of 

monopoly to outweigh the superior local innovation incentives of duopoly, in most cases welfare 

will be greater with duopoly. For welfare to be greater with monopoly, there must be multiple 

duopoly equilibria. Whenever there is just one duopoly equilibrium, the monopolist will 

distribute its product development effort across the two products in the same way as occurs in the 

duopoly equilibrium. Since there is more development in duopoly (and there is too little 

development in both cases), duopoly is necessarily superior. 

When there is at least one duopoly equilibrium where both products are developed and 

one where only one product is developed, however, then sometimes, but not always, the 

monopolist will only develop one product. This is when there is a coordination advantage in 

monopoly. Welfare is almost always greater when a monopolist develops only one product than 

in a duopoly equilibrium where both products are developed (the exception is that welfare is 
23

greater in (2B) than in (M-1I)). Because I assume a finite distribution of consumer types, so there 

are cases where local development incentives are identical in monopoly and duopoly, it is 

possible for the monopoly outcome to be identical to the duopoly equilibrium where only one 

product is developed. When this happens, this is part (a) of the proposition, duopoly can never 

generate more welfare than monopoly. If the monopoly outcome where only one product is 

developed has some, but not all, consumers with binding outside options, however, then there will 

always be a duopoly equilibrium where only one product is developed that is superior to the 

monopoly outcome (because local development incentives differ). Nonetheless, if there are other 

possible duopoly equilibria where both products are developed, part (b) says that monopoly could 

still generate more welfare in expectation provided the probability of these equilibria occurring is 

large enough.

The situations where monopoly can be superior occur when development costs are not 

too large and the products are not too differentiated (ck small) so that a monopolist will want to 

focus all its product development effort on one product. On the other hand, if ck is too small, 

then there will also be no duopoly equilibrium where both products are developed. The 

conditions on the products’ initial value, V, are important because they determine whether the 

point at which outside options are binding or not is interior in each case. (This is true in the 

monopoly case because this proposition examines the case where the value of the third product, v, 

is V-k, the largest it can be and still not affect anyone’s decisions in the duopoly case.) Thus, 

when ck is between a third and a half, V must be small for a duopoly equilibrium where both 

products are sold to exist. For larger ck, a monopolist will choose to develop only one product 

only if enough consumers have a binding outside option, i.e., only if the value of the third product 

is high enough. Since this is bounded by the original value of the dominant products, this value 

must be large also. If I were not explicitly assuming that the value of the third product were at its 

maximum, V-k, there would be an added minimum on its value, v. Thus, the smaller the value of 

the third product the less likely monopoly is to generate more welfare than duopoly.
24

It is important to note that the existence of cases where monopoly can never be worse 

than duopoly is dependent on the assumption of a finite distribution of consumer types. If the 

support of consumer types extended across the real line, then there would never be any regions 

where local development incentives were identical in monopoly and duopoly. If this distribution 

had very small density in the tails, however, there would be cases where the local innovation 

incentives were almost identical. Thus, the probability that duopoly would result in both products 

being developed would not have be much greater than zero for monopoly to generate more 

expected welfare than duopoly. So, while the exact result in part (a) is sensitive to the finite 

support assumption, a very similar result would obtain for most plausible distributions with 

infinite support.

V. Conclusion

Analyzing the welfare effects of market structure in markets where bilateral contracting is 

the norm is very different from analyzing these effects in markets where spot market transactions 

are predominate. With efficient bilateral contracting, there will be no ex post allocation 

inefficiency; market structure will only affect welfare when it affects the hold up problem. 

Because of the holdup problem, firms will have insufficient incentives to improve the quality of 

their products. I show that competition among suppliers, however, will reduce this problem. 

Competition, however, brings with it the problem of equilibrium selection. I show that in 

a duopoly there is the potential for multiple equilibria in development levels. When that is the 

case, the more asymmetric is the development of the two products the greater is welfare. When 

one firm owns both products, however, then there is no such problem; the monopolist chooses the 

outcome that maximizes its profits. If the duopoly market sometimes arrives at an equilibrium 

where both products are developed, even when there is also an equilibrium where only one 

product is developed, then it is possible that, for some parameter values, the coordination benefits 

of monopoly can generate greater welfare than duopoly, despite the inferior local development 

incentives. As Proposition 2 makes clear, however, there are many more parameter values for 
25

which the benefit of added competition in increasing the value of the customers’ outside option 

will result in greater total welfare. Moreover, Proposition 2 considers the case where the value of 

the third option is as large as it can be and still have the original market be a duopoly. When the 

third option is of lower value, then the set of parameter values where monopoly is superior will be 

even more restrictive.

Of course, the paper assumes that a firm’s ability to develop its product(s) (the 

development cost function) is independent of market structure. If innovation synergies occur 

when one firm owns both products, then this makes monopoly more advantageous than the model 

in this paper predicts. In fact, because (efficiently) negotiated trade eliminates the effect of 

market structure on allocation efficiency, it will be much easier for synergies to result in greater 

welfare with less competition than in markets governed by fixed price transactions. 

Clearly, analysis of the holdup problem should not be restricted to bilateral monopoly. I 

extend the holdup model by showing how market structure impacts non-contractible investment 

incentives in the presence of holdup problems. Of course, this paper is just a first step in this 

direction. There is a great deal of room for further research. I assume that the buyer’s valuation 

from the product was independent of the value other buyers received from the product they 

bought. If there were downstream competition among the buyers, this might not be the case. 

Analyzing how this would affect the results of the paper would be quite interesting.

Even more interesting would be to relax the assumption that ε, the relative preference 

parameter, is common knowledge. While the suppliers often know much more about their 

products than customers, and often learn a great deal about how the customer will use the product 

in the sales process, the customer will always know more about its valuation for each product 

than the suppliers. Because I assumed away this type of information asymmetry, there were no 

bargaining failures in the model. Since bargaining failures reduce welfare, the effect of market 

structure on the incidence of bargaining failures will be important. The direction of this effect is 

unclear. On the one hand, under monopoly the disagreement option is worse for the buyer, so it 
26

has a greater incentive to avoid a bargaining failure. On the other hand, the consequences of 

bargaining failure in this situation are more severe.

Bargaining failures could also impact the results to the extent they affect innovation 

incentives. The social planner will always weigh bargaining failures more severely than the 

monopolist since the monopolist only bears part of the cost of a failure. Asymmetric information 

could also affect innovation incentives if it creates a business stealing incentive in duopoly. This 

is unlikely to occur, however, since it can only happen if an indifferent customer must pay a 

positive price for either product. Since the most likely form of bargaining failure from 

asymmetric information is delay, there should be no business stealing incentive even with 

asymmetric information. That said, analyzing more precisely the effect of asymmetric 

information would be quite interesting.
27

APPENDIX

Proof of Proposition 1. To prove this Proposition, I compare welfare in each equilibria to 

welfare in every other equilibria that can exist for the same set of parameter values.

Lemma 1.1. Welfare is greater in (2B) than in (2I) when both exist.

Proof. Evaluating the social welfare function, (18), for each equilibria and subtracting the 

welfare in (2I) from the welfare in (2B) gives:

(A1) 2 2 2

2 3 2

4(1 ) (1 10 12( ) )

( 5 50 108( ) 72( ) )(1 4 2 )

ck ck ck

k ck ck ck ck cV

− − +

− + − + − +

.

This has the sign of ( 5 50 108( ) 72( ) )

2 3

− + ck − ck + ck . Since it is only possible to have both 

equilibria (2I) and (2B) when 

12

5 13

2

1 +

≤ck ≤ , I only need to determine the sign of this 

expression within this region. The expression is convex in ck in this region, so the minimum 

value will be at the value of ck where the first derivative of ( 5 50 108( ) 72( ) )

2 3

− + ck − ck + ck is 

zero. Taking this derivative, setting it equal to zero and solving for ck gives 

18

9 + 6

ck = . At 

this value for ck, ( 5 50 108( ) 72( ) ) 2(27 2 6) 27 0

2 3

− + ck − ck + ck = − > . Thus, (A1) is always 

positive, proving the lemma.

Lemma 1.2. When both exist, welfare is always greater in (1I) than (2I).

Proof. Evaluating the social welfare function, (18), for each equilibria and subtracting the 

welfare in (2I) from the welfare in (1I) gives:

 (A2) 2 2

2 3 4

2 2

4(1 36( ) )

72( ) (3 4 ) 1584( ) 1296( ) } 2 (4 )

{ 19 8 72( ) 4 (25 18( ) )

ck

ck cV ck ck V cV

k cV cV ck cV

−

+ + − + + +

− − + − + −

.

The denominator is positive. The numerator is convex in V, so its minimum is again at the value 

of V that solves its first order condition. This is 

(1 36( ) )

2( 1 36( ) )

2

3

c ck ck

ck ck

V

+ +

− + +

= . At this V, the 
28

numerator is 

(1 36( ) )

(1 36( ) )( 8 35 36( ) )

2

2 3

c ck ck

ck ck ck

+ +

− − + −

. This has the sign of 3− 8 + 35ck − 36(ck) . 

Since these two equilibria can both exist only when

12

5 13

3

1 +

≤ck ≤ , and 3− 8 + 35ck − 36(ck) is 

always positive in this interval, the numerator of (A2) is always positive, proving the lemma.

Lemma 1.3. Welfare is greater in (1NB) than in (2I) when both exist.

Proof. Evaluating the social welfare function, (18), for each equilibria and subtracting the 

welfare in (2I) from the welfare in (1NB) gives:

(A3) 2 2 2

2 3

8 (1 6 ) (1 10 12( ) )

3 30 92( ) 72( ) 8 (1 4 )

c ck ck ck

ck ck ck cV ck cV

− − +

− + − + − +

.

This has the sign of the numerator, which is decreasing in V whenever (2I) exists. So, the 

smallest the numerator can be is when 

c

ck V

2

4 − 1

= . At this V, the numerator is 

(1 6 ) (1 2 )

2

− ck − ck , which is positive since (1NB) requires that ck<1/2. Thus, the numerator is 

always positive, proving the lemma.

Lemma 1.4. Welfare is greater in (2B) than in (2BB) when both exist.

Proof. Evaluating the social welfare function, (18), for each equilibria and subtracting the 

welfare in (2BB) from the welfare in (2B) gives:

(A4)

2 2

2 2

4 (1 10 12( ) )

(1 3 2( ) )(1 4 )

c ck ck

ck ck ck cV

− +

− − + − +

.

This has the sign of (1 3 2( ) )

2

− − ck + ck , which is always negative when 1

2

1

< ck ≤ , as must be 

true when (1B) exists.

Lemma 1.5. Welfare is greater in (1I) than in (2B) when both exist.

Proof. Evaluating the social welfare function, (18), for each equilibria and subtracting the 

welfare in (2B) from the welfare in (1I) gives:
29

(A5) 2 2 2

2 3

2 3 4 5

2(1 6 ) (1 10 12( ) )

8 (1 38 120( ) 72( ) )}

( 3 (1 2 )){4 29 8( ) 420( ) 792( ) 432( )

ck ck ck

cV ck ck ck

V k ck ck ck ck ck ck

+ − +

+ + − +

+ − − − + − +

This has the sign of the term in curly braces (when both (1I) and (2B) exist the first term in the 

numerator is positive). The term in curly braces is decreasing in V since 

1 38 120( ) 72( ) 0

2 3

+ ck − ck + ck < whenever 

12

5 13

2

1 +

≤ck ≤ (which must be rue when both 

(1I) and (2B) exist). When (2B) exists and 

12

5 13

2

1 +

≤ck ≤ , we know that 

c

ck V

2

4 − 1

≤ . Thus, 

the curly braces term cannot be any smaller than it is at 

c

ck V

2

4 − 1

= , where it is:

(A6) (1 10 12( ) )(7 22 48( ) 72( ) )/ 2

2 2 3

− ck + ck − ck − ck + ck

The first term is negative when 

12

5 13

2

1 +

≤ck ≤ and it is easy to verify that the second is also 

negative in this region. Thus, (A5) is always positive, proving the lemma.

Lemma 1.6. Welfare is always greater in (1I) than (2BB) when both exist.

Proof. Evaluating the social welfare function, (18), for each equilibria and subtracting the 

welfare in (2BB) from the welfare in (1I) gives:

(A7) 2

2

8 (1 6 )

(2 1)( 1 18( ) ) 2 (2 4 )

c ck

ck ck ck cV ck cV

+

− − + + + + −

.

This is positive since 1

2

1

≤ck ≤ and 

c

ck V2 + 1

≤ when these two equilibria both exist.

Lemma 1.7. Welfare is always greater in (1B) than (2BB) when both exist.

Proof. Evaluating the social welfare function, (18), for each equilibria and subtracting the 

welfare in (2BB) from the welfare in (1B) gives (1 − ck) 4c > 0 when (1B) exists.

Table 1 shows that these seven situations exhaust all the possibilities for multiple equilibria in the 

duopoly model. Q.E.D.
30

Proof of Proposition 2. By combining the information in Tables 1 and 2, and setting v=V-k, one 

can get the following list of feasible duopoly outcomes and the monopoly outcome for various 

parameter values. The list is divided into eight cases based on the value of ck. The feasible 

duopoly equilibria are given immediately after the parameter restriction and the monopoly 

outcome then follows the semi-colon. Where there is a (***) this denotes parameter values that 

are covered by part (a) of the Proposition. The (**) denotes parameter values covered by part (b) 

of the Proposition. All other parameter values are covered by part (c).

1. If ck < 1•3then:

V <

1- 2ck

2c

:H1 NBL;HM - 1 NBL

1- 2ck

2c

£ V <

2ck + 1

2c

:H1IL;HM - 1 NBL

2ck + 1

2c

£ V <

2ck + 1

c

:H1IL;HM - 1IL

2ck + 1

c

£ V :H1BL;HM - 1BL

2. If 1•3 £ ck < 1•2then:

V <

1- 2ck

2c

:H1 NBLorH2IL;HM - 1 NBLH***L

1- 2ck

2c

£ V <

4ck - 1

2c

:H1ILorH2IL;HM - 1 NBLH**L

4ck - 1

2c

£ V <

2ck + 1

2c

:H1IL;HM - 1 NBL

2ck + 1

2c

£ V <

2ck + 1

c

:H1IL;HM - 1IL

2ck + 1

c

£ V :H1BL;HM - 1BL
31

3. If 1•2 £ ck <

1

12

I5 +

•

13Mthen :

- 1+ 2ck

4c

£ V < 3kH- 1 + 2ckL:H2IL;HM - 2 NBNBL

3kH- 1+ 2ckL£ V <

4ck - 1

2c

:H1ILorH2ILorH2BL;HM - 2 NBNBL

4ck - 1

2c

£ V <

6ck + 1

4c

:H1ILorH2BBL;HM- 2 NBNBL

6ck + 1

4c

£ V <

•

ck

•

•- 1+ 2ck

2 c

+ 2k :H1ILorH2BBL;HM - 2IL •

ck

•

•- 1+ 2ck

2 c

+ 2k £ V <

2ck + 1

c

:H1ILorH2BBL;HM- 1ILH**L

2ck + 1

c

£ V :H1BLorH2BBL;HM- 1BLH***L

4. If

1

12

I5+

•

13M£ ck <

1

24

I9+

•

105Mthen :

- 1+ 2ck

4c

£ V <

4ck - 1

2c

:H2IL;HM - 2 NBNBL

4ck - 1

2c

£ V < 3kH- 1+ 2ckL:H2BLorH2BBL;HM - 2 NBNBL

3kH- 1+ 2ckL£ V <

6ck + 1

4c

:H1ILorH2BBL;HM - 2 NBNBL

6ck + 1

4c

£ V <

•

ck

•

•- 1+ 2ck

2 c

+ 2k :H1ILorH2BBL;HM - 2IL •

ck

•

•- 1+ 2ck

2 c

+ 2k £ V <

2ck + 1

c

:H1ILorH2BBL;HM- 1ILH**L

2ck + 1

c

£ V :H1BLorH2BBL;HM- 1BLH***L
32

5. If

1

24

I9+

•

105M£ ck <

1

4

I1+

•

5Mthen :

- 1+ 2ck

4c

£ V <

4ck - 1

2c

:H2IL;HM - 2 NBNBL

4ck - 1

2c

£ V <

6ck + 1

4c

:H2BLorH2BBL;HM - 2 NBNBL

6ck + 1

4c

£ V < 3kH- 1+ 2ckL:H2BLorH2BBL;HM - 2IL

3kH- 1+ 2ckL£ V <

•

ck

•

•- 1+ 2ck

2 c

+ 2k :H1ILorH2BBL;HM - 2IL •

ck

•

•- 1+ 2ck

2 c

+ 2k £ V <

2ck + 1

c

:H1ILorH2BBL;HM - 1ILH**L

2ck + 1

c

£ V:H1BLorH2BBL;HM - 1BLH***L

6. If

1

4

I1+

•

5M£ ck < .968339then:

- 1+ 2ck

4c

£ V <

4ck - 1

2c

:H2IL;HM - 2 NBNBL

4ck - 1

2c

£ V <

6ck + 1

4c

:H2BLorH2BBL;HM- 2 NBNBL

6ck + 1

4c

£ V <

4ck + 1

2c

:H2BLorH2BBL;HM- 2IL

4ck + 1

2c

£ V < 3kH- 1+ 2ckL:H2BLorH2BBL;HM - 2BBL

3kH- 1+ 2ckL£ V <

4ck + 2-•

2

"

1+ ck - 2c

2 k2

2c

:H1ILorH2BBL;HM - 2BBL

4ck + 2-•2

"

1+ ck - 2c

2 k2

2c

£ V <

2ck + 1

c

:H1ILorH2BBL;HM - 1ILH**L

2ck + 1

c

£ V :H1BLorH2BBL;HM- 1BLH***L
33

7. If .968339£ ck < 1then:

- 1+ 2ck

4c

£ V <

4ck - 1

2c

:H2IL;HM - 2 NBNBL

4ck - 1

2c

£ V <

6ck + 1

4c

:H2BLorH2BBL;HM- 2 NBNBL

6ck + 1

4c

£ V <

4ck + 1

2c

:H2BLorH2BBL;HM- 2IL

4ck + 1

2c

£ V <

4ck + 2-•2

"

1 + ck - 2c

2 k2

2c

:H2BLorH2BBL;HM - 2BBL

4ck + 2-•2

"

1+ ck - 2c

2 k2

2c

£ V < 3kH- 1 + 2ckL:H2BLorH2BBL;HM- 1ILH**L

3kH- 1+ 2ckL£ V <

2ck + 1

c

:H1ILorH2BBL;HM - 1ILH**L

2ck + 1

c

£ V :H1BLorH2BBL;HM- 1BLH***L

8. If 1 £ ck then :

- 1+ 2ck

4c

£ V <

4ck - 1

2c

:H2IL;HM - 2 NBNBL

4ck - 1

2c

£ V <

6ck + 1

4c

:H2BBL;HM - 2 NBNBL

6ck + 1

4c

£ V <

4ck + 1

2c

:H2BBL;HM - 2IL

4ck + 1

2c

£ V :H2BBL;HM - 2BBL

First, I prove part (c). Where there are no stars next to an outcome, the following 

observations establish that all of the possible duopoly equilibria provide at least as much welfare 

as the monopoly outcome. First, whenever a duopoly equilibria and a monopoly outcome involve 

either equal development of both products or only development of one product, then the duopoly 

outcome will provide equal welfare if the outside options are never binding or always binding and 

greater welfare when they bind for some customers and not for others (the interior cases). If 

development is distributed identically across the products in the two regimes, then, since in both 

regimes there is (weakly) too little development, the regime that provides more development 

generates more welfare. Development is identical when the outside options never bind or always 
34

bind but is strictly greater in duopoly when the cutoff point is interior. Second, when comparing 

monopoly outcomes where an equal number of products are developed, welfare is greater the 

greater the fraction of the customers with a binding outside option (this obviously follows from 

the first observation). Third, by Proposition 1, duopoly will always be (weakly) superior to 

monopoly when the least asymmetric equilibrium is (weakly) superior to the monopoly outcome. 

In every case above when there are no stars, the monopoly outcome develops the same number of 

products and has the same or fewer number of customers with binding outside options as the least 

asymmetric duopoly equilibrium. Thus, the three observations guarantee that duopoly provides 

(at least weakly) greater welfare than duopoly in all of these cases.

Now, I prove part (a). Monopoly and duopoly development levels are identical between 

(1B) and (M-1B) and between (1NB) and (M-1NB) (see Tables 1 and 2). Thus, welfare must be 

identical in these cases also. In every parameter set with (***), there are more than one duopoly 

equilibria, the best of which (by Proposition 1) is either (1B) or (1NB). The monopoly outcome 

in these cases is always (M-1B) (when the best duopoly outcome is (1B) or (M-1NB) (when the 

best duopoly outcome is (1NB)). This proves (a).

To prove part (b), I show that in every case with (**), the monopoly outcome provides 

strictly greater welfare than at least one possible duopoly equilibrium and strictly less welfare 

than one possible duopoly equilibrium. The observations for part (c) establish the second part of 

this contention in every case with (**) except the third to last case of 7, where I need to show that 

welfare is greater in (2B) than (M-1I). 

Lemma 2.1. In Case 7 welfare is greater in (2B) than (M-1I).

Proof. Using the social welfare function, (18), and substituting in for development levels in each 

case using Tables 1 and 2, I can write the welfare difference between (2B) and (M-1I) as follows:
35

(A8)

kH- 4+ 31ck - 26c

2 k2

- 64c

3 k3 + 72c4 k4L

2H1- 10ck + 12c

2 k2L2

- H2- 25ck + 124c

2 k2

- 220c

3 k3 + 128c4 k H4LV

1+ 2ckLH1- 10ck + 12c2 k2L2

+

cH- 1- 22ck + 136c

2 k2

- 232c

3 k3 + 128c4 k4LV2

2H1+ 2ckL2H1- 10ck + 12c

2 k2L2

Taking the second derivative of (A8) with respect to V gives the following:

(A9) 2cI- 1- 22ck + 136c

2

k

2

- 232c

3

k

3

+ 128c

4

k

4M

This is always positive when .968339 ≤ck < 1, so the first derivative of (A8) is increasing in ck, 

meaning it can be no larger than when V = 3k(2ck − 1) , the upper bound in (2B) in case 7. At 

this V, the first derivative of (A8) is:

(A10) 4H- 1+ ckLI1- 10ck + 12c

2

k

2MI1- ck - 26c2

k

2

+ 32c

3

k

3M

This is negative for .968339 ≤ck < 1, so the welfare difference, (A8), is smallest at this same, 

maximal value of V. Evaluating (A8) at this V gives the following:

(A11) 8kH1+ 4ckLI1- 11ck + 22c

2

k

2

- 12c

3

k

3M2

This is clearly positive, proving the lemma.

Now I prove that wherever there is a (**) that the monopoly outcome generates strictly 

greater welfare than one duopoly equilibrium. Doing so requires proving the following lemmas.

Lemma 2.2. In cases 3 through 7 welfare is greater in (M-1I) than in (2BB).

Proof. Using the social welfare function, (18), and substituting in for development levels in each 

case using Tables 1 and 2, I can write the welfare difference between (2BB) and (M-1I) as 

follows:

(A12)

1+ 4c

3 k3 + cH5k - 4VL+ 2c2H- 2k + VL2

4cH1+ 2ckL2

This has the sign of the numerator, which is increasing in V since its derivative with respect to V

is:
36

(A13) 4 c (c (V-2 k)-1)

This is positive whenever (2BB) exists. So the welfare difference, (A12), can be no larger than it 

is when 

c

ck V2 + 1

= , the maximum value for V when (M-1I) exists (at v=V-k). At this V, the 

welfare difference is:

(A14) (1 2 ) ( 1)

2

+ ck ck −

This is negative since ck<1 in these cases, proving the lemma.

Lemma 2.3. In case 2 welfare is greater in (M-1NB) than (2I).

Proof. Using the social welfare function, (18), and substituting in for development levels in each 

case using Tables 1 and 2, I can write the welfare difference between (2I) and (M-1NB) as 

follows:

(A15)

- 3+ 72c

3 k3 + 2cH15k - 4VL- 4c2H23k2

- 8kV + 2V

2L

8cH1- 6ckL2

.

This has the sign of the numerator. The derivative of the numerator with respect to V is:

(A16) 8c(2c(2k − V ) − 1)

When (2I) exists, this is positive. So the numerator of (A15) is at its maximum at 

c

ck V

2

4 − 1

= , 

the maximum V for (2I). At this V, the numerator of (A15) is:

(A16) (1 6 ) (2 1)

2

− ck ck −

This is negative in case 2 since ck<1/2, proving the lemma.

Lemmas 2.2 and 2.3 together establish that when the conditions of (b) hold, the 

monopoly outcome provides more welfare than one duopoly equilibrium. This completes the 

proof. Q.E.D.
37

REFERENCES

Arrow, Kenneth J. “Economic Welfare and the Allocation of Resources for Invention.” In The 

Rate and Direction of Innovative Activity: Economic and Social Factors. Washington: National 

Bureau of Economic Research. 1962.

Bolton, Patrick and Michael D. Whinston. “Incomplete Contracts, Vertical Integration, and 

Supply Assurance.” The Review of Economic Studies, January 1993, 60(1), pp.121-48.

Bond, Eric W. and Larry Samuelson. “Durable Goods, Market Structure and the Incentives to 

Innovate.” Economica, February 1987, 54 (213), pp. 57-67.

Bulow, Jeremy I. “Durable Good Monopolists.” Journal of Political Economy, April 1982, 90 

(2), pp. 314-32.

_____, Geanakoplos, John D., and Klemperer, Paul D. “Multimarket Oligopoly: Strategic 

Substitutes and Complements.” Journal of Political Economy, June 1985, 93 (3), pp. 488-511.

Che, Yeon-Koo and Ian Gale. “The Optimal Design of Research Contests.” Mimeo, 2000.

_____ and Donald B. Hausch. “Cooperative Investments and the Value of Contracting.” 

American Economic Review, March 1999, 89 (1), pp. 125-47.

Chiu, Stephen Y. “Noncooperative Bargaining, Hostages, and Optimal Asset Ownership.” 

American Economic Review, September 1998, 88 (4), pp.882-901

Cole, Harold L., Mailath, George J., and Postlewaite, Andrew. "Efficient Non-Contractible 

Investments in Finite Economies." Advances in Theoretical Economics, 2001, 1(1), Article 2.

Dasgupta, Partha and Josesph Stiglitz. “Industrial Structure and the Nature of Innovative 

Activity.” Economic Journal, June 1980, 90 (358), pp. 266-93.

de Meza, David and Lockwood, Ben. “Does Asset Ownership Always Motivate Managers? 

Outside Options and the Property Rights Theory of the Firm.” Quarterly Journal of Economics, 

May 1998, 113(2), pp.361-86.

Felli, Leonardo and Roberts, Kevin. “Does Competition Solve the Hold-up problem.” Caress 

Working Paper #00-04, 2000.

Greenstein, Shane and Garey Ramey. “Market Structure, Innovation and Vertical Product 

Differentiation.” International Journal of Industrial Organization, May 1998,16 (3), pp. 285-

311.

Grossman, Sanford J. and Hart, Oliver. “The Costs and Benefits of Ownership: A Theory of 

Vertical and Lateral Integration.” Journal of Political Economy, August 1986, 94 (4), pp. 691-

719
38

Hart, Oliver and Moore, John. “Incomplete Contracts and Renegotiation.” Econometrica, July 

1988, 56 (4), pp. 755-85.

MacLeod, W. Bentley and Malcomson, James M. “Investments, Holdup, and the Form of Market 

Contracts.” American Economic Review, Septmember 1993, 83(4), pp.811-837.

Rubinstein, Ariel. “Perfect Equilibrium in a Bargaining Model.” Econometrica, January 1982, 

50(1), pp.97-109.

Shaked, Avner and Sutton, John. “Involuntary Unemployment as a Perfect Equilibrium in a 

Bargaining Model.” Econometrica, November 1984, 52(6), pp.1351-64.

Tirole, Jean. “Procurement and Renegotiation.” Journal of Political Economy, April 1986, 94 

(2), pp. 235-59.

Williamson, Oliver. The Economic Institutions of Capitalism. New York: Free Press, 1985.