Price collusion and stability of research partnerships

Kaz Miyagiwa

Emory University

Abstract: This paper examines the age-old question whether R&D cooperation leads to

price collusion. Without R&D cooperation, new technology gives rise to an inter-firm

cost asymmetry, which makes it difficult to collude. The prospect that collusion fails in

post-discovery periods destabilizes pre-discovery collusion. Sharing the technology can

avert this chain reaction. Thus, R&D cooperation facilitates collusion. However, the

inability to monitor partners’ R&D efforts undermines cooperation. One way to curb the

opportunism is to dissolve the partnership with a positive probability every time R&D

fails. Such strategies are consistent with high mortality rates of R&D partnerships (20%

in one estimate).

JEL Classifications System Numbers: L12, L13

Keywords:Oligopoly, Collusion, Research Joint Ventures, Innovation, R&D

Correspondence: Kaz Miyagiwa, Department of Economics, Emory University, Atlanta,GA 30322, U.S.A.; e-mail: kmiyagi@emory.edu; telephone: 404-727-6363, fax: 404-727-4639

1. Introduction

This paper addresses two issues with regard to R&D cooperation among firms in

related industries. One issue concerns the age-old suspicion among antitrust authorities

that legalizing R&D cooperation gives firms a chance to collude in product markets in

which they should compete. That suspicion for a long time motivated U.S. antitrust

authorities to apply full forces of antitrust laws to squash any attempt to form research

joint projects in that country. Their attitude towards cooperative R&D however had to

soften during the 1960s and early 1970’s when key American industries began to lose the

competitive edge to Japanese and European rivals which had made considerable

technological progress through government-sponsored joint research projects.1 Although

cooperative R&D enjoys governmental support and encouragement almost everywhere,

the age-old suspicion lingers: does cooperation in R&D facilitate product market

collusion?

This paper is also concerned with the fact that R&D partnerships break up at

surprisingly high rates. For example, Harrigan (1988) shows that only 45% of R&D

alliances are successful, with the average life span of 3.5 years. Similar results are

obtained by Kogut (1988), who finds the average mortality rate to be 20 percent, with a

peak in five or six years after formation of joint projects.2 If cooperative R&D is

profitable, what explains high mortality rates?

1 See Calogirou, Ioannides and Vonortas (2003).

2 These studies are quoted in Veugelers (undated).

3

This paper shows how these two issues are related. Consider a group of firms

engaged in R&D competition. Without R&D cooperation, a discovery of a new cost-

reducing technology gives rises to an inter-firm cost asymmetry. The firm with the new

technology has a strong incentive to lower the price but price collusion stands in its way.

Thus, innovation makes price post-discovery collusion difficult to sustain. But the

prospect that collusion ends with discovery of new technology makes it harder to sustain

collusion in pre-discovery periods, too. One way to put a stop to this collusion meltdown

is to form an R&D partnership and to share new technology. R&D partnerships allow

firms to commit legally to share new technology and thereby abate the incentive to lower

the price unilaterally. Thus, we conclude that R&D cooperation facilitates price collusion.

However, if firms cannot perfectly monitor partner’s R&D inputs, the inherently

uncertain nature of the R&D process makes it impossible to disentangle failures due to

exogenous causes from those due to non-compliance of partners, giving firms an

incentive to free ride on partners’ R&D efforts. One way to curb this opportunism is to

adopt a strategy that calls for dissolution of the partnership with positive probability

every time R&D fails to bear fruit. Such equilibrium strategies are consistent with

observations of high rate at which R&D partnership break up in the real world.

Now is the time to outline our model. It adopts an infinitely repeated game

framework. In each period, two firms set prices and invest in R&D. R&D outcomes are

stochastic. We consider only a single cost-reducing innovation so R&D investments are

conditional on the innovation not having been occurred to date. In the basic setup we

assume firms try to collude in price without formation of an R&D partnership, and we

identify the critical (i.e., minimum) discount factor above which price collusion is

4

possible both in pre- and post-discovery periods. The assumption that firms can collude

implicitly but cannot form an R&D partnership illegally seems reasonable, because the

innovating firms cannot be forced to share new technology with partners under an illegal

R&D partnership. One way to eschew this problem is to do R&D in a single laboratory,

which however is also difficult to maintain unless it is legal.

In the first variation of the model we allow firms to form an R&D partnership to

share innovation and monitor R&D inputs perfectly. We find that the critical discount

factors for implicit price collusion fall both in pre- and post-discovery periods. We thus

conclude that R&D cooperation facilitates price collusion.

In the second variation, we assume monitoring of partner’s R&D inputs to be

impossible. To alleviate the free-rider problem alluded to above, firms chooses a

probability with which they dissolve the partnership. We investigate the conditions under

which the partnership breaks up with a positive probability in equilibrium.

We now relate this paper to the literature. Despite a plethora of literature both on

R&D cooperation and on collusion, the linkage between the two has received little

attention to date.3 Exceptions are works of Martin (1995) and Lambertini, Poddar and

Sasaki (2002). Lambertini et al. analyze a three-stage game, in which two firms first

decide whether to form a joint venture, then choose horizontal locations in product space

and finally compete or collude in prices over an infinite time horizon. The firms are

allowed to select different locations (or differentiate products) freely to abate the degree

3 Kesteloot and Veugelers (1995) use a repeated-game setup to analyze the stability of R&D cooperation

without price collusion. Their focus is on the role of R&D input spillovers.

5

of price competition when acting individually but constrained to choose a single location

when cooperating in R&D. Having to produce an identical product intensifies price

competition, and makes collusion more difficult to sustain under R&D cooperation, a

result that contrasts sharply with ours. Their model also differs in that there is no

uncertainty in R&D and that pre-discovery collusion is ignored.

By contrast Martin (1995) assumes R&D to be stochastic and studies pre-

discovery price collusion. In his model, time is continuous and the detection date of non-

collusive behavior is not immediate but follows a stochastic process.4 Moreover,

collusion is assumed to end with the innovation. Thus, there is no discussion of post-

discovery collusion. Despite these differences from our model, however, Martin reaches

the same conclusion as ours: namely, pre-discovery price collusion is more likely to

succeed if firms cooperate in R&D.

The standard literature on collusion typically assumes symmetric firms and

focuses on collusion at the price that maximizes industry profit, the natural focal point

under assumed firm symmetry. However, in models of innovation firms are not

symmetric in post-discovery periods, a fact that makes it difficult to select the natural

focal point for collusive behavior. Martin (1995) eschews this problem by assuming

collusion to end with a discovery of new technology. Lambertini et al. (2002) avoid it by

assuming simultaneous and deterministic R&D so as to keep a firm symmetry in post-

discovery periods.

4 In a continuous time setting there is no defection if it is detected immediately. Thus this probabilistic

detection process is necessary.

6

In this paper we address the asymmetry problem by an appeal to the notion of

balanced temptation equilibrium of Friedman (1971), who was the first economist to

tackle the general problem of collusion under asymmetry in repeated games. The

balanced temptation equilibrium requires that all firms be equally tempted to defect from

collusion. However, there is still infinity of collusive equilibria that fulfill the Friedman

requirement, so an additional condition is typically imposed to select a unique collusive

equilibrium. For example, Bae (1987) focuses on a collusive equilibrium that maximizes

total industry profits for homogeneous-goods, asymmetric-cost Bertrand duopoly, while

Verboven (1997), Rothschild (1999) and Collie (2004) do the same for Cournot

oligopoly.5 In this paper, however, since we are interested in the sustenance of collusion,

we select as the focal point the most collusive equilibrium, that is, an equilibrium that

yields the lowest critical discount factor at which collusion is sustainable under a given

cost asymmetry. In a completely different context and motivation, Collie (2004) studies

such a discount factor for Cournot duopoly.

The remainder of the paper is organized in eight sections. Section 2 sets up the

model, focusing on non-drastic innovation.6 Sections 3 and 4 examine the possibility of

collusion in post- and pre-discovery periods, respectively. Section 5 analyzes the effect of

R&D partnerships under perfect monitoring of R&D inputs. Section 6 studies the

5 The equilibrium selection based on the balanced temptation requirement is not without criticism. For

example, Harrington (1991) preferred Nash bargaining in his study of collusion in homogeneous-goods

Bertrand duopoly.

7

consequence of imperfect monitoring. Sections 7 and 8 extend the model to the case of a

single-track lab and drastic innovation, respectively. Section 9 concludes.

2. Model setup and non-collusive equilibrium

This section presents the model in the absence of price collusion. Consider two

price-setting firms interacting over an infinite number of periods. The firms initially

employ the identical (old) technology that enables them to produce a homogeneous

product at constant unit cost, c- > 0. The demand function is time-invariant and is denoted

by D(p). D(p) is differentiable, with the first derivative D’(p) < 0.7 We also assume D(c-)

> 0, or a sufficiently large market size.

In each period t = 1, 2, … the two firms choose prices simultaneously. The firm

offering a lower price p captures the entire market and sells D(p) units. If both firms offer

the same price, consumers buy from each seller with an equal probability. In each period,

the firms also decide whether to invest in R&D. When investing in R&D, the firm incurs

fixed investment cost k and discovers a new technology with a probability µ (0 < µ < 1).8

The new technology is known to decrease the unit cost to c- (< c-), and becomes applicable

to production in the following period. If one firm makes a discovery, that firm is granted

6 Innovation is called non-drastic when the innovator cannot earn the monopoly profit unchallenged by

competitors. Otherwise, innovation is drastic.

7 Primes indicate differentiation.

8

permanent patent protection. If both firms make a discovery in one period, each has the

equal chance of getting a patent.9 Either way, in each period at most one firm has access

to the new technology in the absence of cooperation in R&D.

Until section 8 we assume the innovation to be non-drastic. For non-drastic

innovation the following strategies form a subgame-perfect Nash equilibrium: each firm

chooses a price equal to c- until there is the innovation. With a discovery of the new

technology, the innovator sets a price equal to c- (minus an epsilon) to limit-price the non-

innovator, who may set any price above c-. This is referred to as the limit-price strategy. If

both firms adopt the limit-price strategy, the innovator receives

pL ≡ D(c-)(c- - c-) > 0

and the non-innovator receives zero profit per period, and both firms lose k per period

until there is the innovation.10

We now want to find the discounted sum of profits in the limit-price strategy

equilibrium, which we denote by vo. Since each firms has probability µ(1 – µ) of

innovating alone and probability µ2 of innovating jointly, each firm has total probability

[µ2/2 + µ(1 - µ)] of gaining access to the new technology in each period and hence of

earning the flow profit πL forever. On the other hand, with probability (1 - µ)2 there is no

8 The assumption of fixed-intensity R&D, adopted in Bloch and Markowitz (1996) and others, simplifies

the analysis.

9 The assumption is also adopted in Cardon and Sasaki (1998), for example.

9

innovation and the R&D race continues, Given the stationary environment, however, the

firms face exactly the same situation in the following period as in the current one. Thus,

we can write vo as

(1) vo = - k + d[µ2/2+ µ(1 – µ)]pL/(1 - d) + d(1 – µ)2vo,

where d is a common discount factor per period (0 < d < 1). Collecting terms,

vo = -"k"+"[µ2/2"+"µ(1"–"µ)]dpL/(1"-"d)

"1 - d(1"–"µ)2

We assume vo to be positive or that investment in R&D is worthwhile.

3. Collusion in post-innovation periods

In this section we examine the possibility of price collusion in post-discovery

periods; i.e., for a subgame that begins at the (stochastic) date of innovation. We first

define additional notation. Let p-m and p-m be the (unconstrained) monopoly price under

the old and new technology, respectively; i. e.,

p-m ≡ argmaxp

D(p)(p - c-) and p-m ≡ argmax

p D(p)(p - c-),

10 This assumes k to be sufficiently small (discussed later).

10

and let M- and M- be the associated maximum monopoly profit, respectively. Assume

concavity so that these monopoly prices are unique. It is easy to check that p-m < p-m.

Innovation is non-drastic if and only if p-m > c-.

To solve the post-discovery subgame, we focus on the following trigger strategies

with Nash reversions (to the limit-price strategies). The firms begin the post-discovery

period with a selection of price p and market share a to the innovator, and repeat that

selection as long as they have observed p and a in every past post-discovery period. If

another selection was made in at least one period, the firms switch to the limit-price

strategy described in section 2.

If both firms adopt this trigger strategy, the innovator earns aD(p)(p - c-) and the

non-innovator receives (1 - a)D(p)(p - c-) per period along the equilibrium path. Suppose

p £ p-m for the moment and consider a one-period deviation from the equilibrium. A

defection yields the innovator at most D(p)(p - c-) in one period and pL in every

subsequent period. Thus, a defection is unprofitable if

aD(p)(p - c-)/(1 - d) ≥ D(p)(p - c-) + dpL/(1 - d)

or

(2) d ≥ (1 - a)D(p)(p - c-)/{D(p)(p - c-) - pL}.

11

Likewise, a defection yields to the non-innovator the one-time profit D(p)[p - c-] but

nothing else as he gets limit-priced forever. Thus, a defection is unprofitable for the non-

innovator if

(1 - a)D(p)(p - c-)/(1 - d) ≥ D(p)(p - c-).

or

(3) d ≥ a.

Collusion is sustainable if d satisfies both (2) and (3).

We now find the most collusive trigger-strategy equilibrium (i.e., the minimum

critical discount factor) among the {p, a} pairs that satisfy the balanced temptation

requirement of Friedman (1971), which in the present case is equivalent to:

(4)(1"-"a)D(p)(p"-"c-)""D(p)(p"-"c-)""-"pL" = a.

We now seek p and a that minimize d subject to (4) and p ≤ p-m and. For given a the

right-hand side of (2) is minimized at p = p-m. Substituting p = p-

m into (4) yields the

associated market share to the innovator:

a- = M- /(2M- - pL).

Now consider the case in which p-m > p ≥ p-m. Then, the no-defection condition for

the innovator becomes

aD(p)(p - c-)/(1-d) ≥ M- + dpL/(1-d)

while that for the non-innovator is unchanged. The balanced temptation requirement

becomes:

12

[M- - aD(p)(p - c-)]/(M- - pL) = a.

It is straightforward to show that {p-m, a- } also is the solution to this case.11 Then by (4)

the critical discount factor is

(5) dA = a- = M- /(2M- - pL).

We have established our first result.

Proposition 1.

(A) The collusive equilibrium price p and market share a to the innovator that satisfy the

balanced temptation requirement of Friedman and that minimize the critical

discount factor are uniquely given by (p-m, a- ) where a- = M- /(2M- - pL).

(B) The critical discount factor is given by dA = a- .

(C) The collusive equilibrium profits per period are given by pi ≡ a- M- to the innovator

and pn ≡ (1 - a- )D(p-m)(p-

m - c-) to the non-innovator.

For d < dA it is impossible to maintain collusion in post-innovation periods, and hence the

limit-price equilibrium is the only subgame-perfect equilibrium of the model.

11 We ignored the case p ≥ p-m, for this case fails to satisfy the balanced temptation requirement.

13

Evidently dA depends on the magnitude of cost saving under the new technology.

Differentiating (5) yields ∂dA/∂c- = - D(c-)[M- - D(p-m)(c- - c-)]/(2M- - pL)2 < 0 as long as p-

m

> c- or the innovation is non-drastic. Thus,

Corollary 1: Hold c-. As c- is decreased, dA increases.

This result is understood intuitively. The lower the unit cost under the new technology,

the greater is πL, and hence the greater is the temptation for the innovator to defect. To

curb such a temptation, future benefits to the innovator from collusion must be raised.

That is accomplished by increasing the market share a- to the innovator or equivalently by

raising the critical discount factor dA.

The relationship between dA and the unit cost c- under the new technology is

depicted in figure 1 for given c-. Two observations are noteworthy. First, dA approaches

1/2 as the new technology becomes less cost-saving. Second, at c* indicated in figure 1,

dA reaches unity, implying a- = 1. It follows that any innovation that lowers the unit cost

below c* is drastic.

4. Collusion in pre-innovation periods

14

We next examine the possibility of price collusion in pre-innovation periods.

Consider the following strategy that applies to the whole game. Begin the initial (pre-

innovation) period by charging p-m. Choose this monopoly price as long as no other price

was observed and there was no innovation in any past period. If no other price was

observed but there was the innovation, switch to the post-discovery trigger strategy

described in section 3. If another price was observed in any past period, switch to the

limit-price strategy of section 2.

We now look for the conditions under which the above strategies form a

subgame-perfect equilibrium. We initially restrict our search to d ≥ dA > 1/2. If both

firms follow the equilibrium strategy, each firm receives M- /2 – k at the end of period t.

During that period, one firm innovates with probability m(1 – m), securing the future per-

period profit pi for itself and pn for the non-innovator. During the same period both firms

innovate with probability m2, and earns the expected per-period profit (pi + p n)m2/2.

Finally, with probability (1 – m)2 firms fail to innovate and engage in the R&D race in

period t + 1. Thus, we can express the discounted sum vc of per-firm profits under

symmetric collusive equilibrium as:

(6) vc = M- /2 – k + d[m2/2 + m(1 – m)](pi + pn)/(1 - d) + d(1 – m)2vc.

Collecting terms yields

(7) vc = d[m2/2"+"m(1"–"m)](pi"+pn)/(1"-"d)"+"M- /2"–"k

1"-"d(1"-"m)2 .

15

Consider now a one-period defection from the equilibrium strategy. A defector

earns the one-shot profit (M- – k) in period t, and the expected discounted future profits

equal to [m2/2 + m(1 – m)]πL/r + (1 – m)2vo as the firms will switch to the trigger-price

strategy in period t + 1. The first term is the discounted sum of profits to the innovator

weighted by the probability of becoming one. The second term is the sum of profits in the

limit-price strategy equilibrium weighted by the continuation probability of the R&D

race. Thus, a defection yields

(8) (M- – k) + d[m2/2 + m(1 – s)]pL/(1 - d) + d(1 – m)2vo

= M- + vo

where we used (1). A defection is unprofitable if (6) exceeds (8) or

(9) vc – vo ≥ M-

where

vc – vo = d[m2/2"+"m(1"–"m)](pi"+"pn"-"pL)/(1"-"d)"+"M- /2

1"-"d(1 - m)2 .

Since (pi + pn - pL) is positive for d ≥ dA, vc – vo increases monotonically in d

without bounds. Therefore, (9) holds for a high enough d. But (9) does not hold for all

d ≥ dA because as c- approaches c-, vc – vo becomes arbitrarily small, violating (9). It

follows that for small enough c- - c-, there exists a unique d (> dA), at which (9) holds with

equality. Denote that d by dB.

16

By definition dB is a function of the unit cost c- under the new technology. The

relationship between the two is illustrated in figure 1. A fall in c- raises pL more than (pi +

pn), thereby increasing the incentive to defect (see Appendix A). To curb such a

temptation the discount factor dB must rise. Direct computation shows that (9) holds with

strict inequality at d = 1/[1 + (1 – m)2], suggesting that the locus dB is trapped between

that of dA and the horizontal line at 1/[1 + (1 – m)2] as depicted in figure 1.

Proposition 2: Collusion is sustainable both in pre-discovery and post-discovery periods

if d ≥ max {dB, dA}, where dB is given implicitly by (9) with strict equality.

The area AB in figure 1 represents the set (d, c-) such that proposition 2 holds. The

area NA, on the other hand, represents the set (d, c-) that enable the firms to collude in

post-discovery periods only. In the latter area, the firms set p = c- in pre-discovery periods

regardless of past prices, and switch to the post-discovery collusive strategy described in

section 3 whenever there is the innovation.

If d < dA, the firms cannot collude in post-discovery periods but may still be able

to collude in pre-discovery periods. A procedure similar to the one leading to (9) gives

the following condition for collusion in pre-discovery periods only:

(10) v~c – vo ≥ M- ,

where

17

v~c =

[m2/2"+"m(1"–"m)]pLd/(1"-"d)"+"M- /2"–"k1"-"d(1 - m)2

is the discounted sum of profits from collusion in pre-discovery periods only. Simplifying

(10) yields

(11) d ≥ 1/[2(1 – m)2].

The area BN in figure 1 contains the pairs (d, c-) satisfying (11).

Finally, if d is less than min{1/[2(1 – m)2], dA} there is no collusion in any

period.12 Given that min {1/[2(1 – m)2], dA} > 1/2, and the fact collusion is sustainable for

d ≥ 1/2 in non-innovating industries yields the following conclusion.

Proposition 3: Collusion is more difficult to sustain in dynamic R&D-intensive

industries than in static industries without chance of innovation.

We conclude this section by examining the relationship between collusion in pre-

discovery periods and probability of discovery, µ. First, it can be shown that (vc – vo) in

(9) is increasing in µ. This means that the locus of dB in figure 1 shifts down as

µincreases, thereby expanding the area AB. It follows that when d ≥ dA an increase in

probability of discovery makes it easier to maintain collusion in pre-innovation periods.

On the other hand, the right-hand side of (11) is decreasing in µ, implying that the

18

sustenance of collusion gets more difficult as probability of discovery increases. The next

proposition restates this finding. The intuition is straightforward.

Proposition 4:

(A) When d ≥ dA so collusion in post-innovation is sustainable, an increase in the

probability of discovery makes it easier to maintain collusion in pre-innovation periods.

(B) When d < dA so collusion in post-innovation is unsustainable, an increase in the

probability of discovery makes it harder to maintain collusion in pre-innovation periods.

5. R&D Partnership

Still focusing on non-drastic innovation, we now allow the firms to form an R&D

partnership and share the new technology. R&D partnerships are defined broadly enough

to encompass research joint ventures as well as royalty-free cross-licensing agreements,

under which each firm incurs own R&D and gains free access to any innovations made

by partners.

In this section we make two additional assumptions, which will be relaxed in later

sections. One is that the firms retain their own R&D laboratories. This assumption is

common in the literature on research joint ventures [e.g., Kamien, Muller and Zang

(1992)]. The other is that the firms can perfectly monitor each other’s R&D input.

12 When full collusion is impossible to maintain, it may be still possible to collude at less than the full

monopoly price p-m, the issue we do not pursue here.

19

Now, suppose that the firms adopt the following strategy. (i) Begin the initial

(pre-innovation) period by setting p = p-m. (ii) Choose p-m in every subsequent period as

long as no other price was observed and there was no innovation in the past. (iii) When

there is the innovation and all past prices were set to p-m, set p = p-m and choose the latter

until another price is observed, at which time choose p = c- forever. (iv) If a price other

than p-m was observed before the innovation, dissolve the R&D partnership and set p = c-

until there is the innovation; at which time switch to the limit-pricing strategy.

If the above strategy is adopted, the firms share the new technology so the cost

asymmetry never appears; hence collusion in post-discovery periods is sustainable for d ≥

1/2. The fact that collusion becomes easier to sustain in post-discovery periods makes it

easier to do so before the innovation occurs.

To formally demonstrate this result, let vJ denote the equilibrium expected

discounted sum of per-firm profits under the R&D partnership. Since the innovation

occurs at least in one lab with probability [m2 + 2m(1 - m)] in every period (conditional

on the innovation not having occurred yet), we can express vJ as

(12) vJ = M- /2 – k + d[m2/2 + m(1 – m)]M- /(1 - d) + d(1 – m)2vJ,

or

(13) vJ = d[m2/2"+"m(1"–"m)]M- /(1"-"d)"+"M- /2"–"k

1"-"d(1 - m)2 .

20

A one-period defection from the equilibrium strategy yields (M- – k). At the end of

the period there is the innovation with probability [m2 + 2m(1 – m)] = 1 – (1 – m)2. This

innovation is to be shared by both firms under the partnership agreement. However, since

each firm sets p = c-, each firm earns zero profit. If there is no innovation, the firms switch

to the non-cooperative strategies forever, earning the expected sum of profits d(1 – m)2vo.

Thus, a defection yields:

(M- – k) + d(1 - m)2vo.

A defection is unprofitable if that sum is less than vJ. By (12), the no-defection

condition is written

M- /2 – k + d[m2/2 + m(1 – m)]M- /(1 - d) + d(1 – m)2vJ

≥ (M- – k) + d(1 - m)2vo,

which simplifies to

(14) d[m2/2 + m(1 – m)]M- /(1 - d) + d(1 – m)2(vJ – vo) – M- /2 ≥ 0.

where

vJ – vo = d[m2/2"+"m(1"-"m)](M- "-"pL)/(1"-"d)"+"M- /2

1"-"d(1 - m)2 .

In Appendix B we show that (14) holds for all d ≥ 1/2. Thus,

21

Proposition 5: Formation of R&D partnerships facilitates collusion in pre-innovation and

post-innovation periods.

6. Imperfect monitoring

In this section we relax the assumption that the firms can monitor each other’s

R&D activity perfectly. In the absence of perfect monitoring, the inherently uncertain

nature of the R&D process makes it impossible to disentangle failures due to exogenous

sources from those due to non-compliance of the partner. Thus, the firms have the

incentive to free-ride on each other’s R&D effort, undermining the firms’ ability to

collude.

To curb this opportunism, we modify part (ii) of the strategy in section 5 as

follows.

(ii’) As long as no other price than p-m was observed and there was no innovation in the

past, dissolve the partnership with probability (1 - q) and switch to the non-collusive

strategy (iv) of section 5. (ii”) If the partnership continues go to step (i) of section 5.

Let va be the equilibrium profit per firm. Then, each firm’s expected sum of

profits after R&D efforts fail to make a discovery is qva + (1 - q)vo. Therefore,

va = M- /2 – k + [m2 + 2m(1 –m)](M- /2)d/(1 - d)

+ d(1 – m)2{qva + (1 - q)vo}.

Collecting terms, we obtain

22

va = M- /2"–"k"+"[m2/2"+"m(1"–m)]M- d/(1"-"d)"+"(1"–"m)2(1"-"q)dv0"

"1"-"(1"–"m)2qd" .

Now, consider a one-period defection in R&D. As its partner invests in R&D, the

innovation occurs with probability m and does not occur with probability (1 - m). In the

latter case, the R&D partnership breaks down with probability (1 - q). Thus, a defection

in R&D yields

vd = M- /2 + m(M- /2)d/(1 - d) + (1 – m){qdva + (1 - q)dvo}.

Observe that when there is the innovation the firms are better off colluding, which

explains the second term on the right-hand side. A defection is unprofitable if

(15) va – vd = – k + m(1 – m)d{(M- /2)/(1 - d) - qva - (1 - q)vo} ≥ 0

We suppose that the R&D partnership chooses the continuation probability, q, to

maximize the present value va of cooperation subject to the incentive-compatibility

constraint (15). It is easy to check that va is increasing in q while (va – vd) is decreasing

in q. Therefore, the partnership selects the greatest q that satisfies the constraint (15) with

equality. It is found, after manipulation (shown in Appendix C) that the optimum q is

equal to

qopt = A/B

where

A ≡ – k/[m(1 – m)d] + (M- /2)/(1 - d) - vo

B ≡ M- /2 – k/m + (M- /2)d/(1 - d) – vo.

23

The next proposition gives the condition for the partnership to break up with

positive probability in equilibrium.

Proposition 6: If

(16)k[1 - (1 - m)d]"

"m(1"–"m)d" > M- "-"M- "

"2" ,

the R&D partnership is dissolved with strictly positive probability after every period with

no innovation.

The left-hand side of (16) is obviously greater, (a) the smaller d and (b) the greater k. The

less important the future profits, or the greater the cost of R&D investment, the greater is

the temptation to free ride. To curb such opportunism, the firms punish each other by

dissolving the partnership with positive probability.

The left-hand side of (16) also depends on the probability of discovery, m. Its

graph is U-shaped in m on (0, 1), and is unbounded as m approaches zero or unity.13 Thus,

the inequality in (16) holds for m being sufficiently small or sufficiently large. With high

probabilities of discovery, the new technology is very likely to be discovered in the

current period even if only one lab is used; therefore, each firm has little incentive to

contribute to the partnership. On the other hand, with low probabilities of discovery, there

is little hope of innovating. This sense of hopelessness prompts each firm to cut back on

24

R&D investment. In either case, the fear of losing the R&D partnership prompts each

firm to invest in R&D despite those temptations to free ride.

7. Single R&D Laboratory

When R&D costs are high enough, the firms may use just one research laboratory

to share them. In our model that would be the case if the expected discounted sum of

profits from having a single lab exceeds that from having two labs:

vSJ = mM- d/[2(1"-"d)]"+"M- /2"–"k/2

1"-"d(1 - m) > vJ,

where vJ is given in (13). Of course this makes sense only for d ≥ 1/2, for otherwise

collusion is impossible after the innovation.

Doing R&D in a single laboratory does not completely solve the monitoring

problem discussed in section 6, as firms are known not to send top scientists and

engineers to joint projects. It is also difficult to monitor individual contributions in team

projects. However, in this section we assume there is perfect monitoring of R&D inputs

and consider defections in prices only.

As before, if a defection is detected, the firms dissolve the partnership

immediately. Thus, a defection is unprofitable if:

M- /2 – k/2 + dmM- /[2(1 - d)] + d(1 – m)vSJ > M- – k/2 + d(1 - m)vo

13 For m Π[0, 1] h(m) = k[1 Р(1 Рm)d]/[m(1-m)d] = (k/d){1/[m(1-m)] Р1/m} reaches its minimum at m = {(1

- d)1/2 – (1 - d)}/d.

25

or

(17) dmM- /[2(1 - d)] + d(1 – m)(vSJ – vo) – M- /2 ≥ 0.

This expression clearly holds when d ≥ 1/2, regardless of the new unit cost, implying that

cooperation in R&D facilitates price collusion.

8. Drastic innovation

Innovation becomes drastic if the unit cost falls below c* in figure 1. Consider

R&D competition. Since the innovator becomes a monopolist, there is no reason to

collude in post-discovery periods. Thus we look at the possibility of pre-discovery

collusion for d ≥ 1/2. The expected profit per firm without collusion is

wo = [m2/2"+"m(1"-"m)]dM- /(1"-"d)"–"k

1"-"d(1 - m)2

and with collusion it is

wc = [m2/2"+"m(1"-"m)]dM- /(1"-"d)"+"M- /2"–"k

1"-"d(1 - m)2 .

The non-defection condition is written:

(18) wc – wo > M- .

Straightforward computation shows that (18) holds for

(19) d ≥ 1/[2(1 – m)2] > 1/2.

The area denoted by BD in figure 1 depicts this case.

26

Under the R&D partnership each firm acquires the new technology with

probability {m2 + 2m(1 – m)} every period and earns M- /2 in each post-discovery period if

they collude. If they also collude in pre-discovery periods, the per-firm expected sum of

profits is wc.

A defection in (pre-discovery) period t yields M- – k in that period. Although the

innovation occurs with probability {m2 + 2m(1 – m)} in period t, it yields zero profits as

the both firms employ the new technology. Thus, a defection is unprofitable if the

following analog of (14) holds:

d[m2/2 + m(1 – m)]M- /(1 - d) + d(1 – m)2(wc – wo) – M- /2 ≥ 0.

It can be shown that this inequality holds for d ≥ 1/2.14 It follows from (19) that formation

of an R&D partnership facilitates collusion.

9. Concluding remarks

In this paper we developed a model of a repeated game to show that cooperation

in R&D facilitates collusion in product markets. The model was also used to explain why

R&D partnerships break up at high rates.

Several extensions suggest themselves for future research. First, it seems

straightforward to extend the analysis to allow for the licensing of the new technology,

14 We have that wc – wo = (M

-/2)/[1 - d(1 – m)2]. Substituting shows that (17) holds with strict inequality at

d =1/2. The desired result then follows from the fact that the left-hand side of (17) increases in d.

27

finite patent life or technological spillovers.15 Extensions to Cournot behavior and

multiple innovations also seem worthwhile. Thirdly, although we assumed that R&D

partnerships can potentially last forever, actual R&D partnership agreements often

specify how long the partnership should last. It is interesting to investigate the purpose of

such agreements in the light of sustaining collusion. Finally, our study suggests that

R&D-intensive industries are less collusive than stagnant industries. This proposition

seems intuitive but needs confirmation with empirical testing.

15 See Miyagiwa and Ohno (2002) for extensions in the different context.

28

Appendices

Appendix A: We show the negative relationship between dBand c- for given c-. The

defining equation of dB is g(dB, c-) ≡ vc – vo – M- = 0. It is straightforward to show that

∂g/∂dB > 0. Therefore it is sufficient to show ∂g/∂c- > 0 to conclude ddB/dc- < 0. Observe

that

sgn {∂g/∂c-} = sgn {∂(vc – vo)/∂c-}

= sgn {∂(pi+ pn - pL)/∂c-}dc- = 0 = - sgn{∂(pi+ pn - pL)/∂c-}dc-=0,

We thus need to show that the last derivative is negative. Since

(pi+ pn - pL) = a- D(p-)(p- - c-) + (1 - a- )D(p-)(p- - c-) - D(c-)(c- - c-).

differentiating the right-hand side with respect to c- yields,

(A1) - (1 - a- )D(p-) - D(p-)(p- - c-)∂a- /∂c- - [D’(c-)(c- - c-) + D(c-)].

Since ∂πL/∂c- = [D’(c-)(c- - c-) + D(c-)] > 0 at c- < p-m, the third term is positive. Since a- = M-

/(2M- - pL), ∂a- /∂c- is also positive. Thus the expression in (A1) is negative.o

Appendix B: It is straightforward to show that d[m2/2 + m(1 – m)]M- /(1 - d) and d(1 –

m)2(vJ – vo) are both increasing in d.Thus, we need only to show that (14) holds with

strict inequality at d = 1/2. Using the equality:

29

(1/2)(1 – m)2(vJ – vo) = (vJ – vo) – (1/2)[1 – (1 – m)2](M- - pL) - M- /2,

we can rewrite the left-hand side of (14) at d = 1/2 as

(1/2)[1 - (1 – m)2]M- /(1 - d) + (vJ – vo) – (1/2)[1 – (1 – m)2](M- - pL) – M-

= (1/2)[1 – (1 – m)2]pL + (vJ – vo) – M- > 0.

The inequality holds because

(vJ – vo) – M- = (1/2)[1 – (1 – m)2](M- - pL – M- )/[1 - (1/2)(1 – m)2] > 0

and

M- - pL – M- = D(p-m)( p-

m - c-) - D(c-)(c- - c-) - D(p-m)(p-m - c-)

> D(p-m)(p-

m - c-) - D(c-)(c- - c-) - D(c-)(p-m - c-)

= D(p-m)(p-

m - c-) - D(c-)(p-m - c-) > 0. o

Appendix C: Steps leading to Proposition 6 are as follows. The non-defection constraint

is

– k + m(1 – m)d[(M- /2)/(1 - d) - qva - (1 - q)vo] ≥ 0,

which can be written:

(A2) – k/[m(1 – m)d] + (M- /2)/(1 - d) - vo > q(va – vo).

We find, after manipulation, that

va – vo = {M- /2 – k + [m2 + 2m (1 –m)](M- /2)d/(1 - d)

30

+ (1 – m)2(1 - q)dvo}/{1 - (1 – m)2qd} – vo

= {M- /2 – k + [m2 + 2m(1 –m)](M- /2)d/(1 - d)

- [1 - (1 – m)2d]vo}/{1 - (1 – m)2qd}.

Substituting into (A2) and multiplying both sides through by {1 - (1 – m)2qd} > 0 yields:

{1 - (1 – m)2qd}{– k/[m (1 – m)d] + (M- /2)/(1 - d) - vo}

> q{M- /2 – k + [m2 + 2m(1 –m)](M- /2)d/(1 - d) - [1 - (1 – m)2d]vo}

Working on both sides, we simplify this inequality as

{– k/[m (1 – m)d] + (M- /2)/(1 - d) - vo}

> q{M- /2 – k/m + (M- /2)d/(1 - d) - vo}.

Since both sides of the inequality are positive, we get:

q < A/B = q*

where A and B are given in the text. q* is less than unity if

B – A = k[1 - ( 1 - s)d]/[s(1 – s)d] - (M- - M- )/2 > 0.

This leads to the condition in Proposition 6. o

31

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Figure 1

d

BD

BN

AB

NA

c_

c*0

1/[1 + (1 + m)2]

1/[2(1 - m)2]

1

c

_

1/2

Ad

Bd