JOURNAL OF ECONOMIC THEORY 40, 349-363 (1986)

Research and Development with Publicly Observable Outcomes*

MICHAEL PETERS AND RALPH WINTER

Universify of Toronto, Department of Economics, 150 St. George St., Toronto, Ontario, MSS IAl. Canada

Received November 13, 1984; revised December 13, 1985

This paper studies the simultaneous evolution of costs and research incentives in a model in which two firms research each period by purchasing a draw from a stationary distribution of possible production costs. Research decisions each period are based on the results of previous research which are common knowledge. When research stops a Bertrand pricing game occurs. We give conditions under which the perfect equilibrium of this game involves a race by both tirms to reach some endogenously determined critical cost level. The first firm to reach this critical level wins the race, capturing the entire market, and continues to research until a second critical cost level is reached. Journal of Economic Literature Classification Numbers: 022, 621. ( 1986 Academic Press. Inc.

1. INTRODUCTION

The theory of search, which is well developed for the case of a single economic agent, has recently been extended by Reinganum [2, 31 to the “strategic” case of simultaneous search by many economic agents. In the context of an R & D model in which firms search for new production processes, Reinganum proves that a Nash equilibrium exists and that the reservation property of optimal search rules generalizes to the strategic case. These results are based on the assumption that the outcome of research by each agent is private information, observed by that agent alone. An important element of the strategic interaction among firms undertaking R&D-that each firm’s research decisions depend upon its

* We would like to thank N. T. Gallini for useful discussions, J. Reinganum for valuable communication on a related paper. and an associate editor and referee of this journal for their particularly insightful comments.

349 0022-0531/86 163.00

Copyright t 1986 by Academs Press. lnc All nghla of reproducuon in any form reserved.

350 PETERS AND WINTER

perception of how its rivals will react to changes in its technology-is left unstudied in the extension of search theory to the multi-agent case.

This paper characterizes a perfect equilibrium of an R & D model in which the outcome of individual research investments is common knowledge. In this model, research decisions at any time depend on the evolution of all firms’ technologies to that point in time. The model adopts special assumptions: Payoffs to research decisions are derived from Ber- trand duopoly in a market with a demand that is perfectly inelastic (up to some price). But we find that the simple generalizations cause the obvious solution technique to break down. Some of the difficulties associated with extending the results to more general environments are discussed in the final section of the paper.

Section 2 outlines the basic model of R & D and the equilibrium concept. Section 3 derives an equilibrium for the competitive R & D game. Section 4 isolates the impact of rivalry on the extent of research in our model by comparing the equilibrium with the case in which firms split the market instead of competing. The final section summarizes briefly our results and considers the extensions of our technique for the state-space charac- terization of an equilibrium to more general models.

2. THE BASIC ASSUMPTIONS

We consider a market with two firms in which competition takes place in two stages. In the first stage, the firms conduct research. This research results in some pair of production costs c = (c,, c,), where ci is firm i’s (constant) per unit production cost at which the production phase takes place in the second stage. Profits are given by n,(c) for firm i. The profit function can be given various interpretations depending upon the com- petition that takes place at the production phase. For most of the paper n,(c) will represent the profits earned by firm i in a Bertrand pricing equilibrium with its competitor. However, we will also consider the case where there is a market sharing agreement in the production phase. For most of the paper it is assumed that demand for the product is perfectly price inelastic and equal to one unit up to some cutoff price p, at which it falls to zero.

The R & D phase of the game is assumed to take place in discrete stages. At the first stage, firms share a technology that yields a per-unit production cost C, and at each stage, a firm may investigate a new research avenue or not. Researching costs K dollars and yields a random unit cost of produc- tion distributed on the interval [0, C] according to a stationary distribution F, At each stage subsequent to the first, the results of the research at the previous stage are realized and made known to both firms before the next

RESEARCH AND DEVELOPMENT 351

research decisions are taken. When both firms terminate research, produc- tion takes place at the pair of minimum costs discovered to date and the profits 17, (. ) and Z7,(. ) are realized.

The two firms research simultaneously at each stage. Each firm’s strategy at any stage is allowed to depend on the entire history of play up to that stage. The set of histories is defined inductively. Let

H, = { {O,O> x (c, F}, (0, l} X {c} X [O, Cl,

(LO} x co, 4 x {T}, { 1,1$ x co, d2)

represent the set of possible histories after one stage. For example, if firm 1 researches in the first period and firm 2 does not, then h, = { 1, 0, c”, , C}, where ?, is the actual draw from the cost distribution acquired by firm 1 during its research. Define H, = H, _ I x H, and let H = U ,“= , H,. A strategy si for firm i is a mapping from H into { 0, 1 } and S denotes the set of all strategies. Generalizations of these, and the following definitions to allow for mixed strategies are immediate. Since we consider only pure strategy equilibria we will ignore them.

Let ?,(A,) be minimum cost drawn by firm i through t periods when the actual history is II,. Given h,-, the payoff to firm 1 when its uses strategy s, and firm 2 uses strategy s2 is defined to be

u,cs,, S?, h,-,I

= lim &TIS,.S2,h,-, T-cc { 17[CP,(h.), b@T)l- i ~,uc3.~ 3

!%=I I

where Irk’ is the projection of h, onto H,. In particular, if with probability one both firms terminate research after a finite number of stages under the strategy pair (sr, sZ), the payoffs are the expected values of II, and Ii’, at the minimum costs discovered up to the (random) terminating stage minus the expected research expenditures.’

A perfect equilibrium for this game is a pair of strategies ST, s? satisfying

~,C~:,~:,~~-ll~~IC~I,~:,~r~11

for all sI ES, h, , E H, _ , and t = 1, 2 ,...,

u,cs:, $7 A,- ,I 2 bcs:, s2, A, ,I

for all s2 ES, h, ~ 1 E H,- , and t = 1, 2 ,... .

(1)

(2)

’ Of course, we expect that any equilibrium strategies will involve, with probability one, ter- mination after a finite number of steps. The payoff functions defined in (1) are complicated because of the requirement that they be delined on arbitrar~~ admissible strategies.

352 PETERS AND WINTER

We will construct a perfect equilibrium for this game. In fact, the solution will satisfy a stronger definition of equilibrium. Define a perfect state-space equilibrium to be a pair of strategies ST and ST such that the pair (ST, ST) constitutes a perfect equilibrium and

whenever

s)fyh,) = s:(h;) for i-1,2, (3)

?;(h,) = e,(h;) for i= 1, 2.

This merely requires that all histories that lead to the same minimum cost pair give rise to the same strategies. Notice that this restricts the set of per- fect equilibria that will be admissible as equilibria but it does not restrict the firms strategy spaces per se; an alternative formulation, the state-space game, would restrict the strategy spaces to the set of mappings from the state space (of minimum cost pairs) to (0, 1 }.

In a perfect state-space equilibrium, the values of the game can be characterized completely by the lowest costs so far discovered by the com- petitors. Thus we can define

s!qc,, c2)=s,*(P,(h,), ?,(h,))

V,*(C,) c-2) = zjp[s,*, s;, h,]

for all h, satisfying E,(h,) = c,, ?,(A,) = c,. Now let

if c’<c, and S,*(c,,c?)=O

G[c’; S,*(c,, cz), (c,, cl)] = if c’< c, and ST(c,, c2) = 1

if c’>c,,

then q

WC,, c,)= SI ” Ws,, s,)dGCs,; WC,, c,), Cc,, cz)l 0 0

x~GCs~;S:(c,,c,),(c,,c~)l--S:(c,,c~).K. (4)

3. STATE-SPACE EQUILIBRIA : THE COMPETITIVE CASE

Characterization of an equilibrium of this game involves the simultaneous construction of a pair of value functions and a pair of strategies that satisfy Eq. (4) and cannot be improved upon at any minimum cost pair. In this section we construct an equilibrium for the case where both firms are Bertrand price competitors in the production phase.

RESEARCH AND DEVELOPMENT 353

The first step in the construction is to show that at sufficiently low costs the pair of strategies “don’t research” for each player, together with value functions given by 17, and ZZ2 constitute an equilibrium, where n, and 17, give the Bertrand equilibrium profits of each firm. We then proceed to extend this equilibrium over the remainder of the state space using a simple iterative procedure. It should be noted at the outset that the efficacy of this procedure is very sensitive to our assumptions about the L7,.

The equilibria of the state-space game are not unique. For example, when costs are close and intermediate in value, research by either firm may be a pure strategy equilibrium and a mixed strategy equilibrium may exist as well. We restrict our attention to a pure strategy equilibrium that is sym- metric, leaving aside the complete characterization of all equilibria.

The nature of the pricing outcome when both firms stop is simple in the Bertrand case. At c the low cost firm sells at a price equal to the price of the high cost firm. The profit function is then defined simply as

if c1 <c2 otherwise.

Firm two’s profit function is defined similarly. To construct an equilibrium, suppose that there is a region where both

firms have very low costs and where both firms find it best to stop at all lower cost levels. The boundary of this region can be found by computing the payoff to a firm that deviates to its “continue” strategy for one period, then reverts subsequently to the equilibrium strategy stop.2

‘It is clear in this, and subsequent tests of equilibrium strategies, that any proposed strategy that cannot be improved upon with a one-step deviation cannot be improved upon by any finite-step deviation. As two referees have pointed out to us, however, because we do not have discounting we must also consider the possibility of dominating strategies that deviate from the proposed strategies infinitely many times. The following argument, along the lines suggested by an associate editor, proves that any proposed strategy that can be improved upon in an infinite-step deviation, can also be improved upon in a finite-step deviation. Let s; improve upon sr for tirm 1 so that u,[s;, s2. h, t] > e,[s,, s2. 11,~ ,I. Define the strategy .s[ such that ,s: follows s; up to period T and reverts to s, thereafter. Then

= lim EhT15,.+-, I-,

u,[s,,s2,hT]- i: .Y;(h:).K

n=,

354 PETERS AND WINTER

Take c2 as fixed. If firm 1 deviates for one period to its “continue” strategy, its expected payoff is given by

J (‘I

n,(s,C,)ms)+ Cl -Qc,)l n,(C)--. 0

The gain to researching instead of stopping is then given by

G,(c) = Ji’ z7,( $7 c2)dF(s)-F(c,)l7,(c)-K

_! J (‘2 o (C?-s)dF(s)-K

J ci i o (C*-S)dF(S)-F(c,)(cz-c,)-K

for cl bc2

I =~(c,).c,- J).w(s)-~ for c, dc?. (5)

This gain is increasing (strictly) in c2 if c, > c2 but is independent of c2 if c1 -L c2. Consider, therefore, states where c, = c2 and imagine increasing c,

x Ii’,[f,(h,).c:,(h,)]- i s’,(h;).K i “=I I

In each case that we test for equilibrium here, firm 2’s strategy has two properties: first, once two stops it never restarts; second, since two stops on an open neighbourhood of 0,

lim,,, Prob{hT: s2(/z,.)=O} = 1 for all S,. The second to last line in this expression then follows because if hT is such that 2 stops, then 1 can do no worse using s than it could by stopping forever and taking cl(h,.) against c,(h.) into the second stage. If 2 continues after hT, 1 can stop forever and do no worse than zero. But if limr,,X u,[s~. So, h,-,I> V, [s;, sz, h,- ,] > v1 [s,, s2. h,- t], then there exists some T large enough such that S, can be dominated by a finite set of deviations from the equilibrium path.

RESEARCH AND DEVELOPMENT 355

and c2 simultaneously. If c, = c2 = 0, then from (5) the gain to firm 1 from researching is negative. Since the gain is increasing in the joint cost, there is at most one critical value c* such that both firms are satisfied to stop when their (equal) costs are less than or equal to c*.

Now if we begin at any common cost c1 = cZ d c* and cut firm I’s cost unilaterally, then by (5) the gain to continuing for firm 1 falls below zero (since this gain is increasing in c,, in this range). On the other hand if firm 2’s costs are cut unilaterally, the gain to 1 from researching, which is at most zero, cannot increase (since G, is nondecreasing in cZ). Hence we conclude that 1 prefers “stop” to “continue researching for one draw, then stop” given that 2 stops, and vice versa. It is easy to extend this argument to show that the strategy “stop” dominates any other finite-step departure from “stop” (such as “continue for two draws, then stop”); by the argument in footnote 2, “stop” is therefore optimal. Thus research ter- minates at or below c* and nj give the appropriate value functions for this range of the state space.

This argument can be extended a bit further. Begin again at any comon cost less than or equal to c*. It is easily verified that raising either c, or c2 unilaterally does not affect the gain to research to either player. Therefore if either c, or cz is below c*, a steady state equilibrium will arise with both firms stopping and agreeing to charge their Bertrand prices. (See Fig. 1.) It is clearly possible that c* 3 C. In this simple case the solution to the game is trivial. But for small fixed costs c* will be less than c.

Now suppose both firms’ costs are slightly above c* and that c2 > c,. At such a point at least one of the firms must choose the strategy “continue” in equilibrium. We will concentrate on the equilibrium where firm 1 con- tinues at (c,, c?) while firm 2 stops. At the point (c?, c, ) (where the costs of the firms are interchanged) we will analyze the equilibrium where firm 2 continues. It will become apparent that we could also construct an equilibrium where firm 2 continues when c, < c?. (Though we do not analyze these, there should also be mixed strategy equilibria where each firm “continues” to research with positive probability.)

What we wish to show is that there is an equilibrium in which another critical cost level c** defines a “winner” for the state-space game in the sense that the first firm to reach c** captures the entire market in the ex post Bertand equilibrium. We do this by showing that in the region defined by c* < L’ < c** and c, < c2 an equilibrium exists which has firm 1 continuing to research while firm 2 stops. To show that these strategies are indeed equilibrium strategies, we need only construct a value function that has the properties that these strategies are best replies to each other.

We begin by observing that if the equilibrium has firm 1 continuing and firm 2 stopping, then firm 1 will continue until it attains a cost level c* or lower. Hence let us define the following value functions:

356 PETERS AND WINTER

if c,dc* or c,<c*

V,(c,, c,)= dF(s) = n,(c*, c2) (6)

if c*<c, <c,<c**

0 if c*<c~<c, Cc**.

The equality in the second part of the definition (a standard result in single-agent search theory) is clear from the observations that the left-hand side is independent of cr in the corresponding region and V, is continuous and equals I7,(c*, c2) at c, = c*.

A value function, V,, for firm 2 is defined symmetrically. We have established already that firm l’s best reply to stop is to continue. Hence to show that the Vi’s as defined are value functions for the problem, we need show that “stop” is optimal for 2 given l’s strategy of continuing on {c,,cI!Ic*~c~~c**,c~~c~, 1 for some c**. As before it suffices to con- sider the plan for 2 of deviating from its “stop” strategy by researching only once; if researching once does not pay, researching any number of times does not pay.

A positive payoff to firm 2, if it follows the “continue” strategy, arises only if firm 2 draws a lower cost than firm 1 on the first draw, for then the candidate strategies imply that firm l’s best reply will be to stop forever. The payoff to firm 2 if it continues for one more period (given that one continues) is then

1" is I7,[s, t] dF(f)dF(s)+ j"S‘ V2[s, t] dF(t)dF(s) 0 0 c* 0

+ [I -F(c,)]so" V2[cl,f]dF(t)-K (7)

This is independent of c,; this independence arises from the fact that the payoff to the high cost firm in the final equilibrium is zero. Hence if it is best to stop when c, = c2, it will be best for firm 2 to stop for any higher cost level.

Evaluating (7) at c, = c* establishes immediately that if both costs are c*, firm 2 strictly prefers to stop if firm 1 continues. Since (7) is increasing m c,, there is at most one value c** at which firm 2 becomes indifferent between the options continue and stop. Symmetric arguments cover the converse case.

To this point, the set of costs with at least one firm below c** has been partioned into sets where one firm researches alone or where both firms

RESEARCH AND DEVELOPMENT 351

terminate research in equilibrium. This leaves a final region, A = (c 1 c,, c2 > c**} over which the equilibrium must be defined. (A will be non-empty if the cost of researching is sufficiently small.) An appropriate guess for the equilibrium strategies on this region is that both firms research. To show that this is indeed the equilibrium, we derive the expected payoff to each player on A under the supposition that both players continue until c** is reached and then follow the equilibrium strategies. We then verify that this pair of expected payoffs supports the continue strategies as individually optial on A.

The value functions on A are derived in the Appendix. To prove the individual optimality of the proposed strategies on A, however, it suffices to express the value functions as follows. Let x:, .x: ,... and s:, ,x: ,... represent independent random variables with distribution F. For any (c,, c2) in A, let the random variables T and c: be defined as T= min{ t 1 X; or .x; d c** 1 and c: = min{ .yf ,..., s;, c,:. Then let

V~(~)-E[V;(CT,C~)-K(T- l)],

where the expectation is over the joint distribution of T and cf, I$,..., i= 1,2. (Note that the term Vi(cT, CT) is well defined since (cf, CT) is in the region over which the equilibrium has been constructed.) The value function Vi defined on A by (7) inherits the monotonicity properties of V, on the lower regions; it is nonincreasing in ci and nondecreasing in c,, for j# i.

The value functions V, and V, on A have been constructed to satisfy condition (1). To demonstrate individual optimality, i.e., to show that con- tinue is a rational response to continue on A, it suffices to consider the payoff (say to firm 1) from stopping alone for one period. Using the fact that if firm 2 draws a cost below c ** it is optimal for 1 to stop and earn zero; this payoff is

s ‘* V,(c,, 3) @s) + Cl - F(c,)l f’,(c,, cz). <**

Since VI is increasing in c?, the payoff to the stop strategy is the mean of payoffs that cannot exceed V,(c,, cZ). Hence “continue” will always dominate “stopping” for one period (and, by an extension of this argument, for any number of periods) for firm 1 on A. Symmetric arguments apply for firm 2.

This completes the characterization of equilibrium in the competitive (Bertrand) case: In the equilibrium derived, both firms research until at least one reaches c** or below, then the lower cost firm continues until c* is reached. The resulting evolution of firm costs is depicted in Fig. 1.

358 PETERS AND WINTER

=2

C*

FIG. 1. The evolution of costs in equilibrium. (In regions marked SS. (stop, stop) is an equilibrium pair of strategies.)

4. COOPERATIVE EQUILIBRIUM

A traditional question in the R & D literature is: What is the impact of rivalry on the equilibrium levels of R & D? One result established along this line is that for R & D models in which the research game is assumed to take the form of a “race” with a single winner (of a patent), rivalry’ increases equilibrium research (e.g., Dasgupta and Stiglitz (1)). In the model of this paper, the contest-like payoff structure to the research game, in which one firm makes positive profits while the other makes zero profits, is endu~e~aus rather than assumed. An additional distinguishing feature of the model is that the reward to the winner depends upon the best technology discovered to date by the rival.

The issue of the impact of rivalry in our model can be addressed by com- paring the non-cooperative equilibrium of the previous section with an equilibrium in which firms split the market (say, geographically) prior to research. Splitting the market is equivalent to cooperating perfectly on market shares (set equal to one-half) and price (set at p) but deciding individually on research.

The result of this comparison is that rivalry expands research efforts in this model. The essential reason behind this result is that in the cooperative equilibrium, research serves only to reduce (stochastically) the cost of production of a known quantity of output. In the non-cooperative case, firms rationally anticipate the impact of research on market shares as well as costs.

To prove this result, note that since the price and market share that each firm receives are independent of the actions of the other firm, the state space in this case can be broken down into separate state spaces [0, F] for

RESEARCH AND DEVELOPMENT 359

each firm and includes two symmetric regions, the region where both firms stop, denoted by [0, Z], and the region where both firms continue [?, C]. The cost level ? is determined by the condition that the gain to researching once (for either firm) be zero at S. To show T > c*, we need show that at the cost pair (S, c”) the gain to researching once, in the non-cooperative research game, is positive.

The benefit of researching once in the non-cooperative game (ignoring the cost K) is twice that in the cooperative game since in the former the successful researcher captures the entire market, not just half the market. Because at (?, T) the benefit equals K in the cooperative game, the nrt benefit from researching at (2, S) in the non-cooperative case must therefore be 2K - K = K. The market sharing agreement induces the ,firms to cease research hcfore the-v would in the strictly non-cooperative case.

5. EXTENSIONS

The technique used to solve for the state-space equilibrium in this research game is recursive. Once the equilibrium has been established on some set in c,-cZ space (which includes the origin and has a nonincreasing boundary), a guess is made about the equilibrium strategies on a neighbourhood of the set. If the neighbourhood has the property that under the candidate strategies costs evolve within the neighbourhood or original set, then the value functions yielded by the strategies can be calculated; one then verifies that the value functions support the candidate strategies as individually optimal. The procedure is then repeated on a new neighbourhood and so on, until the state space is exhausted.

This technique does not appear to work in the two most natural exten- sions of the model: Bertrand competition with downward-sloping (rather than L-shaped) demands and Cournot competition in the post-research production phase. To consider the first of these, note that in the equilibrium of the Bertrand pricing game the low cost firm charges a price equal to the minimum of the high cost and the monopoly price at the low cost. As the high cost is raised, provided it is below the monopoly price, the market equilibrium quantity decreases when demand slopes downward. The incentive for the low cost firm to reduce its cost further through research (which depends only upon this quantity) therefore falls.

This effect means that the set where “terminate research” is a best response for firm 1 to “terminate research” for firm two, has an upward sloping boundary in the region where c2 > c, (Fig. 2). The corresponding set for firm 2, however, has a vertical boundary above c* (defined as the intersection of these boundaries with the 45” line). This means that in con- trast to the model analyzed in Section 3, the strategy pair “continue, ter-

360 PETERS AND WINTER

C*

IiIIIiIlIllllIl C* =1

FE. 2. Sets where “Terminate Research” is a best response to “Terminate Research” for each firm, under downward-sloping demand curves. Horizontal lines mean: Terminate is a best response for 1. Vertical lines mean: Terminate is a best response for 2.

minate” for the respective players cannot be an equilibrium pair near the boundary of the set (c: cz 3 c,, c* 3 c , ). The natural strategy pair to select as a candidate for equilibrium is “terminate, continue” but the evolution of firm 2’s costs under this strategy pair leads outside the set where the equilibrium has already been established. The value function on this neighbourhood must be determined simultaneously with the value function on a neighbourhood where a different equilibrium strategy pair prevails. Establishing the existence of an equilibrium then requires the solution of a difficult fixed point problem.

We do not yet know whether the problem where firms play an ex post Cournot game is amenable to solution using the recursive technique. The difficulties in this case are quite different from those that arise with Ber- trand competition and downward sloping demand. Some of these dif- ficulties can be seen using the results of Reinganum [4]. These results indicate that it is natural that the boundary of the set of cost pairs where firm l’s best reply to stop is to stop, be downward sloping in the space of (c,, c2) pairs. Once this boundary is crossed firm l’s strategy must revert to continue and remain this way until another boundary is crossed. At this new boundary it is again conceptually simple to see what the new equilibrium strategy pair must be, and the recursive technique can be applied. The success of the recursive technique, however, depends upon the behaviour of the boundaries. They are always weakly monotonic, but may have flat or vertical segments that create discontinuities in a value function defined at higher costs. Furthermore, application of the recursion of techni- que from the stop-stop region does not obviously fill up the space of costs. The existence of equilibrium in the case of an ex post Cournot game, a natural model of the evolution of costs and research incentives, remains unproved.

RESEARCH AND DEVELOPMENT 361

APPENDIX: THE DERIVATION OF THE VALUE FUNCTION ON A

Let F,,, with density ,f;,, be the distribution of the minimum order statistic from a sample of size II drawn from F. We show here that the expected payoffs to firm 1 at any cost pair (cl, c,), assuming that both firms research simultaneously until c** is reached by at least one and follow equilibrium strategies therefater, is given by

- K(2F(c**)- [F(c**)]‘\ ’ (Al 1

(and similarly for firm 2) where g( ., ), which is integrated as the density over the lowest of the costs drawn subsequent to the current position (C,, c,), up to the realization of one cost below c**, is given by the following:

on (s, ,.s2<c**;,

g(s,, .s2)= .,f(s,).f’(s,)/{2F(c**)- F’(c**)j,

on s, 6~ **, .Y2 > c**,

g(s,,s2)= 2 fjs,)j;Js,l. [I -F’ ‘(c**,], ,I= I

and similarly on S, > c* *, s7 d c**. The function I’,[., .] defined for s, or _ s2 6 c** is defined by (6).

To prove this let c: be the rth draw from a sequence of pairs of indepen- dent draws from the distribution F, and let cl’ = min,,,,, cj. The dis- tribution of c?: is given by F,,,(x) = [ 1 - [ 1 - F(Y)]“‘]. Define the random variable

N=min{mlc’;Gc** or cpldc**i.

The probability that either c;‘d c** or cy< c** or both is equal to 2F( c** ) - [ F(c**)]~ s p. Therefore the probability distribution of N is given by Prob{N=m}=(l -p)‘“-’ p. The expected cost of research for any firm is then EN.N.K=K/p=K/{2F(c**)-[F(c**)]~) in any state (c, 3 c,), where c, >c** and c2 > c**. This is the final term in expression (61.

362 PETERS AND WINTER

The probability distribution over terminal states is then given by

Prob{C;Y<s,;Cf<s,J

% = c Prob{c;‘~sl;c;‘6s,;~;‘-‘3c**;E4’3c**)

n, = I

if s,<c** and s,<c**,

or c**<c;“-‘<s,})

if s, > c**, s2 < c**,

= i: F(s,)Fb,) F,,,- ,(c**)F,,, ,tc**j

>>I = I

if s, <c** and s,dc**,

% = c F(s,)F,,,--l(C **){Fb,)Cl -F,,~,bz)l

171 = I

+ CF,~~,(.~,)-F,,,~,(c**)li

if s,dc**,s,>,c**,

= f F(s,) F,,-,(c** ){F(s,)Cl -Fm~,(s,)l

m=l

+ CFwIts,)-Fe ,(c**)l)

if s, 3 c**, s2 < c**.

RESEARCH AND DEVELOPMENT 363

To get (Al) note that when s, <c** and szdc**,

f Fb,) F(s2) FM- ICC**) F,-,(c**) n,= I

For s,>c** or s,>c** note that the derivative of

Fb,)L-1 -F,v,b,)l+ CFmm ,(sz)-Fw ,(c**)l

is equal to the derivative of

F(s,)Cl -FM- i( +F,,,-,(S,)=F,h,).

REFERENCES

1. P. DASGUPTA AND J. E. STIGLITZ, Uncertainty, industrial structure and the speed of R & D. Bell J. Econ. (1980). l-28.

2. J. REINGANUM, Strategic search theory, Int. Econ. Reu. (1982), l-17. 3. J. REINGANUM. Nash equilibrium search for the best alternative, J. Econ. Theory 30 (1983).

139- 152. 4. J. REINGANUM. Technology adoption under imperfect information, Be// J. Econ. 14 ( 1983).

57-69.