Strategic commitment with R&D

375

Economics Letters 21 (1986) 375-378 North-Holland

STRATEGIC COMMITMENT WITH R&D The Case of Bertrand Competition Tom K. LEE National Uniuersity of Singapore, Kent Ridge, Singapore 051 I Received 11 March 1986 Final version received 24 April 1986

When there is multiplicatively separable demand and firms compete in the price dimension, there is no room for strategic use of R&D by firms. Under a different demand situation, strategic use of R&D by firms competing in the price dimension reduces R&D spending, reduces output, increases price, and increases profits of each firm.

1. Introduction The purpose of this paper is to show that the equilibrium levels of R&D of a firm in an industry is the result of the interaction of economic forces such as product demand, the form of competition in the product market and the form of competition in the R&D market. The basic economic points are as follows. R&D is derived demanded. Different demand structures, different forms of competition in the product market and the different forms of competition in the R&D market yield different levels of derived demand for R&D. It is the interaction of these economic forces of the derived demand for R&D that determines the equilibrium R&D spending of a firm in an industry for a given cost structure of R&D. The rest of the paper is organized as follows. In section 2 we specify a two stage game model where firms choose R&D levels, these R&D levels are made known to each firm, and then price levels are determined. A subgame perfect equilibrium of this game is characterized and then compared to an equilibrium of a simultaneous game where firms simultaneously choose price and R&D levels. All the results are derived in this section. 2. The models Following Brander and Spencer (1983), we consider a symmetric duopoly producing substitutes. Our point of departure is to consider price and R&D competition instead of output and R&D competition. We shall derive results opposite to theirs. Let xi be the level of R&D spending chosen by firm i, and pi be firm i’s price level. The profits from production for firm i are denoted by R’(pi, p2, xi), i = 1, 2. 2.1. The simultaneous

game model

Without production

loss of generality, consider net of R&D spending,

WPI,

x1)

P2,

0165-1765/86/$3.50

firm 1. Firm

1 chooses

-x1.

0 1986, Elsevier Science Publishers

B.V. (North-Holland)

p,

and x1 to maximize

profits

from

376

T. K. Lee / Strategic commitment

Assuming conditions R: =0

the existence of a symmetric are given by and

Ri-

Nash equilibrium

with R&D

of the simultaneous

game, the equilibrium

1 =O,

(1) ,(2)

where Rf is the first partial derivative of R’ ( pl, p2, xi) with respect to its jth argument, j = 1, 2, 3, and i = 1, 2. Eqs. (1) and (2) together determine the symmetric equilibrium values of p and x for the simultaneous game. 2.2. The two stage game model Without loss of generality, consider firm 1. Firm 1 chooses pl, p2, x1 and the multipliers a and b to maximize production profits net of R&D spending subject to Nash equilibrium conditions in the product market, R’(p,,

pz, x,)-x,+aR;+bR;.

Assuming the existence of a symmetric stage game are given by eqs. (1) and

Nash

equilibrium,

the equilibrium

conditions

for the two

aR;, + bR;, = 0,

(3)

R\ + aR& + bR& = 0,

(4)

R’, - 1+ aR& = 0,

(5)

of R’(p,, p2, xi) with respect to its jth and kth where Rik is the second partial derivative arguments, j = 1, 2, 3, k = 1, 2, 3, and i = 1, 2. From eqs. (3) and (4), we can solve for the value of the multiplier a, and substitute it into eq. (5) to obtain the following:

R:,R:R2,,

R;-l+

R;,R&

- R&R&

0.

(5’)

=

Eqs. (1) and (5’) together determine the symmetric equilibrium values of p and x for the two stage game. Stability of the Nash equilibrium in the product market requires that the denominator of the third term on the left-hand side of eq. (5’) be positive. The difference between the equilibria of the simultaneous game and the two stage game depends on the difference between eqs. (2) and (5’) which in turn depends crucially on the sign of the numerator of the third term on the left-hand side of eq. (5’) that is, whether Rt3R\R& is positive, zero or negative. 2.3. A comparison of the two models Consider 4, = 4,(Pl?

the system of demand P2),

i= 1,2.

functions

given by

T. K. Lee / Strategic commitment

Then the profit

from production

R’= (P,- +iMP,?

P2)?

We have the following

Ri3 =

-d(x,)$

371

with R&D

to firm i is given by i = 1,2.

values for eq. (5’):

co,

R& = -c”(xl)ql

CO,

2

R;=(p,-c(x,))+O. 2

If demand

functions

4;=hi(Pl)ki(P2),

are of the multiplicatively

separable

form,

i= 1,2,

then we can show that R & = 0. With R&R\R&

= 0 in eq. (5’), we conclude:

Proposition I. If demand is multiplicatively separable, then the equilibria of the Bertrand two stage game and the Bertrand simultaneous game are equivalent. There is no room for strategic use of R&D, On the other hand Ri3RiRi1 < 0 if Rzl > 0, i.e., the best response function of firm 2 is upward sloping. An increase in the price of firm 1 increases the best response price of firm 2. Now consider eqs. (1) and (2) for this alternate case. In the price and R&D expenditure space as depicted in fig. 1, the slope of the graph of eq. (2) is negative. The slope of the graph of eq. (1) is also negative by a second-order condition for profit maximization with respect to price choice. For stability, the graph of eq. (1) has to cut the graph of eq. (2) from below in the price and R&D expenditure space. The intersection point of the graphs of eqs. (1) and (2) determine the symmetric equilibrium values of the price and R&D expenditure of the Bertrand simultaneous game which are denoted by pbS and xbs respectively in fig. 1. Since Ri3R;R& c 0, the graph of eq. (5’) must lie below that of eq. (2). The intersection of the graphs of eqs. (1) and (5’) determine the equilibrium values of,price and R&D expenditure of the Bertrand two stage game which are denoted by pb’ and xbt respectively in fig. 1. We have the following results: Proposition 2. Strategic commitment with R&D required to minimize the cost of output it produces. Proposition 3. neous game.

Each Bertrand firm undertakes

induces each Bertrand firm

less R&D

Proposition 4. Each Bertrand firm produces less output, the two stage game than in the simultaneous game.

to use less R&D

than

in the two stage game than in the simulta-

sets a higher price,

and earns more profit in

378

T. K. Lee / Strategic commitment

with R&D

PRICE

P”

Pb’

I

L

L

Xb Fig. 1. The Bertrand

competition

R & 0 EXPENDITURE *X

xt 1s

case.

References Brander, J.A. and no. 1, Spring, Dasgupta, P. and 266-293. Kamien, ML and

B.J. Spencer, 1983, Strategic commitment with R&D: The symmetric case, Bell Journal of Economics 14, 225-235. J.E. Stiglitz, 1980, Industrial structure and the nature of innovative activity, The Economic Journal 90, June, N.L. Schwartz,

1975, Market

structure

and innovation:

A survey, Journal

of Economic

Literature

13, l-37.