Two-stage R&D competition: an elasticity characterization.

• Author: Joshi, Sumit

1. Introduction

   The literature on strategic investment in R&D in atemporal multistage games has focused on comparisons of the R&D expenditure levels obtained from models ranging from complete non-cooperation, and through varying degrees of cooperation, to complete cooperation [2; 3; 4; 6; 8; 9; 11; 13; 14]. This literature has tended to downplay the impact of three important factors on the obtained solutions: (a) the possibility that the R&D expenditures build on some pre-existing stock of technological knowledge [7]; (b) the possible differences between firms in terms of their capability to absorb knowledge in the public domain [1]; and, (c) the various possible functional forms describing the transformation of R&D inputs into technology outputs [10] and those into cost savings.

   This paper builds on the usual model specification in much of the existing literature to include initial stocks of technological knowledge and differentiated rates of spillovers. In addition, various specifications of the production cost function and the production function of knowledge are being considered. The paper offers a new characterization of the non-cooperative subgame perfect equilibrium. It is shown that the effect of a change in the initial stock of knowledge and own and rival spillover rates on the firm's equilibrium level of R&D investment depends crucially on the relationship between (a) the elasticity of Nash equilibrium outputs with respect to R&D expenditure; (b) the elasticity of technical knowledge with respect to R&D expenditure; and (c) the degree of convexity of the unit cost function.

   The rest of the paper is divided into three sections. The following (second) section sets up the model and characterizes the solutions at a general level of specification. The third section specifies parametric functional forms for market demand, the cost function (four alternative specifications) and the production function for technological knowledge (two specifications). It then discusses the solutions in the resulting eight parametric representations of the setup. The fourth section concludes.

2. The Non-Cooperative Duopoly

   Consider a Cournot duopoly consisting of firms i and j producing a homogeneous product and facing a negatively sloped demand P(q) = p([q.sup.i] + [q.sup.j]) in the product market. Competition takes place in two stages. The first determines the expenditures on R&D and their impact on the available technical knowledge ([K.sup.i], [K.sup.j]). The second stage determines production costs and the equilibrium levels of output. As usual, the model is solved recursively.(1)

   Stage 2: The production cost function for firm i is given by:

   [C.sup.i]([K.sup.i], [q.sup.i]) = [F.sup.i]([K.sup.i])[q.sup.i] (1)

   where [F.sup.i.[prime]] [less than] 0, [F.sup.i.[prime]] [greater than] 0. The firms play a Cournot quantity game in the product market. The objective of firm i is to choose its output [q.sup.i] to maximize its second-stage profits:

   [[Pi].sup.i] = P(q)[q.sup.i] - [F.sup.i]([K.sup.i])[q.sup.i]. (2)

   This quantity game determines the Cournot-Nash equilibrium quantities as a function of the tuple ([K.sup.i], [K.sup.j]):

   [q.sup.i] = [Q.sup.i]([K.sup.i], [K.sup.j]), [q.sup.j] = [Q.sup.j]([K.sup.i], [K.sup.j]) (3)

   Stage 1: Given some initial stock of knowledge ([Mathematical Expression Omitted]), the two firms undertake R&D expenditures ([x.sup.i], [x.sup.j]) in a non-cooperative manner. These expenditures are transformed into new technical knowledge for Firms i and j via the production functions [g.sup.i]([x.sup.i], [x.sup.j], [[Theta].sup.j]) and [g.sup.j]([x.sup.i], [x.sup.j], [[Theta].sup.j]) respectively. Here, [[Theta].sup.i]([[Theta].sup.j]) denotes the rate at which spillovers accrue to Firm i (Firm j) from the R&D expenditure of the rival firm. The total...