Innovation, imitation, and social welfare.

• Author: Pepall, Lynne M.

1. Introduction

   This paper focuses on the impact of competition by imitation, more specifically copying, for innovative activity and social welfare. In part, it is a response to recent empirical work |10; 12~ that has tried to measure the relationship between innovation and imitation costs. These papers include a direct call for theoretical modeling that takes explicit account of such cost relationships, and we try to answer this challenge. Specifically, we examine the implications for product quality and social welfare when innovators anticipate that a successful product will be copied by rivals.

   Obviously, previous empirical research is not the only source of interest in these issues. The potential effects of imitation on innovative activity have long been both a theoretical and practical concern. In recent years, a variety of authors have addressed the appropriability issue raised by imitation and its implications for innovative activity |4; 7; 2~. There has also been work on explicit copying, i.e., photocopying, |8; 11~. In the policy arena, the protection of product design and intellectual property rights is the focus of much recent actual and proposed U.S. legislation(1) and a major issue in the Uruguay round of trade talks |1; 21~.

   Our analysis of this topic differs from that of previous work in a number of important respects. First, we focus explicitly on product as opposed to process innovations. Second, and perhaps even more important, we do not restrict the firm's choice to be to innovate or not to innovate. Instead, we allow the innovator to choose what kind, or quality, of new good to introduce into the market. The higher the quality, the higher is the development cost. Third, when the innovating firm decides what kind of new product to market, it does so under uncertainty about demand. The popularity of the new product is not known until after the innovator has chosen the level of quality and sunk its product development costs. Fourth, at the initial time at which the innovator chooses the quality of product to develop, it does so with the knowledge that its innovation, if it is popular, will be copied by later rival entrants. Finally, in direct response to the empirical work mentioned above, we explicitly focus on the ratio of imitation to innovation costs.

   Our approach yields a number of insights not the least of which is a more precise identification of the potential gains and losses from imitative competition. Indeed, using a quadratic development cost function, we explicitly calculate the impact of copying on social welfare. This explicit welfare analysis further distinguishes our analysis from most previous work.

   We present our model and examine the innovator's optimal strategy in the next section. Then, in section III, we consider the model's predictions regarding quality, price and social welfare. We also discuss the consistency of these predictions with empirical evidence. A brief summary and some concluding remarks follow in section IV.

2. Uncertain Demand, Potential Imitation, and Product Choice

   We consider a new good whose quality is vertically differentiated. That is, the good embodies a characteristic (or a weighted combination of several characteristics), z, increases in which all consumers agree enhance product quality. Consumers disagree, however, regarding the value of increments in z. More specifically, consumer preferences are generated by the utility function:

   |Mathematical Expression Omitted~.

   U is the utility derived from purchasing one unit of a good with quality z at price p. |Theta~ |is an element of~ R+ is a taste parameter that varies over consumers. A consumer type is indexed by |Theta~, and we assume that |Theta~ is continuously and uniformly distributed over the interval |0, |Beta~~. Thus, the fraction of consumers with taste parameter less than |Theta~ is |Theta~/|Beta~. One may interpret |Theta~ as an index of taste for quality. The higher is |Theta~ the greater is the consumer's willingness to pay for a well-designed product. Alternatively, one may interpret |Theta~ as the inverse of the marginal rate of substitution between income and quality, z |22, 97~. In this view, all consumers derive the same benefit from the good. But because they have different incomes they have different marginal rates of substitution. If the marginal utility of income is diminishing, wealthier consumers will have lower marginal rates of substitution and greater willingness to pay for quality, i.e., they will have higher |Theta~ values.

   The demand for the new good with quality level z and price p is equal to the number of consumers whose taste parameter |Theta~ satisfies z|Theta~ |is greater than or equal to~ p. This implies the following inverse demand function:

   P(Q, z) = z|Beta~(1 - Q). (2)

   The most straightforward way to model firm uncertainty about the strength of market demand for its new good is to assume that the firm does not know the value of |Beta~, the strongest taste for quality in the consumer population. As |Beta~ increases, the demand curve rotates outward. Hence, as |Beta~ increases, the number of individuals willing to buy a good of quality z at a given price p also rises. We assume then that the parameter |Beta~ is unknown to the would-be innovating firm. However, the firm does know the distribution of possible |Beta~ values. This distribution is taken to be uniform, and for simplicity we normalize it to the unit interval so that |Beta~ lies between 0 and 1.(2) The product innovation decision facing the firm is how high a quality product it should develop and market. A number of factors influence this choice. To begin with there are development costs of increasing quality which we denote as K(z). We assume that these costs are sunk. We also assume that K(z) is convex with both K|prime~(z) and K|double prime~(z) positive. Marginal production costs, on the other hand, are assumed constant and for simplicity set to zero. Viewed only from the standpoint of demand uncertainty, developing a high quality product is a risky innovation strategy because it is less likely that demand will be sufficient to cover the greater (sunk) development costs.(3) However, consideration of potential competition from imitators may make such a choice more attractive. This is because we assume that copying costs are proportional to initial development costs, with the proportionality factor denoted by |Lambda~, 0 |is less than~ |Lambda~ |is less than~ 1. Thus, the copying cost borne by an imitator, |Lambda~K(z), also increases with the innovator's quality choice. Hence, by choosing a high quality design, the innovator increases not only its costs but its rivals' costs as well. This makes entry into the innovator's market less likely.(4)

   Innovation and imitation is a sequential process. We model this process as a three stage one. In the first stage, the innovator chooses the level of quality z of the new product and incurs the associated sunk costs K(z) under uncertain demand. In the second stage, market demand strength, or the value of |Beta~, is revealed and, if sufficiently strong, "copycat" rivals enter the market producing clones of the innovator's good. To produce the same quality of good a copycat firm incurs an imitation cost equal to |Lambda~K(z). In the third stage we assume that the firms compete in quantities and that, as first entrant in the market, the innovator acts as a Stackelberg leader-dominant firm. Production decisions are realized and the market clears. The assumption of a Stackelberg equilibrium in stage three is a simple way to incorporate the abundant literature that finds some market power advantages for first entrant firms |6; 18~. Indeed, without such advantages, the uncertainty and free-rider problems here may well preclude such innovation from occurring altogether. Moreover, the Stackelberg equilibrium is especially attractive when there is, as here, an historical differentiation in the age of firms |16~.(5) Two other assumptions, that imitation costs rise with the level of product quality and that the follower...