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Innovation, imitation, and social welfare.

I. 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.

II. 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 firms are literally copycats mimicking the quality choice of the original innovation may also be defended. The first of these is supported by previous empirical work. In a sample of innovations in the chemical, drug, electronics and machinery industries, it has been found that imitation costs go up as the innovator's product development and start up costs rise |12~. This is often because regulation requires that the reliability and safety, i.e., the quality, of the imitative product be tested in much the same way as the innovative product with which it competes.(6)

We justify the assumption of quality mimicry on two grounds. To begin with, this assumption allows us to explicitly model the role of imitation costs in the innovative process just as called for in the empirical work mentioned at the start of the paper. Second, the empirical fact is that such duplication occurs quite frequently and may well be the dominant form of imitation |1; 13~. This is rather obvious in the case of photocopying or reprints. But it is also true, for example, for many household products, pharmaceuticals, and computer software programs, which are qualitatively identical. Such quality duplication is especially common when, as here, the innovator is a dominant firm.(7) The innovator's problem then is to choose the quality, z, that maximizes expected profits in light of both demand uncertainty and the prospect of rival imitators. To solve this problem, we start with stage three. At this time the demand parameter |Beta~ is known and, given the chosen quality z, n rival firms have entered the market. The Stackelberg assumption implies that the equilibrium output of the innovator, |q.sub.L~, and of each copycat rival, |q.sub.F~, along with the industry price, P, in stage three, are:

|q.sub.L~ = 1/2; |q.sub.F~ = 1/2(n + 1); and P = z|Beta~/2(n + 1). (3)

Innovator profits depend on the strength of market demand and the number of copycats entering the market in stage two. The equilibrium number of copycats, n*, is given by the zero profitability condition in stage three. Let ||Pi~.sub.F~ denote the profits of a copycat firm. n* |is greater than~ 1 implies:

||Pi~.sub.F~ = z|Beta~/{4|(n* + 1).sup.2~} - |Lambda~K(z) = ||Pi~.sub.F~(|Beta~, z, n*). (4)

There may be no entry of copycat firms, especially if market demand, as indicated by |Beta~, is weak. Define |Mathematical Expression Omitted~ to be the level of market demand such that a single rival entrant earns zero profit, i.e., |Mathematical Expression Omitted~. If |Mathematical Expression Omitted~, no entry occurs. From equation (4) we may then obtain:

|Mathematical Expression Omitted~.

Observe that the innovator can always choose a level of quality, |Mathematical Expression Omitted~, that completely deters entry for all states of demand. In particular, |Mathematical Expression Omitted~ solves:

|Mathematical Expression Omitted~.

For |Mathematical Expression Omitted~, the innovator is a monopolist earning monopoly revenue, |R.sub.M~:

|R.sub.M~ = z|Beta~/4. (7)

On the other hand, if |Mathematical Expression Omitted~, at least one copycat firm enters and, by competition, earns zero profit. When n* |is greater than or equal to~ 1, innovator revenue is |R.sub.L~:

|R.sub.L~ = z|Beta~/4(n* + 1). (8)

If the number of copycat firms, n* |is greater than~ 1, equations (4), (5) and (8) imply:

|Mathematical Expression Omitted~.

Substitution of (9) into (8) yields:

|Mathematical Expression Omitted~.

We may now write the innovator's expected profit E(|Pi~) for any choice of z:

|Mathematical Expression Omitted~.

Integration then yields:

|Mathematical Expression Omitted~

where, from (5), |Mathematical Expression Omitted~, and where |Mathematical Expression Omitted~.

Clearly, a key parameter influencing the profitability of the innovator's quality choice is the state of demand |Mathematical Expression Omitted~ at which entry occurs. The innovator firm can affect this state through its choice of product quality. However, the innovator does not have complete control over |Mathematical Expression Omitted~. This demand state also depends on |Lambda~, the ratio of imitation to development costs. In order to find the innovator's profit-maximizing quality choice z*, we must solve the first order condition of the maximand in expression (12). To do so in a tractable manner, we assume a simple quadratic cost function, i.e., K(z) = |Alpha~|z.sup.2~. This assumption is not implausible. Moreover, the main points that we wish to make do not depend on this particular functional form. When K(z) = |Alpha~|z.sup.2~, it follows that |Mathematical Expression Omitted~, and hence, we can rewrite (12), the innovator's expected profit function, as follows: E|Pi~ = |z.sup.2~(32||Alpha~.sup.2~||Lambda~.sup.2~z/3 + (||Alpha~.sup.1/2~||Lambda~.sup.1/2~|z.sup.-1/2~)/3 - |Alpha~). (13)

Of course, the innovator can always choose the deterrent quality |Mathematical Expression Omitted~ equal to 1/16|Lambda~|Alpha~ and earn expected profit |Mathematical Expression Omitted~. But it is more profitable to choose a quality |Mathematical Expression Omitted~, if the solution to the first order condition:

|Delta~E|Pi~/|Delta~z = 32||Lambda~.sup.2~||Alpha~.sup.2~|z.sup.2~ + (1/2)||Lambda~.sup.1/2~||Alpha~.sup.1/2~|z.sup.1/2~ - 2|Alpha~z = 0, (14)

leads to a greater profit, or when |Mathematical Expression Omitted~. If one defines x = |z.sup.1/2~ and factors, the first order condition may be rewritten as:

x(32||Lambda~.sup.2~||Alpha~.sup.2~|x.sup.3~ + (1/2)||Lambda.sup.1/2~||Alpha~.sup.1/2~ - 2|Alpha~x) = 0. (15)

From this it is clear that the first order condition is met by a choice of x (or z) such that the parenthetical expression of (15) vanishes. In turn, it is also clear that this requires solving a cubic equation of the general form:

|x.sup.3~ - n = mx or |x.sup.3~ + mx = n. (16)

There is a rich history regarding the discovery of the solution to such cubic equations |9~. One particularly interesting aspect of this solution is that, for the case of three real and distinct roots, the solution will require working with complex numbers and then using a trigonometric transformation. That is, if (16) has three real and distinct roots, obtaining a solution will require TABULAR DATA OMITTED working with complex numbers even though the solution values themselves are real. As it turns out, the case of three real and distinct roots is exactly the one we have here. Fortunately, however, two of the roots are negative and hence not economically sensible. The expression for the remaining meaningful root, in terms of z, is:

z* = ||Alpha~.sup.-1~(1/12||Lambda~.sup.2~) sin||(1/3) arcsin ((|3.sup.3/2~||Lambda~.sup.3/2~)/2)~.sup.2~. (17)

In Table I, we present the innovator's expected profit for all values of |Lambda~ between 0.00 and 1.00 at both the deterrent choice, |Mathematical Expression Omitted~, and at the interior solution of z*. In constructing this table we have, without loss of generality, set the parameter |Alpha~ in the cost function at 1/64. As shown below, this normalization transforms the expected welfare level associated with complete optimality to unity. Hence, the welfare outcome for all other cases may be interpreted as a fraction of the welfare achieved under optimality.

As the Table I results indicate, there is a critical value of |Lambda~ at which the innovator switches from an interior solution z* to the deterrent solution |Mathematical Expression Omitted~. For the quadratic cost function assumed above this value is 0.52. At this point, there is a dramatic rise in the profit-maximizing quality choice as the innovator selects a quality level that deters all subsequent entry.

On the other hand, for values of |Lambda~ such that 0 |is less than~ |Lambda~ |is less than~ 0.52, the innovator chooses a quality level that will invite some imitation if demand strength is revealed to be sufficiently strong. In this case, the cost of deterring imitation is simply too great relative to the monopoly gains from doing so. Note too that when entry occurs the number of copycat firms, n*, increases as |Lambda~ decreases in the range 0 |is less than~ |Lambda~ |is less than~ .52.

III. Product Quality and Social Welfare

We now turn to the welfare implications of potential competition on new product development. It is a well known result that when, as here, marginal production costs are unaffected by quality, a pure monopolist chooses a product quality that is too low relative to the social optimum |19; 20~. Welfare is lower because the monopolist both chooses too low a product quality and too high a price.

In this context, the foregoing analysis suggests a number of important questions. One of these is the impact of imitative competition on product quality. Does such competition raise or lower quality relative to the monopoly outcome? A second question is whether imitative competition raises or lowers welfare. These two questions are conceptually distinct. For example, potential imitation may lead to a lower quality choice, and yet still be welfare improving. This is because entry by copycat firms induces some competition in the industry and keeps price closer to marginal cost than would be the case under pure monopoly. The gain from a lower price may outweigh any loss of reduced product quality. Indeed, this example suggests a third question namely, do the benefits of competition, if any, stem primarily from enhanced product quality or from lower prices?

We begin by describing the socially optimal and monopoly outcomes. The socially optimal quality level is that z that maximizes the sum of consumer and producer surplus less sunk development cost. Since marginal production cost is zero, total surplus for any quality, z, and demand state, |Beta~, is simply the area under the demand curve. Hence the optimal quality, |z.sup.e~, solves: |Mathematical Expression Omitted~.

From which it follows that:

|z.sup.e~ = 1/(8|Alpha~) and E|W.sup.e~ = 1/(64|Alpha~) (19)

where E|W.sup.e~ is the optimal expected social welfare.

The monopolist quality, |z.sup.M~, maximizes producer surplus only. Hence:

|Mathematical Expression Omitted~.

In turn, this implies:

|z.sup.M~ = 1/(16|Alpha~) and E|W.sup.M~ = 1/(128|Alpha~) (21)

where E|W.sup.M~ is the welfare under monopoly equal to the expected sum of producer and consumer surplus net of development costs. As noted, a monopolist chooses too low a product quality and, of course, welfare is not maximized.(8)

To understand the impact of imitative competition we begin by recalling that when imitation is costly, 0.52 |is less than~ |Lambda~ |is less than~ 1, the innovator deters entry by choosing a quality level of |Mathematical Expression Omitted~. Hence, in this range of |Lambda~, the innovator is alone in the market. Yet while the innovator has monopoly power, it has only acquired such power by choosing a high quality good. The potential for imitation has induced the innovator to behave differently from a secure monopolist. Indeed, as Table I indicates, the entry-deterring quality, |Mathematical Expression Omitted~, always exceeds the monopoly choice, |z.sup.M~ (a proof is found in the appendix).

An opposite outcome occurs when the imitation parameter lies in the range, 0 |is less than or equal to~ |Lambda~ |is less than or equal to~ 0.52. Here it does not pay to try and deter imitation. Faced with the prospect of either weak demand, on the one hand, or strong demand and many rivals, on the other, the innovator chooses a low quality design that reduces its sunk costs. As Table I shows, the quality choice, z*, made in this range of |Lambda~ is always less than the pure monopoly choice, |z.sup.M~.

The welfare outcome in each case is more complex. For the first case, when the innovator deters entry, the higher quality product design that this entails represents a gain relative to what would occur under a secure monopoly. But at the same time, this higher quality means that the innovator's demand curve is rotated up and out relative to the monopoly case. So, its price, p*, also exceeds the monopoly price, |p.sup.M~. Given a marginal production cost of zero, this implies a greater price distortion that may offset the welfare gain from higher quality. Similarly, the fact that when 0 |is less than or equal to~ |Lambda~ |is less than or equal to~ 0.52 the product quality is below the monopolist's choice, |z.sup.M~, does not necessarily imply a net welfare reduction. In this range, entry and price competition occur, leading to welfare gains that may offset the losses due to lower quality.

Table II shows the quality, price, and welfare results for all values of |Lambda~, again with |Alpha~ = 1/64. (The formal mathematics underlying these results is described in the appendix.) These results are interesting in a number of respects. To begin with, the findings show that the market failure induced by the inability to use marginal cost pricing is serious. The minimum reduction in welfare from the optimal level is 44 percent. When copying is relatively easy, the welfare loss becomes very serious and, in the limit, approaches 100 percent.

Generally, increasing the difficulty of imitation, |Lambda~, improves social welfare. But two points should be made in this connection. First, so long as |Lambda~ is such that some entry is expected to occur, welfare remains below the level obtained under pure monopoly (0.50). Second, the relation between welfare and the difficulty of copying is not monotonic. Welfare rises as copying costs rise to 65 or 70 percent of original development costs. But further increases in |Lambda~ so increase the monopoly power of the innovator that welfare declines. In the limit, when imitation is just as expensive as original development, the market outcome in terms of price, quality, and welfare is identical to that of the pure monopoly case.

An important overall implication of the results in Table II is that quality effects dominate the price effects. Prices are much lower when imitation is easy and entry occurs than when imitation is sufficiently difficult that the innovator chooses the entry-deterrent quality level. But the gain in quality that such deterrent behavior requires always raises welfare relative to these low-price outcomes. This result is broadly consistent with the Schumpeterian view that competition over product design and not over prices is the true source of welfare gains. If, as a Schumpeterian adherent might recommend, innovators are granted some protection from imitative rivals, then the quest for monopoly power will induce such a high quality product that welfare will rise despite the attendant price-cost distortions such power creates |14~. However, the lack of monotonicity in the relationship between |Lambda~ and welfare suggests that policy makers should not aim for the pure monopoly outcome.

The foregoing results derive from a fairly stylized analytical model. Obviously, we do not hold this model out as a literal description of reality. Still, it is worth noting that the model's predictions are consistent, both qualitatively and quantitatively, with a number of empirical observations.

TABULAR DATA OMITTED

To begin with, the strategy of raising rivals' costs by choosing a high quality product design has not escaped the attention of actual practitioners. Consider, for example, the following remark of a corporate officer for Apple, Inc.," . . . we've built a lot of custom chips, designed for us to specifically enhance the operation and performance of |our~ computers. That, in itself, creates a tremendous barrier to anyone who wants to copy the hardware because the investment in research, development, and custom manufacturing is in the millions and millions of dollars" |3~.

A second confirmed prediction of the model is that low sunk costs (easy entry) and low quality will often be observed together along with deleterious effects on social welfare. Rashid has presented evidence that this is precisely the relationship one observes historically |15~. In markets where the sunk costs of entry and exit have been low--such as English cloth, Chinese silk, and rural milk markets--product quality has been distinctly inferior, and social welfare has suffered. Note that in our model, the low sunk cost-low quality linkage is a compound one. When imitation costs are low, the innovator selects a low quality good. This not only lowers the innovator's sunk costs but it also lowers the imitators' sunk costs, which, in any case, are already low precisely because imitation is cheap.(9)

The model's predictions are also consistent with many of the findings of Mansfield, et al. |12~. For example, they found both that the probability of entry was increased and that industry concentration decreased as imitation costs declined.(10) Both findings are predicted by our model. Moreover, in 40 percent of the cases studied, the innovator's product had not been imitated even after four years had elapsed since the good's introduction. This is consistent with the prediction that if imitation costs are sufficiently high the innovator selects a high quality design meant to prevent copying.

IV. Summary and Conclusions

An important part of the innovation decision concerns the quality of product to develop. This decision is taken before the innovator knows whether the innovation will be a commercial success. Demand uncertainty and the possible entry of copycat rivals weaken an innovator's incentive to incur the higher sunk development costs of a better product. Such costs may not be recovered when demand is weak. They may also not be recovered when it is strong because it is precisely this outcome that attracts rivals. But if imitators' expenses also rise with the quality of the good being copied, then choosing a high quality product can serve to raise rivals' costs and limit entry.

We have examined these issues and their welfare implications in a model of vertical product differentiation. We find that as the cost of copying relative to original development, |Lambda~, rises, the quality choice also rises. Indeed, in our model, values of |Lambda~ greater than 0.51 induce the innovator to select so high a quality design that social welfare rises relative to what would occur if, say by regulation, the innovator were guaranteed a monopoly. But as relative imitation costs fall below the 0.51 level, free rider problems become too great. Here, both product quality and social welfare decline relative to the monopoly outcome despite the much lower prices such competition brings. Our analysis suggests that some of the traditional debates in patent and regulatory policy need to be recast. Rather than focus on the tradeoff between lower prices and the decision to innovate at all, such debates should address the tension between lower prices and the kind or quality of innovation that changes in the ease of imitation imply.

Appendix. Product Quality and Expected Social Welfare Analytics

For the cost function, K(z) = |Alpha~|z.sup.2~, equations (19) and (21) give the optimal and pure monopoly quality choices, |z.sup.E~ and |z.sup.M~, respectively, as:

|z.sup.E~ = 1/(8|Alpha~) |z.sup.M~ = 1/(16|Alpha~). (A1)

Similarly, equation (5) implies that the entry-deterring quality level, |Mathematical Expression Omitted~, for this cost function satisfies: 16|Lambda~|Alpha~z = 1. Hence,

|Mathematical Expression Omitted~.

Clearly, since |Lambda~ is a positive fraction, |Mathematical Expression Omitted~ always exceeds |z.sup.M~. If the potential for imitative competition induces the innovator to adopt the deterrent quality level, |Mathematical Expression Omitted~, then quality clearly rises relative to the pure monopoly case. However, as Table I shows, the choice of |Mathematical Expression Omitted~ is only made when |Lambda~ lies between 0.52 and 1. A comparison of (A1) and (A2) reveals that within this range, |Mathematical Expression Omitted~. The choice of |Mathematical Expression Omitted~ does not lead to as high a quality as optimality requires.

To calculate the welfare consequences of the adoption of the deterrent quality, |Mathematical Expression Omitted~, recall that such a quality choice leaves the innovator as a monopolist. Therefore, the sum of expected consumer and producer surplus less development costs is:

|Mathematical Expression Omitted~.

From equation (21) we have that the expected welfare under monopoly is 1/128|Alpha~. Comparing this result with that of equation (A3), it is straightforward to show that |Mathematical Expression Omitted~ for all values of |Lambda~ for which |Mathematical Expression Omitted~ is chosen, i.e., for 0.52 |is less than~ |Lambda~ |is less than~ 1.

For 0 |is less than~ |Lambda~ |is less than or equal to~ 0.52, imitative entry occurs as the innovator chooses a profit-maximizing quality level, |Mathematical Expression Omitted~. It is clear from above that z* is also less than the socially optimal quality choice and, from Table I, that it is less than the pure monopoly choice, |z.sup.M~. To consider the social welfare for this range of |Lambda~ we need to consider the market price and quality results for two different cases. The first occurs when demand strength is weak, i.e., when |Mathematical Expression Omitted~, so that no entry occurs even though the innovator did not try to prevent it. The second case occurs when demand strength is sufficiently strong that some entry does occur, i.e., when |Mathematical Expression Omitted~. However, even when entry occurs the entrant(s) earns zero profits. So, under both regimes, social welfare is just the profit earned by the innovator plus consumer surplus. Thus, for the range of |Lambda~ in which the innovator chooses a less-than-entry-deterring quality level, z*, we may calculate expected social welfare, EW*, as follows:

|Mathematical Expression Omitted~.

From equations (5) and (9), |Mathematical Expression Omitted~, and |Mathematical Expression Omitted~. Again, we assume |Mathematical Expression Omitted~, so |Mathematical Expression Omitted~. By substituting for n and |Mathematical Expression Omitted~, and then integrating, the expression above for EW* becomes:

EW* = 48||Alpha~.sup.2~||Lambda~.sup.2~|z*.sup.3~ + (z*/4) - ((||Alpha~.sup.1/2~||Lambda~.sup.1/2~|z*.sup.3/2~)/3) + ((|Lambda~ - 2)/2)|Alpha~|z*.sup.2~ - (152/3)||Alpha~.sup.2~||Lambda~.sup.2~|z*.sup.3~. (A5) Together, equations (A3) and (A5) are used to generate the expected welfare results of Table II in the text (again with |Alpha~ = 1/64).

1. The legislation includes the Computer Software Act of 1980, the Semi-conductor Chip Protection Act of 1984, and the proposed HR 3017-90, all of which impose penalties on copycats.

2. In short, the firm knows the number of customers that would buy the good at a zero price, but not the reservation price of any potential customers.

3. AT&T's Picturephone, Nimslo's 3D camera, and Polaroid's instant home movie system, Polavision, are all innovations that failed for lack of demand.

4. To our knowledge, Benoit was the first to note the possibility of deterring entry by choosing a costly and/or risky innovation |2~. Our work differs from his in that we link development costs to quality and explicitly model consumer demand. These changes allow welfare conclusions to be drawn from the analysis.

5. We do not consider the timing of imitation per se. One may interpret our model as positing a given optimal speed of imitation, but that to achieve such timely copying requires more expense for a high quality good than for a low quality one. The parameter |Lambda~ then captures these relationships. We also collapse the time dimension by specifying three stages rather than three separate periods. But allowing the innovator to enjoy monopoly profits for a short while before imitators enter would not qualitatively alter our results.

6. Patents may seem an obvious way to raise imitation costs. Yet most research finds patent protection ineffective in this role. Mansfield, Schwartz and Wagner found that patents raise imitation costs by only 6 percent, and that over 60 percent of the patented innovations they studied were imitated within four years |12~. Similarly, Levin et al. report that patents are ineffective both because copiers are clever at getting around them and because successful prosecution of infringement is difficult |10~. Indeed, seeking patent protection may facilitate imitation since the documents accompanying patent application may reveal secret information |17~.

7. The dominant firm's ability to set the industry quality standard has been explicitly noted by Tirole |22, 405~. Network externalities may also be a reason for the high frequency of product duplication. In addition, it seems likely that developing a different quality than the innovator increases imitation costs. This too gives an incentive to copy the innovator's quality.

8. If |z.sup.M~ exceeds the deterrent quality choice, |Mathematical Expression Omitted~, then the monopolist chooses |Mathematical Expression Omitted~. In this case, the good is effectively a natural monopoly. We do not consider this case to be interesting. Further, for the quadratic cost function used here it does not occur.

9. Fears of quality deterioration and a consequent decline in social welfare as a result of eased entry conditions have also been prominent in recent debates over deregulation |5~.

10. From equation (9) we have:

n* = ((||Beta~.sup.1/2~||Lambda~.sup.-5/2~)/24) sin|(1/3) arcsin((|3.sup.3/2~||Lambda~.sup.3/2~)/2)~.

From this, tedious algebra yields:

|Delta~n*/|Delta~|Lambda~ = (|3.sup.1/2~||Beta~.sup.12~) cos|(1/3) a

.3~))

- (|3.sup.1/2~||Beta~.sup.12~) sin |(1/3) arcsin ((|3.sup.3/2~||Lambda~.sup.3/2~)/2)~/48||Lambda~.sup.7/2~.

It is easy to verify that this expression is negative for all values of |Lambda~ in the range 0 |is less than~ |Lambda~ |is less than~ 0.52. Both Mansfield, Schwartz and Wagner and Levin et al. find that easier imitation (lower |Lambda~) leads to less concentration |10; 12~.

References

1. Andrews, Edward, "When Imitation Isn't the Sincerest Form of Flattery." New York Times, August 5, 1990, E20.

2. Benoit, Jean-Pierre, "Innovation and Imitation in a Duopoly." Review of Economic Studies, January 1985, 99-106.

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Author:Richards, Daniel J.
Publication:Southern Economic Journal
Date:Jan 1, 1994
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