Imperfect competition and quality signalling.

The fact that H-type firms (at least for some parameter configurations) price below the full-information cartel price is yet more significant when one considers the case of a monopolist ([gamma] = 0) with private information, which would involve pricing above the full-information monopoly price (due to the signalling distortion). Thus, in this sense, the presence of competitors moderates the distortionary effects of the presence of incomplete information. Moreover, the proposition implies that there are parameter configurations (high-value markets with a sufficiently high proportion of H-types) for which both types of firm prefer playing an incomplete information game involving product quality to the full-information counterpart. This provides another contrast with the monopoly version of the model, in which an L-type firm would charge its full-information price and obtain its full-information profits, whereas an H-type firm would distort its price upward and necessarily receive lower profits than under full information.

The degree of "competitiveness" and the consumer's maximum willingness to pay also affect the equilibrium, in an intuitively expected manner, as described in the following proposition.

Proposition 5 (Effects of product value, product substitutability, and the number of firms).

(i) An increase in the consumer's willingness to pay ([alpha]) yields an increase in both L-type and H-type equilibrium prices, quantities, and profits, and an increase in the difference in equilibrium prices and the difference in equilibrium quantities; that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(ii) In high-value markets, an increase in the degree of product substitutability ([gamma]) yields a decrease in both L-type and H-type equilibrium prices, quantities, and profits, and a decrease in the differences both in equilibrium prices and in equilibrium quantities; that is,

[there exists][[alpha].sub.5ii] < [infinity] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. and [Q.sub.L] - [Q.sub.H][up arrow].

(iii) In high-value markets, an increase in the number of firms (n) yields a decrease in both L-type and H-type equilibrium prices, quantities, and profits, and a decrease in the differences both in equilibrium prices and in equilibrium quantities; that is,

[there exists][[alpha].sub.5iii] < [infinity] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and [Q.sub.L] - [Q.sub.H][down arrow].

Thus, higher-value markets have higher prices, greater sales volumes, and higher profits, whereas increases in competition (brought about either via less product heterogeneity or via the presence of more competitors in the market) reduce prices, sales volumes, and profits when the markets are high value.

Finally, the loss ([delta]) suffered by a consumer of a low-quality product affects the equilibrium, but in a surprising manner. In the full-information model, an increase in [delta] reduces the equilibrium price, output, and profits of the low-quality firm and increases the equilibrium price, output, and profits of the high-quality firm. However, in the model with incomplete information, an increase in the loss [delta] may increase the equilibrium price, output, and profits of the L-type firm. This is because the increase in the consumer's loss increases the incentive for L-type firms to mimic H-type firms. To deter mimicry, the H-type's price must rise; indeed, the gap between the prices increases with any increase in the amount of the loss that consumers of L-type products must bear. This could lead to increased demand for the L-type good, and therefore higher price and profits for the L-type firm. The following proposition characterizes the results of an exogenous increase in the consumer loss parameter, [delta].

Proposition 6 (Effects of low-quality loss).

(i) The difference in prices is always increasing in the loss associated with low quality, [delta]; that is, [delta][up arrow] [??] [P.sub.H] = [P.sub.L[up arrow]. For high-value markets, the difference in quantities is always increasing in the loss associated with low quality, [delta]; that is, [delta][up arrow][??][Q.sub.L] - [Q.sub.H][up arrow].

(ii) As the proportion of H-types becomes arbitrarily small, an increase in the loss [delta] results in a decrease in the L-type's price, quantity, and profits. More formally,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(iii) For high-value markets, there are values of the proportion of H-types such that an increase in the loss [delta] results in an increase in the L-type's price, quantity, and profits, and an increase in the H-type's price as well; that is,

[there exists][[alpha].sub.6iii] < [infinity] and [[lambda].sujb.6iii] [member of] (0, 1) such that [for all][alpha].6iii] and [for all][lambda] > [[lambda].sub.6iii], [delta][up arrow][??] [P.sub.L][up arrow], [Q.sub.L][up arrow], [[PI].sub.L][up arrow], and [P.sub.H][up arrow].

The above proposition paints a surprising picture: for an industry that is composed of a sufficiently large proportion of H-type firms, an increase in the loss actually can make the L-type firms better off. Examples of this are illustrated in Figure 1. (17) The figure illustrates five computations of the curve [partial derivative][P.sub.L]/[partial derivative][delta], for different values of the relevant parameters (for convenience, we set [beta] = [delta] = 1). The heaviest line illustrates the case wherein n = 3, [alpha]/[delta] = 15, and [gamma]/[beta] = 0.5. Variations on this case include increasing n, decreasing [alpha]/[delta], or modifying [gamma]/[beta]; the underlined number in the triple indicates which parameter was changed to obtain the curve. We have included one parameter set, (n, [alpha]/[delta], [gamma]/[beta]) = (3, 15, 0.35), to illustrate that for some portions of the parameter space, [partial derivative][P.sub.L]/[partial derivative][delta] is always negative. However, the primary point of the figure is that there are large portions of the parameter space where, for sufficiently high (but still fractional) values of [lambda], [partial derivative][P.sub.L]/[partial derivative][delta] is strictly positive. In these portions of the parameter space, [Q.sub.L] and [PI].sub.L] increase as well.

[FIGURE 1 OMITTED]

We have illustrated this effect in the computations for the figure for a triopoly, but there are duopoly configurations that also provide this effect. Moreover, as one might conjecture from Proposition 6, higher values of n (consistent with Assumptions 1 and 2) also produce such curves. Thus, both smaller and larger industries could satisfy the conditions of the proposition. We again emphasize that this result is due to the interplay between imperfect competition and incomplete information; absent either, we would not see such a result. (18) In Section 4, we discuss the implications of this finding in two applications, tort reform and professional licensing.

Summary of comparative statics results. Table 1 summarizes the results provided in Propositions 2, 5, and 6. The following notation is used: (i) a global (i.e., parameter-independent) positive sign on a derivative is denoted as [direct sum]; (ii) a positive (or negative) sign for a derivative which requires that a be sufficiently large will be denoted as + (or -), whereas an unknown sign is denoted by a question mark (?); (iii) the positive sign for the influence of [delta], which is dependent upon both the size of a and of [lambda], is denoted +[lambda].

Discussion of interim profits, quality, and entry. In this analysis, we have assumed that quality is determined exogenously by Nature, and we have taken the number of firms as given. In a more comprehensive model, one might allow both of these assumptions to be relaxed. Although a full analysis of the implications of relaxing these assumptions is beyond the scope of this article, we briefly discuss each of these issues in turn.

As mentioned above, it is common in models wherein high quality is signalled by a high price to find that the interim profits for an H-type firm are lower than those for an L-type firm. This would seem to provide perverse incentives regarding the provision of quality if we allowed firms to influence the quality of their products. However, if this interim stage is placed in a larger context, there are several effects that can reverse this inference.

First, although a firm producing a high-quality product may have a higher marginal cost, it may have lower fixed costs. For instance, if high quality is due to having a better-educated and more reliable work force, then wages may be higher but training and turnover costs may be lower for a high-quality firm. The nature of the workforce available to the firm may be a function of attributes of the firm's geographic location, including the extent of funding for public education and the workers' possibilities for alternative employment. The firm's location is most likely a longer-run decision than either public support for education or the strength of the local economy (indeed, changes in these attributes could result in quality shocks; see below). Thus, a firm that wanted to affect its quality (in either direction) might do so by relocating, but this is costly and it does not guarantee that local conditions will remain the same.