Imperfect competition and quality signalling.

In this equilibrium, [P.sub.H] is the lowest price that an H-type firm can charge and still deter mimicry by an L-type firm; a higher price also signals high quality, but is less profitable, whereas a lower price provides a profitable deviation for an L-type firm, if consumers were to infer that the firm that is charging that price is an H-type firm. As is discussed in Proposition 3 below, these separating-equilibrium prices exceed their full-information counterparts (see the Appendix for the relevant expressions). Moreover, unlike the full-information equilibrium, wherein a high-quality firm would produce more than a low-quality firm, in the refined asymmetric-information equilibrium, a low-quality firm produces more output than a high-quality firm: [Q.sub.L] - [Q.sub.H] > 0. This reflects the distortion in prices due to signalling quality: the high-quality firm's price is so much higher than the price charged by the low-quality firm that consumers are redistributed toward the L-type firm. (14)

It is straightforward to show that an L-type firm's interim profits exceed the interim profits of an H-type firm; this is a common finding in models wherein high quality is signalled by a high price (see, e.g., Bagwell and Riordan, 1991; Daughety and Reinganum, 2005). This is not problematical in this model, because firms are unable to choose their quality levels. However, if one were to contemplate a more general model that allows firms to influence their product quality, this suggests that they would all prefer to produce low-quality products. We believe that this inference ignores some other important considerations, and we provide a detailed discussion of this issue at the end of this section.

Some of the results to follow are global, that is, independent of the parameter values (given that they satisfy Assumptions 1 and 2), whereas others can be demonstrated via sufficient conditions concerning one or more of the parameters. In particular, a number of our results employ a sufficient condition on the maximum willingness to pay, [alpha]. For convenience, we use the terminology "high value" to denote markets wherein [alpha] satisfies a given sufficiency condition (that is, it is "large enough"), which will be made more precise in each result. In particular, we will employ the following convention: a lower bound on a parameter that is specific to a result in a proposition will be subscripted by that proposition number (and subpart, if needed). Thus, for example, there will be an [[alpha].sub.2] asserted in Proposition 2, but no [[alpha].sub.1], as Proposition 1 does not require any other lower bound than is implied by Assumptions 1 and 2.

Definition 2. A "high-value" market indicates the presence of a finite lower bound for [alpha].

Interestingly, a perusal of the formulas (in the Appendix) for [P.sub.H] and [P.sub.L] reveals that both prices are functions of [lambda], the prior probability that a firm is of type H. Thus, unlike separating equilibria in most other models, where only the support of the prior affects the equilibrium, (15) here the prior probabilities themselves influence the separating equilibrium through the expected prices for the rival firms. As the following proposition indicates, increasing [lambda] raises both prices, the gap between the prices, the gap between the quantities, the L-type quantities and profits, and (when dealing with a high-value market) the quantities and profits for the H-type firms.

Proposition 2 (Effect of prior probability distribution on prices, quantities, and profits).

(i) An increase in X increases:

(a) the equilibrium prices [P.sub.L] and [P.sub.H];

(b) the difference between the equilibrium prices [P.sub.H] - [P.sub.L];

(c) the difference between the equilibrium output levels [Q.sub.L] - [Q.sub.H]; and

(d) the L-type's quantity, [Q.sub.L], and profits, [[PI].sub.L].

(ii) In high-value markets, an increase in [lambda]. results in an increase in the H-type's quantity, [Q.sub.H], and profits, [[PI].sub.H]. More formally, [there exists][[alpha].sub.2] < [infinity] such that [for all] [alpha] > [[alpha].sub.2], [lambda][up arrow] [??] [Q.sub.H] [up arrow] and [[PI].sub.H] [up arrow]. (16)

The source of this effect can be understood by reconsidering the definition of equilibrium provided earlier. The incentive compatibility conditions (items (i) and (ii) in Definition l) depend upon the expected price of the competitors, E([p.sup.*]). Because the equilibrium involves the H-type firm posting a higher price than the L-type firm, an increase in the proportion of firms that are likely to be of type H shifts the incentive compatibility constraints. Moreover, because firms' prices are strategic complements ([gamma] > 0), best-response functions are upward sloping, so that an increase in the expected price of a firm's competitors encourages each firm to increase its price.

This effect turns out to be parameter independent for the L-type's price, quantity, and profits, and for the H-type's price; moreover, the difference between the high and low price (and between the low and high quantity) also increases as [lambda]. increases, again, for all portions of the parameter space. Although we cannot provide the same global result for the H-type's quantity and profit, if the market is high value, then increasing [lambda] also increases the H-type's quantities and profits.

The above results suggest that incomplete information about the quality of the good may act to soften competition; because the firms are choosing prices in a noncooperative manner, incomplete information may allow them to achieve higher prices. This leads us to the next proposition, which provides a comparison of equilibrium prices under incomplete versus full-information. Let [P.sup.F]([[theta].sub.i], [[theta].sub.-i]) denote the full-information price for firm i if its true quality is [[theta].sub.i] and the vector of its rivals' true qualities is [[theta].sub.-i]; the formula for the full-information price function is provided in the Appendix. Likewise, let [Q.sup.F]([[theta].sub.i], [[theta].sub.-i]) and [[PI].sup.F]([[theta].sub.i], [[theta].sub.-i]) denote the corresponding full-information outputs and profits. It is straightforward to show that a firm's full-information price, output, and profits are highest when all of its rivals have low-quality products. We will sometimes use the notation [0.sub.-i] to denote an (n - 1)-dimensional vector of zeros, and [1.sub.-i] to denote an (n -1)-dimensional vector of ones.

Proposition 3 (Prices under alternative information structures). The equilibrium price under incomplete information is higher, for both firm types, than the corresponding price under full information, regardless of the rivals' realized qualities; that is, for all [lambda] [member of] (0, 1), [P.sub.L] > [P.sup.F](0, [[theta].sub.-i]) and [P.sub.H] > [p.sup.F](1, [[theta].sub.-i]) for all [[theta].sub.-i].

Thus, there is an upward price distortion for both types of firm, because the equilibrium price for each type under incomplete information exceeds the highest price that type of firm would charge (independent of the realizations of the firm's rivals' types) if quality were observable.

Next, the following proposition (also proved in the Appendix) provides a comparison of profits under incomplete information with those under alternative information structures. In particular, we find that an L-type firm always benefits from incomplete information. We obtain a weaker result for an H-type firm by comparing the profits of an H-type firm under incomplete information with a natural analog: the ex ante expected value of an H-type firm's profits when pricing occurs under full information (that is, [E.sub.-i]{[[PI].sup.F](1, [[theta].sub.-i])}).

Proposition 4 (Profits under alternative information structures).

(i) The L-type's profits under incomplete information are strictly higher than its full-information profits for any realization of the rivals' types; that is, for all [lambda] [member of] (0, 1), [[PI].sub.L] > [[PI].sup.F](0, [[theta].sub.i]) for all [[theta].sub.-i].

(ii) As the proportion of H-type firms becomes arbitrarily close to one, the H-type price converges to a value that is higher than the full-information equilibrium price, but (for high-value markets) lower than the full-information price, denoted [P.sup.F.sub.C], that would be set by a cartel in an industry comprised only of H-type firms. More formally, [lim.sub.[lambda][right arrow]1] [P.sub.H] > [P.sup.F] (1, [1.sub.-i]) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, for high-value markets with a sufficiently high proportion of H-types, an H-type firm's profits under incomplete information exceed the ex ante expected value of an H-type firm's profits when pricing occurs under full information.