Imperfect competition and quality signalling.

Other (more tangentially related) literatures include those involving quality-guaranteeing prices and those involving disclosure of quality. In the quality-guaranteeing price literature (dating back to the early 1980s; see, e.g., Klein and Leffler, 1981), firms choose their qualities, as well as their prices, whereas consumers observe only the prices. Equilibrium is characterized by a price premium that is sufficient to induce firms subsequently to provide high quality. Thus, unobservable quality relaxes price competition. A recent example is Bester (1998), who relates the magnitude of this effect to the degree of endogenous horizontal product differentiation. Levin, Peck, and Ye (forthcoming) provide a model in which two firms have private information about the quality of their respective products, but can engage in costly disclosure. Consumers are located along a line between the two products, reflecting horizontal product differentiation. The cost of production is independent of quality and thus no signalling is possible. In equilibrium, firms engage in socially excessive disclosure. The current article differs from the quality-guaranteeing price literature because Nature chooses firm quality in our model, and differs from the disclosure literature because firms cannot credibly disclose quality and must instead resort to signalling.

Finally, there is also a small literature on noncooperative signalling when each firm has private information about its cost of production. The most closely related paper is Mailath (1989), which provides an n-firm oligopoly model with linear demand and constant marginal costs in which firms produce horizontally differentiated products and engage in noncooperative price competition across two periods. (6) A firm's first-period price can signal its (privately observed) marginal cost of production, which influences its rivals' pricing behavior in the second period. (7) Consumers have no inference problem, because they care only about prices, not marginal costs. Mailath finds that firms' prices are upward distorted (in order to persuade rivals to price higher in the second period) relative to the "non-signalling benchmark," which retains incomplete information in the first period but assumes that the firms' types are exogenously revealed prior to the second period (so the signalling motive is removed).

Although we also use a horizontally differentiated products model with linear demand and constant marginal costs, our model differs from that of Mailath in other ways. First, we consider a one-shot (three-period) model wherein each firm signals its quality to consumers, rather than to its rivals. Second, a firm's product quality (type) affects both its constant marginal cost of production and the demand curve it faces, because product quality also reflects vertical differentiation. Finally, in our model, the nonsignalling benchmark is the full-information outcome in which both rivals and consumers observe product quality directly. Like Mailath, we find that equilibrium prices are upward distorted relative to our (full-information) benchmark prices.

3. Model setup and results

Our model employs a representative consumer, who consumes some of each product, and n firms, each of whom produces one of the products, under conditions of constant marginal costs. The products are horizontally and vertically differentiated, where the quality of the product (the vertical attribute) takes on two possible levels (high and low). In period one, Nature independently draws a type for each firm from a common distribution and each firm observes its type. In period two, firms simultaneously choose prices. Finally, in period three, the representative consumer observes all prices and buys quantities of the products accordingly. In the incomplete-information model, firms do not observe the types of other firms, and consumers do not observe directly the type of any firm. In the full-information model, firms and consumers observe all the types in period two before firms choose prices. In all settings, we restrict the analysis to interior equilibria.

Consumer model. To keep things as simple as possible, we consider a single consumer (8) who consumes a variety of goods; products 1, 2, ..., n are differentiated substitute goods and good n + 1 is a numeraire good. Each product is made by a different firm, and we assume there are n [greater than or equal to] 2 products. Products 1, 2, ..., n may be of either high or low quality (signified by H or L, respectively). Let [[theta].sub.i] be an indicator function which takes on the value 1 when product i is of high quality and the value 0 when product i is of low quality. We assume that the consumer derives utility from the product, less a loss per unit consumed, which is zero for the high-quality good and [delta] > 0 for the low-quality good. (9) The occurrence of this loss is unverifiable (e.g., an uncomfortable mattress, a lazy real estate agent, or a mediocre meal) and therefore cannot be covered by a warranty. Nature determines product quality independently for each firm, and Pr{H} is given by [lambda] [member of] (0, 1). (10) The consumer receives higher utility from a high-quality product i than a low-quality product i, but both versions of product i are worthwhile. In particular, we assume the consumer's utility function is quadratic in the n differentiated products, with the parameters [alpha] > 0, [beta] > 0, and [gamma] > 0:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [gamma] is the degree of product substitution between any two products in the class of interest. We take [gamma] to lie in the interval (0, [beta]). (11) Product quality enters through the linear coefficient on [q.sub.i]; this coefficient is [alpha] if product i is of high quality but falls to [alpha]--[delta] if product i is of low quality.

The consumer with income I chooses ([q.sub.1], ..., [q.sub.n]) so as to maximize her utility of consumption (the consumption of the numeraire good is found as the residual):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, for positive demands, the inverse demand function for product i is

[p.sub.i]([q.sub.i], ..., [q.sub.n]) = [alpha] - (1 - [[theta].sub.i])[delta] - [[beta]q.sub.i]-[gamma][summation over (j[not equal to]i)][q.sub.j].

Because we are interested in firms using price strategies to signal their product quality, we solve for the ordinary demand functions,

[q.sub.i]([p.sub.1], ..., [p.sub.n]) = a - b(1 - [[theta].sub.i])[delta] + g [summation over (j[not equal to]i)] (1 - [[theta].sub.j])[delta] - [bp.sub.i] + g [summation over (j[not equal to]i] [p.sub.j], (1)

where a [equivalent to] [alpha]/([beta] + (n - 1)[gamma]), b = ([beta] + (n - 2)[gamma])/([beta] - [gamma])([beta] + (n - 1)[gamma]), and g [equivalent to] [gamma]/([beta] - [gamma])([beta] + (n - 1)[gamma]).

These represent the consumer's demand functions when quality is observable. When quality is unobservable to the consumer, she will have perceptions of product quality, which we will denote by [[??].sub.j], j = 1, 2, ..., n. Then equation (1), modified by substituting perceived for true qualities, still describes the consumer's demand functions. We will later discuss in greater detail how the consumer's perceptions are formed based on observed prices.

Firm and industry model. For simplicity, we assume that each firm has constant marginal costs which depend on the quality of its product. The cost of producing a unit of a low-quality product is normalized to zero, (12) and the cost of producing a unit of a high-quality product is k > 0. We also assume that [delta] > k, so that the additional utility generated by a unit of a high-quality product justifies its incremental production cost (i.e., a consumer would be willing to pay k to receive higher quality, thus avoiding the loss [delta]).

A firm's profits can be written as a function of its product's true quality, its product's perceived quality (from the consumer's point of view), and its price, given the perceived qualities and prices of its rivals. If quality were observable, then the perceived qualities would coincide with the true qualities. However, perceived quality may differ from true quality if quality is not observable. Then profits for firm i, when it charges price [p.sub.i], its true quality is [[theta].sub.i], and its perceived quality is [[??].sub.i] (and the vector of other firms' prices is [p.sub.-i] and the vector of other firms' perceived qualities is [[??].sub.-i]) can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, firm i's profits are a product of the true price-cost margin and the consumer's demand for product i, which is based on prices and her perceptions of quality.

We want to characterize a symmetric separating perfect Bayesian equilibrium for this game, wherein each firm's product quality is its private information; that is, firm i's product quality is unknown both to the consumer and to firm i's rivals. (13) Suppose that all other firms employ the same separating pricing rule [p.sup.*]([theta]); that is, [p.sup.*](1) [not equal to] [p.sup.*](0). Then because this is a separating strategy, firm i predicts that the consumer's perception of all rival firms' product qualities will be correct. Moreover, firm i also predicts that each of its rivals will charge the price [p.sup.*](1) with probability [lambda] and the price [p.sup.*](0) with probability 1 - [lambda]. Thus, firm i's expected profits, when it charges price [p.sub.i], its true quality is [[theta].sub.i], and its perceived quality is [[??].sub.i] (and all rival firms use the separating strategy [p.sup.*](x)) can be written as