Imperfect competition and quality signalling.

Professional licensing requirements. In many industries, especially service industries, producers must meet certain licensing requirements. For instance, real estate brokers, healthcare providers, lawyers, accountants, professional engineers, architects, public school teachers, barbers, and restaurants are typically licensed by the state. One effect of licensing is (arguably) to provide a higher floor on consumer satisfaction with the product (in our model, a smaller value of [delta]) by providing specific training requirements for future practitioners, along with certification examinations. (22) Moreover, meeting the licensing requirement involves a fixed cost, leaving variable costs largely unaffected. In the full-information model, the impact of a licensing requirement that lowers [delta], leaving marginal costs unchanged, is to raise the low-quality firm's price, output, and profits and to lower the high-quality firm's price, output, and profits. However, in the model with incomplete information, such a licensing requirement can have a double benefit to consumers in high-value markets with a sufficiently high proportion of H-type firms. This is because, in addition to the direct benefit of reducing the loss due to low quality, there is also an indirect benefit because both H-type and L-type prices fall, and H-type output increases relative to L-type output. (23)

The conflicting effects that lead to these results are as follows. A reduction in [delta] would, in principle, allow an L-type firm to charge a higher price for its product. This would reduce the L-type firm's incentive to mimic, which would allow the H-type firm to lower its price, to which an L-type firm would respond by lowering its price as well. If the fraction of H-type firms is sufficiently high, then the second incentive outweighs the first, and the L-type firm's price falls in equilibrium, as does its quantity, [Q.sub.L], and its profits, [[PI].sub.L].

5. Conclusions

In this article, we combine two relatively well known models from industrial organization and find new and unexpected results. We employ a signalling model in which the quality of a firm's product is its private information; the firm's choice of price may signal its quality to consumers. We integrate this signalling aspect into a model of imperfect competition in a product market with horizontally differentiated substitute goods. Thus, in choosing its price, a firm must play a best response to its rivals' price strategies and, at the same time, deter mimicry by its own alter ego.

We generate a variety of results that do not occur in the separate portions of the model. For instance, we find that a low-quality firm produces more output than a high-quality firm under incomplete information; this does not occur under full information (though it does occur in a monopoly model with incomplete information).

We find that incomplete information always raises prices for both types of firm. Moreover, there are circumstances under which incomplete information also raises equilibrium profits in the case of imperfect competition, whereas incomplete information only lowers (or leaves unchanged) equilibrium profits in the case of a monopoly. Under imperfect competition, the need to signal high quality acts as a credible commitment to higher prices, which allows rival firms to price higher as well, and can raise equilibrium profits. Under monopoly, the need to signal high quality causes the monopolist to price higher than the full-information monopoly price, thereby reducing profits.

In our model, the parameter representing the proportion of high-quality firms is an important determinant of equilibrium prices, quantities, and profits, all of which are increasing in this parameter. A higher proportion of high-quality rivals implies a higher expected rival price and, because prices are strategic complements, a higher own price. At the same time, a higher expected rival price shifts demand toward the firm so that it also produces a higher quantity of output (in high-value markets). Combining these two effects clearly implies higher profits. This parameter does not matter in a monopoly version of the model, nor in the full-information version of the model.

We find that (for some parameter values) an increase in the loss associated with low quality can have the perverse effect of increasing the price, quantity, and profits of a low-quality firm. A higher loss increases the incentive for a low-quality firm to mimic a high-quality firm, causing the high-quality firm to raise its price even further to signal its quality. This in turn shifts demand toward the low-quality firm and allows the low-quality firm to increase its price. This cannot occur in a monopoly version of the model with incomplete information, nor can it occur in the full-information version of the model.

It is reassuring that some cherished results do carry over in this model (at least for highvalue markets). In particular, the incomplete-information imperfect-competition model behaves as expected with respect to the variables that measure market size, product substitutability, and the number of firms. A higher-value market corresponds to higher prices, outputs, and profits for both quality levels, whereas an increase in the substitutability of the products or an increase in the number of firms causes prices, output (per firm), and profits to fall.

We employ the model to address two applications. In the case of tort reform, our results on the effect of the increase in consumer loss on low-quality firm prices, quantities, and profits suggest that such reforms as caps on damages or increased evidentiary standards may backfire, possibly leading to more harmed consumers and more lawsuits. In the case of professional licensing, we find that such requirements may actually enhance competitiveness and lead to reduced prices, in addition to their straightforward effect on consumer satisfaction.

Appendix

This Appendix contains the derivation of the unique (refined) separating equilibrium, the full-information price and profit formulas, and proofs of selected propositions.

Derivation of the symmetric separating equilibrium price function.

Derivation of best-response functions. Recall the function describing firm i's profit as a function of its price, [p.sub.i], its actual type, [[theta].sub.i], and the type the consumer believes it to be, [[??].sub.i]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note that for any given price, it is always more profitable to be perceived as type H, regardless of true type. If there were no signalling considerations, then [[PI].sub.st] [equivalent to] (p - [c.sub.s])b([d.sub.t] - p) would be maximized by [[rho].sub.st] = ([c.sub.s] + [d.sub.t])/2, and the resulting profits would be [[PI].sub.st] = b[([d.sub.t] - [c.sub.s]).sup.2]/4. These prices (actually, "best responses" to E([p.sup.*])) are ordered as follows: [[rho].sub.HH] > [[rho].sub.LH] > [[rho].sub.HL] > [[rho].sub.LL]. The only nonobvious case is [[rho].sub.LH] > [[rho].sub.HL]; this holds if and only if [d.sub.H] - [d.sub.L] > [c.sub.H] - [c.sub.L], which is ensured by the assumption that [delta] > k.

Our method of deriving the separating equilibrium prices is to first derive a best-response function for firm i that reflects the need to signal its type. This will consist of a pair of prices ([[rho].sub.L](E([p.sup.*])), [[rho].sub.H](E([p.sup.*]))). We will then impose the equilibrium condition that E([p.sup.*]) = [lambda][[rho].sub.H] (E([p.sup.*])) + (1 - [lambda])[[rho].sub.L](E([p.sup.*])) and solve for a fixed point. Finally, the resulting solution is substituted into ([[rho].sub.L](E([p.sup.*])), [[rho].sub.H](E([p.sup.*]))) to obtain the equilibrium interim prices.

No firm is willing to distort its price away from its best response (were its type known) in order to be perceived as type L (because this is the worst type to be perceived to be). Thus, if a firm of type L is perceived as such, its best response is [[rho].sub.LL], which yields profits of b[([d.sub.L] - [c.sub.L]).sup.2]/4. If a firm of type H is perceived as being of type L, its best response is [[rho].sub.HL], which yields profits of b[([d.sub.L] - [c.sub.H]).sup.2]/4.

However, either firm would be willing to distort its price away from its best response (were its type known) in order to be perceived as type H. Thus, a candidate for a revealing equilibrium must involve a best response for type H that satisfies two conditions. First, it must deter mimicry by the type L firm (who thus reverts to [[rho].sub.LL]); and second, it must be worthwhile for the type H firm to use this price rather than to allow itself to be perceived as a type L firm (and thus revert to [[rho].sub.HL]). Formally, a separating best response for the type H firm is a member of the following set:

{p|(p - [c.sub.L])b([d.sub.H] - p) [less than or equal to] b[([d.sub.L] - [c.sub.L]).sup.2]/4 and (p - [c.sub.H])b([d.sub.H] - p) [greater than or equal to] b[([d.sub.L] - [c.sub.H]).sup.2]/4}.

The first inequality says that the type L firm prefers to price at [[rho].sub.LL] (and be perceived as type L) than to price at p (and be perceived as type H). The second inequality says that the type H firm prefers to price at p (and be perceived as type H) than to price at [[rho].sub.HL] (and be perceived as type L). Solving these two inequalities implies that the H-type firm's best response belongs to the interval

[.5{[d.sub.H] + [c.sub.L] + [(([d.sub.H] - [c.sub.L]).sup.2] - [(d.sub.L] - [c.sub.L]).sup.2]).sup.1/2]}, .5{[d.sub.H] + [c.sub.H] + ([([d.sub.H] - [c.sub.H]).sup.2] - [([d.sup.L] - [c.sub.H]).sup.2]).sup.1/2]}].