9.5: A Formal Model of Signals
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Consider a market with two kinds of firms. The first type is a high-quality firm. This firm is capable of producing high-quality products but must incur extra production costs if it decides to produce high quality. Because of asymmetric information, the high-quality firm might cheat customers by charging a high-quality price for low-quality products. The other type of firm is a low-quality firm. This firm is only capable of producing low quality. Because of asymmetric information, the low-quality firm might deceive customers by offering its low-quality products at a high-quality price. Finally, assume that there is a demand for both high-quality and low-quality products. The high-quality demand is comprised of quality-sensitive customers who are willing to pay a premium for quality. The low-quality market is comprised of quality-insensitive customers who are unwilling to pay a premium for quality.
Let us define the following profit outcomes in terms of the firm’s choice of price and/or quality level.
- \(\pi(P_{H}, H)\) is the profit level of a high-quality firm that sells at the high-quality price and truthfully provides high quality (and incurs extra production costs).
- \(\pi(P_{H}, L)\) is the profit level of a high- or low-quality firm that is untruthful. It sells at the high-quality price but actually provides customers with a low-quality product.
- \(\pi(P_{L}, L)\) is the profit level of a high- or low-quality firm that truthfully provides low quality.
- \(\pi(P_{L}, H)\) is the profit of a high-quality firm who provides high quality (and incurs extra production costs) but sells at the low-quality price.
Let us rule out the the fourth case by assuming \(\pi(P_{L}, H) < 0\). That is, it is never profitable for a high-quality firm to produce a high-quality product but sell at the low-quality price. Thus, cases 1 to 3 are the only ones of interest in our model.
Conditions for a Lemons Market
Given asymmetric information, a lemons market could occur if
- There is an incentive for high-quality firms to cheat by providing low quality at the high-quality price. In other words, \(\pi(P_{H}, L) > \pi (P_{H}, H)\). In this case, the incentive to cheat is the difference \(\pi (P_{H}, L) - \pi (P_{H}, H) > 0\),
and/or
- There is an incentive for low-quality firms to deceive by pretending to be high-quality firms because \(\pi (P_{H}, L) > \pi (P_{L}, L)\). In this case, the incentive to deceive is \(\pi (P_{H}, L) - \pi (P_{L}, L) > 0\).
With either or both of these incentives in place, customers would be wary of any firm that charges the high-quality price. Even if the customer cares a great deal about quality, he/she would naturally be suspicious of cheating or deception by sellers and would want to avoid paying a premium for quality that is not provided. If customers were very suspicious of cheating or deception, none may be willing to pay the high-quality price. The result is a market outcome where only low quality is provided and all firms get \(\pi (P_{L}, L)\). It may be true that \(\pi (P_{H}, H) > \pi (P_{L}, L)\) and high-quality firms would be better off economically by truthfully providing high quality. However, customer suspicions prevent high-quality firms from accessing the high-quality market. This is the lemons problem mentioned earlier in the chapter.
A Signal to Correct the Lemons Problem
An economic signal could correct the lemons problem if the cost of the signal was lower when a firm provided high quality and higher when the firm provided low quality. Let us consider the case of a warranty. Define \(C\) as the warranty cost of a firm that provides high quality and define \(C'\) to be the warranty cost of a firm that provides low quality. High-quality products should have fewer warranty claims, so it is reasonable to assert that \(C' > C\).
Could offering a warranty serve as the entry ticket for a high-quality firm that wants to truthfully sell on the high-quality market? The answer is yes, provided that customers understand the signal and each of the following conditions hold.
Condition 1. Access to the high-quality market is economically attractive in the presence of a truthful signal. If warranties are to be a successful signal, then high-quality firms must be willing to truthfully provide high quality in the presence of warranty costs. It must be that
1A. \(\pi (P_{H}, H)-C>\pi (P_{L}, L),\)
which implies
1B. \(\pi (P_{H}, H) - \pi(P_{L}, L) > C\)
The left side of 1B represents the benefits of truthfully accessing the high-quality market and selling at a high-quality price. The right side is the cost of offering warranties on high-quality products.
Condition 2: The signal removes the incentive for high-quality firms to cheat. If warranties are to be a successful signal, then the warranty must remove the incentive for high-quality firms to cheat. It must be that
2A. \(\pi (P_{H}, H) - C > \pi (P_{H}, L) - C',\)
which implies
2B. \(C' - C > \pi(P_{H}, L) - \pi (P_{H}, H).\)
The left side of inequality 2B represents the additional signal cost the firm would face if it cheated. The right side represents the incentive to cheat. Thus, if 2B holds, the cost of cheating is larger than the benefits of cheating.
Condition 3: The signal removes the incentive for low-quality firms to deceive. Finally, if warranties are to be a successful signal then they must make it unattractive for low-quality firms to deceive customers by pretending to be high-quality firms. It must be that
3A. \(\pi (P_{L}, L) > \pi(P_{H}, L) - C',\)
which implies
3B. \(C' > \pi(P_{H}, L) - \pi (P_{L}, L).\)
The left side of inequality 3B is the cost of offering a warranty on low-quality products, and the right hand side represents the incentive to deceive. Thus if 3B holds, the costs of deception exceed the benefits.
If all three of these conditions hold, there will be a separating equilibrium in the model where high-quality firms will signal and truthfully provide high-quality and low-quality firms will not signal and will provide low quality. Consumers who are willing to pay a premium for quality can safely purchase from firms with the signal. Those consumers who are not quality conscious will buy from firms that do not signal. In short, the signal corrects the lemons market problem and allows for the high-quality market to exist. If any one of these three conditions do not hold, however, there is no separating equilibrium and the signal is ineffective.
Advertising Cost as Quality-Dependent
The example of a warranty was chosen to explain the separating equilibrium because it is simple to see why warranty costs in the presence of false quality claims would be higher than warranty costs in the presence of true quality claims. Let us now consider why and how advertising might also be an effective signal. Given the model presented above, advertising could be an effective signal if advertising had a lower cost when quality claims were true than when quality claims were false.
To see why this might be the case, let \(A\) represent brand equity and suppose that access to the high-quality market in any given time period, \(t\) requires that \(A_{t} > \bar{A}\). Firms with brand equity greater than or equal to \(\bar{A}\) are able to charge the high-quality price. Firms with brand equity below \(\bar{A}\) cannot and must charge the low-quality price. Suppose further that brand equity depreciates with time and needs to be replenished with advertising. The dynamics of brand equity when high quality is truthfully provided are
\(A_{t} = \delta A_{t -1} + a_{t},\)
where \(0 < \delta < 1\) is the rate of depreciation in brand equity and \(a_{t}\) is advertising expenditure in the current period.
The dynamics of brand equity when a firm provides low quality are
\(A_{t} = \delta' A_{t-1} + a_{t},\)
where \(0 \leq \delta ' < \delta \). If a firm puts low-quality products on the market, a smaller portion, \(\delta ' < \delta \), of its brand equity carries over from the prior period. This means that brand equity depreciates at a faster rate when low quality is provided. This is a reasonable assertion.
Given these dynamics, the minimal steady-state expenditure required to maintain continuous access to the high-quality market for a firm that truthfully provides high quality would be \(a_{t} = (1-\delta) \bar{A}\). If a firm falsely claimed high quality, its cost of accessing the high-quality market would be higher at \(a_{t} = (1 - \delta ') \bar{A}\). Thus, advertising has the feature that it is less expensive when high-quality is provided than when not. In the parlance of the signalling model above, we have a truthful signal cost, \(C\), and a false signal cost, \(C'\), that satisfy
\(C = (1 - \delta) \bar{A} < C' = (1- \delta ') \bar{A}.\)
This quality-dependent difference in advertising cost could bring about a separating equilibrium provided that \(C\) and \(C'\) are of appropriate magnitudes to satisfy conditions 1 through 3 above.