Chapter 10 Asymmetric information: adverse selection

10.1 Introduction

For a wide range of products (for example, things you buy from a grocery store) the markets are large and decentralised, and exchange is anonymous. There are, however, important exceptions. Consider the market for used cars. It is difficult to tell how good or bad a used car is unless you use it for some time. The seller, on the other hand, knows how good the car is.

In such situations, you would care about the identity of the seller. In fact, in such cases anonymous exchange is often impossible, as that would lead to a breakdown of the market. The customers are willing to pay a price based on the average quality. However, this price is not acceptable to the better-than-average quality car owner. This leads the owners of better-than-average cars to leave the market. Knowing this, customers would revise average quality downwards and consequently be willing to pay a lower price. But the same phenomenon then applies again, leading to further falls in average quality and willingness to pay, and so on.

The problem is called the ‘lemons problem’ or ‘adverse selection’.

The same problem arises in selling insurance. Consider the problem of selling health insurance to older people who know much more about their state of health than an insurance company. If the insurance company sets a price based on the average medical expenditure, this would typically lead to a market price (in this case, the insurance premium) that is too high for the healthiest individuals – and only the people who feel
that they are very likely to claim the insurance would buy insurance. This would lead, as in the case of used cars above, the market to fail.

Banks (or other lenders) face the same problem in credit markets. If a high rate is set in expectation of high average risk, safer borrowers will exit the market and only the riskiest borrowers will want to borrow. Again, as in the market for lemons, this causes a market failure.

A different sort of problem arises when you, as the manager of a company, are trying to hire some salespeople. As the job requires a door-to-door sales campaign, you cannot supervise them directly. And if the workers choose not to work very hard, they can always blame it on the mood of the customers. If you pay them a fixed wage independent of the sales they achieve, they are unlikely to work hard.

This problem is known as ‘moral hazard’.

Note the difference between adverse selection and moral hazard. In the first case, the asymmetry in information exists before you enter into the exchange (buy a used car, or sell insurance). In the latter, however, the asymmetry in information arises after the wage contract is signed. This is why another name for adverse selection is ‘hidden information’ and another name for moral hazard is ‘hidden action’.

In this chapter and the next, we will consider certain adverse selection and moral hazard problems and remedies in some detail.

In analysing markets so far, we have assumed that consumers and firms either have all relevant information about each other, or, if something is not known (the outcome of a random event, say), all parties face the same uncertainty. However, in many important markets, one party to a transaction is better informed compared to the other party. This chapter and the next aim to introduce you to this class of problems. The analysis in this chapter is concerned with one particular class of problems of this sort called adverse selection. The chapter aims to introduce you to this class of problems and offer some insights about potential solutions.

* explain the different types of asymmetric information problems
* explain how the scope of enquiry of economics is expanded by considering information asymmetry problems
* explain the types of problems that arise under adverse selection
* analyse the problem of adverse selection in the market for lemons
* analyse the problem of price discrimination by a monopolist under adverse selection
* analyse separating and pooling equilibria in the signalling model of education.

The chapter starts with a discussion of the scope of economic theory, then introduces some concepts useful for the analysis in this chapter. The chapter then covers the first model of adverse selection: Akerlof’s model of the market for lemons. Next, the chapter covers the second model of adverse selection: the problem of price discrimination when consumer types are unknown. Finally, the chapter covers the third model of adverse selection: the job market signalling model of Spence.

10.3 The scope of economic theory: a general comment

As you study the topics under asymmetric information, try to see what this analysis, combined with game theory, adds to economic theory. If you think about this question, you should be able to see that many different types of transactions and institutional arrangements cannot be understood by using the tools developed under the study of competitive markets.

The study of competitive markets and general equilibrium theory will not help you much in answering these. Indeed, in a general equilibrium environment you would not even be able to explain why banks exist, let alone why or how they need to be regulated.

However, with the tools of asymmetric information theory and game theory, we can address all of these, and many other questions. We will address only a small subset of these questions here, but as you read around these topics, try to see how vast the scope of the enquiry is once you learn to wield these tools properly. Indeed, such methods cover many types of questions across several fields of social sciences, evolutionary biology and the design of tort law.

These tools have expanded the scope of economic enquiry so dramatically that it is really a hopeless task to try to define it. Some textbooks still contain statements like ‘economics studies how scarce resources are allocated among competing uses’. This is indeed correct as a description of the study of general competitive equilibrium, but as a definition of the whole of economic theory such statements are entirely out-of-date. Modern tools have ensured that it is no longer possible to neatly define the scope of economics.

10.4 Akerlof’s (1970) model of the market for lemons

This is based on the famous article that gave rise to the field of asymmetric information: George A. Akerlof (1970) ‘The market for ‘lemons’: quality uncertainty and the market mechanism’, Quarterly Journal of Economics, volume 84, pages 488–500. Suppose there are four kinds of cars – there are new cars and old cars, and in each of these two categories there may be good cars and bad cars. Buyers of new cars purchase them without knowing whether they are good or bad. After using a car for some time, the owners can find out whether the car is good. This leads to an asymmetry: sellers of used cars have more information about the quality of the specific car they offer for sale.

The market for lemons: an example with two qualities

Consider a market for used cars. There are some low quality cars and some high quality cars. Potential sellers have a car each, and there are many more buyers than possible sellers in the market. A high quality car rarely breaks down. A low quality car provides a poorer ride quality over longer journeys and also breaks down with higher probability compared to high quality cars.

A seller values a high quality car at 1,000 and a low quality car at 300. A buyer values a high quality car at 1,300 and a low quality car at 400. All agents are risk-neutral. Assume that the sellers get the entire surplus from trade. Market failure

Suppose quality is observable to sellers but not to buyers. Buyers only know that a fraction of 1/2 of the cars in the market are high quality and the rest are low quality. Let us analyse the market outcome in this case.

The average value of buyers is 850. If all cars are in the market, this is the most buyers would pay. But at this price high quality cars withdraw. So only low quality cars would sell. Knowing this, buyers would be willing to pay at most 400. Assuming sellers get the entire surplus, the market price is 400 and only low quality cars exchange hands. The market outcome is not efficient since the gains from trading high quality cars are not exploited.

Separation through refunds

Suppose low quality cars break down with probability 0.8, and high quality cars break down with probability 0.1. Suppose the sellers have an option of promising a refund of R if the car breaks down.

Let us see if such a policy can lead to a separating equilibrium. In a separating equilibrium, it must be the case that sellers of high quality cars sell at 1,300 and promise a refund of R in the event of a breakdown, while sellers of low quality cars sell at 400 without any refund promise.

For high quality sellers to sell at 1,300 with a refund promise, it must be that they like this better than not selling, which means:

1300 − 0.1R > 1000

which implies R 6 3000. This constraint is called the participation constraint (PC) of high quality sellers.

Also, the low quality sellers must prefer to sell at 400 without a guarantee rather than imitate the high quality sellers (offer a refund of R and sell at 1,300). This implies:

400 > 1300 − 0.8R

which implies R > 1125. This is called the incentive compatibility constraint (IC) of low quality sellers.

The value of R must be such that the PC of high quality sellers and the IC of low quality sellers are both satisfied. This implies that the range of values of R for which the separating equilibrium holds is 1125

Forced refunds

Suppose the government decides to force each seller to offer a full refund if the car sold by the seller breaks down. How does this change the market outcome? Is the market outcome efficient?

Note that the maximum possible price in the market is 1,300. But even at a price of 1,300, 0.2 × 1300 = 260 is below 300. It follows that low quality sellers would not participate in the market, and only high quality sellers would remain and sell at 1,300.

But this is not efficient as the gain from trade of low quality cars is not realised. In other words, forcing sellers to issue a refund destroys the role of a refund policy in separating the two qualities. Since all sellers must issue a refund if the car breaks down, low quality sellers prefer to withdraw from the market, making it impossible to exploit gains from trading low quality cars. Policy choices that are not sensitive to the role of an instrument in providing information to uninformed agents do not necessarily promote efficiency.

10.5 A model of price discrimination

There is a seller and a buyer. The seller sells a unit of quality q at price t. The cost of producing quality q is c(q) = q2 .

The profit of the seller is given by:
π(t, q) = t − q 2
and the utility of the buyer is given by:
u(t, q, θ) = θq − t.

Here θ is a parameter that reflects how much the buyer cares about quality. θ is usually referred to as the buyer’s ‘type’. This is the buyer’s private information.

In this model we suppose θ can take two values, θ 1 and θ 2 , where θ 2 > θ 1 .

A contract is a quality and price pair (q, t) offered by the seller. We assume, for simplicity, that the seller has all the bargaining power.

The buyer gets 0 utility if they do not buy. Therefore, any contract must give the buyer at least 0. This is called the participation constraint (PC) of the buyer.

The indifference curve of a buyer of type θ i , i ∈ {1, 2} is given by: θ i q − t = constant.

The slope of an indifference curve is given by dt/dq = θ i .

Next, an iso-profit curve for the seller is given by:
t − q 2 = constant.

The slope of the iso-profit curve at quality q is 2q. Note that the slope does not depend on t, only q. So if we fix any q and change t, the slope does not change. In other words, all iso-profit curves have the same slope at points along any vertical line (at all values of t). Figure 10.1 shows the indifference map of the buyer of type θ 2 and that of the seller.

The arrows show the direction of improvement.

The full information benchmark

Under full information, the seller offers a contract (q_1^* , t_1^∗ ) to type θ_1 and another
contract (q_2^∗ , t_2^* ) to type θ_2.

To determine the optimal contract for each type θ_i , i ∈ {1, 2}, the seller solves:

max(q_i t_i) t_i − q_i^2 subject to θ_i q_i − t_i > 0.

However, since the seller has all the bargaining power, there is no reason to give the buyer any more than 0, and thus the constraint holds with equality. Thus the seller solves:

max(q_i t_i) t_i − q_i^2 subject to θ_i q_i − t_i = 0

which can be rewritten as:
max(q_i) θ_i q_i − q_i^2 .

Thus q_i^∗ = θ_i / 2, and t_i^* = θ_i q_i^∗ = θ_i^2 /2.

The optimal contract for type θ_i is obtained at the point of tangency between the iso-profit curve and the indifference curve of type θ_i at the reservation utility level.

Contracts under asymmetric information

Under asymmetric information, the full information contracts are not incentive compatible; type θ_1 has no incentive to take the contract meant for type θ_2. Doing so would yield a strictly negative payoff for type θ_1. However, type θ_2 does want to take the contract meant for type θ_1. ndeed, given any choice of (q_1 , t_1 ), incentive compatibility for type θ_2 requires that
(q_2 , t_2 ) must be on the indifference curve of type θ_2 passing through (q_1 , t_1)

Which point should the seller choose on the solid indifference curve for type θ 2 ? Where
the seller’s iso-profit curve is tangent to this line. The line has a slope of θ 2 , while the
slope of the iso-profit curve is 2q. Therefore, the optimal q 2 equals θ 2 /2, which is the
same as q 2 ∗ under full information.

The lesson that you should take away from this is the difference that asymmetric information makes. Under full information, each contract can be determined separately by satisfying the participation constraint of each type. Under asymmetric information, incentive constraints need to be considered. Sellers who sell different quality versions of a product (household appliances, say) at different prices need to solve this type of problem all the time. If the high quality version is very expensive, most consumers might select the lower quality but cheaper product. The seller must take this into account when setting the price of the high quality version.

Spence’s (1973) model of job market signalling

In previous applications, the uninformed party was trying to screen the informed. A different type of solution arises if the informed party can signal their information to the uninformed party. Spence’s
(1973) model of job market signalling studies precisely such a scenario. There are two types of workers – some with a high productivity (the high type) and some with a low productivity (the low type). High type workers have a cost of acquiring education of c ` per year of education, whereas the
low types have a cost of acquiring education of c_h per year of education, where c_h > c_l. Under full information, the competitive yearly wage for a high type is w h whereas the competitive yearly wage for a low type is w_l. Suppose employment lasts for n years.

A separating equilibrium has the following structure. There is a threshold number y ∗ of years of education such that if years of education y of a worker is at least y ∗ , they are considered to be a high type, and any worker with fewer than y ∗ years of education is considered to be a low type.

Thus, for a separating equilibrium, we must be able to choose a y ∗ such that B/c_h

A second type of equilibrium is a pooling equilibrium. Consider a pooling equilibrium in which both types of workers acquire an education. Suppose the firm requires y b years of education and suppose all types acquire this much education. Then the firm only knows that all types are in the market and pays an average wage w̄, reflecting the average productivity. Acquiring more education does not help, so there is no such deviation possible. But is it possible for a type to deviate to lower than y b years of education? Clearly, if anyone would want to deviate, it is the low type for whom the cost of education is higher. But in equilibrium the firm expects all types of workers to get y b years of education. If some worker deviates to no education, what would the firm believe about this worker’s type? the firm believes this worker is a low type and would pay w_l. In this case, deviation is not beneficial

Reading: Nicholson, W., Synder, C., Intermediate Microeconomics and its application (eleventh edition), South-Western, Cengage Learning, 2010

Chapter 15: Asymmetric Information

In contrast to the moral-hazard problem, in which the agent has private information about an action he or she chooses after the contract is signed, with the adverse-selection problem, the agent has private information about his or her type (an innate characteristic) before the contract is signed.

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Consumers differ in how much they value the good, but these valuations are not observable to the monopo- list. The monopolist offers the customer a menu of different-sized bundles at different prices. With second-degree price discrimination, the monopolist is not restricted to a constant price per unit but rather offers a menu of bundles at different prices, perhaps involving price discounts for large purchases, and has the consumers select bundles from the menu themselves

Whatever the size of the bundle the monopolist chooses to offer, it may as well ask the highest price
for the bundle that the consumer would be willing to pay. The most the consumer would pay for the bundle rather than doing without it is called gross consumer surplus. Gross consumer surplus is related to (ordinary) consumer surplus. Both are measures of consumers’ valuation for a good. Whereas consumer surplus subtracts the amount the consumer pays for the bundle from the amount the consumer would be willing to pay, gross consumer surplus does not. Gross consumer surplus equals the whole area under the demand curve, the shaded triangle and rectangle. Consumer surplus subtracts the
amount paid for the good (the shaded rectangle), leaving just the shaded triangle.


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If the monopolist has full information about types and can act on this information (that is, can require a consumer to buy only the bundle directed at his or her particular type and not some other bundle and can prevent consumers from selling repackaged bundles among themselves), the analysis of two consumer types adds nothing new to the analysis of one consumer type.

The profit-maximizing bundle for the low-value type involves q L units, given by the intersection between the low type’s marginal consumer surplus and the monopolist’s marginal cost. The bundle price equals the area of the shaded regions (A and B). The monopolist’s profit equals the area of region A. Similarly, the profit-maximizing bundle for the high-value type involves q H units given by the intersection between the high type’s marginal consumer surplus and the monopolist’s marginal cost. The bundle price equals the area of the entire shaded region A, B, C, and D, and the monopolist’s profit equals the areas of A and C.

The menu of bundles that maximized profit in the full information case will not work if the monopolist cannot observe the consumer’s types. The q H -unit bundle meant for the high-value type is priced to extract all of his or her consumer surplus. The high type would obtain positive surplus from instead purchasing the q L -unit bundle meant for the low-value type. The q H -unit bundle sold at a price that extracts all of the high type’s consumer surplus is not incentive-compatible.
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The monopolist can do even better than this. The monopolist can reduce the quantity associated with the bundle meant for the low-value type. On the one hand, reducing quantity reduces the profit from the sale of the bundle to low-value consumers. But a bigger effect is that the bundle meant for the low-value type becomes much less attractive to the high-value type. The high-value type places a high value on quantity, and a reduction in quantity ‘‘scares him or her off’’ from choosing the low-value bundle. As a result, the monopolist does not need to leave the high type with as much surplus, and can raise the price charged for the q H -unit bundle.
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By reducing the quantity associated with the low type’s bundle, the monopolist reduces the profit from sales to low types by the area of the triangle, E. This loss is more than offset by the fact that the low type’s bundle is less attractive to high types, and so the price charged to high types for the q H -unit bundle can be increased (by the area of F).
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Adding agents to the basic principal-agent model can make the moral-hazard problem better or worse, depending on the details of the setting. Suppose first that a single principal needs to hire a team of several agents to perform a task. The moral-hazard problem may be more severe in this setting. Each of the agents may slack off, relying on the efforts of the others. In large teams, it may be difficult to identify who is working hard and who is not, possibly leading all of them to slack. It is hard to provide a large number of agents with high-powered incentives because even if the firm is sold to the team of them, each would only obtain a small fraction of the firm’s gross profit.

On the other hand, if there are many agents in the market, but each works for a separate firm/principal, moral hazard may be less of a problem than it would be with one agent. By comparing the performance of their own firms with that of others’, uncertainty about agents’ efforts can be reduced. If a firm’s gross profit is low, but so are the gross profits of similar firms, it can be inferred that the poor performance was due to random market forces rather than the agent’s slacking off. On the other hand, if all firms but one perform well, it becomes increasingly clear that the one agent had slacked off.
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Competition among consumers in an auction can help the monopolist solve the adverse-selection problem. High-value consumers are pushed to bid high to avoid losing the good to another bidder. The exact outcome of the auction depends on the nature of the economic environment (which consumers know what information when) and the auction format.

A powerful and somewhat surprising result due to Vickery is that in simple settings (risk-neutral bidders who each know their valuation for the good perfectly, no collusion, and so forth), many of the different auction formats listed previously (and more besides) provide the monopolist with the same expected revenue in equilibrium
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Separating equilibrium Each type chooses a different action in a signaling game.
Pooling equilibrium All types choose the same action in a signaling game
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The presence of private information typically leads to inefficiency in signaling games. In the Spence education model, depending on the equilibrium, one or the other type of worker, or even sometimes both, obtained an education even though education had no social benefit in terms of raising productivity. In the standard model in which firms had full information about worker productivity, there would be no need for workers to seek wasteful education. This is a typical finding in signaling games. Players with private information depart from the efficient action choice to provide an informative signal to other players.
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