Models in Microeconomic Theory

Part I (Chapters 1–7) presents models of an economic agent, discussing abstract models of preferences, choice, and decision making under uncertainty, before turning to models of the consumer, the producer, and monopoly. Part II (Chapters 8–14) introduces the concept of equilibrium, beginning, unconven� onally, with the models of the jungle and an economy with indivisible goods, and con� nuing with models of an exchange economy, equilibrium with ra� onal expecta� ons, and an economy with asymmetric informa� on. Part III (Chapters 15–16) provides an introduc� on to game theory, covering strategic and extensive games and the concepts of Nash equilibrium and subgame perfect equilibrium. Part IV (Chapters 17–20) gives a taste of the topics of mechanism design, matching, the axioma� c analysis of economic systems, and social choice.


expectations
In the models of markets we have discussed so far, equilibrium prices make the individuals' decisions compatible. Each individual takes the prices as given when deciding on her action, and at the equilibrium prices the demand and supply of every good are equal.
In this chapter, an individual's behavior is affected not only by the prices but also by her expectations regarding other parameters. Each individual takes these expectations, like the prices, as given. In equilibrium, each individual behaves optimally, the supply and demand for each good are equal, and the expectations of individuals are correct.
We present three models. In the first model, each individual chooses one of two bank branches. Her decision is affected only by her belief about the expected service time in each branch. In the second model, potential buyers of a used car, who cannot observe the quality of the cars for sale, take into account their expectation of the average quality of these cars as well as the price. In the third model, the unit cost of catching fish depends on the total amount of fish caught. Each fisher makes her decision taking as given both the price of fish and her expectation about the unit cost she will incur.

Introduction
Individuals live on the long main street of a town. At each end of the street there is a branch of a bank. Each individual cares only about the amount of time she spends dealing with the bank, which is the sum of her travel time and waiting time. The waiting time in each branch depends on the number of individuals who patronize the branch; each individual forms expectations about these waiting times. We are interested in the distribution of the individuals between the branches in an equilibrium in which each individual's expectations are correct.

188
Chapter 13. Equilibrium with prices and expectations interval [0, 1], with the interpretation that individual z resides at point z . Thus for each z ∈ [0, 1], the fraction z of individuals reside to the left of z and the fraction 1 − z reside to the right of z . The assumption that the set of individuals is infinite aims to capture formally a situation in which the number of individuals is very large and each individual's behavior has a negligible effect on the waiting times in the branches, even though these waiting times are determined by the aggregate behavior in the population.
The waiting time in each branch depends on the number of individuals who use that branch. Specifically, if the fraction of individuals who use branch j (i.e. the branch located at j , which is 0 or 1) is n j , then the waiting time in that branch is f j (n j ). We assume that each function f j is increasing and continuous, with f j (0) = 0 (i.e. if there are no customers in a branch, the waiting time in that branch is zero).
We assume, for simplicity, that an individual's travel time from x to branch z is the distance d (z , x ) = |z − x | between x and z . Every individual prefers the branch for which the sum of the travel time and the waiting time is smallest.

Definition 13.1: Service economy
waiting time technology continuous increasing functions f j : [0, 1] → with f j (0) = 0 for j = 0, 1, where f j (n j ) is the waiting time at branch j when the fraction of individuals who choose branch j is n j preferences each individual i ∈ I prefers a smaller loss to a larger one, where the loss from choosing branch j when t j is the waiting time in that branch Note that the bank branches are not decision-makers in this model: their locations and service technologies are fixed. The only decision-makers are the individuals.

Equilibrium
We define an equilibrium in the spirit of competitive equilibrium. Each individual has beliefs about the waiting times and assumes that her action does not 13.1 Distributing customers among bank branches 189 affect these waiting times. This assumption is analogous to our earlier assumption when defining competitive equilibrium that consumers and producers take prices as given, ignoring the effect of their own actions on the prices. Each individual chooses the branch that minimizes the time she spends dealing with the branch, given her beliefs about the waiting times. In equilibrium the individuals' beliefs are correct. Behind this definition is the assumption that agents' holding incorrect beliefs is a source of instability in the interaction; for stability, we need not only the individuals' actions to be optimal but also their beliefs to be correct.
A candidate for equilibrium consists of two numbers, t 0 and t 1 , the individuals' (common) beliefs about the waiting times in the branches, and a function l : [0, 1] → {0, 1}, assigning to every individual at point x the branch l (x ) (either 0 or 1) that she chooses.
To be an equilibrium, a candidate has to satisfy two conditions.
• The decision of each individual is optimal given her beliefs about the waiting times in the branches.
• The individuals' decisions and beliefs are consistent in the sense that the belief about the waiting time in each branch is correct, given the service technology and the fraction of individuals who select that branch.

Definition 13.2: Equilibrium of service economy
An equilibrium of the service economy 〈B, I , ( f j ) j ∈B , d 〉 is a pair ((t 0 , t 1 ), l ), consisting of a pair of numbers (t 0 , t 1 ) (the waiting times in the branches) and a function l : I → B (an assignment of each x ∈ I to a branch), such that optimality of individuals' choices (each individual is assigned to a branch for which the travel time plus waiting time for that branch is at most the travel time plus waiting time for the other branch) consistency where α(l , j ) is the fraction of individuals assigned to branch j by the function l .

Analysis
We now prove the existence of an equilibrium in this model, characterize it, and show that it is Pareto stable. We start by showing that there is a unique point z * such that if all individuals to the left of z * use branch 0 and all individuals to the right of z * use branch 1 then individual z * is indifferent between the two branches.

Lemma 13.1
There is a unique number z * such that

Proof
The function z + f 0 (z ) is continuous and increasing in z and takes the value 0 at the point 0 and the value 1 + f 0 (1) at the point 1. The function 1 − z + f 1 (1 − z ) is continuous and decreasing in z and takes the value 1 + f 1 (1) at 0 and the value 0 at 1. So the graphs of the functions have a unique intersection.
Next we show that for any expected waiting times, if for an individual at x branch 0 is at least as good as branch 1, then all individuals to the left of x prefer branch 0 to branch 1 (and analogously for an individual for whom branch 1 is at least as good as branch 0).

Lemma 13.2
For any pair of expected waiting times, if branch 0 is at least as good as branch 1 for an individual at x then branch 0 is better than branch 1 for every individual y with y < x , and if branch 1 is at least as good as branch 0 for an individual at x then branch 1 is better than branch 0 for every individual y with y > x .

Proof
Denote by t 0 and t 1 the expected waiting times in the branches. For branch 0 to be at least as good as branch 1 for an individual at x we need If y < x then d (y , 0) < d (x , 0) and d (y , 1) > d (x , 1), so that t 0 + d (y , 0) < t 1 + d (y , 1). A similar argument applies to the other case.
We can now prove the existence and uniqueness of an equilibrium in a service economy.

Proposition 13.1: Equilibrium of service economy
Every service economy has a unique equilibrium (up to the specification of the choice at one point).

Proof
We first show that every service economy has an equilibrium. Let z * be the number given in Lemma 13.1.
) and let l * be the function that assigns 0 to all individuals in [0, z * ] and 1 to all individuals in (z * , 1]. We now argue that ((t * 0 , t * 1 ), l * ) is an equilibrium.

Optimality of individuals' choices
Individual z * is indifferent between the two branches since z 1 (using the definition of z * ). By Lemma 13.2, all individuals on the left of z * prefer 0 to 1 and all on the right of z * prefer branch 1 to 0.
We now show that the equilibrium is unique. First note that a service economy has no equilibrium in which one branch is not used since if there were such an equilibrium, the waiting time at the unused branch would be 0 while the waiting time at the other branch would be positive, and hence individuals who are located close to the unused branch would prefer that branch to the other one.
Let ((t 0 , t 1 ), l ) be an equilibrium. By Lemma 13.2, there is a point z such that all individuals to the left of z choose 0 and all individuals to the right of z choose 1. Thus an individual at z is indifferent between the branches, so that z + t 0 = 1 − z + t 1 , and hence z = z * by Lemma 13.1. Therefore l is identical to l * up to the assignment at z * . By the consistency condition for equilibrium, t 0 = f 0 (z * ) and t 1 = f 1 (1 − z * ).
We now define the notion of Pareto stability for a service economy and show that the equilibrium of such an economy is Pareto stable.

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Chapter 13. Equilibrium with prices and expectations

Definition 13.3: Pareto stability
Consider a service economy 〈B, I , ( f j ) j ∈B , d 〉. For any assignment l and individual x ∈ I define L x (0, l ) = x + f 0 (α(l , 0)), the loss of x from choosing 0 given that all other individuals behave according to l . Similarly define , 1)) .
An assignment l is Pareto stable if there is no assignment l that Pareto dominates l in the sense that L x (l (x ), l ) ≤ L x (l (x ), l ) for all x ∈ I , with strict inequality for some x ∈ I .

Proposition 13.2: Pareto stability of equilibrium of service economy
Every equilibrium of a service economy is Pareto stable.

Proof
Let ((t * 0 , t * 1 ), l * ) be an equilibrium of the service economy 〈B, I , ( f j ) j ∈B , d 〉. Let l be an assignment. If the proportions of individuals at each branch are the same in l * and l , then the waiting times induced by the two assignments are the same. Since all individuals make the optimal choices in l * , the assignment l does not Pareto dominate l * .
If more individuals are assigned to branch 0 (say) by l than l * , then some individuals who are assigned to branch 1 by l * are assigned to branch 0 by l . In the equilibrium such individuals like branch 1 at least as much as branch 0. Under l , branch 0 is less attractive for each of them since the waiting time at that branch is greater than it is under l * . Hence l does not Pareto dominate l * .

Introduction
Second-hand cars of a particular model may differ substantially in quality. Each owner knows the quality of her car, but no buyer knows the quality of any given car. Because cars are indistinguishable to buyers, the price of every car is the same. Each owner decides whether to offer her car for sale, given this price. The decision of each potential buyer depends on her expectation of the quality of the cars offered for sale. A buyer may believe that the quality of the cars offered for sale is low, because owners of high quality cars are not likely to want to sell, given the uniform price. The fact that the cars selected for sale by the owners have low quality is often called adverse selection.

Model
The set of individuals in the market consists of a finite set S of owners and a larger finite set B of potential buyers. Each i ∈ S owns a car of quality Q(i ) ∈ (0, 1], which she knows. The utility of an owner of a car of quality q is q if she keeps it and p if she sells it at the price p . Each potential buyer obtains the utility αq − p , where α > 1, if she purchases a car of quality q at the price p , and the utility 0 if she does not purchase a car. The assumption that α > 1 implies that mutually beneficial trade is possible: every car is valued more highly by every potential buyer than by its owner.
A potential buyer does not know and cannot determine the quality of any specific car before purchasing it, and no owner can credibly communicate the quality of her car to a potential buyer. Thus for a potential buyer, purchasing a car is a lottery with prizes equal to the possible qualities of the car. We assume that a buyer maximizes her expected utility, so her decision depends on her expectationq of the quality of the cars for sale; she wishes to purchase a car if the amount she pays for it is less than αq . preferences the owner of a car of quality q prefers to sell it if in exchange she gets an amount of money p > q and prefers not to sell it if she gets an amount of money p < q a potential buyer prefers to buy a car than not to do so if αq > p , prefers not to buy it if αq < p , and is indifferent between the two options if αq = p , where α > 1 and p is the amount she pays andq is her belief about the expected quality of the cars for sale.

Equilibrium
Two parameters determine the behavior of the buyers and owners: the price of a car and the belief of the potential buyers about the expected quality of the cars 194 Chapter 13. Equilibrium with prices and expectations 0 q → p * = αq * q * Figure 13.1 Equilibrium of a second-hand car market. Each small disk represents a car; the red ones are offered for sale. for sale. An equilibrium consists of a price p * , a (common) belief q * of the potential buyers about the expected quality of cars for sale, a specification of the owners who offer their cars for sale, and a specification of the potential buyers who purchase cars, such that • the decision of every owner and potential buyer is optimal, given p * and q * • the number of cars offered for sale is equal to the number of buyers who wish to purchase a car • if at least one car is traded, the buyers' belief about the expected quality of the cars offered for sale is correct (if there is no trade the belief is not restricted).

Definition 13.5: Equilibrium of second-hand car market
An equilibrium (p * ,q * ,S * , B * ) of a second-hand car market (S, B,Q, α) consists of a number p * ≥ 0 (the price of a car), a number q * ≥ 0 (the potential buyers' common belief about the expected quality of the cars offered for sale), a set S * ⊆ S (the set of owners who offer their cars for sale), and a set B * ⊆ B (the set of potential buyers who purchase a car) such that optimality of choices for potential buyers: if B * = ∅ then p * ≤ αq * and if B \ B * = ∅ then p * ≥ αq * for owners: if i ∈ S * then p * ≥ Q(i ) and if i ∈ S \ S * then p * ≤ Q(i ) consistency |S * | = |B * | (the number of owners who sell their cars is equal to the number of potential buyers who buy a car) if S * = ∅ then q * = i ∈S * Q(i )/|S * |, the average quality of the cars owned by the members of S * (the potential buyers' belief about the expected quality of the cars offered for sale is correct).
An equilibrium in which ∅ ⊂ B * ⊂ B , so that p * = αq * , is illustrated in Figure 13.1.

Analysis
We now show that every second-hand car market has an equilibrium in which trade occurs (the set of owners who sell their cars is nonempty).

Proposition 13.3: Equilibrium of second-hand car market
Let 〈S, B,Q, α〉 be a second-hand car market. Name the owners so that S = {s 1 , . . . , s |S| } with Q(s 1 ) ≤ Q(s 2 ) ≤ · · · ≤ Q(s |S| ). The market has an equilibrium (p * ,q * ,S * , B * ) with S * = ∅. In any equilibrium the quality of every car that is sold is no greater than the quality of every other car.
The last claim in the proposition follows from the optimality of the owners' equilibrium choices. The quality of the cars of owners who sell is at most p * and the quality of the other owners' cars is at least p * .
Every second-hand car market has also an equilibrium in which no car is traded. Let p * be a positive number less than Q(s 1 ), the lowest quality, and let q * be such that αq * < p * . Then (p * ,q * , ∅, ∅) is an equilibrium: no potential buyer is willing to pay p * for a car, given her belief that the average quality of the cars for sale is q * , and no owner has a car whose quality is low enough to justify her selling it for p * . In this equilibrium, the potential buyers expect that the average quality of cars for sale is less than the lowest quality of all owners' cars. Note that the definition of equilibrium does not restrict the belief of the potential buyers when no owner offers a car for sale. We might regard the belief q * that we have assumed to be unreasonable. For example, if potential buyers know the range of qualities of the owners' cars, then their expectation should reasonably lie within this range, in which case an equilibrium in which no trade occurs does not exist.
Note that the equilibrium constructed in the proof of Proposition 13.3 is not Pareto stable unless S * = S. If S * ⊂ S, suppose that the owner of a car of quality q who has not sold the car transfers it to a potential buyer who has not purchased a car, in exchange for an amount of money between q and αq . Then both the owner and the buyer are better off.

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Chapter 13. Equilibrium with prices and expectations For some second-hand car markets, in all equilibria with trade only the lowest quality car is traded. Suppose for example that the set of car qualities is {1, 2, . . . , |S|} and α < 4 3 . In an equilibrium there is a number m * such that S * consists of the owners of cars with qualities 1, 2, . . . , m * and m * ≤ αq * , where q * is the average quality of the cars for sale, which is 1 2 (1 + m * ). That is, m * ≤ 1 2 α(1 + m * ) < 2 3 (1 + m * ), which is satisfied only by m * = 1.

Introduction
A community of fishers and consumers lives near a lake. Each fisher decides how many fish to catch and each consumer decides how many fish to buy, given the price of fish. The cost of catching fish increases with the number of fish caught.
In an equilibrium, the total amount of fish the fishers decide to catch is equal to the total amount the consumers decide to buy. Will the fishers catch too much in the sense that if they reduced their catch the price would adjust in such a way that everybody would be better off?

Model
The set of individuals in the economy consists of a set I of consumers and a set J of fishers. Each fisher decides how many fish to catch, up to a limit of L. If the total amount of fish caught by all fishers is T then the cost for a fisher to catch x fish is c (T )x , where c is a continuous, increasing function with c (0) = 0. That is, the larger is the total catch the more costly it is to fish. Each consumer decides how much fish to consume, up to a limit of one unit. Each consumer's preferences are represented by the function v x + m , with v > 0, where m is the amount of money she has and x is the amount of fish she consumes.
To make the main point of this section we analyze the model under the additional assumptions that (i) c (| J |L) > v (if all fishers operate at full capacity then their unit cost exceeds the value of a unit to consumers), (ii) c (0) < v (if all fishers are idle then their unit cost is less than the value of a unit to the consumers), and (iii) | J |L ≤ |I | (if all fishers operate at full capacity, their total output is less than the maximum possible total amount the consumers can consume).

Equilibrium
A candidate for an equilibrium of a fishing economy consists of a price for a unit of fish, the fishers' common expectation about the unit cost of fishing, the amount of fish that each fisher decides to catch, and the amount of fish chosen by each consumer, such that • every fisher chooses the amount of fish she catches to maximize her profit given the price and her expectation of the cost of fishing • every consumer chooses her consumption optimally given the price • the expectations of the fishers about the cost of fishing are correct • the total amount of fish caught is equal to the total amount the consumers choose to consume.

Definition 13.7: Competitive equilibrium of fishing economy
A competitive equilibrium (p * , c * , y * , x * ) of the fishing economy 〈I , J , v, L, c 〉 consists of a positive number p * (the price of a unit of fish), a nonnegative number c * (the fishers' belief about the unit cost of fishing), a non-negative number y * (the amount of fish caught by each fisher), and a non-negative number x * (the amount of fish chosen by each consumer) such that optimality of choices for consumers: x * maximizes the utility v x − p * x over [0, 1] for fishers: y * maximizes the profit p * y − c * y over [0, L] feasibility |I |x * = | J |y * (the total amount of fish consumed is equal to the total amount of fish caught) consistency c * = c (| J |y * ) (the fishers' expectation about the unit fishing cost is correct).

Proof
First, given c (0) < v , c (| J |L) > v , and the continuity of c there exists a number y * such that c (| J |y * ) = v . Now, given that c (| J |y * ) = v , our assumptions that c (| J |L) > v and c is increasing imply that y * < L and our assumption that L ≤ |I |/| J | implies that x * < 1. The tuple (p * , c * , y * , x * ) is a competitive equilibrium because all consumers and fishers are indifferent between all their possible actions, total production is equal to total consumption, and the fishers' expectation about the unit cost is correct.
To prove that the economy has no other equilibrium, suppose that (p , c , y , x ) is an equilibrium.
If p > v then the optimal choice of every consumer is 0, so that x = y = 0. But then c = c (0) < v , so that the optimal choice of every fisher is L, violating feasibility.
If p < v then the optimal choice of every consumer is 1, so that x = 1 and by the feasibility condition y = |I |/| J |. By the consistency condition c = c (|I |) and by our assumption that | J |L ≤ |I | we have c (|I |) ≥ c (| J |L) > v , so that catching a positive amount of fish is not optimal for any fisher.
Therefore p = v . It now suffices to show that c = p , since then by consistency we have v = c (| J |y ) and by feasibility | J |y = |I |x . If c > p then the optimality of the fishers' choices implies that y = 0; hence x = 0, so that the optimality of the consumers' choices requires p ≥ v . But now by consistency c = c (0) < v , a contradiction. A similar argument shows that c < p is not possible.