Mathematical Economics Some simple useful examples Herb Scarf

Mathematical Economics: Some simple, useful examples Herb Scarf Whitney Humanities Center February 25, 2009

I am a Mathematical Economist. What do I do for a living? • I prove theorems about economic theory. • What’s a theorem? • What’s a proof? • What’s economic theory? Hard to say. Some Examples Will help.

I will try to select some examples that are important, practically useful, and can be appreciated with simple numerical illustrations. But I would like to digress with some general themes in economic theory.

One of the central themes of economic theory is that people (countries? ) exchange the goods and services that they happen to own for goods and services that other economic agents own. Exchange takes place because we like other peoples' goods more than we like the goods we happen to possess.

Sometimes the goods that other agents own are the result of some manufacturing process involving labor, raw materials or machinery to produce. For example, China exchanges clothing that it manufactures for oil from Saudi Arabia. The U. S. exchanges computer software and Caterpillar tractors for grapes produced in Chile. The U. S. exchanges Treasury Bills (claims on future consumption) and wheat for automobiles produced in Japan.

Now let’s turn to the first elementary, useful, topic. The Matching (Marriage) Problem • We consider 3 men, Alan, Bob and Chuck • and 3 women, Alice, Beth and Clair. • The men have preferences for the women, and • the women have preferences for the men. • Who shall marry?

Preferences Men 1 2 3 Alan: Alice Beth Clair Bob: Beth Alice Clair Chuck Alice Beth Clair Women 1 2 3 Alice: Bob Alan Chuck Beth: Bob Alan Chuck Clair: Alan Bob Chuck Suppose that the men and women are paired as follows: Alan Alice Bob Beth Chuck Clair Is this assignment stable? The assignment is unstable if a man and a woman prefer each other to their assigned mates. This assignment is unstable because Alan and Alice prefer each other to their assigned partners.

A Stable Marriage (with the same Preferences) Men 1 2 3 Alan: Alice Beth Clair Bob: Beth Alice Clair Chuck Alice Beth Clair Alan Alice Women 1 2 3 Bob Beth Alice: Bob Alan Chuck Clair Beth: Bob Alan Chuck Clair: Alan Bob Chuck Suppose that the men and women are paired as follows: This pairing is stable. Alan and Bob are assigned their first choice. The only objection can be by Chuck and one of the women. But neither Alice nor Beth prefer Chuck to her assigned partner. Is there always a stable marriage? We’ll see.

The Roommate Problem is the Marriage Problem without Gender. Is there a stable pairing? Suppose that there are 4 roommates who are to be paired to share two rooms. • If the assignments are {Alan, Bob}, {Chuck, Dan} then Bob and Chuck will object. The preferences of the roommates are as follows: • If the assignments are {Alan, Chuck}, {Bob, Dan} then Bob and Alan will object. Alan: Bob Chuck Dan Bob: Chuck Alan Dan • If the assignments are {Alan, Dan}, {Bob, Chuck} then Alan and Chuck will object. Chuck: Alan Bob Dan • There is no stable assignment. Dan: Alan Bob Chuck

Polygamous Matching • An important generalization of the Matching Problem arises when the two sets to be matched don't have the same size. • For example, the two sets can consist of a large number of Residents (R) who are to be assigned to a small number of Hospitals (H). • Each Hospital has a limited number of places. For simplicity let us assume that the number of places at the various Hospitals is equal to the total number of Residents. • We assume that each Resident ranks all of the Hospitals, and each Hospital ranks all of the Residents. • The problem is to assign each Resident to a Hospital in a stable fashion.

Stability • The assignment is stable if there is no Resident R and Hospital H such that – Resident R prefers Hospital H to the hospital to which she is assigned, and – Hospital H prefers Resident R to one of the residents who has been assigned to H. • Is there a stable assignment? • Theorem: Yes! (Gale, Shapley, 1962) • This includes the Marriage Problem as a special case.

Observations The proof of the Theorem asserting that there always is a stable matching is based on an elegant, but not totally obvious, algorithm. In general, there are many different matchings, some of which favor the Residents and others favoring the Hospitals. There are many interesting variations: A Resident may refuse to work at a particular Hospital. (An easy modification. ) A pair of Residents (say a husband wife) may insist on being assigned to Hospitals in the same city. (There may be no such matchings. )

Applications • These techniques have been used for many years in the National Residents Matching Program (MRMP). • They have been used in the assignments of students to schools in New York, Boston, Budapest and Singapore. And in college admissions in Turkey. • In the allocation of graduating students from Hebrew Union College to congregations. • The assignment of students to houses at MIT. • Matching advertisers to advertising slots on Google. • I know of applications to the assignment of packets on the Internet.

• Now I would like to chat with you about a variation in the basic model that has been used to discuss, among other things, kidney transplants. • The issue is to assign the kidneys offered by willing donors to recipients in need of a functioning kidney. • The problem is that the friendly donor’s kidney may be not be suitable for the particular recipient and may have to be exchanged for the kidney of another donor. • “A Kidney Exchange Clearinghouse in New England, ” Alvin E. Roth, Tayfun Sönmez, M. Utku Ünver (2004) • The model to be discussed is called the House Problem. It is very simple, but it is a serious introduction to the economics of exchange.

The House Problem Preferences of Owners for Houses O-1 O-2 O-3 O-4 O-5 H-1 5 5 4 2 2 H-2 2 4 2 1 4 H-3 1 3 3 5 1 H-4 3 1 5 4 5 H-5 4 2 1 3 3 How shall we exchange these houses? If Owners 3 and 5 exchange their houses, they will each receive their favorite house. They will certainly not object. O-1 O-2 O-3 O-4 O-5 H-1 H-2 H-3 H-4 H-5 1. Owner 1’s favorite house is H-3. Let’s give it to him. 2. We have to compensate Owner 3. His favorite house is H-5. Let’s give him H-5. 3. What shall we give Owner 5? Let’s give him his favorite house H-3. 4. We seem to have a cycle of size 2. (It could be size 3 or more. )

How to Continue? Preferences of Owners for Houses Since Owners 3 and 5 exchange their houses, let’s remove them. O-1 O-2 O-3 O-4 O-5 H-1 5 5 4 2 2 H-2 2 4 2 1 4 H-3 1 3 3 5 1 H-4 H-5 3 4 1 2 5 1 4 3 5 3 O-1 O-2 O-4 H-1 5 5 2 H-2 2 4 1 H-4 3 1 4 How shall we distribute these 3 houses?

O-1 O-2 O-4 H-1 5 5 2 H-2 2 4 1 H-4 3 1 4 1. Owner 1’s favorite is House 2. Give it to him. 2. We have to compensate Owner 2. His favorite house is H-4. Give it to him. 3. But Owner 4’s favorite is H-2. We give H-2 to Owner 4. O-1 O-2 O-3 O-4 O-5 H-1 H-2 H-3 H-4 H-5 This sets up a second cycle In which Owners 2 and 4 exchange their houses. Owner 1 seems to be left with his own House which he values less than any other House.

The Solution Preferences of Owners for Houses O-1 O-2 O-3 O-4 O-5 H-1 5 5 4 2 2 H-2 2 4 2 1 4 H-3 1 3 3 4 1 H-4 3 1 5 4 5 H-5 4 2 1 3 3 O-1 O-2 O-3 O-4 O-5 H-1 H-2 H-3 H-4 H-5 This assignment has the property that no set of owners can make themselves better off by exchanging the houses they originally own in a different way. WHY?

We are given a set of Owners, each with a single House. Each Owner has specific preferences for the Houses. Is there always an assignment of the Houses to the Owners that is stable in the sense that no collection of Owners can make themselves better off by swapping their original Houses in a different way? Theorem: Yes. (Shapley, Scarf, 1974) Proof: You’ve just seen it. (Gale) Oddly enough, there is only one such assignment! Even with 7000 Houses.

The particular assignment that we have reached is called the Core of the House Market. Gale’s algorithm constructs a Top Trading Cycle (of arbitrary size), removing the Owners and Houses in this cycle, solving the smaller House Market by finding a new Top Trading Cycle, and repeating the procedure until all of the Houses have been assigned. This is called a “mechanism”: We select an arbitrary set of Owners’ preferences for Houses and obtain a specific redistribution of the Houses. Is it ever advantageous for a specific Owner to misrepresent his preferences if this mechanism is used? Can he obtain a better outcome for himself by lying? Theorem: No. (Can you see the argument? ) We have the following very surprising result: Theorem (Jin. Peng Ma, 1994): Gale’s procedure is the only mechanism (subject to minor conditions) for which lying is disadvantageous!

Some Reflections We have examined several elementary examples illustrating the exchange of spouses, roommates and houses. Example Existence of a Solution Uniqueness of the Solution Marriage Market Yes Typically Many Solutions Roommate Problem Not Guaranteed House Market Yes Unique The existence of a solution and its uniqueness are basic questions to be raised about any economic model.

These examples are very special. In a mature economy such as ours, vast numbers of economic choices are being made at every moment of time. Economists see these choices as mediated by competitive prices which simultaneously equate supply and demand in a great multiplicity of markets. (Our very special examples haven’t used prices at all, but they were selected to be simple. ) Investment decisions and the levels of manufacturing outputs are governed by considerations of profitability, using market prices. Consumer choices depend on income and the prices of available goods and services, today and in the future. These decisions are linked together in an extraordinarily complex way.

This discussion can be captured by a formal mathematical model which allows us to analyze the issues of existence of equilibrium prices, uniqueness of market clearing prices, and the stability of the market system after a disruption in its functioning. There algorithms for calculating market clearing prices, if we are given a description of consumer preferences, the assets held by the agents in the economy and the technical processes available for production. These algorithms are used extensively in discussing the consequences of changes in economic policies - such as the establishment of NAFTA.

This conceptual apparatus has many flaws: • It isn’t very good with economies of scale, • or in dealing with an unforeseen future. • And we are certainly not in equilibrium at the present time. Our old equilibrium has been given a swift kick in the rear end, and we are trying to find our way to a new equilibrium. But, imperfect as it is, the conceptual framework of economic theory is indispensible as a tool of analysis and repair. It ‘s the only game in town!