<\body> The foundation of all choice theory in economics is something called a . The idea is that if I present you with a pair of alternatives, then you could tell me which one you prefer, or possibly that you are indifferent between them. The word 'prefer' has different meanings in different contexts. For example, if I ask you whether you would prefer to see a movie or go to a hockey game your preference is expressing something about which you would enjoy. If I ask you whether you would like to have the Olympics in your local city, your preference may express something about what you think is best for everyone, or possibly something about what you think you are supposed to say. Sometimes you really can't say that one alternative is better than another. For example you might be equally happy with a ham sandwich or a tuna sandwich. If I allow for that possibility, it is hard to imagine a situation where you wouldn't be able to say something. If I am trying to think about your choice behavior and how I might understand it, I could begin by trying to imagine all the alternatives that you could possibly choose. I would collect them together in a big set . Then I could go about choosing different pairs of alternatives in and asking you to express your opinion about which of the two you prefer. Eventually (as long as you didn't get tired of answering questions) I could learn which alternative you preferred among any pair of alternatives in . This collection of information is your preference relation over . The set could be very general. For example, you might have guessed that we are going to be talking about preference relations over collections of possible consumption bundles. There is no need to stop there. Much of modern microeconomic theory arises from thinking about preferences over things like political parties, environmental policies, business strategies, location decisions, and so on. There are many kinds of preference relations you will encounter if you continue studying economics, but the most widely applied reasoning in economics assumes that preference relations have two properties - first, they must be in the sense that for pair of alternatives in , either you prefer one or the other, or are indifferent. There are some interesting preference relations that are incomplete, but lets leave that for the moment and concentrate on another problem. Your preference relation could be 'odd'. For example, suppose you like the Liberals more than the Conservatives because they are more socially progressive. You might like the NDP more than the Liberals for the same reason. However, you may prefer the Conservatives to the NDP because they are more fiscally responsible. Ignoring any other parties, then you have just expressed a complete and reasonable preference relation over the political parties. It does present something of a problem when you are trying to vote. You can't vote Conservative because you prefer the Liberals to the Conservatives, you can't vote Liberal because you prefer the NDP to the Liberals. Unfortunately you can't vote NDP either because you prefer the Conservatives to the NDP. We have a word for this kind of preference relation in economics, it is called an preference relation. To put is another way, a preference relation is one such that for any 3 alternatives , , and in , if is preferred to and is preferred to , then it must be the case that is preferred to . A complete transitive preference relation is called a . In fact, I have just described to you what rationality means in economics. A person is said to be rational in a particular economic environment if they have a complete and transitive preference relation over the alternatives that they face in that environment. In particular, it doesn't mean that people are greedy or self interested. It doesn't mean that they are super sophisticated calculators. It just means that they can express opinions about pairs of alternatives. So how do economists go about predicting what people will do? All they say is that whatever alternative is actually chosen from , then there cannot be another alternative in that is preferred to . It is true that in experiments, people sometimes exhibit intransitive preferences (though they quickly change their behavior when this is pointed out to them). There are also situations in which it seems impossible for people to make a choice. For the most part though, assuming that people are rational (have a complete transitive preference relation) is pretty innocuous. It might also occur to you that if you accept that people are rational decision makers, then you can't really get yourself in too much trouble. I never said what these preference relations had to look like. To assert that an individual chooses the alternative that he or she most prefers is almost tautological. The real content of economic theory involves restrictions it imposes on and on the preference relation over . Its failures and successes having nothing to do with the assumption of rationality. I want to show you the theorem that has transformed economics into a social science that is unlike most others. Before this, let me digress on another important idea. Introductory economics courses focus on consumption and consumption bundles. A consumption bundle is a pair x,y> where the first component of this vector is some quantity that you consume of one good (just call it good for short), the second component is the quantity you consume of the second good. Consumption doesn't generate happiness or utility or utils or anything like that. If we follow your first year course, and imagine that good has a price and good has a price , and that you have to spend, then the consumer is faced with a set of alternatives which consists of all pairs x,y> whose cost is less than or equal to , i.e., <\equation*> X\x,y\:p*x+q*y\W Here > is the set of all vectors with two non-negative components. Read the colon to mean ''such that''. Well, since we have a set of alternatives, it is pretty safe to assume that for any pair of alternatives (a pair of alternatives is a pair of vectors x,y> and x,y> here), the consumer can express a preference between them. Suppose for the moment that we could get the consumer to tell us what his or her preference relation is. But now we face a small problem. Suppose the consumer tells us the he prefers x,y> to x,y>. Suppose that we now look at another budget set > where prices are > and >, and maybe income is >. Lets pick this new set so that it contains both x,y> and x,y>. Do we really need to ask the consumer if he prefers x,y> to x,y> in this new set? Of course his preference could well change. People have no use for telephones unless other people have telephones. The change in income might mean that others can buy phones. The price changes might signal changes in quality of the goods that he is buying (suppose and are stocks or bonds or something like that). Now we begin to impose some restrictions of preferences and economic theory begins to have some content (of course, we also study what happens when preference relations change with prices and income). We are going to assume that if x,y> and x,y> are in both and > and if x,y> is preferred to x,y> in the preference relation relative to , then it must also be preferred in the preference relation relative to >. The important point is that the assumption that our consumer was rational imposed no restriction whatsoever on his behavior. This added assumption does restrict what we should expect to see him do. For example, suppose that we could run a long series of experiments in which our consumer is repeatedly asked to choose something from and that he consistently chooses x,y>. If our assumption is true, then it would be highly unlikely that if we had him choose repeatedly from > that he would consistently pick x,y>. The predictive content of the theory comes from the assumption that his preference relation is independent of the prices and income that he faces, not from the assumption that he is rational. You will see this repeatedly in economics - we will impose restrictions on and the preference relation over it, then make predictions (and test them). If you want to argue about economics the idea is to understand these restrictions and criticize them. It is a waste of time to argue about whether or not consumers are rational. So lets continue with first year economics. Since preference relations (lets' just say preferences from now on) are assumed to be independent of prices and income, we could sensibly take the consumer's preference relation and collect together the consumption bundles x,y>> which are indifferent to some bundle x,y>.x,y> is indifferent to x,y> if x,y> is at least as good as x,y> and at the same time x,y> is at least as good as x,y>.> As you remember from your first year course, this collection of consumption bundles is called an . Please note that the indifference curve comes directly from the preference relation and has nothing to do with utils or satisfaction of anything like that. Since we can construct an indifference curve for any consumption bundle, there is really a of indifference curves. Pick two indifference curves in this family, say > and > and choose a bundle x,y> from > (which is itself a set) and x>,y> from >. If x,y> is preferred to x,y>> then we say that the indifference curve > is >. Then of course, any bundle in > will be preferred to any bundle in >. There isn't much that can be said about indifference curves at this point except that when a consumer is rational, two distinct indifference curves can't have any point in common. To see this suppose that > is higher than >. Let x\>,y\>> be the point that the curves have in common, with x,y> in > and x,y> in >. Then x,y> is at least as good as x\>,y\>> since both are in >. x\>,y\>> is at least as good as x,y> since both are in >. Now transitivity requires that x,y> be at least as good as x,y> which is false if the consumer is rational. isn't true, then A can't be true either. This is called a proof Here we wanted to show that if a preference relation is transitive (A) then a pair of indifference curves couldn't cross (B). We showed that if the curves did cross, the preference relation couldn't be transitive.> At this point, we could try to describe graphic properties of the indifference curves. If we started to do that, we would end up spending considerable time trying to absorb graphic formalism and end up saying what we could have said with words. So it is time for me to introduce the theorem that makes economics work. Write the preference relation as >, meaning that x,y\x,y> whenever x,y> is preferred to x,y>. A is a relation that converts each bundle x,y> into a corresponding utility value or number. The utility function represents the preference relation > as long as x,y\ux,y> if and only if x,y\x,y>. If we happened to be able to find a utility function to represent a preference relation then we would have a big leg up. To predict what a consumer will do so far, we need to scan all pairs of consumption bundles until we find a bundle such that no other bundle is preferred to it. This makes for a lot of tedious pairwise comparisons. There isn't any obvious reason why this sort of reasoning is going to help us understand behavior. If preferences are represented by a utility function, we could take the function and find the bundle that produced the highest utility number in the set of alternatives. That would be relatively easy because we could use all the standard mathematical tricks we know about maximizing functions (like setting derivatives to zero and so on). Yet the utility function yields something far more important. As I mentioned above, the content of economic theory doesn't come from the rationality assumption. It comes from imposing restrictions on the preference relation and the feasible set. It is difficult to formulate ideas about preference relations since they are relative complex objects. On the other hand, it is much easier to impose and understand restrictions on utility functions. Assuming that people have utility functions which they maximize is just about the last thing we want to do. If we did that, then all the people who accuse economists of being irrelevant because they assume that consumers are 'rational' would have a good point. We would be guilty of predicting behavior by assuming that people do something that they obviously don't. So why use a utility function? We need to add one important restriction on preference relations, and one simplifying restriction. The simplifying restriction is that our consumer likes more of both goods - i.e., if x,y> and x,y> are such that x>> and y> and at least one of these inequalities is strict, then x,y\x,y> but not the other way around. Having more of any good makes the consumer strictly better off. For short, lets say that such a preference relation is . Now for the important restriction. Let x>,y\:x,y>\x,y> and x,y\:x,y\x,y>. Then both and are sets. If these sets are closed for any x,y\> then the preference relation is said to be Now the following important theorem is true: <\theorem> Let > be a continuous and monotonic rational preference relation. Then there exists a utility function which represents the preference relation >. <\proof> We are going to prove this constructively by actually making up the function. First some preliminaries. Let represent the 45> line (i.e., the set of all points in > which have the same horizontal and vertical coordinate). Let x,y> be any consumption bundle. Let \0> be a small positive number. The bundle maxx,y+\,maxx,y+\> is in and is strictly preferred to x,y> by the fact that preferences are monotonic. Similarly x,y> is preferred to minx,y-\,minx,y-\> by monotonicity. So the sets and are both non-empty. As preferences are continuous, these sets are both closed. This lets us deduce that the sets =B\Z> and =W\Z> are both closed as the intersection of closed sets. In Figure the set > is marked in red. It is the intersection of the 45> line and the set consisting of all bundles that are preferred to x,y>. The set > is marked in blue in the figure. Now the sets > and > are made up of bundles (in >) that have the same horizontal and vertical component. So we can associated each bundle in with this common component, which is just a positive real number. Since each bundle Z> either has x,y> or x,y\z> by the completeness of preferences, (recall that completeness is part of rationality) each point in is either in > or >. Each point in > or > is also in by construction, so \P>. By happy coincidence > and > share exactly one point in common. Part of the argument for this is an arcane point in set theory. Since \P> is all of , if they don't share a common point, then > must be the complement of > in . Since the complement of a closed set is open, > would have to be open which it cannot be. So there must be at least one common point. Could there be two? Again, suppose there were, say and >. They are both in so they are both on the 45> line. If they are distinct then, say, \z> (meaning each component of is strictly larger than the corresponding component of >). Then by monotonicity z> but not the other way around. Then by transitivity x,y\z> but not the other way around. But this can't be since \P>. All this work leads to the conclusion that for every bundle x,y> we can find a point on the 45> line which is indifferent to it. Lets call the common coordinate of this point the x,y> associated with the bundle x,y> (this probably emphasizes the point that utility is measured as some number of goods, not as utils or satisfaction). Finally, all we need to do is check that this utility function x,y> actually represents preferences. This is pretty straightforward. For example if x,y\ux,y>> then the associated with x,y> has a bigger common component than the > associated with x>,y>. Then x,y\z> (since P> for x,y>) z> (by monotonicity) x,y> (since \P> for x,y>). The other direction is just as easy. |The Sets > and >> So lets' collect our thoughts for a moment. When a consumer chooses a bundle from some budget set she picks something such that if we offer her some other bundle from the same budget set, she will not want it. If her preferences are transitive and complete (and continuous) it will that she is choosing a bundle to maximize a utility function subject to the budget constraint. In the consumer's own mind, there is no such thing as utility, rational utility maximization is an implication of simpler properties of consumer behavior. Nor is it assumed that there is any numerical way to measure happiness or satisfaction. These simply aren't parts of modern microeconomic theory. Why was this theorem so important? Well it shows first that economic methodology itself doesn't rely on grand assumptions about human behavior. Of course, when we impose restrictions on the preference relation or the set of feasible alternatives, we are making assumptions. These assumptions are part of what we call economic . When we formulate an economic model, we try to extract all the implications of the restrictions. These restrictions are predictions the model makes. We can collect data about the choices consumers actually do make, to check whether these predictions are right. When they are wrong, we know we need to reformulate the model (or change some of the restrictions). The second thing is shows is that we can extract these restrictions using some fairly basic mathematical tools, like the theory of optimization (and of course, the dreaded calculus). The mathematization of economics occurred in the late 50's and has had a remarkable impact on the way economists interact. To use mathematics, it is necessary that the concepts, sets, and functions involved be very precisely defined. There is no room for interpretation (though certainly there is room to fine tune and modify concepts). An economic concept must mean the same thing to everyone. This has had an impact that you might not expect. Anyone who understands basic mathematics should be able to understand the most advanced ideas in economic theory. Oddly enough mathematics makes economics very inclusive. This has had great benefits for economist, since other fields have been moving in much the same direction. Computer science, biology, ecology, environmental science, all use methods similar to those used by economists. The level of interaction among practitioners in these different fields is increasing to the enrichment of all. Most of this course tries to develop the mathematical and conceptual tools you need to formulate and analyze economic models on your own. As we go about this, you will see some models that have worked out pretty well in the sense that they give very good insight into some pretty applied problems. You will also get a chance to see some models that don't work so well. These 'failures' give a good deal of insight into how theoretical and empirical work interact. Though these applications are important in the overall scheme of things, they are not the main focus of the course. It is the art of building the models themselves that is the concern here. Once you begin to appreciate this approach, your subsequent studies in more applied areas will make more sense. <\references> <\collection> > > > > > > > > > > > > > > <\auxiliary> <\collection> <\associate|figure> |P> and |P>|> <\associate|toc> |math-font-series||1Preferences> |.>>>>|> |1.1Behavior |.>>>>|> > |1.2Indifference Curves |.>>>>|> > |1.3Economic Modelling |.>>>>|> >