#LyX 1.5.5 created this file. For more info see http://www.lyx.org/ \lyxformat 276 \begin_document \begin_header \textclass amsart \begin_preamble \usepackage{graphicx} \end_preamble \language english \inputencoding latin1 \font_roman palatino \font_sans default \font_typewriter default \font_default_family default \font_sc false \font_osf false \font_sf_scale 100 \font_tt_scale 100 \graphics default \paperfontsize 12 \spacing single \papersize default \use_geometry false \use_amsmath 1 \use_esint 0 \cite_engine basic \use_bibtopic false \paperorientation portrait \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \defskip medskip \quotes_language english \papercolumns 1 \papersides 1 \paperpagestyle plain \tracking_changes false \output_changes false \author "" \author "" \end_header \begin_body \begin_layout Title Preferences \end_layout \begin_layout Author Michael Peters \end_layout \begin_layout Date \begin_inset ERT status collapsed \begin_layout Standard \backslash today{} \end_layout \end_inset \end_layout \begin_layout Section Introduction \end_layout \begin_layout Standard The foundation of all choice theory in economics is something called a \shape italic preference relation \shape default . 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. \end_layout \begin_layout Standard 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 \begin_inset Formula $X$ \end_inset . Then I could go about choosing different pairs of alternatives in \begin_inset Formula $X$ \end_inset 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 \begin_inset Formula $X$ \end_inset . This collection of information is your preference relation over \begin_inset Formula $X$ \end_inset . \end_layout \begin_layout Standard The set \begin_inset Formula $X$ \end_inset 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. \end_layout \begin_layout Standard 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 \shape italic complete \shape default in the sense that for \shape italic any \shape default pair of alternatives in \begin_inset Formula $X$ \end_inset , either you prefer one or the other, or are indifferent. There are some interesting preference relations that are incomplete, but let's 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 Conservative s, 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. \begin_inset Foot status collapsed \begin_layout Standard I suppose this could explain why so many people don't vote. \end_layout \end_inset \end_layout \begin_layout Standard We have a word for this kind of preference relation in economics, it is called an \shape italic intransitive \shape default preference relation. To put this another way, a \shape italic transitive \shape default preference relation is one such that for any 3 alternatives \begin_inset Formula $x$ \end_inset , \begin_inset Formula $y$ \end_inset , and \begin_inset Formula $z$ \end_inset in \begin_inset Formula $X$ \end_inset , if \begin_inset Formula $x$ \end_inset is preferred to \begin_inset Formula $y$ \end_inset and \begin_inset Formula $y$ \end_inset is preferred to \begin_inset Formula $z$ \end_inset , then it must be the case that \begin_inset Formula $x$ \end_inset is preferred to \begin_inset Formula $z$ \end_inset . A complete transitive preference relation is called a \shape italic rational preference relation \shape default . \end_layout \begin_layout Standard 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. \end_layout \begin_layout Subsection Behavior \end_layout \begin_layout Standard So how do economists go about predicting what people will do? All they say is that whatever alternative \begin_inset Formula $x$ \end_inset is actually chosen from \begin_inset Formula $X$ \end_inset , then there cannot be another alternative in \begin_inset Formula $X$ \end_inset that is preferred to \begin_inset Formula $x$ \end_inset . It is true that in experiments, people sometimes exhibit intransitive preferenc es (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. \end_layout \begin_layout Standard 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 \begin_inset Formula $X$ \end_inset and on the preference relation over \begin_inset Formula $X$ \end_inset . Its failures and successes having nothing to do with the assumption of rationality. \end_layout \begin_layout Standard Introductory economics courses focus on consumption and consumption bundles. A consumption bundle is a pair \begin_inset Formula $\left(x,y\right)$ \end_inset where the first component of this vector is some quantity that you consume of one good (just call it good \begin_inset Formula $x$ \end_inset for short), and 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 \begin_inset Formula $x$ \end_inset has a price \begin_inset Formula $p$ \end_inset and good \begin_inset Formula $y$ \end_inset has a price \begin_inset Formula $q$ \end_inset , and that you have \begin_inset Formula $W$ \end_inset to spend, then the consumer faces a set of alternatives \begin_inset Formula $X$ \end_inset which consists of all pairs \begin_inset Formula $\left(x,y\right)$ \end_inset whose cost is less than or equal to \begin_inset Formula $W$ \end_inset , i.e., \begin_inset Formula \[ X\equiv\left\{ \left(x,y\right)\in\mathbb{R}_{+}^{2}:px+qy\leq W\right\} \] \end_inset Here \begin_inset Formula $\mathbb{R}_{+}^{2}$ \end_inset is the set of all vectors with two non-negative components. Read the colon to mean \begin_inset Quotes erd \end_inset such that \begin_inset Quotes erd \end_inset . \end_layout \begin_layout Standard 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 \begin_inset Formula $\left(x,y\right)$ \end_inset and \begin_inset Formula $\left(x',y'\right)$ \end_inset 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 that he prefers \begin_inset Formula $\left(x,y\right)$ \end_inset to \begin_inset Formula $\left(x',y'\right)$ \end_inset . Suppose that we now look at another budget set \begin_inset Formula $X'$ \end_inset where prices are \begin_inset Formula $p'$ \end_inset and \begin_inset Formula $q'$ \end_inset , and maybe income is \begin_inset Formula $W'$ \end_inset . Let's pick this new set so that it contains both \begin_inset Formula $\left(x,y\right)$ \end_inset and \begin_inset Formula $\left(x',y'\right)$ \end_inset . Do we really need to ask the consumer if he prefers \begin_inset Formula $\left(x,y\right)$ \end_inset to \begin_inset Formula $\left(x',y'\right)$ \end_inset 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 \begin_inset Formula $x$ \end_inset and \begin_inset Formula $y$ \end_inset are stocks or bonds or something like that). \end_layout \begin_layout Standard 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 \begin_inset Formula $\left(x,y\right)$ \end_inset and \begin_inset Formula $\left(x',y'\right)$ \end_inset are in both \begin_inset Formula $X$ \end_inset and \begin_inset Formula $X'$ \end_inset and if \begin_inset Formula $\left(x,y\right)$ \end_inset is viewed by the consumer to be at least as good as \begin_inset Formula $\left(x',y'\right)$ \end_inset in the preference relation relative to \begin_inset Formula $X$ \end_inset , then it must also be at least as good as \begin_inset Formula $(x',y')$ \end_inset in the preference relation relative to \begin_inset Formula $X'$ \end_inset . \begin_inset Foot status collapsed \begin_layout Standard This assumption is called the \emph on weak axiom of revealed preference. \end_layout \end_inset \end_layout \begin_layout Standard The important point is that the assumption that our consumer was rational imposed no restriction whatsoever on his behavior. The added assumption about how his or her preferences are related across different budget sets 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 \begin_inset Formula $X$ \end_inset and that he consistently chooses \begin_inset Formula $\left(x,y\right)$ \end_inset . If our assumption is true, then it would be highly unlikely that if we had him choose repeatedly from \begin_inset Formula $X'$ \end_inset that he would consistently pick \begin_inset Formula $\left(x',y'\right)$ \end_inset . \begin_inset Foot status collapsed \begin_layout Standard He might do this once if he were indifferent, but would probably not do it consistently if he were indifferent. \end_layout \end_inset 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. \end_layout \begin_layout Standard You will see this repeatedly in economics - we will impose restrictions on \begin_inset Formula $X$ \end_inset 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 restrictio ns and criticize them. It is a waste of time to argue about whether or not consumers are rational. \end_layout \begin_layout Subsection Indifference Curves \end_layout \begin_layout Standard So let's continue with first year economics. Since preference relations (let's 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 \shape italic all \shape default the consumption bundles \begin_inset Formula $\left(x',y^{\prime}\right)$ \end_inset which are indifferent to some bundle \begin_inset Formula $\left(x,y\right)$ \end_inset . \begin_inset Foot status collapsed \begin_layout Standard To be formal, we could say that \begin_inset Formula $\left(x,y\right)$ \end_inset is indifferent to \begin_inset Formula $\left(x',y'\right)$ \end_inset if \begin_inset Formula $\left(x,y\right)$ \end_inset is at least as good as \begin_inset Formula $\left(x',y'\right)$ \end_inset and at the same time \begin_inset Formula $\left(x',y'\right)$ \end_inset is at least as good as \begin_inset Formula $\left(x,y\right)$ \end_inset . \end_layout \end_inset As you remember from your first year course, this collection of consumption bundles is called an \shape italic indifference curve \shape default . 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 \shape italic family \shape default of indifference curves. \end_layout \begin_layout Standard Pick two indifference curves in this family, say \begin_inset Formula $C_{1}$ \end_inset and \begin_inset Formula $C_{2}$ \end_inset and choose a bundle \begin_inset Formula $\left(x,y\right)$ \end_inset from \begin_inset Formula $C_{1}$ \end_inset (which is itself a set) and \begin_inset Formula $\left(x^{\prime},y'\right)$ \end_inset from \begin_inset Formula $C_{2}$ \end_inset . If \begin_inset Formula $\left(x,y\right)$ \end_inset is preferred to \begin_inset Formula $\left(x',y^{\prime}\right)$ \end_inset then we say that the indifference curve \begin_inset Formula $C_{1}$ \end_inset is \shape italic higher than \shape default \begin_inset Formula $C_{2}$ \end_inset . Then of course, any bundle in \begin_inset Formula $C_{1}$ \end_inset will be preferred to any bundle in \begin_inset Formula $C_{2}$ \end_inset . 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 \begin_inset Formula $C_{1}$ \end_inset is higher than \begin_inset Formula $C_{2}$ \end_inset . Let \begin_inset Formula $\left(x^{\prime\prime},y^{\prime\prime}\right)$ \end_inset be the point that the curves have in common, with \begin_inset Formula $\left(x,y\right)$ \end_inset in \begin_inset Formula $C_{1}$ \end_inset and \begin_inset Formula $\left(x',y'\right)$ \end_inset in \begin_inset Formula $C_{2}$ \end_inset . Then \begin_inset Formula $\left(x',y'\right)$ \end_inset is at least as good as \begin_inset Formula $\left(x^{\prime\prime},y^{\prime\prime}\right)$ \end_inset since both are in \begin_inset Formula $C_{2}$ \end_inset . \begin_inset Formula $\left(x^{\prime\prime},y^{\prime\prime}\right)$ \end_inset is at least as good as \begin_inset Formula $\left(x,y\right)$ \end_inset since both are in \begin_inset Formula $C_{1}$ \end_inset . Now transitivity requires that \begin_inset Formula $\left(x',y'\right)$ \end_inset be at least as good as \begin_inset Formula $\left(x,y\right)$ \end_inset which is false if the consumer is rational. \begin_inset Foot status collapsed \begin_layout Standard A small digression - this simple argument is an example of a line of reasoning that you will see often in economics. If you want to show that some property A implies that another property B must be true, try to show that if \begin_inset Formula $B$ \end_inset isn't true, then A can't be true either. This is called a proof \shape italic by contradiction. \shape default 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. \end_layout \end_inset \end_layout \begin_layout Standard 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. \end_layout \begin_layout Standard Write the preference relation as \begin_inset Formula $\succeq$ \end_inset , meaning that \begin_inset Formula $\left(x,y\right)\succeq\left(x',y'\right)$ \end_inset whenever \begin_inset Formula $\left(x,y\right)$ \end_inset is preferred to \begin_inset Formula $\left(x',y'\right)$ \end_inset . A \shape italic utility function \shape default is a relation that converts each bundle \begin_inset Formula $\left(x,y\right)$ \end_inset into a corresponding utility value or number. The utility function \begin_inset Formula $u$ \end_inset represents the preference relation \begin_inset Formula $\succeq$ \end_inset as long as \begin_inset Formula $u\left(x,y\right)\geq u\left(x',y'\right)$ \end_inset if and only if \begin_inset Formula $\left(x,y\right)\succeq\left(x',y'\right)$ \end_inset . 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 mathematica l tricks we know about maximizing functions (like setting derivatives to zero and so on). \end_layout \begin_layout Standard 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. \end_layout \begin_layout Standard 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. \end_layout \begin_layout Standard So why use a utility function? We need to add one important restriction on preference relations, and one simplifying restriction. \begin_inset Foot status collapsed \begin_layout Standard Simplifying means that I could make the same argument I am about to make without the restriction, but it would take me a lot longer. \end_layout \end_inset The simplifying restriction is that our consumer likes more of both goods - i.e., if \begin_inset Formula $\left(x,y\right)$ \end_inset and \begin_inset Formula $\left(x',y'\right)$ \end_inset are such that \begin_inset Formula $x\geq x^{\prime}$ \end_inset and \begin_inset Formula $y\geq y'$ \end_inset then \begin_inset Formula $\left(x,y\right)\succeq\left(x',y'\right)$ \end_inset . Furthermore, if one of the inequalities is strict, we will assume that it is not true that \begin_inset Formula $\left(x^{\prime},y^{\prime}\right)\succeq\left(x,y\right)$ \end_inset . Having more of any good makes the consumer strictly better off. For short, let's say that such a preference relation is \shape italic monotonic \shape default . Since it is awkard to say that \begin_inset Formula $\left(x,y\right)\succeq\left(x',y'\right)$ \end_inset but not \begin_inset Formula $\left(x^{\prime},y^{\prime}\right)\succeq\left(x,y\right)$ \end_inset , lets just say that \begin_inset Formula $\left(x,y\right)\succ\left(x',y'\right)$ \end_inset , which means that \begin_inset Formula $\left(x,y\right)$ \end_inset is strictly preferred to \begin_inset Formula $\left(x^{\prime},y^{\prime}\right)$ \end_inset . \end_layout \begin_layout Standard Now for the important restriction. The set of bundles that are at least as good as \begin_inset Formula $(x,y)$ \end_inset is given by \begin_inset Formula $B=\left\{ \left(x^{\prime},y^{\prime}\right)\in\mathbb{R}_{+}^{2}:\left(x^{\prime},y^{\prime}\right)\succeq\left(x,y\right)\right\} $ \end_inset . The set of consumption bundles that are no better than \begin_inset Formula $(x,y)$ \end_inset is given by \begin_inset Formula $W=\left\{ \left(x',y'\right)\in\mathbb{R}_{+}^{2}:\left(x,y\right)\succeq\left(x',y'\right)\right\} $ \end_inset . The important assumption is that both \begin_inset Formula $B$ \end_inset and \begin_inset Formula $W$ \end_inset are \shape italic closed \shape default sets \begin_inset Foot status collapsed \begin_layout Standard A closed set is one for which any convergent sequence of points in the set converges to a point in the set. The set \begin_inset Formula $\{x:0\leq x\leq1\}$ \end_inset is closed, the set \begin_inset Formula $\{x:0>z'$ \end_inset (meaning each component of \begin_inset Formula $z$ \end_inset is strictly larger than the corresponding component of \begin_inset Formula $z'$ \end_inset ). Then by monotonicity \begin_inset Formula $z\succ z'$ \end_inset . Since \begin_inset Formula $z\in P^{-}=Z\cap W$ \end_inset , \begin_inset Formula $\left(x,y\right)\succeq z$ \end_inset . Since \begin_inset Formula $z\succ z^{\prime}$ \end_inset , we must have \begin_inset Formula $\left(x,y\right)\succ z'$ \end_inset since preferences are transitive. But this is inconsistent with \begin_inset Formula $z'\in P^{+}$ \end_inset . \end_layout \begin_layout Proof All this work leads to the conclusion that for every bundle \begin_inset Formula $\left(x,y\right)$ \end_inset we can find a point on the 45 \begin_inset Formula $^{0}$ \end_inset line which is indifferent to it. Let's simply call the common coordinate of this point the \shape italic utility \shape default \begin_inset Formula $u\left(x,y\right)$ \end_inset associated with the bundle \begin_inset Formula $\left(x,y\right)$ \end_inset (this emphasizes the point that utility is measured as some number of goods, not as utils or satisfaction). \end_layout \begin_layout Proof Finally, all we need to do is check that this utility function \begin_inset Formula $u\left(x,y\right)$ \end_inset actually represents preferences. This is pretty straightforward. For example if \begin_inset Formula $u\left(x,y\right)\geq u\left(x',y^{\prime}\right)$ \end_inset then the \begin_inset Formula $z$ \end_inset associated with \begin_inset Formula $\left(x,y\right)$ \end_inset has a bigger common component than the \begin_inset Formula $z'$ \end_inset associated with \begin_inset Formula $\left(x^{\prime},y'\right)$ \end_inset . Then \begin_inset Formula $\left(x,y\right)\succeq z$ \end_inset (since \begin_inset Formula $z\in P^{-}$ \end_inset for \begin_inset Formula $\left(x,y\right)$ \end_inset ) \begin_inset Formula $\succeq z'$ \end_inset (by monotonicity) \begin_inset Formula $\succeq\left(x',y'\right)$ \end_inset (since \begin_inset Formula $z'\in P^{+}$ \end_inset for \begin_inset Formula $\left(x',y'\right)$ \end_inset ). The other direction is just as easy. \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Standard \begin_inset Graphics filename preferences_fig1.eps \end_inset \end_layout \begin_layout Standard \begin_inset Caption \begin_layout Standard \begin_inset LatexCommand label name "g1" \end_inset The sets \begin_inset Formula $P^{+}$ \end_inset and \begin_inset Formula $P^{-}$ \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Standard So let's 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 \shape italic appear to be the case \shape default 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. \end_layout \begin_layout Subsection Economic Modeling \end_layout \begin_layout Standard 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 \shape italic models \shape default . 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). \end_layout \begin_layout Standard 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. \end_layout \begin_layout Standard 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. \begin_inset Foot status collapsed \begin_layout Standard You might like to compare the definition of utility I have given above with definitions you will hear for important concepts like capitalism or post modernism. \end_layout \end_inset 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. \end_layout \begin_layout Standard 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. \end_layout \begin_layout Subsection Addendum: Are People Rational in the sense that Economists use the term? \end_layout \begin_layout Standard Since people always make choices, it is pretty find a violation of the assumptio n that 'preferences' are complete. It is possible to 'test' transitivity. This test will be discussed later. However, the implications of rationality are always part of a joint hypothesis. For example, in the standard consumer model, predictions come from the assumption that people are rational \series bold and \series default from the assumption that their preferences don't change when you present them with different budget constraints. The classical theory of demand and markets that is taught in first year economics courses comes more from assumptions that are tacked on in addition to rationality. \end_layout \begin_layout Standard I'll mention a few famous arguments. One example, due to Tversky and Kahneman (1981) refers to something that is now referred to as a 'framing effect'. Subjects are presented with a hypothetical situation in which they need to make a medical decision in response to a new disease. There are 600 people who have been exposed to a new and lethal virus. One of two vaccines can be produced. The first will save 200 of them for sure. The other has a \begin_inset Formula $\frac{1}{3}$ \end_inset chance of saving all 600, but a \begin_inset Formula $\frac{2}{3}$ \end_inset chance of being completely ineffective. Call the first vaccine \begin_inset Formula $a$ \end_inset , and the second vaccine \begin_inset Formula $b$ \end_inset . Some people choose \begin_inset Formula $a$ \end_inset some choose \begin_inset Formula $b$ \end_inset (there is no answer here, it is just a choice). In the second experiment, the same subjects are offered a choice between the following two vaccines. Adopting vaccine \begin_inset Formula $c$ \end_inset will result in 400 people dying for sure. With vaccine \begin_inset Formula $d$ \end_inset there is a \begin_inset Formula $\frac{2}{3}$ \end_inset chance that all 600 will die, but a \begin_inset Formula $\frac{1}{3}$ \end_inset chance that none of them will die. In their experiments, most people who chose vaccine \begin_inset Formula $a$ \end_inset over vaccine \begin_inset Formula $b$ \end_inset proceeded to choose vaccine \begin_inset Formula $d$ \end_inset over vaccine \begin_inset Formula $c$ \end_inset . \end_layout \begin_layout Standard If you think about it for a moment, you should see that in terms of physical outcomes, vaccines \begin_inset Formula $a$ \end_inset and \begin_inset Formula $c$ \end_inset are identical, while vaccines \begin_inset Formula $b$ \end_inset and \begin_inset Formula $d$ \end_inset are identical. In terms of consumer theory, people who were offered the same hypothetical budget made different choices in the two situations. One of the most basic assumptions of consumer theory seems to have been violated. Of course, since no evidence of intransitivity is presented, this isn't a contradiction of rationality. \begin_inset Foot status collapsed \begin_layout Standard It might have occurred to you that asking people what they would do in a hypothetical situation is not likely to elicit much useful information. People are more likely to tell you what they think you want to hear, than what they actually prefer. The experiment was also run with the vaccine story replaced by monetary bets - similar results applied. \end_layout \end_inset \end_layout \begin_layout Standard Another famous example, again due originally to Khaneman and Tversky, has to do with something called an \emph on anchoring \emph default effect. The choice experiment was carried out by Ariely Lowenstein and Prelec (maybe 2003). MBA students were asked whether they were willing to buy consumer items (computer keyboards, wireless mice, wine, chocolates, etc) for a dollar price which was equal to the last two digits of their social security number. This is the same as asking whether you would be willing to pay a price equal to the last two digits of of your student number. If the last two digits were 25, then the computer keyboard was yours for $25 (US of course). Their answer was simply recorded. No transaction actually occurred at this point. Then the actual price they were willing to pay was elicited in a manner that made them report the price truthfully. \begin_inset Foot status collapsed \begin_layout Standard I am not sure exactly what method was used, but one way to do this is as follows: you ask the student for a price between \begin_inset Formula $0$ \end_inset $ and \begin_inset Formula $100\$$ \end_inset , and then draw a price randomly in the same range using a computer. If the price named by the student is higher, then the student can buy the article and pays the price that was chosen by the computer. If the computer's price is higher, then there is no transaction. I leave it to you to see if you can figure out why the student should name a price that is equal to their true willingness to pay. \end_layout \end_inset \end_layout \begin_layout Standard The interesting result - a strong correlation between the last two digits of the students id, and their willingness to pay. In one example, students whose last two digits were below 50 ended up, on average willing to pay 11$ for a bottle of wine, while students whose last two digits were above 50 wanted to pay almost $20 for the same wine. I suppose it is possible that numbers have karma, and people with large digits at the end of their social security numbers end up being richer (and so can afford to pay more for wine). More mundanely, a preference ordering over money and wine seems to depend on things that appear irrelevant. \end_layout \begin_layout Standard I point these things out for two reasons. First, to illustrate that rationality itself is not typically at the root of the problem in these studies, it is much stronger assumptions about the nature of preferences that these studies call into question. We will come upon other examples like this as we go along. \end_layout \begin_layout Standard Second, I want to point out that, though we won't get to much of it in this course, economic theory has presented models to deal with these kinds of behavior, all based on exactly the assumption of complete and transitive preferences that we will use here. Most important, economic theory has recognized that almost all decisions are made with very limited information. The examples above are a bit like that. How to weigh the value of 400 lives in the first example against 600 uncertain lives; what exactly does it mean \begin_inset Formula $\frac{1}{3}$ \end_inset probability. In these kind of situations people make choices even though they don't really understand what the choices involve. We touch on this issue later when we discuss the expected utility theorem. \end_layout \begin_layout Standard One of the most important kinds of problems that economists study are those in which the best action for me depends on what other people do. I don't want to buy a theatre ticket unless my friend wants to go to the theatre with me. The motivations of others are perhaps the most uncertain thing of all. This is the purview of game theory, which we will again be forced to visit briefly when we discuss externalities below. \end_layout \begin_layout Standard It is the methods and art of economic modelling that the material in this course is meant to illustrate. It is exactly these methods that will later make it possible to understand situations that are more complex than those discussed in consumer theory. \end_layout \end_body \end_document