Microeconomic Theory 2 - Lecture - Economics, Lecture notes of Economics

Download Microeconomic Theory 2 - Lecture - Economics and more Lecture notes Economics in PDF only on Docsity! Notes on Microeconomic Theory Nolan H. Miller August 18, 2006 Contents 1 The Economic Approach 1 2 Consumer Theory Basics 5 2.1 Commodities and Budget Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Demand Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Three Restrictions on Consumer Choices . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 A First Analysis of Consumer Choices . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4.1 Comparative Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.5 Requirement 1 Revisited: Walras’ Law . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.5.1 What’s the Funny Equals Sign All About? . . . . . . . . . . . . . . . . . . . . 12 2.5.2 Back to Walras’ Law: Choice Response to a Change in Wealth . . . . . . . . 13 2.5.3 Testable Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.5.4 Summary: How Did We Get Where We Are? . . . . . . . . . . . . . . . . . . 15 2.5.5 Walras’ Law: Choice Response to a Change in Price . . . . . . . . . . . . . . 15 2.5.6 Comparative Statics in Terms of Elasticities . . . . . . . . . . . . . . . . . . . 16 2.5.7 Why Bother? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.5.8 Walras’ Law and Changes in Wealth: Elasticity Form . . . . . . . . . . . . . 18 2.6 Requirement 2 Revisited: Demand is Homogeneous of Degree Zero. . . . . . . . . . . 18 2.6.1 Comparative Statics of Homogeneity of Degree Zero . . . . . . . . . . . . . . 19 2.6.2 A Mathematical Aside ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.7 Requirement 3 Revisited: The Weak Axiom of Revealed Preference . . . . . . . . . . 22 2.7.1 Compensated Changes and the Slutsky Equation . . . . . . . . . . . . . . . . 23 2.7.2 Other Properties of the Substitution Matrix . . . . . . . . . . . . . . . . . . . 27 i Nolan Miller Notes on Microeconomic Theory ver: Aug. 2006 6.2.1 Measuring Risk Aversion: Coefficients of Absolute and Relative Risk Aversion173 6.2.2 Comparing Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 6.2.3 A Note on Comparing Distributions: Stochastic Dominance . . . . . . . . . . 176 6.3 Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 6.3.1 Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 6.3.2 Investing in a Risky Asset: The Portfolio Problem . . . . . . . . . . . . . . . 180 6.4 Ex Ante vs. Ex Post Risk Management . . . . . . . . . . . . . . . . . . . . . . . . . 182 7 Competitive Markets and Partial Equilibrium Analysis 185 7.1 Competitive Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 7.1.1 Allocations and Pareto Optimality . . . . . . . . . . . . . . . . . . . . . . . . 186 7.1.2 Competitive Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 7.2 Partial Equilibrium Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 7.2.1 Set-Up of the Quasilinear Model . . . . . . . . . . . . . . . . . . . . . . . . . 191 7.2.2 Analysis of the Quasilinear Model . . . . . . . . . . . . . . . . . . . . . . . . 192 7.2.3 A Bit on Social Cost and Benefit . . . . . . . . . . . . . . . . . . . . . . . . . 195 7.2.4 Comparative Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 7.3 The Fundamental Welfare Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 7.3.1 Welfare Analysis and Partial Equilibrium . . . . . . . . . . . . . . . . . . . . 201 7.4 Entry and Long-Run Competitive Equilibrium . . . . . . . . . . . . . . . . . . . . . 205 7.4.1 Long-Run Competitive Equilibrium . . . . . . . . . . . . . . . . . . . . . . . 206 7.4.2 Final Comments on Partial Equilibrium . . . . . . . . . . . . . . . . . . . . . 209 8 Externalities and Public Goods 211 8.1 What is an Externality? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 8.2 Bilateral Externalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 8.2.1 Traditional Solutions to the Externality Problem . . . . . . . . . . . . . . . . 217 8.2.2 Bargaining and Enforceable Property Rights: Coase’s Theorem . . . . . . . . 219 8.2.3 Externalities and Missing Markets . . . . . . . . . . . . . . . . . . . . . . . . 221 8.3 Public Goods and Pure Public Goods . . . . . . . . . . . . . . . . . . . . . . . . . . 222 8.3.1 Pure Public Goods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 8.3.2 Remedies for the Free-Rider Problem . . . . . . . . . . . . . . . . . . . . . . . 227 8.3.3 Club Goods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 iv Nolan Miller Notes on Microeconomic Theory ver: Aug. 2006 8.4 Common-Pool Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 9 Monopoly 233 9.1 Simple Monopoly Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 9.2 Non-Simple Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 9.2.1 Non-Linear Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 9.2.2 Two-Part Tariffs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 9.3 Price Discrimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 9.3.1 First-Degree Price Discrimination . . . . . . . . . . . . . . . . . . . . . . . . . 241 9.3.2 Second-Degree Price Discrimination . . . . . . . . . . . . . . . . . . . . . . . 243 9.3.3 Third-Degree Price Discrimination . . . . . . . . . . . . . . . . . . . . . . . . 247 9.4 Natural Monopoly and Ramsey Pricing . . . . . . . . . . . . . . . . . . . . . . . . . 249 9.4.1 Regulation and Incentives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 9.5 Further Topics in Monopoly Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 9.5.1 Multi-Product Monopoly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 9.5.2 Intertemporal Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 9.5.3 Durable Goods Monopoly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 v Nolan Miller Notes on Microeconomic Theory ver: Aug. 2006 These notes are intended for use in courses in microeconomic theory taught at Harvard Univer- sity. Consequently, much of the structure is inherited from the required text for the course, which is currently Mas-Colell, Whinston, and Green’s Microeconomic Theory (referred to as MWG in these notes). They also draw on material contained in Silberberg’s The Structure of Economics, as well as additional sources. They are not intended to stand alone or in any way replace the texts. In the early drafts of this document, there will undoubtedly be mistakes. I welcome comments from students regarding typographical errors, just-plain errors, or other comments on how these notes can be made more helpful. I thank Chris Avery, Lori Snyder, and Ben Sommers for helping clarify these notes and finding many errors. vi Nolan Miller Notes on Microeconomic Theory: Chapter 1 ver: Aug. 2006 preferences stay stable, whether or not I actually had chocolate chip cookies last week. Hence if we define the notion of preferences over a rich enough set of allocations, we can usually find preferences that are stable. The second problem with the four-step marginalist approach outlined above is more trouble- some: Based on these four steps, you really can’t say anything about what is going to happen in the world. Merely knowing that I optimize with respect to stable preferences over the groceries I buy, and that any changes in what I buy are due to changes in the constraints I face does not tell me anything about what I will buy, what I should buy, or whether what I buy is consistent with this type of behavior. The solution to this problem is to impose structure on my preferences. For example, two common assumptions are to assume that I prefer more of an item to less3 (monotonicity) and that I spend my entire grocery budget in the store (Walras’ Law). Another common assumption is that only real opportunities matter. If I were to double all of the prices in the store and my grocery budget, this would not affect what I can buy, so it shouldn’t affect what I do buy. Once I have added this structure to my preferences, I am able to start to make predictions about how I will behave in response to changes in the environment. For example, if my grocery budget were to increase, I would buy more of at least one item (since I spend all of my money and there is always some good that I would like to add to my grocery cart).4 This is known as a testable implication of the theory. It is an implication because if the theory is true, I should react to an increase in my budget by buying more of some good. It is testable because it is based on things which are, at least in principle, observable. For example, if you knew that I had walked into the grocery store with more money than last week and that the prices of the items in the store had not changed, and yet I left the store with less of every item than I did last week, something must be wrong with the theory. The final step in economic analysis is to evaluate the tests of the theories, and, if necessary, change them. We assume that people follow steps 1 - 4 above, and we impose restrictions that we believe are reasonable on their preferences. Based on this, we derive (usually using math) predic- tions about how they should behave and formulate testable hypotheses (or refutable propositions) about how they should behave if the theory is true. Then we observe what they really do. If 3Or, we could make the weaker assumption that no matter what I have in my cart already, there is something in the store that I would like to add to my cart if I could. 4The process of deriving what happens to people’s choices (the stuff in the cart) in response to changes in things they do not choose (the money available to spend in the store) is known as comparative statics. 3 Nolan Miller Notes on Microeconomic Theory: Chapter 1 ver: Aug. 2006 their behavior accords with our predictions, we rejoice because the real world has supported (but not proven!) our theory. If their behavior does not accord with our predictions, we go back to the drawing board. Why didn’t their behavior accord with our predictions? Was it because their preferences weren’t like we though they were? Was it because they weren’t optimizing? Was it because there was an additional constraint that we didn’t understand? Was it because we did not account for a change in the environment that had an important effect on people’s behavior? Thus economics can be summarized as follows: It is the social science that attempts to account for human behavior as arising from consistent (often maximizing — more on that later) behavior subject to one or more constraints. Changes in behavior are attributed to changes in the con- straints, and the test of these theories is to compare the changes in behavior predicted by the theory with the changes that actually occur. 4 Chapter 2 Consumer Theory Basics Recall that the goal of economic theory is to account for behavior based on the assumption that actors have stable preferences, attempt to do as well as possible given those preferences and the constraints placed on their resources, and that changes in behavior are due to changes in these constraints. In this section, we use this approach to develop a theory of consumer behavior based on the simplest assumptions possible. Along the way, we develop the tool of comparative statics analysis, which attempts to characterize how economic agents (i.e. consumers, firms, governments, etc.) react to changes in the constraints they face. 2.1 Commodities and Budget Sets To begin, we need a description of the goods and services that a consumer may consume. We call any such good or service a commodity. We number the commodities in the world 1 through L (assuming there is a finite number of them). We will refer to a “generic” commodity as l (that is, l can stand for any of the L commodities) and denote the quantity of good l by xl. A commodity bundle (i.e. a description of the quantity of each commodity) in this economy is therefore a vector x = (x1, x2, ..., xL). Thus if the consumer is given bundle x = (x1, x2, ..., xL), she is given x1 units of good 1, x2 units of good 2, and so on.1 We will refer to the set of all possible allocations as the commodity space, and it will contain all possible combinations of the L possible commodities.2 Notice that the commodity space includes some bundles that don’t really make sense, at least 1For simplicity of terminology - but not because consumers are more or less likely to be female than male - we will call our consumer “she,” rather than “he/she.” 2That is, the commodity space is the L-dimensional real space RL. 5 Nolan Miller Notes on Microeconomic Theory: Chapter 2 ver: Aug. 2006 Exercise 1 Here is a task to show that you understand budget sets: Show that the effect on a budget set of doubling p1 and p2 is the same as the effect of cutting w in half. This is an illustration of the key economic concept that only relative prices matter to a consumer, which we will see over and over again.6 Now that we have defined the set of consumption bundles that the consumer can afford, the next step is to try to figure out which point the consumer will choose from the budget set. In order to determine which point from the budget set the consumer will choose, we need to know something about the consumer’s preferences over the commodities. For example, if x1 is onions and x2 is chocolate, the consumer may prefer points with relatively high values of x2 and low values of x1 (unless, of course, p2 is very large relative to p1). If we knew exactly the trade-offs that the consumer is willing to make between the commodities, their prices, and the consumer’s income, we would be able to say exactly which consumption bundle the consumer prefers. However, at this point we do not want to put this much structure on preferences. 2.2 Demand Functions Now we need to develop a notation for the consumption bundle that a consumer chooses from a particular budget set. Let p = (p1, ..., pL) be the vector of prices of the L commodities. We will assume that all prices are non-negative. When prices are p and wealth is w, the set of bundles that the consumer can afford is given by the Walrasian budget set Bp,w. Assume that for any price vector and wealth (p,w) there is a single bundle in the budget set that the consumer chooses. Let xi (p,w) denote the quantity of commodity i that the consumer chooses at these prices and wealth. Let x (p,w) = (x1 (p,w) , ..., xL (p,w)) ∈ Bp,w denote the bundle (vector of commodities) that the consumer chooses when prices are p and income is w. That is, it gives the optimal consumption bundle as a function of the price vector and wealth. To make things easier, we will assume that xl (p,w) is single-valued (i.e. a function) and differentiable in each of its arguments. Exercise 2 How many arguments does xl (p,w) have? Answer: L+ 1 : L prices and wealth. Functions xl (p,w) represent the consumer’s choice of commodity bundle at a particular price and wealth. Because of this, they are often called choice functions. They are also called demand 6The idea that only relative prices matter goes by the mathematical name “homogeneity of degree zero”, but we’ll return to that later. 8 Nolan Miller Notes on Microeconomic Theory: Chapter 2 ver: Aug. 2006 functions, although sometimes that name is reserved for choice functions that are derived from the utility-maximization framework we’ll look at later. Generally, I use the terms interchangeably, except when I want to emphasize that we are not talking about utility maximization, in which case I’ll use the term “choice function.” At this point, we should introduce an important distinction, the distinction between endoge- nous and exogenous variables. An endogenous variable in an economic problem is a variable that takes its value as a result of the behavior of one of the economic agents within the model. So, the consumption bundle the consumer chooses x (p,w) is endogenous. An exogenous variable takes its value from outside the model. Exogenous variables determine the constraints on the consumer’s behavior. Thus in the consumer’s problem, the exogenous variables are prices and wealth. The consumer cannot choose prices or wealth. But, prices and wealth determine the budget set, and from the budget set the consumer chooses a consumption bundle. Hence the consumption bundle is endogenous, and prices and wealth are exogenous. The consumer’s demand function x (p,w) therefore gives the consumer’s choice as a function of the exogenous variables. One of the main activities that economists do is try to figure out how endogenous variables depend on exogenous variables, i.e., how consumers’ behavior depends on the constraints placed on them (see principles 1-4 above). 2.3 Three Restrictions on Consumer Choices So, let’s begin with the following question: What are the bare minimum requirements we can put on behavior in order for them to be considered “reasonable,” and what can we say about consumers’ choices based on this? It turns out that relatively weak assumptions about consumer behavior can generate strong requirements for how consumers should behave.7 We will start by enumerating three requirements. • Requirement 1: The consumer always spends her entire budget (Walras’ Law). Requirement 1 is reasonable only if we are willing to make the assumption that “more is better.” That is, for any commodity bundle x, the consumer would rather have a bundle with at least as much of all commodities and strictly more of at least one commodity. Actually, we can get away 7A "weak assumption" imposes less restriction on the behavior of an economic agent than a "strong assumption" does, so when designing a model, we prefer to use weaker assumptions if possible. 9 Nolan Miller Notes on Microeconomic Theory: Chapter 2 ver: Aug. 2006 with a weaker assumption: Given any bundle x, there is always a bundle that has more of at least one commodity that the consumer strictly prefers to x. We’ll return to this later. For now, just remember that the consumer spends all of her budget. • Requirement 2: Only real opportunities matter (demand is homogeneous of degree zero). The essence of requirement 2 is that consumers care about wealth and prices only inasmuch as they affect the set of allocations in the budget set. Or, to put it another way, changes in the environment that do not affect the budget set should not affect the consumer’s choices. So, for example, if you double each price and wealth, the budget set is unchanged. Hence the consumer can afford the same commodity bundles as before and should choose the same bundle as before. • Requirement 3: Choices reveal information about (stable) preferences. So, suppose I offer you a choice between an apple and a banana, and you choose an apple. Then if tomorrow I see you eating a banana, I can infer that you weren’t offered an apple (remember we assume that your preferences stay constant). Requirement 3 is known as the Weak Axiom of Revealed Preference (WARP). The essence is this. Suppose that on occasion 1 you chose bundle x when you could have chosen y. If I observe that on occasion 2 you choose bundle y, it must be because bundle x was not available. Put slightly more mathematically, suppose two bundles x and y are in the budget set Bp,w and the consumer chooses bundle x. Then if at some other prices and wealth (p0, w0) the consumer chooses y, it must be that x was not in the budget set Bp0,w0 . We’ll return to WARP later, but you can think of it in this way. If the consumer’s preferences remain constant over time, then if x is preferred to y once, it should always be preferred to y. Thus if you observe the consumer choose y, you can infer from this choice that x must not have been available. Or, to put it another way, if you observe the consumer choosing x when x and y were available on one day and y when x and y were available on the next day, then your model had better have something in it to account for why this is so (i.e., a reason why the two days were different). 2.4 A First Analysis of Consumer Choices In the rest of this chapter, we’ll develop formal notation for talking about consumer choices, show how the three requirements on consumer behavior can be represented using this notation, and determine what imposing these restrictions on consumer choices implies about the way consumers 10 Nolan Miller Notes on Microeconomic Theory: Chapter 2 ver: Aug. 2006 you do this, you get 2 dz (a) da = 1 dz da = 1 2 That is, if you increase a by 1, z increases by 12 . (If you don’t believe me, plug in some numbers and confirm.) It may seem to you like I’m making a big deal out of nothing, but this is really a critical point. We are interested in determining how endogenous variables change in response to changes in exogenous variables. In this case, z is our endogenous variable and a is our exogenous variable. Thus, we are interested in things like dz(a)da . The only way we can determine these things is to get identities that depend only on the exogenous variables and then differentiate them. Even if you don’t quite believe me, you should keep this in mind. Eventually, it will become clear. 2.5.2 Back to Walras’ Law: Choice Response to a Change in Wealth As we said, Walras’ Law is defined by the identity: p · x (p,w) ≡ w or LX l=1 plxl (p,w) ≡ w. where the vector x(p,w) describes the bundle chosen: x (p,w) = (x1 (p,w) , ..., xL (p,w)) Suppose we are interested in what happens to the bundle chosen if w increases a little bit. In other words, how does the bundle the consumer chooses change if the consumer’s income increases by a small amount? Since we have an identity defined in terms of the exogenous variables p and w, we can differentiate both sides with respect to w: d dw à LX l=1 plxl (p,w) ! ≡ d dw w X l pl ∂xl (p,w) ∂w ≡ 1. (2.1) 13 Nolan Miller Notes on Microeconomic Theory: Chapter 2 ver: Aug. 2006 So, now we have an expression relating the changes in the amount of commodities demanded in response to a change in wealth. What does it say? The left hand side is the change in expenditure due to the increase in wealth, and the right-hand side is the increase in wealth. Thus this expression says that if wealth increases by 1 unit, total expenditure on commodities increases by 1 unit as well. Thus the latter expression just restates Walras’ Law in terms of responses to changes in wealth. Any change in wealth is accompanied by an equal change in expenditure. If you think about it, this is really the only way that the consumer could satisfy Walras’ Law (i.e. spend all of her money) both before and after the increase in wealth. Based only on this expression, P i pi ∂xi(p,w) ∂w ≡ 1, what else can we say about the behavior of the consumer’s choices in response to income changes? Well, first, think about ∂xi(p,w)∂w . Is this expression going to be positive or negative? The answer depends on what kind of commodity this is. Ordinarily, we think that if your wealth increases you will want to consume more of a good. This is certainly true of goods like trips to the movies, meals at fancy restaurants, and other “normal goods.” In fact, this is so much the normal case that we just go ahead and call such goods - which have ∂xi(p,w)∂w > 0 - “normal goods.” But, you can also think about goods you want to consume less of as your wealth goes up - cheap cuts of meat, cross-country bus trips, nights in cheap motels, etc. All of these are things that, the richer you get, the less you want to consume them. We call goods for which ∂xi(p,w)∂w < 0 “inferior goods.” Since x (p,w) depends on w, ∂xl(p,w) ∂w depends on w as well, which means that a good may be inferior at some levels of wealth but normal at others. So, what can we say based on P i pi ∂xi(p,w) ∂w ≡ 1? Well, this identity tells us that there is always at least one normal good. Why? If all goods are inferior, then the terms on the left hand side are all negative, and no matter how many negative terms you add together, they’ll never sum to 1. 2.5.3 Testable Implications We can use this observation about normal goods to derive a testable implication of our theory. Put simply, we have assumed that consumers spend all of the money they have on commodities. Based on this, we conclude that following any change in wealth, total expenditure on goods should increase by the same amount as wealth. If we knew prices and how much of the commodities the consumer buys before and after the wealth change, we could directly test this. But, suppose that we don’t observe prices. However, we believe that prices do not change when wealth changes. What should we conclude if we observe that consumption of all commodities decreases following 14 Nolan Miller Notes on Microeconomic Theory: Chapter 2 ver: Aug. 2006 an increase in wealth? Unfortunately, the only thing we can conclude is that our theory is wrong. People aren’t spending all of their wealth on commodities 1 through L. Based on this observation, there are a number of possible directions to go. One possible explanation is that there is another commodity, L+1, that we left out of our model, and if we had accounted for that then we would see that consumption increased in response to the wealth increase and everything would be right in the world. Another possible explanation is that in the world we are considering, it is not the case that there is always something that the consumer would like more of (which, you’ll recall, is the implicit assumption behind Walras’ Law). This would be the case, for example, if the consumer could become satiated with the commodities, meaning that there is a level of consumption beyond which you wouldn’t want to consume more even if you could. A final possibility is that there is something wrong with the data and that if consumption had been properly measured we would see that consumption of one of the commodities did, in fact, increase. In any case, the next task of the intrepid economist is to determine which possible explanation caused the failure of the theory and, if possible, develop a theory that agrees with the data. 2.5.4 Summary: How Did We Get Where We Are? Let’s review the comparative statics methodology. First, we develop an identity that expresses a relationship between the endogenous variables (consumption bundle) and the exogenous variable of interest (wealth). The identity is true for all values of the exogenous variables, so we can differentiate both sides with respect to the exogenous variables. Next, we totally differentiate the identity with respect to a particular exogenous variable of interest (wealth). By rearranging, we derive the effect of a change in wealth on the consumption bundle, and we try to say what we can about it. In the previous example, we were able to make inferences about the sign of this relationship. This is all there is to comparative statics. 2.5.5 Walras’ Law: Choice Response to a Change in Price What are other examples of comparative statics analysis? Well, in the consumer model, the endoge- nous variables are the amounts of the various commodities that the consumer chooses, xi(p,w). We want to know how these things change as the restrictions placed on the consumer’s choices change. The restriction put on the consumer’s choice by Walras’ Law takes the form of the budget constraint, and the budget constraint is in turn defined by the exogenous variables — the prices of the various commodities and wealth. We already looked at the comparative statics of wealth 15 Nolan Miller Notes on Microeconomic Theory: Chapter 2 ver: Aug. 2006 quantities and prices. In particular, budget shares and elasticities do not depend on price levels, but only on relative prices. Consequently it can be much easier to apply Walras’ Law when it is written as (2.3) than when it is written as (2.2). 2.5.8 Walras’ Law and Changes in Wealth: Elasticity Form Not to belabor the point, but we can also write (2.1) in terms of elasticities, this time using the wealth elasticity, εiw = ∂xi∂w · w xi . Multiplying (2.1) by xiwxiw yields:X i pixi w w xi ∂xi (p,w) ∂w ≡ 1 (2.4)X i bi (p,w) εiw = 1. The wealth elasticity εiw gives the percentage change in consumption of good i induced by a 1% increase in wealth. Thus, in response to an increase in wealth, total spending changes by εiw weighted by the budget share bi (p,w) and summed over all goods. In other words, if wealth increases by 1, total expenditure must also increase by 1. Thus, equation (2.4) is yet another statement of the fact that the consumer always spends all of her money. 2.6 Requirement 2 Revisited: Demand is Homogeneous of Degree Zero. The second requirement for consumer choices is that “only real opportunities matter.” In mathe- matical terms this means that “demand is homogeneous of degree zero,” or: x (αp, αw) ≡ x (p,w) Note that this is an identity. Thus it holds for any values of p and w. In words what it says is that if the consumer chooses bundle x (p,w) when prices are p and income is w, and you multiply all prices and income by a factor, α > 0, the consumer will choose the same bundle after the multiplication as before, x (αp,αw) = x (p,w). The reason for this is straightforward. If you multiply all prices and income by the same factor, the budget set is unchanged. Bp,w = {x : p · x ≤ w} = {x : αp · x ≤ αw} = Bαp,αw. And, since the set of bundles that the consumer could choose is not changed, the consumer should choose the same bundle. There are two important points that come out of this: 18 Nolan Miller Notes on Microeconomic Theory: Chapter 2 ver: Aug. 2006 1. This is an expression of the belief that changes in behavior should come from changes in the set of available alternatives. Since the rescaling of prices and income do not affect the budget set, they should not affect the consumer’s choice. 2. The second point is that nominal prices are meaningless in consumer theory. If you tell me that a loaf of bread costs $10, I need to know what other goods cost before I can interpret the first statement. And, in terms of analysis, this means that we can always “normalize” prices by arbitrarily setting one of them to whatever we like (often it is easiest to set it equal to 1), since only the real prices matter and fixing one commodity’s nominal price will not affect the relative values of the other prices. Exercise 3 If you don’t believe me that this change doesn’t affect the budget set, you should go back to the two-commodity example, plug in the numbers and check it for yourself. If you can’t do it with the general scaling factor α, you should let α = 2 and try it for that. Most of the time, things that are hard to understand with general parameter values like α, p,w are simple once you plug in actual numbers for them and churn through the algebra. 2.6.1 Comparative Statics of Homogeneity of Degree Zero We can also perform a comparative statics analysis of the requirement that demand be homogeneous of degree zero, i.e. only real opportunities matter. What does this imply for choice behavior? The homogeneity assumption applies to proportional changes in all prices and wealth: xi (αp, αw) ≡ xi (p,w) for all i, α > 0. To make things clear, let the initial price vector be denoted p0 = ¡ p01, ..., p 0 L ¢ and let w0 original wealth, and (for the time being) assume that L = 2. For example, ¡ p0, w0 ¢ could be p0 = (3, 2) and w0 = 7. Before we differentiate, I want to make sure that we’re clear on what is going on. So, rewrite the above expression as: xi ¡ αp01, αp 0 2, αw 0 ¢ ≡ xi ¡ p01, p 0 2, w 0 ¢ . (2.5) Now, notice that on the left-hand side for any α > 0 the price of good 1 is p1 = αp01, the price of good 2 is p2 = αp02, and wealth is w = αw 0. That is, given α. We are interested in what happens to demand as α changes, so it is important to recognize that the prices and wealth are functions of α. 19 Nolan Miller Notes on Microeconomic Theory: Chapter 2 ver: Aug. 2006 We are interested in what happens to demand when, beginning at original prices p0 and wealth w0, we scale up all prices and wealth proportionately. To do this, we want to see what happens when we increase α, starting at α = 1. Because the prices and wealth are functions of α, we have to use the Chain Rule in evaluating the derivative of (2.5) with respect to α. Differentiating (2.5) with respect to α yields: ∂xi ¡ αp01, αp 0 2, αw 0 ¢ ∂p1 ∂p1 ∂α + ∂xi ¡ αp01, αp 0 2, αw 0 ¢ ∂p2 ∂p2 ∂α + ∂xi ¡ αp01, αp 0 2, αw 0 ¢ ∂w ∂w ∂α ≡ 0. Since p1 = αp01, ∂p1 ∂α = p 0 1, and similarly ∂p2 ∂α = p 0 2, and ∂w ∂a = w 0, so this expression becomes: ∂xi ¡ αp01, αp 0 2, αw 0 ¢ ∂p1 p01 + ∂xi ¡ αp01, αp 0 2, αw 0 ¢ ∂p2 p02 + ∂xi ¡ αp01, αp 0 2, αw 0 ¢ ∂w w0 ≡ 0. (2.6) Notice that the first line takes the standard Chain Rule form: for each argument (p1, p2, and w), take the partial derivative of the function with respect to that argument and multiply it by the derivative with respect to α of “what’s inside” the argument.12 Finally, notice that (2.6) has prices and wealth ¡ αp01, αp 0 2, αw 0 ¢ . We are asking the question “what happens to xi when prices and wealth begin at ¡ p01, p 0 2, w 0 ¢ and are all increased slightly by the same proportion?” In order to make sure we are answering this question, we need to set α = 1, so that the partial derivatives are evaluated at the original prices and wealth. Evaluating the last expression at α = 1 yields the following expression in terms of the original price-wealth vector (s1, s2, v) : ∂xi ¡ p01, p 0 2, w 0 ¢ ∂p1 p01 + ∂xi ¡ p01, p 0 2, w 0 ¢ ∂p2 p02 + ∂xi ¡ p01, p 0 2, w 0 ¢ ∂w w0 ≡ 0. (2.7) Generalizing the previous argument to the case where L is any positive number, expression (2.7) becomes: ∂xi ¡ p0, w0 ¢ ∂w w0 + LX j=1 ∂xi ¡ p0, w0 ¢ ∂pj p0j = 0 for all i. (2.8) This is where we need to face an ugly fact. Economists are terrible about notation, which makes this stuff harder to learn than it needs to be. When you see (2.8) written in a textbook, it will look like this: ∂xi (p,w) ∂w w + LX j=1 ∂xi (p,w) ∂pj pj = 0 for all i. 12 If you are confused, see the next subsection for further explanation. 20 Nolan Miller Notes on Microeconomic Theory: Chapter 2 ver: Aug. 2006 if you observe her choose y, it must be that z was not available. I apologize for repeating the same definition over and over, but a) it helps to attach words to the math, and b) if you wanted math without explanation you could read a textbook. In its basic form, WARP does not generate any predictions that can immediately be taken to the data and tested. But, if we rearrange the statement a little bit, we can get an easily testable prediction. So, let me ask the WARP question a different way. Suppose the consumer chooses z when prices and wealth are (p,w) , and z is affordable when prices and wealth are (p0, w0). What does WARP tell us about which bundles the consumer could choose when prices are (p0, w0)? There are two choices to consider: either x (p0, w0) = z. This is perfectly admissible under WARP. The other choice is that x (p0, w0) = y 6= z. In this case, WARP will place restrictions on which bundles y can be chosen. What are they? By virtue of the fact that z was chosen when prices and wealth were (p,w), we know that y /∈ Bp,w, since if it were there would be a violation of WARP. Thus it must be that if the consumer chooses a bundle y different than x at (p0, w0), y must not have been affordable when prices and wealth were (p,w). This is illustrated graphically in figure 2.F.1 in MWG (p. 30). In panel a, since x (p0, w0) is chosen at (p0, w0), when prices are (p00, w00) the consumer must either choose x (p0, w0) again or a bundle x (p00, w00) that is not in Bp0,w0 . If we assume that demand satisfies Walras’ Law as well, x (p00, w00) must lay on the frontier. Thus if x (p0, w0) is as drawn, it cannot be chosen at prices (p00, w00). The chosen bundle must lay on the segment of Bp00,w00 below and to the right of the intersection of the two budget lines, as does x (p00, w00). Similar reasoning holds in panel b. The chosen bundle cannot lay within Bp0,w0 if WARP holds. Panel c depicts the case where x (p0, w0) is affordable both before and after the change in prices and wealth. In this case, x (p0, w0) could have been chosen after the price change. But, if it is not chosen at (p00, w00), then the chosen bundle must lay outside of Bp00,w00 , as does x (p00, w00) . In panels d and e, x (p00, w00) ∈ Bp0,w0 , and thus this behavior does not satisfy WARP. 2.7.1 Compensated Changes and the Slutsky Equation Panel c in MWG Figure 2.F.1 suggests a way in which WARP can be used to generate predictions about behavior. Imagine two different price-wealth vectors, (p,w) and (p0, w0), such that bundle z = x (p,w) lies on the frontier of both Bp,w and Bp0,w0 . This corresponds to the following hypothetical situation. Suppose that originally prices are (p,w) and you choose bundle z = x (p,w). I tell you that I am going to change the price vector to p0. But, I am fair, and so I tell you that in 23 Nolan Miller Notes on Microeconomic Theory: Chapter 2 ver: Aug. 2006 order to make sure that you are not made worse off by the price change, I am also going to change your wealth to w0, where w0 is chosen so that you can still just afford bundle z at the new prices and wealth (p0, w0). Thus w0 = p0 · z. We call this a compensated change in price, since I change your wealth to compensate you for the effects of the price change. Since you can afford z before and after the price change, we know that: p · z = w and p0 · z = w0. Let y = x (p0, w0) 6= z be the bundle chosen at (p0, w0). Since you actually choose y at price-wealth (p0, w0), assuming your demand satisfies Walras Law we know that p0 · y = w0 as well. Thus 0 = w0 − w0 = p0 · y − p0 · z so, p0 · (y − z) = 0. Further, since z is affordable at (p0, w0), by WARP it must be that y was not affordable at (p,w) : p · y > w p · y − p · z > 0 p · (y − z) > 0. Finally, subtracting p · (y − z) > 0 from p0 · (y − z) = 0 yields: ¡ p0 − p ¢ · (y − z) < 0 (2.11) Equation (2.11) captures the idea that, following a compensated price change, prices and de- mand move in opposite directions. Although this takes a little latitude since prices and bundles are vectors, you can interpret (2.11) as saying that if prices increase, demand decreases.14 To put it another way, let ∆p = p0 − p denote the vector of price changes and ∆x = x (p0, w0) − x (p,w) denote the vector of quantity changes. (2.11) can be rewritten as ∆p ·∆xc ≤ 0 where we have replaced the strict inequality with a weak inequality in recognition that it may be the case that y = z. Note that the superscript c on ∆xc is to remind us that this is the compensated change in x. This is a statement of the Compensated Law of Demand (CLD): If the price of 14This is especially true in the case where p and p0 differ only in the price of good j, which changes by an amount dpj . In this case, p0 − p = (0, 0, ..., dpj , 0, ..., 0) , and (p0 − p) • (y − z0) = dpjdxj . 24 Nolan Miller Notes on Microeconomic Theory: Chapter 2 ver: Aug. 2006 a commodity goes up, you demand less of it. If we take a calculus view of things, we can rewrite this in terms of differentials: dp · dxc ≤ 0. We’re almost there. Now, what does it mean to give the consumer a compensated price change? Let x̂ be the initial consumption bundle, i.e., x̂ = x (p,w), where p and w are the original prices and wealth. A compensated price change means that at any price, p, bundle x̂ is still affordable. Hence, after the price change, wealth is changed to ŵ = p · x̂. Note that the x̂ in this expression is the original consumption bundle, not the choice function x (p,w). Consider the consumer’s demand for good i xci = xi (p, p · x̂) following a compensated change in the price of good j: d dpj (xi (p, p · x̂)) = ∂xi ∂pj + ∂xi ∂w ∂ (p · x̂) ∂pj dxci dpj = ∂xi ∂pj + ∂xi ∂w x̂j . Since x̂j = x (p,w), we’ll just drop the “hat” from now on. If we write the previous equation as a differential, this is simply: dxci = µ ∂xi ∂pj + ∂xi ∂w xj ¶ dpj = sijdpj where sij = ³ dxi dpj + dxidw xj ´ . If we change more than one pj , the change in xci would simply be the sum of the changes due to the different price changes: dxci = LX j=1 µ ∂xi ∂pj + ∂xi ∂w xj ¶ dpj = si · dp where si = (si1, ..., sij , ..., siL) and dp = (dp1, ..., dpL) is the vector of price changes. Finally, we can arrange the dxci into a vector by stacking these equations vertically. This gives us: dxc = Sdp where S is an L× L matrix with the element in the ith the row and jth column being sij . Now, return to the statement of WARP: dp · dxc ≤ 0 Substituting in dxc = Sdp yields dp · Sdp ≤ 0. (2.12) 25 Nolan Miller Notes on Microeconomic Theory: Chapter 2 ver: Aug. 2006 These can be derived from the comparative statics implications of Walras’ Law and homogeneity of degree zero. Their effect is to impose additional restrictions on the set of admissible demand functions. So, suppose you get some estimates of ∂xi∂pj , p, w, and ∂xi ∂w , which can all be computed from data, and you are concerned with whether you have a good model. One thing you can do is compute S from the data, and check to see if the two equations above hold. If they do, you’re doing okay. If they don’t, this is a sign that your data do not match up with your theory. This could be due to data problems or to theory problems, but in either case it means that you have work to do.16 16The usual statistical procedure in this instance is to impose these conditions as restrictions on the econometric model and then test to see if they are valid. I leave it to people who know more econometrics than I do to explain how. 28 Chapter 3 The Traditional Approach to Consumer Theory In the previous section, we considered consumer behavior from a choice-based point of view. That is, we assumed that consumers made choices about which consumption bundle to choose from a set of feasible alternatives, and, using some rather mild restrictions on choices (homogeneity of degree zero, Walras’ law, and WARP), were able make predictions about consumer behavior. Notice that our predictions were entirely based on consumer behavior. In particular, we never said anything about why consumers behave the way they do. We only hold that the way they behave should be consistent in certain ways. The traditional approach to consumer behavior is to assume that the consumer has well-defined preferences over all of the alternative bundles and that the consumer attempts to select the most preferred bundle from among those bundles that are available. The nice thing about this approach is that it allows us to build into our model of consumer behavior how the consumer feels about trading off one commodity against another. Because of this, we are able to make more precise predictions about behavior. However, at some point people started to wonder whether the predictions derived from the preference-based model were in keeping with the idea that consumers make consistent choices, or whether there could be consistent choice-based behavior that was not derived from the maximization of well-defined preferences. It turns out that if we define consistent choice making as homogeneity of degree zero, Walras’ law, and WARP, then there are consistent choices that cannot be derived from the preference-based model. But, if we replace WARP with a slightly stronger but still reasonable condition, called the Strong Axiom of Revealed Preference (SARP), 29 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 then any behavior consistent with these principles can be derived from the maximization of rational preferences. Next, we take up the traditional approach to consumer theory, often called “neoclassical” con- sumer theory. 3.1 Basics of Preference Relations We’ll continue to assume that the consumer chooses from among L commodities and that the commodity space is given by X ⊂ RL+. The basic idea of the preference approach is that given any two bundles, we can say whether the first is “at least as good as” the second. The “at-least-as- good-as” relation is denoted by the curvy greater-than-or-equal-to sign: º. So, if we write x º y, that means that “x is at least as good as y.” Using º, we can also derive some other preference relations. For example, if x º y, we could also write y ¹ x, where ¹ is the “no better than” relation. If x º y and y º x, we say that a consumer is “indifferent between x and y,” or symbolically, that x ∼ y. The indifference relation is important in economics, since frequently we will be concerned with indifference sets. The indifference curve Iy is defined as the set of all bundles that are indifferent to y. That is, Iy = {x ∈ X|y ∼ x}. Indifference sets will be very important as we move forward, and we will spend a great deal of time and effort trying to figure out what they look like, since the indifference sets capture the trade-offs the consumer is willing to make among the various commodities. The final preference relation we will use is the “strictly better than” relation. If x is at least as good as y and y is not at least as good as x, i.e., x º y and not y º x (which we could write y ² x), we say that x  y, or x is strictly better than (or strictly preferred to) y. Our preference relations are all examples of mathematical objects called binary relations. A binary relation compares two objects, in this case, two bundles. For instance, another binary relation is “less-than-or-equal-to,” ≤. There are all sorts of properties that binary relations can have. The first two we will be interested in are called completeness and transitivity. A binary relation is complete if, for any two elements x and y in X, either x º y or y º x. That is, any two elements can be compared. A binary relation is transitive if x º y and y º z imply x º z. That is, if x is at least as good as y, and y is at least as good as z, then x must be at least as good as z. The requirements of completeness and transitivity seem like basic properties that we would like any person’s preferences to obey. This is true. In fact, they are so basic that they form economists’ 30 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 x2 x1 Ix Figure 3.1: Thin Indifferent Sets x2 x1 Ix x Figure 3.2: Thick Indifference Sets 33 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 better than x, Lx = {y ∈ X|x º y}. Just as monotonicity told us something about the shape of indifference sets, we can also make assumptions that tell us about the shape of upper and lower level sets. Recall that a set of points, X, is convex if for any two points in the set the (straight) line segment between them is also in the set.1 Formally, a set X is convex if for any points x and x0 in X, every point z on the line joining them, z = tx + (1− t)x0 for some t ∈ [0, 1], is also in X. Basically, a convex set is a set of points with no holes in it and with no “notches” in the boundary. You should draw some pictures to figure out what I mean by no holes and no notches in the set. Before we move on, let’s do a thought experiment. Consider two possible commodity bundles, x and x0. Relative to the extreme bundles x and x0, how do you think a typical consumer feels about an average bundle, z = tx + (1− t)x0, t ∈ (0, 1)? Although not always true, in general, people tend to prefer bundles with medium amounts of many goods to bundles with a lot of some things and very little of others. Since real people tend to behave this way, and we are interested in modeling how real people behave, we often want to impose this idea on our model of preferences.2 Exercise 7 Confirm the following two statements: 1) If º is convex, then if y º x and z º x, ty + (1− t) z º x as well. (2) Suppose x ∼ y. If º is convex, then for any z = ty + (1− t)x, z º x. Another way to interpret convexity of preferences is in terms of a diminishing marginal rate of substitution (MRS), which is simply the slope of the indifference curve. The idea here is that if you are currently consuming a bundle x, and I offer to take some x1 away from you and replace it with some x2, I will have to give you a certain amount of x2 to make you exactly indifferent for the loss of x1. A diminishing MRS means that this amount of x2 I have to give you increases the more x2 that you already are consuming - additional units of x2 aren’t as valuable to you. The upshot of the convexity and local non-satiation assumptions is that indifference sets have to be thin, downward sloping, and be “bowed upward.” There is nothing in the definition of convexity 1This is the definition of a convex set. It should not be confused with a convex function, which is a different thing altogether. In addition, there is such thing as a concave function. But, there is no such thing as a concave set. I sympathize with the fact that this terminology can be confusing. But, that’s just the way it is. My advice is to focus on the meaning of the concepts, i.e., “a set with no notches and no holes.” 2 It is only partly true that when we assume preferences are convex we do so in order to capture real behavior. In addition, the basic mathematical techniques we use to solve our problems often depend on preferences being convex. If they are not (and one can readily think of examples where preferences are not convex), other, more complicated techniques have to be used. 34 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 that prevents flat regions from appearing on indifference curves. However, there are reasons why we want to rule out indifference curves with flat regions. Because of this, we strengthen the convexity assumption with the concept of strict convexity. A preference relation is strictly convex if for any distinct bundles y and z (y 6= z) such that y º x and z º x, ty+(1− t) z  x. Thus imposing strict convexity on preferences strengthens the requirement of convexity (which actually means that averages are at least as good as extremes) to say that averages are strictly better than extremes. 3.2 From Preferences to Utility In the last section, we said a lot about preferences. Unfortunately, all of that stuff is not very useful in analyzing consumer behavior, unless you want to do it one bundle at a time. However, if we could somehow describe preferences using mathematical formulas, we could use math techniques to analyze preferences, and, by extension, consumer behavior. The tool we will use to do this is called a utility function. A utility function is a function U (x) that assigns a number to every consumption bundle x ∈ X. Utility function U () represents preference relation º if for any x and y, U (x) ≥ U (y) if and only if x º y. That is, function U assigns a number to x that is at least as large as the number it assigns to y if and only if x is at least as good as y. The nice thing about utility functions is that if you know the utility function that represents a consumer’s preferences, you can analyze these preferences by deriving properties of the utility function. And, since math is basically designed to derive properties of functions, it can help us say a lot about preferences. Consider a typical indifference curve map, and assume that preferences are rational. We also need to make a technical assumption, that preferences are continuous. For our purposes, it isn’t worth derailing things in order to explain why this is necessary. But, you should look at the example of lexicographic preferences in MWG to see why the assumption is necessary and what can go wrong if it is not satisfied. The line drawn in Figure 3.3 is the line x2 = x1, but any straight line would do as well. Notice that we could identify the indifference curve Ix by the distance along the line x2 = x1 you have to travel before intersecting Ix. Since indifference curves are downward sloping, each Ix will only intersect this line once, so each indifference curve will have a unique number associated with it. Further, since preferences are convex, if x  y, Ix will lay above and to the right of Iy (i.e. inside Iy), and so Ix will have a higher number associated with it than Iy. 35 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 0 2 4 6 8 10 x1 0 2 4 6 8 10 x2 0 1 2 3 ux Figure 3.4: Function u (x) 38 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 0 2 4 6 8 10 0 2 4 6 8 10 Figure 3.5: Level sets of u (x) Notice the curvature of the surface. Now, consider Figure 3.5, which shows the level sets (Ix) for various utility levels. Notice that the indifference curves of this utility function are convex. Now, pick an indifference curve. Points offering more utility are located above and to the right of it. Notice how the contour map corresponds to the 3D utility map. As you move up and to the right, you move “uphill” on the 3D graph. Quasiconcavity is a weaker condition than concavity. Concavity is an assumption about how the numbers assigned to indifference curves change as you move outward from the origin. It says that the increase in utility associated with an increase in the consumption bundle decreases as you move away from the origin. As such, it is a cardinal concept. Quasiconcavity is an ordinal concept. It talks only about the shape of indifference curves, not the numbers assigned to them. It can be shown that concavity implies quasiconcavity but a function can be quasiconcave without being concave (can you draw one in two dimensions). It turns out that for the results on utility 39 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 0 2 4 6 8 10 x1 0 2 4 6 8 10 x2 0 250 500 750 1000 vx Figure 3.6: Function v (x) maximization we will develop later, all we really need is quasiconcavity. Since concavity imposes cardinal restrictions on utility (which is an ordinal concept) and is stronger than we need for our maximization results, we stick with the weaker assumption of quasiconcavity.4 Here’s an example to help illustrate this point. Consider the following function, which is also of the Cobb-Douglas form: v (x) = x 3 2 1 x 3 2 2 . Figure 3.6 shows the 3D graph for this function. Notice that v (x) is “curved upward” instead of downward like u (x). In fact, v (x) is a not a concave function, while u (x) is a concave function.5 But, both are quasiconcave. We already saw that u (x) was quasiconcave by looking at its level 4As in the case of convexity and strict convexity, a strictly quasiconcave function is one whose upper level sets are strictly convex. Thus a function that is quasiconcave but not strictly so can have flat parts on the boundaries of its indifference curves. 5See Simon and Blume or Chiang for good explanations of concavity and convexity in multiple dimensions. 40 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 0 20 40 60 80 100 0 20 40 60 80 100 Figure 3.9: Level Sets of h(x) 43 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 ensure that the problem is “well-behaved,” we will assume that preferences are rational, continuous, convex, and locally nonsatiated. These assumptions imply that the consumer has a continuous utility function u (x), and the consumer’s choices will satisfy Walras’ Law. In order to use calculus techniques, we will assume that u () is differentiable in each of its arguments. Thus, in other words, we assume indifference curves have no “kinks.” The consumer’s problem is to choose the bundle that maximizes utility from among those avail- able. The set of available bundles is given by the Walrasian budget set Bp,w = {x ∈ X|p · x ≤ w}. We will assume that all prices are strictly positive (written p >> 0) and that wealth is strictly positive as well. The consumer’s problem can be written as: max x≥0 u (x) s.t. : p · x ≤ w. The first question we should ask is: Does this problem have a solution? Since u (x) is a continuous function and Bp,w is a closed and bounded (i.e., compact) set, the answer is yes by the Weierstrass theorem - a continuous function on a compact set achieves its maximum. How do we find the solution? Since this is a constrained maximization problem, we can use Lagrangian methods. The Lagrangian can be written as: L = u (x) + λ (w − p · x) Which implies Kuhn-Tucker first-order conditions (FOC’s): ui (x ∗)− λ∗pi ≤ 0 and xi (ui (x∗)− λ∗pi) = 0 for i = (1, ..., L) w − p · x∗ ≥ 0 and λ∗ (w − p · x∗) = 0 Note that the optimal solution is denoted with an asterisk. This is because the first-order conditions don’t hold everywhere, only at the optimum. Also, note that the value of the Lagrange multiplier λ is also derived as part of the solution to this problem. Now, we have a system with L+1 unknowns. So, we need L+1 equations in order to solve for the optimum. Since preferences are locally non-satiated, we know that the consumer will choose a consumption bundle that is on the boundary of the budget set. Thus the constraint must bind. This gives us one equation. The conditions on xi are complicated because we must allow for the possibility that the consumer chooses to consume x∗i = 0 for some i at the optimum. This will happen, for example, if the relative 44 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 price of good i is very high. While this is certainly a possibility, “corner solutions” such as these are not the focus of the course, so we will assume that x∗i > 0 for all i for most of our discussion. But, you should be aware of the fact that corner solutions are possible, and if you come across a corner solution, it may appear to behave strangely. Generally speaking, we will just assume that solutions are interior. That is, that x∗i > 0 for all commodities i. In this case, the optimality condition becomes ui (x ∗ i )− λ∗pi = 0. (3.1) Solving this equation for λ∗ and doing the same for good j yields: − ui (x ∗ i ) uj ³ x∗j ´ = − pi pj for all i, j ∈ {1, ..., L} . This turns out to be an important condition in economics. The condition on the right is the slope of the budget line, projected into the i and j dimensions. For example, if there are two commodities, then the budget line can be written x2 = −p1p2x1 + w p2 . The left side, on the other hand, is the slope of the utility indifference curve (also called an isoquant or isoutility curve). To see why − ui(x ∗ i ) uj(x∗j) is the slope of the isoquant, consider the following identity: u (x1, x2 (x1)) ≡ k, where k is an arbitrary utility level and x2 (x1) is defined as the level of x2 needed to guarantee the consumer utility k when the level of commodity 1 consumed is x1. Differentiate this identity with respect to x1:8 u1 + u2 dx2 dx1 = 0 dx2 dx1 = −u1 u2 So, at any point (x1, x2), −u1(x1,x2)u2(x1,x2) is the slope of the implicitly defined curve x2 (x1) . But, this curve is exactly the set of points that give the consumer utility k, which is just the indifference curve. As mentioned earlier, we call the slope of the indifference curve the marginal rate of substitution (MRS): MRS = −u1u2 . Thus the optimality condition is that at the optimal consumption bundle, the MRS (the rate that the consumer is willing to trade good x2 for good x1, holding utility constant) must equal the ratio of the prices of the two goods. In other words, the slope of the utility isoquant is the same as the slope of the budget line. Combine this with the requirement that the optimal bundle be on 8Here, we adopt the common practice of using subscripts to denote partial derivatives, ∂u(x)∂xi = ui. 45 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 isoquants will have flat parts. If the budget line has the same slope as the flat part, an entire region may be optimal. However, we can say that if preferences are convex, the optimal region will be a convex set. Further, we can add that if preferences are strictly convex, so that u () is strictly quasiconcave, then x (p,w) will be a single point for any (p,w). This is because strict quasiconcavity rules out flat parts on the indifference curve. A Note on Optimization: Necessary Conditions and Sufficient Conditions Notice that we derived the first-order conditions for an optimum above. However, while these conditions are necessary for an optimum, they are not generally sufficient - there may be points that satisfy them that are not maxima. This is a technical problem that we don’t really want to worry about here. To get around it, we will assume that utility is quasiconcave and monotone (and some other technical conditions that I won’t even mention). In this case we know that the first-order conditions are sufficient for a maximum. In most courses in microeconomic theory, you would be very worried about making sure that the point that satisfies the first-order conditions is actually a maximizer. In order to do this you need to check the second order conditions (make sure the function is “curved down”). This is a long and tedious process, and, fortunately, the standard assumptions we will make, strict quasiconcavity and monotonicity, are enough to make sure that any point that satisfies the first-order conditions is a maximizer (at least when the constraint is linear). Still, you should be aware that there is such things as second-order conditions, and that you either need to check to make sure they are satisfied or make assumptions to ensure that they are satisfied. We will do the latter, and leave the former to people who are going to be doing research in microeconomic theory. A Word on Nonconvexities It is worthwhile to spend another moment on what can happen if preferences are not convex, i.e. utility is not quasiconcave. We already mentioned that with nonconvex preferences it becomes necessary to check second-order conditions to determine if a point satisfying the first-order condi- tions is really a maximizer. There can also be other complications. Consider a utility function where the isoquants are not convex, shown in Figure 3.11. When the budget line is given by line 1, the optimal point will be near x. When the budget line is line 2, the optimal points will be either x or y. But, none of the points between x and y on line 48 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 x y 1 2 3 Figure 3.11: Nonconvex Isoquants 2 are as good as x or y (a violation of the idea that averages are better than extremes). Finally, if the budget line is given by line 3, the only optimal point will be near y. Thus the optimal point jumps from x to y without going through any intermediate values. Now, lines 1, 2, and 3 can be generated by a series of compensated decreases in the price of good 1 (plotted on the horizontal axis). And, intuitively, it seems like people’s behavior should change by a small amount if the price changes by a small amount. But, if the indifference curves are non-convex, behavior could change a lot in response to small changes in the exogenous parameters. Since non-convexities result in predictions that do not accord with how we feel consumers actually behave, we choose to model consumers as having convex preferences. In addition, non-convexities add complications to solving and analyzing the consumer’s maximization problem that we are very happy to avoid, so this provides another reason why we assume preferences are convex. Actually, the same sort of problem can arise when preferences are convex but not strictly convex. It could be that behavior changes a lot in response to small changes in prices (although it need not do so). In order to eliminate this possibility and ensure a unique maximizing bundle, we will generally assume that preferences are strictly convex and that utility is strictly quasiconcave (i.e., has strictly convex upper level sets). 3.3.2 The Lagrange Multiplier You may recall that the optimal value of the Lagrange multiplier is the shadow value of the con- straint, meaning that it is the increase in the value of the objective function resulting from a slight relaxation of the constraint. If you don’t remember this, you should reacquaint yourself with the 49 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 point by looking in the math appendix of your favorite micro text or “math for economists” book.9 If you still don’t believe this is true, I present you with the following derivation. In addition to showing this fact about the value of λ∗, it also illustrates a common method of proof in economics. Consider the utility function u (x). If we substitute in the demand functions, we get u (x (p,w)) which is the utility achieved by the consumer when she chooses the best bundle she can at prices p and wealth w. The constraint in the problem is: p · x ≤ w. So, relaxing the constraint means increasing w by a small amount. If this is unfamiliar to you, think about why it is so: The budget set Bp,w+dw strictly includes the budget set Bp,w, and so any bundle that could be chosen before the wealth increase could also be chosen after. Since there are more feasible points, the constraint after the wealth increase is a relaxation of the constraint before. We can analyze the effect of this by differentiating u (x (p,w)) with respect to w : d dw u (x (p,w)) = LX i=1 dui dxi dxi dw = LX i=1 λpi dxi dw = λ LX i=1 pi dxi dw = λ. The transition from the first line to the second line is accomplished by substituting in the first-order condition: duidxi − λpi = 0. The transition from the second line to the third line is trivial (you can factor out λ since it is a constant). The transition from the third line to the fourth line comes from the comparative statics of Walras’ Law that we derived in the choice section. Since p ·x (p,w) ≡ w,P pi dxl dw = 1 (you could rederive this by differentiating the identity with respect to w if you want). 3.3.3 The Indirect Utility Function and Its Properties The Walrasian demand function x (p,w) gives the commodity bundle that maximizes utility subject to the budget constraint. If we substitute this bundle into the utility function, we get the utility 9 If you don’t have a favorite, I recommend “Mathematics for Economists” by Simon and Blume. 50 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 The definition of the indirect utility function implies that the following identity is true: v (p,w) ≡ u (x (p,w)) . Differentiate both sides with respect to pl : ∂v ∂pl = LX i=1 ∂u ∂xi ∂xi ∂pl . But, based on the first-order conditions for utility maximization, which we know hold when u () is evaluated at the optimal x, x (p,w) (see equation (3.1)): ∂u ∂xi = λpi. And, we also know (from Section 3.3.2) that the Lagrange multiplier is the shadow price of the constraint: λ = ∂v∂w . Hence: ∂v ∂pl = LX i=1 ∂u ∂xi ∂xi ∂pl = LX i=1 λpi ∂xi ∂pl = λ LX i=1 pi ∂xi ∂pl = ∂v ∂w LX i=1 pi ∂xi ∂pl . Now, recall the comparative statics result of Walras’ Law with respect to a change in pl : xl (p,w) = − LX i=1 pi ∂xi ∂pl . Substituting this in yields: ∂v ∂pl = − ∂v ∂w xl (p,w) xl (p,w) = − ∂v ∂pl ∂v ∂w . The last equation, known as Roy’s identity, allows us to derive the demand functions from the indirect utility function. This is useful because in many cases it will be easier to estimate an indirect utility function and derive the direct demand functions via Roy’s identity than to derive x (p,w) directly. Estimating Roy’s identity involves estimating a single equation. Estimating x (p,w), on the other hand, amounts to finding for every value of p and w the solution to a set of L+ 1 first-order equations, which themselves may have unknown parameters. 53 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 3.3.5 The Indirect Utility Function and Welfare Evaluation Consider the situation where the price of good 1 increases from p1 to p01. What is the impact of this price change on the consumer? One way to measure it is in terms of the indirect utility function: impact = v ¡ p0, w ¢ − v (p,w) . That is, the impact of the price change is equal to the difference in the consumer’s utility at prices p0 and p. While this is certainly a measure of the impact of the price change, it is essentially useless. There are a number of reasons why, but they all hinge on the fact that utility is an ordinal, not a cardinal concept. As you recall, the only meaning of the numbers assigned to bundles by the utility function is that x  y if and only if u (x) > u (y). In particular, if u (x) = 2u (y), this does not mean that the consumer likes bundle x twice as much as bundle y. Also, if u (x)− u (z) > u (s)− u (t), this doesn’t mean that the consumer would rather switch from bundle z to x than from t to s. Because of this, there is really nothing we can make of the numerical value of v (p0, w) − v (p,w). The only thing we can say is that if this difference is positive, the consumer likes (p0, w) more than (p,w). But, we can’t say how much more. Another problem with using the change in the indirect utility function as a measure of the impact of a policy change is that it cannot be compared across consumers. Comparing the change in two different utility functions is even more meaningless than comparing the change in a single person’s utility function. This is because even if both utility functions were cardinal measures of the benefit to a consumer (which they aren’t), there would still be no way to compare the scales of the two utility functions. This is the “problem of interpersonal comparison of utility,” which arises in many aspects of welfare economics. As a possible solution to the problem, consider the following thought experiment. Initially, prices and wealth are given by (p,w). I am interested in measuring the impact of a change in prices to p0. So, I ask you the following question: By how much would I have to change your wealth so that you are indifferent between (p0, w) and (p,w0)? That is, for what value w0 does v ¡ p0, w ¢ = v ¡ p,w0 ¢ . The change in wealth, w0 − w, in essence gives a monetary value for the impact of this change in price. And, this monetary value is a better measure of the impact of the price change than the utility measurement, because it is, at least to a certain extent, comparable.11 You can compare 11 I say to a certain extent, because the value of an additional ∆w dollars of wealth will depend on the initial state. 54 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 the impact of two different changes in prices by looking at the associated changes in wealth needed to compensate the consumer. Similarly, you can compare the impact of price changes on different consumers by comparing the changes in wealth necessary to leave them just as well off.12 Finally, although using the amount of money needed to compensate the consumer is an imperfect measure of the impact of a policy decision, it has one huge benefit for the neoclassical economist, and that is that it is observable, at least in principle. There is nothing we can do to observe utility scales. However, we can often elicit from people the amount of money they would find equivalent to a certain policy change, either through experiments, surveys, or other estimation techniques. 3.4 The Expenditure Minimization Problem (EMP) In the previous section, I argued that a good measure of the impact of a change in prices was the change in wealth necessary to make the consumer as well off at the old prices and new wealth as she was at the new prices and old wealth. However, this is not an easy exercise when all you have to work with is the indirect utility function. If we had a function that tells you how much wealth you would need to have in order to achieve a certain level of utility, then we would be able to do this much more efficiently. There is such a function. It is called the expenditure function, and in this section we will develop it. The expenditure minimization problem (EMP) asks the question, if prices are p, what is the minimum amount the consumer would have to spend to achieve utility level u? That is: min x p · x s.t. : u (x) ≥ u. Before we go on, let’s take a moment to figure out what the endogenous and exogenous variables are here. The exogenous variables are prices p and the reservation (or target) utility level u. The endogenous variable is x, the consumption bundle. So, in words, the expenditure minimization bundle amounts to finding the bundle x that minimizes the cost of achieving utility u when prices are p. The Lagrangian for this problem is given by: LEMP = p · x− λ (u (x)− u) . For instance, poor people presumably value the same wealth increment more than rich people. 12Again, this measure is imperfect because it assumes that the two consumers have the same marginal utility of wealth. 55 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 x* p1x1 + p2x2 = w u(x)=u* Figure 3.13: The Utility Maximization Problem x* p1x1 + p2x2 = w u(x)=u* Figure 3.14: The Expenditure Minimization Problem 58 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 to which we will return later: h (p, v (p,w)) ≡ x (p,w) x (p, e (p, u)) ≡ h (p, u) . These identities restate the principles discussed previously. The first says that the commodity bundle that minimizes the cost of achieving the maximum utility you can achieve when prices are p and wealth is w is the bundle that maximizes utility when prices are p and wealth is w. The second says that the bundle that maximizes utility when prices are p and wealth is equal to the minimum amount of wealth needed to achieve utility u at those prices is the same as the bundle that minimizes the cost of achieving utility u when prices are p. Similar identities can be written using the indirect utility function and expenditure function: u ≡ v (p, e (p, u)) w ≡ e (p, v (p,w)) . Note to MWG readers: There is a mistake in Figure 3.G.3. The relationships on the horizontal line connecting v (p,w) and e (p, u) should be the ones written directly above. The main implication of the previous analysis is this: The expenditure function contains the exact same information as the indirect utility function. And, since the indirect utility function can be used (by Roy’s identity) to derive the Walrasian demand functions, which can, in turn, be used to recover preferences, the expenditure function contains the exact same information as the utility function. This means that if you know the consumer’s expenditure function, you know her utility function, and vice versa. No information is lost along the way. This is another expression of what people mean when they say that the UMP and EMP are dual problems - they contain exactly the same information. 3.4.2 Properties of the Hicksian Demand Functions and Expenditure Function In this section, we refer both to function u (x) and to a particular level of utility, u. In order to be clear, let’s put a bar over the u when we are talking about a level of utility, i.e., ū. Just as we derived the properties of x (p,w) and v (p,w), we can also derive the properties of the Hicksian demand functions h (p, ū) and expenditure function e (p, ū). Let’s begin with h (p, ū) . We will assume that u () is a continuous utility function representing a locally non-satiated preference relation. 59 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 Properties of the Hicksian Demand Functions The Hicksian demand functions have the following properties: 1. Homogeneity of degree zero in p : h (αp, ū) ≡ h (p, ū) for p, ū, and α > 0. NOTE: THIS IS HOMOGENEITY IN P , NOT HOMOGENEITY IN P AND U ! Homogeneity of degree zero is best understood in terms of the graphical presentation of the EMP. The solution to the EMP is the point of tangency between the utility isoquant u (x) = ū and one of the budget lines. This is determined by the slope of the expenditure lines (lines of the form p · x = k, where k is any constant). Any change that doesn’t affect the slope of the budget lines should not affect the cost-minimizing bundle (although it will affect the expenditure on the cost minimizing bundle). Since the slope of the expenditure line is determined by relative prices and since scaling all prices by the same amount does not affect relative prices, the solution should not change. More formally, the EMP at prices αp is min x αp · x s.t. : u (x) ≥ ū. But, this problem is formally equivalent to: minα (p · x) : s.t. : u (x) ≥ x which is equivalent to: αmin x p · x : s.t. : u (x) ≥ x which is just the same as the EMP when prices are p, except that total expenditure is multiplied by α, which doesn’t affect the cost minimizing bundle. 2. No excess utility: u (h (p, ū)) = ū. This follows from the continuity of u (). Suppose u (h (p, ū)) > ū. Then consider a bundle h0 that is slightly smaller than h (p, ū) on all dimensions. By continuity, if h0 is sufficiently close to h (p, ū), then u (h0) > ū as well. But, then h0 is a bundle that achieves utility ū at lower cost than h (p, ū), which contradicts the assumption that h (p, ū) was the cost minimizing bundle in the first place.15 From this we can conclude that the constraint always binds in the EMP. 15This type of argument - called “Proof by Contradiction” - is quite common in economics. If you want to show a implies b, assume that b is false and show that if b is false then a must be false as well. Since a is assumed to be true, this implies that b must be true as well. 60 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 Now, substitute in the first-order conditions, pj = λuj ∂e ∂pi ≡ hi (p, ū) + λ X j uj ∂hj ∂pi . (3.4) Since the constraint binds at any optimum of the EMP, u (h (p, ū)) ≡ ū Differentiate with respect to pi : X j uj ∂hj ∂pi = 0 and substituting this into (3.4) yields: ∂e ∂pj ≡ hj (p, ū) . (3.5) That is, the derivative of the expenditure function with respect to pj is just the Hicksian demand for commodity j. The importance of this result is similar to the importance of Roy’s identity. Frequently, it will be easier to measure the expenditure function than the Hicksian demand function. Since we are able to derive the Hicksian demand function from the expenditure function, we can derive something that is hard to observe from something that is easier to observe. From (3.5) we can derive several additional properties (assuming u () is strictly quasiconcave and h () is differentiable): 1. (a) ∂hi∂pj = ∂2e ∂pi∂pj . This one follows directly from the fact that (3.5) is an identity. Let Dph (p, ū) be the matrix whose ith row and jth column is ∂hi∂pj . This property is thus the same as saying that Dph (p, ū) ≡ D2pe (p, ū), where D2pe (p, ū) is the matrix of second derivatives (Hessian matrix) of e (p, ū). (b) Dph (p, ū) is a negative semi-definite (n.s.d.) matrix. This follows from the fact that e (p, ū) is concave, and concave functions have Hessian matrices that are n.s.d. The main implication is that the diagonal elements are non-positive, i.e., ∂hi∂pi ≤ 0. (c) Dph (p, ū) is symmetric. This follows from Young’s Theorem (that it doesn’t matter what order you take derivatives in): ∂hi∂pj = ∂2e ∂pi∂pj = ∂hj ∂pi . The implication is that the cross-effects are the same — the effect of increasing pj on hi is the same as the effect of increasing pi on hj . 63 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 x x' p p' Figure 3.15: Compensated Demand (d) P j ∂hi ∂pj pj = 0 for all i . This follows from the homogeneity of degree zero of h (p, ū) in p. Consider the identity: h (ap, ū) ≡ h (p, ū) . Differentiate with respect to a and evaluate at a = 1, and you have this result. The Hicksian demand curve is also known as the compensated demand curve. The reason for this is that implicit in the definition of the Hicksian demand curve is the idea that following a price change, you will be given enough wealth to maintain the same utility level you did before the price change. So, suppose at prices p you achieve utility level ū. The change in Hicksian demand for good i following a change to prices p0 is depicted in Figure 3.15. When prices are p, the consumer demands bundle x, which has total expenditure p · x = w. When prices are p0, the consumer demands bundle x0, which has total expenditure p0 · x0 = w0. Thus implicit in the definition of the Hicksian demand curve is the idea that when prices change from p to p0, the consumer is compensated by changing wealth from w to w0 so that she is exactly as well off in utility terms after the price change as she was before. Note that since ∂hi∂pi ≤ 0, this is another statement of the compensated law of demand (CLD). When the price of a good goes up and the consumer is compensated for the price change, she will not consume more of the good. The difference between this version and the previous version we saw (in the choice based approach) is that here, the compensation is such that the consumer can achieve the same utility before and after the price change (this is known as Hicksian substitution), and in the previous version of the CLD the consumer was compensated so that she could just afford the same bundle as she did before (this is known as Slutsky compensation). It turns out that the 64 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 two types of compensation yield very similar results, and, in fact, for differential changes in price, they are identical. 3.4.4 The Slutsky Equation Recall that the whole point of the EMP was to generate concepts that we could use to evaluate welfare changes. The purpose of the expenditure function was to give us a way to measure the impact of a price change in dollar terms. While the expenditure function does do this (you can just look at e (p0, u)− e (p, u)), it suffers from another problem. The expenditure function is based on the Hicksian demand function, and the Hicksian demand function takes as its arguments prices and the target utility level u. The problem is that while prices are observable, utility levels certainly are not. And, while we can generate some information by asking people over and over again how they compare certain bundles, this is not a very good way of doing welfare comparisons. To summarize our problem: The Walrasian demand functions are based on observables (p,w) but cannot be used for welfare comparisons. The Hicksian demand functions, on the other hand, can be used to make welfare comparisons, but are based on unobservables. The solution to this problem is to somehow derive h (p, u) from x (p,w). Then we could use our observations of p and w to derive h (p, u), and use h (p, u) for welfare evaluation. Fortunately, we can do exactly this. Suppose that u (x (p,w)) = u (which implies that e (p, u) = w), and consider the identity: hi (p, u) ≡ xi (p, e (p, u)) . Differentiate both sides with respect to pj : ∂hi ∂pj ≡ ∂xi (p, e (p, u)) ∂pj + ∂xi (p, e (p, u)) ∂e(p, u) ∂e (p, u) ∂pj ≡ ∂xi (p,w) ∂pj + ∂xi (p,w) ∂w hj (p, u) ≡ ∂xi (p,w) ∂pj + ∂xi (p,w) ∂w xj (p, e (p, u)) ≡ ∂xi (p,w) ∂pj + ∂xi (p,w) ∂w xj (p,w) . The equation ∂hi (p, v (p,w)) ∂pj ≡ ∂xi (p,w) ∂pj + ∂xi (p,w) ∂w xj (p,w) is known as the Slutsky equation. Note that it provides the link between the Walrasian demand functions x (p,w) and the Hicksian demand functions, h (p, u). Thus if we estimate the right-hand 65 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 Part 0: Recall that ∂hi∂pi ≤ 0 because the Slutsky matrix is negative semi-definite. We’ll assume, as is typical, that ∂hi∂pi < 0. To keep things simple, I’ll omit the subscripts for the rest of the subsection, since we’re always talk about a single good. Part 1: Relationship between Hicksian and Walrasian Demand. By duality, we know that through any point on Walrasian demand there is a Hicksian demand curve through that point. How do their slopes compare? This is given by the Slutsky equation. But, keep in mind two things. First, since we put p on the vertical axis and x on the horizontal axis, when we draw a graph, the derivatives of the demand functions aren’t slopes. They’re inverse slopes. That is, the slope of the Walrasian demand is 1∂x/∂p and the slope of the Hicksian is 1 ∂h/∂p . Second, these derivatives are usually negative. So, we have to be a bit careful about thinking about quantities that are larger (i.e., further to the right on the number line) and quantities that are larger in magnitude (i.e., are further from zero on the number line). Since slopes are negative, a “larger” slope corresponds to a flatter curve. You’lls ee why this is important in a minute. To figure out whether x (p,w) or h (p, u) through a point is steeper, use the Slutsky equation. ∂h ∂p = ∂x ∂p + ∂x ∂w x. The answer will depend on whether x is normal or inferior. So, begin by considering a normal good. In this case, ∂x∂px > 0, so: ∂h ∂p > ∂x ∂p¯̄̄̄ ∂h ∂p ¯̄̄̄ < ¯̄̄̄ ∂x ∂p ¯̄̄̄ 1¯̄̄ ∂h ∂p ¯̄̄ > 1¯̄̄ ∂x ∂p ¯̄̄ The first line comes from the Slutsky equation and the fact that the income effect is positive. The second comes from the fact that, for a normal good, both sides are negative, and hence if ∂h∂p > ∂x ∂p , ∂h ∂p is smaller in magnitude (absolute value) than ∂x ∂p . The third follows from the second sinxe if x < y and both are positve, then 1/x > 1/y. Hence, for normal goods, the Hicksian Demand through a point is steeper than the Walrasian Demand through that point. For an inferior good, things reverse. To simplify, suppose x is inferior but not Giffen (so that 68 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 ∂x ∂p < 0 — you can do the Giffen case on your own). In this case ∂x ∂px < 0, so: ∂h ∂p < ∂x ∂p¯̄̄̄ ∂h ∂p ¯̄̄̄ > ¯̄̄̄ ∂x ∂p ¯̄̄̄ 1¯̄̄ ∂h ∂p ¯̄̄ < 1¯̄̄ ∂x ∂p ¯̄̄ and so Hicksian demand is flatter than Walrasian Demand. Part 2: Dependence of Hicksian demand on u. How does changing u shift the Hicksian Demand curve? Again, the answer depends on whether the good is normal or inferior. To see how, use duality: h (p, u) ≡ x (p, e (p, u)) , and differentiate both sides with respect to u : ∂h ∂u ≡ ∂x ∂w ∂e ∂u . By the properties of the expenditure function, we know that ∂e∂u > 0 (see MWG Prop 3.E.2, p. 59), so that ∂h∂u has the same sign as ∂x ∂w . Hence, when the good is normal, increasing u increases Hicksian demand for any price. Thus, increasing u shifts the Hicksian demand curve to the right. Similarly, when the good is inferior, increasing u decreases Hicksian demand for any price, and thus increasing u shifts the Hicksian demand for an inferior good to the left. The intuition is that in order to achieve a higher utility level, the consumer must spend more, and consumption increases with expenditure for a normal good and decreases with expenditure for an inferior good. A couple of pictures. These pictures depict Walrasian and Hicksian demand before and after a price decrease for a normal good and for an inferior good. Note that p1 < p0 so that u1 > u0. For the normal good, Hicksian demand is steeper than Walrasian, and shifts to the right when the price decreases. For the inferior good, Hicksian demand is flatter than Walrasian and shifts to the left when the price decreases. Substitutes and Complements Revisited Remember when we studied the UMP, we said that goods i and j were gross complements or substitutes depending on whether ∂xi∂pj was negative or positive? Well, notice that we could also classify goods according to whether ∂hi∂pj is negative or positive. In fact, we will call goods i and j complements if ∂hi∂pj < 0 and substitutes if ∂hi ∂pj > 0. That 69 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 p x x(p,w) h(p,u1) h(p,u0) p0 p1 NORMAL GOOD Figure 3.16: p x x(p,w) h(p,u1) h(p,u0) p0 p1 INFERIOR GOOD Figure 3.17: 70 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 If prices change to p1, the consumer’s utility at the new prices is given by: v ¡ p1, w ¢ . Thus the consumer’s utility increases, stays constant, or decreases depending on whether: v ¡ p1, w ¢ − v ¡ p0, w ¢ is positive, equal to zero, or negative. While looking at the change in utility can tell you whether the consumer is better off or not, it cannot tell you how much better off the consumer is made. This is because utility is an ordinal concept. The units that utility is measured in are arbitrary. Thus it is meaningless to compare, for example, v ¡ p1, w ¢ − v ¡ p0, w ¢ and v (p2, w) − v (p3, w). And, if v () and y () are the indirect utility functions of two people, it is also meaningless to compare the change in v to the change in y. However, suppose we were to compare, instead of the direct utility earned at a particular price- wealth pair, the wealth needed to achieve a certain level of utility at a given price-wealth pair. To see how this works, let u1 = v ¡ p1, w ¢ u0 = v ¡ p0, w ¢ . We are interested in comparing the expenditure needed to achieve u1 or u0. Of course, this will depend on the particular prices we use. It turns out that we have broad latitude to choose whichever set of prices we want, so let’s call the reference price vector pref , and we’ll assume that it is strictly greater than zero on all components. The expenditure needed to achieve utility level u at prices pref is just e ³ pref , u ´ . Thus, if we want to compare the expenditure needed to achieve utility u0 and u1, this is given by: e ³ pref , u1 ´ − e ³ pref , u0 ´ e ³ pref , v ¡ p1, w ¢´ − e ³ pref , v ¡ p0, w ¢´ . This expression will be positive whenever it takes more wealth to achieve utility u1 at prices pref than to achieve u0. Hence this expression will also be positive, zero, or negative depending on 73 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 whether u1 > u0, u1 = u0, or u1 < u0. However, the units now have meaning. The difference is measured in dollar terms. Because of this, e ¡ pref , v (p,w) ¢ is often called a money metric indirect utility function. We can construct a money metric indirect utility function using virtually any strictly positive price as the reference price pref . However, there are two natural candidates: the original price, p0, and the new price, p1. When pref = p0, the change in expenditure is equal to the change in wealth such that the consumer would be indifferent between the new price with the old wealth and the old price with the new wealth. Thus it asks what change in wealth would be equivalent to the change in price. Formally, define the equivalent variation, EV ¡ p0, p1, w ¢ , as EV ¡ p0, p1, w ¢ = e ¡ p0, v ¡ p1, w ¢¢ − e ¡ p0, v ¡ p0, w ¢¢ = e ¡ p0, v ¡ p1, w ¢¢ − w. since e ¡ p0, v ¡ p0, w ¢¢ = w. Equivalent variation is illustrated in MWG Figure 3.I.2, panel a. Notice that the compensation takes place at the old prices — the budget line shifts parallel to the one for¡ p0, w ¢ . Since w = e ¡ p1, v ¡ p1, w ¢¢ , an alternative definition of EV would be: EV ¡ p0, p1, w ¢ = e ¡ p0, v ¡ p1, w ¢¢ − e ¡ p1, v ¡ p1, w ¢¢ . In this form, EV asks how much more money does it take to achieve utility level v ¡ p1, w ¢ at p0 than at p1. Note: if EV < 0, this means that it takes less money to achieve utility v ¡ p1, w ¢ at p0 than p1 (which means that prices have gone up to get to p1, at least on average). When considering the case where the price of only one good changes, EV has a useful interpre- tation in terms of the Hicksian demand curve. Applying the fundamental theorem of calculus and the fact that ∂e(p,u)∂pi = hi (p, u), if only the price of good 1 changes, we have: 22 e ¡ p0, v ¡ p1, w ¢¢ − e ¡ p1, v ¡ p1, w ¢¢ = Z p01 p11 h1 ¡ s, p0−1, v ¡ p1, w ¢¢ ds Thus the absolute value of EV is given by the area to the left of the Hicksian demand curve between p01 and p 1 1. If p 0 1 < p 1 1, EV is negative - a welfare loss because prices went up. If p 0 1 > p 1 1, EV is 22Often when we are interested in a particular component of a vector - say, the price of good i - we will write the vector as (pi, p−i), where p−i consists of all the other components of the price vector. Thus, (p∗i , p−i) stands for the vector (p1, p2, ..., pi−1, p∗i , pi+1, ..., pL). It’s just a shorthand notation. Another notational explanation - in an expression such as p01, the superscript refers to the timing of the price vector (i.e. new or old prices), and the subscript refers to the commodity. Thus, p01 is the old price of good 1. 74 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 positive - a welfare gain because prices went down. The relevant area is depicted in MWG Figure 3.I.3, panel a. The other case to consider is the one where the new price is taken as the reference price. When pref = p1, the change in expenditure is equal to the change in wealth such that the consumer is indifferent between the original situation ¡ p0, w ¢ and the new situation (p1, w+∆w). Thus it asks how much wealth would be needed to compensate the consumer for the price change. Formally, define the compensating variation (depicted in MWG Figure 3.I.2, panel b) CV ¡ p0, p1, w ¢ = e ¡ p1, v ¡ p1, w ¢¢ − e ¡ p1, v ¡ p0, w ¢¢ = w − e ¡ p1, v ¡ p0, w ¢¢ . Again, when only one price changes, we can readily interpret CV in terms of the area to the left of a Hicksian demand curve. However, this time it is the Hicksian demand curve for the old utility level, u0. To see why, note that w = e ¡ p0, v ¡ p0, w ¢¢ , and so (again assuming only the price of good 1 changes): CV ¡ p0, p1, w ¢ = e ¡ p0, v ¡ p0, w ¢¢ − e ¡ p1, v ¡ p0, w ¢¢ = Z p01 p11 h1 ¡ s, p0−1, v ¡ p0, w ¢¢ ds, which is positive whenever p01 > p 1 1 and negative whenever p 0 1 < p 1 1. The relevant area is illustrated in MWG Figure 3.I.3, panel b. Recall that whenever good i is a normal good, increasing the target utility level u shifts hi (pi, p̄−i, u) to the right in the (xi, pi) space. This is because in order to achieve higher utility the consumer will need to spend more wealth, and if the good is normal and the consumer spends more wealth, more of the good will be consumed. Thus when the good is normal, EV ≥ CV . On the other hand, if the good is inferior, then increasing u shifts hi (pi, p̄−i, u) to the left, and CV ≥ EV . When there is no wealth effect on the good, i.e., ∂xi(p,w)∂w = 0, then CV = EV. Figure 3.I.3 also shows theWalrasian demand curve. In fact, it shows it crossing h ¡ p1, p 0 −1, v ¡ p1, w ¢¢ at p11 and h1 ¡ p1, p 0 −1, v ¡ p0, w ¢¢ at p01. This results from the duality of utility maximization and expenditure minimization. Formally, we have the equalities h1 ¡ p0, v ¡ p0, w ¢¢ = x ¡ p0, w ¢ h1 ¡ p1, v ¡ p1, w ¢¢ = x ¡ p1, w ¢ , which each arise from the identity hi (p, v (p,w)) ≡ xi (p,w). The result of this is that the Walrasian demand curve crosses the Hicksian demand curves at the two points mentioned above, and that the area to the left of the Walrasian demand curve lies somewhere between the EV and CV. There are a number of comments that must be made on this topic: 75 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 a toll each time they use the road. The lump-sum tax is non-distortionary, but it must be paid by people who don’t drive, even people who can’t afford to drive. The gasoline tax is paid by all drivers, including people who don’t use the particular roads being repaired, and it is distortionary in the sense that people will generally reduce their driving in response to the tax, which induces a deadweight loss. Charging a toll to those who use the road places the burden of paying for repairs on exactly those who are benefiting from having the roads. But like the gasoline tax, it is also dis- tortionary (since people will tend to avoid toll roads). And, since the tolls are focused on relatively few consumers, the tolls may have to be quite high in order to raise the necessary funds, imposing a large burden on those people who cannot avoid using the toll roads. These are just some of the issues that must be considered in deciding which commodities should be taxed and how. 3.4.8 Bringing It All Together Recall the basic dilemma we faced. The UMP yields solution x (p,w) and value function v (p,w) that are based on observables but not useful for doing welfare evaluation since utility is ordinal. The EMP yields solution h (p, u) and value function e (p, u), which can be used for welfare evaluation but are based on u, which is unobservable. As I have said, the link between the two is provided by the Slutsky equation ∂hi (p, v (p,w)) ∂pj = ∂xi (p,w) ∂pj + ∂xi (p,w) ∂w xi (p,w) . We now illustrate how this is implemented. Suppose the price of good 1 changes. EV is given by: EV ¡ p0, p1, w ¢ = Z p01 p11 h1 ¡ s, p0−1, u 1 ¢ ds. We can approximate h1 ¡ s, p0−1, u 1 ¢ using a first-order Taylor approximation. Recall, a first-order Taylor approximation for a function f (x) at point x0 is given by: f̃ (x) ∼= f (x0) + f 0 (x0) (x− x0) . This gives a linear approximation to f (x) that is tangent to f (x) at x0 and a good approximation for x that are not too far from x0. But, the further x is away from x0, the worse the approximation will be. See Figure 3.18. Now, the first-order Taylor approximation to h ¡ s, p0−1, u 1 ¢ is given by: h1 ¡ s, p0−1, u 1 ¢ ∼= h1 ¡p11, p0−1, u1¢+ ∂h1 (p, v (p,w))∂p1 ¡s− p11¢ (3.6) 78 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 x0 x f(x) ( )f x Figure 3.18: A First-Order Taylor Approximation Note that we have taken as our original point p = ¡ p11, p 0 −1 ¢ . That is, the price vector after the price change? Why do we do this? The reason is that we are using the Hicksian demand curve for u1, the utility level after the price change. Because of that, we also want to use the price after the price change. We know that, at p = ¡ p11, p 0 −1 ¢ , h ¡ p11, p 0 −1, u 1 ¢ = x1 ¡ p11, p 0 −1, w ¢ . This fact, along with the Slutsky equation, allows us to rewrite (3.6) as: h̃1 ¡ s, p0−1, u 1 ¢ ∼= x1 ¡p11, p0−1, w¢ (3.7) + à ∂x1 ¡ p11, p 0 −1, w ¢ ∂p1 + ∂x1 ¡ p11, p 0 −1, w ¢ ∂w x1 ¡ p11, p 0 −1, w ¢!¡ s− p11 ¢ . The last equation provides an approximation for the Hicksian demand curve based only on observ- able quantities. That is, we have eliminated the need to know the (unobservable) target utility level. Finally, note that demand x1 ¡ p11, p 0 −1, w ¢ and derivatives ∂x1(p11,p0−1,w) ∂p1 and ∂x1(p11,p0−1,w) ∂w can be observed or approximated using econometric techniques. Note that the difference between this approximation and one based on the Walrasian demand curve is the addition of the wealth-effect term, ∂x1(p11,p0−1,w) ∂w x1 ¡ p11, p 0 −1, w ¢ . Figure 3.19 illustrates the first-order Taylor approximation to EV. Since the “original point” in our estimate to the Hicksian demand function is p = ¡ p11, p 0 −1 ¢ , estimated Hicksian demand h̃1 is coincides with and it tangent to the actual Hicksian demand h1 at this point. As you move to prices that are further away from p11, the approximation is less good. True EV is the area left of h1. Thus, the estimation “error”, the difference between true EV and estimated EV, is given by the area between h̃1 and h1 between prices p10 and p 1 1. In the case of CV, CV is computed as the area left of the Hicksian demand curve at the original 79 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 x1(p1,p-1,w) p11 p10 h1(p1,p-1,u1) ( )11 1, ,h p p u− EV Estimation “error” p1 x1 Figure 3.19: The First-Order Taylor Approximation to EV 80 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 Another thing we could do is figure that the harm done to the consumer is just the change in price times the original consumption of this good, i.e., 2.5 (100) = 250. However, if we gave the consumer 250 additional dollars, we would be overcompensating for the price increase. 3.4.9 Welfare Evaluation for an Arbitrary Price Change The basic analysis of welfare change using CV and EV considers the case of a single price change. However, what should we do if the policy change is not a single price change? For changes in multiple prices, we can just compute the CV for each of the changes (i.e., changing prices one by one and adding the CV (or EV) from each of the changes along a “path” from the original price to the new price). If price and wealth change, we can add the change in wealth to the CV (or EV) from the price changes (see below). But, what if the policy change involves something other than prices and wealth, such a change in environmental quality, roads, etc. How do we value such a change? The answer is that, if we have good estimates of Walrasian demand, we can always represent the change as a change in a budget set. After doing so, we can compute the CV is the usual way. Part 1: Any arbitrary policy change can be thought of as a simultaneous change in p and w. To illustrate, suppose that we have a good estimate of consumers’ demand functions (i.e., we fit a flexible functional form for demand using high-quality data). Let x (p,w) denote demand. Suppose that initially prices and wealth are ¡ p0, w0 ¢ and the consumer chooses bundle x ¡ p0, w0 ¢ . Now, suppose that “something happens” that leads the consumer to choose bundle x0 instead of x0. What is the CV (or EV) of this change? The first step is to note that, if demand is quasiconcave, there is some price-wealth vector for which x0 and x0 are optimal choices. You can find these price-wealth vectors, which we’ll call¡ p0, w0 ¢ and (p0, w0), by solving the equations x0 = ¡ p0, w0 ¢ and x0 = x (p0, w0). (In reality you probably already know ¡ p0, w0 ¢ and have an observation of x0 or estimate of.) Remember, we have a good estimate of x (p,w). Once we find (p0, w0), then we know that the change in the consumer’s utility in going from x0 to x 0 is just v (p0, w0)− v ¡ p0, w0 ¢ , and so the impact of the policy change reduces to computing the EV or CV for this simultaneous change in p and w. Let v (p0, w0) = u0 and v ¡ p0, w0 ¢ = u0. Part 2: Compute the EV or CV for a simultaneous change in p and w. So, we’ve recast the policy change as a change from ¡ p0, w0 ¢ to (p0, w0), letting u0 and u0 denote 83 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 the utility levels before and after the change. To compute EV , return to the definition of EV we used before. EV = e ¡ p0, u0 ¢ − e ¡ p0, u0 ¢ Adding and subtracting e (p0, u0), we get: EV = £ e ¡ p0, u0 ¢ − e ¡ p0, u0 ¢¤ + £ e ¡ p0, u0 ¢ − e ¡ p0, u0 ¢¤ . But, note that e (p0, u0) = w0 and e ¡ p0, u0 ¢ = w0, so EV = £ e ¡ p0, u0 ¢ − e ¡ p0, u0 ¢¤ + w0 −w0, (*) and note that £ e ¡ p0, u0 ¢ − e (p0, u0) ¤ is as in the definition of EV when only a price changes. So, if only the price of good 1 changes, EV can be written as: EV = Z p01 p01 h1 ¡ s, p−1, u 0¢ ds+ ¡w0 − w0¢ , and this can be estimated in the usual way from the estimated Walrasian demand curve. If multiple prices change, we change them one by one and add up the integral from each change, and then we add the change in wealth. That is, if prices change from ¡ p01, p 0 2, ..., p 0 L ¢ to (p01, p 0 2, ..., p 0 L) and wealth changes from w0 to w0, the EV is: EV = Z p01 p01 h1 ¡ s, p02, ..., p 0 L ¢ ds+ Z p02 p02 h2 ¡ p01, s, p 0 3, ..., p 0 L ¢ ds+...+ Z p0L p0L h2 ¡ p01, p 0 2, ..., p 0 L−1, s ¢ ds+w0−w0. If you replace each Hicksian demand with an estimate based on Marshallian demand and the Slutsky equation, you can estimate this using only observables. It is tedious, but certainly possible. This is a diagram that illustrates the whole thing. Suppose a policy change moves the con- sumer’s consumption bundle from x0 to x0. To compute the EV, the first thing you do is find the (p,w) for which x0 = x ¡ p0, w0 ¢ . This budget set is labeled B ¡ p0, w0 ¢ . Then, you find the (p,w) for which x0 is optimal, which we call (p0, w0). This budget set (red) is labeled B (p0, w0). Denote the initial utility level u0 and the final utility level u0, and note that neither the utility levels nor the indifference curves (which are drawn in as dotted lines for illustration) are observed. Next, we decompose the change from ¡ p0, w0 ¢ to (p0, w0) into two parts. Part 1 is a change in wealth holding prices fixed at p0. Let y denote the point the consumer chooses at ¡ p0, w0 ¢ , and let uy denote the utiltiy earned. This point and the associated budget set are in blue. Note that moving from budget set B ¡ p0, w0 ¢ to budget set B ¡ p0, w0 ¢ is just like losing w0−w0 dollars (since prices don’t change this is, in fact, exactly what happens). This is where the ¡ w0 −w0 ¢ term comes 84 Nolan Miller Notes on Microeconomic Theory: Chapter 3 ver: Aug. 2006 from in expression (*) above. Distance w0−w0 is denoted on the left. (Note that in the diagram, these distances are scaled by p2, since we are showing them on the x2-axis.) Part 2 of the decomposition is the change in prices from p0 to p0 when wealth is w0. But, note that this just the kind of EV we computed in the simple case. That is, prices change but wealth remains constant. Let z denote the point that offers the same utility as x0 but is chosen at prices p0. That is, z = x ¡ p0, w0 +EV ¡ p0, p0.w0 ¢¢ . The budget line supporting z is denoted in green, and the EV for the price change from p0 to p0 at wealth w0 is just the distance that the budget shifts up from blue B ¡ p0, w0 ¢ to green B ¡ p0, w0 +EV ¡ p0, p0.w0 ¢¢ denoted EV ¡ p0, p0, w0 ¢ on the left. Since EV ¡ p0, p0, w0 ¢ = e ¡ p0, u0 ¢ − e ¡ p0, uy ¢ = e ¡ p0, u0 ¢ −w0 = e ¡ p0, u0 ¢ − e ¡ p0, u0 ¢ , this is just like the EV’s we computed when only prices changed. This is where the £ e ¡ p0, u0 ¢ − e (p0, u0) ¤ comes from in expression (*) above. The total EV is the sum of these two parts. The distance is denoted Total EV in the diagram. Note that since the consumer ends up worse off overall, the total EV should be negative. You could also do something similar for CV. CV = e ¡ p0, u0 ¢ − e ¡ p0, u0 ¢ = £ e ¡ p0, u0 ¢ − e ¡ p0, u0 ¢¤ + h e ¡ p0, u0 ¢ − e ³ p 0 , u0 ´i = w0 − w0 + h e ¡ p0, u0 ¢ − e ³ p 0 , u0 ´i , and note that once again £ e ¡ p0, u0 ¢ − e ¡ p0, u0 ¢¤ is as in our original definition of CV . So, this term can be rewritten in terms of integrals of Hicksian demand curves at utility level u0. 85 Nolan Miller Notes on Microeconomic Theory: Chapter 4 ver: Aug. 2006 x2 x1 Figure 4.1: Homothetic Preferences preference relation º is homothetic if and only if it can be represented by a utility function that is homogeneous of degree one. In other words, homothetic preferences can be represented by a function u () that such that u (αx) = αu (x) for all x and α > 0. Note that the definition does not say that every utility function that represents the preferences must be homogeneous of degree one — only that there must be at least one utility function that represents those preferences and is homogeneous of degree one. EXAMPLE: Cobb-Douglas Utility: A famous example of a homothetic utility function is the Cobb-Douglas utility function (here in two dimensions): u (x1, x2) = x a 1x 1−a 2 : a > 0. The demand functions for this utility function are given by: x1 (p,w) = aw p1 x2 (p,w) = (1− a)w p2 . Notice that the ratio of x1 to x2 does not depend on w. This implies that Engle curves (wealth expansion paths) are straight lines (see MWG pp. 24-25). The indirect utility function is given by: v (p,w) = µ aw p1 ¶aµ(1− a)w p2 ¶1−a = w µ a p1 ¶aµ1− a p2 ¶1−a . Another restriction on preferences that can allow us to draw inferences about all indifference curves from a single curve is called quasilinearity. A preference relation is quasilinear if there is one commodity, called the numeraire, which shifts the indifference curves outward as consumption 88 Nolan Miller Notes on Microeconomic Theory: Chapter 4 ver: Aug. 2006 of it increases, without changing their slope. Indifference curves for quasilinear preferences are illustrated in Figure 3.B.6 of MWG. Again, we can extend this definition to utility functions. A continuous preference relation is quasilinear in commodity 1 if there is a utility function that represents it in the form u (x) = x1 + v (x2, ..., xL). EXAMPLE: Quasilinear utility functions take the form u (x) = x1 + v (x2, ..., xL). Since we typically want utility to be quasiconcave, the function v () is usually a concave function such as log x or √ x. So, consider: u (x) = x1 + √ x2. The demand functions associated with this utility function are found by solving: maxx1 + x 0.5 2 s.t. : p · x ≤ w or, since x1 = −x2 p2p1 + w p1 , max−x2 p2 p1 + w p1 + x0.52 . The associated demand curves are x1 (p,w) = − 1 4 p1 p2 + w p1 x2 (p,w) = µ p1 2p2 ¶2 and indirect utility function: v (p,w) = 1 4 p1 p2 + w p1 . Isoquants of this utility function are drawn in Figure 4.2. 4.2 Aggregation Our previous work has been concerned with developing the testable implications of the theory of the consumer behavior on the individual level. However, in any particular market there are large numbers of consumers. In addition, often in empirical work it will be difficult or impossible to collect data on the individual level. All that can be observed are aggregates: aggregate consumption of the various commodities and a measure of aggregate wealth (such as GNP). This raises the 89 Nolan Miller Notes on Microeconomic Theory: Chapter 4 ver: Aug. 2006 2 4 6 8 10 x2 2 4 6 8 10 12 14 x1 Figure 4.2: Quasilinear Preferences natural question of whether or not the implications of individual demand theory also apply to aggregate demand. To make things a little more concrete, suppose there are N consumers numbered 1 through N , and the nth consumer’s demand for good i is given by xni (p,w n), where wn is consumer n’s initial wealth. In this case, total demand for good i can be written as: D̃i ¡ p,w1, ..., wN ¢ = NX n=1 xni (p,w n) . However, notice that D̃i () gives total demand for good i as a function of prices and the wealth levels of the n consumers. As I said earlier, often we will not have access to information about individuals, only aggregates. Thus we may ask the question of when there exists a functionDi (p,w) , where w =PN n=1w n is aggregate wealth, that represents the same behavior as D̃i ¡ p,w1, ..., wN ¢ . A second question is when, given that there exists an aggregate demand function Di (p,w), the behavior it characterizes is rational. We ask this question in two ways: First, when will the behavior resulting from Di (p,w) satisfy WARP? Second, when will it be as if Di (p,w) were generated by a “representative consumer” who is herself maximizing preferences? Finally, we will ask if there is a representative consumer, in what sense is the well-being of the representative consumer a measure of the well-being of society? 4.2.1 The Gorman Form The major theme that runs through our discussion in this section is that in order for demand to aggregate, each individual’s utility function must have an indirect utility function of the Gorman 90