Week 6: Consumer Theory Part 1 (Jehle and Reny, Chapter 1)

Transcription

1 Week 6: Consumer Theory Part 1 (Jehle and Reny, Chapter 1) Tsun-Feng Chiang* *School of Economics, Henan University, Kaifeng, China November 2, / 28

2 Primitive Notions 1.1 Primitive Notions Consumer theory is the foundation of economics, and a study of consumer choice. There are four building blocks in any model of consumer choice. They are the consumption set, the feasible set, the preference relation, and the behavioral assumption. A consumption set or choice set represents the set of all alternatives, or complete consumption plans, that the consumer can conceive, whether some of them will be achievable or not. Let each commodity be measured in some infinitely divisible units. Let x i R represent the number of units of good i. We assume that only nonnegative units of each good are meaningful. We let x = (x 1,, x n ) be a vector containing different quantities of each of the n commodities and call x a consumption bundle or consumption plan. A consumption bundle is x X is thus represented by a point x R n +. To simplify things, there are basic requirements of a consumption set X: 2 / 28

3 Primitive Notions Assumption 1.1 Minimal Properties of the Consumption Set, X X R n +. X is closed. X is convex. 0 X When the consumption bundle x is conceivable and achievable given the economic realities the consumer faces, then we say x is in a feasible set, denoted by B. It is clear B is a subset of X that remains after we have accounted for any constraints on the consumer s access to commodities due to the practical, institutional, or economic realities of the world. A preference relation typically specifies the limits, if any, on the consumer s ability to perceive in situations involving choice, the form of consistency or inconsistency in the consumer s choices, and information about the consumer s tastes for the different objects of choice. 3 / 28

4 Primitive Notions Finally, the model is closed by specifying some behavioral assumption which supposes that the consumer seeks to identify and select an available alternative that is most preferred in the light of his personal tastes. 4 / 28

5 1.2 The section explains how to derive utilty by preference relations. Consumer preferences are characterized axiomatically. Formally, we represent the consumer s preferences by a binary relation,, defined on the consumption set, X. If x 1 x 2, we say that "x 1 is at least as good as x 2," for this consumer. The following two axioms set forth basic criteria with which those binary comparisons must conform. AXION 1: Completeness. For all x 1 and x 2 in X, either x 1 x 2 or x 2 x 1. Axiom 1 formalizes the notion that the consumer has the ability to discriminate and the necessary knowledge to evaluate alternatives. It says the consumer can examine any two distinct consumption plans x 1 and x 2 and decide whether x 1 is at least as good as x 2 or x 2 is at least as good as x 1. 5 / 28

6 AXION 2: Transitivity. For any three elements x 1, x 2 and x 3 in X, if x 1 x 2 and x 2 x 3, then x 1 x 3 Axiom 2 gives a very particular form to the requirement that the consumer s choices be consistent. These two axioms together imply that the consumer can completely rank finite number of elements in the consumption set, X, from best to worst. We summarize the view that preferences enable the consumer to construct such a ranking by saying that those preferences can be represented by a preference relation. Definition 1.1 Preference Relation The binary relation on the consumption set X is called a preference relation if it satisfies Axiom 1 and 2. There are two additional relations that we will use in our discussion of consumer preferences. Each is determined by the preference relation,, and they formalize the notions of strict preference and indifference. 6 / 28

7 Definition 1.2 Strict Preference Relation The binary relation on the consumption set X is defined as follows: x 1 x 2 if and only if x 1 x 2 and x 2 x 1. The relation is called the strict preference relation induced by, or simply the strict preference relation when is clear. The phrase x 1 x 2 is read "x 1 is strictly preferred to x 2." Definition 1.3 Indifference Relation The binary relation on the consumption set X is defined as follows: x 1 x 2 if and only if x 1 x 2 and x 2 x 1. The relation is called the indifference relation induced by, or simply the indifference relation when is clear. The phrase x 1 x 2 is read "x 1 is indifferent to x 2." Using these two supplementary relations, we can establish something 7 / 28

8 very concrete about the consumer s ranking of any two alternatives. For any pair x 1 and x 2, exactly one of three mutually exclusive possibilities exists: x 1 x 2, or x 2 x 1, or x 1 x 2. Now we use the preference relation to define some related sets. These sets focus on the single alternative in the consumption set and examine the ranking of all other alternatives relative to it. Definition 1.4 Sets in X Derived from the Preference Relation Let x 0 be any point in the consumption set, X. Relative to any such point, we define the following subsets of X: 1. (x 0 ) {x x X, x x 0 }, called the "at least as good as" set. 2. (x 0 ) {x x X, x 0 x}, called the "no better than" set. 3. (x 0 ) {x x X, x x 0 }, called the "preferred to" set. 4. (x 0 ) {x x X, x 0 x}, called the "worse than" set. 5. (x 0 ) {x x X, x x 0 }, called the "indifference" set. 8 / 28

9 A hypothetical set of preferences satisfying Axiom 1 and 2 is sketched in Figure 1.1 (see the next slide) for X = R 2 +. We pick any point x 0 = (x1 0, x 2 0 ) representing a consumption plan consisting of a certain amount x1 0 of commodity, together with a certain amount x 2 0 of commodity 2. From Axiom 1, it is assumed the consumer knows how to compare different points (or consumption bundles) with x 0. From Axiom 2, it is assumed that he knows how to rank his preferences of these points. Given Definition 1.4, Axioms 1 and 2 tell us that the consumer must place every point in X into one of three mutually exclusive categories relative to x 0 : (x 0 ), (x 0 ), or (x 0 ) The preference in Figure 1.1 may seem rather odd. First we see the the gap, or the open area. This irregularity can be ruled out by imposing an additional requirement on preferences: AXION 3: Continuity. For all x R n +, the "at least as good as" set, (x), and the "no better than" set, (x), are closed in x R n +. 9 / 28

10 Figure 1.1 (Left): Hypothetical preferences satisfying Axioms 1 and 2; Figure 1.2 (right): Hypothetical preferences satisfying Axioms 1, 2 and 3 10 / 28

11 Because (x 0 ) and (x 0 ) are closed, so is (x 0 ) because the latter is the intersection of the former two. Consequently, Axiom 3 rules out the open area in the difference set in Figure 1.1. Thus we have Figure 1.2 (see the last slide). Another irregularty is the thick region for the set (x 0 ) in Figure 1.2. We use the following weak axiom to deal with it: AXION 4 : Local Nonsatiation. For all x 0 R n +, and for all ɛ > 0, there exists some x B ɛ (x 0 ) R n + such that x x 0. Axiom 4 says that within any vicinity of a given point x 0, no matter how small that vicinity is, there will always be at least one other point x that the consumer prefers to x 0. It rules out the possibility of having "zones of indifference," such as that surrounding x 1 in Figure 1.2. The preferences depicted in Figure 1.3 (see the next slide) do satisfy Axiom 4 as well as Axioms 1 to 3. After applying Axiom 4 to the cases of (x 0 ) and (x 0 ), the thick regions are all eliminated as shown in Figure 1.4 (see the next slide). 11 / 28

12 Figure 1.3 (Left): Hypothetical preferences satisfying Axioms 1, 2, 3 and 4 ; Figure 1.4 (right): Hypothetical preferences satisfying Axioms 1, 2, 3, and 4 12 / 28

13 AXION 4: Strict Monotonicity. For all x 0, x 1 R n +, if x 0 x 1 then x 0 x 1, while if x 0 x 1, then x 0 x 1. Axiom 4 says that if one bundle contains at least as much of every commodity as another bundle, then the one is at least as good as the other. Moreover, it is strictly better if it contains strictly more of every good. Axiom 4 eliminates the possibility that the indifference sets in R 2 + "bend upward," or contain positively sloped segments. It also requires that the "preferred to" sets be "above" the indifference sets and that "the worse than" sets be "below" them. For example, in Figure 1.4, points like x 2 located in southwest quadrant of x 0 should be worse to x 0 because x 0 x 2 ; and points like x 1 located in southwest quadrant of x 0 should be preferred to x 0 because x 1 x 0. Therefore, a set of preferences satisfying Axioms 1, 2, 3, and 4 is given in Figure 1.5 (see the next slide). 13 / 28

14 The indifference set in Figure 1.5 has both regions convex and concave to the origin. To have the familiar shape of preference, we need one final axiom on tastes. We will state two different versions of the axiom and then consider their meaning and purpose. Figure 1.5: Hypothetical preferences satisfying Axioms 1, 2, 3 and 4 14 / 28

15 AXION 5 : Convexity. If x 1 x 0, then tx 1 + (1 t)x 0 x 0 for all t [0, 1]. A slightly stronger version of this is the following: AXION 5: Strict Convexity. If x 1 x 0 and x 1 x 0, then tx 1 + (1 t)x 0 x 0 for all t (0, 1). Now we pick x 1 and x 2 on the indifference set (x 0 ) in Figure 1.5. and find a point x t which is a convex combination of x 1 and x 2. Axiom 5 says x t should be either in (x 0 ) or (x 0 ). Axiom 5 says x t should be in (x 0 ) only. Both Axioms 5 and 5 rule out the cases where the indifference set concave to the origin. Thus we can obtain an indifference set depicted in Figure 1.6. (see the next slide). (What would the graph look like as long as Axiom 5 is imposed but Axiom 5 isn t?) The thrust of Axiom 5 or Axiom 5 is to forbid the consumer from preferring extremes in consumption. Axiom 5 requires that any relatively balanced bundle as x t be no worse than either of the two extremes between which the consumer is indifferent. 15 / 28

16 Axiom 5 goes a bit further and requires that the consumer strictly prefer any such relatively balanced consumption bundle to both of the extreme s between which she is indifferent. Figure 1.6: Hypothetical preferences satisfying Axioms 1, 2, 3, 4, and 5 or 5 16 / 28

17 Another way to describe the implications of convexity for consumer s tastes focuses attention on the "curvature" of the indifference sets themselves. When X = R 2 +, the (absolute value of the) slope of an indifference curve is called the marginal rate of substitution, or MRS. The slope measures, at any point, the rate at which the consumer is willing to give up x 2 in exchange for x 1, such that he remains indifferent after the exchange. Axiom 5 requires that the MRS to be either constant or decreasing, where the latter one means the consumer wants to have more x 1 to compensate a given amount of loss in x 2 because he has relatively much x 1 and little x 2. Axiom 5 goes a bit further and requires that the MRS to be strictly diminishing. The indifference curve in Figure 1.6 display this property, sometimes called the principle of diminishing marginal rate of substitution in consumption. 17 / 28

18 The Utility Function In modern theory, a utility function is simply a convenient device for summarizing the information contained in the consumer s preference relation. When we would like to employ calculus methods, it is easier to work with a utility function. A utility function is defined formally as follows. Definition 1.5 A Utility Function Representing the Preference Relation A real-valued function u : R n + R is called a utility function representing the preference relation, if for all x 0, x 1 R n +, u(x 0 ) u(x 1 ) x 0 x 1. Thus a utility function represents a consumer s preference relation if it assigns higher number to preferred bundles. 18 / 28

19 Having a continuous real-valued utility function can simplify the analysis of many problems. The preference relations used in the last section can guarantee the existence of a continuous real-valued utility function. Theorem 1.1 Existence of a Real-Valued Function Representing the Preference Relation If the binary relation is complete, transitive, continuous, and strictly monotonic, there exists a continuous real-valued function, u : R n + R, which represents. With Theorem 1.1, we can represent preferences in terms of a continuous utility function. But this utility representation is never unique. If some function u represents a consumer s preferences, then so too will the function v = u + 5, or the function v = u 3, because each of these functions ranks bundles the same way u does. For preference representation, the rankings of consumption bundles are more meaningful than the value of utility associated with these consumption 19 / 28

20 bundles. If we transform the function u to another function v and the order of every consumption bundle is preserved, we call it monotonic transform. The following theorem tells us an utility function still represents the original preference relations after the monotonic transforms. Theorem 1.2 Invariance of the Utility Function to Positive Monotonic Transforms Let be a preference relation on R n + and suppose u(x) is an utility function that represents it. Then v(x) also represents if and only if v(x) = f (u(x)) for every x, where f : R R is strictly increasing on the set of values taken on by u. Any additional axioms we impose on preferences will be reflected as additional structure on the utility function representing them. The following theorems are some of them, 20 / 28

21 Theorem 1.3 Properties of Preferences and Utility Functions Let be represented by u : R n + R. Then: u(x) is strictly increasing if and only if is strictly monotonic. u(x) is quasiconcave if and only if is convex. u(x) is strictly quasiconcave if and only if is strictly convex. We had known there exist continuous real-valued utility functions by axioms imposed on preferences. However, to use calculus tools for the utility functions, they should be differentiable. This implies a more strict restriction should be imposed on preferences. To our convenience, we are content to simply assume that the utiltiy function is differentiable whenever necessary. Once the utility function is differentiable, the first-order partial derivative of u(x) with respect to x i is called the marginal utility of good i. For the case of two goods, we can derive the MRS in terms of the two goods marginal utilities. To see this, consider any bundle 21 / 28

22 x 1 = (x1 1, x 2 1). We can write x 2 1 as a function of x 1 1, that is, x 2 1 = f (x 1 1). Therefore, as x1 1 varies, the bundle (x 1 1, x 2 1) = (x 1 1, f (x 1 1 )) traces out the difference curve through x 1. Consequently, for all x1 1 u(x1 1, f (x 1 1 )) = constant (eq.1) Now the MRS of good one for good two at the bundle x 1 = (x1 1, x 2 1), denoted MRS 12 (x1 1, x 2 1 ), is the absolute value of the slope of the indifference curve through (x1 1, x 2 1 ). That is, MRS 12 (x 1 1, x 1 2 ) f (x 1 1 ) = f (x 1 1 ) (eq.2), because f < 0. Take the derivative of (eq.1) with respect to x 1 1, u(x 1 1, x 1 2 ) x 1 1 (eq.2) and (eq.3) together give + u(x 1 1, x 2 1) x2 1 f (x1 1 ) = 0 (eq.3) 22 / 28

23 MRS 12 (x 1 ) = u(x1 )/ x 1 1 u(x 1 )/ x 1 2 Be more general, if we have x R n +, we can define the MRS of good i for good j as the ratio of their marginal utilities, MRS ij (x) = u(x)/ x i u(x)/ x j Again this formula tells us the rate at which x i can be substituted for x j with no change in the consumer s utility. When u(x) is differentiable and preferences are strictly monotonic, the marginal utility of every good is virtually always strictly positive. That is, u(x)/ x i > 0 for all i = 1,, n. When preferences are strictly convex, the MRS between two goods is always strictly diminishing along any level surface of the utility function. 23 / 28

24 1.3 The Consumer s Problem A consumer have viewed the consumption set X = R n +. Her inclination and attitudes toward consumption bundles x are described by the preference relation. The consumer s circumstances limit the alternatives she is actually able to achieve, and we collect these all together into a feasible set B R n +. The consumer s problem is to find the most preferred alternatives in the fesible set according to her preference relation. Formally, the consumer seeks x B such that x x for all x B (Pr.1) To make further progress, we make the following assumptions that will be maintained unless stated otherwise. Assumption 1.2 Consumer Preferences The consumer s preference relation is complete, transitive, continuous, strictly monotonic, and strictly convex on R n +. Therefore, by Theorem 1.1 and 1.3 it can be represented by a real-valued utility function, u, that is continuous, strictly increasing, and strictly 24 / 28

25 Assumption 1.2 (Continued) function, u, that is continuous, strictly increasing, and strictly quasiconcave on R n +. In the two-good case, preferences like these can be represented by an indifference map whose level sets are nonintersecting, strictly convex away from the origin, and increasing northeasterly, as depicted in Figure 1.8.(see the next slide). Next we assume the consumer is in the market economy. By a market economy, we mean an economic system in which transactions between agents are mediated by markets. The transactions are so numerous that no individual agent has power to affect the prices. To the consumer, a price p i for each commodity i is fixed to him. We suppose that prices are strictly positive, so p i > 0, i = 1,, n, or p 0 by vector notation. 25 / 28

26 Figure 1.8: Indifference map for preferences satisfying Assumption 1.2 The consumer is endowed with a fixed money income y 0. The purchase of x i units of commodity i at price p i per unit requires an expenditure of p i x i dollars, the requirement that expenditure not exceed income can be stated as n i=1 p ix i y, or more compactly, as p x y. 26 / 28

27 A feasible set B is the set of consumption bundle afford to the consumer given his income y, B = {x x R n +, p x y}. B is also called budget set. In the two-good case, B consists of all bundles lying inside or on the boundaries of the shared region in Figure 1.9. Figure 1.9: Budget set, B = {x x R n +, p x y}, in the case of two commodities. 27 / 28

28 By Assumption 1.2 and the assumptions on the feasible set, total expenditure must not exceed income. We can recast the consumer s problem in (Pr. 1) as max x R n + u(x) s.t. p x y. (Pr.2) Note that if x solves this problem, then u(x ) u(x) for all x B, which means that x x for all x B. That is, solution to (Pr.2) are indeed solutions to (Pr.1). The converse is also true. 28 / 28