Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2

Transcription

1 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 Prof. Dr. Oliver Gürtler Winter Term 2012/ Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

2 1. Introduction Microeconomic theory covers the analysis of individual economic behavior (e.g., a consumer or rm) and the aggregation of individuals actions in an institutional framework (e.g., a price mechanism in an impersonal market place). Doing this, we intend to get a better understanding of economic activity and outcomes. This is useful in two distinct senses: positive sense: we obtain a better understanding of individual behavior in certain situations. normative sense: we understand when to intervene, both at the government level and at the institutional level. However, the models we will analyze are highly simpli ed and sometimes too simple to be realistic. Still, they have some general predictive power and represent the building blocks of more complex and realistic testable models. 2 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

3 Structure of the course: 2. Consumer Theory 3. Theory of the Firm 4. Partial Equilibrium 5. General Equilibrium 6. Social Choice and Welfare Literature: Jehle, Reny (2011): Advanced Microeconomic Theory, Pearson Education. MasColell, Whinston, Green (1995): Microeconomic Theory, Oxford University Press. 3 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

4 Organizational matters: The course consists of a lecture (10am) and an integrated exercise course (2pm; starting in the fth week). Slides and exercise sets are available at ILIAS. Students are expected to prepare the exercises and to present their solutions. Some mathematical concepts are needed, most of which should be known to everyone. The concepts that are probably not known to everyone are brie y addressed when they are encountered for the rst time. The nal exam takes place on Jan 31, 2013 at 10 am. 4 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

5 2. Consumer Theory In consumer theory, we focus on an individual s (consumer s) decision to consume a set of commodities (goods and services). There are four building blocks in any such model: consumption set feasible set preference relation behavioral assumption By specifying each of these in a given problem, many di erent situations involving choice can be formally analyzed. 5 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

6 We focus on n commodities i = 1; :::; n. De nition 1 The consumption set X represents the set of all consumption bundles that the consumer can conceive. De nition 2 A consumption bundle (or consumption plan) x = (x 1 ; :::; x n ) is a vector specifying the amounts of each of the di erent commodities. Note: Time and location are included in the de nition of a commodity. 6 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

7 Assumption 1 The consumption set X satis es: 1. X R n + (nonnegativity) 2. X is closed, it includes its own boundary. 3. X is convex: if x 2 X and y 2 X then z = x + (1 ) y 2 X (for every 2 [0; 1]) X: De nition 3 The feasible set B X represents those consumption plans that the consumer can conceive and a ord. The behavioral assumption speci es the ultimate objectives in choice (typically utility maximization). 7 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

8 2.1 Preferences and Utility Each consumer is endowed with a binary relation, %, de ned on the consumption set X. The expression x % y means that x is at least as good as y for the consumer. This relation gives us information about the consumer s tastes for the di erent objects of choice. We impose several axioms that the relation should ful ll. These axioms formalize the view that the consumer can choose and that choices are consistent in a particular way. Moreover, they serve to characterize the consumer s tastes over the consumption bundles. 8 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

9 Axiom 1 Completeness. For all x; y 2 X, either x % y or y % x (or both).! Consumer can always compare two consumption bundles and decide which is (weakly) preferred. Axiom 2 Transitivity. For any three elements x; y; z 2 X, if x % y and y % z, then x % z.! Axiom links pairwise comparisons in a consistent way.! Consumer can rank any nite number of elements in X from best to worst (ties are possible). De nition 4 The binary relation % on the consumption set X is called a preference relation if it satis es Axioms 1 and 2. 9 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

10 De nition 5 The binary relation on the consumption set X is de ned as follows: x y if and only if x % y and not y % x. The relation is called the strict preference relation induced by %. De nition 6 The binary relation on the consumption set X is de ned as follows: x y if and only if x % y and y % x. The relation is called the indi erence relation induced by %. Note: and do not necessarily have the same properties as %. For instance, while both are transitive, neither is complete. 10 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

11 De nition 7 Let x 0 be any point in X. Relative to this point, we can de ne the following subsets of X: 1. % x 0 = x x 2 X; x % x 0, called the "at least as good as" set x 0 = x x 2 X; x 0 % x, called the "no better than" set. 3. x 0 = x x 2 X; x 0 x, called the "worse than" set. 4. x 0 = x x 2 X; x x 0, called the "preferred to" set. 5. x 0 = x x 2 X; x x 0, called the "indi erence" set. Note: For any bundle x 0, the three sets x 0, x 0 and x 0 partition X. 11 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

12 We introduce some additional axioms to put more structure on preferences (from now on X = R n +). Axiom 3 Continuity. For all x 2 R n +, the sets % (x) and - (x) are closed in R n +. Technical assumption ensuring that there are no sudden preference reversals.! Rules out "open" area in the indi erence set. Axiom 4 Strict Monotonicity. For all x; y 2 R n +, if x y, then x % y, while if x y, then x y. Consumer always prefers a consumption bundle involving more to one involving less.! Rules out the possibility of having "zones of indi erence".! Ensures that indi erence sets have negative slope. 12 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

13 Axiom 5 Strict Convexity. If x 6= y and x % y, tx+(1 t) y y for all t 2 (0; 1). The consumer prefers "balanced" bundles to more extreme ones.! The (absolute value of the) slope of the indi erence curve (the marginal rate of substitution) is decreasing. (Weak) convexity is de ned analogously. 13 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

14 De nition 8 A real-valued function u : R n +! R is called a utility function representing the preference relation % if for all x; y 2 R n +, u (x) u (y), x % y. Utility function represents consumer s preference relation if it assigns higher numbers to preferred bundles. Utility function is a device for summarizing the information contained in the preference relation. Is it always possible to represent the preference relation by a continuous realvalued function? Theorem 1 If the binary relation % is complete, transitive, continuous and strictly monotonic, there exists a continuous real-valued function, u : R n +! R, which represents %. 14 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

15 Proof. Idea: For given preferences, nd one function representing these preferences! Let e = (1; :::; 1) 2 R n + and consider the mapping u : R n +! R de ned by u (x) e x In words: assign to any bundle x the number u (x) such that the consumer is indi erent between x and a bundle with u (x) units of every commodity. Does such number always exist? If so, is it uniquely determined? 15 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

16 To tackle the rst question, de ne A ft 0 jte % xg B ft 0 jte - xg We have to show that A \ B is always nonempty. By continuity of %, both A and B are closed in R +. By strict monotonicity, t 2 A ) t 0 2 A for all t 0 t.! A is a closed interval of the form ^t; 1. Similarly, B is a closed interval of the form 0; ~t. 16 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

17 By completeness, te - x or te % x; i.e., t 2 A [ B.! A [ B = R +. Since A [ B = 0; ~t [ ^t; 1, it must be that ^t ~t.! A \ B is nonempty. To tackle the second question, suppose there were two numbers t 1 and t 2 satisfying t 1 e x and t 2 e x. By transitivity, we have t 1 e t 2 e. By strict monotonicity, t 1 = t Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

18 It remains to show that the function we have constructed represents % and is continuous. Consider two bundles x and y and their associated numbers u (x) and u (y). Let x % y. By the de nition of u, u (x) e x and u (y) e y. By transitivity, u (x) e u (y) e. By strict monotonicity, u (x) u (y).! The function we have constructed represents % : To show that the function is continuous, we show that the inverse image under u of every open ball in R is open in R n + (by Theorem A1.6 in the mathematical appendix of Jehle/Reny). 18 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

19 Since open balls in R are open intervals, we must show that u 1 ((a; b)) is open in R n + (for a < b). By de nition, the inverse image u 1 ((a; b)) is x 2 R n + ja < u (x) < b, or x 2 R n + jae u (x) e be (by strict monotonicity), or x 2 R n + jae x be (by u (x) e x), or (ae) \ (be) By continuity of, the sets % (be) and - (ae) are closed in R n +.! (ae) and (be), being the complements of closed sets, are open. u 1 ((a; b)) is open. Q.E.D. 19 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

20 According to the theorem, we can describe the consumer s preferences in two ways, either by the preference relation or a continuous utility function. The latter approach is often more convenient. The preference relation ranks consumption bundles.! There is more than one utility function representing a particular ranking. Theorem 2 Let be a preference relation on R n + and suppose u (x) is a 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. Utility function is invariant to positive monotonic transforms. Intuition: positive monotonic transform does not change the ranking of bundles. 20 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

21 Theorem 3 Let be represented by u : R n +! R. Then 1. u (x) is strictly increasing if and only if is strictly monotonic. 2. u (x) is quasiconcave if and only if is convex. 3. u (x) is strictly quasiconcave if and only if is strictly convex. The proof follows easily from the de nitions involved. Consider part 2, for example. By de nition a function u () is quasiconcave if and only if the set y 2 R n + ju (y) k is convex for every k 2 R. If x is chosen such that u (x) = k, the de nition of quasiconcavity of u () coincides with the de nition of convexity of %. In the following we additionally assume u () to be di erentiable. 21 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

22 De nition 9 The rst-order partial derivative of u (x) with respect to x i is called the marginal utility of good i. Assume n = 2. Let x 2 = f (x 1 ) be the function describing an indi erence curve.! u (x 1 ; f (x 1 )) = const: Di erentiating with respect to x 1 f 0 (x 1 ) = 2, MRS = f 0 (x 1 @x 2 The marginal rate of substitution equals the ratio of the marginal utilities of the two goods. As indicated before, convexity of % may be interpreted as diminishing M RS. 22 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

23 2.2 The Consumer s Problem Formally, the consumer seeks x 2 B such that x % x for all x 2 B Consumer is assumed to operate within a market economy. His impact on the market is negligible.! The vector of prices p = (p 1 ; :::; p n ) 0 is xed from the consumer s point of view. The feasible set (or budget set) B is de ned by B = x 2 R n + jpx m where m 0 denotes the consumer s xed money income. 23 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

24 The consumer s maximization problem can be stated equivalently as max u (x) x2r n + s.t. px m Under the assumptions made u (x) is continuous and real-valued, while B is nonempty and compact (closed and bounded).! By the Weierstrass theorem, a solution to the maximization problem always exists. Since B is convex and u (x) strictly quasiconcave, the solution is unique. The solution vector x depends on p and m: The optimal quantities, viewed as functions of p and m, are known as ordinal or Marshallian demand functions. We write as x (p; m) the vector of these quantities. 24 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

25 Derivation of the solution: To solve this nonlinear programming problem with inequality constraint, we form the Lagrangian L = u (x) + (m px) We assume x 0. Then the solution satis es the following optimality (x ) p i = 0, i = 1; :::; i m px 0 (m px ) = Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

26 Suppose = 0. i! > 0 and, hence, m px = 0. = 0; 8i, contradicting strict monotonicity. Combining the conditions for di erent commodities i 6= j j = p i p j! At the optimum, the marginal rate of substitution between any two goods must be equal to the ratio of the goods prices. But: the conditions we stated are merely necessary. What about su cient conditions? Under the assumptions imposed, the conditions are also su cient for optimality. 26 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

27 Proof: Because u () is quasiconcave, we have ru (x) (y u (x) and x; y 0. x) 0 whenever u (y) The optimality conditions can be restated as ru (x ) = p px = m If x is not utility-maximizing, there must be some z 0 such that u (z) > u (x ) pz m 27 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

28 Because u () is continuous and m > 0, there exists some t 2 [0; 1] close enough to one such that u (tz) > u (x ) ptz < m Setting y = tz, it follows ru (x ) (y x ) = p (y x ) = (py px ) < 0 This condition contradicts the condition from the beginning of the proof. Q.E.D. 28 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

29 2.3 Indirect Utility and Expenditure The Indirect Utility Function The ordinary utility function u (x) represents the consumer s preferences directly.! It is called the direct utility function. De nition 10 The function obtained by substituting the Marshallian demands in the consumer s utility function is the indirect utility function: v(p; m) = u(x(p; m)) Alternatively: v(p; m) = max u (x) s:t: px m x2r n + 29 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

30 Theorem 4 If u (x) is continuous and strictly increasing on R n +, then v (p; m) is 1. continuous on R n ++ R + ; 2. homogeneous of degree zero in (p; m) ; 3. strictly increasing in m; 4. decreasing in p, 5. quasiconvex in (p; m). Moreover, it satis es Roy s identity: If v (p; m) is di erentiable 0 ;m 0 6= 0, then p 0 ; m 0 and x i p 0 ; m 0 0 ;m 0 i, i = 1; :::; 0 ;m 0 30 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

31 Proof of parts 2, 3, 5, 6: To prove 2, we must show that v (p; m) = v (tp; tm) for all t > 0. By de nition, v(tp; tm) = max u (x) s:t: tpx tm x2r n + = max u (x) s:t: px m = v (p; m) x2r n + To prove 3, note (x (p; m)) = = i Using the i 1, we obtain = p i and px (p; m) = m ) P i (p; = > 0 31 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

32 The proof of part 5 is more elaborate. Let us consider the following three sets B 1 = x p 1 x m 1 B 2 = x p 2 x m 2 B t = x p t x m t with p t tp 1 + (1 t) p 2 and m t tm 1 + (1 t) m 2, t 2 [0; 1]. v (p; m) is quasiconvex in (p; m) if and only if v p t ; m t max v p 1 ; m 1 ; v p 2 ; m 2 ; 8t 2 [0; 1] It su ces to show that every choice a consumer can make when facing B t could also be made when facing either B 1 or B Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

33 The latter condition holds trivially for t = 0 or t = 1. t 2 (0; 1). So, let us assume The proof is by contradiction. Therefore, assume that there exists some t 2 (0; 1) and x 2 B t such that x =2 B 1 and x =2 B 2. Then p 1 x > m 1 and p 2 x > m 2 Multiplying the rst inequality by t and the second by (1 t) yields tp 1 x > tm 1 and (1 t) p 2 x > (1 t) m 2 Adding the two conditions yields the desired contradiction x =2 B t :! If x 2 B t, then x 2 B 1 or x 2 B 2 ; 8t 2 [0; 1]. 33 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

34 To prove the last part, note (p; i (x (p; i = i Using the j = p j and px (p; m) = m ) x i + Pj i = 0 (p; m) = x i Roy s identity is obtained by solving the condition for x i for. Q.E.D. 34 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

35 2.3.2 The Expenditure Function The expenditure function is derived from a di erent kind of optimization problem. We ask: What is the minimum level of money expenditure the consumer must make to achieve a given level of utility? Formally, the expenditure function is de ned as e (p; u) = min px s:t: u (x) u x2r n + Let U = u (x) x 2 R n + denote the set of attainable utility levels.! The domain of e () is R n ++ U. 35 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

36 The solution to the minimization problem is denoted by x h (p; u).! e (p; u) = px h (p; u) Idea: We imagine a process where the consumer s income is changed to compensate him for changes in prices such that he can achieve exactly the same utility level as before. Still, the optimal quantities of the goods will typically change.! The optimal quantities, viewed as functions of p and u, are known as compensated or Hicksian demand functions. x h (p; u) is the vector of these demands. 36 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

37 Theorem 5 If u (x) is continuous and strictly increasing on R n +, then e (p; u) is 1. zero when u takes on the lowest level of utility in U, 2. continuous on its domain R n ++ U; 3. for all p 0; strictly increasing and unbounded above in u; 4. strictly increasing in p; 5. homogeneous of degree one in p, 6. concave in p. Moreover, if u (x) is strictly quasiconcave, we have Shephard s lemma: e (p; u) is di erentiable in p at p 0 ; u 0 with p 0 0 p 0 ; u i = x h i p 0 ; u 0, i = 1; :::; n: 37 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

38 Proof of parts 1, 4, 6, 7: To prove 1, note that the lowest value in U is u (0) (since u (x) is strictly increasing). x = 0 requires an expenditure of 0, hence e (p; u (0)) = 0. Part 4 follows from the proof of part 7. To prove 6, note that the expenditure function will be concave in prices if te p 1 ; u + (1 t) e p 2 ; u e tp 1 + (1 t) p 2 ; u for any two price vectors p 1 and p 2 and t 2 [0; 1]. 38 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

39 Denote the bundles minimizing expenditure to achieve u in the three situations by x 1, x 2 and x t. By de nition, p 1 x 1 p 1 x p 2 x 2 p 2 x for any other bundle x achieving utility u: In particular p 1 x 1 p 1 x t p 2 x 2 p 2 x t 39 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

40 Multiplying the rst inequality by t and the second by (1 t) yields tp 1 x 1 tp 1 x t and (1 t) p 2 x 2 (1 t) p 2 x t Adding the two conditions, we obtain tp 1 x 1 + (1 t) p 2 x 2 tp 1 x t + (1 t) p 2 x t, tp 1 x 1 + (1 t) p 2 x 2 tp 1 + (1 t) p 2 x t This is the same as te p 1 ; u + (1 t) e p 2 ; u e tp 1 + (1 t) p 2 ; u 40 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

41 To prove 7, we di erentiate e (p; u) with respect to p (p; u) pxh (p; u) = x h i + h j p j i From u (x) = u, we have h i = 0. Deriving the optimality conditions and combining j Together, these conditions imply P j h j i 7. = p i. = 0 completing the proof of part Since x h i 0, part 4 follows immediately. Q.E.D. 41 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

42 2.3.3 Relations Between The Two There are close relations between the indirect utility function and the expenditure function. From the de nitions of e and v, we know e (p; v (p; m)) m; 8 (p; m) 0; v (p; e (p; u)) u; 8 (p; u) 2 R n ++ U: One can further show that both of these inequalities, in fact, must be equalities. This means that knowing the indirect utility function enables us to calculate the expenditure function (and vice versa). Accordingly, we do not have to solve both optimization problems to derive the indirect utility function and the expenditure function. 42 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

43 There are also close relations between the Marshallian and Hicksian demand functions. Theorem 6 We have the following relations between the Hicksian and Marshallian demand functions for p 0; m 0, u 2 U and i = 1; :::; n : 1. x i (p; m) = x h i (p; v (p; m)) 2. x h i (p; u) = x i (p; e (p; u)) If x solves the utility-maximization problem at (p; m), it also solves the expenditure-minimization problem at (p; u), where u = u (x ). Conversely, if x solves the expenditure-minimization problem at (p; u), it also solves the utility-maximization problem at (p; m), where m = px. In this sense x has a dual nature. 43 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

44 2.4 Properties Of Consumer Demand The theory developed so far leads to a number of predictions about behavior in the marketplace. In the following, we present some of these predictions. Theorem 7 The consumer demand function x i (p; m) ; i = 1; :::; n; is homogeneous of degree zero in all prices and income, and it satis es budget balancedness, px (p; m) = m for all (p; m). Proof: We have shown before that v (p; m) is homogeneous of degree zero in (p; m), i.e., v (p; m) = v (tp; tm) ; 8t > 0, or u (x (p; m)) = u (x (tp; tm)) ; 8t > Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

45 Since budget sets at (p; m) and (tp; tm) are the same, x (p; m) was feasible when x (tp; tm) was chosen (and vice versa). From the last equality on the previous slide and strict quasiconcavity of u it then follows that x (p; m) = x (tp; tm). This condition says that x i (p; m) ; i = 1; :::; n; is homogeneous of degree zero in prices and income. The property of budget balancedness has been proven before. Q.E.D. Homogeneity implies that we could focus on relative prices and real income. For instance, if good n serves as numéraire, we have p1 x (p; m) = x (tp; tm) = x ; :::; p n 1 ; 1; m p n p n p n 45 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

46 An important consideration is the response in quantity demanded when prices change. Typically, we expect a consumer to buy more of a good when its price declines, but this is not always true. To get an understanding of the relevant e ects, we consider the Slutsky equation. Theorem 8 Let x (p; m) be the consumer s Marshallian demand system. Let u be the level of utility the consumer achieves at prices p and income m. i (p; m) i (p; u ) x j (p; i (p; m) ; i; j = 1; :::; The substitution e ect is the rst, the income e ect the second term on the RHS. 46 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

47 Proof: Start from the identity x h i (p; u ) = x i (p; e (p; u )) Di erentiating both sides with respect to p j, h i (p; u j i (p; e (p; u )) i (p; e (p; u j (p; u j From Shephard s lemma, we j = x h j (p; u ) = x j (p; e (p; u )). Moreover, we know u = v (p; m) and m = e (p; v (p; m)) = e (p; u ). 47 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

48 Using these conditions, we obtain Q.E.D. h i (p; u j i (p; j i (p; x j (p; m) i (p; m) i (p; u ) x j (p; i (p; Substitution e ect: (hypothetical) change in consumption that would occur after prices were set to the new level but where income was adapted such the maximum level of utility the consumer can achieve were kept the same as before the price change (Hicks). Income e ect: change in consumption if prices were kept at the new level but income was set back to its real value. 48 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

49 The decomposition of the total e ect of a price change into the substitution and income e ect is useful since we are able characterize the single e ects in more detail. Theorem 9 Let x h i (p; u) be the Hicksian demand for good i. i i 0, i = 1; :::; n: Proof: In words, own-substitution terms are negative. From Shephard s lemma, we have x h i Di erentiating with respect to p i, h i (p; i. (p; u) e (p; 2 i We have shown before that e (p; u) is concave in p, 2 i Q.E.D Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

50 De nition 11 A good is called normal if consumption of it increases as income increases, holding prices constant. De nition 12 A good is called inferior if consumption of it declines as income increases, holding prices constant. From the two theorems derived before and the de nitions, it is easy to derive the so-called "law of demand". Theorem 10 A decrease in the own price of a normal good will cause quantity demanded to increase. If an own price decrease causes a decrease in quantity demanded, the good must be inferior. The law of demand connects concepts about the reactions of quantity demanded to income changes with concepts about the reactions of quantity demanded to price changes. 50 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

51 In the remainder of this section, we derive some useful elasticity relations. These relations can be derived from the budget balancedness constraint, px (p; m) = m. De nition 13 Let x i (p; m) be the consumer s Marshallian demand for good i. Denote as j 3. s i = pixi(p;m) m good i. m x i(p;m) p j x i(p;m) the income elasticity of demand for good i; the price elasticity of demand for good i and 0 (with P n i=1 s i = 1) the income share spent on purchases of 51 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

52 Theorem 11 Let x (p; m) be the consumer s system of Marshallian demands. Then the following relations must hold among income shares, price, and income elasticities of demand: 1. Engel aggregation: P n i=1 s i i = 1: 2. Cournot aggregation: P n i=1 s i ij = s j ; j = 1; :::; n: Proof: Di erentiate both sides of the budget constraint px (p; m) = m with respect to income to obtain i p = 1 The LHS can be rewritten as nx i=1 i=1 nx i=1 p i x i m p i x i i x i i m x i nx s i i i=1 52 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

53 Di erentiating both sides of the budget constraint px (p; m) = m with respect to p j, yields i x j + p i = j This condition is equivalent to, nx i=1 nx i=1 p i p j m p i x i m j = p j x j = p jx j m p jx j m Using the respective de nitions, gives us the second condition of the theorem. Q.E.D. Because of budget balancedness, all consumer demand responses to price and income changes must add up in a way that preserves the budget constraint. 53 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

54 2.5 Uncertainty Until now, we have assumed that consumers act in a world of perfect certainty. In what follows, we relax this assumption and consider uncertain situations.! Consumers are assumed to have a preference relation over gambles (instead of consumption bundles). Let A = fa 1 ; :::; a n g denote a nite set of outcomes. The a i s involve no uncertainty, but it is not clear which of these outcomes is realized. Let p i denote the probability that outcome a i is realized. 54 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

55 De nition 14 Let A = fa 1 ; :::; a n g be the set of outcomes. Then G S, the set of simple gambles (on A), is given by ( ) nx G S (p 1 a 1 ; :::; p n a n ) p i 0; p i = 1 : Note: When one or more of the p i s is zero, we can drop those components from the expression. i=1 It is sometimes the case that gambles have prizes that are themselves gambles. Such gambles are called compound gambles. Example: In some lotteries, one can win monetary prizes, but also tickets for the lottery. For simplicity, we rule out in nitely layered compound gambles. 55 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

56 Let G denote the set of all (simple and compound) gambles. A gamble g 2 G can then be written as g = p 1 g 1 ; :::; p k g k, k 1, where g i might be a compound gamble, a simple gamble or an outcome. As indicated before, the consumer (or decision maker) is assumed to have preferences, %, over G. We impose certain axioms the preference relation is assumed to ful ll, called axioms of choice under uncertainty. Axiom 6 Completeness. For any two gambles g; g 0 2 G, either g % g 0 or g 0 % g (or both). Axiom 7 Transitivity. For any three gambles g; g 0 ; g 00 2 G, if g % g 0 and g 0 % g 00, then g % g Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

57 Note that each a i 2 A is represented in G as a degenerate gamble.! The two axioms imply that the elements in A can be ordered by %. Assume that these elements have been indexed such that a 1 % a 2 % ::: % a n. Axiom 8 Continuity. For any gamble g 2 G, there is some probability 2 [0; 1] such that g ( a 1 ; (1 ) a n ). Axiom 9 Monotonicity. For all probabilities ; 2 [0; 1], ( a 1 ; (1 ) a n ) % ( a 1 ; (1 ) a n ) if and only if. Monotonicity implies a 1 a n.! Decision maker is not indi erent between all outcomes in A. 57 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

58 Axiom 10 Substitution. If g = p 1 g 1 ; :::; p k g k and h = p 1 h 1 ; :::; p k h k are in G, and if g i h i for every i, then g h. If decision maker is indi erent between the realizations of one gamble and another, and the realizations occur with the same probabilities, then he is indi erent between the two gambles. In a compound gamble, one can calculate the e ective probabilities of the single outcomes in A: For any gamble g 2 G, if p i denotes the e ective probability assigned to a i by g, then we say that g induces the simple gamble (p 1 a 1 ; :::; p n a n ) 2 G S. The nal axiom states that the decision maker cares only about the e ective probabilities a gamble assigns to the single outcomes in A. Axiom 11 Reduction to Simple Gambles. For any gamble g 2 G, if (p 1 a 1 ; :::; p n a n ) is the simple gamble induced by g, then (p 1 a 1 ; :::; p n a n ) g. By this axiom and transitivity, a decision maker s preferences over all gambles are completely determined by the preferences over simple gambles. 58 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

59 We can show that preferences, obeying the above axioms, can be represented with a continuous, real-valued function. Let u : G! R describe a utility function representing % on G.! For every g 2 G, u (g) denotes the utility number assigned to g. Similarly, u (a i ) denotes the utility number assigned to the degenerate gamble (1 a i ) (called the utility of outcome a i ). De nition 15 The utility function u : G! R has the expected utility property if, for every g 2 G, nx u (g) = p i u (a i ) ; i=1 where (p 1 a 1 ; :::; p n a n ) is the simple gamble induced by g.! u assigns to each gamble the expected value of utilities that might result! 59 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

60 A utility function possessing the expected utility property is referred to as von Neumann-Morgenstern (VNM) utility function. Theorem 12 Let preferences % over gambles in G satisfy the above axioms. Then there exists a utility function u : G! R representing % on G, such that u has the expected utility property. Proof: Implication: Under the axioms imposed, one can assign utility numbers to the outcomes in A such that the decision maker prefers one gamble over another if and only if it has a higher expected utility. We construct a function and show that it possesses the desired properties. Consider an arbitrary gamble g 2 G and let u (g) be the number satisfying g (u (g) a 1 ; (1 u (g)) a n ) 60 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

61 By continuity, such number exists and, by monotonicity, it is unique.! We have derived a real-valued function u on G. To show that u represents %, consider two arbitrary gambles g; g 0 2 G and suppose g % g 0. By transitivity and the de nition of u, this is equivalent to (u (g) a 1 ; (1 u (g)) a n ) % (u (g 0 ) a 1 ; (1 u (g 0 )) a n ) By monotonicity, this is equivalent to u (g) u (g 0 ). 61 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

62 It remains to show that u has the expected utility property. To do this, let g 2 G be an arbitrary gamble and g s (p 1 a 1 ; :::; p n a n ) 2 G S the simple gamble it induces. By reduction to simple gambles, we have g g s and, hence, u (g) = u (g s ). We therefore must show that u (g s ) = P n i=1 p iu (a i ). By de nition, u (a i ) satis es a i (u (a i ) a 1 ; (1 u (a i )) a n ) q i By substitution, g s (p 1 a 1 ; :::; p n a n ) p 1 q 1 ; :::; p n q n g 0. The simple gamble induced by g 0 is! nx gs 0 = p i u (a i ) a 1 ; 1 i=1!! nx p i u (a i ) a n i=1 62 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

63 By reduction to simple gambles and transitivity, g s g 0 gs, 0 or!!! nx nx g s p i u (a i ) a 1 ; 1 p i u (a i ) a n i=1 i=1 By de nition, u (g s ) is the unique number satisfying g s (u (g s ) a 1 ; (1 u (g s )) a n ) Comparing the two expressions, we obtain u (g s ) = Q.E.D. nx p i u (a i ) i=1 63 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

64 VNM utility functions are not unique. However, because of the expected utility property not every positive monotonic transformation is possible. Theorem 13 Suppose the VNM utility function u () represents %. Then the VNM utility function v () represents those same preferences if and only if for some scalar and some scalar > 0 v (g) = + u (g) ; for all gambles g. Proof: In words: VNM utility functions are unique up to positive a ne transformations. For simplicity, we suppose g (p 1 a 1 ; :::; p n a n ) is a simple gamble and prove only necessity. 64 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

65 Since u () represents %, we have u (a 1 ) u (a 2 ) ::: u (a n ) and u (a 1 ) > u (a n ) : Hence, for any i there is a unique i 2 [0; 1] such that u (a i ) = i u (a 1 ) + (1 i ) u (a n ) : Because of the expected utility property, it follows that u (a i ) = u ( i a 1 ; (1 i ) a n ) ; or a i ( i a 1 ; (1 i ) a n ) : If v () represents % as well, we have v (a i ) = v ( i a 1 ; (1 i ) a n ) : If v () has the expected utility property, this condition can be transformed into v (a i ) = i v (a 1 ) + (1 i ) v (a n ) : 65 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

66 Combining the expressions for u (a i ) and v (a i ), yields u (a 1 ) u (a i ) u (a i ) u (a n ) = 1 i i = v (a 1) v (a i ) v (a i ) v (a n ) for every i such that a i a n and, thus, i > 0. Rearranging, we obtain (u (a 1 ) u (a i )) (v (a i ) v (a n )) = (v (a 1 ) v (a i )) (u (a i ) u (a n )), v (a i ) = + u (a i ) ; with u (a 1) v (a n ) u (a n ) v (a 1 ) ; v (a 1) v (a n ) u (a 1 ) u (a n ) u (a 1 ) u (a n ) > Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

67 Using this condition, it follows that nx nx v (g) = p i v (a i ) = p i ( + u (a i )) i=1 = + Q.E.D. i=1 nx p i u (a i ) = + u (g) : i=1 67 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

68 VNM utility functions are not completely unique, nor are they entirely ordinal. Still, we should not attach undue signi cance to the absolute level of a gamble s utility. It is, for instance, not possible to use VNM utility functions for interpersonal comparisons of well-being. In the remainder of this section, we analyze the relationship between a VNM utility function and the decision maker s attitude toward risk. We con ne attention to gambles whose outcomes consist of di erent amounts of wealth w. We assume A = R + and, hence, wealth levels to be nonnegative. Moreover, we consider only gambles giving nitely many outcomes a strictly positive e ective probability. Finally, we assume u () to be strictly increasing and twice di erentiable for all wealth levels. 68 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

69 De nition 16 Let u () be a decision maker s VNM utility function for gambles over nonnegative levels of wealth. Then for the simple gamble g = (p 1 w 1 ; :::; p n w n ) ; the decision maker is said to be 1. risk averse at g if u (E (g)) > u (g) ; 2. risk neutral at g if u (E (g)) = u (g) ; 3. risk loving at g if u (E (g)) < u (g) : If for every nondegenerate simple gamble, g, the decision maker is, for example, risk averse at g, then he is said simply to be risk averse. Similarly, a decision maker can be de ned to be risk neutral and risk loving. Note that! nx u (E (g)) = u p i w i i=1 nx u (g) = p i u (w i ) i=1 and Intuition: The gamble entails risk, while both options lead to the same expected value. 69 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

70 A decision maker is risk averse (risk neutral, risk loving) if and only if his VNM utility function is strictly concave (linear, strictly convex) over the appropriate domain of wealth. Illustration for two wealth levels w 1 and w 2 : With two wealth levels, we have u (E (g)) = u (p 1 w 1 + (1 p 1 ) w 2 ) and u (g) = p 1 u (w 1 ) + (1 p 1 ) u (w 2 ) Consider a graph, where w is measured on the abscissa and u () on the ordinate. If we connect the points (w 1 ; u (w 1 )) and (w 2 ; u (w 2 )), we obtain a line segment. u (g) is the ordinate of a particular point on this line segment. If the decision maker is risk averse, we have u (E (g)) > u (g).! The utility function must proceed above the line segment and, thus, be strictly concave. 70 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

71 De nition 17 The certainty equivalent of any simple gamble g over wealth levels is an amount of wealth, CE, o ered with certainty, such that u (g) u (CE). The risk premium is an amount of wealth, P, such that u (g) u (E (g) P ). Clearly, P E (g) CE. When a person is risk averse and prefers more money to less, we have CE < E (g) and, thus, P > 0.! A risk averse decision maker will "pay" some positive amount of wealth to avoid the gamble s inherent risk. Similarly, we have CE = E (g) and CE > E (g) for risk neutral and risk loving decision makers. Often, we do not only want to know whether a decision maker is risk averse, but also how risk averse he is. De nition 18 The Arrow-Pratt measure of absolute risk aversion is given by R a (w) = u00 (w) u 0 (w). 71 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

72 R a (w) is positive, zero or negative as the decision maker is risk averse, risk neutral or risk loving. R a (w) is una ected by any positive a ne transformation of the utility function (u 00 (w) is not). Decision makers with a larger Arrow-Pratt measure have a lower certainty equivalent and are willing to accept fewer gambles. Proof of the last statement: Consider two decision makers and denote their VNM utility functions by u (w) and v (w), with u 0 (w) ; v 0 (w) > 0. Let R 1 a (w) = u00 (w) u 0 (w) > v00 (w) v 0 (w) = Ra 2 (w), for all w 0. Assume that v (w) takes on all values in [0; 1) and write h (x) = u v 1 (x), for all x 0. Di erentiating h twice with respect to x, one can show that h 0 (x) > 0 and h 00 (x) < Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

73 Consider a gamble (p 1 w 1 ; :::; p n w n ) over wealth levels and denote by CE 1 and CE 2 the two decision makers certainty equivalents for the gamble. By de nition, nx p i u (w i ) = u (CE 1 ) ; i=1 nx p i v (w i ) = v (CE 2 ) : i=1 Substituting v (w) by x and using h (x) = u v 1 (x), we obtain nx u (CE 1 ) = p i h (v (w i )) i=1 By Jensen s inequality (because h is strictly concave), this expression is strictly smaller than! nx h p i v (w i ) = h (v (CE 2 )) = u (CE 2 ) i=1 73 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

74 Because u () is strictly increasing, it follows that CE 1 < CE 2. If both decision makers have the same initial wealth, this nding implies that 2 will accept any gamble that 1 will accept. Q.E.D. Note that h (x) = u v 1 (x) implies u (w) = h (v (w)).! u () is more concave than v (), because it is a concave function of v (). Arrow-Pratt measure of absolute risk aversion typically varies with wealth. A VNM utility function is said to display constant, decreasing or increasing absolute risk aversion over some domain of wealth if, over that interval, R a (w) remains constant, decreases or increases with an increase in wealth. Decreasing absolute risk aversion (DARA) is often a sensible assumption.! The greater wealth, the less averse one becomes to accepting the same gamble. 74 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J