Demand Theory. Lecture VI and VII Utility Maximization, Choice, Demand (Varian Ch. 7, 8, 9, 10) Federico Trionfetti

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1 Demand Theory Lecture VI and VII Utility Maximization, Choice, Demand (Varian Ch. 7, 8, 9, 10) Federico Trionfetti Aix-Marseille Université Faculté d Economie et Gestion Aix-Marseille School of Economics October 18, 2017

2 Table of Content 1 The reserch question 2 Preferences 3 Consumer behaviour 4 Pirouettes 5 Money metric utility functions 6 Choice 7 Demand 8 Compnsating and equivlent variation ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 18, School 2017of Economics 2 / 30

3 The reserch question The research question The objective is to build a theory of demand, that is, a relationship between tastes (or preferences), prices, and income so as to explain consumers choices. ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 18, School 2017of Economics 3 / 30

4 Preferences Preferences Preferences are defined on a consumption set X R k`. We assume preferences to be rational: Rationality 1 Completeness. For all x and y in X, either x ľ y or x ĺ y or x y. 2 Transitivity. For all x, y, and z in X, if x ľ y and y ľ z, then x ľ z. Proposition 1 If preferences are rational then they can be expressed by a function u : R k` Ñ R. Proof. Skipped Henceforth we assume rationality. ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 18, School 2017of Economics 4 / 30

5 Preferences Given the existence of a utility function we can define two useful concepts: Definition 1 Marginal Utility. The m.u. of a good is the change in utility obtained from an infinitesimal increase in the consumption of that good. Bupxq Bx i (1) Marginal rate of substitution: the rate at which one good must substitute for the other in order to give the same level of utility: dx j dx i Bupxq Bx i Bupxq Bx j (2) ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 18, School 2017of Economics 5 / 30

6 Consumer behaviour Consumer behaviour Consumers are utility maximizers. max x upxq s.t. px m. (3) The solution to this problem is a function vpp, mq known as the indirect utility function. ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 18, School 2017of Economics 6 / 30

7 Consumer behaviour The Lagrangian of the problem above is: Lpλ, xq upxq λppx mq, (4) The F.O.C. characterizing an interior solutions (for smooth utility function) are: BLpλ, xq Bx i BLpλ, xq Bλ 0 ùñ Bupxq Bx i λp (5) 0 ùñ px m. (6) The solution of this system is a vector x and a scalar λ. The vector x is known as the vector of demanded goods because each element is the quantity of a good demanded. The functions x i pp, mq are the demand functions. Sometimes they are called the Marshallian or market demand functions. ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 18, School 2017of Economics 7 / 30

8 Consumer behaviour The F.O.C for utility maximization have a straightforward interpretation. Taking the ratio of any two BLp.q{Bx i we obtain p i p j lomon Economic R.S. Bupxq{Bx i Bupxq{Bx j looooomooooon MRS (7) Note that the l.h.s. is equal to the ratio Bpx{Bx i Bpx{Bx j called the Economic Rate of Substitution. and, as such, it may be Reflection: what is the margin of improvement if (7) is not satisfied? ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 18, School 2017of Economics 8 / 30

9 Consumer behaviour Properties of indirect utility function Plugging x pp, mq into upxq gives a function known as the indirect utility function, denoted vpp, mq. The indirect utility function has the following properties: Non-increasing in p. Homogeneous of degree 0 in (p, mq. Convex in p. Continuous in p. ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 18, School 2017of Economics 9 / 30

10 Consumer behaviour Expenditure minimization We can pose the consumer problem as that of minimizing the cost of acheiving a given level of utility. min x px, s.t. upxq u. (8) The solution to this problem is called the expenditure function and is denoted epp, uq. It is analogous to the cost function and its properties with respect to pp, uq are the same as those of the cost function with respect to pw, yq: Non-decreasing in p. Homogeneous of degree 1 in p. Concave in p. Continuous in p. ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 18, School 2017 of Economics 10 / 30

11 Consumer behaviour Furthermore the expenditure-minimizing bundle necessary to achieve utility u, denoted hpp, uq is hpp, uq Bepp, uq Bp (9) The function hpp, uq is called the Hicksian demand function or compensated demand function. ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 18, School 2017 of Economics 11 / 30

12 Consumer behaviour Expenditure Minimization is equivalent to Utility Maximization Proposition 2 Assume that upxq is continuous and that preferences satisfy local non-satiation. Then max x Proof. upxq s.t. px m. ô min x px, s.t. upxq u. (10) 1) U-max ñ E-min Let x be the solution of U-max and let u upx q. We need to show that x solves E-min. Suppose not and let x 1 be the solution to E-min; therefore by the statement of the E-min problem px 1 ă px and upx 1 q u upx q. But by the assumption of non-satiation there exists a bundle x 2 close enough (slightly preferred) to x 1 such that px 2 ă px m and that upx 2 q ą upx q. But this contradicts the assumption that x is the solution of U-max. ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 18, School 2017 of Economics 12 / 30

13 Consumer behaviour Expenditure Minimization is equivalent to Utility Maximization Proof. 2) U-max ð E-min Let x be the solution of E-min and let m px. We need to show that x solves U-max. Suppose not and let x 1 be the solution to U-max; therefore by the statement of the U-max problem upx 1 q ą upx q and px 1 px m. Since, px ą 0 and utility is continuous, there exist a t P p0, 1q such that ptx 1 ă px m and that uptx 1 q ą upx q. But this contradicts the assumption that x is the solution of E-min. ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 18, School 2017 of Economics 13 / 30

14 Pirouettes Pirouettes (Some identities) Expenditure functdions and indirect utility functions are linked. This can be seen by writing the two problems in a compatible way. vpp, mq max x upxq s.t. px m. (11) Let x be the solution to this problem and let u upx q be the associated utility. Now let s write the cost-minim problem as epp, u q min x px, s.t. upxq u. (12) At first sight, these two problems should have the same solution x ; and in fact thay do. ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 18, School 2017 of Economics 14 / 30

15 Pirouettes A first set of identities: 1 e pp, vpp, mqq m. The minimum expenditure necessary to reach utility vpp, mq is m. 2 v pp, epp, uqq u. The maximum utility from income epp, uq is u. 3 x i pp, mq h i pp, vpp, mqq. The Marshallian demand at income m is the same as the Hicksian demand at utility vpp, mq. 4 h i pp, uq x i pp, epp, uqq. The demand at utility u is the same as the Marshallian demand at income epp, uq. ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 18, School 2017 of Economics 15 / 30

16 Pirouettes Proposition 3 Roy s identity. If xpp, mq is the Marshallian demand function, then x i pp, mq Bvpp, mq{bp i Bvpp, (13) Proof. Suppose that x yields utility u at pp, m q. Then we know from the second identity above that v pp, epp, u qq u (14) Identity (14) says that for whatever p the maximum utility that can be obtained from spending the least as possible to achieve u at prices p is exactly u.... ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 18, School 2017 of Economics 16 / 30

17 Pirouettes Proof Cont d. Differentiating totally (14) we obtain Bvpp, m q Bp i ` Bvpp, m q Bepp, u q 0 (15) Bm Bp i From (9) we know that the last multiplicand in (15) is the Hicksian demand function. Therefore h i pp, u q Bvpp, m q{bp i Bvpp, m q{bm. (16) And then from the third identity above we know that which proves the theorem. x i pp, m q h i pp, u q (17) ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 18, School 2017 of Economics 17 / 30

18 Money metric utility functions Money metric utility functions The problem with the indirect utility is that we are unable to measure the level of utility, the problem with the expenditure function is that it depends on utility, which is unobservable. Thus, a consumer can tell us whatever s/he wants about the level of utility s/he is having. Quite clearly, interpersonal comparisons become not-credible. We now ask agents to put their wallet where their mouth is by measuring the willingness to pay. How much money would a consumer need at prices p to be as well of as s/he could be by consuming the bundle of goods x. ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 18, School 2017 of Economics 18 / 30

19 Money metric utility functions Note that the consumer does not need to consume x to be as happy as consuming x, s/he may be consuming any other bundle on the same indifference curve. But then how much money is needed for such happyness? Formally, the problem is min z pz, s.t. upzq upxq. (18) The solution to this problem is the expendituire function epp, upxqq. This particular expenditure function is known as the money metric utility function and it is usually denoted by mpp, xq; i.e., mpp, xq epp, upxqq (19) As you see it is measurable and it is defined over observable variables pp, xq. Note also that mpp, xq is a monotonic transformation of upxq through the vector p. Fig. 7.5 here. ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 18, School 2017 of Economics 19 / 30

20 Money metric utility functions There is a similar concept: the money metric indirect utility function, defined as: µpp; q, mq epp, vpq, mqq (20) That is, µpp; q, mq measures how much money one would need at prices p to be as happy as one would be facing prices q and income m. Fig. 7.6 here. Cobb-Douglas and CES examples here. ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 18, School 2017 of Economics 20 / 30

21 Choice Choice (Chapter 8) Proposition 4 Slutsky Equation The effect of a price change on demand can be decomposed in a substitution effect and an income effect as follows: Bx i pp, mq Bp i Bh ipp, vpp, mqq Bp i Bx ipp, mq Bm x ipp, mq (21) Fig. 8.4 here. Cobb-Douglas example here ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 18, School 2017 of Economics 21 / 30

22 Demand Demand: separability (Chapter 9, Sect. 3) In some circumstances we want to think of consumer maximizing behaviour as separable between groups of goods (food, cloths, vacation). Think of a decomposition of a consumption bundle into sub-bundles: px, zq (22) where, for instance, x could be food, and z could be cloths; and let be the associated price vectors. pp, qq (23) ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 18, School 2017 of Economics 22 / 30

23 Demand Then the U-max problem is: max x,z upx, zq s.t. px ` qz m. (24) The question is under what conditions we can treat x-goods as a group in the U-max problem, without having to worry about how income is allocated among them. If we can, than this implies that we can find two indices (or single-valued functions) P f ppq a price index (25) X gpxq a quantity index or aggregate (26) that can adequately replace x and p in the U-max problem. ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 18, School 2017 of Economics 23 / 30

24 Demand Formally the problem is as follows. Consider the following utility maximization problem: max X,z UpX, zq s.t. P x ` qz m. (27) Let XpP, q, mq be the demand function for the aggregate X. The problem is to find under what conditions the two distinct U-max problems (24) and (27) have the same solution. That is, under what conditions XpP, q, mq Xpfppq, q, mq gpxpp, q, mqq. (28) It turns out that this occurs in two cases: the case of Hicksian separability and the case of functional separability. We study the latter. ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 18, School 2017 of Economics 24 / 30

25 Demand Functional separability. Of course, if such condition is to exist, it must require something about the form of the original utility function. What is required is known as weak separability. Assume that the underlying preferences ordering has the property of independence between sub-bundles, that is: px, zq ą px 1, zq iff px, z 1 q ą px 1, z 1 x, z, x 1, z 1. (29) If preferences satisfy such condition and non-satiation then we may write upx, zq Upvpxq, zq (30) where Upvpxq, zq is increasing in vpxq and the latter is called the sub-utiliy. A function up.q that satisfies (30) is said to be weakly separable. ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 18, School 2017 of Economics 25 / 30

26 Demand Proposition 5 Assume up.q is weakly separable and let xpp, q, mq and zpp, q, mq be the demand functions stemming from the original U-max problem. Furthermore, let m x pxpp, q, mq (31) then the optimal choice of the little x-goods can be found by solving the sub-utility maximization problem Proof. Left to students. max x Note that q is in problem (32) only through m x. vpxq s.t. px m x. (32) ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 18, School 2017 of Economics 26 / 30

27 Demand Let epp, vq the the expenditure function of the Sub-U-Max problem (32). Then we can write the overall problem as max v,z Upv, zq s.t. epp, vq ` qz m. (33) This is almost what we want: the part of the problem that says Upv, zq is what we want because vpxq is a suitable index for the max v,z x-goods - it is suitable because it makes up.q Up.q - but epp, vq does not look very much like P X, which is what we want. Now, what would make epp, vq P vp.q P X?... mumble mumble,... ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 18, School 2017 of Economics 27 / 30

28 Demand Homotheticity! If preferences are homothetic then the expenditure function may be written as eppqv. Then by chosing fpxq vpxq, ñ X vpxq (34) P epxq (35) and our utility function to be UpX, zq the two problems max X,z UpX, zq s.t. P X ` qz m (36) max x have the same solution. Cobb-Douglas-CES example. upvpxq, zq s.t. px m (37) ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 18, School 2017 of Economics 28 / 30

29 Compnsating and equivlent variation Compensating and equivalent variation (Ch. 10, sect.1) Consider a consumer facing a change in price and income from pp 0, m 0 q to pp 1, m 1 q. What is the effect of such change on welfare? The obvious measure of the welfare change is vpp 1, m 1 q vpp 0, m 0 q (38) The problem is that this welfare difference is only ordinal. One way to put a unit of measure on it is to use the money metric indirect utility functions. Recall that µpq; p, mq is defined as epq, vpp, mqq, that is, the minimal expenditure needed to reach utility level vpp, mqq at prices q. Then the the welfare effects of the change from pp 0, m 0 q to pp 1, m 1 q can be measured as µpq; p 1, m 1 q µpq; p 0, m 0 q (39) Now it remains to choose the base price q with respect to which we compute the money metric indirect welfare change.... ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 18, School 2017 of Economics 29 / 30

30 Compnsating and equivlent variation Compensating and equivalent variation (Ch. 10, sect.1)... There are two obvious choices, p 0 or p 1, and each has a name: EV µpp 0 ; p 1, m 1 q µpp 0 ; p 0, m 0 q µpp 0 ; p 1, m 1 q m 0 (40) CV µpp 1 ; p 1, m 1 q µpp 1 ; p 0, m 0 q m 1 µpp 1 ; p 0, m 0 q (41) Figure 10.1 here ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 18, School 2017 of Economics 30 / 30