Demand Theory Lecture IX and X Utility Maximization (Varian Ch. 7) Federico Trionfetti Aix-Marseille Université Faculté d Economie et Gestion Aix-Marseille School of Economics October 5, 2018
Table of Content 1 The research question 2 Preferences 3 Consumer behavior 4 Indirect utility 5 Expenditure minimization 6 Exp. Minimization is equivalent to U. Maximization 7 Some identities 8 Money metric utility functions ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October School 5, 2018of Economics 2 / 28
The research 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. Textbook sections for Mini-Lecture: 1 Discuss Fig. 7.2 and 7.3. 2 Discuss Fig. 7.4. 3 Discuss Fig. 7.6. 4 Cobb-Douglas example at the end of Ch. 7. 5 CES example at the end of Ch. 7. ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October School 5, 2018of Economics 3 / 28
Preferences Preferences (Sect. 7.1) Preferences are defined on a consumption set X R k`. We assume preferences to be rational: Rationality 1 Complete. For all x and y in X, either x ľ y or x ĺ y or x y. 2 Reflexive. For all x in X, x ľ x 3 Transitive. For all x, y, and z in X, if x ľ y and y ľ z, then x ľ z. We shall also assume continuity because it allows us to use continuous functions. Continuity. For all y in X the sets x : x ľ y and x : x ĺ y are closed sets. Therefore the sets x : x ą y and x : x ă y are open sets. ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October School 5, 2018of Economics 4 / 28
Preferences Proposition 1 If preferences are rational and continuous then there exist a utility function u : R k` Ñ R. Proof. Skipped Henceforth we assume rationality and continuity. ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October School 5, 2018of Economics 5 / 28
Preferences There are some other assumptions often used for their convenience in economic modeling. Weak Monotonicity. If x ŕ y then x ľ y. At least as much of everything is at least as good. Strong Monotonicity. If x ě y and (x y) then x ą y. At least as much of everything and strictly more of some thing is better. 1 Local non-satiation. Given any x in X and any ɛ ą 0 then there is a vector y with x y ă ɛ such that y ą x. Note that strong monotonicity implies local non-satiation. Convexity. Given x, y, and z in X such that x ľ z and y ľ z, then it follows that tx ` p1 tqy ľ z for all t P p0, 1q. Strict convexity. Given x, y, and z in X such that x ľ z and y ľ z, then it follows that tx ` p1 tqy ą z for all t P p0, 1q. 1 Notation. x ŕ y means that each component of vector x is less than or equal to each corresponding component of vector y. Instead, x ě y necessarily impliesx y. ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October School 5, 2018of Economics 6 / 28
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 School 5, 2018of Economics 7 / 28
Consumer behavior Consumer behavior (Sect. 7.2) 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. Incidentally, you notice by the way we posed that problem that we are working under the assumption of strict monotonicity. ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October School 5, 2018of Economics 8 / 28
Consumer behavior 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 0 ùñ Bupxq λp i @i (5) Bx i Bx i BLpλ, xq 0 ùñ px m. (6) Bλ The solution of this system is a vector x and a scalar λ. The vector x is known as the vector of demands 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 School 5, 2018of Economics 9 / 28
Consumer behavior 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 5, School 2018 of Economics 10 / 28
Indirect utility Indirect utility (Sect. 7.3) 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 four properties: Non-increasing in p and non increasing in m. That is: vpp, mq ě vpp 1, mq for any p 1 ě p. Proof. Let B tx : px ď mu and B 1 tx : p 1 x ď mu for p 1 ě p. Then B 1 Ă B. 2 But then the maximum utility obtained by choosing in B must be at least as good as the maximum utility obtained by choosing in B 1. Analogously for m. Homogeneous of degree 0 in (p, mq. Proof. The budget set is unaltered by the multiplication of p and m by any t ą 0. 2 Ă = proper subset. ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 5, School 2018 of Economics 11 / 28
Indirect utility Quasi-convex in p. Recall that a function is quasi-convex if the lower contour set is convex. The lower contour set is: tp : vpp, mq ď ku (8) Therefore, to prove quasi-convexity we have to show that the lower contour set is a convex set for any k ą 0. Proof. Consider two vectors p and p 1 such that vpp, mq ď k and vpp 1, mq ď k. Let p 2 tp ` p1 tqp 1. We want to show that vpp 2, mq ď k. ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 5, School 2018 of Economics 12 / 28
Indirect utility Proof (Cont ). Graphically our objective is to show that along any p 2 tp ` p1 tqp 1 we have vpp 2, mq ď k. Figure 1 : The vectors p, p 1, and p 2 In Fig. 1 the dots represent p and p 1, and the line represents p 2. ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 5, School 2018 of Economics 13 / 28
Indirect utility Proof (Cont ). To this purpose define the following two budget sets and their union: B tx : px ď mu. B 1 x : p 1 x ď m (. BUB B Y B 1. (9) Figure 2 : The sets B, B 1 and B Y B 1 ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 5, School 2018 of Economics 14 / 28
Indirect utility Proof (Cont ). We also define a third set based on p 2 : Three examples of such set are in the Fig. 3 B 2 x : p 2 x ď m (. (10) Figure 3 : The set B 2 for different t and the set B Y B. As a preliminary step towards proving that the lower contour is convex we show that any x in B 2 must also be in BUB. We do this in the next page. ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 5, School 2018 of Economics 15 / 28
Indirect utility Proof (Cont ). Memo: B tx : px ď mu; B 1 x : p 1 x ď m ( ; B 2 x : p 2 x ď m (. Proof that any x in B 2 must also be in BUB. Suppose not. Then x is such that tpx ` p1 tqp 1 x ď m but px ą m and p 1 x ą m. The latter two inequalities can be written as Summing them up gives: tpx ą tm (11) p1 tqp 1 x ą p1 tqm (12) tpx ` p1 tqp 1 x ą m (13) which is a contradiction because x belongs to B 2. ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 5, School 2018 of Economics 16 / 28
Indirect utility We can now take the final step of the poof. vpp 2, mq max upxq s.t. x P B 2 (14) ď max upxq s.t. x P B Y B 1 since B 2 Ď B Y B 1 (15) ď k since vpp, mq ď k and vpp 1, mq ď k. (16) Which proves the quasi-convexity of vpp, mq. The lines above say: 1 The indirect utility we obtain from choosing x in B 2 (vpp 2, mq ) cannot be larger that the indirect utility we obtain by choosing in BUB because the former set is smaller than the latter. This is said in lines (14) and (15). 2 But then vpp 2, mq must be no larger than k because some there are surely two indirect utilities taken from BUB that are smaller than k; namely, vpp, mq and vpp 1, mq are both smaller than k. This is said in line (16). Which proves the quasi-convexity of vpp, mq. Continuous in p. END OF LIST OF PROPERTIES ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 5, School 2018 of Economics 17 / 28
Expenditure minimization Expenditure minimization (Sect. 7.3) We can pose the consumer problem as that of minimizing the cost of achieving a given level of utility. min x px, s.t. upxq u. (17) 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 5, School 2018 of Economics 18 / 28
Expenditure minimization Furthermore the expenditure-minimizing bundle necessary to achieve utility u, denoted hpp, uq is hpp, uq Bepp, uq Bp i @i. (18) 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 5, School 2018 of Economics 19 / 28
Exp. Minimization is equivalent to U. Maximization Exp. Min is equivalent to U. Max (App. to Ch. 7) 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. (19) 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 5, School 2018 of Economics 20 / 28
Exp. Minimization is equivalent to U. 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 5, School 2018 of Economics 21 / 28
Some identities Some identities (Sect. 7.4) The expenditure function and the indirect utility function are linked. This can be seen by writing the two problems in a compatible way. vpp, mq max x upxq s.t. px m. (20) 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. (21) 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 5, School 2018 of Economics 22 / 28
Some identities 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 5, School 2018 of Economics 23 / 28
Some identities Proposition 3 Roy s identity. If xpp, mq is the Marshallian demand function, then x i pp, mq Bvpp, mq{bp i Bvpp, mq{bm @i. (22) 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 (23) Identity (23) 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 5, School 2018 of Economics 24 / 28
Some identities Proof Cont d. Differentiating totally (23) we obtain Bvpp, m q Bp i ` Bvpp, m q Bepp, u q 0 (24) Bm Bp i From (18) we know that the last multiplicand in (24) is the Hicksian demand function. Therefore h i pp, u q Bvpp, m q{bp i Bvpp, m q{bm. (25) And then from the third identity above we know that which proves the theorem. x i pp, m q h i pp, u q (26) ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 5, School 2018 of Economics 25 / 28
Money metric utility functions Money metric utility functions (Sect. 7.5) 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 5, School 2018 of Economics 26 / 28
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 happiness? Formally, the problem is min z pz, s.t. upzq upxq. (27) The solution to this problem is the expenditure 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 (28) 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 5, School 2018 of Economics 27 / 28
Money metric utility functions There is a similar concept: the money metric indirect utility function, defined as: µpp; q, mq epp, vpq, mqq (29) 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. ederico Trionfetti (Aix-Marseille Université Faculté Demand d Economie Theory et Gestion Aix-Marseille October 5, School 2018 of Economics 28 / 28