Lecture 4
1 Lecture 4
There is a fundamental equivalence between utility maximization and expenditure minimization captured in: MWG Proposition 3.E.1: Suppose that u(.)is a continuous utility function representing a locally non- satiated preference relation defined on the consumption set X=R and that the price vector is p>>0. Then:
2 There is a fundamental equivalence between utility maximization and expenditure minimization captured in: MWG Proposition 3.E.1: Suppose that u(.) is a continuous utility function representing a locally nonsatiated preference relation defined on the consumption set X= and that the price vector is . Then: R L + p 0
(i)if x'maximizes utility for w>0,then x'minimizes expenditure when the required utility level is u(x").Moreover the minimized expenditure level is exactly w. (ii)if x minimizes expenditure then the required utility level is u then x maximzes utility when wealth equals p.x.Moreover the maximized utility level is exactly u. 3
3 • (i) if maximizes utility for w>0, then minimizes expenditure when the required utility level is u( ). Moreover the minimized expenditure level is exactly w. • (ii) if minimizes expenditure then the required utility level is u then maximzes utility when wealth equals .Moreover the maximized utility level is exactly u. * x * x * x * x * x * p• x
Proof(i):Suppose that maximizes utility and does not minimize expenditure,this implies that there exists an such that u(x )zu(x)and p●xu(x)andp●x<p●x,and this contradicts maximization. The proof to the second part is quite similar
4 • Proof (i): Suppose that maximizes utility and does not minimize expenditure, this implies that there exists an such that u( )≥u( ) and . But then local nonsatiation implies that by spending a little more than on the nonsatiated good, we can find an such that and , and this contradicts maximization. • The proof to the second part is quite similar. * x ' x ' '' p • x p • x ' p • x ( ) ( ) '' * u x u x ' * p • x p • x
Properties of the Expenditure Function MWG Proposition 3.E.2: Suppose that u(.)is a continuous utility function representing a locally non- satiated preference relation defined on the consumption set X=R..The expenditure function e(p,u)is: (i)homogeneous of degree one in p. (ii)strictly increasing in u and nondecreasing in P for any j. (iii)Concave in p. (iv)Continuous in p and u
5 • Properties of the Expenditure Function MWG Proposition 3.E.2: • Suppose that u(.) is a continuous utility function representing a locally nonsatiated preference relation defined on the consumption set X= . The expenditure function e(p, u) is: (i) homogeneous of degree one in p. (ii) strictly increasing in u and nondecreasing in for any j. (iii) Concave in p. (iv) Continuous in p and u. R L + j p
Proof:(i)The problem Minimize px subject to u(x)zu and minimize apx subject to u(x)z u,yields exactly the same optimal value for the x vector, denoted.As the expenditure function equals prices times quantities,when prices are multiplied by a,the expenditure function must be multiplied by the same amount. 6
6 • Proof: (i) The problem Minimize p·x subject to u(x)≥u and minimize αp·x subject to u(x)≥ u , yields exactly the same optimal value for the x vector, denoted . As the expenditure function equals prices times quantities, when prices are multiplied by α, the expenditure function must be multiplied by the same amount
(ii)Assume that e(p,u)is not strictly increasing in u,and let and denote optimal consumption bundles for utility levels and respectively,,where u>u and p●x≥p●x. Continuity ensures that there exists a value of a which is less than one but sufficiently close to one so that u(ax)>u(x)and p●x<p●x≤p●x but then the bundle ax yields more utility at less cost than the Bundle x sox is not expenditure minimizing and we have a contradiction
7 • (ii) Assume that e(p, u) is not strictly increasing in u , and let and denote optimal consumption bundles for utility levels and respectively, where and . Continuity ensures that there exists a value of α which is less than one but sufficiently close to one so that u( )＞ u( ) and but then the bundle yields more utility at less cost than the Bundle so is not expenditure minimizing and we have a contradiction. '' ' u u ' '' p • x p • x '' x ' x '' ' ' p •x p • x p • x '' x ' x ' x
(iii)Fix a utility level and consider two price level p andand let p"=ap+(1-a)p forae01].Let x'denote the bundle that minimizes expenditures and achieves utility level when prices arep".If so then e(p,w)=p●x=p●x"+(1-)p·x"≥ oe(p,w)）+(1-x)e(p,u）
8 • (iii) Fix a utility level , and consider two price level p and and let for . Let denote the bundle that minimizes expenditures and achieves utility level , when prices are . If so then '' ' p =p + (1−) p 0,1 '' x _ u '' p ( , ) (1 ) ( , ) ( , ) (1 ) ' '' '' '' ' '' _ '' e p u e p u e p u p x p x p x + − = • = • + − •
Concavity of the expenditure function with respect to prices is the natural parallel of convexity of the indirect utility function with respect to prices.Under some circumstances,price variation is good,not bad
9 • Concavity of the expenditure function with respect to prices is the natural parallel of convexity of the indirect utility function with respect to prices. Under some circumstances, price variation is good, not bad
Again consider the utility function x+y+z, and think about the expenditures need to yield one unit of utility.In the case where the price of all three goods is one,the one unit of currency is needed. In the case where one of the goods costs 1.5 and the other 0.5,then only 0.5 units of currency are needed. 10
10 • Again consider the utility function x+y+z, and think about the expenditures need to yield one unit of utility. In the case where the price of all three goods is one, the one unit of currency is needed. • In the case where one of the goods costs 1.5 and the other 0.5, then only 0.5 units of currency are needed