哈佛大学：《中级微观经济学》（英文版）Lecture 3 Comparative Statics

provide utility level u given prices p, The value of e(p, u equals p. x* where x* mInimIzes p· x subject to u(x)≥l

provide utility level u given prices p, The value of ep,u equals p  x where x minimizes p  x subject to ux  u

There is a fundamental equivalence between utility maximization and expenditure minimization captured in MWG Proposition 3. E1: Suppose that u() is a continuous utility function representing a locally non-satiated preference relation >defined on the consumption set X=Rl and that the price vector is p≥0.Then 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 (i if x* minimizes expenditure then the required utility level is u then x* maximizes utility when wealth equals

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   L and that the price vector is p  0 . Then: (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*.M oreover the maximized utility level is exactly u Proof (: Suppose that x* maximizes utility and does not minimize expenditure this implies that there exists an x' such that l(x)>l(x*)andp·x<p·x*, But then local nonsatiation implies that by spending a little more than on the nonsatiated good, we can find an x"such that l(x)>l(x*)andp·x<p·x*≤W,and this contradicts maximization The proof to the second part is quite similar

p  x. Moreover the maximized utility level is exactly u. Proof (i): Suppose that x maximizes utility and does not minimize expenditure, this implies that there exists an x such that ux   ux  and p  x  p  x. But then local nonsatiation implies that by spending a little more than p  x on the nonsatiated good, we can find an x such that ux  ux and p  x  p  x  w, and this contradicts maximization. The proof to the second part is quite similar

Properties of the Expenditure Function MWG Proposition 3. E2 Suppose that u() is a continuous utility function representing a locally non-satiated preference relation defined on the consumption set X=4. The expenditure function e(p, u) is homogeneous of degree one in p (i strictly increasing in u and nondecreasing in p; for any j (ti Concave in p (iv) Continuous in p and u Proof: The problem Minimize p.x subject to u(x)≥ u and minimize ap·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   L . The expenditure function ep,u is: (i) homogeneous of degree one in p. (ii) strictly increasing in u and nondecreasing in pj for any j. (iii) Concave in p. (iv) Continuous in p and u. Proof: (i) The problem Minimize p  x subject to ux  u and minimize p  x

subject to u(x)> u, yields exactly the same optimal value for the x vector, denoted x*. As the expenditure function equals prices times quantities, when prices are multiplied by a, the expenditure function must be multiplied by the same amount (1 Assume that e(p, u) is not strictly increasing in u, and let x and x"denote optimal consumption bundles for utility levels u and u"respectively, where andp·x 0. 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·ax"<p·x≤p·x' but then the le bundle ax"yields more utility at less cost than the bundle x' so x' is not expenditure minimizing and we have a contradiction

subject to ux  u , yields exactly the same optimal value for the x vector, denoted x. As the expenditure function equals prices times quantities, when prices are multiplied by , the expenditure function must be multiplied by the same amount. (ii) Assume that ep,u is not strictly increasing in u , and let x and x denote optimal consumption bundles for utility levels u and u respectively, where u  u and p  x  p  x  0. Continuity ensures that there exists a value of  which is less than one but sufficiently close to one so that ux  ux  and p  x  p  x  p  x but then the bundle x yields more utility at less cost than the bundle x so x is not expenditure minimizing and we have a contradiction