中级微观经济学课件(第4章).pdf

Chapter FourUtility效用What Do We Do in This Chapter?• We create a mathematical measure of preference in order to advance our analysis.Utility Functions• A preference relation that is complete, reflexive, transitive and continuous can be represented by a continuous utility function.Utility Functions• A utility function U(x) represents a preference relation if and only if:x’ x” U(x’) > U(x”)x’ x” U(x’) 0 and b > 0 is called a Cobb-Douglas utility function.• E.g. U(x1,x2) = x11/2x21/2(a = b = 1/2)V(x1,x2) = x1 x23 (a = 1, b = 3)Cobb-Douglas Indifference Curvesx2x1All curves are hyperbolic,asymptoting to, but nevertouching any axis.Marginal Utilities• Marginal means “incremental”.• The marginal utility of commodity i is the rate-of-change of total utility as the quantity of commodity i consumed changes; i.e.MUUxii=∂∂Marginal Utilities and Marginal Rates-of-Substitution• The general equation for an indifference curve isU(x1,x2) ≡ k, a constant.Totally differentiating this identity gives∂∂∂∂UxdxUxdx11220+=Marginal Utilities and Marginal Rates-of-Substitution∂∂∂∂UxdxUxdx11220+=∂∂∂∂UxdxUxdx2211=−rearranged isMarginal Utilities and Marginal Rates-of-Substitution∂∂∂∂UxdxUxdx2211=−rearranged isAnddxdxU xUx2112=−∂ ∂∂∂//.This is the MRS.Marg. Rates-of-Substitution for Quasi-linear Utility Functions• A quasi-linear utility function is of the form U(x1,x2) = f(x1) + x2.so∂∂Uxfx11= ′()∂∂Ux21=MRSdxdxU xUxfx==− =−′21121∂ ∂∂∂//().Marg. Rates-of-Substitution for Quasi-linear Utility Functions•MRS = -f(x1) does not depend upon x2so the slope of indifference curves for a quasi-linear utility function is constant along any line for which x1is constant. What does that make the indifference map for a quasi-linear utility function look like?′Marg. Rates-of-Substitution for Quasi-linear Utility Functionsx2x1Each curve is a vertically shifted copy of the others.MRS is a constantalong any line for which x1isconstant.MRS =- f(x1’)MRS = -f(x1”)x1’x1”Monotonic Transformations & Marginal Rates-of-Substitution• More generally, if V = f(U) where f is a strictly increasing function, thenMRSV xVxf U U xfU U x=− =−′ ××∂ ∂∂∂∂ ∂∂∂//() /'( ) /1212=−∂ ∂∂∂U xUx//.12So MRS is unchanged by a positivemonotonic transformation.The Key to this Chapter• The indifference curve of a consumer preference can be represented by a utility function based equation:U(x1, x2) = k, a constant.。