2020/6/21,中级微观经济学,Chapter Four,Utility 效用,2020/6/21,中级微观经济学,Structure,Utility function (效用函数） Definition Monotonic transformation （单调转换） Examples of utility functions and their indifference curves Marginal utility （边际效用） Marginal rate of substitution 边际替代率 MRS after monotonic transformation,2020/6/21,中级微观经济学,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) U(4,1) = U(2,2) = 4. Call these numbers utility levels.,p,2020/6/21,中级微观经济学,Utility Functions each is the other.,2020/6/21,中级微观经济学,Utility Functions,There is no unique utility function representation of a preference relation. Suppose U(x1,x2) = x1x2 represents a preference relation. Again consider the bundles (4,1), (2,3) and (2,2).,2020/6/21,中级微观经济学,Utility Functions,U(x1,x2) = x1x2, so U(2,3) = 6 U(4,1) = U(2,2) = 4; that is, (2,3) (4,1) (2,2).,p,2020/6/21,中级微观经济学,Utility Functions,U(x1,x2) = x1x2 (2,3) (4,1) (2,2). Define V = U2.,p,2020/6/21,中级微观经济学,Utility Functions,U(x1,x2) = x1x2 (2,3) (4,1) (2,2). Define V = U2. Then V(x1,x2) = x12x22 and V(2,3) = 36 V(4,1) = V(2,2) = 16 so again (2,3) (4,1) (2,2). V preserves the same order as U and so represents the same preferences.,p,p,2020/6/21,中级微观经济学,Utility Functions,U(x1,x2) = x1x2 (2,3) (4,1) (2,2). Define W = 2U + 10.,p,2020/6/21,中级微观经济学,Utility Functions,U(x1,x2) = x1x2 (2,3) (4,1) (2,2). Define W = 2U + 10. Then W(x1,x2) = 2x1x2+10 so W(2,3) = 22 W(4,1) = W(2,2) = 18. Again, (2,3) (4,1) (2,2). W preserves the same order as U and V and so represents the same preferences.,p,p,2020/6/21,中级微观经济学,Utility Functions: Monotonic Transformation,If U is a utility function that represents a preference relation and f is a strictly increasing function, then V = f(U) is also a utility function representing .,2020/6/21,中级微观经济学,Goods, Bads and Neutrals,A good is a commodity unit which increases utility (gives a more preferred bundle). A bad is a commodity unit which decreases utility (gives a less preferred bundle). A neutral is a commodity unit which does not change utility (gives an equally preferred bundle).,2020/6/21,中级微观经济学,Goods, Bads and Neutrals,Utility,Water,x,Units of water are goods,Units of water are bads,Around x units, a little extra water is a neutral.,Utility function,2020/6/21,中级微观经济学,Some Other Utility Functions and Their Indifference Curves,Perfect substitute V(x1,x2) = x1 + x2. Perfect complement W(x1,x2) = minx1,x2 Quasi-linear U(x1,x2) = f(x1) + x2 Cobb-Douglas Utility Function U(x1,x2) = x1a x2b What do the indifference curves for these utility functions look like?,2020/6/21,中级微观经济学,Perfect Substitution Indifference Curves,5,5,9,9,13,13,x1,x2,x1 + x2 = 5,x1 + x2 = 9,x1 + x2 = 13,All are linear and parallel.,V(x1,x2) = x1 + x2.,2020/6/21,中级微观经济学,Perfect Complementarity Indifference Curves,x2,x1,45o,minx1,x2 = 8,3,5,8,3,5,8,minx1,x2 = 5,minx1,x2 = 3,All are right-angled with vertices on a ray from the origin.,W(x1,x2) = minx1,x2,2020/6/21,中级微观经济学,Quasi-Linear Utility Functions,A utility function of the form U(x1,x2) = f(x1) + x2 is linear in just x2 and is called quasi-linear （准线性）. E.g. U(x1,x2) = 2x11/2 + x2.,2020/6/21,中级微观经济学,Quasi-linear Indifference Curves,x2,x1,Each curve is a vertically shifted copy of the others.,2020/6/21,中级微观经济学,Cobb-Douglas Utility Function,Any utility function of the form U(x1,x2) = x1a x2b with a 0 and b 0 is called a Cobb-Douglas utility function. E.g. U(x1,x2) = x11/2 x21/2 (a = b = 1/2) V(x1,x2) = x1 x23 (a = 1, b = 3),2020/6/21,中级微观经济学,Cobb-Douglas Indifference Curves,x2,x1,2020/6/21,中级微观经济学,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.,2020/6/21,中级微观经济学,Marginal Utilities,If U(x1,x2) = x11/2 x22 then,2020/6/21,中级微观经济学,Marginal Utilities and Marginal Rates-of-Substitution,The general equation for an indifference curve is U(x1,x2) k, a constant. Totally differentiating this identity gives,2020/6/21,中级微观经济学,Marginal Utilities and Marginal Rates-of-Substitution,rearranged is,This is the MRS.,2020/6/21,中级微观经济学,Marg. Utilities An example,Suppose U(x1,x2) = x1x2. Then,so,2020/6/21,中级微观经济学,Marg. Utilities An example,MRS(1,8) = - 8/1 = -8 MRS(6,6) = - 6/6 = -1.,x1,x2,8,6,1,6,U = 8,U = 36,U(x1,x2) = x1x2;,2020/6/21,中级微观经济学,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,2020/6/21,中级微观经济学,Marg. Rates-of-Substitution for Quasi-linear Utility Functions,x2,x1,MRS = - f (x1) does not depend upon x2.,MRS is a constant along any line for which x1 is constant.,MRS = - f(x1),MRS = -f(x1”),x1,x1”,2020/6/21,中级微观经济学,Monotonic Transformations i.e. V(x1,x2) = x12x22. What is the MRS for V? which is the same as the MRS for U.,2020/6/21,中级微观经济学,Monotonic T