哈佛大学：《中级微观经济学》（英文版）Lecture 4 Welfare Analysis and other Issues

Economics 2010a Fa2003 Edward L. Glaeser Lecture 4

Economics 2010a Fall 2003 Edward L. Glaeser Lecture 4

4. Welfare Analysis and other Issues Measuring Welfare b. First Order and Second Order Losses C. Taxes and Welfare d. Household production(did last class) The Hedonic Approach

4. Welfare Analysis and Other Issues a. Measuring Welfare b. First Order and Second Order Losses c. Taxes and Welfare d. Household Production (did last class) e. The Hedonic Approach

f. The Theory of Equalizing Differentials g. Application Land Prices and Consumption

f. The Theory of Equalizing Differentials g. Application: Land Prices and Consumption

A few useful utility functions to think about (1)Quasi-Linear Preferences, i.e U(x1, x2,.xL=x1+o(x2, ...xL) First order conditions are then pi tor all i >1 This means that consumption of all goods except for good one is independent of income and depends only on prices. Good one just takes up the residual income

A few useful utility functions to think about: (1) Quasi-Linear Preferences, i.e. Ux1, x2,... xL   x1  x2,... xL  First order conditions are then:  xi  pi for all i  1. This means that consumption of all goods, except for good one is independent of income and depends only on prices. Good one just takes up the residual income

2)Homothetic Preferences(MWG Definition 3. B6 )A monotone preference relation on X=Rl is homothetic if x-y then ax~ ay for any a≥0 (Parallel indifference curves)-homothetic preferences can be represented by a utili ity function u(x) that is homogeneous of degree one, i. e au(x=u(ax) for all positive a Does that mean that utility functions that are not homogeneous of degree one cant be homothetic?

(2) Homothetic Preferences (MWG Definition 3.B.6) A monotone preference relation on X   L is homothetic if x  y then x  y for any   0. (Parallel indifference curves)– homothetic preferences can be represented by a utility function ux that is homogeneous of degree one, i.e. ux  ux for all positive . Does that mean that utility functions that are not homogeneous of degree one can’t be homothetic?

(3)Leontief/Fixed Proportions u(x1, x2,.xL)= min(ax1, a2x2,.a3xL) n this case, people always consume these proportions are independent of snd goods in exactly the same proportions prices (4)Cobb-Douglas( fixed budget shares u( x1,x℃2,XL ∏x"or∑a,log(x)

(3) Leontief/Fixed Proportions ux1, x2,... xL  min1x1,2x2,...3xL In this case, people always consume goods in exactly the same proportions and these proportions are independent of prices. (4) Cobb-Douglas (fixed budget shares) ux1, x2,... xL   i1 L xi i or  i1 L i logxi

Whee∑a;=1 Maximization yields:a=λp;,Or当 i=pixi with>pixi= w, this means 1=w and this yields fixed budgets shares - what are income and price elasticities in this case?

where  i1 L i  1 Maximization yields: i xi  pi, or i   pixi with  i1 L pixi  w, this means 1   w and iw  pixi this yields fixed budgets shares– what are income and price elasticities in this case?

(5) Separable utility (x12X2,L ∑ This yields first order conditions (xi= api so consumption depends only on the price for your own good and the marginal utility of income. There are cross-price elasticities but they only work through the marginal utility of income, or another way to think about this is if we normalize the price of good one to one, we have VI(x1)="bi for all i

(5) Separable utility ux1, x2,... xL   i1 L vixi This yields first order conditions vi  xi  pi, so consumption depends only on the price for your own good and the marginal utility of income. There are cross-price elasticities but they only work through the marginal utility of income, or another way to think about this is if we normalize the price of good one to one, we have v1  x1  vi  xi pi for all i

a particularly standard case of this function u(x1,x2,.IL ∑a which yields f oc pi Again if the price of good 1 is one, this tells us: no xo=no1xi- for all i.(n. b. this is actually the same as the C.E.s. utility function wai p oi =o, then xi L

A particularly standard case of this function is ux1, x2,... xL   i1 L ixi i which yields f.o.c. iixi i1  pi Again if the price of good 1 is one, this tells us: ii pi xi i1  11 p1 x1 11 for all i. (n.b. this is actually the same as the C.E.S. utility function) If i  , then xi  wi 1 1 pi 1 1 j1 L pj  1 j 1 1

Separability will particularly show up when thinking about time and uncertainty. In the temporal context, L (x1,x2x)=∑B-1x is a particularly usual formulation where the i subscripts generally refer to some time periods and of course this means B-PLxo-1=xo-1 Or Xi=xI PiBi-I pi Consumption rises or falls depending on Whether the change in prices is greater or less than the discount factor the extent to Which consumption rises depends on the elasticity term

Separability will particularly show up when thinking about time and uncertainty. In the temporal context, ux1, x2,... xL   i1 L i1xi  is a particularly usual formulation where the i subscripts generally refer to some time periods. and of course this means: i1p1 pi xi 1  x1 1 or xi  x1 p1i1 pi 1 1 Consumption rises or falls depending on whether the change in prices is greater or less than the discount factor– the extent to which consumption rises depends on the elasticity term