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

Quasi-convexity gives us the idea that certain types of variation in prices are actually good, not bad for consumers As an obvious example, consider the utility function X+y+z(three goods),where consumption is contrained to be non-negative The price of x is always one. Income is always one What is utility if the price of y and z are both one? What is utility if the price of y equals 1.5 and the price of z equals 5?

Quasi-convexity gives us the idea that certain types of variation in prices are actually good, not bad for consumers. As an obvious example, consider the utility function xyz (three goods), where consumption is contrained to be non-negative. The price of x is always one. Income is always one. What is utility if the price of y and z are both one? What is utility if the price of y equals 1.5 and the price of z equals .5?

What is utility if the price of y equals 5 and the price of z equals 1.5?

What is utility if the price of y equals .5 and the price of z equals 1.5?

Roy's Identity MWG Proposition 3.G. 4: Suppose that u() is a continous utility function representing a locally non-satiated and strictly convex preference relation > defined on the consumption set X=RL Suppose also that the indirect utility function is differentiable at(p, w)>0 Then x(万,W) P v(p w that is for every j=1, 2,.L, x;(p, w) av(p,w),Ovp, w) opj In words, marshallian demand for a good

Roy’s Identity: MWG Proposition 3.G.4: Suppose that u. is a continous utility function representing a locally non-satiated and strictly convex preference relation  defined on the consumption set X   L. Suppose also that the indirect utility function is differentiable at p,w  0. Then xp,w   1 wvp,w pvp,w that is for every j1,2,...L, xjp,w   vp,w pj / vp,w w In words, marshallian demand for a good

equals the ratio of the derivative of indirect utility with respect to the price of that good divided by the derivative of indirect utility with respect to wealth Proof: My favorite is the envelope theorem argument v(D,v)=U(x*)+(-p·x*) Ov(p, w) ax But the second term is zero(thats the envelope argument) at the maximum Likewise avp, w)=2 and we're done

equals the ratio of the derivative of indirect utility with respect to the price of that good divided by the derivative of indirect utility with respect to wealth. Proof: My favorite is the envelope theorem argument: vp,w  Ux   w  p  x  vp,w pj  xj   xj  pj U xj  p But the second term is zero (that’s the envelope argument) at the maximum. Likewise vp,w w   and we’re done

The expenditure function Previously, we have discussed the utility maximization problem (i.e. maximize utility subject to a fixed budget constraint). In many cases, it is valuable to discuss the expenditure minimization problem (i.e minimize expenditure subject to a fixed utility level) Formally, this problem is Minimize x subject to u(x)≥a We define the expenditure function e(p, u) as the lowest level of income needed to

The expenditure function. Previously, we have discussed the utility maximization problem (i.e. maximize utility subject to a fixed budget constraint). In many cases, it is valuable to discuss the expenditure minimization problem (i.e. minimize expenditure subject to a fixed utility level). Formally, this problem is Minimize p  x subject to ux  u We define the expenditure function ep,u as the lowest level of income needed to