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Topic: Exercises 1.29 & 1.46
Conf: Chapter 1, Consumer Theory, Msg: 13541
From: Martin Caley (martin.caley@economics.treasury.gov.im)
Date: 10/28/2003 07:17 AM
Exercises 1.29 & 1.46 Martin Caley MCaley martin.caley@economics.treasury.gov.im
1.29 Slope:- Holding v and y constant, differentiate totally to get dp2/dp1= -Dv/Dp1/Dv/Dp2<0. (D denotes partial differentiation.) Theorem 1.6.4.
Direction of increasing utility:- Decreasing p1 and p2. Theorem 1.6.4
Curvature:- Pick two points on the price-indifference curve and use quasiconvexity to show the curve is bowed towards the origin. Theorem 1.6.5.
1.46 (a) e(p,1)=min px subject to u(x)>=1 so, using homogeneity of u(x),
e(p,1)=1/u* min pu*x subject to u*u(x)=u(u*x)>=u*.
Substituting z=u*x gives u*e(p,1)=min pz subject to u(z)>=u* and this equals e(p,u*).
(b)e(p,u)=ue(p,1) and so de(p,u)/du=e(p,1).