Exercise 1.52 Martin Caley MCaley martin.caley@economics.treasury.gov.im

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Topic: Exercise 1.52
Conf: Chapter 1, Consumer Theory, Msg: 15013
From: Martin Caley (martin.caley@economics.treasury.gov.im)
Date: 1/19/2005 11:42 AM

Exercise 1.52 Martin Caley MCaley martin.caley@economics.treasury.gov.im What you want to show is whether or not the Marshallian demands are the same given the conditions on the expenditure functions. Marshallian demands, x(p,y), are observable; Hicksian demands, h(p,u), are not.
(a) eA(p,2u)=eB(p,u)=y so hA(p,2u)=hB(p,u) using Shephard's lemma. Hence xA(p,eA(p,2u))=xB(p,eB(p,u))as x(p,e(p,u))=h(p,u) by Theorem 1.9. Thus xA(p,y)=xB(p,y).
(b) In a similar fashion to part (a), 2xA(p,y)=xB(p,2y). Now use the result of Exercise 1.47 to get xA(p,2y)=2xA(p,y) and then deduce that xA(p,y)=xB(p,y).