Christensen, Jorgenson and Lau (1975) defined the translog indirect utility function u(p,m)

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Problem 2

Christensen, Jorgenson and Lau (1975) defined the translog indirect utility function u(p,m)

as:

(1)

Inu(p,m) = a+a, In()+Ź3, In (P/m) In("/m)

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where p; is the price of good j and m is income. Assume that we have symmetric price effects, i.e., B = B. Use the translog function and assume there are only two goods when you answer the questions below.

a) Show that the translog indirect utility function (1) is homogenous of degree zero in prices and income.

b) What restrictions must be imposed on the parameters of the translog indirect utility function (1) to ensure strict convexity in prices?

c) Derive the Marshallian demand functions expressed as a budget shares, w (instead of quantities).

d) Show that the derived Marshallian demand functions are homogenous of degree zero in prices and income. What are the practical implications of homogeneity of degree zero in income and prices for Marshallian demand functions?

Problem 3

In this problem, we look at the almost ideal demand system (AIDS) of Deaton and Muellbauer (1980). AIDS is defined by the expenditure function e(p,u):

Ine(p,u) = a +α, ln p, +ŹŹ, Inp, Inp, +u

We assume that jk-kj.

a) Find the associated indirect utility function (p.x).

j-l

b) Find the Hicksian demand functions h(p,u) expressed as expenditure shares, wi, by using Shephard's lemma (i.e., differentiating the expenditure function).

c) Substitute the indirect utility function into the Hicksian demand functions to obtain the Marshallian demand functions.

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