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Marshallian/ Hicksian Demand Function

An individuals preferences over goods x=(x1,x2) can be represented by the following utility function:

u(x)= ln(x1-b)+ ln(x2)

The individual faces prices p=(p1,p2)>>0 and has income m>p1b>0

Why is it important that m>p1b? What is the interpretation of the coefficient b? Do the demand functions satisfy the relevant homogeneity conditions? Derive the indirect utility function v(p,w).

b) Using the result from part a), show the expenditure function is

e(p,u)=2e^(u/2)p1^(1/2)p2(1/2)+hp1

From the expenditure function derive the Hicksian demand functions, h1(p,u) and h2(p,u). Do the Hicksian demand functions satisfy their relevant homogeneity conditions?

Solution Preview

a) Why is it important that m>p1b? What is the interpretation of the coefficient b? Do the demand functions satisfy the relevant homogeneity conditions? Derive the indirect utility function v(p,w).

From the structure of the utility function, b is the minimum consumption of x1, because x1>b is the necessary condition for the ln(x1-b) to be meaningful.

m>p1b means that the total budget should cover the spending on b units of x1. If and only if m> p1b, the individual can afford more x1 and x2.

A function f(x) is homonegous of degree k if f(sx) = s^k f(x).
To test the homogeneity conditions, we multiply x1 and x2 by a scalar s and substitute into the utility function:
U(sx) = ln(sx1 - b) + ln(sx2)
Since we cannot write the function into s^k ...

Solution Summary

Marshallian/ Hicksian Demand Functions are performed.

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