Question

Given the following utility function U(X,Y)=X^0.4 Y^0.6 subject to the constraint

I=XPx+YPy

find:

a the marshallian demand functions for X and Y

b the indirect utility function

c the expenditure function

d the compensated demand functions for X and Y

e from 1 and 2 establish whether x and y are inferior or normal goods

f discuss the compensating variation, and distinguish it from equivalent variation, for a normal and inferior good. give real life examples

Answer 1

First to maximize utility subject to the Budget constraint – I = xP_x + yp_y , we set up a lagrangean function -

Partially differentiate this with respect to x, y and lambda to get first order conditions -

Divide equations 1) and 2) -

On simplifying we get

Writing this condition in terms of y we get -

a) Marshallian demand of a good is a function of price and Income (I) .

We know the budget constraint -

Put y from the first order condition , found above, in this equation -

Writing in terms of x -

This is the Marshall demand for good x.

Put this in the the first order condition -

This is the Marshall demand for good y.

b) Indirect utility function is given by putting marshallian demand function into the utility function -

c) We can get expenditure function by writing Indirect utility function in terms of Income -

This is the Expenditure function.

d) To get compensated demand function we put the first order condition (FOC)

into the utility function -

Writing this in terms of good x -

This is the compensated demand function for x .

Similarly for good y - put FOC written in terms of good x

into the utility function -

Writing in terms of y -

This is the compensated demand function for y .