Question

Suppose, there is a consumer who derives utility from the consumption of two goods, X & Y, her utility function is U (X,Y) = X0.75 Y0.25. Initially, the price of X & Y are given byP_x&P_y. Her total income is given by m.

a) Form the Lagrangian Function for the the expenditure minimization problem.
b) Derive the Hicksian demand functions for X & Y.
c) If U=20, P_x=6 &P_y=2, find the values of X & Y.
d) Based on your answers in part (c) what can you conclude about the relation between the two goods?

Answer 1

$U(x.y) = X$ ^0.75 $Y$ ^0.25

A) $h = min P1X +P1 Y - \lambda(X$ ^0.75$Y$ ^0.25 $- \bar U$ $)$

$\frac{dh}{dx} = P1-$ $\lambda 0.75 X$ ^-0.25$Y$ ^0.25 ----------- 1

$\frac{dh}{dx} = P2-$ $\lambda 0.25 X$ ^-0.75$Y$ ^0.75 $= 0$ ----------- 2

$\frac{dx}{d\lambda} = - (X$ ^0.75 $Y$ ^0.25$-$ $\bar U)$ $= 0$ --------------- 3

using equation 1 and 2

$\frac{P1}{P2} =\frac{3Y}{X}$ --------------------------------------------------- 4

B)- Put equation 4 and 3

$X$^c $= \bar U(\frac{3P2}{P1})$ ^0.25 , $Y$^C $= \bar U(\frac{P1}{3P2})$^0.75

C) Putting $\bar U = 20 ,P1 = 6 , P2 = 2$ in $X$^C and $Y$ ^C

We get $X$ ^C $= 20$ ,$Y$ ^C $= 20$

D) part C shows that for any price level of X and Y ,The Hickson demand for X and Y would be same