Question

Assume there is consumer, his utility function is u(x,y) =8*x^0.5+y , and his budget constraint is px*x +y = m, which implies py = 1.

a.Please derive the Marshallian demand function of x.

b.Please derive the indirect utility function.

c.Please derive the expenditure function.

Answer 1

a) Marshallian demand function of x

Max $8x^{0.5} + y$

Subject to $Px^{x} + y = 1$

use budget constrain to determine y

$y = 1 - px^{x}$

Substituting in the utility function

max $f(x)= 8x^{0.5} + (1-Px^{x})$

Take first derivative using product rule

f(x) = $4x^{-0.5} + Px^{2}$

Set $f(x) = 0$ and obtain

$4x^{-0.5} = - Px^{2}$

Solving for x

$x= \frac{-Px^{2.5} }{4}$

b) Indirect utility function

$U^*(Px,y,M) = max U(x,y)| px*x +y = m$

$=U(x^*,y^*)$

$= U(Dx(Px,Py,M), Dy(Px,Py,M))$

c) Expenditure function

$u = U^*(Px.Py,M) \iff M=M^*(Px,Py,u)$

$$M^*(Px,Py,u) = min{Px^X+Py^y|U(x,y)}\ge u$$

$x^* = D_x^H (Px,Py, u)$ $y^* = D_y ^H (Px,Py,u)$