Question

1.    If an individual’s preferences are described by the utility function U(X₁ , X₂ ) = X₁² + X₂²,

a)    graph the indifference curve for U = 20 and U= 40.

b)    Find the optimal consumption quantities if P₁ = US$2.50 ;                     P₂ = US$ 7.50; and M = US$ 60.

Answer 1

Part a

U(X₁ , X₂ ) = X₁² + X₂²

The indifference curve is as shown below

If $P_1=2.5; P_2=2.5; m=60$ , the budget line will be as follows

$P_1x_1+P_2x_2 =m$

$2.5x_1+2.5x_2 =60$

Also from U(X₁ , X₂ ) = X₁² + X₂²

$MR_{(x_1,x_2)}=\frac{mUx_1}{mUx_2}$ as $mUx_1 = 2x_1 \& mUx_2 = 2x_2$

So, $MR_{(x_1,x_2)}=\frac{2x_1}{2x_2} =\frac{x_1}{x_2}$

Part b

Equating the slope of budget line and MRs

$\frac{P_1}{P_2} =MRs_{(x_1,x_2)} \implies \frac{2.5}{7.5}=\frac{x_1}{x_2}$

$x_2=3x_1$

Therefore, $2.5x_1+7.5(3x_1)=60 \implies x_1= \frac{60}{25}=2.4; x_2= 7.2$