Question

4. 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

a)

$U(X_1,$ $X_2)\ =\ X_1^2+X_2^2$

$U=20$

$20=X_1^2+X_2^2$

$X_1^2=20-X_2^2$

For U=40

$40=X_1^2+X_2^2$

$X_1^2=40-X_2^2$

b)

Budget line equation= $P_1X_1\ +P_2X_2=M$ .................Equation 1

Substituting the values given to Equation 1 above. We get,

$2.50X_1+7.50X_2=60..........................2$

$U(X_1,X_2)=X_1^2+X_2^2$

$MRS(X_1,X_2)=\frac{MUx_!}{MUx_2}$

$MUX_1=X_1,MUX_2$ $=2X_2$

$MRS=X_1,X_2=\frac{X_1}{X_2}$

Optimal Bundle=

$\frac{P_1}{P_2}=MRS$$(X_1,X_2)$

$\frac{2.5}{7.5}=\frac{X_1}{X_2}$

$2.5X_2=7.5X_1$

$X_2=3X_1.......................................3$

Substituting $X_2$ in equation 2

We get,

$X_1=\frac{60}{25}$

$=2.4$

Substituting $X_1$ in equation 3

We get,

$X_2=3X_1$

$=3\times2.4=7.2$

$X_1=2.4,\ X_2=7.2$