Question

If an individual’s preferences are described by the utility function U(X1 , X2 ) = X12 + X22, graph the indifference curve for U = 20 and U= 40.Find the optimal consumption quantities if P1 = US$2.50 ;         P2 = US$ 7.50; and M = US$ 60.

Answer 1

Utility function

$U(x_1,x_2)=x_1^2+x_2^2$

so,

The indifference curve

$U=20\\V=40$

Given information

$p_1=US\$2.50\\p_2=US\$7.50$

$M=US\$60$

Budget line equation

$p_1x_1+p_2x_2=M..............................................(1)$

substitute these values in the above equation 1

$\$2.50x_1+\$7.50x_2=60....................................(2)\\From\space U(x_1,x_2)=x_1^2+x_2^2\\MRS(x_1,x_2)=\frac{MUx_1}{MUx_2}\\MUx_1=2x_1,MUx_2=2x_2\\MRS(x_1,x_2)=\frac{2x_1}{2x_2}\\=\frac{x_1}{x_2}$

For optimal bundle

equation slope of budget line and MRS

$\frac{p_1}{p_2}=MRS_{(x_1,x_2)}\\$

$\frac{2.5}{7.5}=\frac{x_1}{x_2}$

$x_2(2.5)=(7.5)x_1\\x_2=3x_1....................................................(3)$

putting x2 value in eq 2

we get

$\$2.50x_1+\$7.50x_2=60\\\$2.50x_1+\$7.50\times3x_1=60\\\$2.50x_1+\$22.5x_1=60\\25x_1=60\\x_1=\frac{60}{25}\\x_1=2.4$

putting x1 value in eqn 3

$x_2=3x_1\\=3\times2.4\\x_2=7.2$

x1=2,4

x2=7.2