Question

If a consumers objective function is given by U=X₁^aX₂^b subject to P₁X₁ + P₂X₂ = m. Establish the levels of X₁ and X₂ that maximizes the consumers satisfaction.

Answer 1

Solution

U=X₁^aX₂^b

M=P₁X₁+P₂X₂

MUx₁=$\frac{DU}{DX1}=aX$₁^a-1X₂^b

^MUX2⁼$\frac{DU}{DX_2}=bX_1^aX^{b-1}$

maximizing utility

$\frac{MU x1}{MU x2}$ =$\frac{P 1}{P2}$

$\frac{aX_1^{a-1}x_2^b}{bX_1^aX_2^{b-1}}$ =$\frac{P_1}{P_2}$

solve for X₂

$\frac{aX_2^bX_2^{-b+1}}{ bX_1^aX_1^{-a+1}}$ =$\frac{P_1}{P_2}$

$\frac{aX_2}{bX_1}$ =$\frac{P_1}{P_2}$ X₂₌ $\frac{bP_1}{aP_2}$

M=P₁X₁+P₂X₂

substituting X₂

M= P₁X₁+P₂ $\frac{bP_1}{aP_2}$ X₁

_M=$\frac{aP_1X_1}{a}$ +$\frac{bP_1X_1}{a}$

M=$\frac{(a+b)P_1}{a}$ X₁

X₁= $\frac{aM}{(a+b)P_1}$

X₂= $\frac{bP_1}{aP2}$ x $\frac{aM}{(a+b)P_1}$

X₂₌$\frac{bM}{(a+b)P_2}$