Answer to Question #201208 in Microeconomics for Ishaq Mubarak

Question #201208

If a consumers objective function is given by U=X1aX2b subject to P1X1 + P2X2 = m. Establish the levelsbof X1 and X2 that maximizes the consumers satisfaction.


1
Expert's answer
2021-06-01T12:08:53-0400

Solution

U=X1aX2b

M=P1X1+P2X2

MUx1=DUDX1=aX\frac{DU}{DX1}=aX1a-1X2b-1

maximizing utility


MUx1MUx2\frac{MU x1}{MU x2} =P1P2\frac{P 1}{P2}


solve for X2

aX2bX2b+1bX1aX1a+1\frac{aX_2^bX_2^{-b+1}}{ bX_1^aX_1^{-a+1}} =P1P2\frac{P_1}{P_2}


aX2bX1\frac{aX_2}{bX_1} =P1P2\frac{P_1}{P_2} X2= bP1aP2\frac{bP_1}{aP_2}


M=P1X1+P2X2

substituting X2


M= P1X1+P2 bP1aP2\frac{bP_1}{aP_2} X1


M=aP1X1a\frac{aP_1X_1}{a} +bP1X1a\frac{bP_1X_1}{a}


M=(a+b)P1a\frac{(a+b)P_1}{a} X1


X1= aM(a+b)P1\frac{aM}{(a+b)P_1}


X2= bP1aP2\frac{bP_1}{aP2} x aM(a+b)P1\frac{aM}{(a+b)P_1}


X2=bM(a+b)P2\frac{bM}{(a+b)P_2}


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