Question

Given the utility function, U = x²y², with the budget constraint , M = P₁ x + P₂ y , where U is the utility, x is the quantity of commodity one and Y is the quantity of commodity two and M is income of the consumer .

a. find the utility maximizing quantities of both commodities

b. if the P₁= 2 and P₂ and M =50, determine the specific quantities of both commodities

c determine whether utility is maximized or minimized

Answer 1

$Max U=X^2Y^2$

$St. P_1X+P_2Y=M$

$L=X^2Y^2-\lambda$$(P_1X+P_2Y-M)$

$dL/dX=2XY^2-\lambda$$P_1=0 ......i$

$dL/dY=2X^2Y-\lambda$$P_2 =0......ii$

$dL/d\lambda$$=P_1X+P_2Y-M=0....iii$

Rearranging equation i and ii and dividing

$2XY^2/2X^2Y=P_1/P_2$

$Y=P_1X/P_2.......iv$

Hence; $X=P_2Y/P_1.....v$

Replacing equation iv and v in equation iii then

$P_1(P_2Y/P_1)+P_2Y=M$

$2P_2Y=M$

$Y=M/2P_2$ .......Utility maximizing quantity of Y

Hence; $X=M/2P_1.......$ Utility maximizing quantity of X

b) Specific Maximizing quantities given that P₁ =2 P₂ and M=50

$Y=50/2(50) =0.5$

$X=50/2(2)=12.5$

c) It is maximized since the consumer is consuming certain amounts of each commodity and not one commodity hence achieving maximum utility.