Question

Utility function of an individual is given by begin mathsize 12px style U equals f left parenthesis x comma y right parenthesis equals X cubed Y to the power of 4 end style. find out the optimal quantities of the two goods, if it is given that price of good X is N$8 per unit and price of good Y is N$4 per unit and income of the individual is equal to N$1200?

Answer 1

Ans. Utility function, U = X^3 * Y^4

Marginal Utility of X, MUX = dU/dX = 3X^2 * Y^4

Marginal Utility of Y, MUY = dU/dY = 4X^3 * Y^3

=> Marginal rate of substitution, MRS = MUX/ MUY = (3/4)*(Y/X)

At utility maximizing level,

MRS = Price of X/ Price of Y

=> (3/4)*(Y/X) = 8/4

=> Y = (8/3)*X --> Eq1

Substituting Eq1 in budget constraint, 8X + 4Y = 1200, we get

=> 8X + (32/3)*X = 1200

=> X = 64.2857 units

and from Eq 1, Y = 171.42857 units

Thus, the optimal quantity of X is about 64.3 units and optimal quantity of Y is about 171.4 units

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