Question

A consumer derives utility from good X and Y according to the following utility function:
U(X, Y) = X^(3/4)Y^(1/4) The price of good X is $15 while good Y is priced $10. The consumer’s budget is $160. What is the utility maximizing bundle for the consumer?

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Answer 1

Given,

U = X^3/4 Y^1/4 = X^0.75 Y^0.25

MUX = Partial derivative of U with respect of X

         = 0.75Y^0.25 / X^0.25

MUY = Partial derivative of U with respect of Y

          = 0.25X^0.75 / Y^0.75

Budget constraint: 160 = 15X + 10Y …..(1)

MUX/PX = MUY/PY

(0.75Y^0.25 / X^0.25) / 15 = (0.25X^0.75 / Y^0.75) / 10 ……… (2)

In order to get the demand function of X, the above equation (2) should be solved for Y.

10Y^0.75 × 0.75Y^0.25 = 15X^0.25 × 0.25X^0.75

7.5Y = 3.75X

Y = (3.75 / 7.5) X

Y = 0.5X

Now Y = 0.5X should be placed in the equation (1)

160 = 15X + 10Y

160 = 15X + 10 × 0.5X

160 = 15X + 5X

160 = 20X

X = 8

In order to get the demand function of Y, the above equation (1) should be solved as below:

160 = 15X + 10Y

Now, by placing (X = 8),

160 = 15X + 10Y

160 = 15 × 8 + 10Y

160 = 120 = 10Y

40 = 10Y

Y = 4

Answer: Required bundle, X = 4, Y = 8