Question

Derive the demand function for X in terms of the price of the goods and income given the utility function U = X^1/5 Y^4/5

Answer 1

Answer - We have to derive demand function for good X in terms of price of good X(Px) and income (I). The utility function is given, U = X^1/5 . Y^4/5

We will need marginal utility of good X and good Y. Thus we can find it from utility function.

MUx = U /X

MUx = 1/5 * X^-4/5 * Y^4/5

and

MUy = U / Y

MUy = 4/5 * X^1/5 *Y^-1/5

We know that a consumer will try to get maximum utility from the utility function.

MUx / Px = MUy/Py

(1/5 * X^-4/5 * Y^4/5) / (Px) = (4/5 * X^1/5 *Y^-1/5) / (Py)

(1/5 * X^-4/5 * Y^4/5)/ (4/5 * X^1/5 *Y^-1/5) = Px / Py

1/4 (Y / X) = Px / Py

Y = (4Px. X ) / Py

Place value of 'Y' into budget constraint. Budget constraint is given below where income is represented by 'I'

I = Px. X + Py. Y (Y = 4Px. X / Py)

I = Px .X + Py (4Px. X / Py)

I = Px. X + 4Px. X

I = 5Px. X

X = I / 5Px

This is demand function of good X. Demand for good is positively related to income (I) and negatively related to price of good X(Px)