Question

Suppose an individual had a utility function given by: U=X^0.4*Y^5. The price of Good X is $5 and the price of Good Y is $1. The individual has a budget of $472.50.

Solve the optimization condition for Y given the values above and fill in the blank below.

Y = ____ X?

Answer 1

Answer : Given,

Utility function : U = X^0.4 * Y^5

Budget equation : M = Px * X + Py * Y

Here, M = Budget; Px = Price of good X; Py = Price of good Y; X and Y are quantities of good X and good Y respectively.

By putting all given values in budget equation, we get,

472.50 = 5X + 1Y

=> 472.50 = 5X + Y

Now, the Lagrangian function becomes :

L = U + $\lambda$ (472.50 - 5X - Y)

=> L = X^0.4 * Y^5 + $\lambda$ (472.50 - 5X - Y)

The first order conditions with respect to X and Y are :

$\partial$L / $\partial$X = 0.4X^(0.4 - 1) * Y^5 - 5$\lambda$ = 0

=> 0.4X^(-0.6) * Y^5 = 5$\lambda$

=> [0.4X^(-0.6) * Y^5] / 5 = $\lambda$ ......... (i)

$\partial$L / $\partial$Y = X^0.4 * 5Y^(5 - 1) - $\lambda$ = 0

=> X^0.4 * 5Y^4 = $\lambda$ ............ (ii)

From equation (i) and (ii) we get,

[0.4X^(-0.6) * Y^5] / 5 = X^0.4 * 5Y^4

=> 0.4X^(-0.6) * Y^5 = [X^0.4 * 5Y^4] × 5

=> 0.4X^(-0.6) * Y^5 = 25 * X^0.4 * Y^4

=> Y^5 / Y^4 = (25 / 0.4) * [X^0.4 / X^(-0.6)]

=> Y^5 × Y^(-4) = 62.5 * [X^0.4 × X^0.6]

=> Y^[5 + (-4)] = 62.5 * X^(0.4 + 0.6)

=> Y = 62.5 X

Therefore, Y = 62.5 X.