Question

Consider a consumer consuming two goods X, and Y, faces the following utility function: U(X,Y)=30X4/5Y1/5 assume further that price X is $5 per unit, price of Y is $10 per unit and income of the consumer is $2000. based on the above information, find the optimum combination of X and Y which maximize the utility

Answer 1

Utility is maximized at the point where the marginal utility per dollar spent on each good is the same. That is:

$$\dfrac{MU_x}{MU_y} = \dfrac{P_x}{P_y}$$

The utility function is given as:

$$U = 30X^{1/5}Y^{4/5}$$

The marginal utility for good X is:

$$MU_x = 6X^{-4/5}Y^{4/5}$$

And the marginal utility for good Y is:

$$MU_y = 24X^{1/5}Y^{-1/5}$$

Therefore:

$$\dfrac{6X^{-4/5}Y^{4/5}}{ 24X^{1/5}Y^{-1/5}} = \dfrac{5}{10}$$

$$\dfrac{Y}{X}= 2$$

Solving for X and Y each at a time:

$$Y = 2X.........(i)$$

$$X = 0 .5Y..........(ii)$$

The consumer has an income of $2000. Therefore, the budget line is:

$$2000 = 5X + 10Y$$

Substituting each of the equations above into the budget constraint:

$$2000 = 5X + 10(2X)$$

$$2000 = 25X$$

$$X ^*= \dfrac{2000}{25} = 80$$

$$2000 =5(0.5Y) + 10Y$$

$$2000 =12.5Y$$

$$Y^* = \dfrac{2000}{12.5} = 160$$

Therefore:

$$\color{red}{(X^*, Y^*) = (80, 160)}$$