Question

Suppose a consumers preferences can be represented by the utility function U (x,y)=x*y .also suppose the consumer has 150 dollar to spend and the price of good x is px =2 dollar and the price of good y is py=3dollar .if the consumer maximize their utility subject to their budget constraints how much of good x and how much of good y will be the consumer purchase?

Answer 1

Solution:

Derive the budget constraint:

I = PxX + PyY

Where: I = Income = 320

           Px = Price of X = 2

           Py = Price of Y = 4

            X = Good X

            Y = Good Y

320 = 2X + 4Y

The condition for utility maximizing bundle is where $\frac{MU_{x} }{MU_{y}} = \frac{P_{x} }{P_{y}}$

MUx = $\frac{\partial U} {\partial Y} = Y$

MUy = $\frac{\partial U} {\partial Y} = X$

$\frac{X} {Y} = \frac{2} {4} = 0.5$

Y = 0.5X

Substitute in the budget constraint:

320 = 2X + 4(0.5X)

320 = 2X + 2X

320 = 4X

X = 80

Substitute to derive Y:

Y = 0.5X

Y = 0.5(80) = 40

Y = 40

Utility maximizing bundle (Uxy) = (80,40)

If the consumer maximizes their utility subject to their budget constraints, the consumer will purchase 80 units of good X and 40 units of good Y.