Question

Justin has the utility function U = xy, with the marginal utilities MUx = y and MUy = x. The price of x is $2, the price of y is p_y, and his income is 40. When he maximizes utility subject to his budget constraint, he purchases 5 units of y.

(a)  What must be the price of y and the amount of x consumed? (1 marks).

(b)  Prove that this allocation follows the equi-marginal principle (2 marks).

(c)  What would be the new bundles of x, y if Px was $3 (2 marks).

Answer 1

Solution:

a.). U = xy

MUx = y

MUy = x

Px = 2

Py = py

Income = 40

Derive the budget constraint: I = PxX + PyY

Where: I = Income = 40

           Px = Price of X = 2

           Py = Price of Y = py

           Y = 5

40 = 2X + 5y

We know that: $\frac{MU_{x} }{MU_{y}} = \frac{P_{x} }{P_{y}} = \frac{y}{x} = \frac{2 }{5} = 5y = 2x$

$y = \frac{2 }{5}x$

Substitute in the budget constraint:

40 = 2X + 5(2/5x)

40 = 2X + 2X

40 = 4X

X = 10 Units

The amount of x consumed = 10 units

Substitute to get the price of y:

$y = y = \frac{2 }{5}x = y = \frac{2 }{5}\times 10 = 4$

The price of y = $4

b.). The equi-marginal principle states that consumers will select a combination of goods to maximize their total utility. Therefore, this will occur where $\frac{MU_{x} }{MU_{y}} = \frac{P_{x} }{P_{y}}$.

 

MUx = y = 5

MUy = x = 10

Px = 2

Py = 4

$\frac{5 }{2} = \frac{10}{4}$

2.5 = 2.5

This signifies that the allocation in question adheres to the equi-marginal principle since it satisfies the condition to maximize total utility.

c.). If Px = 3:

$\frac{MU_{x} }{MU_{y}} = \frac{P_{x} }{P_{y}} = \frac{y }{x} = \frac{3 }{5} = 5y = 3x$

y = $\frac{3 }{5}x$

Substitute for x in the budget constraint:

40 = 2X + 5($\frac{3 }{5}x)$

40 = 2X + 3X

40 = 5X

X = 8 Units

The amount of x consumed = 8 units

Substitute to get the price of y:

$y = \frac{3 }{5}x = \frac{3 }{5}\times 8 = \frac{24 }{5} = 4.80$

The price of y = $4.80

 

The new bundles of x and y will be as follows: U(x,y) = (8, 5)