Question

Given utility function U= P x X+P y Y

where P X = 12 Birr, Birr, P Y = 4 Birr and the income

of the consumer is, M= 240 Birr.

Answer 1

Solutions:

A. Find the utility maximizing combinations of X and Y.

A. Utility function is given:

U= PxX+PyY

Putting the values of Px and Py in the utility function:

U= 12X + 4Y

Differentiate with respect to X:

$\frac{dU}{dX}=MUx=12$

Differentiate with respect to Y:

$\frac{dU}{dY}=MUy=4$

At the equilibrium level:

$\frac{MUx}{MUy}=\frac{Px}{Py}$

$\frac{12}{4}=\frac{12}{4}$

3 = 3

Thus, consumer is in equilibrium condition

Budget function:

M = PxX + PyY

240 = 12X + 4Y ......Divide by 4

60 = 3X + Y

Thus, the equilibrium utility maximizing combinations of X and Y is :

3X + Y = 60

B. Calculate marginal rate of substitution of X for Y (MRSX,Y) at equilibrium and interpret your result

At the equilibrium level:

$MRSxy= \frac{Px}{Py}$

$\frac{12}{4}$ = 3

= 3

Thus, the marginal rate of substitution of X and Y (MRSxy) is 3.

Therefore, it means that the consumer is willing to give up 3 units of X to obtain an additional unit of Y at the same utility level.