Question

David’s utility function for good X and Y is given by ( , )= 

2 

3

. Where , and I are the price of 

good X, price of good Y and consumer income respectively. 

a. Write the budget Constraint of the consumer. 

b. Drive the demand functions for good X and Y 

c. What combination of X and Y maximizes the consumer’s at I=100, = 4, =5 

d. Calculate the marginal rate of substitution between X and Y at equilibrium and interpret your 

results.

Answer 1

Solution:

a.). Budget constraint:

I = PxX + PyY

 

b.). Demand equations for good X and Y:

Demand equation for good X = MU_x

U (X, Y) = X²Y³

MU_x =$\frac{\partial U} {\partial X} = 2XY^{3}$

Demand equation for good X = 2XY³

MU_y = $\frac{\partial U} {\partial Y} = 3X^{2} Y^{2}$

Demand equation for good Y = 3X² Y²

 

c.). Budget line: I = PxX + PyY

100 = 4X + 5Y

To derive combinations of X and Y:

We set: $\frac{MU_{x} }{MU_{y} } = \frac{Px }{Py }$

MU_x = 2XY³

MU_y = 3X² Y²

$\frac{2XY^{3} }{3X^{2} Y^{2} } = \frac{2Y }{3X }$

Therefore:

Px = 4

Py = 5

$\frac{2Y}{3X} = \frac{4}{5}$

Y = 1.2X

Substitute in the budget line:

100 = 4X + 5Y

100 = 4X + 5(1.2X)

100 = 4X + 6X

100 = 10X

X = 10

Substitute to derive Y:

Y = 1.2X = 1.2(10) = 12

The combinations of good X and good Y that will maximize the consumer's income:

U (X, Y) = (10, 12)

 

d.). MRS_x,y = $\frac{MU_{x} }{MU_{y} } = \frac{2XY^{3} }{3X^{2} Y^{2} } = \frac{2Y }{3X }$

2Y = 2 $\times$ 12 = 24

3X = 3 $\times$ 10 = 30

 

MRS_x,y = $\frac{24}{30} = 0.8$

MRS_x,y = 0.8

Since MRS = 0.8 and positive, then for a given level of utility and, as x rises MRS falls. Convex indifference curves.