Question

1.     Suppose we have the utility function, U = XY + X + Y.

a.      Find the function for the marginal rate of substitution.

1. If prices are P_x = $2 and P_y = $4, and if income is M = $18, find the utility maximizing consumption bundle.

Answer 1

1.). a.). MRS = $\frac{MU_{x} } {MU_{y}} = \frac{P_{x} } {P_{y}}$

MUx = $\frac{\partial U} {\partial X}$ = Y + 1

MUy = $\frac{\partial U} {\partial Y}$ = X + 1

 

MRS = $\frac{MU_{x} } {MU_{y}}$ = Y + 1/ X + 1

MRS Function = Y + 1/ X + 1

 

1.). Budget constraint: I = PxX + PyY

18 = 2X + 4Y

Set MRS = $\frac{P_{x} } {P_{y}}$

$Y + \frac{1} {X} = \frac{2} {4}$

$Y = \frac{X} {2} -\frac{1} {2}$

Substitute in the budget constraint to get X:

18 = 2X + 4Y

18 = 2X + 4($\frac{X} {2} -\frac{1} {2}$)

X = 5

$Y = \frac{X} {2} -\frac{1} {2}$

Y = 2

The utility maximizing consumption bundle = (5, 2)