Answer to Question #267527 in Economics of Enterprise for RAHEL

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

Solution:

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

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

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

 

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

MRS Function = $\frac{Y+1}{X+1 }$

 

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

18 = 2X + 4Y

Set MRS = Px$\div$ Py

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

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 = $\frac{X}{2 } - \frac{1}{2} = \frac{5}{2 } - \frac{1}{2} = 2.5 - 0.5 = 2$

Y = 2

The utility-maximizing consumption bundle (U_X,Y) = (5, 2)