Question

Martha s preference over two goods x and y is represented by the utility function U = 5xy. Her budget constraint is given by 5x + y = 30

a) Find the MRS

b)Find MRT

c)Given her budget how much of each goods should she consume in order to maximise her utility? show your work.

Answer 1

A).

U=5xy

budget line;

5x+y=30

MRS_xy=$\frac{MUx}{MUy}$ =$\frac{px}{Py}$ ,from the budget line p_x=$5,P_y=$1

=$\frac{5}{1}$

MRS_xy=5

B).

U=5xy

budget line;

5x+y=30

MRT_xy=$\frac{MCx}{MCy}$ =$\frac{px}{Py}$ ,from the budget line p_x=$5,P_y=$1

=$\frac{5}{1}$

MRT_xy=5

C).

At optimal utlity :

$\frac{MUx}{MUy}$ =$\frac{Px}{Py}$ ,from budget line P_x=$5,P_y=$1

MU_X=$\frac{du}{dx}$ 5xy

MU_X=5Y

MU_Y=$\frac{du}{dy}$ 5XY

MU_Y=5X

Therefore,

$\frac{5Y}{5X}=\frac{5}{1}$

$\frac{Y}{X}=5$

Y=5X (Substitute in the budget line)

5x+y=30

5x+(5x)=30

10x=30

x=3

5(3)+y=30

15+y=30

y=30-15

y=15

at maximum utility X=3,Y=15