Answer to Question #227469 in Microeconomics for jooo

Question

1.     Suppose that the total utility function of a consumer is given by TU(x,y) = 3x2 y and the prices of X and Y are 1 Birr and 2 Birr per unit, respectively. If the income of the consumer is 600 Birr and if he spends all of his income on the consumption of commodities of X and Y, find the optimum amount of X and Y that the consumer will consume at equilibrium and find MRTSx,y.

Answer 1

Solution:

Derive the budget constraint:

I = PxX + PyY

600 = X + 2Y

The utility maximizing rule is where (MUx/MUy=Px/Py):

TU(x,y) = 3x²y

MUx = $\frac{\partial U} {\partial x}$ = 6xy

MUy = $\frac{\partial U} {\partial y}$ ∂ U/∂y = 3x²

Px/Py = 1/2

6xy/3x² =1/2

2y/x = 1/2

Y = x/4

Substitute in the budget constraint:

600 = X + 2Y

600 = X + 2(X/4)

Multiply both sides by 4:

2400 = 4X + 2X

2400 = 6X

X = 400

Y = X/4 = 400/4 = 100

TU(x,y) = (400,100)

The optimum amount of X and Y that the consumer will consume at equilibrium = 400 and 100

 

MRTSxy = MUx/MUy

MUx = 6xy

MUy = 3x²

MRTSxy = 6xy/3x² = 2Y/X = 2(100)/400 = 1/2

MRTSxy = 1/2