Question

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

Derive the budget constraint:

I = PxX + PyY

600 = X + 2Y

The utility maximizing rule is where ($\frac{MUx}{MUy}) = (\frac{Px}{Py})$:

TU(x,y) = 3x²y

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

MUy = $\frac{\partial U} {\partial y} = 3x^{2}$

$\frac{Px}{Py} = \frac{1}{2}$

$\frac{6xy}{3x^{2} } = \frac{1}{2}$

$\frac{2y}{x } = \frac{1}{2}$

Y = $\frac{x}{4}$

Substitute in the budget constraint:

600 = X + 2Y

600 = X + 2($\frac{x}{4}$)

Multiply both sides by 4:

2400 = 4X + 2X

2400 = 6X

X = 400

Y = $\frac{x}{4}$ = $\frac{400}{4}$ = 100

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

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

 

MRTSxy = $\frac{MUx}{MUy}$

MUx = 6xy

MUy = 3x²

MRTSxy = $\frac{6xy}{3x^{2} }$ = $\frac{2y}{x }$ = 2(100)/400 = $\frac{1}{2 }$

MRTSxy = $\frac{1}{2 }$