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:

$I=Px\times X+Py\times Y\\600=X+2\times Y$

The utility maximizing rule is where

$(\frac{MUx}{MUy})=(\frac{Px}{Py})$

$TU(x,y)=3\times x^2\times y$

$MUx=\frac{\partial U}{\partial x}=6\times x\times y$

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

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

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

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

Substitute in the budget constraint:

$600=X+2\times Y$

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

Multiply both sides by 4:

$2400=4\times X+2\times X\\2400=6\times X\\X=400\\Y=\frac{X}{4}=\frac{400}{4}=100$

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

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

$MRTSxy =\frac{MUx}{MUy}\\MUx=6\times x \times y\\MUy=3\times x^2$

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