Question

What is the MRS of the utility function, u = x₁ ^2/4 ∙ x₂ ^1/4?

Answer 1

MRS = Marginal utility of x1/Marginal utility of x2.

Marginal utility of x1 = $\frac{\partial u}{\partial x1} = \frac{2}{4}x_{1}^{\frac{2}{4}-1} = \frac{1}{2}x_{1}^{\frac{1}{2}-1} = \frac{1}{2}x_{1}^{\frac{-1}{2}}$

Marginal utility of x2 = $\frac{\partial u}{\partial x2} = \frac{1}{4}x_{2}^{\frac{1}{4}-1} = \frac{1}{4}x_{2}^{\frac{-3}{4}}$

So, MRS = Marginal utility of x1/Marginal utility of x2 = $\frac{\frac{1}{2}x_{1}^{-1/2}}{\frac{1}{4}x_{2}^{\frac{-3}{4}}}$ = $\frac{4x_{2}^{3/4}}{2x_{1}^{1/2}} = \frac{2x_{2}^{3/4}}{x_{1}^{1/2}}$

The MRS of the utility function is $\frac{2x_{2}^{3/4}}{x_{1}^{1/2}}$