Answer to Question #204458 in Financial Math for Thabiso Mohlakoana

Question

6. Consider the following utility function: 𝒖 = 𝟒𝟎𝟎 (𝟏 + 𝟏)−𝟏. 𝒙𝒚

a. Is this utility function homogenous? Briefly explain. [3]

b. Use implicit differentiation to find the 𝑀𝑅𝐶𝑆. [2]

c. Write down an expression for the indifference curve if 𝒖 = 𝟐𝟎. [2]

Answer 1

(a) The function is not homogeneous

as $u=400[\frac{1}{x}+\frac{1}{y}]^{-1}$

$=\frac{400}{(y)^{-1}}[1-\frac{y}{x}+\frac{y^3}{x^2}-\frac{y^3}{x^3}+......]$

$=40y[1-\frac{y}{x}+\frac{y}{x}^2-\frac{y}{x}^3+......]$

not homogeneous.

(b)$mu=\frac{\delta u}{\delta x}(x,y)=400(-1)[(\frac{1}{x}+\frac{1}{y})]^{-2}$

$=(\frac{-1}{x^2})$

similarly

$mu_2=\frac{\delta u}{\delta x}(x,y)=400(-1)[(\frac{1}{x}+\frac{1}{y})]^{-2}[\frac{-1}{y^2}]$

so $MRS=\frac{\delta u}{\delta x}=\frac{\delta u/\delta x}{\delta u/\delta y}$

$=\frac{-Mu_1}{Mu_2}=\frac{-y^2}{x^2}$

$MRS=\frac{y^2}{ x^2}$

(c)given that u=20

so, $u=20=400[(\frac{1}{x}+\frac{1}{y})]^{-1}$

the required indifference curve is

$\frac{1}{x}+\frac{1}{y}=20$