Question

Check if the following utility functions represent the same preferences: u = x+y, v = x3 +

y3, w = -1/(x+y). Give reasons for your answer.

Answer 1

Using marginal rate of substitution approach which says that if the different utility function gives us the Same rate of substitution when they represent the same preferences

$U=X+Y, V=X^3+Y^3, W=-\frac{1}{X+Y}$

$U=X+Y\\ MUX=\frac{\delta U}{\delta x}=1\\ MUY=\frac{\delta U}{\delta Y}=1\\ MRS=\frac{-MUX}{MUY}\\MRS=-1....(1)$

$V=X^3+Y^3\\MUX=3X^2\\MUY=3Y^2$

$MRS=\frac{-MUX}{MUY}\\MRS=\frac{-X^2}{Y^2}\\MRS=\frac-(\frac{X}{Y})^2....(2)$

Now for

$W=-1(X+Y)^{-1}\\ MUX=\frac{\delta W}{\delta X}=(X+Y)^{-2}\\MUY=\frac{\delta W}{\delta Y}(X+Y)^{-2}\\ MRS=\frac{-MUX}{MUY}\\ MRS=\frac{-(X+Y)^{-2}}{(X+Y)^{-2}}\\MRS=-1...(3)$

Equition 1 and 3 show that U and W represent the same preferences but V does not.