Answer to Question #257927 in Microeconomics for seileza

Question #257927

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.


1
Expert's answer
2021-10-29T11:13:38-0400

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=X3+Y3,W=1X+YU=X+Y, V=X^3+Y^3, W=-\frac{1}{X+Y}

U=X+YMUX=δUδx=1MUY=δUδY=1MRS=MUXMUYMRS=1....(1)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=X3+Y3MUX=3X2MUY=3Y2V=X^3+Y^3\\MUX=3X^2\\MUY=3Y^2

MRS=MUXMUYMRS=X2Y2MRS=(XY)2....(2)MRS=\frac{-MUX}{MUY}\\MRS=\frac{-X^2}{Y^2}\\MRS=\frac-(\frac{X}{Y})^2....(2)

Now for

W=1(X+Y)1MUX=δWδX=(X+Y)2MUY=δWδY(X+Y)2MRS=MUXMUYMRS=(X+Y)2(X+Y)2MRS=1...(3)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.




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