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=−X+Y1
U=X+YMUX=δxδU=1MUY=δYδU=1MRS=MUY−MUXMRS=−1....(1)
V=X3+Y3MUX=3X2MUY=3Y2
MRS=MUY−MUXMRS=Y2−X2MRS=(−YX)2....(2)
Now for
W=−1(X+Y)−1MUX=δXδW=(X+Y)−2MUY=δYδW(X+Y)−2MRS=MUY−MUXMRS=(X+Y)−2−(X+Y)−2MRS=−1...(3)
Equition 1 and 3 show that U and W represent the same preferences but V does not.
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