We saw in class that since we are only interested in the ordering of bundles, a positive monotonic transformation (one that does not change the order) of a utility function also represents the same preferences. Mathematically, a positive monotonic transformation preserves the marginal rate of substitution (MRS) of the original utility at any point (X,Y). Prove that the following utility functions represent the same preferences by showing that the MRS of (a), (b) and (c) is exactly the same.
(a) U(X,Y ) = X1/4.Y 3/4
(b) U(X,Y ) = 1/4.ln(X) + 3/4.ln(Y )
(c) U(X,Y ) = X.Y 3.