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I'm being asked to verify that two utility functions, $U(x_1,x_2)$ and $V(x_1,x_2)$ have the same indifference curves and the same MRS.

There is no problem in verifying the MRS to be the same, but how can it be verified that the two utility functions have the same I.C.? Should I assign a random level of utility and set them equal?

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Here is a way to approach the problem:

An indifference curve is a set of consumption bundles all of which yield the same utility (the consumer being, of course, indifferent to which she consumes). Nonetheless, the numerical level of utility is inconsequential, it is just a placeholder for the more fundamental (and ordinal) preference--any monotone transformation of a utility function represents the preferences.

So let $I_U(x_1,x_2)$ denote the indifference curve (of $U$) that contains $(x_1,x_2)$. That is $$I_U(x_1,x_2) = \{(x'_1,x'_2) \in X \mid U(x'_1,x'_2) = U(x_1,x_2)\}$$ Define $I_V$ in an analogous way.

So showing the indifference curves are the same amounts to showing $I_U(x_1,x_2)= I_V(x_1,x_2)$ for all $(x_1,x_2) \in X$.

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