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Building on @Art's answer, you can also think of it in terms of a dynamic process. Given some initial endowment of goods X and Y, everyone in the economy will want to trade goods X and Y back and forth until they are indifferent to further trading - which is to say, until their private MRS is equal to the price ratio. That's why "marginal rates of ...

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One explanation I could think of: MRS (which is equal to $MU_x/MU_y$) is equal to the price ratio $P_x/P_y$. In equilibrium, all consumers will consume where MRS = price ratio, and hence MRS will be "identical" across consumers. I agree that this might not be a terribly good way of putting it.

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I'm going to answer my own question. Consider multiplying all prices and income (i.e., the wage) by a common factor $\lambda>0$. The new ideal price index is just $P' = \lambda P$. The new wage is just $W' = \lambda W$. So real expenditure, $C' = W'/P' = W/P = C$, is invariant. The "numeraire" should now be $C' (P')^{\eta} = \lambda^{\eta}$, hence prices ...

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