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Consider a static model with CES demand. Real aggregate demand is $$ C = \left(\sum_j c_j^{(\eta-1)/\eta}\right)^{\eta/(\eta-1)} $$ and labor supply is inelastic, so the budget constraint is $ \sum_j p_j c_j = W $ and the demand function is $$ c_j^{*} = \frac{C P^{\eta}}{p_j^{\eta}} $$ where $$ P = \left(\sum_j p_j^{1-\eta} \right)^{1/(1-\eta)} $$ is the ideal price index. Can I set $C P^{\eta} = 1$ as my numeraire so that $c_j^{*} = p_j^{-\eta}$? More generally, what is required of a numeraire?

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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 relative to the new "numeraire" should now be $p_j' = p_j \lambda^{1-\eta}$, not $\lambda p_j$. That's a contradiction---$CP^{\eta}$ is not a stable unit of account.

Another way to see the problem is that demand is no longer homogeneous of degree zero in prices and income under this normalization: $(c_j^{*})' = \lambda^{-\eta} p_j^{-\eta}$, so $(c_j^{*})' / c_j^{*} = \lambda^{-\eta}$.

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