# Can you set $C / P^{\eta}$ to be the numeraire in a NK model?

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?

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 $$(c_j^{*})' / c_j^{*} = \lambda^{-\eta}$$.