<p>I'm going to answer my own question. Consider multiplying all prices and income (i.e., the wage) by a common factor <span class="math-container">$\lambda&gt;0$</span>. The new ideal price index is just <span class="math-container">$P' = \lambda P$</span>. The new wage is just <span class="math-container">$W' = \lambda W$</span>. So real expenditure, <span class="math-container">$C' = W'/P' = W/P = C$</span>, is invariant. The "numeraire" should now be <span class="math-container">$C' (P')^{\eta} = \lambda^{\eta}$</span>, hence prices relative to the new "numeraire" should now be <span class="math-container">$p_j' = p_j \lambda^{1-\eta}$</span>, not <span class="math-container">$\lambda p_j$</span>. That's a contradiction---<span class="math-container">$CP^{\eta}$</span> is not a stable unit of account.</p>

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

CC BY-SA 4.0
2019-10-18T12:56:15.123Z
2019-10-18T12:56:15.123Z
Levi Crews
<p>Consider a static model with CES demand. Real aggregate demand is
<span class="math-container">$$
C = \left(\sum_j c_j^{(\eta-1)/\eta}\right)^{\eta/(\eta-1)}
$$</span>
and labor supply is inelastic, so the budget constraint is
<span class="math-container">$
\sum_j p_j c_j = W
$</span>
and the demand function is
<span class="math-container">$$
c_j^{*} = \frac{C P^{\eta}}{p_j^{\eta}}
$$</span>
where
<span class="math-container">$$
P = \left(\sum_j p_j^{1-\eta} \right)^{1/(1-\eta)}
$$</span>
is the ideal price index. Can I set <span class="math-container">$C P^{\eta} = 1$</span> as my numeraire so that <span class="math-container">$c_j^{*} = p_j^{-\eta}$</span>? More generally, what is required of a numeraire?</p>

<general-equilibrium><new-keynesian-economics><ces-function>
Can you set $C / P^{\eta}$ to be the numeraire in a NK model?
Levi Crews
# Can you set $C / P^{\eta}$ to be the numeraire in a NK model?

(Asked by: username_0 on 2019-10-16)

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?

**Tags:** general-equilibrium, new-keynesian-economics, ces-function


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## 1 answers

### Answer by username_0 on 2019-10-18. Score: 1 | Accepted answer

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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