Exercise 9.11

The CPI Index

⬅ Return

Problem

Show that if the technology for producing the composite consumption given in (9.2) is of the CES form, then the consumption price level, \(P_t\), can be expressed as a CES function of the nominal prices of tradable and nontradables, \(\mathcal{E}_t\) and \(P^N_t\), respectively.

Answer

The Aggretator is:

\[ c = \left[ a (c^T)^{1-\frac{1}{\xi}} + (1-a) (c^N)^{1-\frac{1}{\xi}} \right]^{\frac1{1-\frac1{\xi}}} \tag{*} \]

Define the CPI index as the solution to

\[ P c = P^T c^T + P^Nc^N \]

where \(\{c^T, c^N\}\) solves

\[ \min_{c^T, c^N} P^T c^T + P^N c^N \]

subject to (*)

The solution to this problem is

\[ P = \left[a^{\xi}{P^T}^{1-\xi} + (1-a)^{\xi}{P^N}^{1-\xi} \right]^{\frac{1}{1-\xi}} \]