Derivation of the Ideal Price Index

Question

I'm studying Chapter 15 of Acemoglu's growth book, which is about skill-biased technological change.

The unique final good is produced in perfect competition by combining the output of the two intermediate sectors, $Y_L$ and $Y_H$, according to the following technological constraint:

\begin{equation} Y(t) = \left[\gamma_L Y_L(t)^{\frac{\epsilon - 1}{\epsilon}} + \gamma_H Y_H(t)^{\frac{\epsilon - 1}{\epsilon}}\right]^{\frac{\epsilon}{\epsilon - 1}} \end{equation}

where $\epsilon \in [0, \infty)$ is the elasticity of substitution between the two intermediates.

Then, the book says that the price of the final good is normalized to one at each $t$, which is equivalent to setting the ideal price index of the two intermediates equal to one, that is,

\begin{equation} \left[\gamma_L^{\epsilon} p_L(t)^{1 - \epsilon} + \gamma_H p_H(t)^{1 - \epsilon}\right]^{\frac{1}{1 - \epsilon}} =1 \end{equation}

Can you show me how to derive the last expression, please?

Answer 1

The price index is given by $c(p_L, p_H,1)$ which is the minimal cost of producing 1 unit of output. It is given by: $$ \min p_L Y_L + p_H Y_H \text{ s.t. } \left[\gamma_L Y_L^{(\varepsilon -1)/\varepsilon} + \gamma_H Y_H^{(\varepsilon-1)/\varepsilon}\right]^{\frac{\varepsilon}{\varepsilon-1}} = 1. $$ The first order conditions give: $$ \begin{align*} &p_L = \lambda Y^{-1} \gamma_L Y_H^{-1/\varepsilon}\\ &p_H = \lambda Y^{-1} \gamma_H Y_L^{-1/\varepsilon}. \end{align*} $$ This gives: $$ Y_H = \left(\frac{p_L}{p_H}\right)^\varepsilon \left(\frac{\gamma_H}{\gamma_L}\right)^\varepsilon Y_L. $$ Substituting into the constraint gives: $$ Y_L = \gamma_L^{\varepsilon} p_L^{-\varepsilon} \left[\gamma_L^\varepsilon p_L^{1- \varepsilon} + \gamma_H^\varepsilon p_H^{1 - \varepsilon} \right]^{-\frac{\varepsilon}{\varepsilon-1}}.\\ Y_H = \gamma_H^{\varepsilon} p_H^{-\varepsilon} \left[\gamma_L^\varepsilon p_L^{1- \varepsilon} + \gamma_H^\varepsilon p_H^{1 - \varepsilon} \right]^{-\frac{\varepsilon}{\varepsilon-1}}. $$ As such, $$ c(p_L, p_H, 1) = p_L Y_L + p_H Y_H = \left[\gamma_L^\varepsilon p_L^{1 - \varepsilon} + \gamma_H^\varepsilon p_H^{1 - \varepsilon}\right]^{\frac{1}{\varepsilon - 1}}. $$