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

CC BY-SA 4.0
2024-06-10T10:40:54.630Z
Thanks for answering. Why the exponent of the technological constraint is $(1- \epsilon)/\epsilon$ and not $\epsilon/(\epsilon -1)$ ? Also, why the equation for $p_L$ has $\gamma_H$ and the one for $p_H$ has $\gamma_L$?
Maximilian
CC BY-SA 4.0
2024-06-10T13:16:52.030Z
@Maximilian typos :-( Thanks for noticing them.
tdm
CC BY-SA 4.0
2024-06-10T13:22:30.963Z
Thanks, but why the exponent of the technological constraint is (1−ϵ)/ϵ
 and not ϵ/(ϵ−1)?
Maximilian
CC BY-SA 4.0
2024-06-10T06:06:09.327Z
2024-06-11T17:11:58.050Z
2024-06-11T17:11:58.050Z
tdm
<p>I'm studying Chapter 15 of Acemoglu's growth book, which is about skill-biased technological change.</p>
<p>The unique final good is produced in perfect competition by combining the output of the two intermediate sectors, <span class="math-container">$Y_L$</span> and <span class="math-container">$Y_H$</span>, according to the following technological constraint:</p>
<p><span class="math-container">\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}</span></p>
<p>where <span class="math-container">$\epsilon \in [0, \infty)$</span> is the elasticity of substitution between the two intermediates.</p>
<p>Then, the book says that the price of the final good is normalized to one at each <span class="math-container">$t$</span>, which is equivalent to setting the ideal price index of the two intermediates equal to one, that is,</p>
<p><span class="math-container">\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}</span></p>
<p>Can you show me how to derive the last expression, please?</p>

<microeconomics><macroeconomics><economic-growth>
Derivation of the Ideal Price Index
Maximilian
# Derivation of the Ideal Price Index

(Asked by: username_0 on 2024-06-03)

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?

**Tags:** microeconomics, macroeconomics, economic-growth

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## 1 answers

### Answer by username_1 on 2024-06-10. Score: 3

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,
$$
c(p\_L, p\_H, 1) = p\_L Y\_L + p\_H Y\_H = \left[\gamma\_L^\varepsilon p\_L^{1 - \varepsilon} + \gamma\_H^\varepsilon p\_H^{1 - \varepsilon}\right]^{\frac{1}{\varepsilon - 1}}.
$$

> **Comments:**
> **username_0**: Thanks for answering. Why the exponent of the technological constraint is $(1- \epsilon)/\epsilon$ and not $\epsilon/(\epsilon -1)$ ? Also, why the equation for $p_L$ has $\gamma_H$ and the one for $p_H$ has $\gamma_L$?  
> **username_1**: @Maximilian typos :-( Thanks for noticing them.  
> **username_0**: Thanks, but why the exponent of the technological constraint is (1−ϵ)/ϵ  and not ϵ/(ϵ−1)?  

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