<p>It is going to be easier to use the log form for the Cobb-Douglas: <span class="math-container">$\ln(u) = \sum_i \alpha_i \ln(q_i)$</span>. We can also assume that <span class="math-container">$\sum_j \alpha_j = 1$</span>.
The expenditure function is given by:
<span class="math-container">$$
\min_{q} \sum_i p_i q_i \text{ s.t. } \sum_i \alpha_i \ln(q_i) = \ln(u).
$$</span>
The first order conditions give:
<span class="math-container">$$
\begin{align*}
&amp;p_i = \lambda \frac{\alpha_i}{q_i}.\\
&amp;\ln(u) = \sum_j \alpha_j \ln(q_j).
\end{align*}
$$</span>
Substituting the first into the second is given by:
<span class="math-container">$$
\ln(u) = \sum_j \alpha_j \ln(\lambda) + \sum_j \alpha_j \ln(\alpha_j/p_j).
$$</span>
Then (as <span class="math-container">$\sum_j \alpha_j = 1$</span>):
<span class="math-container">$$
\ln(\lambda) = \ln(u) - \sum_j \alpha_j \ln(\alpha_j/p_j).
$$</span></p>
<p>So:
<span class="math-container">$$
q_i = \frac{\alpha_i}{p_i} u \prod_j \left(p_j/\alpha_j\right)^{\alpha_j}.
$$</span>
Multiplying both sides by <span class="math-container">$p_i$</span> and adding over all <span class="math-container">$i$</span> (taking into account <span class="math-container">$\sum_j \alpha_j = 1$</span>) gives:
<span class="math-container">$$
\sum_j p_j q_j = u \prod_j (p_j/\alpha_j)^{\alpha_j}
$$</span>
Let <span class="math-container">$s_i = \frac{p_i q_i}{\sum_j p_j q_j}$</span> be the expenditure share for good <span class="math-container">$i$</span>. From the above two equations, we get:
<span class="math-container">$$
s_i = \alpha_i.
$$</span>
This gives the conventional knowledge that Cobb-Douglas utility functions lead to constant expenditure shares (equal to <span class="math-container">$\alpha_i$</span>).</p>
<p>Now,
<span class="math-container">$$
c(p,u) = \sum_{j} p_j q_j = u \prod_j (p_j/\alpha_j)^{\alpha_j}
$$</span>
So:
<span class="math-container">$$
\ln(c(p,u)) = \ln(u) + \sum_j \alpha_j\left[\ln(p_j) - \ln(\alpha_j)\right].
$$</span></p>
<p>Now, take two price levels <span class="math-container">$p^0$</span> and <span class="math-container">$p^1$</span>. Then the log price index is given by:
<span class="math-container">$$
\ln\left(\frac{c(p^1,u)}{c(p^0,u)}\right) = \ln(c(p^1,u)) - \ln(c(p^0,u)) = \sum_j \alpha_j \left[\ln(p^1_j) - \ln(p^0_j)\right] = \sum_j s_j \ln\left(\frac{p^1_j}{p^0_j}\right).
$$</span>
Equivalently:
<span class="math-container">$$
\frac{c(p^1,u)}{c(p^0,u)} = \prod_i \left(\frac{p^1_j}{p^0_j}\right)^{s_j}.
$$</span></p>

CC BY-SA 4.0
2024-02-26T08:16:02.873Z
2024-02-26T08:16:02.873Z
tdm
<p>how do you get the final form of the price index for a cobb-douglas with n discrete goods? I can't get to the solution where e(p,1) = logP = the sum of (shares * log(P))</p>

<cobb-douglas>
Cobb-Douglas utility function. Deriving the price index with N discrete goods functions
gow
# Cobb-Douglas utility function. Deriving the price index with N discrete goods functions

(Asked by: username_0 on 2024-02-25)

how do you get the final form of the price index for a cobb-douglas with n discrete goods? I can't get to the solution where e(p,1) = logP = the sum of (shares \* log(P))

**Tags:** cobb-douglas


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## 1 answers

### Answer by username_1 on 2024-02-26. Score: 1

It is going to be easier to use the log form for the Cobb-Douglas: $\ln(u) = \sum\_i \alpha\_i \ln(q\_i)$. We can also assume that $\sum\_j \alpha\_j = 1$.
The expenditure function is given by:
$$
\min\_{q} \sum\_i p\_i q\_i \text{ s.t. } \sum\_i \alpha\_i \ln(q\_i) = \ln(u).
$$
The first order conditions give:
$$
\begin{align\*}
&p\_i = \lambda \frac{\alpha\_i}{q\_i}.\\
&\ln(u) = \sum\_j \alpha\_j \ln(q\_j).
\end{align\*}
$$
Substituting the first into the second is given by:
$$
\ln(u) = \sum\_j \alpha\_j \ln(\lambda) + \sum\_j \alpha\_j \ln(\alpha\_j/p\_j).
$$
Then (as $\sum\_j \alpha\_j = 1$):
$$
\ln(\lambda) = \ln(u) - \sum\_j \alpha\_j \ln(\alpha\_j/p\_j).
$$

So:
$$
q\_i = \frac{\alpha\_i}{p\_i} u \prod\_j \left(p\_j/\alpha\_j\right)^{\alpha\_j}.
$$
Multiplying both sides by $p\_i$ and adding over all $i$ (taking into account $\sum\_j \alpha\_j = 1$) gives:
$$
\sum\_j p\_j q\_j = u \prod\_j (p\_j/\alpha\_j)^{\alpha\_j}
$$
Let $s\_i = \frac{p\_i q\_i}{\sum\_j p\_j q\_j}$ be the expenditure share for good $i$. From the above two equations, we get:
$$
s\_i = \alpha\_i.
$$
This gives the conventional knowledge that Cobb-Douglas utility functions lead to constant expenditure shares (equal to $\alpha\_i$).

Now,
$$
c(p,u) = \sum\_{j} p\_j q\_j = u \prod\_j (p\_j/\alpha\_j)^{\alpha\_j}
$$
So:
$$
\ln(c(p,u)) = \ln(u) + \sum\_j \alpha\_j\left[\ln(p\_j) - \ln(\alpha\_j)\right].
$$

Now, take two price levels $p^0$ and $p^1$. Then the log price index is given by:
$$
\ln\left(\frac{c(p^1,u)}{c(p^0,u)}\right) = \ln(c(p^1,u)) - \ln(c(p^0,u)) = \sum\_j \alpha\_j \left[\ln(p^1\_j) - \ln(p^0\_j)\right] = \sum\_j s\_j \ln\left(\frac{p^1\_j}{p^0\_j}\right).
$$
Equivalently:
$$
\frac{c(p^1,u)}{c(p^0,u)} = \prod\_i \left(\frac{p^1\_j}{p^0\_j}\right)^{s\_j}.
$$


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