<p>Given that preferences are homothetic, we have that <span class="math-container">$e(p_1, u) = e(p_1,1) u$</span> and that <span class="math-container">$e(p_0, u) = e(p_0,1) u$</span> (the expenditure function is linear in <span class="math-container">$u$</span>)</p>
<p>So,
<span class="math-container">$$
\frac{e(p_1, u_0)}{e(p_0, u_0)} = \frac{e(p_1,1)}{e(p_0,1)}\frac{u_0 u_1}{u_0 u_1} = \frac{e(p_1, u_1)}{e(p_0, u_1)}
$$</span>
Next, <span class="math-container">$e(p_0, u_1)$</span> is the minimal expenditure at prices <span class="math-container">$p_0$</span> to reach utility level <span class="math-container">$u_1$</span>. We also know that buying <span class="math-container">$x_1$</span> will give us at least utility level <span class="math-container">$u_1$</span>. As such,
<span class="math-container">$$
p_0 x_1 \ge e(p_0, u_1).
$$</span>
Finally, <span class="math-container">$e(p_0, u_0)$</span> is the minimal expenditure at prices <span class="math-container">$p_0$</span> to reach utilty level <span class="math-container">$u_0$</span>. This is reached by buying bundle <span class="math-container">$x_0$</span> so,
<span class="math-container">$$
p_0 x_0 = e(p_0, u_0).
$$</span></p>
<p>Combining all this, we get:
<span class="math-container">$$
\frac{p_1 x_1}{p_0 x_1} \le \frac{e(p_1, u_1)}{e(p_0, u_1)} = \frac{e(p_1, u_0)}{e(p_0, u_0)}.
$$</span></p>

CC BY-SA 4.0
2023-09-15T05:03:45.423Z
2023-09-15T05:03:45.423Z
tdm
<p>Given that
<span class="math-container">$$CPI=\frac{e(p^1,u^0)}{e(p^0,u^0)}$$</span>
and the Paasche index
<span class="math-container">$$PPI=\frac{p^1\cdot x^1}{p^0\cdot x^1}$$</span>
How do I show that <span class="math-container">$PPI\leq CPI$</span> if preferences are homothetic?</p>
<p>Here's what I've done:
Want to show that
<span class="math-container">\begin{align}
&amp;\frac{p^1\cdot x^1}{p^0\cdot x^1}\leq \frac{e(p^1,u^0)}{e(p^0,u^0)}\\
\implies&amp; p^0x^1 e(p^1,u^0)\geq p^1 x^1 e(p^0,u^0)\\
\implies&amp; p^0x(p^1,w^1) e(p^1,u^0)\geq p^1 x(p^1,w^1) e(p^0,u^0)\\
\implies&amp; p^0w^1x(p^1,1) e(p^1,u^0)\geq p^1 w^1x(p^1,1) e(p^0,u^0)\\
\implies&amp; p^0x(p^1,1) e(p^1,u^0)\geq p^1x(p^1,1) e(p^0,u^0)
\end{align}</span>
Since <span class="math-container">$p^1x(p^1,1)=w_1=1$</span>, and <span class="math-container">$e(p^0,u^0)=w^0$</span>, we want to show that
<span class="math-container">$$p^0x(p^1,1) e(p^1,u^0)\geq w^0$$</span></p>
<p>How exactly do I proceed?</p>

<demand>
How to prove Paasche index is less or equal to CPI if preferences are homothetic?
Ludwig Gershwin
# How to prove Paasche index is less or equal to CPI if preferences are homothetic?

(Asked by: username_0 on 2023-09-14)

Given that
$$CPI=\frac{e(p^1,u^0)}{e(p^0,u^0)}$$
and the Paasche index
$$PPI=\frac{p^1\cdot x^1}{p^0\cdot x^1}$$
How do I show that $PPI\leq CPI$ if preferences are homothetic?

Here's what I've done:
Want to show that
\begin{align}
&\frac{p^1\cdot x^1}{p^0\cdot x^1}\leq \frac{e(p^1,u^0)}{e(p^0,u^0)}\\
\implies& p^0x^1 e(p^1,u^0)\geq p^1 x^1 e(p^0,u^0)\\
\implies& p^0x(p^1,w^1) e(p^1,u^0)\geq p^1 x(p^1,w^1) e(p^0,u^0)\\
\implies& p^0w^1x(p^1,1) e(p^1,u^0)\geq p^1 w^1x(p^1,1) e(p^0,u^0)\\
\implies& p^0x(p^1,1) e(p^1,u^0)\geq p^1x(p^1,1) e(p^0,u^0)
\end{align}
Since $p^1x(p^1,1)=w\_1=1$, and $e(p^0,u^0)=w^0$, we want to show that
$$p^0x(p^1,1) e(p^1,u^0)\geq w^0$$

How exactly do I proceed?

**Tags:** demand


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## 1 answers

### Answer by username_1 on 2023-09-15. Score: 1 | Accepted answer

Given that preferences are homothetic, we have that $e(p\_1, u) = e(p\_1,1) u$ and that $e(p\_0, u) = e(p\_0,1) u$ (the expenditure function is linear in $u$)

So,
$$
\frac{e(p\_1, u\_0)}{e(p\_0, u\_0)} = \frac{e(p\_1,1)}{e(p\_0,1)}\frac{u\_0 u\_1}{u\_0 u\_1} = \frac{e(p\_1, u\_1)}{e(p\_0, u\_1)}
$$
Next, $e(p\_0, u\_1)$ is the minimal expenditure at prices $p\_0$ to reach utility level $u\_1$. We also know that buying $x\_1$ will give us at least utility level $u\_1$. As such,
$$
p\_0 x\_1 \ge e(p\_0, u\_1).
$$
Finally, $e(p\_0, u\_0)$ is the minimal expenditure at prices $p\_0$ to reach utilty level $u\_0$. This is reached by buying bundle $x\_0$ so,
$$
p\_0 x\_0 = e(p\_0, u\_0).
$$

Combining all this, we get:
$$
\frac{p\_1 x\_1}{p\_0 x\_1} \le \frac{e(p\_1, u\_1)}{e(p\_0, u\_1)} = \frac{e(p\_1, u\_0)}{e(p\_0, u\_0)}.
$$


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