<p>The income effect is given by:
<span class="math-container">$$
x^\ast \frac{\partial x}{\partial I} = \frac{0.1 I}{P_x} \frac{0.1}{P_x} = 0.1\left(0.1\frac{I}{(P_x)^2}\right).
$$</span>
The Price effect is:
<span class="math-container">$$
\frac{\partial x}{\partial P_x} = -0.1 \frac{I}{(P_x)^2}.$$</span>
The first is 10% of the second (in absolute terms).</p>
<p>Another way to see this, is to start from the fact that for the Cobb-Douglass, the expenditure share is constant:
<span class="math-container">$$
\frac{p x(p,I)}{I} = \alpha.
$$</span>
Taking derivative with respect to <span class="math-container">$p$</span> gives:
<span class="math-container">$$
\begin{align*}
&amp;\frac{x(p,I) + p \frac{\partial x}{\partial p}}{I} = 0,\\
\to &amp;\frac{\partial x}{\partial p} = -\frac{x(p,I)}{p}
\end{align*}
$$</span>
Taking the partial derivative with respect to <span class="math-container">$I$</span> gives:
<span class="math-container">$$
\begin{align*}
&amp;\frac{p \frac{\partial x(p,I)}{\partial I} I - p x(p,I)}{I^2} = 0,\\
\to &amp;\frac{\partial x(p,I)}{\partial I} = \frac{p x(p,I)}{pI} = \frac{\alpha}{p}
\end{align*}
$$</span>
So the income effect is:
<span class="math-container">$$
\frac{\partial x(p,I)}{\partial I} x(p,I) = \alpha \frac{x(p,I)}{p}
$$</span>
This is <span class="math-container">$\alpha$</span> times the absolute value of the price effect.</p>

CC BY-SA 4.0
2023-11-13T19:39:52.373Z
This makes sense, but is there any specific mathematical reason as to why the price effect is *exactly the same* as the share of income spent on the good? Or does it just happen to work that way?
Aa Z
CC BY-SA 4.0
2023-11-14T06:07:26.950Z
@Aa Z I added some additional explanation.
tdm
CC BY-SA 4.0
2023-11-13T06:35:29.137Z
2023-11-14T06:06:51.877Z
2023-11-14T06:06:51.877Z
tdm
<p>Let's say you have a Cobb-Douglas utility function, U = x^.1*y^.9</p>
<p>This will result in Marshallian demand functions x* = .1I / Px, and y* = .9I / Py.</p>
<p>If you perform a Slutsky decomposition for x* as Px changes; the income effect will be 10% of the total effect.</p>
<p>Can someone help me understand why the effect exactly matches the percent of income spent on the good?</p>

<microeconomics><slutsky-equation>
Explaining the relative share of the income & substitution effects of a price change
Aa Z
# Explaining the relative share of the income & substitution effects of a price change

(Asked by: username_0 on 2023-11-12)

Let's say you have a Cobb-Douglas utility function, U = x^.1\*y^.9

This will result in Marshallian demand functions x\* = .1I / Px, and y\* = .9I / Py.

If you perform a Slutsky decomposition for x\* as Px changes; the income effect will be 10% of the total effect.

Can someone help me understand why the effect exactly matches the percent of income spent on the good?

**Tags:** microeconomics, slutsky-equation


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## 1 answers

### Answer by username_1 on 2023-11-13. Score: 1

The income effect is given by:
$$
x^\ast \frac{\partial x}{\partial I} = \frac{0.1 I}{P\_x} \frac{0.1}{P\_x} = 0.1\left(0.1\frac{I}{(P\_x)^2}\right).
$$
The Price effect is:
$$
\frac{\partial x}{\partial P\_x} = -0.1 \frac{I}{(P\_x)^2}.$$
The first is 10% of the second (in absolute terms).

Another way to see this, is to start from the fact that for the Cobb-Douglass, the expenditure share is constant:
$$
\frac{p x(p,I)}{I} = \alpha.
$$
Taking derivative with respect to $p$ gives:
$$
\begin{align\*}
&\frac{x(p,I) + p \frac{\partial x}{\partial p}}{I} = 0,\\
\to &\frac{\partial x}{\partial p} = -\frac{x(p,I)}{p}
\end{align\*}
$$
Taking the partial derivative with respect to $I$ gives:
$$
\begin{align\*}
&\frac{p \frac{\partial x(p,I)}{\partial I} I - p x(p,I)}{I^2} = 0,\\
\to &\frac{\partial x(p,I)}{\partial I} = \frac{p x(p,I)}{pI} = \frac{\alpha}{p}
\end{align\*}
$$
So the income effect is:
$$
\frac{\partial x(p,I)}{\partial I} x(p,I) = \alpha \frac{x(p,I)}{p}
$$
This is $\alpha$ times the absolute value of the price effect.

> **Comments:**
> **username_0**: This makes sense, but is there any specific mathematical reason as to why the price effect is *exactly the same* as the share of income spent on the good? Or does it just happen to work that way?  
> **username_1**: @username_0 I added some additional explanation.  

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