<p>You interpreted the passage you read as essentially saying that <strong>"with log preferences, the Marshallian demand for the good <em>does not depend on its price</em>."</strong>
That doesn't sound very plausible, don't you think?</p>

<p>But the passage clearly refers to an intertemporal choice problem, it is about log-preferences over a single good and over time, it does not relate to the Slutsky decomposition of the static Marshallian demand function. </p>

<p>I would suggest to continue reading the notes up to page 14.</p>

CC BY-SA 4.0
2019-09-01T16:09:09.920Z
2019-09-01T16:09:09.920Z
Alecos Papadopoulos
<p>Try taking derivative of x* w.r.t. py (which is 0)
And verify with Slutsky decomposition.</p>

CC BY-SA 4.0
2021-10-15T12:14:46.180Z
2021-10-15T12:52:18.913Z
2021-10-15T12:52:18.913Z
Charlene677
<p>I've <a href="https://eml.berkeley.edu/~webfac/gourinchas/e202a_f14/Notes_Consumption_pog.pdf" rel="nofollow noreferrer">read</a> (pg 10) in Gourinchas' notes on consumption that the income and substitution effects cancel for log preferences, and I tried to prove this to myself doing the Slutsky decomposition for the simple case of <span class="math-container">$U(x,y) = \ln(x) + \ln(y)$</span> s.t. <span class="math-container">$p_x x + p_y y = I$</span>. I get that the Hicksian and Marshallian demand functions are <span class="math-container">$x^* = \frac{p_y}{p_x} y = \frac{I}{2 p_x} $</span> respectively. Plugging these in gives me this for the decomposition:</p>

<p><span class="math-container">$$ \frac{\partial x^*}{\partial p_x} = - \frac{p_y}{p_x^2} y - \frac{1}{2 p_x} \frac{p_y}{p_x} y $$</span></p>

<p>which is definitely not zero. Am I misunderstanding? </p>

CC BY-SA 4.0
2019-08-31T07:18:54.073Z
Hicksian demand is not a function of income.
Giskard
CC BY-SA 4.0
2019-08-31T19:05:16.427Z
Thanks--edited for clarity. My math is done for $h(\mathbf{p}, \bar{u}) = \frac{p_y}{p_x} y \implies \frac{\partial h}{\partial p_x} = - \frac{p_y}{p_x^2} y$
aintgeorge
CC BY-SA 4.0
2019-09-01T06:12:58.823Z
When you have $h(\textbf{p},\bar{u})$, shouldn't the stuff on the other side be only a function of the prices and $\bar{u}$? Yet you also have $y$ in $\frac{p_y}{p_x}y$. Perhaps it is this and the missing $\bar{u}$ that is causing the problem?
Giskard
<macroeconomics><mathematical-economics><consumption><slutsky-equation>
Why do the income and substitution effects cancel for log preferences? Trouble reconciling Slutsky decomposition
aintgeorge
# Why do the income and substitution effects cancel for log preferences? Trouble reconciling Slutsky decomposition

(Asked by: username_0 on 2019-08-31)

I've [read](https://eml.berkeley.edu/~webfac/gourinchas/e202a_f14/Notes_Consumption_pog.pdf) (pg 10) in Gourinchas' notes on consumption that the income and substitution effects cancel for log preferences, and I tried to prove this to myself doing the Slutsky decomposition for the simple case of $U(x,y) = \ln(x) + \ln(y)$ s.t. $p\_x x + p\_y y = I$. I get that the Hicksian and Marshallian demand functions are $x^\* = \frac{p\_y}{p\_x} y = \frac{I}{2 p\_x} $ respectively. Plugging these in gives me this for the decomposition:

$$ \frac{\partial x^\*}{\partial p\_x} = - \frac{p\_y}{p\_x^2} y - \frac{1}{2 p\_x} \frac{p\_y}{p\_x} y $$

which is definitely not zero. Am I misunderstanding?

**Tags:** macroeconomics, mathematical-economics, consumption, slutsky-equation

> **Comments:**
> **username_1**: Hicksian demand is not a function of income.  
> **username_0**: Thanks--edited for clarity. My math is done for $h(\mathbf{p}, \bar{u}) = \frac{p_y}{p_x} y \implies \frac{\partial h}{\partial p_x} = - \frac{p_y}{p_x^2} y$  
> **username_1**: When you have $h(\textbf{p},\bar{u})$, shouldn't the stuff on the other side be only a function of the prices and $\bar{u}$? Yet you also have $y$ in $\frac{p_y}{p_x}y$. Perhaps it is this and the missing $\bar{u}$ that is causing the problem?  

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## 2 answers

### Answer by username_2 on 2019-09-01. Score: 2

You interpreted the passage you read as essentially saying that **"with log preferences, the Marshallian demand for the good *does not depend on its price*."**
That doesn't sound very plausible, don't you think?

But the passage clearly refers to an intertemporal choice problem, it is about log-preferences over a single good and over time, it does not relate to the Slutsky decomposition of the static Marshallian demand function.

I would suggest to continue reading the notes up to page 14.


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### Answer by username_3 on 2021-10-15. Score: 1

Try taking derivative of x\* w.r.t. py (which is 0)
And verify with Slutsky decomposition.


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