<p>From <a href="http://www.econ.yale.edu/smith/econ510a/book.pdf" rel="nofollow noreferrer">these notes</a> by Per Krussel on page 16, the author writes that</p>
<blockquote>
<p>&quot;With logarithmic utility, a relative price change between two goods
will make the consumption of the favored good go up whereas the
consumption of other good will remain at the same level. The unfavored
good will not be consumed in a lower amount since there is a positive
income effect of the other good being cheaper, and that effect will be
spread over both goods.&quot;</p>
</blockquote>
<p>I am not quite sure I follow. Imagine that the price of good 1 is 2 and that of good 2 is 1. The relative price of good 1 and good 2 is 2 and income is 10 (we can have money or good 1 can be thought of as the numeraire). The optimal solution, say is 3 units of good 2 and 4 units of good 1. Imagine now, that the relative price changes to <em>20</em>, or that the price of good 1 is 1 and good 2 is 20. Of course the unfavoured good here would now be consumed in a lower amount, as there is simply not enough income in terms of the numeraire! What is going on?</p>
<p>Thanks!</p>

CC BY-SA 4.0
2022-08-30T03:46:32.613Z
Does this answer your question? [Why do the income and substitution effects cancel for log preferences? Trouble reconciling Slutsky decomposition](https://economics.stackexchange.com/questions/30709/why-do-the-income-and-substitution-effects-cancel-for-log-preferences-trouble-r)
Herr K.
CC BY-SA 4.0
2022-08-31T00:14:17.743Z
Your example is a little confused.  But if you start with the price of good A is 1 and of good B is 2 and with income of 10 and logarithmic utility you have consumption of 4 of A and 3 of B, then if the price of good A stays at 1 and the price of good B changes to 20 and income stays at 10 then you have will have consumption of 4 of A and 0.3 of B; similarly if if the price of good A reduces to 0.1 and the price of good B is 2 and income stays at 10 then you have will have consumption of 40 of A and 3 of B.
Henry
<income-effect>
Logarithmic utility and and cancellation of income and sub effects
Kwame Brown
# Logarithmic utility and and cancellation of income and sub effects

(Asked by: username_0 on 2022-08-29)

From [these notes](http://www.econ.yale.edu/smith/econ510a/book.pdf) by Per Krussel on page 16, the author writes that

> "With logarithmic utility, a relative price change between two goods
> will make the consumption of the favored good go up whereas the
> consumption of other good will remain at the same level. The unfavored
> good will not be consumed in a lower amount since there is a positive
> income effect of the other good being cheaper, and that effect will be
> spread over both goods."

I am not quite sure I follow. Imagine that the price of good 1 is 2 and that of good 2 is 1. The relative price of good 1 and good 2 is 2 and income is 10 (we can have money or good 1 can be thought of as the numeraire). The optimal solution, say is 3 units of good 2 and 4 units of good 1. Imagine now, that the relative price changes to *20*, or that the price of good 1 is 1 and good 2 is 20. Of course the unfavoured good here would now be consumed in a lower amount, as there is simply not enough income in terms of the numeraire! What is going on?

Thanks!

**Tags:** income-effect

> **Comments:**
> **username_1**: Does this answer your question? [Why do the income and substitution effects cancel for log preferences? Trouble reconciling Slutsky decomposition](https://economics.stackexchange.com/questions/30709/why-do-the-income-and-substitution-effects-cancel-for-log-preferences-trouble-r)  
> **username_2**: Your example is a little confused.  But if you start with the price of good A is 1 and of good B is 2 and with income of 10 and logarithmic utility you have consumption of 4 of A and 3 of B, then if the price of good A stays at 1 and the price of good B changes to 20 and income stays at 10 then you have will have consumption of 4 of A and 0.3 of B; similarly if if the price of good A reduces to 0.1 and the price of good B is 2 and income stays at 10 then you have will have consumption of 40 of A and 3 of B.  

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## 0 answers

