<p>First, the fact that <span class="math-container">$MRS=\frac{1}{4}$</span> does not tell you by itself that the consumer will only buy <span class="math-container">$x_2$</span>.
We need to go back to the 2nd Gossen's law:
<span class="math-container">$\frac{Um{x_1}}{p_1}=\frac{Um{x_2}}{p_2}$</span> (this is from where the <span class="math-container">$MRS$</span> comes by the way) which is not the case since <span class="math-container">$\frac{1}{3}&lt;\frac{4}{8}=\frac{1}{2}$</span>.</p>
<p>So you are right the consumer in the first place will not consume any <span class="math-container">$x_1$</span>. This is also due to the linear form of your function since you do not have a decreasing marginal utility with respect to <span class="math-container">$x_2$</span> even if you get more and more <span class="math-container">$x_2$</span> (which is quite a common form to model money for instance).</p>
<p>After a change in the price <span class="math-container">$p_2$</span> the decision of consuming only <span class="math-container">$x_2$</span> remains since <span class="math-container">$\frac{1}{3}&lt;\frac{4}{10}=\frac{2}{5}$</span>. With such reasoning, you will easily understand that there is no substitution effect for an increase in the price <span class="math-container">$p_2$</span>.
If you want a formal demonstration:</p>
<p><strong>Substitution effect:</strong> how much does the consumer substitute a good from another when the price varies with a constant purchasing power?
With the demand for the good 2: <span class="math-container">$x_2=\frac{m}{p_2}$</span>, and denoting <span class="math-container">$m'$</span> and <span class="math-container">$p_2'$</span> the &quot;new&quot; revenue and the new price we have:
<span class="math-container">$\Delta x_2^s= x_2(p_2',m')-x_2(p_2,m)$</span>.
The <span class="math-container">$m'$</span> is the &quot;new&quot; revenue such that I can buy the same amount of <span class="math-container">$x_2$</span> before the change in the price ie an augmentation of:
<span class="math-container">\begin{aligned}
\Delta m&amp;=15(10-8)=30 \\
m'&amp;=m+\Delta m=150 \\
x_2^s&amp;=\frac{150}{10}-\frac{120}{8}=0
\end{aligned}</span></p>
<p>Nevertheless, it remains the <strong>income effect</strong> which is the variation of the demand due to the modification of purchasing power.</p>
<p><span class="math-container">\begin{aligned}
\Delta x_2^i=&amp; x_2(p_2',m)-x_2(p_2',m') \\
=&amp; x_2(10,120)-x_2(10,150) \\
=&amp; 12-15=-3
\end{aligned}</span></p>
<p>So the total variation of the demand is only related to the income effect and is -3.</p>

CC BY-SA 4.0
2021-09-09T19:00:20.263Z
thanks a lot for your detailed response, it really helped a lot and I know understand this topic much more. Just to see if I grasp everything correctly, in the case that p2 increases to 18, I should calculate the substitution effect by doing 270/18 - 120/8 = 0, but I feel that there should be a substitution effect as 1/3 > 4/18. Income effect with this price is -8.33. Am I on the right track to solving this ? Is there something I am missing ? Again, thanks a ton for your comment it really helped a lot.
Coco Garazzo
CC BY-SA 4.0
2021-09-10T02:20:39.900Z
I managed to do it again, which resulted in a substitution effect of - 25. for the price change of p2 to 18. I also did another method and got -15, any idea on which one could be the correct one ?
Coco Garazzo
CC BY-SA 4.0
2021-09-10T11:31:24.510Z
I do not clearly understand what you are doing, but if the inequality is reversed as you see then the consumer will only buy $x_1$ this time. Just remind you that it is a very particular case where goods are perfect substitutes (see the link of the previous comment).
Alexandre
CC BY-SA 4.0
2021-09-10T19:31:37.293Z
I am unable to see the previous comment, could you please send the link again. So sorry for this and thanks for your help
Coco Garazzo
CC BY-SA 4.0
2021-09-09T13:48:14.143Z
2021-09-09T13:48:14.143Z
Alexandre
<p>Utility function <span class="math-container">$U(x_1 , x_2) = x_1 + 4 * x_2 $</span></p>
<p><span class="math-container">$ p_1 = 3, p_2 = 8, m =120 $</span></p>
<p><span class="math-container">$p_2$</span> changes from <span class="math-container">$8$</span> to <span class="math-container">$10 $</span></p>
<p>How can I calculate the income and substitution effect.</p>
<p>I first thought about calculating the MRS making the partial derivatives of the Utility function which resulted in <span class="math-container">$MRS = 1/4$</span> which means that the consumer will always prefer at any given moment good <span class="math-container">$x_2$</span> thus only buying that good. At first he can only buy <span class="math-container">$m/p_2 = 120/8 = 15$</span> so the bundle would be <span class="math-container">$(0,15)$</span> and after the price change it would be <span class="math-container">$(0,12)$</span> (not sure about that).</p>
<p>I also calculated the utility at the previous point by plugging <span class="math-container">$(0,15)$</span> into <span class="math-container">$U(x_1 , x_2) = x_1 + 4 * x_2 $</span> which resulted in <span class="math-container">$60$</span></p>
<p><span class="math-container">$60/p'_2 - 60/p_2 $</span> =  <span class="math-container">$60/10 - 60/8= - 1.5$</span> = substitution effect</p>
<p>I am not sure if what I have done is correct and if the substitution effect is 1.5. Is there any way to visualize this graphically ? I thought about doing linear equations with <span class="math-container">$m/p_2$</span> and <span class="math-container">$m/p'_2$</span> as seen below:</p>
<p><a href="https://i.sstatic.net/kz7zp.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/kz7zp.png" alt="Graph" /></a></p>
<p>Am I missing something ? Any help is appreciated</p>

CC BY-SA 4.0
2021-09-09T13:38:18.503Z
Does this answer your question? [Income and substitution effect for perfect substitutes](https://economics.stackexchange.com/questions/25384/income-and-substitution-effect-for-perfect-substitutes)
Herr K.
<microeconomics><utility><consumer-theory><slutsky-equation><income-effect>
Calculate income and sustitution effect from utility funcion
Coco Garazzo
# Calculate income and sustitution effect from utility funcion

(Asked by: username_0 on 2021-09-09)

Utility function $U(x\_1 , x\_2) = x\_1 + 4 \* x\_2 $

$ p\_1 = 3, p\_2 = 8, m =120 $

$p\_2$ changes from $8$ to $10 $

How can I calculate the income and substitution effect.

I first thought about calculating the MRS making the partial derivatives of the Utility function which resulted in $MRS = 1/4$ which means that the consumer will always prefer at any given moment good $x\_2$ thus only buying that good. At first he can only buy $m/p\_2 = 120/8 = 15$ so the bundle would be $(0,15)$ and after the price change it would be $(0,12)$ (not sure about that).

I also calculated the utility at the previous point by plugging $(0,15)$ into $U(x\_1 , x\_2) = x\_1 + 4 \* x\_2 $ which resulted in $60$

$60/p'\_2 - 60/p\_2 $ = $60/10 - 60/8= - 1.5$ = substitution effect

I am not sure if what I have done is correct and if the substitution effect is 1.5. Is there any way to visualize this graphically ? I thought about doing linear equations with $m/p\_2$ and $m/p'\_2$ as seen below:

[![Graph](https://i.sstatic.net/kz7zp.png)](https://i.sstatic.net/kz7zp.png)

Am I missing something ? Any help is appreciated

**Tags:** microeconomics, utility, consumer-theory, slutsky-equation, income-effect

> **Comments:**
> **username_1**: Does this answer your question? [Income and substitution effect for perfect substitutes](https://economics.stackexchange.com/questions/25384/income-and-substitution-effect-for-perfect-substitutes)  

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## 1 answers

### Answer by username_2 on 2021-09-09. Score: 1 | Accepted answer

First, the fact that $MRS=\frac{1}{4}$ does not tell you by itself that the consumer will only buy $x\_2$.
We need to go back to the 2nd Gossen's law:
$\frac{Um{x\_1}}{p\_1}=\frac{Um{x\_2}}{p\_2}$ (this is from where the $MRS$ comes by the way) which is not the case since $\frac{1}{3}<\frac{4}{8}=\frac{1}{2}$.

So you are right the consumer in the first place will not consume any $x\_1$. This is also due to the linear form of your function since you do not have a decreasing marginal utility with respect to $x\_2$ even if you get more and more $x\_2$ (which is quite a common form to model money for instance).

After a change in the price $p\_2$ the decision of consuming only $x\_2$ remains since $\frac{1}{3}<\frac{4}{10}=\frac{2}{5}$. With such reasoning, you will easily understand that there is no substitution effect for an increase in the price $p\_2$.
If you want a formal demonstration:

**Substitution effect:** how much does the consumer substitute a good from another when the price varies with a constant purchasing power?
With the demand for the good 2: $x\_2=\frac{m}{p\_2}$, and denoting $m'$ and $p\_2'$ the "new" revenue and the new price we have:
$\Delta x\_2^s= x\_2(p\_2',m')-x\_2(p\_2,m)$.
The $m'$ is the "new" revenue such that I can buy the same amount of $x\_2$ before the change in the price ie an augmentation of:
\begin{aligned}
\Delta m&=15(10-8)=30 \\
m'&=m+\Delta m=150 \\
x\_2^s&=\frac{150}{10}-\frac{120}{8}=0
\end{aligned}

Nevertheless, it remains the **income effect** which is the variation of the demand due to the modification of purchasing power.

\begin{aligned}
\Delta x\_2^i=& x\_2(p\_2',m)-x\_2(p\_2',m') \\
=& x\_2(10,120)-x\_2(10,150) \\
=& 12-15=-3
\end{aligned}

So the total variation of the demand is only related to the income effect and is -3.

> **Comments:**
> **username_0**: thanks a lot for your detailed response, it really helped a lot and I know understand this topic much more. Just to see if I grasp everything correctly, in the case that p2 increases to 18, I should calculate the substitution effect by doing 270/18 - 120/8 = 0, but I feel that there should be a substitution effect as 1/3 > 4/18. Income effect with this price is -8.33. Am I on the right track to solving this ? Is there something I am missing ? Again, thanks a ton for your comment it really helped a lot.  
> **username_0**: I managed to do it again, which resulted in a substitution effect of - 25. for the price change of p2 to 18. I also did another method and got -15, any idea on which one could be the correct one ?  
> **username_2**: I do not clearly understand what you are doing, but if the inequality is reversed as you see then the consumer will only buy $x_1$ this time. Just remind you that it is a very particular case where goods are perfect substitutes (see the link of the previous comment).  
> **username_0**: I am unable to see the previous comment, could you please send the link again. So sorry for this and thanks for your help  

---

