<p>The <a href="https://is.muni.cz/el/1456/podzim2011/MPE_AMI2/um/27655259/L3.pdf" rel="nofollow noreferrer">usual definition of Substitution Effect (pg. 30; also found in Varian)</a> tells that the Slutsky SE is <span class="math-container">$x(p_x', I') - x(p_x,I) = x(2,15) - x(1,10) = \frac{15}{2 \cdot 2} - \frac{20}{2 \cdot 1} = -6.25$</span> where <span class="math-container">$I$</span> is the initial income and <span class="math-container">$I'$</span> is the new (imaginary) income. Similarly, <span class="math-container">$p_x$</span> is the initial price of good <span class="math-container">$X$</span> and <span class="math-container">$p_x'$</span> is the new price.</p>
<p>There's <a href="https://i.sstatic.net/8vtdJ.png" rel="nofollow noreferrer">another definition of the SE which we derive from the Slutsky equation</a> and it says that the SE (substitution effect) is <span class="math-container">$\frac{\partial x^c}{\partial p_x} = \frac{\partial x}{\partial p_x} \Big|_{u = \overline{u}}$</span>.</p>
<p>Following are my questions related to the second definition of SE:</p>
<ol>
<li><p>The compensated demand is defined as <span class="math-container">$x^c(p_x, \overline{p_y}, \overline{u}) := x(p_x, \overline{p_y}, E(p_x, \overline{p_y}, \overline{u}))$</span>. How does <span class="math-container">$\frac{\partial x^c}{\partial p_x} = \frac{\partial x}{\partial p_x} \Big|_{u = \overline{u}}$</span> follow from this? Also, do we assume <span class="math-container">$u = \overline{u}$</span> first and then calculate the partial derivative or calculate the partial derivative first and then fix the <span class="math-container">$u$</span> values to <span class="math-container">$\overline{u}$</span>?</p>
</li>
<li><p>As I understand, Hick's SE can be written as <span class="math-container">$x(p_x', p_y, E(p_x', p_y, \overline{u})) - x(p_x, p_y, I)$</span> which is equal to <span class="math-container">$x(p_x', p_y, I') - x(p_x, p_y, I)$</span>, the Slutsky SE when the income is set as the expenditure. But how is any of these equal to the other definition of SE: <span class="math-container">$\frac{\partial x^c}{\partial p_x}$</span>?</p>
<p>One definition talks about the change in demand while the other talks about the slope of demand curve. How can they be equal in any way?</p>
</li>
<li><p>This brings me to my final question. Consider the following example problem:</p>
<blockquote>
<p>Suppose a cup of coffee and a plate of beans are sold at € 1 and € 3 respectively during the winter. In summer, the government decides to remove the subsidy on coffee and its new price per cup goes up to € 2. If a customer has an income of € 10 and the utility function <span class="math-container">$u(x,y)=xy$</span>, what is the substitution effect?</p>
</blockquote>
<p>Using the usual definition, we can calculate it as <span class="math-container">$x(p_x', p_y, I') - x(p_x, p_y, I) = -6.25$</span>. It's the Cobb-Douglas utility function, so I skipped the calculations.</p>
<p>How can I calculate the SE using the second definition: <span class="math-container">$\frac{\partial x}{\partial p_x} \Big|_{u = \overline{u}}$</span>? I don't see where/how to consider <span class="math-container">$u = \overline{u}$</span>.</p>
</li>
</ol>

CC BY-SA 4.0
2022-08-18T07:35:16.333Z
You asked a similar question earlier also. I think procedure is clearly explained here: https://economics.stackexchange.com/a/52138/11824
Amit
CC BY-SA 4.0
2022-08-18T10:03:48.597Z
@Amit Yes, I used the same example so it's easier for the answerer to refer to, if needed (as it's linked on the right side). I am interested in calculating using partial derivates which is not mentioned there. I also wanted to know how $\frac{\partial x^c}{\partial p_c} = \frac{\partial x}{\partial p_c} \Big|_{u = u_0}$ and how it's the Slutsky SE, hoping that it is.
Isa
CC BY-SA 4.0
2022-08-18T10:20:05.067Z
If I have understood your question correctly, are asking the difference between the derivative of a function $\frac{df}{dx}$ and the total change $\Delta f$ due to $\Delta x$ change in $x$?
Amit
CC BY-SA 4.0
2022-08-19T06:01:00.110Z
@Amit I partially understood certain things, so I changed my questions (somewhat). I am mainly curious about 1 and 2. Maybe after that, I will understand 3 myself.
Isa
<microeconomics><choice-theory><slutsky-equation>
How to find the substitution and income effects?
Isa
# How to find the substitution and income effects?

(Asked by: username_0 on 2022-08-18)

The [usual definition of Substitution Effect (pg. 30; also found in Varian)](https://is.muni.cz/el/1456/podzim2011/MPE_AMI2/um/27655259/L3.pdf) tells that the Slutsky SE is $x(p\_x', I') - x(p\_x,I) = x(2,15) - x(1,10) = \frac{15}{2 \cdot 2} - \frac{20}{2 \cdot 1} = -6.25$ where $I$ is the initial income and $I'$ is the new (imaginary) income. Similarly, $p\_x$ is the initial price of good $X$ and $p\_x'$ is the new price.

There's [another definition of the SE which we derive from the Slutsky equation](https://i.sstatic.net/8vtdJ.png) and it says that the SE (substitution effect) is $\frac{\partial x^c}{\partial p\_x} = \frac{\partial x}{\partial p\_x} \Big|\_{u = \overline{u}}$.

Following are my questions related to the second definition of SE:

1. The compensated demand is defined as $x^c(p\_x, \overline{p\_y}, \overline{u}) := x(p\_x, \overline{p\_y}, E(p\_x, \overline{p\_y}, \overline{u}))$. How does $\frac{\partial x^c}{\partial p\_x} = \frac{\partial x}{\partial p\_x} \Big|\_{u = \overline{u}}$ follow from this? Also, do we assume $u = \overline{u}$ first and then calculate the partial derivative or calculate the partial derivative first and then fix the $u$ values to $\overline{u}$?
2. As I understand, Hick's SE can be written as $x(p\_x', p\_y, E(p\_x', p\_y, \overline{u})) - x(p\_x, p\_y, I)$ which is equal to $x(p\_x', p\_y, I') - x(p\_x, p\_y, I)$, the Slutsky SE when the income is set as the expenditure. But how is any of these equal to the other definition of SE: $\frac{\partial x^c}{\partial p\_x}$?

   One definition talks about the change in demand while the other talks about the slope of demand curve. How can they be equal in any way?
3. This brings me to my final question. Consider the following example problem:

   > Suppose a cup of coffee and a plate of beans are sold at € 1 and € 3 respectively during the winter. In summer, the government decides to remove the subsidy on coffee and its new price per cup goes up to € 2. If a customer has an income of € 10 and the utility function $u(x,y)=xy$, what is the substitution effect?

   Using the usual definition, we can calculate it as $x(p\_x', p\_y, I') - x(p\_x, p\_y, I) = -6.25$. It's the Cobb-Douglas utility function, so I skipped the calculations.

   How can I calculate the SE using the second definition: $\frac{\partial x}{\partial p\_x} \Big|\_{u = \overline{u}}$? I don't see where/how to consider $u = \overline{u}$.

**Tags:** microeconomics, choice-theory, slutsky-equation

> **Comments:**
> **username_1**: You asked a similar question earlier also. I think procedure is clearly explained here: https://economics.stackexchange.com/a/52138/11824  
> **username_0**: @username_1 Yes, I used the same example so it's easier for the answerer to refer to, if needed (as it's linked on the right side). I am interested in calculating using partial derivates which is not mentioned there. I also wanted to know how $\frac{\partial x^c}{\partial p_c} = \frac{\partial x}{\partial p_c} \Big|_{u = u_0}$ and how it's the Slutsky SE, hoping that it is.  
> **username_1**: If I have understood your question correctly, are asking the difference between the derivative of a function $\frac{df}{dx}$ and the total change $\Delta f$ due to $\Delta x$ change in $x$?  
> **username_0**: @username_1 I partially understood certain things, so I changed my questions (somewhat). I am mainly curious about 1 and 2. Maybe after that, I will understand 3 myself.  

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## 0 answers

