<ol start="0">
<li>The traditional formula for <span class="math-container">$CS$</span> would actually be <span class="math-container">$CS = \int_{0}^{x_0} [p_x(x,\overline{p_y},\overline{m})] dx - p_0 x_0$</span>. In traditional integration with respect to the horizontal axis, we integrate the vertical axis variable with respect to the horizontal axis variable. This implies we need to get the (partial) inverse demand with respect to <span class="math-container">$x$</span>, i.e. solve for <span class="math-container">$p_x$</span> as a function of <span class="math-container">$x$</span> and the other variables as fixed.</li>
</ol>
<p>For example: <span class="math-container">$x(p_x, p_y, m) = \frac{m p_y}{p_x} \implies x = \frac{m p_y}{p_x} \implies p_x = \frac{m p_y}{x} \implies p_x(x,p_y,m) = \frac{m p_y}{x}$</span>.</p>
<ol>
<li>The change in consumer surplus can actually be calculated as <span class="math-container">$\Delta CS = \int_{p_{x_0}}^{p_{x_1}} [x(p_x,\overline{p_y},\overline{m})] dp_x$</span>.</li>
</ol>
<p>The lengthier version is wrong as it is based on the wrong traditional formula for <span class="math-container">$CS$</span>. Also, if <span class="math-container">$x_0$</span> corresponds to the original point and <span class="math-container">$x_1$</span> to the point after the price change, the lengthier version should actually correspond to <span class="math-container">$\Delta CS = CS(x_1) - CS(x_0)$</span>, rather than <span class="math-container">$\Delta CS = CS(x_0) - CS(x_1)$</span>, as the latter one would flip the sign of the change.</p>
<ol start="2">
<li>We can integrate like that with respect to the <span class="math-container">$y$</span> axis as long as the integrand is the dependent variable <span class="math-container">$x$</span> as a function of <span class="math-container">$y$</span>, i.e. <span class="math-container">$x(y)$</span>.</li>
</ol>
<p>To integrate with respect to the <span class="math-container">$x$</span> axis, we need to have <span class="math-container">$y(x)$</span>.</p>
<ol start="3">
<li>What you need to be careful about is:</li>
</ol>
<ul>
<li>Integrating with respect to the <span class="math-container">$x$</span> axis, if we had a function <span class="math-container">$x(y)$</span>, we would need to solve for <span class="math-container">$y$</span> in terms of <span class="math-container">$x$</span>, i.e. <span class="math-container">$y(x)$</span>.</li>
<li>Integrating with respect to the <span class="math-container">$y$</span> axis, if we had a function <span class="math-container">$y(x)$</span>, we would need to solve for <span class="math-container">$x$</span> in terms of <span class="math-container">$y$</span>, i.e. <span class="math-container">$x(y)$</span>.</li>
</ul>

CC BY-SA 4.0
2023-02-16T04:14:31.360Z
Thank you very much for the detail Nicolas, I will go through tomorrow and then accept your answer once I have understood! On the brief skim I wasn't sure why my ∆CS calculation was wrong so I need to check that! Thanks for laying the answer out in such clarity!
CormJack
CC BY-SA 4.0
2023-02-15T12:04:39.117Z
2023-02-15T12:20:30.067Z
2023-02-15T12:20:30.067Z
Nicolas Torres
<p>The traditional formulae for consumer surplus is: <span class="math-container">$\text{CS} = \int_{0}^{x_0} [x(p_x,\overline{p_y}, \overline{m})]dx - x_0P_{x_0}$</span>.</p>
<p>This is the area under the Marshallian demand curve, that is only the triangle above <span class="math-container">$P_x$</span></p>
<p>However I noticed that the formulae for the compensating variation, which is a sliver underneath the Hicksian demand curve (<span class="math-container">$h$</span>) between <span class="math-container">$P_{x_0}$</span> and <span class="math-container">$P_{x_1}$</span> is calculated using <span class="math-container">$\text{CS} = \int_{p_{x_0}}^{p_{x_1}}[h(p_x,\overline{p_y}, \overline{u})]dp_x$</span></p>
<p><strong>Questions:</strong></p>
<ol>
<li><p>Could we calculate ∆CS the same way i.e. <span class="math-container">$\text{∆CS} = \int_{p_{x_0}}^{p_{x_1}} [x(p_x,\overline{p_y}, \overline{m})]dp_x$</span>
instead of using the lengthier: <span class="math-container">$\text{∆CS} = [\int_{0}^{x_0} [x(p_x,\overline{p_y}, \overline{m})]dx - x_0P_{x_0}]$</span> - <span class="math-container">$[\int_{0}^{x_1} [x(p_x,\overline{p_y}, \overline{m})]dx - x_1P_{x_1}]$</span></p>
</li>
<li><p><strong>More generally can we chose to integrate with respect to the Y axis like this</strong>, when we want a sliver of the graph as per these examples?</p>
</li>
<li><p>If i can integrate with respect to the y axis is there anything I need to be careful of?</p>
</li>
</ol>
<p>I have attached an image to give additional clarity to my question. Thank you!</p>
<p><a href="https://i.sstatic.net/6Y5Rv.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/6Y5Rv.jpg" alt="enter image description here" /></a></p>

<utility><demand><welfare-economics>
Integrating over the Y Axis: (∆CS)
CormJack
# Integrating over the Y Axis: (∆CS)

(Asked by: username_0 on 2023-02-08)

The traditional formulae for consumer surplus is: $\text{CS} = \int\_{0}^{x\_0} [x(p\_x,\overline{p\_y}, \overline{m})]dx - x\_0P\_{x\_0}$.

This is the area under the Marshallian demand curve, that is only the triangle above $P\_x$

However I noticed that the formulae for the compensating variation, which is a sliver underneath the Hicksian demand curve ($h$) between $P\_{x\_0}$ and $P\_{x\_1}$ is calculated using $\text{CS} = \int\_{p\_{x\_0}}^{p\_{x\_1}}[h(p\_x,\overline{p\_y}, \overline{u})]dp\_x$

**Questions:**

1. Could we calculate ∆CS the same way i.e. $\text{∆CS} = \int\_{p\_{x\_0}}^{p\_{x\_1}} [x(p\_x,\overline{p\_y}, \overline{m})]dp\_x$
   instead of using the lengthier: $\text{∆CS} = [\int\_{0}^{x\_0} [x(p\_x,\overline{p\_y}, \overline{m})]dx - x\_0P\_{x\_0}]$ - $[\int\_{0}^{x\_1} [x(p\_x,\overline{p\_y}, \overline{m})]dx - x\_1P\_{x\_1}]$
2. **More generally can we chose to integrate with respect to the Y axis like this**, when we want a sliver of the graph as per these examples?
3. If i can integrate with respect to the y axis is there anything I need to be careful of?

I have attached an image to give additional clarity to my question. Thank you!

[![enter image description here](https://i.sstatic.net/6Y5Rv.jpg)](https://i.sstatic.net/6Y5Rv.jpg)

**Tags:** utility, demand, welfare-economics

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## 1 answers

### Answer by username_1 on 2023-02-15. Score: 1 | Accepted answer

0. The traditional formula for $CS$ would actually be $CS = \int\_{0}^{x\_0} [p\_x(x,\overline{p\_y},\overline{m})] dx - p\_0 x\_0$. In traditional integration with respect to the horizontal axis, we integrate the vertical axis variable with respect to the horizontal axis variable. This implies we need to get the (partial) inverse demand with respect to $x$, i.e. solve for $p\_x$ as a function of $x$ and the other variables as fixed.

For example: $x(p\_x, p\_y, m) = \frac{m p\_y}{p\_x} \implies x = \frac{m p\_y}{p\_x} \implies p\_x = \frac{m p\_y}{x} \implies p\_x(x,p\_y,m) = \frac{m p\_y}{x}$.

1. The change in consumer surplus can actually be calculated as $\Delta CS = \int\_{p\_{x\_0}}^{p\_{x\_1}} [x(p\_x,\overline{p\_y},\overline{m})] dp\_x$.

The lengthier version is wrong as it is based on the wrong traditional formula for $CS$. Also, if $x\_0$ corresponds to the original point and $x\_1$ to the point after the price change, the lengthier version should actually correspond to $\Delta CS = CS(x\_1) - CS(x\_0)$, rather than $\Delta CS = CS(x\_0) - CS(x\_1)$, as the latter one would flip the sign of the change.

2. We can integrate like that with respect to the $y$ axis as long as the integrand is the dependent variable $x$ as a function of $y$, i.e. $x(y)$.

To integrate with respect to the $x$ axis, we need to have $y(x)$.

3. What you need to be careful about is:

* Integrating with respect to the $x$ axis, if we had a function $x(y)$, we would need to solve for $y$ in terms of $x$, i.e. $y(x)$.
* Integrating with respect to the $y$ axis, if we had a function $y(x)$, we would need to solve for $x$ in terms of $y$, i.e. $x(y)$.

> **Comments:**
> **username_0**: Thank you very much for the detail Nicolas, I will go through tomorrow and then accept your answer once I have understood! On the brief skim I wasn't sure why my ∆CS calculation was wrong so I need to check that! Thanks for laying the answer out in such clarity!  

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