<p>You are both right. Varian doesn't include the <span class="math-container">$f$</span>-argument in his <span class="math-container">$MC_f$</span> function, so his graph is correct if that function is your red one. Note that he nowhere claims that his <span class="math-container">$MC_f$</span> function shows the one of the fishery <em>before</em> integration.</p>

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2021-10-18T08:24:48.673Z
2021-10-18T09:09:26.053Z
2021-10-18T09:09:26.053Z
VARulle
<p>I wouldn't say that Varian's diagram is wrong: it's more that his explanation of the <span class="math-container">$MC_F$</span> function (see 34.3 on p 652) is incomplete (and to be fair textbooks do sometimes need to simplify matters in order to focus on key points).</p>
<p>This is how I would analyse the effect on the fishery of changes in the quantity of pollution. The fishery's profit function is:</p>
<p><span class="math-container">$$\Pi_f=p_ff-c_f(f,x)\qquad(1)$$</span></p>
<p>Taking the total differential with respect to pollution <span class="math-container">$x$</span>:</p>
<p><span class="math-container">$$\dfrac{d\Pi_f}{dx}=\dfrac{\partial \Pi_f}{\partial f}\dfrac{df}{dx}+\dfrac{\partial \Pi_f}{\partial x}=\bigg(p_f-\dfrac{\partial c_f(f,x)}{\partial f}\bigg)\dfrac{df}{dx}-\dfrac{\partial c_f(f,x)}{\partial x}\qquad(2)$$</span></p>
<p><span class="math-container">$$\dfrac{d\Pi_f}{dx}=p_f\frac{df}{dx}-\dfrac{\partial c_f(f,x)}{\partial f}\dfrac{df}{dx}-\dfrac{\partial c_f(f,x)}{\partial x}\qquad (3)$$</span></p>
<p>Within the right hand side of (3), the third component is the direct effect of a change in pollution on cost (which is what Varian's 34.3 shows).  The other two components are indirect effects of a change in pollution through their effect in changing the profit-maximising volume of fish production <span class="math-container">$f$</span>. That there must be such an effect can be inferred from the profit-maximising condition for the fishery (cf Varian p 651):</p>
<p><span class="math-container">$$p_f=\dfrac{dc_f(f,x)}{df}\qquad(4)$$</span></p>
<p>Here <span class="math-container">$p_f$</span> is fixed, so on varying <span class="math-container">$x$</span> the only way to maintain equality is to vary <span class="math-container">$f$</span>.  The first component on the right hand side of (3) is the indirect effect of a change in <span class="math-container">$x$</span>, via <span class="math-container">$f$</span>, on revenue, and the second component is the indirect effect on cost.</p>
<p>If the intersection of the lines in Varian's diagram is to determine the Pareto-optimal quantity of pollution, then the <span class="math-container">$MC_f$</span> line should be interpreted as in (3) above.</p>

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2021-10-18T20:46:13.237Z
Genuinely appreciate for the answer
LJNG
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2023-08-16T15:16:10.787Z
@Adam Bailey I wonder why you did not substitute (3) into (4) to get $d\Pi_f/dx=-\partial c_f/\partial x$. The marginal change in the fishery's profit from a marginal increase in pollution is equal to the the direct effect on profit (i.e. just the negative of the increase in costs) from a marginal increase in pollution (this follows from the envelope theorem).
smcc
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2021-10-18T12:09:50.677Z
2021-10-18T12:09:50.677Z
Adam Bailey
<p>Here is the capture from <strong>Hal Varian's Intermediate Microeconomics chapter <em>Externality</em></strong>.<a href="https://i.sstatic.net/iKqxI.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/iKqxI.png" alt="enter image description here" /></a></p>
<p>I remember clearly my lecturer said this graph <strong>is wrong</strong> several years ago, but I cannot remember the argument exactly why is this wrong. Then I'm trying to replicate the reasoning with my own graph attached.</p>
<p><a href="https://i.sstatic.net/4irme.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/4irme.jpg" alt="enter image description here" /></a></p>
<p>Please have a look if my argument makes any sense to you.</p>
<p>Steel firm's private solution will be <span class="math-container">$x^*$</span> and this also determine the <span class="math-container">$f^*$</span> in <span class="math-container">$MC_{f}(f^*,x^*)$</span>, <span class="math-container">$f^*$</span> exogenous. When firms integrated together, another social optimal outcome of <span class="math-container">$\hat{x}^*$</span> should determined another outcome of <span class="math-container">$\hat{f^*}$</span> shown as <span class="math-container">$MC_{f'}(\hat{f^*},\hat{x^*})$</span>, and <span class="math-container">$\hat{f^*}&gt;f^*$</span></p>
<p>Therefore, the true social optimal pollution should be <span class="math-container">$\hat{x}^*$</span> in my graph.</p>
<p>Thank you for any comments</p>
<p><a href="https://faculty.ksu.edu.sa/sites/default/files/microeco-_varian.pdf" rel="nofollow noreferrer">Hal Varian Text</a> from page <span class="math-container">$650$</span> to <span class="math-container">$653$</span></p>

CC BY-SA 4.0
2021-10-17T11:32:29.563Z
I think this is difficult to answer without more context.  Is there just one steel firm whose pollution is affecting one fishery (as the Varian diagram seems to suggest)?  How should the $-MC_S$ line be understood?  Is it about cost changing with volume of steel production and consequent volume of pollution?  Or is it about changing cost of abatement of pollution while production volume remains constant?  Also, what is the significance, if any, of showing negative marginal cost rather than marginal benefit?
Adam Bailey
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2021-10-17T11:43:13.923Z
@AdamBailey Thank you for your reply. I've just included a link for the text.
LJNG
<microeconomics><public-economics><welfare-economics><environmental-economics>
Is Hal Varian's intermediate microeconomic externality graph wrong?
LJNG
# Is Hal Varian's intermediate microeconomic externality graph wrong?

(Asked by: username_0 on 2021-10-17)

Here is the capture from **Hal Varian's Intermediate Microeconomics chapter *Externality***.[![enter image description here](https://i.sstatic.net/iKqxI.png)](https://i.sstatic.net/iKqxI.png)

I remember clearly my lecturer said this graph **is wrong** several years ago, but I cannot remember the argument exactly why is this wrong. Then I'm trying to replicate the reasoning with my own graph attached.

[![enter image description here](https://i.sstatic.net/4irme.jpg)](https://i.sstatic.net/4irme.jpg)

Please have a look if my argument makes any sense to you.

Steel firm's private solution will be $x^\*$ and this also determine the $f^\*$ in $MC\_{f}(f^\*,x^\*)$, $f^\*$ exogenous. When firms integrated together, another social optimal outcome of $\hat{x}^\*$ should determined another outcome of $\hat{f^\*}$ shown as $MC\_{f'}(\hat{f^\*},\hat{x^\*})$, and $\hat{f^\*}>f^\*$

Therefore, the true social optimal pollution should be $\hat{x}^\*$ in my graph.

Thank you for any comments

[Hal Varian Text](https://faculty.ksu.edu.sa/sites/default/files/microeco-_varian.pdf) from page $650$ to $653$

**Tags:** microeconomics, public-economics, welfare-economics, environmental-economics
> **Comments:**
> **username_1**: I think this is difficult to answer without more context.  Is there just one steel firm whose pollution is affecting one fishery (as the Varian diagram seems to suggest)?  How should the $-MC_S$ line be understood?  Is it about cost changing with volume of steel production and consequent volume of pollution?  Or is it about changing cost of abatement of pollution while production volume remains constant?  Also, what is the significance, if any, of showing negative marginal cost rather than marginal benefit?  
> **username_0**: @AdamBailey Thank you for your reply. I've just included a link for the text.  

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## 2 answers

### Answer by username_1 on 2021-10-18. Score: 2 | Accepted answer

I wouldn't say that Varian's diagram is wrong: it's more that his explanation of the $MC\_F$ function (see 34.3 on p 652) is incomplete (and to be fair textbooks do sometimes need to simplify matters in order to focus on key points).

This is how I would analyse the effect on the fishery of changes in the quantity of pollution. The fishery's profit function is:

$$\Pi\_f=p\_ff-c\_f(f,x)\qquad(1)$$

Taking the total differential with respect to pollution $x$:

$$\dfrac{d\Pi\_f}{dx}=\dfrac{\partial \Pi\_f}{\partial f}\dfrac{df}{dx}+\dfrac{\partial \Pi\_f}{\partial x}=\bigg(p\_f-\dfrac{\partial c\_f(f,x)}{\partial f}\bigg)\dfrac{df}{dx}-\dfrac{\partial c\_f(f,x)}{\partial x}\qquad(2)$$

$$\dfrac{d\Pi\_f}{dx}=p\_f\frac{df}{dx}-\dfrac{\partial c\_f(f,x)}{\partial f}\dfrac{df}{dx}-\dfrac{\partial c\_f(f,x)}{\partial x}\qquad (3)$$

Within the right hand side of (3), the third component is the direct effect of a change in pollution on cost (which is what Varian's 34.3 shows). The other two components are indirect effects of a change in pollution through their effect in changing the profit-maximising volume of fish production $f$. That there must be such an effect can be inferred from the profit-maximising condition for the fishery (cf Varian p 651):

$$p\_f=\dfrac{dc\_f(f,x)}{df}\qquad(4)$$

Here $p\_f$ is fixed, so on varying $x$ the only way to maintain equality is to vary $f$. The first component on the right hand side of (3) is the indirect effect of a change in $x$, via $f$, on revenue, and the second component is the indirect effect on cost.

If the intersection of the lines in Varian's diagram is to determine the Pareto-optimal quantity of pollution, then the $MC\_f$ line should be interpreted as in (3) above.

> **Comments:**
> **username_0**: Genuinely appreciate for the answer  
> **username_3**: @Adam Bailey I wonder why you did not substitute (3) into (4) to get $d\Pi_f/dx=-\partial c_f/\partial x$. The marginal change in the fishery's profit from a marginal increase in pollution is equal to the the direct effect on profit (i.e. just the negative of the increase in costs) from a marginal increase in pollution (this follows from the envelope theorem).  

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### Answer by username_2 on 2021-10-18. Score: 1

You are both right. Varian doesn't include the $f$-argument in his $MC\_f$ function, so his graph is correct if that function is your red one. Note that he nowhere claims that his $MC\_f$ function shows the one of the fishery *before* integration.


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