Is Hal Varian's intermediate microeconomic externality graph wrong?

Question

Here is the capture from Hal Varian's Intermediate Microeconomics chapter Externality.

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.

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 from page $650$ to $653$

Answer 1

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.

Answer 2

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.