# Is Hal Varian's intermediate microeconomic externality graph wrong?

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.

Hal Varian Text from page $$650$$ to $$653$$

• 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? Oct 17, 2021 at 11:32
– LJNG
Oct 17, 2021 at 11:43

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.

• Genuinely appreciate for the answer
– LJNG
Oct 18, 2021 at 20:46
• @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
Aug 16 at 15:16

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.