# Externalities - First order conditions

I am currently reading the book "Microeconomics: Principles and Analysis" by Cowell on my own. I'm reading the externalities chapter, and i found an interesting example:

There are just two firms: firm 1 is a polluter and firm 2 the victim. Firm 2 (the victim) makes an offer of a side-payment or bribe to firm 1. The bribe is an amount that is made conditional upon the amount of output that firm 1 generates: the greater the pollution, the smaller is the bribe; so we model the bribe as a decreasing function β(⋅).

The optimization problem is

My question is how did they arrive at those FOC's?

UPDATE:The second part of this optimization is to look at the problem from firm 1 perspective, it follows like this: Now look at the problem from the point of view of firm 1. Once the victim firm makes its offer of a conditional bribe, firm 1 should take account of it. So its profits must look like this

This is from F.A.Cowell - Microeconomics - Principles and Analysis p.444-445

• I deleted my answer because it seems like I misinterpreted what some of the variables in the problem are. Could you include details of what the $i$ and $j$ in $q_i^j$ are? Dec 1 '16 at 11:56
• j=1,2 depending on if its firm 1 producing the output q or firm 2 Dec 1 '16 at 12:01
• i=1 or 2, first order condition wrt to firm 1 and first order wrt to firm 2 , i believe Dec 1 '16 at 12:08
• OP are you familiar with partial derivatives at all? Dec 1 '16 at 14:37

To make sure it's completely explicit: superscripts below are indices referring to either firm $1$ or firm $2$.

The choice variables in this problem are $\mathbf{q}^2$ and $\beta$. Notice that $\mathbf{q}^2$ is a vector of $n$ quantities. That is to say, $\mathbf{q}^2=\left(q_1^2,q_2^2,\ldots,q_n^2\right)$.

$(13.9)$ is just the derivative of the objective function $(13.8)$ with respect to $q_i^2$. Notice that you can rewrite the objective function $(13.8)$ as $$p_1q_1^2+p_2q_2^2+\cdots+p_iq_i^2+\cdots+p_nq_n^2-\beta\left(q_1^1\right)-\mu_2\Phi^2\left(q_1^2,\ldots,q_i^2,\ldots,q_n^2 ,q_1^1\right)$$

Differentiating this with respect to $q_i^2$ and setting that equal to $0$ gives us $$p_i - \mu_2 \frac{\partial\Phi^2 \left( \mathbf{q}^2,q_1^1 \right)}{\partial q_i^2} = 0$$

In Cowell's notation, $\Phi^2_i$ is just the derivative of $\Phi^2$ with respect to $q_i^2$.

The second first-order condition is the derivative of the objective function with respect to $\beta$. Since $\beta(\cdot)$ is a decreasing function of $q_1^1$, we can also think of $q_1^1$ as a decreasing function of $\beta$. (Formally, $q_1^1$ is the inverse of $\beta$, which is well-defined since $\beta$ is decreasing. Intuitively, if firm $2$ conditions their bribe on firm $1$'s level of output, then firm $1$'s output choice also depends on the amount of the bribe.)

Thus, applying the chain rule, the derivative of the objective with respect to $\beta$ is $$-1 -\mu_2\frac{\partial\Phi^2 \left( \mathbf{q}^2,q_1^1 \right)}{\partial q_1^1} \frac{dq_1^1}{d\beta}=0$$ which, unfortunately, isn't quite the same as in Cowell. Notice however that $\frac{dq_1^1}{d\beta}<0$, so perhaps he is using the absolute value of that derivative to get rid of the minus sign in front of $\mu_2$.

• can you do the same but now from firm 1's perspective? Dec 1 '16 at 15:34
• I could, but it's essentially exactly the same. If you're actually serious about learning this, you should do it on your own -- you'll learn more that way. Dec 1 '16 at 15:41
• but i want to know if im doing it right , so if you cud that would be great Dec 1 '16 at 15:45
• Why don't you post what you've done (possibly in a separate question) so that other people can comment on whether you've done it correctly? Dec 1 '16 at 15:47
• just by looking at the foc's for firm 1, in ( 13:14 ) ∂q11 and in (13:15) ∂q12 , although im not sure how that would give p1q1i in (13:14) Dec 1 '16 at 16:22