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

<p>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, $\mathbf{q}^2=\left(q_1^2,q_2^2,\ldots,q_n^2\right)$.</p>

<p>$(13.9)$ 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 $$ p_1q_1^2+p_2q_2^2+\cdots+p_iq_i^2+\cdots+p_nq_n^2-\beta\left(q_1^1\right)-\mu_2\Phi^2\left(q_1^2,\ldots,q_i^2,\ldots,q_n^2 ,q_1^1\right) $$</p>

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

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

<p>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.)</p>

<p>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 $\frac{dq_1^1}{d\beta}&lt;0$, so perhaps he is using the absolute value of that derivative to get rid of the minus sign in front of $\mu_2$.</p>

CC BY-SA 3.0
2016-12-01T15:34:34.523Z
can you do the same but now from firm 1's perspective?
user10699
CC BY-SA 3.0
2016-12-01T15:41:00.747Z
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.
Theoretical Economist
CC BY-SA 3.0
2016-12-01T15:45:00.077Z
but i want to know if im doing it right , so if you cud that would be great
user10699
CC BY-SA 3.0
2016-12-01T15:47:43.070Z
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?
Theoretical Economist
CC BY-SA 3.0
2016-12-01T16:22:41.253Z
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)
user10699
CC BY-SA 3.0
2016-12-01T16:36:03.650Z
why cant you (in firm 1 case) differentiate  as ∂q1i  ,  ? but rather ∂q11 and ∂q12
user10699
CC BY-SA 3.0
2016-12-01T18:03:30.570Z
$(13.14)$ appears to be a typo. $q_i^1$ should not appear there.
Theoretical Economist
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2016-12-01T18:38:52.163Z
really ? is it a typo, its exactly like that in the book, so what should 13:14 look like?
user10699
CC BY-SA 3.0
2016-12-01T19:18:45.117Z
Yes. I believe there is a typo in the book. What do you think the derivative of the objective function with respect to $q_1^1$ should look like? Write it out and I'll tell you if I agree.
Theoretical Economist
CC BY-SA 3.0
2016-12-01T23:05:02.843Z
$p_{1}$ + the rest of expression (13:14)? But why do you believe its a typo, would really like to see your solution ,  and also i dont understand why cant you (in firm 1 case) differentiate as ∂q1i , ? but rather ∂q11 and ∂q12 ?
user10699
CC BY-SA 3.0
2016-12-01T15:24:49.493Z
2016-12-01T15:24:49.493Z
Theoretical Economist
<p>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:</p>

<p>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(⋅).</p>

<p>The optimization problem is</p>

<p><a href="https://i.sstatic.net/ZX2kO.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/ZX2kO.png" alt="enter image description here"></a></p>

<p>My question is how did they arrive at those FOC's? </p>

<p>UPDATE:The second part of this optimization is to look at the problem from firm 1 perspective, it follows like this:
<em>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</em></p>

<p><a href="https://i.sstatic.net/5bwPz.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/5bwPz.png" alt="enter image description here"></a></p>

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

CC BY-SA 3.0
2016-12-01T11:56:36.473Z
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?
Ubiquitous
CC BY-SA 3.0
2016-12-01T12:01:42.827Z
j=1,2  depending on if its firm 1 producing the output q or firm 2
user10699
CC BY-SA 3.0
2016-12-01T12:08:47.047Z
i=1 or 2, first order condition wrt to firm 1 and first order wrt to firm 2 , i believe
user10699
CC BY-SA 3.0
2016-12-01T14:37:09.553Z
OP are you familiar with partial derivatives at all?
Lee Sin
<microeconomics><mathematical-economics><optimization><externalities>
Externalities - First order conditions
user10699
# Externalities - First order conditions

(Asked by: username_0 on 2016-12-01)

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

[![enter image description here](https://i.sstatic.net/ZX2kO.png)](https://i.sstatic.net/ZX2kO.png)

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*

[![enter image description here](https://i.sstatic.net/5bwPz.png)](https://i.sstatic.net/5bwPz.png)

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

**Tags:** microeconomics, mathematical-economics, optimization, externalities
> **Comments:**
> **username_1**: 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?  
> **username_0**: j=1,2  depending on if its firm 1 producing the output q or firm 2  
> **username_0**: i=1 or 2, first order condition wrt to firm 1 and first order wrt to firm 2 , i believe  
> **username_2**: OP are you familiar with partial derivatives at all?  

---

## 1 answers

### Answer by username_3 on 2016-12-01. Score: 2 | Accepted answer

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 $\mu\_2$.

> **Comments:**
> **username_0**: can you do the same but now from firm 1's perspective?  
> **username_3**: 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.  
> **username_0**: but i want to know if im doing it right , so if you cud that would be great  
> **username_3**: 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?  
> **username_0**: 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)  
> **username_0**: why cant you (in firm 1 case) differentiate  as ∂q1i  ,  ? but rather ∂q11 and ∂q12  
> **username_3**: $(13.14)$ appears to be a typo. $q_i^1$ should not appear there.  
> **username_0**: really ? is it a typo, its exactly like that in the book, so what should 13:14 look like?  
> **username_3**: Yes. I believe there is a typo in the book. What do you think the derivative of the objective function with respect to $q_1^1$ should look like? Write it out and I'll tell you if I agree.  
> **username_0**: $p_{1}$ + the rest of expression (13:14)? But why do you believe its a typo, would really like to see your solution ,  and also i dont understand why cant you (in firm 1 case) differentiate as ∂q1i , ? but rather ∂q11 and ∂q12 ?  

---

