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November 30th, 2016, 04:51 PM   #1
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EXTERNALITIES - First order conditions

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 $\displaystyle \beta ( \cdot )$.

The optimisation problem is:

$\displaystyle \mathop {\max }\limits_{\{ {q^2},\beta \} } \sum\limits_{i = 1}^n {[{p_i}q_i^2 - \beta ]} - {\mu _2}{\Phi ^2}({q^2},q_1^1)$

The first-order conditions are:

$\displaystyle \eqalign{
& {p_i} - {\mu _2}\Phi _i^2({q^2},q_1^1) = 0 \cr
& - 1 + {\mu _2}{{d\Phi _i^2({q^2},q_1^1)} \over {dq_1^1}}{{dq_1^1} \over {d\beta }} = 0 \cr} $


How did they arrive at this expression for the FOC's?

Last edited by Ku5htr1m; November 30th, 2016 at 04:53 PM.
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November 30th, 2016, 04:56 PM   #2
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This is from F.A.Cowell - Microeconomics - Principles and Analysis p.444-445
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