My Math Forum EXTERNALITIES - First order conditions

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 November 30th, 2016, 03:51 PM #1 Member   Joined: Sep 2013 Posts: 83 Thanks: 0 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 03:53 PM.
 November 30th, 2016, 03:56 PM #2 Member   Joined: Sep 2013 Posts: 83 Thanks: 0 This is from F.A.Cowell - Microeconomics - Principles and Analysis p.444-445

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