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ajaz_eco_204_2012_2013_chapter_14_PMP_Algebra

# We assume the firm is in the short run and has solved

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Unformatted text preview: n in game theory). We assume the firm is in the short run and has solved and its short run Cost Minimization Problem (CMP) for an arbitrary target output level and therefore knows its short run cost function . We also assume that the firm has a finite capacity and a minimum production constraint . Under these assumptions, the firm’s Profit Maximization Problem (PMP) is: 1 Thanks: Asad Priyo, Adam Michael Lavecchi, and especially Akber Nafeh for typing practice problems and solutions. For feedback, comments and typos please e-mail [email protected] Thanks to the following for feedback: Corrado Vindigni, Arshaq Meraj. 2 Formally, the CMP was: 1 ECO 204 Chapter 14: The Mathematics of the Profit Maximization Problem (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. ⏟ ⏟ [ ⏟ Since ⏟] [ we have: [ ⏟ [ ⏟] This is the PMP Lagrangian equation for any company, whether it is perfectly competitive or a monopoly, etc. The FOCs and KT conditions are are: ⏟ ⏟ ⏟ See below for interpretation of “Total ” as [ ⏟ [ ⏟] Intuitively, we know that the PMP will have several possible solutions (“cases”). Before solving for these cases, let’s use the envelope theorem to establish some general results about profit maximization. To apply the envelope theorem we note that after we solve the PMP, its Lagrange multipliers will tell us something (don’t they always?). To see what the Lagrange multipliers tell us, notice that if you solved the PMP and substituted into the Lagrangian equation, you will get the “optimal Lagrangian equation” where some terms will be zero (why?): [⏟ ⏟ [ ⏟] That is, the optimal value of the Lagrangian is the optimal (maximal) value of profits. We can use the envelope theorem to calculate the change in due to a small change in a parameter (without resolving the problem): 2 ECO 204 Chapter 14: The Mathematics of the Profit Maximization Problem (this version 2012-2013) University of...
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