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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 email eco.204@utoronto.ca. 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 20122013) 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 20122013) University of...
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This document was uploaded on 01/19/2014.
 Fall '14

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