ajaz_eco_204_2012_2013_chapter_14_PMP_Algebra

# Eco 204 s ajaz hussain do not distribute express the

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Unformatted text preview: Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. ❶ Express the Lagrangian equation in terms of parameters. We don’t know the specific functional form of the demand and cost function and so the “parameterized Lagrangian” is: [ [ ⏟] ❷ Differentiate the Lagrangian equation with respect to the parameter. The only parameters so far are and to find the change in due to a small change in a parameter (ceteris paribus) we differentiate respect to the parameter. Ceteris paribus these are: and with ❸ To find the change in -- and therefore -- due to a small change in a parameter (ceteris paribus) we evaluate the derivative in step ❷ at the optimal solution (which we’ll know after we have solved the PMP): That is, that a \$1 increase in From KT conditions we know that will reduce optimal profits by \$1. which tells us that a small expansion in the firm’s capacity will never decrease total profits. From KT conditions we know that which tells us that a small increase in the minimum output requirement will never increase total profits. We now use the FOCs and KT conditions to derive and characterize the various possible solutions to the general PMP: ⏟ ⏟ ⏟ See below for interpretation of “Total [ ” as 3 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. ⏟ [ Starting with the “animal” expression ⏟] ⏟] [ followed by the “animal” [ we see that there are 4 possible solutions to the general PMP: ⏟ ⏟ When is When is ⏟ ? ⏟ When is ⏟ When is When is ⏟ When is Case A ⏟ Case B If/when is Case C ⏟ ⏟? Case D If/when is If/when is If/when is If/when is ? If/when is ⏟? If/when is ? If/when is We now examine each case separately. Case A ⏟ When is When is This says the company should produce an output of zero and at capacity (w...
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