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then we must have
so that
Note (come back to this after we have
done CMPs for specific production functions): the expression/value for
must be identical to the expression
for
derived by deriving the cost function and then differentiating it with respect to (since the latter is also marginal
cost). This is illustrated below:
8
ECO 204 Chapter 13: The Short Run Cost Minimization Problem (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute.
Fixed
Input Initial Target
Output
New Target
Output
Initial isocost line Fixed
Input New isocost line 0 1 Variable
Input ( ) That is, with a higher required minimum amount of input 1, total cost must increase or stay the same (why?). This is
illustrated below for the case when the minimum constraint doesn’t bind (can you draw the graph for the case when the
minimum constraint does bind?):
Fixed
Input Initial and new
isocost line Target
Output Fixed
Input
0 Variable
Input
New Initial Having established some general results by using the envelope theorem we now discuss ways to simplify the short run
CMP (ways to cut down the number of KT cases one has to check):
{ } (
⏟ ) ⏟⏟ 9
ECO 204 Chapter 13: The Short Run Cost Minimization Problem (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. { } [ ( ) ∑
⏟ If the only way to produce a positive target output level is by using strictly positive amounts of a variable input, then we
drop that variable input’s nonnegativity constraint (only inequality constraints need to be in the Lagrangian equation).
__________________________________________________________________________ ________________________
Example: Suppose a company uses and
to produce output in the short run according to a CobbDouglas
production function. Its short run CMP is (note how the firm chooses only): {
{ Noting that }
} the production function [ ⏟ implies that: The short run CobbDouglas CMP becomes:
{ } [ By the way, we know that when we solve this problem that
.
__________________________________________________________________________________________________
Example: suppose a company uses
and
to produce output in the short run according to a CobbDouglas
production function. Its short run CMP is (note how the firm chooses
only): { }
{ Noting that the production function } [ ⏟ implies that: 10
ECO 204 Chapter 13: The Short Run Cost Minimization Problem (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. As such the short run CobbDouglas CMP becomes:
{ } [ By the way, we know that when we solve this problem that
.
__________________________________________________________________________________________________
Example: suppose a company uses and
to produce output in the short run according to a linear production
function. It’s short run CMP is:
{ }
} { { { Noting that }
}...
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 Fall '09
 AJAZHUSSAIN
 Economics, Microeconomics

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