If is always then we must have so that note come back

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Unformatted text preview: s 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 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. Fixed Input Initial Target Output New Target Output Initial iso-cost line Fixed Input New iso-cost 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 iso-cost 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 2012-2013) 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 non-negativity 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 Cobb-Douglas production function. Its short run CMP is (note how the firm chooses only): { { Noting that } } the production function [ ⏟ implies that: The short run Cobb-Douglas 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 Cobb-Douglas 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 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. As such the short run Cobb-Douglas 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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