Ch 13 SR CMP - University of Toronto Department of...

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University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. 1 ECO 204 Chapter 13: The Short Run Cost Minimization Problem (this version 2012-2013) Department of Economics (STG), ECO 204, Sayed Ajaz Hussain _________________________________________________________________________________________________ C HAPTER 13: The Short Run Cost Minimization Problem 1 Updated: 3/8/2013 Fixed typos are shaded yellow 1. Introduction In the previous chapter, we modeled a firm’s long run cost minimization problem in which it chooses the cost minimizing amounts of all inputs required to produce a target level of output in the long run: ∑( ) ( ) We saw that solving the long run CMP in parametric form yields demand functions for each input in terms of parameters (“factor demand functions”) as well as the total cost of inputs in terms of parameters (“long run cost function”) . In this chapter we will setup and analyze the general and Cobb-Douglas short run cost minimization problem where the firm chooses the cost minimizing amounts of all variable inputs required to produce a target level of output in the short run: ∑( ) ( ) 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] Note to self: (1) Develop stronger connection between returns to scale and economies of scale to show the concepts are the same when the inputs expansion path is such that ratio of inputs is constant (FIX CHAPTER 11 and 12); (2) Do linear and other short run CMPs; (3) Maybe cut down detailed analysis?; (4) More envelope theorem applications like HWs and tests; (5) Highlight importance of free disposal by doing linear CMP where y-axis of iso-quant for (you would have to dispose of the excess ) which BTW is also an excellent example of a problem where one would have to check SOC for the KT cases (because we can’t get conditions for each case)
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University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. 2 ECO 204 Chapter 13: The Short Run Cost Minimization Problem (this version 2012-2013) Before we start , it’s a good idea to contrast the long run CMP with the short run CMP with a simple example. Suppose a company is in the long run and uses the Cobb-Douglas production function where The following graph shows the optimal ( ) bundle to produce a target output in the long run (you should solve this problem and verify that ): Target output Long run CMP Long run = Short run iso-cost In the long run all inputs are variable and the long run CMP gives the optimal amounts of all inputs with which the firm can produce the target output. In contrast, the short run CMP gives the optimal amounts of all variable inputs with which, given positive amounts of the fixed inputs, the firm can produce the target output. Now, t here’ s no reason why the amount of a fixed input in the short run should be equal to the amount of that amount in the long run. For example, in the long run CMP above we saw that . Now suppose the firm is in the short run and capital is fixed: it may be that or it may
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