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Unformatted text preview: Chapter 9 Long and Short Run Costs As described in the previous chapter to describe firm behavior it is preferable to work from production functions, to cost functions, to profits functions as a means of describing the decisions of the firm. In the one variable case, finding the costs associated with a given production function is straightforward as there is only one amount of the input that will produce a given level of output. If the production function has more than one input, the task is not so simple because there may be many combinations of inputs that produce the same level of output. To make this point clear, consider a case in which the production function has the Cobb-Douglas form: x = L 1 2 K 1 2 . Suppose further that the cost of a unit of labor is w (the wage), while the cost of a unit of capital is r , the hourly rental rate on capital. 1 If we were to try to find a cost function for this production function, what we would do is first find the amount of capital necessary to produce a given x- x ( K ), say - the amount of labor necessary to produce x- x ( L ), and then plug this into the expression describing how much these inputs cost: i.e., C ( x ) = wL ( x ) + rK ( x ). The computational difficulty is that there are many different ways to produce the same level of output. Consider the level of output x = 10. Then, our production function 1 If this way of describing the costs of machinery confuses you, we will discuss it more in a moment, but remember why it is necessary: we have to make sure all of our variables are measured with respect to time in the same way! 191 192 CHAPTER 9. LONG AND SHORT RUN COSTS tells us that our inputs have to satisfy 10 = L 1 2 K 1 2 . There are many different combinations of K and L that produce this level of output. For example, K = 10 ,L = 10 does this trick, but so do K = 100 ,L = 1, K = 1 ,L = 100, or even K = 25 ,L = 4. Which input combination are we to choose to plug into wL + rK ? The answer comes from the idea that our firms are rational actors in- terested in maximizing their profits. This suggests that the best input combination producing a given level of output is the cheapest one. The rest of this chapter will be devoted to developing a methodology for figuring out the cost functions of firms given that firms prefer the cheapest combination of inputs that produce a given x . A tool useful in studying the cheapest combination of inputs problem is the Isoquant . An isoquant describes all possible combinations of inputs that produce a given level of output. In the previous paragraph, I was basically appealing to a description of the points along the x = 10 isoquant corresponding to the production function x = L 1 2 K 1 2 . The x = 10 isoquant for this production function is described by the equation K = x 2 L . We could develop the equations for several isoquants corresponding to the same production function, and in this way create a sort of topographical map of the production function. An example of somesort of topographical map of the production function....
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- Fall '09