Lecture_16_Cost_Minimization

Lecture_16_Cost_Minimization - Lecture 16 Cost Minimization...

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Lecture 16: Cost Minimization. c ° 2008 Je/rey A. Miron Outline 1. Introduction 2. Cost Minimization 3. Returns to Scale and the Cost Function 4. Long-Run and Short-Run Costs 5. Fixed, Quasi-Fixed, and Sunk Costs 1 Introduction In the previous lecture we discussed how a °rm would choose inputs and outputs to maximize pro°ts, given the output price, input prices, and technology. This lecture focuses on a distinct, but related question: how should a °rm choose inputs to minimize the costs of producing a given level of output, again taking inputs prices as given (the output price is irrelevant). Cost-minimization is important for two reasons. First, in many instances economic entities do not necessarily want to maximize pro°ts, but they should still want to minimize costs. Non-pro°t institutions, such as universities, charities, some hospitals, and so on, are standard examples. Harvard is not in the business of maximizing pro°ts: it is by design a non-pro°t that seeks to promote other objectives such as learning, truth, and so on. Harvard still should want to minimize its costs, however, given the objectives that it pursues, because reducing costs leaves more revenue to promote these objectives. Second, even when we are focusing on a pro°t-maximizing entity, any pro°t maximization solution must also minimize costs. 1
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Why? Assume not. That is, consider a set of choices for the inputs and outputs that allegedly maximizes pro°ts, but does not minimize the cost of producing the chosen level of output. Then, consider holding output constant, but changing the mix of inputs so that the °rm is minimizing costs given that level of output. Costs then decline, so pro°ts go up - thus, the original choices cannot have been pro°t-maximizing in the °rst place. In other words, a necessary condition to be maximizing pro°ts is that the °rm be minimizing costs. And, in many contexts, we gain substantial insight into the °rm±s behavior by focusing just on cost-minimization. We therefore spend a couple of lectures characterizing the cost-minization prob- lem and related issues. 2 Cost Minimization Suppose a °rm has access to a production function that relates two inputs, x 1 and x 2 , to one output, y . Assume the °rm wants to °nd the cheapest way to produce a given level of output. Then the problem to be solved is min f x 1 ;x 2 g w 1 x 1 + w 2 x 2 subject to f ( x 1 ; x 2 ) = y: This is a problem of minimization subject to a constraint. It is therefore more involved mathematically than the pro°t-maximization problem we did earlier. So, we start with a graphical presentation and then discuss several variations on a calculus approach. (NB: the fact that this is a minimization rather than a maximization is not the reason for the added di¢ culty; that di/erence simply changes the sign of the appropriate second-order condition.) So, consider the following graph: 2
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Graph: Solution to the Cost-Minimization Problem 4 = x 1 = 2 y 1 = 2 y = 8 ° x 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 x1 x2 isoquant cost line slope = -w1/w2
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