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notes_6 - Cost Minimization and Perfect Competition March 8...

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Cost Minimization and Perfect Competition March 8, 2009 1 Cost Minimization The problem of the firm is to maximize profits and this is accomplished by deciding how much to produce, and how to produce that specific quantity. We are going to focus on the second part for the moment: we assume that the quantity to produce is fixed, the only thing that has to be decided is the combination of labor and capital that is most efficient (in the sense of Minimizing the Costs). the locus of points { L, K } that allow the production of the same quan- tity Q is called the Isoquant: Q = F ( L, K ) the locus of points defined by the line wL + rK = TC represents the total cost of production. The objective of the firm is to minimize cost given the isoquant: min L,K wL + rK s.t. F ( L, K ) = Q Solution is given by the tangency point: this correspond to the following two conditions 1. MRTS = w r or MP L MP K = w r ⇐⇒ MP L w = MP K r 2. The quantity constrain F ( L, K ) = Q 1
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L K Q = 17 Figure 1: Isoquant and Isocosts Example: Cobb-Douglas production function: Find the total cost corresponding to the production of 10 units, given the production function Q = LK and prices w = 1 , r = 2. The total cost is the cost of the optimal combination of labor and capi- tal: let’s impose the two conditions. 1. We know that for the Cobb-Douglas case the MRTS is simply MRTS = α β K L = K L = 1 2 Let’s find L as a function of everything else: L = 2 K 2. Quantity constraint: 10 = LK = 2 K 2 = K 2 = K = 10 2 and L = 20 2 Now that we have the optimal bundle we can find its cost: TC (10) = 20 2 + 2 10 2 = 40 2 1.1 Long Run Total Cost Curve To get the total cost function of the firm, we need to repeat this computation for every possible quantity that can be produced. Therefore, instead of having a specific number for Q , we leave Q as a parameter of the problem. 2
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Example - Continued : find the total cost for the above production function, given the same prices.
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