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review - Review 1 Chapter 7 Cost Minimization The cost...

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Review April 13, 2009 1 Chapter 7: Cost Minimization The cost minimization problem of the firm is that of finding the optimal combination of productive inputs to minimize total cost given the target of producing a certain amount of output. TC ( Q ) = min K,L wL + rK s.t. Q = F ( L, K ) To solve this problem you need to minimize the correspondent Lagrangian function: if you have just two inputs, the solution of cost minimization problem is the solution of the following two conditions MRTS MP L MP K = w r Tangency Condition Q = F ( L, K ) Quantity Constraint This solution corresponds to the Tangency between the Isocost lines (take the TC = wL + rK , write it in the following way K = TC/r - w/rL ) and the Isoquant curve (derived from Q = F ( K, L )). If K is on the vertical axes and L on the horizontal axes, then the slope of the isocosts is the price ratio - w/r . Another useful way to write the tangency condition is MP L w = MP K r If MP L w > MP K r then L is marginally more productive than K in dollar terms, so we should employ more of it and less of K If MP L w < MP K r then K is marginally more productive than L in dollar terms, so we should employ more of it and less of L 1
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X Axis Y Axis Expansion Path Figure 1: Expansion Path The Expansion Path: is the curve that connects all the tangency (op- timal) points between the isocosts and the isoquants (for all the possible quantity levels). This can be found just by using the condition MRTS = w r and expressing it in terms of “ K as a function of L ”. Example: Consider the production function (Cobb-Douglas) Q = 4 LK and prices w, r . Find the Expansion Path assuming that K is mea- sured on the vertical axes. Impose the tangency condition MRTS = K L = w r The expansion path is simply K = L w r If you are in the SHORT RUN than one of the inputs is fixed and you CANNOT impose tangency. Labor and Capital DEMAND: these demands express the quantity of labor and capital employed for any given level of OUTPUT. Therefore they must be functions of Q . 2
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Example: From the example above, Q = 4 LK . Find labor and capital demands. We already have the tangency condition, which can be written as K = L w r . Now we need the quantity constraint: Q = 4 L ( Lw/r ) = 4 L 2 w r = L = r Q 4 r w and K = w r r Q 4 r w = r Q 4 w 2 r 2 r 2 = r Q 4 w r 2 Chapter 8: Long Run Cost Curves Remember that cost curves are functions of Q , so they have to be written in the form TC ( Q ) , AC ( Q ) , MC ( Q ). Relationship Between Average Curves and Marginal Curves: Anytime the MC ( Q ) is below an AC ( Q ) curve, the AC ( Q ) is decreas- ing. If AC ( Q ) is decreasing there are Economies of Scale Anytime the MC ( Q ) is above an AC ( Q ) curve, the AC ( Q ) is increas- ing. If AC ( Q ) is increasing there are Diseconomies of Scale When the MC ( Q ) = AC ( Q ) we have the minimum of the AC ( Q ). The minimum of the AC ( Q ) is the point where all the economies of scale have been exploited and it’s also called the Minimum Efficient Scale If the AC ( Q ) is constant over a certain range, then it must be equal to the MC ( Q ) (in fact is neither increasing nor decreasing, so MC cannot be neither above nor below). In this case the
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