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Unformatted text preview: 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 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 (CobbDouglas) 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 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....
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This note was uploaded on 05/09/2009 for the course ECON 200 taught by Professor Junnie during the Spring '08 term at NYU.
 Spring '08
 JunNie

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