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380 L13-Cost Functions

# 380 L13-Cost Functions - Cost Functions R Pope Probably no...

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Cost Functions R. Pope Probably no concept in Econ. 380 is as useful as cost or expenditure. In Economics 110, you learned about opportunity cost and sunk costs both powerful concepts. Here we formalized costs a bit further and greater understanding is obtained. Modern microeconomics emphasizes a duality between the production function and costs. That is, the cost function can be used to recover the economically relevant portion (where a cost minimizer would be) of an isoquant. In this lecture, we develop fewer examples because cost minimization and expenditure minimization (which you have already studied) are formally identical. See page 342 of the text for worked examples. As you know, we distinguish costs in the short-run where one or more inputs are fixed from costs in the long run where all inputs are considered variable. Continuing to work with two inputs, capital and labor as our workhorse, cost, (1) C = vk wl + . So cost is simple, we just sum expenditures on each of the inputs where v is the rental price of capital and w is the wage rate. The rental price of capital v contains depreciation and the opportunity cost of capital (more in class). When k , capital, is fixed we will say we are in the short fun. When and k l are both freely chosen, this is the long run. Though the equation in (1) is cost, it isn’t the cost function. The cost function embodies the least cost quantity of and k l to produce a given level of output. The economists view will specify the cost equation as a minimization problem whereas (1) would be more like an accounting identity. The economists state cost minimization as: (2) , ( , ; ) min { : ( , )} k l C v w q vk wl q f k l = + = i.e., choose the level of capital and labor so as to minimize cost given a particular isoquant. One then thinks of doing this for all isoquants. Thus, the cost function in the long run depends in general on input prices and ouput. The factor demands from (2) are called variously “cost minimizing factor (input) demands” or “conditional factor (input) demands”. Costs in the Short-Run When plant and equipment, k is fixed, the problem becomes: (3) ( , ; ) min { : ( , )} l SC vk w q FC VC vk wl q f k l = + = + = where vk =total fixed cost or TFC and the c wl is total variable cost, TVC where c l is the cost minimizing level of labor. Fortunately, it is incredibly easy with two inputs to explain what c l is. With k fixed at k , there is only one level of l that generates a given q:

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we just solve for l . To illustrate, suppose 5, 10, k q = = 2 100 1 or, solving for , 20. 5 q q k l l l k = = = = Thus, 20. SC vk wl vk w = + = + If the wage were one and \$6 v = , SC=\$30+\$20=\$50. The cost of producing 10 units of output is \$50: \$30 of fixed costs and \$30 of variable costs. This represents points on the three econ. 110 curves: total costs, fixed costs, and variable costs. To get another point on the curves, change q to 15. At this output, labor must be 2 15 5 =45 and total cost is \$30+\$45=\$75 with total variable cost is \$45. And so on.
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380 L13-Cost Functions - Cost Functions R Pope Probably no...

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