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deriving cost functions

# deriving cost functions - 1 Deriving cost functions This...

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1 Deriving cost functions This note provides step-by-step details on how to solve for the short-run and long-run cost functions for a °rm. I will °rst discuss the solution in short-run and then the solution in the long-run. For the purposes of illustration, suppose that the production technology of the °rm is given by Q = 15 K 1 = 4 L 3 = 4 ; the wage rate of labor is w and the rental rate of capital is r: There are no additional costs of production. 1.1 Short-run total and marginal cost functions Remember that a cost function, C ( Q ) ; tells you how much it costs to a °rm to produce Q units of output, using the production method that minimizes its overall costs of production. In the short-run, the level of capital is °xed. As a result, the only input that the °rm is able to adjust is the amount of labor hired and the associated variation in the output. We can break the derivation of the cost function into three steps: Step 1: Write down the cost of production in terms of labor and capital The total cost of production is simply the amount of labor and capital that the °rm hires. This is given, by de°nition, by C = rK + wL: In the short run, the level of capital is given, so K = K: The °rm can change the amount of output produced only by altering its level of labor. As a result, we can write the short-run total cost function as STC ( Q ) = r K + wL ( Q ) : Step 2: Find out the relationship between labor and output The second step is to °gure out the relationship between labor and output.

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