## Long run and short run cost functions

In the **long run**, the firm can vary all its inputs. In the **short run**, some of these inputs are fixed. Since the firm is constrained in the short run, and not constrained in the long run, the long run cost TC(*y*) of producing any given output *y* is no greater than the short run cost STC(*y*) of producing that output:
TC(*y*) STC(*y*) for all *y*.

Now consider the case in which in the short run exactly *one* of the firm's inputs is fixed. For concreteness, suppose that the firm uses two inputs, and the amount of input 2 is fixed at *k*. For many (but not all) production functions, there is *some* level of output, say *y*_{0}, such that the firm would *choose* to use *k* units
of input 2 to produce *y*_{0}, even if it were free to choose any amount it wanted. In such a case, for this level of output the short run total cost when the firm is constrained to use *k* units of input 2 is equal to the long run total cost: STC_{k}(*y*_{0}) =
TC(*y*_{0}). We generally assume that for any level at which input 2 is fixed, there is some level of output for which that amount of input 2 is appropriate, so that for any value of *k*,

TC(*y*) = STC_{k}(*y*) for some *y*.

(There are production functions for which this relation is not true, however: see the example of a production function in which the inputs are perfect substitutes.)
For a total cost function with the typical shape, the following figure shows the relations between STC and TC.

Examples of long run and short run cost functions

### Long run and short run average cost functions

Given the relation between the short and long run total costs, the short and long run average and marginal cost functions have the forms shown in the following figure.

**Note**:

- The SMC goes through the minimum of the SAC and the LMC goes through the minimum of the LAC.
- When SAC = LAC we must have SMC = LMC (since slopes of total cost functions are the same there).

In the case that the production function has CRTS, the LAC is horizontal, as in the following figure.

Copyright © 1997 by Martin J. Osborne