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 y0, such that the firm would choose to use k units of input 2 to produce y0, 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: STCk(y0) = TC(y0). 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) = STCk(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