How is production managed with respect to the long run vs the short run?

Question

Assuming perfect competition, I think that firms are price takers in the labor/capital markets as well (in the short and long run), correct?

And I know that the Long-run total cost curve is derived by getting

1. a production function q
2. having a given wage and rental rate (price of labor and capital)
3. optimizing C given q and plugging in w and r.

Then we differentiate LRTC to get Long-run MC and equate that to the price (MR) and that gives us the amount to produce.

Therefore the LRTC function is really a function of 3 inputs, we just keep r and w constant and say that it is a function of only q.

But in the short run, aren't w and r still constant? R certainly is. In that case, how is the LRTC different (mathematically) from the SRTC?

In other words, how is production determined in the short-run? If it's all the same, how is LRTC different from SRTC? Is it also a function of q, as well as w and r?

Answer 1

The way the short run is usually treated is that it is assumed that some decisions have already been made, and in the short run no alteration is possible. E.g.; payment for some input was already made, or the installation takes time thus increasing some inputs quantity is not feasible, etc.

Mathematically this is modelled by making the quantity of some inputs $x_i$ fixed. E.g., in the simple two input case the long run cost function might be $$ C(q) = \min_{x_1,x_2} w_1x_1 + w_2x_2 $$ $$\text{s.t.:} \ \ f(x_1,x_2) = q,$$ while if $x_1$ is fixed in the short run at level $\overline{x}_1$, then the short run cost function would be $$ C_s(q,\overline{x}_1) = \min_{x_2} w_1\overline{x}_1 + w_2x_2 $$ $$\text{s.t.:} \ \ f(\overline{x}_1,x_2) = q.$$ The notation above assumes that the input prices $w_i$ are fixed, and hence they were omitted from the cost functions, but you can also include them as parameters in both the short run and the long run cost functions.