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

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?

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.