I am confused about the Short Run Total Cost function of this problem.
If the firm's production function is $F(K,L) = K+\ln(L)$ derive the short run total cost function.
I was able to solve the Short run total cost function, but my TA had 2 cases for her function whereas I only had 1 part of her function.
I got $TC= W*e^{q-K} + r\bar{K}$ for all $q$ but my TA had 2 cases for $q$. Specifically, when $q<\bar{K}$ and $q>\bar{K}$
Thank you.
I'm just gonna try doing it. The problem is:
$$\text{min } C = rK + wL \\ \text{s.t. } q = K + \ln(L) $$
The implied constraint, of course, is $L \in [1, \infty)$ (non-negativity constraint, as $q > 0$, and $K = 0$ is, of course, possible).
We can try solving this using Lagrange multipliers. Define the Lagrangian:
$$\mathcal{L} = (rK + wL) + \lambda(q - K - \ln(L))$$
The first order conditions are:
$$\mathcal{L}_{K} = r - \lambda = 0$$ $$\mathcal{L}_{L} = w - \lambda(\frac{1}{L}) = 0$$ $$\mathcal{L}_{\lambda} = q - K - \ln(L) = 0$$
Using some algebra, we get $r = wL$ and $L = e^{q-K}$. Substituting into the original cost equation:
$$C = K(we^{q-K}) + we^{q-K} = we^{q-K}(K + 1) $$
I don’t think this can be simplified much further.
That seems to be the same answer that you got (I just expressed it in terms of wages and capital). So not sure – maybe I’m making a mistake.
