General Equilibrium with Linear Production I don't think I understand how optimization problems with a linear function work as of now. If you have a production economy with two agents, two goods and Cobb-Douglas utility representation, and you introduce a firm with a linear production function and fixed costs that basically acts as a transfer from one good (say x), to the other good (say y), as far as I can see, the solution to the firm's maximization is to shut down.

This is what I am talking about specifically:

$$
u_1(x,y)=\alpha\ ln(x)+(1-\alpha)ln(y)
$$
$$
u_2(x,y)=\beta\ ln(x)+(1-\beta)ln(y)
$$
$e_1=(0,1), e_2=(1,0)$
with a linear production function for firm $f(x)=x$ where they use x as an input and produce y, and there is a fixed cost FC>0 only if they produce. My problem lies in how do I solve the firm's maximization problem? It's linear, so I don't think I can use the FOC like you can with the utility maximization problem. Intuitively, I "feel" like it's going to be a corner solution, but I don't know how to systematically solve optimization problems with a linear function.
If the firm takes output price and wage as given, then its supply is infinity if the price is larger than the wage, and it is zero if the price is equal, or lower, than the wage. If the price is larger than the wage, the firm by increasing output can make arbitrarily large profits, and thereby, in particular, cover its fixed cost. If the price is equal or below the wage, then there is no way of making profits, and therefore, the unique best option is to set output equal to zero.