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.