# Maximizing a Cobb-Douglas Function

Suppose that a competitive firm receives a price of $$P$$ for its output, and pays prices of w, r and v for its labor $$(L)$$, capital $$(K)$$ and natural resources $$(R)$$ inputs, respectively. The firm operates with the production function $$Q = L^aK^bR^c$$. The firm chooses labor and capital to maximize profit.

a. Derive the firm’s profit function. $$\pi$$

b. Derive the first-order conditions FOC for profit maximization.

My Question is regarding FOC, since the firm chooses labor and capital to maximize profit, do I look at this as a problem with simply two endogenous variables mainly labor $$(L)$$, capital $$(K)$$, hence the FOC are the partial derivative of the proofit function $$w.r.t. L$$ and $$K$$. ($$\pi_L$$ and $$\pi_K$$) or should I include $$\pi_R$$ in my solution

The maximisation problem is : $$\max_{K,L}$$ $$\pi =PL^{a}K^{b}R^{c}-wL-rK-vR$$
The arguments will be only $$K,L$$,so $$F.O.Cs$$ will only include $$\pi _{K},\pi _L$$
The profit function will be a function of $$(P,w,r,v)$$. Finding $$F.O.Cs$$ will give us the $$\pi (P,w)$$ function where $$w$$ denotes all the input prices.