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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

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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.

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