Question

10. Suppose the firm of Example 8.5.1 earns revenue R(Q) and incurs cost C(O) as functions of output Q0, where R' (Q) > 0, C'Q) > 0, R" (Q) < 0, and C"(Q>0 for allQ 0. The firm maximizes profit (Q) R(Q)- C() subject to Q 0. Write down the first-order conditions for the solution to this problem, and find sufficient conditions for the constraint to bind at the optimum

Answer 1

$\pi$(Q) = R(Q) - C(Q)

$\pi$'(Q) = R'(Q) - C'(Q)

First order condition

put $\pi$'(Q) = 0  

R'(Q) - C'(Q) = 0  

R'(Q) = C'(Q)

Now second order condition

$\pi$"(Q) = R''(Q) - C''(Q)   

for maximum   $\pi$"(Q) < 0  

thus   R''(Q) - C''(Q) < 0  

   R''(Q) < C''(Q) which means slope of revenue function is less than slope of cost function

To sum up, Profit maximization conditions are

(i)    R'(Q) = C'(Q) which is necessary condition

(ii)    R''(Q) < C''(Q) which means slope of revenue function is less than slope of cost function. This is sufficient condition.