The Mathematical Derivation of Optimization Rules
Suppose a firm has a cost function TC=C(q) and a faces an inverse demand curve P=f(q), where q is the firm's output, P is the price it charges, TC is total cost, and C(_) and f(_) are the functional relations for cost and the inverse demand curves. Then for this firm
P = qf(q) - C(q).
Profit maximization requires that we set the derivative of P with respect to quantity equal to zero. Thus
dP /dq = dqf(q)/dq - dC(q)/dq = 0
which is equal to
qf'(q) + f(q) - C'(q) = 0.
Equivalently, we can say
dP /dq = dTR/dq - dTC/dq = 0
so that profit is maximized when
dTR/dq = MR = MC = dTC/dq
that is, when marginal revenue equals marginal cost.
The revenue maximizing firm is somewhat more difficult. Then the goal is to maximize revenue subject to a minimum profit constraint, so we must use a lagrangian maximization problem. In this case set up
L = TR - l P
where we assume the minimum acceptable profit is zero. To assure we reach a maximum we assume that marginal revenue is not increasing at the point of production. Thus the second derivative of the total revenue curve is not positive. By substitution
L = qf(q) - l [qf(q) - C(q)].
Taking the derivative with respect to q and setting equal to zero shows that
¶ L/¶ q = f(q) + qf'(q) - l [f(q) + qf'(q) - C'(q)] = 0
where the first two terms of the derivative equal marginal revenue and
¶ L/¶ l = qf(q) - C(q) = 0.
The first equation can be rearranged to (1-l )MR = l MC. If the constraint is binding, we know that marginal revenue is bigger than zero. But if the constraint is not binding, l =0, and marginal revenue is zero. Moreover, the second order conditions when the constraint is binding will force l to be negative, ensuring that marginal revenue is less than marginal cost.
As an example, let P=100-q and TC= 500 + q - q2 + (1/30)q3. Then
P = q[100-q] - 500 - q + q2 - (1/30)q3.
Maximum profit occurs when
100 - 2q - 1 + 2q - (1/10)q2 = 0
or q2 - 990 = 0.
Thus, we find that q equals just under 31.5. so price is about 68.5, revenue is about 2158, and total cost is about 581. Variable cost is 81, so we surpass the shut down rule, which in fact always happens if profit is positive.
The revenue maximizing firm would produce until marginal revenue equals zero, that is when 100-2q=0 or q=50. At that point price equals 50 as well, total revenue is 2500, and total cost is 2217. With a profit of 283, the firm exceeds the fair rate of return. It also produces more than the profit maximizing firm.
Suppose, however, the profit constraint was not just zero economic profit, but some return greater than the normal rate. Say, for the sake of argument, the firm has promised investors a profit of at least 1000. In this case the firm tries to maximize revenue subject to the constraint that total revenue minus total cost be at least 1000, or, by substitution,
(100-q)q - 500 -q + q2 - (1/30)q3 > 1000.
The nature of our functions assures that this can hold as a strict equality, but for convenience we will require that q be a whole number. At an output of 47 profit equals 1192 while if output is 48 profit falls to about 565. So, the firm would have to produce an output of 47, with a total revenue of 2491. Revenue is higher and profit lower than the profit maximizing output, but is offers a lower revenue and higher profit than the pure revenue maximizing firm.
Application Problems
1. Suppose we have a firm which faces a total cost of TC=0.5q2-10q+200 and a demand curve Q=1500-50P. Find the profit maximizing and revenue maximizing output and price for this firm.
2. If the firm in question 1 is assessed a license fee of 50, what would we expect to happen to its output under each goal?
3. Suppose a new tax law lowers fixed cost to 100. Would the firm's output change? Why or why not?
4. Now suppose the government institutes a labor tax, so the variable cost portion of total cost increases by 10 percent? How would this change output?
5. Finally, suppose demand increases by 5 percent. Find the new profit maximizing and revenue maximizing outputs? Would the firm's revenue maximizing output change if it must return at least the normal rate?