11.cThe Mathematics of Monopolistic Profit Maximization
The profit maximizing price and output of a monopolist may also be derived using elementary calculus. For example, consider a firm facing the following market conditions:
(1)Demand:p = 100 – Q
(2)Total Cost:TC = 100 + 20Q
In other words, the price of the good manufactured will be 100 less the quantity sold. For example, if the market supplies 30 units, consumers will pay $70 per unit. Stated in reverse, if a firm set the price of the good at $70, consumers would demand 30 units. The fixed cost of the good is $100, and the variable cost is $20.
The first step in deriving a monopolist’s profit-maximizing price and output is to generate the firm’s average revenue, total revenue, marginal revenue, average cost and marginal cost. For a firm not engaged in price discrimination, the demand curve and average revenue curve are the same.
(3)Average Revenue [D]:AR = 100 – Q
Total revenue is simply the quantity of goods sold multiplied by the average revenue.
(4)Total Revenue [Q*AR]:TR = 100Q –Q2
Next, marginal revenue can be generated by taking the derivative of the total revenue curve.
(5)Marginal Revenue [dTR/dQ]:MR = 100 – 2Q[1]
Moving on to the supply side, average cost is simply total cost divided by quantity.
(6)Average Cost [TC/Q]:AC = [100/Q] – 20
Marginal cost is the derivative of total cost.
(7)Marginal Cost [dTC/dQ]:MC = 20
Now we have all the formulas we need to calculate the monopolist's profit maximizing price and output, along with monopoly profits.
Step One:Set marginal revenue equal to marginal cost.
(5) = (7)
100 – 2Q = 20
2Q =80
Q = 40
Step Two:Plug the monopolist's quantity of output into the demand equation to calculate price.
p = 100 – Q
p = 60
Thus, the monopolist will set its price at $60, and its output at 40 units.
Next, we can calculate monopoly profits in one of two ways. By solving the problem both ways, the answer can be checked against itself.
Method One:Multiply the difference between price and average cost by quantity
Profit = Q*[p – AC]
Profit = 40*[60 – {(100/40) + 20}]
Profit = 40*[60 – 22.5]
Profit = 40*37.5 = $1,500
Method Two:Subtract total cost from total revenue
Profit = TR – TC
Profit = (100Q-Q2) – (100 + 20Q)
Profit = (4,000 – 1,600) – (100 + 800)
Profit = 2,400 – 900 = $1,500
Since the two methods generate the same profit, we can have confidence that our answer is correct.
More advanced calculus can be used to shorten this process. To do so, we must generate a total profit equation (which equals total revenue less total cost)
(8)Total Profit [TR-TC]:-Q2 + 80 Q –100
Next, take the first and second derivatives to calculate marginal profit.
(9)Marginal Profit [dTP/dQ]:-2Q + 80
(10)Marginal Marginal Profit [d2TP/dQ2]:-2
With these equations, one can perform the following steps to calculate monopoly output and profits:
Step One:Find all values where marginal profit equals zero.
-2Q + 80 = 0
2Q = 80
Q = 40
Step Two:Plug the monopolist's quantity of output into the demand equation to calculate price.
p = 100 – Q
p = 60
Step Three:Verify that this is a local maximum by checking to see if the second-order derivative is negative. Since -2 is negative, 40 is a local maximum
Step Four:Plug Q into the total profit equation:
-1600 + 3200 – 100 = $1,500
[1]Notice that the marginal revenue equation has the same vertical intercept (100) and twice the slope (-0.50) as the demand curve (-0.25). Using simple calculus, this can be proven true for all linear demand curves: (1) p = a – bQ; (2) TR = aQ – bQ2; (3) dTR/dQ = MR = a – 2bQ.