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Reacting to rivals in the Cournot model
Using the demand equation to derive total revenue as a function of q requires the following steps:
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1. Add 200P to both sides of the demand equation.
2. Subtract q from both sides of the equation.
3. Divide both sides of the equation by 200.
4. To determine total revenue, multiply both sides of the demand equation by q.
This equation tells you how much total revenue equals given any value for quantity, q. Thus, total revenue is a function of q.
If your total cost equation is
total profit, π, is determined by subtracting total cost from total revenue, or
After you have the total profit equation, the following steps enable you to determine the profit-maximizing quantity and price:
1. Take the derivative of the total profit equation with respect to quantity.
2. Set the derivative equal to zero and solve for q.
This is your profit-maximizing quantity of output.
Chapter 10: Monopoly: Decision-Making Without Rivals
3. Substitute the profit-maximizing quantity of 2,000 into the demand equation and solve for P.
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Or you should set a price of $40 for the good.
4. Finally, total profit is determined by substituting 2,000 for q in the total-profit equation.
Your total profit equals $18,000.
Maximizing profit with a marginally better method
Quite often it’s easier to determine the profit-maximizing quantity of output by focusing on the last unit you produce, or the marginal unit. In order to add to your profit, an additional or marginal unit of the good must add more to your revenue than it adds to your cost. In other words, marginal revenue is greater than marginal cost. As long as an additional unit adds more to your revenue than it adds to your cost, your profit is increasing. At the output level that maximizes profit, an additional unit of output doesn’t add any more to your total profit. This occurs when marginal revenue equals marginal cost. Stop at this output level because if you go beyond this point and continue to produce more, marginal cost is greater than marginal revenue — so you’re adding more to your cost than you’re adding to your revenue, and your total profit is decreasing.
In monopoly, only one firm is producing the good. Therefore, any customer who wants to buy the good must buy it from the monopoly. Thus, the demand for the firm’s product is the market demand.
In Figure 10-3, the monopolist’s downward-sloping demand curve d is the same as the market demand curve D, or D = d. As previously described in the section “Unable to Charge as Much as You Want: Relating Demand, Price, and Revenue,” given the linear demand curve, the marginal revenue curve has the same intercept on the vertical axis and is twice as steep as the demand curve. Marginal revenue is represented by the curve labeled MR.
Figure 10-3:
Profit maximization with marginal revenue and marginal cost.
Marginal cost, MC, is upward-sloping and passes through the minimum point on average total cost, ATC.
To maximize profits, you produce the output level associated with marginal revenue equals marginal cost, or the output level q0 that corresponds to the point where the marginal revenue and marginal cost curves intersect.
Marginal revenue equals marginal cost maximizes total profit.
To find the price you charge, go from the profit-maximizing quantity of output, up to the demand curve and across. The profit-maximizing price corresponds to P0.
The fact that price is determined by the point where q0 hits the demand curve emphasizes the constraint that market demand places on the monopolist’s ability to set price. If advertising increases demand — shifts the demand curve to the right — you’re able to set a higher price.
If you produce less than q0 output, for example qA, those units still add more to your revenue than they add to your cost. For those units, marginal revenue is greater than marginal cost. You increase profits by producing more output moving toward q0.
If you produce output beyond q0, such as qB, those units add more to your cost than they add to your revenue because marginal cost is greater than marginal revenue. These units reduce your profit so you should cut back production to q0.
