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Chapter 10: Problem 13

Suppose a single-price monopoly's demand curve is given by \(P=20-4 Q,\) where \(P\) is price and \(Q\) is quantity demanded. Marginal revenue is \(M R=20-\) 8Q. Marginal cost is \(M C=Q^{2} .\) How much should this firm produce in order to maximize profit?

Short Answer

Expert verified
This firm should produce 2 units in order to maximize profit.

Step by step solution

01

Set Marginal Revenue Equal to Marginal Cost

To find the quantity that maximizes profit for this firm, we start by setting the marginal revenue (MR) equal to the marginal cost (MC). We have the two equations as \(MR = 20-8Q\) and \(MC = Q^2\). So, we can write it as \(20-8Q = Q^2\).
02

Rearrange the Equation

The equation is quadratic so we want to rearrange it into a standard form. That means we need to have the terms ordered in descending power, with the constant term on the other side. So, we rearrange our equation to result in \(Q^2 + 8Q - 20 = 0\).
03

Solve for Q using Quadratic Formula

We can use the quadratic formula to find the solutions to this equation. The quadratic formula is \(Q = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\), where a, b, and c are the coefficients from the quadratic equation. In this case, a is 1, b is 8, and c is -20. Plugging these values into the quadratic formula gives two solutions: \(Q = -10\) and \(Q = 2\).
04

Discard Non-sensible Solution

Quantity can't be negative in economic context. Hence discard Q = -10. Thus, the quantity that maximizes the firm's profit is \(Q = 2\).

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Most popular questions from this chapter

Below is demand and cost information for Warmfuzzy Press, which holds the copyright on the new best-seller, Burping Your Inner Child. $$\begin{array}{ccc} \begin{array}{c} Q \\ \text { (No. of Copies) } \end{array} & \begin{array}{c} P \\ \text { (per Book) } \end{array} & \begin{array}{c} A T C \\ \text { (per Book) } \end{array} \\ \hline 100,000 & \$ 100 & \$ 20 \\ 200,000 & \$ 80 & \$ 15 \\ 300,000 & \$ 60 & \$ 162 / 3 \\ 400,000 & \$ 40 & \$ 221 / 2 \\ 500,000 & \$ 20 & \$ 31 \end{array}$$ a. Determine what quantity of the book Warmfuzzy should print, and what price it should charge in order to maximize profit. b. What is Warmfuzzy's maximum profit? c. Prior to publication, the book's author renegotiates his contract with Warmfuzzy. He will receive a great big hug from the CEO, along with a onetime bonus of \(\$ 1,000,000,\) payable when the book is published. This payment was not part of Warmfuzzy's original cost calculations. How many copies should Warmfuzzy publish now? Explain your reasoning.

Draw demand, \(M R,\) and \(A T C\) curves that show a monopoly that is just breaking even.

Draw demand, \(M R, M C, A V C,\) and \(A T C\) curves that show a monopolist operating at a loss that would cause it to stay open in the short run, but exit the industry in the long run. Then, show how a technological advance that lowers only the monopolist's fixed costs could cause a change in its long-run exit decision.

In a certain large city, hot dog vendors are perfectly competitive, and face a market price of \(\$ 1.00\) per hot dog. Each hot dog vendor has the following total cost schedule: $$\begin{array}{cc} \begin{array}{c} \text { Number of Hot } \\ \text { Dogs per Day } \end{array} & \text { Total cost } \\ \hline 0 & \$ 63 \\ 25 & 73 \\ 50 & 78 \\ 75 & 88 \\ 100 & 103 \\ 125 & 125 \\ 150 & 153 \\ 175 & 188 \\ 200 & 233 \end{array}$$ a. Add a marginal cost column to the right of the total cost column. (Hint: Don't forget to divide by the change in quantity when calculating \(M C .)\) b. What is the profit-maximizing quantity of hot dogs for the typical vendor, and what profit (loss) will he earn (suffer)? Give your answer to the nearest 25 hot dogs. One day, Zeke, a typical vendor, figures out that if he were the only seller in town, he would no longer have to sell his hot dogs at the market price of \(\$ 1.00\). Instead, he'd face the following demand schedule: $$\begin{array}{cc} \text { Price per Hot Dog } & \begin{array}{c} \text { Number of Hot } \\ \text { Dogs per Day } \end{array} \\ \hline>\$ 6.00 & 0 \\ 6.00 & 25 \\ 5.00 & 50 \\ 4.00 & 75 \\ 3.25 & 100 \\ 2.75 & 125 \\ 2.25 & 150 \\ 1.75 & 175 \\ 1.25 & 200 \end{array}$$ c. Add total revenue and marginal revenue columns to the table above. (Hint: Once again, don't forget to divide by the change in quantity when calculating MR.) d. As a monopolist with the cost schedule given in the first table, how many hot dogs would Zeke choose to sell each day? What price would he charge? e. A lobbyist has approached Zeke, proposing to form a new organization called "Citizens to Eliminate Chaos in Hot Dog Sales." The organization will lobby the city council to grant Zeke the only hot dog license in town, and it is guaranteed to succeed. The only problem is, the lobbyist is asking for a payment that amounts to \(\$ 200\) per business day as long as Zeke stays in business. On purely economic grounds, should Zeke go for it? (Hint: If you're stumped, re-read the section on rent-seeking activity.)

A doctor in a rural area faces the following demand schedule: $$\begin{array}{cc} \begin{array}{c} \text { Price per } \\ \text { Office Visit } \end{array} & \begin{array}{c} \text { Number of Office } \\ \text { Visits per Day } \end{array} \\ \hline \$ 200 & 2 \\ \$ 175 & 3 \\ \$ 150 & 5 \\ \$ 125 & 8 \\ \$ 100 & 12 \\ \$ 75 & 18 \\ \$ 50 & 23 \\ \$ 25 & 25 \end{array}$$ The doctor's marginal cost of seeing patients is a constant \(\$ 50\) per patient. a. If the doctor must charge all patients the same price, what price will she charge, and how many patients will she see each day? b. If the doctor can perfectly price discriminate, how many patients will she see each day?
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