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Chapter 10: Problem 5

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?

Short Answer

Expert verified
a. The optimal price that the doctor will charge all patients is \$100 and she will see 12 patients each day. b. If the doctor can perfectly price discriminate, she will see a total of 63 patients per day.

Step by step solution

01

Identifying the optimal price-quantity pair

To find the profit-maximizing price and quantity, one could calculate the profit for each price level. The profit is obtained by multiplying the price (minus the constant marginal cost of \$50) with the associated quantity. For instance, with a price of \$200, profit would be \((\$200 - \$50) * 2\), which equals \$300 in profit. This process needs to be repeated for each price-quantity pair.
02

Maximizing profit

The price and associated quantity that yield the highest profit would be the most optimal choice. If there is more than one pair yielding the same highest profit, the largest quantity is chosen because it serves more people. This is according to the rule that in action a, the doctor must charge all patients the same price.
03

Calculating for Perfect Price Discrimination

In case of perfect price discrimination, it means that the doctor is charging each patient the maximum price that they are willing to pay. That implies she will treat as many patients as possible such that the price they're willing to pay is at least as high as her marginal cost, which is \$50 in this case. We therefore choose the largest quantity on the demand schedule where the price is equal or above \$50, and sum the associated quantities.
04

Draw conclusions

After carrying out the previous steps, we should now have arrived at an answer to both parts of the question.

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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.

You are thinking about tutoring students in economics, and your research has convinced you that you face the following demand curve for your services: $$\begin{array}{cc} \begin{array}{c} \text { Price per Hour } \\ \text { of Tutoring } \end{array} & \begin{array}{c} \text { Number of Students } \\ \text { Tutored per Week } \end{array} \\ \hline>\$ 50 & 0 \\ \$ 40 & 1 \\ \$ 35 & 2 \\ \$ 27 & 3 \\ \$ 26 & 4 \\ \$ 20 & 5 \\ \$ 15 & 6 \\ <\$ 15 & 6 \end{array}$$ Each student who hires you gets one hour of tutoring per week. You have decided that your time and effort is worth \(\$ 25\) per hour and that you will not tutor anyone for less than that. a. Suppose you are wary that your students might talk to each other about the price you charge, so you decide to charge them all the same price. Determine (1) how many students you will tutor; (2) what price you will charge; and (3) your weekly earnings from tutoring. b. Now suppose you discover that your students don't know each other, and you decide to perfectly price discriminate. Once again, determine (1) how many students you will tutor; (2) what price you will charge; and (3) your weekly earnings from tutoring. Now suppose that your city requires all tutors to get a license, at a cost of \(\$ 1,300\) per year (\$25 per week). c. Does it make sense for you to buy this license and be a tutor if you must charge each student the same price? Explain. d. Does it make sense for you to buy the license and be a tutor if you can perfectly price discriminate? Explain.

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?

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.)

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.
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