1. The graph below depicts the demand curve and cost curves for a particular medical procedure from the Sparrow Medical Clinic.

Transcription

1 Economics MBA Spring Mike Conlin Price Discrimination SOLUTIONS. The graph below depicts the demand curve and cost curves for a particular medical procedure from the Sparrow Medical Clinic. 7 ATC AVC D 7 9 Calculate the clinic s maximum profits if the medical clinic can perfectly price discriminate (i.e. first degree price discriminate). Show your calculations. Profits are maximized at an output of. Total revenue is the area under the demand curve which is.(- )*+*=. Total cost is *= so profits are -=.

2 . The following graph depicts a firm's cost curves and the demand curve the firm faces. Profits ATC AVC D Q MR a) What is the firm's marginal revenue (MR) associated with the th unit of output if the firm sets a single price? At Q=9, TR=.(9)=9. At Q=, TR=.()= Therefore, the firm's marginal revenue (MR) associated with the th unit of output is -9.=-9. b) What are the maximum profits if the firm sets a single price? Show calculations. Profits are maximized at Q=. TR=()= TC=ATC(Q)=()= Profits=-= c) What is the firm's marginal revenue (MR) associated with the th unit of output if the firm can st degree price discriminate? At Q=9, TR=.(9)+.(-.)9=9.7 At Q=, TR=.()+.(-)=7 Therefore, the firm's marginal revenue (MR) associated with the th unit of output is 7-9.7=.. OR you could note that D=MR when you perfectly price discriminate and at a quantity of 9., MR=..

3 ATC AVC TR Q d) What are the firm s profits (approximately) if the firm can st degree price discriminate? If the firm can st degree price discriminate, profits are maximized at Q= (approximately). TR=.*(-)*+*= TC=ATC(Q)=.7* =. Profits=-.=. e) What is the maximum revenue the firm can obtain if it st degree price discriminates? (The firm can select any output it would like.) To maximize revenue when the firm st degree price discriminates, the firm would set Q= and total revenue would be.()()=.

4 . The demand curve for students who have applied for early admissions to John Hopkins University this year is given on the graph to the left. The demand curve for students who have applied to John Hopkins University this year, but not for early admissions, is given on the graph to the right. (Assume these are all the students who have the qualifications to be accepted into the program and that these qualifications are the same.) EARLY ADMISSIONS NO EARLY ADMISSIONS De Dne MR e Suppose tuition at John Hopkins University is set at $, and the marginal cost is constant at $,. (Marginal cost is just the increase in total cost associated with increasing the number of students by one.) If John Hopkins is to maximize profits, how much financial aid (i.e., scholarships) should be offered to students who apply for early admissions and how much to students who do not apply for early admissions? Explain. Assume that John Hopkins must provide the same financial aid to all those who apply early and the same financial aid to all those who do not apply early. However, John Hopkins can provide different amounts of financial aid to those who apply early and those who do not apply early. To maximize profit, John Hopkins sets MR= for early admissions and for no early admissions. John Hopkins would charge $, for those who apply for early admissions by offering them $, in financial aid. John Hopkins would charge $, for those who do not apply for early admissions by offering them $, in financial aid. MR ne

5 . The graphs below represent the demands for the Love Boat cruise from senior citizens and non-senior citizens. Senior Citizens S DS Non-Senior Citizens NS DNS MR S MR NS Let the Love Boat s total capacity be passengers. Suppose total fixed costs (TFC) are $, and marginal cost (at quantities less than and equal to capacity) is constant at $. (The marginal cost is extremely high when quantity is more than capacity.) What are the Love Boat s maximum profits if they are able to third degree price discriminate (i.e., charge senior citizens a different price than non-senior citizens)? SHOW CALCULATIONS. The Love Boat would charge senior citizens a price of $ and have senior citizens as passengers. The Love Boat would charge non-senior citizens a price of $ and have non-senior citizens as passengers. [Note that the total number of passengers () is less than the total capacity of. Note also that the marginal cost curves for senior citizens and non-senior citizens are not precise on the graphs (in terms of at what quantity marginal cost increases/ jumps due to capacity constraints). The reason is that the marginal cost for the senior citizens depends on the number of non-senior citizens who are passengers. The reason this is the case is due to the capacity constraint.] Profits would be *+*-*-, = -,.

6 . Suppose the demand for local telephone service in East Lansing is from two types of individuals. There are type A individuals and type B individuals. Each type s individual demand curve is depicted on the graphs below (where q is the number of local phone calls per month). The telephone company cannot distinguish between type A and type B individuals. The marginal cost of a local phone call is constant at $. and the telephone company s monthly fixed costs are $. Suppose the telephone company charges $. per call and a monthly fixed fee that maximizes their profits. Type A Type B.... DA... 7 Maximum monthly fee Type A is willing to pay Maximum monthly fee Type B is willing to pay DB What are the telephone company s profits from this -part pricing scheme? If the telephone company charges $. per call, Type A is willing to pay a maximum monthly fixed fee of.*(.-.)*=. and Type B is willing to pay a maximum monthly fixed fee of.*(.-.)*=.. Profits if monthly fixed fee is $.. *.+*.*+*.*-.[*+*]- =. Profits if monthly fixed fee is $.. *.+*.*-.[*]- =. Therefore, profits are maximized at $ when the monthly fixed fee is $..

7 . Suppose the website is owned by Mr. Film P. Jector and allows users to download movies from the website. The demand for access to this website is from two types of individuals. There are Type A individuals and Type B individuals. Each type s individual demand curve is depicted on the graphs below (where q is the number of movies downloaded per month). The website cannot distinguish between Type A and Type B individuals. The marginal cost incurred by Mr. Film P. Jector for each downloaded movie is zero and Mr. Film P. Jector s monthly fixed costs are $,. Suppose Mr. Film P. Jector charges $ per movie download and a monthly membership fee that maximizes his profits. The monthly membership fee allows one to access the website. Type A Type B Max Type B is willing to pay in membership fee is.*(-)= Max Type A is willing to pay in membership fee is.*(-)= DA DB 7 9 What are Mr. Film P. Jector s maximum profits from this -part pricing scheme? SHOW ALL CALCULATIONS. Profits if membership fee is *+**+**-= Profits if membership fee is 7 *7+**-= Profits are maximized at a membership fee of $7. 7

8 7. The first graph on the next page depicts a monopolist s cost curves and the market demand curve the monopolist faces. The market demand curve is based on demand from Type A individuals and Type B individuals. The demand curve for each Type A individual is depicted on the next page, along with the demand curve for each Type B individual. Suppose the monopolist decides on a two-part pricing scheme where the monopolist charges a fixed fee (often called membership or entry fee) and variable fee (often called a per unit fee). Assume the monopolist charges a variable fee of $ per unit. a) If the monopolist cannot distinguish between type A and type B individuals, what fixed fee will the monopolist select to maximize profits? SHOW ALL CALCULATIONS. If the monopolist charges a variable fee of $ per unit, Type A is willing to pay a maximum fixed fee of.*(- )*= and Type B is willing to pay a maximum fixed fee of.*(-)*=. Profits if fixed fee is $. *7+**+**-*. =. [Note that total quantity is *+*= and average total cost at an output of is..] Profits if fixed fee is $. *+**-* =. Therefore, profits are maximized at $ when the fixed fee is. b) Now suppose that the monopolist can hire Mike Conlin who can distinguish a Type A individual from a Type B individual. Therefore, if the monopolist hires Mike Conlin, the monopolist can offer Type A individuals a different pricing scheme than Type B individuals. What is the maximum amount the monopolist is willing to pay Mike Conlin? SHOW CALCULATIONS. If can distinguish between type A and type B, the monopolist could charge a $ fixed fee to Type A s and a $ fixed fee to Type B s. This is st degree price discrimination. Let c be the amount paid to Mike Conlin. Profits would be *+*+**+**-*.-c=9-c. Profits are greater than a) if c<.

9 D ATC AVC TYPE A TYPE B 7.. Maximum monthly fee Type A is willing to pay 7.. Maximum monthly fee Type B is willing to pay DB DA. 9

10 . There is demand for a bowling alley from two types of individuals. There are Type A individuals and Type B individuals. Each type s individual demand curve is depicted on the graphs below (where q A is the number of games bowled by a Type A individual and q B is the number of games bowled by a Type B individual). The bowling alley has a constant marginal cost per game of $. and total fixed costs of $. Type A () Type B () DA Maximum Type A is willing to pay for shoes q Maximum Type B is willing to pay for shoes DB A q B Suppose the bowling alley charges $ per game and selects the rental price for bowling shoes so that profits are maximized. (Once an individual pays for the bowling shoes, he/she can wear them to bowl as many games as he/she chooses. An individual cannot bowl at all if he/she does not rent bowling shoes.) What are the bowling alley s profits from this -part pricing scheme if the bowling alley cannot distinguish between Type A and Type B individuals? SHOW ALL CALCULATIONS. If the price per game is $, Type A is going to bowl games and Type B is going to bowl game (assuming the rental price for bowling shoes is not too high). Given the price per game is $, the maximum rental price for bowling shoes that Type A is willing to pay is the red area depicted on the graph. The area equals.*(-)*=. Using a similar analysis, the maximum rental price for Type B is willing to pay is.*(-)*=.. Profits if shoe rental fee is $ *+**-.*(*)-= Profits if shoe rental fee is $..*+**+**-.*(*+*)-=7 Profits are maximized if the bowling alley charges $ to rent the bowling shoes.

11 9. You own an Italian restaurant that serves only a fettuccine alfredo meal. The graphs below depicts the demand for your fettuccine alfredo meal during dinner (D D ) and during lunch (D L ). Suppose your marginal cost for each meal is $9 during dinner and $ during lunch. (The difference in the marginal cost is because you serve larger portions at dinner and labor is more expensive during dinner time.) Also assume that you have a capacity of seats in your restaurant and your fixed costs per day are $ D L DL DD 7 9 MR L MR D What are your daily profits? Show Calculations. If the number of seats is, q L =, p L =, q D = and p D =7. TR= ()+(7) = and TC= TVC L +TVC D +TFC=()+(9)+7=7. Therefore, profits are -7=.

12 . Suppose the graph below depicts the hourly demand curves for the Lincoln Tunnel during peak hours, D P, and during non-peak hours D NP. Q P and Q NP indicate the number of cars per hour during peak hours and non-peak hours, respectively. Assume there are peak hours in a day (-am and -pm) and non-peak hours. Let the marginal cost of a car passing through the Lincoln Tunnel be zero. 9 7 Dp Dnp Qp, Qnp MR np MR p Suppose the Port Authority of New York and New Jersey sets one price per car during peak hours and another price per car during non-peak hours. What are the Lincoln Tunnel s daily profits if the short-run capacity of cars per hour is,? Assume that the daily fixed costs are $,. Show your calculations and use above graph. ()+()-, =,.

13 . Sorry I could not make this question applicable to MSU. If only the football and basketball teams played in the same arena. At Syracuse University, both teams played in the Carrier Dome. The good news is that MSU is better than SU in both football and basketball although basketball is close. Suppose the graph below depicts the demand curves for each Syracuse University football game in the Carrier Dome, D f, and for each Syracuse University basketball game in the Carrier Dome, D b. Assume the football team plays games each year in the Carrier Dome and the basketball team plays games each year in the Carrier Dome. Also assume that the Carrier Dome seats, more people for a football game than for a basketball game because for basketball games SU blocks off several sections of seats. (For example, if the Carrier Dome has, seats, then the number of people that can watch a football game is, and the number that can watch a basketball game is,.) Let the marginal cost of a person attending an SU football or basketball game be constant at $. In addition, suppose the Carrier Dome has, seats and the annual fixed costs be $ million. 9 7 Db 7 Df 9 qb, qf MR b MR f What are Syracuse University s annual total profits from football and basketball games? Show your calculations. [*-*]+[*-*]-,,=7,,.

14 . Below is an excerpt of an article entitled Bon Appetit that appeared in the Wall Street Journal. Apparently, hotels will do anything to get out of cooking. First, midrange hotels started closing their fullservice restaurants, replacing them with fast food kiosks and take-out service. Now, even the higher-end hotels are leasing their restaurants to people in the food business. David Ruggerio, chef of the upscale restaurant Le Chantilly in New York, plans to open Pastis, a French Restaurant, at the Parker Meridien in November. Hoteliers hope mane-brand chefs will attract more than just hotel guests to the restaurant, and turn an unprofitable operation into a money maker. It s a very different experience in Europe, where many of the best restaurants are in hotels, says Mark Lomanno of Smith Travel Research, in Hendersonville, Tenn. That very rarely happens in the U.S.. How many times have you gone out to dinner in a hotel? Suppose you own the Parker Meridien hotel and are deciding whether to pay David Ruggerio to open a restaurant (Pastis) in your hotel. Currently, you do not contract with an individual to operate a restaurant in your hotel. Instead, you operate your own restaurant. The current daily demand for your hotel and the current daily demands for lunch and dinner at your restaurant are depicted on the graphs below. The demand for your restaurant during lunch is denoted by D L and the demand during dinner is D D. The hotel has daily total fixed costs of $, a constant marginal cost of $, and rooms. The restaurant has daily total fixed costs of $ and seats. The marginal cost associated with preparing a lunch or a dinner is constant at $. Assume you cannot price discriminate for hotel rooms and you charge a single price for dinner and a single price for lunch. HOTEL RESTAURANT 7 D DL DD MR MR L MR D Profits without Ruggerio. Profits from hotel are *-*-=. Profits from restaurant are *+*-*-*-=7. Total profits are.

15 Suppose you sign a contract with David Ruggerio for him to operate the Pastis restaurant in your hotel. The contract specifies that David Ruggerio is entirely responsible for the operations of Pastis. He is responsible for paying all labor and material expenses (i.e. all variable costs) and keeps all revenue generated by the restaurant. If Pastis does open in your hotel, it increases the demand for your hotel rooms. Suppose the demand for your hotel if Pastis opens is as depicted on the graph below (and you still cannot price discriminate). If you allow Ruggerio to operate the restaurant, assume the hotel s fixed costs remain the same but the hotel s marginal cost increases from $ to $. What is the minimum you are willing to accept per day or the maximum you are willing per day to pay Ruggerio (for him to open the restaurant in your hotel)? SHOW YOUR CALCULATIONS AND EXPLAIN. *-*--=. 7 9 Therefore, the minimum you should be willing to accept from David Ruggerio is -=. When doing the solutions, I am assuming that David Ruggerio s restaurant will replace the existing restaurant in the hotel. While the large majority of students assumed the same thing, others did not. Because I did not explicitly state this in the problem, students who did not assume this were not penalized. However, students who assumed that the hotel would open a second restaurant operated by David Ruggerio would have to specify their assumptions in terms of the additional costs incurred by the hotel from this second restaurant as well as mention the possible affect on the current restaurant to obtain full credit. MR D

16 . This question is based on my experience buying stuff for my kids. Graco is a manufacturer of car seats for infants. Graco is deciding whether to offer a rebate on its Snugrite car seat and, if a rebate is offered, Graco must decide how long to make the rebate form (i.e., number of pages). Graco has two types of customers: Type A s and Type B s. Assume that Graco cannot distinguish between the types. Type A s are willing to pay a maximum of $ for a Snugrite car seat and Type B s are willing to pay a maximum of $ for a Snugrite car seat (assuming they do not have to fill out the rebate form). There are Type A s and Type B s. Suppose the opportunity cost per page associated with filling out the rebate is $ for Type A s and $ for Type B s. Assume that Graco s marginal cost of a Snugrite car seat is constant at $ and Graco s fixed costs are $,. What price should Graco charge for the Snugrite car seat, what should be the amount of the rebate and how many pages should Graco make the rebate form? SHOW CALCULATIONS. If offer no rebate, Graco maximizes profits by setting a price of $. Profits would be *-*-=. The first thing you should do is think about this problem intuitively. What you would ideally like to do is to have Type A s pay $ and Type B s pay $ for the car seat. You cannot achieve this because you cannot offer Type A s and B s different prices. However, you can offer them the option of filling out a rebate form and choose the length and value of the rebate to be such that Type B s fill in the rebate and Type A s do not fill it in. When thinking about the optimal length of a rebate form, you will want to make it as least costly as possible for Type B s. The more costly you make it for Type B s, the lower the amount you can obtain from them. For example, if you make the cost of filling out the rebate form $ for Type B s, then the maximum they are willing to pay for the car seat given that they use the rebate is -=. While making it as least costly for Type B as possible, you also must make sure that Type A does not have incentive to fill out the rebate. In addition, you would like Type A s to pay the maximum they are willing to pay - $ but this may not be possible. Assuming that Type A s will not use the rebate if they are indifferent between using it and not, the table below calculates Graco s profits for different page lengths. [If you don t like this assumption, then have the rebate value be $9.99 if the rebate form is page, $9.99 if the rebate form is pages, ] After determining that the rebate value should be times the number of pages, you then select the regular price such that Type B s are willing to buy the car seat with the rebate. For a one page rebate form, Type B s are willing to pay a maximum of $ for the car seat given that they plan to use the rebate. If the rebate is $, the maximum Graco can set the regular price at is $ if they want Type B s to buy. A similar logic applies for how the regular price was determined in the table below when the rebate form is more than one page. Pages Rebate Value Regular Price Profits $ $ *(-)+*(-)-= $ $7 *(7-)+*(-)-= $ $7 *(7-)+*(-)-= $ $ *(-)+*(-)-= $ $ *(-)+*(-)-= Therefore, Graco s profits are maximized if they set a regular price of $, have a page rebate form and have the value of the rebate be $. The intuition as to why a page, $ rebate is optimal. Graco wants the hurdle to be as low as possible for Type A s while still being high enough to deter Type B from using it. Graco also wants Type B to pay $.

17 . This question is based on my experience when I visited TC Timber. TC Timber is a company located in Skaneateles, New York that manufactures wooden train sets (similar to those produced by Brio). Suppose TC Timber has two types of customers: grandparents and parents. Assume that TC Timber cannot distinguish grandparents from parents. The grandparents are willing to pay a maximum of $ for a wooden train set and parents are willing to pay a maximum of $ for a wooden train set (assuming they do not have to wait in line in order to purchase the train set). There are grandparents and parents. Every year TC Timber has a sale the first weekend in December where they sell the wooden train set for a sale price. During this weekend, people buying a train set must wait hours in line (there are no lines other than this sale weekend). Suppose the opportunity cost associated with waiting the hours is $ for grandparents and $ for parents. Assume that TC Timber s marginal cost of a wooden train set is constant at $. What price should TC Timber charge for the train set and what should be the sale price? SHOW CALCULATIONS. Let F be TC Timber s total fixed costs. If the parents decide to wait in line and obtain the sale price, the maximum sale price they are willing to pay is $. If the sales price is $, then the maximum TC Timber can set the regular price at so that the grandparents pay the regular price is $ (or you could say $9.99 to make grandparents strictly prefer the regular price to the sale price). TC Timber s profits in this case would be (-)+(-)-F=,-F. If TC Timber did not have a regular price and a sale price and instead had just a regular price, TC Timber s maximum profits would be (-)-F=,-F by setting a price of $. (TC Timber s profits would be (-)- F=,-F if they set a price of.) 7

18 . Sony has an outlet store in Aurora Illinois. On Black Friday, the day after Thanksgiving, the Sony outlet store opened at : am. Sony usually opens the store at : am. Sony s outlet store sells the Bravia Flat-Panel High Definition Television. Suppose Sony has two types of customers for their Bravia Flat-Panel High Definition Television (Type A and Type B). Assume that Sony cannot distinguish between the types. Type A s are willing to pay a maximum of $ for a television and Type B s are willing to pay a maximum of $ (assuming they do not have to wait in line in order to purchase the television). There are Type A s and Type B s. On Black Friday the Sony outlet store decided to offer a sales price for the television if the person purchased the television between : and : am. Assume that customers did not have to wait at all if they purchased the television after am but the sales price was not offered at this time. Therefore, customers purchasing the television would have to pay the regular price after am. Between : and : am, Sony only had two cashiers working so customers had to wait an hour in line in order to purchase the television. Suppose the opportunity cost per hour associated with waiting in line is $ for Type A s and $ for Type B s. Assume that Sony s marginal cost of a Bravia Flat-Panel High Definition Television is constant at $ and their fixed costs are zero. For simplicity, assume each consumer only demands a single television. Type A () Type B () WTP Opportunity Cost First note that the firm could not have a sale from to, set a price at and make (-)= in profits. a) What price should Sony charge for the television after : am (i.e., the regular price) and what price should Sony charge for the television between : and : am (i.e., the sales price)? SHOW CALCULATIONS AND EXPLAIN. Sales price= and regular price= so the profits would be *(-)+*(-)=. The profits are the same as when the firm does not have a sale and just sets a regular price of. b) Now suppose that Sony can choose to have more cashiers (increasing the number of cashiers from to ) work from : to : am. This would decrease the waiting time in line (i.e., to check out) from hour to half an hour. What is the maximum Sony should be willing to pay these two cashiers in order for them to work from : to : am? (Note that Sony can change the sales and regular prices calculated for part a) if they increase the number of cashiers from to.) SHOW CALCULATIONS AND EXPLAIN. Type A () Type B () WTP Opportunity Cost Sales price=9 and regular price= so the profits would be *(-)+*(9-) C =-C where C is the cost of the additional cashiers. Therefore, the maximum Sony should be willing to pay these two cashiers in order for them to work from : to : am is -=.