Finance 360 Probem Set #6 Soutions 1) Suppose that you are the manager of an opera house. You have a constant margina cost of production equa to $50 (i.e. each additiona person in the theatre raises your costs by $50 we wi ignore any fixed costs for now.) You have estimated your demand curve for tickets as foows: Q 150 P a) Cacuate your profit maximizing ticket price. Here, we simpy take the demand curve as given, and sove for price: P 150 Q Now, Tota revenues are Price*Quantity PQ 150Q Q Margina costs are the derivative with respect to Q. MR 150 Q Set margina revenue equa to margina cost and sove for Q 50 150 Q Q 50 Pugging Q50 back into the demand curve, which gives us a price of $100. Profits $100(50) - $50(50) $500 Now, suppose that you re-estimated your demand curve, but this time, you incuded a dummy variabe for gender: Q 175 P 50D 1, if consumer is mae D 0, if consumer is femae b) Given your new information, cacuate your profit maximizing price assuming that you can t distinguish between mae and femae customers (i.e. your tickets are sod onine)
First we need to aggregate the two demand curves 175 P, Q 50 Q 300 P, Q 50 Now, cacuate thee two inverse demand pieces: 175 Q, Q 50 P 150.5Q, Q 50 Now, cacuate TR for each piece 175Q Q, Q 50 TR PQ 150Q.5Q, Q 50 Now, cacuate margina revenue for each piece: 175 Q, Q 50 MR 150 Q, Q 50 Lasty, set margina revenue equa to margina cost. Note that with MC $50, you wi aways se at east 50 tickets, so the ony piece of the demand curve that matters is that with Q>50. 50 150 Q Q 100 P $100 Profits $100(100) - $50(100) $5,000 c) Now, cacuate the prices you woud charge if you coud distinguish between mae and femae consumers (i.e. ticket purchasers show up to the box office to buy tickets.) Why might you be concerned about secondary markets forming for your product? In This probem, just treat the individua demand curves separatey, and sove for the price and quantity separatey. For Femaes:
Q 175 P P 175 Q TR 175 Q Q MR 175 Q Q 6.5 P 11.5 50 For Maes: Q 15 P P 15 Q TR 15Q Q MR 15 Q 50 Q 37.5 P 87.5 Profits $11.5(6.5) + $87.5(37.5) - $50(100) $531.50 ) Continuing with the same exampe, suppose again, that you are faced with the same demand curve(s) 175 P, (women) Q 15 P, (men) a) Suppose that you charged the same price to each consumer (the price cacuated in part (b) above. Cacuate the consumer surpus for both consumer types. Given a $100 price, femaes buy 75 tickets and generate a consumer surpus of (1/)(175-100)(75) $,81.50 whie men buy 5 tickets and generate a CS of (1/)(15-100)(5) $31.50. b) Suppose that you were to set a price equa to your margina cost. Cacuate the consumer surpus derived by both consumers. Given a $50 price, femaes buy 15 tickets and generate a consumer surpus of (1/)(175-50)(15) $781.50 whie men buy 75 tickets and generate a CS of (1/)(15-50)(75) $81.50.
c) If you coud distinguish between the types of consumers, how woud you set up your prices to maximize profits? (i.e. you coud start up an opera over s society and charge a membership fee) The idea situation woud be to set up a membership fee for the opera society. Women woud pay a membership equa to $781.50 which woud aow them to buy opera tickets for $50 apiece whie men woud pay a membership price of $81.50 and woud be abe to buy tickets at $50. To make sure everybody joins, set a ridicuousy high price for non-member ticket prices (say, $1,000 per ticket.) d) How woud your answer to (b) change if you coud not distinguish between customer types? (i.e. you coud se different ticket packages. If we se a package of 75 tickets for $,81.50 + 75($50) $656,50, then men woud buy the package, and get no consumer surpus. Suppose that a woman bought the 75 ticket package: We can figure up the tota wiingness of a women to pay for 75 tickets (using the femae demand curve), we get $10,31.50. Subtract off the $656.50 cost of the 75 ticket package and we find that a women gets a CS of $3750 buying the 75 ticket package. We need to make sure she gets at east a CS of $3750 with the 15 ticket package. Tota Wiingness to pay for 15 tickets: $14,06.50: Minus required surpus: $3750 15 Ticket Package: $10,31.50 3) Suppose that you are George Lucas. You are in the process of packaging the fina triogy (actuay the three preques) of Star Wars for sae to the pubic. Your margina costs of production are $ per movie. Further, you know that there are two types of consumers that you face: chidren under the age of 10 and everybody ese. Chidren under 10: Love Jar Jar Binks Everybody ese: Woud ike to see Jar Jar crushed by a very arge truck Consequenty, wiingness to pay for each of the three movies is based on how many minutes Jar Jar is on the screen. Movie Under 10yrs od Over 10 yrs Od Episode 1 $60 $5 Episode $30 $40 Episode 3 $10 $50
a) If you sod these three movies separatey, what woud your prices be? Take the first movie: Movie Saes Tota Revenues P$5 $10 P$60 1 $60 We shoud charge a price of $60 to maximize revenues. Using simiar ogic, we shoud charge a price of $30 for the second and $50 for the third. Cacuate profits: Profits ($60)(1) + ($30)() + ($50)(1) - $(4) $16 b) If you ony sod these movies as a box set, what shoud you charge? As a box set, we can add up each consumers wiingness to pay over the three movies: Movie Under 10 Over 10 Box Set $100 $95 Setting a box set price equa to $95 get us two saes (6 tota movies sod) Profits ($95) - $(6) $178. 4) Suppose that you have two manufacturers: one company speciaizes in the production of eft shoes (They have a store caed The Left Shoe Emporium ). Another company speciaizes in right shoes ( Right Shoes R Us ). Consumers have a demand for shoes given by: Q 150 P Where P is the price of a pair of shoes: P P L + PR. For simpicity, assume that margina costs for each firm constant and equa to zero. a) Write down each firm s inverse demand curve (They price they can charge as a function of saes and their competitor s price) Take the Left Shoe Emporium First: They face the demand curve ( P R ) PL Q 150 Soving for P, we get
P ( PR ) QL 150 Tota revenues are equa to P*Q TR PQ Margina revenue is the derivative. ( P ) Q Q 150 R L ( 150 P R ) QL MR Setting this equa to zero (we assumed MC0) and soving for Q Q 150 PR Substituting back into the demand curve gives us price: P 150 PR Simiary, repeating the process for Right Shoes are us: P R 150 PL b) Sove each firm s profit maximization probem as a function of their competitor s price. In equiibrium, both of the above expressions for price must be true: Take the first expression and pug it into the second: P R P R $50 Note that the cost of a pair of shoes is $100. Now, suppose that these two companies merged. What woud happen to the cost of a pair of shoes? The monopoist faces a demand equa to Q 150 P Just as in previous exercises, soving for the optimum price gives up P $75.
5) Spatia competition is an exampe of non-price competition. Specificay, in addition to the price firm charges, it aso chooses a ocation in which to set up shop. a) Describe how this firm ocation probem is soved. Basicay, it is assumed that consumers pay a money price of a product as we as a time cost for a product (traveing costs, etc). Therefore, if a consumer s reservation price is equa to the sum of the two, then consumers wi pay higher prices at stores that are cose top them. Firms, therefore, must decide (based of consumer s ocations, where to buids their store to maximize profits). b) Suppose that you are a cosmetic manufacturer. How coud you use a spatia competition mode when choosing your cosmetic ine? The same argument as above, however instead of physica ocation, we are taking about preference ocation (i.e. each consumer has an optima version of a product). The coser a firm can ocate to that optima product, the more the firm can charge. Therefore the firm must decide how many version of a product to make and specificay what types to maximize profits.