1. What is price differentiation?
2. What are the real-world limitations of price differentiation? Name two of them and explain briefly.
3. Name of three of the tactics for price differentiation and explain them briefly.
4. What is product versioning? How is it implemented in real life? Explain by an example.
5. What is consumer surplus?
6. How does price differentiation affect total consumer surplus? Does total consumer surplus increase or decrease under price differentiation? Explain.
7. What is runout price?
8. What is opportunity cost?
9. What is marginal opportunity cost?
10. A supplier is selling hammers in two cities, Pleasantville and Happy Valley. It costs him $5.00 per hammer delivered in each city. Let p1 be the price of hammers in Pleasantville and p2 be the price of hammers in Happy Valley. The price-response curves in each city are:
Pleasantville: d1(p1) = 10,000 - 800p1
Happy Valley: d2(p2) = 8,000 - 500p2
a) Assuming the supplier can charge any prices he likes, what prices should he charge for hammers in Pleasantville and Happy Valley to maximize total contribution? What are the corresponding demands and total contributions?
b) An enterprising arbitrageur discovers a way to transport hammers from Pleasantville to Happy Valley for $0.50 each. He begins buying hammers in Pleasantville and shipping them to Happy Valley to sell. Assuming the supplier does not change his prices from those given in part a, what will be the optimal price for the arbitrageur to sell hammers in Happy Valley? How many will he sell? What will his total contribution be? (Assume that Happy Valley customers will buy hammers from the cheapest vendor.) What will happen to the total sales and contribution for the supplier? (Remember that he is now selling to the arbitrageur too.)
c) The supplier decides to eliminate the arbitrage opportunity by ensuring that his selling price in Happy Valley is no more than $0.50 more than the selling price in Pleasantville (and vice versa). What is his new selling price in each city? What are his corresponding sales and total contribution?
d) From among the Pleasantville buyers, the Happy Valley buyers, and the seller, who wins and who loses from the threat of arbitrage?
11. The football game between Stanford and Berkeley is going to be held at Stanford Stadium, which has 60,000 seats. Customers can be segmented into students (those carrying a student ID) and the general public. We assume that the price-response curves for each of the segments is:
General Public: dg(pg) = (120,000 - 3 000pg)+
Students: ds(ps) = (20,000 - 1,250ps)+
where ps is the price charged to students and pg is the price charged to the general public. Assume that 5% of the general public will masquerade as students (perhaps using borrowed ID cards) in order to save money.
a) Assuming that Stanford knows that 5% of the general public will masquerade as students, what are the optimal prices for student tickets and general public tickets it should set in this case?
b) What is the total revenue in this case, and how does it compare to the case without cannibalization? (Without cannibalization, we already know that the total revenue is $1,256,470.59)
c) What does this say about the amount that Stanford would be willing to pay for such devices as photo ID cards in order to eliminate cannibalization?
12. The football game between Stanford and Berkeley is going to be held at Stanford Stadium, which has 60,000 seats. Customers can be segmented into students (those carrying a student ID) and the general public. We assume that the price-response curves for each of the segments is:
General Public: dg(pg) = (120,000 - 3 000pg)+
Students: ds(ps) = (20,000 - 1,250ps)+
where ps is the price charged to students and pg is the price charged to the general public. The optimal prices for each segment are pg = $22.353 and ps = $10.3528. In the optimal solution, 7,059 tickets sold to students, 52,941 to general public, with total contribution of
$1,256,470.59.
a) An earthquake damages Stanford Stadium so that only 53,000 seats are available for the Big Game. What are the optimal single price and the total revenue?
b) What are the optimal separate prices to charge for students and the general public and the corresponding total revenue? What is the "opportunity cost" per seat for the 7,000 unavailable seats in both cases?
13. The football game between Stanford and Berkeley is going to be held at Stanford Stadium, which has 60,000 seats. Customers can be segmented into students (those carrying a student ID) and the general public. We assume that the price-response curves for each of the segments is:
General Public: dg(pg) = (120,000 - 3 000pg)+ Students: ds(ps) = (20,000 - 1,250ps)+
where ps is the price charged to students and pg is the price charged to the general public. Now assume that, on average, each member of the general public will consume $20 worth of concessions, resulting in a $10 contribution margin, while each student only consumes $10, resulting in a $5 contribution margin. Assuming no cannibalization, what are the prices for students and the general public that maximize total contribution margin (including, of course, ticket revenue)?
14. A barber charges $12 per haircut and works Saturday through Thursday. He can perform up to 20 haircuts a day. He currently performs an average of 12 haircuts per day during the weekdays (Monday through Thursday). On Saturdays and Sundays, he does 20 haircuts per day and turns 10 potential customers away each day. These customers all go to the competition. The barber is considering raising his prices on weekends. He estimates that for every $1 he raises his price, he will lose an additional 10% of his customer base (including his turnaways). He estimates that 20% of his remaining weekend customers would move to a weekday in order to save $1, 40% would move to a weekday in order to save $2, and 60% would move to a weekday in order to save $3. Assuming he needs to price in increments of $1, should he charge a differential weekend price? If so, what should the weekend price be? (Assume he continues to charge $12 on weekdays.) How much revenue (if any) would he gain from his policy?
15. Consider a theme park that can serve up to 1,000 customers. The theme park charges a single admission price, and all rides are free after admission. During summer, demand follows a stable and predictable pattern with higher demand on weekends. The demand curves are linear and their parameters are given below:
| Day of week |
Intercept (Di) |
Slope (m,) |
| Sunday |
3,100 |
-62 |
| Monday |
1,500 |
-50 |
| Tuesday |
1,400 |
-40 |
| Wednesday |
1,510 |
-42 |
| Thursday |
2,000 |
-52.6 |
| Friday |
2,500 |
-55.6 |
| Saturday |
3,300 |
-60 |
In addition to the revenue from admissions, the owner determines that visitors to the theme park spend an average of $12 per person on concessions, generating an average concession margin of $5 per person.
a) Assuming theme park has an incremental cost of zero per customer, what is the single- admission price the theme park should charge to maximize total weekly margin (admission price plus concession margin)?
b) Assuming theme park has an incremental cost of zero per customer, what are the individual daily prices he should charge under a variable-pricing policy to maximize total weekly margin, assuming independent daily demands? What is the impact on total weekly admissions? What is the total weekly margin?
c) What is the impact of variable-pricing policy on total weekly margin from explicitly including concessions in the optimization relative to optimizing prices on the basis of admission revenue alone? (Here, you have to calculate the optimal variable-pricing policy on the basis of admission revenue alone as well in order to make the comparison.
16. Assume that the theme park owner invested in expanding his park so that he could accommodate up to 1,500 customers each day. The theme park still charges a single admission price, and all rides are free after admission. During summer, demand follows a stable and predictable pattern with higher demand on weekends. The demand curves are linear and their parameters are given below:
| Day of week |
Intercept (Di) |
Slope (m,) |
| Sunday |
3,100 |
-62 |
| Monday |
1,500 |
-50 |
| Tuesday |
1,400 |
-40 |
| Wednesday |
1,510 |
-42 |
| Thursday |
2,000 |
-52.6 |
| Friday |
2,500 |
-55.6 |
| Saturday |
3,300 |
-60 |
a) What single-price policy would maximize his total revenue, assuming he faces independent demands for each day? What is his corresponding attendance and revenue per day?
b) Under the same assumptions, what would be the variable prices he should charge for each day to maximize expected revenue, and what are the corresponding attendance and revenue per day?
17. The same theme park owner performs some market research and determines that his customers can best be represented by the following model.
| Day of week |
Intercept (Di) |
Slope (m,) |
| Sunday |
3,100 |
-62 |
| Monday |
1,500 |
-50 |
| Tuesday |
1,400 |
-40 |
| Wednesday |
1,510 |
-42 |
| Thursday |
2,000 |
-52.6 |
| Friday |
2,500 |
-55.6 |
| Saturday |
3,300 |
-60 |
i. Base customer demand for each day of the week is linear and is specified by the parameters in the following table:
ii. Weekend (Saturday and Sunday) customers will switch to the other weekend day at the rate of 10 customers for every $1 difference in price. They will not switch to weekdays at any price.
iii. Weekday customers will switch to any other day (including a weekend day) at the rate of 8 customers for every $1 difference in price.
What are the optimal daily prices the theme park should charge?
18. A monopolist sells in two markets. The price-response curve in market 1 is q1 = 200 - p1 while the price-response curve in market 2 is q2 = 300 - p2. The firms total cost function is (q1 + q2)2. The firm is able to price discriminate between the two markets.
a) What quantities will the monopolist sell in the two markets?
b) What price will it charge in each market?
19. Suppose a supplier can identify two distinct groups of customers, students and non- students. The demand by students qS and the demand by nonstudents qN are given by qS = 100 - 8pS and qN = 100 - 4pN, respectively. The total demand, qT = qS + qN, is then qT = 200 - 12pT. The suppliers cost of $2 per unit is constant regardless of the number of units supplied.
a) What price maximizes profits if the firm charges everyone the same price?
b) Show that the firm can secure greater profits by charging different prices for the two groups than it can secure by charging everyone the same price.
20. Consider a seller facing a linear price-response curve d(p) = 1,000 - 100p. The seller can sell an item to these 1,000 potential customers in three periods. Assume that the good being sold has zero incremental cost, and the customers purchase the time (at most once) as soon
as price falls below their willingness to pay in one of the periods. If the seller can charge three different prices at three periods, what three prices, p1, p2, and p3, will maximize total revenue? What if there are four periods and four prices? What is the general formula for n prices?
21. Consider the same seller in the first question. The seller can sell the item to her potential customers in two periods. In total, she has 1000 potential customers who buy the item at most once. In the first period, seller faces a linear price-response curve d(p) = 1,000 - 100p. Assume that customers have a lower willingness to pay for the good in the second period. Specifically, assume that each customer's willingness to pay in the second period is 75% of her willingness to pay in the first period. Moreover, assume that the good being sold has zero incremental cost.
a) What are the optimal prices and corresponding total revenue for the seller, assuming she can charge only a single price in both periods?
b) What are the optimal prices and corresponding total revenue, assuming she can charge different prices in the two periods?
22. A department store has 700 pairs of purple Capri stretch pants that it must sell in the next four weeks. The store manager knows that demand by week for the next four weeks will be linear each week, with the following price-response functions:
Week 1: d1(p1) = 1000 - 100p1 Week 2: d2(p2) = 800 - 100p2 Week 3: d3(p3) = 700 - 100p3 Week 4: d4(p4) = 600 - 100p4
Assume that the demands in the different weeks are independent, that is, that customers who do not buy in a given week do not come back in subsequent weeks.
a) What is the optimum price the retailer should charge per pair if she can only set one price for all four weeks? What is her corresponding revenue?
b) Assume she can charge a different price each week. What are the optimum prices by week she should charge? What is her corresponding revenue?