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ECO3041: Industrial Organisation. (Mock) End of term test

There are two questions, you only need to answer one of them, but you should answer all the

parts of the question you choose. If you answer both questions, the highest of your marks will

be taken, but I do not expect you will have time to do this.

Points will be awarded for your workings as well as your final answer.

I advise you to use a calculator that works with fractions rather than decimals. You may also

find it helps to work with letters rather than numbers for as long as possible.

You have 90 minutes.

In the actual end of term test, I will just change the numbers.

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1. Question 1

Sainsbury’s and Tesco are located at both ends of a one mile long street. There are two types of

shoppers (called “high-value” and “low-value”), with equal numbers of each type. High-value shoppers

obtain a value of £𝑣𝐻𝐼 from buying a tub of Haagen-Dazs ice-cream, and an additional value of £𝑣𝐻𝑆

if they buy a bottle of Sainsbury’s Taste the Difference Maple Syrup as well, or an additional value of

£𝑣𝐻𝑇 if they buy a bottle of Tesco Value Maple Syrup as well. Low-value shoppers obtain a value of

£𝑣𝐿𝐼 from buying a tub of Haagen-Dazs ice-cream, and an additional value of £𝑣𝐿𝑀 from buying either

variety of maple syrup. All shoppers only obtain any value from the first tub of ice-cream and the first

bottle of maple syrup they buy (so they will not buy more than one). No shoppers get any value from

buying maple syrup without ice-cream, but they do still get value from buying ice-cream without

maple syrup.

Shoppers of both kinds are uniformly distributed along this street. Both kinds have linear transport

costs, so that walking 𝑑 miles costs them £𝑡𝑑 in lost value. However, once they have been to either

Sainsbury’s or Tesco, they are laden down with shopping, so they wish to just walk home. (I.e. you

may assume that the transport cost of visiting both Sainsbury’s and Tesco is infinite.)

For both Sainsbury’s and Tesco, Haagen-Dazs has a constant marginal cost of 𝑐𝐼. Sainsbury’s Taste the

Difference Maple Syrup has a marginal cost of 𝑐𝑆 for Sainsbury’s, and Tesco Value Maple Syrup has a

marginal cost of 𝑐𝑇 for Tesco. Sainsbury’s and Tesco cannot differentiate between the two types of

shopper based on observable characteristics.

Sainsbury’s charges 𝑝𝑆𝐼 for Haagen-Dazs and 𝑝𝑆𝑀 for their maple syrup. Tesco charges 𝑝𝑇𝐼 for Haagen-

Dazs and 𝑝𝑇𝑀 for their maple syrup.

You may assume the following parameter values. (These are the only things which will change in the

actual test.) 𝑣𝐻𝐼 = 5, 𝑣𝐻𝑆 = 2, 𝑣𝐻𝑇 =3

2, 𝑣𝐿𝐼 = 3, 𝑣𝐿𝑀 = 1, 𝑐𝐼 = 2, 𝑐𝑆 = 1, 𝑐𝑇 =

1

2, 𝑡 =

2

3.

a) Assume that in equilibrium, all consumers buy both ice cream and maple syrup. How far from

Sainsbury’s is the high-value shopper who is just indifferent between buying from Sainsbury’s and

buying from Tesco’s? How far from Sainsbury’s is the low-value shopper who is just indifferent

between buying from Sainsbury’s and buying from Tesco’s? (Your answers may be a function of firms’

prices.) (10%)

b) Given shoppers are visiting Sainsbury’s or Tesco anyway to buy ice cream, what must be true for

them to want to buy maple syrup as well? (Your answer may differ across types of shopper.) Explain

why this means that Sainsbury’s will price maple syrup at either 𝑣𝐻𝑆 or 𝑣𝐿𝑀 (rather than any other

price), and why Tesco will price it at either 𝑣𝐻𝑇 or 𝑣𝐿𝑀, (rather than any other price). In each case, you

do not have to say which of the two options the supermarket would choose. (10%)

c) Suppose that both shops sell their brands of maple syrup at 𝑣𝐿𝑀. Write down each supermarket’s

profits as a function of prices. (Hint: be careful to include the revenue from selling both ice-cream and

maple syrup, to both high-value and low-value customers. Don’t forget costs either!) Then derive their

optimal price for ice-cream. You should find that Sainsbury’s sell ice-cream at £31

12 and Tesco sell it at

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£9

4. Explain intuitively why Sainsbury’s is charging a higher price for ice-cream despite having identical

costs. (30%)

d) Calculate both supermarkets’ profits at the prices you found in (c). (10%)

e) Now suppose that Sainsbury’s deviates to selling maple syrup at 𝑣𝐻𝑆, with Tesco still selling their

maple syrup at 𝑣𝐿𝑀. Which types of consumers would buy maple syrup from Sainsbury’s, given they

were visiting the store anyway? Which types would buy maple syrup from Tesco’s, given they were

visiting the store anyway? (5%)

f) Under the set-up of (e), how far from Sainsbury’s is the high-value shopper who is just indifferent

between buying from Sainsbury’s and buying from Tesco? how far from Sainsbury’s is the low-value

consumer that is just indifferent between buying from Sainsbury’s and buying from Tesco? Your

answers should be a function of 𝑝𝑆𝐼 and 𝑝𝑇𝐼, not the solution found in (c). (5%)

g) Under the set-up of (e), write down an expression for each supermarket’s profits as a function of

prices. (Again, be careful to include the revenue from selling both ice-cream and maple syrup, to both

high-value and low-value customers.) You do not need to attempt to maximise this expression for

profits. (5%)

h) A kindly mathematician tells you that when you hold 𝑝𝑇𝐼 at the value you found in (c), under the

set-up of (e), by choosing 𝑝𝑆𝐼 optimally Sainsbury’s will make profits of 31

96. They go on to tell you that

if you repeat the exercise of parts (e) to (g) assuming that Tesco instead deviate to charging 𝑣𝐻𝑇 for

syrup, then with 𝑝𝑆𝐼 held at the value you found in (c), and 𝑝𝑇𝐼 chosen optimally, Tesco will make

profits of 27

128. Using your answer to part (d), and the calculations of this friendly mathematician,

explain why both firms selling syrup at 𝑣𝐿𝑀 must be a Nash equilibrium. (5%)

i) Now, suppose that Sainsbury’s decide to introduce a “Brand Match Guarantee”, as they have done

in real life. This is effectively a commitment to never price branded goods higher than Tesco does.

Since both firms only sell non-branded maple syrup, here this promise just applies to the Haagen-Dazs

ice-cream. Given this guarantee (and assuming that Sainsbury’s can react instantly), what effect does

a cut in Tesco’s ice-cream price have on Tesco’s profits? As a result, assuming each firm still sells maple

syrup at 𝑣𝐿𝑀, what price will each firm set for ice-cream in equilibrium? What are profits? (20%)

BONUS QUESTIONS IF YOU HAVE SPARE TIME: YOUR MARK CANNOT GO OVER 100%

SCORES GIVEN HERE ARE ONLY ILLUSTRATIVE, AND EVEN A PERFECT ANSWER MAY ATTRACT EITHER

HIGHER OR LOWER POINTS AT MY DISCRETION. THUS I DO NOT ADVISE YOU TO SKIP PARTS ABOVE

IN ORDER TO ANSWER THESE PARTS.

j) Find the Nash equilibrium for maple syrup prices in the set-up of part (i). (5%)

k) Prove the results of the friendly mathematician from part (h). (20%)

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2. Question 2

The town of Boxton contains four roads arranged in a square, as shown in the picture below:

A newsagent’s shop is located at each of the four corners, and houses are evenly distributed

along each of the roads. Everyone in Boxton is addicted to cigarettes, so they each want to

buy a pack of cigarettes from one of the newsagents each day.

You may assume that each road is of length one, and contains a unit mass of consumers, as

in the standard Hotelling model. Consumers are lazy however, so it costs a consumer 𝑡𝑥 to

walk a distance of 𝑥 to buy their cigarettes, with 𝑡 > 0. (So the total cost of buying from a

newsagent that sells cigarettes at 𝑝 is 𝑝 + 𝑡𝑥 where 𝑥 is the distance from the consumer to

the newsagent.) Assume that consumers value their pack of cigarettes at 𝑣 > 2𝑡, (so 𝑣 is the

most they are prepared to pay in total, including their transport cost) and that all newsagents

can produce cigarettes at zero marginal cost.

Suppose for the moment that firms were choosing a price only once.

(a) Consider any of the four roads. The consumers along that road will want to buy from the

firm at one end or the other. Call the left hand firm 𝐴 and the right hand firm 𝐵. As a

function of the price charged by firm 𝐴 (𝑝𝐴) and the price charged by firm 𝐵 (𝑝𝐵), how far

from firm 𝐴 along the road is the indifferent consumer? (10%)

(b) Now suppose that all firms except one are charging a price 𝑝𝐵. Call the price charged by

the remaining firm 𝑝𝐴. Remembering that firm 𝐴 will be selling to consumers located on

two different roads, what price should firm 𝐴 charge to maximise its profits, as a function

of 𝑝𝐵? You may assume that 𝑣 > 𝑝𝐵 + 𝑡, so consumers always want to purchase a pack

of cigarettes. (20%)

(c) Since all firms are symmetric, in equilibrium all firms will charge the same price. By setting

𝑝𝐴 = 𝑝𝐵, solve for the equilibrium price. (10%)

(d) What is the total amount “spent” on transport costs in this set-up? (To work this out, look

at a single road first, and draw a graph of transport costs as a function of location along

this road (this should be a triangle). The area under this graph is total transport costs for

consumers who live on this road.) (20%)

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The Mayor of Boxton is concerned that there is insufficient competition amongst newsagents, so she decides to build a subway system linking each corner to every other corner, as shown in the dotted lines below:

(You may think of the point in the middle where the lines cross as a central station at which

consumers can change trains.)

The Mayor makes all subway trains free. Additionally, the value consumers get from talking

to their friends on the train exactly offsets their irritation at the length of the journey.

Therefore a consumer located in one of the four corners can buy from any firm without paying

any transport costs at all. (It is as if there were instant teleportation machines located at each

of the corners.)

You may continue to suppose that firms are choosing prices only once.

HINT: You should not need to do any maths for any of the remaining parts.

(e) Suppose a consumer is located at one of the four corners, assuming that the firms are all charging different prices, which firm would the consumer prefer to buy from and why? (4%)

(f) Now suppose that a consumer is located somewhere along one of the roads. Given that

once they reach one end, they can then get a subway to the opposite end at zero cost, to which corner will consumers walk? (They might not necessarily buy from this corner.) (2%)

(g) Given your answers to (e) and (f), describe the behaviour of a consumer located

somewhere along one of the roads, when all firms are charging different prices. (4%) (h) Hence, what prices will firms set? (2%) (i) Suppose that when it does not save them any money, consumers prefer not to use the

subway. (But if they can save even 1p by using it they gladly will.) Does anyone use the subway in equilibrium? Given your answer to (d), what are total transport costs now? (4%)

(j) Has total surplus improved? (And why/why not?) Assuming that building the subway cost

a positive amount, should the Mayor have bothered? (4%)

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Now suppose that rather than just choosing a price just once, firms are able to change their

price each day, based on what they observe other firms doing, for example.

(k) Describe the symmetric equilibrium in which total industry profits are highest. What profits does each firm make on each day in this equilibrium? How are firms able to make profits at all? (You may assume that if all newsagents were owned by the same company,

then that company would set a price of 𝑣 −𝑡

2.) (10%)

(l) Would this equilibrium have been any different in the absence of the subways? Would it

have been any easier to sustain? Does this change your answer to (j)? (10%)

BONUS QUESTIONS IF YOU HAVE SPARE TIME: YOUR MARK CANNOT GO OVER 100%

SCORES GIVEN HERE ARE ONLY ILLUSTRATIVE, AND EVEN A PERFECT ANSWER MAY ATTRACT EITHER

HIGHER OR LOWER POINTS AT MY DISCRETION. THUS I DO NOT ADVISE YOU TO SKIP PARTS ABOVE

IN ORDER TO ANSWER THESE PARTS.

(m) Prove that if all newsagents were owned by the same company, that company would set

a price of 𝑣 −𝑡

2? (25%)