Question

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Consider a small town that has a population of dedicated pizza eaters but is able to accommodate only two pizza shops, Donna’s Deep Dish and Pierce’s Pizza Pies. Each seller has to choose a price for its pizza, but for simplicity, assume that only two prices are available: high and low. If a high price is set, the sellers can achieve a profit margin of $12 per pie; the low price yields a profit margin of $10 per pie. Donna’s has a loyal captive customer base that will buy 11,000 pies per week and Pierce’s has a loyal captive customer base that will buy 3,000 pies per week, no matter what price is charged by either store. There is also a floating demand of 4,000 pies per week. The people who buy these pies are price conscious and will go to the store with the lower price; if both stores charge the same price, this demand will be split equally between them.

a) In equilibrium, Pierce’s earns $_______ in profit.

b) Donna’s loyal captive customer base has to be larger than ________ so that it sets its price at $12/pie if Pierce’s sets its price at $10/pie.

Answer 1

Answer to Qa: 70000
Answer to Qb: 10000

Feel free to use the comment section below for queries-
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If Donna charges high, profit from loyal base = 11000*12 = 132000
If Donna charges low, profit from loyal base = 11000*10 = 110000

If Pierce charges high, profit from loyal base = 3000*12 = 36000
If Pierce charges low, profit from loyal base = 3000*10 = 30000
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This is a Bertrand simultaneous price-setting game;
there are four cases:

i) (HIGH, HIGH) - both Donna and Pierce set price HIGH
ii) (HIGH, LOW) - Donna sets price High, Pierce Low
iii) (LOW, HIGH) - Donna sets price Low, Pierce High
iv) (LOW, LOW) - both set price LOW

Profits in each case:

i)
Donna's (HIGH, HIGH) = 132000 + 2000*12 = 156000
Pierce's (HIGH, HIGH) = 36000 + 2000*12 = 60000

ii)
Donna's (LOW, HIGH)= 110000 (from loyal base) + 4000*10 = 150000
Pierce's (LOW, HIGH) = 36000

iii)
Donna's (HIGH, LOW) = 132000
Pierce's (HIGH, LOW) = 30000 (from loyal base) + 4000*10 = 70000

iv)
Donna's (LOW, LOW) 110000 (from loyal base) + 2000*10 = 130000
Pierce's (LOW, LOW): 30000 (from loyal base) + 2000*10 = 30000 + 20000 = 50000

Arrange this values into a payoff matrix, unit is 1000 (i.e. payoff 70 means profit 70000)
to save us from writing all those 000s:



At equilibrium
Donna will set price HIGH
will Pierece will set price LOW

i.e. equilibrium is (HIGH, LOW)
because Donna is better off charging her existing customers HIGH (and foregoing the floating demand), rather than setting price LOW trying to price-match Pierce

and Pierce's profit is 70000
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Qb) Let Donna's captive base be N.

With Pierce setting price LOW, there are two cases:

Donna's (HIGH, LOW) = 12N (profit of $12 per pie from loyal base only)
Donna's (LOW, LOW) = 10N (profit of $10 per pie from loyal base) + 2000*10 (half of floating demand captured by price matching Pierce)
= 10N +20000

Donna will set price high only if
Donna's (HIGH, LOW) > Donna's (LOW, LOW)
i.e. 12N > 10N + 20000 => 2N > 20000
=> loyal base N > 10000
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