## (Solved) In Tuftsville, everyone lives along Main Street which is 10 miles long. There are 1000 people uniformly spread up and down Main Street, and each day...

Four Competition/Oligopoly questions (Optimization/Equilibrium).

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1. In Tuftsville, everyone lives along Main Street which is 10 miles long. There are 1000 people uniformly spread
up and down Main Street, and each day they each buy one fruit smoothie from one of the two stores located at
either end of Main Street. Customers ride their motor scooters to and from the store and the motor scooters
use \$0.50 worth of gas per mile. Customers buy their smoothies from the store offering the lowest price, which
is the storeâ€™s price plus the customerâ€™s travel costs getting to and from the store. Ben owns the store at the
west end of Main Street and sells smoothies at a price of 19;, per smoothie. Will owns the store at the east end
of Main Street and sells smoothies at a price of pm per smoothie. The marginal cost of a smoothie is \$1. In
addition, each owner pays the city \$250 per day for the right to sell smoothies. Assume that Ben and Will
choose their prices simultaneously. a) Write down the equation that determines the location of the consumer who is indifferent between buying
from Ben or Will. Hint: Make sure you get the transportation cost correct. b) Formulate Benâ€™s optimization problem and derive his best reply to pw.
c) Formulate Willâ€™s optimization problem and derive his best reply to 191,. (1) Find the equilibrium prices, quantities and proï¬ts. 2. George is attracted by the proï¬ts that Ben and Will are earning and decides to open a store at the midpoint
of Main Street. His costs are the same as Ben and Will. Denote the price he charges as pg. a) Write down equations that determine the location of the consumer who is indifferent between buying from
Ben and George and the location of the consumer who is indifferent between buying from George and Will. b) If Will and Ben do not change their prices (your answer to (d) in problem 1), what is Georgeâ€™s best reply?
How much proï¬t would he earn? c) Derive the best replies for Will and Ben and ï¬nd the equilibrium prices and quantities.

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