Question $4$: Spatial Competition: The Circular City Model

$$ Four identical firms $($A$,$ B$,$ C$,$ D$)$ are symmetrically located around a circular street.

$$ The street contains $160$ uniformly distributed households.

$$ Each firm is $40$ houses away from the next firm.

$$ All four firms produce pizza at a marginal cost of $$10.$

$$ Each household demands one pizza per week.

$$ Customers incur transportation costs of $$1$ for every house passed to buy pizza.

$2$

Determining Equilibrium Price: Indifference Condition

A customer located between Firm A and B will be indifferent between buying from either firm if

the price of pizza at Firm A plus the transportation cost to A equals the price of pizza at Firm B

plus the transportation cost to B$.$

Firm AFirm B

Firm C Firm D

Customer

The indifference condition is given by:

PA $+$ x $=$ PB $+(40$ x$)$

where x is the distance from the customer to Firm A$.$ Please compute equilibrium price and quantity

for Firm A$.($Hint: It will be same for B$,$ C$,$ and D$.$ At the equilibrium: PA $=$ PB $=$ PC $=$ PD and

QA $=$ QB $=$ QC $=$ QD $).$