Consider a more general Bertrand model than the one presented in this

chapter. Suppose there are

n

firms that simultaneously and independently

select their prices,

p

1

,

p

2

,

c

,

p

n

in a market. These prices are greater

than or equal to zero. The lowest price offered in the market is defined

as

p

=

min{

p

1

,

p

2

,

c

,

p

n

}. Consumers observe these prices and purchase

only from the firm (or firms) charging

p

,

according to the demand curve

Q

=

a

p

. That is, the firm with the lowest price gets all of the sales. If

the lowest price is offered by more than one firm, then these firms equally

share the quantity demanded. Assume that firms must supply the quantities

demanded of them and that production takes place at a cost of

c

per unit.

That is, a firm producing

q

i

units pays a cost

cq

i

. Assume

a

>

c

>

0.