Question

Two retailers compete on price in a market. Firm 1’s demand depends on both its own price and firm 2’s
price as follows: ?1 = ? - ??1 + ?2. Similarly, firm 2’s demand depends on its own price and firm 1’s
price: ?2 = ? - ??2 + ?1. Their marginal costs of producing one unit of product are both c.
a. Find the expression of firm 1’s equilibrium price.
b. Find the expression of firm 1’s equilibrium profit.

Answer 1

The demand curve for firm 1 is

q1 = b - a.p1 + p2

Hence, total revenue for firm 1 is

R1 = p1.q1 = p1.(b - a.p1 + p2)

or, R1 = b.p1 - a.p1² + p1.p2........(1)

The demand curve for firm 2 is

q2 = b - a.p2 + p1

Hence, the total revenue for firm 2 is

R2 = p2.q2 = p2.(b - a.p2 + p1)

or, R2 = b.p2 - a.p2² + p1.p2........(2)

Marginal Cost for both firms is given as

MC1 = MC2 = c

And, total cost for firm 1 is

C1 = c.q1 = c.(b - a.p1 + p2)

Also, total cost for firm 2 is

C2 = c.q2 = c.(b - a.p2 + p1)

Now, profit of firm 1 is

π1 = R1 - C1

or, π1 = b.p1 - a.p1² + p1.p2 - c.(b - a.p1 + p2)

Now, profit is maximized for firm 1 when

dπ1/dp1 = 0

or, b - 2a.p1 + p2 - a.c = 0

or, 2a.p1 - p2 = b - ac..........(3)

Similarly, profit of firm 2 is

π2 = R2 - C2

or, π2 = b.p2 - a.p2² + p1.p2 - c.(b - a.p2 + p1)

Now, profit is maximized for firm 2 when

dπ2/dp2 = 0

or, b - 2a.p2 + p1 - a.c = 0

or, 2a.p2 - p1 = b - ac..........(4)

Now, solving equations (3) and (4) we get

p1* = p2* = (b - a.c)/(2.a - 1)

Firm 1's equilibrum price is

p1* = (b-ac)/(2a-1) [Answer of a].

Putting p1* and p2* in π1, we get

π1 = (p1* - c).(b - a.p1* + p2*)

or, π1 = [(b-ac)/(2a-1) - c].[b + (1-a).[(b-ac)/(2a-1)]

or, π1* = a.[(b + ac - c)/(2a - 1)]²

Firm 1's equilibrum profit is

π1* = a.[(b+ac-c)/(2a-1)]²

Hope the solution is clear to you my friend.