A monopolist has a cost function $$q = aq (For simplicity I will give you a linear cost function.) and faces a linear demand function, $$D = b - cp. What is the optimal output and price? Consider an output decision:

$$max µ (q) = R(q) - C(q) ______________(1)

which in this case is:

$$ max µ (p,q) = pq - aq

subject to$$ D = b - cp _________________(2)

This problem could be solved using a Lagrange multiplier, however it is easier to solve the problem by substitution. The usual approach is to use the constraint to eliminate $$p by realizing that in market equilibrium the amount supplied by the monopolist equals the demand, or:

$$q = b - cp __________________________(3)

which when solved for p becomes

$$p = b/c - q/c________________________(4)

substituting

_____$$ max µ(q) = (b/c - q/c)q - aq______(5)

Taking the first order conditions and solving

_____$$µ' = 0 = (b/c - a) - 2q/c or q = (c/2)(b/c - a)

_______$$= (1/2)(b - ac)________________(6)

To solve for $$p subsitute the solution for $$q into the constraint to obtain

_____$$p = b/c - 1/c (1/2)(b/c - a)________(7)

2. Duopoly theory

Two firms, A and B, face a linear demand curve $$D = 100 -10p. For simplicity each firm has a linear cost function: $$C(q_A) = 5q_A and $$C(q_B) = 5q_B. For the purpose of having the simplification of symmetry both cost functions will be the same. The profit functions for the two firms are:

_____$$ µ (q_A) = pq_A - 5q_A

_____$$ µ (q_B) = pq_B - 5q_B______________(8)

In the duopoly model like the monopoly model, price is determined by Supply = Demand.

$$ q_A + q_B = 100 - 10p which gives p = 10 - (q_A + q_B)/10_________(9)

Thus

_____$$µ(q_A) = (10 - (q_A + q_B)/10)q_A - 5q_A

_____$$µ(q_B) = (10 - (q_A + q_B)/10)q_B - 5q_B__________(10)

Solving the first order conditions we obtain:

_____$$ µ' = 0 = 5 - q_A/5 - q_B/10 - (q_A/10)(dq_B/dq_A)

_____$$ µ' = 0 = 5 - q_B/5 - q_A/10 - (q_B/10)(dq_A/dq_B)__(11)

Fundamental issue: How does firm A determine how firm B will respond to firm A's action. That is, how doees A determine dq_B/dq_A and B determine dq_A/dq_B.

Solutions:

a. Form a cartel (illegal in the US and many other countries) Suppose firm A and B collude. They decide to act as a monopolist and split production and profits. In this case:

_____$$ µ(q)= (10 - q/10)q - 5q_________(12)

Solving the first order conditions:

_____$$ µ' = 0 = 5 - q/5 ________________(13)

which gives $$q = 25 and p = 7.5

Consequently both firms A and B produce 12.5 units. The profits for the monopoly are:

_____$$ µ = 7.5(25) - 5(25) = 62.5 ________(14)
with each partner obtaining $$31.25

Both firms have an incentive to cheat. For simplicity, this incentive to cheat will be defined as monopoly price - MC (marginal cost) or $$ 7.5 - 5 = 2.5.

b. Cournot

Both firms assume that the other firm will not respond to its actions, thus:

Cournot assumption: $$ dq_B/dq_A = 0 = dq_A/dq_B

With the Cournot assumption the first order conditions become:

_____$$µ_A' = 0 = 5 - q_A/5 - q_B/10________(15)

_____$$µ_B' = 0 = 5 - q_A/10 - q_B/5________(16)

These two first order condtions can be written in the form of reaction functions:

_____$$ q_A = 25 - q_B/2

_____$$ q_B = 25 - q_A/2__________________(17)

Cournot solution: Either solve 2 eqn in 2 unk:

_____$$ 10 - 2q_A/5 - q_B/5 = 0____________(18)

_____$$ 5 - q_A/10 - q_B/5 = 0_____________(19)

Subtracting the second from the first:

_____$$5 - (2/5 - 1/10)q_A = 0 or q_A = 50/3 = 16.67__(20)

A much easier way to solve the two equations is to observe that by symmetry that $$q_A must equal $$q_B. Therefore plugging this fact into one of the reaction functions one obtains the
$$q_A = q_B = 50/3. The profits for firm A and firm B are computed as:

$$ µ_A = µ_B = (10 - (1/10)(50/3 + 50/3))(50/3) - 5(50/3) = 27.78____(21)

c. Stackelberg

Assume firm A in the leader and firm B is the follower. Firm A assumes that firm B will follow his reaction function and hence firm A can exploit this knowledge, that is replace $$q_B with firm B's reaction function, in determining its optimal decision.

_____$$µ_A(q_A)= (10 - (1/10)(q_A + (25 - q_A/2))q_A - 5q_A______(22)

which equals when collecting terms:

_____$$µ_A(q_A)= (25/10)q_A - q_A²/20__________________(23)

Solving the first order conditions:

_____ $$µ_A'= 0 = (25/10) - q_A/10

_____$$q_A = 25 and q_B = 25 - q_A/2 = 12.5_____________(24)

_____ $$p = 10 - (1/10)(37.5) = 6.25__________________(25)

Thus

_____$$µ_A = 6.25(25) - 5(25) = 1.25(25) = 31.25

_____$$µ_B = 6.25(12.5) - 5(12.5) = 15.625_____________(26)

Now suppose both adopt a leader strategy assuming the other will remain a follower:

_____$$q_A = q_B = 25 and p = (10 - 50/10) = 5___________(27)

Thus

_____$$µ_A = µ_B = 0________________________________(28)

This is the pure competition solution.

d. Summary

_____Cartel(Monopoly)

_____ $$p = 7.5, q_A = q_B = 12.5, & µ_a = µ_B = 31.75

_____Cournot

_____$$p = 6.33, q_A = q_B = 16.67, & µ_A = µ_B = 27.78

_____Stackelberg with A leader

_____$$p = 6.25, q_A = 25, µ_A = 31.75, q_B = 12.5, & µ_B = 15.625

_____Both attempt Stackelberg

_____$$p = 5, q_A = q_B = 25, & µ_A = µ_B = 0

Because collusion is per se illegal, there is no stable price determination. Industries with a small number of firms tend to oscilate between price wars and implicit collusion. (The airlines are a good example)

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