I. Three basic models - firms recognize their interdependence but do not collude together - they act in their own best interest.
A. Cournot
B. Bertrand
C. Stackelberg
II. Cournot - first and most widely used (1838)
A. Assumptions
1. No entry
2. Homogenous good
3. Single period
4. Firms set output - output is the strategic variable
5. Each firm assumes that the other firms' output will not change if it changes its output.
B. Example:
Two firms
Q(p)=1000-1000P or P=1-.001Q
MC=AC=$.28
How should firm 1 choose its output level?
By maximizing profit on the residual demand curve q1(p)=Q(p)-q2
Profit1 = Pq1-.28q1
Profit1=(1-.001Q)q1-.28q1 and Q=q1+q2
Profit1=(1-.001(q1+q2))q1-.28q1
d(profit1)/dq1 = 1-.002q1-.001q2-.28=0
.72-.001q2=.001q1
q1=360-q2/2
By symmetry, q2=360-q1/2
These are known as the reaction curves. It gives firm 1's output as a function of firm 2's output. These can be graphed:
The place where the curves intersect is known as the Cournot Equilibrium or Nash equilibrium . It's the point where neither firm has an incentive to adjust its output any further.
To find the intersection, plug one into the other,
q1=360-(360-q1/2)/2=360-180+q1/4=180+q1/4
3/4q1=180
q1=240=q2
Q=q1+q2=480 plugging into demand function
480=1000-1000P
1000P=520
P=.52
profit1=profit2=(.52-.28)240=$57.60 each or 115.20 total industry profit
consumer surplus=1/2hb=1/2(1-.52)(480-0)=115.2
C. How does Cournot compare to PC and Cartel?
PC: P=MC=.28; Q=1000-1000P=1000-280=720; q1=q2=360; profit1=profit2=0
so output is more and price less in PC than cournot; profits less
Consumer surplus = 1/2hb=1/2(1-.28)(720-0)=259.2
Cartel: MR=MC
TR=(1-.001Q)Q
MR=1-.002Q=MC=.28
Q=500(.72)=360
P=1-.001Q=.64
Profit=TR-TC=360*.64-360*.28=129.60
so output is less and price is more in Cartel than Cournot; profits more
consumer surplus = ½hb= 1/2(1-.64)(360-0)=64.8
D. Cournot w/3 or more identical firms
n identical firms
q1(p)=Q(p)-(n-1)q2
profit1=(1-.001Q)q1-.28q1
profit1=(1-.001(q1+(n-1)q2))q1-.28q1=q1-.001(q1)2-.001(n-1)q1q2-.28q1
dprofit1/dq1=1-.002q1-.001(n-1)q2-.28=0
.72-.001(n-1)q2=.002q1
know that q1=q2 in equilibrium
.72=.002q1+.001(n-1)q1
.72=.001(q1)(n-1+2)
.72(1000)/(n+1)=q1
q1=720/(n+1)
industry output = Q=720n/(n+1)
To find price:
Q=1000-1000P
720n/(n+1)=1000-1000P
1000P=1000-720n/(n+1)
P=1-720n/1000(n+1)=(1000(n+1)-720n)/1000(n+1)=(280n+1000)/1000(n+1)
Book is incorrect: proof plug in n=2 and solve
Conclusion: the more firms in a Cournot Oligopoly, the closer the solution is to the competitive one
III. Bertrand Model
A. Joseph Bertrand 1883 reviewed Cournot's book and rejected the notion that firms use output as the strategic variable. Bertrand uses prices as the strategic variable.
B. Assumptions
1. No entry
2. Homogenous good
3. Single period
4. Firms set prices - price is the strategic variable
5. Each firm assumes that the other firms' price will not change if it changes its price.
6. Each firm has "unlimited" capacity
C. Example:
Two firms
Q(p)=1000-1000P or P=1-.001Q
MC=AC=$.28
How should firm 1 choose its price?
Given that the other firm sets a price of X, firm 1 should set its price at just below X and capture all of the demand. Since firm 2 has the same incentive, equilibrium prices won't result until P=MC and the firms have no incentive to price any lower. Thus the Bertrand model results in the PC outcome.
D. Edgeworth Extension of the model
1. Changes assumption about capacity - capacity is limited - i.e. to half the total output at MC.
2. P=MC is no longer an equilibrium because if other firm sets price at MC the other can maximize profit on residual demand curve of those not served by the other firm.
3. In this case there is no equilibrium.
III. Stackelberg Model
A. One firm acts as a leader and chooses its output first. The other (n-1) firms act as followers, knowing the leader's output.
B. Mathematical derivation is too complex for class.
C. Solution results is price that is higher than competitive and lower than Cournot and an output that is higher than Cournot and lower than competitive.
Review Questions
1. Compare and contrast the Bertrand and Cournot models with attention to the assumptions and results.
2. What is a reaction curve and how do you derive one?
3. How do the Cournot and Bertrand models compare to the monopoly and perfect competition outcomes in terms of price, output, profit, and consumer surplus?