Review: Oligopoly Models

Cournot Competition:

Firms compete in quantity of output they put out.

Firms can either produce homogenous goods (beef) or heterogeneous goods (PCs)

Firms do not cooperate

Firms have market power

Firms choose quantity simultaneously: homogenous goods

firm 1: max(q1) { pi=q1(p(q1+q2)-c)}

firm 2: max(q2) { pi=q2(p(q1+q2)-c)}

Solve the FOC of the first equation with respect to q₁(q₂) then substitute it into equation 2.

Bertrand Competition:

Firms compete in price, not quantity.

Firms set prices simultaneously.

Consumers buy entirely from the firm with the lower price.

Result, both firms price at marginal cost (if they share the same cost structure).

If one firm has superior production technology and average cost is lower, that firm will charge just below what the other firm’s marginal cost and put the other one out of the market.

Stackelberg Competition
Leader follower model.
Compete in quantity.  Price is lower than Cournot price but higher than Bertrand price.
If compete in price then price at MC.  Because (P-MC)Q/2 is always less than (P-MC-epsilon)Q so long as P-MC>0 for some epsilon > 0.
Proof by contradiction: 
1) (P-MC)Q/2 >= (P-MC-epsilon)Q
2) (P-MC) >= 2(P-MC-epsilon)
3) 0 >= P-MC-epsilon 
4) epsilon >= P-MC
5) We already said that P-MC>0 if so then there must be a P-MC>epsilon>0 which is a contradiction of (4).  Thus 1 must not be true which means:
6) (P-MC)Q/2 < (P-MC-epsilon)Q