Question

1. Consider a market with inverse demand P(Q) = 100 Q and two firms with cost function C(q) = 20q.

(A) Find the Stackelberg equilibrium outputs, price and total profits (with firm 1 as the leader).

(B) Compare total profits, consumer surplus and social welfare under Stackelberg and Cournot (just say which is bigger).

(C) Are the comparisons intuitively expected?

2. Consider the infinite repetition of the n-firm Bertrand game. Find the set of discount factors for which full collusion (i.e. monopoly pricing) can be sustained by reversion to the one-shot Bertrand equilibrium as a threat.

Answer 1

1)

P = 100 - Q

TC = 20Q

MC = 20

Stackelberg Model: Output of Leader:

= (a-c)/2b

=(100 -20)/2

=40

Output of follower:

= (a-c)/4b

=(100 -20) /4

= 80/4

= 20

Total output = 20+40

= 60

Price = 100 -60

= 40

Total profit = 60*40 - 20*60

= 2400 - 1200

= 1200

b)

Total profit in Cournot is larger than Stackelberg model.

Consumer surplus is larger in Stackelberg model, output in Stackerlberg model is larger than the cournot, thus CS become larger.

Larger output leads to more social welfare, thus stackelberg model increases social welfare.

c)

In stackelberg model, one firm recognizes interdependence, thus output is larger than cournot where no one recognises the interdependence.