Question

Two firms producing a homogeneous good compete in a two-stage game. In stage 1, firm 1 can purchase cost-reducing capital equipment ?. In stage 2, firms compete by simultaneously choosing quantities. Market (inverse) demand is given by the equation ? = 50 − 2?. Firm 1’s total cost (including the cost of the capital equipment) is ?1(2-?/4) + ?^2/18, where ?1 is firm 1’s output. Firm 2’s cost is 2?2.

a. Find the subgame perfect equilibrium quantities. How much investment does firm 1 make?

b. Is this a “top dog” or “puppy dog” investment strategy? Explain

Answer 1

(a) Both firms compete by simultaneously choosing quantities. It is a duopoly as there are 2 firms. So we can apply the Cournot model.

Equilibrium output is Q/3 in the case of the Cournot model where Q is the total market demand.

So,

P=50-2*Q

When P=0 then maximum Q=25

So equilibrium quantity is 25/3 for both firms.

q1=q2=25/3

Total cost for firm1 including investment=q1(2-k/4)+k^2/18^2

Total cost for firm 2= 2q2

The investment made by firms 1= -q1*k/4+k^2/18^2

We know investment made is =k

then, we can say that

k=-q1*k/4+k^2/18^2

substituting q1=25/3

1=-25/12+k/324

k=999

(b) This strategy of competing on quantity simulateously is "puppy dog".