Suppose the demand for pizza in a small isolated town is p = 10 - Q. The only two firms, A and B, behave as Cournot duopolists. Each has a cost function TC = 2 + Q

If the government wants to subsidize firm A to raise its output to that of a Stackelberg leader, how large should the subsidy be?

Firm B's profit is ? = [10 - (qA + qB)]qB - 2 - qB. Maximizing with respect to its own output yields firm B's best response qB = 4.5 - qA/2. Knowing this, firm A substitutes this into demand and maximizes its profit. ? = [10 - qA - (4.5 - qA/2 )]qA - 2 - qA. Maximizing with respect to qA yields qA = 4.5. Firm B responds by producing 2.25. The subsidy (S) must make firm A choose 4.5 units while acting as a Cournot firm. Firm A's reaction function as a Cournot firm is qA = [10 - 1 + S - qB]/2. Setting qA = 4.5 and qB = 2.25 yields a subsidy of 2.25 per unit.