[考題 105台大財金所 經濟分析 第7題]

Firm 1 and firm 2 are the only producers of spring water in the market. The market demand for spring water is given by P = 70 - Q1 - Q2. Firm 1 and firm 2 compete by choosing quantities Q1 and Q2 respectively. Each firm has a marginal cost of 10 and no fixed cost.

(1)Suppose only firm 1 has a chance to bribe the government and get the right to choose the quantity first, the maximum amount of money that firm 1 is willing to pay is 　　.

(2)Suppose the two firms decide to collude and share the profit equally, but both firms do not care about their reputation and might try to take advantage of the other. Foreseeing this, they make a legally enforceable contract saying that if a firm does not produce the quantity agreed, it has to pay some penalty to the other firm. The minimum amount of penalty that ensures each firm producing the right quantity is 　　.

[解答]

(1)兩廠商的邊際成本固定＝10，原為Cournot模型時，兩廠商的產量相同Q1 = Q2 = 20，市場價格P = 30，兩家廠商的利潤相同

若廠商1賄賂政府可以取得領導者地位，則應解出Stackelberg模型的領導者決策，由後向前解可得到Q1 = 30、Q2 = 15，市場價格P = 70 - 30 - 15 = 25，則成為領導者的利潤為

兩種情況的利潤差距即為持為領導者的最大願付賄賂金額

(2)若兩家廠商勾結，即獨占解，故MR = 70 - 2Q = 10 = MC，可解出Q = Q1 + Q2 =30，而Q1 = Q2 = 15，市場價格P = 70 - 30 = 40，兩家廠商的利潤相同

為了避免囚犯困境發生，若要設計罰款以懲罰背叛的廠商，因此要先計算因背叛而獲得的利潤差距是多少？

假設廠商1背叛而廠商2遵守15單位的產量約定，則將Q2 = 15代入廠商1的反應函數：

，市場價格，則背叛的廠商1可獲得利潤

故利潤差距即為最低的懲罰金額