[考題 106年台大財金所甲組]

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 comp 的 by choosing quantities Q1 and Q2 respectively. Each firm has a marginal cost of 10 and no fixed cost. Suppose the two firms decide to collude and share the profit equally. Assume that both firms value their reputation and will behave according to their agreement. What are the quantities they will choose for each firm______? What is the profit of each firm______? Suppose the two firms decide to collude and share 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. What is the minimum amount of penalty that ensures each firm producing the right quantity for colluding and sharing the profit equally?______.

[解答]此題前兩格為標準的Cournot均衡，最後一格比較特別是要透過給予對方「保證金」或「罰款」來確保勾結合作解的穩定。

(1) 對第 1 家廠商而言，其決策為： ，可解出其反應函數為 ，同理，第 2 家廠商的決策與第 1 家廠商對稱，故其反應函數為 ，將兩廠的反應函數聯立求解，可得Q1* = Q2* = 20

(2) 當 Q1* = Q2* = 20 ，則市場均衡價格P* = 70 - 20 - 20 = 30，則兩家廠商的利潤為

(3) 透過支付給對方「保證金」，希望達到勾結解。

STEP 1 ：先解出勾結解 ，可解出Q1* = Q2* = 15。

STEP 2 ：若假設保證金為 T ，則第 1 廠的決策可寫為 ，其反應函數為 ，同理， ，將兩廠的反應函數聯立求解，但希望勾結解發生，故將Q1* = Q2* = 15，可求出 T=15 ，表示將單位保證金設定為 15 時，可使 Cournot 競爭解發生勾結解的結果。