Re: [請益] 一題Stackelberg均衡(請幫我看答案)

看板Economics (經濟學)作者eagleflyfree (結束與開始)時間18年前 (2007/08/21 06:17)推噓0(0推 0噓 0→)

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※ 引述《luckysnow (幸運雪+去冰珍綠)》之銘言： : 我有自己想答案，先附上題目～ : 請板友幫忙 : 感謝 : 題目大意為： : 一島居民買魚，設關稅，利潤為CS＋關稅；<--沒說從量或從價 : 一獨佔者供魚，訂價格， : 捕魚沒成本， : 則：1.島主先訂關稅，獨佔者依關稅設價格，求均衡 : 2.順序顛倒過來的Stackelberg均衡 : 完整題目如下： : The residents of a certain island enjoy consuming a type of fish. : Unfortunately for these people this fish must be bought from a foreign a : foreign firm which has a monopoly over the fish. The leaders on the island : are considering placing a tariff on imports of the fish. Demand on the island : has been determined to be Q=D(p,t)=12-p-t, where p is the price set by the : foreign monopolist and t is the tariff. Model the situation as a same in : which the island leader move first and select a tariff and then given the : choice of the tariff the foreign monopoly selects its price. Assume the costs : of catching and shipping the fish are zero so that the firm sets price to : maximize total revenue. Assume that the island leaders select the tariff to : maximize total island surplus equal to consumer surplus plus tariff revenue. : What is the Stackelberg equilibrium outcome of this game? What is the : Stackelberg equilibrium if the order of moves is reversed? : ----------------------------------------------------- : 我的寫法 : Max profit i = CS＋tariff = 0.5Q^2 + tQ .....island leader （*） : Max profit f = (12-Q)Q-C(Q)-tQ .......foreign follower （**） : （**）一階微分 12-Q-Q=t Q=(12-t)/2 代入 Max profit i （*） : Max profit i = -3t^2/8 + 3t + 18 （***） : （***）一階微分 => t = 4 = Q = P : Max profit i = 0.5*4^2 + 4*4 = 24 : Max profit f = (12-4)*4 - 4*4 = 16 : ～～～～～～～～～～～～～～～ : 但是反過來， : 賣魚廠商先定價， : （*）一階微分 Q + t = 0 這樣好像有點奇怪@@ : 請幫我看一下，可能利潤函數就沒設對了@@ : 我已經問題目到沒人願意回答我了orz : 感謝這題算是變形題因為兩邊所能決定的是價格(稅)而非數量方程式要化成t跟p的函數 ------------------------------------------------ 當島主先訂價時 Max profit i = CS+tariff revenue =0.5Q^2+tQ =0.5(12-p-t)^2+t(12-p-t) =72-12p+0.5p^2-0.5t^2 Max profit f =p(12-p-t) =12p-p^2-pt 對p做一階偏微分=0 得廠商reaction function: p=0.5(12-t) 代入i得 72-72+6t+18-3t+0.125t^2-0.5t^2 =18+3t-0.375t^2 對t做一階微分=0 得 3-0.75t=0 t=4 代入f的reaction function得p=4 p,t代入D(p,t)得Q=4 Stackelberg equilibrium為 p=4,t=4 此時Q=4，profit i=24,CS=8,tariff revenue=16 ----------------------------------------------------------------- 當獨占廠商先訂價時 i對t做一階偏微分=0 得島主reaction function:-t=0 代入f得 12p-p^2 對p做一階微分=0 得 12-2p=0 p=6 代入i的reaction function得t=0 p,t代入D(p,t)得Q=6 Stackelberg equilibrium為 p=6,t=0 此時Q=6，profit i=36,CS=18,tariff revenue=0 -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 140.112.247.163