一題Stackelberg均衡(請幫我看答案) - 經濟

※ 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
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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
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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


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