國際貿易(比較利益) - 經濟

U=F^(2/3) C^(1/3)
則dU= U/3 (2/F dF + 1/C dC)

若在(F1,C1)附近以ΔC1換ΔF1,則 2/F1 ΔF1 + 1/C1 (-ΔC1)需大於0, ΔU1才會>0
ΔF1/ΔC1 > F1/(2C1)
相對方(F2,C2)以ΔF2換ΔC2, 則 2/F2(-ΔF2) + 1/C2 (ΔC2)需大於0
ΔF2/ΔC2 < F2/(2C2)
如果透過同一集中市場交易, ΔF1/ΔC1 = ΔF2/ΔC2 則有
F1/(2C1) < ΔF1/ΔC1 = ΔF2/ΔC2 < F2/(2C2)
若F2/(2C2) > F1/(2C1) 則交易會反向進行, Home以ΔF1換ΔC1

Home內情形:
交易前 F:(6,0) C:(0,3), 若F有h人,C有k人,F總體以食物hkU向C總體換取hkV衣物
交易後 F(6-kU,kV), C(hU,3-hV), 交易比ΔF1/ΔC1 = ΔF2/ΔC2 =(hkU)/(hkV)=U/V
價格均衡下,(6-kU)/(2kV)=(hU)/(2(3-hV))=U/V
(6-kU)/(2kV)=U/V, kU=2, U=2/k
(hU)/(2(3-hV))=U/V,hV=2, V=2/h
代回交易後F(4,2k/h),C(2h/k,1),F效用(32k/h)^(1/3),C效用(2h/k)^(2/3)
若F,C效用不同時,效用差的長期會轉業到效用高的那邊,直到相等
(32k/h)^(1/3)=(2h/k)^(2/3), (h/k)^3=8, h/k=2
再代回交易後F(4,1),C(4,1),發現FC兩種職業交易完成後消費相同都是(4,1)
交換比例: U:V=2/k:2/h=h/k:1=2:1, 2份食物換一份衣物, 衣物價格為食物2倍

Home和Foreign貿易情形:

Home: 每單位2人F,1人C,平均每人可分到F1=6x2/3=4,C1=3x1/3=1
U1=4^(2/3)=2.5198421, F1/(2C1) = 2
Foreign: 每單位2人F,1人C,平均每人可分到F2=1x2/3=2/3,C2=2x1/3=2/3
U2=2/3, F2/(2C2) =1/2
因 F1/(2C1)>F2/(2C2) (比較利益), Home會用食物F換取衣物C,

Foreign的人口是Home的3倍,
令交易量為Home總體以食物900U交換Foreign衣物900V,
Home 300人,每人 (4-3U,1+3V)
Foreign 900人, 每人 (2/3+U,2/3-V)
為使該2處價格均衡, 經2點之效用線切線斜率 F/(2C)要相等
同時價格需符合總交易量比例ΔF1/ΔC1=ΔF2/ΔC2= 900U/900V=U/V
(4-3U)/(2(1+3V))=(2/3+U)/(2(2/3-V))=U/V
(4-3U)/(2(1+3V))=U/V, 4V-3UV= 2U+6UV, 4/U-2/V=9
(2/3+U)/(2(2/3-V))=U/V, 2/3V+UV=4/3U-2UV, 2/U-4/V=-9
解聯立得 1/U=9/2, 1/V=9/2, U=V=2/9, 價格比U/V=1
Home 300人,每人消費 (10/3,5/3), 效用變5/3 x 2^(2/3)=2.6456 增加0.1258
Foreign 900人, 每人消費 (8/9,4/9), 效用變4/9 x 2^(2/3)=0.7055 增加0.0388


※ 引述《betray911015 (回頭太難)》之銘言：
: a country(Home)is populated with workers who produce either food(F)
: or clothing(C). There are 200 workers producing food and 100 producing
: clothing—these numbers are fixed in the short run. Each food worker
: produces 6 units of food and each clothing worker produces 3 of clothing.
: Workers own the output they produce and can trade with another. All
: workers share the same prefereneces over food and clothing represented
: by the utility function.
: U(Dc,DF)=(Dc^1/3)(DF^2/3)
: there also exists another country (Foreign) that looks surprising like the
: Home country except that there are 600 food workers and 300 clothing workers
: in this country. These workers have different(lower) productivity level: a
: food worker can produce only 1 unit of food while a clothing worker can
: only produce 2 units of clothing. The workers in Foreign share the same
: preferences as those in Home.
: now assume that all workers belong to represented"familites"each containing
: 2 food workers and 1 clothing workers.
: Question:
: 1.describe how the consumption pattern(the number of units food and
: and clothing consumed) of the "familites" change between autarky and the
: trade equilibrium. Are the "familites" in both countries unambiguously
: better-off?
: 我想請問，上面題目是指，現在以"familites"為一個單位嗎，而這一個單位，可以
: 生產 2 food and 1 clothing嗎?那這樣的話。那對於Home的clothing的相對價格就為
: 6*2/3*1=4。這樣想對嗎??

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