U=F^(2/3) C^(1/3) 则dU= U/3 (2/F dF + 1/C dC) 若在(F1,C1)附近以ΔC1换ΔF1,则 2/F1 ΔF1 + 1/C1 (-ΔC1)需大於0, ΔU1才会>0 ΔF1/ΔC1 > F1/(2C1) 相对方(F2,C2)以ΔF2换ΔC2, 则 2/F2(-ΔF2) + 1/C2 (ΔC2)需大於0 ΔF2/ΔC2 < F2/(2C2) 如果透过同一集中市场交易, ΔF1/ΔC1 = ΔF2/ΔC2 则有 F1/(2C1) < ΔF1/ΔC1 = ΔF2/ΔC2 < F2/(2C2) 若F2/(2C2) > F1/(2C1) 则交易会反向进行, Home以ΔF1换ΔC1 Home内情形: 交易前 F:(6,0) C:(0,3), 若F有h人,C有k人,F总体以食物hkU向C总体换取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 价格均衡下,(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效用不同时,效用差的长期会转业到效用高的那边,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),发现FC两种职业交易完成後消费相同都是(4,1) 交换比例: U:V=2/k:2/h=h/k:1=2:1, 2份食物换一份衣物, 衣物价格为食物2倍 Home和Foreign贸易情形: Home: 每单位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: 每单位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) (比较利益), Home会用食物F换取衣物C, Foreign的人口是Home的3倍, 令交易量为Home总体以食物900U交换Foreign衣物900V, Home 300人,每人 (4-3U,1+3V) Foreign 900人, 每人 (2/3+U,2/3-V) 为使该2处价格均衡, 经2点之效用线切线斜率 F/(2C)要相等 同时价格需符合总交易量比例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 解联立得 1/U=9/2, 1/V=9/2, U=V=2/9, 价格比U/V=1 Home 300人,每人消费 (10/3,5/3), 效用变5/3 x 2^(2/3)=2.6456 增加0.1258 Foreign 900人, 每人消费 (8/9,4/9), 效用变4/9 x 2^(2/3)=0.7055 增加0.0388 ※ 引述《betray911015 (回头太难)》之铭言： : 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? : 我想请问，上面题目是指，现在以"familites"为一个单位吗，而这一个单位，可以 : 生产 2 food and 1 clothing吗?那这样的话。那对於Home的clothing的相对价格就为 : 6*2/3*1=4。这样想对吗?? --

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