外文翻譯----從支出數(shù)據(jù)中構(gòu)建效用函數(shù)

1、<p>　　附錄2：外文文獻譯文</p><p>　　從支出數(shù)據(jù)中構(gòu)建效用函數(shù)</p><p>　　作者：S.NAFRIAT</p><p>　　在考慮消費者行為，市場假設(shè)提供一定的價格和購買任何數(shù)量的N種貨物。則購買需要支出的錢 e=πξ+...+πξ=px這是一個確定矢量的標量x={ξ，...，ξ}，這是在說明購買的組成，數(shù)量，和 向量當時的價格 ，在

2、大括號{}中表示一個列向量，以及一個基本的位。關(guān)于消費者的古典假設(shè)任何購買東西的行為應(yīng)該是消費者購買根據(jù)實用性來的，根據(jù)其組x，來決定消費者買一系列的Φ（x），大意是購買與組成：R'價格P，因此，需要開支消費模型，e=px以滿足他們最大的效用條件Φ（x）=max {Φ（y）:py≤e}.這種情況下的等效于</p><p>　　Φ（x）=max{φ(y):uy≤1},這里u=p/e是價格向量除以按支出，即用

3、作為單位貨幣和支出是被稱為平衡向量，給出一個合適的價格和支出。根本的原始效用函數(shù)所需的屬性是，給定一個平衡點U，任何構(gòu)成X，這是由最大效用的條件ux=1,決定滿足的，因此有 ：uy≤1Φ(y)≤Φ(x)和 Φ(y)≥Φ(x)uy≥1 這樣的假設(shè)，不能代表的部分必然會引起消費者一些爭議。任何實際的消費是相當不知道CRF附件這樣的功能，甚至可以是任何有此類意圖的附件。“然后如果有適當?shù)拇嬖?，它的模型將會存在?這項工作是研究“消費者

4、分析”和指數(shù)的結(jié)構(gòu)”這個項目的其中一部分，“在計量經(jīng)濟學研究計劃，是在普林斯頓大學和萊斯大學和耶魯大學的考爾斯基金會與國家科學基金會的支持下進行。根據(jù)在股市結(jié)構(gòu)的這些人 投資的假設(shè)，觀測數(shù)據(jù)的分析。在最早的時候。Gossen公司，杰文斯，門格爾和瓦爾拉斯使用的是假設(shè)的商品組成的效用公用事業(yè)的總和獨立的商品，Φ(x)=Φ(x)+...+Φ(x)埃奇沃思然后考慮一個通用函數(shù)Φ(x)=Φ(x,...,x) ，他也考慮了無差異的水平面的效用函數(shù)

5、。但現(xiàn)在所熟悉的方法，是由帕累托提出的脫</p><p>　　這里值得關(guān)注的是，只作為一個優(yōu)先的措施的效用函數(shù)為了達到更好是由商品之間的集合差決定的。但是在這么長的歷史過程都沒有人通過繪制出這樣的關(guān)系來說明這個假設(shè)可以被普遍接受。薩繆爾森[5]顯示性偏好的原則，Houthankker闡述[4]了對于存在的假說的容易排斥的條件。但原則已經(jīng)不在其中，假設(shè)在任何觀察的基礎(chǔ)上都要接受或拒絕消費者的選擇，應(yīng)該他在數(shù)量有限的

6、的情況下接受，一般的方法是一個實現(xiàn)實際需要實用功能，實現(xiàn)數(shù)據(jù)的假設(shè)。 這里將討論這個問題。一般問題當被刪除的時候有限性限制，一個可行的方法是由限制的過程中，在得出結(jié)果的基礎(chǔ)上，進行獲得。這是一個比塞繆爾[5]。霍撒克[4]，Uzawa [6]，Afriat [2]和其他涉及該領(lǐng)域研究更復雜的問題，因此，每一個價格的情況的數(shù)量;就是一個完整的數(shù)據(jù)系統(tǒng)。表單中的數(shù)據(jù)可以假定無限的，但不一定完整，甚至完整的數(shù)據(jù)，通常假設(shè)一個單一的價

7、值需求系統(tǒng)，也可以省略?；蛘?，Uzawa [6]提出如果單值的Lipschitz型條件所承擔的功能是假設(shè)，，因此，那些其他研究此領(lǐng)域的學者的理論可以被丟棄。在熟悉的調(diào)查，假設(shè)已如產(chǎn)量只有一個功能上獨立的效用</p><p>　　Φ(x)=max{φ(x)| ux≤1} (r=1,...,n)在這種情況下的功能，表現(xiàn)出的效用假說稱為E或為E.數(shù)據(jù)的效用函數(shù)，數(shù)據(jù)F可以被定義為實用的財產(chǎn)，如果效用假說展出它的一些功能

8、，換句話說，如果它就存在一個效用函數(shù)。 現(xiàn)在有個問題的決定的說法，對于任何給定的開支配置E，它是否具有實用一致性的財產(chǎn)，如果存在實用效的，可以構(gòu)建它的效用函數(shù)。如果實用程序的一致性為E認可，則它存在一些效用函數(shù)φ，然后ux≤1φ(x)≥ φ(x)和ux≤ 1 Λ φ(x)φ(x) ux= 1所有的r，s= 1，...，K.因此，對于所有r，s，...，q = 1，...，k</p><p>　　ux≤

9、 1 Λ ux≤ 1 Λ ...Λ x ≤ 1</p><p>　　φ(x) ≥ φ(x)≥...≥φ(x)≥φ(x)</p><p>　　φ(x) =φ(x)=...=φ(x) 因此</p><p>　　ux≤1 Λ ux ≤ 1 Λ ...Λ ux ≤ 1</p><p>　　ux=ux=...=ux =

10、 1這種情況定義為E的周期性的一致性的財產(chǎn)，這個被證明是一個實用的一致性明顯的必要條件，要證明他成立也是可以的。為了做到這一點，其他的一些一致性條件為E，并最終可以得到他們?nèi)慷伎梢缘刃У刈C明出來。定義D=ux-1，這可稱之為跨系數(shù)，從Er到Er，跨系數(shù)一起定義跨結(jié)構(gòu)D并決定支出配置E。</p><p>　　附錄3：外文文獻原文</p><p>　　INTERNATIONAL ECONO

11、MIC REVIEW</p><p>　　THE CONSTRUCTION OF UTILITY FUNCTIONS FROM EXPENDITURE DATA</p><p>　　BY S. N. AFRIAT</p><p>　　In considering the behavior of the consumer, a market is assumed wh

12、ich offers some n goods for purchase at certain prices and in whatever quantities.A purchase requires an expenditure of money e=πξ+...+πξ =px,which is determined as the scalar product of the vector x={ξ,...,ξ } of qu

13、antities, which shows the composition of the purchase, and the vector P={π,...,π}of prevailing prices, where braces {} denote a column vector, and a prime its transposition. The classical assumption about the</p&

14、gt;<p>　　according to its composition x, which is the measure of the utility, to the effect that a purchase with composition :r' made at price p and, therefore, requiring an expenditure, e=px, is such as

15、 to satisfy them maximum utility condition</p><p>　　Φ(x)=max{Φ(y):py≤e}.</p><p>　　An equivalent statement of this condition is Φ(x)=max{Φ(y):uy≤1}, Where u=p/e is the vector of prices divided

16、 by expenditure,that is with Expenditure taken as the unit of money and is to be called the balance vector,corresponding to those prices and that expenditure.The fundamental property required for a utility function Φ

17、(x) is that,given a balance u,any composition x, which is determined by the condition of maximum utility satisfies ux=1, so that</p><p>　　uy≤1Φ(y)≤Φ(x)</p><p><b>　　and</b></p>

18、;<p>　　Φ(y)≥Φ(x)uy≥1 </p><p>　　Such an assumption cannot represent necessary deliberations on the part of the consumer. Any actual consumer is quite unaware crf the attachment to s

19、uch a function Φ ，and can even deny by intention and manifest behavior any such attachment. 'then if Φ is to have a proper existence, it would have .</p><p>　　This work is part of a project on 'The

20、Analysis of Consumers' Preferences and</p><p>　　the Construction of Index Numbers," conducted at the Econometric Research Program, Princeton University and at Rice University and the Cowle

21、s Foundation at Yale University with the support of the National Science Foundation.to be in the stock analytical construction of those who entertain the assumption ,and based on data of observation. In the earliest f

22、orm. As the one used by Gossen, Jevons ,Menger and Walras,it was assumed that the utility of a composition of goods was the sum o</p><p>　　Φ(x)=Φ(x)+...+Φ(x)</p><p>　　Edgeworth then considered

23、a general function</p><p>　　Φ(x)=Φ(x,...,x)</p><p>　　and he also considered the indifference surfaces, the level surfaces Φ=constant of the utility function . But the now familiar appro

24、ach which is divorced from numerical utility and deals only with indifference surfaces was established by Pareto. Before this the utility analysis in demand theory dealt with utility and utility differences

25、as measurable quantities. By rendering numerical utility inessential, Pareto brought relief to the discomfort of having to assume a me</p><p>　　Here the concern is with the utility function only as a me

26、asure of preference for deciding for better or worse between a collection of goods. But never through the long drawn out history of the hypothesis has such a function been generally show. The revealed preference princip

27、le of Samuelson[5],elaborated by Houthankker [4], easily given a condition for the rejection of the hypothesis of existence. But the principle has been absent by which the hypothesis can ba accepted or rejected on

28、th</p><p>　　acceptance, a general method is needed for the actual construction of a utility function which will realize the hypothesis for the data.</p><p>　　This problem will be discusse

29、d here. For the general problem which arises when the finiteness restriction is removed, an possible approach is by a limiting process, proceeding on the basis of the results which are going to be obtained. It is

30、more general than the problem considered by samuel-son [5]. Houthakker [4], Uzawa [6], Afriat [2]and others which involves a demand system and, therefore, quantities for every price situation; that is,a complete syste

31、m of data. Form the data cou</p><p>　　function is assumed, the Lipschitz-type condition assumed by Uzawa [6] ,and , therefore, also the differentiability assumed by other writers can be dropped. in t

32、he familiar investigations , the assumptions have been such as to yield just one functionally independent utility function ,In the finite problem , and even in the infinite problem with completeness assumed, ther

33、e is no such essential uniqueness.</p><p>　　While the results for finite data do not immediately give results for complete data, </p><p>　　such as for a demand system, it is also the case that t

34、he familiar investigations on demand systems seeing conditions for the existence of a utility function have no scope for the finite problem now to be considered. Those investigations depend on a continuous, even

35、a differentiable structure , which can have no bearing here in the discrete finite case. They do not take into consideration the problem of establishing criteria by which any finite expenditure data can be taken as arisi

36、ng fr</p><p>　　Let it be supposed that the consumer has been observed on some k occasions of purchase, and the expenditure data obtained for each occasion r(r=1,…，k) provide the pair of vectors(xr，pr) which

37、 gave the composition of purchase and the prevailing prices. Hence the expenditure is e=pxand the balance vector is u=p/e ; and, by definition, ux=1 Let E=(x|u)define the expenditure , finger for occas

38、ion r, and E={E|r=1,...,n} the expenditure configuration, constructed from the data. </p><p>　　The utility hypothesis applied to the configuration E asserts that there exists a utility function φ such tha

39、t Φ(x)=max{φ(x)| ux≤1} (r=1,...,n)</p><p>　　In which case the function，can be said to exhibit the utility hypothesis for E or to be a utility function for E. The data .F can be said to have

40、the property of utility consistency if the utility hypothesis can be exhibited for it by some function, in other words if it has a utility function.</p><p>　　Now there is the problem of deciding, for any giv

41、en expenditure configuration </p><p>　　E , whether or not it has the property of utility consistency , and, if it has, of constructing a utility function for it.</p><p>　　If utility consistency

42、 holds for E, some utility function φ exists for it, and then</p><p>　　ux≤1φ(x)≥ φ(x) And ux≤ 1 Λ φ(x)φ(x) ux= 1</p><p>　　for all r, s = 1,…，k. Hence, for all r, s,…，q=1,…，k</p&

43、gt;<p>　　ux≤ 1 Λ ux≤ 1 Λ ...Λ x ≤ 1</p><p>　　φ(x) ≥ φ(x)≥...≥φ(x)≥φ(x)</p><p>　　φ(x) =φ(x)=...=φ(x)</p><p>　　Hence ux≤1 Λ ux ≤ 1 Λ ...Λ ux ≤ 1</p><p>

44、;　　ux=ux=...=ux = 1</p><p>　　This condition will define the property of cyclical consistency for E ,It has been shown to be an obviously necessary condition for utility consistency, and. it is going to be pr

45、oved also sufficient. In order to do this, some other consistency conditions will be introduced for E , and finally they will all be proved equivalent . Define D=ux-1 ,which may be called the cross-coefficient ,