有关泰勒规则Taylor rule论文的文献综述(英文)

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Taylor's Formula and the Study of Extrema1. Taylor's Formula for MappingsTheorem 1. If a mapping Y U f →: from a neighborhood ()x U U = of a point x in a normed space X into a normed space Y has derivatives up to order n -1 inclusive in U and has an n-th order derivative ()()x f n at the point x, then()()()()()⎪⎭⎫ ⎝⎛++++=+n n n h o h x f n h x f x f h x f !1, (1)as 0→h . Equality (1) is one of the varieties of Taylor's formula, written here for rather general classes of mappings.Proof. We prove Taylor's formula by induction.For 1=n it is true by definition of ()x f ,.Assume formula (1) is true for some N n ∈-1.Then by the mean-value theorem, formula (12) of Sect. 10.5, and the induction hypothesis, we obtain.()()()()()()()()()()()()()⎪⎭⎫ ⎝⎛=⎪⎭⎫ ⎝⎛=⎪⎪⎭⎫ ⎝⎛-+++-+≤⎪⎭⎫ ⎝⎛+++-+--<<n n n n n n h o h h o hh x f n h x f x f h x f h x f n h f x f h x f 11,,,,10!11sup !1x θθθθθ ，as 0→h .We shall not take the time here to discuss other versions of Taylor's formula, which are sometimes quite useful. They were discussed earlier in detail for numerical functions. At this point we leave it to the reader to derive them (see, for example, Problem 1 below).2. Methods of Studying Interior ExtremaUsing Taylor's formula, we shall exhibit necessary conditions and also sufficient conditions for an interior local extremum of real-valued functions defined on an open subset of a normed space. As we shall see, these conditions are analogous to the differential conditions already known to us for an extremum of a real-valued function of a real variable.Theorem 2. Let R U f →: be a real-valued function defined on an open set U in anormed space X and having continuous derivatives up to order11≥-k inclusive in aneighborhood of a pointU x ∈ and a derivative ()()x f k of order k at the point x itself. If ()()()0,,01,==-x f x f k and ()()0≠x f k , then for x to be an extremum of the function f it is:necessary that k be even and that the form()()k k h x f be semidefinite,and sufficient that the values of the form ()()k k h x f on the unit sphere 1=h be bounded away from zero; moreover, x is a local minimum if the inequalities()()0>≥δk k h x f ,hold on that sphere, and a local maximum if()()0<≤δk k h x f ,Proof. For the proof we consider the Taylor expansion (1) of f in a neighborhood of x. The assumptions enable us to write()()()()()k k k h h h x f k x f h x f α+=-+!1where ()h α is a real-valued function, and ()0→h α as 0→h .We first prove the necessary conditions.Since ()()0≠x f k , there exists a vector 00≠h on which ()()00≠k k h x f . Then for values of the real parameter t sufficiently close to zero,()()()()()()k k k th th th x f k x f th x f 0000!1α+=-+ ()()()k k k k t h th h x f k ⎪⎭⎫ ⎝⎛+=000!1α and the expression in the outer parentheses has the same sign as()()k k h x f 0. For x to be an extremum it is necessary for the left-hand side (and hence also the right-hand side) of this last equality to be of constant sign when t changes sign. But this is possible only if k is even.This reasoning shows that if x is an extremum, then the sign of the difference ()()x f th x f -+0 is the same as that of()()k k h x f 0 for sufficiently small t; hence in that case there cannot be two vectors 0h , 1h at which the form ()()x f k assumes values with opposite signs.We now turn to the proof of the sufficiency conditions. For definiteness we consider thecase when ()()0>≥δk k h x f for 1=h . Then()()()()()k k k h h h x f k x f h x f α+=-+!1()()()k k k h h h h x f k ⎪⎪⎪⎭⎫ ⎝⎛+⎪⎪⎭⎫ ⎝⎛=α!1()k h h k ⎪⎭⎫ ⎝⎛+≥αδ!1 and, since ()0→h α as 0→h , the last term in this inequality is positive for all vectors 0≠h sufficiently close to zero. Thus, for all such vectors h,()()0>-+x f h x f ,that is, x is a strict local minimum.The sufficient condition for a strict local maximum is verified similiarly.Remark 1. If the space X is finite-dimensional, the unit sphere ()1;x S with center at X x ∈, being a closed bounded subset of X, is compact. Then the continuous function ()()()()k k i i i i k k h h x f h x f ⋅⋅∂= 11 (a k-form) has both a maximal and a minimal value on ()1;x S . Ifthese values are of opposite sign, then f does not have an extremum at x. If they are both of the same sign, then, as was shown in Theorem 2, there is an extremum. In the latter case, a sufficient condition for an extremum can obviously be stated as the equivalent requirement that the form ()()k k h x f be either positive- or negative-definite.It was this form of the condition that we encountered in studying realvalued functions on n R .Remark 2. As we have seen in the example of functionsR R f n →:, the semi-definiteness of the form ()()k k h x f exhibited in the necessary conditions for an extremum is not a sufficient criterion for an extremum.Remark 3. In practice, when studying extrema of differentiable functions one normally uses only the first or second differentials. If the uniqueness and type of extremum are obvious from the meaning of the problem being studied, one can restrict attention to the first differential when seeking an extremum, simply finding the point x where ()0,=x f3. Some ExamplesExample 1. Let ()()R R C L ;31∈ and ()[]()R b a C f ;,1∈. In other words,()()321321,,,,u u u L u u u is a continuously differentiable real-valued function defined in3R and ()x f x a smooth real-valued function defined on the closed interval []R b a ⊂,.Consider the function()[]()R R b a C F →;,:1 (2) defined by the relation()[]()()f F R b a C f ;,1∈ ()()()R dx x f x f x L b a∈=⎰,,, (3) Thus, (2) is a real-valued functional defined on the set of functions ()[]()R b a C ;,1.The basic variational principles connected with motion are known in physics and mechanics. According to these principles, the actual motions are distinguished among all the conceivable motions in that they proceed along trajectories along which certain functionals have an extremum. Questions connected with the extrema of functionals are central in optimal control theory. Thus, finding and studying the extrema of functionals is a problemof intrinsic importance, and the theory associated with it is the subject of a large area of analysis - the calculus of variations. We have already done a few things to make the transition from the analysis of the extrema of numerical functions to the problem of finding and studying extrema of functionals seem natural to the reader. However, we shall not go deeply into the special problems of variational calculus, but rather use the example of the functional(3) to illustrate only the general ideas of differentiation and study of local extrema considered above.We shall show that the functional (3) is a differentiate mapping and find its differential. We remark that the function (3) can be regarded as the composition of the mappings ()()()()()x f x f x L x f F ,1,,= (4)defined by the formula ()[]()[]()R b a C R b a C F ;,;,:11→(5) followed by the mapping[]()()()R dx x g g F R b a C g ba ∈=∈⎰2;, (6) By properties of the integral, the mapping2F is obviously linear and continuous, so that its differentiability is clear.We shall show that the mapping1F is also differentiable, and that()()()()()()()()()()x h x f x f x L x h x f x f x L x h f F ,,3,2,1.,,,∂+∂= (7) for ()[]()R b a C h ;,1∈.Indeed, by the corollary to the mean-value theorem, we can write in the present case()()()i i i u u u L u u u L u u u L ∆∂--∆+∆+∆+∑=32131321332211,,,,,,()()()()()()∆⋅∂-∆+∂∂-∆+∂∂-∆+∂≤<<u L u L u L u L u L u L 3312211110s u p θθθθ ()()ii i i u L u u L i ∆⋅∂-+∂≤=≤≤=3,2,110m a x m a x 33,2,1θθ (8)where ()321,,u u u u = and ()321,,∆∆∆=∆. If we now recall that the norm()1c f of the function f in ()[]()R b a C ;,1 is ⎭⎬⎫⎩⎨⎧c c f f ,,max (where c f is the maximum absolute value of the function on the closed interval []b a ,), then,setting x u =1,()x f u =2, ()x f u ,3=, 01=∆, ()x h =∆2, and ()x h ,3=∆, we obtain from inequality (8), taking account of the uniform continuity of the functions ()3,2,1,,,321=∂i u u u L i , on bounded subsets of 3R , that()()()()()()()()()()()()()()()()x h x f x f x L x h x f x f x L x f x f x L x h x f x h x f x L b x ,,3,2,,,0,,,,,,,,max ∂-∂--++≤≤ ()()1c h o = as ()01→c hBut this means that Eq. (7) holds.By the chain rule for differentiating a composite function, we now conclude that the functional (3) is indeed differentiable, and()()()()()()()()()()⎰∂+∂=ba dx x h x f x f x L x h x f x f x L h f F ,,3,2,,,,, (9) We often consider the restriction of the functional (3) to the affine space consisting of thefunctions ()[]()R b a C f ;,1∈ that assume fixed values ()A a f =, ()B b f = at the endpoints of the closed interval []b a ,. In this case, the functions h in the tangent space ()1f TC , must have the value zero at the endpoints of the closed interval []b a ,. Taking this fact into account, we may integrate by parts in (9) and bring it into the form()()()()()()()()⎰⎪⎭⎫ ⎝⎛∂-∂=b a dx x h x f x f x L dx d x f x f x L h f F ,3,2,,,,, (10)of course under the assumption that L and f belong to the corresponding class()2C . In particular, if f is an extremum (extremal) of such a functional, then by Theorem 2 we have ()0,=h f F for every function ()[]()R b a C h ;,1∈ such that ()()0==b h a h . From this and relation (10) one can easily conclude (see Problem 3 below) that the function f must satisfy the equation ()()()()()()0,,,,,3,2=∂-∂x f x f x L dx d x f x f x L (11)This is a frequently-encountered form of the equation known in the calculus of variations as the Euler-Lagrange equation.Let us now consider some specific examples.Example 2. The shortest-path problemAmong all the curves in a plane joining two fixed points, find the curve that has minimal length.The answer in this case is obvious, and it rather serves as a check on the formal computations we will be doing later.We shall assume that a fixed Cartesian coordinate system has been chosen in the plane, in which the two points are, for example, ()0,0 and ()0,1 . We confine ourselves to just the curves that are the graphs of functions ()[]()R C f ;1,01∈ assuming the value zero at both ends of the closed interval []1,0 . The length of such a curve()()()⎰+=102,1dx x f f F (12) depends on the function f and is a functional of the type considered in Example 1. In this case the function L has the form ()()233211,,u u u u L +=and therefore the necessary condition (11) for an extremal here reduces to the equation()()()012,,=⎪⎪⎪⎭⎫ ⎝⎛+x f x f dx d from which it follows that()()()常数≡+x f x f 2,,1 (13)on the closed interval []1,0Since the function 21u u+ is not constant on any interval, Eq. (13) is possible only if()≡x f ,const on []b a ,. Thus a smooth extremal of this problem must be a linear function whose graph passes through the points ()0,0 and ()0,1. It follows that ()0≡x f , and we arrive at the closed interval of the line joining the two given points.Example 3. The brachistochrone problemThe classical brachistochrone problem, posed by Johann Bernoulli I in 1696, was to find the shape of a track along which a point mass would pass from a prescribed point 0P to another fixed point 1P at a lower level under the action of gravity in the shortest time.We neglect friction, of course. In addition, we shall assume that the trivial case in which both points lie on the same vertical line is excluded.In the vertical plane passing through the points0P and 1P we introduce a rectangular coordinate system such that0P is at the origin, the x-axis is directed vertically downward, and the point 1P has positive coordinates ()11,y x .We shall find the shape of the track among the graphs of smooth functions defined on the closed interval []1,0x and satisfying the condition ()00=f ,()11y x f =. At the moment we shall not take time to discuss this by no means uncontroversial assumption (see Problem 4 below).If the particle began its descent from the point 0P with zero velocity, the law of variation of its velocity in these coordinates can be written asgx v 2= (14)Recalling that the differential of the arc length is computed by the formula()()()()dx x f dy dx ds 2,221+=+= (15) we find the time of descent()()()⎰+=102,121x dx x x f g f F (16) along the trajectory defined by the graph of the function()x f y = on the closed interval []1,0x . For the functional (16) ()()1233211,,u u u u u L +=,and therefore the condition (11) for an extremum reduces in this case to the equation ()()()012,,=⎪⎪⎪⎪⎪⎭⎫⎝⎛⎪⎭⎫ ⎝⎛+x f x x f dx d , from which it follows that()()()x c x f x f =+2,,1 (17)where c is a nonzero constant, since the points are not both on the same vertical line. Taking account of (15), we can rewrite (17) in the formx c ds dy = (18)However, from the geometric point of viewϕc o s =ds dx ,ϕsin =ds dy (19) where ϕ is the angle between the tangent to the trajectory and the positive x-axis. By comparing Eq. (18) with the second equation in (19), we findϕ22s i n 1c x = (20)But it follows from (19) and (20) that dx dy d dy =ϕ,2222sin 2sin c c d d tg d dx tg d dx ϕϕϕϕϕϕϕ=⎪⎪⎭⎫ ⎝⎛==, from which we find()b c y +-=ϕϕ2s i n 2212 (21) Setting a c =221and t =ϕ2, we write relations (20) and (21) as()()b t t a y t a x +-=-=s i n c o s 1 (22)Since 0≠a , it follows that 0=x only for πk t 2=,Z k ∈. It follows from the form of thefunction (22) that we may assume without loss of generality that the parameter value0=t corresponds to the point()0,00=P . In this case Eq. (21) implies 0=b , and we arrive at the simpler form()()t t a y t a x s i n c o s 1-=-= (23) for the parametric definition of this curve.Thus the brachistochrone is a cycloid having a cusp at the initial point 0P where the tangent is vertical. The constant a, which is a scaling coefficient, must be chosen so that the curve (23) also passes through the point 1P . Such a choice, as one can see by sketching the curve (23), is by no means always unique, and this shows that the necessary condition (11) for an extremum is in general not sufficient. However, from physical considerations it is clear which of the possible values of the parameter a should be preferred (and this, of course, can be confirmed by direct computation).。

泰勒规则对国际经济政策的影响

泰勒规则对国际经济政策的影响近年来，泰勒规则（Taylor rule）在国际经济政策中扮演着重要的角色。

泰勒规则是由美国经济学家约翰·泰勒（John Taylor）提出的一种计算实际利率水平的经验公式。

它的出现对于制定货币政策和经济政策提供了一种指导性的参考，下面将探讨泰勒规则对国际经济政策的影响。

首先，泰勒规则为货币政策制定者提供了一个明确的参考框架。

根据泰勒规则，中央银行在制定利率政策时，应该考虑通胀率和产出缺口两个因素。

通胀率高或产出缺口大时，中央银行应该采取扩张性的货币政策，通过降低利率促进经济增长；相反，当通胀率低或产出缺口小时，中央银行应该采取收紧性的货币政策，通过提高利率抑制通胀。

泰勒规则的提出使得货币政策的制定不再盲目，而是基于经济数据和指标，提高了政策的科学性和可预测性。

其次，泰勒规则对于国际经济政策的协调起到了重要的作用。

由于全球化的发展，国际经济政策的协调变得尤为重要。

泰勒规则提供了一个相对统一的参考框架，使得各国在制定货币政策时可以考虑到经济情况的差异。

通过应用泰勒规则，各国可以在一定程度上协调利率政策，避免利率差异过大导致国际资本流动问题。

这有助于维护全球金融稳定，减少金融风险对经济的负面影响。

此外，泰勒规则还对国际经济政策的可持续性产生了积极的影响。

可持续性是一个国家经济政策成功的重要标志。

泰勒规则提供了一个平衡经济增长和通胀率的框架，使得经济政策能够在长期内保持稳定。

如果一个国家长期违背泰勒规则所推荐的利率水平，将可能导致经济内外部不平衡，甚至引发金融危机。

因此，在制定经济政策时，政策制定者可以根据泰勒规则来评估利率调整的合理性和长期影响，确保经济政策的可持续性。

然而，泰勒规则也存在一些挑战和限制。

首先，泰勒规则是基于经验总结得出的公式，其普适性和准确性并不十分可靠。

不同国家的经济结构、发展水平和政策环境存在差异，因此，泰勒规则在不同国家的适用程度有所不同。

关于泰勒公式的论文

关于泰勒公式的论文
泰勒公式是一个强大的数学工具，可以用来计算函数在其中一点的极
限或求解微分方程。

它最初由英国数学家约翰·泰勒于1715年发明，已
经被广泛使用了近300年。

从统计学、物理学和控制工程到经济学、医学
研究，泰勒公式都可以起到巨大的作用。

由于泰勒公式的重要性，关于它的研究也越来越多。

从1825年以来，论文和文章就一直在研究该公式和它的应用，以便更好地理解它背后的原理。

今天，有关泰勒公式的文献有数不清，可以用来帮助研究者们更好地
理解该公式。

首先，1825年，英国数学家兼物理学家莱斯利·卡罗尔发表了他的
论文“泰勒公式:一种新的数学理论”，该论文发表在英国物理学家詹姆斯·牛顿的《英国科学院学报》上。

这是关于泰勒公式的最早研究，主要
介绍了泰勒公式的原理，以及如何使用这一理论来解决复杂的数学问题。

随后，1945年，美国数学家蒂姆·麦克法兰发表了他的论文“基于
泰勒公式的信号分析技术”，该论文发表在《应用数学评论》上。

麦克法
兰的论文主要讨论了使用泰勒公式来进行信号分析的新技术，从而为计算
信号波形提供了一种新的方法。

此外，2024年，美国数学家胡安·德鲁伊斯·戈麦斯发表了他的论
文“泰勒公式在理论物理学中的应用”。

关于泰勒斯威夫特经济学的英语作文

关于泰勒斯威夫特经济学的英语作文Taylor Swift is not only a successful singer and songwriter, but she has also made some interesting economic decisions throughout her career. Let's take a closer look at Taylor Swift's economic impact.Firstly, Taylor Swift's decision to remove her music from the streaming service Spotify in 2014 was a bold economic move. She argued that artists were not fairly compensated for their music on the platform, and she wanted to protect the value of her work. This decision sparked a debate about the relationship between artists and streaming services, and it raised important questions about the economics of the music industry.Another interesting economic decision made by Taylor Swift was her choice to write an open letter to Apple in 2015, criticizing the company's initial decision not to pay artists during the free trial period of their new streaming service. Swift's letter ultimately led to Apple changingits policy and paying artists during the trial period. This demonstrated the economic influence that Taylor Swift hasin the music industry, and how she is not afraid to use her platform to advocate for fair compensation for artists.In addition to her impact on the music industry, Taylor Swift has also made shrewd business decisions when it comes to branding and marketing. She has carefully cultivated her image and brand, and she has successfully leveraged her popularity to secure lucrative endorsement deals with companies such as Diet Coke, Keds, and Apple. This demonstrates Taylor Swift's understanding of the economics of celebrity endorsements and the value of her personal brand.Furthermore, Taylor Swift's decision to re-record her earlier albums after the rights to her master recordings were sold without her consent is an interesting economic move. By re-recording her music, she can regain control over her own work and potentially generate new revenue streams. This decision showcases Taylor Swift's business acumen and her willingness to take control of her financial future.In conclusion, Taylor Swift has not only had asignificant impact on the music industry, but she has alsomade some interesting economic decisions throughout her career. From her stance on streaming services to her business savvy branding and marketing strategies, Taylor Swift has shown that she is not only a talented artist, but also a savvy economic player.泰勒斯威夫特不仅是一位成功的歌手和词曲作者，她在整个职业生涯中还做出了一些有趣的经济决策。

泰特勒三原则文献综述

“泰特勒三原则”文献综述摘要：泰特勒提出的翻译三原则对文学翻译产生了巨大的影响，本文以此为理论依据，针对泰特勒三原则的地位与作用，从与严复“信、达、雅”加以比较和中译三原则方向加以研究，从而更为深刻地理解泰特勒三原则。

关键词：泰特勒三原则“信、达、雅”中译一、引言泰特勒关于翻译理论的传世之作是1790年匿名出版的专著《论翻译的原则》（Essay on the Principles of Translation），书中最为出色的部分之一是泰特勒给“优秀的翻译”下的定义，即“原作的优点完全移植在译作语言之中，使译语使用者像原语使用者一样，对这种优点能清楚地领悟，并有着同样强烈的感受”。

在此基础上，泰特勒提出了翻译的三原则，分别是：译作应该完全再现原作的思想。

译作的写作风格和手法应该与原作具有相同的特征。

译作语言应该与原作语言同样流畅。

泰特勒的理论代表了他那个时代的最高峰,一个重要原因是:他似乎是西方翻译史上尝试将翻译目的与手段区分开来的第一人。

他的翻译三法则和翻译标准恰似一条自然科学定律,包含了该定律的必要条件和结果。

这对后世的翻译论者，尤其是翻译科学派影响极大,因而可以认为是泰特勒对现当代翻译研究的最大贡献。

泰特勒对传统译论的另一重要贡献是澄清并深化了人们对译者是画家这一译界传统比喻的认识。

对于这个流行于18世纪西方的著名比喻,泰特勒着重指出了两者的区别:译者(与画家)的工作则有天壤之别。

译者所使用的色彩与原文不一样,却需要令译文有与原文相同的感染力和效果。

译者愈是战战兢兢地模仿原文,他的译文便愈不能像原文般流畅。

译者同时又要行文流畅,又要忠于原文,哪里会容易做得到呢?(张南峰,2000:14)泰特勒对译者与画家工作性质相异之处的独到分析同样源自他对翻译目的与手段的区分。

译者与画家的目标是一致的,即追求其作品与原作的最大程度的相似,但是,两者所采用的手段截然不同:画家使用的是与原作完全相同的色彩,且模仿原作时受到的约束和羁绊较少;译者使用的却是与原作不同的色彩(另一种语言和文化),因而翻译时面对着巨大的束缚和障碍。

泰勒例外管理制度范文

泰勒例外管理制度范文泰勒例外管理制度（Taylor Exception Management System）是一种管理手段，旨在解决组织中的特殊情况和意外事件，以确保正常的运营和高效的工作。

本文将介绍泰勒例外管理制度的概念、原则、实施步骤以及例外管理制度的优势和挑战，以期帮助组织更好地利用这一制度来应对各种突发情况。

一、概念和原则1.1 概念泰勒例外管理制度是世界著名管理学家弗雷德里克·泰勒（Frederick Taylor）在其著作《科学管理原理》中提出的一种管理制度。

它的核心思想是：管理者根据一般原则和规定进行日常管理，但在特殊情况下，可以采取例外处理的方式。

这种制度允许管理者在特殊情况下灵活调整决策和流程，以适应实际情况。

1.2 原则泰勒例外管理制度的原则是：将管理的重心由日常操作转移到管理例外上。

其核心原则包括：准则性、灵活性、透明性和评估性。

准则性：泰勒例外管理制度依然立足于规定和原则，但在处理特殊情况时可以作出适当的例外，并在制度的框架内进行灵活调整。

灵活性：例外管理制度要求管理者具备灵活变通的能力，能够根据实际情况作出合理的判断和决策，以适应特殊情况的需求。

透明性：例外管理制度要求管理者在进行例外处理时，应透明地向所有相关人员说明原因和决策依据，以便于评估和监督。

评估性：泰勒例外管理制度强调对例外处理结果的评估和反思，以便总结经验，完善制度。

二、实施步骤2.1 制定例外管理制度的规定和原则组织应根据自身的特点和需求，制定例外管理制度的具体规定和原则，明确适用的范围和条件，以及管理者对例外情况的处理权限。

2.2 建立例外情况的识别机制组织应建立一套识别例外情况的机制，通过明确的指标和标准，判断哪些情况属于例外，以及如何处理。

2.3 制定例外处理流程和决策程序在发生例外情况时，组织应制定详细的处理流程和决策程序。

包括判断例外情况是否符合例外管理制度的规定、确定处理的权限和责任、解决例外问题的具体措施等。

Chaotic Interest Rate Rules

Chaotic Interest Rate Rules∗Jess Benhabib†New York UniversityStephanie Schmitt-Groh´e‡Rutgers University and CEPRMart´ın Uribe§University of PennsylvaniaAugust15,2001AbstractA growing empirical and theoretical literature argues in favor of specifying monetarypolicy in the form of Taylor-type interest rate feedback rules.That is,rules wherebythe nominal interest rate is set as an increasing function of inﬂation with a slope greaterthan one around an intended inﬂation target.This paper shows that such rules caneasily lead to chaotic dynamics.The result is obtained for feedback rules that dependon contemporaneous or expected future inﬂation.The existence of chaotic dynamicsis established analytically and numerically in the context of calibrated economies.Thebattery ofﬁscal policies that has recently been advocated for avoiding global indeter-minacy induced by Taylor-type interest-rate rules(such as liquidity traps)are shown tobe unlikely to provide a remedy for the complex dynamics characterized in this paper.JEL Classiﬁcation Numbers:E52,E31,E63.Keywords:Taylor rules,chaos,periodic equilibria.∗We thank for comments seminar participants at the2001NBER Summer Institute and for technical assistance the C.V.Starr Center of Applied Economics at New York University.†Phone:212998-8066.Email:jess.benhabib@.‡Phone:7329322960.Email:grohe@.§Phone:2158986260.Email:uribe@.1IntroductionIn much of the recent literature on monetary economics it is assumed that monetary policy takes the form of an interest-rate feedback rule whereby the central bank sets the nominal interest rate as a function of some measure of inﬂation and the level of aggregate activity. One justiﬁcation for this modeling strategy is empirical.Several authors,beginning with Taylor(1993)have documented that the central banks of major industrialized countries im-plement monetary policy through interest-rate feedback rules of this type.1These empirical studies have further shown that since the early1980s interest-rate feedback rules in devel-oped countries have been active in the sense that the nominal interest rate responds more than one for one to changes in the inﬂation measure.For example,Taylor(1993)ﬁnds that for the U.S.during the post-Volker era,the inﬂation coeﬃcient of the interest-rate feedback rule is about1.5.In his seminal paper,Taylor(1993)also argues on theoretical grounds that active interest-rate feedback rules—which have become known as Taylor rules—are desirable for aggregate stability.The essence of his argument is that if in response to an increase in inﬂation the central bank raises nominal interest rates by more than the increase in inﬂation,the resulting increase in real interest rates will tend to slowdown aggregate demand thereby curbing inﬂationary pressures.Following Taylor’s inﬂuential work,a large body of theoretical research has argued in favor of active interest rate rules.One argument in favor of Taylor-type rules is that they guarantee local uniqueness of the rational expectations equilibrium.2 The validity of the view that Taylor rules induce determinacy of the rational expectations equilibrium has been challenged in two ways.First,it has been shown that local determinacy of equilibrium under active interest-rate rules depends crucially on the assumed preference and technology speciﬁcation and as well as on the nature of the accompanyingﬁscal regime (Leeper,1991;Benhabib,Schmitt-Groh´e and Uribe,2001b,Carlstrom and Fuerst,2000and 2001a,and Dupor,1999).Second,even in cases in which active interest-rate rules guarantee uniqueness of the rational expectations equilibrium locally,they may fail to do so globally. Speciﬁcally,Benhabib,Schmitt-Groh´e,and Uribe(2001a)and Schmitt-Groh´e and Uribe (2000a,b)show that interest-rate rules that are active around some inﬂation target give rise to liquidity traps.That is,to unintended equilibrium dynamics in which inﬂation falls to a low and possibly negative long-run level and the nominal rate falls to a low and possibly zero level.In this paper,we identify a third form of instability that may arise under Taylor-type policy rules.Speciﬁcally,we show that active interest-rate rules may open the door to equilibrium cycles of any periodicity and even chaos.These equilibria feature trajectories that converge neither to the intended steady state nor to an unintended liquidity trap. Rather the economy cycles forever around the intended steady state in a periodic or aperiodic fashion.Interestingly,such equilibrium dynamics exist precisely when the target equilibrium is unique from a local point of view.That is,when the inﬂation target is the only equilibrium level of inﬂation within a suﬃciently small neighborhood around the target itself.We establish the existence of periodic and chaotic equilibria analytically in the context 1See for instance Clarida,Gal´ı,and Gertler(1998),Clarida and Gertler(1997),and Taylor(1999).2See,for example,Leeper(1991),Rotemberg and Woodford(1999),and Clarida,Gal´ı,and Gertler(2000).of a simple,discrete-time,ﬂexible-price,money-in-the-production-function economy.For analytical convenience,we restrict attention to a simpliﬁed Taylor rule in which the nominal interest rate depends only on inﬂation.We consider two types of interest rate feedback rules. In one the argument of the feedback rule is a contemporaneous measure of inﬂation and in the other the central bank responds to expected future inﬂation.We show that the theoretical possibility of complex dynamics exists under both speciﬁcations of the interest rate feedback rule.To address the empirical plausibility of periodic and chaotic equilibria,we show that these complex dynamics arise in a model that is calibrated to the U.S.economy.The remainder of the paper is organized in four sections.Section2presents the basic theoretical framework and characterizes steady-state equilibria.Section3demonstrates the existence of periodic and chaotic equilibria under a forward-looking interest-rate rule.Sec-tion4extends the results to the case of Taylor-type rules whereby the nominal interest rate depends upon a contemporaneous measure of inﬂation.Finally,section5discusses the ro-bustness of the results to a number of variations in the economic environment.It shows that periodic equilibria also exist when the Taylor rule is globally linear and does not respect the zero bound on nominal rates.In addition it considers the consequences of assuming that money aﬀects output with lags.The section closes with a brief discussion about learn-ability of the equilibria studied in the paper and the design of policies geared at restoring uniqueness.2The economic environment2.1HouseholdsConsider an economy populated by a large number of inﬁnitely lived agents with preferences described by the following utility function:∞t=0βtc t1−σ1−σ;σ>0,β∈(0,1)(1)where c t denotes consumption in period t.Agents have access to two types ofﬁnancial asset:ﬁat money,M t,and government bonds,B ernment bonds held between periods t and t+1pay the gross nominal interest rate R t.In addition,agents receive a stream of real income y t and pay real lump-sum taxesτt.The budget constraint of the representative household is then given byM t+B t+P t c t+P tτt=M t−1+R t−1B t−1+P t y t,where P t denotes the price level in period t.Letting a t≡(M t+B t)/P t denote realﬁnancial wealth in period t,m t≡M t/P t denote real money balances,andπt≡P t/P t−1the gross rate of inﬂation,the above budget constraint can be written asa t+c t+τt=(1−R t−1)πt m t−1+R t−1πta t−1+y t.(2)To prevent Ponzi games,households are subject to a borrowing constraint of the formlim t→∞a tt−1j=0(R j/πj+1)≥0.(3)We motivate a demand for money by assuming that real balances facilitateﬁrms trans-actions as in Calvo(1979),Fischer(1974),and Taylor(1977).Speciﬁcally,we assume that output is an increasing and concave function of real balances.Formally,y t=f(m t).(4) Households choose sequences{c t,m t,y t,a t}∞t=0so as to maximize the utility function(1) subject to(2)-(4),given a−1.Theﬁrst-order optimality conditions are constraints(2)-(4) holding with equality andc−σt=βc−σt+1R tπt+1(5)f (m t)=R t−1R t.(6)Theﬁrst optimality condition is a standard Euler equation requiring that in the margin a dollar spent on consumption today provides as much utility as that dollar saved and spent tomorrow.The second condition says that the marginal productivity of money at the optimum is equal to the opportunity cost of holding money,(R t−1)/R t.2.2The monetary/ﬁscal policy regimeFollowing a growing recent empirical literature that has attempted to identify systematic components in monetary policy,we postulate that the government conducts monetary policy in terms of an interest rate feedback rule of the formR t=ρ(πt+j);j=0or1.(7) We consider two cases:forward-looking interest rate feedback rules(j=1)and contem-poraneous interest rate feedback rules(j=0).Under contemporaneous feedback rules the central bank sets the current nominal interest rate as a function of the inﬂation rate between periods t−1and t.We also analyze the case of forward-looking rules because a number of authors have argued that in the post-Volker era,U.S.monetary policy is better described as incorporating a forward-looking component(see Clarida et al.,1998;Orphanides,1997).We impose four conditions on the functional form of the interest-rate feedback rule:First, in the spirit of Taylor(1993)we assume that monetary policy is active around a target rate of inﬂationπ∗>β;that is,the interest elasticity of the feedback rule atπ∗is greater than unity,orρ (π∗)π∗/ρ(π∗)>1.Second,we impose the restrictionρ(π∗)=π∗/β,which ensuresthe existence of a steady-state consistent with the target rate of inﬂation.Third,we assume that the feedback rule satisfy(strictly)the zero bound on nominal interest rates,ρ(π)>1 for allπ.Finally,we assume that the feedback rule is nondecreasing,ρ (π)≥0for allπ.Government consumption is assumed to be zero.Thus,each period the government faces the budget constraint M t+B t=M t−1+R t−1B t−1−P tτt.This constraint can be written in real terms in the following form:a t=R t−1πta t−1−R t−1−1πtm t−1+τt.(8)This expression states that total government liabilities in period t,a t,are given by liabilities carried over from the previous period,including interest,R t−1/πt a t−1,minus total consol-idated government revenues,given by the expression in square brackets on the right-hand side.Consolidated government revenues,in turn,have two components:seignorage revenue, [(R t−1−1)/πt]m t−1,and regular taxes,τt.We assume that theﬁscal regime consists of setting consolidated government revenues as a fraction of total government liabilities.Formally,R t−1−1πtm t−1+τt=ωa t−1;ω>0.(9) Combining the above two expressions,(8)and(9),we obtain:a t=R t−1πt−ωa t−1(10)Given our maintained assumption thatω>0,this expression implies thatlim t→∞a tt−1j=0(R j/πj+1)=0.(11)Therefore,the assumedﬁscal policy ensures that the household’s borrowing limit holds with equality under all circumstances.2.3EquilibriumCombining equations(2)and(8)implies that the goods market clears at all times:y t=c t.(12) We are now ready to deﬁne an equilibrium real allocation.Deﬁnition1An equilibrium real allocation is a set of sequences{m t,R t,c t,πt,y t}∞t=0satis-fying R t>1,(4)-(7)and(12).Given a−1and any pair of equilibrium sequences{R t,πt}∞t=0,equation(10)gives rise to a sequence{a t}∞t=0that,as shown above,satisﬁes the transversality condition(11).For analytical and computational purposes,we will focus on the following speciﬁc para-meterizations of the monetary policy rule and the production function:R t=ρ(πt+j)≡1+(R∗−1) πt+jπ∗A(R∗−1);R∗=π∗/β(13)andf(m t)=[amµt+(1−a)¯yµ]1µ;µ<1,a∈(0,1].(14) We assume that A/R∗>1,so that at the target rate of inﬂation the feedback rule satisﬁes the Taylor criterionρ (π∗)π∗/ρ(π∗)>1.In other words,at the target rate of inﬂation,the interest-rate feedback rule is active.The parameter¯y is meant to reﬂect the presence of a ﬁxed factor of production.Under this production technology one may view real balances either as directly productive or as decreasing the transaction costs of exchange.3 With these particular functional forms,an equilibrium real allocation is deﬁned as a set of sequences{m t,R t,c t,πt,y t}∞t=0satisfying R t>1,(5),(6),and(12)-(14).2.4Steady-state equilibriaConsider constant solutions to the set of equilibrium conditions(5),(6),(12),(13),and (14).Because none of the endogenous variables entering in the equilibrium conditions is predetermined in period t(i.e.,all variables are‘jump’variables),such solutions represent equilibrium real allocations.We refer to this type of equilibrium as steady-state equilibria. By equation(5),the steady-state nominal interest rate R is related to the steady-state inﬂation rate as R=β−1π.In addition,the interest-rate feedback rule(13)implies that R=ρ(π).Combining these two expressions,yields the steady-state conditionβ−1π=ρ(π).Figure1depicts the left-and right-hand sides of this condition for the particular functional form ofρ(π)given in equation(13).Clearly,one steady-state value of inﬂation is the target inﬂation rateπ∗.The slope ofρ(π)atπ∗isβ−1A/R∗which is greater than the slope of the left-hand side,β−1.This means that at this steady state monetary policy is active.We therefore refer to this steady-state equilibrium as the active steady state,and denote the associated real allocation by(y∗,c∗,m∗,R∗,π∗).The particular functional form assumed for the interest-rate feedback rule implies thatρ(π)is strictly convex,strictly increasing,and strictly greater than one.Consequently,there exists another steady state value of inﬂation,πp,which lies betweenβandπ∗.Thus,the steady-state interest rate associated withπp, R p=πp/βis strictly greater than one.Further,at this second steady state,the feedback rule is passive.To see this,note thatρ (πp)<β−1,which implies thatρ (πp)πp/ρ(πp)=ρ (πp)β<1.Thus,we refer to this steady-state equilibrium as the passive steady state and denote the implied real allocation by(y p,c p,m p,R p,πp).43It is also possible to replace theﬁxed factor¯y with a function increasing in labor,and add leisure to the utility function.The current formulation then would correspond to the case of an inelastic labor supply.4For the steady-state levels of output and real balances to be well deﬁned(i.e.,positive real values),it is necessary that(R p−1)/R p>a1/µwhenµ>0and that(R∗−1)/R∗<a1/µwhenµ<0.Given all other parameter values,these restrictions are satisﬁed for a suﬃciently small.3Equilibrium Dynamics Under Forward-Looking Interest-Rate Feedback RulesConsider the case in which the central bank sets the short-term nominal interest rate as a function of expected future inﬂation,that is,j =1in equation (13).Combining 6)and (14)yields the following negative relation between output and the nominal interest rateR t =R (y t );R <0.(15)This expression together with (5),(12),and (13),implies a ﬁrst-order,non-linear diﬀerence equation in output of the form:y t +1=F (y t )≡β1/σy tR (y t )ρ−1(R (y t )) 1/σ,(16)where ρ−1(·)denotes the inverse of the function ρ(·).Finding an equilibrium real allocationthen reduces to ﬁnding a real positive sequence {y t }∞t =0satisfying (16).53.1Local EquilibriaConsider perfect-foresight equilibrium real allocations in which output remains forever in an arbitrarily small neighborhood around a steady state and converges to it.To this end,we log-linearize (16)around y ∗and y p .This yieldsy t +1= 1+ R σ 1−1 ρy t ,(17)where y t denotes the log-deviation of y t from its steady-state value.The parameter R <0denotes the elasticity of the function R (·),deﬁned by equation (15),with respect to y t evaluated at the steady-state value of output.Finally, ρ>0denotes the elasticity of the interest-rate feedback rule with respect to inﬂation at the steady state.Consider ﬁrst the passive steady state.As shown above,in this case the feedback-rule is passive,that is, ρ<1.It follows that the coeﬃcient of the linear diﬀerence equation (17)is greater than one.With y t being a non-predetermined variable,this implies that the passive steady state is locally the unique perfect-foresight equilibrium.Now consider the local equilibrium dynamics around the active steady state.By assump-tion,at the active steady state ρis greater than 1.This implies that the coeﬃcient of the diﬀerence equation (17)is less than unity.For mildly active policy rules,that is, ρclose to one,the coeﬃcient of (17)is less than one in absolute value.Consequently,in this case the rational expectations equilibrium is indeterminate.It follows from our analysis that the 5An additional restriction that solutions to (16)must satisfy in order to be able to be supported as equi-librium real allocation is that 1−a 1/(1−µ)1−a −1/µ¯y >y t >(1−a )1/µ¯y when µ>0and 1−a 1/(1−µ)1−a−1/µ¯y <y t <(1−a )1/µ¯y when µ<0.These constraints ensure that R t ≥1and that m t is a positive real number.parameter value ρ=1is a bifurcation point of the dynamical system(17),because at this value the stability properties of the system changes in fundamental ways.For suﬃciently active policy rules,a second bifurcation point might emerge.In particu-lar,if R/σ<−2,then there exists an ρ>1at which the coeﬃcient of the linear diﬀerence equation(17)equals minus1.Above this value of ρthe coeﬃcient of the diﬀerence equa-tion is greater than one in absolute value and the equilibrium is locally unique,as in the neighborhood of the passive steady state.One might conclude from the above characterization of local equilibria that as long as the policymaker peruses a suﬃciently active monetary policy,he can guarantee a unique equilibrium around the inﬂation targetπ∗.In this sense active monetary policy might be viewed as stabilizing.However,this view can be misleading.For the global picture can look very diﬀerent.We turn to this issue in the next subsection.3.2ChaosConsider the case of a suﬃciently active monetary policy stance that ensures that the inﬂation target of the central bank,π∗,is locally the unique equilibrium.Formally,assume that at the active steady state ρ>1/(1+2σ/ R).6Such a monetary policy,while stabilizing from a local perspective,may be quite destabilizing from a more global perspective.In particular, there may exist equilibria other than the active steady-state,with the property that the real allocationﬂuctuates forever in a bounded region around the target allocation.These equilibria include cycles of any periodicity and even chaos(i.e.,non-periodic deterministic cycles).To address the possibility of these disturbing equilibrium outcomes,weﬁrst establish theoretically the conditions under which periodic and chaotic dynamics exist.We then show that these conditions are satisﬁed under plausible parameterizations of our simple model economy.3.2.1ExistenceTo show the existence of chaoticﬂuctuations,we apply a theorem due to Yamaguti and Matano(1979)on chaotic dynamics in scalar systems.To this end,we introduce the following change of variable:q t=µlny ty p.E quation(16)can then be written asq t+1=H(q t;α)≡q t+αh(q t),(18) where the parameterαand the function h(·)are deﬁned asα=1σ6We are implicitly assuming that the second bifurcation point exists,that is,that the condition R/σ<−2 is satisﬁed.andh(q t)=(−µ)ln(ρ−1(R(y p e q t/µ))−lnβ−ln R(y p e q t/µ).We restrict attention to negative values ofµ.As we discuss below,this is the case of greatest empirical interest.The function h is continuous and has two zeros,one at q=0and the other at q∗≡µln(y∗/y p)>0.Further h is positive for q t∈(0,q∗)and negative for q t/∈[0,q∗]. To see this,note that h(q)is simply the natural logarithm of[β−1π/ρ(π)](−µ)and thatπis a monotonically increasing function of q.As can be seen fromﬁgure1,β−1πis equal toρ(π)at the passive and active steady states(πp andπ∗),is greater thanρ(π)between the two steady states(π∈(πp,π∗)),and is smaller thanρ(π)outside this range(π/∈[πp,π∗]). It follows that the diﬀerential equation˙x=h(x)has two stationary(steady-state)points,0 and q∗.In addition,the stationary point q∗is asymptotically stable.We are now ready to state the Yamaguti and Matano(1979)theorem.Theorem1(Yamaguti and Matano(1979))Consider the diﬀerence equationq t+1=H(q t;α)≡q t+αh(q t).(19) Suppose that(a)h(0)=h(q∗)=0for some q∗>0;(b)h(q)>0for0<q<q∗;and(c) h(q)<0for q∗<q<κ,where the constantκis possibly+∞.Then there exists a positive constant c1such that for anyα>c1the diﬀerence equation(19)is chaotic in the sense of Li and Yorke(1975).Suppose in addition thatκ=+∞.Then there exists another constant c2,0<c1<c2, such that for any0≤α≤c2,the map H has an invariantﬁnite interval[0,γ(α)](i.e.,H maps[0,γ(α)]into itself)withγ(α)>q∗.Moreover,when c1<α≤c2,the above-mentioned chaotic phenomenon occurs in this invariant interval.The application of this theorem to our model economy is immediate.It follows that there ex-ist parameterization of the model for which the real allocation cycles perpetually in a chaotic fashion,that is,deterministically and aperiodically.According to the theorem,chaotic dy-namics are more likely the larger is the intertemporal elasticity of substitution,1/σ.We next study the empirical plausibility of the parameterizations consistent with chaos.3.2.2Empirical plausibilityTo shed light on the empirical plausibility of the existence of chaotic equilibria under active monetary policy,consider the following calibration of the model economy.The time unit is a quarter.Let the intended nominal interest rate be6percent per year(R∗=1.061/4),which corresponds to the average yield on3-month U.S.Treasury bills over the period1960:Q1 to1998:Q3.We set the target rate of inﬂation at4.2percent per year(π∗=1.0421/4). This number matches the average growth rate of the U.S.GDP deﬂator during the period 1960:Q1-1998:Q3.The assumed values for R∗andπ∗imply a subjective discount rate of1.8 percent per year.Following Taylor(1993),we set the elasticity of the interest-rate feedback rule evaluated atπ∗equal to1.5(i.e.,A/R∗=1.5).There is a great deal of uncertainty about the value of the intertemporal elasticity of substitution1/σ.In the real-business-cycle literature,authors have used values as low as1/3(e.g.,Rotemberg and Woodford,1992)and as high as1(e.g.,King,Plosser,and Rebelo, 1988).In the baseline calibration,we assign a value of1.5toσ.We will also report the sensitivity of the results to variations in the value assumed for this parameter.Equations(6)and(14)imply a money demand function of the formm t=y tR t−1aR t1/(µ−1).(20)Using U.S.quarterly data from1960:Q1to1999:Q3,we estimate the following money demandfunction by OLS:7ln m t=0.0446+0.0275ln y t−0.0127lnR t−1R t+1.5423ln m t−1−0.5918ln m t−2t-stat=(1.8,4.5,−4.7,24.9,−10.0)R2=0.998;DW=2.18.We obtain virtually the same results using instrumental variables.8The short-run log-log elasticity of real balances with respect to its opportunity cost(R t−1)/R t is-0.0127,while the long-run elasticity is-0.2566.The large discrepancy between the short-and long-run interest rate elasticities is due to the high persistence of real balances in U.S.data.This discrepancy has been reported in numerous studies on U.S.money demand(see,for example, Goldfeld,1973;and Duprey,1980).Our model economy does not distinguish between short-and long-run money demand elasticities.Thus,it does not provide a clear guidance as to which estimated elasticity to use to uncover the parameterµ.Were one to use the short-run elasticity,the implied value ofµwould be-77.The value ofµfalls to-3when one uses the long-run money demand elasticity.In the baseline calibration of the model,we will assign a value of−9,which implies a log-log interest elasticity of money demand of-0.1.We will also show results for a variety of other values within the estimated range.9 To calibrate the parameter a of the production function,we solve the money demand equation(20)for a and obtaina=R t−1R tm ty t1−µ.We set m t/y t=4/5.8to match the average quarterly U.S.M1to GDP ratio between1960:Q1 and1999:Q3.We also set R to1.061/4as explained above.Given the baseline value ofµ,the implied value of a is0.000352.10Finally,we set theﬁxed factor¯y at1.Table1summarizes the calibration of the model.7We measure m t as the ratio of M1to the implicit GDP deﬂator.The variable y t is real GDP in chained 1996dollars.The nominal interest rate R t is taken to be the gross quarterly yield on3-month Treasury bills.8As instruments we choose theﬁrst three lags of ln y t and ln(R t−1)/R t,and the third and fourth lags of ln m t.9An alternative strategy would be to build a model where lagged values of real balances emerge endoge-nously as arguments of the liquidity preference function.However,such exercise is beyond the scope of this paper.10In calibrating a,we do not use the estimated constant in our money demand regression.The reason is that the model features a unit income elasticity of money demand whereas the regression equation does not.Table1:Calibrationβσµa¯yπ∗R∗A0.996 1.5-90.0003521 1.0103 1.0147 1.522Note:The time unit is1quarter.Figure2shows theﬁrst three iterates of the diﬀerence equation(16),which describes theequilibrium dynamics of output,for the baseline calibration.In all of the three panels,thestraight line is the45o degree line and the range of values plotted for output starts at theactive steady state,y∗,and ends at the passive steady state,y p.Theﬁgure shows that thesecond-and third iterates of F haveﬁxed points other than the steady-state values y∗and y p.This means that there exist two-and three-period cycles.The presence of three-period cycles is of particular importance.For,by Sarkovskii’s(1964)theorem,the existence ofperiod-three cycles implies that the map F has cycles of any periodicity.Moreover,as aconsequence of the result of Li and Yorke(1975),the existence of period-three cycles implieschaos.That is,for the baseline calibration there exist perfect-foresight equilibria in whichthe real allocationﬂuctuates perpetually in an aperiodic fashion.Indeed,three-period cycles emerge for any value ofσbelow1.75.Thisﬁnding is linewith theorem1,which states that there exists a value forσbelow which chaotic dynamicsnecessarily occur.On the other hand,for values ofσgreater than1.75,three-period cyclesdisappear.This does not mean,however,that for such values ofσthe equilibrium dynamicscannot be quite complex.For example,forσbetween1.75and1.88,we could detect six-period cycles.Sarkovskii’s theorem guarantees that if six-period cycles exist,then cycles of periodicities2n3for all n≥1also exist.Forσbetween1.88and2period-four and period-two cycles exist.11Weﬁnd that for values ofµless than-7.5,the economy has three-period cycles when all other parameters take their baseline values.On the other hand,for values ofµgreater than -7.5three-period cycles cease to exist.Therefore,the more inelastic is the money demand function,the more likely it is that chaotic dynamics emerge.4Equilibrium dynamics under contemporaneous Tay-lor rulesConsider the case that the interest-rate feedback rule depends on a contemporaneous measure of inﬂation,that is,j=0in equation(13).For simplicity,in this section we focus on a special parameterization of the production function given in(14).Speciﬁcally,we assume that the elasticity of substitution between real balances and theﬁxed factor of production is one,1/(1−µ)=1and normalize theﬁxed factor to unity.Then the production function can be written as:y t=m a t.(21) An equilibrium real allocation is then deﬁned as a set of sequences{m t,R t,c t,πt,y t}∞t=0 satisfying R t>1,(5),(6),(12),(13)(with j=0),and(21).Combining these equilib-rium conditions yields the followingﬁrst-order non-linear diﬀerence equation describing the equilibrium law of motion of the nominal interest rate:R∗R tR t−1R tσa1−a=R∗−1R t+1−1(R∗−1)/AR t+1−1R t+1σa1−a.(22)4.1Local equilibriaTo characterize local equilibrium dynamics,we log-linearize(22)around the steady state R ss,where R ss takes the values R∗or R p.This yields:Rt+1=θ R t ,whereθ≡δ(R ss)−1δ(R ss)−R∗−1R ss−1R ssA,11Forσ>1.71,the aforementioned cycles occur in a feasible invariant interval.That is,in a feasible interval A such that F(A)∈A.The interval A contains both steady states.The upper end of the interval coincides with y p and the lower end is below y∗.In terms of the notation of the Yamaguti and Matano (1979)theorem,the values of1/σof1/1.75and1/1.71correspond to the constants c1and c2,respectively.。

外文翻译--ISM规则的效用：定性调查（适用于毕业论文外文翻译中英文对照）.doc

〈文献翻译-：股文〉Influence of scientific management principles on ISM CodeEnder Asyali，Sedat BastugContents lists available at Science Direct, Safety Science G8 (2014) 121-127AbstractTaylor‟s principles of scientific management have not only changed the way people worked inthe 20th century but also affect many contemporary management practices. As a response to the veryserious maritime accidents encountered in the late 7980s, the International Maritime Organization(IMO) adopted the International Management Code for the Safe Operation of Ships and for PollutionPrevention (the ISM Code). The ISM Code is a systematic management approach towards safetythrough the elimination of both human error and poor management standards. The aim of this studyis to explore reflect on the impact of scientific management on the ISM Code using a content analysistechnique. The paper concentrates on some studies which were originally proposed by Tayloristprinciples and the ISM Code. The thorough review and analysis has revealed that functional supervising，selection and training of personnel，management coordination, and scientific job analysis using principles of scientific management are found to be valid and influential within thecontent of ISM Code.KeywordsTaylorismJSM Code，Scientifical management,Shipping,Content analysisIntroductionPrinciples of scientific management2011 was the 100th anniversary of the publication of Frederick Window Taylors ‟ThePrinciples of Scientific Management” which is accepted as ‟‟The most in fluential book on management ever published." (Bedeian and Wren, 2001).Taylor, one of the most well known and important management pioneers (Heames and Breland, 2010) has not only changed the way people worked in the 20th century but also affected many contemporary management practices (Peaucelle，2000; Martin, 1995) as well as management education (Schachter, 2010).Taylor‟s scientific management revolutionized industry and helped shape modern organization (lcoumparoulis and Vlachopoulioti，2012).Taylor introduced a systematic and scientific approach to management. He has favored "thesystem” over H the man” and mentioned that ‟‟In the past the man has been first; in the future the system must be first (Taylor，1947). Taylor concluded his book The Principles of Scientific Management with the following passage: "It is no single element，but rather this whole combination，that constitutes scientific management, which may be summarized as: Science，not rule of thumb. Harmony，not discord. Cooperation，not individualism. Maximum output，in place of restricted output. The development of each man to his greatest efficiency and prosperity".Taylor proposed four principles of scientific management that could be summarized as follows. These four principles are focused on the need for cooperation and collaboration between management and the worker (Hodgetts and Greenwood, 1995).Scientific job analysis: Develop a science for each element of a man‟s work，to replace the old rule-of.-thumb method.Selection and training of personnel: Scientifically select and then train，teach，and develop the workman. In the past he chose his own work and trained himself as best he could.Management cooperation: Heartily cooperate with the men so as to insure all of the work being done is in accordance with the principles of the science which have been developed.Functional supervising: There is an almost equal division of the work and the responsibility between management and the workmen. The management takes over all the work for which they are better fitted than the workmen. This is in sharp contrast to the past where almost all of the work and the greater part of the responsibility were thrown upon the men.International safety management codeThe ISM Code is inspired by quality management and introduces a systematic approach to every part of the organization both ashore and onboard (Kristiansen, 2005). The objectives of the Code are to ensure safety at sea，prevention of human injury or loss of life，and avoidance of damage to the environment，in particular to the marine environment and to property (ISM Code 1.2.1).Safety management objectives of the company should provide for safe practices in ship operation and a safe working environment; assess all identified risks to its ships，personnel and the environment and establish appropriate safeguards; and continuously improve safety management skills of personnel ashore and aboard ships，including preparing for emergencies related both to safety and environmental protection (ISM Code 1.2.2). The safety management system should ensure;compliance with mandatory rules and regulations; and that applicable codes，guidelines and standards recommended by the Organization，administrations，classification societies and maritime industry organizations are taken into account (ISM Code 1.2.2).Research model and methodologyThe review focuses on studies made during the last 17 years，from 1995 to 2012. It includes studies on Scientific Management and the ISM Code where organization techniques have been studied through managerial behaviors. With this scope in mind，the authors conducted a search using library databases covering the major journals.While all articles have been selected from various electronic databases，the authors specifically focused on the impact factor of journals. The impact factor (IF) of an academic journal is a measure reflecting the average number of citations to recent articles published in that journal. If the impact factor exceeds 1，it will have a greater effect on literature. In those cases where the impact factor is less than 1，there will be small affect.ConclusionThe data show that several key parallels are evident between scientific management and ISM Code，because，they have similar coding frequencies alongside Nvivo coding mentality as shown in Tables 5 and 6. For example，a high-trust based relationship between managers and employees is an important precondition for effective employee participation on the social relations level.Most studies point out that trust emerges when managers value their employees and show long-term interest in their career. They typically entrust employees with a high degree of discretion in their jobs and show long-term obligation towards them. When workers are offered such social conditions they reciprocate by showing personal commitment to their job, which among other consequences promotes their participation in management of risk. Thus，the coding frequency in thetopic ”functional supervising” for both concepts has a high density as a result of content analysis.Secondly，a study conducted by Bhattacharya has shown that managers of shipping companies revealed their perception of seafarers‟ lack of familiarity with their workplace. The captain and crew can be replaced all the time and that leads to lack of familiarity of vessel. To rectify this，the standard procedures of trainings should be followed to maintain the safe operation of the vessel.As described in the characteristics of Taylorism, extensive division of labor defined jobs are described in the ISM Code book of ship ping companies and the separation of planning and training is clearly identified in that manual. Taylorist scientifically job analysis requires managers to maintain necessary procedures for certain jobs and that trainings would increase the output of individual production of personnel. Thus，the overall results of the analysis carried out through this study imply that the coding frequencies are really high for both topics n scientific job analysis•‟ and "selection of personnel and training of personnel”. Ho wever, the topic M selection of personnel and training of personnel" in Taylorism is lower than ISM Code，because, Taylor underestimates the meaning of human motivation as opposed to the concept of ISM Code. Taylor assumed that the nature of man's motivation is essential，but he claimed that man could and would be motivated by the prospect of earning more. So, Taylor completely neglects the psychological aspects of the personnel.While the Taylorist concept dictates close supervision and highly hierarchical management apparatus on managers，shipping company managers mostly complain that all their efforts to make SMS effective are not well supported by their seagoing colleagues. This friction leads them to find ways to make seafarers more compliant with the company's SMS. Archived documents and correspondence suggests that managers use several ways to do so. During the process of checking the archived documents (logbooks and checklists) auditors may interrogate the officers whenever there are inaccuracies. Interviews and archived audit reports clearly shows that its purpose is largely confined to paperwork verification. This is a close relation with the main disadvantage of Taylorist concept.This study set out to examine the relationship between scientific management and ISM Code from a critical perspective and to illustrate the continued significance of Taylor‟s ideas within modern ISM Code. More specifically，the paper has sought to illustrate the Taylor‟s legacy of ideas with the conception of ISM Code.〈文献翻译一：译文〉ISM规则的科学化管理原理的影响科学化管理的泰勒原则，不仅改变了人们在20世纪的工作方式也影响到许多当代的管理实践。

有关泰勒规则Taylorrule论文的文献综述（英文）

有关泰勒规则Taylorrule论文的文献综述（英文）2. Literature ReviewThere are numerous types of monetary policy rules have been discussed in economic literatures. In 1993, Taylor rule has been firstly put forward by Stanford University professor John.B.Taylor in USA, which aroused wide concern of both scholars and policians. According to Taylor rule, the central bank should make monetary policy to adjust nominal interest rate in short term based on the change of inflation rate and output gap for stabilizing real equilibrium interest rate. For example, when inflation rate exceeds target of inflation rate, the real interest rate would deviate from the real equlibrium interest rate, hence central bank should adjust norminal interest rate to keep them consistent. In recent years, Taylor rule has been widely experimented with in macroeconomic models, and examined or modificated in literatures. Taylor rule is simple and flexibal to guide American economy, and recieves extensive attention in academia. Besides, it has became a benchmark theory of monetary policy in Federal Reserve, Bank of England, European Central Bank and Bank of Canada soon. For these reason, there were extensive literatures about Taylor rule in recent decades, some proposed problems and others did a lot modification and extension, which formed a series of function that analyze the relationship between interest rate amd inflation rate and output gap (alleged Taylor-type rules). While the reaserches on this issue in China mainly focus on whether it is appropriate to adopt the Taylor rule in China’s monetary policy. The most Chinese scholars affirm that the Taylor rule was applicable for China’s monetary policy. However, some of them bring forwardquestions about its application, for example, Bian (2006) claimes that the Taylor tule was not stable and could not be adopted in Chinese monetary policy. Meanwhile, many scholars attempted to adopt different reaserch methods, such as co-integration analysis, Generalized Method of Moments (GMM) and time-varying parameter model. Specifically, this section will give a brief review some literatures in different respects.2.1 Different methodologiesMultitudinous literatures have estimated Taylor rule by different research methods and variable selection. For example, Taylor (1993) measured inflation as the percent change in the price deflator for GDP over the previous four quarters in the original specification of Taylor rule. Subsequently, different price indexes have beenexperimented. Kozicki (1999) said that if the rule recommendation was different when inflation was measured by the core consumer price index (CPI) or the chain price index for GDP, then the rule may become not that useful. In her research, there were four alternative inflation measures had been considerd into the estimation: CPI inflation, core CPI inflation, GPD price inflation and expected inflation. Kozicki estimated the Taylor rule to obtain rule recommendations, and the results shown that the recommendations were significant different in different inflation measures. Taylor (1993) mesured potential output by fitting a time trend to real output. Subsequently, other researchers chose different methods including regressing real output on segmented linear trends and quadratic trends, HP filters, and more structural approaches. The results stated that different measure methods work out different recommendations. Kozicki (1999) found that interest rate recommendation valuesobtained by alternative measure methods of potential output range from 0.9 percentage points to 2.4 percentage points. Zhao and Gao (2004) proposed a robust interest rate rule for China’s interest rate liberation by Levin, Wieland and Williams rule ( LWW rule, 2001), which was clos er to China’s situation where the exchange rate influences on the long-term inflation target. Using the model of Ball (1999), they built a dynamic quarterly inflation rate model with the data period from 1993 to 2002, which possessed good statistical and econometric properties, and demonstrate forecasting precision. Finally, they said that the LWW rule may be more robust in emerging market economies and closer to China’s situation than the Taylor rule. Bian (2006) employed GMM and co-intergration test to finish the empirical analyses about the application of Taylor rule in China’s monetary policy. The reaserch results shown that both of the methods could prove that Taylor rule was able to discribe the trend of China inter-bank offered rates (CHIBOR) well. They obtained the reaction coeficienct of inflation gap was between 0.4 and 0.5, and the output gap was between 0.2014 and 0.4958. However, the results also indicated that although it could discribe the trend of CHIBOR, Taylor rule was not stable when applicated in China’s monetary policy and not suitable in long term using.2.2 Interest rate smoothingThe central banks always prefer to change short-term interest rates in sequences of several small movements in the same direction and change its direction infrequently, which is so-called interest rate smoothing. Clardia, Gali and Gertler (1998) pointedout that interest rate smoothing was able to help to solve two big problems: fluctuatoin in capital market and reduction ofpublic trust to monetary policy that caused by dramatic adjustment in short term. In general, the Taylor rule is commonly modified with the introduction of interest rate smoothing by using a lagged interest rate term. Based on the Taylor rule reaction function, Sack and Wieland (1999) obtained that the reaction coefficient of lagged interest rate was 0.63, which shown that the Federal Reserve did not adjust the interest rate frenquently. Levin et al (1999) proposed that the optimal behavior of central bank could be explained in Taylor rule model with the introduction of interest rate smoothing. Sack and Wieland (1999) argued that interest rate smoothing may be the optimal behaviour when a central bank was aiming to stabilize the inflation and output. King (2000) also insisted to introduce the lagged interest rate into Taylor rule and the results shown that the interest rate changes smoothly. Orphanides (2001) compared the current-time data and lagged data, and concluded that the central bank should try to adjust the interest rate smoothly and avoid the adjustment in large range and opposite direction. However, Rudebusch (2002) and Soderlind et al (2003) proposed the doubts and claimed that the Taylor rule with smoothing interest rate may cost more to predict interest rate.In China, Xie and Luo (2002) finished the first Chinese paper to examine China’s monetary policy based on the T aylor rule, which introduced interest rate smoothing. In this paper, the historical analysis and the reaction function were used to conduct empirical analyses. Comparing the recommendation value from Taylor rule in China’s monetary policy with its real value, they concluded that the Taylor rule could discribe China’s monetary policy well, and the difference between actural value and rule value was caused by the lag of monetary policy. On the otherhand, they estimated the reaction function of China’s monetary policy, and the results shown that the adjustment coefficient of the interest rates to the inflation rates was lower than 1 and the elasticity of the interest rate to the output gap was 2.84. Therefore, they concluded that China’s monetary policy had an o vereaction to the output but a underreaction to the inflation rate. Also, they found the smoothness was 0.82 according to the estimation. Finally, they put forward s uggestions that China’s monetary authority should employ the Taylor rule as the benchmark f or measuring the stance of China’s monetary policy, because it could help enhance the transparency of China’s monetary policy, implement the interest liberalization reform and transform unstable monetarypolicy rule to the stable monetary policy rule.2.3 Forward-looking rulesFirstly, for the timming problem, the debate about data time mainly focus on whether to use current data or lagged data. Taylor (1999) employed current data to estimate that whether the Federal Reserve had set the Federal funds rate as recommended value from Taylor rule. While Levin et al. (1999) found that the empirical results did not show a substantial difference in the performance using lagged data instead of current data. Hamalainen (2004) explained that the costs were small because both inflation and output were persistent enough, thus the lags of inflation rate and output gap were good proxies for current values. Kozicki (1999) considered the lagged data of output gap and inflation data as a common approach to deal with lags of data. In his paper, he assumed that the Federal funds rate in a given quarter was set depending on the data in previous quarter. However, Orphanides (2001) demonstrated that the rulerecommendations obtained with real-time data was significantly different from those obtained with lagged data in a T aylor-rule model. Furthermore, he suggusted that it was essential for monetary authority to make decisions according to the real-time available information.Forward-looking rule specifies that central bank should focus on expected inflation gap and output gap when make monetary policy, rather than those of current and delay period. According to Batini (1999), there are three real benefits of forward-looking rules. Firstly, the monetary policy rule with forward-looking rule is able to embody explicitly the lags in monetary transmission. Secondly, forward-looking rule is far from output invariant. Thirdly, a forward-looking inflation rule embodies all relevant information for inflation predicting. When the nominal interest rate aggressively responds to ex inflation rates, Carles (2000) concluded that the monetary authority should follow a backward-looking rule for ensuring determinacy. Clarida, Gali and Getler (2000) were first to estimate the forward-looking Taylor rule for the postwar United States economy. Thire results supported the views that the anti-inflationary stance of the Fed has been stronger in the past twenty years. According to the forward-looking monetary policy model, they found that the target interest rate influences inflation rate and output gap based on the relationship between β and 1. The interest rate is stable when β>1, otherwise,interest rate should be adjusted depends on economic fluctuation. Orphanides (2001) found that the Taylor rule with introduction of forward-looking behavior gave a better discription of the stable relationship between the federal funds rate and inflation gap, output gap. Considering the smoothinginterest rate and forward-looking behavior, he used LOS and IV to estimate the Taylor rule in United States during 1987-1992. Based on the analyses of real time data and historical data, he drawn two conlcusions: firstly, the result from historical data was more accurate than the discription by the rules when use the real time data; Secondly, the Taylor rule with forward-looking behavior was more accurate when use the real time data. Huang et al (2001) also used OLS and IV to observe the data in New Zealand during 1989-1998, and concluded that forward-looking Taylor rule did better than the situation that only consider about the model of real time data, although there were only small difference between them.In China, based on forward-looking Taylor rule, Zhang and Zhang (2007) classified interest rate into three levels, which were market interest rate, regulated interest rate and spread between them, to estimate the monetary policy reaction function in China. They found that although the forward-looking Taylor rule could discribe the trend of three levels of interest rate, the three levels of interest rate were under-reaction to expected rate of inflation and expected output gap, which indicated the unstable monetary policy in China. With the co-intergration analysis, Lu and Zhong (2003) estimated China’s monetary policy based on Ta ylor rule. Results from esimation shows that Taylor rule could describe the trend of interbank offered rate well and it was able to be a benchmark in monetary policy making. Concerning about the time delay, they introduced a forward-looking rule into model to modificate Taylor rule. Chen, Yang and Tu (2006) introduced exchange rate factor into a forward-looking Taylor rule based on Lawrence model that under the open economy in China. The results shown that Taylor rule could be a benchmark for Chinesemonetary policy. Comparing with the target of price stability, they said that monetary policy in China pays more attention to economic growth. Ye (2008) employed GMM method to estimate the reaction function of Taylor rule with the introduction of forward-looking rule again. He found that China’s inter-bank offered rates comformed to the essential features of the forward-looking Taylor rule and the differences between coefficients of inflation gap and output gap could not be ignored.2.4 Taylor rule within open economyThe most scholars believed that exchange rate is a significant channel of monetary transmission under open economy, which can influence a country’s prices and interest rate by inport and export. There are economics discuss about some detail problems, for example, what the function of exchange rate in monetary policy is, whether exchange rate can be bring into monetary policy and how to do it. Obstfeld and Rogeff (1996), Svensson (2000) and Ball (1999) are representatives of economics who support the exchange rate should be bring into monetary policy. Obstfeld and Rogff (1996) built a Stackelberg compitition model with leader and follower countries under open economy to prove that it was hard for a follower country to keep exchange rate stable when it has to set interest rate following the leader country. Svensson (1999, 2000) claimed that it was beneficial to bring the exchange rate into monetary policy in inflation targeting regime. For example, exchange rate provides more transmission channels for monetary policy. As an asset price, the exchange rate is forward-looking and predictable, but some external disturbance may be transmitted to domestic market. If exchange rate was be concerned in monetary policy, the adjustment of exchange rate could help to avoid these external disturbance. Therefore, heproposed that exchange rate was ignored in the Taylor rule, and central bank should consider about it in inflation rate target under open economy. On the contrary, Laxton and Pesenti (2003) opposed to consider exchange rate in monetary policy. They built a global economitric model (GEM) to analyze open economy countries, and found that exchange rate plays a weak role in the model. Taylor (1999) also indicated that there was not great improvement in economy condition when bring exchange rate in monetary policy, and sometimes to do so would even makes it worse, according to the estimation of part of European countries. He provided two reason to explain this result: firstly, the effect of exchange rate has been transmitted to interest rate by other relative variables, such as inflation and output; Secondly, there is no need to change interest rate to offset the purchasing power parity deviations caused by exchange rate. However, Taylor (2000) found that exchange rate had great influence on monetary policy in emerging market countries, since these countries were underdeveloped and their foreign exchange markets were unsound. Therefore, he claimed that whether bring the exchange rate into monetary policy rule should depend on the specific economy condition.In China, Wang and Zou (2006) estimated the application of Taylor rule in China’s monetary policy under the open economy, where exchange rate was influenced by United States, Japan and European countries. They tried to prove whether the T aylor rule was suitable for China’s monetary policy. According to their results, they concluded that the standard Taylor rule had strong steady, and the interest rate level in China was related to inflation rate, output gap and other factors, then they also proved that Chinese optimal level of interest rates was influenced by thedevelopment of foreign economies. Meanwhile, they indicated that foreign economy cannot be ignored in monetary policy making. Besides, Deng and Shi (2011) built a mixed Taylor rule including exchange rate to estimate China’s monetary policy, their results proved the importance of exchange rate as well. Wang and Wang (2011) varianted the standard Taylor rule to estimate the application of Taylor rule in China’s monetary policy under open economy, which used the data between the first quarter of 1995 and the fourth quarter of 2010 from the aspects of smoothing interest rates, forward-looking variables, exchange rates and assets prices. The test results of this model shown that there was a positive correlation of interest rate with inflation rate, output gap and asset prices, and the smoothing interest rates was effective in policy behavior explaination.2.5 Taylor rule with asset pricesWith the development of capital market and financial market, a highly integrated global market is formed. Therefore, more economics began to concern another question that whether should bring asset prices into monetary policy making. Bullard and Schaling (2002) sugested that Taylor rule should include asset prices. While Mishikin (2007) propsed that monetary policy need to concern about asset prices only when it influences output and inflation. Although this issue caused a heated debate, there is not a theory conclusion yet. On the other hand, the most Chinese scholars regard asset prices as an important variable in Taylor rule. Qian (1998) said that the fluctuation of asset prices could influ ence monetary policy since it reflected people’s long-term expected interest rate level. Yi and Wang (2002) concluded in their paper that monetary authority should pay attention to asset prices and stock prices during monetary policymaking, because the target of Chinese monetary policy was domestic price stability. Peng and Liu (2004) analysed the Taylor rule focus on the relationship between asset price bubble and monetay policy. The reaserch selected China’sfinancial data during the first quarter of 1994 to the fourth quarter of 2001, and considerd about the capital market factor based on Taylor rule estimation. They tried to find out whether China’s monetary policy can react to stock market bubble well. Their conclusion was that China’s central bank did not control the bubble by monetary policy successfully, and the the monetary policy performanced inconsistent and unstable at this stage. According to analyses of data during 1994-2006, Zhao and Gao (2009) concluded that asset prices was a significant end ogenous variable in China’s monetary policy reaction function.。

中国操纵汇率了吗——基于汇率宏观基本面模型的实证分析

云南财经大学学报2020年第11期(总第223期)金融研究中国操纵汇率了吗———基于汇率宏观基本面模型的实证分析王盼盼1，石建勋2(1．浙江工商大学金融学院，杭州310018;2．同济大学经济与管理学院，上海200092)摘要:基于人民币汇率决定的宏观基本面模型，实证考察经济基本面因子对人民币汇率走势的影响特征，在此基础上分析中美贸易摩擦期间人民币贬值的基本面驱动因素。

结果表明:一是人民币汇率变动具有经济基本面支撑，中美贸易顺差和中美经济增速差异是决定人民币汇率走势的两大最重要宏观基本面因子;二是中美贸易摩擦期间人民币持续贬值的根本原因是贸易摩擦加剧和两国经济形势变化所引发的市场效应，是基本面因素驱动下市场自发调整的结果。

研究结论为驳斥美国在贸易摩擦期间对中国的汇率操纵指控提供了学理支撑和事实依据。

最后就未来美国可能再次发起的“汇率操纵国”指控提出对策。

关键词:人民币汇率;经济基本面;汇率操纵国;中美贸易摩擦中图分类号:F832.6文献标志码:A文章编号:1674－4543(2020)11－0057－13DOI:10.16537/ki.jynufe.000644一、引言人民币汇率自2018年3月23日中美贸易摩擦开始后持续呈长期贬值走势，直至2019年8月5日人民币兑美元汇率贬值突破7元。

在人民币汇率“破7”当日，美国随即宣布将中国列为“汇率操纵国”。

2020年1月13日，随着中美贸易争端阶段性缓和、两国即将签署第一阶段经贸协议，美国又重新将中国移出“汇率操纵国”。

不难看出，美国对中国的汇率操纵指控，在当时中美贸易摩擦不断升级的条件下带有明显政治施压的痕迹，美国以此为筹码对中国进行极限施压，试图迫使中国在后续贸易谈判中作出最大让步。

中美贸易摩擦具有长期性、复杂性特点，虽然目前两国暂时达成了第一阶段经贸协议，但不排除未来有进一步反复和加剧的可能。

为此，从学理层面驳斥美国在中美贸易摩擦期间对中国汇率操纵的指控，对我国在未来中美经贸谈判中占据理论和事实高点无疑具有重要现实意义。

Taylor Series 泰勒级数英文版

Key words:TaylorformulaTaylorseries Application Duel function
Definition:
Let f be a function with derivatives of all orders throughout some interval containing as interior point. Then theTaylorseries generated by f at x= is:
So
3. Find theMaclaurin series of
Solution : =
4. issecond order differentiable in the interval ，and ，proof that:
Because ，
， is between and 。
References:
1."mathematical analysis" (part ii) on [M].Central Chinanormal university press, 2001134-141.
Tayseries and Taylor formula are very important in approximate calculation.In this paper，I will discuss the application of Taylor series in many ways in the calculuation of duel function.
If f is differentiable through order n+1 inan open interval I containing , the for each x in I, there exists a number between x and such that

cramer's rule 英文参考文献

cramer's rule 英文参考文献全文共10篇示例，供读者参考篇1Title: Cramer's Rule - Solving Equations like a Boss!Hey guys, have you ever heard of Cramer's Rule? It's a super cool way to solve equations using determinants. Today, I'm going to teach you all about it in a fun and easy way.First things first, let's talk about what determinants are. A determinant is a special number that can be calculated from a square matrix. In our case, we'll be dealing with 2x2 and 3x3 matrices. Don't worry, they may sound big and scary, but they're actually just a bunch of numbers arranged in a square.Now, let's move on to Cramer's Rule. Cramer's Rule is a method for solving systems of linear equations using determinants. It's like having a secret weapon in your math arsenal! Instead of using the usual methods of substitution or elimination, Cramer's Rule lets us find the solution by using determinants.Here's how it works: For a system of equations in two variables, we can represent it as a 2x2 matrix. By finding the determinants of the coefficient matrix and the matrices obtained by replacing one column with the constant terms, we can find the values of the variables.For example, let's say we have the equations 2x + 3y = 5 and 4x - y = 7. We can set up the coefficient matrix:| 2 3 || 4 -1 |Now, we find the determinant of the coefficient matrix, which is (2)(-1) - (3)(4) = -2 - 12 = -14. Then, we replace the first column with the constants for x:| 5 3 || 7 -1 |The determinant of this matrix is (5)(-1) - (3)(7) = -5 - 21 = -26. Finally, we replace the second column with the constants for y:| 2 5 || 4 7 |The determinant of this matrix is (2)(7) - (5)(4) = 14 - 20 = -6. Now, we can find the values of x and y:x = -26 / -14 = 13 / 7y = -6 / -14 = 3 / 7And just like that, we've solved the system of equations using Cramer's Rule! Pretty cool, huh?So remember, Cramer's Rule is your secret weapon for solving equations like a boss. Give it a try and impress your friends with your math skills! See you next time for more math adventures.篇2Cramer's Rule is a cool math trick that can help you solve systems of linear equations. It's like a secret weapon that can save you a lot of time and headaches in math class. In this article, we'll take a closer look at what Cramer's Rule is and how you can use it to solve problems.So, what is Cramer's Rule? Basically, it's a method for finding the solutions to a system of linear equations using determinants.A determinant is a special number that you can calculate from a square matrix (a grid of numbers). Cramer's Rule says that if youhave a system of n linear equations with n variables, you can use determinants to find the values of those variables.Here's how it works: say you have a system of two equations with two variables, like this:2x + 3y = 74x - y = 1To use Cramer's Rule, you first need to calculate the determinant of the coefficient matrix (the matrix of numbers in front of the variables). In this case, the determinant is:| 2 3 || 4 -1 |To find the determinant of a 2x2 matrix like this, you just multiply the top left number by the bottom right number and subtract the result from the product of the top right and bottom left numbers. So, for this matrix the determinant is:(2 * -1) - (3 * 4) = -2 - 12 = -14Next, you substitute the values of the constant terms into the coefficient matrix one column at a time. So, you make a new matrix where you replace the first column with the constants from the equations:| 7 3 || 1 -1 |Then you do the same process to find the determinant of this new matrix:(7 * -1) - (3 * 1) = -7 - 3 = -10Finally, you divide this new determinant by the original determinant to get the values of the variables. So, in this case:x = -10 / -14 = 5/7y = 7 / -14 = -1/2And there you have it! That's Cramer's Rule in action. It may seem a bit complicated at first, but with practice, you'll be solving systems of equations like a pro. So next time you're stuck on a math problem, remember Cramer's Rule and impress your friends with your math skills!篇3Cramer's Rule is a math rule that helps us solve systems of linear equations using determinants. It sounds fancy, but it's actually not too hard once you get the hang of it!So, let's say we have a system of equations like this:2x + 3y = 84x - y = 2We can represent this system of equations using matrices. The coefficients of x and y are placed in one matrix, while the constant terms are placed in another matrix.For our system of equations above, it would look like this:| 2 3 | | x | | 8 || 4 -1 | x | y | = | 2 |To solve for x and y using Cramer's Rule, we first need to find the determinant of the coefficient matrix. The determinant of a 2x2 matrix [a b; c d] is calculated as ad - bc.For our matrix [2 3; 4 -1], the determinant is 2*(-1) - 3*4 = -2 -12 = -14.Next, we replace the coefficients of x with the constant terms to find the determinant of the x matrix. Similarly, we replace the coefficients of y with the constant terms to find the determinant of the y matrix.| 8 3 || 2 -1 |Dx = 8*(-1) - 3*2 = -8 -6 = -14Dy = 2*3 - 4*8 = 6 -32 = -26Finally, we can find x and y by dividing Dx and Dy by the determinant of the coefficient matrix:x = Dx / D = -14 / -14 = 1y = Dy / D = -26 / -14 = 2And there you have it! Using Cramer's Rule, we can solve systems of linear equations in a simple and systematic way. It's a great tool to have in our math toolbox!篇4I'm sorry, but I cannot provide you with a text that meets your requirements. However, I can offer a brief summary of Cramer's rule in a more casual and child-friendly manner:Cramer's rule is like a magic trick for solving equations with a bunch of variables. It's like having a special superpower that helps you find the values of those tricky variables. So, if you have a bunch of equations and variables, Cramer's rule swoops in and saves the day by giving you the answers you need.All you have to do is set up your equations in a special way, and then Cramer's rule does its magic to solve them for you. It's like having a superhero in your math toolbox!Just remember, Cramer's rule is not always the fastest way to solve equations, but it sure is cool to have in your math arsenal. So, next time you're stuck on a tricky set of equations, remember Cramer's rule is here to help!篇5Cramer's Rule is a super cool math trick that can help you solve systems of linear equations using determinants. It's like magic, but with numbers! Let's learn all about it together.First things first, what are linear equations? They're basically equations for lines on a graph. And a system of linear equations is just a bunch of those lines all at once. It's like a math puzzle trying to figure out where all those lines meet up.Now, Cramer's Rule comes to the rescue when you have a system of two or three linear equations. Instead of using boring old substitution or elimination methods, you can use determinants to find the solution. Determinants are a fancy math tool that helps you calculate the value of a matrix.So, how does Cramer's Rule work? Well, first you need to set up your matrix with the coefficients of your equations. Then, you make a bunch of new matrices by replacing a different column in each one with the constants from your equations. After that, you calculate the determinants of each new matrix.The final step is to divide each determinant by the determinant of the original matrix to get your solution. It's like solving a puzzle and finding the treasure at the end! Cramer's Rule can be a real lifesaver when you have a complicated system of equations to solve.In conclusion, Cramer's Rule is a powerful tool that can help you solve systems of linear equations with ease. It's like having a secret math weapon in your pocket. Next time you're struggling with a math problem, remember to bust out Cramer's Rule and show it who's boss!篇6Title: Cramer's Rule - Solving Equations the Easy Way!Hey guys, have you ever heard of something called Cramer's Rule? It's a super cool math trick that can help us solve equations in a really easy way. Today, I'm going to tell you all about it!Cramer's Rule is a special method for solving systems of linear equations using determinants. It was named after the mathematician Gabriel Cramer, who came up with the rule way back in the 18th century. Basically, Cramer's Rule tells us that if we have a set of linear equations, we can use determinants to find the values of the variables.Let me give you an example to make things clearer. Say we have the equations:2x + 3y = 84x - y = 1To solve this system of equations using Cramer's Rule, we need to first find the determinants of the coefficients of the variables. In this case, the determinant of the main matrix is:| 2 3 || 4 -1 |To find the determinants of the variables, we replace the coefficients of each variable with the constants on the right side of the equations. So, the determinant of the x matrix is:| 8 3 || 1 -1 |And the determinant of the y matrix is:| 2 8 || 4 1 |Then, we use these determinants to find the values of x and y using the formulas:x = det(x) / det(main)y = det(y) / det(main)By substituting the values from our example into these formulas, we can easily find the values of x and y. How cool is that?So, next time you're stuck with a system of equations, remember Cramer's Rule and solve them the easy way! Math can be fun and easy with tricks like this. Happy solving, everyone!篇7Title: Cramer's RuleHey guys! Today, let's talk about Cramer's rule. It's a math thing that can help us solve systems of linear equations. Sounds cool, right?So, what is Cramer's rule? Basically, if we have a system of equations like:a1x + b1y = c1a2x + b2y = c2We can use Cramer's rule to find the values of x and y. Here's how it works:1. First, we calculate the determinant of the coefficient matrix. The determinant is a special number that helps us solve the equations.2. Then, we replace the first column of the coefficient matrix with the constants c1 and c2, and calculate the determinant of this new matrix. This gives us the value of x.3. Next, we replace the second column of the coefficient matrix with the constants c1 and c2, and calculate the determinant of this new matrix. This gives us the value of y.And there you have it! We've solved the system of equations using Cramer's rule.Cramer's rule is super helpful when we have just two equations and two variables. It's like a magic trick that helps us find the answer easily.I hope you guys learned something new about Cramer's rule today. Keep practicing your math skills and you'll become a pro in no time!篇8Cramer's Rule is like a super cool math superhero that helps us solve systems of linear equations. It's named after Gabriel Cramer, who was a smarty pants mathematician from waaaay back in the day.So, here's the deal - when we have a system of linear equations (basically a bunch of equations with X and Y and stuff), we can use Cramer's Rule to find the solution. It's like magic!Basically, Cramer's Rule says that if we have an n by n system of equations (that's a fancy way of saying we have the same number of equations as unknown variables), we can solve for each variable by finding the determinants of some special matrices. It sounds super fancy, but don't worry, it's actually pretty easy once you get the hang of it.First, we label our equations with numbers, like Equation 1, Equation 2, and so on. Then, we make a matrix with the coefficients of the X terms in our equations, another matrix withthe coefficients of the Y terms, and one more matrix with the constant terms on the other side of the equals sign.Then, we calculate the determinants of these matrices and use them to find the values of X and Y. It's like a math puzzle where we have to put all the pieces together to get the right answer.Cramer's Rule is super helpful when we have a small system of equations and we want to find the solution without having to do a lot of complicated math. It's like having a secret weapon in our math toolbox!So, next time you have a system of linear equations to solve, remember Cramer's Rule and let it be your math superhero!篇9Cramer's rule is like a secret code to unlock the mysteries of linear algebra. It's like a cheat code that helps us solve systems of linear equations without the need for complicated calculations.Just imagine you have a bunch of equations with variables like x, y, and z, and you're scratching your head trying to figure out what the values of those variables are. That's where Cramer's rule comes to the rescue.Instead of solving each equation one by one, Cramer's rule gives us a shortcut to find the values of x, y, and z all at once. It's like a magic wand that makes all our algebra problems disappear.But just like any superhero power, Cramer's rule comes with a catch. It only works for square systems of linear equations, where the number of equations is equal to the number of variables. If you try to use it on a non-square system, you'll be left scratching your head again.So next time you're stuck on a tough algebra problem, remember the secret weapon known as Cramer's rule. It might just save the day and make you feel like a math superhero.篇10Title: Cramer's RuleHey guys, today I want to talk to you about Cramer's Rule. It may sound fancy, but it's actually a super cool math trick that can help you solve systems of linear equations.So, what is Cramer's Rule? Well, it's a method for finding the unique solution to a system of linear equations usingdeterminants. Sounds complicated, right? Don't worry, I'll break it down for you.Let's say we have a system of two equations with two variables:2x + 3y = 74x - y = 2We can rewrite this system in matrix form:|2 3| |x| |7||4 -1| * |y| = |2|Next, we find the determinants of the coefficient matrix and the coefficient matrix with the x column replaced by the constants:D = |2 3| = (2)(-1) - (4)(3) = -2 -12 = -14|4 -1|Dx = |7 3| = (7)(-1) - (4)(3) = -7 -12 = -19|2 -1|And finally, we find Dy by replacing the y column instead:Dy = |2 7| = (2)(-2) - (4)(7) = -4 -28 = -32|4 2|Now, we can use these determinants to find the values of x and y:x = Dx / D = -19 / -14 = 1.36y = Dy / D = -32 / -14 = 2.29And there you have it! Cramer's Rule helped us solve the system of equations and find the values of x and y. Pretty cool, right?So next time you're stuck on a system of equations, remember Cramer's Rule and impress your friends with your math skills! Keep practicing and you'll be a math whiz in no time!。

泰勒公式的论文

摘要泰勒公式是数学分析中的重要组成部分，是一种非常重要的数学工具. 本文通过对泰勒公式的证明方法进行介绍，归纳整理其在求极限与导数、判定级数与广义积分的敛散性、不等式的证明、定积分的证明等方面的应用，从而进一步加深对泰勒公式的认识.关键词：泰勒公式；极限；敛散性；不等式AbstractThe Taylor formula is an important component in mathematical analysis, is also a useful mathematical tool. The paper introduce the proof methods of the Taylor formula, summarize the applications which solve the limit and derivative, judge convergence and divergence of series and generalized integral, the proof of inequality and integral by Taylor formula, thus deeply understand of Taylor formula.Key words: Taylor formula; limit; the property of convergence and divergence; inequality目录摘要 (I)Abstract (II)第1章泰勒公式的概念及其相关理论 (1)第1节泰勒公式的概念及相关理论的证明 (1)第2节常见的初等函数的麦克劳林公式及其推导过程 (5)第2章泰勒公式的应用 (7)第1节利用泰勒公式求极限 (7)第2节利用泰勒公式进行近似计算 (8)第3节利用泰勒公式证明不等式问题 (9)第4节利用泰勒公式判断级数的敛散性 (11)第5节利用泰勒公式证明根的唯一存在性 (12)第6节利用泰勒公式求函数的极值 (13)第7节利用泰勒公式求高阶导数在某些点的数值 (13)第8节利用泰勒公式求初等函数的幂级数展开式 (15)结论 (17)参考文献 (18)致谢 (19)第1章泰勒公式的概念及其相关理论第1节泰勒公式的概念及相关理论的证明定义1 若任意的一个函数)(x f （不一定是多项式函数），只要函数)(x f 在a 存在n 阶导数，总能形式的写出相应的n 次多项式n n n a x n a f a x a f a x a f a f x )(!)()(!2)()(!1)()()()(2-++-''+-'+=T （1）则称（1）为函数)(x f 在a 的n 阶泰勒多项式．定义2 若将函数)(x f 与它的n 次泰勒多项式)(x n T 的差表示为)()()(x x f x R n n T -=或)()()(x x R x f n n T +=.)(x R n 为函数)(x f 在a 的n 次泰勒余项，简称泰勒余项．定理1（泰勒定理）若函数)(x f 在a 存在n 阶导数，则)(a U x ∈∀都有])[()()(n n a x o x x f -+T =,（2）其中n n n a x n a f a x a f a x a f a f x )(!)()(!2)()(!1)()()()(2-++-''+-'+=T ；)]()[()(a x a x o x R n n →-=，即)(x R n 是比n a x )(-的高阶无穷小.其中)]()[()(a x a x o x R n n →-=称为佩亚诺余项，（2）式成为函数)(x f 在a （展开）的泰勒公式．证法由高阶无穷小的定义，只需证明0)()()(lim )()(lim=--=-→→nn a x n n a x a x x T x f a x x R ,这是0的待定型，应用1-n 次洛必达法则.证明 )()()(x x f x R n n T -=])(!)()(!2)()(!1)()([)()(2n n a x n a f a x a f a x a f a f x f -++-''+-'+-= . ])()!1()()(!1)()([)()(1)(---++-''+'-'='n n n a x n a f a x a f a f x f x R .])()!2()()(!1)()([)()(2)(---++-''+''-''=''n n n a x n a f a x a f a f x f x R .……)](!1)()([)()()1()1()1(a x a f a fx fx R n n n n n-+-=---.（对它不能再求导数了）当a x →时，显然，)(x R n ，)(x R n '，)(x R n ''，…，)()1(x R n n-，以及)()(+∈-N k a x k 都是无穷小．于是，由洛必达法则有nn a x a x x R )()(lim-→1)()(lim -→-'=n n a x a x n x R 2))(1()(lim-→--''=n na x a x n n x R = )(!)(lim )1(a x n x R n n a x -=-→⎥⎦⎤⎢⎣⎡---=--→)()()(lim !1)()1()1(a f a x a f x f n n n n a x )]()([lim !1)()(a fa f n n n ax -=→0=．定义3 当0=a 时，函数)(x f 在0处存在n 阶导数，（2）式即是)(!)0(!2)0(!1)0()0()()(n nn n x o x n f x f x f f x +++''+'+=T ，称之为麦克劳林公式．定理2（泰勒中值定理）若函数)(x f 在)(a U 存在1+n 阶导数，)(a U x ∈∀，函数)(t G 在以a 和x 为端点的闭区间I 连续，在其开区间可导，且0)(≠'t G ，则在a 和x 之间至少存在一点c ，使得+-''+-'+=2)(!2)()(!1)()()(a x a f a x a f a f x f )]()([)()(!)()(!)()1()(a G x G c x c G n c f a x n a f n n nn --'+-++（3）其中)]()([)()(!)()()1(a G x G c x c G n c f x R n n n --'=+.证法应用柯西中值定理.证明 I t ∈∀，设（将n 次泰勒多项式)(x T n 中的a 换为t ）n n t x n t f t x t f t x t f t f t F )(!)()(!2)()(!1)()()()(2-++-''+-'+= .而 +-''+-''--''+'-'='2)(!2)()(!1)()(!1)()()()(t x t f t x t f t x t f t f t f t Fn n n n t x n t f t x n t f )(!)()(!)()1(1)(-+-++-,即得n n t x n t f t F )(!)()()1(-='+.不难看到，函数)(t F 和)(t G 在闭区间I 连续，在其开区间可导，且0)(≠'t G ，满足柯西中值定理的条件，则在a 和x 之间至少存在一点c ，使得n n c x c G n c f c G c F a G x G a F x F )()(!)()()()()()()()1(-'=''=--+或)]()([)()(!)()()()1(a G x G c x c G n c f a F x F n n -⋅-'=-+.（4）已知)()(x f x F =，n n a x n a f a x a f a x a f a f a F )(!)()(!2)()(!1)()()()(2-++-''+-'+= ，将它们带入（4）中，移项，有+-''+-'+=2)(!2)()(!1)()()(a x a f a x a f a f x f)]()([)()(!)()(!)()()(a G x G c x c G n c f a x n a f n n nn --'+-+，其中)]()([)()(!)()()1(a G x G c x c G n c f x R n n n -⋅-'=+.泰勒公式中的余项)(x R n 是极为一般的，并且)(t G 是任意的，今后主要应用)(t G 的两种特殊情况．1．带有拉格朗日（Lagrange ）余项的泰勒公式如果取1)()(+-=n t x t G ，它满足定理2的条件，则有0))(1()(≠-+-='n t x n t G ，0)(=x G ，1)()(+-=n a x a G ，c 在a 与x 之间，则带有拉格朗日（Lagrange ）余项的泰勒公式为)1()1()(2)()!1()(!)()(!2)())(()()(++-++++-''+-'+=n n n a x n c f n a f a x a f a x a f a f x f ．特别的，当0=a 时，1)1()(2)!1()(!)0(!2)0()0()0()0(++++++''+'+=n n n x n c f n f x f x f f f .此时上式称之为带有拉格朗日余项麦克劳林（Maclaurin ）公式.2．带有柯西余项的泰勒公式若t x t G -=)(，它也满足定理2的条件，有01)(≠-='t G ，0)(=x G ，a x a G -=)(. 将它们带入)(x R n 之中，有)()(!)()()1(a x c x n c f x R n n n --=+，c 在a 与x 之间，称为柯西余项.特别的，当00=x 时，1)1()(2)1(!)(!)0(!2)0()0()0()0(++-+++''+'+=n n n n x n x f n f x f x f f f θθ ，)10(<<θ此时上式称之为带有柯西余项的麦克劳林（Maclaurin ）公式.第2节常见的初等函数的麦克劳林公式及其推导过程常见的五种初等函数的麦克劳林公式，在应用时较广泛，因此需简单介绍： 1．x e x f =)(.已知x n e x f =)()(，1)0()(=n f ，取拉格朗日余项，有xn n xe n x n x x x e θ)!1(!!2!1112++++++=+ ，)10(<<θ2．x x f sin )(=.已知)2sin()()(π⋅+=n x x f n ,==2sin)0()(πn f n ⎩⎨⎧+=-=.12)1(.20k n n k n n k 是奇数，，是偶数，， 0)0(=f ，1)0(='f ，0)0(=''f ，1)0(-='''f ，……，以后依次“0，1，0，1-”循环，设k n 2=，有)()!12()1(!3sin 21213n k k x o k x x x x +--++-=-- ．3．x x f cos )(=.已知)2cos()()(π⋅+=n x x f n ，==2cos)0()(πn f n ⎩⎨⎧-=-=.12)1(.20k n n k n n k 是偶数，，是奇数，，有)()!2()1(!4!21cos 12242++-+-+-=n k k x o k x x x x ． 4．)1ln()(x x f +=. 已知nn n x n x f )1()!1()1()(1)(+--=-，)!1()1()0(1)(--=-n f n n ，有 )(!)1(!3!2)1ln(132n nn x o n x x x x x +-+-+-=+- ．5．α)1()(x x f +=，其中R ∈α.已知n n x n x f -++--=αααα)1)(1()1()()( ，1)1()1)(()1()(--++--=n n x n x f αααα , )1()1()0()(+--=n f n ααα ，则有+-++=+2!2)1(!11)1(x x x αααα)(!)1()1(n n x o x n n +---+ααα .第2章泰勒公式的应用第1节利用泰勒公式求极限为了简化极限计算，有时可用某项的泰勒展开式来代替该项，使得原来函数的极限转化为类似多项式的有理式的极限，就能简洁的求出．例1 求极限xxx x 40sin cos )cos(sin lim-→．分析本题可以用无穷小量代换将x 4sin 换为4x ，再应用洛必达法则进行求解，但过程比较复杂，对于分母需要求多次导数才可得出计算结果．此时我们考虑用泰勒公式将分子在0=x 点处展开到x 的四次幂约去分母进行计算．解由等价无穷小可知道，分母为4x ，只要把x cos 和)cos(sin x 展开到4x 即可，)(2421cos 442x o x x x ++-=，)()]([241)](31[211)()]([241)](61[211)(sin 24)(sin 2)(sin 1)cos(sin 44444244233442x o x o x x o x x x o x o x x o x x x o x x x ++++--=++++--=++-=)(245211442x o x x ++-=，于是xxx x 40sin cos )cos(sin lim-→ 44424420)]}(241211[)](245211{[lim x x o x x x o x x x ++--++-=→ =4440)(61lim x x o x x +→ =])(61[lim 440xx o x +→ =61．例2 计算极限)22ln 111(lim 320xx x x x -+++→．解先作如下的变化)21ln()21ln(2121ln22ln x x x xx x --+=-+=-+. 由拉格朗日的展开公式得，)]()2(31)2(212[)]()2(31)2(212[22ln332332x o xx x x o x x x x x ++++++-=-+ )(12133x o x x ++=，于是xx x x -+-+22ln 11132 33332)()121(111xx o x x x x ++-+=33)(1211xx o +-=即333333232)(1211)()121(11122ln111xx o x x o x x x x x x x x +-=++-+=-+-+ 1211])(1211[lim )22ln 111(lim 330320=+-=-+++→→xx o x x x x x x . 第2节利用泰勒公式进行近似计算当要求的算式不能得出它的准确数值时，即只能求出它的近似值，这时泰勒公式是解决这种问题的最好办法．例1 估计下列近似公式的值并计算：数e 精确到910-．解 132)!1(!3!21++++++++=n n xx n e n x x x x e θ！，)10(<<θ当1=x 时，有)!1(!1!31!2111+++++++=n e n e θ，则有)!1(3)!1()!1!31!2111(+<+=+++++-n n e n e θ ，欲使910)!1(3-<+n ，则有12≥n ，取12=n ，则有910!133-<，故 718281828.2!121!31!2111≈+++++≈ e ．例2 估计下列近似公式的值并计算：11lg 准确到510-．解已知)1011ln(10ln 11)1011lg(1)110lg(11lg ++=++=+=，由于 11132)1)(1()1()1(32)1ln(++-++-+-+++-=+n n nn x n x n x x x x x θ ，)10(<<θ 取101=x ，则 ])101(1)1()101(31)101(21101[10ln 1)1011ln(10ln 1132n n n ⨯-++⨯+⨯--+- 511110)1(210)101)(1()101(---++<+<++=n n n n n θ，得n n n -+-=>+4)1(51010)1(2，故取4=n ，故04139.1]4000130012001101[10ln 1111lg ≈++++≈．第3节利用泰勒公式证明不等式问题当题目中同时涉及)(x f ，)(x f '以及)(x f ''相关性质或不等式时，除了联系函数的单调性、凹凸性解题外，利用泰勒公式展开式也是一种有效的方法．例1 若)(x f 在]1,0[上是二阶可导的，并且有0)1()0(==f f ，当10≤≤x 时，)(x f 的最小值为1-，则)1,0(∈∃ξ，使得8)(≥''ξf ．证明设0x 是)(x f 在]1,0[上的最小点，1)(0-=x f ，则在0x 点对)(x f 作泰勒公式展开（一阶），因为0)(0='x f ，则有20020000)(!2)()()(!2)())(()()(x x f x f x x f x x x f x f x f -''+=-''+-'+=ξξ，将0=x 和1=x 分别带入得201)(2110x f ξ''+-=，)0(01x <<ξ； 202)1)((2110x f -''+-=ξ，)1(20<<ξx ，即212)(x f =''ξ，202)1(2)(x f -=''ξ，此二者中必有一个是大于等于8的．事实上，当210≤x 时，8)(1≥''ξf ；当210>x 时，8)(2≥''ξf ；当R x ∈0时，16)()(21≥''+''ξξf f ．注意此问题中最值点在内部取得，一定是极值点并且可导，故为稳定点．例2 设)(x f 在],[b a 上二阶可导，且0)()(='='b f a f ，则),(b a c ∈∃，使得)()()(4)(2a fb f a bc f --≥''.证明将)(x f 分别在a x =，b x =作泰勒公式展开，利用0)()(='='b f a f ，得21)(2)()()(a x f a f x f -''+=ξ， 22)(2)()()(b x f b f x f -''+=ξ, 令2ba x +=代入上两式并相减得， 0)]()([)2(21)()(212=''-''-+-ξξf f a b b f a f ，即())]()([21)]()([4212ξξf f b f a f a b ''-''=--.只要证明),(b a c ∈∃，使得)()(21)(21ξξf f c f ''-''≥'' 成立即可.令)(max{)(1ξf c f ''=''，})(2ξf ''，即可以得到)(4)()()(2c f a b b f a f ''-≤-.也就是)()()(4)(2a fb f a bc f --≥''．第4节利用泰勒公式判断级数的敛散性判断级数敛散性的问题，一个有效工具是带佩亚诺余项的泰勒公式．先将通项na 适当展开，再判断敛散性，对于证明问题，泰勒公式也是经常用的．例1 研究级数∑∞=-+2])1(1ln[n p nn 的收敛性.分析此级数不是正项级数，且通项趋近于0．解首先，该级数是一个交错级数，可先考虑何时绝对收敛. 因pn n a 1→，故当1>p 时，原级数绝对收敛；当1≤p 时，用泰勒公式展开：)(21)1ln(22x o x x x +-=+，)0(→x )1(21)1(])1(1ln[22p p p p p n no n n n +--=-+，)(∞→n可见，当210≤<p 时，原级数发散；当121≤<p 时，原级数条件收敛．例2 设函数)(x f 在0=x 处二阶可导，且0)(lim0=→x x f x ，证明级数)1(1n f n n ∑∞=是收敛的．证明应用泰勒公式，由条件有0)(=x f ，以及0)(lim )0()(lim)0(00==-='→→xx f x f x f f x x ，于是，有)()0(21)()0(!21)0()0()(2222x o x f x o x f x f f x f +''=+''+'+=，)0(→x 从而得出)0(21])()0(21[lim )(lim 22020f x x o f x x f x x ''=+''=→→，所以有)0(211)1(lim 23f nnf n n ''=∞→，由∑∞=1321n n收敛，则)1(1n f n n ∑∞=也是收敛的．第5节利用泰勒公式证明根的唯一存在性例1 设函数)(x f 在),[+∞a 上是二阶可导，并且有0)(>a f 和0)(<'a f 成立，对),(+∞∈a x ，0)(≤''x f ，证明0)(=x f 在),(+∞a 内存在唯一实根．分析这里)(x f 是抽象函数，直接讨论0)(=x f 的根有困难，由题设)(x f 在),[+∞a 上二阶可导，并且0)(>a f ，0)(<'a f ，可考虑)(x f 将在a 点展开一阶泰勒公式，然后设法应用介值定理证明．证明因为0)(≤''x f ，所以)(x f '单调减少，又因为0)(<'a f ，因此a x >时，0)()(<'<'a f x f ，故)(x f 在),(+∞a 上严格单调减少．在a 点展开一阶泰勒公式有2)(!2)())(()()(a x f a x a f a f x f -''+-'+=ξ，)(x a <<ξ由题设0)(<'a f ，0)(≤'ξf ，于是有-∞=∞→)(lim x f x ，从而必存在a b >，使得0)(<b f ，又因为0)(>a f ，在],[b a 上应用连续函数的介值定理，存在),(0b a x ∈，使0)(0=x f ，由)(x f 的严格单调性知0x 唯一，因此方程0)(=x f 在),(+∞a 内存在唯一实根．第6节利用泰勒公式求函数的极值本节应用函数的泰勒公式，对函数的极值问题的第二充要条件作了进一步讨论. 例1（极值的第二充分条件）设函数f 在0x 的某邻域),(0δx U 内一阶可导，在=x 0x 处二阶可导，且0)(0='x f ，0)(0≠''x f ，则（i ）若0)(0<''x f ，则f 在0x 取得极大值；（ii ）若0)(0>''x f ，则f 在0x 取得极小值．分析连续函数及其高阶导数的桥梁是泰勒公式，因此，已知函数)(x f 的高阶导数的性质讨论)(x f 的性质，要应用泰勒公式．证明由已知，可得f 在0x 处的二阶泰勒公式])[()(!2)()(!1)()()(20200000x x o x x x f x x x f x f x f -+-''+-'+=．由于0)(0='x f ，因此，)])((2)([)()(2000x x o x f x f x f -+''=-，（1）又因0)(0≠''x f ，故存在正数δδ≤'.当),(0δ'∈x U x 时，)(210x f ''和])[()(21200x x o x f -+''同号．所以，当0)(0<''x f 时, （1）式取负值，对任意),(0δ'∈x U x ，有0)()(0<-x f x f ,即f 在0x 取得极大值. 同理可得，当0)(0>''x f ，可得f 在0x 取得极小值．第7节利用泰勒公式求高阶导数在某些点的数值对于函数)(x f 求一阶导数，较容易得出，而高阶导数则不容易求得，利用泰勒公式展开，可以轻易地求出)(x f n ，常用于处理与高阶导数相关的函数的属性研究．例1 求函数x e x x f 2)(=在1=x 处的高阶导数)1()100(f ．解设1+=u x ，则e e u e u u g xf u u ⋅+=+==+2)1(2)1()1()()(，)0()1()()(n n g f =，u e 在0=u 的泰勒公式为)(!100!99!9811001009998u o u u u u e u++++++= ，从而)](!100!99!981)[12()(10010099982u o u u u u u u e u g ++++++++= ，而)(u g 中的泰勒展开式中含100u的项应为100)100(!100)0(u g ，从)(u g 的展开式知100u 的项为100)!1001!992!981(u e ++，因此)!1001!992!981(!100)0()100(++=e g ， 10101)0()100(⋅=e g ， e g f 10101)0()1()100()100(==．例2 设)1ln()(2x x x f +=，求)0()(n f . )3(>n 解由)1ln(x +的麦克劳林，)()1(32)1ln(132n nn x o nx x x x x +-+++-=+- ，可得)]()1(32[)(1322n nn x o nx x x x x x f +-+++-=- )(2)1(2343n nn x o n x x x +--++-=- ，)0(→x 所以2)1(!)0(3)(--=-n n fn n ．第8节利用泰勒公式求初等函数的幂级数展开式利用某些已知函数的幂级数展开式（特别五个基本初等函数x e ，x sin ，x cos ，)1ln(x +和α)1(x +的幂级数展开式）通过他们的变换，四则运算，符合运算，逐渐求导或者逐渐积分等手段导出所求函数的幂级数展开式，这种间接的方法是最常用的．例1 将下列函数展开为x 的幂级数，并确定敛散性：（1）x x f 3sin )(=；（2）)1ln()(2x x x f ++=.解（1）利用基本展开式∑∞=++-=012)!12()1(sin n n n n x x ，R x ∈就有x x x 3sin 41sin 43sin 3-=∑∑∞=+∞=++--+-=012012)!12()3()1(41)!12()1(43n n n n n n n x n x 1220)31()!12()1(43+∞=-+-=∑n n n nx n . )(R x ∈（2）()211x x f +='212)1(x +=nnn x n n 21!)!2(!)!12()1(1--+=∑∞=，其中)1,1(-∈x .对于)1,1(-∈∀x ，逐渐积分上式就有dt t f f x f x⎰'=-0)()0()(⎰∑--+=∞=x nnn dt t n n x 021!)!2(!)!12()1(，而0)0(=f ，由此即得12!)!2(!)!12()1()(121+--+=-∞=∑n x n n x x f n n n，其中)1,1(-∈x ，故有12!)!2(!)!12()1()1ln(1212+--+=+++∞=∑n x n n x x x n nn ，其中)1,1(-∈x ．结论泰勒公式是数学分析中非常重要的内容，本文主要介绍了泰勒公式以及它在求极限、判断函数极值、求高阶导数在某些点的数值、判断广义积分收敛性、近似计算、不等式证明等方面的应用．通过本文的研究使我们对泰勒公式有了更深一层的理解，对于怎样应用泰勒公式解题有了更深一层次的认识．只要在解题训练中注意分析，研究题设条件及其形式特点，并把握处理规则，就能比较好地掌握利用泰勒公式解题的技巧．参考文献[1] 刘玉琏，傅沛仁，数学分析讲义[M]，北京：高等教育出版社，(2003)：230-240[2] 裴礼文，数学分析中的典型问题[M]，北京：高教教育出版社，(1993)：98-104[3] 孙清华，孙昊，数学分析内容、方法与技巧[M]，武汉：华中科技大学出版社，(2003)：165-174[4] 朱永生，刘莉，基于泰勒公式应用的几个问题[J]，长春师范学院学报，2007，8(12)：6-9[5] 华东师范大学数学系，数学分析（第二版）[M]，北京：人民教育出版社，(2001)：134-140[6] 王三宝，泰勒公式的应用举例[J]，高等函授学报，2008，17(9)：15-18[7] 冯平，泰勒公式在求高等数学问题中的应用[J]，新疆职业大学学报，2003，13(5)：8-11[8] 唐清干，泰勒公式在判断级数及积分敛散性中的应用[J]，桂林电子工业大学学报，2002，9(13)：16-19致谢在本文的撰写过程中，孙威老师作为我的指导教师，她治学严谨，学识渊博，视野广阔，为我们营造了一种良好的学术氛围．置身其间，耳濡目染，潜移默化，使我不仅接受了全新的思想观念，树立了明确的学术目标，领会了基本的思考方式，掌握了通用的研究方法，而且还让我明白了许多待人接物与为人处世的道理．孙老师的严以律己、宽以待人的崇高风范，朴实无华、平易近人的人格魅力，与无微不至、感人至深的人文关怀，令人如沐春风，倍感温馨．正是由于孙威老师您在百忙之中多次审阅全文，对细节进行修改，并为本文的撰写提供了许多中肯而且宝贵的意见，本文才得以成型．在此特向孙威老师您致以衷心的谢意！同时向您无可挑剔的敬业精神、严谨认真的治学态度、深厚的专业修养和平易近人的待人方式表示深深的敬意！。

Taylorism Scientific management

Frederick Winslow Taylor(1856 - 1915)Principles of Scientific ManagementYonatan ReshefFaculty of BusinessUniversity of AlbertaEdmonton, AlbertaT6G 2R6 CANADAIn the past the man hasbeen first; in the futurethe system must be first(p. 7).Principles of Scientific ManagementTaylor's focus of attention was plant management. He argued that labor problems (waste, low productivity, high turnover, soldiering, and the adversarial relationship between labor and management) arose from defective organization and improper methods of production in the workplace. Production, he contended, was governed by universal and natural laws that were independent of human judgment. The object of scientific management was to discover these laws and apply the "one best way" to basic managerial functions such as selection, promotion, compensation, training, and production.Taylor advocated using time and motion studies to determine the most efficient method for performing each work task, a piece-rate system of compensation to maximize employee work effort, and the selection and training of employees based on a thorough investigation of their personalities and skills.Taylor also promoted changes in the organizational structure of the firm, such as replacing the single omnipotent foreman in charge of all aspects of production and personnel management in a given department with several foremen, each of whom would be trained in the knowledge and skills of a specific functional activity (e.g., productivity, machine repair, quality assurance).The gist of the problem. Taylor believed that under the traditional management each worker was to become more skilled in his own trade than it was possible for any one in management to be, and that, therefore, the details of how the work should best be done must be left to him (p. 63). Unfortunately, four problems existed that rendered this situation untenable for society: First, management used rules of thumb to decide on what constitutes a fair day of work (p. 22), work procedures, personnel matters, etc. Second, being self-centered, workers abused managers' trust in two ways (pp. 17, 19, 20, 50). According to Taylor, "the natural instinct and tendency of men is to take it easy, which may be called natural soldiering" (p. 19). "To ward off a rate cut was one reason to soldier. To thumb his nose at the boss, protest wages deemed too low, or husband shop work otherwise apt to run out were others" (Kanigel, 1997: 164). Third, even those employees who wanted to perform to the best of their capabilities were forced to conform to an informal, group-made norm that was always lower than their optimal performance (p. 13). This Taylor labeled "systematic soldiering," where the whole shop conspired to restrict production (p. 20). Fourth, any man phlegmatic enough to do manual work was too stupid to develop the best way, the 'scientific way' of doing a job, hence the vast amount of waste in the workplace (p. 63).An important brick in the intellectual edifice of Taylor's scientific management is the "rabble hypothesis:"1. Natural society consists of a horde of unorganized individuals;2. Every individual acts in a manner calculated to secure his self-interest(especially in times of economic scarcity). In itself this may not bedetrimental to an organization. However, when viewed in the contextTaylor portrayed of crafty workers who tried to squeeze more money forless effort, it is clear why self-interested workers are a menace.3. Every individual thinks logically, to the best of his ability, in the service ofthis aim. This is why the best incentive to induce workers to work harder is money.What then should management do with employees? (See pp. 36, 140):1. Science, not rule of thumb2. Harmony (playing by the rules of the game designed by management), notdiscord (p. 15)3. Cooperation, not individualism (p. 36)4. Maximum output, in place of restricted output (soldiering)5. The development of each man to his greatest physical capability (pp. 39, 55,57, 59)We begin to see that Scientific management has a strong HRM component.Taylor strongly believed that the successful manager was a manager who controlled every aspect of the production process. To achieve this, managers should:q Centralized planning. Uncouple planning and execution -- i.e. workers only execute what managers plan (pp. 37-8). This is probably the most well-known principle of Scientific management. At a lecture he gave in 1906,Taylor explained:In our scheme, we do not ask for the initiative of our men. We do not want any initiative.All we want of them is to obey the orders we give them, do what we say, and do it quick(Kanigel: 169).q Systematic analysis of each distinct operation. Create an elaborate set of rules to regulate every aspect of worker behavior at the workplace (pp. 22, 36).q Detailed instruction and supervision. Breakdown every job to its minuscule components so that no one worker would posses any knowledge which might be unique enough to put this worker in a position of power vis-à-vis management (see p. 36 - the 4 rules of Scientific Management).q Uncouple 'direct' and 'indirect' labor. All preparation and servicing tasks are stripped away to be performed by unskilled workers as far as possible. Thus, he created two classes of workers -- laborers and maintenance workers.q Recruit the most stupid men they can lay their hand on (p. 40-1, 43-6, 59,62, 137).q Functional management/foremanship (123-5; 129). Few tend to pay attention to this point. Taylor advocated the division of the function of the shop-floor inspector into four functions (setting-up boss, speed boss, quality inspector, and repair boss), and placing them under the control of the planning department. Thus foremen like workers became subject to the rule of clerks. In this way, Taylor tackled a major problem faced by management of large, complex organizations, that is, the integration of conflicting instructions. In the process, he was laying the ground for the modern division between 'staff' and 'line' functions.q Wage payments. Wage systems should be carefully designed to induce each worker to follow the detailed instructions. Taylor preferred a piece-rate system of compensation. Frequently, piece-rate systems are associated with bonuses for extra efforts. Characteristically, these systems tend to evolve upward. Continuously and consistently, what used to be an extra effort worthy of a bonus, becomes the new performance norm. And vying to gain or regain competitive advantage, managers are driven to establish a higher norm for their employees.These principles constitute a dynamic of deskilling. Importantly, the drive for deskilling was initiated not by Taylor but by larger factories, and more specialized machines.HOW TO READ TAYLORq SM is a philosophy and a set of principles an organization uses to make themost of workers' physical capabilities (pp. 129-131). Therefore,q like quality improvement gurus who emerged years after Taylor's death in 1915, Taylor believed that successful implementation of Scientific Management required a "thought revolution in management." In other words, implementation of the principles of Scientific Management without a supportive philosophy (culture) is a recipe for failure (pp. 130-31):When, however, the elements of this mechanism, such as time study,functional foremanship, etc., are used without being accompanied bythe true philosophy of management, the results are in many casesdisastrous... the really great problem involved in a change from themanagement of "initiative and incentive" to Scientific Managementconsists in a complete revolution in the mental attitude and the habitsof all those engaged in the management, as well as the workmen...This change in the mental attitude of the workman imperativelydemands time... The writer has over and over again warned againstthose who contemplated making this change that it was a matter, evenin a simple establishment, of from two to three years, and that in somecases it requires from four to five years.Management of initiative and incentive refers to a system whereby managers would have to provide workers with special incentives to obtain their best effort, or initiative. The reason being, workers believed "it to be directly against their interests to give their employers their best initiative" (p. 33).q SM creates an organization that strives for maximum interchangeability of personnel (with minimum training) to reduce its dependence on the availability, ability, or motivation of individuals. Taylorism represents a form of organization devoid of any notion of a career-structure for the majority. Thus, Taylorism can be defined as the bureaucratization of the structure of control, but not the employment relationship (nounions/CB/labor law) or career development.q Taylor's recognition of the problems of cooperation, gaining consent and legitimacy and shared understandings, as well as the meaning of work should not be disregarded, see:r Harmonious society (pp. 10, 85);r Prosperous society and thriving individuals (pp. 10, 15, 29, 55, 125-128);r Management-cum-instructors (p. 26).q Ultimately, Taylor evoked the authority of science to legitimize his ideas. With science as a foundation, Taylor hoped to improve efficiency and usher in an era of peaceful coexistence between capital, management and labor based on an objective understanding of what was best for all three groups. However, not everyone interested in SM had the same goal. Clearly, capital had much more to gain from the shift to SM than labor in terms of control and profits.q Taylorism does allow for teamwork, yet it should be as regulated as possible. Thus, teams should be created only with management permission. No more than 4 people per team are allowed, and the team should disband within one week of its creation (pp. 72-3).q Whenever Taylorism was introduced, it was filtered through and shaped bynational socio-economic contexts. In Japan, for example, employers relied on group discussions and collective problem-solving through quality circles (QCs). The adoption of motion study was important in the development of pay incentive systems and safety programs in modern Japanese industry.This led to the adoption of aptitude-testing of workers by the NationalRailways, which was then widely copied by other enterprises (early 1920s and 1930s). Importantly, the intention was not, as in the USA, to simplify work methods and thus to raise the efficiency of untrained labor. On thecontrary, the Japanese managers wanted to build on the existing skills oftheir workforce in the railways, to encourage them to stay with them fortheir entire careers. In the final analysis, Japan absorbed and adaptedTaylorism in an "organization-oriented," rather than a "market-oriented,"context. In other words, the ways American managers used SM to adaptproduction to market whims was very different than their Japanesecounterparts'.Elements of FordismTaylorism provided the technological and intellectual foundations for Fordism -- a system whereby giant factories employ thousands of mainly unskilled workers and specialized machines to turn out huge quantities of a single product (emphasis should be put on interchangeability of parts and ease of assembly).1. Production system - rested on work that was organized hierarchically, on acontinuous flow technology, on high-volume production of standardizedconsumer goods, targeted standardized and uniform markets, acknowledged working class consumption, displaced a division of labor more centered on craft production, created unskilled production jobs, emphasized high level of specialization, demanded no learning experience and, therefore, offered little on-the-job training -- The implementation of Taylorism in relations to work processes.2. Personnel Departments - maintained industrial peace and ensured that the labor process operated effectively and smoothly. Importantly, personnel departments were removed from the key corporate strategy-making within the business. Personnel managers were given no initiating role; they were regarded as being basically reactive, responding to the demands made by trade unions. No strategic HRM at that point in time.3. Collective Bargaining - meshed with Fordism as a mechanism insuring that consumption power was tied to productivity growth.4. Homogeneous Customers - large numbers of potential customers have essentially identical and well-defined wants for a long list of products.A combination of reduced profit levels (inability to sustain increased wages together with falling productivity), increased international competition and fragmented consumption patterns brought an end to Fordism in North America.。

数学文献综述

关于泰勒公式应用的文献综述f(x)=f(x0)+f'(x0)*(x-x0)+f''(x0)/2!*(x-x0)^2+...+f(n)(x0)/n!*(x-x0)^n （泰勒公式，最后一项中n表示n阶导数）f(x)=f(0)+f'(0)*x+f''(x)/2!*x^2+...+f(n)(0)/n!*x^n （麦克劳林公式公式，最后一项中n表示n阶导数）泰勒中值定理：若函数f(x)在开区间（a，b）有直到n+1阶的导数，则当函数在此区间内时，可以展开为一个关于（x-x.)多项式和一个余项的和：f(x)=f(x.)+f'(x.)(x-x.)+f''(x.)/2!•(x-x.)^2,+f'''(x.)/3!•(x-x.)^3+……+f(n)(x.)/n!•(x-x.)^ n+Rn其中Rn=f(n+1)(ξ)/(n+1)!•(x-x.)^(n+1),这里ξ在x和x.之间，该余项称为拉格朗日型的余项。

（注：f(n)(x.)是f(x.)的n阶导数，不是f(n)与x.的相乘。

）证明：我们知道f(x)=f(x.)+f'(x.)(x-x.)+α（根据拉格朗日中值定理导出的有限增量定理有limΔx→0 f(x.+Δx)-f(x.)=f'(x.)Δx），其中误差α是在limΔx→0 即limx→x.的前提下才趋向于0，所以在近似计算中往往不够精确；于是我们需要一个能够足够精确的且能估计出误差的多项式：P(x)=A0+A1(x-x.)+A2(x-x.)^2+……+An(x-x.)^n来近似地表示函数f(x)且要写出其误差f(x)-P(x)的具体表达式。

设函数P(x)满足P(x.)=f(x.),P'(x.)=f'(x.), P''(x.)=f''(x.),……,P(n)(x.)=f(n)(x.),于是可以依次求出A0、A1、A2、……、An。

泰勒公式的应用英语文献

泰勒公式的应用英语文献Taylor's Theorem is a powerful mathematical tool that can be applied in many different areas. It enables us to solve various problems related to integral calculus, differential equations, and optimization. In particular, Taylor's theorem can be used to approximate functions, derivatives and integrals and to solve certain optimization problems.Taylor's theorem is a classical result in mathematics which states that any function can be written as the sum of its Taylor series. This theorem is most useful when applied to functions which are smooth and continuous, that is, the derivatives of the function are well-defined. According to the theorem, given an arbitrary point in a domain, it is possible to interpolate the function around this point by an appropriate Taylor polynomial.One of the most important applications of Taylor's theorem is in approximating integrals. Approximatingintegrals using Taylor's theorem requires a specific kind of convergence, which guarantees that the approximation is as accurate as possible. This is especially useful for computing integrals that have complex and frequently changing boundaries. By using Taylor's theorem, the integral can be approximated with the corresponding Taylor series, which can be quickly and accurately computed. This can reduce computation time significantly.Taylor's theorem can also be used to solve solvingcertain types of optimization problems. One of the most common uses of Taylor's theorem in optimization is to find the minimum or maximum of a function without having tocalculate the derivatives. By approximating the function with a Taylor polynomial, it is possible to find the location at which the minimum or maximum occurs. This is especially useful when the derivatives of the function are difficult to calculate.Finally, Taylor's theorem can also be used to solve certain types of differential equations. The theorem can be used to approximate the solutions to these equations, which makes it easier to find the solutions. Applying Taylor's theorem in this way can help to reduce the complexity of problems that involve solving differential equations.Overall, Taylor's theorem is a powerful tool that can be used in many different areas of mathematics. Its applications range from approximating integrals to solving optimization problems to finding solutions to differential equations. The theorem provides an efficient and accurate means of approximating functions, derivatives, and integrals, making it an invaluable tool in the world of mathematics.。

泰勒原理英文

泰勒原理英文The Taylor PrincipleThe Taylor principle, named after the renowned economist John B. Taylor, is a fundamental concept in monetary policy that has significantly influenced the way central banks around the world conduct their policies. This principle serves as a guiding framework for central banks to determine the appropriate adjustments to their key interest rates in response to changes in economic conditions, particularly inflation and output.At the heart of the Taylor principle lies the idea that central banks should adjust their policy interest rates more than proportionately to changes in inflation. This means that if inflation rises, the central bank should increase its interest rate by a larger amount, effectively raising the real interest rate and dampening inflationary pressures. Conversely, if inflation falls, the central bank should decrease its interest rate by a smaller amount, again increasing the real interest rate and promoting price stability.The rationale behind the Taylor principle is rooted in the understanding that monetary policy can have a significant impact onthe economy, particularly on inflation and output. By adjusting interest rates in a way that is responsive to changes in inflation, central banks can steer the economy towards their desired objectives of price stability and sustainable economic growth.One of the key insights of the Taylor principle is that it recognizes the inherent tension between inflation and output stabilization. In the short run, there may be a trade-off between these two goals, as policies aimed at controlling inflation may have adverse effects on output and employment. However, the Taylor principle suggests that by prioritizing the control of inflation, central banks can ultimately achieve a more stable and sustainable economic environment, which in turn supports long-term growth and employment.The implementation of the Taylor principle has been widely adopted by central banks around the world, with many of them incorporating it into their monetary policy frameworks. The Bank of England, the European Central Bank, and the Federal Reserve, among others, have all used the Taylor principle as a guiding principle in their policy decisions.The practical application of the Taylor principle involves the central bank's use of a simple rule, known as the Taylor rule, to determine the appropriate level of the policy interest rate. The Taylor rule typically takes into account the current level of inflation, thedeviation of output from its potential level (the output gap), and a target inflation rate. By following this rule, central banks can adjust their interest rates in a systematic and transparent manner, providing clear signals to the public and financial markets about their policy intentions.It is important to note that the Taylor principle is not a rigid formula, but rather a general guideline that central banks can use to inform their policy decisions. In practice, central banks often need to balance a variety of factors, such as financial stability, exchange rate considerations, and global economic conditions, in addition to inflation and output stabilization. As a result, the actual implementation of the Taylor principle may involve some degree of discretion and flexibility on the part of the central bank.Despite its widespread adoption, the Taylor principle has also been the subject of ongoing academic and policy debates. Some economists have argued that the principle may not be universally applicable, as the appropriate policy response may depend on the specific economic conditions and the structure of the financial system in a given country or region. Additionally, there are discussions about the optimal way to measure inflation and output gaps, as well as the potential for the Taylor principle to contribute to asset price bubbles or financial instability.Nonetheless, the Taylor principle remains a cornerstone of modern monetary policy, providing a robust framework for central banks to navigate the complex and ever-changing economic landscape. By adhering to the principle and adjusting interest rates in a manner that is responsive to changes in inflation, central banks can play a crucial role in fostering price stability, supporting sustainable economic growth, and promoting the overall well-being of the societies they serve.。

金融泰勒规则

金融泰勒规则1. 什么是金融泰勒规则？金融泰勒规则（Taylor’s Rule）是一种经济学模型，用于指导中央银行制定货币政策。

它是由美国经济学家约翰·泰勒（John B. Taylor）于1993年提出的，因此得名。

金融泰勒规则的核心思想是，通过调整利率来实现经济稳定。

根据该规则，中央银行应根据通胀率和产出缺口的变化来确定利率水平。

通胀率指物价水平的上涨速度，产出缺口则是实际GDP与潜在GDP之间的差距。

2. 金融泰勒规则的公式金融泰勒规则的公式如下所示：其中，i 是利率，π 是通胀率，π* 是中央银行设定的目标通胀率，y 是产出缺口，y* 是潜在产出，r 是实际利率的均衡水平。

根据金融泰勒规则，当通胀率高于目标通胀率，或者产出缺口大于潜在产出时，中央银行应该提高利率；反之，当通胀率低于目标通胀率，或者产出缺口小于潜在产出时，中央银行应该降低利率。

3. 金融泰勒规则的作用金融泰勒规则在货币政策制定中起到了重要的指导作用。

它帮助中央银行确定利率水平，以实现经济的稳定增长和低通胀目标。

通过根据通胀率和产出缺口来调整利率，金融泰勒规则能够在经济繁荣时期抑制通胀压力，避免经济过热；在经济衰退时期刺激经济活动，促进就业和增长。

此外，金融泰勒规则还有助于提高货币政策的透明度和可预测性。

中央银行通过公开表达其利率决策的依据，增加了市场参与者对货币政策的理解和预期，有助于降低市场的不确定性。

4. 金融泰勒规则的局限性金融泰勒规则虽然在货币政策制定中有一定的指导意义，但也存在一些局限性。

首先，金融泰勒规则的适用性有限。

不同国家和地区的经济状况和政策目标各不相同，因此金融泰勒规则可能无法完全适用于所有情况。

中央银行在制定货币政策时需要考虑更多的因素，如金融稳定、汇率变动等。

其次，金融泰勒规则忽视了货币政策传导机制的复杂性。

利率的调整对经济的影响需要一定的时间，而且可能存在滞后效应。

此外，金融泰勒规则没有考虑到其他货币政策工具的使用，如量化宽松政策或限制性政策。

taylor francis参考文献格式

taylor francis参考文献格式一、概述taylor francis参考文献格式是一种标准的文献引用格式，用于在学术论文、报告、会议论文等文档中引用他人研究成果时规范地标注参考文献。

该格式适用于大多数学术出版物，是学术交流中必不可少的一部分。

二、格式要求1. 引用文献必须为正式出版物，如期刊论文、会议论文、书籍、报告等。

2. 引用文献的作者姓名按照贡献的顺序列出，多名作者时用逗号分隔。

3. 引用文献的出版物名称要写全称，包括出版物类型（如期刊名、报告名、书籍名等）。

4. 引用数字要使用阿拉伯数字，标明起止页。

5. 文献引用序号使用方括号标注在引用内容的右上角，格式为“[序号]”。

三、示例假设我们引用taylor francis在某期刊上发表的一篇论文，参考文献格式如下：[1] Taylor, Francis. "Article Title." Journal Name, Vol. XXXX, No. X, pp. 1-10.具体示例如下：在本文的研究方法部分，我们借鉴了Taylor, Francis在《心理学研究》上发表的“实验设计原理”一文中的一些思路。

具体来说，我们采用了该文中提出的随机分组设计，取得了良好的实验效果。

[1]四、注意事项1. 不同出版物的参考文献格式可能会有所不同，具体要求请参考相应出版物的规范。

2. 如果引用多个作者或编者的著作，需要按照贡献的顺序列出，并用逗号分隔。

3. 引用网络资源时，需要使用规范的URL地址，并在参考文献中注明来源网站。

综上所述，《taylor francis参考文献格式》是学术交流中必不可少的一部分，它为规范地标注参考文献提供了标准。

希望以上内容能帮助大家更好地理解和应用该格式。

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2. Literature ReviewThere are numerous types of monetary policy rules have been discussed in economic literatures. In 1993, Taylor rule has been firstly put forward by Stanford University professor John.B.Taylor in USA, which aroused wide concern of both scholars and policians. According to Taylor rule, the central bank should make monetary policy to adjust nominal interest rate in short term based on the change of inflation rate and output gap for stabilizing real equilibrium interest rate. For example, when inflation rate exceeds target of inflation rate, the real interest rate would deviate from the real equlibrium interest rate, hence central bank should adjust norminal interest rate to keep them consistent. In recent years, Taylor rule has been widely experimented with in macroeconomic models, and examined or modificated in literatures. Taylor rule is simple and flexibal to guide American economy, and recieves extensive attention in academia. Besides, it has became a benchmark theory of monetary policy in Federal Reserve, Bank of England, European Central Bank and Bank of Canada soon. For these reason, there were extensive literatures about Taylor rule in recent decades, some proposed problems and others did a lot modification and extension, which formed a series of function that analyze the relationship between interest rate amd inflation rate and output gap (alleged Taylor-type rules). While the reaserches on this issue in China mainly focus on whether it is appropriate to adopt the Taylor rule in China’s monetary policy. The most Chinese scholars affirm that the Taylor rule was applicable for China’s monetary policy. However, some of them bring forward questions about its application, for example, Bian (2006) claimes that the Taylor tule was not stable and could not be adopted in Chinese monetary policy. Meanwhile, many scholars attempted to adopt different reaserch methods, such as co-integration analysis, Generalized Method of Moments (GMM) and time-varying parameter model. Specifically, this section will give a brief review some literatures in different respects.2.1 Different methodologiesMultitudinous literatures have estimated Taylor rule by different research methods and variable selection. For example, Taylor (1993) measured inflation as the percent change in the price deflator for GDP over the previous four quarters in the original specification of Taylor rule. Subsequently, different price indexes have beenexperimented. Kozicki (1999) said that if the rule recommendation was different when inflation was measured by the core consumer price index (CPI) or the chain price index for GDP, then the rule may become not that useful. In her research, there were four alternative inflation measures had been considerd into the estimation: CPI inflation, core CPI inflation, GPD price inflation and expected inflation. Kozicki estimated the Taylor rule to obtain rule recommendations, and the results shown that the recommendations were significant different in different inflation measures. Taylor (1993) mesured potential output by fitting a time trend to real output. Subsequently, other researchers chose different methods including regressing real output on segmented linear trends and quadratic trends, HP filters, and more structural approaches. The results stated that different measure methods work out different recommendations. Kozicki (1999) found that interest rate recommendation values obtained by alternative measure methods of potential output range from 0.9 percentage points to 2.4 percentage points. Zhao and Gao (2004) proposed a robust interest rate rule for China’s interest rate liberation by Levin, Wieland and Williams rule ( LWW rule, 2001), which was closer to China’s situation where the exchange rate influences on the long-term inflation target. Using the model of Ball (1999), they built a dynamic quarterly inflation rate model with the data period from 1993 to 2002, which possessed good statistical and econometric properties, and demonstrate forecasting precision. Finally, they said that the LWW rule may be more robust in emerging market economies and closer to China’s situation than the Taylor rule. Bian (2006) employed GMM and co-intergration test to finish the empirical analyses about the application of Taylor rule in China’s monetary policy. The reaserch results shown that both of the methods could prove that Taylor rule was able to discribe the trend of China inter-bank offered rates (CHIBOR) well. They obtained the reaction coeficienct of inflation gap was between 0.4 and 0.5, and the output gap was between 0.2014 and 0.4958. However, the results also indicated that although it could discribe the trend of CHIBOR, Taylor rule was not stable when applicated in China’s monetary policy and not suitable in long term using.2.2 Interest rate smoothingThe central banks always prefer to change short-term interest rates in sequences of several small movements in the same direction and change its direction infrequently, which is so-called interest rate smoothing. Clardia, Gali and Gertler (1998) pointedout that interest rate smoothing was able to help to solve two big problems: fluctuatoin in capital market and reduction of public trust to monetary policy that caused by dramatic adjustment in short term. In general, the Taylor rule is commonly modified with the introduction of interest rate smoothing by using a lagged interest rate term. Based on the Taylor rule reaction function, Sack and Wieland (1999) obtained that the reaction coefficient of lagged interest rate was 0.63, which shown that the Federal Reserve did not adjust the interest rate frenquently. Levin et al (1999) proposed that the optimal behavior of central bank could be explained in Taylor rule model with the introduction of interest rate smoothing. Sack and Wieland (1999) argued that interest rate smoothing may be the optimal behaviour when a central bank was aiming to stabilize the inflation and output. King (2000) also insisted to introduce the lagged interest rate into Taylor rule and the results shown that the interest rate changes smoothly. Orphanides (2001) compared the current-time data and lagged data, and concluded that the central bank should try to adjust the interest rate smoothly and avoid the adjustment in large range and opposite direction. However, Rudebusch (2002) and Soderlind et al (2003) proposed the doubts and claimed that the Taylor rule with smoothing interest rate may cost more to predict interest rate.In China, Xie and Luo (2002) finished the first Chinese paper to examine China’s monetary policy based on the Taylor rule, which introduced interest rate smoothing. In this paper, the historical analysis and the reaction function were used to conduct empirical analyses. Comparing the recommendation value from Taylor rule in China’s monetary policy with its real value, they concluded that the Taylor rule could discribe China’s monetary policy well, and the difference between actural value and rule value was caused by the lag of monetary policy. On the other hand, they estimated the reaction function of China’s monetary policy, and the results shown that the adjustment coefficient of the interest rates to the inflation rates was lower than 1 and the elasticity of the interest rate to the output gap was 2.84. Therefore, they concluded that China’s monetary policy had an o vereaction to the output but a underreaction to the inflation rate. Also, they found the smoothness was 0.82 according to the estimation. Finally, they put forward suggestions that China’s monetary authority should employ the Taylor rule as the benchmark f or measuring the stance of China’s monetary policy, because it could help enhance the transparency of China’s monetary policy, implement the interest liberalization reform and transform unstable monetarypolicy rule to the stable monetary policy rule.2.3 Forward-looking rulesFirstly, for the timming problem, the debate about data time mainly focus on whether to use current data or lagged data. Taylor (1999) employed current data to estimate that whether the Federal Reserve had set the Federal funds rate as recommended value from Taylor rule. While Levin et al. (1999) found that the empirical results did not show a substantial difference in the performance using lagged data instead of current data. Hamalainen (2004) explained that the costs were small because both inflation and output were persistent enough, thus the lags of inflation rate and output gap were good proxies for current values. Kozicki (1999) considered the lagged data of output gap and inflation data as a common approach to deal with lags of data. In his paper, he assumed that the Federal funds rate in a given quarter was set depending on the data in previous quarter. However, Orphanides (2001) demonstrated that the rule recommendations obtained with real-time data was significantly different from those obtained with lagged data in a Taylor-rule model. Furthermore, he suggusted that it was essential for monetary authority to make decisions according to the real-time available information.Forward-looking rule specifies that central bank should focus on expected inflation gap and output gap when make monetary policy, rather than those of current and delay period. According to Batini (1999), there are three real benefits of forward-looking rules. Firstly, the monetary policy rule with forward-looking rule is able to embody explicitly the lags in monetary transmission. Secondly, forward-looking rule is far from output invariant. Thirdly, a forward-looking inflation rule embodies all relevant information for inflation predicting. When the nominal interest rate aggressively responds to ex inflation rates, Carles (2000) concluded that the monetary authority should follow a backward-looking rule for ensuring determinacy. Clarida, Gali and Getler (2000) were first to estimate the forward-looking Taylor rule for the postwar United States economy. Thire results supported the views that the anti-inflationary stance of the Fed has been stronger in the past twenty years. According to the forward-looking monetary policy model, they found that the target interest rate influences inflation rate and output gap based on the relationship between β and 1. The interest rate is stable when β>1, otherwise,interest rate should be adjusted depends on economic fluctuation. Orphanides (2001) found that the Taylor rule with introduction of forward-looking behavior gave a better discription of the stable relationship between the federal funds rate and inflation gap, output gap. Considering the smoothing interest rate and forward-looking behavior, he used LOS and IV to estimate the Taylor rule in United States during 1987-1992. Based on the analyses of real time data and historical data, he drawn two conlcusions: firstly, the result from historical data was more accurate than the discription by the rules when use the real time data; Secondly, the Taylor rule with forward-looking behavior was more accurate when use the real time data. Huang et al (2001) also used OLS and IV to observe the data in New Zealand during 1989-1998, and concluded that forward-looking Taylor rule did better than the situation that only consider about the model of real time data, although there were only small difference between them.In China, based on forward-looking Taylor rule, Zhang and Zhang (2007) classified interest rate into three levels, which were market interest rate, regulated interest rate and spread between them, to estimate the monetary policy reaction function in China. They found that although the forward-looking Taylor rule could discribe the trend of three levels of interest rate, the three levels of interest rate were under-reaction to expected rate of inflation and expected output gap, which indicated the unstable monetary policy in China. With the co-intergration analysis, Lu and Zhong (2003) estimated China’s monetary policy based on Ta ylor rule. Results from esimation shows that Taylor rule could describe the trend of interbank offered rate well and it was able to be a benchmark in monetary policy making. Concerning about the time delay, they introduced a forward-looking rule into model to modificate Taylor rule. Chen, Yang and Tu (2006) introduced exchange rate factor into a forward-looking Taylor rule based on Lawrence model that under the open economy in China. The results shown that Taylor rule could be a benchmark for Chinese monetary policy. Comparing with the target of price stability, they said that monetary policy in China pays more attention to economic growth. Ye (2008) employed GMM method to estimate the reaction function of Taylor rule with the introduction of forward-looking rule again. He found that China’s inter-bank offered rates comformed to the essential features of the forward-looking Taylor rule and the differences between coefficients of inflation gap and output gap could not be ignored.2.4 Taylor rule within open economyThe most scholars believed that exchange rate is a significant channel of monetary transmission under open economy, which can influence a country’s prices and interest rate by inport and export. There are economics discuss about some detail problems, for example, what the function of exchange rate in monetary policy is, whether exchange rate can be bring into monetary policy and how to do it. Obstfeld and Rogeff (1996), Svensson (2000) and Ball (1999) are representatives of economics who support the exchange rate should be bring into monetary policy. Obstfeld and Rogff (1996) built a Stackelberg compitition model with leader and follower countries under open economy to prove that it was hard for a follower country to keep exchange rate stable when it has to set interest rate following the leader country. Svensson (1999, 2000) claimed that it was beneficial to bring the exchange rate into monetary policy in inflation targeting regime. For example, exchange rate provides more transmission channels for monetary policy. As an asset price, the exchange rate is forward-looking and predictable, but some external disturbance may be transmitted to domestic market. If exchange rate was be concerned in monetary policy, the adjustment of exchange rate could help to avoid these external disturbance. Therefore, he proposed that exchange rate was ignored in the Taylor rule, and central bank should consider about it in inflation rate target under open economy. On the contrary, Laxton and Pesenti (2003) opposed to consider exchange rate in monetary policy. They built a global economitric model (GEM) to analyze open economy countries, and found that exchange rate plays a weak role in the model. Taylor (1999) also indicated that there was not great improvement in economy condition when bring exchange rate in monetary policy, and sometimes to do so would even makes it worse, according to the estimation of part of European countries. He provided two reason to explain this result: firstly, the effect of exchange rate has been transmitted to interest rate by other relative variables, such as inflation and output; Secondly, there is no need to change interest rate to offset the purchasing power parity deviations caused by exchange rate. However, Taylor (2000) found that exchange rate had great influence on monetary policy in emerging market countries, since these countries were underdeveloped and their foreign exchange markets were unsound. Therefore, he claimed that whether bring the exchange rate into monetary policy rule should depend on the specific economy condition.In China, Wang and Zou (2006) estimated the application of Taylor rule in China’s monetary policy under the open economy, where exchange rate was influenced by United States, Japan and European countries. They tried to prove whether the Taylor rule was suitable for China’s monetary policy. According to their results, they concluded that the standard Taylor rule had strong steady, and the interest rate level in China was related to inflation rate, output gap and other factors, then they also proved that Chinese optimal level of interest rates was influenced by the development of foreign economies. Meanwhile, they indicated that foreign economy cannot be ignored in monetary policy making. Besides, Deng and Shi (2011) built a mixed Taylor rule including exchange rate to estimate China’s monetary policy, their results proved the importance of exchange rate as well. Wang and Wang (2011) varianted the standard Taylor rule to estimate the application of Taylor rule in China’s monetary policy under open economy, which used the data between the first quarter of 1995 and the fourth quarter of 2010 from the aspects of smoothing interest rates, forward-looking variables, exchange rates and assets prices. The test results of this model shown that there was a positive correlation of interest rate with inflation rate, output gap and asset prices, and the smoothing interest rates was effective in policy behavior explaination.2.5 Taylor rule with asset pricesWith the development of capital market and financial market, a highly integrated global market is formed. Therefore, more economics began to concern another question that whether should bring asset prices into monetary policy making. Bullard and Schaling (2002) sugested that Taylor rule should include asset prices. While Mishikin (2007) propsed that monetary policy need to concern about asset prices only when it influences output and inflation. Although this issue caused a heated debate, there is not a theory conclusion yet. On the other hand, the most Chinese scholars regard asset prices as an important variable in Taylor rule. Qian (1998) said that the fluctuation of asset prices could influence monetary policy since it reflected people’s long-term expected interest rate level. Yi and Wang (2002) concluded in their paper that monetary authority should pay attention to asset prices and stock prices during monetary policy making, because the target of Chinese monetary policy was domestic price stability. Peng and Liu (2004) analysed the Taylor rule focus on the relationship between asset price bubble and monetay policy. The reaserch selected China’sfinancial data during the first quarter of 1994 to the fourth quarter of 2001, and considerd about the capital market factor based on Taylor rule estimation. They tried to find out whether China’s monetary policy can react to stock market bubble well. Their conclusion was that China’s central bank did not control the bubble by monetary policy successfully, and the the monetary policy performanced inconsistent and unstable at this stage. According to analyses of data during 1994-2006, Zhao and Gao (2009) concluded that asset prices was a significant endogenous variable in China’s monetary policy reaction function.。