## 经济代写|宏观经济学代写Macroeconomics代考|ECON1120

statistics-lab™ 为您的留学生涯保驾护航 在代写宏观经济学Macroeconomics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写宏观经济学Macroeconomics代写方面经验极为丰富，各种代写宏观经济学Macroeconomics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|宏观经济学代写Macroeconomics代考|The Shipbuilding Model and the Lambert Function

In his 1931 article on the “Shipbuilding cycle,” Tinbergen was interested in the increase of tonnage that followed the building of ships with a lag of about one year due to the construction period. From this connection a relation emerged between the increase of total tonnage and the volume of total tonnage two years before (Tinbergen, 1959: 2). Tinbergen proposed to model this relation as a differential equation with a delay, of the form:
$$\dot{f}(t)=-a f(t-\theta),$$
where $f(t)$ is the tonnage as a function of time, $\theta$ represents the delay between the tonnage and its increase in $t$, and $a$ is the intensity of the relation, the volume of increase above the trend (Tinbergen, 1959: 3). Tinbergen assumed a solution to his equation of the form $f(t)=C e^{\lambda t}$, which, inserted into the equation above yields $\lambda=-a e^{-\lambda \theta}$, once we have simplified the $C$ which only depends on the initial conditions. $\lambda$ can be a real or complex number, but because it appears both in the exponential function and alone this equation is transcendental. Now, “transcendental” means that usually the answer will only be found “experimentally” as Tinbergen put it. Indeed a transcendental equation is periodic, in the same sense that the exponential function with an imaginary argument traces a circle repeating itself as the argument increases. To find a general solution, Tinbergen (and after him Frisch and Kalecki) separated the real and the imaginary part of this equation and solved for one of the two in terms of trigonometric functions; for instance Tinbergen obtained the equation:
$$b \frac{\sin (y)}{y}=e^{-\frac{y}{\tan (y)}},$$
where $b=a \theta$ and $\lambda=x+i y, i=\sqrt{-1}$. To find the solutions for $y$ of this type of equations, they took the same approach of plotting both sides and looking for points of intersection, before improving on this solution with simple algorithms. Figure $2.4$ shows in the solid red lines the right hand side of the Eq. $2.2$, while the dashed lines are the left hand side, for three different values of $b$ and both as a function of $y$.

## 经济代写|宏观经济学代写Macroeconomics代考|From Natural Sciences to Economics

Hamburger claimed that it was only with new mathematical tools that economists would be able to account quantitatively and qualitatively for economic processes. Ultimately, the aim was to transform economics into a science similar to biology, ${ }^5 \mathrm{a}$ science capable of understanding the operation of social organisms beset by recurrent “pathologies.” It should be noted that Hamburger was not the only economist interested in business cycles who was showing some discontent with a mechanical analogy. Ernst Wagemann, the German head of the imperial statistical office and of the business cycle research institute of Berlin, in a book published in 1928, called as well for a biological metaphor.

Although Wagemann wrote in German, his book met enough success to warrant its translation in English only two years later under the title Economic Rhythm: A Theory of Business Cycles, with a prefatory note from Wesley C. Mitchell (Wagemann, 1930). In the preface to the English edition, Wagemann presented his contribution as a small step “toward the repayment of the debt which Europe owes to America in the field of research into economic dynamics” (Wagemann, 1930: v). However, the type of dynamics that was applied remained very empirical; although Wagemann was searching for a theory, he steered resolutely away from abstract constructions which were heavily criticized. His review of existing theories led him to propose that “while the American methods are those of engineering, and the Russian those of astronomy, the German institute represents the medical, or, better, the organicbiological point of view” (1930: 10). The “organic-biological principle” which he described (with reference to Menger) was meant to capture both the interconnection of the separate parts of an (economic) organism as well as “a peculiarity which may be defined as consisting in the power to regulate its own movement” (1930: 11), an approach which he emphasized as “anything but mechanical” (1930: 11).

Another radical opinion on the business cycle was that it was only a “myth.” This opinion was shared among American economists and statisticians, such as Carl Snyder (1930) and Irving Fisher (1925). ${ }^6$ While the former based his claim on the fact that compared to the growth of the economy, the amplitude of fluctuations remained within certain limits, the second doubted that “inherent” cyclical regularity in business could be detected. For Fisher, even if there existed a simple self-generating cycle similar to that of a pendulum swinging under the influence of the force of gravity, its tendency to materialize would be necessarily “defeated in practice”

# 宏观经济学代考

## 经济代写|宏观经济学代写Macroeconomics代考|The Shipbuilding Model and the Lambert Function

$$\dot{f}(t)=-a f(t-\theta),$$

$$b \frac{\sin (y)}{y}=e^{-\frac{y}{\tan (y)}},$$

## 经济代写|宏观经济学代写Macroeconomics代考|From Natural Sciences to Economics

Hamburger 声称，只有使用新的数学工具，经济学家才能对经济过程进行定量和定性分析。最终，目标是将经济学转变为类似于生物学的科学，5一种能够理解被反复出现的“病态”困扰的社会有机体运作的科学。应该指出的是，汉堡并不是唯一对商业周期感兴趣并对机械类比表示不满的经济学家。德国帝国统计局局长兼柏林商业周期研究所所长恩斯特·瓦格曼 (Ernst Wagemann) 在 1928 年出版的一本书中也呼吁使用生物学隐喻。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 经济代写|宏观经济学代写Macroeconomics代考|ECON6002

statistics-lab™ 为您的留学生涯保驾护航 在代写宏观经济学Macroeconomics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写宏观经济学Macroeconomics代写方面经验极为丰富，各种代写宏观经济学Macroeconomics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|宏观经济学代写Macroeconomics代考|Supply Lags and Market Movements

Tinbergen proposed a supply lag scheme as early as 1928 , in a paper published in $D e$ Socialistische Gids. The paper was rather long considering that the journal usually published articles of only a dozen pages, and most of it was concerned with problems of exchange related to Cournot’s theories of competition. Incidentally, Tinbergen deplored the fact that Cournot’s work had remained mostly ignored, adding that the mathematical form in which it was published (in 1838) was probably to blame for this (Tinbergen, 1928: 543). ${ }^{14}$

The last part of this paper was concerned with the problem of the temporal element in his analysis, recognizing that this influence was mostly ignored in the previous sections. Tinbergen studied a problem of fluctuations around an equilibrium and based his discussion on a mechanical analogy: solving the “dynamic problem” meant that “as in mechanical dynamics, further data is needed, data which we can denote by inertia and delay, the inertia being related to the effort, the delay to the time of displacement” (Tinbergen, 1928: 544). Tinbergen then proposed to observe the behavior of one market where supply would adjust with a lag, while “demand adapts to supply without delay” (Tinbergen, 1928: 544). Although he did not explicitly present the equations behind his scheme, he did propose a numerical example with a figure of the temporal evolution of the quantities exchanged of a good, which took the form of damped oscillations that were in fact the result of a cobweb mechanism (reproduced in Fig. 2.2).

Obviously this example showed somewhat trivial oscillations from a high point to a low point each period, the only kind allowed by a first-order difference equation. ${ }^{15}$ This did not prevent him from discussing the conclusions one could draw from such a model for the economy as a whole and the observed cycles of three to four years or the longer cycles of seven to ten years, and the problems they posed for the social welfare of workers. Tinbergen also touched upon the problem of the damping of those oscillations, in connection with similar problems in physical system, and briefly considered the case where “the deviations are getting bigger (as in cases of unstable equilibrium),” but he added that these fluctuations could occur “only occasionally” (Tinbergen, 1928: 547).

## 经济代写|宏观经济学代写Macroeconomics代考|The Shipbuilding Model and Mixed Equations

In the introduction of his 1931 paper “Ein Problem der Dynamik,” when Tinbergen distinguished several lines of research that expanded “the static theory of social economy,” he finished his review by remarking that no one had yet tried to combine time derivatives and lags into a “systematic design of the dynamic theory” (Tinbergen, 1931a: 169). A scheme was proposed in the paper but it was still floundering; nevertheless, that same year, Tinbergen proposed a straightforward combination of time derivatives and lags, through his model of the shipbuilding cycle.

The choice of shipbuilding was not anecdotal; in a paper published in De Nederlandsche Conjunctuur, the journal of the Centraal Bureau voor de Statistiek (CBS) where he was working at the time, Tinbergen explained that “[t]he branch of industry concerned with shipbuilding has a number of peculiarities which make a study of the relationship between the business cycle and this branch of industry of great importance, especially for shipbuilding itself and for shipping companies” (Tinbergen, 1931c: 14). Shipbuilding was one of the biggest branches of activity in the Netherlands, and Tinbergen saw at least three characteristics that made its study worthwhile: the Iong production process of building ships, the strong fluctuations in its activity, and the long life of its means of production. This meant that the shipbuilding industry was quite different from the agricultural markets that had spurred the cobweb models of Moore, Schultz, Ricci and Tinbergen. Clearly, trying to explain the fluctuations of a market of durable instead of perishable goods led Tinbergen to important new ideas.

The first study that he published on the question (Tinbergen, 1931c) was very empirical as he tried to determine the lifespan of ships and other characteristics of the cycle. Nevertheless, he was also clearly searching for an explanation of the cycle, for a mechanism that would explain the fluctuations he had found empirically. He argued that a causal mechanism existed in particular between tonnage and its own increase: “[t]here is indeed a clear interaction between tonnage and increase in tonnage, an interaction which will be referred to in the remainder of this article as the ‘own mechanism’ of the development of tonnage” (Tinbergen, 1931c: 17). But he did not try in this paper to give a more mathematical form to this finding, which he reserved for a paper written in German for the Weltwirtschaftliches Archiv, the journal of the Kiel group of economists (Tinbergen 1931b, which was translated in Tinbergen 1959 from which we quote hereafter). ${ }^{24}$

# 宏观经济学代考

## 经济代写|宏观经济学代写Macroeconomics代考|Supply Lags and Market Movements

Tinbergen 早在 1928 年就在一篇发表于丁和Socialistische Gids。考虑到该杂志通常发表的文章通常只有十几页，而且大部分都是与古诺竞争理论相关的交流问题，所以这篇论文相当长。顺便说一句，Tinbergen 对古诺的工作几乎一直被忽视这一事实感到遗憾，并补充说它出版时所用的数学形式（1838 年）可能是造成这种情况的原因（Tinbergen，1928：543）。14

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 经济代写|宏观经济学代写Macroeconomics代考|ECOS3007

statistics-lab™ 为您的留学生涯保驾护航 在代写宏观经济学Macroeconomics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写宏观经济学Macroeconomics代写方面经验极为丰富，各种代写宏观经济学Macroeconomics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|宏观经济学代写Macroeconomics代考|Economic Barometers and “Cumulation”

The hunt for mechanisms responsible for cyclical fluctuations was clearly visible in Tinbergen’s first published article, where he recognized the importance of Aftalion’s theory of crisis, one of the few that was able, in his view, to account for the fact that “in each cycle there is already the seed for the next one” (Tinbergen, 1927: 715). But Tinbergen did not build a model of the economy in this article, which was aimed at the readers of De Economist to inform them of the latest developments in statistical and mathematical economics and business cycle analysis. At this time, what this covered was essentially the latest debates around the barometers, and the critiques they were subjected to.

One debate in particular interested Tinbergen, it was the “new interpretation” of Karl Karsten which had been discussed-among others -by Warren Persons and Alvin Hansen. Karsten was the main proponent in the 1920s of the “quadrature theory,” which had been developed by Charles Edge in 1908 (Karsten, 1924: 14). Being in “quadrature,” a term borrowed from electrical engineering, meant that the fluctuations of one curve corresponded to the fluctuations of the cumulation of another curve, or in other terms, its integral. ${ }^2$ For Karsten, the main advantage of this approach was to show how to obtain cycles of different length or amplitude: “By the quadrature theory, it seems possible to interpret the various phases of the economic cycle as entirely orderly and in accordance with theory, and yet wholly irregular in point of time” (Karsten, 1924: 16).

It was on the basis of this theory that Karsten suggested a new interpretation of the Harvard Business Index in Karsten (1926). While in the “official” interpretation of the barometer, the A curve (speculation) preceded and caused the B curve (production), Karsten thought that the second curve was in fact the causal predecessor of the first and the real driver of the cycle. His theoretical argument was that what counted was not so much business conditions represented by the B curve but the flow of money into the markets that accompanied poor business conditions and inversely the flow of money toward business when it was booming; in terms of “cumulation,” the A curve, representing the prices of securities, “shows the cumulative effects of the flow of money (into the market out of business, or out of the market into business)” (Karsten, 1926: 406). This led him to argue that the B curve was plotted upside down, and he showed that he could obtain a much larger correlation coefficient by following his theory.

## 经济代写|宏观经济学代写Macroeconomics代考|Time Derivatives and the Theories of Maximum Production

By the time he defended his thesis in 1929 , Tinbergen had already published at least two papers in De Economist and one in De Socialistische Gids. The economic appendix of his thesis, which he wrote up under the direction of renowned physicist Paul Ehrenfest, was published the same year in German in the Archiv für Sozialwissenschaft und Sozialpolitik, and the contents of his thesis were reviewed for $D e$

Economist by Ludwig Hamburger (see Chap. 3). This was a strong start for the 26-year-old Tinbergen, although those works bear little resemblance to the large macroeconometric models that he later developed and became known for. In fact, the thesis was taking an approach wholly different from what was hinted at in Tinbergen’s earlier article on barometers, but it opened a research program that he pursued for several years and that connected him to many other researchers in Europe and the USA.

The thesis title was “Minimum problems in physics and in the economy,” although Tinbergen warned from the beginning that the economic problems were only pointed out in the appendix. The title was also misleading in that the object of the work was not so much “minimum problems” but problems involving stationarity, neglecting the problem of knowing whether the solution obtained was a maximum or a minimum, and the latter term was used in this general meaning. Tinbergen argued that he was merely interested in the “formal analogy” between a number of problems that could be represented under this form of a “minimum problem.” This allowed him to avoid the treacherous question of the teleological aspect of such an idea, although he was inclined to think that the “striving for a minimum” was something that lied in the nature of an economic system (Tinbergen, 1929:2), something he shared with Evans, one of the main proponents of maximum principles. ${ }^6$

The thesis was mostly concerned with the demonstration that much of contemporary physics could be derived from extremal principles, Tinbergen taking throughout his work examples of mechanical motions, thermodynamics, optics, electrostatics and electrodynamics. Historically, minimum principles had been developed in optics and mechanics, and had first culminated in the middle of the eighteenth century with Maupertuis’ principle of least action. Maupertuis saw in his principle the proof that God existed and had built the world according to an harmonious principle, a Leibnizian idea that was the subject of Voltaire’s derision in Candide ${ }^7$ While these teleological aspects marked the developments of the idea, the most important works on the question became those that managed to keep a safe distance from it, first in Euler’s work and especially in Lagrange’s analytical mechanics. The works of the latter two formed the basis of the calculus of variation, and the solutions of such problems are still called today Euler-Lagrange equations.

# 宏观经济学代考

## 经济代写|宏观经济学代写Macroeconomics代考|Economic Barometers and “Cumulation”

Tinbergen 特别感兴趣的一场辩论是 Warren Persons 和 Alvin Hansen 讨论过的 Karl Karsten 的“新解释”。Karsten 是 1920 年代“正交理论”的主要支持者，该理论由 Charles Edge 于 1908 年提出 (Karsten, 1924: 14)。处于“正交”（从电气工程中借用的一个术语）意味着一条曲线的波动对应于另一条曲线的累积波动，或者换句话说，它的积分。2对于 Karsten 来说，这种方法的主要优点是展示了如何获得不同长度或幅度的周期：“通过正交理论，似乎可以将经济周期的各个阶段解释为完全有序且符合理论，并且但在时间点上完全不规则”（Karsten，1924：16）。

## 经济代写|宏观经济学代写Macroeconomics代考|Time Derivatives and the Theories of Maximum Production

Ludwig Hamburger 的经济学家（见第 3 章）。对于 26 岁的 Tinbergen 来说，这是一个良好的开端，尽管这些作品与他后来开发并成名的大型宏观计量模型几乎没有相似之处。事实上，这篇论文采用的方法与 Tinbergen 早期关于气压计的文章中暗示的方法完全不同，但它开启了一个他追求了数年的研究计划，并将他与欧洲和美国的许多其他研究人员联系起来。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 经济代写|微观经济学代写Microeconomics代考|ECON20002

statistics-lab™ 为您的留学生涯保驾护航 在代写微观经济学Microeconomics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写微观经济学Microeconomics代写方面经验极为丰富，各种代写微观经济学Microeconomics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|微观经济学代写Microeconomics代考|Criteria for classification of consumer goods

• simple demand price elasticities:
$E_{i i}\left(p_1, p_2, I\right)>0, \quad i=1,2$-Giffen goods or Veblen goods, ${ }^{28}$ (an increase in the price of a given good results in an increase in the demand for this good).
$E_{i i}\left(p_1, p_2, I\right)<0, \quad i=1,2$-ordinary goods, (an increase in the price of a given good results in a decrease in the demand for this good).
• cross price elasticities of demand:
$E_{i j}\left(p_1, p_2, I\right)>0, \quad i, j=1,2, i \neq j$ 一substitute goods, (an increase in the price of $j$-th commodity results in an increase in the demand for $i$-th good),
$E_{i j}\left(p_1, p_2, I\right)=0, \quad i, j=1,2, i \neq j$-independent goods, (an increase in the price of $j$-th good does not affect the demand for $i$-th good),
$E_{i j}\left(p_1, p_2, I\right)<0, \quad i, j=1,2, \quad i \neq j$-complementary goods, (an increase in the price of $j$-th good results in a decrease in the demand for $i$-th good).
• income elasticity of demand
$E_i\left(p_1, p_2, I\right)>0, \quad i=1,2$-normal goods, (an increase in a consumer’s income results in an increase in the demand for $i$-th good).
$E_i\left(p_1, p_2, I\right)<0, \quad i=1$, 2-inferior goods, (an increase in a consumer’s income results in a decrease in the demand for $i$-th good). ${ }^{29}$

Note 2.26 If an increase in the price of an inferior good results in an increase in the demand for this good, then it is called a Giffen good. On the other hand, when an increase in the price of a normal good results in an increase in the demand for this good, then it is called a Veblen good.
Note 2.27 Let us notice that:
(2.122) $\quad E_{i j}\left(p_1, p_2, I\right)=P_{i j}\left(p_1, p_2, I\right) \frac{p_j}{\varphi_i\left(p_1, p_2, I\right)}, \quad i, j=1,2$,
(2.123) $\quad E_i\left(p_1, p_2, I\right)=P_i\left(p_1, p_2, I\right) \frac{I}{\varphi_i\left(p_1, p_2, I\right)}, \quad i=1,2$.
The price and income elasticities of demand have the same sign as the marginal demand for $i$-th good with respect to prices of goods or with respect to a consumer’s income because prices and the demand for consumer goods are positive.

## 经济代写|微观经济学代写Microeconomics代考|Dynamic Approach

Let us introduce a notation $t$ of the time. In a discrete version of a dynamic consumers problem the time changes in jumps, that is we consider the values of the analysed functions in subsequent periods $t=0,1,2, \ldots, T$, where $T$ means a time horizon. For example, assuming we treat periods as months, when $T=30$, this means the time horizon of $2.5$ years. In a continuous version of a dynamic consumer’s problem the time $t \in[0 ; T]$ changes continuously, that is we consider the values of the analysed functions at any moment of the considered time horizon. As in the static approach, we assume that we are interested in bundles composed of two consumer goods ${ }^{30}$ :
$\mathbf{p}(t)=\left(p_1(t), p_2(t)\right) \geq 0$ — a vector of time-varying prices of goods,
$I(t) \geq 0$-a consumer’s income changing over time,
$\mathbf{x}(t)=\left(x_1(t), x_2(t)\right) \geq 0$-a bundle of goods that a consumer is willing to purchase at period/moment $t$ at prices $\mathbf{p}(t)$.

When choosing a bundle $\mathbf{x}(t)$ the consumer takes into account her/his preferences towards bundles of goods, described by a utility function $u(\mathbf{x}(t))$. Over time, it is not the consumer’s preferences that change, but only the bundle of goods that the consumer is willing to buy. This change occurs due to changes over time in the prices of goods and in the consumer’s income.
The consumption utility maximization problem has a form:
$$u(\mathbf{x}(t)) \mapsto \max$$
$$p_1(t) x_1(t)+p_2(t) x_2(t) \leq I(t)$$
$$\mathbf{x}(t) \geq \mathbf{0} .$$

If we assume that an insatiability phenomenon occurs (utility functions are increasing in quantities of goods in a consumption bundle), then as a budget constraint, instead of the inequality of the budget set, we can use the budget line equation:
$$p_1(t) x_1(t)+p_2(t) x_2(t)=I(t) .$$

# 微观经济学代考

## 经济代写|微观经济学代写Microeconomics代考|Criteria for classification of consumer goods

• 简单的需求价格弹性:
$E_{i i}\left(p_1, p_2, I\right)>0, \quad i=1,2$ – 吉芬商品或凡勃伦商品，
28 (给定商品价格的上涨导致对该商品 的需求增加）。
$E_{i i}\left(p_1, p_2, I\right)<0, \quad i=1,2$-普通商品，（给定商品价格的上涨导致对该商品的需求减少)。
• 需求的交叉价格弹性:
$E_{i j}\left(p_1, p_2, I\right)>0, \quad i, j=1,2, i \neq j$ 一替代品，（价格上涨 $j$-th商品导致需求增加 $i$-好)，
$E_{i j}\left(p_1, p_2, I\right)=0, \quad i, j=1,2, i \neq j$ – 独立商品，（价格上涨 $j$-th 商品不影响需求 $i$ – 好)，
$E_{i j}\left(p_1, p_2, I\right)<0, \quad i, j=1,2, \quad i \neq j$ – 互补品，（价格上涨 $j$ – 良好的结果导致对需求的减 少 $i$-很好)。
• 需求收入弹性
$E_i\left(p_1, p_2, I\right)>0, \quad i=1,2$ – 正常商品，（消费者收入的增加导致对商品的需求增加 $i$-很 好)。
$E_i\left(p_1, p_2, I\right)<0, \quad i=1$ ，2-劣质品，(消费者收入的增加导致对 $i$-很好)。 ${ }^{29}$
注释 $2.26$ 如果劣质商品价格上涨导致对该商品的需求增加，则它被称为吉芬商品。另一方面，当一种正 常商品的价格上涨导致对该商品的需求增加时，它被称为凡勃伦商品。
注释 $2.27$ 让我们注意到:
(2.122) $\quad E_{i j}\left(p_1, p_2, I\right)=P_{i j}\left(p_1, p_2, I\right) \frac{p_j}{\varphi_i\left(p_1, p_2, I\right)}, \quad i, j=1,2$ ，
(2.123) $\quad E_i\left(p_1, p_2, I\right)=P_i\left(p_1, p_2, I\right) \frac{I}{\varphi_i\left(p_1, p_2, I\right)}, \quad i=1,2$.
需求的价格弹性和收入弹性与边际需求的符号相同 $i$ – 与商品价格或消费者收入相关的商品，因为价格和对 消费品的需求是正的。

## 经济代写|微观经济学代写Microeconomics代考|Dynamic Approach

$\mathbf{p}(t)=\left(p_1(t), p_2(t)\right) \geq 0$ – 随时间变化的商品价格向量，
$I(t) \geq 0$-消费者的收入随时间变化，
$\mathbf{x}(t)=\left(x_1(t), x_2(t)\right) \geq 0$ – 消费者愿意在一段时间/时刻购买的一览子商品 $t$ 按价格 $\mathbf{p}(t)$.

$$\begin{gathered} p_1(t) x_1(t)+p_2(t) x_2(t) \leq I(t) \ \mathbf{x}(t) \geq \mathbf{0} \end{gathered}$$

$$p_1(t) x_1(t)+p_2(t) x_2(t)=I(t)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 经济代写|微观经济学代写Microeconomics代考|PACC6007

statistics-lab™ 为您的留学生涯保驾护航 在代写微观经济学Microeconomics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写微观经济学Microeconomics代写方面经验极为丰富，各种代写微观经济学Microeconomics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|微观经济学代写Microeconomics代考|Substitute, Independent and Complementary Goods

The substitutability concerns only these consumption bundles $\mathbf{x}=\left(x_1, x_2\right) \in \mathbb{R}_{+}^2$ whose utility is the same.

Any two goods are called substitute goods (substitutes) if in order to keep some given level of utility of a consumption bundle $\mathbf{x} \in \mathbb{R}_{+}^2$ when quantity of one of the goods is reduced (raised) one needs to compensate this change by appropriate increase (decrease) in quantity of the other good in a consumption bundle $\mathbf{x}$.

If in order to keep some given level of utility of a consumption bundle $\mathbf{x} \in \mathbb{R}_{+}^2$ when quantity of one of the goods is reduced (raised) one does need to compensate for this change by any increase (decrease) in quantity of the other good in a consumption bundle $\mathbf{x}$, then such two consumer goods are called independent goods.

Any two goods are called complementary goods (complements) if in order to change the utility of a given consumption bundle $\mathbf{x} \in \mathbb{R}_{+}^2$ one needs to simultaneously raise or reduce quantities of both goods in the bundle.

Note 2.11 When classifying consumption bundles in substitutes or independent goods one takes into account only these goods which are considered in bundles with the same utility level (bundles indifferent to each other).

Note 2.12 To state if any two goods in a consumption bundle $\mathbf{x}=\left(x_1, x_2\right) \in \mathbb{R}{+}^2$ are complements to each other one needs to determine whether an increase (decrease) in the utility of this bundle requires a simultaneous raise (reduction) in quantities of both goods. If there is no such need then the goods are called not complementary. Let us define measures of the substitutability of consumer goods. For this purpose let us assume that we are given: (1) a differentiable utility function $u: \mathbb{R}{+}^2 \rightarrow \mathbb{R}$,

(2) an indifference curve – a set
$$G(u)=\left{\mathbf{x} \in \mathbb{R}_{+}^2 \mid u(\mathbf{x})=u=\text { const. }>0\right} .$$
of all bundles with the same reference utility $u=$ const. $>0$.

## 经济代写|微观经济学代写Microeconomics代考|Marshallian Demand Function

Let us consider a market for two consumer goods where:
$i=1,2$ – consumer goods (products and services),
$X=\mathbb{R}{+}^2-$ a goods space, $\mathbf{p}=\left(p_1, p_2\right) \in \mathbb{R}{+}^2$-a vector of prices of consumer goods,
$\mathbf{x}=\left(x_1, x_2\right) \in B \subset \mathbb{R}{+}^2-$ a bundle of goods that the consumer wants to purchase (a consumption bundle), $B=\left{\mathbf{x}=\left(x_1, x_2\right) \in \mathbb{R}{+}^2 \mid x_1 \leq b_1, x_2 \leq b_2\right}$-a supply set,
$b_i, i=1,2$-supply of $i$-th consumer good, ${ }^{22}$
$I \in$ int $\mathbb{R}{+}-$a consumer’s income, ${ }^{23}$ $u: \mathbb{R}{+}^2 \rightarrow \mathbb{R}-$ a utility function describing the preferences of a consumer (describing a relation of consumer preference).
$D(\mathbf{p}, I)=\left{\mathbf{x} \in \mathbb{R}{+}^2 \mid p_1 x_1+p_2 x_2 \leq I\right} \subset X=\mathbb{R}{+}^2-$ a set of all consumption bundles whose value is not greater than the consumer’s income (a budget set),
Definition 2.26 A bundle $\mathbf{x}$ is called a limit of a sequence $\left{\mathbf{x}^i\right}_{i=1}^{+\infty}$ if a limit of sequence of metric values $\lim {i \rightarrow+\infty} d\left(\mathbf{x}^i, \mathbf{x}\right)=0$, which can be written as $$\lim {i \rightarrow+\infty} \mathbf{x}^i=\mathbf{x} \text { or } \quad \mathbf{x}^i \rightarrow_{i \rightarrow+\infty} \mathbf{x} .$$
Definition 2.27 The budget set $D(\mathbf{p}, I) \subset X=\mathbb{R}{+}^2$ is a closed set because: (2.38) $\forall \mathbf{x}^i \in D(\mathbf{p}, I) \lim {i \rightarrow+\infty} \mathbf{x}^i=\mathbf{x} \Rightarrow \mathbf{x} \in D(\mathbf{p}, I)$.
Definition $2.28$ The budget set $D(\mathbf{p}, I) \subset X=\mathbb{R}_{+}^2$ is a bounded set because:
$$\forall \mathbf{x}^1, \mathbf{x}^2 \in D(\mathbf{p}, I) \quad \exists \mathrm{N}>0 \quad d\left(\mathbf{x}^1, \mathbf{x}^2\right)<N$$

# 微观经济学代考

## 经济代写|微观经济学代写Microeconomics代考|Substitute, Independent and Complementary Goods

(2) 一条无差异曲线一一一个集合
$G(u)=\backslash$ left $\left{\backslash m a t h b f{x} \backslash\right.$ in $\backslash m a t h b b{R}_{-}{+}^{\wedge} 2 \backslash$ mid $u(\backslash \operatorname{mathbf}{x})=u=\backslash \operatorname{text}{$ 常数。 $}>0 \backslash$ 右 $}$

## 经济代写|微观经济学代写Microeconomics代考|Marshallian Demand Function

$i=1,2$ – 消费品（产品和服务)，
$X=\mathbb{R}+{ }^2$ 一货品空间， $\mathbf{p}=\left(p_1, p_2\right) \in \mathbb{R}+{ }^2$-消费品价格的向量，
$\mathbf{x}=\left(x_1, x_2\right) \in B \subset \mathbb{R}+{ }^2$ 一消费者想要购买的一束商品（消费束），
$B=\backslash$ left $\left{\backslash\right.$ mathbf ${x}=\backslash \operatorname{left}\left(x_{-} 1, x_{-} \backslash \backslash\right.$ ight $) \backslash$ in $\backslash m a t h b b{R}{+}^{\wedge} 2 \backslash m i d x_{-} 1 \backslash$ leq b_1, $x_{-} 2 \backslash$ leq b_2\right } } \text { – 一套补给 }

$b_i, i=1,2$-供应 $i$-i肖费品， 22
$I \in$ 整数 $\mathbb{R}+$ 一消费者的收入， ${ }^{23} u: \mathbb{R}+{ }^2 \rightarrow \mathbb{R}$ 一描述消费者偏好的效用函数（描述消费者偏好的关

$\lim i \rightarrow+\infty d\left(\mathbf{x}^i, \mathbf{x}\right)=0$, 可以写成
$$\lim i \rightarrow+\infty \mathbf{x}^i=\mathbf{x} \text { or } \quad \mathbf{x}^i \rightarrow_{i \rightarrow+\infty} \mathbf{x}$$

$$\forall \mathbf{x}^1, \mathbf{x}^2 \in D(\mathbf{p}, I) \quad \exists \mathrm{N}>0 \quad d\left(\mathbf{x}^1, \mathbf{x}^2\right)<N$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 经济代写|微观经济学代写Microeconomics代考|ECON2516

statistics-lab™ 为您的留学生涯保驾护航 在代写微观经济学Microeconomics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写微观经济学Microeconomics代写方面经验极为丰富，各种代写微观经济学Microeconomics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|微观经济学代写Microeconomics代考|Preliminary Terms

For the sake of simplicity we consider rational choices ${ }^4$ made by an individual consumer on a market of two consumer goods ${ }^5$ denoted by subscript $i=1,2$.
Let us introduce basic terms which set the frame of an analysis we conduct in this chapter and Chap. 3.

Definition 2.1 A bundle of consumer goods (a consumption bundle) is a vector $\mathbf{x}=\left(x_1, x_2\right) \in \mathbb{R}_{+}^2$, in which $i$-th component $x_i \geq 0, i=1,2$ indicates a nonnegative expressed in physical units amount of $i$-th good in the consumption bundle $\mathbf{x}$.

Definition 2.2 A consumer goods space is a set $X=\mathbb{R}{+}^2$ of all bundles of goods available on the market along with a metric specified on it ${ }^6$ : (2.1) $\quad d_E\left(\mathbf{x}^1, \mathbf{x}^2\right)=\sqrt{\sum{i=1}^2\left(x_i^1-x_i^2\right)^2}=\sqrt{\left(x_1^1-x_1^2\right)^2+\left(x_2^1-x_2^2\right)^2}$, or
(2.2) $\quad d\left(\mathbf{x}^1, \mathbf{x}^2\right)=\max {i=1,2}\left{\left|x_i^1-x_i^2\right|\right}=\max {i=1,2}\left{\left|x_1^1-x_1^2\right|,\left|x_2^1-x_2^2\right|\right}$
being a measure of distance between two consumption bundles. ${ }^7$
Definition 2.3 A Cartesian product determined on the goods space $X=\mathbb{R}_{+}^2$ is such a set:
$$X \times X=\left{\left(\mathbf{x}^1, \mathbf{x}^2\right) \mid \mathbf{x}^1 \in X, \mathbf{x}^2 \in X\right},$$
of all ordered pairs of consumption bundles in which both bundles (the first one and the second one in the pair) belong to the goods space.

## 经济代写|微观经济学代写Microeconomics代考|Utility Function

Definition 2.12 A consumer’s utility function (defined on the goods space $X=$ $\mathbb{R}{+}^2$ ) is a mapping $u: \mathbb{R}{+}^2 \rightarrow \mathbb{R}$ such that
\begin{aligned} & \forall \mathbf{x}^1, \mathbf{x}^2 \in X=\mathbb{R}{+}^2 \quad \mathbf{x}^1 \succsim \mathbf{x}^2 \Leftrightarrow u\left(\mathbf{x}^1\right) \geq u\left(\mathbf{x}^2\right), \ & \forall \mathbf{x}^1, \mathbf{x}^2 \in X=\mathbb{R}{+}^2 \quad \mathbf{x}^1 \succ \mathbf{x}^2 \Leftrightarrow u\left(\mathbf{x}^1\right)>u\left(\mathbf{x}^2\right), \ & \forall \mathbf{x}^1, \mathbf{x}^2 \in X=\mathbb{R}{+}^2 \quad \mathbf{x}^1 \sim \mathbf{x}^2 \Leftrightarrow u\left(\mathbf{x}^1\right)=u\left(\mathbf{x}^2\right) . \end{aligned} Some properties of the utility function Definition 2.13 A utility function $u: \mathbb{R}{+}^2 \rightarrow \mathbb{R}$ is called continuous at point $\mathbf{x} \in \mathbb{R}{+}^2$ if for any sequence $\left{\mathbf{x}^i\right}{i=1}^{+\infty}$, where $\mathbf{x}^i \in X=\mathbb{R}{+}^2$, it is satisfied: $$\lim {i \rightarrow+\infty} \mathbf{x}^i=\mathbf{x} \Rightarrow \lim {i \rightarrow+\infty} u\left(\mathbf{x}^i\right)=u(\mathbf{x}) .$$ Definition 2.14 A utility function $u: \mathbb{R}{+}^2 \rightarrow \mathbb{R}$ is called continuous on the goods space $X=\mathbb{R}_{+}^2$ if it is continuous at every point of this space.

Definition 2.15 A utility function $u: \mathbb{R}{+}^2 \rightarrow \mathbb{R}$ is called differentiable on the goods space $X=\mathbb{R}{+}^2$ if its partial first-order derivatives:
\begin{aligned} & \frac{\partial u\left(x_1, x_2\right)}{\partial x_1}=\lim {\Delta x_1 \rightarrow 0} \frac{u\left(x_1+\Delta x_1, x_2\right)-u\left(x_1, x_2\right)}{\Delta x_1}, \ & \frac{\partial u\left(x_1, x_2\right)}{\partial x_2}=\lim {\Delta x_2 \rightarrow 0} \frac{u\left(x_1, x_2+\Delta x_2\right)-u\left(x_1, x_2\right)}{\Delta x_2} \end{aligned}
are continuous on this space.
Definition 2.16 A marginal utility of $\boldsymbol{i}$-th good in a consumption bundle $\mathbf{x}=$ $\left(x_1, x_2\right) \in \mathbb{R}_{+}^2$ is a partial first-order derivative of the utility function:
$$\frac{\partial u\left(x_1, x_2\right)}{\partial x_i} \quad i=1,2,$$ which describes by approximately how many units the utility of a consumption bundle $\mathbf{x} \in \mathbb{R}_{+}^2$ changes when quanity of $i$-th good increases by one (notional) unit and quantity of the other good in the bundle does not change.

# 微观经济学代考

## 经济代写|微观经济学代写Microeconomics代考|Utility Function

$$\forall \mathbf{x}^1, \mathbf{x}^2 \in X=\mathbb{R}+^2 \quad \mathbf{x}^1 \succsim \mathbf{x}^2 \Leftrightarrow u\left(\mathbf{x}^1\right) \geq u\left(\mathbf{x}^2\right), \quad \forall \mathbf{x}^1, \mathbf{x}^2 \in X=\mathbb{R}+{ }^2 \quad \mathbf{x}^1 \succ \mathbf{x}^2 \Leftrightarrow \imath$$

$$\lim i \rightarrow+\infty \mathbf{x}^i=\mathbf{x} \Rightarrow \lim i \rightarrow+\infty u\left(\mathbf{x}^i\right)=u(\mathbf{x})$$

$$\frac{\partial u\left(x_1, x_2\right)}{\partial x_i} \quad i=1,2,$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 经济代写|博弈论代写Game Theory代考|ECOS3012

statistics-lab™ 为您的留学生涯保驾护航 在代写博弈论Game Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写博弈论Game Theory代写方面经验极为丰富，各种代写博弈论Game Theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|博弈论代写Game Theory代考|STRATEGIC GAMES

Prisoner’s Dilemma is a canonical example of a strategic game because, as we will see shortly, it typifies many scenarios that confront decision makers. Further, being a simple scenario, it can be used to illustrate many of the fundamental concepts of game theory, and it also clearly demonstrates a fundamental dilemma in our (human) decision-making processes.

We model this scenario as a strategic game in which the two suspects, each confined in a separate interrogation room, are the players. We will often refer to our two players in strategic games as Rose and Colin. (This convention helps later to emphasize the distinction between row and column players and was popularized by Phil Straffin in his book Game Theory and Strategy [110].) They each have two strategies available to them which we name Quiet and Confess. Table $3.1$ lists each of the strategy profiles in the form (Rose, Colin) and the resulting outcome.

We assume that each suspect is primarily concerned about their own sentence and wants to minimize it. Table $3.2$ provides payoffs (a common synonym for utilities) for each player. Here we use the utility function 6 minus the number of years in prison; this is consistent with the player preferences. Based on our assumptions, these payoffs are ordinal. For these payoffs to also be vNM, we would need to assume that the suspects are risk neutral in the number of years to be served in prison.

Tables $3.1$ and $3.2$ complete the construction of the model by identifying the strategies, outcomes, and payoffs. We will refer to this model of the Prisoner’s Dilemma scenario as the Prisoner’s Dilemma strategic game.

We are now ready to look for a solution that maximizes the payoffs to the players. By observing that $5>3$, we see that Confess is the best response strategy for Rose if she knows that Colin will choose Quiet. Further, we can observe that Confess is also a best response for Rose if she knows Colin will choose Confess. We formalize the definition of a best response strategy below.

## 经济代写|博弈论代写Game Theory代考|FINALJEOPARDY

As we observed in Section 3.1, the phrase “Prisoner’s Dilemma” has been used to describe many real-world scenarios; however, not all of these scenarios actually fit the mathematical definition. This can only be revealed by constructing and analyzing a model of the scenario.

We examine a situation in Jeopardy! which fans have identified as a Prisoner’s Dilemma. In the Final Jeopardy round, each contestant makes a wager as to whether they can answer a specific question correctly. When making the wager, contestants know the category of the question, but not the question itself, and the amount of money each of the other contestants has available. Each player’s wager can be between 0 and their current winnings. Depending on whether the contestant answers the question correctly, they win or lose the amount of money wagered. The contestant with the most money after this final round of play wins the game. The winner keeps all of their winnings, and the other two contestants lose essentially all of their money. If there is a tie at the end of the round, a simple, essentially random, tie-breaker rule is applied to identify the winner.

The so-called Prisoner’s Dilemma situation occurs when two contestants are tied for the lead, and the third contestant has less than half of the money of either of the first two contestants. For simplicity we will assume that it is contestants 1 and 2 who are tied with the most money.

In this situation, aficionados of Jeopardy! often refer to “Jeek’s Rule,” which asserts that while they could wager any amount up to their current winnings, contestants 1 and 2 should either wager nothing or everything. We discuss the reasonableness of this rule and then make it an assumption when we define our strategic game.

Let $E$ ‘ be the amount of money contestants 1 and ‘ 2 have each won at the time Final Jeopardy begins. Let $w_i$ denote the wager of contestant $i$ and suppose that contestant 1 ‘s wager satisfies $0<w_1<E$. There are four cases to consider:

Case 1: Both contestants answer the question correctly. In this case, if $w_1<w_2$, contestant 1 regrets not wagering $E$ in order to win. If $w_1 \geq w_2$, then contestant 1 regrets not wagering $E$ to maximize their winnings.

Case 2: Contestant 1 answers the question correctly and contestant 2 does not. Here contestant 1 regrets not wagering $E$ in order to maximize their winnings.

Case 3 : Contestant 1 answers the question incorrectly and contestant 2 answers correctly. Then contestant 1 is indifferent about their bet unless $w_2=0$, in which case they regret not wagering $w_1=0$.

Case 4: Both contestants answer the question incorrectly. Here, if $w_1 \geq w_2$, contestant 1 regrets not wagering $w_1=0$ in order to win. If $w_1<w_2$, then contestant 1 regrets not wagering $w_1=0$ to maximize their winnings.

# 博弈论代考

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 经济代写|博弈论代写Game Theory代考|ECON6025

statistics-lab™ 为您的留学生涯保驾护航 在代写博弈论Game Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写博弈论Game Theory代写方面经验极为丰富，各种代写博弈论Game Theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|博弈论代写Game Theory代考|CONSTRUCTING UTILITIES

We have already suggested how we might construct a utility function to model a player’s choices when there are a finite number of outcomes. Ordinal preferences can be revealed by asking the player to choose among all outcomes and assign those outcomes the highest utility, asking the player to choose among all outcomes not previously chosen and assign those outcomes the second highest utility, and so forth. vNM preferences can be obtained by asking the player to name the highest and lowest ranked outcomes $o_h$ and $o_l$, assign utilities of $u\left(o_h\right)=1$ and $u\left(o_l\right)=0$ to these outcomes, and then for each remaining outcome $o$ determine a probability $p$ for which the player would be willing to choose either the outcome $o$ or the lottery $(1-p) o_l+p o_h$ and assign $u(o)=p$.

In this section, we examine four specific scenarios to illustrate a variety of ways utility functions may be created.

To model Self-Interest and Other-Interest, we simplify our scenario to examine the monthly salaries of the job offers for each spouse. Suppose Scarlett and Regis receive $\$ x$thousand and$\$y$ thousand, respectively; we will denote this by $(x, y)$. Consider the following four possible outcomes: $(7,0),(6,6),(5,7)$, and $(1,6)$. If Scarlett is exclusively self-interested, she would rank order these outcomes in the given order. If Scarlett is primarily interested in Regis receiving money and only secondarily interested in receiving money for herself, Scarlett would rank order the outcomes $(5,7),(6,6),(1,6)$, and $(7,0)$. If Scarlett had a mixture of self-interest, other-interest, and a desire for equity, she might rank order the outcomes $(6,6),(5,7),(7,0)$, and $(1,6)$.

In fact, this last rank order would be obtained if Scarlett considered $\$ 1,000$given to Regis to be worth the same to her as her receiving$\$500$, suggesting the utility function $u(x, y)=x+0.5 y$. Of course, this is only an ordinal utility function unless, at minimum, Scarlett is indifferent between the outcome $(7,0)$ with utility $u(7,0)=7$ and the lottery $L=0.6(6,6)+0.4(1,6)$ with utility
$$u(L)=0.6 u(6,6)+0.4 u(1,6)=0.6(9)+0.4(4)=7 .$$
This example demonstrates how we can incorporate both self-interest and altruistic interests into a player’s utility function. Therefore, maximizing a utility function does not necessarily imply selfishness, but rather achieving the most preferred outcome based on the player’s interests.

## 经济代写|博弈论代写Game Theory代考|DETERMINING RISK

In the duopoly scenario, and in most other economic models, the utility of an outcome is cquivalent to some dollar value associated with the outcome. While we can see how dollar values might capture the intensity of a player’s preferences, dollar values are not necessarily vNM utilities. For example, receiving $\$ 11.00$instead of$\$10.00$ means significantly more than receiving $\$ 1001.00$instead of$\$1000.00$ to most people. To explore this difference, we consider the relationship between the expected utility of a lottery, as given by the Expected Utility Hypothesis, and the utility of the expected value of the lottery.

Consider the following raffle: For $\$ 25$, you can purchase a$\frac{1}{400}$chance for a$\$10,000$ college scholarship. We can represent this lottery with our usual notation
$$\left.\frac{399}{400}(\text { losing } \ 25)+\frac{1}{400} \text { (winning } \ 9,975\right) \text {, }$$ but since the outcomes are numerical, we can calculate the expected monetary value of the raffle as
$$\frac{399}{400}(-\ 25)+\frac{1}{400}(\ 9,975)=\ 0 .$$
The expected monetary value of entering or not entering the raffle is the same, however, entering the raffle involves a small chance of a large gain offset by a large chance of a small loss, while not entering the raffle involves no chance of a gain or a loss. Entering the raffle involves risk while not entering the raffle does not.

Most parents of college students are willing to enter the raffle, but many college students themselves are not. For the college parents,
$$\left.\left.u\left(\frac{399}{400} \text { (losing } \ 25\right)+\frac{1}{400} \text { (winning } \ 9,975\right)\right)>u(\ 0),$$
but for the students themselves,
$$\left.u\left(\frac{399}{400}(\text { losing } \ 25)+\frac{1}{400} \text { (winning } \ 9,975\right)\right)<u(\ 0) .$$
For the parents, the utility of the lottery is greater than the utility of the expected value, making them risk loving in this scenario. On the other hand, the students are risk adverse since the utility of the lottery is less than the utility of the expected value. This principle holds in general, as we describe in the following definition.

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|CONSTRUCTING UTILITIES

$$u(L)=0.6 u(6,6)+0.4 u(1,6)=0.6(9)+0.4(4)=7 .$$

## 经济代写|博弈论代写Game Theory代考|DETERMINING RISK

$$\left.\left.\frac{399}{400} \text { ( losing } \ 25\right)+\frac{1}{400} \text { (winning } \ 9,975\right) \text {, }$$

$$\frac{399}{400}(-\ 25)+\frac{1}{400}(\ 9,975)=\ 0 .$$

$$u\left(\frac{399}{400}(\text { losing } \ 25)+\frac{1}{400}(\text { winning } \ 9,975)\right)>u(\ 0)$$

$$u\left(\frac{399}{400}(\text { losing } \ 25)+\frac{1}{400}(\text { winning } \ 9,975)\right)<u(\ 0) .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 经济代写|博弈论代写Game Theory代考|ECON2112

statistics-lab™ 为您的留学生涯保驾护航 在代写博弈论Game Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写博弈论Game Theory代写方面经验极为丰富，各种代写博弈论Game Theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|博弈论代写Game Theory代考|ORDINAL UTILITIES

In this scenario, there are four possible outcomes: Scarlett leaves with vanilla ice cream, which we will label $V$, leaves with chocolate ice cream, labeled $C$, leaves with strawberry ice cream, labeled $S$, or leaves with no ice cream, labeled $N$. Our outcome set is then ${V, C, S, N}$. The rules dictate that Scarlett may choose any outcome that is available.
To model Scarlett’s preferences and build a utility function, we consider the choices she would make when certain outcomes are available. Supposing that all potential outcomes are available, Scarlett would choose $V$; if $C$ were not an available outcome, Scarlett would choose $S$; if only $V$ and $N$ were available outcomes, Scarlett would flip a coin to determine whether to select $V$ or $N$; and if only $C$ and $S$ were available outcomes, Scarlett would be unable to make a choice. While this behavior would be possible, most would find it bizarre: Scarlett’s choice of $V$ when presented with the outcome set ${V, C, S, N}$ suggests she prefers $V$ to $S$, but Scarlett’s choice of $S$ when presented the outcome subset ${V, S, N}$ suggests she prefers $S$ to $V$. To avoid such absurd possibilities, we will assume that individual behavior is governed by self-consistent internal preferences over the outcomes, which is reflected in the mathematical definition of ordinal preferences below.

Definition 2.1.1. A player $i$ is said to have ordinal preferences among outcomes if there exists a utility function $u_i$ from the set $O$ of outcomes into the real numbers, $\mathbb{R}$, such that whenever presented with a subset $O^{\prime} \subseteq O$ of outcomes, player $i$ chooses any of the outcomes that maximize $u_i$ over all outcomes $o \in O^{\prime}$.

To ensure that ordinal preferences align with the players real-world choice behavior, we note that whenever player $i$ prefers outcome $o_j$ over outcome $o_k$, we should have $u_i\left(o_j\right)>$ $u_i\left(o_k\right)$, and when player $i$ is indifferent between $o_j$ and $o_k$ we should have $u_i\left(o_j\right)=u_i\left(o_k\right)$. We now ask under what conditions Scarlett’s outcome choice behavior can be modeled by ordinal preferences and describe three reasonable properties for a self-consistent set of choices. Since it is usually easier for a player to choose between two rather than among many outcomes, these properties will focus on pairwise choices.

First, we want Scarlett’s pairwise choices to be complete, meaning that whenever she is presented with a pair of outcomes, Scarlett is able to make a choice. Equivalently, for each pair of outcomes, $A$ and $B$, exactly one of the following conditions holds: (a) Scarlett chooses $A$ over $B$, (b) Scarlett chooses $B$ over $A$, or (c) Scarlett is willing to flip a coin to determine which outcome to choose (in this case we will often say Scarlett chooses either $A$ or $B$ ). Hence, this condition excludes the following option as a possibility: (d) Scarlett chooses neither $A$ nor $B$. (When we want the rules to allow a player to choose none of the options, we must include that as an outcome, as we did in the Ice Cream Parlor scenario.) For (a), we will assign utilities so that $u(A)>u(B)$. Likewise for (b) we will assign utilities so that $u(B)>u(A)$. Finally for (c), since Scarlett is willing to flip a coin to determine the outcome, we assume she is indifferent between $A$ and $B$ and assign $u(A)=u(B)$.

## 经济代写|博弈论代写Game Theory代考|VON NEUMANN-MORGENSTERN UTILITIES

A significant limitation of ordinal preferences and their associated utility functions is that they cannot describe the intensity of a player’s preference for a particular outcome. That is, they cannot capture the difference between Scarlett preferring vanilla ice cream over chocolate ice cream and Scarlett so strongly preferring vanilla that she would pay for it rather than have a free serving of chocolate. Notice how we have once again translated our intuitive sense of internal preference intensity into something that is observable (a real-world choice) so we can create a utility function based on these choices. While asking players to choose among outcomes that include the receipt or payment of money would be one observable way to determine intensity of preferences, we will take an approach that does not rely on the availability of money.

We begin by introducing a new, probability-based outcome called a lottery. Suppose that when a second customer, Regis, enters the ice cream parlor, he encounters a college student conducting a taste test involving different flavors of ice cream. The college student offers Regis the choice of either a sample of chocolate ice cream (his second-most favorite) or an unknown sample that is either vanilla (his favorite) or strawberry (his least favorite). The second option in this example is a simple lottery.

Definition 2.2.1. Given a set of outcomes, $O$, a simple lottery is a probability distribution over this set. When $O=\left{o_1, o_2, \ldots, o_m\right}$, a finite set, a simple lottery can be denoted by $p_1 o_1+p_2 o_2+\ldots+p_m o_m$ where $p_i$ is the probability of outcome $o_i$. A compound lottery is a probability distribution over other lotteries. Because an outcome $o \in O$ can be written as the simple lottery $1 o$, we see that $O \subset \mathcal{L}$, the set of all (simple and compound) lotteries.
Tó reveál thè strength of a player’s préference for one outcomé over another, wè must examine not nnly chnires hetween single nutromes hut chnices hetween single nutcomes and lotteries. Suppose Regis prefers vanilla $V$ over chocolate $C$ and prefers $C$ over strawberry $S$. The choice Regis makes between $C$ and the lottery $0.5 S+0.5 V$ tells us about the strength of his preference for $V$ over $C$ and for $C$ over $S$. If he would choose either of the two possibilities, then the strength of Regis’s preference for $C$ is exactly halfway between $S$ and $V$. If Regis were to choose $C$ over $0.5 S+0.5 \mathrm{~V}$, it reveals that his preference intensity for $C$ is closer to $V$ than to $S$. If Regis were willing to choose either $C$ or the lottery $0.1 S+0.9 \mathrm{~V}$, it would reveal that his preference for $V$ over $C$ is very small and his preference for $C$ over $S$ is relatively large. However, if he were instead willing to choose $C$ or the lottery $0.9 S+0.1 \mathrm{~V}$, it would reveal a strong preference for $V$ over $C$ and that his preferences for $C$ over $S$ is small. When a player is willing to choose either of two lotteries, this reveals the player is indifferent between these choices. This motivates the following generalization of the utility function concept.

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|VON NEUMANN-MORGENSTERN UTILITIES

O=\left{0_1，o_2, Vdots, o_m\right } } \text { ，一个有限集，一个简单的彩票可以表示为 }
$p_1 o_1+p_2 o_2+\ldots+p_m o_m$ 在哪里 $p_i$ 是结果的概率 $o_i$. 复合彩票是其他彩票的概率分布。因为一个结 果 $o \in O$ 可以写成简单的彩票 $l o$ ，我们看到 $O \subset \mathcal{L}$ ，所有（简单和复合） 彩票的集合。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 经济代写|宏观经济学代写Macroeconomics代考|ECON1120

statistics-lab™ 为您的留学生涯保驾护航 在代写宏观经济学Macroeconomics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写宏观经济学Macroeconomics代写方面经验极为丰富，各种代写宏观经济学Macroeconomics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|宏观经济学代写Macroeconomics代考|The Shipbuilding Model and the Lambert Function

In his 1931 article on the “Shipbuilding cycle,” Tinbergen was interested in the increase of tonnage that followed the building of ships with a lag of about one year due to the construction period. From this connection a relation emerged between the increase of total tonnage and the volume of total tonnage two years before (Tinbergen, 1959: 2). Tinbergen proposed to model this relation as a differential equation with a delay, of the form:
$$\dot{f}(t)=-a f(t-\theta),$$
where $f(t)$ is the tonnage as a function of time, $\theta$ represents the delay between the tonnage and its increase in $t$, and $a$ is the intensity of the relation, the volume of increase above the trend (linbergen, 1959: 3). Tinbergen assumed a solution to his equation of the form $f(t)=C e^{\lambda t}$, which, inserted into the equation above yields $\lambda=-a e^{-\lambda \theta}$, once we have simplified the $C$ which only depends on the initial conditions. $\lambda$ can be a real or complex number, but because it appears both in the exponential function and alone this equation is transcendental. Now, “transcendental” means that usually the answer will only be found “experimentally” as Tinbergen put it. Indeed a transcendental equation is periodic, in the same sense that the exponential function with an imaginary argument traces a circle repeating itself as the argument increases. To find a general solution, Tinbergen (and after him Frisch and Kalecki) separated the real and the imaginary part of this equation and solved for one of the two in terms of trigonometric functions; for instance Tinbergen obtained the equation:
$$b \frac{\sin (y)}{y}=e^{-\frac{y}{\tan (y)}},$$
where $b=a \theta$ and $\lambda=x+i y, i=\sqrt{-1}$. To find the solutions for $y$ of this type of equations, they took the same approach of plotting both sides and looking for points of intersection, before improving on this solution with simple algorithms. Figure $2.4$ shows in the solid red lines the right hand side of the Eq. 2.2, while the dashed lines are the left hand side, for three different values of $b$ and both as a function of $y$.
It is readily apparent that there will be only one solution in each interval of length $2 \pi$. We can see that the leftmost solution will have the lowest frequency, that is, the largest period, and that all other solutions will have a higher frequency; thus the roots of the characteristic equation above will be ordered by their decreasing period or increasing frequency.

## 经济代写|宏观经济学代写Macroeconomics代考|From Natural Sciences to Economics

Hamburger claimed that it was only with new mathematical tools that economists would be able to account quantitatively and qualitatively for economic processes. Ultimately, the aim was to transform economics into a science similar to biology, ${ }^5$ a science capable of understanding the operation of social organisms beset by recurrent “pathologies.” It should be noted that Hamburger was not the only economist interested in business cycles who was showing some discontent with a mechanical analogy. Ernst Wagemann, the German head of the imperial statistical office and of the business cycle research institute of Berlin, in a book published in 1928, called as well for a biological metaphor.

Although Wagemann wrote in German, his book met enough success to warrant its translation in English only two years later under the title Economic Rhythm: A Theory of Business Cycles, with a prefatory note from Wesley C. Mitchell (Wagemann, 1930). In the preface to the English edition, Wagemann presented his contribution as a small step “toward the repayment of the debt which Europe owes to America in the field of research into economic dynamics” (Wagemann, 1930: v). However, the type of dynamics that was applied remained very empirical: although Wagemann was searching for a theory, he steered resolutely away from abstract constructions which were heavily criticized. His review of existing theories led him to propose that “while the American methods are those of engineering, and the Russian those of astronomy, the German institute represents the medical, or, better, the organicbiological point of view” (1930: 10). The “organic-biological principle” which he described (with reference to Menger) was meant to capture both the interconnection of the separate parts of an (economic) organism as well as “a peculiarity which may be defined as consisting in the power to regulate its own movement” (1930: 11), an approach which he emphasized as “anything but mechanical” (1930: 11).

Another radical opinion on the business cycle was that it was only a “myth.” This opinion was shared among American economists and statisticians, such as Carl Snyder (1930) and Irving Fisher (1925). ${ }^6$ While the former based his claim on the fact that compared to the growth of the economy, the amplitude of fluctuations remained within certain limits, the second doubted that “inherent” cyclical regularity in business could be detected. For Fisher, even if there existed a simple self-generating cycle similar to that of a pendulum swinging under the influence of the force of gravity, its tendency to materialize would be necessarily “defeated in practice” (Fisher, 1925: 192). To show this, he proposed to move away from the pendulum metaphor, toward the “physical analogue” of “the sway of the trees or of their branches.” For instance, after a tree is bended, one observes a swaying movement similar to that of the cycle: but Fisher did not think that such a movement was actually observed in the woods: “in actual experience […] twigs or tree tops seldom oscillate so regularly, even temporarily; they register instead, chiefly the variations in wind velocity” (Fisher, 1925: 192). A steady wind as well as any “outside forces”7 may thus bend the trees for weeks and annihilate completely their tendency to swing back and forth while changes in wind speed or in its direction will simply modify the angle of the tree with the ground.

# 宏观经济学代考

## 经济代写|宏观经济学代写Macroeconomics代考|The Shipbuilding Model and the Lambert Function

$$\dot{f}(t)=-a f(t-\theta),$$

$$b \frac{\sin (y)}{y}=e^{-\frac{y}{\tan (y)}},$$

## 经济代写|宏观经济学代写Macroeconomics代考|From Natural Sciences to Economics

Hamburger 声称，只有使用新的数学工具，经济学家才能对经济过程进行定量和定性分析。最终，目标 是将经济学转变为类似于生物学的科学， ${ }^5$ 一门能够理解被反复出现的”病态” 困扰的社会有机体运作的科 学。应该指出的是，汉堡并不是唯一对商业周期感兴趣并对机械类比表示不满的经济学家。德国帝国统 计局局长兼柏林商业周期研究所所长恩斯特. 瓦格曼 (Ernst Wagemann) 在 1928 年出版的一本书中也呼 吁使用生物学隐喻。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。