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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|What have we learned

The Solow model shows that capital accumulation by itself cannot sustain growth in per capita income in the long run. This is because accumulation runs into diminishing marginal returns. At some point the capital stock becomes large enough – and its marginal product correspondingly small enough that a given savings rate can only provide just enough new capital to replenish ongoing depreciation and increases in labour force. Alternatively, if we introduce exogenous technological change that increases productivity, we can generate long-run growth in income per capita, but we do not really explain it. In fact, any differences in long-term growth rates come from exogenous differences in the rate of technological change – we are not explaining those differences, we are just assuming them! As a result, nothing within the model tells you what policy can do about growth in the long run.

That said, we do learn a lot about growth in the transition to the long run, about differences in income levels, and how policy can affect those things. There are clear lessons about: (i) convergence the model predicts conditional convergence; (ii) dynamic inefficiency – it is possible to save too much in this model; and (iii) long-run differences in income – they seem to have a lot to do with differences in productivity.

Very importantly, the model also points at the directions we can take to try and understand longterm growth. We can have a better model of savings behaviour: how do we know that individuals will save what the model says they will save? And, how does that relate to the issue of dynamic inefficiency?

## 经济代写|宏观经济学代写Macroeconomics代考|The Ramsey problem

We have seen the lessons and shortcomings of the basic Solow model. One of its main assumptions, as you recall, was that the savings rate was constant. In fact, there was no optimisation involved in that model, and welfare statements are hard to make in that context. This is, however, a very rudimentary assumption for an able policy maker who is in possession of the tools of dynamic optimisation. Thus we tackle here the challenge of setting up an optimal program where savings is chosen to maximise intertemporal welfare.

As it turns out, British philosopher and mathematician Frank Ramsey, in one of the two seminal contributions he provided to economics before dying at the age of 26, solved this problem in 1928 (Ramsey (1928)). ${ }^{1}$ The trouble is, he was so ahead of his time that economists would only catch up in the 1960s, when David Cass and Tjalling Koopmans independently revived Ramsey’s contribution. ${ }^{2}$ (That is why this model is often referred to either as the Ramsey model or the Ramsey-Cass-Koopmans model.) It has since become ubiquitous and, under the grand moniker of Neoclassical Growth Model (NGM), it is the foremost example of the type of dynamic general equilibrium model upon which the entire edifice of modern macroeconomics is built.

To make the problem manageable, we will assume that there is one representative household, all of whose members are both consumer and producer, living in a closed economy (we will lift this assumption in the next chapter). There is one good and no government. Each consumer in the representative household lives forever, and population growth is $n>0$ as before. All quantities in small-case letters are per capita. Finally, we will look at the problem as solved by a benevolent central planner who maximises the welfare of that representative household, and evaluates the utility of future consumption at a discounted rate.

At this point, it is worth stopping and thinking about the model’s assumptions. By now you are already used to outrageously unrealistic assumptions, but this may be a little too much. People obviously do not live forever, they are not identical, and what’s this business of a benevolent central planner? Who are they? Why would they discount future consumption? Let us see why we use these shortcuts:

1. We will look at the central planner’s problem, as opposed to the decentralised equilibrium, because it is easier and gets us directly to an efficient allocation. We will show that, under certain conditions, it provides the same results as the decentralised equilibrium. This is due to the so-called welfare theorems, which you have seen when studying microeconomics, but which we should perhaps briefly restate here:
a. A competitive equilibrium is Pareto Optimal.
b. All Pareto Optimal allocations can be decentralised as a competitive equilibrium under some convexity assumptions. Convexity of production sets means that we cannot have increasing returns to scale. (If we do, we need to depart from competitive markets.)

# 宏观经济学代考

## 经济代写|宏观经济学代写Macroeconomics代考|The Ramsey problem

1. 我们将研究中央计划者的问题，而不是去中心化均衡，因为它更容易并且让我们直接获得有效的分配。我们将证明，在某些条件下，它提供与分散均衡相同的结果。这是由于所谓的福利定理，你在学习微观经济学时已经看到了，但我们或许应该在这里简单地重申一下
：竞争均衡是帕累托最优的。
湾。在某些凸性假设下，所有帕累托最优分配都可以作为竞争均衡分散。生产集的凸性意味着我们不能获得规模报酬递增。（如果我们这样做了，我们就需要离开竞争市场。）

## 有限元方法代写

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## 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。

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