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• 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 Lucas island model

The challenge to Keynesian orthodoxy, and hence the initial push from which modern Keynesian theories were built, took the shape of the pioneering model by Lucas (1973) – part of his Nobel-winning contribution. He derived a positively-sloped AS curve in a model founded at the individual level, and where individuals had rational expectations. This model also more explicit the role of expectations in constraining aggregate demand policy. The key idea was that of imperfect information: individuals can observe quite accurately the prices of the goods they produce or consume most often, but they cannot really observe the aggregate price level. This means that, when confronted with a higher demand for the good they produce, they are not quite sure whether that reflects an increase in its relative price – a case in which they should respond by increasing their output – or simply a general increase in prices – a case in which they should not respond with quantities, but just adjust prices. We will see that rational expectations implies that individuals should split the difference and attribute at least part of the increase to relative prices. (How much so will depend on how often general price increases occur.) This yields the celebrated Lucas supply curve, a positively-sloped supply curve in which output increases when the price increases in excess of its expected level.

The model is one with many agents (Lucas’s original specification places each person on a different island, which is why the model is often referred to as the Lucas island model). Each agent is a consumer-producer that every period sees a certain level of demand. The basic question is to figure out if an increase in demand is an increase in real demand, which requires an increase in production levels, or if it is simply an increase in nominal demand, to which the optimal response is just an increase in prices. The tension between these two alternatives is what will give power to the model. In order to solve the model we will start with a specification with perfect information and, once this benchmark case is solved, we will move to the case of asymmetric information, which is where all the interesting action is.

## 经济代写|宏观经济学代写Macroeconomics代考|The model with perfect information

The representative producer of good $i$ has production function
$$Q_i=L_i,$$
so that her feasible consumption is
$$c_i=\frac{P_i Q_i}{P} .$$
Utility depends (positively) on consumption and (negatively) on labour effort. Let’s assume the specification
$$u_i=c_i-\frac{1}{\gamma} L_i^\gamma \quad \gamma>1 .$$
If $P$ is known (perfect information), the problem is easy; the agent has to maximize her utility (15.13) with respect to her supply of the good (which is, at the same time, her supply of labour). Replacing (15.11) and (15.12) in (15.13) gives
$$u_i=\frac{P_i L_i}{P}-\frac{1}{\gamma} L_i^\gamma .$$

The first order condition for $L$ is
$$\frac{P_i}{P}-L_i^{\gamma-1}=0,$$
which can be written as a labour supply curve
$$L_i=\left(\frac{P_i}{P}\right)^{\frac{1}{\gamma-1}},$$
or, if expressed in logs (denoted in lower case letters), as
$$l_i=\left(\frac{1}{\gamma-1}\right)\left(p_i-p\right)$$

# 宏观经济学代考

## 经济代写|宏观经济学代写Macroeconomics代考|The model with perfect information

$$Q_i=L_i,$$

$$c_i=\frac{P_i Q_i}{P} .$$

$$u_i=c_i-\frac{1}{\gamma} L_i^\gamma \quad \gamma>1 .$$

$$u_i=\frac{P_i L_i}{P}-\frac{1}{\gamma} L_i^\gamma .$$
-阶条件为 $L$ 是
$$\frac{P_i}{P}-L_i^{\gamma-1}=0,$$

$$L_i=\left(\frac{P_i}{P}\right)^{\frac{1}{\gamma 1}},$$

$$l_i=\left(\frac{1}{\gamma-1}\right)\left(p_i-p\right)$$

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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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