Posted on Categories:Macroeconomics, 宏观经济学, 经济代写

# 经济代写|宏观经济学代考Macroeconomics代写|EC1B1 A Model with Private Labor Effort

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 经济代写|宏观经济学代考Macroeconomics代写|A Model with Private Labor Effort

The model of the previous section was very useful in thinking about equilibrium in the credit market. However, since in the real world we do not necessarily recognize “type1 ” and “type-2” agents from birth, and we do not readily observe different audit costs, it makes sense to consider a different model. In this model all agents will be identical except for their wealth level. Some agents will be richer, others poorer. There will be no audit cost, and output will be public. However, agents are going to have to work to make the project succeed, and the amount of their labor effort will be private (that is, hidden). It can never be known (cannot be audited). This is another example of moral hazard.

Technology, Endowments, Preferences
All agents will have access to a common “back-yard” technology which will map capital $k$ and labor effort $\ell$ into a probability that the project succeeds. If the project succeeds, output is high, at $q$. If it fails, output is zero. Capital $k$ can take on only two values: $k=0$ or $k=1$. If $k=1$ then the probability of the high output is just given by the level of labor effort, $\ell$. If $k=0$ then the low output occurs with certainty, no matter what the effort level was.

Agents are all endowed with some level of wealth $a$. For a particular agent, if $a<1$, then that agent must get a loan of size $1-a$ to operate the technology. If $a>1$, then the agent can finance the technology alone and lend the surplus $a-1$.
Agents have preferences over consumption $c$ and labor effort $\ell$ given by:
$$c-\frac{q}{\alpha} \frac{\ell^2}{2}$$
Here $0<\alpha<1$ and $q$ is just the high output level.

## 经济代写|宏观经济学代考Macroeconomics代写|Rich Age

Assume that there is some riskless rate of return $r$ on wealth. A rich agent, one with $a>1$, can finance the project from her own wealth and lend the remainder at this interest rate $r$. How much effort does she supply? If the project fails and output is zero, she consumes $c=0+(1+r)(a-1)$. If the project succeeds and output is $q$, she consumes $c=q+(1+r)(a-1)$. The project succeeds with probability $\ell$, her labor effort. Thus her maximization problem is:
$$\max _{\ell}\left{\ell[q+(1+r)(a-1)]+(1-\ell)[(1+r)(a-1)]-\frac{q}{\alpha} \frac{\ell^2}{2}\right}$$
The first-order condition for maximization with respect to $\ell$ is:
$$q-\frac{q}{\alpha} \ell=0$$
This implies that $\ell^{\star}(a)=\alpha$ for $a \geq 1$. That is, agents wealthy enough to finance the project out of their own funds all supply the same labor effort, $\alpha$.

Given that all rich agents supply $\ell=\alpha$ regardless of their wealth, we can easily calculate their expected utility by plugging back in:
$$\begin{gathered} \alpha q+(1+r)(a-1)-\frac{q}{\alpha} \frac{\alpha^2}{2}, \text { or: } \ \frac{\alpha q}{2}+(1+r)(a-1) . \end{gathered}$$

Thus the value from operating the technology is $\alpha q / 2$, and the value from lending any excess capital is $(1+r)(a-1)$. The opportunity cost of the capital used in the productive process is $(1+r) \cdot 1$ (since production requires one unit of capital). Thus for any agent to undertake the project, it must the case that:
$$\frac{\alpha q}{2} \geq 1+r$$
For the rest of this section we are going to assume that the interest rate $r$ is below $\alpha q / 2-1$, so agents will want to use the productive technology.

# 宏观经济学代写

## 经济代写|宏观经济学代考Macroeconomics代写|A Model with Private Labor Effort

$$c-\frac{q}{\alpha} \frac{\ell^2}{2}$$

## 经济代写|宏观经济学代考Macroeconomics代写|Rich Age

$$q-\frac{q}{\alpha} \ell=0$$

$$\alpha q+(1+r)(a-1)-\frac{q}{\alpha} \frac{\alpha^2}{2}, \text { or: } \frac{\alpha q}{2}+(1+r)(a-1)$$

$$\frac{\alpha q}{2} \geq 1+r$$

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

## 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。