Assignment-daixieTM为您提供利物浦大学University of Liverpool PRINCIPLES OF MICROECONOMICS ECON121微观经济学原理代写代考和辅导服务！
Instructions:
This module aims to provide students with a strong understanding of neo-classical microeconomics, which is a fundamental topic in economics. The module is designed to equip students with the mathematical tools necessary to analyze microeconomic problems and to introduce them to theoretical models and their applications. The customized textbook will provide students with a comprehensive resource that covers all aspects of microeconomics.
By the end of this module, students should have a solid understanding of the basic principles of microeconomics, including consumer behavior, producer behavior, market equilibrium, and welfare analysis. They should also be able to apply mathematical tools such as calculus, optimization, and equilibrium analysis to solve microeconomic problems. This knowledge will form the foundation for future courses in microeconomics and related subjects, such as industrial organization, labor economics, and game theory.
To succeed in this module, students should engage fully with the course material, participate actively in class discussions, and complete all assignments on time. They should also seek help from their instructors or tutors if they encounter any difficulties. With a strong foundation in microeconomics, students will be well-prepared to pursue advanced studies in economics or to apply their skills in a wide range of professional settings.
An economy has two agents, Bill and Bob. Bill has $\$ 110$, and Bob has $\$ 200$. Utility of agents in this economy is characterized by the following function of income:
$$
U=u(y)= \begin{cases}\log (y-60) & \text { if } y<160 \ \frac{1}{80} y & \text { if } y \geq 160\end{cases}
$$
The minimum level of income possible in this economy is 60 .
Each agent is about to choose a new business venture, and has a choice between project A and project B. Neither project requires any investment up front. Project A yields revenues of 20 with probability $\frac{1}{2}$ and revenues of -10 with probability $\frac{1}{2}$. Project B yields revenues of 4 with probability one-half and revenues of 5 with probability one-half. Throughout this problem, assume that fractional income is possible.
(a) Which project would each agent choose? Provide intuition for your answer.
Bill is in the risk-averse portion of the utility function, and so he will choose project $B$ where there is no possibility of a loss. Bob is risk-neutral, and so he chooses the pro ject with higher expected returns, which is pro ject $A$. We can verify that Bill receives higher utility from pro ject $B$.
$$
\begin{aligned}
E\left[U_{\text {Bill }}, B\right] & =\frac{1}{2} \log 54+\frac{1}{2} \log 55=\frac{1}{2} \log 2970 \
E\left[U_{\text {Bill }}, A\right] & =\frac{1}{2} \log 40+\frac{1}{2} \log 70=\frac{1}{2} \log 2800
\end{aligned}
$$
Clearly, the first expression is higher.
If Bill and Bob each choose an investment project each year and receive the associated income for 20 years, will the expected gap in their incomes be larger or smaller at the end of this period than it was initially? How does this relate to attitudes toward risk? You do not need to calculate income over 20 years, just provide intuition.
The income gap will be la rger. Eventually Bill will get to a point on his utility function where it is optimal for him to choose the more lucrative project, but because he starts at a lower income level and initially chooses a less lucrative pro ject, the income gap will grow. Because there is risk aversion at lower levels of income, relatively poor a gents will not choose more profitable but riskier sources of income beca use the disutility of a decline of income from an already low level is high.
Now, assume that there is a job available that provides fixed wage income. What salary would the job have to provide in order to induce Bill to take the job rather than entering a new business venture? What salary would the job have to provide in order to induce Bob to take the job? Which is higher, and why? Algebraic expressions are acceptable as answers.
We need to provide each person with his certainty-equivalent level of income as a wage. For Bill, this is equal to $\sqrt{2970}-50$, or around \$4.50. Since Bo $b$ is risk neutral, we need to provide him with the same level of income he could earn in project $A$, which is \$5. The salary is higher for Bob-Bill has higher utility from the job by virtue of it being a steady stream of income as opposed to a gamble, and so a lower salary will make him satisfied. Bob is indifferent to risk, so he needs a higher salary.
