(c) \((5\%)\) Derive the steady-state unemployment rate for the economy.

We usually observe that the less educated workers tend to hold manual and service occupations. Besides, the unemployment rate for the less educated workers is found to be higher than that for more educated workers. Furthermore, the less educated were found to be hurt the most in terms of job loss during the COVID-19 pandemic. (d) \((3\%)\) Which of the sector in the model corresponds to the "manual and service" in the description?

(e) (5%) Suppose that there is a developed North country and a developing South country. Suppose that prior to the pandemic, the two countries have the same sector specific unemployment rate, but their labor force compositions are different, i.e., \( v^{N} \neq v^{S} \). Which country has a higher \( v \), the North or the South? Which country has a higher unemployment rate prior to the pandemic? How the pandemic affects the job separation rates and the job finding rates in the North and the South? Which country has a higher unemployment rate during the pandemic? Briefly explain your answer.

(f) (5%) Up so far, we have assumed that workers have no mobility across sectors. In reality, it is costly for workers to switch sectors. Discuss what could prevent workers from switching sectors. Formulate your answer by writing down an equation (equations) for a worker to be indifferent between switching across sectors.

4. (25%) Consider a simple two-period model where the world lasts only for two periods. An agent has one unit of labor and provides the labor to the market inelastically. The agent's budget constraints in the two periods are: $$c1 + b1 = w_1$$ $$c2 = b1(1 + r1) + w2$$ where $b1$ is the asset position of the agent in period 1, $w1$ and $w2$ are the real wage rate in periods 1 and 2, and $r1$ is the real interest rate from period 1 to period 2. The production technology in this economy is $$Y = AL$$ for both periods, where $L$ is total number of workers or agents in the economy. All markets are competitive. Answer the following questions. (a) \((2\%)\) What is the equilibrium wage rate in period 1 and period 2?

(b) \((3\%)\) Derive the two-period intertemporal budget constraint for the agent. Plot it on the \((c{1}, c{2})\)-space. Suppose that there is a government collecting taxes from and providing services to the agents in the economy. The government can choose to impose a consumption tax, a capital income tax or a labor income tax on the household.

(c) \((10\%)\) Let \(\tau{c}, \tau{k}\) and \(\tau_{w}\) denote the consumption tax rate, the capital income tax rate and the labor income tax rate. How will the taxes affect the agent's budget constraint? Write down the agent's budget constraints in periods 1 and 2 as well as the intertemporal budget constraint. Illustrate the intertemporal budget constraint under consumption tax, capital income tax and labor income tax in figures, respectively.

(d) \((10\%)\) Suppose that the government collects labor income taxes only. In both periods, the government provides services to all the agents in the economy, inclusive of public health services, national defense, and etc. Denote government expenditure by \(G\). Suppose that the technology scaling factor in the two periods, \(A{1}\) and \(A{2}\), are different and \(A{1} > A{2}\). Now the total taxes collected in period 1 is greater than \(G_{1}\). Do you think that the government should return the "excess" tax revenue to the agents in the economy? Discuss.

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