1. Consider the production function Y = AK1/3H 2/3L 2/3 . Here, "Y" is a measure of real GDP, "A" is a measure of technology, "K" is a measure of capital stock, "H" is a measure of human capital stock, and L is a measure of the amount of labor in the economy.

a. Graph Y as a function of K, holding A, H, and L constant (Hint: the easiest way to do this is to set A, H, and L equal to 1).

b. Are there increasing, decreasing, or constant returns to capital?

c. Now, suppose we care more about GDP per capita than we do about total GDP. This is given by the expression y = AH2/3k 1/3 , where y = GDP per capita, k = capital per worker. Graph y as a function of H, holding k constant. Are there increasing or decreasing returns to H?

d. The savings rate in the U.S. was about 7.2% before the pandemic. The depreciation rate was about 3.6%. Do you think the amount of capital in the U.S. was increasing, decreasing, or staying the same?

e. Graph Y as a function of A, keeping ceteris paribus, or all else equal. Are there increasing, decreasing, or constant returns to technology?

f. In class, we learned that the Solow model allows for infinite growth through technology. Let's change our production function to Y = A 1/2K 1/3H 2/3L 2/3 . Does this new production function allow for infinite growth? Why or why not?

2. Consider the trading city of Eutropia, which has 20 people. Their incomes are presented below:

Person

Annual Income

A.J. $54,800

Arjun $25,200

Charmaine $6,600

David $17,400

Elida $14,700

Emily $194,000

Evan $4,900

Frank $75,900

Feliks $25,400

Grant $267,100

Hannah $196,700

Isabella $146,600 J

ohn $23,200

Lola $40,500

Nathan $4,000

Rachel $10,900

Sam $208,700

Tevin $22,900

Tommy $711,800

Zach $961,800

a. What is the mean of the data? What is the median?

b. Remember that $16,300 is considered the poverty line. What percentage of this town is below the poverty line? Considering that the poverty rate is about 13.7%, is this high or low?

c. Consider that average income in the U.S. is about $54,000. Would you say that Eutropia is a wealthy city or a poor city?

d. Now, suppose the citizens of Eutropia voted on a progressive tax code. Income up to $25,000 is not taxed, income from $25,000 to $200,000 is taxed at 30%, and income above $200,000 is taxed at 80%. What percentage of her income does Hannah pay? What percentage of their income does Sam pay?

e. Suppose Sam can work less and only make $200,000. How much do they pay? What is their after-tax income when they work more and when they work less?

f. The citizens of Eutropia are very concerned about fairness. They see owning a home as a key requirement for living comfortably in the middle class. Everyone in Eutropia making less than $75,000 rents, while everyone making $75,000 to $150,000 owns 1 home, everyone making $150,000 to $500,000 owns 2 homes, and Tommy and Zach both own 8 homes each (presumably, other citizens rent from them). Under the Eutropia Home Credit Program, citizens can subtract $75,000 from their total income for each home that they own, and only pay taxes on the remainder. Calculating as a percentage of pre-deduction income, what percentage of her income does Hannah pay? What about Sam? What about Tommy?

3. In Germany and some other countries, there is a type of social insurance, funded through income taxes, where the government lets companies pay workers during recessions and the government itself picks up some of the difference, so that businesses do not have to fire and rehire employees.

a. This targets which type of unemployment?

b. It costs about $4,000 to hire an employee. Germany has a 60% labor force participation rate and a population of 83 million people. If unemployment increases from 5% to 10%, how much would it cost to rehire all of the workers as unemployment comes back down to 5%?

c. With the social insurance program, only 1% of workers need to be rehired (unemployment increases from 5% to 6%). Instead, each worker makes $25,000 and the government gives each worker an additional $12,500. With this plan, each worker makes $12,500 less than they would when the economy is not in recession, but most keep their jobs. If they lost their jobs, they would lose $50,000. Additionally, suppose the program is financed with $5,000 per person per year in taxes, and there are 5 good years for every bad year. Is there a net economic benefit? (Hint: think about what the next best option would be).