Problems

Of course! Here are the remaining Anki cards based on your problems:

Card 9:

Q: How do you calculate national savings, public savings, and private savings in an economy with the following variables?

Y=1000, G=180, T=300, C=160+0.7(Y−T), I=400−25r

A:

  • National Savings = Private Savings + Public Savings

  • Private Savings = Y−T−C

  • Public Savings = T−G

Card 10:

Q: How much richer will the fast-growing country be after 10, 20, 50, and 100 years, given the following growth rates?

  • Country 1: 2% growth rate per year

  • Country 2: 3% growth rate per year

A: Use the formula Future Value=Present Value×(1+growth rate)time

  • After 10 years: (1+0.03)10 and (1+0.02)10

  • After 20 years: (1+0.03)20 and (1+0.02)20

  • After 50 years: (1+0.03)50 and (1+0.02)50

  • After 100 years: (1+0.03)100 and (1+0.02)100

Card 11:

Q: How do you calculate the average annual growth rate of GDP over a 20-year period, given GDP in the present and 20 years ago?

  • GDP today: $3 billion

  • GDP 20 years ago: $2 billion

A: Use the formula:
Growth Rate=(GDP TodayGDP 20 Years Ago)120−1

Card 12:

Q: How long will it take for GDP to quadruple, assuming a constant growth rate of 5% per year?

A: Use the Rule of 70:
Time to Quadruple=log⁡(4)log⁡(1.05)

Card 13:

Q: What was GDP 15 years ago, assuming a constant growth rate of 2% per year, given current GDP is $3 billion?

A: Use the formula:
GDP in the Past=Current GDP÷(1+growth rate)time

Card 14:

Q: How do you decompose the growth rate of GDP into the contributions of technology, capital, and labor in the production function Y(t)=A(t)K(t)0.3L(t)0.7?

A: Derive each factor:

  • Technology: dAA

  • Capital: 0.3⋅dKK

  • Labor: 0.7⋅dLL

Card 15:

Q: If GDP grows at 2% per year, capital grows at 0.5%, and labor grows at 1%, what are the contributions of technology, capital, and labor to GDP growth?

A:

  • Technology contribution: 2%−(0.5%×0.3)−(1%×0.7)

  • Capital contribution: 0.5%×0.3

  • Labor contribution: 1%×0.7

Card 16:

Q: How would you compute the contributions of technology, capital, and labor to GDP growth if output is given by Y(t)=A(t)K(t)αL(t)β and you do not know the values of α and β?

A: Estimate α and β using data for capital and labor shares from national accounts. Use these values to compute the contributions.

These cards should cover the key concepts in the remaining problems. Let me know if you'd like to adjust any of the cards or if you'd like me to generate more!