**Question 1.**

Suppose over recent years, real GDP the US (country A) has been growing at an average rate of 2% per year. Another country B’s real GDP is about 25% of the US real GDP in current year. Here we treat current real GDP of the US as 100 units. Then real GDP of country B is 25 units. Use the data given and finish the blanks in the following table. Keep two or three decimal digits for your answer.

Unit | Growth rate | Current GDP | 5-year GDP | 10-year GDP | 25-year GDP | 50-year GDP |

Country A | 2% | 100.00 | ||||

Country B | 6% | 25.00 | ||||

GDP ratio | 0.25 |

(1) If the US economic growth rate is 2% per year, then how much would be the US real GDP after 5 years, 10 years, 25 years and 50 years. You can use US current GDP (country A) 100 as starting point.

(2) Similarly, for country B, with a long-run growth rate at 6% per year, how much would be Country B’s real GDP in the next 5 years, 10 years, 25 years and 50 years. You can use current GDP of country B 25 as starting point.

(3) Following questions above, how much is the ratio of country B’s real GDP to the US GDP (GDP* _{B}*/GDP

*) in the next 5 years, 10 years, 25 years and 50 years.*

_{A}

**Question 2. **

Review the definitions of constant return to scale (CRS), increasing return to scale (IRS) and decreasing return to scale (DRS) to production functions in your notes. Explain the following production functions are CRS, IRS or DRS. To get full credit, you need show your work via step-by-step derivations.

(1) In country C, the production function determines national output (*Y*) is a function of capital (*K*) and labor (*L*) as below. Is the function CRS, IRS or DRS?

*Y* = A * *K** ^{(1/2)}* *

*L*

^{(2/3}

^{)};

(2) In country D, suppose in its production function, national output (*Y*) is a function of capital (*K*), labor (*L*) and exchange of information (*E*). Is the function CRS, IRS or DRS?

*Y* = A * *K** ^{(1/2)}* *

*L*

^{(1/4}

^{)}*

*E*

^{(1/4}

^{)};

**Question 3.**

The first column of the table below shows you the ratio real GDP per capita relative to the US level, across several countries. In the second column, you can see capital per capita (*k*) relative to the US level. Try to finish the table. Keep three decimal digits for your answer.

Country | GDP per capita (y) |
Capital per capita (k) |
Capital input k ^{(1/3)} |
TFP (A) |

US | 1.000 | 1.000 | ||

Switzerland | 1.147 | 1.416 | ||

UK | 0.733 | 0.833 | ||

Japan | 0.685 | 1.021 | ||

Italy | 0.671 | 1.125 | ||

Spain | 0.615 | 1.128 | ||

Brazil | 0.336 | 0.458 | ||

South Africa | 0.232 | 0.218 |

Note: Both GDP per capita and Capital per capita are the relative ratio to the levels of the US.

With the Cobb-Douglas production function we can calculate the value of national output or GDP (*Y*) with a combination of capital (*K*) and labor (*L*) inputs and total factor productivity (A).

*Y* = A * *K** ^{α}* *

*L*

^{(1-}

^{α)};

At per capita level, we can rewrite Cobb-Douglas function as below. Here *y* = *Y/L* and *k* = *K/L*. Here we have GDP per capita (*y*) and capital per capita (*k*). Also we assume α is the same.

y = A** k** ^{α}*;

(1) Fill in the blanks (green color) of the values of capital input in the fourth column. Now you have the value of parameter α = 1/3.

(2) Following the above question, now use GDP per capita (*y*) and capital per capita (*k*) to derive total factor of productivity (A) of each country. Fill in the blanks of pink color in the last column.

(3) Plot TFP (A) against GDP per capita (*y*). Is there any relationship between these two factors? More exactly, is it a positive or negative trend between them?

(4) Can you derive Cobb-Douglas function at per capita level using *y* and *k* from Cobb-Douglas function at aggregate level?

*Y* = A * *K** ^{α}* *

*L*

^{(1-}

^{α)};