Question

Consider two economies: A and B. Both have identical parameters a =1/3, s =0.2, d = 0.05, L =30, and A =1. However, Economy A starts with K₀ = 100, while Economy B starts with K₀ = 300.

1. Which economy is richer in year 0 (calculate GDP per capita for each country)?

2. Which economy has higher consumption per person in year 0 (calculate C per capita)?

3. After 50 years, which economy is richer in year 0 (calculate GDP per capita for each country)?

4. After 250 years, which economy is richer in year 0 (calculate GDP per capita for each country)?

5. In light of your findings, does it matter – in the long run – where an economy starts (i.e. whether it starts rich or poor). What kind of experiments with the model would you conduct to make sure you can be confident about your conclusion?

Answer 1

Cobb- Douglas production function will be,

Y=A*K^a*L^(1-a)=1*K^1/3*L^2/3=K^1/3*L^2/3

1)

GDP per worker =y(Y/L)=k^1/3, where k is capital per worker.

Country A capital per worker=100/30=10/3

Country B capital per worker=300/30=10

Country A gdp per worker=(10/3)^1/3

Country B gdp per worker=(10)^1/3

So country B is richer as it's gdp per worker is higher.

2)C( country A)=y-sy=y-0.2y=0.8*y=0.8(10/3)^1/3

C( country B)=0.8*(10)^1/3

So country B has higher Consumption per worker

3) depreciation reduce capital stock and so reduce capital per worker.while saving turned into Investment and Increase capital stock and so Increase capital per worker.

Yearly growth of capital per worker,

∆k=sy-dk=0.2*(k)^1/3-0.05k

Steady state ,where ∆k=0

0.2*(k)^1/3=0.05k

4=k^2/3

k=4^3/2=8

50 years is very long run ,in which economies reach to their steady state level of capital per worker and gdp per worker.

Steady state capital per worker=8

Country A capital per worker in year=0 is lower than 8, so it's capital stock will increase and capital per worker will increase and reach to steady state,where

y=(8)^1/3=2

Country B capital per worker in year 0,is higher than 8, so depreciation will be higher than Investment,which will decrease capital stock and capital per worker and reach to 8 where depreciation and Investment will be equal.

y=(8)^1/3=2

In year 50, both countries will be Same gdp per worker,so both countries will at same level of richness.

4) because after reaching in steady state, capital per worker remain Constant and unchanged ,so gdp per worker also remain unchanged.

So both will be equally rich in year 250.

5) If there is no technology progess (positive Change in A) ,then it's doesn't matter country start rich or poor ,they will reach at same level of richness.

So In very long the country growth doesn't depends on investment or saving rate ,it depends on technology progess or productivity Increase.