Question

In: Economics

Consider two economies: A and B. Both have identical parameters a =1/3,s =.2, L =30, and...

Consider two economies: A and B. Both have identical parameters a =1/3,s =.2, L =30, and A =1. However, Economy A has depreciation rate d = 0.03 while Economy B has d = 0.10. Both economies start with K0 = 10.

  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 – how quickly the capital stock depreciates in an economy.

Solutions

Expert Solution

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=10/30=1/3

Country B capital per worker=10/30=1/3

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

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

So both countries are equally rich as both have same gdp per worker.

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

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

So country have same 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

Stead state ,where ∆k=0

county A,

0.2*k^1/3=0.03k

20/3=k^2/3

k=20/3)^3/2

y=k^1/3=(20/3)^3/2}^1/3=20/3)^1/2

Country B,

0.2*k^1/3=0.1k

2=k^2/3

k=2^3/2

y=k^1/3={2^3/2}^1/3=2^1/2

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

So in steady state ,country A is Richer than country B as GDP per worker of country A is higher.

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

So country A will be Richer in 250 years.

5) If there is no technology progess (positive Change in A) ,then In very long or steady state all economies of reach at a constant capital per worker and gdp per worker.The economy with a higher saving or Investment rate or lower depreciation rate will be at a higher level of capital per worker and gdp per worker but in steady state there won't be any growth. So yes it matters,how quickly capital is depreciated.

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


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