Question

(Hybrid Harrod-Domar-Solow Model) An economy has a population of 2 million, the current capital stock of $6 billion, and a current GDP of $3 billion. The savings rate is a constant 8% and depreciation rate is 3%. The population growth rate is 0. Its production function is given by Yt=AtKt, where Yt denotes GDP, Kt denotes capital stock and At denotes productivity of capital in year t. Capital productivity will remain at its current level until the economy achieves a per capita income of $2000. Between per capita income of $2000 and $3000, capital productivity will be at a constant level, which will be 10% lower than what it is currently, owing to some natural resource (energy) constraints. Between per capita income of $3000 and $4000, capital productivity will also be at a constant level, which will be 10% lower what it would be between per capita income of $2000 and $3000. And so on: for every successive range of per capita income of a thousand dollars, capital productivity will be (constant at a level which is) 10% lower what it was for the previous range.

A) Calculate the current and future growth rates of per capita income. how will they differ?

B) What will the growth rate and level of per capita income be in the long-run?

Answer 1

(Hybrid Harrod-Domar-Solow Model) An economy has a population of 2 million the current capital stock of $6 billion, and a current GDP of $3 billion. The savings rate is a constant 8% and depreciation rate is 3%. The population growth rate is 0. Its production function is given by Yt=AtKt, where Yt denotes GDP, Kt denotes capital stock and At denotes productivity of capital in year t.

A) Calculate the current and future growth rates of per capita income. how will they differ?

Production Function Yt = At*Kt------------------ (equation 1)

Yt is GDP which is $3 billion, Kt is Capital stock which is $6 billion and At is Productivity of Capital which is need to calculate

According to the given production fuction we have to place the value of GDP & Capital stock

So Yt = At*Kt

3=At*6

At= 3/6 = 0.5 , Productivity of Capital (At) = 0.5 and most of the time productivity of capital mention as α(Lambda)

So α= 0.5 until per capita income 2000

after that productivity of capital is going fall and 10% lower that mean 0.5* 0.9 = 0.45

Previously the productivity of capital was 0.5 and after that it was 0.45

So α= 0.45 until per capita income 3000

From below & equal to 2000 and α=0.5

then the aggregate growth (g) = 8* (0.5)-3 = 1% ( where g= growth rate, 8 is saving rate , 0.5 is capital productivity and 3 is depreciation)

From 2000 to 3000 and  α=0.5* 0.9 = 0.45

then the aggregate growth (g) = 8* (0.5)*(0.9)-3 = 0.6%

From 3000 to 4000 and  α=0.5* (0.9)2

then the aggregate growth (g) = 8* (0.5)*(0.9)2 -3 = 0.24%

above 4000 and  α=0.5* (0.9)3

then the aggregate growth (g) = 8* (0.5)*(0.9)3 -3 = -0.08%

The above mathematical conclusion shows that when the percapita income above 2000 then the growth rate is falling and one interesting facts comes that if it is above 4000 then the growth rate is going to negative.

B) What will the growth rate and level of per capita income be in the long-run?

Ans:- In the above final output shows that when the per capita income go on and on then growth rate decline and negative when the PCI is above 4000 but in the long run growth rate will be zero and PCI will stabilize at 4000 US $