Question

7. Let K(t) denote the quantity of capital a country has at the beginning of period t. Also, assume that capital depreciates at a constant rate d, so that dK(t) of the capital stock wears out during period t. If investment during period t is denoted I(t), and the country does not trade with the rest of the world, then we can say that the quantity of capital at the beginning of period t+1 is given by K (t1 (1-d) K (t) +I (t) Suppose at the beginning of year 0 that this country has 100 units of capita Investment expenditures are 15 units each in the years0, 1, 2, 3, 4 and 5. The capital stock depreciates at 10% per year. Calculate the quantity of capital at the beginning of years 0, 1, 2, 3, 4 and 5

Answer 1

K(t-1)=(1-d)K(t) + I(t)

Given that depreciation rate is constant at 10% each year and innvestment expenditures are also 15 units each year, we can simplify the above equation as

K(t-1) = (1- 10/100)K(t) + 15 = (100-10)100 * K(t) + 15

= (90/100)K(t) + 15

At the beginning of period 0, I.e. t=0 .K(0)= 100 units

Insubsequent periods capital stock is such

In t+1=0+1,

K(1) = (90/100)K(0) + 15 = (90/100)100 + 15 =105 units.

In t+2= 2 K(2) = (90/100)K(1) + 15 = (90/100)105 + 15 = 109.5 units.

In t+3= 3 . K(3) = (90/100)K(2) + 15 = (90/100)109.5 + 15= 113.55 units

In t+4 = 4 . K(4) = (90/100)K(3) + 15 = (90/100)113.55+ 15= 117.195 units

In t+5=5 K(5) = (90/100)K(4) +15 =(90/100)117.195+15 = 120.4755 units.

Or simply without writing t+1, t+2 etc. we can define it as

In time 0,

K(0) =100

K(1)= (1-10/100)K(0) + I(0) = (90/100)*100 + 15 =105 units

K(2) =(1-10/100)K(1)+I(1)=(90/100)*105 +15=109.5 units

K(3)=(90/100)K(2)+ 15 =(90/100)*109.5 +15= 113.55 units

K(4)=(90/100)113.55 + 15 = 117.195 units

K(5) = (90/100)117.195 + 15 = 120.4755 units