Question

. 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 (t + 1) = (1 − d) K (t) + I (t)

Suppose at the beginning of year 0 that this country has 80 units of capital. Investment expenditures are 10 units each in the years 0, 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

Consider the given problem here “K(t)” be the level of capital stock at the beginning of period “t”. we also have a relation of “K(t+1)” to “K(t)”, which is mentioned below.

=> K(t+1) = (1 – d)*K(t) + I(t), where t = 0, 1, 2, 3, 4 , 5.

We also have given that at the period “0”, the level of capital stock is “K(0)=80”, d=10%=0.1, => (1 – d) =0.9 and I(t) = 10 for t=0, 1, 2, 3, 4, 5.

So, the level of capital stock at the beginning of the 1^st period is given below.

=> K(1) = (1 – d)*K(0) + I(0), => K(1) = 0.9*80 + 10 = 82, => K(1) = 82.

=> K(2) = (1 – d)*K(1) + I(1), => K(2) = 0.9*82 + 10 = 83.8, => K(2) = 83.8.

=> K(3) = (1 – d)*K(2) + I(2), => K(3) = 0.9*83.8 + 10 = 85.42, => K(3) = 85.42.

=> K(4) = (1 – d)*K(3) + I(3), => K(4) = 0.9*85.42 + 10 = 86.88, => K(4) = 86.88.

=> K(5) = (1 – d)*K(4) + I(4), => K(5) = 0.9*86.88 + 10 = 88.19, => K(5) = 88.19.

Where “K(t)” be the level of capital at the beginning of the period “t”.