Question

8. Solve the following macroeconomic model. Let C(t) be the consumption, I(t) the investment of capital and Y(t) the national rent. C(t) = aY (t) + b C(t) + I(t)=Y(t), I(t)= kC'(t), Y(0) = Y., 0<a<1,b,k > 0 Solve for Y(t) & I(t). Calculate the long run value of y(t)/I(t)

Answer 1

Given-

C(t) + I(t) = Y(t)

I(t) = kC’(t) ……………….1

C(t) = aY(t) + b…………..2

Given Assumptions –

Y(0) = Y₀

0 < a <1, b, k > 0

Steps –

C(t) + I(t) = Y(t)

Substituting 1 & 2

aY(t) + b + kC’(t) = Y(t)…….3

on solving equation 3 for Y(t)

b + kC’(t) = Y(t) – aY(t)

b + kC’(t) =Y(t) [1 – a]

Y(t) = b + kC’(t)/(1 – a)……..4

On solving equation 3 for I(t) = kC’(t)

kC’(t) = Y(t) – aY(t) – b

kC’(t) = Y(t)[1 - a] – b……….5

I(t) = Y(t)[1 - a] – b………6

Solving for Y(t)/I(t)

Substituting from equation 4 and 6

Y(t)/I(t) = [b + kC’(t)/(1 – a)]/ Y(t)[1 - a] – b

Substituting from equation 5

Y(t)/I(t) = [b + {Y(t)[1 - a] – b} /(1 – a)]/ Y(t)[1 - a] – b

Y(t)/I(t) = Y(t) (1 - a)/(1 – a) / Y(t)[1 - a] – b

Y(t)/I(t) = Y(t) / Y(t)[1 - a] – b

Y(t)/I(t) = 1/[1 - a] – Y(t)/ b

Therefore Y(0),

Y(0)/I(0) = 1/[1 - a] – Y(0)/ b

Y(0)/I(0) = 1/[1 - a] – 1/ b