Keynes’ simple model for an economy is given by where Y, C, I and G are gross na

Question

Keynes’ simple model for an economy is given by

where Y, C, I and G are gross national product (GNP), consumption, investment and government expenditure for year k. Consumption and investment are modeled by difference equations of the form

where a and b are parameters. The first equation implies that consumption increases with GNP but that the effect is delayed. The second equation implies that investment is proportional to the rate of change of consumption. Show that the equilibrium value of the GNP is given by

Explanation / Answer

Y(k) = C(k) + I(k) + G(k) = aY(k-1) + I(k) + G(k)

Y(k) - Y(k-1) = 1/1-a (I + G)

a=0.25 Y(k) - Y(k-1) = 4(I+G)

To show : C(k+1) = aC(k) + aI(k) + aG(k)

I(k+1) = (ab-a)C(k) + abI(k) + abG(k)

C(k+1) = a(C(k) + I(k) + G(k)) = aC(k) + aI(k) + aG(k)

I(k+1) = b[(aC(k) + aI(k) + aG(k)) - bC(k)]

= (ab-a)C(k) + abI(k) + abG(k)

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