Let X be the vector of outputs by industry and K be the vector of capital goods by type. The required capitals goods for outputs X is given by

K = BX

The capital goods industries and their productions are included as part of X. The demand for additional capital goods depends upon the growth of output from one period to the next ΔX. Thus

ΔK = BΔX

The demand for industrial output is made up of final demands F, interindustry demand AX, Where A is the matrix of technical coefficients, and investment demand for capital goods, B*ΔX, where B* is the matrix B augmented with blocks of zeroes to make it compatible with X.

If production is equal to demand then

X = AX + B*ΔX + F

Labeling the output by time period gives

X_t = AX_t + B*(X_t−X_t-1) + F_t
and thus
X_t = (A+B*)X_t − B*X_t-1 + F_t

The levels of production in period t are given by

X_t = (I − (A+B*))^-1(F_t−B*X_t-1)

The Dynamics

Suppose F_t=e^λtF₀ where λ is possibly a complex number to allow for cycles.

Then assume X_t=e^ktX₀. The condition to be satisfied is that

e^ktX₀ = e^ktAX₀ + e^ktB*X₀ − e^k(t-1)B*X₀ + e^λtF₀

In order for this to be satisfied for all t, k must be equal to λ. Furthermore division by e^λt shows that

X₀ = AX₀ + B*X₀ − e^−λB*X₀ + F₀
which reduces to
X₀ = AX₀ + (1− e^−λ)B*X₀ + F₀
and hence
X₀ = [I−(A + (1− e^−λ)B*)]^-1F₀

(To be continued.)