San José State University
Department of Economics
Thayer Watkins
Silicon Valley
& Tornado Alley

Dynamic Input-Output Analysis

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


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

Xt = AXt + B*(Xt−Xt-1) + Ft
and thus
Xt = (A+B*)Xt − B*Xt-1 + Ft

The levels of production in period t are given by

Xt = (I − (A+B*))-1(Ft−B*Xt-1)

The Dynamics

Suppose Ft=eλtF0 where λ is possibly a complex number to allow for cycles.

Then assume Xt=ektX0. The condition to be satisfied is that

ektX0 = ektAX0 + ektB*X0 − ek(t-1)B*X0 + eλtF0

In order for this to be satisfied for all t, k must be equal to λ. Furthermore division by eλt 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.)