The dynamics for the mean and the covariance matrix elements
The dynamic equation for the two means is given by and
where we assume that g2 > g1, that is, block 1 receives an inflow of resources from block 2. The three dynamic equations for the elements of the covariance matrix are
where x is a vector with elements к1Д, к2,2 , and к12, and m has elements Kj and к2.
The matrix D is defined by
where 2g1 = m ~ W and the matrix H is defined by
We use the Laplace transforms to solve the dynamic equations.
First, we assume that the two blocks of economies are expected to grow at the same rate,
Next, we set
where we recall that g2 > g1 by assumption, in other words, g1 = m~v, and g2=m+v.
Solving the elements of the covariance matrix is straightforward. The cross term к12 goes to zero asymptotically. This indicates that the growth patterns of the two economies asymptotically becomes uncorrelated.
The behaviour of the coefficients of variation
From the dynamic equations for the means, we derive their Laplace transform equations as
where Дх = (s - gjX s - g2) + n2 = (s - m )2.
The matrix Д has two equal roots m=(g2~ gi)/2 by choice.
Collect (k: :, k2 2, к12 ), as a three-dimensional vector x, and (k:, k2 ) as a two-dimensional vector y.
Solving the equation is straightforward. The results are that we have and
In words, block i, which is an exporting sector of capital goods or labor, is self-averaging, but block 2 is non-self-averaging. This result is interesting since the block of economies that is importing the factor of production such as labor or innovations is self-averaging but the exporting block is not self-averaging.