## Murder by the State

No. 211
Monday, September 01, 1997
by Gerald W. Scully

## Technical Appendix

To determine the growth paths in democidal countries, let the economy be described by a simple Cobb-Douglas production function: Y = KßL1-ß. Let the fraction of output saved be s. The annual increment in the capital stock is the amount saved, sY. The annual growth rate of the capital stock, gK, is s(Y/K). Then, in terms of the production function the rate of growth of national output is:

gY = (1 + gK)ß (1 + gL)1-ß -1

= (1 + sY/K)ß (1 + gL)1-ß - 1.                         (1)

The effect of a change in the savings rate on the rate of growth of output is:

dgY = [(K + sY)ß-1 ßY K (1 + gL)1-ß] ds.         (2)

The sign of dgY depends only upon the sign of ds, since the term in brackets is positive.

Since we are using data on per capita RGDP, the relationship is restructured in per capita terms. The corresponding equation for the growth rate of output per head, gy, is:

gy = [(1 + gK)/(1 + gL)]ß -1

= [(1 + sY/K)/ (1 + gL)]ß - 1.                           (3)

The effect of a change in the savings rate on the rate of growth of per capita output is:

dgy = [(K + sY)ß -1ßY K(1 + gL)] ds.            (4)