Elements of Ammunition by Major Theodore C. Ohart

Show description

However we do not know the expression of dk/ds. By differentiating sf…k† ˆ n with respect to s we obtain sf 0 …k†dk=ds ‡ f…k† ˆ n…dk=ds† Since dk=ds‰n � sf 0 …k†Š ˆ f…k†…dk=ds† ˆ f…k†=n � sf 0 …k†, by substituting it follows that df…k†=ds ˆ ‰f…k†f 0 …k†Š=n � sf 0 …k† This expression can be reinterpreted in terms of the elasticity of steady state per capita production with respect to propensity to save "q;s ˆ ‰s=f…k†‰df…k†=dsŠ ˆ s=f…k†‰f 0 …k†f…k†=n � sf 0 …k†Š By substituting for sf(k) its stationary value nk, we get "q;s ˆ nkf 0 …k†=nf…k† � sf…k†f 0 …k† ˆ kf 0 …k†=f…k† � kf 0 …k†; that is "q;s kf 0 …k† f…k† ˆ kf 0 …k† 1� f…k† Descriptive Models of Growth 17 kf 0 …k† is the elasticity of per capita production with respect f…k† to per capita capital.

Let 1 and 2 be the roots of the characteristic equation, and let be 1 > 0. The general solution of the dynamic system (see Mathematical Appendixes) is Since k…t† ˆ k ‡ a1 v11 e1 t ‡ a2 v21 e 2 t c…t† ˆ c ‡ a1 v12 e1 t ‡ a2 v22 e2 t Here a1 and a2 are arbitrary constants, that must be determined. The vij are the components of eigenvectors (v11 , v12 †T and (v21 , v22 †T associated with The set of eigenvectors associated eigenvalues 1 and 2 . with 1 and 2 must contain the (column) eigenvectors   v1 ˆ …1  00  ; 1†; v2 ˆ …2  00  ; 1†:5 By substituting, the solution c f …k † c f …k † to the linearized dynamic system becomes k…t† ˆ k ‡ a1 1  c f 00 …k † e1 t ‡ a2 2  c f 00 …k † e 2 t c…t† ˆ c ‡ a1 e1 t ‡ a2 e2 t Arbitrary constants a1 and a2 are now determined by admitting that it is possible to exclude paths which diverge from steady state from the set of optimal paths.

Empirically (Barro and Sala-I-Martin, 1995) the variability of growth rates for different countries seems due to 1. a positive correlation between growth of per capita income and rate of investment I/K: the greater the I/K is, the greater the q_ =q; 2. a positive correlation between per capita income and education level and skill of workers: the greater the skill, education and workmanship, the greater the q_ =q; 3. a negative correlation between per capita income growth rate and population growth rate: the greater rate of growth of income, the less the rate of growth of population; 4.