On a Useful Capital Growth Matrix
A linear growth system is examined by applying a nonnegative output matrix to an age distribution of capital stock. The elements of the matrix are productivity rates and retention rates (depreciation rates). In this form, the matrix is irreducible and possesses a positive root which is greater in absolute value than any other root. Hence the well-known Perron-Frobenius theory can be applied. Then the matrix is generalized by decomposition into parts. Thes decomposition introduces interactions among capital sectors. Finally, the inverse of the maximal root is interpreted as a discount rate and the row eigenvector associated with that root as a set of implicit prices.