Introduction to Modern Economic Growth Proposition 10.1 In the neoclassical growth model with physical and human capital investments described above, the optimal path of physical capital and consumption are given as in the one-sector neoclassical growth model, and satisfy the following two differential equations c˙ (t) = [fk (k (t) , ξ (k (t))) − δk − ρ] , c (t) εu (c (t)) k˙ (t) = [f (k (t) , ξ (k (t))) − δ h ξ (k (t)) − δ k k (t) − c (t)] , + ξ (k) where εuh (c (t)) = ³−u00 (c (t)) c (t) /u0 (c (t)), together with the transversality condition ´i Rt limt→∞ k (t) exp − fk (k (s) , ξ (k (s))) ds = 0, while the level of human capital at time t is given by h (t) = ξ (k (t)) Proof see Exercise 10.11 Ô What is perhaps more surprising, at first, is that equation (10.24) implies that human and physical capital are always in “balance” Initially, one may have conjectured that an economy that starts with a high stock of physical capital relative to human capital will have a relatively high physical to human capital ratio for an extended period of time However, Proposition 10.1 and in particular, equation (10.24) show that this is not the case The reason for this is that we have not imposed any non-negativity constraints on the investment levels If the economy starts with a high level of physical capital and low level of human capital, at the first instant it will experience a very high level of ih (0), compensated with a very negative ik (0), so that at the next instant the physical to human capital ratio will have been brought back to balance After this, the dynamics of the economy will be identical to those of the baseline neoclassical growth model Therefore, issues of imbalance will not arise in this version of the neoclassical growth model This result is an artifact of the fact that there are no non-negativity constraints on physical and human capital investments The situation is somewhat different when there are such non-negativity or “irreversibility” constraints, that is, if we assume that ik (t) ≥ and ih (t) ≥ for all t In this case, initial imbalances will persist for a while In particular, it can be shown that starting with a ratio of physical to human capital 478