Introduction to Modern Economic Growth and h˙ (t) = ih (t) − δ h h (t) (10.21) where ik (t) and ih (t) are the investment levels in physical and human capital, while δ k and δh are the depreciation rates of these two capital stocks The resource constraint for the economy, expressed in per capita terms, is c (t) + ik (t) + ih (t) ≤ f (k (t) , h (t)) for all t (10.22) Since the environment described here is very similar to the standard neoclassical growth model, equilibrium and optimal growth will coincide For this reason, we focus on the optimal growth problem (the competitive equilibrium is discussed in Exercise 10.12) The optimal growth problem involves the maximization of (10.19) subject to (10.20), (10.21), and (10.22) The solution to this maximization problem can again be characterized by setting up the current-value Hamiltonian To simplify the analysis, we first observe that since u (c) is strictly increasing, (10.22) will always hold as equality We can then substitute for c (t) using this constraint and write the current-value Hamiltonian as H (k (t) , h (t) , ik (t) , ih (t) , µk (t) , µh (t)) = u (f (k (t) , h (t)) − ih (t) − ik (t)) (10.23) +µh (t) (ih (t) − δ h h (t)) + µk (t) (ik (t) − δk k (t)) , where we now have two control variables, ik (t) and ih (t) and two state variables, k (t) and h (t), as well as two costate variables, µk (t) and µh (t), corresponding to the two constraints, (10.20) and (10.21) The necessary conditions for an optimal 476