Economic growth and economic development 490

Introduction to Modern Economic Growth and 10.21 ˙h t = iht− δhh t where ikt and iht are the investment levels in physical and human capital, while δk and δh are the depreciation rates o

Introduction to Modern Economic Growth and

(10.21) ˙h (t) = ih(t)− δhh (t)

where ik(t) and ih(t) are the investment levels in physical and human capital, while

δk and δh are the depreciation rates of these two capital stocks The resource

constraint for the economy, expressed in per capita terms, is

(10.22) c (t) + ik(t) + ih(t)≤ f (k (t) , h (t)) for all t

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))

+µh(t) (ih(t)− δhh (t)) + µk(t) (ik(t)− δkk (t)) , (10.23)

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

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