Economic growth and economic development 491

There are two necessary transversality conditions since there are two state variables and two costate variables.. This observation makes it clear that the model can be reduced to the neo

Introduction to Modern Economic Growth solution are

Hik(k (t) , h (t) , ik(t) , ih(t) , µk(t) , µh(t)) = −u0(c (t)) + µk(t) = 0

Hi h(k (t) , h (t) , ik(t) , ih(t) , µk(t) , µh(t)) = −u0(c (t)) + µh(t) = 0

Hk(k (t) , h (t) , ik(t) , ih(t) , µk(t) , µh(t)) = fk(k (t) , h (t)) u0(c (t))− µk(t) δk

= ρµk(t)− ˙µk(t)

Hh(k (t) , h (t) , ik(t) , ih(t) , µk(t) , µh(t)) = fh(k (t) , h (t)) u0(c (t))− µh(t) δh

= ρµh(t)− ˙µh(t) lim

t→∞exp (−ρt) µk(t) k (t) = 0 lim

t→∞exp (−ρt) µh(t) h (t) = 0

There are two necessary transversality conditions since there are two state variables (and two costate variables) Moreover, it can be shown that

H (k (t) , h (t) , ik(t) , ih(t) , µk(t) , µh(t)) is concave given the costate variables µk(t) and µh(t), so that a solution to the necessary conditions indeed gives an optimal path (see Exercise 10.9)

The first two necessary conditions immediately imply that

µk(t) = µh(t) = µ (t) Combining this with the next two conditions gives

(10.24) fk(k (t) , h (t))− fh(k (t) , h (t)) = δk− δh,

which (together with fkh > 0) implies that there is a one-to-one relationship between physical and human capital, of the form

h = ξ (k) , where ξ (·) is uniquely defined, strictly increasing and differentiable (with derivative denoted by ξ0(·), see Exercise 10.10)

This observation makes it clear that the model can be reduced to the neoclassical growth model and has exactly the same dynamics as the neoclassical growth model, and thus establishes the following proposition:

477