I am reading Robert Lucas (2004), Life Earnings and Rural-Urban Migration, and I came across a rather peculiar optimal control problem that I'd like to ask about. Thank you!
The objective function is $ \int_0^\infty \exp\left(-\int_0^t r(s)\, ds\right) \cdot h(t) \cdot u(t) dt $
The constraints are $ \frac{dh(t)}{dt} = \delta \cdot h(t) \cdot (1 - u(t)), \quad 0 < u(t) < 1 $
The first-order condition provided in the article (Equation 4 in the paper) is $ h(t) = \delta \int_t^\infty \exp\left(-\int_t^\tau r(s)\, ds\right) \cdot h(\tau) \cdot u(\tau)\, d\tau $
However, I cannot derive this result myself. Here's my approach:
I constructed the Hamiltonian
$H = \exp\left(-\int_0^t r(s)\, ds\right) \cdot h(t) \cdot u(t) + \lambda \cdot \delta \cdot h(t) \cdot (1 - u(t)) .
$
Setting $ \frac{\partial H}{\partial u} = 0 $, I obtain
$ \exp\left(-\int_0^t r(s)\, ds\right) \cdot h(t) = \lambda \cdot \delta \cdot h(t) $
After simplification, this yields $ \exp\left(-\int_0^t r(s)\, ds\right) = \lambda \cdot \delta, \quad \text{and} \quad \lambda(0) = \frac{1}{\delta} $
I then have $ \frac{d\lambda}{dt} = - \frac{\partial H}{\partial h} = -\left( \exp\left(-\int_0^t r(s)\, ds\right) \cdot u + \lambda \cdot \delta \cdot (1-u(t)) \right) $.
Substituting the above equations, I get $ \frac{d\lambda}{dt} = -\lambda \cdot \delta $
With the initial value for $ \lambda $, this leads to $ \lambda(t) = \frac{1}{\delta} \cdot \exp(-\delta \cdot t) $
However, I am stuck at this point as there seems to be no further conditions involving $ h(t) $ and $ u(t) $. How is the first-order condition in the paper derived? Could anyone provide some guidance? Thank you!