Recall:
(2.29) $ \frac{dL/dt}{L} = \frac{d\ln(L)}{dt} = n = \beta \left( \frac{y}{\phi y^{sub}}-1 \right) $
(2.39) $ \ln(y) = \theta\ln(\kappa) + \ln(H) -\ln(L)/\gamma $
Substitute:
(2.40) $ \frac{1}{L}\frac{dL}{dt} = \frac{d\ln(L)}{dt} = n = \beta \left( \frac{\kappa^\theta H L^{-\gamma}}{\phi y^{sub}}-1 \right) $
(2.41) $ \frac{d\kappa}{dt} = -(1-\alpha)(h + (1-1/\gamma)n +\delta)\kappa + (1-\alpha)s $
Define population-adjusted-for-ideas $ P $:
(2.42) $ P = L H^{-\gamma} $
Then we have two state variables—capital-intensity $ \kappa $, the capital-output ratio, and population-adjusted-for-ideas $ P $. We have two dynamic equations: The rate of change of population-adjusted-for-ideas $ P $ is a function of the capital-output ratio and itself:
(2.43) $ \frac{1}{P}\frac{dP}{dt} = p = \beta \left( \frac{\kappa^\theta }{P^{1/\gamma} \phi y^{sub}}-1 \right) -h\gamma $
And the rate of change of capital-intensity $ \kappa $ is a function of itself andf of the rate of change of population-adjusted-for-ideas $ P $:
(2.44) $ \frac{d\kappa}{dt} = (1-\alpha)s -(1-\alpha)((1-1/\gamma)p +\delta)\kappa $
With the two parameters $ \alpha $ and $ \theta $ related by:
(2.45) $ \theta = \frac{\alpha}{1-\alpha} $
(2.46) $ \alpha = \frac{\theta}{1+\theta} $
And if we set:
(2.47) $ P = \Pi P^{*mal} $
(2.48) $ \frac{1}{\Pi}\frac{d\Pi}{dt} = p = \beta \left[ \pi^{(-1/\gamma)} \kappa^\theta (s/(\delta + \gamma h))^{-\theta} (1 + \gamma h/\beta) -1 \right] - h \gamma $
(2.44) and (2.48) are then our system...
And, with respect to its dynamics, in Python:
delong_classes.malthusian
at:
https://github/braddelong/LS2019/blob/master/2019-09-06-210a-ancient-intro.ipynb
we can examine how the simulated model behaves dynamically.