Recall our Solow-Malthus model. The rate of growth of the labor force and population depends on luxury-adjusted income per worker $ y/\phi $ divided by subsistence $ y^{sub} $:
(1) $ \frac{dL/dt}{L} = \frac{d\ln(L)}{dt} = n = \beta \left( \frac{y}{\phi y^{sub}}-1 \right) $
Income per worker depends on the capital-output ratio $ \kappa $, the level of the useful ideas stock $ H $, and the amount of resource scarcity induced by the labor force $ L $:
(2) $ \ln(y) = \theta\ln(\kappa) + \ln(H) - \ln(L)/\gamma $
(3) $ \frac{d\kappa}{dt} = (1-\alpha)s - (1-\alpha)(h + (1 - 1/\gamma)n + ֿֿ\delta)\kappa $
(3') $ \frac{dH/dt}{H} = h $
With the parameters $ \alpha $ and $\theta $—the capital share of income and the capital-intensity elasticity of income—related by:
(4) $ \theta = \frac{\alpha}{1-\alpha}$ and $ \alpha = \frac{\theta}{1+\theta} $
Substitute:
(5) $ \frac{1}{L}\frac{dL}{dt} = \frac{d\ln(L)}{dt} = n = \beta \left( \frac{\kappa^\theta H L^{-1/\gamma}}{\phi y^{sub}}-1 \right) $
(6) $ \frac{d\kappa}{dt} = -(1-\alpha)(h + (1-1/\gamma)n +\delta)\kappa + (1-\alpha)s $
Define ideas-adjusted-for-population $ I $:
(7) $ I = H L^{-1/\gamma} $
(8) $ i = h - n/\gamma $
(9) $ \frac{d\kappa}{dt} = -(1-\alpha)(\gamma h - (\gamma-1)i +\delta)\kappa + (1-\alpha)s $
(10) $ \frac{d\kappa}{dt} = (1-\alpha)s -(1-\alpha)(\gamma h +\delta)\kappa + (1-\alpha) (\gamma-1)i\kappa $
(11) $ \frac{1}{I}\frac{dI}{dt} = i = h - n/\gamma = h - \frac{\beta}{\gamma} \left( \frac{\kappa^\theta I}{\phi y^{sub}}-1 \right) $
Then we have two state variables—capital-intensity $ \kappa $, the capital-output ratio, and ideas-adjusted-for-population $ I $. We have two dynamic equations: The rate of change of ideas-adjusted-for-population $ I $ is a function of the capital-output ratio and itself. And the rate of change of capital-intensity $ \kappa $ is a function of itself and of the rate of change of ideas-adjusted-for-population $ I $.
The steady state is then:
(12) $ I^{*mal} = \frac{H}{L^{1/\gamma}} = \phi y^{sub}\left(\frac{\delta}{s}\right)^{\theta}\left(1+ \frac{\gamma h}{\delta}\right)^{\theta}\left( 1 + \gamma h/\beta \right) $
(13) $ \kappa^{*mal} = \frac{s}{\gamma h + \delta} $
Define:
(14) $ I = (1 + \xi) I^{*mal} $
(15) $ \kappa = (1 + k) \kappa^{*mal} = (1 + k) (s/(\delta + \gamma h)) $
Then:
From (11):
(16) $ \frac{1}{1 + \xi}\frac{d\xi}{dt} = h - \frac{\beta}{\gamma} \left( \frac{(1+k)^\theta\left(\kappa^{*mal}\right)^\theta (1+\xi) I^{*mal}}{\phi y^{sub}}-1 \right) $
(17) $ \frac{1}{1+\xi}\frac{d\xi}{dt} = i = h - \frac{\beta}{\gamma} \left(( 1 + \gamma h/\beta )(1+k)^\theta (1+\xi) - 1 \right) $
(18) $ \frac{1}{1+\xi}\frac{d\xi}{dt} = i = h - \left(( h + \frac{\beta}{\gamma})(1+k)^\theta (1+\xi) - \frac{\beta}{\gamma} \right) $
Using the approximation:
$ 1 + \theta k = (1+k)^{\theta} $
(19) $ \frac{1}{1+\xi}\frac{d\xi}{dt} = h - h - \frac{\beta}{\gamma} - h \theta k - \frac{\theta \beta}{\gamma}k - h \xi - \frac{ \beta}{\gamma}\xi + \frac{\beta}{\gamma} $
(20) $ \frac{d\xi}{dt} = \left[ h - h - \frac{\beta}{\gamma} - h \theta k - \frac{\theta \beta}{\gamma}k - h \xi - \frac{ \beta}{\gamma}\xi + \frac{\beta}{\gamma} \right](1+\xi) $
(21) $ \frac{d\xi}{dt} = -(h \theta + \theta \beta / \gamma)k - (h + \beta/\gamma)\xi $
This is our linearized exponential-convergence equation for the deviation of ideas-adjusted-for-the-population $ \xi $.
Now on to the capital-instensity. Recall:
(10) $ \frac{d\kappa}{dt} = (1-\alpha)s -(1-\alpha)(\gamma h +\delta)\kappa + (1-\alpha) (\gamma-1)i\kappa $
And from our definition of $ k $ we get:
(22) $ \frac{d\kappa}{dt} = \frac{dk}{dt}\kappa^{*mal} $
(23) $ \kappa^{*mal}\frac{dk}{dt} = (1-\alpha)s -(1-\alpha)(\gamma h +\delta)(1+k)\kappa^{*mal} + (1-\alpha) (\gamma-1)i(1+k)\kappa^{*mal} $
(24) $ \kappa^{*mal}\frac{dk}{dt} = (1-\alpha)s -(1-\alpha)(\gamma h +\delta)\kappa^{*mal} -(1-\alpha)(\gamma h +\delta)k\kappa^{*mal} + (1-\alpha) (\gamma-1)i(1+k)\kappa^{*mal} $
(25) $ \kappa^{*mal}\frac{dk}{dt} = -(1-\alpha)(\gamma h +\delta)k\kappa^{*mal} + (1-\alpha) (\gamma-1)i(1+k)\kappa^{*mal} $
(26) $ \kappa^{*mal}\frac{dk}{dt} = -(1-\alpha)sk - (1-\alpha) (\gamma -1)(h\theta + \theta \beta / \gamma)k + (h + \beta/\gamma)\xi)\kappa^{*mal} $
(27) $ \frac{dk}{dt} = -(1-\alpha)(\delta + \gamma h)k - (1-\alpha) (\gamma-1)(h\theta + \theta \beta / \gamma)k - (1-\alpha) (\gamma-1)(h + \beta/\gamma)\xi$
(28) $ \frac{dk}{dt} = -(1-\alpha)\left[\delta + \gamma h + (\gamma-1)(h\theta + \theta \beta / \gamma) \right]k - (1-\alpha) (\gamma-1)( h + \beta/\gamma)\xi$
(21) $ \frac{d\xi}{dt} = -(h \theta + \theta \beta / \gamma)k - (h + \beta/\gamma)\xi $
(28) $ \frac{dk}{dt} = -(1-\alpha)\left[\delta + \gamma h + (\gamma-1)(h\theta + \theta \beta / \gamma) \right]k - (1-\alpha) (\gamma-1)( h + \beta/\gamma)\xi$
This is our linearized exponential-convergence system for the deviation of ideas-adjusted-for-the-population $ \xi $ and the deviation of capital-intensity $ k $ from steady-state Malthusian equilibrium.
import numpy as np
import pandas as pd
P_vector = []
kappa_vector = []
P_vector = P_vector + [1.941371*.666]
kappa_vector = kappa_vector + [1.980198*1.166]
for i in range(0,100):
P = (1-0.025-0.0005)*P_vector[i] + 0.025*kappa_vector[i]
kappa = (1-0.02525)*kappa_vector[i]+0.05
P_vector = P_vector + [P]
kappa_vector = kappa_vector + [kappa]
malthus_converge_df = pd.DataFrame()
malthus_converge_df['P'] = P_vector
malthus_converge_df['kappa'] = kappa_vector
print(malthus_converge_df)
P kappa 0 1.292953 2.308911 1 1.317706 2.300611 2 1.341619 2.292520 3 1.364721 2.284634 4 1.387037 2.276947 5 1.408591 2.269454 6 1.429408 2.262151 7 1.449512 2.255031 8 1.468925 2.248092 9 1.487670 2.241327 10 1.505767 2.234734 11 1.523239 2.228307 12 1.540104 2.222042 13 1.556382 2.215936 14 1.572093 2.209983 15 1.587254 2.204181 16 1.601884 2.198526 17 1.615999 2.193013 18 1.629616 2.187639 19 1.642752 2.182401 20 1.655422 2.177296 21 1.667641 2.172319 22 1.679424 2.167468 23 1.690785 2.162739 24 1.701739 2.158130 25 1.712298 2.153637 26 1.722475 2.149258 27 1.732283 2.144989 28 1.741735 2.140828 29 1.750841 2.136772 .. ... ... 71 1.934296 2.033684 72 1.935814 2.032333 73 1.937259 2.031017 74 1.938634 2.029734 75 1.939942 2.028483 76 1.941186 2.027264 77 1.942367 2.026075 78 1.943489 2.024917 79 1.944553 2.023788 80 1.945561 2.022687 81 1.946517 2.021614 82 1.947421 2.020568 83 1.948276 2.019549 84 1.949084 2.018555 85 1.949846 2.017587 86 1.950564 2.016643 87 1.951241 2.015723 88 1.951877 2.014826 89 1.952475 2.013951 90 1.953036 2.013099 91 1.953561 2.012268 92 1.954052 2.011458 93 1.954510 2.010669 94 1.954937 2.009900 95 1.955333 2.009150 96 1.955701 2.008419 97 1.956041 2.007706 98 1.956355 2.007012 99 1.956643 2.006335 100 1.956907 2.005675 [101 rows x 2 columns]