Malthusian Model: Convergence

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.

 

In [3]:
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]
In [4]:
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]