## Monday, January 17, 2011

### A non-ergodic model of economic growth?...

As part of my ongoing quest to learn more about ergodic theory (and to apply a bit of the Python skills that I have been developing), I have been writing code to implement variations of the Solow model of economic growth using John Stachurski's Economic Dynamics: Theory and Computation as my guide.

Below are some plots of simulations from what I think is a non-Ergodic version of the Solow model.  The equation of motion of capital stock in the Solow model is:
Parameters are defined as follows: α is capital's share in production, s is the savings rate, δ is the rate of capital depreciation.  W are shocks which are assumed to be independently and identically log-normal.  The twist is that technology A follows a step function:
My choice of parameters was α=0.5, s=0.25, A1=15, A2=25, δ=1.0, kb=21.6,  the log-normal shocks have mean 0 and variance 0.2.  This parameter choice might seem very specific, but it simply matches with exercises from Stachurski's book.

Time Series Plots: Below is a plot of two time-series drawn from the above model.  The initial conditions are k0 = 1.0 (blue) and 80.0 (green).   Note the persistence...if you start with low levels of capital stock to begin with it can take a long while to get away from the low attractor.  If the system was deterministic, then if you started with low capital stock, you will always have low capital stock. Initial conditions would completely determine long-run outcomes, and we thus have an example of a poverty trap!
Marginal Distributions: Below is a plot of the empirical distribution function at t=100 time steps.  The left-hand plot is for k0 = 1.0 whilst the right-hand plot is for k0 = 80.0.  Note that the the distributions differ depending on the initial conditions.
I will update this post and make the Python code available via GitHub once I have finished working out the code to simulate and solve for the steady-state distribution of this model.  I know that it will be double-humped (b/c of the two attractors)...

But just to confirm, is this version of the Solow Model non-Ergodic?