Technology and the Solow residual...

I recently wrote some Python code to compute paths of technology and the implied Solow residuals using data from the Penn World Tables.  Combining the results with some country metadata (i.e., income groupings) from the World Bank API yields this pretty interesting graphic...

If you don't already have the Penn World Tables data...no worries! The script will download the PWT data, compute the Solow residuals based on a method used by Hall and Jones (1999). Based on this decomposition, high income (i.e., red) countries had higher levels of technology in 1960 and higher subsequent growth rates of technology.  In fact, the low income (i.e., purple) countries have had effectively zero technological progress since 1960!

Enjoy!
  

More on the Solow Model...

In working through the material on the version of the Solow model from Chapter 6 of Economic Dynamics (if you want details of the model see previous posts) I began to wonder how an agent living in this world would go about choosing an optimal policy.  Here is the answer using U(c) = 1 - exp(-θc):

Iterations of the Value Function:

Note the jump up in the optimal savings policy.  

The Optimal Policy:

The plot of output above assumes that the shock takes its average value.  You can clearly see the two steady-state levels of output (they occur where the blue line cuts the 45 degree line from above). 

I am not quite sure what to make of this...I wasn't expecting the output and the optimal policy lines to cross...maybe a bug in my code.  Thoughts and interpretations are welcome!  The code has been posted to my Google Code repository.  Try it with a different (bounded and continuous) utility function and let me know your results...

The Look-Ahead Estimator...

Here is another estimate of the stationary distribution for the version of the Solow growth model detailed in previous posts.  Estimation of this stationary distribution was carried out using the Look Ahead estimator detailed in Economic Dynamics: Theory and Computation.  Although the Look Ahead estimator has superior sampling properties to NPKDE, I have also included the NPKDE version of the stationary distribution for comparison.  I have not yet figured out exactly why the y-axis scales differ (perhaps they are supposed to?).  Many thanks to John Stachurski for help working optimizing my implementation of the Look Ahead estimator!  Buy his book!  Code will be posted to my  GitHub Google Code repository tomorrow...

An ergodic model of economic growth?...

So, I am now thoroughly confused!  I now think that perhaps the version of the Solow model outlined in my previous post is actually an Ergodic model!  I think this despite the fact that the deterministic version of the same model is (I think) non-Ergodic because it has multiple steady-states and long-run outcomes are completely dependent on initial conditions of capital stock.

Below are estimates of the unique pdf and cdf of steady-state distribution of the Solow model from my previous post (with identical parameter values).  John Stachurksi, in his book Economic Dynamics proves that the model is globally stable and thus that the steady-state distribution is unique.  Global stability, I think, implies that a process is Ergodic.  The distribution was created by simulating the model for 100,000 periods and then estimating the pdf using non-parametric kernel density estimation. The cdf was estimated using an estimator based on the empirical cdf.

Threshold Solow PDF:

Note that the distribution is double-humped.  The reason that the left-hand hump (corresponding to the "poor" attractor) is small relative to the right-hand hump is that the threshold level of capital stock, kb, is such that once the economy moves into the "rich" attractor's basin it is unlikely to get shocked out.  If you where to raise the threshold kb, then the left-hand hump would get bigger.

Threshold Solow CDF:

 As promised, I will post a link to Github later today with the Python code...

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:

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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?