Dynamic programming with credit constraints

I am looking for simple examples of economic models with occasionally binding credit constraints.  I would like to find the most straightforward example possible, and then bludgeon it into submission with my various numerical algorithms...suggestions are much appreciated!

Solving a deterministic RBC model

Taking a short break from marking undergraduate economic essays and decided to write a bit of Python code to solve a deterministic RBC model using value function iteration.  Code to replicate the result can be found here.  Below are plots of the optimal policy functions (I included some of the iterates of the policy functions as well).

Again the code is mind-numbingly slow (possibly due to the interpolation scheme I am currently using) and takes roughly 8-10 minutes to finish. Any suggestions for speeding up the code (perhaps by using fancy indexing to avoid the for loop!) would greatly appreciated!  

Assaulting the Ramsey model (numerically!)

Everything (and then some!) that you would ever want to know about using dynamic programming techniques to solve deterministic and stochastic versions of the Ramsey optimal growth model can be found in this paper.

I wrote up a quick implementation of the most basic version of the value function iteration described in the paper (vanilla value iteration with a good initial guess and cubic spline interpolation).  Below is a graphic I produced of the optimal value and policy functions as well as every 50th iterate (to give a sense of the convergence properties).

The Python code is slowish (takes several minutes to compute the above functions).  Suggestions on ways to speed up the code are definitely welcome!

Back to the grind of marking essays...enjoy!

Solving models with heterogenous agents...

This material was previously part of another post on welfare costs of business cycles. I decided that the links on solving Krusell-Smith deserved its own post!

Place to start is definitely Wouter den Haan's website, specifically his introductory slides on heterogenous agent models. Krusell and Smith (2006) is an excellent literature review on heterogenous agent models.

Various solution algorithms: Key difficulty in solving Krusell-Smith type models is to find a way to summarize the cross-sectional distribution of capital and employment status (a potentially infinite dimensional object) with a limited set of its moments. 

• Original KS algorithm: The KS algorithm specifies a law of motion for these moments and then finds the approximating function to this law of motion using a simulation procedure. Specifically, for a given a set of individual policy rules (which can be solved for via value function iterations, etc), a time series of cross-sectional moments is generated and new laws of motion for the aggregate moments are estimated using this simulated data. 
• Xpa algorithm  
• Other algorithms

General issues with implementing Krusell-Smith algorithm:

Den Haan (2010):

This paper compares numerical solutions to the model of Krusell and Smith [1998. Income and wealth heterogeneity in the macroeconomy. Journal of Political Economy 106, 867–896] generated by different algorithms. The algorithms have very similar implications for the correlations between different variables. Larger differences are observed for (i) the unconditional means and standard deviations of individual variables, (ii) the behavior of individual agents during particularly bad times, (iii) the volatility of the per capita capital stock, and (iv) the behavior of the higher-order moments of the cross-sectional distribution. For example, the two algorithms that differ the most from each other generate individual consumption series that have an average (maximum) difference of 1.63% (11.4%).

Concise description of Krusell and Smith (1998) model can be found in the introduction to the JEDC Krusell-Smith comparison project.  It would seem the JEDC Krusell-Smith comparison project is a good place to start thinking about how to implement the Krusell-Smith algorithm in Python.  The above paper suggests that implementation details matter and can substantially impact the accuracy of solution.  Wouter den Haan provides code for JECD comparison project.  Maybe start by implementing the den Haan and Rendahl (2009) algorithm in Python? Followed by Reiter (2009). Given that different algorithms can arrive at different solutions, checking the accuracy of a given solution method against alternatives is important.  Slides on the checking accuracy of the above algorithms.

Other Krusell-Smith related links:

• Anthony Smith's original Fortran implementation of Krusell-Smith.  Code looks completely impenetrable! Not much in the way of comments or documentation.  Probably not worth working out how the code works.
• Sergei Mailar's MatLab code for Krusell-Smith (Mailar provides code for other interesting projects as well!).
• Fortran code for Krusell and Smith (2009).
• Wouter den Haan's slides on additional applications of the Krusell-Smith approach (includes  monetary models with consumer heterogeneity, models with entrepreneurs, turning KS into a matching model, portfolio problem).

New Code Posted...

So I have finally worked out the Python code for generating marginal densities of the evolution of output when following the optimal savings policy for the growth model detailed in Chapter 6 of Economic Dynamics.  The new graphics for the model from the textbook are below and the code has been posted to my repository (fvi_v2.py).

Simulated Trajectories: Not all that interesting...

Simulated Marginal Densities of Log Output: More interesting...notice the convergence.  This is Figure 6.12 from Chapter 6.

Output when following the Optimal Savings Policy: Same plot as before, but know I have added axes labels.  This is Figure 6.11 from Chapter 6.

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...