phase transition diagram for PortfolioConsumers

[sbenthall]

One contribution of our paper will be a numerical/computational buffer stock saving analysis of the portfolio consumer model.

This will build on @llorrac buffer stock theory work. But we have endogenous portfolio choice in our model, so unlike the original buffer stock work, we won't be able to get analytic results (at least not in the next two weeks). However, we can still identify attractor regions/phase transitions in the parameter space.

There are lots of parameters in the model but we only care about some of them. The ones we don't care about we can set to the most trivial possible value, like 1 or 0. Parameters we can vary over might include:

• CRRA, DiscFac -- consumer preferences
• Permanent and transitory labor shocks
• Mean and standard deviation of returns on the risky asset.

We are sensitive to a few aspects of the 'phase space' of these parameters:

• The existence, or not, of a target level of wealth. (finite, 0/borrowing constraint, or infinite)
• The satisfying of the binding conditions for Lucas asset pricing ('subjective return' > 1)

We can plot this on a heatmap over 2 parameters leaving all the rest constant, since both these values are numerically computable.

Which 2 variables shall we plot over?

[sbenthall]

As a start on this:

• This notebook computes the target wealth numerically over a grid of parameterizations of the Portfolio model: https://github.com/sbenthall/SHARKFin/blob/master/macro/Numerical%20Buffer%20Stock%20on%20Portfolio%20Models.ipynb

• This is the CSV of the resulting data. 'roots' are the roots of the expected wealth gain function given optimal strategies:
  roots.csv

Not 100% sure what it means when the root is 1; that may have to do with some details of the computation. But something like this should help us sketch out the buffer stock conditions for the portfolio model. It runs rather quickly and I expect a rather fine-grained grid could be computed on my laptop overnight.

But we'll need to:

• confirm the implementation of the target wealth computation, including getting the edge cases right
• produce a strong dataset
• choose some nice visualizations of it.

[alanlujan91]

I think we need a way to incorporate the equity premium, Eqprem = RiskyAvg - Rfree.

We could do this by just setting Rfree to 1, and then we only care about RiskyAvg.

So I think the most important parameters to focus on are CRRA, DiscFac, Eqprem, RiskyStd