BUG: Including 0 in wealth and consumption grid

[MImmesberger]

Problem description

On this branch, I test the results of lcm against the analytical solution by Iskhakov et al. (2017). In this model, agents face a discrete retirement decision (absorbing state) and a continuous consumption decision.

When testing the value function of lcm, I set up the model in the following way. Note that the wealth and consumption grid starts at 0 and there are T=3 periods.

I included 0 in the consumption and wealth grid because I think this is necessary in order to correctly test the numerical results against an analytical solution. Else, retired agents being in the n-1 lowest possible states (in an n period problem with equal consumption and wealth grid) would necessarily violate their budget constraint, which produces incorrect results that may spill over to the other states.

from lcm.example_models import PHELPS_DEATON_NO_BORROWING
from lcm.entry_point import get_lcm_function

model = PHELPS_DEATON_NO_BORROWING
model["n_periods"] = 3
model["choices"]["consumption"]["start"] = 0
model["choices"]["consumption"]["stop"] = 100
model["choices"]["consumption"]["n_points"] = 10_000
model["states"]["wealth"]["start"] = 0
model["states"]["wealth"]["stop"] = 100
model["states"]["wealth"]["n_points"] = 10_000
solve_model, params_template = get_lcm_function(model=model)

params_template["beta"] = 0.98
params_template["labor_income"]["wage"] = 20.0
params_template["next_wealth"]["interest_rate"] = 0.0
params_template["utility"]["delta"] = 1.0

numerical_solution = solve_model(params=params_template)

The value functions of the first two periods (for the retired agent) seem to be wrong. The last period value function is correct.

Value function in period 2

For the initially retired, I get the following result for the value function on the wealth grid:

[ -inf , -inf, -9.118039 , ..., 7.7455106, 7.7457085, 7.745907 ]

The correct/analytical solution would be:

[-inf, -10.49036936,  -9.11793795, ..., 7.74551444, 7.74571051, 7.74590656]

The -inf in the second entry is due to the value function being computed on a grid, this doesn't show up in the analytical solution (only the first -inf entry is correct). What surprises me is that the following values on the grid are imprecise as well. When working on a grid with 10_000 grid points, only after the 800th or so grid points, results are always close up to 5 or 6 decimals. Maybe the issues at the first few grid points induce numerical imprecision somehow?

The solution for the workers coincides with the analytical solution.

Value function in period 1

I get a list of nan, both for the initially retired and for the agents still working. This problem does not occur if the minimum grid value is larger than 0.

[timmens]

Changing the label from bug to enhancement, as we've noticed that these issued can be circumvented by using a very fine wealth grid. The enhancement must deal with the actual zero case.

[hmgaudecker]

I am not sure we'll ever want this. Why would we want to support a grid value where the utility function is undefined?

[timmens]

I think it matters whether the utility value is actually undefined, or if it is negative infinity. I guess in the former case, LCM should throw an error, but in the latter, LCM should be able to handle the case without introducing NAs.

Nevertheless, I think once we decide on which behavior we expect from LCM we can open a new Issue requesting it and close this Issue.

[timmens]

After talking to @hmgaudecker, we realized that we also do not want to handle the "negative infinity" case, but want to throw an error in both cases. Moved to #55.