Clarification on Bellman and the need for Linear Interp


#1

Hi all,

I’m hoping to get some clarification on numerically solving Bellman equations.

So in the McCall sequential search model (https://lectures.quantecon.org/py/mccall_model_with_separation.html) we just directly go about trying to numerically solve the Bellman’s at hand using the CMP.

But once we get ahead of that (for example Optimal growth https://lectures.quantecon.org/py/optgrowth.html) we start using linear interpolation. I’m not exactly clear about what makes the difference here? What part of the model requires us to put in linear interpolation in one of the models but not the other? I am not able to fully get a grasp on this.

Additionally, just another related clarfication while I have you here: the value function which we input essentially just a vector of numbers each representing the value under a certain state right? So if my V = [v_1, v_2, v_3] then v_i is the value that I’d get under outcome i which has a probability (lets say) p_i?

Thanks. And sorry. I seem to be very confused about the fundamentals here which I once thought i was clear about


#2

Maybe this is too long, but is the interpolation just applied because the number of realizations which y can take is greater than what we can actually compute?

What I am getting at is, is the lecture making a point about how a theoretically continuous function cannot be used in a numerical algorithm, and hence we should apply some linear interpolation?


#3

Hi @Hariharan_Jayasankar,

in the McCall model we are discretizing the wage draw process w, so that the state space is discrete and interpolation becomes unnecessary. We could use linear interpolation instead, and perhaps in a future lecture that will be added too. But this is one of the earlier lectures on DP, so it seemed best to keep things simple.

For the optimal growth lecture we used linear interpolation because now the state is an endogenous process that is a bit more difficult to discretize. Moreover, this is a later DP lecture, where we imagine the reader can handle a more sophisticated analysis — including an interpolation step.

Regarding your last question, I think it’s always possible to fully discretize a model, one way or another. But discretization is a very crude form of approximation. So for continuous state models, piecewise linear interpolation over a grid is generally better.


#4

Hey,

Thanks john. This does clarify things up a bit.

Great lectures!