Monte Carlo Simulator that accepts aggregate blocks about hark HOT 12 OPEN

The simplest model with micro-heterogeneity is possibly the "Aiyagari Model"

Lots of old integrals math in there but it amounts to including the equilibrium restriction that
$$\texttt{Rfree} = F(\int a_{i,t} di)$$ where $F(\cdot)$ can be something simple like $F(x) = 1 + x^{0.33 - 1}$, adequately scaled so that the Rfree ultimately makes sense and is not like... 5.

There are no aggregate shocks and, since with enough agents the integral should be constant, this is a model in which the conditions of every individual agent change (idiosyncratic risk) but the macro aggregates are constant. That is why it is simpler than, e.g., Krusell-Smith

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Intuition is that R is the price of renting capital (the funds that agents save). If there is a lot of saved funds available, renting them should be cheaper.

Think of $F(\cdot)$ as the price that some external entity (in this case a firm) is willing to pay to agents for renting their savings for one period. Impose that and you have an equilibirum model.

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Thanks @Mv77 that's perfect.

I'm linking here to the dolark version of this model just for reference, since it's a succinct representation of the problem:

https://github.com/EconForge/dolark.py/blob/master/examples/ayiagari.yaml

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It's a little hard to unpack the forward transition equations from the original paper, but I believe it, for the individual agent:

$$z = lw + (1+r)a$$

$$c < z + b$$

$$a' = z - c$$

And at the aggregate level:

$$K = \sum_N a$$

$$r = \alpha (K/N) ^ { \alpha - 1}$$

$$w = (1-\alpha)*(K/N)^\alpha$$

EDIT: The equation for $r$ has been edited as per @mnwhite suggestion below.

CRRA utility as before.

I believe this is sufficient for the Monte Carlo simulation for arbitrary decision rules (which may be out of equilibrium).
I wonder if you agree.

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How would you like it expressed, @mnwhite ?

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I agree. Good point, @mnwhite

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I'd suggest setting $N$ to 1 and forgetting about wages. Receiving wages as an input that affects the income of agents might require defining a new agent type. I'd just take income to be whatever the baseline income process of ConsIndShock is.

That way the only real equilibrium restriction to be satisfied would be $F\left ( \int a_{i,t}(\texttt{Rfree}) di \right) = \texttt{Rfree}$. Call it a "simplified Aiyagari" just to start testing the framework with a single simple equation.

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@Mv77 The point of this issue is to develop (a) a python configuration object which contains model equations for aggregation and (b) changes to the (general) Monte Carlo simulator to enable forward simulation with such a configuration object. There are no AgentType objects involved here at all.

So it's the aggregation, not the equilibrium condition, which I'm getting at here.

Out of scope for this issue is to write a solver that can take these equations and derive a solution. That's perhaps far ahead. It would follow on #1438

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