[Question I was emailed:] I want to make a comparison of allocations between the steady state equilibrium of Aiyagari(1994) and the allocations of the steady state equilibrium of it’s complete markets counterpart. I have solved the equilibrium of Aiyagari(1994) using the VFI toolkit. However, there are no instructions about how to solve steady state equilibrium of complete market version using VFI toolkit. How can I get steady state equilibrium of a complete market model using VFI toolkit?
Answer: The Aiyagari (1994) model has idiosyncratic but no aggregate shocks. With complete markets, idiosyncatic shocks effectively disappear, and so there will be no shocks left in the model; more accurately, the idiosyncratic shocks are still present in the complete markets model but everyone has perfect insurance against them (e.g., via Arrow-Debreu securities) and so they have no actual impact on any of the outcomes in the complete markets model.
The Krussell-Smith model is the Aiyagari (1994) plus aggregate shocks. The complete markets version of the Krussell-Smith model is the Stochastic Neo-classical Growth model (the basic Real Business Cycle model, but with exogenous labor). Without aggregate shocks you are just looking at the steady-state of the Stochastic Neo-Classical Growth model. So the complete markets version of Aiyagari (1994) is just the steady-state of the Stochastic Neo-Classical Growth model.
Solving for the steady-state equilibrium of the Stochastic Neo-Classical Growth model can be done by hand (in closed-form equations) and is a standard exercise in many graduate economics lecture notes (actually, it is probably more common that you solve the steady-state of the basic Real Business Cycle model, which is just the same thing but with endogenous labor). Because it can be solve as closed-form equations you don’t need anything like VFI Toolkit to solve it, just derive the equations by hand and then evaluate them using a computer.
You can find something very similar done in Pijoan-Mas (2006). He uses the extension of the Aiyagari (1994) model to have endogenous labor, and so the complete markets version of his model is just the steady-state of the basic Real Business Cycle model. I can’t remember if he has the equations in the paper, I assume so. He also does a transition path but that is pretty irrelevant to the issue.
PS. If you can’t find how to derive the steady-state of the basic Real Business Cycle model (or the Stochastic Neo-Classical Growth model) let me know and I will find a link.