Suppose I have an Aiyagari model (for simplicity, partial equilibrium) with endogenous labor supply. The separable functional form of utility u(c)-v(\ell) allows me to obtain analytically optimal labor supply \ell as a function of consumption but not of next-period assets a' (as in Guerrieri and Lorenzoni 2017 and unlike Conesa and Krueger 1999).
In this case I would like to use c as a decision variable. Given c, labor supply \ell follows from the analytical first order condition and a' follows from the budget constraint. However, since a' does not lie on the grid where the value function is defined, I have to interpolate.
Is this possible to do this in the VFI toolkit with experience assets? In this case aprime depends on decision d (consumption), current period endogenous state a and current period Markov shock z. I have seen (in this document) only
- riskyasset: aprime(d, u)
- experienceasset: aprime(d, a)
- experienceassetu: aprime(d, a, u) where u is an iid shock…
To clarify, this is the model example I have in mind:
V(a,z) = \max_{c,a',\ell} \dfrac{c^{1-crra}}{1-crra}-\chi \dfrac{\ell^{1+\sigma_2}}{1+\sigma_2}+\beta E[V(a',z')|z]
subject to
c+a' = (1+r)a + wz\ell, \quad a'\geq 0.
The first-order condition with respect to \ell gives \ell as a function of c:
\ell = \left( \dfrac{wz}{\chi} \right) ^{\dfrac{1}{\sigma_2}} \times c^{-\dfrac{crra}{\sigma_2}}
Thanks!
I would be happy to write this as an example using experience asset (either infinite or finite horizon), if it is possible.
