Perturbation

Perturbation w.r.t. aggregate state.

Implementation restrictions:

ex ante identical agents
agents idiosyncratic shocks must be discretizable by markov chains
aggregate shock is an AR1 (possibly multivariate)

Notations (vague):

$m_t$ is the aggregate exogenous shock, associated to transition function such that $m_t = \tau(m_{t-1}, \epsilon_t)$
$x_t$ is a vector representing the values of the decision rule of all agents at date t. Denote by $\varphi^{x_t}()$ the decision rule it determines. Denote by $T$ the time iteration operator which pins down optimal decision today as a function of decisions tomorrow.
$y_t$ is the vector of aggregate prices
$\mu_t$ is a vector representing the distributions of agents across endogenous and exogenous idiosyncratic states given $m_t$
$\Pi(m_t, x_t, y_t)$ is the transition matrix, associated to policy $x_t$ across the individual's states

Recall the definition of the equilibrium function:

$\int \mathcal{E}\left(e^i, s, x_t(e^i,s), y_t\right) d \mu_t(e^i,s)= 0$

Knowing the approximate decision rule $\phi^{x_t}$ and distribution $\mu_t$ the above equation is naturally approximated by a function $\mathcal{A}$ such that:

$\mathcal{A}(m_t, \mu_t, x_t, y_t) = \int \mathcal{E}\left(e^i, s, \varphi^{x_t}(e^i,s), y_t\right) d \mu_t(e^i,s)$

The whole economy is characterized by the following equations:

transition ( $G$ ):

$m_t = \tau(m_{t-1}, \eta_t)$

$\mu_t = \mu_{t-1}.\Pi(m_{t-1}, x_{t-1}, y_{t-1})$

equilibrium ( $F$ )

$x_t = T(m_t, y_t, m_{t+1}, x_{t+1}, y_{t+1})$

$\mathcal{A}(m_t, \mu_t, x_t, y_t)=0$

Note that aggregate states (in the sense that they are predetermined) are $m_t$ and $\mu_t$ that is the aggregate shock and the distribution of agents' states. The controls are $x_t$ and $y_t$ that is agent's decisions and aggregate prices.

For the sake of clarity, let us exemplify the notations above. Considering the Aiyagari (1994) model with aggregate productivity shocks presented in this note from the Model-Definition section - $s_t = a_t$ - $x_t = i_{t}$ - $y_t = \left( r_t, w_t \right)$ - $e_t^i = \left(y_t,\epsilon_t^i\right)$ - $m_t = z_t$ - $\mu_t$ is the joint distribution of agents over $s_t$ and $\epsilon_t^i$ given $m_t$

The transition equation associated with $\tau$ is simply

$\log(z_t) = \rho \log(z_{t-1}) + \sigma \eta_t, \quad \eta_t \sim \mathcal{N}(0,1)$

The equilibrium equations defining are $r_t = A \alpha z_t \left( \frac{L}{K_t} \right)^{1 - \alpha} - \delta\\ w_t = A (1-\alpha) z_t \left( \frac{L}{K_t} \right)^{\alpha}\\ K_t = \int \phi^{x_t} \left( \epsilon_t^i, a_t \right) d\mu_t\left( a_t, \epsilon_t^i\right)$

$\mathcal T$ is the time-iteration operator, which solves for $x_t$ in the Euler equation given $m_t$ , $y_t$ , $m_{t+1}$ and $y_{t+1}$