# Perfect-Foresight Transitions of the Aiyagari Economy¶

Soon we will discuss how to analyze models with aggregate uncertainty and the challenges associated with that. We sometimes analyze macroeconomic fluctuations without aggregate uncertainty. This approach is computationally attractive if the economy has a complicated partial-equilibrium problem and relatively few markets that need to be cleared. Moreover, if the economy is linear, this approach gives the same impulse response function as we would obtain with aggregate uncertainty due to certainty equivalence.

Suppose that the economy is subject to a transitory aggregate shock that was previously viewed as a zero-probability event. Going forward, there is no uncertainty over the dynamics of the shock or other aggregate variables, but there can be uncertainty over the idiosyncratic variables.

- Let’s extend our discussion of the Aiyagari economy with a productivity shock \(Z_t\). So now
- \[Y_t = F(K_{t-1},L) = Z_t K_{t-1}^\alpha L^{1-\alpha}.\]
- Suppose \(Z=1\) in steady state and was expected to remain there when news arrives at \(t=0\) that \(Z\) will follow a path
- \(\{Z_t\}_{t=0}^\tau\) and thereafter take a value \(Z=1\).

Because the capital stock will react to the productivity fluctuations, the economy will not be back in a stationary equilibrium at \(t=\tau\). Our goal is to compute the path for \(\{K_t,Y_t,R_t,W_t\}_{t=0}^\infty\). To do so, we pick a length of the transition \(T>\tau\) and we will assume that the economy is back in the stationary equilibrium at \(T\). Also, using the production function and firm first-order conditions we can determine \(\{Y_t,R_t,W_t\}_{t=0}^T\) from \(\{K_t\}_{t=0}^T\). So the problem is to find an equilibrium path \(\{K_t\}_{t=0}^T\).

The algorithm reflects the one we used for the stationary equilibrium of the Aiyagari economy: we guess \(\{K_t\}_{t=0}^T\), solve the household’s problem, simulate the distribution of wealth, and then check market clearing. The difference is that here there is a time dimension to everything.

Given the guess \(\{K_t\}_{t=0}^T\), we compute the return on capital and wage at each date from the firm’s first-order condition. We then solve the household’s problem backwards in time. We can again use the endogenous grid method but now we need to specify which time period the prices and policy rules correspond to.

Like before, suppose the consumer has labor endowment \(e\) at date \(t\) and saves an amount \(A\). Then the Euler equation can be written as

\[c_t^{-\gamma} = \beta R_{t+1} \mathbb E_{e'} \left[ {R_{t+1} A + W_{t+1} e' - g_{t+1}(A,e')}^{-\gamma} \right],\]

where the subscript \(t+1\) indicates that \(g_{t+1}\) is the policy rule at date \(t+1\). Given \(g_{t+1}\) we can compute the right-hand side and solve this equation for \(c_t(a,e)\). Next we use the budget constraint to solve for \(a\)

\[a = \frac{1}{R_t}\left[ c_t + A - W_t e \right].\]

We now have a mapping from \((a,e)\) to \(A\) in period \(t\) which is the policy rule \(g_t(a,e)\). We begin these steps in \(T-1\) using the stationary policy rule as \(g_T(a,e)\). We then iterate backwards in time to arrive at \(\{g_t(a,e)\}_{t=0}^{T-1}\).

Once we have the policy rules, we simulate the distribution of wealth forwards through time. That is, entering date \(0\), the distribution of wealth is the steady state distribution. We then update this with the policy rules from date \(0\). Let \(T(g_t)\) be the non-stochastic simulation transition matrix using the saving policy rule \(g_t\). Let \(D^*\) be the steady state distribution. We then set \(D_{-1} = D^*\) and \(D_{t}' = T(g_t) D_{t-1}\) for :\(t \in 0,...,T\).

- We can now compare the capital stock implied by the distribution of wealth
- \[\tilde K_t \equiv \int a d \Gamma_t(a,e),\]

where \(\Gamma_t\) is the distribution approximated by \(D_t\). Our goal is that \(\tilde K_t = K_t \forall t\), where \(K_t\) is our guess. If that’s not the case we need to adjust our guess. The difficult part of the algorithm is how to update the guess so as to move towards market clearing. A simple alogithm tends to work in this example: make the new guess a convex combination of the previous guess and \(\tilde K_t\).