Assumptions on Aggregate Uncertainty¶

We will assume that there are TFP shocks so the production function becomes

\[Y = Z K^\alpha L^{1-\alpha}.\]

The first order conditions of the firm’s problem are then

\[ \begin{align}\begin{aligned}R = \alpha Z \left( \frac{K}{L} \right)^{\alpha -1} + 1 - \delta\\W = (1-\alpha) Z \left( \frac{K}{L} \right)^{1-\alpha}\end{aligned}\end{align} \]

Log TFP follows an AR(1) process

\[\log Z' = \rho \log Z + \varepsilon',\]

where \(\varepsilon\) is an i.i.d. mean-zero innovation.

Krusell-Smith Algorithm¶

The best-known solution algorithm is the Krusell-Smith algorithm. [1] The main idea is to approximate the distribution of wealth with a small number of moments, e.g. it’s mean \(K\), and specify a reduced-form law of motion for these moments. Given the laws of motion, one can solve the household’s problem and simulate the evolution of the economy and the distribution of wealth over a series of business cycles. One can then ask whether the laws of motion for \(K\) fits well and update the conjectured laws of motion as necessary.

This algorithm has been shown to work well in many applications. In models with complicated market structures, a challenging step in the algorithm is the simulation because along the simulated path one must take care to find market-clearing prices. Doing so can involve repeatedly solvling the household problem for different prices. Avoiding this step is one reason that some of the literature on heterogeneous-agent New Keynesian models adopts an alternative solution technique, which we turn to next.

Reiter Method¶

The Reiter method [2] creates a local approximation to the economy in the neighborhood of the stationary equilibrium that has idiosyncratic shocks but no aggregate shocks. In doing so, the first step is to create a discretized version of the economy that replaces the equilibrium functions (the policy rules and the distribution of wealth) with vectors. We already know how to do this: we will use the matrix \(G\) to represent the policy rule and the vector \(D\) to represent the distribution of wealth. We then specify a set of equations that these variables satisfy. \(G\) must satisfy the Euler equation for each pair of \(A_i\) and \(e_j\). Using our projection method interpretation of the endogenous grid method, we require that the residual function \(R(A_i,e_j;G)\) is zero for all pairs \((A_i,e_j)\). In addition, we require that \(D' = T D\) where \(T\) is the transition matrix implied by the decision rules.

In addition to \(G\) and \(D\), we also have two “aggregate variables” in our system, specifically, the aggregate capital stock and the level of TFP. These variables satisfy two “aggregate equations.” The first is the stochastic process for the evolution of TFP. The second requires that the aggregate capital stock must be consistent with the distribution \(D\). It is good practice to write the Euler equations as depending on \(K\) (or prices) and \(K\) depending on \(D\) as opposed to having the Euler equations depending directly on \(D\) because doing so leads to a sparser system of equations that that the computer can analyze more quickly.

We stack these equations up into a large system and then linearize it using automatic differentiation and solve it using Rendahl’s Time Iteration Algorithm just as we solved the RBC model.

Implementing the Reiter method is not too challenging given that we have already set the stage in our preceding steps. The endogenous grid method essentially already gives us the Euler equation residuals and the non-stochastic simulation method gives us the equations that dictate the evolution of the distribution of wealth.

Typically, there are two challenges with the Reiter method. First, you need to write the codes for these algorithms in such a way that your automatic differentiation algorithm actually works. Exactly what that means will depend on the package. For the automatic differentiation package I use, I have found it easier (and much faster) to provide the derivatives of the transition matrix for the distribution of wealth myself rather than using automatic differentiation.

Second, we need to keep track of objects at different dates. In a stationary environment, the household decision rule is the same in each period, but now there is a decision rule for each date. Similarly for the distribution of wealth. So we could think of indexing \(G\) and \(D\) with a time subscript. We then need to be careful as to the timing of which objects enter the equations in which places. I find it helpful to use the rule that objects dated \(t\) should be known at date \(t\) and not before.

Footnotes