Most dynamic models do not have known analytical (closed-form) solutions. Traditionally, the models are linearized or log-linearized around the steady state to obtain a system of linear policy rules (Kydland and Prescott (1982) and King et al. (1988)). This procedure greatly reduces the computational burden and is suitable for many questions in the neighborhood of steady state. However, it is not suitable when the curvature of the models is important: for instance the questions involving welfare comparison and policy asymmetry. Alternatively, the behavior of dynamic models can be obtained by using the computational methods. The most direct approach is to apply the value function iteration to the social planner’s problem. However, this approach is extremely inefficient due to computational burdens and a strong curse of dimensionality. The other alternative solution methods can be classified into two broad classes: perturbation and projection methods.
The perturbation method involves a local approximation using Taylor’s expansion. The agents’ policy functions are approximated around the steady state and a perturbation pa- rameter. Hall (1971) and Magill (1977) demonstrate the usefulness of using the first-order perturbation method to find the solutions. The first-order method gives linear decision rules. Thus, the dynamic of the model is governed by linear rules. Judd and Guu (1993) extend the methodology to higher orders. The first- and second-order perturbation methods are exten- sively studied by Schmitt-Groh´e and Uribe (2004).
The projection method builds approximated decision rules from basis functions (see Judd (1992) and Miranda and Helmberger (1988)). Basis functions with the optimal parameters are combined to minimize a residual function. There are two common types of basis functions used in the projection method: finite elements (neural network) and spectral (Chebychev orthogonal). While the former’s basis functions are non-zero only locally, the latter’s are non- zero globally. Thus, the projection method with spectral functions is more accurate in general. However, the procedure becomes increasingly complicated as the number of state variables rise.
Aruoba et al. (2006) examine the performances of the perturbation and the projection methods. Both methods have good convergence properties as well as computing time. The perturbation method gains more accuracy when a higher order is used. The only cost of increasing the order of approximation is to take a number of derivatives. However, the accuracy of the perturbation method is limited only to the neighborhood of the steady state since it is a local approximation. The projection method is a global approximation and generally provides more accurate solutions. For the choice of basis function, the finite element method is more robust and gives more accurate solutions along the range of parameters, while the Chebsyshev method shares most preferred features of finite elements if the system is not very highly nonlinear.
This paper uses the second-order perturbation method. The main objective of the paper is to explain asymmetric effects of monetary policy in a DSGE framework. Market frictions that are incorporated in the model will add up the curvature of the model. As a result, the curvature of the model would create the asymmetric responses. Thus, the solution method must be able to capture the nonlinearity of the model. Moreover, since most analysis will be done at the steady state, with the exception of the perturbation method which might not give an accurate result as the projection method, it would be able to capture the effects of nonlinearity if the model has enough curvature.
3.4.1 Solving the Model Using Perturbation Method
The perturbation method applies Taylor approximation around the steady state level (see Schmitt-Groh´e and Uribe (2004) and Fern´andez-Villaverde and Rubio-Ramirez (2006)). The equilibrium conditions for most dynamic models can be written as:
Etf(yt+1,yt,xt+1,xt), (3.39)
whereytis a vector of endogenous variables andxtis a vector of state variables. State variables
For example, a particular endogenous variable consumptionCtis taken in to consideration.
The policy function for consumption,c(xt, σ), is a function of state variablesxtand perturba-
tion parameter σ. Further assumption can be made that there is only one endogenous state variable, capitalkt, and only one exogenous variable, technologyAt. Thus, the vector of state
variables is xt ={kt, At}. If the optimal consumption is smooth at the steady state level of
state variables, (k0, A0, σ= 0), then the approximated function can be written as:
c(kt, At, σ)' X i,j,m 1 (i+j+m)! ∂i+j+mc(kt, At, σ) ∂kti∂Ajt∂σm k0,A0,0 (kt−k0)i(At−A0)jσm. (3.40)
The above equation is essentially a standard Taylor series expansion. The policy functions for other variables would have the same representations. The extension to the system with many endogenous-state and exogenous-state variables is very straightforward. The only cost for the extension is just taking a number of derivatives.
The objective of the method is to obtain the unknown parameters ∂i+j+mc(kt,At,σ)
∂ki t∂A j t∂σm k0,A0,0.
The recursive algorithm is applied for this purpose. The first step is acquiring parameter val- ues for the first-order approximation. The solutions of only the first-order approximation at the steady state are equivalent to those of a linear deterministic model. Once a linear system is obtained, the model is solved by a forward-looking rational expectation procedure. Among other procedures6, Blanchard and Kahn (1980) and King et al. (1988) use the eigenvalue de-
composition. The procedure decouples the solutions to forward-looking and backward-looking parts. Then each part can be separately solved and combined to get the solutions. Next, the second-order approximation is applied around the steady state. With the parameters obtained from the first-order approximation, the second-order approximation can be uniquely solved by using the same procedure. Ultimately, the solutions of the higher order approximation can be solved by iterating this procedure.
6The other solution procedures are (see Fern´andez-Villaverde and Rubio-Ramirez (2006)); linear quadratic
approximation( Kydland and Prescott (1982)), generalized Schur(QZ) decomposition (Klein (2000) and Sims (2002)). These solution methods deliver the same linear policy functions.