Keywords: bifurcation, perturbation, relative price distortion, optimal monetary policy
JEL Classification: C63; C61; E52.
In recent analysis of nonlinear dynamic macroeconomic models, the characterization of their first-order dynamics has been an important step for understanding theoretical implications and evaluating empirical success. However, the presence of a bifurcation in perturbation analysis of nonlinear dynamic systems implies that the first-order behavior of the economy cannot be characterized solely in terms of the first-order derivatives of the model equations.
In this paper, we use two simple macroeconomic models to address several issues regarding bifurcations. In particular, the bifurcation problem would emerge in conjunction with the price dispersion generated by staggered price setting in the part of firms. We then show how to apply l'Hospital's rule to characterize the solution of each model in terms of its higher-order derivatives. We also show that in some cases the bifurcation can be eliminated through renormalization of model variables; furthermore, renormalization may yield a more accurate first-order solution than applying l'Hospital's rule to the original formulation.
Before presenting our results, it is noteworthy that our definition of bifurcation is distinct from the one analyzed in Benhabib and Nishimura (1979). In particular, their analysis on bifurcation is associated with time evolution of dynamic systems. However, our concern with bifurcation arises in the process of approximating nonlinear equations, as discussed in Judd (1998).
We proceed as follows. Section 2 describes the two examples and illustrates how to detect the existence of a bifurcation problem. Section 3 follows the general approach of Judd (1998) and applies l'Hospital's rule to characterize the first-order behavior of each model. Section 4 shows how the bifurcation can be eliminated through renormalization of model variables. Section 5 concludes.
This section discusses how we can detect the existence of bifurcation in two simple economies. In both models, Calvo-style price setting behavior of firms can be summarized by the following law of motion for the relative price distortion:
To discuss the issue of bifurcation, we have to close the model with another equation. In the first example, we simply assume that inflation follows an exogenous stochastic process,
By combining the two equations, we now have a single-equation model:
Woodford (2003) and Benigno and Woodford (2005) pointed out that, when deviations of the (net) inflation rate from its zero steady state are of first order in terms of exogenous variations, deviations of the distortion index from one is of second order. Based on this observation, one can naturally approximate the system with respect to the square root of the logarithm of the relative price distortion index. Note that this distortion index is unity at the steady state with zero inflation rate. We follow the convention of using lower cases for log deviations,
Under the choice of as the approximation variable, (2.2) can be rewritten as follows:
It is noteworthy to find out what would happen if we feed this case into computer codes commonly available for dynamic macroeconomic analysis. The Dynare package (version 3.05) produces an message saying `Warning: Matrix is singular to working precision', and AIM (developed by Gary Anderson and George Moore, and widely used at the Federal Reserve Board) returns a code indicating `Aim: too many exact shiftrights'. The routine developed by Christopher Sims (gensys.m) ends without any output or error message.
The second example is a case with multiple equations. Our example is a prototypical Calvo-style sticky-price model, and the optimal policy problem is to maximize the household welfare subject to the following four constraints: the law of motion for relative price distortions, the social resource constraint, the firms' profit maximization condition, and the present-value budget constraint of the household. However, it is shown in Yun (2005) that the optimal policy problem can be reduced to minimizing the index for relative price distortion (2.1). At the optimum, we have the following relationship:
As in the single-equation case, we start with a normalization according to which and are endogenous variables and is exogenous:
As explained in Judd (1998) and Judd and Guu (2001), the bifurcation problem can be resolved by using l'Hospital's rule.
To understand the approximated behavior of in the singe-equation example, we need to compute and where is defined as an implicit function as follows:
Noting that the derivatives in the numerators are also zero at the steady state, we apply l'Hospital's rule to the two ratios in the form of and obtain the following first-order approximation:1
In this single-equation model, it is easy to avoid the bifurcation problem when we consider the following equation that is equivalent to (2.3),
To illustrate how we can invoke the bifurcation method in the multi-dimensional example, we substitute the second equation in (2.5) into the first to obtain:
However, since both derivatives on the right-hand side are zero at the steady state, we need to apply l'Hospital's rule to compute . The first-order solution for is
The presence of bifurcations is not only related to the economic model in hand, but also to the choice of the variable with respect to which the Taylor approximation is applied. This section shows that the bifurcation can be eliminated through renormalization of model variables; furthermore, renormalization may yield a more accurate first-order solution than applying l'Hospital's rule to the original formulation.
Under this renormalization, the expression for the relative price distortion is richer--and more accurate--than (3.6) derived using l'Hospital's rule. Another renormalization that produces a solution similar to (3.6) is to approximate with respect to (instead of ). This alternative way is based on the interpretation that the initial relative price distortion is of second order. Specifically, we rewrite the model as
In the multi-dimensional case, the two ways of renormalization would correspond to
According to the first renormalization, the second-order approximation of (2.1) is
The alternative renormalization consistent with the timeless perspective is to adopt (instead of ) as an exogenous variable. Based on this choice of an expansion parameter, Woodford (2003) concluded that the optimal inflation rate is zero to the first order in the absence of cost-push shocks. Under this normalization, the two model equations are approximated as follows:
After presenting two different renormalizations, it is natural to compare approximation errors for these two methods.6 For this purpose, we use as a reference point the closed-form solution to the optimal policy problem (2.5). Specifically, as shown in Yun (2005), the exact nonlinear solution for the optimal inflation rate is
The difference between the two methods is that the expansion parameter of the first renormalization is , while that of the second is . Figure 1 compares the accuracy of the two normalizations based on the first-order solution under each normalization.7 The black solid line represents the exact closed-form solution for annualized inflation ( ) in terms of initial relative distortion ( ). The blue line with crosses is the linear approximation of this nonlinear solution. This corresponds to the first-order approximation of when the expansion parameter is --that is, . It is evident that this approximation is more accurate than : the first-order approximation with as the expansion parameter, depicted by the red circles.
We can provide an intuitive understanding about the improved accuracy of the approximation with respect to as follows. Since is the square of , the first-order approximation with respect to is equivalent to the second-order approximation with respect to :
We have illustrated how to detect the existence of a bifurcation and demonstrated how to apply l'Hospital's rule to characterize the solution. We have also shown that the bifurcation can be eliminated through renormalization of model variables; furthermore, renormalization may yield a more accurate first-order solution than applying l'Hospital's rule to the original formulation. This paper has focused on the consequences of renormalization on the treatment of bifurcations. However, the renormalization is also associated with the welfare evaluation of different policies as in Benigno and Woodford (2005).