Controlling chaos in New Keynesian macroeconomics

2.1 Steady states and local stability properties

The long-run properties of system M are well understood. Benhabib, Schmitt-Grohé, and Uribe (2001a, 2001b) showed that if Assumption 1 holds, then, in general, two steady states exist: one where inflation is at the intended rate π = π* and one where π = π ̄ ≠ π * . The unintended steady-state is labeled as a liquidity trap, in which the interest rate is zero or near-zero, and inflation is below the target level and possibly negative. Moreover, at the steady state where inflation is at the intended rate, π = π*, it follows that μ 1 * exists and is unique. The local stability properties around the intended steady state, P * ≡ μ 1 * , π * , a * , are also well described in Barnett et al. (2022a).

2.2 An explicit variant of the model

To proceed, we must first provide specific forms for the implicit functions presented in system M. We assume that the utility function has constant relative risk aversion in a composite good, which in turn is produced with consumption goods and real balances via a CES aggregator as follows:

where 0 < κ < 1 is a share parameter, β measures the intra-temporal elasticity of substitution between the two arguments, c and m, and Φ > 0 is the inverse of the intertemporal elasticity of substitution. Since we have, for now, assumed that consumption and real money balances are Edgeworth substitutes, the following parametric restriction is implied.

The disutility of labor is captured by the following functional form

where ψ > 0 measures the preference weight of leisure in utility. Furthermore, following Carlstrom and Fuerst (2003), we also assume that production is linear in labor, y(l) = Al, with A being the productivity level in the composite goods production. Without loss of generality, we set A = 1.

Additionally, we use the specification of the Taylor principle in Benhabib, Schmitt-Grohé, and Uribe (2001a, 2001b), and assume that the monetary authority observes the inflation rate and conducts market operations to ensure that

where C is a positive constant. Notice that, from (6), our chosen functional form implies that R ( π * ) = R ̄ , and R′(π*) = C.

Finally, to avoid a violation of the Transversality Condition, we assume that the economy satisfies a Ricardian monetary-fiscal regime. More specifically, Eq. (6) is complemented by the fiscal rule

where the marginal tax rate α ≡ τ′(a) ∈ (0, 1).

2.3 Conditions for the existence of Shilnikov chaos

In this section, we provide the mathematical underpinnings that guarantee the existence of a chaotic regime in system M. Consider the Shilnikov (1965) theorem.

A convenient statement of Shilnikov’s famous theorem is in Chen and Zhou (2011). Because obtaining a general result on the critical parametric bifurcation surface is extremely difficult, we propose a numerical strategy based on the parameterization of the US economy from 1960(Q1) to 1998(Q3) proposed by Benhabib, Schmitt-Grohé, and Uribe (2001a, 2001b) and extensively used in the succeeding literature. We show that there are regions in the parameter space where system M may satisfy the conditions of Theorem 1.

Set the pair ( R ̄ , π * ) = ( 0.06 , 0.042 ) to match the (average) three-month Treasury Bill rate and (average) inflation rate observed over the period for the US economy. Therefore, since τ cancels out in the calculations, the characteristic Eq. (A.2) in Appendix A is a function of the remaining policy parameters, C and τ′. Solving the characteristic equation gives

Therefore, since u c m * ≅ 0.0008 < u ̂ c m * ≅ 15.7812 , an active monetary-fiscal regime implies three eigenvalues with positive real parts for any reasonable value of the coefficient C. A fiscal policy switch to a passive rule implies one negative eigenvalue and two eigenvalues with positive real parts. In this example, the saddle quantity equals σ ≡ τ′ − 0.027 + 0.00058C. Therefore, if we set τ′ > 0.027–0.00058C, the saddle quantity is positive.

We are now ready to provide the following result.

Furthermore, we can keep the deep parameters at D ̄ ∈ D, assume π* = 0.042,and study the surface

in the remaining ( C , R ̄ , τ ′ ) parameter space. As shown in Bella, Mattana, and Venturi (2017), the vanishing of Ω corresponds to the critical parametric surface, at which a generic steady state is a saddle-focus equilibrium with null saddle quantity.

Figure 1 depicts the parametric surface in the ( C , R ̄ , τ ′ ) space, such that P* is a saddle-focus equilibrium at the bifurcation point Ω = 0. Above the surface, the saddle quantity is positive. Below the surface, the saddle quantity is negative. Interestingly, the figure shows that a positive saddle quantity can be determined exactly, when the pair ( R ̄ , τ ′ is plausibly low and C > 1.

Figure 1:

Combinations of the ( C , R ̄ , τ ′ ) parameters at which Ω = 0.

We end this section by noticing some further details regarding the form of the eigenvalues in Example 1. It is clear that for C > 1.00046, and irrespective of the stance of fiscal policy, one eigenvalue is real, while the remaining two eigenvalues are complex conjugate. This means that locally, when monetary policy is active, convergence towards P* occurs typically through (damped) oscillating paths.^[4] It is also useful to observe the following.

Once it has been established that there are regions in the parameter space such that P* is a saddle-focus equilibrium with σ > 0, we follow Bella, Mattana, and Venturi (2017) to show that system M admits homoclinic solutions (pre-condition H.2 in Theorem 1).

The procedure leads to the following split function^[5]

where ( Ξ , Ψ , Ω ) ∈ 0,1 3 are free constants, while γ = λ₁, χ = Re(λ_2,3), and ξ = Im(λ_2,3). Then, with given parameters, conditions for the existence of the homoclinic loop, doubly asymptotic to the saddle-focus equilibrium point, rely on the existence of a triplet ( Ξ , Ψ , Ω ) ∈ 0,1 3 satisfying Σ = 0 (admissible solution).

To verify whether there are admissible solutions to (9) in the feasible parameter space, we specify further the calibration of the economy used in this section.

If τ ′ > τ ̃ ′ , then P* is a saddle-focus with positive saddle quantity. Let us now use the marginal tax rate, τ′, as the bifurcation parameter. More precisely, we iteratively increase τ′ above 0.02613 with a grid of 0.01 until a solution for Σ = 0 with Ξ , Ψ , Ω ∈ 0,1 3 emerges. The procedure reveals that there exists an interval I_τ′ ≅ (0.02613, 0.23543) such that, for all τ′ ∈ I_τ′, a family of homoclinic loops doubly asymptotic to the saddle-focus equilibrium point exists.

Figure 2 depicts the combinations of Ξ , Ψ , Ω solving the split function (9) for the case of τ′ = 0.15.

Figure 2:

Coordinates in Ξ , Ψ , Ω that give rise to the homoclinic loop for τ′ = 0.15.

2.4 Existence and properties of the chaotic attractor

Let ν = τ ′ − τ ̄ ′ , where τ ̄ ′ ∈ I τ ′ is the critical value of the marginal tax rate, such that an admissible solution of the split function exists for given coordinates, ( Ξ , Ψ , Ω ) ∈ 0,1 3 . Let V ⊂ R be a small open neighborhood of 0. We have the following result.

The attractor generated by this specific example is represented in Figure 3.

Figure 3:

The chaotic attractor in the w ̃ 1 , w ̃ 2 , w ̃ 3 space.

Figure 3 displays the distinct shape of the Shilnikov attractor. When the phase point approaches the saddle-focus point, the economy’s dynamics along the spiral attractor experience periods of relative calm. When the phase point begins to spiral away from the saddle-focus point, irregular episodes of oscillatory activity begin. The presence of a chaotic attractor is very important to the dynamics implied by NK models and has important economic implications. Because of its presence, small changes in initial conditions can result in large changes in dynamics over time. Moreover, within a chaotic attractor, given the initial value of the predetermined variable, there exists a continuum of initial values of the jump variables giving rise to admissible equilibria. Therefore, the policy options required to recover the uniqueness suggested by the local analysis are exactly those which may cause global indeterminacy of the equilibrium.

The numerical simulations developed in the paper show that if the initial conditions of the jump variables are chosen far enough from the target steady-state, then the emerging aperiodic dynamics tend to evolve for a long time around lower-than-targeted inflation and nominal interest rates. This can be interpreted as a liquidity trap phenomenon, which now depends on the existence of a chaotic attractor and not on the influence of an unintended steady state.

Consider the reconstructed time profile of the inflation rate (Figure 4). The time span is 10 years. First, observe the distinct shape of Shilnikov chaos. The wave train generated by the spiral attractor has long quiescence periods, when the phase point approaches the saddle-focus, followed by bursts of oscillatory activity. The average inflation rate can be persistently higher/lower than the steady-state value. This implies the possibility of long periods, during which inflation is stubbornly high, or long periods during which inflation is stubbornly low (akin to a deflationary equilibrium), and periods during which inflation is volatile. Also observe the downward drift, which was even more evident in Barnett et al. (2021), calibrated with parameter settings based on recent data.

Figure 4:

The time profile of the chaotic inflation rate.

3.1 Ending the chaos

Potential policy solutions to the problems produced by Shilnikov chaos can be divided into two groups. One group of approaches ends the Shilnikov chaos by removing the Taylor rule’s closed-loop interest rate feedback dynamics, while introducing a fundamentally different monetary policy design. The other approach is to retain the Taylor principle and thereby the Shilnikov chaos, while imposing an algorithm to control the chaos. This latter approach requires introduction of a second policy instrument in addition to the interest rate that appears in the Taylor rule. The source of the drift problem, produced by augmenting the system’s dynamics with a myopic Taylor rule, is the lack of a terminal condition. The additional policy instrument attaches a long run anchor to the feedback policy.

Alternatively, to end the chaos, the central bank could adopt any of the policy approaches that do not use interest rate feedback. There are many such policy designs in the literature that could be selected by the central bank in accordance with the central bank’s mechanism design. It is not the purpose of this paper to advocate any one of those alternatives.

Examples could include using an active fiscal policy and a passive monetary policy. That approach produces its own dynamical problems in an NK Model, but not Shilnikov chaos. Another example could include monetary policy without interest rate feedback. An open loop fixed monetary quantity growth rate would be the simplest approach. More sophisticated modern closed-loop approaches could include those using Divisia monetary quantity aggregates, such as those approaches proposed by Belongia (1996), Serletis and Rahman (2013), or Belongia and Ireland (2014, 2017, 2018, 2019, and as advocated by Peter Ireland in his role on the Shadow Open Market Committee. A long literature exists on alternatives to Taylor rule policies, such as Cochrane (2011).

3.2 Controlling the chaos

If the central bank were to decide to retain imposition of the Taylor Principle and thereby the resulting Shilnikov chaos, a second instrument of policy would need to be introduced to deal with the consequent eventual drift into a liquidity trap. The need for such a second instrument, in addition to the interest rate in the Taylor rule, is widely accepted and has been applied by most central banks in recent years. A survey of such new tools of monetary policy, such as forward guidance and intervention in long term bond markets, has been provided by Bernanke (2020) in his Presidential Address to the American Economic Association. But what is less well established is how to design a rule for use of such an instrument of policy. Under those circumstances, we would propose one of the available algorithms for controlling chaos. In addition, we find that when the economy is on a Shilnikov chaos attractor set associated with imposition of a Taylor rule, a second instrument of policy to control the undesirable properties of chaos should be adopted, even if the economy is not yet in a liquidity trap.

We now consider policy to control chaos. Assume the economy is enmeshed in a chaotic attractor. What should a policy maker do to alleviate the implied economic uncertainty, and bring agents’ inflationary expectations back in line with those coherent with the intended steady state? In each such approach, one of the new tools of policy would be adopted as a second instrument of policy to target, as an intermediate target, a long run anchor consistent with an available algorithm for controlling chaos.

The methods of controlling chaotic dynamics in the engineering literature provide useful tools in this regard. Under certain conditions, undesired irregular or even cyclical behavior can be switched off.^[6] An ingenious and well-known method for doing so is proposed by Ott, Grebogi, and Yorke (1990), also known as the OGY algorithm. It enables one to force a chaotic trajectory onto a desired target (a periodic orbit or a steady state of the system) by a correction mechanism. This mechanism has the form of a small, time-dependent perturbation of a certain control parameter. Suppose also that a neighborhood of the desired fixed point can be found, such that the system is guaranteed to be driven to the fixed point. If this neighborhood has points in common with a chaotic attractor, the neighborhood may be used as a controllable target to attain the fixed point.

To achieve control over a chaotic solution trajectory, the control parameter must be accessible to the central bank. This is a relevant point in our NK sticky-price model with Taylor rule interest rate feedback and a Ricardian fiscal policy. If we go back to the form of the eigenvalues in Example 1, it is clear that fiscal policy parameters can only govern the sign of the real eigenvalue.^[7] As a consequence, fiscal policy is ineffective in controlling chaos in the present setting.^[8]

In this regard, we choose to operate through the manipulation of the long-run, steady-state nominal interest rate, R ̄ , at which the steady-state Fisher equation ( R ( π * ) = R ̄ is satisfied. Since that interest rate is not under the direct control of the Central Bank, our algorithm treats manipulation of that interest rate as an intermediate target, rather than as an instrument of policy.

The choice of policy instrument or of market intervention operating procedure to be used in that intermediate targeting could depend upon the mechanism design of the central bank, which is not a topic of this research. An alternative OGY procedure could use the long run inflation rate, π*, instead of the nominal interest rate, R ̄ , as intermediate target. Although we believe that the two procedures would likely prove to be mathematically equivalent, we anticipate that an intermediate targeting OGY procedure using R ̄ would be more easily implemented by a Central Bank than an OGY procedure using the long run inflation rate, which is also a final target of policy.

From a more technical point of view, and to operate an informed choice between π* and R ̄ , we have also computed and evaluated at the intended steady-state, the partial derivatives of G J in (A.5) with regard to π and R. This computation is helpful, since in correspondence of a saddle-focus, the Jacobian of system M presents negative Tr J a n d D e t J at the intended steady-state. Therefore, chaos control according to the OGY mechanism translates into varying R ̄ or π* in such a manner that G J becomes negative. We found that, for parameters as in Example 3, ∂ G J ∂ π P * ≪ ∂ G J ∂ R P * , implying that using R ̄ has proportionally much higher stabilization power than varying π*.

Before proceeding with the implementation of the OGY algorithm, some preliminary steps need to be taken. First, we need to show that system M is controllable. Then, we will need to discuss the region of the parameter space supporting application of the OGY algorithm. Consider the following initial result.

Once controllability of system M can be established, the OGY algorithm requires that the eigenvalues of the controlled system be chosen such that stability is implied. Stabilizing a system is thus translated into searching for values of the nominal, steady state interest rate, R ̄ , such that all eigenvalues exhibit a negative real part (see Appendix B).

We have anticipated that there exists a critical value, u ̂ c m * ( R ̄ ) , such that if u c m * ( R ̄ ) > u ̂ c m * ( R ̄ ) , then an active-passive monetary-fiscal regime implies stability of the intended steady state. For notational convenience, let us define

and denote

We are now ready to prove the following.

Set the triplet ( R ̄ , π * , C ) at (0.06, 0.042, 1.5), as in the preceding examples, and set β = 1.78.^[9]

Re-running the algorithm for the presence of a chaotic attractor, we find that system M has a saddle-focus equilibrium with positive saddle quantity and a family of homoclinic orbits for values of the bifurcation parameter τ′ belonging to the (extended) interval, I τ ′ ≅ ( 0.02613 , 0.27923 ) . Set τ′ = 0.15. For this specific parameter configuration, J has three eigenvalues with negative real parts for any R ̄ ≥ 0.06767 . Let us select R ̄ = 0.07 and apply the OGY algorithm (as described in Appendix C) to obtain the controlled system solution (blue curve), where the control has been initiated at iteration 800. In Figure 5, we superimpose the solution time profile with uncontrolled inflation (red curve).

Figure 5:

Un-controlled and controlled inflation rate (control activated at the 800th iteration).

The result in Example 4 might appear puzzling when compared to the policy prescriptions suggested by Benhabib, Schmitt-Grohé, and Uribe (2002). However, a closer look at the different comparative statics implied by our model provides a full explanation of this apparent contradiction.^[10] Recall that Benhabib, Schmitt-Grohé, and Uribe (2002) maintain throughout their paper the complementarity condition in the utility function between real balances and consumption, u _cm > 0; this implies that a drop of the interest rate increases consumption and real economic activity in the model, via the implied increase in money holdings. In our case, the same stimulus to the real activity of the economy is obtained by an increase of the interest rate, provided that we have assumed u _cm < 0 (see Assumption 2 above).^[11] In Benhabib, Schmitt-Grohé, and Uribe (2002), stability is achieved through the simultaneous modulation of nominal interest rates.^[12] In our case, instead, it is the commitment to a long-run easing or tightening of the monetary policy stance, which is able to re-anchor expectations to the long-run target of inflation.