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The Phillips curve, regime switching, and the NAIRU
Journal of Economic Behavior & Organization Vol. 46 (2001) 23–37
The Phillips curve, regime switching, and the NAIRU Piero Ferri a , Edward Greenberg b,∗ , Richard H. Day c b
a University of Bergamo, Milan, Italy Department of Economics, Washington University at St. Louis, Eliot Hall 205, Campus Box 1208, One Brookings Drive, St. Louis, MO 63130-4899, USA c Department of Economics, University of Southern California, Los Angeles, CA 90089-0253, USA
Received 27 March 2000; received in revised form 19 June 2000; accepted 19 June 2000
Abstract This paper explores the impact of nonlinearities in the Phillips curve and its representation by means of a regime switching model on the meaning and the usefulness of the NAIRU. By implementing a regime switching model that can endogenously generate fluctuations in inflation and in the rate of unemployment, we conclude that the NAIRU is not necessarily a single value, is not necessarily reachable, and is very difficult to measure. The model yields a “neutral rate”, which is more an interval of values than a precise number. Furthermore, by postulating dynamic interdependence between real and monetary aspects, the model weakens the theoretical basis of the NAIRU concept itself. © 2001 Elsevier Science B.V. All rights reserved. JEL classification: E3; J3; C6 Keywords: Phillips curve; Regime switching; NAIRU; Nonlinearities
1. Introduction The recent performance of the US economy, particularly the persistent decline in the rate of unemployment in the face of a steady rate of inflation, has raised questions about the validity of a NAIRU-based Phillips curve. There are defenders of the NAIRU (non-accelerating inflation rate of unemployment) such as Stiglitz (1997), and those claiming that it has never existed (Galbraith, 1997). Others hint that the NAIRU is a time varying concept (see Gordon, 1997 and Phelps, 1999), and others claim that even if it were to exist it would not be ∗ Corresponding author. Tel.: +1-314-935-5670; fax: +1-314-935-4156. E-mail address: [email protected] (E. Greenberg).
0167-2681/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 6 8 1 ( 0 1 ) 0 0 1 8 5 - 8
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important to check the inflationary process (see Staiger et al., 1997). Solow (1998) believes NAIRU is an interval and also misnamed; he would call them “neutral values”. In the present paper we examine these issues from a dynamic point of view. We stress the presence of nonlinearities that usually play a very important role in dynamics. In the case of the Phillips curve, an increasing number of authors refer to a nonlinear relationship between inflation and unemployment (see Fair, 1999), even though there is no consensus on the exact shape. According to Eisner (1997) and Coen et al. (1999) the Phillips curve is S-shaped and fundamentally concave, while others claim that is nonlinear but essentially convex (see, for instance, Clark and Laxton, 1997). Although the policy implications of these hypotheses have been thoroughly discussed, their analytical underpinnings remain relatively unexplored. This is what we intend to pursue in the present paper. In particular, since some forms of nonlinearities are topologically conjugate to regime switching models, 1 we resort to this mathematical device to study its economic implications. In fact, “the economics of drifting parameters” has at least two implications that are worth mentioning. First, one is compelled to study the dynamics of the process. This aspect has been recognized by Sargent (1999, p. 132) when he states: “Regime shifts occur, not from procedures, but from changes in beliefs created by its econometric procedures. The system’s nonlinearities, rather than large shocks, explain this behavior”. In the present paper, we shall look for other reasons behind the drifting parameters. Second, the stability of the process must be studied, which has implications for attempts to estimate the NAIRU. The theses put forward may be briefly summarized. First, a regime switching model implies the presence of multiple equilibria, which implies more than one NAIRU, thereby destroying the convenience of a single number. Second, the interdependence between real and monetary factors weakens the theoretical basis of the NAIRU concept itself. In Section 2 the Phillips curve, the NAIRU, and the problem of its dynamical stability are introduced. In Section 3 a regime switching model is discussed, and its dynamic properties are explored in Section 4. A different environment is considered in Section 5, while an interpretation of the results is offered in Section 6. Conclusions are drawn in Section 7. An appendix (Appendix A) contains the proofs of our principal findings.
2. The NAIRU and its stability To study the NAIRU and its dynamical stability, the model must include the Phillips curve, which in the literature represents a supply side constraint, and a relationship that links inflation and the unemployment rate from the demand side. This latter equation is important for determining the stability of the system, even though it has not always been explicitly considered. Both equations are standard in the macro literature. The Phillips curve (or better, the supply curve because it links inflation to unemployment) can be represented by the following equation: πt = πt−1 − τ (ut − u∗ ),
(1)
1 On this concept, see Holmgren (1996) who studies a relationship between the logistic function and the tent map.
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where πt is the inflation rate at time t, ut the rate of unemployment, and u∗ the NAIRU. When the rate of unemployment is constant and equal to u∗ , the inflation rate is constant. 2 The value of τ , which measures the impact of the labor market conditions on the inflation dynamics, depends on the nature of the model underlying the Phillips curve. Layard et al. (1991) have shown how Eq. (1) can be obtained from two equations, one referring to the firms’ behavior (indexed by f) and the other referring to wage determination (indexed by w): log p − log w = δf − βf u − γf (πt − πt−1 )
(2)
log w − log p = δw − βw u − γw (πt − πt−1 ).
(3)
It follows that τ in Eq. (1) can be considered as a ratio of the parameters reflecting real and nominal rigidities, τ = β/γ , where β = βf + βw measures the intensity with which the product and labor markets respond to the rate of unemployment and γ = γf + γw indicates the impact of price dynamics on both equations. On setting πt = πt−1 in the above equations, we obtain u∗ = δ/β, so that the NAIRU depends only on the real rigidity parameters and the intercepts that reflect exogenous forces (in contrast, we show that all parameters are important for the dynamics of our model). Ceteris paribus, the higher is β, i.e. the smaller is real rigidity, the smaller will be the NAIRU. In view of the above equations, it must be that β > 0, γ > 0, and δ > 0 to maintain a positive NAIRU and a Phillips curve with the traditional negative slope (a general discussion of the values of these parameters in generating regime switching model is in Day and Lin (1991)). This type of Phillips curve reflects constraints originating in the supply side of the economy. In order to generate system relationships, one must introduce a demand side relationship between inflation rate and unemployment. This can represented by the following simplified equation: πt = m +
ut , α
(4)
where m is the rate of growth of money (assumed constant) and α measures the impact of inflation on the dynamics of aggregate demand. Usually, one assumes α > 0, implying that an increase in inflation make the unemployment situation worse. We discuss this hypothesis below. The system of Eqs. (1) and (4) is a causal one: the NAIRU is determined in Eq. (4), and the equilibrium rate of inflation is determined in Eq. (4) when u = u∗ . To study the dynamics, one must combine Eqs. (1) and (4). The dynamics of unemployment are described by ut = (1 − ω)u∗ + ωut−1 ,
(5)
where ω = 1/(1 + τ α) < 1. The dynamics of inflation are given by πt = (1 − ω)π ∗ + ωπt−1 .
(6)
Eqs. (5) and (6) define the dynamics of a NAIRU-type system. It follows that the dynamic process governing both the rate of unemployment and the rate of inflation are asymptotically 2 According to Layard et al. (1991, p. 77), “. . . a more accurate term would be non-increasing rate of unemployment, but the common usage is NAIRU”.
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stable (also see Nickell, 1988). This means that if the initial unemployment is above the equilibrium level, it falls asymptotically to its equilibrium level and vice versa. For ω > 1, the system is unstable; this case is discussed below. The stability of the curve, in turn, has further consequences. First, if the curves shift because of exogenous shocks, the NAIRU will still be reached with a velocity that depends both on ω and the nature of the shocks. Second, stability facilitates the econometric work to estimate its values.
3. Nonlinearities and regime switching The equations introduced above are linear or log-linear. Richer dynamical properties, however, are associated with nonlinear models. An example of a nonlinearity has been described in the following terms by Rowthorn (1977, pp. 221–222): In an era of slow inflation, expectations about future price changes may not be held with any great certainty and, even if they are, it is not particularly important to act on them. By contrast, in an era of fast inflation, the cost of inactivity may be high and workers must do something to protect themselves against the effect of future price changes. Moreover, even though there may be uncertainty about exactly how much prices will rise, everyone can be sure they will rise by a considerable amount. Under these circumstances, wage settlements are likely to contain some provision against future inflation . . . The transition from one kind of behavior may be rather abrupt. At low rates of inflation there may be little or no anticipatory behavior, but then suddenly, when inflation passes a certain critical point, qualitative changes may occur in the whole mechanism of wage bargaining. An example of nonlinearity in expectations concerning the Phillips curve can be found in Sargent (1999, p. 5): “. . . adaptive system can spontaneously generate regime shifts by their learning-inspired nonlinearities”. Nonlinearities, however, do not refer only to the process of expectation formation. They also concern other parameters of the Phillips curve. For instance, an important distinction for the purpose of the present analysis is between the supporters of concavity of the Phillips curve and those in favor of convexity; both issues can be interpreted in terms of asymmetries. According to Eisner (1997), leftward movements of the Phillips curve, when unemployment is falling, are substantially horizontal. However, rightward movements as unemployment rises result in decreased inflation. In this perspective, recessions are indeed disinflationary. But additional high unemployment adds little to disinflation. According to Stiglitz (1997, p. 9): “This concavity is consistent with the literature on asymmetric price adjustment, which shows that in monopolistically competitive markets, producers might adjust prices down to avoid being undercut by a rival but will be more reluctant to raise prices even in the face of generally rising prices”. In contrast, Clark and Laxton (1997, p. 33), stressing the asymmetry in the labor market, come to the conclusion of a convex Phillips curve: “The implication of convexity means that gradual disinflation is less costly in terms of foregone output and employment than a more rapid ‘cold shower’ approach to reducing inflation”.
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In the present paper, we approximate the nonlinearity of the curve with a regime switching device. In doing this, we impose a discontinuity to generate endogenous dynamics and obtain a self-exciting threshold autoregressive (SETAR) model. 3 This model captures the insight of Keynes (1936, p. 301): “In actual experience the wage-unit does not change continuously in terms of money in response to every small change in effective demand, but discontinuously. These points of discontinuity are determined by the psychology of the workers and by the policies of employers and trade unions”. Three channels can be hypothesized through which regime switching can be manifested and which summarize the literature just mentioned: (i) changes in the parameters measuring nominal rigidities; (ii) changes in the parameters measuring the impact of real economic activity; (iii) changes in the exogenous factors captured by the intercepts of the Eqs. (2) and (3). Of course, changes can refer to either the price or the wage equation or both. While Eisner’s main finding is that the accelerationist Phillips curve appears to be a valid econometric relation only for unemployment rates above the neutral rate, Solow’s view “. . . is that the relation just becomes mushy” (p. 15). And this is what happens in a version of our model.
4. The dynamics To interpret Eqs. (5) and (6) as a regime switching model, we have to introduce a two-phase dynamic system of the following type (see Day (1994, chapter 6): ut ≤ us (1 − ω)u∗ + ωut ut+1 = θ (ut ) := (7) (1 − ω )u∗ + ω ut ut > us , where us is a threshold level of unemployment at which switches take place, while u∗ and u∗ are respectively, the NAIRU for Regimes I and II. In other words, system (7) assumes that when the threshold is passed, the reaction of either or both price and wage-setting strategies change, and that this threshold is represented by some value of the unemployment rate, us . This threshold divides Regime I of low unemployment from Regime II of high unemployment. It is not an equilibrium value of unemployment. It is just a level separating an economy with “high” unemployment rates from one with “low” rates. For unemployment rates below us , the parameters shaping ω prevail. For unemployment rates above us , the parameters forming ω become operative. 4 A dual two-phase system holds for the rate of inflation: πt ≤ π s (1 − ω)π ∗ + ωπt πt+1 = Θ(πt ) := (8) (1 − ω )π ∗ + ω πt πt > π s , where π s = m + us /α. There are two regimes described by Eqs. (7) and (8): Regime I is the virtuous one, with low unemployment and low inflation, while Regime II is the vicious one, 3 4
For this definition in a stochastic environment, see Tong (1990). Empirical estimates of these parameters may be found in Ferri and Greenberg (1992).
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characterized by high inflation and unemployment. The threshold is given by a particular rate of unemployment (us ) or of the inflation rate (π s ). The dynamics of a system like Eqs. (7) and (8) depend fundamentally on three factors: (i) the value of the threshold; (ii) the size of the coefficients ω and ω ; and (iii) the relative position of the two equilibria. To simplify the exposition, we concentrate on the unemployment equation; reference to the inflation equation will be made either when its pattern differs from the one established for the rate of unemployment or when it can help to deepen the underlying economic meaning. Two cases characterized by different parameter values are considered: Case I: 0 < ω < ω < 1, u∗ < us < u∗ Case II: ω, ω > 1, u∗ > us > u∗ Case I arises when the slope of the Phillips curve in Regime II is larger than in Regime I, i.e. τ > τ . Since ω = 1/(1 + τ α), we have ω < ω < 1, and so the first condition of Case I is fulfilled. There are several ways to satisfy the second condition of Case I. 5 What is important to stress is that this condition implies that the long-run equilibrium of each regime is not reachable, because it lies in the opposite regime. This is what Day (1994) calls the switching condition. Given these assumptions, we obtain the picture shown in Fig. 1 for the adjustment process in terms of the rate of unemployment. Suppose we take up the process in the high unemployment Regime II. Initially, unemployment declines. As shown in the example in the diagram, these changes would be gradual until the switching threshold us is crossed and Regime I is entered; then unemployment increases. Analogous movements can be shown for the rate of inflation. Observe that there is no stationary state: a point which begins in one phase zone is drawn toward the equilibrium value it would have if it were the only regime, but when the switching threshold is crossed, the other phase equation takes over and variables are drawn in the opposite direction, so that the process moves cyclically from one regime to the other. As long as the conditions of Case I hold, neither equilibrium can be reached — even in the long-run. The qualitative behavior is generic, that is, the cycles alternate, but the “speed” of the changes and the duration within each regime depends on the parameter values. But where do trajectories go? Consider iterates of the map θ(·). This map is piecewise linear and, as ω and ω are positive and less than one, the slope in any piece must involve products of these slopes. This means that any two trajectories that come close enough to each other will converge. Indeed the following result can be established. Proposition 1. Given the conditions of Case I, there exists a unique, asymptotically stable cycle in unemployment of period k ≥ 2. This result, a proof of which is given in the Appendix A, means that any trajectory converges closer and closer to a sequence of k distinct unemployment levels {u1 , . . . , uk }. The same can be said for inflation. To fulfill the second condition, it is sufficient to suppose that β > β; i.e. real rigidity is smaller in Regime II (see Eqs. (2) and (3)). Given the definition of NAIRU, this implies that u∗ < u∗ . Alternatively, one could allow changes in the nominal rigidity parameters, as suggested by Rowthorn (1977), and in the values of the intercepts. In this case if γ < γ , then ω < ω, while at the same time, if δ < δ, then the NAIRU in Regime II is smaller than in Regime I. 5
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Fig. 1. Case I, stable cycles: all trajectories enter the trapping set or box.
If random shocks were added to the model, irregular fluctuations would follow, exacerbated by the underlying regime switching mechanism. Undoubtedly, one would want to add such a term in empirical work. But irregular or chaotic fluctuations can also be brought about purely by the intrinsic forces represented in the model as can be seen by considering Case II.
5. A different environment When one turns to Case II, the most important possibility to be considered is that ωω > 1. For simplicity we assume that they are both greater than one. We have to show that even if each regime is unstable if considered in isolation, it may still produce some sort of stability within a regime switching model. 6 To obtain this kind of stability, we change the sign of 6 As stressed by Tong (1990), the instability of a regime does not imply the instability of a system in the case of a regime switching model.
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Fig. 2. Case II, strongly ergodic, chatoic fluctuations: all trajectories enter the trapping set or box and are nonperiodic.
α. If we suppose that α < 0, then ω = 1/(1 + ατ ) > 1; the same holds true for ω . If this hypothesis is accepted, then the phase has the appearance shown in Fig. 2. Can this hypothesis be justified from an economic point of view? Fazzari et al. (1998) show how the relation between prices and aggregate demand is more problematic than is usually recognized. One line of argument that implies α < 0 is the Minsky (1982) debt–deflation hypothesis, in which deflation results in an increased real burden of debt that inhibits investment, offsetting the Pigou and Keynes effects reflected in α > 0 (Case I). 7 In this case, the inequality u∗ > us > u∗ can be justified by making a piece-wise approximation of a convex Phillips curve. 8 All trajectories become trapped in the interval [(1 − ω )u∗ + ω us , (1 − ω)u∗ + ωus ]. 7 8
For definitions of these effects, see Flashel et al. (1991). If one refers to Eqs. (2) and (3), it is enough to suppose that δ < δ.
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Proposition 2. Given the hypotheses of Case II, then for almost all positive initial unemployment levels u0 that belong to (u∗ , u∗ ), unstable nonperiodic (chaotic) fluctuations occur. The proportion of time the process is in Regime I converges to a constant fraction, say p, and that in Regime II converges to a constant fraction 1 − p. The proof is in Appendix A. This proposition is also true when π is substituted for u. Regime switching of the SETAR variety implies a discontinuity at the threshold level. Although at the micro level there is evidence in support of discontinuous decision processes, 9 at the macro level these discontinuities are rarely observed. An alternative is to specify regime switching of the smooth threshold autoregressive (STAR) variety (see Granger and Terasvirta, 1993). 10
6. Discussion We first discuss the economics of changing parameters. According to Sargent (1999, p. 16): “Lucas left the drift in coefficients unexplained. Neither the macroeconomic theory nor the rational expectations econometrics constructed after Lucas’s critique explain such drift”. In the same place, Sargent (p. 105) argues that a change in the coefficients of the estimated Phillips curve “. . . is the key to the behavior of inflation”. Sargent’s analysis differs from ours in two respects. First, in Sargent’s analysis, changes in beliefs drive the dynamics of the model. These changes in beliefs are mainly those of the Federal Reserve System that tries to learn the exact shape of the Phillips curve. In turn, these beliefs are driven by the so called “induction hypothesis” that refers to a particular kind of adaptive expectation formulation. Second, it is not only monetary policy that is important in determining the dynamics of the inflation–unemployment tradeoff; phenomena that affect the product and the labor market also play key roles. Our analysis is based upon the interplay of real and nominal rigidities; both real and nominal parameters change. Some of these changes are simply cyclical and reflect the different pressures that a regime closer to full employment exert on the behavior of both entrepreneurs and workers. Others are more structural and reflect such phenomena 9 10
On this point see Cooper and Haltiwanger (1993). The SETAR system can be written as
u(t + 1) = (1 − ω)u∗ + ωu(t) + [(1 − ω )u∗ + ω ut ]I (u), where I (u) =
0
if
ut ≤ us
1
if
ut > u s .
(9)
If I (u) is replaced by a smooth function such as (see Tong, 1990, p. 107) F (u) =
1 , 1 + exp[(ut − us )/η]
(10)
the transition around the threshold becomes smooth. This is called the STAR model. As η → ∞, the SETAR model is obtained.
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as the globalization of markets, new information technology, and the increased flexibility of the labor market that can modify the dynamics by changing both the threshold and the shape of the equations. Next, we compare our results with those obtained from a NAIRU-based model. As we have seen, the dynamics of a NAIRU model imply, for ω < 1, convergence to constant inflation and unemployment rates. For ω ≥ 1, however, the system is unstable. Propositions 1 and 2 have shown that our regime switching model implies bounded dynamics whether ωω is greater or less than one. 7. Concluding remarks The policy implications of a NAIRU model on the one hand and a regime switching model on the other are somewhat different. Policy makers who think in terms of a NAIRU model believe that intervention is necessary to prevent unacceptably high rates of inflation when unemployment falls below the NAIRU. But, since a NAIRU model with ω < 1 is stable, why is intervention necessary? Both inflation and unemployment will converge to their steady-state values monotonically without intervention. One possibility is that policymakers believe that convergence of the system will be so slow that the economy will suffer high rates of inflation for an excessive length of time. Another possibility is that policymakers believe ω > 1, in which case continuous intervention is necessary because the system is unstable. In the latter case, even a small departure from NAIRU must be offset as soon as possible because of the instability. In contrast, the regime switching model we have discussed is less sensitive to parameter values. We have seen that the system may exhibit limit cycles or more complex patterns, but unemployment and inflation rates remain bounded. Although policymakers may dislike the behavior of the variables, intervention becomes problematic because there is no NAIRU to indicate whether and what kind of intervention is necessary. Moreover, a feature of our model is an interaction between real and monetary phenomena that weakens the theoretical foundations of the NAIRU concept. Some generalizations of the model should be considered. For instance, to understand the implications of different values of α, it is necessary to study the role of a changing supply of money and credit conditions. Other aspects — for example, capital accumulation, the role of real interest on investment, the effect of government and private sector debts — should be considered. Indeed, nonlinear effects of the interest rate on the demand for liquidity and the demand for goods can generate complex dynamics when the product market adjusts with a lag to disequilibrium, as has been shown by Day and Lin (1991). A logical next step would be to put such quantity adjustments into the model together with price–wage adjustments. Finally, the role of expectations should be explicitly examined. Acknowledgements A financial contribution from the Italian Ministry of University and Research is acknowleged. We thank Ara Mentcherian for checking our calculations and for a number of useful comments.
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Appendix A Proof of Proposition 1. Let U1 = [um , us ],
U2 = (us , uM ],
where
um = (1 − ω )u∗ + ω us ,
uM = (1 − ω)u∗ + ωus .
The interval U = [um , uM ] is a trapping set: all trajectories must enter this set. Of course, U = U1 ∪ U2 . To see how cycles arise, consider the simplest possibility first. Assume θ (um ) > us and θ (uM ) < us ; then it is easy to see that for all u ∈ U1 , θ1 (u) ∈ U2 and for all u ∈ U2 , θ2 (u) ∈ U1 . Therefore, θ 2 (u) = θ2 · θ1 (u) ∈ U1 all u ∈ U1 , and θ 2 (u) = θ1 · θ2 (u) ∈ U2 all u ∈ U2 . That is, θ 2 (U1 ) ⊂ U1 and θ 2 (U2 ) ⊂ U2 . By standard fix point arguments this implies fix points, say x1 , x2 of the map θ 2 (·) which must be two-cyclic, i.e. θ 2 (x2 ) = θ2 · θ1 (x1 ) = x1 . This is shown in Fig. 3. In general we proceed as follows. First, we observe Lemma 1. Without loss of generality, assume us − um < uM − us , then θ (U1 ) ⊂ U2 . Proof. By assumption ω < 1, so ω=
uM − θ (um ) < 1. us − u m
Fig. 3. Sufficent conditions for two-period cycles.
(11)
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Fig. 4. Existence of convergent (m + 1) period cycles (m = 2).
Combining (11) with this expression gives θ(um ) > us , but θ (u) = (1 − ω)u∗ + ωu > θ (um ) > us all u ∈ U1 . This gives the desired result. The lemma is illustrated in Fig. 3. It implies that if ut ∈ U1 , then ut+1 ∈ U2 and ut ∈ U2 ⇒ ut+1 ∈ U1 . Now we consider two cases. Assume (11) Case I. Assume there exists a smallest integer m such that θ2−m (U1 ) ⊃ θ1 (U1 ).
(12)
Then U1 ⊃ θ2m · θ1 (U1 ) = θ m+1 (U1 ), so U1 is nonexpanding under θ m+1 (·) and by a standard theorem a fix point exists. 11 This result is illustrated in Fig. 4. Although a cycle of period 3 exists in this example, this does not imply chaos, because the map θ(·) is discontinuous at us . Indeed, the cycle must be asymptotically stable. To see this, we note that a p = m + 1 cyclic point satisfies (u∗ )p = θ p ((u∗ )p ) and that the slope of θ p (·), which is defined everywhere except at us , is the product ω1 ω2m < 1, which implies asymptotic stability. Case II. If (12) does not hold, then there is a smallest integer m such that U2 := θ (U1 ) ∩ θ −m (U1 ) ∩ U ⊂ U2 = ∅, U2 = θ (U1 ) ∩ θ −(m+1) (U1 ) ⊂ U2 = ∅, i.e. θ (U1 ) ⊂ θ −m (U1 ) ∪ θ −(m+1) (U1 ). Now choose U1 , U1 so that θ (U1 ) = U2 , 11
θ (U1 ) = U2 .
See Day (1994, theorem 5.2).
(13)
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Fig. 5. Existence of 2m + 3 period cycles: (a) m = 1, five-period cycle exits (θ 5 (u) = ω12 · ω23 ); (b) m = 2, seven-period cycle exits (θ 7 (u) = ω12 · ω25 ).
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Applying θ m to both sides of the first of these, we get θ m+1 (U1 ) = θ m (U2 ) = U1 . Applying θ again gives θ m+2 (U1 ) = θ (U1 ) = U2 . Applying θ m+1 to both sides of this expression gives θ 2m+3 (U1 ) = θ m+1 (U2 ) = U1 , which gives a cycle of 2m + 3 that must be asymptotically stable for reasons analogous to that given for Case I. This completes the proof of Proposition 1. Fig. 5a gives an example for m = 1 and Fig. 5b for m = 2. Proof of Proposition 2. By assumption ω1 , ω2 > 1. Consequently, θ(·) is expansive and, by the Lasota–Yorke theorem, there exists a unique absolutely continuous invariant probability measure µ(·) such that µ(S) =
S
f (u) du = lim
T →∞
T χs (θ i (u)) i=t
T
,
where f is an integrable density function and supp f > 0 on a set of intervals such that U1 ∩ suppf = ∅, U2 ∩ suppf = ∅. Let p = U1 f (u) du. Then 1 − p = U2 f (u) du. Absolute continuity means that trajectories are dense in supp f . It is easy to see in this case that there exist Li–Yorke points so that chaotic trajectories exist. Since supp f is a basin of attraction for all trajectories with θ(suppf ) = suppf , the scrambled set S ⊂ suppf , so that µ− almost all trajectories will be nonperiodic and chaotic, passing as close as you like to every chaotic point in S (for further discussion, see Day, 1994, chapters 7 and 8). References Clark, P., Laxton, D., 1997. Phillips curves, Phillips lines and the unemployment costs of overheating. Centre for Economic Performance Discussion Paper 344. Coen, R.M., Eisner, R., Marlin, J.T., Shah, S.N., 1999. The NAIRU and wages in local markets. American Economic Review Papers and Proceedings 89, 52–57. Cooper, R., Haltiwanger, J., 1993. The aggregate implications of machine replacement: theory and evidence. American Economic Review 83, 360–382. Day, R.H., Lin, T.-Y., 1991. A Keynesian business cycle. In: Nell, E.J., Semmler, W. (Eds.), Nicholas Kaldor and Mainstream Economics, Confrontation and Convergence. St. Martin’s Press, New York, pp. 282–305. Day, R.H., 1994. Complex economic dynamics. In: An Introduction to Dynamical Systems and Market Mechanisms, Vol. I. MIT Press, Cambridge. Eisner, R., 1997. A new view of the NAIRU. In: Davidson, P., Kragel, J.A. (Eds.), Improving the Global Economy: Keynesianism and the Growth in Output and Employment. Elgar, Cheltenham, pp. 196–230. Fair, R.C., 1999. Does the NAIRU have the right dynamics? American Economic Review Papers and Proceedings 89, 58–62. Fazzari, S.M., Ferri, P., Greenberg, E., 1998. Aggregate demand and firm behavior. Journal of Post Keynesian Economics 20, 527–558.
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Ferri, P., Greenberg, E., 1992. Wages, Regime Switching and Cycles. Springer, New York. Flashel, P., Franke, R., Semmler, W., 1991. Dynamic Macroeconomics. MIT Press, Cambridge. Galbraith, J.K., 1997. Time to ditch the NAIRU. Journal of Economic Perspectives 11, 93–108. Gordon, R.J., 1997. The time-varying NAIRU and its implications for economic policy. Journal of Economic Perspectives 11, 11–32. Granger, C.W.J., Terasvirta, T., 1993. Modeling Nonlinear Economic Relations. Oxford University Press, Oxford. Holmgren, R.A., 1996. A First Course in Discrete Dynamical Systems. Springer, New York. Keynes, J.M., 1936. The general Theory of Employment, Interest and Money. Macmillan, London. Layard, P.R.G., Nickell, S., Jackman, R., 1991. Unemployment. Oxford University Press, Oxford. Minsky, H.P., 1982. Can “It” Happen Again? Sharpe, New York. Nickell, S.J., 1988. Wages and economic activity. In: Eltis, W., Sinclair, P. (Eds.), Keynes and Economic Policy: The Relevance of the General Theory After Fifty Years. Macmillan, London, pp. 65–75. Phelps, E.S., 1999. Behind this structural boom: the role of asset valuations. American Economic Review Papers and Proceedings 89, 63–68. Rowthorn, R.E., 1977. Conflict, inflation and money. Cambridge Journal of Economics 1, 215–239. Sargent, T.J., 1999. The Conquest of American Inflation. Princeton University Press, Princeton. Solow, R.M., 1998. How cautious must the Fed be? In: Friedman, B.M. (Ed.), Inflation, Unemployment, and Monetary Policy. MIT Press, Cambridge, pp. 1–28. Staiger, D., Stock, J.H., Watson, M.W., 1997. The NAIRU, unemployment and monetary policy. Journal of Economic Perspectives 11, 33–50. Stiglitz, J., 1997. Reflections on the natural rate hypothesis. Journal of Economic Perspectives 11, 3–10. Tong, H., 1990. Non-Linear Time Series. Clarendon Press, Oxford.
The Phillips curve, regime switching, and the NAIRU Piero Ferri a , Edward Greenberg b,∗ , Richard H. Day c b
a University of Bergamo, Milan, Italy Department of Economics, Washington University at St. Louis, Eliot Hall 205, Campus Box 1208, One Brookings Drive, St. Louis, MO 63130-4899, USA c Department of Economics, University of Southern California, Los Angeles, CA 90089-0253, USA
Received 27 March 2000; received in revised form 19 June 2000; accepted 19 June 2000
Abstract This paper explores the impact of nonlinearities in the Phillips curve and its representation by means of a regime switching model on the meaning and the usefulness of the NAIRU. By implementing a regime switching model that can endogenously generate fluctuations in inflation and in the rate of unemployment, we conclude that the NAIRU is not necessarily a single value, is not necessarily reachable, and is very difficult to measure. The model yields a “neutral rate”, which is more an interval of values than a precise number. Furthermore, by postulating dynamic interdependence between real and monetary aspects, the model weakens the theoretical basis of the NAIRU concept itself. © 2001 Elsevier Science B.V. All rights reserved. JEL classification: E3; J3; C6 Keywords: Phillips curve; Regime switching; NAIRU; Nonlinearities
1. Introduction The recent performance of the US economy, particularly the persistent decline in the rate of unemployment in the face of a steady rate of inflation, has raised questions about the validity of a NAIRU-based Phillips curve. There are defenders of the NAIRU (non-accelerating inflation rate of unemployment) such as Stiglitz (1997), and those claiming that it has never existed (Galbraith, 1997). Others hint that the NAIRU is a time varying concept (see Gordon, 1997 and Phelps, 1999), and others claim that even if it were to exist it would not be ∗ Corresponding author. Tel.: +1-314-935-5670; fax: +1-314-935-4156. E-mail address: [email protected] (E. Greenberg).
0167-2681/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 6 8 1 ( 0 1 ) 0 0 1 8 5 - 8
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important to check the inflationary process (see Staiger et al., 1997). Solow (1998) believes NAIRU is an interval and also misnamed; he would call them “neutral values”. In the present paper we examine these issues from a dynamic point of view. We stress the presence of nonlinearities that usually play a very important role in dynamics. In the case of the Phillips curve, an increasing number of authors refer to a nonlinear relationship between inflation and unemployment (see Fair, 1999), even though there is no consensus on the exact shape. According to Eisner (1997) and Coen et al. (1999) the Phillips curve is S-shaped and fundamentally concave, while others claim that is nonlinear but essentially convex (see, for instance, Clark and Laxton, 1997). Although the policy implications of these hypotheses have been thoroughly discussed, their analytical underpinnings remain relatively unexplored. This is what we intend to pursue in the present paper. In particular, since some forms of nonlinearities are topologically conjugate to regime switching models, 1 we resort to this mathematical device to study its economic implications. In fact, “the economics of drifting parameters” has at least two implications that are worth mentioning. First, one is compelled to study the dynamics of the process. This aspect has been recognized by Sargent (1999, p. 132) when he states: “Regime shifts occur, not from procedures, but from changes in beliefs created by its econometric procedures. The system’s nonlinearities, rather than large shocks, explain this behavior”. In the present paper, we shall look for other reasons behind the drifting parameters. Second, the stability of the process must be studied, which has implications for attempts to estimate the NAIRU. The theses put forward may be briefly summarized. First, a regime switching model implies the presence of multiple equilibria, which implies more than one NAIRU, thereby destroying the convenience of a single number. Second, the interdependence between real and monetary factors weakens the theoretical basis of the NAIRU concept itself. In Section 2 the Phillips curve, the NAIRU, and the problem of its dynamical stability are introduced. In Section 3 a regime switching model is discussed, and its dynamic properties are explored in Section 4. A different environment is considered in Section 5, while an interpretation of the results is offered in Section 6. Conclusions are drawn in Section 7. An appendix (Appendix A) contains the proofs of our principal findings.
2. The NAIRU and its stability To study the NAIRU and its dynamical stability, the model must include the Phillips curve, which in the literature represents a supply side constraint, and a relationship that links inflation and the unemployment rate from the demand side. This latter equation is important for determining the stability of the system, even though it has not always been explicitly considered. Both equations are standard in the macro literature. The Phillips curve (or better, the supply curve because it links inflation to unemployment) can be represented by the following equation: πt = πt−1 − τ (ut − u∗ ),
(1)
1 On this concept, see Holmgren (1996) who studies a relationship between the logistic function and the tent map.
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where πt is the inflation rate at time t, ut the rate of unemployment, and u∗ the NAIRU. When the rate of unemployment is constant and equal to u∗ , the inflation rate is constant. 2 The value of τ , which measures the impact of the labor market conditions on the inflation dynamics, depends on the nature of the model underlying the Phillips curve. Layard et al. (1991) have shown how Eq. (1) can be obtained from two equations, one referring to the firms’ behavior (indexed by f) and the other referring to wage determination (indexed by w): log p − log w = δf − βf u − γf (πt − πt−1 )
(2)
log w − log p = δw − βw u − γw (πt − πt−1 ).
(3)
It follows that τ in Eq. (1) can be considered as a ratio of the parameters reflecting real and nominal rigidities, τ = β/γ , where β = βf + βw measures the intensity with which the product and labor markets respond to the rate of unemployment and γ = γf + γw indicates the impact of price dynamics on both equations. On setting πt = πt−1 in the above equations, we obtain u∗ = δ/β, so that the NAIRU depends only on the real rigidity parameters and the intercepts that reflect exogenous forces (in contrast, we show that all parameters are important for the dynamics of our model). Ceteris paribus, the higher is β, i.e. the smaller is real rigidity, the smaller will be the NAIRU. In view of the above equations, it must be that β > 0, γ > 0, and δ > 0 to maintain a positive NAIRU and a Phillips curve with the traditional negative slope (a general discussion of the values of these parameters in generating regime switching model is in Day and Lin (1991)). This type of Phillips curve reflects constraints originating in the supply side of the economy. In order to generate system relationships, one must introduce a demand side relationship between inflation rate and unemployment. This can represented by the following simplified equation: πt = m +
ut , α
(4)
where m is the rate of growth of money (assumed constant) and α measures the impact of inflation on the dynamics of aggregate demand. Usually, one assumes α > 0, implying that an increase in inflation make the unemployment situation worse. We discuss this hypothesis below. The system of Eqs. (1) and (4) is a causal one: the NAIRU is determined in Eq. (4), and the equilibrium rate of inflation is determined in Eq. (4) when u = u∗ . To study the dynamics, one must combine Eqs. (1) and (4). The dynamics of unemployment are described by ut = (1 − ω)u∗ + ωut−1 ,
(5)
where ω = 1/(1 + τ α) < 1. The dynamics of inflation are given by πt = (1 − ω)π ∗ + ωπt−1 .
(6)
Eqs. (5) and (6) define the dynamics of a NAIRU-type system. It follows that the dynamic process governing both the rate of unemployment and the rate of inflation are asymptotically 2 According to Layard et al. (1991, p. 77), “. . . a more accurate term would be non-increasing rate of unemployment, but the common usage is NAIRU”.
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stable (also see Nickell, 1988). This means that if the initial unemployment is above the equilibrium level, it falls asymptotically to its equilibrium level and vice versa. For ω > 1, the system is unstable; this case is discussed below. The stability of the curve, in turn, has further consequences. First, if the curves shift because of exogenous shocks, the NAIRU will still be reached with a velocity that depends both on ω and the nature of the shocks. Second, stability facilitates the econometric work to estimate its values.
3. Nonlinearities and regime switching The equations introduced above are linear or log-linear. Richer dynamical properties, however, are associated with nonlinear models. An example of a nonlinearity has been described in the following terms by Rowthorn (1977, pp. 221–222): In an era of slow inflation, expectations about future price changes may not be held with any great certainty and, even if they are, it is not particularly important to act on them. By contrast, in an era of fast inflation, the cost of inactivity may be high and workers must do something to protect themselves against the effect of future price changes. Moreover, even though there may be uncertainty about exactly how much prices will rise, everyone can be sure they will rise by a considerable amount. Under these circumstances, wage settlements are likely to contain some provision against future inflation . . . The transition from one kind of behavior may be rather abrupt. At low rates of inflation there may be little or no anticipatory behavior, but then suddenly, when inflation passes a certain critical point, qualitative changes may occur in the whole mechanism of wage bargaining. An example of nonlinearity in expectations concerning the Phillips curve can be found in Sargent (1999, p. 5): “. . . adaptive system can spontaneously generate regime shifts by their learning-inspired nonlinearities”. Nonlinearities, however, do not refer only to the process of expectation formation. They also concern other parameters of the Phillips curve. For instance, an important distinction for the purpose of the present analysis is between the supporters of concavity of the Phillips curve and those in favor of convexity; both issues can be interpreted in terms of asymmetries. According to Eisner (1997), leftward movements of the Phillips curve, when unemployment is falling, are substantially horizontal. However, rightward movements as unemployment rises result in decreased inflation. In this perspective, recessions are indeed disinflationary. But additional high unemployment adds little to disinflation. According to Stiglitz (1997, p. 9): “This concavity is consistent with the literature on asymmetric price adjustment, which shows that in monopolistically competitive markets, producers might adjust prices down to avoid being undercut by a rival but will be more reluctant to raise prices even in the face of generally rising prices”. In contrast, Clark and Laxton (1997, p. 33), stressing the asymmetry in the labor market, come to the conclusion of a convex Phillips curve: “The implication of convexity means that gradual disinflation is less costly in terms of foregone output and employment than a more rapid ‘cold shower’ approach to reducing inflation”.
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In the present paper, we approximate the nonlinearity of the curve with a regime switching device. In doing this, we impose a discontinuity to generate endogenous dynamics and obtain a self-exciting threshold autoregressive (SETAR) model. 3 This model captures the insight of Keynes (1936, p. 301): “In actual experience the wage-unit does not change continuously in terms of money in response to every small change in effective demand, but discontinuously. These points of discontinuity are determined by the psychology of the workers and by the policies of employers and trade unions”. Three channels can be hypothesized through which regime switching can be manifested and which summarize the literature just mentioned: (i) changes in the parameters measuring nominal rigidities; (ii) changes in the parameters measuring the impact of real economic activity; (iii) changes in the exogenous factors captured by the intercepts of the Eqs. (2) and (3). Of course, changes can refer to either the price or the wage equation or both. While Eisner’s main finding is that the accelerationist Phillips curve appears to be a valid econometric relation only for unemployment rates above the neutral rate, Solow’s view “. . . is that the relation just becomes mushy” (p. 15). And this is what happens in a version of our model.
4. The dynamics To interpret Eqs. (5) and (6) as a regime switching model, we have to introduce a two-phase dynamic system of the following type (see Day (1994, chapter 6): ut ≤ us (1 − ω)u∗ + ωut ut+1 = θ (ut ) := (7) (1 − ω )u∗ + ω ut ut > us , where us is a threshold level of unemployment at which switches take place, while u∗ and u∗ are respectively, the NAIRU for Regimes I and II. In other words, system (7) assumes that when the threshold is passed, the reaction of either or both price and wage-setting strategies change, and that this threshold is represented by some value of the unemployment rate, us . This threshold divides Regime I of low unemployment from Regime II of high unemployment. It is not an equilibrium value of unemployment. It is just a level separating an economy with “high” unemployment rates from one with “low” rates. For unemployment rates below us , the parameters shaping ω prevail. For unemployment rates above us , the parameters forming ω become operative. 4 A dual two-phase system holds for the rate of inflation: πt ≤ π s (1 − ω)π ∗ + ωπt πt+1 = Θ(πt ) := (8) (1 − ω )π ∗ + ω πt πt > π s , where π s = m + us /α. There are two regimes described by Eqs. (7) and (8): Regime I is the virtuous one, with low unemployment and low inflation, while Regime II is the vicious one, 3 4
For this definition in a stochastic environment, see Tong (1990). Empirical estimates of these parameters may be found in Ferri and Greenberg (1992).
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characterized by high inflation and unemployment. The threshold is given by a particular rate of unemployment (us ) or of the inflation rate (π s ). The dynamics of a system like Eqs. (7) and (8) depend fundamentally on three factors: (i) the value of the threshold; (ii) the size of the coefficients ω and ω ; and (iii) the relative position of the two equilibria. To simplify the exposition, we concentrate on the unemployment equation; reference to the inflation equation will be made either when its pattern differs from the one established for the rate of unemployment or when it can help to deepen the underlying economic meaning. Two cases characterized by different parameter values are considered: Case I: 0 < ω < ω < 1, u∗ < us < u∗ Case II: ω, ω > 1, u∗ > us > u∗ Case I arises when the slope of the Phillips curve in Regime II is larger than in Regime I, i.e. τ > τ . Since ω = 1/(1 + τ α), we have ω < ω < 1, and so the first condition of Case I is fulfilled. There are several ways to satisfy the second condition of Case I. 5 What is important to stress is that this condition implies that the long-run equilibrium of each regime is not reachable, because it lies in the opposite regime. This is what Day (1994) calls the switching condition. Given these assumptions, we obtain the picture shown in Fig. 1 for the adjustment process in terms of the rate of unemployment. Suppose we take up the process in the high unemployment Regime II. Initially, unemployment declines. As shown in the example in the diagram, these changes would be gradual until the switching threshold us is crossed and Regime I is entered; then unemployment increases. Analogous movements can be shown for the rate of inflation. Observe that there is no stationary state: a point which begins in one phase zone is drawn toward the equilibrium value it would have if it were the only regime, but when the switching threshold is crossed, the other phase equation takes over and variables are drawn in the opposite direction, so that the process moves cyclically from one regime to the other. As long as the conditions of Case I hold, neither equilibrium can be reached — even in the long-run. The qualitative behavior is generic, that is, the cycles alternate, but the “speed” of the changes and the duration within each regime depends on the parameter values. But where do trajectories go? Consider iterates of the map θ(·). This map is piecewise linear and, as ω and ω are positive and less than one, the slope in any piece must involve products of these slopes. This means that any two trajectories that come close enough to each other will converge. Indeed the following result can be established. Proposition 1. Given the conditions of Case I, there exists a unique, asymptotically stable cycle in unemployment of period k ≥ 2. This result, a proof of which is given in the Appendix A, means that any trajectory converges closer and closer to a sequence of k distinct unemployment levels {u1 , . . . , uk }. The same can be said for inflation. To fulfill the second condition, it is sufficient to suppose that β > β; i.e. real rigidity is smaller in Regime II (see Eqs. (2) and (3)). Given the definition of NAIRU, this implies that u∗ < u∗ . Alternatively, one could allow changes in the nominal rigidity parameters, as suggested by Rowthorn (1977), and in the values of the intercepts. In this case if γ < γ , then ω < ω, while at the same time, if δ < δ, then the NAIRU in Regime II is smaller than in Regime I. 5
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Fig. 1. Case I, stable cycles: all trajectories enter the trapping set or box.
If random shocks were added to the model, irregular fluctuations would follow, exacerbated by the underlying regime switching mechanism. Undoubtedly, one would want to add such a term in empirical work. But irregular or chaotic fluctuations can also be brought about purely by the intrinsic forces represented in the model as can be seen by considering Case II.
5. A different environment When one turns to Case II, the most important possibility to be considered is that ωω > 1. For simplicity we assume that they are both greater than one. We have to show that even if each regime is unstable if considered in isolation, it may still produce some sort of stability within a regime switching model. 6 To obtain this kind of stability, we change the sign of 6 As stressed by Tong (1990), the instability of a regime does not imply the instability of a system in the case of a regime switching model.
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Fig. 2. Case II, strongly ergodic, chatoic fluctuations: all trajectories enter the trapping set or box and are nonperiodic.
α. If we suppose that α < 0, then ω = 1/(1 + ατ ) > 1; the same holds true for ω . If this hypothesis is accepted, then the phase has the appearance shown in Fig. 2. Can this hypothesis be justified from an economic point of view? Fazzari et al. (1998) show how the relation between prices and aggregate demand is more problematic than is usually recognized. One line of argument that implies α < 0 is the Minsky (1982) debt–deflation hypothesis, in which deflation results in an increased real burden of debt that inhibits investment, offsetting the Pigou and Keynes effects reflected in α > 0 (Case I). 7 In this case, the inequality u∗ > us > u∗ can be justified by making a piece-wise approximation of a convex Phillips curve. 8 All trajectories become trapped in the interval [(1 − ω )u∗ + ω us , (1 − ω)u∗ + ωus ]. 7 8
For definitions of these effects, see Flashel et al. (1991). If one refers to Eqs. (2) and (3), it is enough to suppose that δ < δ.
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Proposition 2. Given the hypotheses of Case II, then for almost all positive initial unemployment levels u0 that belong to (u∗ , u∗ ), unstable nonperiodic (chaotic) fluctuations occur. The proportion of time the process is in Regime I converges to a constant fraction, say p, and that in Regime II converges to a constant fraction 1 − p. The proof is in Appendix A. This proposition is also true when π is substituted for u. Regime switching of the SETAR variety implies a discontinuity at the threshold level. Although at the micro level there is evidence in support of discontinuous decision processes, 9 at the macro level these discontinuities are rarely observed. An alternative is to specify regime switching of the smooth threshold autoregressive (STAR) variety (see Granger and Terasvirta, 1993). 10
6. Discussion We first discuss the economics of changing parameters. According to Sargent (1999, p. 16): “Lucas left the drift in coefficients unexplained. Neither the macroeconomic theory nor the rational expectations econometrics constructed after Lucas’s critique explain such drift”. In the same place, Sargent (p. 105) argues that a change in the coefficients of the estimated Phillips curve “. . . is the key to the behavior of inflation”. Sargent’s analysis differs from ours in two respects. First, in Sargent’s analysis, changes in beliefs drive the dynamics of the model. These changes in beliefs are mainly those of the Federal Reserve System that tries to learn the exact shape of the Phillips curve. In turn, these beliefs are driven by the so called “induction hypothesis” that refers to a particular kind of adaptive expectation formulation. Second, it is not only monetary policy that is important in determining the dynamics of the inflation–unemployment tradeoff; phenomena that affect the product and the labor market also play key roles. Our analysis is based upon the interplay of real and nominal rigidities; both real and nominal parameters change. Some of these changes are simply cyclical and reflect the different pressures that a regime closer to full employment exert on the behavior of both entrepreneurs and workers. Others are more structural and reflect such phenomena 9 10
On this point see Cooper and Haltiwanger (1993). The SETAR system can be written as
u(t + 1) = (1 − ω)u∗ + ωu(t) + [(1 − ω )u∗ + ω ut ]I (u), where I (u) =
0
if
ut ≤ us
1
if
ut > u s .
(9)
If I (u) is replaced by a smooth function such as (see Tong, 1990, p. 107) F (u) =
1 , 1 + exp[(ut − us )/η]
(10)
the transition around the threshold becomes smooth. This is called the STAR model. As η → ∞, the SETAR model is obtained.
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as the globalization of markets, new information technology, and the increased flexibility of the labor market that can modify the dynamics by changing both the threshold and the shape of the equations. Next, we compare our results with those obtained from a NAIRU-based model. As we have seen, the dynamics of a NAIRU model imply, for ω < 1, convergence to constant inflation and unemployment rates. For ω ≥ 1, however, the system is unstable. Propositions 1 and 2 have shown that our regime switching model implies bounded dynamics whether ωω is greater or less than one. 7. Concluding remarks The policy implications of a NAIRU model on the one hand and a regime switching model on the other are somewhat different. Policy makers who think in terms of a NAIRU model believe that intervention is necessary to prevent unacceptably high rates of inflation when unemployment falls below the NAIRU. But, since a NAIRU model with ω < 1 is stable, why is intervention necessary? Both inflation and unemployment will converge to their steady-state values monotonically without intervention. One possibility is that policymakers believe that convergence of the system will be so slow that the economy will suffer high rates of inflation for an excessive length of time. Another possibility is that policymakers believe ω > 1, in which case continuous intervention is necessary because the system is unstable. In the latter case, even a small departure from NAIRU must be offset as soon as possible because of the instability. In contrast, the regime switching model we have discussed is less sensitive to parameter values. We have seen that the system may exhibit limit cycles or more complex patterns, but unemployment and inflation rates remain bounded. Although policymakers may dislike the behavior of the variables, intervention becomes problematic because there is no NAIRU to indicate whether and what kind of intervention is necessary. Moreover, a feature of our model is an interaction between real and monetary phenomena that weakens the theoretical foundations of the NAIRU concept. Some generalizations of the model should be considered. For instance, to understand the implications of different values of α, it is necessary to study the role of a changing supply of money and credit conditions. Other aspects — for example, capital accumulation, the role of real interest on investment, the effect of government and private sector debts — should be considered. Indeed, nonlinear effects of the interest rate on the demand for liquidity and the demand for goods can generate complex dynamics when the product market adjusts with a lag to disequilibrium, as has been shown by Day and Lin (1991). A logical next step would be to put such quantity adjustments into the model together with price–wage adjustments. Finally, the role of expectations should be explicitly examined. Acknowledgements A financial contribution from the Italian Ministry of University and Research is acknowleged. We thank Ara Mentcherian for checking our calculations and for a number of useful comments.
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Appendix A Proof of Proposition 1. Let U1 = [um , us ],
U2 = (us , uM ],
where
um = (1 − ω )u∗ + ω us ,
uM = (1 − ω)u∗ + ωus .
The interval U = [um , uM ] is a trapping set: all trajectories must enter this set. Of course, U = U1 ∪ U2 . To see how cycles arise, consider the simplest possibility first. Assume θ (um ) > us and θ (uM ) < us ; then it is easy to see that for all u ∈ U1 , θ1 (u) ∈ U2 and for all u ∈ U2 , θ2 (u) ∈ U1 . Therefore, θ 2 (u) = θ2 · θ1 (u) ∈ U1 all u ∈ U1 , and θ 2 (u) = θ1 · θ2 (u) ∈ U2 all u ∈ U2 . That is, θ 2 (U1 ) ⊂ U1 and θ 2 (U2 ) ⊂ U2 . By standard fix point arguments this implies fix points, say x1 , x2 of the map θ 2 (·) which must be two-cyclic, i.e. θ 2 (x2 ) = θ2 · θ1 (x1 ) = x1 . This is shown in Fig. 3. In general we proceed as follows. First, we observe Lemma 1. Without loss of generality, assume us − um < uM − us , then θ (U1 ) ⊂ U2 . Proof. By assumption ω < 1, so ω=
uM − θ (um ) < 1. us − u m
Fig. 3. Sufficent conditions for two-period cycles.
(11)
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Fig. 4. Existence of convergent (m + 1) period cycles (m = 2).
Combining (11) with this expression gives θ(um ) > us , but θ (u) = (1 − ω)u∗ + ωu > θ (um ) > us all u ∈ U1 . This gives the desired result. The lemma is illustrated in Fig. 3. It implies that if ut ∈ U1 , then ut+1 ∈ U2 and ut ∈ U2 ⇒ ut+1 ∈ U1 . Now we consider two cases. Assume (11) Case I. Assume there exists a smallest integer m such that θ2−m (U1 ) ⊃ θ1 (U1 ).
(12)
Then U1 ⊃ θ2m · θ1 (U1 ) = θ m+1 (U1 ), so U1 is nonexpanding under θ m+1 (·) and by a standard theorem a fix point exists. 11 This result is illustrated in Fig. 4. Although a cycle of period 3 exists in this example, this does not imply chaos, because the map θ(·) is discontinuous at us . Indeed, the cycle must be asymptotically stable. To see this, we note that a p = m + 1 cyclic point satisfies (u∗ )p = θ p ((u∗ )p ) and that the slope of θ p (·), which is defined everywhere except at us , is the product ω1 ω2m < 1, which implies asymptotic stability. Case II. If (12) does not hold, then there is a smallest integer m such that U2 := θ (U1 ) ∩ θ −m (U1 ) ∩ U ⊂ U2 = ∅, U2 = θ (U1 ) ∩ θ −(m+1) (U1 ) ⊂ U2 = ∅, i.e. θ (U1 ) ⊂ θ −m (U1 ) ∪ θ −(m+1) (U1 ). Now choose U1 , U1 so that θ (U1 ) = U2 , 11
θ (U1 ) = U2 .
See Day (1994, theorem 5.2).
(13)
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35
Fig. 5. Existence of 2m + 3 period cycles: (a) m = 1, five-period cycle exits (θ 5 (u) = ω12 · ω23 ); (b) m = 2, seven-period cycle exits (θ 7 (u) = ω12 · ω25 ).
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Applying θ m to both sides of the first of these, we get θ m+1 (U1 ) = θ m (U2 ) = U1 . Applying θ again gives θ m+2 (U1 ) = θ (U1 ) = U2 . Applying θ m+1 to both sides of this expression gives θ 2m+3 (U1 ) = θ m+1 (U2 ) = U1 , which gives a cycle of 2m + 3 that must be asymptotically stable for reasons analogous to that given for Case I. This completes the proof of Proposition 1. Fig. 5a gives an example for m = 1 and Fig. 5b for m = 2. Proof of Proposition 2. By assumption ω1 , ω2 > 1. Consequently, θ(·) is expansive and, by the Lasota–Yorke theorem, there exists a unique absolutely continuous invariant probability measure µ(·) such that µ(S) =
S
f (u) du = lim
T →∞
T χs (θ i (u)) i=t
T
,
where f is an integrable density function and supp f > 0 on a set of intervals such that U1 ∩ suppf = ∅, U2 ∩ suppf = ∅. Let p = U1 f (u) du. Then 1 − p = U2 f (u) du. Absolute continuity means that trajectories are dense in supp f . It is easy to see in this case that there exist Li–Yorke points so that chaotic trajectories exist. Since supp f is a basin of attraction for all trajectories with θ(suppf ) = suppf , the scrambled set S ⊂ suppf , so that µ− almost all trajectories will be nonperiodic and chaotic, passing as close as you like to every chaotic point in S (for further discussion, see Day, 1994, chapters 7 and 8). References Clark, P., Laxton, D., 1997. Phillips curves, Phillips lines and the unemployment costs of overheating. Centre for Economic Performance Discussion Paper 344. Coen, R.M., Eisner, R., Marlin, J.T., Shah, S.N., 1999. The NAIRU and wages in local markets. American Economic Review Papers and Proceedings 89, 52–57. Cooper, R., Haltiwanger, J., 1993. The aggregate implications of machine replacement: theory and evidence. American Economic Review 83, 360–382. Day, R.H., Lin, T.-Y., 1991. A Keynesian business cycle. In: Nell, E.J., Semmler, W. (Eds.), Nicholas Kaldor and Mainstream Economics, Confrontation and Convergence. St. Martin’s Press, New York, pp. 282–305. Day, R.H., 1994. Complex economic dynamics. In: An Introduction to Dynamical Systems and Market Mechanisms, Vol. I. MIT Press, Cambridge. Eisner, R., 1997. A new view of the NAIRU. In: Davidson, P., Kragel, J.A. (Eds.), Improving the Global Economy: Keynesianism and the Growth in Output and Employment. Elgar, Cheltenham, pp. 196–230. Fair, R.C., 1999. Does the NAIRU have the right dynamics? American Economic Review Papers and Proceedings 89, 58–62. Fazzari, S.M., Ferri, P., Greenberg, E., 1998. Aggregate demand and firm behavior. Journal of Post Keynesian Economics 20, 527–558.
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