An Analytical Critique of ‘New Keynesian’ Dynamic Model

Post Keynesian Review Vol. 2, No. 1, Pages 1–28 (the 31st August, 2013) ISSN 2187-4875 An Analytical Critique of ‘New Keynesian’ Dynamic Model ASADA, Toichiro A BSTRACT. In this paper, we present an analytical critique of New Keynesian dynamic model. It is shown analytically that the prototype New Keynesian dynamic model produces the paradoxical behaviors that are inconsistent with the empirical facts. We also present more traditional alternative approach that is consistent with the empirical facts, which is called the Old Keynesian dynamic model. Keywords: New Keynesian dynamic model, (in)determinacy, exogenous disturbance, Old Keynesian dynamic model, (in)stability, Hopf bifurcation, endogenous fluctuation. JEL Classification: E12, E31, E32, E52 1. Introduction Recently, the so-called New Keynesian(NK) dynamic model, which is represented by Woodford(2003), Bénassy(2007), Galí(2008) and others, is becoming more and more influential in the mainstream macroeconomic analysis. The method of modeling that is adopted in this approach is based on the dynamic optimization of the representative agents with perfect foresight or rational expectations, but it contradicts Keynes’(1936) own vision on the working of the market economy that is based on the bounded rationality due to the inherent fundamental uncertainty. In this paper, we present an internal and analytical critique of the NK dynamic model. The adjectives internal and analytical mean that our critique is based on the detailed mathematical study of the prototype NK dynamic model. It is shown that this model produces the paradoxical behaviors of the main variables which contradict the empirical facts. We also propose an alternative approach that is called the Old Keynesian dynamic model, which is immune from such kind of anomaly. In Section 2, we formulate the simplest version of the prototype NK dynamic model in terms of a linear system of difference equations. In Section 3, we present a detailed mathematical analysis of such a model without exogenous disturbances. Section 4 is devoted to critical considerations of the prototype NK dynamic model, Professor Dr. Asada Toichiro, Faculty of Economics, Chuo University, Tokyo, Japan. ©2013 The author and The Japanese Society for Post Keynesian Economics 1 2 ASADA, Toichiro which is largely based on the analysis in Section 3. Section 5 provides a detailed analysis of the effects of the exogenous but non-stochastic disturbances in the NK setting. The considerations up to Section 5 have rather critical and negative implications, although they are largely based on the detailed mathematical analyses. In Section 6, however, we propose an alternative positive and constructive approach, which is called the Old Keynesian dynamic model in contrast to the NK dynamic model. Section 7 is devoted to concluding remarks. Mathematical appendices supplement the descriptions in the text. 2. Formulation of the Prototype NK Dynamic Model The simplest version of the prototype NK dynamic model may be formulated as follows.1 (2.1)  t D  teC1 C ˛.y t (2.2) y t D y teC1 (2.3) r t D o C N C (2.4) ˇ.r t  teC1 D E t  t C1 ; y/ N C "t ;  teC1 1 . t o / C  t ; / N C 2 .y t y teC1 D E t y tC1 ; y/; N where  t the rate of price inflation at period t ,  teC1 expected rate of price inflation at period t C 1, i.e. rate of price inflation at period t C 1 that is expected at period t , Y t real national income at period t , y t D log Y t , Y teC1 expected real national income at period t C 1 (real national income at period t C 1 that is expected at period t ), y teC1 D log Y teC1 , r t nominal rate of interest at period t , o equilibrium value of the real rate of interest (fixed), YN equilibrium value of the real national income (fixed), yN D log YN , and N target rate of inflation that is set by the central bank (fixed). All of the parameter values ˛, ˇ, 1 and 2 are positive. The terms " t and  t represent the (stochastic or non-stochastic) exogenous disturbances. Note that r t  teC1 expresses the expected real rate of interest at period t . Equations (2.1) and (2.2) represent NK Phillips curve and NK IS curve respectively. According to the NK convention that sticks to the so-called microeconomic foundation, (2.1) is derived from the optimizing behavior of the imperfectly competitive firms with costly price changes, and (2.2) is derived from the first order 1 As for the textbooks of the New Keynesian dynamic model by the original authors, see Woodford(2003), Bénassy(2007) and Galí(2008). Romer(2006) contains some useful expositions. The formulation in this section is the simplified version of the formulation by Asada, Chiarella, Flaschel and Proaño(2007). See also Flaschel, Franke and Proaño(2008) and Asada, Chiarella, Flaschel and Franke(2010) as for similar formulations. The Japanese Society for Post Keynesian Economics A NALYTICAL C RITIQUE OF NK DYNAMICS 3 condition of the inter-temporal optimization for the representative consumers, that is called the Euler equation of consumption.2 This means that the real aggregate demand becomes a decreasing function of the real rate of interest even if the firms’ investment expenditure is neglected in this model. (2.3) represents the interest rate monetary policy rule of the central bank by means of the Taylor rule, which is also called the flexible inflation targeting that considers both of the rate of price inflation and the employment(real output).3 (2.4) is the formalization of the rational expectation hypothesis concerning the rate of price inflation and the real output.4 We can rewrite equations (2.1) and (2.2) as follows. (2.5)  teC1 (2.6) y teC1  t D ˛.yN yt / " t ; .˛ > 0/  teC1 y t D ˇ.r t o /  t ; .ˇ > 0/: Substituting (2.3) into these two equations, we have the following system of equations. (2.7a)  teC1 (2.7b) y teC1 (2.7c)  t D ˛.yN y t D ˇf.  teC1 yt / 1 " t D F1 .y t I " t /; 1/. t / N C .˛ C 2 /.y t y/ N C "t g t D F2 . t ; y t I " t ;  t /; D E t  t C1 ; y teC1 D E t y t C1 : 3. Mathematical Analysis of the System without Exogenous Disturbance The system (2.7) is called the DSGE (Dynamic Stochastic General Equilibrium) model if at least one of the exogenous disturbances " t and  t is a stochastic variable. On the other hand, this system is called DGE (Dynamic General Equilibrium) model if there is no stochastic disturbance.5 In this section, we shall consider the case without exogenous disturbance, that is to say, " t D  t D 0: (3.1) In this case, the rational expectation hypothesis, that is expressed by (2.7c) , is reduced to the following perfect foresight hypothesis. (3.2)  teC1 D  t C1 ; y teC1 D y t C1 : 2 In fact, these equations are log-linear approximations of the optimal conditions of the economic agents. 3 See Taylor(1993) as for the original formulation of the Taylor rule. 4 E t stands for the operator of the mathematical expectation. 5 See Woodford(2003), Bénassy(2007), Galí(2008), and Chiarella, Flaschel and Semmler(2013). Post Keynesian Review Vol. 2 4 ASADA, Toichiro In such a case, equations (2.5) and (2.6) become (3.3)
 t D  t C1  t D ˛.yN y t /; (3.4)
y t D y t C1 y t D ˇ.r t  t C1 o /; and the system (2.7) becomes as follows. (3.5a)  t C1 D  t C ˛.yN (3.5b) y t C1 D y t C ˇf. y t / D G1 . t ; y t /; 1/. t 1 / N C .˛ C 2 /.y t y/g N D G2 . t ; y t /: The equilibrium solution .  ; y  / of this system becomes simply   D ; N y  D y; N (3.6) and the Jacobian matrix J1 of this system becomes as follows. ! 1 ˛ : (3.7) J1 D ˇ. 1 1/ 1 C ˇ.˛ C 2 / Its characteristic equation is given by J1 j D 2 C a1  C a2 D 0; '1 ./  jI (3.8) where (3.9) a1 D tr J1 D 2 ˇ.˛ C (3.10) a2 D jJ1 j D 1 C ˇ.˛ C 2/ 2/ < 0; C ˛ˇ. 1/ D 1 C ˇ.˛ 1 1 C 2/ > 1: Then, we have the following set of relationships. (3.11a) A1  1 C a1 C a2 D ˛ˇ. (3.11b) A2  1 a2 D (3.11c) A3  1 a1 C a2 D 4 C ˛ˇ C 2ˇ ˇ.˛ 1 1/; 1 C 2/ < 0; 2 C ˛ˇ 1 > 0: The equilibrium point of the system of difference equations (3.5) is dynamically stable if and only if jj j < 1, where j , j D 1; 2, are two characteristic roots of (3.8). It is well known that such a stability condition is in fact satisfied if and only if the following set of inequalities are satisfied.6 Aj > 0; j D 1; 2; 3: (3.12) In addition, the discriminant of the characteristic roots is given by (3.13) D  a12 4a2 D ˇfˇ.˛ C 2/ 2 C 4˛.1 1 /g: 6 See Gandolfo(2009, p.60). The set of inequalities (3.12) is the two-dimensional version of the so-called Schur-Cohn stability conditions of difference equations. The Japanese Society for Post Keynesian Economics A NALYTICAL C RITIQUE OF NK DYNAMICS 5 That is to say, the characteristic equation (3.8) has a pair of conjugate complex roots if and only if D < 0. Now, we can obtain the following result. L EMMA 3.1. (1) At least one root of (3.8) has the absolute value that is greater than 1 for all 1 > 0. In other words, equilibrium of (3.5) is unstable. (2) The characteristic equation (3.8) has a pair of conjugate complex roots, if and only if the inequality 1 >1C ˇ.˛ C 2 /2 is satisfied. 4˛ Proof. (1) Inequality (3.11b) means that at least one of the Schur-Cohn stability conditions (3.12) is violated. (2) It follows from (3.13) that we have D < 0 if and only if the inequality 1 >1C ˇ.˛ C 2 /2 is satisfied. 4˛ □ The equilibrium point of the system (3.5) is called totally unstable if and only if both of two roots of (3.8) have the absolute values that are greater than 1. In what follows, we write D ˇ.˛ C 2 /2 : 4˛ To study whether the system is totally unstable or not, it is convenient to consider the inverse matrix of J1 , that is, ! 1 1 C ˇ.˛ C 2 / ˛ : (3.14) K  J1 1 D ˇ.1 1 1 C ˇ.˛ 1 C 2 / 1/ Then, let us consider the following characteristic equation. (3.15) ./  jI Kj D 2 C b1  C b2 D 0; where (3.16) (3.17) 2 ˇ.˛ C 2 / < 0: 1 C ˇ.˛ 1 C 2 / 1 b2 D jKj D < 1: 1 C ˇ.˛ 1 C 2 / b1 D tr K D Therefore, we obtain the following relationships. ˇ˛. 1 1/ ; 1 C ˇ.˛ 1 C 2 / ˇ.˛ 1 C 2 / > 0; b2 D 1 C ˇ.˛ 1 C 2 / 4 C ˇf˛.1 C 1 / C 2 2 g > 0: b1 C b2 D 1 C ˇ.˛ 1 C 2 / (3.18a) B1  1 C b1 C b2 D (3.18b) B2  1 (3.18c) B3  1 Post Keynesian Review Vol. 2 6 ASADA, Toichiro The following lemma is a direct consequence of the relationships (3.18). L EMMA 3.2. Absolute values of all of two roots of (3.8) are greater than 1, if and only if 1 > 1. In other words, the equilibrium point of (3.5) is totally unstable if and only if 1 > 1. Proof. It follows from the Schur-Cohn stability condition with respect to the system of difference equations that we have a set of inequalities with respect to (3.18) (3.19) Bj > 0; .j D 1; 2; 3/ if and only if 1 > 1. (Cf. Footnote 6). This means that all of the absolute values of the roots of the characteristic equation (3.15) are less than 1 if and only if 1 > 1. This result together with Theorem A.1 in Appendix A implies Lemma 3.2. □ Lemmas 3.1 and 3.2 entail the following proposition. P ROPOSITION 3.1. Let 1 and 2 stand for two roots of (3.8). (1) If 0 < 1 < 1, then 1 and 2 are real with 0 < 1 < 1 < 2 . In this case, the equilibrium of (3.5) becomes a saddle point. (2) If 1 D 1, then 1 and 2 are real, with 1 D 1 < 2 . (3) If 1 < 1 < 1 C , then 1 and 2 are real, with 1 < 1 < 2 . In this case, the equilibrium of (3.5) becomes totally unstable, and any solution path that starts from points off equilibrium diverges monotonically. (4) If 1 D 1 C , then two roots are duplicated such that 1 < 1 D 2 . (5) If 1 C  < 1 , then the two roots are a pair of conjugate complex roots such p that j1 j D j2 j D  2 C ! 2 > 1, where 1 D  C i!; 2 D  i !, ! 6D 0 and p 1. In this case, the equilibrium of (3.5) becomes totally unstable, and any i D solution path that starts from points off equilibrium diverges cyclically. As for proof, see Appendix B. 4. Some Critical Assessments The prototype NK dynamic model that was taken up in the previous sections has several problematical features. In this section, we shall try to provide some critical assessments of such a model. 4.1. Anomalous dynamics with sign reversals. The most notorious problem of the prototype NK dynamic model is the phenomenon that is called the sign reversal in New Keynesian Phillips curve, pointed out by Mankiw (2001) clearly.7 7 As for the related topics, see also Asada, Chiarella, Flaschel and Franke(2010), Flaschel and Schlicit(2006), Franke(2007) and Romer(2006). The Japanese Society for Post Keynesian Economics A NALYTICAL C RITIQUE OF NK DYNAMICS 7 It follows from (3.3) that
 t becomes a decreasing function of y t . That is, the rate of price inflation continues to decrease whenever the actual real output level is greater than the natural real output level. This contradicts the empirical fact of the most of countries. Mankiw(2001) writes as follows. “Although the New Keynesian Phillips curve has many virtues, it also has one striking vice: It is completely odds with the facts. In particular, it cannot come even close to explaining the dynamic effects of monetary policy on inflation and unemployment. This harsh conclusion shows up several places in the recent literature, but judging from the continued popularity of this model, I think it is fair to say that its fundamental inconsistency with the facts is not very appreciated.” (Mankiw, 2001, p. C52) It is worth noting that there is another difficulty of sign reversal in this model, which Mankiw(2001) does not refer to. That is, it follows from (3.4) that
y t becomes an increasing function of r t  t C1 . This means that the real output level continues to increase whenever the actual real interest rate is greater than its equilibrium level. It is evident that this fact also contradicts the empirical fact of the most of countries. 4.2. Problematical jump variable technique. In the traditional economic dynamic models, that conforms with the orthodox mathematical concept of stability, the initial values of the endogenous variables are historically given. In this setting, the equilibrium point is considered to be stable if and only if all of the characteristic roots are stable roots, and it is considered to be unstable if at least one characteristic root is an unstable root.8 Lemma 3.1(1) means that the prototype NK model is accompanied by equilibrium that is unstable in the mathematically orthodox sense. The equilibrium of NK dynamic model will never be reached. This undermines the basis of the NK theory, however. Hence, the NK literature adopts the trick that makes the unstable system try to mimic a ‘stable’ system by using the so-called jump variable technique. In the NK dynamic model, both of two endogenous variables  t and y t are considered to be jump variables or not-predetermined variables, the initial values of which are freely chosen by the economic agents. Furthermore, it is assumed that 8 A stable (resp. unstable) root is the root which has the absolute value that is less (resp. greater) than 1 in case of the system of difference equations, and it is the root which has the negative (resp. positive) real part in case of the system of differential equations. Post Keynesian Review Vol. 2 8 ASADA, Toichiro by the economic agents only the initial conditions of the endogenous variables are chosen so as to ensure the convergence to the equilibrium point.9 Hence, in place of the concept of stability and instability, NK literature introduces the following concept of the determinacy and indeterminacy.10 D EFINITION 4.1. (1) Suppose that a set of the initial values of the jump variables, which ensures the convergence to the equilibrium point, is uniquely determined. Then, the system is called determinate. (2) Suppose that there are multiple sets of the initial values of the jump variables which ensure the convergence to the equilibrium point. Then, the system is called indeterminate. P ROPOSITION 4.1. (1) If 0 < 1 < 1, then the system (3.5) is indeterminate. (2) If 1 < 1 , the system (3.5) is determinate. The meanings of Proposition 4.1 are as follows. The general solution of the system of difference equations (3.5) becomes ! ! ! 2 X N Cj t t ; j C D (4.1) Dj yN yt j D1 where the constants Cj and Dj must satisfy the following relationships ! ! ! 0 Cj j 1 ˛ ; .j D 1; 2/: D (4.2) 0 Dj ˇ. 1 1/ j f1 C ˇ.˛ C 2 /g (See Gandolfo, 2009, Ch.5). Substituting (4.2) into (4.1) we obtain ! ! ! ! 1 t 1  N D 1 1 C1 t1 C 1 2 C2 t2 C (4.3) ; yt yN ˛ ˛ where constants C1 and C2 are determined by a set of initial values in the following way. (4.4a) o D C1 C C2 C ; N yo D 1 1 C1 C 1 2 C2 C y: N ˛ ˛ First, suppose that 0 < 1 < 1. In this case, we can consider that 1 and 2 are real values such that 0 < 1 < 1 and 1 < 2 without loss of generality in view of (4.4b) 9 Usually, this assumption is rationalized on the ground that the transversality condition, that requires the convergence to the equilibrium point, must be satisfied in order to ensure the behavior of the representative agents to be truly optimal. 10 See Woodford(2003), Bénassy(2007), and Galí(2008). See Blanchard and Kahn(1980) as for the sophisticated mathematical treatment that is related to this concept. The Japanese Society for Post Keynesian Economics A NALYTICAL C RITIQUE OF NK DYNAMICS t ✻ 9 B ✠ Er✠ N r ✒ ✒ A ✲ r yN 0 yt F IGURE 1. Case of 0 < 1 < 1 (Indeterminate). Proposition 3.1(1). In this situation, only the pair of the initial values .o ; yo / that satisfies the condition C2 D 0 can ensure the convergence to the equilibrium point. This condition is reduced to the following set of equations in view of (4.4). (4.5a) o D C1 C ; N (4.5b) yo D 1 1 ˛ C1 C y: N Substituting (4.5b) into (4.5a), we obtain the following relationship between o and yo that ensures the convergence to the equilibrium point. (4.6) o N D ˛ 1 1 .yo y/; N 0 < 1 < 1: Any pair .o ; yo / that is on the line in Figure 1 satisfies the relationship (4.6). Therefore, there are infinite numbers of the pairs of the initial values .o ; yo / which ensure the convergence to the equilibrium point.11 In other words, the system becomes indeterminate in case of 0 < 1 < 1. Next, suppose that 1 > 1. In this case, both of the absolute values of 1 and 2 become greater than 1 (see Lemma 3.2). In this situation, only the pair of the initial values .o ; yo / that satisfies the condition C1 D C2 D 0 can ensure the convergence 11 In fact, the line in Figure 1 is the saddle path. Post Keynesian Review Vol. 2 10 ASADA, Toichiro t ✻ N r Er ✲ r 0 yN F IGURE 2. Case of yt 1 > 1 (Determinate). to the equilibrium point. Substituting this condition into (4.4), we obtain (4.7) o D ; N yo D y: N This means that the system is determinate in case of 1 > 1, but in this case, the economic agents in this model select only the solution that is stuck to the equilibrium point E in Figure 2 for all times. In this case, the economic fluctuation does not occur unless there are exogenous disturbances, and the dynamic model is reduced to the static model virtually. We think that the adoption of such a jump variable technique in the NK dynamic model is unconvincing because of the following three reasons. Firstly, as Mankiw(2001) pointed out, actual economic data suggest that the rate of price inflation is not the jump variable that conveniently jumps discontinuously, but it is a state variable that moves sluggishly. Mankiw(2001) writes as follows. “In these models of staggered price adjustment, the price level adjusts slowly, but the inflation rate jumps quickly. Unfortunately for the model, that is not what we see in the data.” (Mankiw, 2001, p. C54) A proposal to resolve such a problem is to introduce the inertia into the model by considering the backward-looking factors. Typical method of reformulation is to replace (2.1) by the following new equation that includes the backward-looking The Japanese Society for Post Keynesian Economics A NALYTICAL C RITIQUE OF NK DYNAMICS factor  t (4.8) 1 11 as well as the forward-looking factor  teC1 .12  t D  teC1 C .1  / t 1 C ˛.y t y/ N C " t ; 0 <  < 1: However, such a reformulation is rather ad hoc and it is not clear how it can be derived from the analytical framework of the NK dynamic model consistently. Secondly, as we already pointed out, the prototype NK dynamic model cannot produce the economic fluctuations unless we introduce the exogenous disturbances into the model. This conclusion is not confined to the case of determinacy (the case of 1 > 1). It also applies to the case of indeterminacy (the case of 0 < 1 < 1), because even in case of indeterminacy, the solution converges to the equilibrium point monotonically (see Figures 1 and 2). In other words, the endogenous economic fluctuation is impossible in this model. Thirdly, some economists criticize the methodology of the representative agent approach of the NK dynamic model that is based on the analysis of the behavior of the omniscient and homogeneous representative agent. For example, Kirman(1992), Asada, Chiarella, Flaschel and Franke(2010) and Chiarella, Flaschel and Semmler(2013) proposed the approach that considers the interactions of the heterogeneous agents who behave bounded-rationally.13 Let us quote from their papers. “A tentative conclusion, at this point, would be that the representative agent approach is fatally flawed because it attempts to impose order on the economy through the concept of an omniscient individual. In reality, individuals operate in very small subsets of the economy and interact with those with whom they have dealings. It may well be that out of this local but interacting activity emerges some sort of self organization which provides regularity at the macroeconomic level.





The equilibria of the worlds described by any of these approaches may be conceptually very different from those implied by the artifact of the representative individual. Cycles and fluctuations emerge not as the result of some substantial exogenous shock and the reaction to it of one individual, but as a natural result of interaction, together with occasional small changes or ‘mutations’ in the behavior of some individuals.” (Kirman, 1992, pp.132-3) 12 See, for example, Furler(1997). Romer(2006, Ch.6) and Franke(2007) contain the useful interpretations of this topic. 13 Such an approach is somewhat similar to the approach of Akerlof and Shillar(2009) that is influenced by the discourse of Keynes(1936). Post Keynesian Review Vol. 2 12 ASADA, Toichiro “While the microfoundation of economic behavior is per se an important desideratum to be reflected also by behaviorally oriented macrodynamics, the use of ‘representative’ consumers and firms for the explanation of macroeconomic phenomena is too simplistic and also too narrow for a proper treatment of what is really interesting on the economic behavior of economic agents – the interaction of heterogeneous agents –, and it is also not detailed enough to discuss the various feedback channels present in the real world.





Indeed, agents are heterogeneous, form heterogeneous expectations along other lines than suggested by the rational expectations theory, and have different short and long term views about the economy.” (Asada, Chiarella, Flaschel and Franke, 2010, pp.205-6) “Central to our approach is the use of non-market clearing adjustment process that are indeed the basis for the consideration of the various Keynesian macroeconomic feedback effect, the Pigou real balance effect, the Tobin price expectation effect (where an expected price fall will trigger instabilities), and the Keynes effect in the context of a dynamic multiplier model. These have all been characteristic in the traditional Keynesian approaches to macrodynamics but have mostly been forgotten in modern macroeconomics.” (Chiarella, Flaschel and Semmler, 2013, p.110) 5. Mathematical Analysis of the Effects of the Exogenous Disturbance in a Prototype New Keynesian Dynamic Model In this section, we introduce the exogenous disturbance into the model and study its effects mathematically.14 In (2.7), we assume that " t D 0 and  t are exogenous but non-stochastic disturbances, that is, (5.1a)  t C1 D  t C ˛.yN (5.1b) y t C1 D y t C ˇf. y t /; 1 1/. t / N C .˛ C 2 /.y t y/g N t : Furthermore, let us assume as follows. (5.2) For all t 2 f0; 1; 2; : : : ; to 1g;  t D ; N y t D y: N 14 As we see in this section, the existence of the exogenous disturbance is essential for the occurrence of the economic fluctuations in the prototype NK dynamic model, but the stochastic element does not play the essential role. The Japanese Society for Post Keynesian Economics A NALYTICAL C RITIQUE OF NK DYNAMICS (5.3) 8̂ ˆ <0  t D  to 6D 0 ˆ :̂  .t to / 13 for t 2 f0; 1; 2; : : : ; to 1g, for t D to , for t 2 fto C 1; to C 2; : : :g, to where  to is given, and 0 <  < 1. Then, we have the following three-dimensional system of difference equations for all time periods t 2 fto ; to C 1; : : :g: (5.4a)  t C1 D  t C ˛.yN (5.4b) y t C1 D y t C ˇf. y t / D G1 . t ; y t /; 1 1/. t / N C .˛ C 2 /.y t y/g N t D G2 . t ; y t ;  t /; (5.4c)  t C1 D  t : The Jacobian matrix of this system becomes 0 1 ˛ B (5.5) J2 D @ˇ. 1 1/ 1 C ˇ.˛ C 0 0 2/ and the characteristic equation of this system is given by (5.6) '2 ./  jI J2 j D jI J1 j. 1 0 C 1A ;  / D 0; where J1 is defined by (3.7). This characteristic equation has a real root 3 D  and other two roots are determined by (3.8). To simplify the analysis, we consider only the case of 1< (5.7) 1 < 1 C : It follows from Proposition 3.1(3) that (3.8) has two real roots such that 1 < 1 < 2 . In this case, the general solution of (5.4) for t  to becomes 0 1 0 1 0 1 C N t 3 j B C XB C t B C (5.8) @Dj A j C @ yN A ; @yt A D j D1 0 Ej t where 1 < 1 < 2 , 0 <  < 1, and the relationships that, for j D 1; 2; 3, 10 1 0 1 0 j 1 ˛ 0 0 Cj CB C B C B (5.9) 1 A @Dj A D @0A ; @ ˇ. 1 1/ j f1 C ˇ.˛ C 2 /g 0 Dj 0 0 j  must be satisfied (see Gandolfo, 2009, Ch.9). Post Keynesian Review Vol. 2 14 ASADA, Toichiro Rewriting (5.8) by using (5.9), we obtain the following relationship. 0 1 0 1 0 1 0 1 0 1 ˛ 1 t 1 N B C B 1 1 C B 1 2 C B1  C B C t to t to t to (5.10) @ y t A D @ ˛ A C1 1 C @ ˛ A C2 2 C @  A E3  C @ yN A ; t 0 0 1 0 where  D ˛ˇ. 1/ C .1 1 /f1 C ˇ.˛ C 2/ g > 0; and C1 ; C2 and E3 are the constants which are determined by the initial values . to ; y to ;  to /. In fact, we have the following relationships from (5.10). (5.11a) (5.11b) (5.11c) ˛ N  t C ;  o 1 1 1 2 1  y to D C1 C C2 C  to C y; N ˛ ˛   to D E3 :  to D C1 C C2 C We can determine the values of C1 ; C2 and E3 by solving (5.11) if a triplet of the initial values . to ; y to ;  to / is predetermined. However, in the NK setting, the initial values  to and y to are considered to be not-predetermined variables which can be chosen freely by the economic agents, although the initial value of the exogenous disturbance . to / is a predetermined variable. Furthermore, it is assumed that the economic agents choose only the pair of the initial values . to ; y to / which ensures that the solution (5.10) converges to the equilibrium point .; N y/. N It is apparent that only the initial values that satisfy the condition C1 D C2 D 0 (5.12) can ensure the convergence to the equilibrium point. Substituting (5.12) into (5.11), we obtain the following requirement. ˛  t C ; N  o 1   to C y: N y to D   to D (5.13a) (5.13b) From the assumptions (5.14) 1 > 1, 0 <  < 1 and (5.13), we have the relationships sign. to / N D sign.y to y/ N D sign  to ; The Japanese Society for Post Keynesian Economics A NALYTICAL C RITIQUE OF NK DYNAMICS 15 and the special solution that ensures the convergence to the equilibrium point becomes as follows. ˛  t D  to  t to C ; (5.15a) N  1  yt D (5.15b)  to  t to C y; N  (5.15c)  t D  to  t to : Figure 3 is the so-called impulse response functions that are produced by the special solution (5.15) in case of  to < 0. In this figure, it is supposed that the system is at the equilibrium point for 0  t < to , the exogenous disturbance is caused at the time period t D to , and then it diminishes continuously. Responding to this exogenous disturbance, the pair of the values of  and y suddenly jumps to the initial condition . to ; y to / that ensures the convergence to the equilibrium point conveniently. The jump variable postulate manages to conceal the paradoxical behavior of the main variables. However, it is not clear why the economic agents in this model can choose the special initial values that are given by (5.13) immediately after the exogenous shock occurs. What occurs if another combination of the initial conditions is chosen by mistake? As a thought experiment, let us assume that  to D ; N y to D y; N (5.16) even if  to 6D 0, which means that the economic variables cannot conveniently jump even if an exogenous shock occurs. Substituting (5.16) into (5.11), we have the following system of equations. ! ! !  ˛ 1 1 C1 to (5.17) D : 1 1 1 2 C2  1  ˛ ˛ Solving this equation with respect to C1 and C2 , we obtain the following result. (5.18a) (5.18b) ˛.2 / t ; .2 1 / o ˛.1 / C2 D t : .2 1 / o C1 D Substituting (5.18) and E3 D  to into (5.10), we obtain the special solution of the system when (5.16) is assumed. Since it is assumed that 0 <  < 1 < 1 < 2 , (5.18) implies that (5.19) Post Keynesian Review sign C1 D sign C2 D sign  to : Vol. 2 16 ASADA, Toichiro t ✻ (a) t❜ o 0  to r ✲ t r t ✻ N r (b) ❜ to 0  to ✲ t r yt ✻ r ❜ (c) y to 0 r to ✲ t F IGURE 3. The special solution (5.15) in case of  to < 0. The Japanese Society for Post Keynesian Economics A NALYTICAL C RITIQUE OF NK DYNAMICS 17 t ✻ (a) t❜ o 0  to r ✲ t r t ✻ N r (b) r 0 ❜ to ✲ t yt ✻ (c) yN r r 0 ❜ to ✲ t F IGURE 4. The special solution (5.10) in case of  to D ; N y to D y; N  to < 0. Post Keynesian Review Vol. 2 18 ASADA, Toichiro In this case, the propagation process of the effects of the exogenous disturbance becomes as follows. For simplicity, we consider only the case of  to < 0. If (5.16) is satisfied, we have the following from (5.4) in case of  to < 0. (5.20a) to C1 D N D  to ; (5.20b) yto C1 D yN  to > yN D y to ; (5.20c) to C2 D N C ˛.yN yto C1 / D N C ˛ to < N D to C1 ; (5.20d) yto C2 D yto C1 C .˛ C 2 /.yto C1 y/ N D yto C1 .˛ C 2 / to > yto C1 : A set of relationships (5.20) means that the fall of  and the rise of y are caused soon after this exogenous shock. This tendency is not restricted to the periods soon after the shock. We have C1 > 0; C2 < 0 and 1 < 1 < 2 where 2 is the dominant root that governs the dynamic behavior of the system in case of C2 6D 0. Therefore, (5.10) implies that  continues to fall and y continues to rise ultimately, which means that the deflationary prosperity rather than the deflationary depression is caused by this exogenous shock. Figure 4 illustrates the typical impulse response functions of such unconvincing and paradoxical behaviors of the main variables that are caused by the exogenous disturbance. 6. Toward an Alternative Old Keynesian Approach Up to the previous section, we presented an analytical critique of the prototype NK dynamic model. The point is that the prototype NK dynamic model produces the anomalous behaviors that are inconsistent with the empirical facts. In this section, we develop a constructive and alternative approach that was presented by Asada(2012), which is called the Old Keynesian dynamic model after Tobin(1969, 1994).15 Asada’s(2012) Old Keynesian dynamic model consists of the following set of equations.16 (6.1)  t D x t C f .y t / ; f .y/ N D 0; fy D f 0 .y t / > 0; (6.2) y t D z t C g.r t (6.3) x t / ; gr x t D  teC1 ; z t D y teC1 ; x D g 0 .r t x t / < 0; 15 The work, such as Asada, Chiarella, Flaschel and Franke(2003, 2010), Chiarella, Flaschel and Franke(2005), and Asada, Flaschel, Mouakil and Proanõ(2011), is also based on the Old Keynesian tradition of the modeling methodology. 16 In this formulation, the exogenous disturbance is neglected for simplicity. The Japanese Society for Post Keynesian Economics A NALYTICAL C RITIQUE OF NK DYNAMICS ( 1 . t / N C 19 2 .y t y/; N if r t > 0; (6.4)
r t D r t C1 rt D (6.5)
x t D x t C1 x t D ıŒ1 .N x t / C .1 1 /. t x t /; 0  1  1; (6.6)
z t D z t C1 z t D Œ2 .yN z t / C .1 2 /.y t z t /; 0  2  1; maxŒ0; 1 . t / N C 2 .y t y/; N if r t D 0; where x t stands for the expected rate of price inflation, z t the expected real national income, and r t x t the expected real interest rate; 1 ; 2 ; ı and  are positive parameters. Meanings of other symbols are the same as those in the previous sections; Equations (6.1) and (6.2) are the expectation-augmented Phillips curve and the expectation-augmented IS curve respectively. In these formulations, it is allowed for that the functions f .y t / and g.r t x t / are nonlinear functions.17 (6.4) is a Taylor-rule-type interest rate monetary policy rule. In this formulation, the natural nonlinearity that is due to the nonnegative constraint of the nominal interest rate is explicitly considered. Equations (6.5) and (6.6) are the expectation formation equations concerning the rate of price inflation and the real national income. They are the mixtures of the forward-looking and backward-looking (adaptive) expectation formations, although they are different from the NK rational expectation formations. We can consider that 1 is the parameter that reflects the credibility of the central bank’s inflation targeting, and 2 is the parameter that reflects the credibility of the central bank’s employment targeting. We can consider that the formulations (6.5) and (6.6) are the simplest versions of the heterogeneous expectation hypothesis. Incidentally, we have the following equations by substituting (6.1) and (6.2) into (6.5) and (6.6). (6.7)
x t D ıŒ1 .N x t / C .1 1 /f .y t //; f 0 .y t / > 0; (6.8)
z t D Œ2 .yN z t / C .1 2 /g.r t x t /; g 0 .r t x t / < 0: These equations imply that the rate of change of the expected rate of inflation
x t is an increasing function of the actual real national income .y t / and the rate of change of the expected real national income
z t is a decreasing function of the expected real interest rate .r t x t /. In other words, this Old Keynesian dynamic model is immune from the notorious NK sign reversal problems. Asada(2012) showed that this system can be reduced to the following three dimensional system of nonlinear difference equations. First, put D 17 1 fx t C f .z t C g.r t x t // g N C 2 fz t C g.r t xt / The empirical data of the Japanese economy suggests that fyy D f 00 .y t / > 0. Post Keynesian Review Vol. 2 yg; N 20 ASADA, Toichiro and we have the following. (6.9a)
r t D r t C1 r t D H1 .r t ; x t ; z t I (6.9b)
x t D x t C1 x t D H2 .r t ; x t ; z t I ı; 1 / D ıŒ1 .N (6.9c)
z t D z t C1 x t / C .1 1; 2/ D ( ; if r t > 0; maxŒ0; ; if r t D 0; 1 /f .z t C g.r t x t //; 0  1  1; z t D H3 .r t ; x t ; z t I ; 2 / D Œ2 .yN z t / C .1 2 /g.r t x t /; 0  2  1: Asada(2012) studied the following differential equation version of (6.9). (6.10a) rP D H1 .r; x; zI 1; 2 /; (6.10b) xP D H2 .r; x; zI ı; 1 /; (6.10c) zP D H3 .r; x; zI ; 2 /: Now, we shall outline Asada’s(2012) analysis. The equilibrium solution of this system that ensures rP D xP D zP D 0 can be written as follows, if we ignore the nonnegative constraint of r. r  D o C ; N x  D   D ; N z  D y  D y: N (6.11) It is clear that we have r  > 0 if and only if the inequality N > (6.12) o is satisfied. In this case, the equilibrium real interest rate becomes r (6.13)   D o : We assume that the inequality (6.12) is in fact satisfied.18 Then, the Jacobian matrix of this system at the equilibrium point becomes 0 1 H11 H12 H13 B C (6.14) J3 D @H21 H22 H23 A ; H31 H32 H33 where (6.15) H11 D . 1 fy C H13 D H22 D 18 1 fy C 2 /gr x 2 < 0; H12 D > 0; ıŒ1 C .1 1 /fy gr 1 .1 H21 D ı.1 x ; fy gr 1 /fy gr H23 D ı.1 x/ x 2 gr x > 0;  0; 1 /fy  0; This inequality is automatically satisfied if o > 0 and N > 0. The Japanese Society for Post Keynesian Economics A NALYTICAL C RITIQUE OF NK DYNAMICS H31 D .1 2 /gr x  0; H32 D 21 ˛.1 2 /gr x  0; H33 D 2  0: The characteristic equation of this system at the equilibrium point becomes as follows. (6.16) '3 ./  jI J3 j D 3 C d1 2 C d2  C d3 D 0; where (6.17a) (6.17b) (6.17c) d1 D tr J3 D H11 H22 H33 ; ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇH ˇ 11 H12 ˇ ˇH11 H13 ˇ ˇH22 H23 ˇ d2 D ˇ ˇCˇ ˇ; ˇCˇ ˇH21 H22 ˇ ˇH31 H33 ˇ ˇH32 H33 ˇ d3 D jJ3 j > 0; if 0  1 < 1 and 0  2 < 1: In this Old Keynesian model, the traditional concept of (in)stability is adopted. In other words, the jump variables are not allowed for in this model, and the equilibrium point is considered to be locally stable if and only if all of the roots of the characteristic equation (6.16) have negative real parts.19 It is well known that such local stability condition is satisfied if and only if the following Routh-Hurwitz conditions are satisfied (see Gandolfo, 2009, Ch.16). (6.18) dj > 0 .j D 1; 2; 3/; and d1 d2 d3 > 0: By utilizing this result, Asada(2012) proved the following two propositions. P ROPOSITION 6.1. Suppose that the parameters 1 and 2 are fixed at any positive values. Then, the equilibrium point of (6.10) is unstable if either of the following two conditions (a) or (b) is satisfied. (a) ı > 0 is sufficiently large and 1 is sufficiently small (close to zero). (b) The condition 0  1 < 1 is satisfied, ı > 0 and  > 0 are sufficiently large, and 2 is sufficiently small (close to zero). P ROPOSITION 6.2. (1) Suppose that the parameters 1 , 2 , ı and  are fixed at any positive values. The equilibrium point of (6.10) is locally asymptotically stable, if both of 1 and 2 are close to 1 (including the cases of 1 D 1 and 2 D 1). (2) Suppose that the parameter values ı; ; 1 and 2 are fixed at any values such that ı > 0;  > 0; 0  1 < 1, and 0  2 < 1. Then, the equilibrium point of (6.10) is locally asymptotically stable, if either of 1 > 0 or 2 > 0 is sufficiently large. 19 The same traditional concept of stability/instability is adopted also in Post Keynesian literature, such as Keen(2000), Asada(2001), and Charles(2008). Post Keynesian Review Vol. 2 22 ASADA, Toichiro These two propositions show that the increases (resp. decreases) of the parameter values 1 ; 2 ; 1 and 2 have the stabilizing (resp. destabilizing) effects in the traditional sense in this Old Keynesian model. In other words, the equilibrium point of the dynamic system (6.10) is stable (resp. unstable) if the credibility of the inflation targeting and the employment targeting by the central bank is high (resp. low) and the central bank’s monetary policy is active (resp. inactive). Let us select one of such parameters, for example, 1 as a bifurcation parameter, and suppose that the equilibrium point of the system is unstable for all sufficiently small values of 1 , and it is locally stable for all values of 1 that are sufficiently close to 1. Then, there exists at least one bifurcation point 1o 2 .0; 1/ at which the discontinuous switch from the unstable system to the stable system occurs as the parameter value 1 increases. It is apparent that at least one real part of the roots of the characteristic equation (6.16) must be zero at the bifurcation point. However, we cannot have the real root  D 0 at such a bifurcation point, because we have (6.19) '3 .0/ D d3 6D 0; if 0  1 < 1 and 0  2 < 1 from equations (6.16) and (6.17c). This means that the characteristic equation (6.16) has a pair of pure imaginary roots at the bifurcation point. This situation is enough to apply the Hopf bifurcation theorem (Appendix C) to the dynamic system (6.10), and we have the following proposition. P ROPOSITION 6.3. The bifurcation point of the dynamic system (6.10), which is the point at which the discontinuous switch from the unstable system to the stable system occurs as one of the parameter values 1 ; 2 ; 1 and 2 increases, is in fact the Hopf bifurcation point. In other words, the non-constant closed orbits exist at some range of the parameter values that are sufficiently close to the bifurcation point. Proposition 6.3 shows that the endogenous cyclical fluctuations occur at some range of intermediate values of the parameters 1 ; 2 ; 1 and 2 , even if there is no exogenous disturbance, contrary to the prototype NK dynamic model that cannot produce the economic fluctuations without the exogenous disturbance. 7. Concluding Remarks In this paper, we showed that the prototype New Keynesian(NK) dynamic model produces several paradoxical and anomalous behaviors, which are inconsistent with the empirical facts. Sign reversals of NK Philips curve and NK IS curve are typical examples of such anomaly. We also observed that the problematical trick of The Japanese Society for Post Keynesian Economics A NALYTICAL C RITIQUE OF NK DYNAMICS 23 the jump variable technique is an unconvincing device. Finally, we presented the outline of the alternative Old Keynesian dynamic model that is consistent with the empirical facts and immune from the NK anomalies. Appendix A. The Total Instability Theorem The following purely mathematical result is useful for the proof of Lemma 3.2 in the text. T HEOREM A.1. Let A be an n  n matrix with the property jAj 6D 0. Then, the absolute values of all roots of the characteristic equation jI Aj D 0 are greater than 1 if and only if the absolute values of all roots of the characteristic equation jI A 1 j D 0 are less than 1. Proof. The assumption jAj 6D 0 means that 0 cannot be a characteristic root of the equation jI Aj D 0. Let A 6D 0 be a characteristic root of this equation that satisfies jA I Aj D 0. Then, I A D .A 1 I /A D I.A 1  1 I /A, so that jI Aj D . 1/n n j 1 I A 1 jjAj: Hence, jI Aj D 0 if and only if j 1 I A 1 j D 0. This means that B  1 is a characteristic root of the equation jI Bj D 0, A where B  A 1 . Suppose that A is a real number. In this case, it is obvious that jA j > 1 if and only if jB j < 1. Next, suppose that A is a complex number such that A D  C i !; i D p  i! 1 D 2 . In this case, we have 1; ! 6D 0: Then, we have B D  C i!  C !2 p 1 jA j D  2 C ! 2 and jB j D p 2 . This means that jA j > 1 if and only if 2  C! jB j < 1 even if A is a complex number. Appendix B. Proof of Proposition 3.1. The characteristic equation (3.8) has the following properties. (B.1) (B.2) (B.3) (B.4) (B.5) '10 ./ D 2 C a1 , '1 .0/ D a2 > 0, '10 .0/ D a1 < 0, '1 .1/ D 1 C a1 C a2 D ˛ˇ. 1 1/, '10 .1/ D 2 C a1 D ˇ.˛ C 2 / < 0. These properties entail that the following statements hold. Post Keynesian Review Vol. 2 □ 24 ASADA, Toichiro (1) Suppose that 0 < 1 < 1. Then, we have '1 .0/ > 0; '10 .0/ < 0; '1 .1/ < 0 and '10 .1/ < 0. In this case, we have Figure 5, which implies that we have two real roots, 1 and 2 , such that 0 < 1 < 1 < 2 . (2) Suppose that 1 D 1. Then, we have '1 .0/ > 0; '10 .0/ < 0; '1 .1/ D 0 and '10 .1/ < 0. In this case, we have Figure 6, which implies that we have two real roots, 1 and 2 , such that 1 D 1 < 2 . (3) Suppose that 1 < 1 < 1 C . Then, we have '1 .0/ > 0; '10 .0/ < 0; '1 .1/ > 0 and '10 .1/ < 0, with D > 0. In this case, we have Figure 7, which implies that we have two real roots, 1 and 2 , such that 1 < 1 < 2 . (4) Suppose that 1 D 1 C . Then, we have '1 .0/ > 0; '10 .0/ < 0; '1 .1/ > 0 and '10 .1/ < 0, with D D 0. In this case, we have Figure 8, which implies that we have a pair of multiple roots such that 1 < 1 D 2 . (5) Suppose that 1 C  < 1 . Then, it follows from Lemma 3.2 and D < 0 that the characteristic equation (3.8) has a pair of conjugate complex roots and their absolute values are greater than 1. '1 ./ ✻ a2 r r 0 1 1 C a1 C a2 r 1r ✲ r 2  r F IGURE 5. Case of 0 < 1 < 1. The Japanese Society for Post Keynesian Economics A NALYTICAL C RITIQUE OF NK DYNAMICS 25 '1 ./ ✻ a2 r 1r 1 0 ✲ r 2 F IGURE 6. Case of 1  D 1. '1 ./ ✻ a2 r 1 C a1 C a2 r r r 0 r 1 1 F IGURE 7. Case of 1 < Post Keynesian Review Vol. 2 ✲ r 2 1 <1C  26 ASADA, Toichiro '1 ./ ✻ a2 r 1 C a1 C a2 r r r 0 1 ✲ r 1 D 2 F IGURE 8. Case of 1  D1C Appendix C. Hopf Bifurcation Theorem Proposition 6.3 is a direct consequence of the following version of the Hopf bifurcation theorem (see Asada, Chiarella, Flaschel and Franke 2003, 2010, Mathematical appendices, and Gandolfo 2009, Ch.24). T HEOREM C.1. Let xP D f .xI "/; x 2 Rn ; " 2 R be a system of differential equations with a parameter ". Suppose that the following properties (i)–(iii) are satisfied. (i) This system has a smooth curve of equilibria given by f .x  ."/I "/ D 0. (ii) The characteristic equation jI Df .x  ."o /I "o /j D 0 has a pair of pure imaginary roots, ."o / and S ."o /, and no other roots with zero real parts, where  Df .x ."o /I "o / represents the Jacobian matrix of the above system at .x  ."o /; "o /. ˇ ˇ d Re ."/ ˇ (iii) 6D 0, where Re ."/ is the real part of ."/. d" ˇ "D"o Then, there exists a continuous function ". / with ".0/ D "o , and for all sufficiently small values of 6D 0 there exists a continuous family of non-constant periodic solutions x.t; / for the above dynamic system, which collapses to the equilibrium point x  ."o / as ! 0. The period of the cycle is close to 2 , Im ."o / where Im ."o / is the imaginary part of ."o /. The Japanese Society for Post Keynesian Economics A NALYTICAL C RITIQUE OF NK DYNAMICS 27 Bibliography [1] Akerlof, G. A. and R. J. Shiller(2009), Animal Spirits: How Human Psychology Drives the Economy, and Why It Matters for Global Capitalism, Princeton University Press. 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Post Keynesian Review Vol. 2 28 ASADA, Toichiro [19] Gandolfo, G.(2009), Economic Dynamics, Fourth Edition, Springer. [20] Keen, S.(2000), “The Nonlinear Economics of Debt Deflation,” in Barnett, W. A., C. Chiarella, S. Keen, R. Marks and H. Scnabl (eds.), Commerce, Complexity, and Evolution, Cambridge University Press, pp.83-110. [21] Keynes, J. M.(1936), The General Theory of Employment, Interest and Money, Macmillan. [22] Kirman, A. P.(1992), “Whom or What Does the Representative Individual Represent?,” Journal of Economic Perspective 6(2), pp.117-36. [23] Mankiw, G.(2001), “The Inexorable and Mysterious Tradeoff between Inflation and Unemployment,” Economic Journal 111, C45–C61. [24] Romer, D.(2006), Advanced Macroeconomics, Third Edition, McGraw-Hill. [25] Taylor, J. B.(1993), “Discretion versus Policy Rules in Practice,” Carnegie-Rochester Conference Series on Public Policy 39, pp.195-214. [26] Tobin, J.(1969), “A General Equilibrium Approach to Monetary Theory,” Journal of Money, Credit, and Banking 1, pp.15-29. [27] Tobin, J.(1994), “Price Flexibility and Output Stability: An Old Keynesian View,” in Semmler, W. (ed.), Business Cycles: Theory and Empirical Methods, Kluwer Academic Publishers, pp.165-95. [28] Woodford, M.(2003) , Interest and Prices: Foundations of a Theory of Monetary Policy, Princeton University Press. Acknowledgments. This research was financially supported by the Japan Society for the Promotion of Science (Grant-in Aid (C) 25380238), Chuo University Grant for Special Research, 2011–2013, and MEXT-Supported Program for the Strategic Research Foundation at Private Universities, 2013–2017. Correspondence. Faculty of Economics, Chuo University, 742-1 Higashinakano, Hachioji, Tokyo, 192-0393 Japan. Email: asada@tamacc.chuo-u.ac.jp The Japanese Society for Post Keynesian Economics