The Taylor principle in a New Keynesian model with capital accumulation, government debt dynamics and non-Ricardian consumers Campbell Leith∗
Leopold von Thadden†
February 15, 2004 (First, preliminary version)
Abstract This paper develops a small, analytically tractable New Keynesian model with capital accumulation and government debt dynamics. We show that, in the absence of Ricardian equivalence, the latter channel is of conceptual importance for the design of monetary and Þscal policy rules consistent with determinate equilibrium dynamics. Our model can be used to see how deviations from Ricardian equivalence lead to potentially signiÞcant modiÞcations in the design of such rules, compared with a New Keynesian benchmark economy in which Þscal policy typically plays no prominent role. Using a Blanchard-Yaari structure, non-Ricardian features of our economy result from intertemporally ‘disconnected’ decisions of consumers. This friction is policy-invariant, which distinguishes our approach from recent contributions by Woodford. Keywords: Monetary policy, Fiscal regimes. JEL classiÞcation numbers: E52, E63.
∗
Campbell Leith, Department of Economics, Adam Smith Building, University of Glasgow, GlasgowG12 8RT, e-mail:
[email protected] † Current Address: European Central Bank, Division Monetary Policy Strategy, Kaiserstrasse 29, 60311 Frankfurt/Main, Germany, e-mail:
[email protected];
[email protected].
1
Introduction
This paper studies dynamic properties of stylized rules of monetary and Þscal policy in a tractable New Keynesian economy with physical capital accumulation and ‘nontrivial’ government debt dynamics. In line with the New Keynesian paradigm, the model economy is subject to imperfect competition and nominal price stickiness and monetary policy uses an interest rate rule of the Taylor-type. Deviating from the bulk of the analytical contributions to the paradigm, however, our model gives Þscal policy a prominent role by allowing for deviations from Ricardian equivalence in a Blanchard-Yaari framework. We take it for granted that policy rules should be consistent with (locally) determinate equilibrium dynamics, thereby ruling out the occurrence of non-fundamental episodes of inßation.1 The design of any such rules needs to reßect that, from a general equilibrium perspective, movements in the real interest rate (which are at the heart of the Taylor principle) trigger not only well investigated changes in aggregate demand and aggregate supply, but they also affect the evolution of government debt dynamics. Our paper shows that, in the absence of Ricardian equivalence, the latter channel is a source of considerable conceptual importance, compared with a New Keynesian benchmark economy in which Þscal policy typically plays no prominent role. In particular, our analysis stresses that the requirements for such rules may substantially differ between economies characterized by low and high levels of (steady-state) government debt. To assess how Taylor-type interest rate rules affect the three margins of aggregate demand, aggregate supply, and government debt dynamics in a general equilibrium framework with capital accumulation is not straightforward. Hence, it is common practice in the literature to make strong simplifying assumptions regarding Þscal policy, with the aim to establish a recursive dynamic structure. In particular, by restricting taxation to lump sum taxes, inÞnite horizon economies of the Ramsey-type have typically the feature that pure Þscal variables do not affect the dynamics of the other endogenous variables. When combined with the assumption of real balances being sufficiently ‘small’, this feature ensures that government debt dynamics can be speciÞed as recursive to the other dynamic equations. This recursive structure gives conveniently rise to a two-step-procedure which establishes whether overall dynamics are determinate or not.2 1
For details, see Taylor (1993) and, among others, Clarida, Gali, Gertler (1999, 2000). Along the lines of Benhabib, Schmitt-Grohé and Uribe (2001) there is a large literature which studies local vs. global (in)determinacy issues that are not addressed in this paper. For an authoritative treatment and further references, see, in particular, Woodford (2003). 2 For a closely related sketch of this logic, see Sims (1999) and Woodford (2003, chapter 4.4). As will become clear below, our approach differs from Woodford in one key aspect. Woodford uses a Ramsey-type inÞnite horizon framework and reserves the term ‘non-Ricardian’ for a policy scenario in which ‘it is not the case that real government liabilities remain bounded for all possible paths for the other endogenous variables’ (p. 315). By contrast, we use a Blanchard-Yaari overlapping
1
First, there are Þscal aspects. SpeciÞcally, if the dynamic system without Þscal policy is determinate, there is no ‘active’ role for Þscal policy in the sense of Leeper (1991), i.e. overall determinacy remains preserved if government debt dynamics evolve ‘passively’ with a stable root. If, however, the dynamic system without Þscal policy exhibits indeterminacy of degree one, then, consistent with the view of the ‘Þscal theory of the price level’ expressed in Woodford (1995) and Sims (1994), unstable (‘active’) debt dynamics are needed to restore overall determinacy. Second, isolated from Þscal aspects, there are monetary aspects, as witnessed by the large literature which establishes conditions for determinacy directly (and typically solely) in terms of monetary policy parameters. Evidently, of particular prominence in this context is the Taylor principle. According to this principle, determinacy of the (remaining) dynamic system requires that monetary policy reacts ‘actively’ to inßation by changing nominal interest rates by more than one-to-one. Conversely, indeterminacy is likely to obtain if monetary policy reacts passively (with an inßation coefficient less than one). Combined with the Þscal aspects discussed above, this reasoning gives rise to a classiÞcation which we call the Taylor-Leeper conjecture: ‘determinacy requires active/passive combinations of Þscal and monetary policy, i.e. one agent has to be active, while the other has to be passive’.3 However, there are also contributions which stress the strong model dependency of the Taylor principle. Dupor (2001), for example, completely overturns the Taylor principle in a model which establishes a strong supply-side link between real interest rates and the equilibrium return on physical capital by introducing investment dynamics without adjustment costs.4 In Dupor’s framework, this supply-side channel completely dominates the conventional demand-side effect, leading to the provocative result that dynamics of the economy (without Þscal policy) can only be determinate if monetary policy responds passively, while under active responses of monetary policy dynamics exhibit at best indeterminacy of degree one. Combined with Þscal aspects, this reasoning generations economy with intertemporally ‘disconnected’ decisions of consumers. Accordingly, the notion of ‘non-Ricardianess’ used in this paper is policy-invariant and results from a real friction of the economy, in line with the more traditional meaning of the term, as used, for example, in Barro (1974). In short, we have a structure with ‘non-Ricardian consumers’. When talking about policies we prefer in the following the active vs. passive-classiÞcation of Leeper (1991), which can be linked, however, to the non-Ricardian vs. Ricardian-classiÞcation of Woodford, as discussed in his book on p. 314. 3 The use of the terms active and passive is not uniform across the literature. SpeciÞcally, we stress that in Leeper’s analysis two (potentially different) notions of active monetary policy coincide, in the sense that i) a policy with a strong feedback on inßation is able to ii) deliver determinate dynamics when combined with a stable speciÞcation of government debt dynamics. In our set-up, these two notions do not necessarily coincide and our classiÞcation of active monetary policy is based on the Þrst one (i.e. in line with the literature on Taylor rules). 4 Alternatively, Carlstrom and Fuerst (2001) establish the fragility of the Taylor-principle in a set-up which allows for different timing conventions.
2
leads to a classiÞcation which is completely orthogonal to the Taylor-Leeper conjecture and which we call the Dupor conjecture: ‘determinacy can never prevail under active/passive combinations of Þscal and monetary policy’. Against this background, it is the purpose of this paper to show within a small, fully tractable model that these two polar characterizations of determinacy features are completely driven by the recursiveness assumption of Þscal policy and that they disappear in a more general framework in which government debt dynamics fully interact with the remaining dynamic equations. To this end, we embed a New Keynesian structure in a model of the Blanchard-Yaari type in which consumers are short-sighted and Ricardian equivalence ceases to hold whenever consumers face a positive probability of death. As a general feature, aggregate consumption dynamics as given by the Euler equation depend under this assumption on the aggregate level of debt. Through this channel government debt dynamics are no longer recursive and we show that now determinacy, generically, is equally compatible with active/passive combinations of monetary and Þscal policy (as suggested by the Taylor-Leeper conjecture) as well as with active/active or passive/passive combinations of policy (as suggested by the Dupor conjecture). The exact ranges of activism and passivism of monetary and Þscal policy that are consistent with determinate dynamics depend critically on the steady-state level of government debt. SpeciÞcally, we illustrate that the conditions for determinate dynamics are substantially different in economies characterized by low and high levels of steady-state debt. In the limiting case of a zero probability of death, however, the dependence between aggregate consumption dynamics and the aggregate level of debt vanishes and our model economy conveniently converges towards a standard Ramsey-economy, characterized by Ricardian equivalence and recursive Þscal dynamics. As we demonstrate, (only) in this limiting case the Taylor-Leeper conjecture and the Dupor conjecture are consistent with mutually exclusive scenarios. In related work, Leith and Wren-Lewis (2000) also employ a Blanchard-Yaari structure to discuss the role of government debt dynamics for interactions of monetary and Þscal policy. Using a speciÞcation without capital stock dynamics, however, supply-side patterns are less rich. This prevents the economy from behaving differently under low and high levels of steady-state debt, as established in this paper.5 Benassy (2003) considers an endowment economy in which, in terms of our classiÞcation, under Ricardian consumers compliance with the Taylor principle is a necessary condition for a determinate price level, while this is not the case under non-Ricardian consumers. Lubik (2003) considers a framework with Ricardian consumers in which the degree of competition is of key importance whether active monetary policy should be combined with passive or active Þscal policy to deliver determinate dynamics. Davig, Leeper, and Chung (2003) reconsider the stabiliz5
For an analysis closely related to Leith and Wren-Lewis, see also Chadha and Nolan (2002).
3
ing properties of the Taylor principle in a framework (with Ricardian consumers) in which policies are perceived to switch occasionally between constellations of i) active monetary policy/passive Þscal policy and ii) passive monetary policy/active Þscal policy. Because of the switching feature, even when policies are of the Þrst type, Þscal policy matters for the dynamics of the price level. The remainder of this paper proceeds as follows. Section 2 develops a BlanchardYaari model with New Keynesian features. Section 3 establishes the existence of steady states and summarizes the dynamic equations. Section 4 derives the main results of the paper by considering a linear labour supply speciÞcation which is tractable and qualitatively in line with Dupor. Section 5 checks the robustness of the main Þndings by investigating a polar (and similarly accessible) speciÞcation in which the labour supply is completely Þxed. Section 6 offers some conclusions. Most of the proofs are delegated to the Appendix.
2
The Model
Consumers are assumed to be of the Blanchard-Yaari type (see Blanchard (1985)) in that they face a constant probability of death, denoted by ». If the probability of death is assumed to be positive (» > 0), the effective decision horizon of private agents is shorter than the one of the government and the model differs through this channel from a standard Ramsey-type inÞnite horizon economy. The latter type of economy, however, can be discussed as a special case in our set-up if one considers a zero probability of death (» = 0): In setting up the model, we allow for three distinct assets: physical capital, interest-bearing government debt, and real balances. Later on, when studying the dynamics of the economy, we consider for simple tractability the cashless-limit. To this end, it is convenient to assume that real balances enter the utility function of agents in an additively separable manner. Moreover, physical capital and governments bonds are considered as perfect substitutes in the portfolios of agents. Abstracting from capital adjustment costs, this implies that the return on physical capital must be identical to the (risk-free) return on government bonds.
2.1
Problem of the representative consumer
A consumer born at time, j, has a constant time endowment of unity per period. Mj He chooses consumption cjt , real money balances, ptt and the proportion of time to devoted to work (njt ) in order to maximise his intertemporal utility function, taking as given the rate of time preference µ and a constant probability of death »:6 As to be discussed below, cjt denotes a consumption index of the Dixit-Stiglitz-type, i.e. Þnal output is produced in terms of differentiated goods (along the unit interval) and pt stands for the 6
4
Expected utility at time t reads as Z ∞ Mj Et U (j) = [ln cjs +  ln( s ) − ´njs ] exp(−(» + µ)(s − t))ds; ps t where  governs the share of real balances in the consumer’s portfolio (and  → 0 corresponds to the cashless limit). The labour supply term ´njs allows for a ßexible speciÞcation which can be used to discuss two tractable limiting scenarios: First, if ´ = 0 the labour supply is ‘exogenously’ Þxed at the level njs = 1 for all periods. Second, ´ > 0 generates an endogenous linear labour supply, similar to Dupor (2001). The consumer holds Þnancial (or nun-human) wealth (ajt ) in real terms ajt = ktj + mjt + bjt ; consisting of physical capital ktj , real balances mjt ; and bonds bjt . The consumer’s ßow budget constraint is given by, dajt = rt (ajt − mjt ) + »ajt + wt njt − ¿ jt − cjt − ¼t mjt + Ωjt :
(1)
Both physical capital and government bonds earn a risk-less real rate of return rt , while real balances depreciate at the rate of inßation ¼ t . As consumers do not live forever, competitive insurance companies are prepared to pay a premium »ajt in each period in return for obtaining the Þnancial wealth of consumers in the case of death. The individual is paid a real wage of wt and is subject to a lump-sum tax ¿ jt . Consumers also receive a share of the proÞts of Þnal goods producers of Ωjt ; to be derived below. The Þrst-order condition for consumption is given by, cjt =
1 ¸jt
where ¸jt denotes the costate variable from the current value Hamiltonian used to solve the consumer’s problem. Assuming ´ > 0; the Þrst order-condition pinning down the labour supply is given by ´ = wt ¸jt =
wt : cjt
Holdings of real money balances must satisfy 1 Â j = ¸jt (rt + ¼t ); mt which can be solved to give the usual money demand equation, mjt = Â
cjt : rt + ¼ t
corresponding aggregate price index.
5
The costate variable evolves over time according to d¸jt = −(rt − µ)¸jt : Given the Þrst order condition for consumption this implies dcjt = (rt − µ)cjt : By integrating the ßow budget constraint and imposing the no-Ponzi game condition regarding non-human wealth, the intertemporal budget constraint can be written as, Z ∞ Z s 1 j cs(exp − (r¹ + »)d¹))ds = (ajt + hjt ); 1+Â t t where hjt is the individual’s human wealth, given by Z ∞ Z s j j j (wsns + Ωs − ¿ s )(exp − (r¹ + »)d¹))ds: ht = t
t
Integrating the equation of motion for consumption and substituting it into the intertemporal budget constraint we obtain the individual consumer’s consumption function, »+µ j cjt = (at + hjt ): 1+Â
2.2
Aggregate behaviour of consumers
Following Blanchard (1985), a convenient normalization of the total population size can be obtained by assuming that at any instant in time a large cohort is born of size »: Any such cohort born at t has a size, as of time T; of »(exp R t −»(T − t)): Then, the size of the total population at any time t will be unity, since −∞ »(exp −»(t−s)ds) = 1: Integrating over individuals of all cohorts in the usual way yields the aggregate consumption function and money demand function, respectively »+µ (at + ht ) 1+Â ct = Â ; rt + ¼t
ct = mt
(2) (3)
with variables without the j-index denoting aggregates. Assuming ´ > 0; aggregate labour supply must satisfy the equation ´= 6
wt ; ct
while nt = 1 for all t in the special case of ´ = 0: The evolution of aggregate human wealth follows dht = (rt + »)ht − wt nt + ¿ t − Ωt :
(4)
Aggregate non-human wealth is given by at = kt + bt + mt
(5)
dat = rt at − (rt + ¼t )mt + wt nt − ct + Ωt − ¿ t = rt at + wt nt − (1 + Â)ct + Ωt − ¿ t
(6)
and follows the law of motion.
For further reference it is convenient to express the dynamics of the aggregate behaviour of consumers in terms of c and a: To this end, we differentiate (2) with respect to time and obtain upon substituting (4) and (6) dct = (rt − µ)ct − »
»+µ at : 1+Â
(7)
Equation (7) captures the well-known feature that, whenever » > 0; the growth rate of individual consumption (rt −µ) exceeds the growth rate of aggregate consumption, despite a constant propensity to consume out of wealth for all individuals. The reason for this is given by the fact that at any moment in time agents with a high level of non-human wealth are replaced by new-borns with zero non-human wealth. Because of this generational turnover effect aggregate consumption dynamics in the Euler equation (7) depend on the aggregate level of non-human wealth (which includes the outstanding level of government liabilities), in contrast to a standard Ramsey-economy with » = 0:
2.3
The Firm’s Problem
We have two types of Þrm in our model. 2.3.1
Capital rental Þrms
There is a competitive continuum of Þrms which accumulate capital for rental to Þnal goods producers. Let it denote the real investment of the Þrm, using a mix of Þnal goods which is identical to the private consumption pattern. Moreover, assume that capital depreciates at the rate ± > 0: Then, the equation of motion for the capital stock is given by
7
dkt = it − ±kt :
Capital rental Þrms are owned by private households. The return on capital will be identical to the risk-free rate rt if the rental rate of capital (pkt ) charged to the Þnal goods sector satisÞes the zero-proÞt condition pkt = rt + ±: 2.3.2
Final Goods Producers
We assume that Þnal goods are produced by imperfectly competitive Þrms which are subject to the constraints implied by Calvo (1983) contracts, such that at any point in time Þrms are only able to change prices with instantaneous probability ®. Firms are lined up along the unit interval and a typical Þrm, with index z; produces according to a Cobb-Douglas function y(z)t = (n(z)t )° k(z)1−° : t Input markets are perfectly competitive. Cost-minimisation implies that the combination of labour and capital employed by the Þrm is given by, n(z)t nt ° pkt = = k(z)t kt 1 − ° wt which is common across Þrms because of price-taking behaviour in the input markets. Because of the Cobb-Douglas assumption, the cost function is linear in output, with marginal cost of production being given by MCt = (pkt )1−° (wt )° ° −° (1 − °)°−1 :
In period t Þrm z is assumed to face a demand schedule y(z)t = (
p(z)t −½ ) yt ; pt
(8)
where yt is the total demand for Þnal goods and ½ > 1 denotes the constant elasticity of demand.7 The objective of a Þrm which has the chance to reset its price in period t can be written as ¶1−½ µ ¶−½ Z ∞µ Z s p(z)t p(z)t Vt = [ ys − MCs ys ](exp − (r¹ + ®)d¹))ds: ps ps t t 7 The demand schedule (8) is consistent with the Dixit-Stiglitz consumption aggregator ct = ½−1 R1 R1 ½ 1 dz] 1−½ : Moreover, in line with [ 0 c(z)t ½ dz] ½−1 and the aggregate price level pt = [ 0 p(z)1−½ t these aggregators, it is assumed that investment and public consumption have the same demand structure as private consumption.
8
The optimal price implied by the optimisation of this objective function is given by R ∞ ³ 1 ´−½ Rs ½ ps MCs ys (exp − t (r¹ + ®)d¹))ds t p(z)t = R ³ ´1−½ Rs ∞ 1 (½ − 1) y (exp − (r¹ + ®)d¹))ds s t t ps which represents the forward-looking generalization of the familiar static (or steadystate) mark-up pricing rule p(z)=p = ½=(½ − 1)MC: Moreover, the aggregate price index prevailing at any moment in time can be seen as a weighted average of prices set in the past, where the weights reßect the proportion of those prices that are still in existence, 1 ·Z t ¸ 1−½ pt = ®(p(z)s )1−½ exp(−®(t − s))ds −∞
Finally, aggregate proÞts earned in the Þnal goods sector can be written as Z t p(z)s 1−½ p(z)s −½ Ωt = ®[( ) − MCt ( ) ]yt exp(−®(t − s))ds; pt pt −∞
(9)
with proÞts distributed to the private sector as speciÞed in (1).
2.4
The government
Let lt denote aggregate real liabilities of the public sector, consisting of real balances (mt ) and bonds (bt = lt − mt ): Substituting out for real balances from (3), ßow dynamics of public sector liabilities are given by dlt = rt lt − (rt + ¼t )mt + gt − ¿ t = rt lt − Âct + gt − ¿ t
(10)
where gt denotes some exogenous stream of government expenditures in terms of aggregate Þnal output, which does not affect the utility of consumers. The behaviour of both the monetary and the Þscal agent is assumed to follow deliberately simple feedback rules. First, regarding the monetary agent, we consider an inßation target (¼) of zero, i.e. ¼ = 0: Given the nominal stickiness due to Calvo pricing, the monetary agent has in the short-run some leverage over the real interest rate and we consider a feedback rule of the form rt = r + f M (¼t − ¼);
¼ = 0;
(11)
where r stands for a certain steady-state level of the real interest rate established below. SpeciÞcally, in (11), a positive value of the feedback parameter f M amounts to an ‘active’ response of monetary policy in line with the Taylor principle, as discussed in the literature on interest rate rules inspired by Taylor (1993). Conversely, 9
a negative value of f M corresponds to a passive response of monetary policy. Second, the Þscal agent also follows a simple feedback rule which aims at stabilizing government liabilities at some target level l: To achieve this target, lump-sum taxes get adjusted according to ¿ t = ¿ + f F (lt − l); (12)
while we keep gt from now onwards Þxed at some constant level g > 0: The idea behind this particular assumption is to make deviations from Ricardian equivalence as transparent as possible. From (10), in any steady-state equilibrium real government bonds are given by ¿ −g ¼ + m: (13) b= r r With ¼ = 0; steady-state seigniorage income will be zero, implying that the real value of outstanding government bonds must be completely backed by the present value of future primary Þscal surpluses. We assume from now on ¿ ≥ g; ensuring that steady state government bonds b are non-negative and that total government liabilities l remain non-negative in the cashless limit  → 0: There are two further comments worth making regarding the cashless limit to be considered below. First, as  → 0, the feedback rule (12) turns into a pure Þscal rule, in the sense that it targets the long-run bond level of the economy. Second, with government debt growing at the real interest rate r; the rule (12) will be stabilizing if the feedback parameter f F exceeds r: More speciÞcally, following the logic outlined by Leeper (1991), we call the Þscal rule (12) passive if f F > r; while it is active if f F < r. In sum, this gives rise to the following classiÞcation of monetary and Þscal policy rules: ClassiÞcation of monetary and Þscal policy rules active monetary policy passive monetary policy
2.5
active Þscal policy passive Þscal policy f M > 0; f F > r f M > 0; f F < r f M < 0; f F < r f M < 0; f F > r
Summary of equilibrium conditions
For further reference, this section compactly summarizes the set of equations which characterize dynamic equilibria at the aggregate level. 2.5.1
Consumers ´ =
wt ct
or
n=1
dct = (rt − µ)ct − » at = kt + lt 10
»+µ at 1+Â
(14) (15) (16)
2.5.2
Government dlt = rt lt − Âct + g − ¿ t rt = r + f M (¼t − ¼); ¼ = 0 ¿ t = ¿ + f F (lt − l)
2.5.3
2.5.4
(17) (18) (19)
Capital rental Þrms pkt = rt + ± dkt = it − ±kt
(20) (21)
yt = n°t kt1−° nt ° pkt = kt 1 − ° wt MCt = (pkt )1−° (wt )° ° −° (1 − °)°−1 Rs R ∞ ³ 1 ´−½ ½ ps MCs ys (exp − t (r¹ + ®)d¹))ds t p(z)t = R ³ ´1−½ Rs ∞ 1 (½ − 1) ys (exp − t (r¹ + ®)d¹))ds ps t 1 ·Z t ¸ 1−½ 1−½ pt = ®(p(z)s) exp(−®(t − s))ds
(22)
Final goods
(23) (24) (25)
(26)
−∞
2.5.5
Income identities
Mutual consistency of all plans of consumers, Þrms, and the government requires that in equilibrium aggregate Þnal output yt satisÞes the market clearing condition yt = ct + gt + it :
(27)
Corresponding to (27), in equilibrium aggregate output must also satisfy the income identity yt = wt nt + pkt kt + Ωt ; (28) with aggregate proÞts Ωt of the Þnal goods sector deÞned as in equation (9). By the law of Walras, however, (27) and (28) are not independent. SpeciÞcally, one can verify that if (14)-(27) are satisÞed, (28) will be satisÞed as well if aggregate proÞts Ωt follow (9). Accordingly, because of the residual character of Ωt ; we can drop (28) and (9) from the following analysis. 11
3
Steady states
The Þrst part of this section discusses brießy the existence conditions of steady states. In the second part, we develop, in general terms, the set of linearized dynamic equations that govern the local stability behaviour around steady states and that will be further examined for the two special labour supply speciÞcations in subsequent sections.
3.1
Existence
Apart from the nominal rigidities in the Þnal goods sector, which are of considerable importance for the off-steady state dynamics of the economy to be discussed below, our economy is in many ways standard. Hence, this section is kept deliberately brief. In steady state, p(z) = p; which we normalize to p = 1: Then, from (25), mark-up pricing in steady state implies (½−1)=½ = MC: Combining (20) and (22)-(24), factor prices can be rewritten as a function of the capital intensity (k=n) ½−1 (29) w = °(k=n)1−° = w(k=n) ½ ½−1 r = (1 − °)(k=n)−° − ± = r(k=n); (30) ½ where, according to (29) and (30), factor prices tend towards the standard marginal productivity expressions as the economy becomes perfectly competitive (½ → ∞). Using these expressions for factor prices, the remaining equations can be arranged as a system in c; k; and n: First, the steady-state versions of the consumption Euler equation (15) and the government’s budget constraint (17), when combined with (16), can be arranged as c=
»+µ »+µ ¿ −g » 1+Â k + » 1+Â r(k=n) Â »+µ r(k=n) − µ − » 1+Â r(k=n)
(31)
Second, from (14), the labour supply satisÞes 1 c = w(k=n) or n = 1 (32) ´ Third, combining the steady state version of capital stock dynamics (21), the technology condition (22), and the income identity (27), gives c = n° k 1−° − ±k − g;
(33)
Upon inspecting (31)-(33), one can show that the economy is generically characterized by two steady states with non-negative levels of consumption, capital and production, as long as the amount of outstanding government debt is not too large. We consider the two labour supply speciÞcations in turn. 12
c
(34)
6
c
»=0
(37)
6
»=0
EH
EH (36)
(35) - k
k¯
-
¯k
k n
n
(a) Fixed labour supply
(b) Linear labour supply
Figure 1: Steady-state conÞgurations 3.1.1
Fixed labour supply (´ = 0)
With the labour supply being Þxed at n = 1; the two remaining equations reduce immediately to a system in c and k; with: −g k + ¿r(k) »+µ · c = » »+µ Â 1 + Â r(k) − µ − » 1+Â r(k)
c = k 1−° − ±k − g;
(34) (35)
which, similar to Blanchard (1985), can be graphed as in Figure 1a: 3.1.2
Linear labour supply (´ > 0)
Upon isolating n in (33) as n = (c + g)=[(k=n)1−° − ±k=n] and using the identity k = n · (k=n) in (31), the equations (31)-(33) can be arranged as c =
1 w(k=n) ´
(36) −°
−± g + (¿ − g) (k=n) »+µ r(k=n) c = » · ; »+µ 1 + Â [r(k=n) − µ][(k=n)−° − ±] − » 1+Â
(37)
giving a tractable system in c and k=n: Equations (36) and (37) can be qualitatively graphed as in Figure 1b; with n being recursively determined in (33).8 −°
−± According to (29) and (30), w(k=n) and (k=n) are strictly rising functions in k=n; while r(k=n) r(k=n) and (k=n)−° are strictly falling in k=n: Moreover, (k=n)−° > r(k=n) + ± for all k=n > 0: 8
13
In either case, ´ = 0 or ´ > 0; the non-Ricardian features of the economy can be readily inferred from Figures 1a and 1b; in the sense that for a given value of g variations in ¿ change the steady states because of wealth effects in consumption. The reason for this is well understood in the literature: as long as » > 0; government bonds are perceived as net wealth by currently alive consumers, since the tax burden, backing these bonds, is partly borne by members of future generations.9 Moreover, as is well known, in the Ricardian limit (» = 0) the economy has a unique steady state and this steady state is characterized by the modiÞed golden rule r = µ: By contrast, if » > 0 there are generically two steady states with non-negative activity levels. At any of the two steady states, r > µ:10 To facilitate a straightforward comparison of our results with studies using Ricardian consumers, we concentrate in the following, however, exclusively on the high activity steady state.
3.2
Local dynamics
Upon combining (14)-(27), it is possible to establish a dynamic system in l; c; k, and ¼: Given the non-linearity of the system, we consider a Þrst-order approximation of the system around the (high-activity) steady state and specify the local dynamics of variables in terms of percentage deviations, whenever the steady state level is non-zero. SpeciÞcally, with x denoting some non-negative steady steady value, we · use the notation: xbt = xtx−x ; dxbt = @@txbt = xxt for x = l; k, c: Since the steady-state inßation target ¼ = 0; we express inßation dynamics in terms of level changes, i.e. · we consider ¼ t and d¼ t = ¼, respectively. 3.2.1
Dynamics of government liabilities
Starting out from (17)- (19) it is straightforward to obtain dlt = (r − f F + f M ¼t )lt − Âct + g − ¿ + f F l; which simpliÞes upon a Þrst-order approximation around the steady state to Âc db lt = (r − f F )b lt − b ct + f M ¼t : l
(38)
9 For details, see, in particular, Weil (1991). The crucial mechanism for this result is not the probability of death, but rather the ‘disconnectedness’ between consumers currently alive and those born at some point in the future. 10 The generic multiplicity of steady states in Blanchard-Yaari frameworks is discussed in further detail in Heijdra and van der Ploeg (2002)
14
3.2.2
Consumption dynamics
Using (16) and (18) within (15) gives for the evolution of consumption dct = (r + f M ¼t − µ)ct −
»(» + µ) (kt + lt ); 1+Â
which can be approximated to the Þrst order as db ct = − 3.2.3
»(» + µ) l b »(» + µ) k b lt + (r − µ)b kt + f M ¼ t : ct − 1+Â c 1+Â c
(39)
Capital stock dynamics
Consider the dynamics of the capital stock in (21). To substitute out for it we combine (22) and (27), yielding dkt = n°t kt1−° − ±kt − ct − g; which can be approximately written as y c y db kt = ° n bt − b ct + [(1 − °) − ±]b kt : k k k
(40)
bt = 0 and dynamics will be described by If the labour supply is Þxed at nt = 1, n (40). To allow also for the case of an elastic labour supply (with n bt 6= 0), we need to substitute out for n bt within (40). To this end, we linearize the labour supply condition (14) as b ct = w bt
and establish for w bt from (18), (20) and (23)
w bt = b kt − n bt +
(41)
fM ¼t r+±
(42)
Using (42) within (41), labour supply dynamics can be expressed as fM b n bt = kt − b ct + ¼t : r+±
(43)
Finally, inserting (43) in (40) gives
y y y fM c db kt = −[° + ]b ct + [ − ±]b kt + [° ]¼t k k k kr+± 15
(44)
3.2.4
Inßation dynamics
To establish a linearized equation describing the inßation dynamics we proceed in four steps. First, starting out from (25), the evolution of optimally adjusted prices p(z)t can be approximated to the Þrst order as11 Z ∞ d d s ] exp[(−(r + ®)(s − t))ds p(z)t = (r + ®)[b ps + MC (45) t
Differentiating this expression with respect to time gives, using the Leibnitz rule, d − pbt − MC d = (r + ®)(p(z) d t ): dp(z) t t
Second, by similar reasoning, the evolution of the aggregate price level (26) can be approximated as Z t d exp(−®(t − s))ds: pbt = ®p(z) s −∞
Note the identities db pt ≡ ¼ t and d2 pbt ≡ d¼ t : Differentiating the last expression for pbt with respect to time gives d − pbt ); db pt = ¼t = ®(p(z) t
i.e. inßation will be positive whenever ‘newly adjusted prices rise relatively more strongly than average prices’. To describe changes in inßation the last expression needs to be differentiated once more with respect to time. Intuitively, at any moment in time, inßation will accelerate (d¼ t > 0) whenever the ‘inßation rate of newly d ) exceeds the (average) inßation rate (db pt = ¼t ); with d¼ t to be set prices’ (dp(z) t calculated as d − pbt − MC d − pbt )] = r¼t − ®(r + ®)MC d t ) − ®(p(z) d t: d¼ t = ®[(r + ®)(p(z) t t
Third, from (24), marginal cost evolves approximately according to, d t = °w MC bt + (1 − °)b pkt
which, using (42) established above as well as (23), further simpliÞes to M d t = f ¼t + °(b MC kt − n bt ): r+±
To see the rationale behind this approximation, remember that in steady state p(z)=P = ½=(½− 1)M C: Assume that P and M C change permanently in percentage terms by certain amounts Pb d respectively. Then, (45) simply predicts that p(z) changes, ceteris paribus, in percentage and MC; d = Pb + M d terms by p(z) C 11
16
Then, under a Þxed labour supply (b nt = 0); inßation dynamics can be approximated as fM d¼ t = −[®(r + ®)°]b kt + {r − ®(r + ®) }¼t (46) r+± By contrast, under a linear labour supply (b nt 6= 0)), using the expression (43) for n bt , inßation dynamics follow d¼ t = −[®(r + ®)°]b ct + {r − ®(r + ®)[1 − °]
3.2.5
fM }¼ t r+±
(47)
Summary of dynamic equations (Â → 0)
The equations (38), (39), (40) or (44), and (46) or (47) constitute four-dimensional ct ; b kt ; and ¼t for the two regimes of a linear and a Þxed labour linear systems in b lt ; b supply. In either case, the system is characterized by two state variables (b lt ; b kt ) and 12 two forward-looking jump variables (b ct ; ¼t ) with free initial conditions. To enhance the analytical tractability of the system, we consider from now on the cashless limit (Â → 0); i.e. the weight of real balances in total government liabilities in (38) is assumed to be negligible such that the dynamics in l can be reinterpreted as the dynamics in government bonds b: Let J denote the Jacobian matrix which governs the linearized dynamics in b lt ; b ct ; b kt , and ¼t ; i.e.: b db lt lt db ct : ct = J · b (48) db b kt kt d¼ t ¼t Then, the dynamics of the two labour supply regimes can be compactly summarized as follows. Linear labour supply r − fF 0 0 fM −»(» + µ) l r−µ −»(» + µ) kc fM c J = fM y 0 −[° ky + kc ] −± ° ky r+± k fM 0 −®(r + ®)° 0 r − ®(r + ®)[1 − °] r+± 12
(49)
The notion of a predetermined stock of real government debt can be justiÞed as follows. First, because a fraction of Þrms sets prices in a forwardlooking manner, the inßation rate ¼ t is a jump variable. Second, assume that the stock of nominal government bonds is predetermined. Then, real government debt must be counted as a state variable, because it cannot move independently of the jump in inßation.
17
Fixed labour supply r − fF 0 0 fM −»(» + µ) cl r − µ −»(» + µ) kc fM J = y c 0 − k (1 − °) k − ± 0 fM 0 0 −®(r + ®)° r − ®(r + ®) r+±
(50)
The next two sections address in turn the implications of (49) and (50), respectively, for the classiÞcation of equilibrium dynamics.
4 4.1
Dynamics: linear labour supply Ricardian consumers (» = 0)
By assuming a zero probability of death (» = 0); there is no disconnect between the consumption decisions of agents over time and the model turns into a standard Ramsey-economy with inÞnitely lived agents. At the unique steady state, the modiÞed golden rule is satisÞed, i.e. » = 0 implies r = µ: Hence, under » = 0 the matrix J in (49) turns into µ − fF 0 0 fM 0 0 0 fM M J = (51) : y y y f c 0 −[° + ] − ± ° k k k k µ+± fM 0 −®(µ + ®)° 0 µ − ®(µ + ®)[1 − °] µ+±
The assumption of » = 0 simpliÞes the dynamics of (49) in two important ways, both linked to the (aggregate) consumption Euler equation (described by the second row in (49)). First, since in this special situation government bonds are not perceived as net wealth of consumers, consumption dynamics are not affected by government debt dynamics (i.e. »(» + µ) cl = 0). In fact, government debt dynamics do not affect any of the other three dynamic equation. As a result of this recursive structure, one eigenvalue of (51) is given by ¸3 = µ − f F : Second, the absence of the generational turnover effect also implies that consumption dynamics are not affected by capital stock dynamics (i.e. »(»+µ) kc = 0). Hence, there exists a simple relationship between the stance of monetary policy (measured by f M ) and the proÞle of consumption close to the steady state. Assume that, in response to some shock, ¼ t is positive (i.e. above target). Then, consumption increases (decreases) on impact, whenever monetary policy is active (passive), while it stays ßat if monetary policy is neutral (f M = 0): Moreover, combined with the assumption of a linear labour supply, this implies that the three-dimensional sub-dynamics of (51) in c; k, ¼ are qualitatively similar to Dupor, in the sense that capital stock dynamics not only leave the dynamics of 18
consumption, but also of inßation unaffected. This introduces a second recursive element into the overall dynamics, ensuring that a second eigenvalue of (51) is given by ¸4 = ky − ± > 0:13 Owing to these strong recursive features, it is possible to derive a simple set of necessary and sufficient conditions for the stability behaviour of the system which covers the entire parameter space in terms of f M and f F . The remaining two roots ¸1 and ¸2 can be assessed from the sub-dynamics in c and ¼; which are governed by the 2x2- sub-matrix · ¸ 0 fM c;¼ J = ; fM −®(µ + ®)° µ − ®(µ + ®)[1 − °] µ+± with determinant and trace Det(J)c;¼ = f M ®(µ + ®)° T r(J)c;¼ = µ − ®(µ + ®)[1 − °]
fM µ+±
Let q = (µ + ±)µ=[®(µ + ®)(1 − °)] > 0 denote the critical value of f M at which T r(J)c;¼ equals zero. Then, the signs of the eigenvalues ¸1 and ¸2 of J c;¼ can be classiÞed as follows:14 Lemma (Eigenvalues of J c;¼ ) i) if f M < 0; then ¸1 > 0, ¸2 < 0; ii) if 0 < f M < q; then ¸1 > 0, ¸2 > 0; iii) if f M > q; then ¸1 < 0, ¸2 < 0: Exploiting this recursive set-up, we distinguish in the following between two classiÞcations of dynamic equilibria that have been prominently discussed in the literature and that are completely orthogonal to each other. 4.1.1
Dynamics according to the Taylor-Leeper conjecture
The literature inspired by Taylor (1993) typically sees an active stance of monetary policy (f M > 0) as conducive to determinate dynamics.15 One often invoked explanation of the stabilizing properties of the Taylor principle goes as follows. Assume 13
Dupor introduces nominal rigidities through utility-based price adjustment costs, while this paper uses Calvo contracts. This difference, however, does not affect the the qualitative feature of recursive capital stock dynamics. 14 The sign restrictions are implied by Det(J c;¼ ) = ¸1 · ¸2 and T r(J c;¼ ) = ¸1 + ¸2 : In the following, we do not distinguish explicitly between real and conjugate complex eigenvalues, i.e. our classiÞcation refers to the real part of any eigenvalue which is crucial for the stability behaviour. 15 For a representative discussion, see, in particular, Clarida, Gali and Gertler (2000).
19
fF 6
I
II
III
µ VI
V
q
IV
- fM
II, VI: Determinacy
III: 2-Dim. Indeterminacy
V: 1-Dim. Instability
I, IV: 1-Dim. Indeterminacy
Figure 2: Taylor-Leeper conjecture ( » = 0) that the private sector expects that inßation might be temporarily above target. The active response of monetary policy would lead to a higher real interest rate. A higher real interest rate, however, is likely to reduce aggregate demand and to dampen inßationary pressure, invalidating thereby the possibility of non-fundamental, selffulÞlling inßationary episodes. By contrast, a passive reaction of monetary policy which lowers the real interest rate and tends to stimulate aggregate demand - could well be a source of such episodes. Moreover, a regime in which monetary policy actively relies on movements in the real interest rate to stabilize inßation needs an appropriate backing by Þscal policy, as discussed by Leeper. SpeciÞcally, to support an active monetary policy, Þscal policy should behave passively, in the sense that government debt dynamics cannot become unstable because the Þscal agent is committed to deliver sufficiently strong reactions of primary Þscal surpluses over time. By symmetry, Leeper shows that there is an alternative role assignment conducive to determinate dynamics. Whenever monetary policy is passive it will be too weak, in a fundamental sense, to anchor inßation dynamics, but this degree of freedom can be Þlled by active Þscal policy, i.e. the requirement of non-explosive debt dynamics can then be used to constrain inßation dynamics to follow a particular path. This rationalization of the Taylor principle stresses strongly adverse demand-side effects induced by a rise in the real interest rate.16 In the modelling context at 16
Of course, any equilibrium at any moment in time must be consistent with both the demand side and the supply side of the model economy. The notion that a rise in the real interest rate dampens inßationary pressure may be classiÞed, loosely speaking, as a negative demand side effect.
20
hand, this demand side channel will always dominate if one makes the simplifying assumption of a constant capital stock, i.e. kt = k in all periods. Under this assumption, because of the recursive structure, the dynamic equation for k in (51) can be dropped without loss of generality. Then, as summarized in Figure 2; local dynamics of the remaining sub-system in l; c; and ¼ around the steady state can be completely described in terms of f M and f F : In Figure 2, regions II, III, and VI correspond to constellations in which one agent behaves actively and the other passively, while dashed areas represent determinate dynamics. This leads to the result: Proposition 1 Taylor-Leeper conjecture Consider the dynamic system in l; c; ¼: 1) Determinacy requires active/passive combinations of Þscal and monetary policy, i.e. one agent is active, while the other is passive. 2) SpeciÞcally, i) passive monetary policy and active Þscal policy always yield determinacy, ii) active monetary policy and passive Þscal policy yield determinacy if monetary policy is not too aggressive (0 < f M < q): Remark: The proposition ‘adds’ ¸3 = µ − f F (which can be positive or negative, depending on the stance of Þscal policy) to the sign restrictions established for ¸1 and ¸2 in the above given Lemma and uses the fact that, neglecting capital, the economy is characterized by one predetermined variable and two jump variables.17 4.1.2
Dynamics according to the Dupor conjecture
The reasoning summarized in the previous section has been challenged by Dupor (2001). In his analysis Dupor shows that the introduction of (recursive) capital stock dynamics may well invalidate the logic underlying the Taylor principle. As a general feature, by adding a further state variable (kt ) with an unstable root (¸ > 0), regimes which have been previously consistent with determinate dynamics now become generically unstable, while regimes which have previously exhibited indeterminacy of degree one, now become generically determinate. Because of this feature, Dupor establishes in his his set-up - which does not consider, however, aspects of Þscal policy - the strong result that ‘active monetary policy leads to either indeterminacy or no equilibria that converge to a stationary steady state’, In principle, however, the total effect is ambiguous because substitution and income effects work in opposite directions. The latter channel becomes relevant under the Dupor conjecture discussed below. 17 The Þnding that under a combination of active monetary policy and passive Þscal policy (forwardlooking) monetary policy should not be too aggressive is in line with related literature. In particular, see Bernanke and Woodford (1997).
21
fF 6
I
II
III
µ VI
I, IV: Determinacy
V
q
IV
- fM
V: 2-Dim. Instability
II, VI: 1-Dim. Instability III: 1-Dim. Indeterminacy Figure 3: Dupor conjecture ( » = 0) i.e. active monetary policy is not compatible with locally determinate dynamics. In our set-up, given the recursive role of Þscal policy, this logic naturally overturns the Taylor-Leeper conjecture, as illustrated in Figure 3: In other words, the reasoning of Dupor, when combined with recursive government debt dynamics, gives rise to a result which is completely orthogonal to the Taylor-Leeper conjecture in terms of dynamic properties of equilibria: Proposition 2 Dupor conjecture Consider the dynamic system in l; c; k; ¼: 1) Determinacy can never prevail under active/passive combinations of monetary and Þscal policy in which one agent is active, while the other is passive. 2) Determinacy requires a policy mix that is active/active or passive/passive. Specifically, i) passive monetary policy and passive Þscal policy always yield determinacy, ii) active monetary policy and active Þscal policy yield determinacy if monetary policy is aggressive (f M > q > 0): Remark: The proposition ‘adds’ ¸4 = ky − ± > 0 to the sign restrictions established in Proposition 1 and uses the fact that the economy is characterized by two predetermined and two jump variables. Essentially, the Dupor conjecture allows for supply-side reactions that ‘dominate’ the demand-side channel sketched above. The key difference is that changes in the real interest rate now need to be matched by corresponding changes in the rental 22
rate of capital. For illustration, assume that inßation is expected to be above target, triggering under active monetary policy a rise in the real interest rate. For the rental rate to rise as well, this requires, for a predetermined level of the capital stock, a combination of a rise in the labour supply and a lower mark-up charged in the Þnal goods sector. Both channels are on impact expansionary. And the associated rise in inßation may well become self-fulÞlling under a consumption proÞle that jumps upward on impact and follows temporarily a rising path (reßecting the rise in the real interest rate), before gradually returning to the initial steady state. In sum, neglecting Þscal policy, active monetary policy is not compatible with determinate dynamics, but it may be consistent with indeterminacy of degree 1 (f M > q > 0). In the latter case, as indicated in region IV in Figure 3, active Þscal policy can be used to invalidate such self-fulÞlling expectations within the overall system of fourdimensional dynamics. Conversely, combinations of passive monetary and passive Þscal policy are also in line with determinate dynamics (region I).
4.2
Non-Ricardian consumers (» > 0)
Assuming » > 0; dynamics are governed, as derived in (49) above, by r − fF 0 0 fM −»(» + µ) l r−µ −»(» + µ) kc fM c J = fM y 0 −[° ky + kc ] −± ° ky r+± k fM 0 −®(r + ®)° 0 r − ®(r + ®)[1 − °] r+±
:
(52)
Owing to the non-Ricardian structure, consumption dynamics are now, in general, affected by the dynamics of government debt and the dynamics of the capital stock, implying that the 4x4−system is no longer recursive. Moreover, the relevant weights of these margins within the Euler equation depend crucially on the ‘position’ of the (high-activity) steady state, which now depends on Þscal policy in a non-trivial manner. First, one can infer from Figure 1 that the steady state levels of k and r depend on the level of government expenditures g:18 Second, for any given level of g; the levels of k and r also depend on the Þnancial mix between bonds and taxes. SpeciÞcally, since bonds are perceived as net wealth, a higher level of outstanding bonds has crowding out effects, leading to a lower level of k and a higher level of r: Both effects need to be taken into account if one attempts to track the dynamics implied by (52). Because of these features, it is elusive to characterize the dynamics of (52) by a set of necessary and sufficient conditions. Instead, by invoking a tractable sufficient condition, this section has the purpose to show that in a non-Ricardian structure the 18
Under » = 0; changes in g lead to a complete crowding out of c; leaving k and r unaffected.
23
two conjectures presented above are no longer mutually exclusive. To demonstrate this, it is convenient to proceed in two steps. 4.2.1
Balanced budgets
To start out with, this section considers the special scenario of a permanently balanced budget, with ¿ = g in all periods, and a zero level of initial debt. Upon this assumption, government debt dynamics can never take off and overall dynamics reduce to the three-dimensional sub-dynamics in c; k, ¼: fM r−µ −»(» + µ) kc fM y −± ° yk r+± J c;k;¼ = −[° yk + kc ] (53) : k fM −®(r + ®)° 0 r − ®(r + ®)[1 − °] r+±
Essentially, (53) embeds Dupor’s model in an overlapping generations framework, by introducing generational turnover effects associated with the capital stock (»(» + µ) kc > 0). Because of this latter feature, Dupor’s strong result (‘active monetary policy is not compatible with locally determinate dynamics’) is no longer, in general, valid. This can be established by evaluating the determinant and the trace of (53), which are linear functions of f M . The Trace is simply given by T r(J )c;k;¼ = ! 0 − ! 1 · f M ; with: y ®(r + ®)(1 − °) ! 0 = 2r − µ + − ± > 0; ! 1 = > 0: k r+± The determinant has the generic structure Det(J )c;k;¼ = Á0 + Á1 · f M ;
(54)
(55)
and one can derive two meaningful conditions that determine the signs of Á0 and Á1 : First, Á1 > 0 links the analysis one-to-one to the high activity steady state (EH ) in Figure 1b; i.e. Á1 > 0 is satisÞed if and only if (37) intersects (36) from below. Second, Á0 < 0 will be satisÞed whenever government expenditures are not too high in the following sense:19 (A 1) Assume g < °y: Then, Á0 < 0: Assuming that (A 1) is satisÞed, one can show: Proposition 3 Balanced-budget dynamics in c; k, ¼ Consider the high activity steady state EH : Then, there exists a value q1 > 0 such that at EH the system is determinate if f M < q1 : In particular, this implies that active monetary policies, satisfying 0 < f M < q1 ; yield determinacy. Recall from above, that ° is the Cobb-Douglas coefficient on labour in the production function of the Þnal good sector. Hence, (A 1) describes a very mild restriction. 19
24
Proof: The two properties: a) (A 1) ⇒ Á0 < 0 and b) Á1 > 0 ⇔ slope (37) < slope (36) at EH in Figure 1b are derived in Appendix 1. Exploiting the sign restrictions on ! 0 ; !1 ; Á0 ; and Á1 ; there must exist a value 0 < q1 such that, whenever f M < q1 , i) T r(J )c;k;¼ > 0 and ii) Det(J)c;k;¼ < 0: This implies ¸1 > 0; ¸2 > 0; ¸3 < 0; i.e. dynamics are determinate, since there is one state variable and two jump variables.20 ¤ Intuitively, this Þnding reßects that the generational turnover effect associated with the capital stock may act like a brake on the consumption dynamics. Because of this feature, consumption dynamics may no longer sufficiently support the positive supply-side response which active monetary policy induces in Dupor’s set-up. SpeciÞcally, for this mechanism to be strong enough to restore determinacy under active monetary policy, two things are needed: i) the capital stock itself must be sufficiently large, as implicitly ensured by assumption (A 1), and ii) monetary policy must not be too active (0 < f M < q1 ). 4.2.2
Government debt dynamics: general case
Extending the previous analysis, this section allows for unbalanced budget dynamics and positive target levels of government debt. As a general feature, the existence of generational turnover effects of governments bonds creates a feedback between debt dynamics and consumption dynamics which lead to non-trivial dynamic effects of Þscal policy. SpeciÞcally, to establish determinate dynamics for a given stance of monetary policy (measured by f M ); it no longer suffices to know whether Þscal policy is active or passive. Instead, depending on the level of steady state debt, the degree of activism (passivism) matters, and the strong dichotomy between the Taylor-Leeper conjecture and the Dupor conjecture breaks down. To derive this result, we establish a tractable sufficient condition for determinate dynamics in terms of Det(J) and T r(J) with a simple graphical representation. In particular as shown in Appendix 1, the matrix (52) has the convenient feature that Det(J) > 0; T r(J) < 0 is a sufficient condition for dynamics being determinate.21 Calculating the trace of (52) yields By similar reasoning one can show that there exists also some value q2 > q1 such that, whenever f > q2 ; i) T r(J)c;k;¼ < 0 and ii) Det(J)c;k;¼ > 0; implying indeterminacy of degree 1. This means that, under balanced budget dynamics, Dupor’s intuition remains valid for strongly active monetary policies. 21 If Det(J) > 0 and T r(J) < 0; eigenvalues must follow the pattern 1) ¸1 > 0; ¸2 > 0; ¸3 < 0; ¸4 < 0 or, alternatively, 2) ¸1 < 0; ¸2 < 0; ¸3 < 0; ¸4 < 0: We show that the structure of J in (52) is such that the second pattern of four negative eigenvalues can never occur. 20
M
25
fF
fF
6
6
r
r
-f M
(a)
fF 6
r -f M
(b)
- fM
(c)
Determinacy: active/passive Determinacy: active/active or passive/passive
Figure 4: Dynamics if consumers are non-Ricardian ( » > 0) T r(J) = ! 0 − f F − !1 · f M ; with: y ®(r + ®)(1 − °) > 0; ! 0 = 3r − µ + − ± > 0; ! 1 = k r+±
(56)
which can be rearranged as T r(J) = 0 ⇔ f F = ! 0 − ! 1 · f M :
(57)
The determinant has the generic structure l Det(J) = [r − f F ][Á0 + Á1 f M ] + »(» + µ) Á2 f M ; c
(58)
with Á0 ; Á1 ; Á2 being derived in Appendix 1. Rearranging (58), one obtains Det(J ) = 0 ⇔ f F = r(1 +
Á3 f M 1 l ); with: Á3 = »(» + µ) Á2 M Á0 + Á1 f r c
(59)
The critical line (57) describes a linear downward-sloping function in f M -f F -space, while the determinant-condition (59) represents a pair of hyperbolas, with the exact conÞguration depending on the coefficients Á0 ; Á1 ; Á2 ; Á3 : Evaluating the system at the high activity steady state EH ; however, imposes a clear structure on these 26
coefficients. To connect the analysis to the balanced-budget analysis of the previous section (with ¿ = g; l = 0), we consider the ‘non-Ricardian experiment’ to raise ¿ ; while keeping g Þxed. As ¿ increases, this experiment gives rise to 3 qualitatively distinct conÞgurations of the hyperbolas characterizing (59) in f M -f F -space, as graphed in Figure 4; i.e. Figures (a), (b), and (c), describe, respectively, scenarios of low, medium and high steady-state values of government debt. Shaded areas in Figure 4 denote regions where the sufficient condition for determinate dynamics is satisÞed.22 In all 3 cases the linearity of (57) and the non-linearity of (59) ensure that these regions break the dichotomy between the Taylor-Leeper conjecture and the Dupor conjecture. Proposition 4 Dynamics in l; c; k, ¼ Consider the high activity steady state EH : Then, 1) there exist active/passive combinations of monetary and Þscal policy ensuring determinate dynamics; 2) similarly, there exist constellations of active/active or passive/passive monetary and Þscal policy ensuring determinate dynamics. Proof: see Appendix 1. Figure 4 points at signiÞcant non-linearities in the dynamics describing the economy, depending on the level of government debt to be rolled over between generations. In general, as government debt rises, the associated wealth effect becomes a signiÞcant margin which challenges the ‘Ricardian demand and supply-side effects’ discussed above. Note that, in general, a higher level of government debt affects consumption dynamics adversely (i.e. the generational turnover effect of governments bonds has a negative sign). This reßects that a higher level of government debt, when perceived as private sector net wealth, leads at the aggregate level in steady-state comparison to a lower level of aggregate consumption because of the crowding out of physical capital. For a tentative illustration, consider Þrst a scenario with high government debt, as described in Figure 4c. Assume monetary policy is active. In principle, Þscal policy may be either active or passive to yield determinate dynamics. As monetary policy becomes more strongly active, however, the required Þscal discipline increases (since the Det(J) = 0−locus, which is the lower bound of the necessary condition Allowing for a positive probability of death (» > 0); Figure 4 generalizes Figure 3: However, by invoking Det(J) > 0 and T r(J) < 0; Figure 4 is based on a condition that is not necessary and sufficient. SpeciÞcally, in Figure 4; as » → 0; the hyperbolas collapse to the conditions f F = r and f M = 0 used in Figure 3 and all regions of determinacy in Figure 3 would also support determinacy in Figure 4: 22
27
for determinacy, slopes upward).23 Intuitively, under active monetary policy, expectations of high inßation would lead on impact to a higher debt burden. This burden acts like a brake on consumption, preventing that the supply-side dynamics of the Dupor conjecture are supported by appropriate consumption dynamics. If Þscal policy is (arbitrarily) passive, debt dynamics tend to be stable, leading to overall determinate dynamics. By contrast, if Þscal policy is active, debt dynamics tend to be unstable. Unlike the logic underlying the Þscal theory of the price level, however, this unstable feature is no longer separated from the other equations. As suggested by Figure 4c; at best a very weak degree of activism (‘f F being only slightly smaller than r’) is in line with determinate dynamics, while otherwise the system becomes locally unstable. Because of the strong wealth effects associated with a high level of government debt, this scenario bears a certain resemblance to the Taylor-Leeper conjecture. SpeciÞcally, for a large subset of the parameter space, determinacy is consistent with active/passive combinations of monetary and Þscal policy. By contrast, consider in a parallel manner a scenario with very low government debt, as described in Figure 4a: If monetary policy is active, Þscal policy may, again, be either active or passive to yield determinate dynamics. However, as monetary policy becomes more strongly active, the required Þscal discipline decreases (since the Det(J) = 0−locus slopes downward). Expectations of high inßation, again, lead on impact to a higher debt burden. Given the rather low weight of the wealth effect, this debt burden may no longer prevent per se that self-fulÞlling expectations stimulate inßationary consumption dynamics, in line with the Dupor conjecture. One way to break such expectations is the Þscal commitment to embark on (arbitrarily) active debt dynamics, as suggested by the logic underlying the Þscal theory of the price level. Alternatively, if Þscal policy is only weakly passive (‘f F being only slightly larger than r’), debt dynamics return only slowly to the initial steady state. This slow return to the steady state may ensure that the debt burden imposed on consumers in the transitional phase is substantial enough to slow down consumption dynamics. Hence, to deliver determinacy, Þscal policy can at best be weakly passive. In general, however, because of the weak wealth effect associated with a low level of government debt, this scenario bears resemblance to the Dupor conjecture, i.e. for a large subset of the parameter space determinacy is consistent with active/active or passive/passive combinations of monetary and monetary policy. In sum, assuming that consumers are non-Ricardian, Figure 4 indicates that dynamic properties of equilibria depend signiÞcantly on the steady-state level of government debt. In an environment of low government debt, conditions ensuring determinate equilibrium dynamics are qualitatively close to the Dupor conjecture. By contrast, 23
This reasoning is consistent with the analysis of Leith and Wren-Lewis (2000). Using a model without dynamics in the capital stock, however, the supply-side dynamics in their model is different, i.e. essentially in line with the Taylor-Leeper conjecture.
28
in an environment of high government debt the associated wealth effects change these conditions signiÞcantly, leading to a pattern that qualitatively resembles the Taylor-Leeper conjecture.
5
Dynamics: Þxed labour supply
This section brießy shows that the qualitative Þndings established in Section 4 do not depend on the assumption of a linear labour supply. They remain valid even under the polar (and equally accessible) assumption of a completely inelastic labour supply that is Þxed at a constant level (n = 1). This statement is subject to the caveat, however, that the particular illustration of the Taylor-Leeper conjecture chosen above is no longer feasible, since in any meaningful analysis at least one of the two productive margins, labour or capital, needs to be ßexible. From equation (50) derived above, recall that the crucial Jacobian matrix will now, in general, be given by the expression r − fF 0 0 fM −»(» + µ) cl r − µ −»(» + µ) kc fM : J = (60) y c 0 − k (1 − °) k − ± 0 M f 0 0 −®(r + ®)° r − ®(r + ®) r+±
5.1
Ricardian consumers (» = 0)
Assuming that consumers are Ricardian, in steady state, again, the modiÞed golden rule (r = µ) will be satisÞed, leading to very tractable dynamics in (60). SpeciÞcally, similar to Proposition 2; it is straightforward to see that the 4x4−system with recursive debt dynamics lends support to the Dupor conjecture. Proposition 5 Dupor conjecture with Þxed labour supply Dynamics in l; c; k, ¼ 1) Determinacy can never prevail under active/passive combinations of monetary and Þscal policy in which one agent is active, while the other is passive. 2a) Passive monetary policy and passive Þscal policy always yield determinacy; 2b) There exists some qe > 0 such that active monetary policy and active Þscal policy yield determinacy if f M > qe > 0: Proof: See Appendix 2.
Essentially, Proposition 5 differs from Proposition 2 only to the extent that, assuming » = 0; dynamics are no longer recursive in k: Hence, Proposition 5 is based on a sufficient condition, while Proposition 2 (and Figure 3) exploit a more rewarding condition which is necessary and sufficient. 29
5.2
Non-Ricardian consumers (» > 0)
If one allows for non-Ricardian consumers, the Dupor conjecture is no longer uncontested. Similar to Proposition 4 derived in Section 4 one can rather show: Proposition 6 Dynamics in l; c; k, ¼ (Fixed labour supply) Consider the high activity steady state EH : Then, 1) there exist active/passive combinations of monetary and Þscal policy ensuring determinate dynamics; 2) similarly, there exist constellations of active/active or passive/passive monetary and Þscal policy ensuring determinate dynamics. Proof: see Appendix 2. As derived in Appendix 2, the dynamic classiÞcation depends, again, critically on the steady-state level of government debt. Through the same channels as discussed above, outcomes resembling the Taylor-Leeper conjecture become more likely as the debt level of the economy increases, while the Dupor conjecture tends to prevail at low debt levels, i.e. Figure 4 extends qualitatively to the speciÞcation of a Þxed labour supply.
6
Conclusion
This paper starts out from the observation that in the New Keynesian paradigm Þscal policy typically plays no prominent role. SpeciÞcally, inspired by the contribution by Taylor (1993) there is a large literature that studies the appropriate degree of monetary feedback (in terms of the Taylor coefficient) which is needed to ensure determinate equilibrium dynamics. However, the appropriate degree of Þscal feedback is typically not addressed, reßecting implicitly a recursive speciÞcation of government debt dynamics. Motivated by this asymmetric treatment of monetary and Þscal policy, this paper considers a Blanchard-Yaari structure in which government bonds are perceived as net wealth by private agents. Because of this wealth channel, government debt dynamics interact with the other equilibrium conditions in a non-trivial manner. Enriching an otherwise standard New Keynesian model with capital accumulation by this particular channel, we show that the requirements for (locally) determinate equilibrium dynamics depend signiÞcantly on the level of steady-state debt. In particular, in our set-up in an environment of low government debt the required degree of Þscal discipline decreases whenever monetary policy becomes more active. By contrast, in an environment of high government debt the required degree of Þscal discipline increases whenever monetary policy becomes more active. Moreover, 30
we show that these non-linear implications of government debt for the design of policy rules conducive to determinate equilibrium dynamics disappear in a standard Ramsey economy, characterized by Ricardian equivalence, which can be derived as a special case of the more general Blanchard-Yaari structure.
Appendix 1 (Linear labour supply) Preliminaries to the proofs of Propositions 3 and 4: As derived in the main text, the determinant of the Jacobian matrix (52) has the representation (58) l Det(J) = [r − f F ][Á0 + Á1 f M ] + »(» + µ) Á2 f M ; c and encompasses (55)
Det(J)c;k;¼ = Á0 + Á1 · f M ;
as a special case under the 3-dimensional balanced budget regime: From (52); the coefficients Á0 ; Á1; Á2 can be calculated as y y c − ] Á0 = (r − µ)r[ − ± − ° k k+l k+l ®(r + ®)(r − µ) r + ± y y y c Á1 = { °( − ±) − (1 − °)( − ±) + ° + (1 − °) } r+± r−µ k k k+l k+l y Á2 = ®(r + ®)°( − ±) > 0; k where we use from (31) that in any cashless steady state »(» + µ) = (r − µ)c=(k + l) holds. Proof of Proposition 3: With  = 0; the balanced budget assumption amounts to l = 0: Invoking the steady state relationship g = y − ±k − c; the coefficient Á0 reduces to Á0 = (r − µ) kr [g − °y], i.e. g < °y ⇒ Á0 < 0: Moreover, in Figure 1b; the high activity steady state EH is characterized by the fact that the slope of (37) exceeds the slope of (36) in k=n − c−space. To assess the slopes we use from (31) that in any cashless steady state without government debt »(» + µ) = (r − µ)c=k holds. Then, by differentiating (36) and (37) with respect to k=n and comparing derivatives one obtains slope (37) > slope (36) ⇔
r+± y y y c °( − ±) − (1 − °)( − ±) + ° + (1 − °) > 0; r−µ k k k k
¤
which corresponds for l = 0 to Á1 > 0:
31
Proof of Proposition 4: Allowing for l > 0; the slope of (37) exceeds the slope of (36) in k=n − c−space iff
r+± y y y c l y (r + ±) °( − ±) − (1 − °)( − ±) + ° + (1 − °) +° ( − ±) > 0: r−µ k k k+l k+l k+l k r
Moreover, write the critical Det(J) = 0 condition as Det(J) = 0 ⇔ f F = r(1 +
l Á3 f M 1 ); with: Á3 = »(» + µ) Á2 : M Á0 + Á1 f r c
Using »(» + µ) = (r − µ)c=(k + l) and substituting out for Á2 ; Á3 can be expressed as Á3 =
l y (r + ±) ®(r + ®)(r − µ) {° ( − ±) } r+± k+l k r
Hence, slope (37) > slope (36) ⇔ Á1 + Á3 > 0:
Moreover, note that Á3 > 0 is always satisÞed and that Á0 and Á1 are linked as follows Á1 =
®(r + ®)(r − µ) r + ± y y c Á0 { °( − ±) + °( − ± − )− } r+± r−µ k k k+l (r − µ)r
Starting out from the balanced budget scenario, from these restrictions follows that, as ¿ rises for a Þxed value of g; EH reaches the 3 conÞgurations qualitatively graphed in Figure 4: (a) Á0 < 0; Á1 > 0; Á3 > 0; (b) Á0 > 0; Á1 > 0; Á3 > 0; (c) Á0 > 0; Á1 < 0; Á3 > 0: Finally, note that Det(J) > 0; T r(J) < 0 (61) implies that eigenvalues must follow the pattern 1) ¸1 > 0; ¸2 > 0; ¸3 < 0; ¸4 < 0 or, alternatively, 2) ¸1 < 0; ¸2 < 0; ¸3 < 0; ¸4 < 0: However, one can show that the structure of J in (52) is such that the second pattern of four negative eigenvalues can never occur. To prove this, we invoke the ‘quasi-negative deÞniteness’ criterion as stated, for example, in Gandolfo (p.252) which calls for forming the matrix B = (J + J 0 )=2; i.e. 1 M r − fF − 12 »(» + µ) cl 0 f 2 − 1 »(» + µ) l r−µ − 12 [ »(»+µ)k + °y+c ] 12 [f M − ®(r + ®)°] c c k B= 2 : °y+c y 1 y fM 0 − 12 [ »(»+µ)k + ] − ± ° c k k 2 k r+± ®(r+®)(1−°)f M 1 M 1 y fM 1 M f [f − ®(r + ®)°] ° r− 2 2 2 k r+± r+±
Then, a set of necessary and sufficient conditions for all four eigenvalues of J to be negative is that the leading minors of B should alternate in sign, beginning with 32
minus. In our case, the Þrst leading minor will only be negative if r < f F : This implies, however, that the second leading minor must also be negative, ruling out the possibility of four negative eigenvalues. Hence, Det(J) > 0; T r(J) < 0 implies ¸1 > 0; ¸2 > 0; ¸3 < 0; ¸4 < 0 ¤
Appendix 2 (Fixed labour supply) 1) Ricardian consumers (» = 0) Proof of Proposition 5: With » = 0, (60) turns into µ − fF 0 0 fM 0 0 0 fM J= 0 − kc (1 − °) ky − ± 0 fM 0 0 −®(µ + ®)° µ − ®(µ + ®) µ+± with the determinant given by
;
c Det(J) = (µ − f F )f M ®(µ + ®)°: k Part 1) Since there are 2 state and 2 jump variables, determinacy requires Det(J) > 0: This can never be satisÞed by active/passive combinations of policy. Part 2): Note that ¸4 = µ − f F and that the remaining root structure can be assessed from c Detc;k;¼ = f M ®(µ + ®)° k y fM T r(J)c;k;¼ = (1 − °) − ± + µ − ®(µ + ®) : k µ+± 2a) Assume f F > µ and f M < 0: Then, Detc;k;¼ < 0 and T r(J)c;k;¼ > 0; implying ¸1 > 0; ¸2 > 0; ¸3 < 0; ¸4 < 0: 2b) Assume f F < µ. Let qe > 0 denote the unique value of f M such that T r(J)c;k;¼ = 0: If f M > qe, Detc;k;¼ > 0 and T r(J)c;k;¼ < 0; then ¸1 < 0; ¸2 < 0; ¸3 > 0; ¸4 > 0:¤
2) Non-Ricardian consumers (» > 0) Proof of Proposition 6: The logic of the proof is identical to the proof of Proposition 4 in Appendix 1. For brevity, we skip the balanced budget part. From the main text, the critical matrix J (60) is given by
33
0 0 fM r − fF −»(» + µ) cl r − µ −»(» + µ) kc fM : J = y c 0 − k (1 − °) k − ± 0 M f 0 0 −®(r + ®)° r − ®(r + ®) r+±
(62)
The determinant of (62) can be written as
l Det(J) = [r − f F ][ÁF0 ix + ÁF1 ix f M ] + »(» + µ) ÁF2 ix f M ; with: c c y F ix Á0 = (r − µ)r[(1 − °) − ± − ] k k+l c y ®(r + ®)(r − µ) r + ± c { ° + − (1 − °) + ±} ÁF1 ix = r+± r−µ k k+l k c F ix Á2 = ®(r + ®)° > 0; k where we use from (31) that in any cashless steady state »(» + µ) = (r − µ)c=(k + l) holds. The slope of (34) exceeds the slope of (35) in k=n − c−space iff r+± c y c l c (r + ±) ° − (1 − °) + ± + +° > 0: r−µ k k k+l k+lk r
The critical Det(J) = 0 condition can be written as Det(J) = 0 ⇔ f F = r(1 +
ÁF3 ix f M 1 l ); with: ÁF3 ix = »(» + µ) ÁF2 ix : (63) F ix F ix M r c Á0 + Á1 f
Using »(» +µ) = (r − µ)c=(k + l) and substituting out for ÁF2 ix ; ÁF3 ix can be expressed as l c (r + ±) ®(r + ®)(r − µ) ÁF3 ix = {° } r+± k+lk r Hence, slope (34) > slope (35) ⇔ ÁF1 ix + ÁF3 ix > 0: Note that ÁF3 ix > 0 is always satisÞed, while ÁF0 ix and ÁF1 ix are linked as follows ÁF1 ix =
®(r + ®)(r − µ) r + ± c ÁF0 ix { ° − } r+± r − µ k (r − µ)r
In a balanced budget situation with l = 0, ÁF0 ix = (r−µ)r [g − °y]: Again, g < °y ⇒ k F ix F ix Á0 < 0; implying Á1 > 0: Hence, starting out from a balanced budget scenario, it follows that, as ¿ rises for a Þxed value of g; EH reaches 3 distinct conÞgurations which can be graphed in a manner that is qualitatively identical to Figure 4 (that 34
was obtained for the linear labour supply): (a) ÁF0 ix < 0; ÁF1 ix > 0; ÁF3 ix > 0; (b) ÁF0 ix > 0; ÁF1 ix > 0; ÁF3 ix > 0; (c) ÁF0 ix > 0; ÁF1 ix < 0; ÁF3 ix > 0: The trace of (62) is given by T r(J) = ! F0 ix − f F − ! F1 ix · f M ; with: y ®(r + ®) !F0 ix = 3r − µ + (1 − °) − ± > 0; ! F1 ix = > 0; k r+± implying
T r(J) = 0 ⇔ f F = ! F0 ix − ! F1 ix · f M :
(64)
Similar to the proof of Proposition 4; one can establish, by forming the matrix B F ix ; that Det(J) > 0; T r(J) < 0 implies ¸1 > 0; ¸2 > 0; ¸3 < 0; ¸4 < 0; as required for determinate dynamics. Then, the linearity of (64) and the non-linearity of (63) implies that in all 3 conceivable steady-state constellations there must exist active/passive combinations of monetary and Þscal policy as well as combinations which are active/active or passive/passive, in line with determinate dynamics. ¤
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