The Taylor principle fights back, Part II

Journal of Economic Dynamics & Control 46 (2014) 30–49

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The Taylor principle fights back, Part II Edward F. Buffie n Department of Economics, Wylie Hall 105, Indiana University, Bloomington, IN 47405, USA

a r t i c l e in f o

abstract

Article history: Received 8 August 2013 Received in revised form 11 April 2014 Accepted 21 May 2014 Available online 14 June 2014

The existing literature holds that the Taylor principle often leads to indeterminacy in New Keynesian models that allow for capital accumulation and limited asset market participation. This conclusion is special, however, to the case of continuous full employment. When the assumption of perfect wage flexibility is relaxed very slightly so that the labor market clears quickly but not instantaneously, determinacy is the norm. The threat of indeterminacy is limited to a tiny, irrelevant corner of the parameter space where the elasticity of labor supply is unusually high and real wage adjustment is unbelievably fast. Everywhere else, the Taylor principle guarantees a unique rational expectations equilibrium. The dramatic difference in results reflects the sensitivity of the monetary transmission mechanism to the speed of adjustment in the labor market. & 2014 Elsevier B.V. All rights reserved.

JEL classification: E52 E58 Keywords: Inflation Taylor principle Indeterminacy Limited asset market participation

1. Introduction Some type of Taylor rule guides the conduct of monetary policy in most developed countries and an increasing number of less developed countries. The exact form of the rule depends on the nature of information lags, the structure of the economy, and policy makers' preferences. All Taylor rules, however, share a common feature: ceteris paribus, they instruct the central bank to increase the real interest rate when inflation exceeds its target level. The rationale for the Taylor principle is that monetary policy should contract aggregate demand to neutralize incipient inflationary pressures. This requires the central bank to react aggressively, increasing the nominal interest rate more than the inflation rate. Passive policy, by contrast, accommodates adverse shocks by allowing higher inflation to reduce the real interest rate and increase aggregate demand. Taylor rules are simple and intuitively appealing, but they also carry some risk. Any rule that mechanically links policy adjustment to changes in endogenous variables may introduce indeterminacy into an economy that would otherwise have a unique equilibrium. This is thought to be an especially serious problem in models that allow for capital accumulation. Dupor (2000) proved in a continuous-time New Keynesian (NK) model with perfect substitution between government bonds and capital that the Taylor rule engenders indeterminacy. The causal explanation for indeterminacy is that higher interest rates immediately increase the capital rental and marginal costs. Since an increase in real marginal cost translates directly into higher inflation in NK models, arbitrary expectations of higher inflation prove to be self-fulfilling.

n

Tel.: þ1 8123398682. E-mail address: [email protected]

http://dx.doi.org/10.1016/j.jedc.2014.05.016 0165-1889/& 2014 Elsevier B.V. All rights reserved.

E.F. Buffie / Journal of Economic Dynamics & Control 46 (2014) 30–49

31

In continuous time the absence of adjustment costs to changes in the capital stock forces the current capital rental to move in lockstep with the real interest rate. This “extra restriction” is inconsequential. Carlstrom and Fuerst (2005) and Kurozumi and Van Zandweghe (2008) have confirmed that Dupor's results in similar models cast in discrete time.1 The central issue therefore is whether the results in all three papers are an artifact of the assumption that the elasticity of investment with respect to Tobin's q is infinite. At present the answer in the literature is a tentative no (Table 1). Although the contradictory results in Gali et al. (2004) and Carlstrom and Fuerst (2005),2 do not justify the claim that “the presence of capital makes determinacy essentially impossible” (Carlstrom and Fuerst, 2005, p.10), it is clear that the Taylor principle is problematic in a large part of the potentially relevant parameter space. The canonical NK model assumes full participation of agents in asset markets. In this model, a slight tweak of the Taylor rule may convert pessimism into optimism. GLV (short for Gali et al., 2004), Kurozumi and Van Zandweghe (2008), and Huang and Meng (2007) all show that the indeterminacy problem suddenly vanishes when the interest rate rule includes a very small positive coefficient on real output. This comforting result is sensitive, however, to the elasticity of labor supply (Huang et al., 2009). Moreover, the latest refinement to the NK model brings the indeterminacy problem back in full force. In models featuring limited asset market participation (LAMP), the determinacy region decreases steadily with the share of households that live check to check (GLV, 2004; Bilbiie, 2008) until, when the share enters the neighborhood of its estimated value,3 the Taylor principle and standard aggregate demand logic get stood on their head: higher interest rates cause aggregate demand to increase and passive policy (allowing higher inflation to reduce the real interest rate) is necessary and sufficient for a unique equilibrium. Attaching a large positive coefficient to real output in the Taylor rule does not solve the problem or even alter the threshold level of non-savers at which active policy triggers indeterminacy. The upshot of all this is that theory still withholds its endorsement; doubts about the general validity of the Taylor principle are, if anything, greater than before. In the companion paper The Taylor Principle Fights Back, Part I, I proved that the Taylor principle is valid in LAMP models without investment as long as the real wage is not exceptionally flexible. This paper completes the defense of the Taylor principle by demonstrating that the results in models with capital accumulation are also highly sensitive to small departures from the assumption of perfect wage flexibility. Aiming for exact comparisons, I investigate the outcome for alternative degrees of wage flexibility with and without LAMP. In the base case of the flex-wage full-participation model, the equilibrium is indeterminate when the q-elasticity of investment exceeds .97, a value close to the median estimate in empirical studies. The determinacy region shrinks further if some households do not save and disappears altogether when the share of non-savers exceeds the same threshold level as in Bilbiie (2008). Thus the Taylor principle struggles in pure flex-wage models. The validity of the principle may be a judgment call in the canonical NK model, but indeterminacy dominates the landscape in models that allow for LAMP. Temporary wage rigidity changes everything. In the rigid-wage full-participation model, the borderline value of the q-elasticity of investment jumps to 3–20 in runs where the labor market clears in 3–6 months. The results in GLV suggest that indeterminacy is a greater threat in LAMP models. But this is true only for the polar case of perfect wage flexibility. With high but imperfect wage flexibility, the determinacy region expands very rapidly as the share of non-saving households increases. For example, when the labor market clears in six months and 40% of households do not save, the borderline value of the q-elasticity (base case) is 39.4 vs. .35 in the pure flex-wage model. The dramatic difference reflects the sensitivity of the monetary transmission mechanism to the speed of adjustment in the labor market. Under perfect wage flexibility, higher interest rates reduce investment more the larger the share of non-saving households. The opposite relationship holds when the wage is temporarily rigid. Since sunspot equilibria stem from large near-term reductions in the capital stock that increase marginal costs, temporary wage rigidity changes LAMP from a powerful force for indeterminacy into an equally powerful force for uniqueness. This is not the only paper to analyze how temporary wage rigidity affects the viability of the Taylor principle. Two other papers, Huang et al. (2009) and Colciago (2011), carry out similar analyses in models with sticky nominal wages and capital accumulation. Colciago sets the q-elasticity at unity in a model that allows for limited asset market participation.4 Huang et al. assume full participation in asset markets but fix the q-elasticity at three. In contrast to the strong, positive results I obtain, the results in these models are discouraging: Colciago (2011) cautions that forward-looking rules ”should be implemented with care” as the determinacy region is “severely restricted with respect to the case of a contemporaneous rule,” while Huang et al. find that the determinacy region is completely empty. A lot rides therefore on the distinction between real and nominal wage rigidities; in one case the (forward-looking) Taylor principle is highly robust; in the other case it is exceedingly fragile. The rest of the paper is organized in five sections. In Section 2 I develop flex- and rigid-wage variants of a model with LAMP. Sections 3–5 quantify how the region of indeterminacy depends on the speed of real wage adjustment and

1 In a continuous-time model that treats inflation as a jump variable, i ¼ ρ þ απ is a forward-looking interest rate rule. Confusion on this point has led to confusion about the extent to which the results differ in continuous vs. discrete-time models. Under the correct interpretation of Dupor's interest rate rule, there is no contradiction between his results and those in Carlstrom and Fuerst (2005). Both conclude that indeterminacy is the norm for a forwardlooking rule. 2 Contradictory is probably not the right word. Presumably differences in calibration values and the form of the utility function explain the opposing results. 3 Macroeconomic estimates and microeconomic studies put the share of non-saving households at 30–60% in the United States and the Euro zone. See Campbell and Mankiw (1989), Mankiw (2000), Wolff and Caner (2002), Johnson et al. (2004), Muscatelli et al. (2004), Di Bartolomeo and Rossi (2007), Forni et al. (2009), and Di Bartolomeo et al. (2010). 4 Colciago does not state the value of the q-elasticity used to calibrate the model. But since the calibration is based on the parameter values in GLV, the q-elasticity probably equals unity.

32

E.F. Buffie / Journal of Economic Dynamics & Control 46 (2014) 30–49

Table 1 Determinacy results for the forward-looking Taylor rule in New Keynesian models with capital accumulation and flexible wages. Study

q-elasticity of investment

Coefficient on output in the Taylor rule

Asset market participation

Elasticity of labor supply

Determinacy region

Dupor (2000) Carlstrom and Fuerst (2005)

1 1 2 1 1 1 1 1 1

0 0 0 0 Very small 0 Very small 0 .5–1

Full Full Full Full Full Full Full Limited Limited

1 1 1 1 1 1 1 1 1

1 3, 1 3,1

Very small 0–.44 .45–1

Full Full Full

1 .1 .1

Empty Virtually empty Large Virtually empty Large Virtually empty Large Virtually empty Empty for high share of non-saving households Large Empty Empty, small, or fairly large depending on the degree of price stickiness

Kurozumi and Van Zandweghe (2008) Gali et al. (2004)

Huang and Meng (2007) Huang et al. (2009)

the degree of asset market participation. Section 6 explains, with the aid of impulse response functions, that quantitative differences in the monetary transmission mechanism underlie the different stability properties of the flex- and rigid-wage models. Section 7 concludes. 2. The model I work with a cashless deterministic model in which the real wage is either perfectly flexible or partially rigid. The economy is populated by a continuum of households indexed by i A ½0; 1, a continuum of monopolistically competitive firms that produce differentiated consumer goods indexed by j A ½0; 1, and a central bank that controls the path of the nominal interest rate. Prices are sticky à la Calvo (1983) and lump-sum taxes adjust in the background to satisfy the government budget constraint. 2.1. Households All households have an instantaneous utility function of the form 1  1=τ

Ui ¼

1 þ 1=ψ

i i Ci L  ai i ; 1 1=τi 1 þ 1=ψ i

ð1Þ

where "Z Ci ¼

1 0

ðϵ  1Þ=ϵ

cji

#ϵ=ðϵ  1Þ ;

dj

cji is the consumption of good j; Li is the labor supply; τi is the intertemporal elasticity of substitution; ϵ is the elasticity of substitution between differentiated consumer goods; and ψi is the Frisch elasticity of labor supply. The cji are chosen to minimize the cost of purchasing the composite consumption good. This yields the demand functions  ϵ pj C ð2Þ cj ¼ P and the solution for the exact consumer price index "Z #1=ð1  ϵÞ P¼

1

0

p1j  ϵ dj

;

ð3Þ

R1 R1 where cj ¼ 0 cji di and C ¼ 0 C i di. The technology for production of the composite capital good is the same as for production of the composite consumer good. Parallel to (2), new orders for investment good j are #    ϵ" pj ðI=K  δÞ2 K ; ð4Þ I þf Ij ¼ P 2 where δ is the depreciation rate and I þ f ðI=K  δÞ2 K=2 is the aggregate investment inclusive of adjustment costs incurred in changing the capital stock. A fraction λ of households are non-savers who consume all their wage income each period. These households choose labor supply Ln to maximize (1) subject to C n ¼ wLn , where w is the real wage (the nominal wage deflated by P). The first-

E.F. Buffie / Journal of Economic Dynamics & Control 46 (2014) 30–49

33

order condition for Ln reads 1=ψ n

an Ln;t

 1=τn

¼ C n;t

wt ;

or τ ψ n =ðψ n þ τn Þ

ann

ψ n ðτn  1Þ=ðψ n þ τn Þ

Ln;t ¼ wt

;

after substituting for Cn. The other group of households, savers, solves the problem " 1  1=τ # 1 þ 1=ψ s s 1 L t C s;t  as s;t max ∑ β ; 1  1=τs 1 þ 1=ψ s t¼0

ð5Þ

ð6Þ

subject to  2 Bt þ 1 f I s;t Bt þ C s;t þI s;t þ  δ K s;t ¼ wt Ls;t þr K;t K t þg t þ Rt  1 ; 2 K s;t Pt Pt

ð7Þ

K s;t þ 1 ¼ I s;t þ ð1  δÞK s;t :

ð8Þ

where B is the nominal stock of government bonds; β is the discount factor; K is the capital stock; rK is the real capital rental; g is the sum of lump-sum taxes and rebated monopoly profits; and R is the gross nominal interest rate. On an optimal path,  1=τs

C s;t

1=ψ s

as Ls;t

 1=τ

¼ βRt C s;t þ 1s P t =P t þ 1 ;  1=τ s

¼ C s;t

wt ;

    2   Rt P t I s;t f I s;t þ 1 I s;t þ 1 1þf δ δ þf δ ¼ r K;t þ 1 þ1  δ þ 2 K s;t þ 1 Pt þ 1 K s;t K s;t þ 1

ð9Þ ð10Þ ð11Þ

Eq. (9) is the familiar Euler equation, while Eqs. (10) and (11) state that the marginal rate of substitution between consumption and leisure equals the real wage and that the capital rental net of adjustment costs and depreciation equals the real interest rate. GLV and Bilbiie (2008) structure their models so that Ln is constant and Ls ¼ Ln at the initial equilibrium. I follow suit, setting τn ¼ 1 and choosing as and an so that Ls ¼ Ln ¼ 1. To economize on notation, henceforth ψ and τ refer to the elasticity of labor supply and the intertemporal elasticity of substitution of savers.5 2.2. Aggregation The aggregate capital stock and the aggregate investment are simply K t ¼ K s;t ð1  λÞ;

ð12Þ

I t ¼ I s;t ð1  λÞ:

ð13Þ

Aggregating labor supply and consumption across households is equally straightforward. Let Lna denote the actual employment of non-savers. (Lna ¼ Ln only when the real wage is at its equilibrium level.) Since Ln ¼ 1 and C n ¼ wLna , total labor supply and total consumption are Lt ¼ ð1  λÞLs;t þ λ;

ð14Þ

C t ¼ ð1  λÞC s;t þ λwt Lna;t :

ð15Þ

2.3. Firms Firms hire capital and labor in competitive economy-wide markets. Assuming that constant returns technology, the real unit cost function for each differentiated good is Hðw; r K Þ. By Shepherd's lemma,

5

Lj;t ¼ H w ðwt ; r K;t Þðcj;t þI j;t Þ;

ð16Þ

K j;t ¼ H r ðwt ; r K;t Þðcj;t þ I j;t Þ:

ð17Þ

If the assumption that τs a τn is considered awkward, then set ψ n ¼ 0 in (4).

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E.F. Buffie / Journal of Economic Dynamics & Control 46 (2014) 30–49

Prices change when firms receive an informative signal about the state of the market. Each period a fraction 1  ω of firms receive the requisite signal. The average price quote lasts ð1  ωÞ  1 periods and the probability of receiving a new signal is independent of other firms and the duration of the current price quote. Upon receiving a signal, the firm chooses its price so as to maximize the present discounted value of real profits. Real profits per period are ½pj =P  Hðw; r K Þðcj þ I j Þ ¼ ½pj =P Hðw; r K Þðpj =PÞ  ϵ Q , where Q t ¼ C t þ I t þf ðI t =K t  δÞ2 K t =2. z The discount factor, β ðC s;t þ z =C s;t Þ  τ , incorporates the marginal rate of substitution of the firm's owners (savers), and future profits are weighted by ωz (the probability that the price chosen at time t will still be in force at time z). Thus    τ "    # 1 pj;t 1  ϵ pj;t  ϵ z C s;t þ z  Ht þ z ð18Þ Q t þ z: pj;t ¼ arg max ∑ ωz β C s;t Pt þ z Pt þ z z¼0 In a symmetric equilibrium, all firms that receive a signal choose the same price. The optimal pt and the overall price level satisfy τ ϵ z pt ϵ ∑1 z ¼ 0 ω β C s;t þ z H t þ z ðP t þ z =P t Þ Q t þ z ¼ z τ ϵ1 z P t ϵ 1 ∑1 Qtþz z ¼ 0 ω β C s;t þ z ðP t þ z =P t Þ

ð19Þ

P 1t  ϵ ¼ ð1  ωÞp1t  ϵ þ ωP 1t 1ϵ :

ð20Þ

z

and

2.4. Monetary policy There are many plausible interest rate rules. For simplicity, I restrict the analysis to the pure forward-looking rule: Rt ¼ RðP t þ 1 =P t Þα ;

ð21Þ

where R is the steady-state gross nominal interest rate. The above rule assumes that the central bank targets zero inflation. When α 4 1, monetary policy abides by the Taylor principle. 2.5. Aggregate factor demands and real wage adjustment After summing over firm demands in (16) and (17) and substituting for cj and Ij from (2) and (4), we have "Z #" # 1   ðI t =K t  δÞ2 Kt ; ðpj;t =P t Þ  ϵ dj C t þI t þ f K t ¼ H r wt ; r K;t 2 0 

Ld;t ¼ H r wt ; r K;t



"Z

1 0

#" ðpj;t =P t Þ

ϵ

dj

ðI t =K t  δÞ2 Kt C t þI t þ f 2

ð22Þ

# ð23Þ

where Ld denotes the aggregate employment. In the main variant of the model, the real wage adjusts gradually in the direction of its market-clearing level. Following Blanchard and Gali (2007), L Lt wt  wt  1 ¼ v d;t ; wt  1 Lt

v 4 0;

or wt ¼ wt  1 þ vwt  1

Ld;t Lt ; Lt

v 4 0;

ð24Þ

where the parameter v reflects the weight of outstanding contracts and the state variable xt ¼ wt  1

ð25Þ 6,7

tracks the inertial component of wage adjustment. The rationale for the specification in (24) is threefold. First, as Blanchard and Gali (2007) show, a similar specification can be derived from Calvo-type staggered real wage setting. Second, the empirical evidence for real wage rigidity is extensive and, if anything, greater than that for nominal wage rigidity; moreover, the evidence strongly supports the specification in (24) while casting doubt on insider-outsider and search-theoretic models of unemployment.8 Third, given 6 Blanchard and Gali use mrs  w (where mrs is the marginal rate of substitution between leisure and consumption) to measure excess supply in the labor market. I consider the specification in (24) to be slightly more natural. 7 Eq. (24) assumes that workers temporarily supply overtime hours when labor demand exceeds labor supply. Replacing vðLd;t  Lt Þ=Lt in (24) with v½un  ðLt  Ld;t Þ=Lt , where un is the natural rate of unemployment, gives the same results while ensuring Ld o L. See Blanchard and Gali (2007). 8 See Buffie (2013) for a review of empirical studies of different types of wage rigidities.

E.F. Buffie / Journal of Economic Dynamics & Control 46 (2014) 30–49

35

the sharp disagreements among macroeconomists about the degree of real wage flexibility, it is desirable to work with a general specification that can be finely calibrated to different speeds of adjustments in the labor market. Eq. (24) meets this criterion: it can accommodate any degree of real wage flexibility, including the polar cases of completely rigid (v ¼0) and perfectly flexible (v ¼ 1) wages.9 2.6. The flex-wage model For purposes of comparison, I need the solution for the pure flex-wage model. In this case, Eq. (24) is replaced by Ld;t ¼ Lt

ð26Þ 10

and Lna ¼ 1 in Eq. (15).

3. Preliminary solutions This section prepares the ground for the stability analysis by linearizing the model around the zero-inflation stationary equilibrium. Since lower-case letters have already been assigned to the real capital rental, the real wage, etc., I use a hat to signify the log deviation of a variable from its steady-state value. The logical starting point is the firm's pricing equation. Log-linearizing (19) and (20) leads to the familiar NK Phillips Curve:

π t ¼ βπ t þ 1 þ κ H^ t ;

ð27Þ

where π is the inflation rate and κ  ð1  ωβÞð1  ωÞ=ω. Consider next the dynamics for spending and the capital stock. The linearized versions of (9) and (15) are C^ s;t þ 1 ¼ C^ s;t þ τðα 1Þπ t þ 1 ;

ð28Þ

 C s λw ^ ^ w t þ L na;t : C^ t ¼ 1  λ C^ s;t þ C C

ð29Þ

To link Is and Ks to I and K, use (12) and (13) to rewrite (8) and (11) as     2   Rt P t It f It þ 1 It þ 1 1þf δ δ þf δ ; ¼ r K;t þ 1 þ1  δ þ 2 Kt þ 1 Pt þ 1 Kt Kt þ 1 K t þ 1 ¼ I t þ ð1  δÞK t :

ð30Þ ð31Þ

We need pseudoreduced-form solutions that relate marginal cost, labor demand, non-savers employment (Lna), labor supply, and capital rental to the aggregate consumption, the aggregate investment, the aggregate capital stock, and the real wage. Eqs. (9), (14), and (29) provide the intermediate solution for labor supply:   λw ψλw ^ ψC ^ L^ t ¼ 1  λ þ L  ψ w^ t þ ð32Þ C ; τC s τC s na;t τC s t while (22), (23), and direct differentiation of the unit cost function give11,12 ^ tþ r^ K;t ¼ w

K^ t C ^ I ^ ; Ct þ It  Q σθL Q σθL σθL

  C I^ ^ t þ C^ t þ I; L^ d;t ¼ σθK r^ K;t  w Q Q C ^ I ^ θK ^ It  K t; ⟹L^ d;t ¼ Ct þ θL Q θL Q θL

ð33Þ

ð34Þ

9 In the older literature on wage rigidity [see Blanchard and Fischer, 1989, Section 10.5 in Chapter 10], it was common to treat the real wage as a state variable that does not respond contemporaneously to the unemployment rate. (The assumption is unavoidable in models framed in continuous time.) The specification wt þ 1 ¼ wt þ vwt ðLd;t  Lt Þ=Lt generally yields stronger results (i.e., results more favorable to determinacy) than (24) because the real wage is completely rigid in the first period. The drawbacks of the specification are twofold: (i) it does not nest the pure flex-wage model and (ii) in some runs where vψ is large, the lagged response of the real wage to unemployment results in an unstable cobweb-type adjustment cycle. 10 The specification in (24) nests the pure flex-wage model. It is easier, however, to solve the full-employment case using (26) than by solving the general model and evaluating the solution at v ¼ 1. 11 The Allen–Uzawa partial elasticity of substitution between factors i and j is σ ij ¼ H ij H=Hi Hj . This formula and the adding-up conditions σ LK θK þ σ LL θL ¼ σ KL θL þ σ KK θK ¼ 0 are used to derive the solutions in Eqs. (24) and (25) (where σ corresponds to σ LK ). R1 12 The term that multiplies Hr in (22) and Hw in (23) is Q 1  0 ðcj þ I j Þ dj. It is readily shown that dQ 1 =Q 1  dQ =Q is second-order small when linearization occurs around a steady state with zero inflation (where there is no relative price dispersion).

36

E.F. Buffie / Journal of Economic Dynamics & Control 46 (2014) 30–49

^ t þ θK r^ K;t ; H^ t ¼ θL w ^ tþ ⟹H^ t ¼ w





θK C ^ I ^ ^ C þ I Kt ; θL σ Q t Q t

ð35Þ

where Q  C þ I (evaluated at a steady state), θj denotes the cost share of factor j, and σ is the elasticity of substitution between capital and labor. Finally, L^ na depends on how job losses (or temporary overtime hours) are divided between savers and non-savers in periods where aggregate labor demand is less (greater) than aggregate labor supply. Although it is not necessary for most of the ensuing results, I assume that employment losses are proportional to the group's share in total labor supply [i.e., λL^ na;t ¼ λðL^ d;t  L^ t Þ].13,14 Eqs. (32) and (34) then deliver   λ C ^ I θK ^ ; k1 C t þ I^t  K^ t  k2 w λL^ na;t ¼ ð36Þ wð1 þ μÞ ko w θL where μ  1=ðϵ 1Þ is the markup rate and15 ko  1 þ λwψ =τC s ; k1  1=ð1 þ μÞ þ ψ w=τC s ;

k2  ψ ð1  λ þ λw=τC s Þ:

3.1. Linking adjustment costs to the q-elasticity of investment The first-order condition for investment reads 1 þ f ðI s;t =K s;t  δÞ ¼ ϕ2;t =ϕ1;t  qt ; where ϕ1 and ϕ2 are the multipliers attached to the constraints in Eqs. (7) and (8). The ratio ϕ2;t =ϕ1;t is Tobin's q, the ratio of the shadow price to the supply price of capital.16 The elasticity of investment with respect to q is Ω  I^ t =q^ t ¼ 1=f δ, evaluated at a steady state (where I ¼ δK and q ¼1). The higher values of Ω correspond therefore to lower values of the adjustment cost parameter f. 4. The flex-wage solution In the flex-wage model, L^ na;t ¼ 0. Hence Eqs. (26), (29), (32), and (34) yield C I θ ^ t ¼ e1 C^ t þe2 I^t  K K^ t ; w w w θ L k2 C^ s;t ¼

1  λe1 ^  λIe2 ^ λwθK Ct  It þ K^ t ; ð1  λÞC s =C ð1  λÞC s ð1  λÞC s θL k2

where e1  k1 =k2 and e2  1=k2 ð1 þ μÞ. Substituting for H^ t in (27) and for C^ s in (28) now produces       π κ C e1 θK ^ κ I e2 θK ^ κθK 1 1 ^ πt þ 1 ¼ t  þ þ þ Ct  It þ K ; β βQ 1 þ μ θL σ βQ 1 þ μ θL σ βθL k2 σ t τðα 1Þð1  λÞC s =C λe I=C ^ λθK w=C  ^ I t þ 1  I^t  K πt þ 1 þ 2  K^ t : C^ t þ 1 ¼ C^ t þ 1  λe1 1  λe1 θL k2 ð1  λe1 Þ t þ 1 From (30), (31), (33), and (37),         rK C e1 1 ^ rK I e2 1 rK θK 1 I^t þ 1 þ Rf δI^ t  Rf δK^ t þ f δ þ K^ t þ 1 ; þ þ þ C t ¼ Rðα  1Þπ t þ 1  f δ þ θL Q 1 þ μ θL σ θL Q 1 þ μ θL σ θL k2 σ K^ t þ 1 ¼ δI^ t þ ð1  δÞK^ t :

ð37Þ ð38Þ

ð39Þ

ð40Þ

ð41Þ ð42Þ

13 This is the most natural assumption in which new hires are distributed randomly among the unemployed. (Firms do not favor one group over the other.) It is also required in order for the model that treats the real wage as a jump variable to nest the pure flex-wage model. LAMP models with Calvo-staggered nominal wages use the same rule for allocating scarce jobs. The models postulate that an optimizing union maximizes a weighted average of the utilities of savers and non-savers. Since the weight assigned to each group's utility equals its share in total labor supply, the share of each group in total employment also equals its share in total labor supply. See Ascari et al. (2010) and Colciago (2011). 14 Ωn , the threshold value of the q-elasticity of investment spending compatible with determinacy, is lower when employment losses fall disproportionately on non-savers. Results for this case are available from the author on request. They do not differ substantively from the results presented in Section 5.2. (See footnotes 27 and 28.) 15 In deriving (36), I have made use of the fact that θL ¼ wð1þ μÞ=Q evaluated at a steady state. 16 ϕ2 is the shadow price of capital measured in utils. Dividing by ϕ1 converts the shadow price into a price measured in dollars. The real supply price of capital equals unity.

E.F. Buffie / Journal of Economic Dynamics & Control 46 (2014) 30–49

1.4

1.4

1.2

1.2

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

37

0.2 0.5

1.0

1.5

2.0

2.5

3.0

ψ

0.3

1.2

1.2

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2 0.5

1.0

1.5

τ

2.0

0.2

1.5

1.5

1.0

1.0

0.5

0.5

1.5

2.0

2.5

3.0

3.5

4.0

α

0.4

0.4

0.4

0.6

0.5

0.8

0.6

1.0

ω

0.7

1.2

σ

1.4

0.5

0.6

0.7

0.8

0.10

0.15

0.20

0.25

θL

1.0

3.0 2.5

0.8

2.0

0.6

1.5 0.4 1.0 0.2

0.5 0.970

0.975

0.980

0.985

0.990

β

0.05

μ

Fig. 1. Borderline value of Ω in the flexwage model with full asset market participation.

Eqs. (39)–(42) comprise a self-contained system of difference equations in which C, π, and I are jump variables and K is a state variable. The stationary equilibrium is a saddle point iff one of the system's four eigenvalues lies inside the unit circle. Before moving on, it is worthwhile to draw attention to one feature of the solution in (40). Observe that an increase in the real interest rate increases consumption when 1  λe1 o 0, or, equivalently, when

λ 4 λn 

ψ ð1 þ μÞ : 1 þ ψ ð1 þ μÞ

This is Bilbiie's (2008) condition for “inverted aggregate demand logic.” As will become apparent in Section 4.2, it is also a necessary and sufficient condition for inversion of the Taylor principle. Thus one-half of Bilbiie's results generalize exactly to the model with capital accumulation. What does not generalize is the conclusion that the Taylor principle is unconditionally

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E.F. Buffie / Journal of Economic Dynamics & Control 46 (2014) 30–49

valid for λ o λ . In models with capital accumulation and perfectly flexible wages, the equilibrium path may be n indeterminate even when λ is far below λ . n

4.1. The full-participation case I start by analyzing the outcome in the standard NK model that assumes continuous full employment and positive saving by all agents. The results in this case constitute the benchmark for evaluating the effects of LAMP and imperfect wage flexibility. In the standard model and the more complicated models examined in later sections, many parameters play a role in determining the outcome. One parameter, however, the q-elasticity of investment spending (Ω), is very much primus inter pares. To see why, return to the solution for real marginal cost in (35)   θ C^ I^ ^ ^ tþ K H^ t ¼ w ð35Þ Ct þ It K t ; Q θL σ Q and suppose the central bank counters an increase in expected inflation by increasing the real interest rate. In the short run, real output (C þ I) and marginal cost decline. Ceteris paribus, this eases inflationary pressures. But when Ω is large, the short run is quickly followed by decreases in the capital stock that increase marginal cost. With Calvo pricing, the impact on inflation today depends on the entire future path of marginal cost. Consequently, if Ω exceeds some critical value, inflation will increase immediately even though real output and current marginal cost decline. This completes the circle of indeterminacy; inadvertently, the Taylor principle validates expectations of higher inflation unrelated to changes in economic fundamentals. Although the q-elasticity commands the spotlight, it does not work alone. Other parameters are important to the extent that they condition the impact of the capital stock on marginal cost. In the standard model, the algebra simplies just enough to permit identification of the supporting cast and its contribution to the indeterminacy problem. For λ ¼ 0,

  1 1 θK H^

þ : dC;dI ¼ 0 ¼ 

ψ σ θL K^ Low values for ψ and σ and a high value for θK increase the risk of indeterminacy by increasing the elasticity of real marginal cost with respect to the capital stock. It is easy to break down the solution and isolate the influence of each parameter. When the capital stock decreases, the capital rental rises and firms substitute labor for capital in production. The increase in labor ^ Thus the induced increase in the market-clearing demand is L^ d ¼  ðθK =θL ÞK^ , while the increase in labor supply is L^ ¼ ψ w. ^ ¼  ðθK =θL ψ ÞK^ , is larger the higher the cost share of capital and the lower the elasticity of labor supply. In partial wage, w equilibrium, the increase in rK required to clear the market for capital goods depends on the cost share of labor and the curvature of the production isoquant as measured by σ: r^ K jdw ¼ 0 ¼  K^ =σθL . After taking into account the feedback effect of a higher wage on the demand for capital, the general equilibrium solution reads r^ K ¼  ðθK =ψ þ 1=σ ÞK^ =θL . The total increase ^ þ θK r^ K ¼  ð1=ψ þ 1=σ ÞðθK =θL ÞK^ . in marginal cost is therefore H^ ¼ θL w 4.1.1. Numerical solutions n Fig. 1 shows how Ω , the borderline value of Ω that separates the region of unique equilibria from the region of indeterminacy, depends on the various parameters that enter into the core dynamic system. In the base case,

δ ¼ :015;

θL ¼ :72;

β ¼ :99;

τ ¼ ψ ¼ σ ¼ 1;

ϵ ¼ 6;

ω ¼ :75;

α ¼ 1:5:

17

The annual depreciation rate is 6% and the cost share of labor is 72%. The other seven parameters take the same values as in GLV. A quick scan of Fig. 1 encounters steep slopes, flat slopes, and a mix of small and big numbers on the vertical axes. Four results stand out:

 As expected, Ωn is highly sensitive to the elasticity of labor supply. In the base case where ψ ¼ 1, Ωn is .97. But when ψ ¼ :1 – the midpoint of empirical estimates informed by microeconomic data – Ωn drops to .19. Going in the other direction and imposing ψ ¼ 1 as in Dupor (2000), Carlstrom and Fuerst (2005), and Kurozumi and Van Zandweghe n (2008) yields Ω ¼ 1:83, a value above most empirical estimates of the q-elasticity of investment (Table 2).  Technology matters. Lowering the elasticity of substitution between capital and labor from unity to one-half reduces Ωn from .97 to .65 in the base case. In scenarios with temporary wage rigidity, the solutions diverge much more. Given this and the weight of the empirical evidence, which now favors σ ¼.4–.6 in developed countries (Klump et al., 2007; 17 With a markup rate of 20%, a cost share of 72% for labor translates into an income share of 60%. This is close to labor's share in GDP (59%). A depreciation rate of 6% is in line with empirical estimates of the depreciation rate for physical capital (Blundell et al., 1992; Nadiri and Prucha, 1996) and with data on service lives of equipment and structures reported by the Bureau of Economic Analysis (Musgrave, 1992). Papers that use a higher depreciation rate of 10% explicitly or implicitly count consumer durables as part of investment. While this is conceptually correct, it is inappropriate in a model that focuses on the feedback effects running from changes in the physical capital stock to marginal cost and inflation.

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39

Table 2 Estimates of the elasticity of investment with respect to Tobin's q (Ω). Studya

Ωb

Craine (1975) Engel and Foley (1975)c

.94 .78–.87 2–2.3 .3–.5 .1–.3 .30–1.02 .71 0–1.85 .28–.32 .89–1.33 1.50–2.98 .25 .36–.65 .20

Abel (1980) Abel and Blanchard (1986)d Summers (1981) Hayashi (1982) Malkiel et al. (1979) Schaller (1990) Galeoti and Schiantarelli (1991) Eberly (1997) Blundell et al. (1992)e Erickson and Whited (2000) Smets and Wouter (2003)

a See Schaller (1990) and Chirinko (1993) for the results in other studies, most of which estimate Ω to be smaller than unity. b In studies that estimate the adjustment cost parameter, Ω is computed at I ¼ δK and δ ¼ .06. c The two ranges given for Ω are for equipment and non-residential structures. d The elasticity with respect to the marginal profit component of q exceeds unity, however. e Short-run elasticity.





Chirinko et al., 2009) and is highly mixed in LDCs, it is preferable to employ a flexible specification of technology. The reflex assumption of Cobb–Douglas technology is not innocuous. Ωn is highly insensitive to the intertemporal elasticity of substitution (τ), the markup rate (μ), the degree of price stickiness (ω), and the Taylor coefficient. This reflects the fact that the prospects for determinacy turn mainly on the magnitude of the elasticity of real marginal cost with respect to the capital stock. Since none of the aforementioned n parameters affect this key elasticity, none strongly affect Ω . n The sensitivity of Ω to β and θL suggests that the severity of the indeterminacy problem may differ across the development spectrum. Whether the problem is judged more or less severe in LDCs depends partly on one's priors. Doubtless the time preference rate (1=β) is higher in LDCs, but there is disagreement about the reliability of national income accounts data that purport to show that θL is significantly lower in less developed than in developed countries.18 Overall, the Taylor principle seems to be more robust with LDC parameter values.19

4.2. Limited asset market participation Fig. 2 confirms the thesis of Gali et al. that LAMP greatly increases the likelihood of indeterminacy. Ω decreases n monotonically with λ, and for λ 4 λ (.545 in the base case) the equilibrium path is indeterminate for any positive value of Ω. (Even at conservative estimates of λ, the region of indeterminacy is much smaller than in the full-participation model. In the n base case, Ω decreases from .97 to .35–.54 at λ ¼ :3–:4.) This result has several important corollaries. Most notably, there is no longer any presumption that the Taylor principle is more robust in LDCs. Since higher values for λ counterbalance the effects of lower β, the determinacy region could be smaller than in developed countries. In the run with β ¼ :975, for n example, Ω plunges from 2.49 at λ ¼ 0 to just .29 at λ ¼ :50. All the runs in Fig. 2 assume active policy (the Taylor coefficient is 1.5). Under passive policy, the equilibrium is n n indeterminate for λ o λ but unique for λ 4 λ . Thus Bilbiie's (2008) condition for full inversion of the Taylor principle n generalizes to the flex-wage model with capital accumulation. It should be emphasized in this connection that λ is not n 20 particularly large. Many empirical studies place ψ between .20 and .50. The associated range for λ is .19–.37, below the consensus best guess for λ in the United States (.4), let alone the higher λ likely to prevail in LDCs.21 Summing up, the Taylor principle is struggling to survive in Fig. 2. If the principle is not dead, it has at least one foot in the grave. n

18 The United Nation's National Income Accounts data understate labor's share in income in LDCs. Gollin (2002) contends that after correcting for the bias labor's share is similar to that in developed countries. On the other hand, firm-level data from industrial censuses suggests that labor's share could be 10–20 percentage points lower than in developed countries. n 19 With β ¼ :975 and θL ¼ :55–:72, for example, Ω ranges from 1.19 to 2.49 vs. .97 in the base case (where β ¼ :99 and θL ¼ :72). 20 See Mankiw et al. (1985), Pencavel (1986), Altonji (1986), Card (1994), and Smets and Wouter (2003). 21 There is not much hard data on the share of non-savers in LDCs. A recent survey (Steadman Group, 2009), however, found that 60% of households in Uganda do not have access to a financial institution.

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E.F. Buffie / Journal of Economic Dynamics & Control 46 (2014) 30–49

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.1

0.2

0.3

0.4

0.5

0.6

λ

0.7

0.0

0.1

0.2

Base case

0.6

0.25

0.5

0.20

0.4

0.15

0.3

0.10

0.2

0.05

0.1 0.2

0.3

0.4

0.4

0.5

0.6

0.7

0.5

0.6

0.7

0.5

0.6

0.7

λ

ψ = .5

0.30

0.1

0.3

0.5

0.6

0.7

λ

0.1

0.2

ψ = .2

0.3

0.4

λ

σ = .5 1.2

2.5

1.0

2.0

0.8

1.5

0.6 1.0

0.4

0.5

0.2 0.1

0.2

0.3

0.4

.975

0.5

0.6

0.7

λ

0.1

0.2

0.3

0.4

λ

.975 and ӨL .55

Fig. 2. Borderline value of Ω in the flexwage model with limited asset market participation.

5. Temporary wage rigidity Recall that wt ¼ wt  1 þ vwt  1

xt ¼ wt  1

Ld;t Lt ; Lt

v 4 0;

ð24Þ

ð25Þ

when the real wage is temporarily rigid. The same solution procedure as in Section 4 leads to a system of five first-order difference equations in C, I, π, K, and x (see Appendix A). Since the core dynamic system now includes x as a second state variable, saddlepoint stability requires two eigenvalues inside the unit circle. The numerical runs in this section allow v to vary from .25 to 3. This range encompasses slow, extremely fast, and intermediate speeds of adjustment in the labor market. The times required for 80% adjustment to an exogenous permanent

E.F. Buffie / Journal of Economic Dynamics & Control 46 (2014) 30–49

41

shock to labor demand are22 Time to 80% adjustment ðyearsÞ

v ¼ :25

v ¼ :5

v¼1

v¼3

1:80

:99

:58

:29

It should be emphasized here that the empirical evidence disputes the label “intermediate speed of adjustment” for v¼ .5–1. Runs with v ¼.5 imply that the real wage decreases at an annual rate of 2% in response to a one percentage point increase in the unemployment rate. This is 2–4 times larger than the response found in the great majority of empirical studies, and slightly higher than the largest estimated response.23 The decision to include runs for v Z:5 is a debating tactic. I want to show that the Taylor principle survives even when the value assigned to v is unrealistically large and strongly biases the results against determinacy. I proceed as in Section 4, presenting results for the full-participation case before examining the outcome under LAMP. 5.1. The full-participation case Several new effects come into play when the wage is temporarily rigid. Given the results in Buffie (2013), it is to be expected that the determinacy region will expand in the case of full asset market participation. What is striking in Fig. 3 and Table 3 is how much it expands. The borderline value of Ω exceeds 15 in 35 of 42 cases where v r1. Nor are the numbers n invariably small when v ¼3 and the labor market clears very quickly; when ψ ¼ :5, for example, Ω ¼ 14:3 vs. .66 in the pure flex-wage model. Despite the many large numbers in Table 3, it is not clear whether the results support any firm policy conclusions. The problem is that macroeconomists have extraordinarily diffuse priors over both v and Ω. The weak co-variation of real wages and employment over the business cycle can be interpreted either as evidence of persistent involuntary unemployment and low values of v or as evidence of elastic intertemporal substitution of leisure and high values of v (although micro-level studies strongly favor the first interpretation). Econometric estimates of Ω cluster between .3 and 1.5 (see Table 2), but three is a popular choice for calibration, and values of five or higher are not uncommon.24 In light of the disparate views in the literature, I propose two strict tests for validity of the Taylor principle: Test 1 ðT1Þ: Ω Z10 when v r 1; n

Test 2 ðT2Þ: Ω Z10 when v ¼ 3: n

Unfortunately, these tests result in a lot of failing grades. The pass rate is 86% for T1 but a woeful 29% for T2. The progress relative to the flex-wage solution is substantial; it is insufficient, however, to eliminate disagreement about the general validity of the Taylor principle. 5.1.1. Raising the pass rates More active monetary policy delivers higher pass rates. The drawback of the solution is that fully satisfactory scores obtain only at very high values of the Taylor coefficient. Increasing α from 1.5 to 2 raises the pass rate to 98% on T1, but seven runs still fail T2 (Table 4). The pass rate jumps to 86% for a Taylor coefficient of three, but this requires unrealistically large increases in the real interest rate to combat inflation. There is another, much better way to increase the pass rates. In a flexible IT regime, the central bank targets current output as well as inflation25: R^ t ¼ απ t þ 1 þ ηQ^ t ;

  C^ I ⟹R^ t ¼ απ t þ 1 þ η C t þ I^t ; Q Q

η 40:

ð43Þ

Flexible IT fails to resolve the indeterminacy problem in flex-wage models when labor supply is inelastic and/or price quotes are short-lived. (See Table 1 and the discussion in the introduction.)26 But this limitation is unique to pure flex-wage models. 22 Let wo and wn denote the initial and the new equilibrium wage. In the base case of the full-participation model, Eq. (24) then yields t ¼ lnð:20Þ=ln½1=ð1 þ vÞ as the time at which w has traveled 80% of the distance from wo to wn . 23 Table 3 in Buffie (2013) summarizes the findings of empirical studies that estimate the response of the real wage to unemployment. 24 The values assumed for Ω in Dotsey (1999), GLV, 2004, Sveen and Weinke (2005), Dupor (2002), Woodford (2003), and Baxter and Crucini (1993) are .25, 1, 3, 5, 13.3 and 15, respectively. King and Watson (1995) observe that proponents of RBC models typically ignore adjustment costs or assume a very large value for the q-elasticity, while Ω ¼ 1 is frequently seen in sticky-price models. The decision of many authors to set Ω above two presumably reflects the view that empirical estimates of the q-elasticity are unrealistically low. Shapiro (1986), Schaller (1990), Chirinko (1993), and Erickson and Whited (2000) argue that many estimates are biased downward by aggregation of heterogeneous firm adjustment costs and by measurement error in the proxies for Tobin's q. 25 I am indebted to an anonymous referee for the suggestion to analyze the case of flexible IT. 26 Let ω ¼ :33, ψ ¼ :20 and the other parameters take their values in the base case. The equilibrium is then indeterminate for η o :74 when Ω ¼ 1 and for η o 1:39 when Ω ¼ 3. The threshold value of η is also highly sensitive to small departures from the assumption of full asset market participation. In runs with λ ¼ :25; Ω ¼ 1, and ψ ¼ :20, the equilibrium is indeterminate for η o 3:92 when ω ¼ :33 and for η o 1:92 when ω ¼ :50.

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E.F. Buffie / Journal of Economic Dynamics & Control 46 (2014) 30–49

50

40

30

20

10 Flex wage

0.5

1.0

1.5

2.0

2.5

3.0

v

Fig. 3. Borderline value of Ω with temporary wage rigidity and full participation in asset markets (base case). Table 3 Borderline value of Ω with full asset market participation and temporary wage rigidity. v

Scenario

Base casea ψ¼ .20 ψ¼ .50 ψ¼ 2 ψ¼ 3 ω¼ .33 ω¼ .5 ω¼ .83 τ¼ .20 τ¼ .50 σ¼ .50 θL ¼ .60 LDC1b LDC2c a b c

Flex-wage

.25

.5

1

3

52.0 87.0 71.9 37.7 16.9 83.7 94.9 26.8 48.6 49.9 18.1 25.6 47.6 18.9

40.7 101.6 72.8 25.2 8.9 63.9 122.0 14.9 35.7 37.7 15.7 21.6 34.5 15.3

20.4 90.5 52.4 7.3 4.9 46.0 120.6 6.9 15.0 17.2 9.3 11.8 18.7 9.1

5.4 43.1 14.3 3.0 2.5 27.4 59.2 2.6 3.5 4.2 3.2 3.3 7.0 3.6

.97 .34 .66 1.27 1.42 .97 .97 .97 .92 .95 .65 .57 2.49 1.19

In the base case, τ ¼σ ¼ψ ¼1, θL ¼ .72, β ¼ .99, ϵ ¼6, α ¼ 1.5, ω¼ .75, and δ¼ .015. Run with β ¼.975. Run with β ¼.975 and θL ¼ .55.

Troublesome cases disappear when real wage adjustment is fast but not instantaneous. Keeping the Taylor coefficient at 1.5 and assigning a value of .1 to η raises the pass rate to 100% for both T1 and T2. 5.2. Limited asset market participation In the model with perfect wage flexibility, the determinacy region shrinks rapidly as the share of non-saving households increases. Because of this, the Taylor principle found itself in deep trouble at the end of Section 4. For shares of non-savers suggested by the data, the determinacy region either did not exist or was confined to a small part of the parameter space where the q-elasticity of investment was .30 or less (see Fig. 2). If there was an equally strong inverse relationship between LAMP and determinacy in the model with temporary wage n rigidity, the Ω  v schedule in Fig. 3 would move much closer to the origin and the failure rate in tests T1 and T2 would be much higher. But this is emphatically not the case. It turns out that the relationship between determinacy and asset market participation is the exact opposite of the relationship depicted in Fig. 2: under temporary wage rigidity, the size of the determinacy region is strongly increasing in the share of non-saving households. Fig. 4, and Tables 5 and 6 compare the n solutions for Ω in the flex-wage and rigid-wage models. Remarkably, when λ ¼ :4–:5 – the estimated range for both the United States and the Euro zone27– all but two runs pass T1.28 A sizeable minority (39%) fail T2, but this problem is readily solved either by adding a half point to the Taylor coefficient (Table 7) in the case of strict IT or by assigning a small value to 27 See Campbell and Mankiw (1989), Mankiw (2000), Wolff and Caner (2002), Johnson et al. (2004), Muscatelli et al. (2004), Di Bartolomeo and Rossi (2007), Forni et al. (2009), and Di Bartolomeo et al. (2010). 28 The results are robust to the assumption that job losses for non-savers are proportional to their share in total labor supply. In the base case in n Table 6, for example, Ω decreases from 23.9–51.7 to 19.1–41.6 when λL^ na;t ¼ gλðL^ d;t  L^ t Þ and g increases from 1 to 1.5.

E.F. Buffie / Journal of Economic Dynamics & Control 46 (2014) 30–49

43

Table 4 Borderline value of Ω with full asset market participation and temporary wage rigidity: v¼ 3 and either (i) α¼ 2–3 strict IT or (ii) η ¼.1 for flexible IT.a Strict IT

Base caseb ψ¼ .20 ψ¼ .50 ψ¼ 2 ψ¼ 3 ω¼ .33 ω¼ .5 ω¼ .83 τ¼ .20 τ¼ .50 σ¼ .50 θL ¼ .60 LDC1c LDC2d

Flexible IT

α ¼2

α¼ 3

η ¼ .1

12.4 P P 5.0 3.7 P P 4.6 8.2 9.9 6.4 7.4 12.7 6.4

35.0 P P 10.1 6.2 P P 9.9 28.5 31.2 14.3 19.5 27.6 13.3

U U U U U U U U U U U U U 52.5

a α is the Taylor coefficient. P is entered when the run already passes the Strong Test for validity of the Taylor principle in Table 1 (where α¼ 1.5). In the solutions for flexible IT, U for unique is entered when the threshold value of Ω exceeds 100. b In the base case, τ ¼σ ¼ψ ¼1, θL ¼ .72, β¼ .99, ϵ¼ 6, α ¼1.5, ω¼ .75, and δ ¼ .015. c Run with β¼ .975. d Run with β¼ .975 and θL ¼ .55.

v .5 v 1

60

v .25

50

v 3

40 30 20 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

λ

Fig. 4. Borderline value of Ω with temporary wage rigidity and limited asset market participation (base case).

the coefficient on the output target in the case of flexible IT.29 The efficacy of flexible IT derives, again, from its synergy with temporary wage rigidity; in flex-wage LAMP models, determinacy often requires implausibly large values for η.30 These results may not lay to rest all doubts about the general validity of the Taylor principle. They do, however, greatly narrow the zone of potential disagreement. Macroeconomists may disagree about whether Ω equals 1, 5, or 10, about whether ψ equals .2, 1, or 3, etc.; but if they can agree that v is less than three, then they should also agree that the Taylor principle is effectively necessary and sufficient for a stable, unique equilibrium path. 6. LAMP and the monetary transmission mechanism For some time now, an important question has lingered in the background: namely why is the size of the determinacy region so sensitive to the degree of real wage flexibility? The answer, developed below, is that very slight deviations from perfect wage flexibility radically change the nature of the monetary transmission mechanism. 29 A small value of η also solves all the problem cases when employment losses fall disproportionately on non-savers. In runs with λL^ na;t ¼ g λðL^ d;t  L^ t Þ n and g ¼ 1.5–1.75, a value of .1 for η ensures that Ω exceeds 100 in all but two of the scenarios in Table 6. The exceptions are the runs LDC2 with v¼ 3, where Ωn ¼ 63:9–64:7, and ω ¼ :33 with v¼3, where Ωn ¼ 1:4–4:1. In the latter case, Ωn ¼ 22:5–27:4 for η ¼ :35. 30 When Ω ¼ 1, λ ¼ :50, ψ ¼ :50 and the other parameters take their values in the base case, the equilibrium is indeterminate for η o 1:42. Lowering ψ to .25 increases the threshold value of η to 4.79.

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E.F. Buffie / Journal of Economic Dynamics & Control 46 (2014) 30–49

Table 5 Borderline value of Ω with limited asset market participation and temporary wage rigidity, λ¼ .4. v

Base case ψ¼ .20 ψ¼ .50 ψ¼ 2 ψ¼ 3 ω¼ .33 ω¼ .5 ω¼ .83 τ¼ .20 τ¼ .50 σ¼ .50 θL ¼ .60 LDC1d LDC2e

a

Flex-wage

.25

.5

1

3

51.8 74.7 65.4 31.2 18.6 52.8 78.2 33.0 50.1 50.7 18.9 25.0 48.5 18.5

52.3 94.5 76.5 22.2 10.6 38.0 109.0 26.6 49.9 50.9 21.6 27.0 44.7 18.9

39.4 98.2 71.8 10.9 5.4 24.3 113.3 15.9 36.3 37.5 18.5 21.7 31.7 14.7

13.6 78.0 45.4 3.4 2.4 9.7 51.9 4.6 10.1 11.6 7.9 8.3 11.9 6.2

.35 Ib Ib .80 1.05 Ic .35 .35 .35 .35 .26 .20 .88 .42

a

In the base case, τ ¼σ ¼ψ ¼1, θL ¼ .72, β ¼ .99, ϵ ¼6, α ¼ 1.5, ω¼ .75, and δ¼ .015. Equilibrium is indeterminate because λ 4λn . Equilibrium is indeterminate because the upper bound on the Taylor coefficient is below 1.5 for any positive value of Ω. d Run with β ¼.975. e Run with β ¼ .975 and θL ¼.55. b c

Table 6 Borderline value of Ω with limited asset market participation and temporary wage rigidity, λ¼ .5. v

Base casea ψ¼ .20 ψ¼ .50 ψ¼ 2 ψ¼ 3 ω¼ .33 ω¼ .5 ω¼ .83 τ¼ .20 τ¼ .50 σ¼ .50 θL ¼ .60 LDC1d LDC2e

Flex-wage

.25

.5

1

3

51.7 71.2 63.4 33.1 20.4 44.8 74.2 35.1 50.4 50.9 19.4 25.1 48.8 18.6

56.1 92.3 77.2 26.8 12.6 31.2 105.9 32.1 54.3 55.0 24.1 29.0 48.7 20.5

47.7 99.9 77.3 14.7 6.3 18.5 88.3 23.7 45.4 46.4 23.5 26.3 39.1 17.9

23.9 87.9 58.5 4.2 2.5 4.9 32.7 10.1 21.3 22.3 14.2 14.2 19.5 9.9

.12 Ib Ib .60 .89 Ic Ic .12 .12 .12 .09 .07 .29 .14

a

In the base case, τ ¼σ ¼ψ ¼1, θL ¼ .72, β ¼ .99, ϵ ¼6, α ¼ 1.5, ω¼ .75, and δ¼ .015. Equilibrium is indeterminate because λ 4λn . Equilibrium is indeterminate because the upper bound on the Taylor coefficient is below 1.5 for any positive value of Ω. d Run with β ¼.975. e Run with β ¼ .975 and θL ¼.55. b c

Higher interest rates depress the demand for capital and investment spending through two distinct channels. First, the decrease in aggregate demand and real output reduce the demand for capital directly. Second, under perfect wage flexibility, layoffs elicit an immediate reduction in the nominal and real wage. As firms try to substitute away from capital toward labor, the capital rental and investment decrease further. These first-round effects are amplified by a powerful multiplier: the decline in investment induced by a lower wage causes output to fall more, which causes the wage to fall more, which causes output to fall more, etc. Tossing LAMP into the mix makes for an even larger multiplier because non-savers decrease spending by the full amount of the reduction in their wage income. Everything conspires to maximize the contraction in investment and the threat that large near-term decreases in the capital stock will increase future marginal costs so much that forward-looking firms will find it optimal to markup prices more today even though current sales and costs are lower. Consequently, when the q-elasticity of investment is not extremely small, blind allegiance to the Taylor principle leads the central bank to validate arbitrary expectations of higher inflation. The problem, in a nutshell, is that the instrument chosen to combat inflation does not do so. Higher interest rates are stagflationary; they reduce aggregate demand, real output, and wages, but increase inflation.

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45

Table 7 Borderline value of Ω with limited asset market participation and temporary wage rigidity: v¼3 and either (i) α ¼ 2 for strict IT or (ii) η ¼ .10 for flexible IT. Strict IT with α¼ 2

a

Base case ψ ¼.20 ψ ¼.50 ψ ¼2b ψ ¼3 ω ¼.33c ω ¼.5 ω ¼.83 τ ¼ .20 τ ¼ .50 σ ¼ .50 θL ¼.60 LDC1 LDC2 Pass rate for T2

Flexible IT with η ¼.10

λ¼ .4

λ ¼.5

λ¼ .4

λ¼ .5

P P P 7.8 3.8 18.3 P 11.0 P P 16.6 18.6 P 12.3 86%

P P P 10.7 4.3 10.6 P P P P 24.7 P P 17.0 93%

U U U U U 14.0 U U U U U U U 52.4 100%

U U U U U 7.6d U U U U U U U 62.4 93%

α is the Taylor coefficient. P is entered when the run already passes the Strong Test for validity of the Taylor principle at α ¼1.5 in Table 5 or Table 6. In the solutions for flexible IT, U for unique is entered when the threshold value of Ω exceeds 100. a In the base case, τ ¼σ ¼ψ ¼1, θL ¼ .72, β ¼.99, ϵ¼ 6, ω ¼.75, and δ ¼.015. b Solution for α¼ 1.35. c Solution for α¼ 2.1. d Borderline value of Ω¼ 15.9 for η ¼.25.

Temporary wage rigidity short-circuits the feedback loops through which LAMP and lower wages magnify the decrease in investment spending. To fix ideas, suppose that the real wage is completely rigid (v¼0). The capital rental then decreases less as firms do not substitute away from capital toward labor to the same extent. In addition, knock-on effects of the multiplier are smaller. With perfect wage flexibility, the second-round effects of lower wage income and reduced spending by non-savers result in larger decreases in aggregate demand and investment than in the full-participation model. Similar second-round effects operate in the rigid-wage model, but they run through variations in employment of non-savers. This difference is crucial. Multiplier effects are much weaker when variations in employment mediate the reduction in non-savers income. Because substitution and multiplier effects are much weaker, both output and investment decrease much less than in the pure flex-wage model. And since investment decreases much less, the likelihood that higher marginal costs in the near future will increase inflation today is also much less. The general case mixes the pure flex-wage and pure rigid-wage solutions in proportions determined by v=ð1 þ vÞ.31 The explanation proffered above needs to be supplemented therefore with a demonstration that a small degree of wage rigidity has a large quantitative effect on the monetary transmission mechanism. This is provided by Figs. 5 and 6. The plots show how the coefficients X Q jt ¼ 0 ¼ 

Q^ ð0Þ ; ^ Rð0Þ  π ð1Þ

X C jt ¼ 0 ¼ 

C^ ð0Þ ; ^ Rð0Þ  π ð1Þ

and

X I jt ¼ 0 ¼ 

^ Ið0Þ ^ Rð0Þ  π ð1Þ

vary with λ and the degree of wage flexibility when the Taylor coefficient is 1.5 and the initial equilibrium is perturbed by a two percent cost-push shock with persistence parameter ρ ¼ :75. In Fig. 5, β ¼ :975 (think LDC), Ω ¼ 1, ψ ¼ 1:5, and τ ¼ :5. In Fig. 6, β ¼ :99, Ω ¼ 3, ψ ¼ :5, and τ ¼ :5. All other parameters take their base case values. Consistent with intuition, the solid line lies below the dashed line in each panel in Fig. 5. Moreover, the vertical gap between the two lines increases rapidly as the share of non-saving households rises, with the XI schedule sloping downward for v r :5.32 At λ ¼ :50, XI is 6.9 in the flex-wage case but only 3.3–4.4 in the runs with temporary wage rigidity and v r1. n This large difference maps into huge differences in the value of Ω in the flex- and rigid-wage models. I chose the LDC calibration β ¼ :975 in Fig. 5 because the prevalence of indeterminacy in the flex-wage model restricts comparisons with the rigid-wage model to small values of λ and Ω when β ¼ :99. Fig. 6 drops the comparison with flexwage solution in order to highlight three additional points about how wage flexibility conditions the effects of LAMP in NK developed country models:

 There is a good “fit,” especially at low values of v, between the results and the stylized fact that investment is much more volatile than either consumption or output. For λ ¼ :4–:5, XI is 2.3–3.3 times larger than XC and 1.9–2.5 times larger than XQ. The corresponding numbers when v ¼3 are 1.5–2.3 and 1.4–1.9.

31 In the base case of the full-participation model, v=ð1þ vÞ equals the ratio of the change in the wage to the change in the wage in the flexwage solution in the first quarter following a permanent exogenous shock to labor demand. 32 The scale on the axis makes it hard to see the downward slope. As λ increases from zero to .5, XI decreases from 3.59 to 3.27 for v¼ .25 and from 4.01 to 3.70 for v¼ .5.

46

E.F. Buffie / Journal of Economic Dynamics & Control 46 (2014) 30–49 XQ

XI

7

6

6

5

5

4

4

3

3

2

2

1

1

0.1

0.2

0.3

0.4

0.5

λ

0.1

v = .25

0.2

0.3

0.4

0.5

0.4

0.5

0.4

0.5

0.4

0.5

λ

v = .25

XQ

XI

7

6

6

5

5

4

4

3

3

2

2

1

1

0.1

0.2

0.3

0.4

0.5

λ

0.1

v = .50

0.2

0.3

λ

v = .50

XQ

XI

7

6

6

5

5

4

4

3

3

2

2

1

1

0.1

0.2

0.3

0.4

0.5

λ

0.1

v=1

0.2

0.3

λ

v=1

XQ

XI

7

6

6

5

5

4

4

3

3

2

2

1

1

0.1

0.2

0.3

v=3

0.4

0.5

λ

0.1

0.2

0.3

λ

v=3

Fig. 5. Comparison of aggregate demand and investment coefficients in the LAMP model when the wage is temporarily rigid instead of perfectly flexible (- - -).

 In the pure flex-wage model, standard aggregate demand logic gives way to inverted aggregate demand logic at λn ¼ :375. There is no such break point in the model with temporary wage rigidity until v is very far out toward the pure 

flex-wage end of the spectrum. Even when the labor market clears very fast (v ¼3) and 70% of households do not save, higher real interest rates reduce aggregate demand. Indeterminacy in full-employment LAMP models stems from two sources: (i) inverted aggregate demand logic and (ii) highly interest-elastic investment spending. As just noted, a small degree of real wage rigidity eliminates the first source. It also disarms the second source by greatly decreasing the interest-elasticity of investment spending at values of λ in the relevant range (.4–.5). LAMP is friend not foe as long as the real wage is not unbelievably flexible (v b3).

E.F. Buffie / Journal of Economic Dynamics & Control 46 (2014) 30–49

8

47

8

6

6

4

4

2

2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

λ

0.1

0.2

v = .25

0.3

0.4

0.5

0.6

0.7

0.4

0.5

0.6

0.7

λ

v = .5 14

10

12

8

10

6

8 6

4

4 2

2 0.1

0.2

0.3

v=1

0.4

0.5

0.6

0.7

λ

0.1

0.2

0.3

λ

v=3

Fig. 6. Semi-elasticities of investment (solid), consumption (dashed), and real output (dash þ dot), with respect to the real interest rate in the LAMP model with temporary wage rigidity.

7. Concluding remarks Most central banks subscribe to the Taylor principle, raising the real interest rate to contract aggregate demand when inflation exceeds its target level. The logic of fighting inflation with a reduction in aggregate demand seems to be unassailable. If investment is highly sensitive to the interest rate, however, the Taylor principle may lead to indeterminacy. The risk is unavoidable because sharp contractions in investment spending are followed in short order by large decreases in the capital stock that exacerbate inflationary pressures. In NK models that assume continuous full employment and full participation in asset markets, the crossing point from determinacy to indeterminacy occurs at values of the q-elasticity of investment in the .5–1.2 range. This is a grey area in empirical estimates. But when the NK model is refined to allow for limited participation in asset markets, the Taylor principle is driven to the edge of extinction: determinate equilibria survive only in a narrow toehold where very low values of the q-elasticity intersect with elastic labor supply (ψ 4 :5). In this paper I have argued that the pessimistic results in the existing literature are sui generis. The critiques of the forward-looking Taylor principle in Dupor (2000), GLV (2004), Carlstrom and Fuerst (2005), and Bilbiie (2008) rely on results obtained from models that assume continuous full employment. When this strong assumption is relaxed very slightly, the table turns and the Taylor principle regains its swagger. Active monetary policy does not guarantee a unique rational expectations equilibrium, but it comes very close, relegating indeterminacy to a tiny, irrelevant corner of the parameter space where the elasticity of labor supply is unusually high and the real wage is unbelievably flexible.33 Everywhere else, the rational expectations equilibrium is unique. Central banks can stop worrying; the forward-looking Taylor principle is alive and well. Appendix A In the model with temporary wage rigidity, the economy's equilibrium path is driven by a 5  5 system of first-order difference equations in C, I, π , K, and x. Routine manipulations similar to those in Section 4 yield     π κ θ C ^ κ ^ κ θK I ^ C t  k4 x t  I π t þ 1 ¼ t  k3 þ K k5 þ β β σθL Q σθL Q t β β   κθK v 1 ^ K ; þ þ ðA:1Þ βθL ko þ vk2 σ t 33

Moreover, determinacy holds even in this corner of the parameter space when the interest rate rule includes a small coefficient on current output.

48

E.F. Buffie / Journal of Economic Dynamics & Control 46 (2014) 30–49

  N N λI 1  ψ ð1  λÞ ^ 1þv I t þ 1  I^t C^ t þ 1 ¼ C^ t þ ko þ vk2 ko þ vk2 Cð1 þ μÞko ko þ vk2  C s     λw=C  1  ψ 1  λ x^ t þ 1  x^ t þ τðα 1Þ 1  λ π t þ 1 þ ko þ vk2 C   λwθK 1  ψ ð1  λÞ  ^  1þv K t þ 1  K^ t ; ko þ vk2 C θ L ko 

   rK I þ r K k5 I^t þ 1 þ k3 r K C^ t þ 1 ¼ Rðα  1Þπ t þ 1 r K k4 x^ t þ 1  g δ þ Q σθL σθL Q    rK r K vθ K ^ ^ K^ þ Rgδ I t  K t þ g δ þ þ ; σθL θL ðko þ vk2 Þ t þ 1

ðA:2Þ

C

K^ t þ 1 ¼ δI^ t þ ð1  δÞK^ t ; ^ t ¼ k3 C^ t þ k4 x^ t þ k5 I^t  x^ t þ 1 ¼ w

ðA:3Þ ðA:4Þ

vθ K K^ ; θL ðko þ vk2 Þ t

ðA:5Þ

where Cvk1 ; wðko þ vk2 Þ ko k4  ; ko þvk2 vI ; k5  wð1 þ μÞðko þ vk2 Þ     λ λ þv ψ 1  λ  : N  1 1þμ 1þμ

k3 

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