Levantar una Bola y Ponerla Otra Vez en Juego (Regla 14) Regla 14 – Procedimientos para la Bola: Marcar, Levantar y Limpiar, Reponer

The connection between the models is as follows.

Proposition 4. Consider the Phillips curve slope parameterizations {w, b−1, ˜w} for Calvo, Lucas and Rotemberg models defined by (36)-(37), (69)-(70) and (77)-(78) re- spectively along with the set of common parameters γc= {σ, β, η, Vm} and model spe-

cific parameters γs = (α, cp, Vz) There exist settings for the common parameters that

Phillips curve slopes and for the Lucas model also provided that the slope parameter exceeds unity.

Proof. Write a general Phillips curve as πt= gyet where g is the slope of the Phillips

curve. For the Calvo model first of all apply proposition 1 to show that the expected inflation term will always be zero. Then consider the slope as a function of the reset parameter α note crucially this parameter is unique to the Calvo model. Note further limα→1ω(α) = 0 and limα→0ω(α) = ∞. As ω(α) is a composition of power and linear

functions it must be continuous. Hence, apply the intermediate value theorem to show that for every g > 0 there is a parametization γm such that ω = g.

Likewise for Rotemberg note that cp is its unique parameter with limcp→∞ ω = 0 and˜

lim_cp→0 ω = ∞ and apply the intermediate value theorem for ω(c˜ p) to complete the

proof that for every g > 0 there is a parametization γm such that ω = ˜ω = g and

apply proposition 1- to complete the proof. Its implementation is cp= _{(1−α)(1−αβ)}α(θ−1) .

Turning to the Lucas model. The proposition follows if I can show that a unique

b exists for every possible implicit form (69) and that 0 < b < 1 with continuity

in parameters (69) note that by the parameter restrictions the right hand side is strictly positive for all possible b hence b > 0 next note that θ > 1 ensures both terms in the denominator are strictly positive for all possible b hence we can only have 0 < b < 1. Also note this bound cannot be tightened since for all other parameter values lim bVz→0= 0 and lim bη→1,Vz→∞= 1

The proof is completed by an implicit function theorem argument completed with the identical intermediate value theorem argument. Define

F = b −1 η  _V z Vz+ (θ+b) 2 (1+b)2Vm  ≡ b −1 η (1 + b)2Vz (1 + b)2_V z+ (θ + b)2Vm = 0

and its partial derivatives can be characterized as follows

Fb= 1 − 2 η (1 + b)V_z (1 + b)2_V z+ (θ + b)2Vm +1 η(1 + b) 2_V z [2(1 + b)V_z+ 2(θ + b)V_m] ((1 + b)2_V z+ (θ + b)2Vm)2

Now the task at hand is to give Fb a definitive sign converting to a common denomi-

nator and a little cancellation gives

η(1 + b)4V_z2+ 2η(1 + b)2(θ + b)2VzVm+ η(θ + b)4Vm2 − 2(θ − 1)(1 + b)(θ + b)VzVm

((1 + b)2_V

z+ (θ + b)2Vm)2

Finally combining the second and fourth terms yields a strictly positive expression

Fb =

η(1 + b)4V_z2+ 2η[(η − 1 + b)θ + 1 + η(1 + b)b](1 + b)(θ + b)VzVm+ η(θ + b)4Vm2

((1 + b)2_V

z+ (θ + b)2Vm)2

This proves F_b > 0. The other partial derivatives can be characterized as fol-

lows Fη = _η12Vz/(Vz+ (θ+b_1+b)2)2 > 0, Fθ = 2_η(θ+b) 2 1+b Vz/(Vz+ ( θ+b 1+b) 2_V m)2 > 0, FVm = 1 η( θ+b 1+b)2Vz/(Vz + ( θ+b 1+b)2Vm)2 > 0 and FVz = − 1 η( θ+b 1+b)2/(Vz + ( θ+b 1+b)2)2 < 0 all are

non-zero as required by the implicit function theorem.

As a composition of smooth functions F is itself smooth. Hence by the implicit function theorem b exists is unique and continuously differentiable in parameters (η, θ, Vz, Vm). It is therefore continuous in these arguments, as is b−1 by preservation

of continuity under function composition and the intermediate value theorem on 1 <

b−1(η, θ, Vz) < ∞ proves that as with Rotemberg and Calvo in the relevant range I can

always find common parameters such that b−1 = g. Since I used different members of γs for each parametization- I am able to vary them independently. Hence they are observationally equivalent for every possible setting for the common parameters provided g > 1.

The interpretation is that provided the Phillips curve is sufficiently steep then the methods of macroeconometrics cannot be used to distinguish between these three underlying designs for the Phillips curve. Vm has been assigned to the common pa-

rameters because it would be easy to add a monetary sector to either of the New Keynesian models without affecting topological properties beyond dimensionality if we cared to add a money demand shock. Moreover, it can be estimated in principle from monetary aggregates in a model free fashion.27 _{The model specific parameters}

27

In practice there are issues with how money is defined and measured. SeeHendry and Ericsson

[1991], Friedman and Schwartz [1991], Ericsson et al. [2016] and Lucas and Nicolini [2015] for a window into this debate. Note there are also issues to do with the construction and availability of data- for monetary aggregates.

are those that only feature in a single model and need to be estimated from microeco- nomic data. This accords with the conventional practice in macroeconometric model comparison of calibrating common parameters and estimating parameters specific to a single model.

Without microeconometric evidence the standard New Keynesian model is equivalent to the Rotemberg model which contradicts the findings of 2.4 and 2.7 concerning their non-linear forms. Furthermore, if there is sufficient price flexibility it is also equiva- lent to its New Classical counterpart. This should be a surprising result given that the express purpose of the New Keynesian framework was to differentiate itself from earlier Classical monetary theory. Moreover, the fact that there can be equivalence with a non-dynamic model is indicative that the benchmark New Keynesian model has a weak claim to be the pinnacle of Dynamic Stochastic General Equilibrium mod- eling. Intuitively, if one misses out on the ’stochastic’ part of the equilibrium then one loses the ’dynamics’ that ought to a define a successful Dynamic Stochastic General Equilibrium.

Chapter 3

The Role of Price Dispersion

This Chapter builds on Section 2.3. It begins by analyzing the global properties of ∆ as an aggregator which will prove crucial to subsequent chapters. I general- ize this notion across a collection of sticky price models then compare the notion of price dispersion in a Keynesian context to a leading counterpart from New Classical macroeconomics. The difference lies with rigidity of the aggregate price level caused by the absence of selection as to which firms change price. I consider dynamic prop- erties specific to the Calvo setting and by way of example application to zero bound models. I link the condition that ∆ ≥ 1 to underlying non-negativity constraints. I link coordination failure on the production side before discussing extensions that can incorporate idiosyncratic shocks and some state dependence.

3.1

Lower Bound on Price Dispersion

The following powerful result derived directly from the construction of the price level tells us that the measure of price dispersion ∆ defined by equation (18) in Section 2 is strictly greater than unity unless all firms set the same price.

Proposition 5. ∆ ≥ 1 with ∆ = 1 if and only if p_t(i) = P_t, ∀ i.

Where p_t(i) is the price set by any firm i at time t. The lengthy proof is contained in Appendix A, the first part is a familiar application of Jensen’s inequality, possible because the demand system is sufficiently convex, and the second a small extension exploiting strict convexity. Versions of this result are well known in the literature- see for example Schmitt-Grohe et al. [2007] or Damjanovic and Nolan [2010], although full proofs are omitted and analysis is usually restricted to non-stochastic steady

states. The extension to other New Keynesian models that use the constant elastic- ity of substitution preference scheme is straightforward. I do so in Appendix A by modifying the probability measure used to aggregate the various prices to obtain the price level to correspond to three common pricing models: the basic Calvo model used here, the Calvo model with indexation to trend inflation used byYun[1996] and the General Taylor Economy ofTaylor[1993b], Coenen et al.[2007],Dixon and Kara

[2010], Dixon and Kara [2011] andDixon and Le Bihan [2012] which encompasses a wide range of pricing models and can be fitted exactly to match cross-section price distributions. 1

The economic force driving this result is preference for variety- where the individual prefers averages to extremes. It ensures that variations in price unrelated to marginal costs make the individual worse-off. This is a very weak economic assumption. A strict preference against variety would make it difficult to ensure interior demand sys- tem for all products. The only cases in widespread usage where this argument would not prove widely applicable are those with homogeneous products as with Bertrand, perfect or Cournet competition. Therefore the result is certainly not specific to one demand system.

To allow for heterogeniety between firms I can redefine ∆ to normalize each firm’s price relative to its optimal reset price.Yun and Levin [2011], Fuhrer [2000], Den- nis [2009], Ravn et al. [2010], Givens [2013], Santoro et al. [2014], Lewis and Poilly

[2012], Lewis and Stevens [2015] and Etro and Rossi [2015] consider various alterna- tive demand systems with a variety of motivations. I could alter the source of the price dispersion from staggered to for example information-constrained price-setting.

Mankiw and Reis[2002], Mankiw and Reis [2006], Mankiw and Reis [2007], Loren- zoni[2009],Lorenzoni[2010],Nimark[2008],Nimark[2014],Adam[2007],Mackowiak et al.[2008] andPaciello and Wiederholt[2014] are papers where this price dispersion is present but not accounted for. All that is required is a motivation for firms to set different prices when in a flexible price world it would be efficient if they all set the same.

1_{Dixon [2012] shows that the Generalized Taylor model can approximate arbitrarily well the}

Generalized Calvo used by authors such asWolman[1999],Dotsey and King[2006],Sheedy[2010], the multiple Calvo associated withCarvalho [2006] and de Carvalho [2011], as well as the familiar simple Taylor and Calvo.

Price dispersion would also come about where there are physical costs of price chang- ing provided that firms face idiosyncratic shocks or differing adjustment costs. See for example, Gertler and Leahy [2008], Nakamura and Steinsson [2008], Reiff et al.

[2014], Bouakez et al. [2009] and Bouakez et al. [2014]. In these cases the relevant interpretation of the price dispersion variable ∆ is the difference between the actual and flexible price. Suppose for example fixed adjustment costs of a price change varying across firms with an aggregate shock - ∆ > 1 will come about if some firms adjustment costs are below and some above the common adjustment threshold. Sim- ilarly, with common fixed adjustment cost but idiosyncratic shocks ∆ > 1 will occur if some firms keep their price constant because their idiosyncratic shock ’cancels out’ the aggregate shock i.e. they remain inside their band of price inaction. The behavior of price dispersion and its dynamics are integral to all the analysis which follows. All results that do not refer to a specific specification of sticky price setting (e.g. Calvo or Taylor) generalize to all models covered by this lemma. The fundamental mechanism in this paper is that inflation causes price dispersion.