another look at sticky prices and output persistence

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Journal of Economic Dynamics & Control 30 (2006) 2533–2552

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Another look at sticky prices andoutput persistence

Peng-fei Wanga, Yi Wenb,�

aDepartment of Economics, Cornell University, Ithaca, NY, USAbResearch Department, Federal Reserve Bank of St. Louis, P.O. Box 442, St. Louis, MO 63166-0442, USA

Received 31 December 2004; accepted 9 August 2005

Available online 8 November 2005

Abstract

Price rigidity is the key mechanism for propagating business cycles in traditional Keynesian

theory. Yet the new Keynesian literature has failed to show that sticky prices by themselves

can effectively propagate business cycles. We show that price rigidity in fact can (by itself) give

rise to a strong propagation mechanism in standard models, provided that investment is also

subject to a cash-in-advance constraint. Reasonable price stickiness can generate highly

persistent, hump-shaped movements in output under either monetary or non-monetary

shocks. Hence, whether or not price rigidity is responsible for output persistence is not a

theoretical question, but an empirical one.

r 2005 Elsevier B.V. All rights reserved.

JEL classification: E52; E41; E32

Keywords: Sticky prices; Output persistence; New-Keynesian models; Cash-in-advance; Financing

constraints

1. Introduction

Sticky prices are the key mechanism assumed in traditional Keynesian theory forpropagating the impact of monetary shocks as well as other aggregate shocks

see front matter r 2005 Elsevier B.V. All rights reserved.

.jedc.2005.08.002

nding author. Tel.: +1314 444 8559; fax: +1 314 444 8731.

dress: [email protected] (Y. Wen).

www.elsevier.com/locate/jedc

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P.-f. Wang, Y. Wen / Journal of Economic Dynamics & Control 30 (2006) 2533–25522534

throughout the economy. Yet to demonstrate a persistent output effect of stickyprices in a fully-specified new Keynesian dynamic-general-equilibrium model hasproven to be very difficult, as recently stressed by Chari et al. (2000, hereafter, CKM,2000). CKM (2000) show that an empirically plausible degree of price rigiditygenerates only a modest degree of output persistence in responding to monetaryshocks, far from enough to account for the estimated output persistence in the U.S.economy. The usefulness of the sticky price assumption, one of the corner stones oftraditional Keynesian business cycle theory, is thus under serious challenge.1

The persistence problem raised by CKM (2000) along with others has ledresearchers to explore other types of rigidities or economic forces, in conjunctionwith sticky prices, to explain the persistent effects of monetary shocks. For example,Christiano et al. (2005) obtain more persistent output responses to monetary shockby combining both sticky prices and sticky wages on the nominal side, aided by habitformation, adjustment costs, limited participation in money markets and variablecapital utilization on the real side. In a model without capital, Jeanne (1998) showsthat adding real-wage rigidity into sticky price models can significantly increase thepropagation of monetary shocks in output. Dotsey and King (2001) show thatoutput persistence can be increased by features such as a more important role forproduced inputs, variable capacity utilization, and labor supply variability throughchanges in employment. Together, these elements can reduce the elasticity ofmarginal cost with respect to output, thus improving the persistence of output.Bergin and Feenstra (2000) emphasize interactions between input–output productionstructures and translog preferences to increase output persistence under sticky prices.Similar results based on production chains can also be found in the work of Huangand Liu (2001). Other researchers such as Mankiw and Reis (2002), Woodford(2001), and Erceg and Levin (2003) have emphasized the important role of imperfectinformation in helping sticky prices to generate persistent output responses tomonetary shocks.2

Adding a large number of building blocks, such as real rigidities and complexinformation structures, into the standard sticky-price model can improve the model’sfit in terms of output persistence, but it comes at the expense of simplicity. Often,more than one factor is added to mix with nominal rigidities such that it becomeshard to disentangle exactly which factor is doing what in generating output

1For a review of the New Keynesian literature, see Clarida et al. (1999). For empirical literature on the

persistent effects of monetary shocks, see Sims (1992), Christiano et al. (1996) and Strongin (1995), among

others.2The literature has also explored the implications of sticky nominal wages for output persistence.

Models based on staggering wages such as those in Andersen (1998), Erceg (1997), and Huang and Liu

(2002) are still not able to generate as much real persistence as is seen in the data, though they do alleviate

the problem to some extent. Edge (2002) recently establishes conditions under which wage and price

staggering are equivalent regarding their effects on output persistence, showing that the persistence

problem is similar in both sticky-wage and sticky-price models. Also see Dotsey and King (2005) for the

recent literature on state-dependent pricing in general equilibrium. This literature shows that state-

dependent pricing can have dramatically richer propagation mechanisms than time-dependent pricing in

generating output and inflation persistence. Benhabib and Farmer (2000) show that externalities can also

give rise to output persistence in a monetary model.

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P.-f. Wang, Y. Wen / Journal of Economic Dynamics & Control 30 (2006) 2533–2552 2535

persistence. In addition, while sticky or imperfect information proves to be effectivein giving rise to output persistence, it is often modeled in the literature within apartial equilibrium framework. It was shown recently by Keen (2004), for example,that the business cycle implications of sticky information proposed by Mankiw andReis (2002) may not be robust to general equilibrium extensions.3

This paper takes a step back and asks whether a canonical sticky price modelwithout any additional frictions or real rigidities can generate a reasonable degree ofoutput persistence. Put another way, this paper asks why sticky prices by themselvesmay fail to provide a strong propagation mechanism for the business cycle. This is anintriguing question because intuitively there is no reason price rigidity would notlead to output persistence, since it could turn i.i.d. money shocks into seriallycorrelated movements in real balances just as effectively as any type of real rigidity.Real balances in turn could affect aggregate spending and production. Yet despitethe exploding literature trying to overcome the persistence problem, what exactlyfails the Keynesian sticky price propagation mechanism in general equilibriummodels remains unclear. CKM (2000), for example, show the inability of stickyprices to generate output persistence mainly via model simulations. The reasonsbehind the failure are less obvious when capital is included.

We show in this paper that sticky prices by themselves can in fact generate highlypersistent output movements, contrary to the findings of the existing literature. Inparticular, we show that an empirically plausible degree of price stickiness cangenerate hump-shaped output responses to monetary shocks in a way very similar tothose seen in the data. Thus, sticky prices are indeed a useful assumption inexplaining business cycles as far as theory is concerned. Whether they are in factresponsible for business cycles in the real world, however, is an empirical question.

The key to our finding is a cash-in-advance (CIA) constraint on aggregate demand(consumption plus investment). CIA constraints can significantly limit the initialincrease in aggregate demand after a money injection, in sharp contrast to thepopular money-in-utility (MIU) specification, because agents are forced toaccumulate real balances before they can fully raise spending, which leads to moresmoothed output responses. Consider a model without capital. Under a money-in-utility specification, demand for goods and demand for money are only looselylinked. Households can therefore raise consumption significantly beyond the initialincrease in money injection in anticipation of future money increases, which leads tovolatile impulse responses in output. But a CIA constraint limits the rise inconsumption to the current rise in money, thus forcing households to wait for futuremoney injections to fully raise consumption. Hence, CIA can lead to hump-shapedoutput persistence under serially correlated money shocks while MIU cannot. Whenthere is capital in the model, if both consumption and investment are subject to CIAconstraints (a specification without equivalent form under MIU), the increase inaggregate demand is again limited to the current rise in money, giving rise to moresmoothed, hump-shaped output responses to shocks.

3Erceg and Levin (2003) is an exception.

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In addition, with a CIA constraint on aggregate spending (consumption plusinvestment), sticky prices can lead to hump-shaped output persistence not onlyunder monetary shocks, but also under non-monetary shocks, such as technologyshocks and preference shocks. The intuition is similar: CIA postpones the maximumimpact of shocks on aggregate demand because agents are forced to intertemporallysmooth aggregate spending via real balance accumulation over time. A smoothedaggregate demand thus dictates a smoothed aggregate supply (production).4

The key assumption driving our results is that investment must be subject to a CIAconstraint. Given the availability of complicated financial markets, this seems to be adifficult assumption to defend. However, this assumption may be defended on atleast three grounds. First, firms’ investment projects are often subject to financingconstraints due to capital market imperfections (moral hazard, asymmetricinformation, incomplete markets, and so on). Consequently, firms’ internal cashflows are often crucial in determining their investment level. A vast empiricalliterature in the past twenty years has documented a strong link between firms’internal cash flow and investment. This literature shows that such a closerelationship is due to financial constraints rather than to good performances insales (see, most notably, Fazzari et al., 1988).5

Second, the so called ‘‘money’’ in a standard CIA model can be understood inmore general terms as M1 or M2, rather than just as cash. Given that the broadmoney supply (M1 or M2) is proportional to the monetary base (B) according toMt ¼ mBt, the above interpretation is valid as long as the money multiplier (m) isrelatively constant. Thus, a monetary shock in the model can be reinterpreted as ashock to the availability of liquidity or credit in terms of M1 or M2, which is just aslikely to affect firms as to affect consumers.6

4It is thus not surprising that our findings also contradict a branch of the existing literature that assumes

CIA constraints. For example, Yun (1996) studies a CIA constrained sticky price model with capital and

finds that money shocks have no persistent effect on output. Ellison and Scott (2000) use the same model

and demonstrate that sticky prices not only fail to produce persistent output fluctuations but also generate

extremely volatile output at very high frequencies. This is because both papers assume a CIA constraint on

consumption only. When there is capital in the model, intertemporal substitution between current

consumption and future consumption can be achieved through capital accumulation. In this case,

imposing a CIA constraint only on consumption spending is not effective for generating persistent output,

since investment becomes very volatile by serving as the buffer for consumption, and consequently

investment will dictate output dynamics. Thus, even if consumption is hump-shaped, output is not. This

suggests that a CIA constraint on investment is crucial for generating output persistence, as consumption

can always be smoothed by capital accumulation.5See Hubbard (1998) for a comprehensive review of this large literature. Also see Fazzari et al. (2000) for

the recent debate on this issue.6There is strong evidence that firms hold a substantial fraction of money in the economy. Using data

from the Federal Reserve’s demand deposit ownership survey (DDOS), which separately reports the

ownership of demand deposits at commercial banks by financial firms, nonfinancial firms, households and

foreigners, Mulligan (1997) found that nonfinancial firms hold at least 50%more demand deposits than do

households. By the 1980s firms had accumulated almost twice as many demand deposits as households

had. On the other hand, the Federal Reserve’s flow of funds (FOF) reports that households may hold more

M1 than firms do, but Mulligan convincingly argued that the DDOS data are more accurate than the FOF

data.

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Third, the subjection of both consumption and investment to money is moreconsistent with the estimation of aggregate money demand function. For example,Mulligan and Sala-i-Martin (1992) find that aggregate income is a better scalevariable than aggregate consumption in estimating money demand. Indeed, mostempirical work in money demand estimation has adopted income rather thanconsumption as a scale variable. Classical examples include Friedman (1959) andGoldfield (1973, 1976). This may explain why CIA constraints on both consumptionand investment are widely used in the theoretical monetary literature, such asStockman (1981), Abel (1985), Gong and Zou (2001) and Fuerst (1992), to name justa few.

The above justifications notwithstanding, we show that output continues to behump-shaped even when as little as 30% of firms’ investment is subject to CIAconstraint in our model. Therefore, theoretically speaking, sticky prices have notrouble generating output persistence as long as investment is partially constrainedby money holdings.

The rest of the paper proceeds as follows. Section 2 demonstrates outputpersistence under CIA in a simple model without capital. We show in this model thatsticky prices can give rise to hump-shaped output responses to money shocks underCIA, but not under MIU.7 Section 3 studies a fully specified general equilibriummodel with capital. It is shown that under either monetary or non-monetary shocks,output exhibits hump-shaped persistence as long as a certain fraction of investmentis subject to a money-in-advance constraint. Hence, introducing capital into themodel does not destroy the persistence mechanism of sticky prices, in contrast toCKM (2000). Section 4 concludes the paper.

2. The basic model

2.1. Households

A representative household chooses sequences of consumption, fCtg1t¼0, labor

supply, fNtg1t¼0, and money demand, fMtg

1t¼0, to solve

max E0

X1t¼0

bt½logCt � aNt�

s:t: Ct þMt

Pt

pMt�1 þ X t

Pt

þ wtNt þPt

and the CIA constraint, CtpMt=Pt, where X is the money injection, P is theaggregate goods price in terms of money, w is the real wage, and P is the profitincome contributed from firms which the household owns. Since the current moneyholdings, Mt, enter the CIA constraint, there is no inflation tax on consumption.

7Jeanne (1998) shows that real wage rigidity must be introduced into a CIA model without capital in

order to generate persistence. Here we show that this is not necessary.

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Hump-shaped output persistence remains if the inflation tax effect is allowed. Notethat a linear leisure function is assumed for simplicity. Making the leisure functionnonlinear has little effect on the results. Denoting l1 and l2 as the Lagrangemultipliers for the budget constraint and the CIA constraint respectively, the firstorder conditions can be summarized by

1

Ct

¼ l1t þ l2t, (1)

a ¼ l1twt, (2)

l1t ¼ bEt

Pt

Ptþ1l1tþ1 þ l2t. (3)

2.2. Firms

The final goods, Y t, are produced by a perfectly competitive firm according to the

technology Y t ¼R 10 ytðiÞ

ðs�1Þ=sdih is=ðs�1Þ

, where s41 measures the elasticity of

substitution among the intermediate goods, yðiÞ. Let ptðiÞ denote the price ofintermediate good i. Thus the demand for intermediate goods is given byytðiÞ ¼ ½ptðiÞ=Pt�

�sY t, and the relationship between final goods price and inter-

mediate goods prices is given by Pt ¼R 10

ptðiÞ1�sdi

h i1=ð1�sÞ.

Each intermediate good i is produced by a single monopolistically competitive firmaccording to the following technology: ytðiÞ ¼ ntðiÞ. Intermediate good firms faceperfectly competitive factor markets, and are hence price takers in the factormarkets. Profits are distributed to household at the end of each time period. The costfunction for firm i can be derived from minimizing wtntðiÞ subject to ntðiÞXy.Denoting ft as the Lagrange multiplier, which is also the real marginal cost, the firstorder condition for cost minimization is given by wt ¼ ft. Consequently, the realprofit in period t is given by ððptðiÞ=PtÞ � ftÞytðiÞ.

Following Calvo (1983) in assuming that each firm has a probability of 1� y ofadjusting its monopoly price in each period, a firm’s intertemporal profitmaximization problem is to choose the optimal price, pn

t , to maximize

Et

X1s¼0

ðbyÞsLt;tþs

pnt

Ptþs

� ftþs

� �yt;tþsðiÞ, (4)

where Lt;tþs � ½Ctþs=Ct��1 is the ratio of marginal utilities taken as exogenous by the

firm; and yt;tþs denotes the firm’s output level in period tþ s given its optimal price inperiod t: yt;tþsðiÞ ¼ ½p

nt ðiÞ=Ptþs�

�sY tþs. The first order condition for optimalmonopoly price implies the following pricing rule:

pn

t ¼sP1

s¼0 ðbyÞsEtLtþsP

stþsY tþsftþs

ðs� 1ÞP1

s¼0 ðbyÞsEtLtþsP

s�1tþs Y tþs

. (5)

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Because all firms that can adjust their prices face the same problem, all monopolistfirms will set their prices in the same way, as indicated above. The average price offirms that do not adjust prices is simply last period’s price level, Pt�1. Given that onlya fraction of 1� y can adjust their prices in each period, the final goods price indexcan be written as Pt ¼ ½yP1�s

t�1 þ ð1� yÞPn1�st �1=ð1�sÞ.

2.3. Equilibrium dynamics

In equilibrium, households’ first order conditions and firms’ profit maximizationconditions are satisfied, all markets clear, and the CIA constraint binds. We studysymmetric equilibrium only. The model is solved by log-linearization around a zero-inflation steady state. Using circumflex lower-case letters to denote percentagedeviations around the steady state, the log-linearized optimal price and the priceindex are given respectively by pn

t ¼ ð1� byÞP1

s¼0 ðbyÞsEtðftþs þ ptþsÞ and

pt ¼ ypt�1 þ ð1� yÞpn

t , which together imply the new Keynesian Phillips relationship

pt ¼ bEtptþ1 þð1� yÞð1� byÞ

yft, (6)

where pt � pt � pt�1 is the inflation rate.The log-linearized aggregate production function is given by yt ¼ nt, so around the

steady state the aggregate production function is the same as the individual firm’sproduction function. Note that the CIA constraint can be expressed asyt � yt�1 ¼ xt � pt, where x � logðX t=Mt�1Þ denotes the growth rate of the nominalmoney stock. We assume that the monetary authority follows a money growth rulegiven by xt ¼ rxt�1 þ �t. The household’s first-order conditions are thus reduced to:ð2� bÞyt � 2ft ¼ �bðptþ1 þ ftþ1Þ. Substituting out pt in this equation and in thenew Keynesian Phillips curve using the CIA constraint, the system of equations forsolving fyt; ftg is given by

xt þ yt�1 � ð1þ bÞyt ¼ bEtðxtþ1 � ytþ1Þ þð1� yÞð1� byÞ

yft, (7)

2yt � 2ft ¼ bytþ1 � bxtþ1 � bftþ1, (8)

which can be arranged more compactly as

Et

ytþ1

yt

ftþ1

xtþ1

0BBBB@

1CCCCA ¼

1þ bb

�1

bð1� yÞð1� byÞ

by�1þ rb

b1 0 0 0

�1þ bb

�1

b1� byþ yþ by2

by�1

b0 0 0 r

0BBBBBBB@

1CCCCCCCA

yt

yt�1

ft

xt

0BBBB@

1CCCCA. (9)

The eigenvalues of the Jacobian matrix are given by f2=b; 1=by; y;rg. Note that thefirst two of the eigenvalues are larger than unit, hence they can be utilized to solvethe system forward to determine fyt; ftg as functions of the state fyt�1;xtg. Clearly,the other two smaller roots, fy;rg, determine the propagation mechanism of output.

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The decision rule of output takes the form: yt ¼ yyt�1 þ axt, where a � ð2yð1� rbÞþrbðbry� 1þ by� by2ÞÞ=ð2� rbÞð1� rbyÞo1 is the elasticity of output with respectto money growth shocks.8 Clearly, the persistence of output is determined jointlyby the degree of price stickiness, y, and the persistence of shocks. If monetaryshocks follow an AR(1) process ðxt ¼ rxt�1 þ �tÞ, then output becomes an AR(2)process:

yt ¼ ðyþ rÞyt�1 � yryt�2 þ a�t, (10)

which implies a hump-shaped impulse response function. Supposing that the averageprice stickiness is about four quarters in the economy, the probability of notadjusting prices is y ¼ 0:75. Assuming that money growth shocks are autocorrelatedwith r ¼ 0:6, a value common in the literature (e.g., CKM, 2000),9 the degree ofoutput persistence implied by the model matches the contract multiplier of the U.S.economy estimated by CKM (2000) almost exactly. The maximum impact of amoney injection on output is delayed for three quarters after the shock. Thesimulated impulse responses of output are graphed in Fig. 1 (top window).

When the money demand arises from MIU instead, under standard assumptionsregarding the elasticity of substitution between consumption and money, no hump-shaped output persistence can be generated from the model. To demonstrate, let thehousehold solve

max E0

X1t¼0

bt logCt þ Z logMt

Pt

� aNt

� �

s:t: Ct þMt

Pt

pMt�1 þ X t

Pt

þ wtNt þPt.

Letting all parameters take the same values as in the previous CIA model, thebottom panel in Fig. 1 shows that output does not have hump-shaped persistence.10

Technically speaking, the log-linearized first order conditions of the MIU modelcan be reduced to the following system:

Et

ytþ1

ptþ1

mt

xtþ1

0BBBB@

1CCCCA ¼

1

bþð1� yÞð1� byÞ

by�1

�1þ bb

�1þ bb

�1

bð1� yÞð1� byÞ

y1

b0 0

0 �1 1 1

0 0 0 r

0BBBBBBB@

1CCCCCCCA

yt

pt

mt�1

xt

0BBB@

1CCCA,

(11)

8a40 if y is large enough.9Also see our calibration using post-war data in the next section.10Again, the linear leisure function is assumed for simplicity. Making it nonlinear has little effect on the

results. CKM (2000) argue that perfect substitutability between consumption and leisure is crucial for

generating output persistence under the Taylor (1980) type of price rigidity. Here we find that this

requirement is not necessary under the Calvo (1983) type of price rigidity. See Kiley (2002) for discussion

regarding the differences between the Taylor type and the Calvo (1983) type of price rigidities.

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0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.00 4 8 12 16 20 24 28 32

2.01.81.61.41.21.00.80.60.40.20.0

0 4 8 12 16 20 24 28 32

Money-In-Utility

Cash-In-Advance

Fig. 1. Impulse responses of output to a money shock.


which has the following analytical solution for the decision rule of output: yt ¼

yyt�1 þ ðyð1� b2yrÞ=ð1� brÞð1� ybrÞÞxt � ðry=ð1� brÞÞxt�1. When xt is ARð1Þ,we have

yt ¼ ðyþ rÞyt�1 � yryt�2 þyð1� b2yrÞ

ð1� brÞð1� ybrÞ�t �

ryð1� brÞ

�t�1. (12)

Note that output is no longer an ARð2Þ process as in Eq. (10), but an ARMAð2; 1Þprocess. The crucial difference this makes is that one of the autoregressive roots (thepoles) and the moving average root (the zeros) almost cancel each other in the MIUmodel, reducing the ARMAð2; 1Þ process in Eq. (12) to an ARð1Þ process. An ARð1Þprocess cannot exhibit hump-shaped dynamics. To see the pole-zero cancellation, letb ¼ 1.11 Eq. (12) reduces to

yt ¼ yyt�1 þy

1� r�t. (13)

11This near pole-zero cancellation will take place regardless of the value of b. The reader can check this

by setting b ¼ 0, for example.

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This also explains why consumption (output) can be very volatile in MIU modelsdue to the value of r typically assumed in the literature. For example, letting y ¼0:75 and r ¼ 0:6, consumption will increase by 0:75

0:4 ¼ 1:875% when money growthincreases by just 1%. This is consistent with the graph in the lower panel in Fig. 1.

The intuition is that CIA prevents consumption from rising too much in theimpact period since agents are not able to increase consumption beyond the currentcash injections. This smooths demand and hence production. On the other hand,when consumption is not cash-in-advance constrained (as in MIU models),households can raise consumption significantly beyond the initial increase in moneyinjection, in anticipation of future money increases. This mechanism of outputsmoothing due to a CIA constraint on aggregate demand continues to work in moregeneral models that include capital, as the following section shows.

3. The full model

3.1. Households

The representative household chooses consumption (C), hours worked (N), capitalstock (K), money demand (M), and bond holdings (B) to solve

max E0

X1t¼0

bt Y logCt � aN

1þgt

1þ g

" #

s:t: Ct þ ½Ktþ1 � ð1� dÞKt� þMt þ Bt=Rt

pt

¼Mt�1 þ Bt�1 þ X t

pt

þ rtKt þ wtNt þPt, ð14Þ

Ct þ Ktþ1 � ð1� dÞKtpMt

pt

, (15)

where rt and wt denote the real rental rate and real wage rate that prevail incompetitive factor markets; R denotes nominal returns to bonds; d denotes the rateof capital depreciation. At the end of each period, the household receives wages fromhours worked, rental payments from capital lending, and nominal bond returns, aswell as profits Pt from all firms the household owns. If consumption is the only cashgood, then our model reduces to that of Yun (1996) and Ellison and Scott (2000).12

Denoting the Lagrange multipliers for (14) and (15) as l1 and l2 respectively, thefirst order conditions with respect to Ct;Nt;Ktþ1;Mt;Bt

� �can be summarized by

Yt

Ct

¼ l1t þ l2t, (16)

12An alternative specification of the model would be to impose CIA constraints on individual firms’

investment instead of on aggregate investment. However, doing so gives rise to a non-trivial aggregation

problem under sticky prices, which is beyond the scope of the current paper to solve. We hope to tackle

this aggregation problem in a future project.

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aNgt ¼ l1twt, (17)

l1t þ l2t ¼ bð1� dÞEtðl1tþ1 þ l2tþ1Þ þ bEtl1tþ1rtþ1, (18)

l1t ¼ bEtl1tþ1Pt

Ptþ1þ l2t, (19)

l1t

Rt

¼ bEtl1tþ1Pt

Ptþ1. (20)

3.2. Firms

The final goods sector is the same as described previously. Hence, the demand ofintermediate goods is given by yðiÞ ¼ ½pðiÞ=P��sY , and the price index for final goods

is given by P ¼R 10 pðiÞ1�s di

h i1=ð1�sÞ. The production technology for intermediate

good i is given by yðiÞ ¼ AkðiÞanðiÞ1�a, where 0oao1 and A denotes aggregatetechnology shocks to productivity. The cost function of firm i is derived by

minimizing rkðiÞ þ wnðiÞ subject to AkðiÞanðiÞ1�aXy. The first order conditions aregiven by r ¼ faðyðiÞ=kðiÞÞ;w ¼ fð1� aÞðyðiÞ=nðiÞÞ, where f denotes the real marginalcost. Given the production function, the real marginal cost can be written as

ft ¼1

At

rt

a

� �a wt

1� a

� �1�a. (21)

Note that, since the total cost equals ftytðiÞ, the marginal cost equals average cost.Let the probability of a price adjustment in each period for any intermediate firm be1� y. Then a firm’s optimal price is again to choose pn to maximizeEt

P1s¼0 ðbyÞ

sLt;tþs½pnt =Ptþs � ftþs�½p

nt =Ptþs�

�sY tþs, which yields the same pricingrule as before:

pn

t ¼sP1

s¼0ðbyÞsEtLtþsP

stþsY tþsftþs

ðs� 1ÞP1

s¼0ðbyÞsEtLtþsP

s�1tþs Y tþs

. (22)

3.3. Equilibrium and calibration

In a symmetric equilibrium near the steady state, the aggregate productionfunction can still be expressed as Y t ¼ AtK

at N1�a

t and the aggregate profit is stillgiven by Pt ¼ Y t � wtNt � rtKt. Hence the equilibrium market clearing conditionsand constraints are

Ct þ Ktþ1 � ð1� dÞKt ¼ Y t, (23)

Mt ¼Mt�1 þ X t, (24)

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Bt ¼ Bt�1 ¼ 0, (25)

Ct þ Ktþ1 � ð1� dÞKt ¼Mt

Pt

. (26)

The optimal pricing rule in (22) in conjunction with the law of motion of theaggregate price index, Pt ¼ ½yP1�s

t�1 þ ð1� yÞPn1�st �1=ð1�sÞ, leads to the same

relationship for the dynamics of inflation around the steady state as before: pt ¼

bEtptþ1 þ ð1� yÞð1� byÞ=yft, except that the marginal cost function is nowdifferent.

In a zero-inflation steady state, it can be shown that the following relationshipshold:

f ¼s� 1

s, (27)

K

Y¼ f

bað2� bÞ½1� bð1� dÞ�

. (28)

Note that, compared to a standard RBC model in which K=Y ¼ ba=ð1� bð1� dÞÞ,there are two distortions on the steady state capital-output ratio in the sticky pricemodel. First, monopolistic competition gives rise to a markup of ð1� fÞ=f%40,which approaches zero only if the elasticity of substitution s!1 (i.e., f! 1). Apositive markup implies a lower steady state capital-output ratio. Second, due to thefact that money is needed to facilitate transactions, an inflation tax is imposed oninvestment returns, which lowers the steady state capital-output ratio by a factor ofð2� bÞ. If b ¼ 1, this effect disappears.13

The exogenous shocks are assumed to be orthogonal to each other and followAR(1) processes in log

xt ¼ rxxt�1 þ �xt,

logAt ¼ rA logAt�1 þ �At,

logYt ¼ rY logYt�1 þ �Yt, (29)

where x � logðX t=Mt�1Þ denotes the money growth rate. The model is calibrated atquarterly frequency. We choose the time discounting factor b ¼ 0:99, the rate ofcapital depreciation d ¼ 0:025, the capital elasticity of output a ¼ 0:3, the inverselabor supply elasticity g ¼ 0 (Hansen’s indivisible labor), and the elasticity ofsubstitution parameter s ¼ 10 (implying a markup of about 10%).14 The pricerigidity parameter y is set to 0:75, and persistence parameters for technology andpreference shocks are set to rA ¼ rY ¼ 0; 9. These parameter values are quitestandard in the literature. To calibrate money growth shocks, we estimate an AR(1)

13See Stockman (1981) for more discussion on this issue.14The results are robust to the values of these parameters. For example, very similar results are obtained

even when the markup approaches zero and when the utility function on leisure has the log form (i.e.,

g ¼ 0:5).

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model for the growth rate of monetary base ðM0Þ in the U.S. (1950:1–2003:4), andwe obtain rx ¼ 0:6 and s�x ¼ 0:006.

3.4. Model evaluation

The impulse responses of output (Y), consumption (C), investment (I) andemployment (N) to a one-standard-deviation shock to money growth are graphed inFig. 2. Several features are worth noting. First, a monetary growth shock can causesignificant increases in economic activity. On impact, investment increases by 2:6%and output increases by 0:56%, while consumption increases by only 0:06%. Theoverall standard deviation of investment is about four times that of output, and theoverall standard deviation of consumption is about half that of output. Thesedifferent magnitudes suggest that monetary shocks can explain one of the mostprominent business cycle facts emphasized by the real business cycle literature;namely, that consumption is less volatile than output and that investment is morevolatile than output.

Second and most strikingly, the impulse responses of output (Y), employment (N),and investment (I) are all hump-shaped, with a peak response reached around thethird quarter after the shock. This suggests a richer propagation mechanism of the

0.90.80.70.60.50.40.30.20.10.0

0 2 4 6 8 10 12 14 16 18 20

I

Y0.500.450.400.350.300.250.200.150.100.050.00

0 2 4 6 8 10 12 14 16 18 20

N

C

4.03.53.02.52.01.51.00.50.0

-0.50 2 4 6 8 10 12 14 16 18 20

1.4

1.0

0.6

0.2

-0.20 2 4 6 8 10 12 14 16 18 20

Fig. 2. Impulse responses to a money shock.

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0.012

0.010

0.008

0.006

0.004

0.002

0.0000 4 8 12 16 20 24 28 32

Quarters

ModelData

0.00024

0.00020

0.00016

0.00012

0.00008

0.00004

0.000000.000.050.100.150.200.250.300.350.400.450.50

Cycles per Quarter

ModelData

Power SpectrumImpulse Response

Fig. 3. Output dynamics of model and data.


model than a standard RBC model or a sticky-price model with MIU. This richerpropagation mechanism induced by sticky prices and the CIA constraint enables themodel to match the observed output persistence in the U.S. economy quite well. Forexample, if we estimate an AR(2) process for the logarithm of real GDP of theUnited States (1950:1–2003:4) with a quadratic time trend, then the fitted equation is

logðytÞ ¼ 1:3 logðyt�1Þ � 0:37 logðyt�2Þ þ vt, (30)

where the standard deviation of the residual is sv ¼ 0:0088.15 Using this estimatedstandard deviation to simulate the U.S. output by equation (30), Fig. 3 (left window)shows that the shape of the impulse response function of the U.S. output looks verymuch like that implied by the model (where the standard deviation of money shockin the model is s�x ¼ 0:006), except that the volatility of the model output is onlyabout one third that of the data.

CKM (2000) propose to measure the persistence of output by its half-life. Whenthe half-life is measured starting from the initial response at impact period, the half-life of output in the model is 10, while that in the data is 11. When it is measuredstarting from the peak of the response after a shock, the half-life is 8 in the modeland 9 in the data.

15CKM (2000) obtain similar estimates.

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Table 1

Selected moments

sx=sy corðxt; ytÞ corðxt; xt�1Þ

x ct it nt ct it nt yt ct it nt

U.S. data 0.53 3.36 0.97 0.92 0.96 0.82 0.90 0.86 0.89 0.82

Model 0.49 4.18 1.26 0.66 0.93 0.96 0.94 0.99 0.92 0.92


Ellison and Scott (2000) show that sticky price models cannot explain the businesscycle since sticky prices tend to generate too much variations in output at the highfrequencies but not enough variations at the business cycle frequencies. Here we showthat this is not true if investment spending is subject to a CIA constraint. The right-hand side window of Fig. 3 shows that the power spectrum of output growth in themodel matches that in the data quite closely in terms of variance distribution acrossfrequencies. However, in terms of total variance (proportional to the area underneaththe spectral density function), the model explains only about 16% of the data.16

The intuition for the persistent output effect of sticky prices in the full model withcapital is similar to that in the basic model without capital. CIA acts to smoothaggregate spending across time; by requiring cash, the maximum impact of shockson demand (and hence supply) is postponed until enough real balances areaccumulated. Thus the CIA constraint serves essentially like an intertemporal formof adjustment cost, which is well-known for generating hump-shaped outputdynamics. However, if only consumption goods are subject to CIA, output cannothave enough persistence since shocks immediately impact investment spending,which dictates aggregate demand and supply, thus making output very volatile at thehigh frequencies (see, e.g., Ellison and Scott, 2000).17

Since it is well known that investment is much more volatile than output in thedata, to make sure that a CIA constraint on investment does not lead to too littleinvestment volatility relative to output, Table 1 reports the standard business cyclestatistics of the model. It shows that, among other things, the model is able toexplain the large volatility of investment relative to output despite investment beingsubject to a CIA constraint.

Sticky prices under a CIA constraint can also effectively propagate non-monetaryshocks. Fig. 4 plots the impulse responses of output and employment to a one-standard-deviation technology shock and preference shock respectively. It showsthat non-monetary shocks can also generate hump-shaped output persistence in themodel (windows A and C). This feature of the model is worth emphasizing since it iswell known that standard RBC models lack an endogenous propagation mechanism

16Introducing capacity utilization could improve the model in this regard.17Inflation in the model behaves like an AR(1) process, indicating certain degree of persistence, but not

hump-shaped persistence. Hence the model cannot explain the well known fact that inflation lags output.

However, its volatility relative to output matches the U.S. data quite well. For the issue of inflation

persistence and its relation to output, see Fuhrer and Moore (1995), Ireland (2003), Mankiw and Reis

(2002) and Wang and Wen (2005), among others.

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0.180.160.140.120.100.080.060.040.020.00

0 4 8 12 16 20 24 28 32 36 40

0.360.320.280.240.200.160.120.080.040.00

0 4 8 12 16 20 24 28 32 36 40

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.00 4 8 12 16 20 24 28 32 36 40

0.4

0.2

0.0

-0.2

-0.4

-0.6

-0.8

-1.00 4 8 12 16 20 24 28 32 36 40

Preference Shock--Y Preference Shock--N

Technology Shock--Y Technology Shock--N

Fig. 4. Output and employment under non-monetary shocks.


to explain the hump-shaped, trend reverting output response to transitory shocks(Cogley and Nason, 1995; Watson, 1993). Here it is shown that sticky prices alonecan do the job.18

One more feature of the model to notice is that employment responds negatively totechnology shocks (see Window D in Fig. 4). Because sticky prices and the CIAconstraint render aggregate demand rigid in the short run, higher total factorproductivity induces cost-minimizing firms to lower employment. This negativeeffect of technology shocks on employment as a result of sticky prices has beenempirically documented and explained by Gali (1999).19 However, in a money-in-utility general equilibrium model, technology shocks generate positive employmenteven if prices are sticky, since investment can increase to absorb the shocks.

3.5. Sensitivity analysis

The assumption that investment is subject to a CIA constraint is crucial inobtaining our results. In reality, not necessarily all firms’ investments are subject to

18For other mechanisms that can also generate hump-shaped output dynamics, see Wen (1998a–c) and

Benhabib and Wen (2004).19Whether empirical data supports the view that technology shocks generate negative employment

movements is controversial. See Chari et al. (2005) for a debate.

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Table 2

Sensitivity analysis

c 1 0:9 0:8 0:7 0:6 0:5 0:4 0:3 0:2 0:1 0:0

Hump (at nth quarter) 3 3 3 3 3 2 2 2 1 1 1

Half-life 10 10 10 9 9 8 7 6 5 4 2


financing constraints and hence they may not all be tight to internal cash flows. Weshow here that even if as little as 30% of aggregate investment is subject to a money-in-advance constraint, aggregate output remains hump-shaped.

To demonstrate, modify the CIA constraint in the model to

Ct þ c½Ktþ1 � ð1� dÞKt�pMt

Pt

, (31)

where c 2 ½0; 1� measures the degree of financial constraint on aggregate investment.The model reduces to the previous model if c ¼ 1 and it reduces to the model of Yun(1996) and Ellison and Scott (2000) if c ¼ 0.

Table 2 shows that output remains hump-shaped and highly persistent undermoney shocks even for small values of c. For example, when c ¼ 0:6, the peak ofoutput response is not reached until three quarters after the shock, a fact consistentwith the U.S. data. The half-life is 9 quarters, only slightly shorter than the case ofc ¼ 1. As we reduce c further to 0.3, the half-life is still 6 quarters long and the peakof output response is still postponed beyond the impact period of the shock to thesecond quarter, indicating a hump shape. Hump-shaped output disappears whenc ¼ 0:2, but the half-life of output is still more than twice as long as the case ofc ¼ 0.20

4. Conclusion

We showed in this paper that sticky prices alone can generate strong outputpersistence if the cash-in-advance constraint is extended to investment. Outputexhibits a hump-shaped response pattern even when as little as 30% of aggregateinvestment is subject to a CIA constraint. In such a model monetary shocks seemcapable of explaining a broad range of business cycle facts better than, or at least aswell as, a standard RBC model driven by technology shocks. Hence, whether or notsticky prices are responsible for the business cycle is not a theoretical question, butan empirical one. Given that multiple mechanisms can give rise to hump-shapedoutput persistence (see, e.g., Wen, 1998a–c; Benhabib and Wen, 2004, among

20Suppose that investment consists of two parts, I ¼ I1 þ I2, where I1 is subject to CIA constraint and

I2 is not. Then the elasticity of investment to cash flow is given by ðqI=IÞ=qm=m ¼ I1=I ¼ c. Using annual

data, Worthington (1995) found that the elasticity of investment to cash flow is between 0:2 and 0:65. Theimplied quarterly elasticity of investment to cash flow should be even higher. Thus in a quarterly model,

cX0:6 is a reasonably good approximation.

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others), it remains to empirically test which mechanism is the main culprit inpropagating the business cycle in the real world. Bils et al. (2003), for example, findsome empirical evidence against the sticky price propagation mechanism ofmonetary shocks. Baharad and Eden (2004) find that the staggered price settingassumption is not favored by their micro data. Dittmar et al. (2005) and Wang andWen (2005) show that endogenous monetary policy, rather than sticky prices, aremore likely to be responsible for the inflation dynamics found in the U.S. data. Thus,to establish sticky prices as a key propagation mechanism of the business cycle, moreempirical work is obviously needed.

Acknowledgements

We would like to thank Jess Benhabib, Bill Gavin, Patrick Kehoe, Karl Shell, twoanonymous referees and the editor, Wouter DenHaan, for helpful comments. Wealso thank the CAE at Cornell for financial support. The views expressed in thepaper and any errors that may remain are the authors’ alone.

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another look at sticky prices and output persistence

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