Nominal Rigidities Matthias Kehrig UT Austin February 11, 2014

1

Introduction

While the stochastic neoclassical growth model did fairly well in matching moments of real data, it didn’t do as well matching the overall sluggishness of nominal data such as inflation and wages. There are multiple ways to think how rigidities at the micro level may emerge: Nominal contracts for example may prevent firms from changing a wage. Firms might be reluctant to change their prices for fear of loss of customers (Kleshchelski and Vincent (2009)) or because it may encounter price adjustment costs, esp. menu costs. Some empirical accounts of price stickiness are Nakamura and Steinsson (2008); Bils and Klenow (2004); Eichenbaum, Jaimovich and Rebelo (2011). How can we model nominal rigidities while still maintaining analytical tractability? The workhorse model used to capture the notion of sticky prices is Calvo (1983) (Rotemberg (1982) is another one; Erceg, Henderson and Levin (2000) have extended Calvo’s model of sticky prices to sticky wages) which is described below. Sticky price and sticky wage models have an important implication for monetary policy as they break monetary neutrality. Thus, nominal rigidities have the potential to explain a sluggish response of inflation and wages to aggregate disturbances that cause business cycles. In fact, models that incorporate nominal rigidities perform quite well in empirical estimates of the U.S. and European economy (see Smets and Wouters (2003); Christiano, Eichenbaum and Evans (2005)). Mankiw and Reis (2002); Ma´ckowiak and Wiederholt (2009) show that models with imperfect information can equally generate sluggishness in nominal variables that are similar to rigidities that directly affect nominal rigidities.

2 2.1

The Calvo Model of Sticky Prices Firms

We suppose that a representative final good producer manufactures final output using the following linear homogenous technology: Z Yt = 0

1

σ−1 σ

Yit

1

σ  σ−1

,σ ≥ 1

(1)

Intermediate good j is produced by a price-setting monopolist according to the following technology: ( 1−α t Kitα L1−α − Φ if t Kitα Lit >Φ it Yit = ,0 < α < 1 (2) 0 otherwise where Φ is a fixed cost and Kit and Lit denote the services of capital and homogenous labour. Capital and labour services are hired in competitive markets at nominal prices, Pt rtk , and Wt , respectively. In (2), the shock to technology, t , has the time series representation, log t = ρ log t−1 + ut , where ut is iid. The first order necessary conditions for profit maximization by final good producers are:  − 1 σ Yit , i ∈ [0, 1]. Pit = Pt Yt One can verify that the price of final goods satisfies the following relation: 1  1−σ

1

Z Pt = 0

Pit1−σ di

.

Given the Cobb-Douglas specification of the production function of intermediate good producers, marginal cost divided by Pt is: mct =

1 t



wt 1−α

1−α 

rtk α

α (3)

where wt = Wt /Pt denotes the real wage and rtk denotes the real rental rate of capital. In addition, real marginal cost must be equal to the cost of renting one unit of capital divided by the marginal productivity of capital: rtk mct = (4)  1−α Lit αt K it

We adopt a variant of Calvo sticky prices. In each period t, a fraction of intermediate-goods firm, 1 − ξ, can adjust their price while the remaining fraction ξ is stuck with its price from the previous period indexed by some aggregate inflation measure. So we can write the price of the ith firm in time t as ( P˜t if adjust; probability 1 − ξ Pit = (5) Pit−1 π ˜t if no adjust; probability ξ. The indexation for non-adjusters π ˜t is a convex combination of steady-state inflation and last period’s inflation: ι π ˜t = πt−1 π ¯ 1−ι ι ∈ [0, 1] For stability it is necessary that the indexation in inflation, π ˜t , has constant returns to scale in its two components; otherwise there are problems to obtain a balanced steady state. Alternatively, one could also assume that there is no indexation at all (as in Gal´ı (2008), Ch. 3), so non-updating

2

firms are stuck with the price level they set whenever they got the chance to adjust the price for the last time. When the ith firm sets its price, it considers the possibility that it is possibly stuck with that price for many periods, so it maximises the net present value of a stream of profits in those future periods in which it is stuck with the price it is setting now. Nominal flow profits1 in a given period t + j are Pit+j Yit+j − Pt+j mct+j Yt+j − Pt+j mct+j Φ and the net present value of real profits is " # ∞ X P it+j Et (βξ)j λt+j Y − mct+j Yit+j − mct+j Φ . Pt+j it+j j=0

Here, λt+j is the multiplier on firm profits in the household’s budget constraint (expressed in real terms). Since households own firms, they discount future profits using the marginal utility the firm’s owner derives from future profits. The monopolist takes into account his demand curve which we can substitute into the expression for profits:   !1−σ !−σ ∞ X P P it+j it+j Et (βξ)j λt+j  Yt+j − mct+j Yt+j − mct+j Φ . Pt+j Pt+j j=0

Also, Pit+j denotes the price of a firm in period t + j that sets Pit = P˜it today and does not reoptimise between t + 1, . . . , t + j. Thus, we can make use of the indexing rule of non-adjusters (eq. (5)) to write Pit+j = Pit+j−1 π ˜t+j = Pit+j−2 π ˜t+j π ˜t+j−1 = ... = Pit π ˜t+1 π ˜t+2 ...˜ πt+j−1 π ˜t+j Similarly, Pt+j = Pt+j−1 πt+j = Pt+j−2 πt+j πt+j−1 = ... = Pt πt+1 πt+2 ...πt+j−1 πt+j Thus, Pit+j P = it Xtj = pit Xtj Pt+j Pt 1

Note that the fixed cost are denoted in terms of the output the firm has to produce itself, that means firms have to hire labour and capital (for which they pay Pt+j mct+j ) to produce Φ. If the fixed cost were denoted in terms of aggregate output, the fixed cost term would be Pt+j Φ; if the fixed cost were denoted in labour, then the fixed cost term would be Wt+j Φ.

3

where Xtj ≡

( π˜

πt+1 t+j ...˜ πt+j ...πt+1

if j > 0

1

if j = 0.

We can now conveniently rewrite the net present value of real profits as   !1−σ !−σ ∞ X P P it+j it+j Et (βξ)j λt+j  Yt+j − mct+j Yt+j − mct+j Φ Pt+j Pt+j j=0

∞ h i X Et (βξ)j λt+j Yt+j (Xtj pit )1−σ − mct+j (Xtj pit )−σ − mct+j Φ j=0

The ith firm maximises this expression by choice of pit . Note that all updating firms in period t face the same problem, so they will all have the same solution and pick the same price, denoted by p˜t . Even if a firm has not updated its price for thousands of periods it will choose the same price p˜t as a firm that updated its price last period. This is because only the future (when the firm can expect to reoptimise its price again) matters for the optimal price set today. Therefore, this equation is forward-looking. The first-order condition that maximises the net discounted profits is:   ∞ X σ j Et (βξ)j Ψt+j p˜−σ−1 p ˜ X − mc = 0, t t t σ − 1 t+j j=0

where Ψt+j is exogenous from the point of view of the firm: Ψt+j = λt+j Yt+j (Xtj )−σ . We can rearrange the first-order condition for the optimal price: P∞ j Kp,t σ Et j=0 (βξ) Ψt+j mct+j p˜t = ≡ P∞ j j σ − 1 Et j=0 (βξ) Ψt+j Xt Fp,t

(6)

Equation (6) demonstrates how the optimal price today is a markup over the discounted steam of future marginal cost adjusted for inflationary changes. If every firm gets to adjust its price every σ period (ξ = 0), then prices are perfectly flexible and the optimal price collapses to p˜t = σ−1 mct which is the optimal price from the real model with monopolistic competition. If few firms get to adjust their price every period (ξ → 1), then prices are very sticky because they are determined by expected marginal costs far out in the future. For convenience, we define ∞

Kp,t ≡

X σ Et (βξ)j Ψt+j mct+j σ−1 j=0

Fp,t

∞ X ≡ Et (βξ)j Ψt+j Xtj . j=0

4

such that p˜t =

Kp,t Fp,t

(6’)

You may verify that these objects have convenient recursive representations: " #   π ˜t+1 1−σ = 0 Et λt Yt + βξ Fp,t+1 − Fp,t πt+1     σ π ˜t+1 σ Et Kp,t+1 − Kp,t = 0. mc λ Y + βξ σ−1 t t t πt+1

(7) (8)

Expressing the optimal price as a function of Kp,t and Fp,t is a convenient way to get rid of the infinite sum in the firm problem by exploiting the recursive nature of the problem. This is a convenient trick to keep heterogeneity nice and tractable. The aggregate price index, aggregate inflation and price dispersion Z Pt = 0

1  1−σ

1

Pit1−σ di

Z = i adjust

Z = i adjust

Pit1−σ di P˜t1−σ di

 = (1 − ξ)P˜t1−σ ⇔ πt πt

⇔ p˜t

Z + i don’t adjust

+ +

π ˜t1−σ

Pit1−σ di

1 1−σ



Z

π ˜t1−σ ξ



i don’t adjust

Z i

1−σ Pit−1 di

  1 = (1 − ξ)˜ pt1−σ πt1−σ + ξ˜ πt1−σ 1−σ   1 1−σ ξ = π ˜t 1−σ 1 − (1 − ξ)˜ pt 1   1−σ  1−σ π ˜t  1 − ξ πt  =   1−ξ



1−σ Pit−1 di

1 1−σ

1 1−σ

(9)

(10)

Equation (9) demonstrates how inflation depends on the price adjustment parameter. If virtually ι π no firm gets to update its price (ξ = 1), inflation is highly autocorrelated: πt = π ˜t = πt−1 ¯ 1−ι . If most firms get to update their price, inflation jumps around like p˜. Consider the following measure of price dispersion: p∗t =

"Z 0

1

Pit Pt

#− 1

−σ

σ

di

.

(11)

By itself, this measure does not have a particular meaning, but it will be required when we examine the effects of price rigidity on misallocation in the aggregate resource constraint below. You may 5

verify that this expression can be written recursively: " p∗t

=

(1 − 

ξ)˜ p−σ t 

 +ξ

∗ π ˜t Pt−1 Pt

−σ #− σ1 − 1

σ  1−σ  σ−1

π ˜t πt

 1 − ξ =  (1 − ξ)   1−ξ

σ

  

+ξ

π ˜t p∗t−1

−σ

πt

  

(12)

The seven equilibrium conditions associated with the firms are (3), (4), (6), (7), (8), (10) and (12). This setup can be incorporated into a DSGE model. To close the model in general equilibrium one has to specify a household sector that buy the firms’ products and some way how the interest rate is determined, i.e. a monetary authority.

2.2

Households

 d ,T ,i ,k The household chooses ct , lt , Mt+1 t t t+1 to maximise utility   1−ψq    Pt+l ct+l     1+ψL ∞   c1−γ d X M L t+l t+l t+l l − φL −v Et β ,  1 − γ 1 + ψL 1 − ψq   l=0    

v > 0,

(13)

s.t. the budget constraint: d Pt (ct + it ) + Mt+1 − Mtd + Tt ≤ Wt Lt + Pt rtk kt + (1 + Rt−1 )Tt−1 + Xt .

(14)

Also, Mtd denotes the household’s beginning-of-period stock of money and Tt denotes nominal bonds issued in period t − 1, which earn interest, Rt , in period t. This nominal interest rate is known at t − 1. Finally, Xt represents a lump sum transfer from the government: Xt = Mt+1 − Mt , where Mt denotes the beginning-of-period stock of money in the economy. In equilibrium, we require that Mtd = Mt . The household’s problem is to maximise (13) subject to (14) and the capital accumulation technology linking investment, i, to capital: kt+1 ≤ (1 − δ)kt + it .

(15)

The household problem generates first order conditions for consumption (this gives us an equation that can be used to determine the multiplier on (14), λt , capital, investment, hours worked, money and bonds.

6

Note: If one were to consider Calvo-type sticky wages as in Erceg, Henderson and Levin (2000), one would introduce another agent, called a labour contractor, that demands differentiated labour hours from each household and uses a Dixit-Stiglitz aggregator to produce a homogeneous labour good that firms can use in their input. Each household would be a monopolistic supplier of its “type” of labour and demand its own wage. With Calvo-sticky wages the household considers how to set this wage taking into account that it might not have the chance to update that wage with a certain probability. Analogously to prices, there would be wage dispersion and wage rigidity that by itself would make monetary policy have real effects.

2.3

Monetary Authority

The monetary authority controls the supply of money, Mts . When policy is exogenous, it does so to implement a following Taylor rule. The target interest rate is Rt∗ :   π ¯ Yt Rt∗ = − 1 + απ [Et (πt+1 ) − π , ¯ ] + αy log β Y+ where Y + denotes steady state output. The monetary authority manipulates the money supply to ensure that the equilibrium nominal rate of interest, Rt , satisfies: Rt = ρi Rt−1 + (1 − ρi )Rt∗ + ηt ,

(16)

where ηt is an iid monetary policy shock.

2.4

Resource Constraint

We follow Yun (1996) in our development of the aggregate resource constraint for this economy, in terms of the aggregate stock of capital and the aggregate supply of labour by households. The issue at hand is that some firms post low prices not because they are more productive and it would be efficient to buy more of their variety, but because they just haven’t updated their price for a long time. We now take this inefficiency into account. Let Y ∗ denote the unweighted integral of output of the intermediate good producers: Z 1 ∗ Yt = Yit dj 0 Z 1   = t Kitα L1−α − Φ dj it 0  α Z 1 Kt = t Lit dj − Φ, Lt 0 Where Kt is the economy-wide average of stock of capital and Lt is the economy-wide average level of labour. This expression exploits the fact that all firms confront the same factor prices, and so they adopt the same capital to labour ratio. In equilibrium, this ratio must coincide with the economy-wide aggregate capital to homogenous labour ratio. Thus, Yt∗ = t Ktα Lt1−α − Φ.

7

The RHS of this equation denotes the aggregate inputs (minus those inputs needed to produce the fixed cost Φ); Yt∗ on the LHS is this potential output that can be produced in an economy with resources K and L. How much of that potential output gets actually used (consumed and invested) in this monetary economy? Using the demand curve, we can rewrite potential output Z 1 ∗ Yit dj Yt ≡ 0 Z 1  σ Pt ∗ Yt = Yt dj Pit 0 Z 1  σ −1 Pt dj Yt∗ = Yt P 0 it (p∗t )

 σ

(p∗t )σ Yt∗ = Yt  εt Ktα Lt1−α − Φ = Yt = ct + it

(17)

The law of motion of p∗t is provided in (12). Looking at the functional form that defines p∗t in equation (11) you can notice that p∗t is never greater than unity. The economics behind this is the inefficiency caused by the nominal rigidity: some intermediate firms post very low prices, not because they are very productive, but because they didn’t get to update their price in a long time. Profit-maximising behaviour induces final goods producers to buy too much of varieties produced by firms that haven’t updated their price in a long time. If some shock brings the economy outside of steady state, prices become more dispersed thus worsening misallocation and the loss of potential output.

2.5

Equilibrium Conditions

There are seven equilibrium conditions associated with the firm, six with households and one having to do with the aggregate resource constraint. The 15th equation is the monetary policy rule. The 15 date-t endogenous variables to be solved for are ct , Lt , p˜t , p∗t , Kt+1 , wt , rtk , mt+1 = Mt+1 /Pt , mct , πt , λz,t , Kp,t , Fp,t , Rt , Pk0 ,t .

2.6

Empirical match

See Figure 1. A monetary policy shock has a response of nominal and real variables that lasts over several periods (in fact, it builds up over several periods). In the basic RBC model, real variables would not respond to distortions in the money supply and nominal variables would adjust immediately. None of that is what we see in the data, however.

8

22

journal of political economy

Output  

Infla0on  

Money   supply  

Fig. 2.—Response of price level, output, and money stock to an expansionary monetary policy shock in the benchmark model.

At the same time, there is a prolonged boom in output that lasts even after the boom in the money supply is over. The peak in output is almost twice as big as the peak in the money supply, with the former occurring one-half of a year after the latter. Returning to figure 1, notice that the model is able to account for the dynamic response of the interest rate to a monetary policy shock.

This content downloaded from 198.213.249.164 on Mon, 29 Apr 2013 12:21:45 PM All use subject to JSTOR Terms and Conditions

Fig. 1.—Continued

Fig. 1.—Model- and VAR-based impulse responses. Solid lines are benchmark model impulse responses; solid lines with plus signs are VAR-based impulse responses. Grey areas are 95 percent confidence intervals about VAR-based estimates. Units on the horizontal axis are quarters. An asterisk indicates the period of policy shock. The vertical axis units are deviations from the unshocked path. Inflation, money growth, and the interest rate are given in annualized percentage points (APR); other variables are given in percentages.

7

Figure 1: Response of price level, output, and money stock to an expansionary monetary policy shock (Fig. 1 from6Christiano, Eichenbaum and Evans (2005))

Model- and VAR-based impulse responses. Solid lines are benchmark model impulse responses; This content downloaded from 198.213.249.164 on Mon, 29 Apr 2013 12:21:45 PM All use subject to JSTOR Terms and Conditions solid lines with plus signs are VAR-based impulse responses. Gray areas are 95 percent confidence intervals about VAR-based estimates. Units on the horizontal axis are quarters. An asterisk indicates the period of policy shock. The vertical axis units are deviations from the unshocked path. Inflation, money growth, and the interest rate are given in annualised percentage points This content downloaded from 198.213.249.164 on Mon, 29 Apr 2013 12:21:45 PM (APR);Allother given in percentages. use subjectvariables to JSTOR Termsare and Conditions

9

References Bils, Mark, and Peter J. Klenow. 2004. “Some Evidence on the Importance of Sticky Prices.” Journal of Political Economy, 112: 947–985. Calvo, Guillermo A. 1983. “Staggered Prices in Utility-Maximising Framework.” Journal of Monetary Economics, 12(3): 383–398. Christiano, Lawrence J., Martin Eichenbaum, and Charles L. Evans. 2005. “Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy.” Journal of Political Economy, 113(1): 1–45. Eichenbaum, Martin, Nir Jaimovich, and Sergio Rebelo. 2011. “Reference Prices, Costs and Nominal Rigidities.” American Economic Review, 101(1): 234–262. Erceg, Christopher J., Dale W. Henderson, and Andrew Levin. 2000. “Optimal Monetary Policy with Staggered Wage and Price Contracts.” Journal of Monetary Economics, 46: 281–313. Gal´ı, Jordi. 2008. Monetary Policy, Inflation, and the Business Cycle: An Introduction to the New Keynesian Framework. Princeton, NJ:Princeton University Press. Kleshchelski, Isaac, and Nicolas Vincent. 2009. “Market Share and Price Rigidity.” Journal of Monetary Economics, 56(3): 344–352. Ma´ ckowiak, Bartosz, and Mirko Wiederholt. 2009. “Optimal Sticky Prices under Rational Inattention.” American Economic Review, 99(3): 769–803. Mankiw, Nicholas G., and Ricardo Reis. 2002. “Sticky Information versus Sticky Prices: A Proposal to Replace the New Keynesian Phillips Curve.” Quarterly Journal of Economics, 117(4): 1295–1328. Nakamura, Emi, and J´ on Steinsson. 2008. “Five Facts About Prices: A Reevaluation of Menu Cost Models.” Quarterly Journal of Economics, 123(4): 1415–1464. Rotemberg, Julio J. 1982. “Monopolistic Price Adjustment and Aggregate Output.” Review of Economic Studies, 49(4): 517–531. Smets, Frank, and Rafael Wouters. 2003. “An Estimated Dynamic Stochastic General Equilibrium Model of the Euro Area.” Journal of the European Economic Association, 1(5): 1123–1175. Yun, Tack. 1996. “Nominal Price Rigidity, Money Supply Endogeneity, and Business Cycles.” Journal of Monetary Economics, 37(2): 345–370.

10