chapter 14 a model of imperfect competition and staggered ... · a model of imperfect competition...

George Alogoskoufis, Dynamic Macroeconomics, 2016

Chapter 14 A Model of Imperfect Competition and Staggered Pricing

The realization of the instability of the original Phillips curve, as well as the weak microeconomic foundations of traditional keynesian models, have gradually led to a paradigm shift in the macroeconomics of aggregate fluctuations, from the largely ad hoc keynesian models that we examined in Chapter 12 to dynamic stochastic general equilibrium (DSGE) models with explicit microeconomic foundations.

The perfectly competitive “new classical” models that we analyzed in Chapters 11 and 13 are examples of such DSGE models, in which wages and prices are perfectly flexible and equilibrate both the product and labor markets. In “new classical” models, only real shocks, such as shocks to productivity, can affect the fluctuations of output, employment and other real variables. Monetary shocks only affect nominal variables, such as the price level and inflation, and there is no short run relation between employment and inflation. In addition, employment fluctuations are based on intertemporal substitution and, thus, there is no involuntary unemployment in the “new classical” models.

In this chapter we present the structure of an imperfectly competitive model based on staggered price setting. The model is a DSGE model based on monopolistic competition in product markets. We initially analyze it assuming both full flexibility of wages and prices and, subsequently, assuming staggered price adjustment. The model with full flexibility of wages and prices is comparable to the perfectly competitive “new classical model”, presented in Chapter 13. The model with staggered pricing allows for a “Phillips curve” type relation between inflation and deviations of output and employment from their “natural rate”, and thus for real effects of nominal shocks and monetary policy. It thus differs from the “new classical” model, in that nominal shocks and monetary policy have real effects.

The imperfectly competitive model introduced in this chapter has two important differences from the typical perfectly competitive “new classical” model.

First, instead of fully competitive markets for goods and services, it assumes that markets are characterized by conditions of monopolistic competition. Firms do not take prices as given, but determine prices so as to maximize profits. Because of the market power of firms, and monopolistic competition, the “natural rate” of employment, real output, consumption and real wages is determined at a lower level than in the corresponding competitive model, even when there is complete flexibility in prices and wages. However, by itself this difference does not result in major differences from the “new classical” competitive model regarding the nature of aggregate fluctuations.

George Alogoskoufis, Dynamic Macroeconomics, 2016 Chapter 14

Second, this imperfectly competitive model can account for a “Phillips type” relation, if it is assumed that firms adjust prices only gradually. Two observationally equivalent versions of gradual price adjustment have dominated the literature since the early 1980s. The one is the Rotemberg (1982a,b) model of gradual price adjustment, and the second is the Calvo (1983) model of staggered pricing. In the Rotemberg model, firms balance the costs of adjusting prices against the costs of deviating from the profit maximizing optimal price. They end up gradually adjusting prices towards the profit maximizing price. In the Calvo model, it is assumed that only a fixed proportion of firms have the freedom to adjust prices in any given period. This results in the remaining firms not being able to adjust prices. Although optimal pricing takes this restriction into account in advance, the aggregate price level can only adjust gradually to aggregate shocks, because not all firms have the option of adjusting prices in every period.

These two alternative assumptions lead to models with price level stickiness, whose properties differ from “new classical” models, and share some of the properties of “keynesian” models.

In the case of staggered pricing, it turns out that deviations of output from its “natural” level cause deviations of inflation from expected future inflation, as higher aggregate demand and output causes an increase in nominal marginal costs and hence prices. This results in a Phillips curve type relation, called the new keynesian Phillips curve. This differs from the original expectations augmented Phillips curve that we analyzed in Chapter 12, as current inflation depends on current expectations about future inflation, and not on prior expectations of current inflation.

The imperfectly competitive model analyzed in this chapter has the following structure:

Deviations of inflation from the target of the central bank are determined by the “new keynesian” Phillips curve, and depend of expected future deviations of inflation from the central bank target, and deviations of real aggregate demand and output from its “natural” level, as the latter cause an increase in nominal marginal costs and hence optimal prices.

The deviations of aggregate demand from the “natural” level of real output depend on the “new keynesian” IS curve, and are driven by deviations of the current real interest rate from its “natural” level.

The nominal interest rate is determined by the central bank, which follows a Taylor interest rate rule. According to the Taylor rule, the nominal interest rate reacts positively to deviations of current inflation from the central bank target, as well as deviations of real output from its “natural” level.

After presenting the properties of this model, we analyze the effects of monetary and real shocks on fluctuations in real output and inflation.

The imperfectly competitive new keynesian model with staggered prices can, unlike the classical model, explain monetary cycles, i.e aggregate fluctuations caused by monetary shocks. These shocks are transmitted to real variables, and, to the extent that they persist over time, have persistent real effects.

However, the weakness of this model is that it cannot account for involuntary unemployment but only for under-employment. In this model, because of the assumed flexibility of nominal wages, the labor market clears continuously, in the sense that labor demand is always equal to labor supply.

!2


Because of imperfect competition, the equilibrium real wage is lower than in a fully competitive model, and thus, equilibrium employment is lower than in a fully competitive economy. However, there is no unemployment, and fluctuations in employment are due to intertemporal substitution in labor supply, exactly as in the “new classical” model. In this respect, this is a “quasi classical” model of aggregate fluctuations, not unlike the Lucas (1972) imperfect information model alluded to in the previous chapter.

Thus, the analysis of this “sticky price” new keynesian model eventually leads us to the conclusion that in order to account for unemployment, one has to investigate models with distortions in the labor market.

14.1 An Imperfectly Competitive Model of Aggregate Fluctuations

In this section we examine in detail the structure of an imperfectly competitive model of aggregate fluctuations. The basic model that we analyze has much in common with the “new” classical model of Chapter 13. It is a dynamic stochastic general equilibrium model with two important differences from the “new classical” model. 1

First, instead of perfectly competitive markets for goods and services we assume that markets are characterized by conditions of imperfect (monopolistic) competition. Firms do not take prices as given, but have the power to determine prices that maximize profits. Because of imperfect competition, in equilibrium, employment, real output, consumption and real wages are determined at a lower level than in the corresponding competitive model, even when there is complete flexibility in prices and wages. However, by itself this difference does not result in material differences from the competitive classical model with respect to the nature of aggregate fluctuations. If this was the only difference, we could well talk about an imperfectly competitive “new classical” model.

Second, we assume that there is staggered price adjustment, i.e. that firms do not have the ability to change their prices at all times. This assumption is what makes the model “new keynesian”, as it leads to a model in which the price level adjusts gradually towards the equilibrium price level. As a result of gradual price adjustment, real variables deviate from their “natural” rates, and monetary shocks can have real effects.

14.1.1 The Representative Household

The problem of the representative household under monopolistic competition has one difference from the corresponding problem under perfect competition. The difference is that because of monopolistic competition, the household consumes differentiated products.

The representative household maximizes,

! (14.1) E01

1+ ρ⎛⎝⎜

⎞⎠⎟

t

u(Ct ,Nt )t=0

∞∑

See Gali (2008) for a fuller presentation and analysis of this model.1

!3


where C is consumption, N is labor supply, and ρ is the pure rate of time preference. Consumption consists of all produced goods, which are defined on the basis of a constant index j in the interval [0,1]. Aggregate consumption is thus given by,

! (14.2)

where ε is also a parameter of the preferences of the representative household, and more precicely, the elasticity of substitution between goods. We assume that ε>1.

The sequence of budget constraints under which the household maximizes intertemporal utility is given by,

(14.3)

The household must also satisfy the transversality condition,

! (14.4)

where P(j) is the price of good j, W the nominal wage, i the nominal interest rate, B a nominal one period bond, and T an exogenous transfer of nominal income to the household (dividends, government transfers or taxes).

Apart from the decision about aggregate consumption and labor supply, which we have analyzed in the relevant section of Chapter 13, the household must now decide on the distribution of its consumption expenditure among the various goods. This requires the maximization of the consumption bundle (14.2) for any level of monetary expenditure. One can easily deduce that this implies,

! (14.5)

for any good j in the interval [0,1], where P is the average price level, defined as,

! (14.6)

In addition, when the household follows this optimal allocation policy, we also have that,

! (14.7)

(14.7) suggests that total consumption expenditure can be written as the product of the aggregate consumption index and the aggregate price index. Substituting (14.7) in the sequence of budget constraints (14.3), we get,

Ct = Ct ( j)ε−1ε dj

j=0

1

∫⎛⎝⎜

⎞⎠⎟

εε−1

Bt+1 = (1+ it )Bt +WtNt +Tt − Pt ( j)Ct ( j)djj=0

1

∫

limT→∞

EtBT ≥ 0

Ct ( j) =Pt ( j)Pt

⎛⎝⎜

⎞⎠⎟

−ε

C(t)

Pt = Pt ( j)1−ε dj

j=0

1

∫( )11−ε

Pt ( j)Ct ( j)djj=0

1

∫ = PtCt

!4


! (14.8)

This sequence of budget constraints is the same as the sequence of budget constraints of the representative household in the competitive classical model.

As a result, the first order conditions for consumption and labor supply are analogous to the ones of the new classical model we analyzed in Chapter 13.

! (14.9)

! (14.10)

We assume, as in the “new classical” model without capital of Chapter 13, that the utility function is defined by,

! (14.11)

where 1/θ is the intertemporal elasticity of substitution in consumption, and λ the Frisch elasticity of labor supply.

Assuming that preferences take the form of (14.11), the first order conditions (14.9) and (14.10) can be written in log-linear form as,

! (14.12)

! (14.13)

where lower case letters denote the logarithms of the corresponding variables. π is the rate of inflation. (14.12) and (14.13) are analogous to the ones in the “new” classical model without capital in Chapter 13.

14.1.2 The Representative Firm and Optimal Pricing

We assume that output is produced by a set of firms denoted by a continuous index j defined in the interval [0,1]. Each firm produces a differentiated product under conditions of monopolistic competition. All firms have access to the same production technology, denoted by the production function,

! (14.14)

PtCt +11+ it

Bt ≤ Bt−1 +WtNt −Tt

−uNtuCt

= Wt

Pt

11+ it

= 11+ ρ

Et

uCt+1uCt

PtPt+1

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

U(Ct ,Nt ) =C

t

1−θ

1−θ− Nt

1+λ

1+ λ

wt − pt = θct + λnt

ct = Et (ct+1)−1θit − Et (π t+1)− ρ( )

Yt ( j) = AtLt ( j)1−α

!5


where Α>0 and 0<α<1 are exogenous technological parameters, common to all firms. L(j) is employment of labor by firm j. The parameter α is constant, while A is assumed to follow an exogenous stochastic process.

The optimal price of each firm, if it can choose its price in every period, is given by the maximization of its profits, under the constraint of the production function (14.14) and the demand function for its product (14.5). Each firm takes the average price P, the average wage W and the level of total demand C as given.

The per period profits of firm j are given by,

! (14.15)

From the first order conditions for a maximum of (14.15), under the constraints (14.14) and (14.5), the optimal price is determined as,

! (14.16΄)

The optimal price is a fixed markup on the firm’s marginal cost, which equals the expression in brackets. The markup depends on the elasticity of substitution between goods in the preferences of consumers, which determines the price elasticity of demand of their product, and therefore the profit margin of the firm. In the case of perfect competition that we examined in Chapter 13, the elasticity of substitution tends to infinity, and the price tends to marginal cost. In the case of monopolistic competition with ε > 1, as we have assumed, the optimal price is higher than the marginal cost of labor.

As all firms have the same production function and face the same nominal wage and the same demand function for their product, they will all choose the same price. Consequently, the price level is defined as,

! (14.16)

Taking the logarithm of the production function (14.14) for the representative firm, and equation (14.16) for the optimal price, we get,

! (14.17)

! (14.18)

where,

! , ! .

Pt ( j)Yt ( j)−WtLt ( j)

Pt ( j) =ε

ε −1Wt

(1−α )AtLt ( j)−α

⎛⎝⎜

⎞⎠⎟

Pt =ε

ε −1Wt

(1−α )AtLt−α

⎛⎝⎜

⎞⎠⎟

yt = at + (1−α )lt

wt − pt = at −αlt − µ

at = lnAt µ = ln εε −1

⎛⎝⎜

⎞⎠⎟ − ln(1−α )

!6


a is the logarithm of the exogenous productivity shock, and the constant µ is the logarithm of the markup on marginal cost, minus the logarithm of the coefficient of decreasing returns to labor.

14.1.3 Equilibrium with Full Price Flexibility

Solving the model under the assumption of full flexibility of prices, one can show that fluctuations in employment, output, consumption and real wages are a function only of the exogenous shocks to productivity, while fluctuations in the real interest rate are a function of the expected change in productivity, just as in the classical model with the assumption of perfect competition.

In the basic form of this model we shall assume that there is no investment or public consumption. Thus, in equilibrium, labor supply would be equal to labor demand by firms, and consumption will be equal to output.

! (14.19)

! (14.20)

The model consists of equations (14.12), (14.13), (14.17) and (14.18) and the equilibrium conditions (14.19) and (14.20). The model determines employment, output, consumption, real wages and the real interest rate as functions of the exogenous shock to productivity a.

The real interest rate is determined by the Fisher equation as,

! (14.21)

Solving the model for the five endogenous variables, we get,

! (14.22)

where, ! and, ! .

! (14.23)

where, ! and, ! .

! (14.24)

where, ! and, ! .

! (14.25)

nt = lt

yt = ct

rt = it − Et (π t+1)

ltN = nt

N = φat + n_

φ = 1−θθ(1−α )+α + λ

n_= − µ

θ(1−α )+α + λ

ytN = ct

N =ψ at + y_

ψ = 1+ (1−α )φ = 1+ λθ(1−α )+α + λ

y_= (1−α )n

_

w − p( )tN = χat +ω

_

χ = 1−αφ = θ + λθ(1−α )+α + λ

ω_= θ(1−α )+ λ( )n

_

rtN = ρ +θψ Et (Δat+1)

!7


(14.22), (14.23), (14.24) and (14.25), along with the equilibrium conditions (14.19) and (14.20), determine the five endogenous real variables as functions of the exogenous productivity shock. Superscript N (natural) denotes the equilibrium values of the relevant variables, which, according to the Friedman definition are their “natural” rates.

Output, consumption and real wages are positive functions of the productivity shock a, while employment is a positive function of the productivity shock only if θ<1, i.e. only if the elasticity of intertemporal substitution is greater than one. If θ>1 employment is a negative function of productivity, while if θ=1 employment is independent of productivity. This applies because if θ<1 the intertemporal substitution effect dominates on the income effect, following a change in productivity and real wages. If θ>1 the income effect dominates on the intertemporal substitution effect, which in the case where θ=1 the two effects cancel each other out, and employment is not affected.

No other factor affects fluctuations in real variables. We see that, as in the competitive real business cycle model, monetary factors such as the money supply and nominal interest rates have no effect on the evolution of real variables.

14.1.4 The Inefficiency of the “Natural” Rate

However, in this model there is a significant difference from the competitive model of Chapter 13. Because of monopolistic competition, which implies a positive margin of prices over marginal costs of firms, both employment and output, as well as consumption and real wages, are determined at a lower level than in the case of perfect competition. Monopolistic competition implies a distortion in the market of goods and services, which leads to lower equilibrium employment and output and to lower real wages than with perfect competition. 2

If the productivity shock follows a stationary stochastic process with mean zero, then, from (14.22), the log of the steady state employment level will be equal to,

!

If ε>1, the steady state employment level will be lower than in the case of perfect competition.

Under perfect competition, goods are perfect substitutes in the preferences of consumers. Thus, steady state employment would be equal to,

!

Thus, because of imperfect competition, this model implies under employment relative to a fully competitive model, even when there is full flexibility of prices and wages. Through (14.23) and (14.24), this under employment implies that steady state output and steady state real wages will also be lower compared to perfect competition.

n_= − µ

θ(1−α )+α + λ= ln(1−α )− ln(ε / (ε −1))

θ(1−α )+α + λ

limε→∞

n_= ln(1−α )θ(1−α )+α + λ

See Akerlof and Yellen (1985), Mankiw (1985), Blanchard and Kiyotaki (1987) and Ball and Romer (1990) for the 2

first generation of “new keynesian” models that relied on monopolistic competition.!8


In all other respects, this model resembles the “new classical” competitive real business cycle model analyzed in Chapter 13.

14.2 Staggered Price Adjustment and Aggregate Fluctuations

In contrast to the “new classical” model, in “new keynesian” models one assumes gradual and not full adjustment of wages and prices towards their equilibrium values. In this chapter we shall assume gradual adjustment of prices and fully flexible wages in a competitive labor market.

A number of alternative “new keynesian” models of gradual price adjustment under monopolistic competition have been developed in the literature. We shall concentrate on one of them, the Calvo (1983) model, which is based on staggered pricing. 3

Following Calvo (1983), we shall assume that firms cannot freely adjust their prices in every period. For each firm, the probability of adjusting prices in any period is equal to 1-γ, which is constant and independent of the length of time that has elapsed since the last price adjustment by the firm. Thus, in each period, a proportion 1-γ of all firms adjust their prices, and the remaining proportion γ do not adjust their prices. This assumption has critical implications for the properties of the model, the nature of aggregate fluctuations and the effects of monetary shocks and monetary policy. 4

Under this assumption, in period t, the expected future duration of any price contract is given by,

!

From the definition of the price level, and the fact that all firms that reset their prices in period t set the same price, it follows that,

! (14.26)

where ! is the price set by the firms that reset their prices in the current period.

From (14.26) one can show that the dynamic adjustment of the price level is given by,

! (14.27)

(1− γ ) sγ ss=0

∞∑ = γ1− γ

Pt = γ Pt−1( )1−ε + (1− γ ) P_t

⎛⎝

⎞⎠

1−ε⎛⎝⎜

⎞⎠⎟

11−ε

P_

PtPt−1

⎛⎝⎜

⎞⎠⎟

1−ε

= γ + (1− γ ) Pt_

Pt−1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

1−ε

An observationally equivalent model, the Rotemberg (1982 a,b) model of quadratic costs of adjusting prices, is 3

analyzed in the Annex to this Chapter.

See Yun (1996) for the first analysis of the “new keynesian” dynamic stochastic general equilibrium model under the 4

assumption that prices are set as postulated by Calvo (1983).!9


In the steady state with inflation equal to π* we have that,

! (14.28)

A linear logarithmic approximation of (14.27) around a steady state inflation rate of π* yields,

! (14.29)

From (14.29) it follows that inflation exceeds its steady state level if firms that set prices in the current period set them at a higher level than the average price of the previous period adjusted for steady state inflation.

14.2.1 Optimal Pricing with Staggered Price Adjustment

In order to analyze the adjustment of inflation, one thus has to examine how firms decide on their optimal price, taking into account the fact that for a period in the future they may not be able to readjust their prices, while their competitors have the option of readjusting their own prices.

The problem of the firm that decides on the price it is about to set in period t is to set the price that maximizes the expected present value of its profits, given that the probability of readjusting its price in any future period is equal to 1-γ. Thus, all firms that readjust their prices in period t maximize,

! (14.30)

under the constraints of the production function,

! (14.31a)

and the demand function,

! (14.31b)

where, ! and ! is the volume of output and employment in period t+s, of the firm that has set its prices in period t. The higher the relative price of the firm in any period, the lower the demand for its product and thus the volume of its output and employment.

From the first order conditions for a maximum it follows that,

Pt = (1+π*)Pt−1 = Pt_

pt − pt−1 ! γπ *+(1− γ ) pt_− pt−1

⎛⎝

⎞⎠ = π *+(1− γ ) pt

_− pt−1 −π *

⎛⎝

⎞⎠

γ sEt1

1+ it+z

⎛⎝⎜

⎞⎠⎟z=0

s∏ P_t Yt+s

t −Wt+sLt+st⎛

⎝⎞⎠

⎛

⎝⎜⎞

⎠⎟s=0

∞∑

Lt+st = Yt+s

t

At+s

⎛⎝⎜

⎞⎠⎟

11−α

Yt+st = Pt

Pt+s

_⎛

⎝⎜⎜

⎞

⎠⎟⎟

−ε

Yt+s

Yt+st Lt+s

t

!10


!

(14.32)

(14.32) implies that the expected present value of revenues from the optimal price is equal to the expected present value of the marginal cost of production, augmented by the profit margin ε/(ε-1) of the firm.

It is worth noting that, as we have already shown (equation (14.16)), if the firm could determine its prices in every period, the price of the product in each period would be equal to the marginal cost of production plus the same profit margin. However, if the firm cannot adjust prices in every period, as is assumed in the Calvo (1983) model, pricing follows the rule (14.32).

Assuming that in the steady state inflation is equal to π*, (14.32) can be transformed in logarithmic deviations from the steady state equilibrium, using a log linear Taylor approximation. Thus, in logarithms we shall have that,

! (14.33)

where,

! and ! .

Consequently, firms that reset their prices in period t will choose a price which corresponds to a weighted average of the current and expected future price levels, plus a margin µ on a weighted average of the current and expected future level of real marginal costs. The discount factor of a future period t+s depends on the probability that the firm will not be able to reset its price in the future period t+s, which equals γs,, times the discount rate βs. Furthermore, the part of pricing which depends on the expected marginal cost of the firm depends negatively on the elasticity of demand for the product of the firm, through the parameter ω.

Using the future mathematical expectations operator F, (14.33) can be written as,

! (14.34)

Substituting (14.34) in the equation for the adjustment of the average price level (14.29) we get that,

! (14.35)

γ sEt1

1+ it+z

⎛⎝⎜

⎞⎠⎟z=0

s∏ (ε −1) P_t

Pt+s

⎛

⎝⎜⎜

⎞

⎠⎟⎟

−ε

Yt+s −ε

(1−α )P_t

Pt+s

⎛

⎝⎜⎜

⎞

⎠⎟⎟

−1−α+ε1−α

Wt+s

Pt+s

⎛⎝⎜

⎞⎠⎟Yt+sAt+s

⎛⎝⎜

⎞⎠⎟

11−α

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

s=0

∞∑ = 0

p_

t ! 1− βγ( ) βγ( )s=0

∞∑sEt pt+s +ω µ +wt+s − pt+s +

11−α

α yt+s − at+s( )⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

β = 11+ ρ

ω = 1−α1−α + ε

<1

p_

t !1− βγ1− βγ F

pt +ω µ + 1− βγ1− βγ F

wt − pt +1

1−αα yt − at( )⎛

⎝⎜⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

pt = π *+γ pt−1 + (1− γ )1− βγ1− βγ F

pt +ω µ + 1− βγ1− βγ F

wt − pt +1

1−αα yt − at( )⎛

⎝⎜⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

!11


Multiplying both sides of (14.35) by 1-βγF , after some rearrangements, we get that,

! (14.36)

(14.36) is the equation of adjustment of the price level towards the steady state price level, which is a constant markup on the marginal cost of production.

In order to examine the short run behavior of the model, we must introduce the equilibrium conditions in the markets for goods and services, labor and money.

14.2.2 Equilibrium in the Market for Goods and Services and the “New Keynesian” IS Curve

Equilibrium in the market for good j implies that,

!

As a result, equilibrium in the market for all goods requires that,

! (14.37)

where Υ is total output, defined in the same way as total consumption C in equation (14.2).

Substituting the Euler equation for consumption (14.13) in the equilibrium condition (14.37), the logarithm of real output is determined by,

! (14.38)

(14.38) is often referred to as the new keynesian IS curve, as it is derived from the equilibrium condition for the market for goods and services. Compared to the conventional IS curve, (14.38) contains the rational expectation about the future volume of output and depends on the real and not just the nominal interest rate. Its advantage over the conventional IS curve is that it has been derived from explicit microeconomic foundations, and that its parameters depend on deep structural parameters, such as the pure rate of time preference of the representative household ρ, and the intertemporal elasticity of substitution in consumption 1/θ.

14.2.3 Labor Market Equilibrium and the “New Keynesian” Phillips Curve

We next turn to the equilibrium condition in the labor market. We assume that in contrast to product prices that adjust gradually, nominal wages adjust immediately in order to equate the demand and supply of labor in each period. This assumption is made for reasons of analytical simplicity. Thus, the only nominal stickiness which is analyzed in this version of the new Keynesian model is the gradual adjustment of prices rather than wages. This means that fluctuations in employment are the result of intertemporal substitution by households and that no involuntary unemployment exists.

(1+ β )pt − pt−1 − βEt pt+1 =1− βγγ

π *+ (1− γ )(1− βγ )γ

ω µ + wt − pt +1

1−αα yt − at( )⎛

⎝⎜⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

Yt ( j) = Ct ( j)

Yt = Ct

yt = Et (yt+1)−1θit − Etπ t+1 − ρ( )

!12


Gali (2008, 2011) and others have analyzed this model with the additional assumption of rigidity not only in prices but also in nominal wages. In this case there are fluctuations in the unemployment rate due to the fact that wages are not equate the demand with the supply of labor in each period, as is assumed here.

Due to the gradual adjustment of prices, firms produce so as to satisfy aggregate demand at the given prices in each period. Aggregate output is determined at the level which is determined by aggregate demand, and differs from its “natural level”, which is the level that would prevail if there was immediate price adjustment by all firms.

As a result, aggregate output, employment, consumption, real wages and the real interest rate, differ from their “natural levels” and display fluctuations which depend on nominal as well as real disturbances.

From the price adjustment equation (14.36), we can deduce an equation for fluctuations in inflation. Expressing (14.36) as an inflation equation we have that,

! (14.39)

where π is the rate of inflation, defined as,

! .

(14.39) implies that deviations of current inflation from steady state inflation are greater than discounted expected deviations of future inflation from steady state inflation, if the current marginal cost of labor, plus the margin µ is higher than the current price level p. The reason is that firms setting prices in the current period post larger price increases than (discounted) expected future inflation, in order to offset the higher current marginal cost of labor.

The assumption of equilibrium in the labor market means that we can substitute the real wage in (14.39) from the first order condition (14.12) for the representative household. Using (14.12), the condition for equilibrium in the market for goods and services c=y, and the production function (14.17), (14.39) can be rewritten as,

! (14.40)

where yN is the “natural rate” of real output, i.e. the output that would be produced if there was full flexibility of prices, and is given by (14.23). The parameter κ is defined as,

! > 0.

(14.40) is referred to as the new keynesian Phillips curve, and constitutes the second important behavioral equation of the imperfectly competitive new keynesian dynamic stochastic general equilibrium model.

π t = (1− β )π *+βEtπ t+1 +(1− γ )(1− βγ )

γω µ + wt − pt +

11−α

α yt − at( )⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

π t = pt − pt−1

π t = (1− β )π *+βEtπ t+1 +κ yt − ytN( )

κ = (1− γ )(1− βγ )γ

θ(1−α )+ λ +α1−α + ε

!13


The reason that deviations of output from its “natural rate” cause inflation to rise relative to expected future inflation is that higher output implies a higher current real marginal cost of labor, and thus induces firms that have the opportunity to change their current price, to post price increases which exceed discounted expected future inflation.

Like the new keynesian IS curve, the new keynesian Phillips curve has been derived from explicit microeconomic foundations, and its parameters are functions of the “deep” structural parameters describing the preferences of households, the technology of production and the price setting technology.

14.2.4 The Structure of the Imperfectly Competitive Model with Staggered Pricing

Equations (14.38) and (14.40), along with equations (14.23) and (14.25) for the “natural level” of real output and the real interest rate constitute the basic structure of the imperfectly competitive “new keynesian” model.

Deviations of inflation from discounted expected future inflation are determined by the new keynesian Phillips curve (14.40), as a function of deviations of real aggregate demand and output from the “natural level” of output.

Deviations of real output from its “natural level” are determined by the new keynesian IS curve, which depends on deviations of the real interest rate from its “natural level”. The new keynesian IS curve can be expressed as,

! (14.41)

where the natural levels of output and the real interest rate yN, rN are determined by (14.23) and (14.25).

In order to close the model we must consider the determination of the nominal interest rate. In contrast to the “new classical” model, due to staggered price adjustment, fluctuations in real variables cannot be determined without reference to monetary factors. Monetary factors and monetary policy determine not only the price level and inflation, as in the “new classical” model, but also fluctuations in real variables such as real output, consumption and employment, real wages and the real interest rate.

We shall analyze the model under the assumption that the central bank follows a Taylor (1993) rule of the form,

! (14.42)

yt − ytN = Et yt+1 − yt+1

N( )− 1θ it − Etπ t+1 − rtN( )

it = ρ +π *+φπ π t −π *( )+φy(yt − ytN )+ vt

!14


where φπ and φy are positive coefficients, and v is an exogenous stochastic disturbance in the nominal interest rate. It is worth noting that because the constant in this rule is equal to ρ+π*, this rule is consistent with steady state inflation equal to π*. 5

This rule implies a countercyclical monetary policy. When inflation is above π*, the central bank increases nominal interest rates in order to reduce it. When employment is low, i.e. when output is lower than its “natural” level, the central bank reduces nominal interest rates in order to increase employment and nudge output towards its “natural” level. This feedback interest rate rule does not result in inflation and price level indeterminacy if the Taylor principle is satisfied, i.e. if the reaction of nominal interest rates to inflation is sufficiently strong. 6

The stochastic disturbance in the nominal interest rate is assumed to follow an AR(1) process of the form,

! (14.43)

where 0<ηv<1, and εv is a white noise process.

The productivity shock a is also assumed to follow an AR(1) process of the form,

! (14.44)

where 0<ηa<1, and εa is a white noise process.

14.2.5 Real and Monetary Shocks and Aggregate Fluctuations

Having now fully determined the “new keynesian” model with staggered pricing, we can analyze how nominal and real disturbances produce aggregate fluctuations.

The full model consists of the “new” Phillips and IS curves (14.41) and (14.40), the Taylor rule (14.42), and the equation for the determination of the “natural” real interest rate (14.25). Thus, the model can be written as,

! (14.45)

! (14.46)

vt =ηvvt−1 + ε tv

at =ηaat−1 + ε ta

π^

t = βEt π^

t+1+κ y^

t

y^

t = Et y^

t+1−1θ

it −π *−Et π^

t+1− rtN⎛

⎝⎞⎠

Note that in the Taylor rule (14.42), the nominal interest rate does to react to shocks that change the “natural” real rate 5

of interest, such as productivity shocks, and thus productivity shocks turn out to affect deviations of output from its “natural rate” and inflation. Obviously, one can also analyze the imperfectly competitive new keynesian model under the assumption that monetary policy follows a rule for the money supply and not nominal interest rates. See Gali (2008).

See Chapters 10, 11 and 13 for discussions of the properties of interest rate rules, as well as Woodford (2003), for a 6

more extensive and complete analysis.!15


! (14.47)

where, ! , ! denote the deviations between current and target inflation, and between current (log) real output and its “natural” level.

The percentage deviation between current real output and its “natural” level is often referred to as excess output, and its opposite is referred to as the output gap. When excess output is positive, the economy produces more than its “natural” level, while when it is negative (the output gap is positive) it produces less than its “natural” level.

The “natural” real rate of interest rN is determined by (14.25) and depends only on the expected change in the productivity shock and the pure rate of time preference.

The model can be solved by substituting the Taylor rule for the nominal interest rate in the “new” IS curve (14.46) and solving for excess output. This substitution results in an aggregate demand function of the form,

! (14.48)

Combined with the “new” Phillips curve (14.45), the model can determine excess output and inflation, as functions of the productivity shocks affecting the “natural” real rate of interest, and the monetary shocks, affecting the nominal interest rate.

One can then solve the two equation system (14.45) and (14.48) by substituting out for excess output from (14.48) into (14.45). This will, after some re-arrangement, result in an inflation equation of the form,

! (14.49)

As can be seen from (14.49), deviations of inflation from steady state inflation depend on expectations about future inflation deviations, productivity shocks, which affect the “natural” real rate of interest, and nominal interest rate shocks. Recalling the definition of κ in (14.40), one can see that the parameters determining the inflation process depend on the preferences of the representative household, (θ, λ, ε and ρ), the technology of production (α), market structure (ε), the price adjustment mechanism (γ) and the parameters of the Taylor rule (φπ and φy).

Since inflation is an non predetermined variable, both roots of (14.49) must be less than unity (inside the unit circle), which requires that the sum of the coefficients of the future expectations of inflation in (14.49) is less than one. Thus, for a stable inflation process a necessary and sufficient condition is that,

! (14.50)

it = ρ +π *+φπ π^

t+φy y^

t+ vt

π^

t = π t −π * y^

t = yt − ytN

y^

t =θ

θ +φyEt y

^

t+1−ηπ

θ +φyπ^

t+1

θ +φyEt π

^

t+1−1

θ +φyθψ (1−ηa )at + vt( )

π^

t =β(θ +φy )+θ +κθ +φy +κφπ

Et π^

t+1−βθ

θ +φy +κφπ

Et π^

t+2−κ

θ +φy +κφπ

θψ (1−ηa )at + vt( )

β(θ +ηy )+θ +κ − βθθ +ηy +κηπ

<1

!16


Under the assumption that the coefficients of the Taylor rule φπ and φy are positive, after some re-arrangement of (14.50), one can see that a necessary and sufficient condition for a stable inflation process is,

! (14.51)

Thus, for a stable inflation process, the Taylor principle, as implied by (14.51) must hold. The reaction of nominal interest rates to current inflation must be sufficiently high to satisfy (14.51). For example, if the reaction of the nominal interest rate to excess output is zero (φy =0), then the reaction of nominal interest rates to inflation must exceed unity.

Assuming that the central bank policy satisfies (14.51), (14.49) can be solved forward for inflation, which is a non pre-determined variable. Inflation will be determined by,

! (14.52)

where 1/λ1 and 1/λ2 are the two roots of the inflationary process (14.49), which are both lower than one, if the Taylor principle (14.51) is satisfied. λ1 and λ2 will both be higher than one if the Taylor principle is satisfied, and will be defined by,

!

(14.53)

!

Inflation will depend on future expectations of deviations of the “natural” real rate of interest from ρ and future expectations about nominal interest rate shocks v. The solution for inflation, assuming that the two shocks follow AR(1) processes as in (14.43) and (14.44), takes the form,

! (14.54)

where,

! (14.55)

! (14.56)

Note that the coefficients Λa and Λv depend negatively on the Taylor rule parameters φπ and φy which are in the denominator of both fractions. The higher the Taylor rule parameters, the lower the impact of real and nominal shocks on deviations of inflation from the central bank target.

φπ >1−(1− β )κ

φy

π^

t = − κβθ

1λ1λ2

Et1λ1

⎛⎝⎜

⎞⎠⎟s=0

∞∑s

1λ2

⎛⎝⎜

⎞⎠⎟q=0

∞∑q

θψ (1−ηa )at+q + vt+q( )

λ1 + λ2 =β(θ +φy )+θ +κ

βθ> 2

λ1λ2 =θ +φy +κφπ

βθ>1

π^

t = −θψκ (1−ηa )Λaat −κΛvvt

Λa =1

(1− βηa )(φy +θ(1−ηa ))+κ (φπ −ηa )

Λv =1

(1− βηv )(φy +θ(1−ηv ))+κ (φπ −ηv )

!17


By substituting the solution for inflation in the “new” Phillips curve (14.45), one can get the corresponding solution for the evolution of excess output. Excess output will be given by,

! (14.57)

From (14.57), because the coefficients Λa and Λv depend negatively on the Taylor rule parameters φπ and φy , the higher are the Taylor rule parameters, the lower the impact of real and nominal shocks on deviations of output from its “natural” level.

The thing to note however is that both monetary and real shocks affect fluctuations of real output and inflation in this model. Thus, the “classical” dichotomy does not hold. Persistent shocks to either nominal interest rates or productivity, result in persistent deviations of inflation from the target of the central bank, and persistent deviations of output from its “natural” rate.

This “new keynesian” model, unlike “new classical” models, can explain aggregate fluctuations caused by monetary shocks. These shocks are transmitted to real variables and persist over time through staggered pricing. Such monetary cycles cannot result from models with immediate and full adjustment of wages and prices, unless there is imperfect current information as in the Lucas (1972) model.

14.2.6 A Dynamic Simulation of the Model

In order to evaluate the quantitative properties of the model summarized in (14.45)-(14.47), it is worth simulating it for particular parameter values.

Figures 14.1 and 14.2 present the dynamic effects of both monetary and real shocks, for commonly used values of the parameters.

In Figure 14.1 we present the dynamic effects of a 0.25 shock εv in nominal interest rates. This shock by 25 basis points leads to an increase of the nominal and the real interest rate and reduces excess production and inflation. Because this shock does not affect the "natural" level of output, real output, employment and real wages decline. The economy gradually returns to long-run equilibrium, as the effects of the monetary shock gradually peter out.

In Figure 14.2 we present the dynamic effects of a 0.25 shock εa to productivity. This shock leads to a prolonged rise of the “natural” level of output, a reduction of excess output and inflation, an increase in the real wage, and a decline in nominal and real interest rates. The reduction in real interest rates leads to an increase in actual output, which however is smaller than the increase of the “natural” level of output. This is the reason that excess output falls. Again, the economy gradually returns to its steady state equilibrium, as the effects of the real disturbance gradually peter out. 14.3 Conclusions

In this chapter we have analyzed the structure and the properties of a “new keynesian” model, based on monopolistic competition and staggered pricing. Unlike traditional Keynesian models, in which the basic relations are not derived explicitly from microeconomic foundations, this new Keynesian

y^

t = −θψ (1−ηa )(1− βηa )Λaat − (1− βηv )Λvvt

!18


model is a dynamic stochastic general equilibrium model based on explicit microeconomic foundations, analogous to those of “new classical” models.

After presenting the properties of this model, we analyzed the effects of monetary and real shocks with regard to fluctuations in excess output and inflation.

This “new keynesian” model, unlike “new classical” models, can explain aggregate fluctuations caused by monetary shocks. These shocks are transmitted to real variables and persist over time through staggered pricing. Such monetary cycles cannot result from models with immediate and full adjustment of wages and prices, unless there is imperfect information as in the Lucas (1972) model.

As in the fully competitive “new classical” models, even under monopolistic competition, when there is full adjustment of prices and wages, monetary shocks affect only nominal and not real variables such as output, consumption, employment, real wages and real interest rates.

It is worth noting that there is no unemployment in this model. Output deviates from its “natural rate”, because of staggered pricing, but fluctuations in employment are due to intertemporal substitution, since the labor market is assumed fully competitive. This is a significant weakness of this particular model, a weakness shared with the Lucas and Rapping (1969) and the Lucas (1972) model. This weakness can be addressed if one were to allow for labor market distortions that can account for unemployment. In the next chapter we present a model with labor market distortions that can account for both involuntary unemployment and the effects of nominal shocks and monetary policy.

!19


Annex to Chapter 14 The Rotemberg Model of Convex Costs of Price Adjustment

An alternative model of sluggish price adjustment is the Rotemberg (1982 a,b) model of costly price changes.

For the representative monopolistically competitive firm, as the one examined in section 14.1.2, the optimal price is given by,

! (A14.1)

The optimal price is a constant markup on marginal costs. Marginal costs are equal to wage costs over the marginal productivity of labor. Note that because of decreasing returns to employment, increasing employment and output implies declining marginal productivity of labor and increasing marginal costs of production. Using the production function to substitute out for labor, (A14.1) can also be expressed as,

! (A14.1΄)

An increase in output increases the marginal costs of production for given wages, because of the declining marginal productivity of labor. Hence, with higher output the optimal price must rise.

In logs, (A14.1) and (A14.1΄) imply,

! (A14.2)

where,

! , ! .

a is the logarithm of the exogenous productivity shock, and the constant µ is the logarithm of the markup on marginal cost, minus the logarithm of the coefficient implying decreasing returns to labor.

All firms, are assumed to be facing convex costs of adjusting prices. Rotemberg (1982 a,b) assumes that firms balance the costs of deviating from their optimal price against the costs of adjusting prices. In the model that follows, following Rotemberg, we assume that firms set current prices minimizing a quadratic cost function which penalizes both deviations of prices from the optimal price, and the adjustment of prices over steady state inflation from period to period. This takes the form,

P_t =

εε −1

Wt

(1−α )AtLt−α

⎛⎝⎜

⎞⎠⎟

P_t =

εε −1

Wt Yt( )α1−α

(1−α ) At( )11−α

⎛

⎝⎜⎜

⎞

⎠⎟⎟

p_

t = µ +wt − at +αlt = µ +wt +1

1−αα yt − at( )

at = lnAt µ = ln εε −1

⎛⎝⎜

⎞⎠⎟ − ln(1−α )

!20


! (A14.3)

where p is the log of the actual price of the representative firm. ξ is a parameter measuring the cost of price adjustment relative to the cost of deviations from the optimal price.

From the first order conditions for the minimization of (A14.3), it follows that,

! (A14.4)

The current price, in logs, is a weighted average of the optimal price, the past price and the expected future price. The firm is forward looking, and anticipates the future costs of adjusting prices, so its current price depends not only on its past price, but on its expected future price as well. Since this is the representative firm, we can take in to be equal to the log of the price level.

Expressing (A14.4) as an inflation equation, one gets,

! (A14.5)

where πt=pt-pt-1 is the rate of inflation.

Inflation deviates from expected future inflation, to the extent that the optimal price exceeds the current price. Substituting for the optimal price from (A14.2), one gets,

! (A14.6)

where, ! .

From (A14.6), deviations of current inflation differ from expected deviations of future inflation, to the extent that the marginal cost of production plus the optimal price markup exceeds the current price. Using the labor and product market equilibrium conditions to substitute out for the real wage and employment, as well as the definition of the “natural rate” of output, we can express (A14.6) as,

! (A14.7)

where ! and ! .

(A14.7) has exactly the same form as the “new keynesian” Phillips curve (14.40) derived from the Calvo (1983) model of staggered pricing. The only difference is in the definition of κ which is now in terms of the parameter ξ of the Rotemberg model, instead of the parameter γ of the Calvo model. Thus, the two models of sluggish price adjustment, the Rotemberg model of costs of adjustment of

Λt = Et β ss=0

∞∑ 12(pt+s − p

_

t+s )2 + ξ2(pt+s − pt+s−1 −π*)

2⎛⎝⎜

⎞⎠⎟

pt =1

1+ ξ(1+ β )p_

t+ξ

1+ ξ(1+ β )pt−1 +

ξβ1+ ξ(1+ β )

Et pt+1 +ξ(1+ β )1+ ξ(1+ β )

π *

π t −π*= β Etπ t+1 −π *( )+ 1ξ

p_

t− pt⎛⎝

⎞⎠

π^

t = βEt π^

t+1+1ξ

µ +wt − at +αlt − pt( )

π^

t = π t −π *

π^

t = βEt π^

t+1+κ y^

t

κ = θ(1−α )+α + λξ(1−α )

> 0 y^

t = yt − ytN

!21


prices and the Calvo model of staggered pricing are observationally equivalent at the aggregate level.

!22


References

Akerlof G. and Yellen J. (1985), “A Near-Rational Model of the Business Cycle with Wage and Price Inertia”, Quarterly Journal of Economics, 100, Supplement, pp. 823-838.

Ball L. and Romer D. (1990), “Real Rigidities and the Non-Neutrality of Money”, Review of Economic Studies, 57, pp. 183-203.

Blanchard O.J. and Kahn C. (1980), “The Solution of Linear Difference Equations under Rational Expectations”, Econometrica, 48, pp. 1305-1311.

Blanchard O.J. and Kiyotaki N. (1987), “Monopolistic Competition and the Effects of Aggregate Demand”, American Economic Review, 77, pp. 647-666.

Bullard J. and Mitra K. (2002), “Learning about Monetary Policy Rules”, Journal of Monetary Economics, 49, pp. 1005-1129.

Calvo G. (1983), “Staggered Prices in a Utility Maximizing Framework”, Journal of Monetary Economics, 12, pp. 383-398.

Gali J. (2008), Monetary Policy, Inflation and the Business Cycle, Princeton N.J., Princeton University Press.

Gali J. (2011), Unemployment Fluctuations and Stabilization Policies: A New Keynesian Perspective, Cambridge Mass., The MIT Press.

Lucas R.E. Jr (1972), “Expectations and the Neutrality of Money”, Journal of Economic Theory, 4, pp. 103-124.

Mankiw G. (1985), “Small Menu Costs and Large Business Cycles: A Macroeconomic Model of Monopoly”, Quarterly Journal of Economics, 100, pp. 529-539.

Rotemberg J. (1982a), “Monopolistic Price Adjustment and Aggregate Output”, Review of Economic Studies, 44, pp. 517-531.

Rotemberg J. (1982b), “Sticky Prices in the United States”, Journal of Political Economy, 90, pp. 1187-1211.

Taylor J.B. (1993), “Discretion versus Policy Rules in Practice”, Carnegie-Rochester Conference Series on Public Policy, 39, pp. 195-214.

Woodford M. (2003), Interest and Prices: Foundations of a Theory of Monetary Policy, Princeton N.J., Princeton University Press.

Yun T. (1996), “Nominal Price Rigidity, Money Supply Endogeneity and Business Cycles”, Journal of Monetary Economics, 37, pp. 345-370.

!23


Figure 14.1 The Dynamic Behavior of the Staggered Pricing Model

Following a Contractionary Monetary Shock

Note: The parameter values for this simulation are: θ=1, λ=1, ρ=0.01, α=0.333, ε=6, γ=0.667, φπ=1.50, φy=0.125, ηa=0.90, ηv=0.50.

!24


Figure 14.2 The Dynamic Behavior of the Staggered Pricing Model

Following a Positive Productivity Shock

Note: The parameter values for this simulation are: θ=1, λ=1, ρ=0.01, α=0.333, ε=6, γ=0.667, φπ=1.50, φy=0.125, ηa=0.90, ηv=0.50.

!25

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