Sticky Prices versus Sticky Information A Horse Race - Fabrice Collard

May 11, 2004 - where the gross growth rate of the money supply, µt, is assumed to ... Definition 1 An equilibrium of this economy is a sequence of prices {Pt}∞ .... But few would accept nowadays that money supply misperceptions play a.
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Sticky Prices versus Sticky Information A Horse Race Fabrice Collard∗and Harris Dellas† May 11, 2004

Abstract To be written . . . Key Words: JEL Class.:



CNRS-GREMAQ, Manufacture des Tabacs, bˆ at. F, 21 all´ee de Brienne, 31000 Toulouse, France. Tel: (33-5) 61–12–85–42, Fax: (33–5) 61–22–55–63, email: [email protected], Homepage: http://fabcol.free.fr † Department of Economics, University of Bern, CEPR, IMOP. Address: VWI, Gesellschaftsstrasse 49, CH 3012 Bern, Switzerland. Tel: (41) 31–631–3989, Fax: (41) 31–631-3992, email: [email protected], Homepage: http://www-vwi.unibe.ch/amakro/dellas.htm

1

Sticky Prices versus Sticky Information: A Horse Race

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Introduction To be written . . .

1

The Model

The set up is the standard NNS model. The economy is populated by a large number of identical infinitely–lived households and consists of two sectors: one producing intermediate goods and the other a final good. The intermediate good is produced with capital and labor and the final good with intermediate goods. The final good is homogeneous and can be used for consumption (private and public) and investment purposes.

1.1

The Household

Household preferences are characterized by the lifetime utility function:1 ∞ X τ =0

Et β τ U

  Mt+τ , ℓt+τ Ct+τ , Pt+τ

(1)

where 0 < β < 1 is a constant discount factor, C denotes the domestic consumption bundle, M/P is real balances and ℓ is the quantity of leisure enjoyed by the representative household. The  utility function,U C, M P , ℓ : R+ × R+ × [0, 1] −→ R is increasing and concave in its arguments. The household is subject to the following time constraint ℓt + ht = 1

(2)

where h denotes hours worked. The total time endowment is normalized to unity. In each and every period, the representative household faces a budget constraint of the form Bt+1 + Mt + Pt (Ct + It + Tt ) ≤ Rt−1 Bt + Mt−1 + Nt + Πt + Pt Wt ht + Pt zt Kt

(3)

where Wt is the real wage; Pt is the nominal price of the domestic final good;.Ct is consumption and I is investment expenditure; Kt is the amount of physical capital owned by the household and leased to the firms at the real rental rate zt . Mt−1 is the amount of money that the household brings into period t, and Mt is the end of period t money holdings. Nt is a nominal lump–sum transfer received from the monetary authority; Tt is the lump–sum taxes paid to the government and used to finance government consumption. 1 Et (.) denotes mathematical conditional expectations. Expectations are conditional on information available at the beginning of period t.

Sticky Prices versus Sticky Information: A Horse Race

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Capital accumulates according to the law of motion 2  ϕ It − δ Kt + (1 − δ)Kt Kt+1 = It − 2 Kt

(4)

where δ ∈ [0, 1] denotes the rate of depreciation. The second term captures the existence of capital adjustment costs. ϕ > 0 is the capital adjustment costs parameter. The household determines her consumption/savings, money holdings and leisure plans by maximizing her utility (1) subject to the time constraint (2), the budget constraint (3) and taking the evolution of physical capital (4) into account.

1.2

Final sector

The final good is produced by combining intermediate goods. This process is described by the following CES function Yt =

Z

1

0

 θ1 Yt (i) di θ

(5)

where θ ∈ (−∞, 1). θ determines the elasticity of substitution between the various inputs. The producers in this sector are assumed to behave competitively and to determine their demand for each good, Xt (i), i ∈ (0, 1) by maximizing the static profit equation Z 1 Pt (i)Yt (i)di max Pt Yt − {Xt (i)}i∈(0,1)

(6)

0

subject to (5), where Pt (i) denotes the price of intermediate good i. This yields demand functions of the form: Yt (i) = and the following general price index



Pt =

Pt (i) Pt Z



1 θ−1

1

Pt (i) 0

Yt for i ∈ (0, 1)

θ θ−1

 θ−1 θ di

(7)

(8)

The final good may be used for consumption — private or public — and investment purposes.

1.3

Intermediate goods producers

Each firm i, i ∈ (0, 1), produces an intermediate good by means of capital and labor according to a constant returns–to–scale technology, represented by the Cobb–Douglas production function Yt (i) = At Kt (i)α ht (i)1−α with α ∈ (0, 1)

(9)

where Kt (i) and ht (i) respectively denote the physical capital and the labor input used by firm i in the production process. At is an exogenous stationary stochastic technology shock, whose

Sticky Prices versus Sticky Information: A Horse Race

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properties will be defined later. Assuming that each firm i operates under perfect competition in the input markets, the firm determines its production plan so as to minimize its total cost min

{Kt (i),ht (i)}

Pt Wt ht (i) + Pt zt Kt (i)

subject to (9). This leads to the following expression for total costs: Pt St Yt (i) where the real marginal cost, S, is given by

Wt1−α ztα χAt

with χ = αα (1 − α)1−α

Intermediate goods producers are monopolistically competitive, and therefore set prices for the good they produce. We consider two price setting behavior. The first one assumes that only a fraction of the firms can reset their prices for a fixed number of periods, leading to a Taylor type price setting behavior, as investigated by Chari, Kehoe and McGrattan (2001). In the second setup, we follow Reis and Mankiw (2001) and assume that firms make their decisions based on imperfect information. Every price setter sets a new price every period, but they collect information about the economy and update their decisions slowly over time.

Sticky Prices:

We follow Chari, Kehoe and McGrattan (2001) and assume that intermediate

producers set prices for N periods of time in a staggered fashion. In each and every period, a fraction 1/N of producers choose a new price pet (i) that will remain operative for the N next periods. We assume that while this price does not react to shock, it however grows at the steady state inflation rate, such that

Pt+j (i) = π j pet (i) for j = 1, . . . , N

We assume that intermediate producers are indexed such that producers i ∈ [0, 1/N ] set new prices in 0, N , 2N , . . . , those indexed by i ∈ [1/N, 2/N ] set prices in 1, N + 1, 2N + 1, . . . Prices are set so as to maximize the expected sum of discounted profits from period t to period t+N −1, that is Et

N −1 X τ =0

Φt+τ ((π τ pet (i) − Pt+τ St+τ )Yt+τ (i))

subject to the total demand it faces

Yt (i) =



Pt (i) Pt



1 θ−1

Yt

Sticky Prices versus Sticky Information: A Horse Race

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where Φt+τ is an appropriate discount factor related to the way the household values future as opposed to current consumption. This leads to the price setting equation

pet (i) =

1 θ

Et

N −1 h X

1

π θ−1

τ =0 N −1 h X

Et

π



θ θ−1

τ =0

2−θ 1−θ St+τ Yt+τ Φt+τ Pt+τ



(10) 1 θ−1

Φt+τ Pt+τ Yt+τ

Since the price setting scheme is independent of any firm specific characteristic, all firms that reset their prices will choose the same price. Hence, from (8), the aggregate intermediate price index is given by 1 N

Pt =

Sticky Information:

N −1 X i=0

π i pet−i

θ  θ−1

! θ−1 θ

(11)

We follow Reis and Mankiw (2001) and assume information diffuses

slowly through the population of price setters. More specifically, we assume that in each an every period, all intermediate good producers reset their price, but using only a restricted piece of information. More specifically, we assume that the population of firms can be split into N parts, each of them indexed by i ∈ [0, N − 1]. Then each fraction of firm i/N is able to use information available in period t − i. This implies that a firm i sets its price maximizing its profit Et−i ((e pt (i) − Pt St )Yt (i)) subject to the total demand it faces Yt (i) =



Pt (i) Pt



1 θ−1

Yt

which leads to the price setting behavior for firm i

pet (i) = Et−i



Pt St θ



(12)

Hence, from (8), the aggregate intermediate price index is given by Pt =

1.4

1 N

N −1  X i=0

Et−i



Pt St θ



θ θ−1

! θ−1 θ

(13)

The monetary authorities

Monetary policy are assumed to follow an exogenous money supply rule, such that Mt = exp(µt )Mt−1

(14)

Sticky Prices versus Sticky Information: A Horse Race

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where the gross growth rate of the money supply, µt , is assumed to follow an exogenous stochastic process whose properties will be defined later.

1.5

The government

The government finances government expenditure on the domestic final good using lump sum taxes. The stationary component of government expenditures is assumed to follow an exogenous stochastic process, whose properties will be defined later.

1.6

The equilibrium

We now turn to the description of the equilibrium of the economy. Definition 1 An equilibrium of this economy is a sequence of prices {Pt }∞ t=0 = {Wt , zt , Pt , Rt , F ∞ H ∞ ∞ Pt (i), i ∈ (0, 1)}∞ t=0 and a sequence of quantities {Qt }t=0 = {{Qt }t=0 , {Qt }t=0 } with ∞ {QH t }t=0 = {Ct , It , Bt , Kt+1 , ht , Mt } ∞ ∞ {QH t }t=0 = {Yt , Yt (i), Kt (i), ht (i); i ∈ (0, 1)}t=0

such that: H ∞ (i) given a sequence of prices {Pt }∞ t=0 and a sequence of shocks, {Qt }t=0 is a solution to the

representative household’s problem; F ∞ (ii) given a sequence of prices {Pt }∞ t=0 and a sequence of shocks, {Qt }t=0 is a solution to the

representative firms’ problem; ∞ (iii) given a sequence of quantities {Qt }∞ t=0 and a sequence of shocks, {Pt }t=0 clears the markets

Yt = Ct + It + Gt Z 1 ht (i)di ht = 0 Z 1 Kt (i)di Kt =

(15) (16) (17)

0

Gt = Tt

(18)

and the money market. (iv) Prices satisfy either (10) and (11), or (12) and (13) depending on the price setting behavior.

Sticky Prices versus Sticky Information: A Horse Race

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Parametrization

The model is parameterized on US quarterly data for the post WWII period. The data are taken from the Federal Reserve Database.2 The parameters are reported in table 1. β, the discount factor is set such that households discount the future at a 4% annual rate, implying β equals 0.988. The instantaneous utility function takes the form   !1−σ ν    η η Mt 1  M t ℓ1−ν − 1 U Ct , , ℓt = Ctη + ζ t Pt 1−σ Pt

where ζ capture the preference for money holdings of the household. σ, the coefficient ruling

risk aversion, is set equal to 1.5. ν is set such that the model generates a total fraction of time devoted to market activities of 31%. η is borrowed from Chari et al. (2000), who estimated it on postwar US data (-1.56). The value of ζ, 0.0649, is selected such that the model reproduces the average ratio of M1 money to nominal consumption expenditures. γ, the probability of price resetting is set in the benchmark case at 0.25, implying that the average length of price contracts is about 4 quarters. The nominal growth of the economy, µ, is set such that the average quarterly rate of inflation over the period is π = 1.2% per quarter. The quarterly depreciation rate, δ, is set equal to 0.025. θ in the benchmark case is set such that the level of markup in the steady state is 15%. α, the elasticity of the production function to physical capital, is set such that the model reproduces the US labor share — defined as the ratio of labor compensation to GDP — during the sample period (0.575). The stochastic technology shock, at = log(At /A), is assumed to follow a stationary AR(1) process of the form at = ρa at−1 + εa,t with |ρa | < 1 and εa,t ; N (0, σa2 ). We set ρa = 0.95 and σa = 0.008.

The government spending shock3 is assumed to follow an AR(1) process log(gt ) = ρg log(gt−1 ) + (1 − ρg ) log(g) + εg,t with |ρg | < 1 and εg,t ∼ N (0, σg2 ). The persistence parameter is set to, ρg , of 0.97 and the standard deviation of innovations is σg = 0.02. The government spending to output ratio is set to 0.20. 2 3

URL: http://research.stlouisfed.org/fred/ The –logarithm of the– government expenditure series is first detrended using a linear trend.

Sticky Prices versus Sticky Information: A Horse Race

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Table 1: Calibration: Benchmark case Preferences Discount factor Relative risk aversion Parameter of CES in utility function Weight of money in the utility function CES weight in utility function Technology Capital elasticity of intermediate output Capital adjustment costs parameter Depreciation rate Parameter of markup Length of contracts/information stickiness Shocks and policy parameters Persistence of technology shock Standard deviation of technology shock Persistence of government spending shock Volatility of government spending shock Persistence of money growth Volatility of money shock Steady state money supply growth (gross) Share of government spending

β σ η ζ ν

0.988 1.500 -1.560 0.065 0.344

α ϕ δ θ N

0.281 2.000 0.025 0.850 2 or 4

ρa σa ρg σg ρm σm µ g/y

0.950 0.008 0.970 0.020 0.500 0.007 1.012 0.200

Sticky Prices versus Sticky Information: A Horse Race

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When the monetary authorities are assumed to stick to an exogenous money supply rule, gross money growth is assumed to be of the form µt = (1 − ρm )µ + ρm µt−1 + ǫmt where |ρm | < 1, µ = E(µt ) and ǫmt is a gaussian white noise process. When an HTM rule is assumed, we use the values of ρr = 0.75, kπ = 1.5 and ky = 0.15 suggested by Clarida, Gali and Gertler, 2000.

In order to investigate the role of information imperfection, we will assume that some of the variables are observed with error by the agents. In particular, for mis–measured variable x x⋆t = xT t + ξt where xT t denotes the value of the variable under perfect information and ξt is a noisy process that satisfies E(ξt ) = 0 for all t; E(ξt εa,t ) = E(ξt εg,t ) = 0; and E(ξt ξk ) =



σξ2 if t = k 0 Otherwise

We assume symmetric information for the government and the private agents. Their learning is based on the Kalman filter. In order to facilitate the interpretation of σξ we set its value in relation to the volatility of the technology shock. More precisely, we define ς as ς = σξ /σa . We experiment with different values but end up reporting results with two values of ς, namely, ς = {3}. The latter value is close to that used elsewhere in the literature, for instance by Woodford, 2002. One expects that the choice of the noisy variables would affect the properties of the model. While some variables are more likely to be observed with error than others, there is nothing in the literature that could help us operationalize the incidence and degree of mis-measurement. In the case of activistic policy, it seems natural to assume that potential output is observed with noise. This is a standard assumption in the literature. In the case of monetary targeting, potential output does not enter the system at all, so other signal(s) must be selected. Imperfect observation of aggregate money was the key building block of the imperfect information rational expectations model. But few would accept nowadays that money supply misperceptions play a key role in business cycles so we have decided against using it. In the analysis below we have used alternative noisy variables. Encouragingly, the results are quite robust with regard to the location of the noise, which indicates that the key element is the existence of noise in the system rather than its exact location.

Sticky Prices versus Sticky Information: A Horse Race

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Results

Figure 1 and 2 report the impulse response functions of output, inflation and the nominal interest rate to, respectively a technology, a fiscal and a money supply shock. Figure 1 considers the case of price contracts of 2 periods, as well as an information stickiness of 2 periods (N =2). Figure 2 considers the case N =4. The plain dark line corresponds to the IRF under sticky information, the plain grey line is the IRF under price stickiness and the dashed dark line refers to price stickiness with imperfect information. Table 2: Second Order Moments Data σ(·) ρ(·, y)

σ(·)

1.49 0.80 6.03 0.43 0.16 0.40

1.56 0.89 5.61 1.61 1.64 0.06

SI ρ(·, y)

SPFI σ(·) ρ(·, y)

σ(·)

SPII ρ(·, y)

N=2 y c i h π R

1.00 0.86 0.92 0.69 0.32 0.21

1.00 0.80 0.97 0.75 0.54 0.58

2.07 1.03 7.88 2.66 1.15 0.05

1.00 0.86 0.98 0.87 0.31 0.66

3.45 1.74 13.67 4.81 2.42 0.25

1.00 0.90 0.96 0.95 0.38 0.13

3.38 1.41 13.51 4.99 0.80 0.07

1.00 0.93 0.99 0.95 0.55 0.92

4.26 2.12 16.26 6.32 1.03 0.19

1.00 0.93 0.97 0.97 0.60 0.44

N=4 y c i h π R

1.49 0.80 6.03 0.43 0.16 0.40

1.00 0.86 0.92 0.69 0.32 0.21

2.67 1.17 10.61 3.81 1.27 0.08

1.00 0.89 0.99 0.92 0.86 0.91

Note: All series are first HP–filtered. All data cover the period 1960:1–2002:4, except for hours worked that cover the period 1964:1–2002:2.

Table 3: Serial Correlation of the inflation rate

N=2 N=4

Data 0.33 0.33

SI 0.08 0.14

SPPI 0.22 0.36

SPII 0.31 0.40

Note: The series are first HP–filtered and cover the period 1960:1–2002:4.

Sticky Prices versus Sticky Information: A Horse Race

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Figure 1: Impulse Response Functions (N = 2) (a) Technology Shock Output

Inflation

2

Nominal Interest Rate

0.5

0.02

S.I. S.P. I.S.P.

1.5

0

1

% dev.

% dev.

% dev.

0

−0.5 0.5

−0.02 −0.04 −0.06

0

5

10 15 Quarters

−1

20

5

10 15 Quarters

−0.08

20

5

10 15 Quarters

20

(b) Fiscal Shock Output

Inflation

S.I. S.P. I.S.P.

0.2

% dev.

% dev.

0.3

0.1 0

Nominal Interest Rate

0.4

0.05

0.3

0.04

0.2

0.03

% dev.

0.4

0.1 0

−0.1

5

10 15 Quarters

20

−0.1

0.02 0.01

5

10 15 Quarters

20

0

5

10 15 Quarters

20

(c) Money Supply Shock Output

Inflation

S.I. S.P. I.S.P.

% dev.

% dev.

2 1 0 −1

Nominal Interest Rate

2

0.08

1.5

0.06 0.04

1

% dev.

3

0.5

0

0 5

10 15 Quarters

20

−0.5

0.02

−0.02 5

10 15 Quarters

20

−0.04

5

10 15 Quarters

20

Sticky Prices versus Sticky Information: A Horse Race

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Figure 2: Impulse Response Functions (N = 4) (a) Technology Shock Output

Inflation

2 S.I. S.P. I.S.P.

1.5

0.01 0.005

0

0.5

% dev.

0

1

% dev.

% dev.

Nominal Interest Rate

0.2

−0.2

−0.005 −0.01

−0.4

0

−0.015

−0.5

5

10 15 Quarters

20

−0.6

5

10 15 Quarters

20

−0.02

5

10 15 Quarters

20

(b) Fiscal Shock Output

Inflation

0.6

Nominal Interest Rate

0.15 S.I. S.P. I.S.P.

0.4

0.05 0.04

0.1

0.2

% dev.

% dev.

% dev.

0.03 0.05

0.02 0.01

0

0 0

−0.2

5

10 15 Quarters

20

−0.05

5

10 15 Quarters

20

−0.01

5

10 15 Quarters

20

(c) Money Supply Shock Output

Inflation

S.I. S.P. I.S.P.

% dev.

% dev.

4 2 0 −2

Nominal Interest Rate

1.5

0.15

1

0.1 % dev.

6

0.5 0

5

10 15 Quarters

20

−0.5

0.05 0

5

10 15 Quarters

20

−0.05

5

10 15 Quarters

20

Sticky Prices versus Sticky Information: A Horse Race

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Table 4: The acceleration Phenomenon

Data S.I. S.P. P.I. S.P. I.I. S.I. S.P. P.I. S.P. I.I.

ρ(yt , πt+2 − πt−2 ) ρ(yt , πt+4 − πt−4 ) 0.39 0.51 N=2 0.10 0.07 -0.07 0.04 0.02 0.08 N=4 0.04 0.04 0.37 -0.18 0.39 -0.11

Table 5: Second Order Moments (Taylor rule) Data σ(·) ρ(·, y)

σ(·)

SI ρ(·, y)

y c i h π R

1.49 0.80 6.03 0.43 0.16 0.40

1.00 0.86 0.92 0.69 0.32 0.21

1.28 0.94 3.91 0.68 0.56 0.29

1.00 0.83 0.95 0.66 -0.69 -0.98

y c i h π R

1.49 0.80 6.03 0.43 0.16 0.40

1.00 0.86 0.92 0.69 0.32 0.21

1.09 0.82 3.50 1.13 0.43 0.25

1.00 0.76 0.93 0.45 -0.51 -0.89

SPFI σ(·) ρ(·, y)

σ(·)

SPII ρ(·, y)

1.42 0.97 4.61 1.22 0.55 0.29

1.00 0.84 0.96 0.69 -0.66 -0.93

1.33 1.04 5.63 0.83 0.56 0.30

1.00 0.72 0.79 0.65 -0.72 -0.91

1.44 0.91 5.17 1.90 0.42 0.26

1.00 0.81 0.96 0.73 -0.53 -0.80

1.14 0.94 4.62 1.60 0.34 0.23

1.00 0.70 0.69 0.58 -0.57 -0.76

N=2

N=4

Note: All series are first HP–filtered. All data cover the period 1960:1–2002:4, except for hours worked that cover the period 1964:1–2002:2.

Table 6: Serial Correlation of the inflation rate (Taylor rule)

N=2 N=4

Data 0.33 0.33

SI 0.30 0.37

SPPI 0.40 0.53

SPII 0.45 0.67

Note: The series are first HP–filtered and cover the period 1960:1–2002:4.

Sticky Prices versus Sticky Information: A Horse Race

Table 7: The acceleration Phenomenon (Taylor rule)

Data S.I. S.P. P.I. S.P. I.I. S.I. S.P. P.I. S.P. I.I.

ρ(yt , πt+2 − πt−2 ) ρ(yt , πt+4 − πt−4 ) 0.39 0.51 N=2 0.36 0.21 0.27 0.18 0.33 0.19 N=4 0.46 0.34 0.53 0.16 0.49 0.23

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