Sentiments and Aggregate Fluctuations · The informational structure is simple: trades take place in centralized ... Relation to Global Games ... Sentiments and Aggregate Fluctuations

Sentiments and Aggregate Fluctuations

Jess Benhabib Pengfei Wang Yi Wen

March 15, 2013

Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations March 15, 2013 1 / 60

Introduction

The model tries to capture the Keynesian idea that employment andproduction decisions are based on expectations of aggregate demanddriven by consumer sentiments, while realized demand follows fromthe production and employment decisions of firms.

We study a rational expectations equilibrium where all agents knowcorrect distributions, all prices are flexible, all markets clear andconsumer expectations about aggregate consumption, employmentand real wages are correct in each period.

Our work is inspired by Angeletos and Lao (2011), where sentimentscan drive output, and by the Lucas Island model.


A Model of Sentiments and Fluctuations

Consumers make consumption and labor supply plans based on their"sentiments" or expectations about aggregate demand and realwages. Nominal wages are normalized to one.

Each firm must make a production decision on the basis of signalsabout what its demand will be, before demand is realized.

Since the real wages and employment have not yet been determined,and production has not yet taken place, these signals captureconsumer sentiment.

The signals are based on firms’market research about their demand,early orders, initial inquiries, as well as public signals of aggregatedemand/consumer sentiments.


Signals

In simplest benchmark model, for the simplest exposition, the signal issimply a weighted sum of the firm’s idiosyncratic demand shock and ashock to aggregate demand, both of which shift the firm’s demandcurve.We can also add an iid firm-specific noise to each firm’s signal.Later we generalize further and introduce a second noisy but publicsignal of aggregate demand. This signal may represent publicforecasts of aggregate demand, and is available to all firms in theeconomy.A second model replaces the idiosyncratic demand shock with anaggregate shock. Even if producers observe aggregate demand, theycannot separately identify the magnitudes due to its components: thesentiment shock or the fundamental aggregate shock.Furthermore, we also introduce heterogeneous but correlatedconsumer sentiments, so that surveying a subset of consumers yields anoisy signal on the common component of sentiments.


Model cont’d

The informational structure is simple: trades take place in centralizedmarkets rather than bilaterally through random matching, and at theend of each period all history can become public knowledge.Informational asymmetries obtain only within the period: firmsoptimally decide on how much to produce on the basis of their privatebut correlated signals.Firms act on the signal to maximize profits: they hire labor atnominal wages, and produce. Only then aggregate output is realizedand prices clear all markets.In equilibrium all agents "know" the correct distribution of theidiosyncratic and aggregate demand shocks.The realized real wage is equal the wage that the consumers expectedgiven their sentiments, and aggregate output equals to thehouseholds’planed consumption. So households can in factimplement their consumption plans.Thus we have rational expectations equilibria.


Model cont’d

We show that in this simple model, there can be two distinct rationalexpectations equilibria: one with constant output and one withstochastic output driven by self-fulfilling sentiments.

We can then easily introduce a Markov Sunspots, transitioningbetween stretches of constant equilibria, punctuated by periods ofvolatile, sentiment-driven equilibria with lower mean output. (I willnot dwell on this).

But note that the self-fulfilling stochastic equilibrium is not arandomization over multiple equilibria.


Relation to Global Games

In models of global games multiple coordination equilibria canbecome unique once agents face some private uncertainty. Whenprivate signals about fundamentals are relatively precise but diverse,agents put heavy weight on them and multiple coordination equilibriacan be ruled out.

In our simple model, we start with a unique equilibrium that hasconstant output, but when we introduce uncertainty about aggregatedemand perceived through signals, we obtain multiple equilibria.


The Benchmark Model: Household

maxE0 ∑ βt [log(Ct )− ψNt ]

subject to

Ct ≤Wt

PtNt +

Πt

Pt,

where Wt denotes nominal wage and Πt aggregate profit income fromfirms, all measured in final goods. Denoting Λt as the Lagrangianmultiplier for the budget constraint, the first-order conditions imply

Λt =1Ct= ψ

PtWt

Pt =Wt

ψCt;

Wt

Pt= ψCt (1)

Households have expectations/sentiments about aggregate output Ct ,so given a nominal wage Wt = 1, they expect a price Pt and a realwageWt

Pt, consistent with 1.


Final Good Aggregator

The final-good firms (or a representative consumer) produce a finalgood according to

Ct = Yt =[∫

ε1θjtY

θ−1θ

jt dj] θ

θ−1

where θ > 1 and log εjt are iid zero mean shocks, and maximizesprofit

maxPt

[∫ε1θjtY

θ−1θ

jt dj] θ

θ−1−∫pjtYjtdj .

The demand function:

pjtPt= Y

− 1θ

jt (εjtYt )1θ

Yjt =(Ptpjt

)θ

εjtYt .


Intermediate Goods

Each intermediate firm produces good Yjt without perfect knowledgeeither of εjt or of aggregate demand Yt , which could also be random andsentiment driven. Instead, as in the Lucas island model, they have a noisyindication of what their demand will be from a signal sjt ,

sjt = λ log εjt + (1− λ) logYt ,

where the parameter λ reflects the weights of the idiosyncratic andaggregate components of demand. Based on the signal, the firm choosesto produce output and maximize profits.

This is our simplest signal, to be generalized later.


Intermediate Goods, Cont’d

An intermediate goods producer j has the production function

Yjt = Anjt .

So the firm maximizes expected nominal profits Πjt = pjtYjt − WtA Yjt :

maxYjt

Et

[(PtY

− 1θ

jt (εjtYt )1θ

)Yjt −

Wt

AYjt

]|sjt .

with constant markup (based on expected price and marginal cost):(1− 1

θ

)=1A

Et [Wt |sjt ]Y− 1

θjt Et

[Pt (εjtYt )

1θ |sjt

]After simplifications using 1

Ct= 1

Yt= ψ Pt

Wt, and Wt = 1 :

Yjt =(

1− 1θ

)AψEt[(εjt )

1θ Y

1θ−1t

]|sjtθ

.

Note that in equilibrium, since θ > 1, Yjt is negatively related to Yt .Ifaggregate output increases, real wages rise and firms reduce their output.


Certainty Equilibrium

There exists a fundamental certainty equilibrium with constant aggregateoutput Ct = Yt = Y ∗ and Pt = P. Information is perfect, consumers andfirms expect Pt = P, Ct = Yt = Y ∗, and the signal fully reveals the firm’sown idiosyncratic demand εjt :

Y1θjt =

(1− 1

θ

)Aψ

ε1θjtY

1−θθ

t , (2)

Without loss of generality set(1− 1

θ

) Aψ = 1. The final good output is:

Y1− 1

θt =

∫ε1θjtY

1− 1θ

jt dj (3)

Combine (2) and (3):

Yθ−1

θt = Y

(1−θ) θ−1θ

t

∫εjtdj ,

or, if εjt ≡ log εjt has zero mean and variance σ2ε ,

φ0 = logYt =1

θ − 1 log E exp(εjt ) =1

2 (θ − 1)σ2ε (4)Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations March 15, 2013 12 / 60

Self-Fulfilling Equilibrium

We conjecture there exists an another equilibrium, such that aggregateoutput is not a constant. In particular all agents "know" output follows

logYt = φ0 + zt , (5)

The noisy signal received by each firm (now defined net of the constantterm φ0) is

sjt = λεjt + (1− λ)zt . (6)

where zt ∼ N(0, σ2z

). With fluctuations in aggregate output, the signal is

not fully revealing.


Sentiment-Driven REE

We may view zt as a sentiment held by households about aggregatedemand, perceived by firms through their signal sjt . Using (5) and (6)and the distributions of εjt and zt each intermediade good firm willchose its output optimally.

We will show that in our self-fulfilling equilibrium the distribution ofthe perceived sentiment z will be consistent with the realizeddistribution of aggregate output Y , so the sentiment zt held byhouseholds in logYt = φ0 + zt will be the realized zt , and marketswill clear each period.


Self-Fulfilling Equilibrium, Cont’d

Proposition

If λ ∈(0, 12), there exists a self-fulfilling rational expectations equilibrium

with stochastic aggregate output Yt . Furthermore log Yt is normallydistributed with mean

φ0 =(1− λ) + (θ − 1) λ

θ(1− λ)φ0 < φ0

and variance

σ2z =λ (1− 2λ)

(1− λ)2θσ2ε

Welfare? Note that in this case (but not necessarily in models thatfollow) the mean of the constant output equilibrium φ0 > φ0, so inthis case it Pareto dominates the sentiment driven stochasticequilibrium.



In the Certainty Equilibrium with σ2z = 0, Yjt = εjtY 1−σt , and since

σ > 1, equilibrium firm-level outputs depend negatively on aggregateoutput as in the case of strategic substitutability. Hence, thecertainty or fundamental equilibrium in the model will be unique as aresult of this strategic substitutability.

If firms believe that the signal contains information about changes inaggregate demand, zt , then this belief will partially coordinate theoutput response of firms, and sustain self-fulfilling fluctuationsconsistent with the agents’beliefs about the distribution of output.

However, the optimal supply of the firm’s output positively dependson firm-level demand shocks. Consequently, if firms cannot distinguishfirm-level shocks from aggregate demand, informational strategiccomplementarities can arise, giving rise to self-fulfilling equilibria.



Under strategic substitutability the optimal output of an intermediategoods firm declines with σ2z as the firm attributes more of the signalto an aggregate demand shock. In the self-fulfilling equilibrium, σ2z isdetermined at a value that will clear markets for all z .

Given λ and the variance of the idiosyncratic shock σ2ε , for markets toclear for all possible realizations of the sentiment zt , the variance σ2zhas to be precisely pinned down, as in the Proposition above.

Instead, in the next model, we will get an interval for σ2z for which REequilibria obtain with aggregate as opposed to idiosyncraticfundamental shocks.

In this model if the signal gives too low a weight to aggregate asopposed to idiosyncratic demand, that is if λ ∈ [0.5, 1], then wecannot find a positive variance σ2z that will clear the markets for everyzt .


Generalizing the signalsImperfect signal with firm-specific noise

So far we assumed that firms can get an initial signal for the overalldemand for their product, but cannot disaggregate it into itscomponents arising from idiosyncratic and from aggregate demand.They only observe their sum.

Since the signals are based on early and initial demand indications foreach of the firms, they may well contain additional firm-specific noisecomponents. Suppose then that the signal takes the slightly moregeneral form,

sjt = vjt + λεjt + (1− λ) zt , (7)

where vjt is a pure firm-specific iid noise with zero mean and varianceσ2v .


Imperfect signal with firm-specific noise, Cont’d

As before we definelogYt = yt = φ0 + zt

In this setup, both the certainty equilibrium and the self-fulfillingequilibrium will be different than those of the benchmark setting of theprevious Proposition. We first state the result for the certainty equilibrium.


Imperfect signal with firm-specific noise, Cont’dThere is a Certainty Equilibrium

Let µ =1θ λσ2ε+

1−θθ (1−λ)σ2z

σ2v+λ2σ2ε+(1−λ)2σ2z.

Proposition

There is a constant certainty equilibrium, yt = φ0, given by

φ0 =12

[(θ + θµλ (θ − 1) + (θµλ (θ − 1))2

θ2 (θ − 1)

)σ2ε + (θ − 1) (θµ)2 σ2v

]

= φ0

(θ + θµλ (θ − 1) + (θµλ (θ − 1))2

θ2

)+ (θ − 1) (θµ)2 σ2v

Note that if σ2v = 0, then µ = 1θλ and φ0 = φ0, so the certainty solution

reduces to the previous benchmark case.


Imperfect signal with firm-specific noise, Cont’dThe self-fulfilling equilibrium

We had definedlogYt = yt = φ0 + zt

Proposition

Let λ < 12 , and σ2v < λ (1− 2λ) σ2ε . In addition to the certainty

equilibrium, there also exists a self-fulfilling rational expectationsequilibrium with stochastic aggregate output, log Yt that has a mean

φ0 =12

((1− λ+ (θ − 1) λ)

θ(1− λ)

1(θ − 1)

)σ2ε −

(θ − 1) σ2v2θ2(1− λ)2

and a variance

σ2z =λ (1− 2λ)

(1− λ)2θσ2ε −

1(1− λ)2θ

σ2v .


Imperfect signal with firm-specific noise, Cont’d

If either λ ≥ 12 , or if σ2v > λ (1− 2λ) σ2ε , then σ2z < 0, suggesting

that the only equilibrium is z = 0. Hence, to have a self-fulfillingexpectations equilibrium, we require λ ∈

(0, 12)and

σ2v < λ (1− 2λ) σ2ε . This pins down the equilibrium σ2z > 0, thevariance of z or of output as a function of σ2ε and σ2v .

Introducing the extra noise vjt into the signal makes output in theself-fulfilling equilibrium less volatile compared to our benchmark case.

The reason for the smaller volatility of output when σ2v > 0 is thatthe signal now is more noisy, and firms attribute a smaller fraction ofthe signal to demand fluctuations.


Multiple signals

The government and public forecasting agencies as well as newsmedia often release their own forecasts of the aggregate economy.Such public information may influence and coordinate outputdecisions of all firms and affect the equilibria. Suppose firms receivetwo independent signals, sjt and spt .The firm-specific signal sjt is based on firm’s own preliminaryinformation about its demand and is as before

sjt = vjt + λεjt + (1− λ) zt

The public signal isspt = zt + et

where we can interpret et as a common noise in the public forecast ofaggregate demand with mean 0 and variance σ2e .In a later model we will model the noisy public signal withheterogenous but correlated sentiments across consumers, soobserving a subset of consumers will reveal a noisy signal of theaverage sentiment.


Multiple signals, Cont’d

To establish the existence of the certainty equilibrium, we also assumethat σ2e = γσ2z , where γ > 0. This assumption states that thevariance of the forecast error of the public signal for aggregatedemand is proportional to the variance of z , or of equilibrium output.Then in the certainty equilibrium where output is constant over time,the public forecast of output is correct and constant as well.



Proposition

If λ < 12 , and σ2v < λ (1− 2λ) σ2ε , then there exists a self-fulfilling rational

expectations equilibrium with stochastic aggregate output

logYt = yt = zt + ηet + φ0 = zt + φ0,

which has mean φ0 =12

((1−λ+(θ−1)λ)

θ(1−λ)1

(θ−1)

)σ2ε −

(θ−1)σ2v2θ2(1−λ)2

and variance

σ2y = σ2z =λ (1− 2λ)

(1− λ)2θσ2ε −

1(1− λ)2θ

σ2v > 0,

and where η = − σ2zσ2e= − 1

γ . In addition, with σ2z = γσ2e = 0, there is a"certainty" equilibrium with constant output identical to the certaintyequilibrium in the previous Proposition with a single signal.

Note: the public forecast error et affects output.Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations March 15, 2013 25 / 60


In the self-fulfilling equilibrium of the Proposition above, whereσ2z =

λ(1−2λ)(1−λ)2θ

σ2ε − 1(1−λ)2θ

σ2v and η = − σ2zσ2e= − 1

γ , the optimalweight that firms place on the public signal is zero. Neverthelessaggregate output

logYt = yt = zt + ηet + φ0 = zt + φ0

is stochastic, and driven by the volatility of z = zt + ηet .

It is easy to see that the certainty equilibrium of Proposition 2 withσ2z = 0 will also apply in the certainty equilibrium of this case sincewe would have σ2e = γσ2z = 0. Namely the public signal also becomesa constant. We can then directly apply Proposition 2 to find thecertainty equilibrium output.


PersistencePersistence and Sunspots

We now construct a persistent sunspot equilibrium with Markovtransitions between the certainty and the stochastic self-fulfillingequilibrium.

To construct such an equilibrium, we introduce a sunspot St = 1 or 0.

We have the transition probabilities Pr(St = 1|St−1 = 1) = ρ andPr(St = 0|St−1 = 0) = ξρ.

Then the stationary distribution is Pr(S∞ = 1) =1−ξρ

1−ρ+1−ξρ ,

Pr(S∞ = 0) =1−ρ

1−ρ+1−ξρ .

The agents observe the sunspots first and if St = 1, they cooperateon the certainty equilibrium but if St = 0, then they cooperate on theuncertainty equilibrium.


Persistence and Sunspots, Cont’d

A simulated path of aggregate output is obtained in Figure 1 below. Weset θ = 4, σ2ε = 1, ρ = 0.95, ξ = 0.7. For these parameters the certaintyequilibrium is logYt = 1

2(θ−1)σ2ε = φ0 = 0.1667.

0 50 100 150 200 250 300 350 4000.8

0.6

0.4

0.2

0

0.2

0.4

0.6

Simulated Path of Output


Persistence: Autoregressive productivity

If productivity, At , is a stochastic process that firms can observe, thenin parallel to our benchmark case we can express output as

logYt = φ0 + zt + logAt .

Setting(1− 1

θ

) 1ψ instead of

(1− 1

θ

) Atψ to unity, as before market

clearing would require the sum of log ouputs of firms to add toaggregate log output for every zt .

Later we will explore a model where At is stochastic, cannot beobserved by firms.


A Simple Abstract Model

To set the basic intuition consider the following three equation model.

Assume for simplicity that the economy is log-linear, so optimal logoutput (or investment, price, labor, etc.) of firms coming from alinear quadractic objective, is given by the rule

yjt = Et[β0εjt + βyt ] |sjt (8)

where εjt is zero mean, iid .

The coeffi cient β can be either negative or positive, so we can haveeither strategic substitutability or strategic complementarity in firms’actions.


A Simple Abstract Model cont’d

The signal sjt is given by

sjt = vjt + λεjt + (1− λ) yt , (9)

where both the exogenous noise vjt and the idiosyncratic demandshock εjt are iid and normally distributed with a zero mean.

Market clearing requires

yt =∫yjtdj . (10)



We have, from a linear quadratic objective, for β < 1,

yjt = Et[β0εjt + βyt ] |sjt (11)

sjt = vjt + λεjt + (1− λ) yt , (12)

yt =∫yjtdj . (13)

In the Certainty Equilibrium yt is constant, so equation (11) yields

yjt = βyt +λβ0σ

2ε

σ2v + λ2σ2ε(vjt + λεjt ) . (14)

Substituting into equation (13) and integrating give

yt =∫yjtdj = βyt (15)

So unless β = 1, in which case there is a continuum of certaintyequilibria, the unique certainty equilibrium is given by

yt = 0



In the Self-fulfilling Stochastic Equilibrium, assume that yt is normallydistributed with zero mean and variance σ2y . Based on the simple responsefunction given by equation (11), signal extraction implies

yjt =λβ0σ

2ε + (1− λ)βσ2y

σ2v + λ2σ2ε + (1− λ)2σ2y[vjt + λεjt + (1− λ) yt ] . (16)

Then market clearing requires

yt =∫yjtdj =

λβ0σ2ε + (1− λ)βσ2y

σ2v + λ2σ2ε + (1− λ)2σ2y(1− λ)yt . (17)

Since this relationship has to hold for every realization of yt , we need

λβ0σ2ε + (1− λ)βσ2y

σ2v + λ2σ2ε + (1− λ)2σ2y(1− λ) = 1 (18)



We haveλβ0σ

2ε + (1− λ)βσ2y

σ2v + λ2σ2ε + (1− λ)2σ2y(1− λ) = 1

This implies

σ2y =λ(β0 − (1+ β0)λ)σ

2ε − σ2v

(1− λ)2(1− β)

Thus, σ2y is pinned down uniquely and it defines the self-fulfillingequilibrium.

Note that if β < 1, for σ2y > 0 we would requireλ(β0 − (1+ β0)λ)σ

2ε − σ2v > 0. Then a necessary condition for σ2z to

be positive is λ ∈(0, β01+β0

). If β0 = 1, this restriction becomes

λ ∈ (0, 0.5).


Model with Aggregate Shocks

A representative household derives utility from a final good andleisure. The final good is produced by a representative consumer/finalgoods producer using a continuum of intermediate goods indexed byj ∈ [0, 1].Each intermediate good is produced using labor. The nominal wagerate is fixed at 1. The real wage (in term of the final goods) can ofcourse fluctuate with the price of the final goods.

The households are subject to aggregate "preference" shocks At andsentiment shocks zt in each period. In all equilibria of the model thehouseholds have perfect foresight (rational expectations in theextension with heterogenous sentiments).

Namely, conditional on the aggregate shock and their sentiments,they can perfectly forecast the price level. Based on the forecastedprice, they make their consumption and labor supply decisions.


The aggregate consumption decisions made by the households arethe source of "possibly" noisy demand signals for the intermediategoods producers. Based on their signal, obtained through marketresearch, intermediate goods producers decide how much to produce.

Prices of each intermediate good adjusts to equalize demand andsupply. These prices then determine the price of the final good. Inequilibrium this realized price coincides with the price expected byhouseholds based on their sentiments.

The results extend to the case where consumer sentiments areheterogenous but correlated.


More precisely,

Firms get a "signal" that reveals aggregate demand ct (At , zt ) ,possibly with some iid noise vjt .

There is still a signal extraction problem since their optimal outputresponds differentially to At and zt .

For example in equilibrium, they might like to decrease output inresponse to zt , but the otimal response of output to At is ambiguous.

There is a constant output equilibrium ct = c .

In addition, there are a continuum of sunspot equilibria parametrizedby σ2z .


Households

Ut = AtC 1−γt

1− γ−Nt , (19)

where Ct is consumption of the final good, At is the preference shock(could easily be a productivity shock-see below) and Nt is labor.Household’s budget constraint as

PtCt ≤ Nt +Πt . (20)

Here Pt is the price of the final goods and Πt is the profit collected fromall intermediate firms and the wage rate is normalized to 1. The first ordercondition for Ct is

AtC−γt = Pt . (21)


Firms

The supply side has a repesentative final good producer and a continuumof intermediate goods producers indexed by j ∈ [0, 1]. The final goodsproducer serves a convenient aggregator of all intermediate goods. Theydo not play an active role in the model. We assume final goods producersmake decisions after all shocks realize so their decisions are not subject toany uncertainty.


The final good aggregation:

maxCjt

PtCt −∫PjtCjtdj , (22)

where Ct is produced by a continuum of intermediate goods according tothe Dixit-Stiglitz production function, with θ > 1,

Ct =[∫ 1

0C

θ−1θ

jt dj] θ

θ−1. (23)

The final goods producer’s profit maximization problem yields the inversedemand curve for each individual intermediate goods,

PjtPt= C

− 1θ

jt C1θt , where Pt =

[∫ 1

0P1−σjt

] 11−σ

(24)


Firms Cont’d

The Intermediate goods firms: The intermediate goods productionfunction is

Cjt = Njt . (25)

A intermediate goods producer j solves the following problem.

maxE [(PjtCjt − Cjt )|Sjt ] (26)

Substituting out Pjt , we have

(1− 1θ)C− 1

θjt E [PtC

1θt |Sjt ] = 1 (27)

So we have

Cjt =

E [PtC

1θt |Sjt ](1−

1θ)

θ

(28)

Cjt =

(1− 1

θ)E [AtC

1θ−γt |Sjt ]

θ

. (29)


The Signal

log Sjt = sjt = logCt + vjt = ct + vjt .

In what follows, the noise vjt will not be essential for our results: wecould have set σ2v = 0. In that case the signal sjt would fully revealaggregate consumption ct to the intermediate goods firms.

Nevertheless, we will see that sentiment-driven rational expectationsequilibria would still exist. This is so because firms set their optimaloutputs under imperfect information.

In the sentiment driven equilibria we will show later that output willbe

logCt − c = ct = φat + σzzt

Since firms do not observe at and zt independently, they formconditional expectations of them based on their signal sjt = ct .


Equilibrium Again

(a) Based on the preference shock At and sentiments Zt , householdsconjecture that the aggregate price is given by Pt = P(At ,Zt );(b) Based on the conjectured price Pt , the households tentatively choosetheir consumption plan Ct = C (At ,Zt ) to maximize their utility;(c) The consumption decisions create signals to firms as log Sjt = ct + vjt ;(d) Based on the signal Sjt , firm j produces Cjt to maximize its expectedprofit;(e) Given the production of Cjt , price Pjt adjusts to equalize demand andsupply;(f) The total production of final goods Ct , equals to the households’planed consumption. Hence the realized price is also equal to theconjectured price Pt .


Equilibrium in logs

Without loss of generality assume At = exp(at/θ)(

θθ−1) 1

θ , where at isnormally distributed with mean 0 and variance σ2a. Aggregate consumptionis given by

Ct =[∫ 1

0C

θ−1θ

jt dj] θ

θ−1. (30)

Then we can write down the system as

cjt = logCjt − c = E[at + (1− γθ)ct ]|sjt (31)

ct = logCt − c =∫ 1

0cjtdj . (32)

sjt = ct + vjt (33)

where c are c are constants to be determined. Note that in equation (31)β = 1− γθ < 1 and can be negative. If β < 0,we are in the case of grosssubstitutes.


Constant Fundamental Equilibrium

PropositionIf the consumers conjecture that the aggregate price is given by

pt = logPt − p = at/θ. (34)

where p = 1θ log

(θ

θ−1)− γc , and c = c = 1

21γ1θ2

σ2a. Then

cjt = ct = 0. (35)

is always an equilibrium.


Stochastic Fundamental EquilibriaDefinitions

Let µ = σ2vσ2a, and assume 0 < µ < 1

4(1−β)where β = 1− γθ < 1 where

γ > 0 is the curvature of utility of consumption and θ > 0 is the elasticityof substitution in final good production.Define the constant terms:

c =121

θ2γ[(θ − 1)σ2v + (1+ βφ)(1− (1− β)φ)σ2a ], (36)

p =1θlog(

θ

θ − 1

)− γc (37)

c = (1− θγ)c +12(1+ βφ)(1− (1− β)φ)σ2a

θ(38)


Stochastic Fundamental Equilibria, cont’d

PropositionLet consumers conjecture that the aggregate price and consumption follow:

logPt − p =

(1θ− γφ

)at (39)

logCt − c = φat (40)

Let

φ =1

2(1− β)±√

14(1− β)2

− µ

1− β> 0. (41)

where µ = σ2vσ2a. Given the consumption expenditures of the households, in a

rational expectations equilibrium each firm j produces

logCjt − c = cjt = φat + vjt .

Note: if σ2v → 0 so that vjt ≡ 0, φ→(0, (1− β)−1

)Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations March 15, 2013 47 / 60

Sentiment-Driven Equilibria

Let output expectations be driven by sentiment shocks as well as at :

logCt − c = ct = φat + σzzt . (42)

The firm’s signal now becomes

sjt = (ct + vjt ) = (φat + σzzt + vjt )

Given aggregate consumption, production of the individual firm j is

cjt = E (at + βct )|(ct + vjt ) (43)

= E (at + βφat + βσzzt )|(φat + σzzt + vjt ) (44)

=(φ+ βφ2)σ2a + βσ2z

φ2σ2a + σ2v + σ2z(φat + σzzt + vjt ).


Define the constant terms c, p be

c =12γ[θ − 1

θ2σ2v +

(1+ βφ)(1− (1− β)φ)σ2a − βσ2z (1− β)

θ2] (45)

p =1θlog(

θ

θ − 1

)− γc (46)


Proposition

Suppose that 0 < σ2v <1

4(1−β)σ2a and µ = σ2v+σ2z (1−β)

σ2a< 1

4(1−β). There

exists a continuum of sentiment-driven equilibra indexed by σ2z ,

0 < σ2z <1

4(1− β)2σ2a −

σ2v1− β

where prices and optimal consumption expenditures follow:

logPt − p = pt = (1θ− γφ)at − γσzzt


where φ is given by

φ =1

2(1− β)±√

14(1− β)2

− µ

1− β(47)


We can plot the coeffi cients φ for the fundamental equilibria (σ2z = 0 )and the corresponding coeffi cients φ for the sentiment-driven equilibriaagainst variance of the noise σ2v . We calibrate θ = 10, γ = 1, the varianceof log At at 4.5, and plot φ against feasible σ2v for various variances ofsentiments σ2z = (0.25, 0.5, 1) and:

0 2 4 6 8 10 120

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 .1

σ2 ( v )

φ

σz

2 = 1

σz

2 = 0.5

σz

2 = 0

σz2 =0.25


Extensions: Persistence

Let aggregate shock follow at = ρat−1 + σaεt where ρ = 0 is the specialcase we considered before. There is a fundamental equilibrium wherect = 1

1−βat . In addition assume zt = ρzt−1 + εz ,t where the variance ofεzt is normalized to unity .We assume that each firm can observe theentire history of aggregate production (or aggregate demand ct−k ,k = 0, 1, 2, ...) but not the history of preference shocks separately. Namely,

sjt = [ct , ct−1, ct−2, ...ct−∞]

We conjecture the existence of sentiment-driven equilibria where zt isserially correlated and aggregate output takes the form

ct = φat + σzzt


Persistence Cont’d

At a sentiment-driven equilibrium the past realizations of aggregateconsumption cannot help firms pin down the innovations in fundamentalshock εt−k , k = 1, 2, ... The history, however, can reveal the weighted sumof εt−k and εz ,t−k for k ≥ 1. So the signal for firm j is

sjt = [φεt + σz εz ,t , ..., φεt−k + σz εz ,t−k , ...].

Then the effective signal for a firm’s decision making can be simplified to

sjt = [φεt + σz εz ,t , φat−1 + σzzt−1]


Persistence Cont’d

PropositionThere exists an continuum of sentiment-driven equilibria for0 < σ2z/σ2a <

14(1−β)2

. Given σ2z we can solve for two φ,

φ =1

2(1− β)±√

14(1− β)2

− σ2zσ2a, (48)

where aggregate production is given by

logCt − c = ct = φat + σzzt , (49)

c =12γ

1

θ2σ2z

φ2σ2a + σ2z(50)


Extensions: Heterogenous Sentiments

Households only observe At and the noisy signal shit = zt + eit andconjecture

logCt − c = ct = φat + σzztlogPt − p = pt = φpaat + φpz zt

They choose labor supply to maximize

EiAtC 1−γt

1− γ−Nt|[At , shit ]. (51)

subject to PtCt ≤ Nit +Πt , (52)

where Πt is the total profits accruing from all the firms. The first ordercondition for consumers now changes to

Ct =

1E (Pt |shit )

(exp(at/θ)

(θ

θ − 1

)) 1γ

. (53)

Aggregating across consumers, ct = logCt = log(∫ 10 Citdi).


Extensions Cont’d

As before, we assume that each firm receives a noisy signallog Sjt = ct + vjt . The production decision by the firms is given by

Cjt =E [PtC

1θt |Sjt ](1−

1θ)

θ

. (54)

(Note σ2v ≥ 0, can be 0.) We have the following Proposition:


Extensions Cont’d

Proposition

Suppose σ2v <1

4(1−β)σ2a and let κ = 1

1+σ2e. There exists a continuum of

sentiment-driven equilibria indexed by σ2z ∈(0, κ4(1−β)2

σ2a −κσ2v1−β

). At each

equilibrium consumers "correctly" conjecture that aggregate price will be

pt = logPt − p = φpaat + φpz zt ≡(1θ− γφ

)at −

γ

κσzzt (55)

and that aggregate consumption (output) is


φ =1

2(1− β)±√

14(1− β)2

− µ

1− β

and µ = σ2v+σ2z (1−β)/κσ2a

.Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations March 15, 2013 57 / 60

Proposition Cont’d

PropositionConsumers’demand for each intermediate good is:

logC cjt − c = cit = φat + σz (zt + eit )

But each individual firm’s optimal production is:

logC fjt − c = cjt = φat + σzzt + vjt

The constant terms are given by

p = log(θ

θ − 1 )−θ − 12θ2

σ2v −12

Ωs (56)

c =1γ[θ − 12θ2

σ2v +12

Ωs ]−γ

2

(1κ

σz

)2(1− κ) +

12

σ2z1− κ

κ(57)

c = c − 12

σ2zσ2e , c = c − 12

θ − 1θ

σ2v . (58)


Conclusions

We explored a Keynesian model where changing sentiments (orexpectations) about aggregate demand can generate output andemployment fluctuations under rational expectations.

We showed that when production decisions must be made underuncertain demand conditions, optimal decisions based on sentimentscan generate self-fulfilling rational expectations equilibria in simpleproduction economies without persistent informational frictions.

These sentiment-driven equilibria are not the result of randomizationsacross multiple fundamental equilibria.

Obviously our model is very simple, but could serve as a benchmarkfor more complicated equilibrium models with additional features.


Conclusions Cont’d

"But public thinking is inscrutable. We can keep trying to understand it,but we’ll be puzzled again the next time the markets or the economy makemajor moves."

Yes, We’re Confident, but Who Knows Why

Robert J. Shiller, New York Times, March 9, 2013


Documents

Sentiments and Aggregate Fluctuations · The informational structure is simple: trades take place in centralized ... Relation to Global Games ... Sentiments and Aggregate Fluctuations