Sentiments and Aggregate FluctuationsSentiments and Aggregate Fluctuations Jess Benhabib Pengfei Wang Yi Wen October 15, 2013 Jess Benhabib Pengfei Wang Yi Wen Sentiments and Aggregate

Sentiments and Aggregate Fluctuations

Jess Benhabib Pengfei Wang Yi Wen

October 15, 2013

Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 1 / 43

Introduction

We try to capture the Keynesian notion that animal spirits orsentiments, unconnected to fundamentals, can drive employment andoutput fluctuations under rational expectations.

The key Keynesian feature of our model is 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.


Introduction

Nevertheless, in equilibrium all agents know correct distributions, allprices are flexible, all markets clear and consumer expectations aboutaggregate consumption, employment and real wages are correct eachperiod.

So we have rational expectations equilibria.


Introduction

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

It is related to Cass and Shell (1983), and to correlated equilibria viaAumann (1974, 1987), and to Maskin and Tirole (1987).

In the absence of sentiments, the models that we study have uniqueequilibria.

But sentiments and beliefs can amplify employment, production andconsumption decisions, and lead to multiple rational expectationsequilibria


A Model of Sentiments and Fluctuations

Consumers make consumption and labor supply plans based on their"sentiments" about aggregate demand, and real wages. Nominalwages are normalized to one.

Each firm must make optimal production decisions on the basis ofsignals about what its demand will be, before demand is realized.The signals can be based on firms’market research about theirdemand, early orders, initial inquiries, as well as public signals ofaggregate demand/consumer sentiments.

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


Signals

In the first simplest benchmark model, the signal is a weighted sum ofthe firm’s idiosyncratic demand shock and a shock to aggregatedemand/sentiments, both of which enter the firm’s demand curve.

Later in the paper we provide the explicit microfoundations for howthe signals are generated.

We also introduce a second noisy but public signal of aggregatedemand. This signal may represent public forecasts of aggregatedemand/sentiments.


Model cont’d

Trades take place in centralized markets rather than bilaterallythrough random matching. At the end of each period all history canbecome public knowledge.

Firms optimally decide on how much to produce on the basis of theirprivate but correlated signals about demand. Only then aggregateoutput is realized and 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 consumersexpected, given their sentiments.

Aggregate output equals to the households’planed consumption. Sohouseholds can in fact implement their consumption plans.

Thus we have rational expectations equilibria.


Model cont’d

We show that in the simple benchmark model, there can be twodistinct rational expectations equilibria: one with constant output andone with stochastic output driven by self-fulfilling sentiments.

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

When we discuss the microfoundations of signals, we will see that infact we have a continuum of equilibria, parametrized by the varianceof sentiment shocks.


The Basic Benchmark Model: Household

Households make a consumption and labor supply plan:

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. The first-order conditions for labor imply

1Ct

Wt

Pt= ψ (1)

Households have (so far) common point expectations/sentimentsabout aggregate Ct , so given a nominal wage Wt = 1, they can infera price Pt and a real wage Wt

Pt, consistent with (1) and supply labor.


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 firm-specific shocks, andmaximizes profit

maxPt

[∫ε1θjtY

θ−1θ

jt dj] θ

θ−1−∫pjtYjtdj .

The demand function depends on both εjt and Yt :

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 be random andsentiment-driven. Instead, as in the Lucas island model, firms have anoisy indication of what their demand will be from a signal

sjt = λ log εjt + (1− λ) log Yt

The parameter λ reflects the weights of the idiosyncratic andaggregate components of demand.

This signal is our simplest signal. In the paper the simplest signal hasiid shocks vjt ∼ N

(0, σ2v

)as well.

Signal microfoundations will determine λ later.

In a REE aggregate demand Yt that generates the signal will be equalto actual aggregate output Yt .


Intermediate Goods, Cont’d

An intermediate goods producer j has the production function

Yjt = Anjt

The firm maximizes expected nominal profits Πjt = pjtYjt −WtYjtA :

maxYjt

Et

[(PtY

− 1θ

jt (εjtYt )1θ

)Yjt −Wt

YjtA|sjt .]

After simplifications using 1Ct= 1

Yt= ψ Pt

Wt, and Wt = 1 :

Yjt ={(

1− 1θ

)AψEt[(εjt )

1θ Y

1θ−1t

]|sjt}θ

.

Strategic Substitutabilty: since θ > 1, Yjt is negatively related to Yt .


Certainty Equilibrium

Information is perfect and sentiments have no role.The signal sj fullyreveals the firm’s own idiosyncratic demand εjt . (σy = 0)There exists a fundamental certainty equilibrium with constantaggregate output Ct = Yt = Y ∗ and Pt = P and firm output is:

Y1θjt =

(1− 1

θ

)Aψ

ε1θjtY

1−θθ

t

Without loss of generality set(1− 1

θ

) Aψ = 1. Final good output is:

Yt =[∫

ε1θjtY

θ−1θ

jt dj] θ

θ−1

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


Self-Fulfilling Equilibrium

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

logYt = φ0 + zt ,

The noisy signal received by each firm (now defined net of theconstant term φ0) is

sjt = λεjt + (1− λ)zt

where zt ∼ N(0, σ2z

).

With fluctuations in aggregate output, the signal is not fully revealing.


Sentiment-Driven REE

In the self-fulfilling equilibrium:

Each period the sentiment zt held by households in logYt = φ0 + ztwill be the realized zt in logYt .

So the distribution of the perceived sentiment {z} will be consistentwith the realized distribution of aggregate output {Y }.Prices will clear al markets each period.


Self-Fulfilling Equilibrium

Proposition

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

with stochastic aggregate output Yt = φ0 + zt . Furthermore log Yt isnormally distributed 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.


Self-Fulfilling Equilibrium, Cont’d

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, thisfundamental equilibrium is unique.

Given λ in sjt = λεjt + (1− λ)zt and the variance of the idiosyncraticshock σ2ε , for markets to clear for all possible realizations of thesentiment zt , the variance σ2z has to be precisely pinned down, as inthe Proposition above.

If σ2z is too high, output is too low relative to aggregate demand byconsumers, and vice-versa.


Intuition

If firms believe that the signal contains information about changes inaggregate demand, zt , then this belief will induce all agents to amplifytheir output response, up or down, and sustain self-fulfilling fluctuationsconsistent with the agents’beliefs about the distribution of output. Notethat both the variance of the sentiment shock σ2z and λ affect the firms’optimal output responses through their signal extraction. Given λ and thevariance of the idiosyncratic shock σ2ε , for markets to clear for all possiblerealizations of the sentiment zt , the variance σ2z has to be precisely pinneddown, as in the Proposition. If however the signal gives a low weight toaggregate as opposed to idiosyncratic demand, that is if λ ∈ [0.5, 1], thenwe cannot find a positive variance σ2z that will clear the markets for everyzt .


Generalizing the signals (SKIP)Imperfect 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 , (2)

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


Imperfect signal with firm-specific noise, Cont’d (SKIP)The 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 = φ0 + zt that has amean

φ0 =12

((1− λ+ (θ − 1) λ)

θ(1− λ)

1(θ − 1)

)σ2ε −

(θ − 1) σ2v2θ2(1− λ)2

and variance

σ2z =λ (1− 2λ)

(1− λ)2θσ2ε −

1(1− λ)2θ

σ2v .


Multiple signals

The government and public forecasting agencies as well as newsmedia often release their own forecasts of the aggregate economy.

Suppose firms receive two independent signals, sjt and spt .

The firm-specific signal sjt is based on firm’s own preliminaryinformation about its demand:

sjt = vjt + λεjt + (1− λ) zt

The public signal is:spt = zt + et

where et ∼ N(0, σ2e

)is noise in the public forecast of aggregate

demand.

We can also model the noisy public signal with heterogenous butcorrelated sentiments across consumers, so observing a subset ofconsumers will reveal a noisy signal of the average sentiment.


Multiple signals, Cont’d (SKIP)

To establish the existence of the certainty equilibrium, we also assumethat σ2e = γσ2z , where γ > 0.

This assumption states that the variance of the forecast error of thepublic signal for aggregate demand is proportional to the variance ofz , 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.


Multiple signals, Cont’d

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

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

σ2z =λ (1− 2λ)

(1− λ)2θσ2ε −

1(1− λ)2θ

σ2v > 0,

and where η = − σ2zσ2e= − 1

γ . In addition, there is a "certainty" equilibriumwith constant output identical to the certainty equilibrium in the previousProposition with a single signal.

Note: the public forecast error et affects output.Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 23 / 43

A Simple Abstract Model

To set up the intuition consider the following simple model.

Assume for simplicity that the economy is log-linear, so optimal logoutput of firms coming from a linear quadractic objective, is given bythe rule

yjt = Et{[β0εjt + βyt ] |sjt}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.

Market clearing requires

yt =∫yjtdj .


Abstract Model Cont’d:

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

yjt = Et{[β0εjt + βyt ] |sjt} (3)

sjt = vjt + λεjt + (1− λ) yt (4)

yt =∫yjtdj (5)

Assume that yt is normally distributed with zero mean and varianceσ2y . Based on (3), signal extraction gives:

yjt =λβ0σ

2ε + (1− λ)βσ2y

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

Market clearing implies:

yt =∫yjtdj =

[λβ0σ

2ε + (1− λ)βσ2y

σ2v + λ2σ2ε + (1− λ)2σ2y(1− λ)

]yt .


Abstract Model Cont’d: Certainty and Self-FulfillingStochastic Equilibria

Market clearing implies

yt =∫yjtdj =

λβ0σ2ε + (1− λ)βσ2y

σ2v + λ2σ2ε + (1− λ)2σ2y(1− λ)yt . (6)

The Certainty Equilibrium is yt ≡ 0.But (6) holds for all yt if

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-FulfillingStochastic Equilibrium.Note that if β < 1, a necessary condition for σ2y to be positive is

λ ∈(0, β01+β0

).


Stability Under Learning

Our model is essentially static, but we can investigate whether theequilibria of the model are stable under adaptive learning.

For simplicity we will confine our attention to the simplified abstractmodel of section with σ2v = 0, where without loss of generality we setβ0 = 1. So the model is

sjt = λεjt + (1− λ) logYt

yjt = Et{εjt + βyt |sjt}

yt =∫yjtdj


Stability Under Learning Cont’d

We can renormalize our model so that the sentiment or sunspotshock zt has unit variance by redefining output as yt = logYt = σzzt .The variance of output yt then is still σ2z . .

Suppose that agents understand that equilibrium yt is proportional tozt and they try to learn σz .

If agents conjecture at the beginning of the period t that theconstant of proportionality is σzt =

ytzt, then the realized output is

yt =λσ2ε + (1− λ)βσ2ztλ2σ2ε + (1− λ)2σ2zt

(1− λ)σztzt .

Under adaptive learning with constant gains g = 1− α, agentsupdate σzt :

σzt+1 = σzt + (1− α)

(ytzt− σzt

)≡ h(σzt )


Stability Under Learning Cont’d

We have the adaptive learning updating rule

σzt+1 = ασzt + (1− α)

(ytzt

)= σzt + (1− α)

(ytzt− σzt

)≡ h(σzt )

For any initial σzt > 0, we can show that σzt does not converge to 0,the certainty equilibrium.

By contrast the sentiment-driven sunspot equilibrium is locally stableunder learning provided the gain g = 1− α is not too large.

In particular h′(0) > 1 and |h(σz )| < 1 at the sentiment drivenequilibrium σz =

(λ(1−2λ)σ2ε(1−λ)2(1−β)

)0.5.


Microfoundations of the signal

So far we simply assumed that firms receive signals

sjt = vjt + λεjt + (1− λ) yt

based on their market research, market surveys, early orders, initialinquiries and advanced sales, to form such expectations.

In particular, we assumed that the signals weights λ and (1− λ) areexogenous. It is therefore desirable to spell out in more detail themicrofoundations for how firms obtain these signals.

If the signal reveals, except for iid noise, the precise weighted sum ofthe fundamental and the sentiment shocks that the firms need toforecast, then the signal extraction problem of the firm disappears.

This then excludes the possibility of sentiment-driven equilibria.


Microfoundations of the signal, Cont’d

For example, suppose firm j can post a hypothetical price pjt and aska subset of consumers about their intended demand given thishypothetical price.The firms then obtain a signal about the intercept of their demandcurve, possibly with some noise vjt if consumers have heterogenoussentiments, sjt = εjt + (1− θ) yt + vjt .To see this note that the demand curve of firm j is given by

Yjt =(PjtPt

)−θYtεjt . Since from the labor market first order

conditions we have Pt = 1ψYt, the logarithm of the demand curve,

ignoring constants that can be filtered, becomesyjt = εjt + (1− θ) yt − θpjt .Suppose firm j can post a hypothetical price pjt and ask a subset ofconsumers about their intended demand given this hypothetical price.The firms can then obtain a signal about the intercept of theirdemand curve, possibly with some noise vjt if consumers haveheterogenous sentiments, sjt = εjt + (1− θ) yt + vjt .



The optimal output decision of firms is given by

yjt = E [εjt + (1− θ)yt ]| sjt =σ2s − σ2v

σ2s[εjt + (1− θ)yt + vjt ]

where σ2s is the variance of the signal.

Note that σ2s − σ2v > 0 is the variance of εjt + (1− θ)yt , orCov (εjt + (1− θ)y , sjt ) .

Integrating across firms, yt =∫yjtdj , and equating coeffi cients of yt

yields 1 = σ2s−σ2vσ2s

(1− θ). This equality is impossible (even if σ2v = 0)

since by construction σ2s > σ2v and θ > 1.

In other words, a constant output with yt = 0 is the only equilibrium.



To obtain the possibility of sentiment-driven equilibria, we can eitherslightly complicate the signal extraction problem of the firm by addingan extra source of uncertainty,

Or we can modify the signal so that it does not eliminate the signalextraction problem faced by the firm.

We provide microfoundations for both of these approaches below.


Signal with Consumer Preference Uncertainty

First we study a model with an additional source of uncertainty.We still allow a firm to post a hypothetical price pjt and ask a subsetof consumers about their intended demand at that price.However at the time of the survey the preference shock is not yetrealized with certainty: each consumer i receives a signal for his/herpreference shock εjt : s

jht = εjt + hijt which forms the basis of their

response to the posted hypothetical price.

Let utility of consumers be C 1−γt −11−γ − ψNt , and let σ2h be the variance

of hijt and σ2ε be the variance of εjt .

Then we can show the weight λ in the firm’s signalsjt = vjt + λεjt + (1− λ) zt is uniquely determined if 1− θγ > σ2ε

σ2ε+σ2h:

λ ≡σ2ε

σ2ε+σ2hσ2ε

σ2ε+σ2h+ (1− θγ)

∈(0,12

)Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 34 / 43

A Quantity Signal

Instead of introducing additional sources of uncertainty to establishsentiment-driven equilibria, we instead modify the signals so that theydo not eliminate the signal extraction problem faced by firms.

Suppose the intermediate-good firms receive a signal from consumersabout the quantity of the demand Yjt for their good.

We drop the final-good sector and assume instead that therepresentative household purchases a variety of goods Cjt to maximize

utility logCt − ψNt , where Ct =[∫

ε1θjtC

θ−1θ

jt dj] θ

θ−1.


A Quantity Signal, Cont’d

The households have utility utility logCt − ψNt and choose theirdemand Cjt and labor supply Njt based on the sentiment shock Ztand the preference shocks εjt .

In equilibrium household demand function for each variety is:

Cjt =(PtPjt

)θ

εjtCt =(PtPjt

)θ

εjtYt .

Since demand depends on the consumers’conjectures about theirprice Pjt = Pt (εjt ,Zt ), the demand Cjt is a function of preferenceshocks and the sentiment, Cjt = C (εjt ,Zt ).



The intermediate-good firms, based on the signal sjt = cjt choosetheir production according to the first order condition given by

Cjt = Yjt ={(

1− 1θ

)AψEt[

ε1θjtY

1θ−1t

∣∣∣ Sjt]}θ

,

We conjecture that in equilibrium,

cjt = φεjt + φzzt

where φ and φz are undetermined coeffi cients.



PropositionThere is a continuum of sentiment-driven equilibria cjt = φεjt + φzzt withi) φz = 1ii) φ ∈ [0, 1]iii) σ2z =

φ(1−φ)1−β σ2ε ∈

[0, 14(1−β)

σ2ε

]PropositionThe sentiment-driven equilibria of this model with signal sjt = cjt can bemapped one-to-one to the sentiment-driven equilibria of our benchmarkmodel with the signal sjt = λεjt + (1− λ)yt where

λ =φ

φ+ 1∈[0,12

].


A Quantity Signal, Cont’d (SKIP)

PropositionUnder the signal sjt = cjt , there also exists another type ofsentiment-driven equilibria with firm-level output driven not only by thefundamental shock εjt and aggregate sentiment shock zt , but also by afirm-specific iid shock vjt with zero mean and variance σ2v :

cjt = φεjt + zt + (1+ φ)vjt .

Furthermore the signal sjt = cjt is isomorphic to the signalsjt = λ log εjt + (1− λ)yt + vjt .


Signal with Cost Shocks (SKIP)

Utility is C1−γt −11−γ − ψNt where Ct =

[∫ε1θjtC

θ−1θ

jt dj] θ

θ−1.

Each firm j’s total production cost (or labor productivity) is affectedby an idiosyncratic shock that is correlated with the demand shockεjt .

For example marketing costs may be lower under favorable demandconditions: for a higher amount of sales, labor becomes moreproductive so that labor demand Njt = Yjtε−τ

jt is lower, where τ > 0is a parameter.

Alternatively, if marketing costs increase with sales and labor demandis higher, we may have τ < 0.

Firm j’s optimal output, given consumer demands, is

yjt = Et [ ((1+ θτ)εjt + (1− θγ)yt )| sjt ] .


Signal with Cost Shocks Cont’d (SKIP)

Let firms get a signal sjt = εjt + (1− θγ)y : the intercept of theirdemand curve.Optimal output, after taking logs, is

yjt =(1+ θτ)σ2ε + (1− θγ)2σ2z

σ2ε + (1− θγ)2σ2z(εjt + (1− θγ)yt )

Define β = (1− θγ) < 1. Since in equilibrium Cjt = Yjt ,integratingfor market clearing and equating coeffi cients we get

σ2z =[(1+ θτ)β− 1]

β2(1− β)σ2ε

.For σ2z ≥ 0 we need (1+ θτ)β > 1 so we must have τ 6= 0.If β > 0 (the case where firm output and aggregate output arecomplements) sentiment-driven equilibria exist if τ > 1−β

βθ .If β < 0 (the case where firm output and aggregate output aresubstitutes) we need τ < 1−β

βθ and τ can be negative.


Extensions

Sunspots can be idiosyncratic across consumers.

Price setting instead of quantity setting before demand is realized.

Persistence in output can be obtained with : a) Markov sunspots, b)Autocorrelated productivity shocks.


Conclusion

When production decisions must be made under uncertain demandconditions, optimal decisions based on sentiments can generateself-fulfilling rational expectations equilibria in simple productioneconomies without persistent informational frictions.


Sentiments and Aggregate FluctuationsSentiments and Aggregate Fluctuations Jess Benhabib Pengfei Wang Yi Wen October 15, 2013 Jess Benhabib Pengfei Wang Yi Wen Sentiments and Aggregate

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