Sentiments and Aggregate Fluctuations

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 or sentiments, unconnected to fundamentals, can drive employment and output fluctuations under rational expectations. The key Keynesian feature of our model is that employment and production decisions are based on expectations of aggregate demand driven by consumer sentiments, while realized demand follows from the production and employment decisions of firms. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 2 / 43

Introduction Nevertheless, in equilibrium all agents know correct distributions, all prices are flexible, all markets clear and consumer expectations about aggregate consumption, employment and real wages are correct each period. So we have rational expectations equilibria. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 3 / 43

Introduction Our work is inspired by Angeletos and Lao (2011), where sentiments can drive output, by the Lucas Island model It is related to Cass and Shell (1983), and to correlated equilibria via Aumann (1974, 1987), and to Maskin and Tirole (1987). In the absence of sentiments, the models that we study have unique equilibria. But sentiments and beliefs can amplify employment, production and consumption decisions, and lead to multiple rational expectations equilibria Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 4 / 43

A Model of Sentiments and Fluctuations Consumers make consumption and labor supply plans based on their "sentiments" about aggregate demand, and real wages. Nominal wages are normalized to one. Each firm must make optimal production decisions on the basis of signals about what its demand will be, before demand is realized. The signals can be based on firms market research about their demand, early orders, initial inquiries, as well as public signals of aggregate demand/consumer sentiments. Since the real wages and employment have not yet been determined, and production has not yet taken place, these signals capture consumer sentiment. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 5 / 43

Signals In the first simplest benchmark model, the signal is a weighted sum of the firm s idiosyncratic demand shock and a shock to aggregate demand/sentiments, both of which enter the firm s demand curve. Later in the paper we provide the explicit microfoundations for how the signals are generated. We also introduce a second noisy but public signal of aggregate demand. This signal may represent public forecasts of aggregate demand/sentiments. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 6 / 43

Model cont d Trades take place in centralized markets rather than bilaterally through random matching. At the end of each period all history can become public knowledge. Firms optimally decide on how much to produce on the basis of their private but correlated signals about demand. Only then aggregate output is realized and prices clear all markets. In equilibrium all agents "know" the correct distribution of the idiosyncratic and aggregate demand shocks. The realized real wage is equal the wage that the consumers expected, given their sentiments. Aggregate output equals to the households planed consumption. So households can in fact implement their consumption plans. Thus we have rational expectations equilibria. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 7 / 43

Model cont d We show that in the simple benchmark model, there can be two distinct rational expectations equilibria: one with constant output and one with stochastic output driven by self-fulfilling sentiments. But note that the self-fulfilling stochastic equilibrium is not a randomization over multiple equilibria. When we discuss the microfoundations of signals, we will see that in fact we have a continuum of equilibria, parametrized by the variance of sentiment shocks. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 8 / 43

The Basic Benchmark Model: Household Households make a consumption and labor supply plan: subject to max E 0 β t [log(c t ) ψn t ] C t W t P t N t + Π t P t, where W t denotes nominal wage and Π t aggregate profit income from firms, all measured in final goods. The first-order conditions for labor imply 1 C t W t P t = ψ (1) Households have (so far) common point expectations/sentiments about aggregate C t, so given a nominal wage W t = 1, they can infer a price P t and a real wage W t P t, consistent with (1) and supply labor. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 9 / 43

Final Good Aggregator The final-good firms (or a representative consumer) produce a final good according to [ ] θ C t = Y t = ɛ 1 θ jt Y θ 1 θ 1 θ jt dj where θ > 1 and log ɛ jt are iid zero mean firm-specific shocks, and maximizes profit [ ] θ max P t ɛ 1 θ 1 θ θ dj p jt Y jt dj. jt Y θ 1 jt The demand function depends on both ɛ jt and Y t : p jt P t Y jt = = Y 1 θ jt (ɛ jt Y t ) 1 θ ( Pt p jt ) θ ɛ jty t Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 10 / 43

Intermediate Goods Each intermediate firm produces good Y jt without perfect knowledge either of ɛ jt or of aggregate demand Ỹ t, which could be random and sentiment-driven. Instead, as in the Lucas island model, firms have a noisy indication of what their demand will be from a signal s jt = λ log ɛ jt + (1 λ) log Ỹ t The parameter λ reflects the weights of the idiosyncratic and aggregate components of demand. This signal is our simplest signal. In the paper the simplest signal has iid shocks v jt N ( 0, σ 2 v ) as well. Signal microfoundations will determine λ later. In a REE aggregate demand Ỹ t that generates the signal will be equal to actual aggregate output Y t. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 11 / 43

Intermediate Goods, Cont d An intermediate goods producer j has the production function Y jt = An jt The firm maximizes expected nominal profits Π jt = p jt Y jt W t Y jt [ ( max E t P t Y ) ] 1 θ Y jt (ɛ jt Y t ) 1 Y jt θ Y jt W t jt A s jt. After simplifications using 1 C t = 1 Y t = ψ P t W t, and W t = 1 : Y jt = {( 1 1 ) A [ θ ψ E t (ɛ jt ) 1 θ Y 1 θ 1 t ] s jt } θ. Strategic Substitutabilty: since θ > 1, Y jt is negatively related to Y t. A : Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 12 / 43

Certainty Equilibrium Information is perfect and sentiments have no role.the signal s j fully reveals the firm s own idiosyncratic demand ɛ jt. (σỹ = 0) There exists a fundamental certainty equilibrium with constant aggregate output C t = Y t = Y and P t = P and firm output is: Y 1 θ jt = ( 1 1 θ ) A ψ ɛ 1 θ jt Y 1 θ θ t Without loss of generality set ( 1 1 θ ) A ψ = 1. Final good output is: [ Y t = ɛ 1 θ jt Y θ 1 θ jt dj ] θ θ 1 or, if ε jt log ɛ jt has zero mean and variance σ 2 ε, φ 0 = log Y t = 1 θ 1 log E exp(ε jt) = 1 2 (θ 1) σ2 ε Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 13 / 43

Self-Fulfilling Equilibrium We conjecture there exists an another equilibrium, such that aggregate output is not a constant. In particular all agents "know" output follows log Y t = φ 0 + z t, The noisy signal received by each firm (now defined net of the constant term φ 0 ) is where z t N ( 0, σ 2 z ). s jt = λε jt + (1 λ)z t With fluctuations in aggregate output, the signal is not fully revealing. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 14 / 43

Sentiment-Driven REE In the self-fulfilling equilibrium: Each period the sentiment z t held by households in log Y t = φ 0 + z t will be the realized z t in log Y t. So the distribution of the perceived sentiment {z} will be consistent with the realized distribution of aggregate output {Y }. Prices will clear al markets each period. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 15 / 43

Self-Fulfilling Equilibrium Proposition If λ ( 0, 1 2 ), there exists a self-fulfilling rational expectations equilibrium with stochastic aggregate output Y t = φ 0 + z t. Furthermore log Y t is normally distributed with mean φ 0 = (1 λ) + (θ 1) λ θ(1 λ) φ 0 < φ 0 and variance σ 2 z = λ (1 2λ) (1 λ) 2 θ σ2 ε Welfare? Note that in this case (but not necessarily in models that follow) the mean of the constant output equilibrium φ 0 > φ 0, so in this case it Pareto dominates the sentiment driven stochastic equilibrium. ess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 16 / 43

Self-Fulfilling Equilibrium, Cont d In the Certainty Equilibrium with σ 2 z = 0, Y jt = ɛ jt Yt 1 σ, and since σ > 1, equilibrium firm-level outputs depend negatively on aggregate output as in the case of strategic substitutability. Hence, this fundamental equilibrium is unique. Given λ in s jt = λε jt + (1 λ)z t and the variance of the idiosyncratic shock σ 2 ε, for markets to clear for all possible realizations of the sentiment z t, the variance σ 2 z has to be precisely pinned down, as in the Proposition above. If σ 2 z is too high, output is too low relative to aggregate demand by consumers, and vice-versa. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 17 / 43

Intuition If firms believe that the signal contains information about changes in aggregate demand, z t, then this belief will induce all agents to amplify their output response, up or down, and sustain self-fulfilling fluctuations consistent with the agents beliefs about the distribution of output. Note that both the variance of the sentiment shock σ 2 z and λ affect the firms optimal output responses through their signal extraction. Given λ and the variance of the idiosyncratic shock σ 2 ε, for markets to clear for all possible realizations of the sentiment z t, the variance σ 2 z has to be precisely pinned down, as in the Proposition. If however the signal gives a low weight to aggregate as opposed to idiosyncratic demand, that is if λ [0.5, 1], then we cannot find a positive variance σ 2 z that will clear the markets for every z t. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 18 / 43

Generalizing the signals (SKIP) Imperfect signal with firm-specific noise So far we assumed that firms can get an initial signal for the overall demand for their product, but cannot disaggregate it into its components arising from idiosyncratic and from aggregate demand. They only observe their sum. Since the signals are based on early and initial demand indications for each of the firms, they may well contain additional firm-specific noise components. Suppose then that the signal takes the slightly more general form, s jt = v jt + λε jt + (1 λ) z t, (2) where v jt is a pure firm-specific iid noise with zero mean and variance σ 2 v. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 19 / 43

Imperfect signal with firm-specific noise, Cont d (SKIP) The self-fulfilling equilibrium We had defined log Y t = y t = φ 0 + z t Proposition Let λ < 1 2, and σ2 v < λ (1 2λ) σ 2 ε. In addition to the certainty equilibrium, there also exists a self-fulfilling rational expectations equilibrium with stochastic aggregate output, log Y t = φ 0 + z t that has a mean φ 0 = 1 ( ) (1 λ + (θ 1) λ) 1 σ 2 ε (θ 1) σ2 v 2 θ(1 λ) (θ 1) 2θ 2 (1 λ) 2 and variance σ 2 z = λ (1 2λ) 1 (1 λ) 2 θ σ2 ε (1 λ) 2 θ σ2 v. ess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 20 / 43

Multiple signals The government and public forecasting agencies as well as news media often release their own forecasts of the aggregate economy. Suppose firms receive two independent signals, s jt and s pt. The firm-specific signal s jt is based on firm s own preliminary information about its demand: s jt = v jt + λε jt + (1 λ) z t The public signal is: s pt = z t + e t where e t N ( 0, σ 2 e ) is noise in the public forecast of aggregate demand. We can also model the noisy public signal with heterogenous but correlated sentiments across consumers, so observing a subset of consumers will reveal a noisy signal of the average sentiment. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 21 / 43

Multiple signals, Cont d (SKIP) To establish the existence of the certainty equilibrium, we also assume that σ 2 e = γσ 2 z, where γ > 0. This assumption states that the variance of the forecast error of the public signal for aggregate demand 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. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 22 / 43

Multiple signals, Cont d Proposition If λ < 1 2, and σ2 v < λ (1 2λ) σ 2 ε, then there exists a self-fulfilling rational expectations equilibrium with stochastic aggregate output which has mean φ 0 = 1 2 log Y t = y t = z t + ηe t + φ 0 ( (1 λ+(θ 1)λ) θ(1 λ) ) 1 σ (θ 1) 2 ε (θ 1)σ2 v 2θ 2 (1 λ) 2 and variance σ 2 ẑ = λ (1 2λ) 1 (1 λ) 2 θ σ2 ε (1 λ) 2 θ σ2 v > 0, and where η = σ2 z = 1 σ 2 e γ. In addition, there is a "certainty" equilibrium with constant output identical to the certainty equilibrium in the previous Proposition with a single signal. Note: the public forecast error e t affects output. ess 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 log output of firms coming from a linear quadractic objective, is given by the rule y jt = E t {[β 0 ε jt + βy t ] s jt } where ε jt is zero mean, iid. The coeffi cient β can be either negative or positive, so we can have either strategic substitutability or strategic complementarity in firms actions. Market clearing requires y t = y jt dj. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 24 / 43

Abstract Model Cont d: We have, from a linear quadratic objective, for β < 1, y jt = E t {[β 0 ε jt + βy t ] s jt } (3) s jt = v jt + λε jt + (1 λ) y t (4) y t = y jt dj (5) Assume that y t is normally distributed with zero mean and variance σ 2 y. Based on (3), signal extraction gives: y jt = λβ 0 σ2 ε + (1 λ)βσ 2 y σ 2 v + λ 2 [v σ 2 ε + (1 λ) 2 σ 2 jt + λε jt + (1 λ) y t ] y Market clearing implies: [ ] λβ0 σ 2 ε + (1 λ)βσ 2 y y t = y jt dj = σ 2 v + λ 2 (1 λ) y σ 2 ε + (1 λ) 2 σ 2 t. y Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 25 / 43

Abstract Model Cont d: Certainty and Self-Fulfilling Stochastic Equilibria Market clearing implies y t = y jt dj = λβ 0 σ2 ε + (1 λ)βσ 2 y σ 2 v + λ 2 (1 λ)y σ 2 ε + (1 λ) 2 σ 2 t. (6) y The Certainty Equilibrium is y t 0. But (6) holds for all y t if σ 2 y = λ(β 0 (1 + β 0 )λ)σ2 ε σ 2 v (1 λ) 2 (1 β) Thus, σ 2 y is pinned down uniquely and it defines the Self-Fulfilling Stochastic Equilibrium. Note that if β < 1, a necessary condition for σ 2 y to be positive is λ ( 0, β 0 1+β 0 ). Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 26 / 43

Stability Under Learning Our model is essentially static, but we can investigate whether the equilibria of the model are stable under adaptive learning. For simplicity we will confine our attention to the simplified abstract model of section with σ 2 v = 0, where without loss of generality we set β 0 = 1. So the model is s jt = λε jt + (1 λ) log Y t y jt = E t {ε jt + βy t s jt } y t = y jt dj Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 27 / 43

Stability Under Learning Cont d We can renormalize our model so that the sentiment or sunspot shock z t has unit variance by redefining output as y t = log Y t = σ z z t. The variance of output y t then is still σ 2 z.. Suppose that agents understand that equilibrium y t is proportional to z t and they try to learn σ z. If agents conjecture at the beginning of the period t that the constant of proportionality is σ zt = y t z t, then the realized output is y t = λσ2 ε + (1 λ)βσ 2 zt λ 2 σ 2 ε + (1 λ) 2 σ 2 (1 λ)σ zt z t. zt Under adaptive learning with constant gains g = 1 α, agents update σ zt : ( ) yt σ zt+1 = σ zt + (1 α) σ zt h(σ zt ) z t Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 28 / 43

Stability Under Learning Cont d We have the adaptive learning updating rule ( ) yt σ zt+1 = ασ zt + (1 α) = σ zt + (1 α) z t ( ) yt σ zt h(σ zt ) z t For any initial σ zt > 0, we can show that σ zt does not converge to 0, the certainty equilibrium. By contrast the sentiment-driven sunspot equilibrium is locally stable under learning provided the gain g = 1 α is not too large. In particular h (0) > 1 and h(σ z ) < 1 at the sentiment driven ( ) equilibrium σ z = λ(1 2λ)σ 2 0.5 ε (1 λ) 2 (1 β). Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 29 / 43

Microfoundations of the signal So far we simply assumed that firms receive signals s jt = v jt + λε jt + (1 λ) y t based on their market research, market surveys, early orders, initial inquiries and advanced sales, to form such expectations. In particular, we assumed that the signals weights λ and (1 λ) are exogenous. It is therefore desirable to spell out in more detail the microfoundations for how firms obtain these signals. If the signal reveals, except for iid noise, the precise weighted sum of the fundamental and the sentiment shocks that the firms need to forecast, then the signal extraction problem of the firm disappears. This then excludes the possibility of sentiment-driven equilibria. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 30 / 43

Microfoundations of the signal, Cont d For example, suppose firm j can post a hypothetical price p jt and ask a subset of consumers about their intended demand given this hypothetical price. The firms then obtain a signal about the intercept of their demand curve, possibly with some noise v jt if consumers have heterogenous sentiments, s jt = ε jt + (1 θ) y t + v jt. To see this note that the demand curve of firm j is given by ( ) Pjt θ Y jt = P Yt t ɛ jt. Since from the labor market first order conditions we have P t = 1 ψy t, the logarithm of the demand curve, ignoring constants that can be filtered, becomes y jt = ε jt + (1 θ) y t θp jt. Suppose firm j can post a hypothetical price p jt and ask a subset of consumers about their intended demand given this hypothetical price. The firms can then obtain a signal about the intercept of their demand curve, possibly with some noise v jt if consumers have heterogenous sentiments, s jt = ε jt + (1 θ) y t + v jt. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 31 / 43

Microfoundations of the signal, Cont d The optimal output decision of firms is given by y jt = E [ε jt + (1 θ)y t ] s jt = σ2 s σ 2 v σ 2 s [ε jt + (1 θ)y t + v jt ] where σ 2 s is the variance of the signal. Note that σ 2 s σ 2 v > 0 is the variance of ε jt + (1 θ)y t, or Cov (ε jt + (1 θ)y, s jt ). Integrating across firms, y t = y jt dj, and equating coeffi cients of y t yields 1 = σ2 s σ 2 v (1 θ). This equality is impossible (even if σ 2 σ 2 v = 0) s since by construction σ 2 s > σ 2 v and θ > 1. In other words, a constant output with y t = 0 is the only equilibrium. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 32 / 43

Microfoundations of the signal, Cont d To obtain the possibility of sentiment-driven equilibria, we can either slightly complicate the signal extraction problem of the firm by adding an extra source of uncertainty, Or we can modify the signal so that it does not eliminate the signal extraction problem faced by the firm. We provide microfoundations for both of these approaches below. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 33 / 43

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 p jt and ask a subset of consumers about their intended demand at that price. However at the time of the survey the preference shock is not yet realized with certainty: each consumer i receives a signal for his/her preference shock ɛ jt : s j ht = ε jt + hjt i which forms the basis of their response to the posted hypothetical price. Let utility of consumers be C 1 γ t 1 1 γ ψn t, and let σ 2 h be the variance of hjt i and σ2 ε be the variance of ε jt. Then we can show the weight λ in the firm s signal s jt = v jt + λε jt + (1 λ) z t is uniquely determined if 1 θγ > λ σ 2 ε σ 2 ε +σ 2 h σ 2 ε + (1 θγ) σ 2 ε +σ 2 h ( 0, 1 ) 2 σ2 ε : σ 2 ε +σ 2 h 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 establish sentiment-driven equilibria, we instead modify the signals so that they do not eliminate the signal extraction problem faced by firms. Suppose the intermediate-good firms receive a signal from consumers about the quantity of the demand Y jt for their good. We drop the final-good sector and assume instead that the representative household purchases a variety of goods C jt to maximize [ ] θ 1 utility log C t ψn t, where C t = ɛ θ jt C θ 1 θ 1 θ jt dj. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 35 / 43

A Quantity Signal, Cont d The households have utility utility log C t ψn t and choose their demand C jt and labor supply N jt based on the sentiment shock Z t and the preference shocks ɛ jt. In equilibrium household demand function for each variety is: C jt = ( Pt P jt ) θ ɛ jtc t = ( Pt P jt ) θ ɛ jty t. Since demand depends on the consumers conjectures about their price P jt = P t (ɛ jt, Z t ), the demand C jt is a function of preference shocks and the sentiment, C jt = C (ɛ jt, Z t ). Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 36 / 43

A Quantity Signal, Cont d The intermediate-good firms, based on the signal s jt = c jt choose their production according to the first order condition given by C jt = Y jt = {( 1 1 ) A [ θ ψ E t ɛ 1 θ We conjecture that in equilibrium, c jt = φε jt + φ z z t jt Y 1 θ 1 t where φ and φ z are undetermined coeffi cients. ] } θ S jt, Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 37 / 43

A Quantity Signal, Cont d Proposition There is a continuum of sentiment-driven equilibria c jt = φε jt + φ z z t with i) φ z = 1 ii) φ [0, 1] iii) σ 2 z = φ(1 φ) 1 β σ2 ε Proposition [ ] 1 0, 4(1 β) σ2 ε The sentiment-driven equilibria of this model with signal s jt = c jt can be mapped one-to-one to the sentiment-driven equilibria of our benchmark model with the signal s jt = λε jt + (1 λ)y t where λ = φ [ φ + 1 0, 1 ]. 2 ess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 38 / 43

A Quantity Signal, Cont d (SKIP) Proposition Under the signal s jt = c jt, there also exists another type of sentiment-driven equilibria with firm-level output driven not only by the fundamental shock ε jt and aggregate sentiment shock z t, but also by a firm-specific iid shock v jt with zero mean and variance σ 2 v : c jt = φε jt + z t + (1 + φ)v jt. Furthermore the signal s jt = c jt is isomorphic to the signal s jt = λ log ɛ jt + (1 λ)y t + v jt. ess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 39 / 43

Signal with Cost Shocks (SKIP) [ Utility is C 1 γ t 1 1 1 γ ψn t where C t = ɛ θ jt C θ 1 θ jt dj ] θ θ 1. Each firm j s total production cost (or labor productivity) is affected by an idiosyncratic shock that is correlated with the demand shock ɛ jt. For example marketing costs may be lower under favorable demand conditions: for a higher amount of sales, labor becomes more productive so that labor demand N jt = Y jt ɛjt τ is lower, where τ > 0 is a parameter. Alternatively, if marketing costs increase with sales and labor demand is higher, we may have τ < 0. Firm j s optimal output, given consumer demands, is y jt = E t [ ((1 + θτ)ε jt + (1 θγ)y t ) s jt ]. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 40 / 43

Signal with Cost Shocks Cont d (SKIP) Let firms get a signal s jt = ε jt + (1 θγ)y : the intercept of their demand curve. Optimal output, after taking logs, is y jt = (1 + θτ)σ2 ε + (1 θγ) 2 σ 2 z σ 2 ε + (1 θγ) 2 σ 2 (ε jt + (1 θγ)y t ) z Define β = (1 θγ) < 1. Since in equilibrium C jt = Y jt,integrating for market clearing and equating coeffi cients we get σ 2 z = [(1 + θτ)β 1] β 2 σ 2 ε (1 β). For σ 2 z 0 we need (1 + θτ)β > 1 so we must have τ = 0. If β > 0 (the case where firm output and aggregate output are complements) sentiment-driven equilibria exist if τ > 1 β βθ. If β < 0 (the case where firm output and aggregate output are substitutes) we need τ < 1 β βθ and τ can be negative. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 41 / 43

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. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 42 / 43

Conclusion When production decisions must be made under uncertain demand conditions, optimal decisions based on sentiments can generate self-fulfilling rational expectations equilibria in simple production economies without persistent informational frictions. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations October 15, 2013 43 / 43