Sentiments and Aggregate Fluctuations

Transcription

1 Sentiments and Aggregate Fluctuations Jess Benhabib Pengfei Wang Yi Wen June 15, 2012 Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

2 Introduction We construct a model to capture the Keynesian insight that employment and production decisions are based on expectations of aggregate demand driven by consumer sentiments, and that realized demand follows from the production and employment decisions of firms. Our work is inspired by Angeletos and Lao (2011), where sentiments can drive output, and the Lucas Island model. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

3 A Model of Sentiments and Fluctuations Each firm must make a production decision before demand is realized on the basis of a signal about what its demand will be. Since the real wages and employment have not yet been determined and production has not yet taken place, these signals capture consumer sentiment based on market research about demand, early orders, initial inquiries... In the simplest benchmark model each firm s signal also reflects an idiosycratic shock to the firm s demand. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

4 Model cont d In simplest benchmark model the signal is simply a weighted sum of the firm s idiosyncratic demand shock and a shock to aggregate demand, both of which shift the firm s demand curve. We can also add an iid firm-specific noise to the signal that each firm receives. Later we generalize further and introduce a second noisy but public signal of aggregate demand. This signal may represent public forecasts of aggregate demand, and is available to all firms in the economy. A second model, presented later, replaces the idiosyncratic demand shock with an aggregate preference/productivity shock. Even if producers observe aggregate demand, they cannot precisely identify the magnitudes of its components: the sentiment shock or the fundamental aggregate shock. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

5 Model cont d Firms act on the signal and produce, and only then aggregate output and demand is realized and markets clear. In equilibrium the all agents "know" the correct distribution of the idiosyncratic and aggregate demand shocks. The informational structure in our model is simple: trades take place in centralized markets rather than bilaterally through random matching, and at the end of each period all history can become public knowledge. Informational asymmetries obtain only within the period as firms optimally decide on how much to produce on the basis of their signals. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

6 We show that in this simple model, there can be two distinct rational expectations equilibria: one with constant output and one with stochastic output driven by self-fulfilling sentiments. We can then easily introduce a Markov Sunspots, transitioning between stretches of constant equilibria, punctuated by periods of volatile, sentiment-driven equilibria with lower mean output. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

7 Relation to Global Games In models of global games multiple coordination equilibria can become unique once agents face some private uncertainty. When private signals about fundamentals are relatively precise but diverse, agents put heavy weight on them and multiple coordination equilibria can be ruled out. In our model, we start with a unique equilibrium that has constant output, but when we introduce uncertainty about aggregate demand perceived through signals, we obtain multiple equilibria. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

8 The Benchmark Model: Household 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. Denoting Λ t as the Lagrangian multiplier for the budget constraint, the first-order conditions imply Λ t = 1 C t = ψ P t W t P t = W t ψc t Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

9 Final Good The final-good firms (or a representative consumer) produce a final good according to [ ] θ 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 shocks, and maximizes profit [ ] θ max P t ɛ 1 θ 1 θ θ dj p jt Y jt dj. The demand function: jt Y θ 1 jt p jt P t = Y 1 θ jt (ɛ jt Y t ) 1 θ Y jt = ( Pt p jt ) θ ɛ jty t. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

10 Intermediate Goods Each intermediate firm produces good j to meet its demand Y jt without perfect knowledge either of ɛ jt or of aggregate demand Y t, which could also be random. Instead, as in the Lucas island model, they have a noisy indication of what their demand will be from a signal s jt, s jt = λ log ɛ jt + (1 λ) log Y t, where λ reflects the weights assigned to the idiosyncratic and aggregate components of demand. Based on the signal, the firm chooses to produce output and maximize profits. This is our simplest signal, to be generalized later. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

11 Intermediate Goods, Cont d An intermediate goods producer j has the production function Y jt = An jt. So the firm maximizes expected nominal profits Π jt = p jt Y jt W t A Y jt by solving: [ ] max E t P t Y 1 1 θ Y jt (ɛ jt Y t ) 1 W t θ jt A Y jt s jt. After simplifications using 1 C t = 1 Y t = ψ P t W t, Y jt = {( 1 1 ) A [ θ ψ E t (ɛ jt ) 1 θ Y 1 θ 1 t ] s jt } θ. Note that in equilibrium, since θ > 1, Y jt is negatively related to Y t.if aggregate output increases, real wages rise and firms reduce their output. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

12 Certainty Equilibrium There exists a fundamental certainty equilibrium in this economy with constant aggregate output C t = Y t = Y and P t = P. In this case information is perfect and the signal fully reveals the firm s own idiosyncratic demand. We then have Y 1 θ jt = ( 1 1 θ ) A ψ ɛ 1 θ jt Y 1 θ θ t, (1) Without loss of generality set ( 1 1 ) A θ ψ = 1. The final good output is: Y 1 1 θ t = ɛ 1 θ jt Y 1 1 θ jt dj (2) Combine (1) and (2): Y θ 1 θ t = Y θ 1 (1 θ) θ t ɛ jt dj, 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 (3) Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

13 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, (4) where z t is a normally distributed random variable with zero mean and variance σ 2 z. The noisy signal received by each firm is (factoring out the constant term φ 0 ), s jt = λε jt + (1 λ)z t. (5) With fluctuations in aggregate output, the signal is not fully revealing. We may view z t as a sentiment held by households about aggregate demand, perceived by firms through their signal s jt. Using (4) and (5) and the distributions of ε jt and z t each intermediade good firm will chose its output optimally. In our self-fulfilling equilibrium the distribution of the perceived sentiment {z} will be consistent with the realized distribution of aggregate output {Y }, the sentiment z t held by households will be the realized z t in (4), and markets will clear each period. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

14 Self-Fulfilling Equilibrium, Cont d Proposition If λ ( 0, 1 2 ), there exists a self-fulfilling rational expectations equilibrium with stochastic aggregate output Y 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 ess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

15 Self-Fulfilling Equilibrium, Cont d If firms believe that the signal contains information about changes in aggregate demand, z t, then this belief will partially coordinate the output response of firms, up or down, and sustain self-fulfilling fluctuations consistent with the agents beliefs about the distribution of output. In the Certainty Equilibrium Y jt = ɛ jt Yt 1 σ and since σ > 1, in equilibrium firm-level outputs depend negatively on aggregate output because intermediate goods are strategic substitutes. Hence, the certainty or fundamental equilibrium in the model will be unique as a result of this strategic substitutability. However, the optimal supply of the firm s output positively depends on firm-level demand shocks. Consequently, if firms cannot distinguish firm-level shocks from aggregate demand, informational strategic complementarities can arise, giving rise to self-fulfilling equilibria. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

16 Self-Fulfilling Equilibrium, Cont d Under strategic substitutability the optimal output of an intermediate goods firm declines with σ 2 z as the firm attributes more of the signal to an aggregate demand shock. In the self-fulfilling equilibrium, σ 2 z is determined at a value that will clear markets for all z. 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 above. (We will get an interval for σ 2 z for which RE equilibria obtain in the next model with aggregate as opposed to idiosyncratic fundamental shocks.) If however the signal gives too low a 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 June 15, / 59

17 Generalizing the signals 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, (6) 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 June 15, / 59

18 Imperfect signal with firm-specific noise, Cont d As before we define log Y t = y t = φ 0 + z t In this setup, both the certainty equilibrium and the self-fulfilling equilibrium will be different than those of the benchmark setting of the previous Proposition. We first state the result for the certainty equilibrium. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

19 Imperfect signal with firm-specific noise, Cont d We first establish the Certainty Equilibrium. Let µ = 1 θ λσ2 ε + 1 θ θ (1 λ)σ 2 z. σ 2 v +λ 2 σ 2 ε +(1 λ) 2 σ 2 z Proposition There is a constant certainty equilibrium, y t = φ 0, given by [( ) ] φ 0 = 1 θ + θµλ (θ 1) + (θµλ (θ 1)) 2 2 θ 2 σ 2 ε + (θ 1) (θµ) 2 σ 2 v (θ 1) ( ) θ + θµλ (θ 1) + (θµλ (θ 1)) = φ (θ 1) (θµ) 2 σ 2 v θ 2 Note that if σ 2 v = 0, then µ = 1 θλ and φ 0 = φ 0, so the certainty solution reduces to the previous benchmark case. ess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

20 Imperfect signal with firm-specific noise, Cont d We had defined log Y t = y t = φ 0 + z t The self-fulfilling equilibrium is now given by the following Proposition. 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 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 a variance σ 2 y = 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 June 15, / 59

21 Imperfect signal with firm-specific noise, Cont d If either λ 1 2, or if σ2 v > λ (1 2λ) σ 2 ε, then σ 2 z < 0, suggesting that the only equilibrium is z = 0. Hence, to have a self-fulfilling expectations equilibrium, we require λ ( 0, 1 2 ) and σ 2 v < λ (1 2λ) σ 2 ε. This pins down the equilibrium σ 2 z > 0, the variance of z or of output as a function of σ 2 ε and σ 2 v. Introducing the extra noise v jt into the signal makes output in the self-fulfilling equilibrium less volatile compared to our benchmark case. The reason for the smaller volatility of output when σ 2 v > 0 is that the signal now is more noisy, and firms attribute a smaller fraction of the signal to demand fluctuations. However the existence of the self-fulfilling equilibrium does require the additional restriction that the variance of the extra noise is not too big, σ 2 v < λ (1 2λ) σ 2 ε, to assure that σ 2 z > 0. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

22 Multiple signals The government and public forecasting agencies as well as news media often release their own forecasts of the aggregate economy. Such public information may influence and coordinate output decisions of all firms and affect the equilibria. 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 and is as before s jt = v jt + λε jt + (1 λ) z t The public signal is s pt = z t + e t where we can interpret e t as a common noise in the public forecast of aggregate demand with mean 0 and variance σ 2 e. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

23 Multiple signals, Cont d 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 June 15, / 59

24 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 = ẑ 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 y = 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 that given in Proposition 2 with σ 2 z = γσ 2 e = 0. ess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

25 Multiple signals, Cont d In the self-fulfilling equilibrium of the Proposition above, where σ 2 ẑ = λ(1 2λ) (1 λ) 2 θ σ2 ε 1 (1 λ) 2 θ σ2 v and η = σ2 z = 1 σ 2 e γ, the optimal weight that firms place on the public signal is zero. Nevertheless aggregate output log Y t = y t = z t + ηe t + φ 0 = ẑ t + φ 0 is stochastic, and driven by the volatility of ẑ = z t + ηe t. It is easy to see that the certainty equilibrium of Proposition 2 with σ 2 z = 0 will also apply in the certainty equilibrium of this case since we would have σ 2 e = γσ 2 z = 0. Namely the public signal also becomes a constant. We can then directly apply Proposition 2 to find the certainty equilibrium output. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

26 Persistence Persistence and Sunspots We now construct a persistent sunspot equilibrium with Markov transitions between the certainty and the stochastic self-fulfilling equilibrium. To construct such an equilibrium, we introduce a sunspot S t = 1 or 0. We have the transition probabilities Pr(S t = 1 S t 1 = 1) = ρ and Pr(S t = 0 S t 1 = 0) = ξρ. Then the stationary distribution is Pr(S = 1) = Pr(S = 0) = 1 ρ 1 ρ+1 ξρ. 1 ξρ 1 ρ+1 ξρ, The agents observe the sunspots first and if S t = 1, they cooperate on the certainty equilibrium but if S t = 0, then they cooperate on the uncertainty equilibrium. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

27 Persistence and Sunspots, Cont d A simulated path of aggregate output is obtained in Figure 1 below. We set θ = 4, σ 2 ε = 1, ρ = 0.95, ξ = 0.7. For these parameters the certainty equilibrium is log Y t = 1 2(θ 1) σ2 ε = φ 0 = Simulated Path of Output Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

28 Persistence: Autoregressive productivity If productivity, A t, is a stochastic process that firms can observe, then in parallel to our benchmark case we can express output as log Y t = φ 0 + z t + log A t. Setting ( 1 1 ) 1 θ ψ instead of ( 1 1 ) At θ ψ to unity, as before market clearing would require the sum of log ouputs of firms to add to aggregate log output for every z t. Later we will explore a model where A t is stochastic, cannot be observed by firms. ess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

29 A Simple Abstract Model To set the basic intuition before turning to macro models consider the following three equation model. Assume for simplicity that the economy is log-linear, so optimal log output (or investment, price, labor, etc.) 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 } (7) 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. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

30 A Simple Abstract Model cont d The signal s jt is given by s jt = v jt + λε jt + (1 λ) y t, (8) where both the exogenous noise v jt and the idiosyncratic demand shock ε jt are iid and normally distributed with a zero mean. Market clearing then requires y t = y jt dj. (9) Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

31 A Simple Abstract Model cont d We have (from a linear quadratic objective) y jt = E t {[β 0 ε jt + βy t ] s jt } (10) s jt = v jt + λε jt + (1 λ) y t, (11) y t = y jt dj. (12) with β < 1. In the Certainty Equilibrium y t is constant, so equation (10) yields λβ 0 σ2 ε y jt = βy t + σ 2 v + λ 2 (v σ 2 jt + λε jt ). (13) ε Substituting the above solution into equation (12) and integrating give y t = y jt dj = βy t (14) So unless β = 1, in which case there is a continuum of certainty equilibria, the unique certainty equilibrium is given by y t = 0. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

32 A Simple Abstract Model cont d In the Self-fulfilling Stochastic Equilibrium, assume that y t is normally distributed with zero mean and variance σ 2 y. Based on the simple response function given by equation (10), signal extraction implies y jt = λβ 0 σ2 ε + (1 λ)βσ 2 y σ 2 v + λ 2 σ 2 ε + (1 λ) 2 σ 2 y [v jt + λε jt + (1 λ) y t ]. (15) Then market clearing requires 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. (16) y Since this relationship has to hold for every realization of y t, we need λβ 0 σ 2 ε + (1 λ)βσ 2 y σ 2 v + λ 2 (1 λ) = 1 (17) σ 2 ε + (1 λ) 2 σ 2 y Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

33 A Simple Abstract Model cont d This implies λβ 0 σ 2 ε + (1 λ)βσ 2 y σ 2 v + λ 2 (1 λ) = 1 (18) σ 2 ε + (1 λ) 2 σ 2 y 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 equilibrium. Note that if β < 1, for σ 2 y > 0 we would require λ(β 0 (1 + β 0 )λ)σ 2 ε σ 2 v > 0. Then a necessary condition for σ 2 z to be positive is λ ( 0, (19) β 0 1+β 0 ). If β 0 = 1, this restriction becomes λ (0, 0.5). Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

34 Model with Aggregate Shocks A representative household derives utility from a final good and leisure. The final good is produced by a representative consumer/final goods producer using a continuum of intermediate goods indexed by j [0, 1]. Each intermediate good is produced using labor. The nominal wage rate is fixed at 1. The real wage (in term of the final goods) can of course fluctuate with the price of the final goods. The households are subject to aggregate "preference" shocks A t and sentiment shocks z t in each period. In all equilibria of the model the households have perfect foresight. Namely, conditional on the aggregate shock and their sentiments, they can perfectly forecast the price level. Based on the forecasted price, they make their consumption and labor supply decisions. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

35 The aggregate consumption decisions made by the households are the source of "possibly" noisy demand signals for the intermediate goods producers. Based on their signal, obtained through market research, intermediate goods producers decide how much to produce. Prices of each intermediate good adjusts to equalize demand and supply. These prices then determine the average cost of the final good and hence the price of the final good. In equilibrium this realized price coincides with the price expected by households based on their sentiments. The results extend to the case where consumer sentiments are heterogenous but correlated. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

36 More precisely, Firms get a "signal" that reveals aggregate demand c t (A t, z t ), possibly with some iid noise v jt. There is still a signal extraction problem since their optimal output responds differentially to A t and z t. For example in equilibrium, they might like to decrease output in response to z t, but the effect of A t on output is ambiguous because of effects both on real wages and the complementarity of intermediate goods (in production or preferences). There is a constant output equilibrium c t = c. In addition, there are a continuum of sunspot equilibria parametrized by σ 2 z, or more precisely, two sunspot equilibria (in logs) c jt = φ i a t + σ z z t, i = 1, 2, for each σ 2 z in 0 < σ 2 z < 1 σ 4(1 β) 2 2 a σ2 v 1 β (with CRRA preferences we have β < 1, could be negative). Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

37 Households C 1 γ t U t = A t 1 γ N t, (20) where C t is consumption of the final good, A t is the preference shock (could easily be a productivity shock-see below) and N t is labor. Household s budget constraint as P t C t N t + Π t. (21) Here P t is the price of the final goods and Π t is the profit collected from all intermediate firms and the wage rate is normalized to 1. The first order condition for C t is A t C γ t = P t. (22) Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

38 Firms The supply side has a repesentative final good producer and a continuum of intermediate goods producers indexed by j [0, 1]. The final goods producer serves a convenient aggregator of all intermediate goods. They do not play an active role in the model. We assume final goods producers make decisions after all shocks realize so their decisions are not subject to any uncertainty. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

39 The final goods firms: max P t C t C jt P jt C jt dj, (23) where C t is produced by a continuum of intermediate goods according to the Dixit-Stiglitz production function, [ 1 C t = 0 ] θ C θ 1 θ jt dj θ 1. (24) The final goods producer s profit maximization problem yields the inverse demand curve for each individual intermediate goods, P jt P t [ = C 1 θ jt C 1 1 ] 1 θ t, P t = Pjt 1 σ 1 σ 0 (25) Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

40 Firms Cont d The Intermediate goods firms: The intermedite goods production function is C jt = N jt. (26) A intermediate goods producer j solves the following problem. Substituting out P jt, we have max E [(P jt C jt C jt ) S jt ] (27) (1 1 θ )C 1 θ jt E [P t C 1 θ t S jt ] = 1 (28) So we have C jt = C jt = { E [P t C 1 θ t S jt ](1 1 } θ θ ) (29) { (1 1 θ θ )E [A tc 1 θ γ t S jt ]}. (30) Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

41 The Signal log S jt = s jt = log C t + v jt = c t + v jt. In what follows, the noise v jt will not be essential for our results: we could have set σ 2 v = 0. In that case the signal s jt would fully reveal aggregate consumption c t to the intermediate goods firms. Nevertheless, sentiment-driven rational expectations equilibria would still exist. this is so because firms set their optimal outputs under imperfect information. Since they do not observe a t and z t directly, they form conditional expectations of them based on their signal s jt = c t. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

42 Equilibrium Again (a) Based on the preference shock A t and sentiments Z t, households conjecture that the aggregate price is given by P t = P(A t, Z t ); (b) Based on the conjectured price P t, the households choose their consumption plan C t = C (A t, Z t ) to maximize their utility; (c) The consumption decisions create signals to firms as log S jt = c t + v jt ; (d) Based on the signal S jt, firm j produces C jt to maximize its expected profit; (e) Given the production of C jt, price P jt adjusts to equalize demand and supply; (f) The total production of final goods C t, equals to the households planed consumption. Hence the realized price is also equal to the conjectured price P t. Without loss of generality assume A t = exp(a t /θ) ( θ θ 1 ) 1 θ, where a t is normally distributed with mean 0 and variance σ 2 a. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

43 Equilibrium in logs Aggregate consumption is given by [ 1 C t = Then we can can write down the systems as 0 ] θ C θ 1 θ jt dj θ 1. (31) c jt = log C jt c = E {[a t + (1 γθ)c t ] s jt } (32) c t = log C t c = s jt = c t + v jt 1 0 c jt dj. (33) (34) where c are c are constants to be determined. Note that in equation (32) β = 1 γθ < 1 and can be negative. If β < 0,we are in the case of gross substitutes. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

44 Constant Fundamental Equilibrium Proposition If the consumers conjecture that the aggregate price is given by p t = log P t p = a t /θ. (35) where p = 1 θ log ( ) θ θ 1 γ c, and c = c = γ σ 2 θ a. Then 2 is always an equilibrium. c jt = c t = 0. (36) ess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

45 Stochastic Fundamental Equilibria Definitions Let µ = σ2 v, and assume 0 < µ < 1 where β = 1 γθ < 1 where σ 2 a 4(1 β) γ > 0 is the curvature of utility of consumption and θ > 0 is the elasticity of substitution in final good production. Define the constant terms: c = θ 2 γ [(θ 1)σ2 v + (1 + βφ)(1 (1 β)φ)σ 2 a], (37) p = 1 ( ) θ θ log γ c (38) θ 1 c = (1 θγ) c (1 + βφ)(1 (1 β)φ)σ 2 a θ (39) Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

46 Stochastic Fundamental Equilibria, cont d Proposition Let consumers conjecture that the aggregate price and consumption follow: log P t p = ( ) 1 θ γφ a t (40) log C t c = φa t (41) Let φ = 1 2(1 β) ± 1 4(1 β) 2 µ > 0. (42) 1 β where µ = σ2 v. Given the consumption expenditures of the households, in a σ 2 a rational expectations equilibrium each firm j produces log C jt c = c jt = φa t + v jt. Note: if σ 2 v 0 so that v jt 0, φ (0, (1 β) 1) ess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

47 Sentiment-Driven Equilibria Suppose consumer expectations on the final good are driven by sentiment shocks as well as the shock log A t : The firm s signal now becomes Let the constant terms c, p be c = 1 2γ [ θ 1 p = 1 θ log ( θ θ 1 log C t c = c t = φa t + σ z z t. (43) s jt = (c t + v jt ) = (φa t + σ z z t + v jt ) θ 2 σ 2 v + (1 + βφ)(1 (1 β)φ)σ2 a βσ 2 z (1 β) θ 2 ] (44) ) γ c (45) Also assume µ = σ2 v +σ 2 z (1 β) σ 2 a < 1 4(1 β). Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

48 Proposition Suppose that 0 < σ 2 v < 1 4(1 β) σ2 a. There existing a continuum of sentiment-driven equilibra indexed by σ 2 z with: 0 < σ 2 z < 1 σ 4(1 β) 2 2 a σ2 v In such equilibrium prices follow 1 β. log P t p = p t = ( 1 θ γφ)a t γσ z z t Optimal consumption expenditure is given by where φ is given by φ = log C t c = c t = φa t + σ z z t 1 2(1 β) ± 1 4(1 β) 2 µ 1 β, (46) ess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

49 We can plot the coeffi cients φ for the fundamental equilibria (σ 2 z = 0 ) and the corresponding coeffi cients φ for the sentiment-driven equilibria against variance of the noise σ 2 v. We calibrate θ = 10, γ = 1, the variance of log A t at 4.5, and plot φ against feasible σ 2 v for various variances of sentiments σ 2 z = (0.25, 0.5, 1) and: Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

50 Extensions: Persistence Let aggregate shock follow a t = ρa t 1 + σ a ε t where ρ = 0 is the special case we considered before. There is a fundamental equilibrium where c t = 1 1 β a t. In addition assume z t = ρz t 1 + ε z,t Proposition There exists an continuum of sentiment-driven equilibria for 0 < σ 2 z /σ 2 a < 1 4(1 β) 2. Given σ 2 z we can solve for two φ, φ = where aggregate production is given by 1 2(1 β) ± 1 4(1 β) 2 σ2 z σ 2, (47) a log C t c = c t = φa t + σ z z t, (48) c = 1 2γ 1 σ 2 z θ 2 φ 2 σ 2 a + σ 2 z (49) ess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

51 Extensions: Heterogenous Sentiments Households only observe A t and the noisy signal s h it = z t + e it. They choose labor supply to maximize C 1 γ t E i {A t 1 γ N t} [A t, sit]. h (50) subject to P t C it N it + Π t, (51) where Π t is the total profits accruing from all the firms. The first order condition for consumers now changes to { C it = 1 E (P t z it ) ( ( ))} 1 θ γ exp(a t /θ). (52) θ 1 Aggregating across consumers, we obtain the aggregate consumption c t = log C t = log( 1 0 C itdi). Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

52 Extensions Cont d As before, we assume that each firm receives a noisy signal log S jt = c t + v jt. The production decision by the firms is given by C jt = { E [P t C 1 θ t S jt ](1 1 θ ) } θ. (53) (Note σ 2 v 0, can be 0.) Conjecturing that equilibrium consumption and production (in logs) can be written as log C jt = c + c jt, log C t = c + c t, and log C it = ĉ + c it we have the following Proposition: Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

53 Extensions Cont d Proposition Suppose σ 2 v < 1 4(1 β) σ2 a and let κ = 1. There exists a continuum of 1+σ 2 e ( ) sentiment-driven equilibria indexed by σ 2 κ z 0, σ 4(1 β) 2 2 a κσ2 v 1 β. At each equilibrium consumers conjecture that aggregate price will be ( ) 1 p t = log P t p = φ p a a t + φ p z z t θ γφ a t γ κ σ z z t (54) and their idiosyncratic consumption demand is φ = where µ = σ2 v +σ 2 z (1 β)/κ. σ 2 a c it = φa t + σ z (z t + e it ) (55) 1 2(1 β) ± 1 4(1 β) 2 µ 1 β, (56) ess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

54 Proposition Cont d Proposition Aggregate consumption (output) is given by Each individual firm s optimal production is and the constant terms are given by c t = φa t + σ z z t (57) c jt = φa t + σ z z t + v jt (58) θ p = log( θ 1 ) θ 1 2θ 2 σ2 v 1 2 Ω s (59) c = 1 γ [ θ 1 2θ 2 σ2 v Ω s ] γ ( ) κ σ z (1 κ) κ 2 σ2 z κ ĉ = c 1 2 σ2 z σ 2 e, c = c 1 2 (60) θ 1 σ 2 v. (61) θ ess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

55 Conclusions We explored a Keynesian model where changing sentiments (or expectations) about aggregate demand can generate output and employment fluctuations under rational expectations. We showed that 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. These sentiment-driven equilibria are not the result of randomizations across multiple fundamental equilibria. In our first model however we also introduced Markov sunspots across equilibria, generating stretches of time when the economy exhibits calm and steady output, alternating with occasional periods of sentiment driven high volatility and low average output. Obviously our model is very simple, but could serve as a benchmark for more complicated equilibrium models with additional features. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

56 Firms set prices before observing demand Here firms commit to fulfill demand. As in our benchmark case, the signal is s jt = λ log ɛ jt + (1 λ) log Y t We obtain self-fulfilling equilibria with constant return to scale only if λ = 1, so the firm knows its own idiosyncratic technology shocks. But they do not know the aggregate demand when setting their prices. In that case the variance of aggregate demand, σ 2 z, is not pinned down by σ 2 ε but is free, subject to some constraints involving the mean. The reason is that in the Dixit-Stiglitz model the idiosyncratic demand component does not change the optimal price rule and with the final good price normalized to unity, firms will set the same price, P jt = EY t in response to the signal. Under market clearing however this is optimal either in the certainty equilibrium with σ 2 z = 0, or in the self-fulfilling stochastic equilibrium if λ = 1. Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

57 Proof Sketch From firm s optimization [ ] y jt θ log E t (ɛ jt ) 1 θ Y 1 θ θ t s jt [ = (1 θ) φ 0 + θ log E t exp( 1 θ ε jt + 1 θ ] z t ) s jt θ [ E t exp( 1 θ ε jt + 1 θ ] z t ) s jt θ ( ( 1 = exp E θ ε jt + 1 θ ) z t s jt + 1 ( 1 θ 2 var θ ε jt + 1 θ )) z t s jt θ where ε jt log ɛ jt has zero mean and variance σ 2 ε. From signal extraction E ( 1 θ ε jt + 1 θ z t s jt ] = θ 1 θ λσ2 ε + 1 θ θ (1 λ)σ 2 z λ 2 (λε σ 2 ε + (1 λ) 2 σ 2 jt + (1 λ)z t ). z Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

58 Proof Sketch, Cont d Ω s = [ (1 θ Ω s = var( 1 θ ε jt + 1 θ z t s jt ] θ ) ] [ 2 (1 1 ) 1 θ 2 1 λ θ 2 λ σ 2 ε + + θ ] 1 θ θ 2 σ 2 z y jt = 1 θ (1 θ)φ 0 + θ ε + 1 θ θ (1 λ)σ 2 z λ 2 (λε σ 2 ε + (1 λ) 2 σ 2 jt + (1 λ)z t ) + θ z 2 Ω s ϕ 0 + θµ(λε jt + (1 λ)z t ) Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59

59 Proof Sketch, Cont d Since Y 1 1 θ t = ɛ 1 θ jt Y 1 1 θ jt dj, Market Clearing for every z : (φ 0 + z t )(1 1 1 θ ) = log 0 ɛ 1 θ jt Y 1 1 θ jt = (1 1 θ )ϕ [1 θ + (1 1 θ )θµλ]2 σ 2 ε + θµ(1 λ)z t (1 1 θ ) This put two constraints by matching the coeffi cient and constant terms: dj 1 θ θµ = θ λσ2 ε + 1 θ θ (1 λ)σ 2 z λ 2 σ 2 ε + (1 λ) 2 σ 2 z = 1 1 λ which implies σ 2 λ (1 2λ) z = (1 λ) 2 θ σ2 ε if λ < 1 2, and matching the coeffi cient for the constant term, after simplifications, φ 0 = (1 θ)φ 0 + θ 2 Ω s + θ 1 θ 1 2 [1 θ + (1 1 θ )θµλ]2 σ 2 ε, (62) Jess Benhabib Pengfei Wang Yi Wen () Sentiments and Aggregate Fluctuations June 15, / 59