Excessively Volatile Stock Markets - NYU Stern School of Business

Excessively Volatile Stock Markets: 

Equilibrium Computation and Policy Analysis 

Thomas M. Mertens ∗ 

November 14, 2008 

JOB MARKET PAPER 

Abstract 

This paper incorporates excess volatility in stock prices into a standard general equilibrium 

model and finds large welfare gains from stabilizing policies. Stock prices in this model 

aggregate information about fundamentals which is dispersed in the economy but also reflect 

excess volatility stemming from correlated distortions in beliefs. To solve the model, this paper 

develops a novel solution method for nonlinear models with dispersed information which 

can be applied to a large class of dynamic general equilibrium models. The innovation lies in 

a nonlinear change of variables which, when combined with perturbation methods, yields the 

nonlinear price function consistent with equilibrium expectation operators. The main positive 

result shows that dispersion of information allows arbitrarily small distortions in beliefs 

to generate large amounts of excess volatility and renders arbitrage infeasible. The government 

cannot observe whether a given stock price movement originates from information or 

noise. As a normative result, price stabilizing policies lead to a higher level of consumption: 

a fall in the risk premium lowers the marginal product of capital and raises the capital stock 

and production. History-dependent policies may improve the information content of prices 

and result in even higher welfare gains. 

JEL classification: C63, E44, E61, G18, H21 

I am indebted to John Y. Campbell, Nicola Fuchs-Schündeln, Kenneth L. Judd, N. Gregory 

Mankiw, and Andrei Shleifer for many discussions and suggestions. I also thank George-Marios 

Angeletos, Robert Barro, Emmanuel Farhi, Tarek Hassan, David Laibson, and seminar partic- 

ipants at Harvard University, MIT, and the Federal Reserve Bank of New York for valuable 

comments. 

∗ Department of Economics, Harvard University, e-mail: mertens@fas.harvard.edu 

1

1 Introduction 

Stock markets around the world display high levels of aggregate volatility. Since Shiller (1981) 

and LeRoy and Porter (1981), a large body of literature has demonstrated that stock prices 

fluctuate more than their ex-post realized net present value of dividends. This finding has gen- 

erated some concern among economists about the possible adverse effects on welfare. However, 

policymakers have been reluctant to stabilize asset prices partially for fear of negative effects on 

the aggregation of information in financial markets. 

This paper incorporates excess volatility in stock prices into a standard general equilibrium 

model by assuming distorted beliefs under dispersed information. The model provides a frame- 

work to study the welfare consequences of price stabilizing policies and their effects on the 

aggregation of information. However, the literature does not provide a method for solving this 

type of model. Therefore this paper develops a solution method for general equilibrium models 

with dispersed information. 

The starting point of the analysis is a standard neoclassical economy with capital accumula- 

tion. As in the q-theory of investment, the link between stock prices and real investment arises 

from capital adjustment costs. The streams of dividends and labor income are stochastic due to 

shocks to total factor productivity which is the only source of fundamental uncertainty in the 

model. 

When forming expectations, agents learn from two sources about the economy’s future path. 

First, they receive an exogenous private signal about next period’s total factor productivity 

(a “news shock”) that contains idiosyncratic noise. Information about fundamentals is thus 

dispersed across the economy. Second, agents observe the resulting equilibrium stock price which 

acts as an endogenous public signal. In equilibrium, the stock price contains the news shock as 

well as aggregate noise which inhibits full revelation of information. The optimal forecast solves 

the signal extraction problem to get the best prediction of next period’s productivity given both 

private and public signals. 

Agents are assumed to deviate from optimal forecasts by slight distortions in beliefs (“belief 

shocks”) which are perfectly correlated across households. The equilibrium stock price influences 

households’ expectations which, in turn, determine the stock price. In equilibrium, belief shocks 

enter prices and, as a solution to this fixed point, constitute the source of aggregate noise. I 

assume distortions in beliefs (as in Cecchetti, Lam and Mark (2000)) and suggest one among 

many possible microfoundations. 

Typically, simplifying assumptions guarantee linearity of policy rules in models with dis- 

persed information and learning from prices, so called Noisy Rational Expectations equilibria. 

In linear models, we can obtain the price function as a solution to the fixed point problem in 

2

closed form. This paper shows that capital accumulation and resulting nonlinearities are the 

driving forces of welfare gains from policy intervention and hence it is essential to quantify them. 

However, to the best of my knowledge, there are no tools available to solve the model. 

Therefore, I develop a general solution technique that is applicable to nonlinear Noisy Ra- 

tional Expectations equilibria in open and closed economies. The innovation lies in a specific 

nonlinear change of variables that allows me to solve the signal extraction problem for house- 

holds in closed form. The solution technique delivers a closed-form approximation of arbitrary 

accuracy not only for the equilibrium variables but also for social welfare. 

The method proceeds in five steps. The first step obtains an approximation to the solution 

using perturbation methods which are discussed in Judd (1998) and Jin and Judd (2002) and 

applied in Fernández-Villaverde and Rubio-Ramírez (2006). The basic idea of perturbation is to 

find the deterministic steady-state of the system and expand the solution in all state variables 

around it to get a higher-order approximation to equilibrium variables. The second step is the 

important part in the procedure. I show that there is a nonlinear change of variables that 

allows me to solve for an equilibrium with dispersed information. To implement the change of 

variables, I build on the method in Judd (2002). It directly leads to step three in which I obtain 

a guess for the equilibrium price function. Given the guess, it is possible to carry out the signal 

extraction problem in closed form in step four. Step five imposes market clearing and verifies the 

guess for prices. With the solution to competitive equilibrium at hand, we can analyze positive 

predictions and potential gains from policy intervention. 

As the main positive result, I show that in equilibrium a given level of excess volatility can 

be sustained by arbitrarily small distortions in beliefs if information is sufficiently dispersed. 

Intuitively, a higher level of idiosyncratic noise in private signals leads households to rely more 

on stock prices when forming expectations. Aggregate noise thus increasingly feeds into optimal 

beliefs, households’ expectations, and hence back into prices (similar to the reasoning suggested 

by Black (1986)). This feedback effect amplifies the impact of distortions in beliefs on aggregate 

noise in prices. 

In the limiting case where information is fully dispersed, the same price function arises as 

in a model with noise traders where agents hold optimal beliefs as in Grossman and Stiglitz 

(1980) and Hellwig (1980). In that sense, the assumption of distorted beliefs merely serves as 

a modeling tool to generate noise trader risk while retaining a framework with a single class of 

investors with heterogenous information. 

I then turn to the normative analysis. Distorted beliefs constitute the only departure from 

the first welfare theorem in this model and thus the only possible reason for intervention. The 

government aims at raising expected utility of households through intervention while not having 

superior information. Thus the government cannot condition its policy on belief shocks and has 

3

to rely on observable quantities. I study two general types of policies that aim at improving on 

competitive market allocations. 

First, I look at stabilization policies that condition on the current period’s stock price. De- 

mand for stocks might change unexpectedly for two reasons: either information about next 

period’s productivity arrives and alters demand for capital or distortions in beliefs lead house- 

holds to misjudge next period’s productivity. While the first movement in demand is desirable, 

the government wants to diminish movements due to the latter. Owing to its inability to distin- 

guish sources of price movements, the government conditions its policy on prices. It can therefore 

only reduce the response to all sources of price movements simultaneously and faces a tradeoff 

between stabilizing fluctuations due to aggregate noise and maintaining efficient responses to 

true information. The optimal solution to this stability-efficiency tradeoff determines optimal 

intervention. 

Policy intervention conditioning on prices raises the level of aggregate consumption after 

optimal policies are put in place. The higher level of consumption is supported by a higher level 

of capital in steady-state. Since lowering excess volatility translates into a lower risk premium, 

the now smaller expected return on stocks has to be matched by a lower marginal product of 

capital in equilibrium. Consequently, the steady-state level of capital rises in response to policy 

intervention. In this sense, the standard thought experiment for the cost of business cycles which 

is often used as an upper bound on gains from stabilization (see for example Barlevy (2005)) 

does not apply to this economy. Lucas (1987) computes the cost of fluctuations in consumption 

by comparing welfare under a stochastic stream of consumption to utility under its mean as a 

deterministic consumption stream. The cost of fluctuations around mean consumption turn out 

to be very small. Contrary to this result, policy intervention in this paper leads to a higher level 

of consumption and thus to significant gains from stabilization. 

The second general type of policy conditions on the previous period’s misjudgment, once it 

becomes public information, in addition to the current stock price. With this policy in place, 

the government can influence the way people use information in the economy. When forecasting 

asset payoffs, agents no longer only predict next period’s productivity to determine the payoff 

but now also take next period’s policy intervention into account. If the policy leads asset payoffs 

to fall in response to previous period’s overpricings, stock prices have less power in predicting 

stock payoffs. Households consequently put relatively more weight on private signals when 

forming expectations and thus improve the information content of prices in equilibrium. The 

mechanism that amplifies distortions in beliefs is mitigated, leading to a lower level of excess 

volatility. The government thus faces an improved stability-efficiency tradeoff. 

Having derived implications of stabilizing policies in a general framework, I turn to a calibra- 

tion for several concrete policy instruments. Quantification of welfare gains depends crucially 

on the ability of the solution technique to capture changes of the risk premium for stocks. 

4

I study the effects of two policy interventions for financial markets: open market operations 

and an interest rate policy. In this paper, open market operations refer to a policy of equity 

purchases and sales by the government as in Kiyotaki and Moore (2008). I implement open 

market operations as a time-invariant rule that conditions on current period’s stock demand. 

The government sells stocks whenever current demand is unexpectedly high and thus reduces 

stock prices. This policy enables the government to stabilize stock prices. The optimal policy 

intervention is determined by the stability-efficiency tradeoff. For the benchmark economy, 

optimally designed open market operations lead to welfare gains of 0.24% of consumption. These 

benefits from stabilization are several orders of magnitude larger than what the standard upper 

bound on stabilization policies would predict. 

The other type of intervention, an interest rate policy conditioning on stock prices, depends 

on an “asset price gap” similar to an output gap in a Taylor rule. Under this policy, interest rates 

rise with higher stock prices. This policy is a blunter instrument than open market operations as 

it distorts the intertemporal margin for bonds which results in higher variance of consumption 

over time. Yet these fluctuations imply only small reductions in utility of agents. Welfare gains 

from interest rate policy amount to an equivalent of 0.23% of consumption and are thus of 

similar magnitude as gains under open market operations. 

To implement the second class of policies, I choose a backward-looking interest rate rule that 

conditions on an asset price gap as well as on the previous period’s misjudgment. It influences 

the way households use information and reduces the level of excess volatility in stock prices. 

With a more favorable stability-efficiency tradeoff, welfare gains almost double compared to 

previous policies and achieve an equivalent of a 0.55% rise in consumption. 

This paper relates to a strand of literature that studies the welfare cost of excess volatility. 

For example, DeLong, Shleifer, Summers and Waldmann (1990), Stein (1987), Kurz (2005), 

and Lansing (2008) analyze the cost of volatility. In closely related work, Hassan and Mertens 

(2008) study a capitalist-worker economy where excess volatility is generated by near-rational 

investment, and find the welfare effects to be high due to effects on capital accumulation. In 

a similar mechanism to the one in this paper, near-rational errors get magnified in equilibrium 

to generate excess volatility in stock prices. The setup of the present paper starts out from a 

first-best economy to isolate the intervention due to distortions in beliefs. 

Another related literature studies costs and gains from stabilizing non-fundamental price 

movements. This research analyzes distortions in expectations in a New-Keynesian framework, 

as for example in Bernanke and Gertler (2000), Bernanke and Gertler (2001), Cecchetti, Genberg 

and Wadhwani (2002), Woodford (2002), Woodford (2005), Dupor (2005), Bullard, Evans and 

Honkapohja (2007), Christiano, Ilut, Motto and Rostagno (2007), and Gilchrist and Saito (2008). 

Relative to this strand of literature, the present paper compares various policies in a frictionless 

economy and quantifies their welfare gains. 

5

A third evolving strand of literature studies the optimal use of information. How policy 

can change the information content of prices through changing the use of information has been 

explored by King (1982). More recently, following Morris and Shin (2002), several papers have 

studied the social value of information and policies to correct for informational inefficiencies, 

among them Hellwig (2005), Amador and Weill (2007), Amador and Weill (2008), Angeletos and 

Pavan (2007), Angeletos and Pavan (2008), Angeletos and La’O (2008), and Lorenzoni (2008). 

Most closely related to this paper is Angeletos, Lorenzoni and Pavan (2007) who analyze the 

interaction between asset prices and investment in an economy where complementarities for 

investors lead to a suboptimal use of information. Price stabilization turns out to be an effective 

strategy to improve welfare. While previous papers assume (log-) linear models, this paper 

shows the importance of the interplay between dispersed information and nonlinearities. 

The structure in the body of the paper follows that of the introduction: section 2 presents 

the model and the definition of equilibrium. Section 3 provides tools for obtaining a solution 

analyzed in section 4. Section 5 introduces a government and derives implications of intervention. 

Section 6 contains a calibration for particular policy instruments whose robustness is laid out in 

section 7. Section 8 concludes. 

2 Model 

I study the effects of distortions in beliefs and remedies for their adverse impacts on asset markets 

in a decentralized small open economy 1 . The economy can borrow and lend from abroad at a risk- 

free return Rbt b,t . Foreign direct investment and international contracts contingent on technology 

shocks are not permitted. Consequently, domestic agents bear all risk originating within their 

borders. Economic actors in this model are households and a representative firm. Households 

consume, work, and hold stocks and bonds. The representative firm rents capital services and 

hires labor from households. 

2.1 Households 

There is a continuum of households indexed by i ∈ [0,1]. Agents supply their endowment of L 

units of labor inelastically. They invest in a risky and a riskless asset and choose their portfolio 

to maximize expected utility. Agents take prices and the distribution of returns as given when 

choosing their optimal actions. When forming expectations about future returns, agents observe 

the equilibrium asset price for the risky asset Pt which acts as an endogenous public signal and 

1 The assumption of a small open economy merely serves as a convenient modeling device to study excess 

volatility. In a small open economy, consumption, investment, and stock prices comove in response to information 

about future productivity (for a discussion, see Beaudry and Portier (2004) and Jaimovich and Rebelo (2006)). 

With standard capital adjustment costs, this comovement is necessary to generate excess volatility in stock prices. 

6

a noisy private signal st(i) about the state of the economy next period. The signal has the 

logarithm of next period’s true aggregate productivity zt+1 as its mean clouded by idiosyncratic 

noise θt(i) and is defined as 

st(i) = zt+1 + θt(i). 

The logarithm of productivity z and idiosyncratic noise θ are normally distributed with mean 

z ∗ resp. 0 and variances σ 2 z and σ 2 θ . Household i chooses optimal consumption Ct(i) and stock 

holdings Kt+1(i) to solve the maximization problem 

max ˜Et 

Ct(i),Kt+1(i) 

 

∞ 

β t 

 

u(Ct(i)) 

st(i),Pt 

t=0 

subject to the budget constraint I discuss below. The utility function features constant relative 

risk aversion, u(C) = C1−γ 

1−γ , and the expectation operator ˜E corresponds to agents’ beliefs. 

The probability density function of agents’ beliefs deviates from that of the optimal fore- 

cast. Let ϕ(zt+1|st(i),Pt) denote the conditional density of the best possible forecast given a 

private signal st(i) and the market clearing price Pt that contains true information zt+1 and 

aggregate noise. Agents’ forecast with density ˜ϕ deviates from the optimal forecast by a shift εt 

in expectations where the distortion εt ∼ N(− 1 

2σ2 ε ,σ2 ε ) is i.i.d over time. Hence, the expectation 

operator ˜ Et used by households is defined as 

 

˜Et[zt+1|st(i),Pt] = 

 

zt+1 ˜ϕi(zt+1|st(i),Pt)dzt+1 = 

ϕ(zt+1|st(i), Pt) 

z t+1 

(1) 

zt+1ϕi(zt+1 − εt|st(i),Pt)dzt+1. (2) 

Figure 1: Distortions in beliefs result in beliefs with the density (dashed line) being a shifted 

version of the optimal forecast (solid line). 

Figure 1 plots the shape of distortions in beliefs for one particular private signal st(i). All 

agents deviate symmetrically from the best forecast. Since the equilibrium stock price is de- 

7

termined by expectations, it not only contains true information but also distortions in beliefs 

acting as aggregate noise. Agents understand the economy perfectly but they neither know the 

realization of the news shock zt+1 nor the current distortion in beliefs εt. They try to infer both 

shocks from their signals. Note that all higher moments under optimal forecasts and distorted 

beliefs are identical which rules out any role for higher-order expectations. 

I follow Cecchetti, Lam and Mark (2000) in assuming distortions in beliefs rather than mod- 

eling their precise nature. There are several appealing features of distorted beliefs. First, their 

effects on individual losses are tiny. Yet each agent with distorted beliefs imposes a risk exter- 

nality on society by adding noise to prices and inhibiting them from being perfectly revealing. 

Distortions in beliefs are in that sense rather a modeling tool to introduce noise into prices than 

a deviation from rationality as in Hellwig (1980) or Grossman and Stiglitz (1980). Section 4 

makes this statement more precise. Furthermore, they allow for a variety of microfoundations 

through which they can be linked to previous models of excess volatility as, for example, over- 

confidence (similar to Odean (1998)) or noise trader risk (see e.g. DeLong, Shleifer, Summers 

and Waldmann (1988)). Shiller (2003) provides a survey of the literature. One possible mi- 

crofoundation for the present setup is the presence of aggregate noise in private signals that 

agents are unaware of. Redefining the private signal shows that the microfoundation leads to 

an equivalent setup to the one in this paper. Lastly, as Cecchetti, Lam and Mark (2000) show, 

appropriately designed distortions of beliefs can account for numerous asset pricing anomalies. 

Besides buying stocks and bonds, households trade contingent claims on the information prior 

to its arrival which are settled at the beginning of the following period. Trading of contingent 

claims merely serves the purpose of eliminating the wealth distribution from the state space in 

equilibrium. Contingent claim holdings Q i t(zt+1,εt,θt(i)) denote the security holdings of agent 

i that pay off if the household receives idiosyncratic noise θt(i) when the news shock is zt+1 and 

the belief shock εt. Agents can purchase contingent claims at price ωt(zt+1,εt,θt(i)). Households 

face the budget constraint 

Bt(i) + PtKt+1(i) =R at 

c,tPt−1Kt(i) + R at 

b,t−1Bt−1(i) − Ct(i) + wtL 

+ Q i t−1 − 

 

ωtQ i t − τg + Tt(i) 

where I suppress arguments for contingent claims Q i and their prices ω. wt denotes wages in pe- 

riod t, Bt(i) bond holdings, and Rat c,t and Rat b,t the post-intervention returns on capital and bonds 

respectively. Households deliver distortionary payments τg defined below, and receive lump-sum 

transfers Tt(i) from the government. The first-order conditions with respect to consumption and 

stock holdings as choice variables yield the optimality conditions 

u ′ (Ct(i)) = β ˜ E u ′ (Ct+1(i))R at 

c,t+1 |st(i),Pt 

 

8 

(3) 

(4)

and 

u ′ (Ct(i)) = βR at 

b,t ˜ E u ′ 

(Ct+1(i))|st(i),Pt . (5) 

Appendix A contains details on the optimal choices for contingent claims trading. 

2.2 Representative Firm 

A representative firm produces a single consumption good which serves as the numeraire. Inputs 

in the production process are capital Kt and labor Lt. The production possibility set is charac- 

terized by a Cobb-Douglas production function f(·, ·) and the shock to total factor productivity 

zt. We write production of output as 

e zt f(Kt,Lt) = e zt K α t L1−α 

t 

with α being the capital share of output. Capital depreciates at rate δ and evolves according to 

Kt+1 = (1 − δ)Kt + It 

where It denotes aggregate investment. There are convex adjustment costs to capital investment 

where χ is a positive constant 2 . 

ACt = 1 

2 χ I2 t 

Kt 

A representative firm rents capital and labor services at competitive prices from households 

to produce the consumption good. It converts It units of output, where It can be negative, 

into capital goods to sell them at price Pt while incurring adjustment costs. The remaining 

output goods are sold on the goods market at a price of one. The firm’s problem reduces to a 

period-by-period maximization problem given by 3 

max e 

Kt,Lt,It 

zt f(Kt,Lt) − It − wtLt − (Dt + δPt)Kt + PtIt − 1 

2 χ I2 t 

Kt 

where Dt denotes dividend payments. First-order conditions with respect to capital, investment, 

2 

Models of capital accumulation of that form have found empirical support (see e.g. Eberly, Rebelo and 

Vincent (2008) for a recent contribution). 

3 

The firm is assumed to be purely equity financed here. However, the same first-order conditions arise in 

models where the firm has access to bonds and shares the beliefs of agents. 

9

and labor determine the optimal choices 

e zt fK(Kt,Lt) + 1 

2 χ I2 t 

K 2 t 

e zt fL(Kt,Lt) = wt 

− δPt = Dt 

(6) 

(7) 

It = Kt 

χ (Pt − 1) (8) 

Every period, the proceeds of putting capital to work are distributed among shareholders whereas 

the marginal product of labor determines the wage. The third optimality condition requires some 

attention: the firm does arbitrage between consumption and investment goods. The price of 

capital is given by Pt and the firm decides how much to buy or sell at that given price. At 

the same time, the price of consumption goods is fixed at one. Since the firm pays a quadratic 

adjustment costs, of converting goods, it makes a non-trivial investment decision. As a result, 

the third necessary condition determines the supply of capital. The link between stock prices, 

i.e. the price of capital, and real investment is tight. Whenever stock prices move up, it becomes 

lucrative for the firm to convert more goods. 

2.3 Resources 

The economy is subject to the resource constraint 

Ct + Bt + Kt+1 ≤ e zt f(Kt,Lt) + R bt 

b,t Bt−1 + (1 − δ)Kt − 1 

2 χ I2 t 

Kt 

where X denotes aggregate variables X = C,K,B,I as opposed to X(i) denoting individual 

variables for agent i. In order to guarantee a deterministic steady-state, the risk-free return 

pre-intervention Rbt b,t is endogenized via 

R bt 

b,t = 1 + r + ψeB∗ −Bt . (10) 

The world interest rate r, a parameter for decreasing returns to bond holdings ψ, and the 

deterministic steady-state level of bond holdings B ∗ are all constants. Endogenizing the interest 

rate ensures a unique deterministic steady-state and is one among several alternatives to close 

a small open economy (for details, see Schmitt-Grohé and Uribe (2003)). 

2.4 Definition of Equilibrium 

The choices of households and the firm define a competitive equilibrium. Optimality conditions 

for households (4) and (5) have to hold as well as their budget constraints (3), and capital 

10 

(9)

market clearing given by 

 

Kt+1(i)φθ(θt(i))di = (1 − δ)Kt + Kt 

χ (Pt − 1) (11) 

which follows directly from the equation of motion for capital. φθ(·) denotes the probability 

density function for the normal distribution representing the cross-section of households in the 

economy. Market clearing in the labor market Lt = L yields wages and dividends and investment 

are determined by the firm’s first-order conditions. Furthermore, returns to stocks are defined 

in the usual way as 

R bt 

c,t+1 = Pt+1 + Dt+1 

Pt 

ensuring that aggregating households’ budget constraints yields the resource constraint. 

Since contingent claims trading eliminates the wealth distribution from the set of state vari- 

ables, we are left with only five state variables, namely capital, bonds, current and next period’s 

logarithm of total factor productivity, and the belief shock. Let St = (Kt,Bt−1,zt,εt,zt+1) de- 

note the set of state variables. Except for the distortion of beliefs and next period’s total factor 

productivity, the state variables S k t = (Kt,Bt−1,zt) are common knowledge among households. 

Agents try to infer the remaining two states Su t = (εt,zt+1) from prices and private signals. 

Lastly, S∗ = (K∗ ,B ∗ , −1 2σ2 z , −1 

2σ2 ε , −1 

2σ2 z ) denotes steady-state levels of all state variables. 

3 Solution method 

The model of the previous section features dispersion of information in a standard small open 

economy setup. This section develops a solution technique capable of solving this model. Fur- 

thermore, the method is applicable to general nonlinear open and closed economy models with 

dispersed information. The reader only interested in the results might want to skip this section 

and move on to the next. However, there it becomes clear that the results of this paper crucially 

depend on the novel solution method. 

General equilibrium models with capital accumulation and standard preferences are inher- 

ently nonlinear. On the one hand, we want to capture nonlinearities through higher-order 

approximations, on the other hand closed-form solutions to signal extraction problems are only 

available in special (linear) settings. This section provides a tool to bridge the two needs. I show 

that a nonlinear change of variables can bring the problem into a form in which we can avail of 

perturbation methods to solve for the equilibrium in closed form. Perturbation methods have 

previously been used for the study of nonlinear dynamic representative agent models. They 

deliver an expansion in state variables of an arbitrary order and hence accuracy. 

The solution method involves the following five steps. First, we build an expansion around 

11 

(12)

the deterministic steady-state of the system and find an approximation to the optimal policy. 

Second, I show how a nonlinear change of variables brings equilibrium conditions in a form in 

which we can compute conditional expectations in closed form. Third, we propose a form for the 

market price. Given the nonlinear change of variables, there is a natural guess for the functional 

form of the equilibrium price function. Fourth, taking prices as given, we obtain a solution to 

the signal extraction problem. Lastly, we get market clearing and confirm the validity of the 

guess. In summary, the first two steps make the problem amenable to the standard procedure 

for computing linear models with dispersed information carried out in steps three to five. 

Starting points for the solution method are the optimality conditions (4) and (5). We stack 

them in a vector 

F(St,St+1,σ) = 

 

u ′ (Ct(i)) − βu ′ (Ct+1(i))R at 

c,t+1 

u ′ (Ct(i)) − βR at 

b,t u′ (Ct+1(i)) 

 

. (13) 

The expansion in standard deviation of shocks σ allows us to get from the deterministic system 

(σ = 0) to the stochastic system. There, we choose σ = σz as a normalization and scale 

the standard deviation of belief shocks proportionately ( σε 

σ). We plug in for stock returns 

σz 

using their definitions (10) and (12) and for consumption using the budget constraints (3). 

Furthermore, stock markets clear (see equation (11)). The resulting functional equation that we 

ultimately want to solve takes the form 

˜Et[F(St,St+1)|st(i),Pt] = 0. (14) 

Note that St involves unknown state variables that agents need to forecast. 

3.1 Higher-order expansion 

This section describes the first step of the solution method in which we obtain an approximation 

to optimal choices using perturbation methods. 

3.1.1 Intuition 

Perturbation methods exploit the fact that the solution to a functional equation possesses all 

derivatives around a steady-state point. As an example, figure 2 shows optimal stock holdings 

as a function of capital Kt. It displays the steps to derive an approximate solution. In a first 

step, we compute the steady-state value of a deterministic version of the system (14). The 

deterministic steady-state allows us to anchor the expansion. We first expand with respect to 

the state variable Kt on the x-axis leading to the thick solid line. Continuing with the expansion 

in a second state variable causes the approximation of the deterministic system to shift up or 

12

down (plotted as thin solid lines around the thick solid line). The crucial step in the procedure is 

K t+1 

Figure 2: Perturbation methods build an approximation in state variables around the deterministic 

steady-state (thick solid line). Expansions in other state variables than the one plotted 

shift the approximate solution up or down. 

to get to the original problem of interest: in order to get to a stochastic economy, we expand the 

solution in the standard deviation of shocks. In the first step, we follow standard perturbation 

procedures that solve the auxiliary problem to (14) using only unconditional expectations. The 

following step in section 3.2 then shows how to get to the problem of dispersed information 

where we solve for conditional expectations. 

To get from the deterministic to the stochastic economy, we have to build at least a second- 

order expansion with respect to the standard deviation of shocks. Applied to our problem, it 

means that the stochasticity affects the economy through the second moment of shocks and hence 

through a risk premium channel. When returns become risky, agents demand a compensation 

through higher mean returns. Incorporating second-order terms with respect to the standard 

deviation leads the deterministic solutions (solid lines) to shift (here down to the dashed lines). 

The effect shows that a second-order term can have a first-order effect on equilibria through 

risk-premium effects. The 45 ◦ line intersects with the stochastic solution to the left of the 

intersection with the deterministic choice. Hence, the stochastic steady-state level of capital is 

lower than the deterministic. 

3.1.2 Expansion 

For the method to be applicable to a specific problem, we have to make sure that the policies 

are analytic in the neighborhood of the deterministic steady-state. Since all derivatives of our 

equilibrium equations F(·, ·, ·) in equation (13) exist, we have the necessary regularity to carry 

out the expansion. Jin and Judd (2002) discuss appropriate implicit function theorems in detail. 

13 

K t

In a first step, we solve the nonlinear system of equations F(·, ·,0) to get deterministic steady- 

state values. Obtaining first derivatives with respect to states involves solving nonlinear systems 

of equations for coefficients of the approximation. For all higher derivatives, the systems to be 

solved are linear. Appendix B contains details of the solution procedure. 

The resulting approximation to the optimal policies can be written as 

Xt(St, θt(i)) ≈ 

(i1 + i2 + . . . + i6)! 

i1,i2,...,i6 

1 

∂ i1+i2+...+i6Xt(St) 

∂K i1 

t ∂Bi2 t−1∂zi3 t ∂εi4 t ∂zi5 t+1∂σi6 

 

 

 

St=S ∗ 

· (Kt − K ∗ ) i1 (Bt−1 − B ∗ ) i2 (zt − z ∗ ) i3 (εt − ε ∗ ) i4 (zt+1 − z ∗ ) i5 σ i6 + xθ(St, θt(i)) 

for X = Kt+1,Ct. It takes the form of a Taylor approximation around the deterministic steady- 

state. We approximate optimal policies with a polynomial whose coefficients we get from ap- 

plying perturbation methods. For the sake of brevity, we can rewrite the expansion of equation 

(15) as 

ˆX(S,θ(i)) = 

n∈S 

1 

|n|! 

∂ n X 

∂S n 

 

 

St=S ∗ 

 

i 

(Sni − S∗ ni )ni + xθ(S,θ(i)) 

where S = {(1,0,0,0,0,0),(2,0,0,0, 0, 0), ...} is the set of all multi-indices of the expansion, 

|n| = 

i ni, xn = 

i xni , and Sni is the ni-th component of the state vector S. 

3.2 Change of variables 

This section describes the crucial step that enables us to solve nonlinear models with dispersed 

information. To solve the signal extraction problem analytically, we want an expansion of the 

form (15) for all terms in Euler equations (4) and (5). Recall that in the first step we only 

solved for the unconditional expectation in (14). This section demonstrates how we can obtain 

a nonlinear price function that allows us to solve the conditional expectations operator consistent 

with the price function as in equation (14). 

The variables of interest in the Euler equations are marginal utility, returns, and stock 

prices. To get from the optimal choice for consumption to marginal utility, we can perform an 

expansion in the logarithm of consumption. For the logarithm of stock prices, the idea is to map 

individual demand Kt+1(i) into an individually requested “price” p(i) which we can aggregate 

to get market prices. A nonlinear change of variables allows us to get the right transformation. 

The “individual price” p(i) would prevail as a market price if agent i’s capital holdings would 

coincide with average holdings in that period. We define it as 

 

Kt+1(i) 

pt(i) = log χ − (1 − δ) + 1 . (16) 

Kt 

Lower case letters for state variables and aggregate variables denote the logarithm thereof. Judd 

(2002) shows that a nonlinear change of variables can be carried out in a simple operation. In a 

14 

(15)

slight extension, I show that a state-dependent nonlinear change of variables such as the one for 

pt(i) can be computed in the same fashion. Using the chain rule, we see that we have to take 

the derivative of the transformation at the point of expansion into account: 

ˆp(i) = 1 

|n|! 

n∈S 

∂ n p 

∂K n t+1 

∂ n Kt+1 

∂S n 

 

 

St=S ∗ 

 

i 

(Sni − S∗ ni )ni + pθθ(i). (17) 

Hence, we only have to multiply the coefficient in the expansion by the derivative of the trans- 

formation. We also transform consumption into log consumption in the same fashion. To get an 

expansion for the logarithm of returns, we use the definition of stock returns given by equation 

(12) and hence know the nonlinear transformation. The implementation is analogous to stock 

prices and can be found in more detail in appendix B.1. 

3.3 Proposal 

As in any linear noisy rational expectations equilibrium, we guess a form of the price that we 

confirm subsequently. To form the right proposal, it is important to notice that the stock market 

clearing condition can be written as an aggregation over individual choices pt(i) 

 

pt = 

pt(i)φθ(θ(i))di. (18) 

This form of aggregation is equivalent to the previously described market clearing condition (11) 

as can be seen using equation (16). Hence, a natural proposal for the price is the average of all 

“individual prices” p(i). Furthermore, averaging over all prices preserves the structure of the 

expansion and eliminates all idiosyncratic noise from equation (17). Contrary to standard lin- 

ear setups, coefficients are state dependent, there are higher-order terms in the proposed price, 

and we have a nonlinear guess for the equilibrium price function. A higher-order approxima- 

tion in unknown state variables εt and zt+1 incorporates higher moments of their distributions. 

Since higher moments are common knowledge among households, they do not influence signal 

extraction directly. 

3.4 Signal extraction 

Given the functional form of stock prices in (17) that delivers a guess for stock prices, we can 

compute conditional expectations operators. With transformations of consumption to marginal 

utility, returns to log returns, and stock holdings to individual prices, both Euler equations (4) 

and (5) are in a form where all terms that are not common knowledge are split into a normally 

distributed part and higher moments. As derived in subsection 3.2, we have expansions for all 

15

terms in the Euler equations to which we apply the logarithm on both sides of the equation 

and 

−γct = log(β) + r at 

b,t − γ ˜ Et[ct+1|st(i),Pt] + γ2 

2 vart[ct+1|st(i),Pt] (19) 

−γct = log(β) + ˜ Et[r at 

c,t+1 − γct+1|st(i),Pt] + 1 at 

vart[rc,t+1 − γct+1|st(i),Pt] (20) 

2 

When carrying out the signal extraction problem, it becomes clear why it is helpful if distortions 

in beliefs only affect the mean and not higher moments. Given beliefs defined in equation (2), we 

can take a higher-order approximation but only have to carry out the signal extraction problem 

over mean productivity. All higher moments of the distribution are common knowledge. 

Exploiting the normality assumption of shocks, we can solve for the conditional expectation 

in the regular fashion. The signal extraction problem results in expectations of the form 

Et[x|st(i),Pt] = e x S(S k t ) + e x s(S k t )st(i) + e x p(S k t )pt. 

where x can be the logarithm of consumption, returns, or prices. The first part is a known 

state-dependent part independent of the information from private signals and the stock price. 

Expectations put weight on two signals to get the optimal expectation about variable x. 

3.5 Market clearing 

The last step is to make the result of the signal extraction problem compatible with the proposal. 

The perturbation method relies on having fixed volatilities for all shocks. In particular, the noise 

in prices was fixed at some free parameter value (as a multiple of the noise in the system measured 

by σ). In order to get market clearing, the volatility of the noise in prices has to match the 

volatility implied by signal extraction, i.e. the coefficients have to match. Appendix C carries 

out all steps in detail. 

3.6 Value function 

In the same fashion as we compute an approximation to the optimal policies, we can find an 

asymptotically valid approximation for the value function 4 . Define the objective function for 

households as in equation (1); the residual of the recursion for the value function is 

FV (St,St+1) = V (St) − u(Ct(i)) − βV (St+1). (21) 

4 I am indebted to Ken Judd for suggesting the use of computing welfare by approximating value functions. 

16

The value function then has to obey the functional equation ˜ Et[FV |st(i),Pt] = 0. Using the 

same techniques as in section 3.1, we can derive an expansion for the value function 

ˆV = 

l∈S 

1 

|l|! 

∂ l X 

∂S l 

 

 

S=S ∗ 

 

i 

(Sli − S∗ li )li + vθθ. 

We use the true law of motion for all states to derive the value function. Therefore, agents 

understand the economy perfectly and agree on transitions. The disagreement only occurs for 

unobserved states. In order to compute expected utility under distorted beliefs, agents then 

forecast the unknown states S u t = (εt,zt+1) at time t. 

Having an approximation to the value function simplifies welfare comparisons under different 

policies significantly. For a given state, we simply evaluate V for given private and public signals 

to get expected utility of an agent. Furthermore, we can aggregate over all agents to get the 

value of social welfare. 

3.7 Discussion 

The method developed in this section has the advantage of being universally applicable to nonlin- 

ear noisy rational expectations equilibria. In particular, it does not require specific assumptions 

such as constant absolute risk aversion or the absence of labor income risk or intertemporal hedg- 

ing demand as do previously suggested methods. However the functional form for the change 

of variables might have to be adapted to the model at hand. Approximating the logarithm of 

consumption, as suggested here, is useful since it prevents consumption from becoming negative. 

Since the resulting approximation is asymptotically valid, it is up to the user to decide how many 

orders are necessary for a good fit. There is a reliable way of performing error estimation via 

first-order conditions discussed in appendix B.2 which applies to other nonlinear models accord- 

ingly. Most importantly, due to asymptotic validity, we can compute effects on steady-states 

through higher-order expansions in state variables and the standard deviation. A computer al- 

gebra system as for example Mathematica allows for easy implementation and circumvents the 

arduousness of having to compute derivatives by hand. 

4 Equilibrium and Positive Implications 

This section applies the solution method of the previous section to the model and studies prop- 

erties of the competitive equilibrium. 

17

4.1 Equilibrium 

To obtain a solution for all equilibrium variables, I follow the five steps prescribed in the previous 

section. Given the parameters, I compute an expansion of optimal policies for consumption and 

stock holdings to complete the first step and transform the solution into the logarithms of 

consumption, stock prices, and stock returns. Aggregation over transformed stock holdings 

yields a guess for the logarithm of stock prices of the form 

ˆpt = pS(S k t ) + pε(S k t )µεt + pzt+1 (Sk t )zt+1 

where µ is a constant that relates distortions in beliefs to noise in prices. This constant deter- 

mines whether there is amplification of distortions in beliefs when aggregation of information 

takes place. 

The guess for the price function looks intriguingly simple. The key to understanding the 

form lies in the nonlinear expansion carried out in the previous section (see equation (17)). The 

known state-dependent part of the price function pS(S k t 

(22) 

) is a nonlinear function of all known 

state variables and also contains information about higher moments of the distribution which 

are known to all agents. The only unknown parts of the logarithm of stock prices are true 

information zt+1 versus aggregate noise µεt. Thus the equilibrium stock price contains not 

only information about the state of the economy in a given period but also information about 

productivity in the following period as well as noise stemming from distorted beliefs. Hence it 

is excessively volatile. 

Since total factor productivity z and belief shifters ε are normally distributed, expectations 

on the right-hand side of logged Euler equations (19) and (20) are normally distributed. Because 

the state variables capital and bonds are determined this period and tomorrow’s unknown states 

are unforecastable, the only remaining state to be predicted is next period’s productivity. The 

undistorted expectation of next period’s logarithm of productivity zt+1 takes a linear form 

Et[zt+1|st(i),Pt] = eS + esst(i) + epˆpt 

= eS + eppS + (es + ep)zt+1 + epµεt + esθt(i). 

where I suppressed the state variables as an argument to state-dependent constants. 

The state-dependent constant in front is known to all agents. The coefficients on shocks solve 

the signal extraction problem that makes the best prediction about next period’s total factor 

productivity given the two signals. The private signal contains information and idiosyncratic 

noise whereas the public signal contains information and aggregate noise. Optimal expectations 

put positive weight on the private as well as the public signal. Therefore, optimal expectations 

themselves contain noise from belief shifters as it is impossible to filter them out. 

18 

(23)

Yet agents in the economy do not hold optimal beliefs. Their expectation about next period’s 

total factor productivity is distorted by a belief shock ε as defined in (2). It is given by 

˜E[zt+1|st(i),Pt] = Et[zt+1|st(i),Pt] + εt. 

The last step completes the procedure by imposing market clearing and verification of the 

guess. There is a unique solution for the coefficients that solves the signal extraction problem 

as well as market clearing. Consequently, the approximate solution to the equilibrium is unique. 

Appendix C contains a derivation of this result and lays out how to obtain coefficients for optimal 

expectations. 

4.2 Positive Implications of the Model 

As shown in the previous subsection, optimal beliefs use both signals to make the best possi- 

ble forecast about next period’s productivity. Households deviate from optimal forecasts by a 

distortion in beliefs that adds to aggregate noise in prices. This property stems from the fact 

that aggregate noise is a multiple of distortions in beliefs. The equilibrium price function then 

solves the following fixed point problem: it delivers the price which prevails under households’ 

expectations that deviate by their distortion in beliefs from the optimal expectation obtained 

under a given price. 

The following proposition shows that even tiny distortions can create a substantial amount 

of excess volatility in stock prices. Agents with and without belief shifters have the same 

expectations in the limit and the economy displays excess volatility. 

Proposition 4.1 (Amplification) 

An equilibrium with a given amount of excess volatility in stock prices below an upper bound 

of pε(S k t )µεt < γσ2 η 

σ 2 η +σ2 θ 

can be sustained with arbitrarily small distortions in beliefs as informa- 

tion gets more dispersed in the economy. More formally, aggregation of information amplifies 

distortions in beliefs, i.e. 

µ → ∞ for σθ → ∞. 

Proof: Appendix D lays out the proof of the proposition. 

As information gets more dispersed across households, the private signal becomes less in- 

formative. Optimal beliefs adjust by putting relatively more attention to the public signal. 

There are two immediate implications: first, if agents put less weight on private signals, less 

information enters the equilibrium stock price and hence optimal expectations. Second, optimal 

expectations catch more aggregate noise. Households deviate from those optimal expectations 

by their belief shocks, thus adding more noise to expectations and in turn prices. This noise 

19

feeds back into optimal expectations and so forth. Proposition 4.1 shows that, in the limit as 

information becomes more dispersed, amplification becomes arbitrarily small. 

We can use arbitrarily small distortions in beliefs to generate a given level of excess volatility 

in prices. By definition this means that the difference between households’ and optimal expec- 

tations is arbitrarily small. In the limit, both expectations coincide and we get the following 

corollary. 

Corollary 4.2 (Convergence to Noise Trader Equilibrium) 

The limit of completely dispersed information, i.e. σθ = ∞, is an equilibrium with the same 

price function and allocation as an economy with rational agents and noise traders. 

The corollary shows that belief distortions serve as a way of generating noise in prices and 

create excess volatility. The main purpose of an individual distortion in beliefs remains to 

make the equilibrium price function consistent with individual choices. Hence it is an alter- 

native to assuming noise traders as, for example, in Hellwig (1980) or Grossman and Stiglitz 

(1980). Furthermore, as distortions in beliefs can be arbitrarily small, they impose little costs 

on households. Through having the same mistaken beliefs as everybody else, though not costly 

individually, agents impose a risk externality on the economy. Since each household’s distortions 

add to aggregate noise, each individual demand pushes prices even further in the direction of 

noise and makes it more difficult for others to filter out information from prices. 

The difference in expectations between households and optimal beliefs is a measure for how 

much profits optimally acting agents can earn. Since the expectation about next period’s pro- 

ductivity pins down the expectation of the stock return, it determines the difference in portfolio 

decisions between households and a hypothetical agent with optimal beliefs. From proposition 

4.1, it then follows: 

Corollary 4.3 (Absence of Excess Profits for Optimally Acting Agents) 

There are no extra profits to be made for agents with optimal beliefs if information in the economy 

is fully dispersed. 

In other words, if expectations about future returns and state variables coincide, portfolio and 

consumption decisions will be the same, too. But although agents might hold almost optimal 

beliefs, welfare in the economy might be severely affected through noise in prices. The following 

sections explore whether a government can improve on equilibrium allocations. 

5 Policy Intervention 

This section introduces a government into the model which aims at improving on social welfare. 

Distortions in beliefs constitute the only departure from the first welfare theorem. The govern- 

20

ment wants to shield the economy from harmful effects of excess volatility. I study two classes of 

policy intervention capable of stabilizing prices. First, the government can condition its policy 

on prices which generates a tradeoff between reducing volatility of prices and preserving the 

reaction to true information. Second, policy can additionally condition on previous period’s 

misjudgment. While the first class of policies merely treats the symptoms of excess volatility, 

the latter policy is able to alter the information content of prices and even reduce the level of 

excess volatility. 

5.1 Government 

The government maximizes social welfare of agents defined as the sum of all expected discounted 

utilities in the economy. Importantly, the government does not have superior information. The 

government observes equilibrium prices and the state of the economy insofar as it is common 

knowledge. The government can thus be thought of as all agents agreeing on policy intervention 

prior to the arrival of information in a given period. The crucial difference between the govern- 

ment and individuals lies in the ability to change prices. The government therefore has various 

policy interventions τ j at hand which it can install in order to raise social welfare. For now, I 

am intentionally vague about the exact nature of the policy intervention which the next section 

then discusses. The government balances its budget every period by redistributing any revenues 

arising in a lump-sum fashion. The government’s task is to maximize 

max E 

τj 

 

∞ 

˜Et β t 

 

 

u(Ct(i)) s(i),P φθ(θ(i))di 

subject to the economy being in an equilibrium and budget balance. 

t=0 

Using results of section 3.6, we get an approximate closed-form expression for social welfare 

of the form 

(24) 

ˆV (St) = VS(S k t ) + Vε(S k t )µεt + Vzt+1 (Sk t )zt+1 + Vσ(S k t ). (25) 

As a robustness check, I study the social welfare function that takes forecasts under optimal 

beliefs as its argument rather than social welfare under distorted beliefs. The resulting policy 

intervention is less aggressive than the one implied from social welfare in equation (24) but the 

effects are tiny. The reason for very small differences between the two policies lies in the fact 

that distortions are very small. Section 7.3 contains details on the robustness check. 

21

5.2 Policy intervention 

The government’s goal is to shield the economy from negative impacts of excess volatility. At the 

same time, it wants to preserve price responses to true information and let investment respond to 

it. Lastly, the government acknowledges that policies might affect the formation of expectations 

and it does not want to exacerbate amplification of belief shocks to aggregate noise. Since the 

government cannot detect distortions in beliefs, it is left with the options of stabilizing prices 

by conditioning its policy on stock prices or it can additionally use information about previous 

periods. The following sections explore the implications of policies of this sort. 

5.2.1 Policy conditioning on prices 

In a first step, I analyze a policy that allows the government to target the reaction to both 

unknown shocks, true information as well as aggregate noise, simultaneously. Therefore, I study 

a general policy that allows the government to stabilize prices, i.e. a policy that lowers the 

volatility of prices in equation (22). The next section deals with specific instruments as for 

example an interest rate policy. The general policy takes the form 

τ(p,p ref ) = t g (p − p ref ). (26) 

The reference price p ref is the price that would occur given current known state variables if news 

and belief shocks would take on their mean. The policymaker determines the price that would 

occur solely based on the knowledge of known state variables. In other words, it is the known 

state dependent part in equation (22), p ref 

t = pS(S k ). Any deviation of the current period’s 

stock price from the reference price has to stem from current period’s news and belief shocks 

τ(p,p ref ) = t g (pzt+1 zt+1 + pεtµεt). 

Policy conditioning on prices hence affects price reactions to both shocks symmetrically. If 

the reaction to excess volatility is dampened, it also reduces the reaction of prices and hence 

investment to true information. 

Observation 5.1 (Stability-Efficiency Tradeoff) 

The government faces a tradeoff between stabilizing prices and efficiency (since dampened re- 

sponses reduce reactions to shocks to fundamentals). 

This stability-efficiency tradeoff determines the optimal policy intervention. To see the effects 

formally, we can use the closed-form approximation to social welfare of equation (25). Four 

22

terms matter for changes due to policy intervention: 

dV 

= κ1 

dtg ∂Vε ∂Vzt+1 

+ κ2 + κ3 

∂tg ∂tg ∂Cσ2 + κ4 

∂tg ∂Kσ2 . 

∂tg The first term is positive as it describes a correction for Jensen’s inequality due to reduced 

reaction of the economy to aggregate noise. The second term corrects social welfare for a loss 

in efficiency due to a dampened reaction of prices and thus investment to true information. 

The last two terms only appear in a nonlinear equilibrium but have a first-order effect. 

A (log-) linear solution around the deterministic steady-state would not contain the effects of 

volatility on consumption and capital. These two terms represent the influence of the variance 

of shocks on the two margins. The third term captures the impact of policy intervention on 

how volatility affects the consumption-savings decisions while the last term reflects the effect on 

capital accumulation. The consumption-savings margin tilts towards less savings if stock returns 

become more volatile without a compensation in mean returns. Policy intervention mitigates the 

influence of volatility and the consumption-savings margin reacts less to impacts of shocks. The 

main effect comes from the impact of volatility on capital accumulation. In equilibrium, capital 

has to adjust in order to deliver the required return on stocks. Policy intervention shields the 

economy from shocks by dampening the volatility in stock prices. Price stabilizing policies not 

only reduce the price response to shocks but indirectly lower the variance of returns and thus the 

risk premium. Since the now lower expected return must be reflected in lower mean dividends, 

the marginal product of capital falls leading to a higher level of capital and production in the 

stochastic steady-state. Policy thus affects consumption through raising the net present value 

of production. 

Observation 5.2 (Welfare gains) 

The standard upper bound for the gains from stabilization does not apply to the economy. A sta- 

bilizing policy increases mean consumption and can thus bring about considerably larger welfare 

gains. 

Lucas (1987) computes the cost of business cycles as the percentage share of consumption that 

agents would be willing to give up to replace a stochastic consumption path by constant mean 

consumption. Calibrating the consumption stream to the data, Lucas (1987) finds the cost of 

business cycles to be very small. The reduction in the volatility of consumption streams serves 

as a measure for the welfare benefits the best stabilizing policy can achieve. Results in this 

economy suggest otherwise: stabilizing asset markets brings about a higher mean consumption 

and thus large welfare gains. 

The discussion so far dealt with the direct effects of policy intervention on price responses. 

The question is how the use of information and thus the amount of excess volatility in prices 

is affected. The following proposition shows that the use of information does not change in 

23

esponse to policy intervention in this setting. 

Proposition 5.3 

The information content of prices does not change in response to policy intervention. 

Proof: The proof of proposition 5.3 is in Appendix E. 

This proposition states that policy does not alter agents’ expectations about next period’s 

productivity. Intuitively, agents know prices in the setting without intervention and hence do 

not gain any information through policy intervention. Furthermore, policy affects the reaction 

to news and belief shocks symmetrically so that no incentives for a different use of information 

arise. 

The choice of the social welfare function ensures that households would favor policy interven- 

tion prior to arrival of information in a given period. Since every agent has the same expected 

utility in equilibrium after contingent claims have been settled, any policy intervention that 

raises social welfare would necessarily be favored by all agents. The social welfare function thus 

guarantees unanimous support for stabilizing policy. Even after the arrival of private and public 

signals, the median voter would still favor policy intervention. Since we have a shift in mean 

consumption, the government can raise welfare for the majority of households. 

5.2.2 Backward-looking Policy 

The previous class of policies trades off stability for efficiency in the best possible manner 

without affecting the use of information. Policies conditioning only on prices, however, do not 

exploit all information the government has in a given period. Stock prices in period t reflect 

news about next period’s productivity as well as aggregate noise. In the period t + 1, total 

factor productivity becomes public information and so does period t’s misjudgment. Thus the 

government can condition its policy on the misjudgment of the previous period to get a policy 

intervention of the form 

τ(pt,εt−1) = t e p(pt − p ref 

t ) + t e ε(εt−1 − ε ∗ ). (27) 

In order to compute the competitive equilibrium under this policy, the set of state variables has 

to be enlarged. Last period’s distortion in beliefs becomes part of the state space leading to a 

vector of states Se,t = (Kt,Bt−1,εt−1,zt,εt,zt+1). 

If the government lowers the payoff of stocks in response to revelation of previous period’s 

overpricing and conversely for undervaluations, investors not only have to forecast stock payoffs 

but also policy interventions. As a result, stock prices lose some of their predictive power for 

asset payoffs because stock prices depend positively on belief shocks whereas policy leads stock 

24

payouts to decrease in them. Agents thus have an incentive to put relatively more weight on 

the private signal. As a direct consequence, we get: 

Lemma 5.4 (Backward-looking Policy Influences Use of Information) 

By conditioning its policy on previous period’s misjudgments, the government can alter the way 

people use information to form expectations. 

Proof: Appendix F contains the proof. 

When people put relatively more weight on private signals, more information about funda- 

mentals enters the stock price. The thereby improved information content of prices together 

with the fact that agents put less weight on the price to form expectations implies that optimal 

beliefs contain less aggregate noise. Therefore, aggregate noise feeds back less into expectations 

of all agents and amplification partly unravels. 

Proposition 5.5 (Mitigated Amplification) 

Backward-looking policy mitigates amplification of belief shocks in prices and thus reduces excess 

volatility. 

Proof: Please see appendix G for the proof. 

A history-dependent (backward-looking) policy gets to the root of the problem and mitigates 

amplification. The policy rule influences the stability-efficiency tradeoff advantageously and can 

thus bring about larger welfare gains. The next section contains a calibration and the welfare 

effects of policy instruments. 

6 Policy Instruments and their Effects 

We now turn to a discussion on implementable policies of the classes studied in the previous 

section. I first discuss a benchmark calibration for parameters and a welfare measure that allows 

to quantify gains from implementation. I discuss a reference equilibrium without any distortions 

for comparison of welfare gains. The next parts discuss policies conditional on prices, specifically 

open market operations, an interest rate rule, and a backward-looking interest rate rule. 

6.1 Calibration 

Table 1 lists parameter values for the benchmark economy. There seems to be a consensus in 

the literature over some of the parameters, others require further discussion. I choose a utility 

function with relative risk aversion γ = 1 for the benchmark case and carry out robustness checks 

with respect to the value of γ. The depreciation rate δ is set at 10% and the capital share α at 

25

one third. The risk-free world interest rate r is constant at 3% which implies a time preference 

rate β of 1/(1 + r) ≈ 0.97 to pin down constant consumption in deterministic steady-state. The 

risk-free rate moves only slightly with the level of bond holdings. The corresponding parameter 

ψ is set at 10 −5 . I set the parameter for adjustment costs χ to one. In section 7.1, I perform 

robustness checks. A labor force of 1 merely serves as a normalization. As a natural benchmark, 

I look at an economy that holds all wealth inside its boundaries in deterministic steady-state. I 

therefore choose steady-state bond holdings B ∗ to be zero. The remaining parameters require 

further discussion. I match the standard deviation of returns σR to be 0.18 measured for US data 

(see Campbell (2003)). In the benchmark calibration, I set the standard deviation of returns 

due to fundamentals to be two thirds of the standard deviation of returns. The remainder of 

0.06 of the standard deviation of returns is due to noise where the standard deviation of belief 

shifters σε is only 0.000015. A one standard deviation event of agents belief distortions is thus 

marginal and households hold almost optimal beliefs. 

Relative risk aversion γ 1 

Depreciation rate δ 0.1 

Capital share of output α 1 

3 

Endogenized interest rate ψ 0.00001 

Adjustment cost parameter χ 1 

Labor force L 1 

World interest rate r 0.03 

Steady-state bond holdings B ∗ 0 

Time preference rate β 1 

1+r 

Standard deviation of returns σR 0.18 

Std. dev. of excess volatility σε,R 0.06 

Table 1: Parameter values for benchmark case. 

As a welfare measure, I use the percentage increase of consumption that agents would have 

to receive in competitive equilibrium to be indifferent between remaining in equilibrium and 

putting a policy in place and transitioning to a new equilibrium. More formally, it is the 

fraction λ that equates utility from competitive equilibrium consumption streams C ce 

t (i) and 

consumption streams under policies C pi 

t (i) 

˜Et 

∞ 

t=0 

β t log((1 + λ)C ce 

t ) 

 

= ˜ Et 

6.2 Reference equilibrium without distortions 

∞ 

t=0 

β t log(C pi 

t ) 

As a reference point for welfare gains, this section discusses an equilibrium in which all agents 

hold optimal beliefs. The signal extraction problem is trivial as there is no idiosyncratic com- 

26

ponent in expectations because prices reveal next period’s total factor productivity as shown by 

Grossman (1976). Since in this economy only news shocks remain, there is no noise in stock 

prices and returns. Steady-state levels of capital and consumption are thus higher than in the 

benchmark case with distorted beliefs. The first welfare theorem applies for this economy and 

hence there is no welfare improving intervention. Compared to the equilibrium without inter- 

vention, agents would gain 1.13% of consumption if they would start out from the steady-state 

of the competitive equilibrium with distortions in beliefs and beliefs were undistorted from then 

on. 

Welfare gains originate from an initial jump in consumption and the transition to the new 

steady-state level. Higher consumption is justified by a higher steady-state level of capital and 

thus a higher net present value of production. Effects on capital accumulation arise for the same 

reason that I discuss in the previous section: the risk-premium of the economy falls which allows 

a lower expected return on stocks which matches the marginal product of capital. Hence the 

capital stock rises with less volatility and brings about welfare gains. 

6.3 Policies conditional on prices 

This section analyzes the effects of two policies that condition on economic activity. First, I 

discuss open market operations where the government buys and sells securities. Second, I look 

at an interest rate policy which comes closest to monetary policy in a model without frictions. 

6.3.1 Open Market Operations 

The government can influence stock markets in the most direct way by participating in them. 

Stock prices are invariably linked to the amount of investment in the economy. The government 

can influence the amount of investment and thus affect stock prices to shield the economy from 

harmful effects of excess volatility. The form of intervention I consider is a linear time-invariant 

demand schedule for the government of the form 

K gov ref 

t+1 (Kt+1,Kt+1 ) = tk (Kt+1 − K ref 

t+1 

). (28) 

The reference demand corresponds to the reference price discussed in the previous section (as 

demand for stocks and prices are directly linked). It is the demand that would have arisen had 

the economy not received any news or belief shocks. The policy is a time-invariant feedback rule 

as it takes capital demand as its input that prevails only when the policy is in place. Capital 

demand affects the amount of intervention which in turn affects the demand for stocks of agents. 

The intervention works as follows: Whenever demand is above reference demand, the gov- 

ernment suspects a positive belief shock and wants to lean against it. It sells stocks to lower 

27

prices which in turn leads to a reduction in investment. The government thereby stabilizes stock 

prices and real investment. 

2.0 

1.5 

1.0 

0.5 

Marginal gains 

Marginal losses 

0.1 0.2 0.3 0.4 0.5 0.6 0.7 

7.45 

7.40 

7.35 

7.30 

7.25 

7.20 

7.15 

V 

0.2 0.4 0.6 0.8 1.0 tk 

Figure 3: Marginal gains (downward sloping) and losses (upward sloping) from policy intervention. 

To the right: Social welfare function plotted over parameter of intervention t k . 

Since the government reacts to news and belief shocks symmetrically, the previously discussed 

stability-efficiency tradeoff arises. The graph to the left in figure 3 shows the marginal gains 

from reducing excess volatility compared to marginal losses due to a dampened reaction to news. 

Optimal policy intervention lies where the two lines meet. The graph demonstrates that a price 

stabilizing policy is beneficial. The graph to the right shows social welfare as a function of the 

sensitivity of government’s demand for stocks with respect to deviations from reference demand. 

Social welfare rises significantly as a result of policy intervention. 

Ct 

1.35 

1.30 

1.25 

1.20 

1.15 

0 50 100 150 200 t 

0 10 20 30 40 50 

Figure 4: Graph to the left: consumption dynamics in an economy without intervention (dashed 

line) compared to an economy that starts out from the same equilibrium but an optimal price 

stabilizing policy of the form (28) is implemented (solid line). The graph to the right shows the 

dynamics of the capital stock. 

Kt 

4.0 

3.9 

3.8 

3.7 

3.6 

3.5 

3.4 

Welfare gains stem from a level effect on consumption as shown by the left graph in figure 

4. The dashed line shows an economy starting out from stochastic steady-state and following 

28 

t

Volatility effects 

Cross-sectional dispersion 0.002% 

Variance of consumption over time 4.7 × 10 −5 % 

Level effects 

Excess adjustment costs 0.04% 

Capital accumulation 0.20% 

Overall 0.24% 

Table 2: Sources of welfare gains for open market operations. 

a path in which all news and belief shocks are at their mean. The solid line displays the path 

of consumption the economy would follow if it started out in a no-intervention equilibrium, 

implemented the optimal policy in equation (28), and transitioned to the new stochastic steady- 

state. For the path, I again set news and belief shocks to their mean. The graph to the right 

hand side shows the corresponding graphs for the dynamics of the capital stock. 

Welfare can rise significantly after intervention since policy intervention raises the level of 

consumption. Table 2 breaks the overall welfare gains of an equivalent of a 0.24% rise in 

consumption down into its components. Gains due to a reduction in the variance of consumption 

in the cross-section and over time do not contribute significantly to gains. Since there are 

adjustment costs to investment in the economy, stabilizing prices over time reduces the dead- 

weight loss from adjustment costs. With 0.04% of consumption, the contribution to overall 

welfare gains is still small. The bulk of gains lies in the effect on capital accumulation. Hence, 

solving for a nonlinear equilibrium is key to analyzing welfare cost of stabilizing policies as the 

effect on capital accumulation would be absent in a (log-) linearized version of the model. 

6.3.2 Feedback rule on interest rates 

The government has two margins it can tackle with its policies since there are two optimality 

conditions for households, the Euler equations (4) and (5). An interest rate rule distorts the 

Euler equation for bonds. The objective is the same: Stock prices are excessively volatile because 

agents want to revise their portfolio decisions too often compared to the optimum. 

The government conditions changes of the interest rate on asset prices. We might imagine the 

government here to take the form of a Central Bank that sets interest rates 5 . The analysis then 

applies insofar as the Central Bank can move real interest rates. The post-intervention return 

for the risky asset does not display any intervention R at 

c,t = R bt 

c,t whereas the post-intervention 

5 Rigobon and Sack (2003) measure this interaction between monetary policy and asset prices empirically. 

29

eturn of the riskless asset in Euler equation (5) is subject to intervention 

R b,at 

b,t = (1 + rt)(1 − τ b (pt,p ref 

t )). 

Interest rate policy can stabilize asset prices. If the government raises interest rates, the equity 

premium falls and bonds become a relatively more attractive investment opportunity. At the 

same time, the policy affects the intertemporal margin for consumption. Raising real interest 

rates makes savings more attractive in that period. Thus consumption remains slightly more 

volatile after policy intervention compared to the policy using open market operations. The first 

policy intervention I consider conditions on asset prices and takes a linear form in the logarithm 

of stock prices 

τ b (pt,p ref 

t ) = t b (pt − p ref 

t ). (29) 

The time-invariant rule (29) conditional on an “asset price gap” resembles a Taylor rule. The 

choice of the reference price plays a crucial role. In the benchmark intervention, I choose the 

reference price to be the price that would result if beliefs were undistorted and next period’s 

productivity would be at its mean but perform robustness checks in the next section. 

The analogous stability-efficiency tradeoff to the previous sections determines optimal inter- 

est rate policy. Figure 5 shows the resulting interest rate schedule as a function of the size of 

the movement of stock prices measured in standard deviations. The magnitudes of the interven- 

tion are quite large: A one standard deviation event requires a 3.7 percentage points change of 

interest rates. Empirical estimates of how strongly monetary policy affects interest rates can be 

found for instance in Bernanke and Kuttner (2005). 

0.03 

0.02 

0.01 

1.0 0.5 0.5 1.0 Σp 

0.01 

0.02 

0.03 

Τ b 

Figure 5: Tax schedule showing changes in interest rates plotted over standard deviation movements 

in stock prices. 

Since the same stability-efficiency tradeoff arises as before, welfare implications are similar. 

With a welfare gain of the equivalent of a 0.23% increase in consumption, welfare improves 

slightly less than for open market operations. The difference lies in the fact that interest rate 

adjustments distort consumption decisions over time. 

30

6.4 Backward-looking Policy 

The previous policy instruments are of the class that merely depends on stock prices. The post- 

intervention riskless return under the backward-looking policy is history-dependent and takes 

the form 

for which interventions for real interest rates are chosen to be 

R e,at 

b,t = (1 + rt)(1 − τ e (pt,p ref 

t )) (30) 

τ e (pt,p ref 

t ) = t e p(pt − p ref 

t ) + t e ε(εt−1 − ε ∗ ). (31) 

0.003 

0.002 

0.001 

1.0 0.5 0.5 1.0 Σ 

0.001 

0.002 

0.003 

Τ e 

Reaction to price p 

Reaction toΕ t1 

Figure 6: Tax schedules for standard deviation movements in stock prices (solid) and previous 

period’s misjudgments (dashed). 

The first part of (31) is equivalent to the policy in equation (29) with the same reference price 

but the second term is new. The policy takes the deviation of previous period’s misjudgment 

from its mean value into account. Since the previous policy is a special case of the one in this 

section, the welfare gains will be at least as high as before. 

The interest rate policy works as follows: Whenever this period’s stock price is higher than 

the price that would have occurred in the absence of news or belief shocks, policy raises the 

risk-free rate to lower the equity premium. Demand for stocks falls and their price is stabilized 

in equilibrium. New to this policy is that it also raises interest rates in response to overpricings 

in the previous period. Since stock prices and thus stock payoffs in the following period fall 

with this period’s overpricing, stock prices have less predictive power for stock returns than in 

the previous cases. Agents thus put relatively more weight on private signals which reduces 

amplification and improves the information content of prices. 

31

Optimal policy intervention reduces the level of excess volatility and thus gets to the root of 

the problem. Figure 6 plots optimal policy intervention for both terms in interest rate schedule 

(31). Compared to the previous interest rate rule, reactions to the asset price gap are less 

aggressive. Since the reaction to last period’s mispricing reduces amplification by 17% in the 

optimum, there is less need for aggressive intervention with regard to this period’s price. The 

government faces an improved stability-efficiency tradeoff and is able to raise welfare significantly 

more than before. The overall welfare gains amount to an equivalent of a permanent increase of 

consumption of 0.55% . 

7 Robustness 

Having established the main results of the paper, I study three kinds of deviations from previ- 

ous exercises. First, I look at the robustness of results with respect to variations in parameters. 

Second, I study policies that extend previously studied policies or might simplify their imple- 

mentation. And in the last section, I study the welfare effects of policy interventions under the 

social welfare function that takes optimal expectations for agents as its input. 

7.1 Comparative Statics 

Previous results use the baseline calibration of section 6.1. This section shows comparative 

statics with respect to parameter values. 

Welfare gains λ 

0.4 

0.35 

0.3 

0.25 

0.2 

0.15 

0.1 

0.05 

Comparative statics w.r.t. adjustment cost χ 

Open Market Operations 

Interest Rate Rule 

0 

0.8 1 1.2 1.4 1.6 1.8 2 

Adjustment cost parameter χ 


0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 



0 

0.1 0.2 0.3 0.4 0.5 0.6 

Amount of excess volatility σ /σ 

ε,R R 

Figure 7: Comparative statics with respect to adjustment cost parameter χ (left) and the amount 

of excess volatility in returns (right). 

The graph to the left in figure 7 shows how welfare gains from policy intervention change for 

different values of the adjustment cost parameter χ. The link between real investment and stock 

32

prices decreases with higher adjustment costs (see equation (8)). Price volatility affects the real 

economy less and hence policy intervention cannot improve on welfare as much. The graph to 

the right shows the welfare changes due to different levels of excess volatility in stock returns. 

The higher the amount of aggregate noise in prices, the more effective policy becomes. If the 

true information to aggregate noise ratio falls, optimal intervention becomes more aggressive 

and brings about higher welfare gains. 

The last comparative statics concern the level of risk aversion. If the utility function changes 

from logarithmic utility to a general utility displaying constant relative risk aversion γ, two 

opposing effects arise. On the one hand, higher levels of risk aversion γ cause the risk premium to 

rise. Capital accumulation should therefore be affected more strongly and welfare gains should 

rise. On the other hand, a higher level of risk aversion leads the elasticity of intertemporal 

substitution to fall. Since effects from distorted beliefs are temporary, there exists negative 

serial correlation in stock prices. An overpricing in a given period means that stock prices will, 

on average, return to lower values in the next period. A departure from logarithmic utility 

introduces an intertemporal hedging demand on the part of households. Agents realize this 

negative serial correlation and adjust their demand accordingly. Since a given mispricing only 

has temporary effects, the intertemporal hedging lowers the cost of that component of volatility. 

Hence, it works in the opposite direction as a higher risk premium. Figure 8 shows that the 

latter effect slightly dominates and welfare gains from policy intervention fall with risk aversion. 

0.3 

0.25 

0.2 

0.15 

0.1 

0.05 

7.2 Variations on Policies 


Comparative statics w.r.t γ 



0 

1 1.2 1.4 1.6 1.8 2 

Risk aversion γ 

Figure 8: Comparative statics with respect to risk aversion. 

The following three subsections test the robustness of previous policies along several dimensions. 

The first paragraph studies the welfare effects using a different reference price which might be 

easier to implement. The second section demonstrates the ineffectiveness of conditioning on 

33

known state variables only compared to conditioning policy on economic activity measured by 

prices. The third paragraph deals with a policy intervention for each margin of adjustment, i.e. 

both Euler equations. 

7.2.1 Different reference prices 

The previously discussed choice of references prices for the interest rate rule is designed to let 

the government respond to forecasting variables only. It might be the case, however, that the 

reference price and thus the “asset price gap” might be hard to determine. An alternative choice 

would be to use steady-state stock prices p ref 

t 

results remain intact. 

0.010 

0.005 

= log(1 + δχ) as a reference. Qualitatively, all 

1.0 0.5 0.5 1.0 Σ 

0.005 

0.010 

Τ b 

Figure 9: Interest rate policy as a function plotted over standard deviation movements in stock 

prices away from steady-state reference price. 

Since the policy is not as well targeted towards responding to news and belief shocks only, 

optimal intervention is much less aggressive than in the benchmark case. As figure 9 shows, a 

one standard deviation event requires a 1.4% change in real interest rates compared to 3.7% in 

the benchmark case (as in figure 5). Consequently, welfare improvements relative to competitive 

equilibrium without intervention are only 0.05% of consumption. The welfare gains are about 

five times smaller than for the interest rate rule that conditions on an asset price gap. 

7.2.2 State-(in-)dependent interest-rate rules 

An alternative to the form of intervention in the benchmark case (29) would be to condition 

interest rate rules not on economic activity but on state variables only or not to condition on 

anything at all. The first implementation is a constant change in interest rates. The intuition 

for why such a rule might be helpful would be that a permanently higher risk-free rate leads to a 

reduction in the risk premium exogenously imposed by the government. Welfare gains, however, 

34

are tiny at λ = 0.006% for an optimally chosen constant change in interest rates. The reason for 

the result is that if policy changes the level of risk premia, it automatically distorts the margin 

of a consumption-savings choice. Lowering risk premia while not reducing their variance leads to 

inefficiently high returns on risk-free savings for households and lower consumption. Policy thus 

needs to stabilize stock prices rather than adjusting the level of expected returns. An interest 

rate rule depending on capital and bond holdings only reduces welfare. Again, such a policy is 

incapable of reducing volatility in stock prices and cannot bring about welfare gains. 

This observation implies that policy needs to condition on economic activity rather than state 

variables, similar to the findings of Angeletos and Pavan (2008). The reason for intervention, 

however, is very different. While the use of information changes in their setup in which private 

and public signals are exogenous, it does not change in this economy because agents have access 

to market prices when forming expectations. 

7.2.3 Additional tax on capital 

Since there are two optimality conditions, it might prove useful to adjust both margins. A 

feedback rule on interest rates distorts not only the choice between stocks and bonds but also 

the intertemporal margin of consumption. Having an additional capital tax in place allows to 

distort both Euler equations separately. For the purpose of computing optimal interventions, 

we keep the same feedback rule for interest rates (29) in place. We add a tax on capital that 

takes the analogous form 

R c,at 

c,t+1 = Rc,bt c,t+1 (1 − τc (pt,p ref 

t )) = R c,bt 

c,t+1 (1 − tc (pt − p ref 

t )) 

The addition of a second margin of adjustment reduces intertemporal fluctuations in consump- 

tion. Yet, these fluctuations are not very costly for agents as table 2 shows. Thus it comes as 

no surprise that the welfare gains from having both policies in place is 0.24% of consumption 

relative to the case with no intervention compared to 0.23% for the benchmark case with only 

an interest rate policy. 

7.3 Different Social Welfare Function 

The benchmark case uses the average utility of households under distorted beliefs as objective 

function for the government. An alternative specification would specify agents’ expected utility 

levels under optimal beliefs given competitive equilibrium allocations. The government wants to 

lean against prices more aggressively in that case since the government has a stronger aversion 

to distortions in beliefs. As proposition 4.1 states, the difference between optimal and distorted 

beliefs is minimal. Changing the social welfare function consequently does not translate into 

35

large effects on policies. Figure 10 plots optimal policy interventions of the form (29). The 

solid line represents the optimal interest rate rule under a social welfare function with distorted 

beliefs and the dashed line its counterpart under optimal beliefs. The resulting difference in 

welfare gains is minimal: welfare gains under distorted beliefs are by 0.0001 percentage points 

higher than under optimal beliefs. 

0.04 

0.02 

1.0 0.5 0.5 1.0 Σp 

0.02 

0.04 

Τ 

SWF distorted 

SWF optimal 

Figure 10: Comparison between tax schedules under social welfare function using optimal versus 

distorted beliefs. 

8 Conclusion 

This paper presents a model of excess volatility, develops a way of solving this and general 

nonlinear models with dispersed information, and shows the gains from stabilizing policies. Price 

stabilizing policy conditioning on stock prices not only reduces the variance of consumption but 

raises its level through capital accumulation. Since the effect works through the risk premium 

on stock returns, it demonstrates the necessity of the nonlinear solution method to capture 

higher-order terms. Backward-looking policies can further raise welfare by reducing the amount 

of excess volatility in stock prices directly. Excess volatility diminishes after intervention due to 

mitigated amplification of distortions in beliefs which backward-looking policies achieve through 

a change in the use of information. The paper analyzes the effects of two policy instruments: 

stock trades through open market operations and interest rate policy. 

Concerns might arise about the implementability of specific policies in reality. Open market 

operations could require the government to engage in large stock trades. Nevertheless, recent 

academic contributions consider open market operations for equity purchases and sales as pos- 

sible policy interventions (see, for example, Kiyotaki and Moore (2008)). However, this policy 

might not meet with everyone’s approval if the government holds shares which allow it to exercise 

voting rights. 

36

These concerns might explain why the main academic debate has taken place with respect to 

monetary policy reactions to asset prices. Central Banks are established institutions that might 

be able to adopt a policy of the discussed form more easily. Monetary policy works best as a tool 

to correct for mispricings if potential mispricings in different markets (for stocks, housing, etc.) 

are highly correlated. With regard to a history-dependent policy, it is crucial that misjudgments 

become known in a timely manner to react to them. 

The present paper might prove useful for future research for two reasons. First, it provides 

a solution method that is applicable to a large class of models. Using the technique, we can 

incorporate models with dispersed information into standard dynamic general equilibrium set- 

tings. Hence, we can start making quantitative predictions with these models and test them 

empirically. Second, we can merge the present framework with a standard New Keynesian model 

and compute the welfare gains monetary policy can achieve by reacting to asset prices. 

In summary, the paper demonstrates large welfare gains from price stabilizing policies for 

economies with excess volatility due to distorted beliefs. Arbitrarily small distortions in beliefs 

suffice to generate large amounts of excess volatility. The paper presents a novel solution method 

which applies to general nonlinear models with dispersed information. 

37

References 

Amador, Manuel and Pierre-Olivier Weill, “Learning from Private and Public Observa- 

tions of Others’ Actions,” mimeo, Stanford University, 2007. 

and , “Learning from Prices: Public Communication and Welfare,” NBER Working Paper 

No. W14255, 2008. 

Angeletos, George-Marios and Alessandro Pavan, “Efficient Use of Information and So- 

cial Value of Information,” Econometrica, 2007, 75 (4), 1103 – 1142. 

and , “Policy with Dispersed Information,” forthcoming in Journal of the European Eco- 

nomic Association, 2008. 

and Jennifer La’O, “Dispersed Information over the Business Cycle: Optimal Fiscal and 

Monetary Policy,” mimeo, Massachusetts Institute of Technology, 2008. 

, Guido Lorenzoni, and Alessandro Pavan, “Wall Street and Silicon Valley: A Delicate 

Interaction,” mimeo, Massachusetts Institute of Technology, 2007. 

Barlevy, Gadi, “The Cost of Business Cycles and the Benefits of Stabilization,” Economic 

Perspectives, Federal Reserve Bank of Chicago, 2005. 

Beaudry, Paul and Franck Portier, “An Exploration into Pigou’s Theory of Cycles,” Journal 

of Monetary Economics, 2004, 51 (6), 1183–1216. 

Bernanke, Ben S. and Kenneth N. Kuttner, “What Explains the Stock Market’s Reaction 

to Federal Reserve Policy?,” Journal of Finance, 2005, 60 (3), 1221 – 1257. 

Bernanke, Benjamin S. and Mark Gertler, “Monetary Policy and Asset Price Volatility,” 

NBER Working Paper No. W7559, 2000. 

and , “Should Central Banks Respond to Movements in Asset Prices?,” American Eco- 

nomic Review, Papers and Proceedings, 2001, 91 (2), 253–257. 

Black, Fischer, “Noise,” The Journal of Finance, 1986, 41 (3), 529–543. 

Bullard, James, George W. Evans, and Seppo Honkapohja, “Monetary Policy, Judgment 

and Near-Rational Exuberance,” Federal Reserve Bank of St. Louis, Working Paper 2007- 

008A, 2007. 

Campbell, John Y., “Consumption-based Asset Pricing,” in George Constantinides, Milton 

Harris, and Rene Stulz eds. Handbook of the Economics of Finance Vol. IB, North-Holland, 

Amsterdam, 2003, IB, 803–887. 

38

Cecchetti, Stephen G., Hans Genberg, and Sushil Wadhwani, “Asset Prices in a Flexible 

Inflation Targeting Framework,” NBER Working Paper No. W8970, 2002. 

, Pok-Sang Lam, and Nelson C. Mark, “Asset Pricing with Distorted Beliefs: Are Equity 

Returns Too Good to Be True?,” American Economic Review, 2000, 90 (4), 787–805. 

Christiano, Lawrence, Cosmin Ilut, Roberto Motto, and Massimo Rostagno, “Mon- 

etary Policy and Stock Market Boom-Bust Cycles,” mimeo, Northwestern University, 2007. 

DeLong, J. Bradford, Andrei Shleifer, Lawrence H. Summers, and Robert J. Wald- 

mann, “The Size and Incidence of the Losses from Noise Trading,” Journal of Finance, 1988, 

44 (3), 681–696. 

, , , and , “Noise Trader Risk in Financial Markets,” Journal of Political Economy, 

1990, 98 (4), 703–738. 

Dupor, Bill, “Stabilizing Non-Fundamental Asset Price Movements under Discretion and Lim- 

ited Information,” Journal of Monetary Economics, 2005, 52 (4), 727–747. 

Eberly, Janice, Sergio Rebelo, and Nicolas Vincent, “Investment and Value: A Neoclas- 

sical Benchmark,” NBER Working Paper No. W13866, 2008. 

Fernández-Villaverde, Jesús and Juan F. Rubio-Ramírez, “Solving DSGE Models with 

Perturbation Methods and a Change of Variables,” Journal of Economic Dynamics and Con- 

trol, 2006, 30 (12), 2509–2531. 

Gilchrist, Simon and Masahi Saito, “Expectations, Asset Prices, and Monetary Policy: The 

Role of Learning,” Asset Prices and Monetary Policy, edited by John Y. Campbell, 2008. 

Grossman, Sanford J., “On the Efficiency of Competitive Stock Markets Where Traders Have 

Diverse Information,” Journal of Finance, 1976, 31 (2), 573–585. 

and Joseph E. Stiglitz, “On the Impossibility of Informationally Efficient Markets,” Amer- 

ican Economic Review, 1980, 70 (3), 393–408. 

Hassan, Tarek A. and Thomas M. Mertens, “The Social Cost of Near-Rational Invest- 

ment,” mimeo, Harvard University, 2008. 

Hellwig, Christian, “Heterogeneous Information and the Welfare Effects of Public Information 

Disclosures,” mimeo, University of California, Los Angeles, 2005. 

Hellwig, Martin, “On the Aggregation of Information in Competitive Markets,” Journal of 

Economic Theory, 1980, 22, 477–498. 

39

Jaimovich, Nir and Sergio Rebelo, “Can News about the Future Drive Business Cycles?,” 

NBER Working Paper No. W12537, 2006. 

Jin, Hehui and Kenneth L. Judd, “Perturbation Methods for General Dynamic Stochastic 

Models,” mimeo, Hoover Institution, Stanford University, 2002. 

Judd, Kenneth L., Numerical Methods in Economics, The MIT Press, 1998. 

, “Perturbation Methods with Nonlinear Change of Variables,” mimeo, Hoover Institution, 

Stanford University, 2002. 

King, Robert G., “Monetary Policy and the Information Content of Prices,” Journal of Po- 

litical Economy, 1982, 90 (2), 247–279. 

Kiyotaki, Nobuhiro and John Moore, “Liquidity, Business Cycles, and Monetary Policy,” 

mimeo, Princeton University, 2008. 

Kurz, Mordecai, “Measuring the Ex-Ante Social Cost of Aggregate Volatility,” SIEPR Dis- 

cussion Paper No. 04-06, 2005. 

Lansing, Kevin J., “Speculative Growth and Overreaction to Technology Shocks,” Federal 

Reserve Bank of San Francisco, Working Paper 2008-08, 2008. 

LeRoy, Stephen F. and Richard D. Porter, “The Present-Value Relation: Tests Based on 

Implied Variance Bounds,” Econometrica, 1981, 49 (3), 555–574. 

Lorenzoni, Guido, “Optimal Monetary Policy with Uncertain Fundamentals and Dispersed 

Information,” mimeo, Massachusetts Institute of Technology, 2008. 

Lucas, Robert E., Models of Business Cycles, Oxford: Basil Blackwell, 1987. 

Morris, Stephen and Hyun Song Shin, “Social Value of Public Information,” The American 

Economic Review, Dec 2002, 92 (5), 1521–1534. 

Odean, Terrance, “Volume, Volatility, Price, and Profit When All Traders Are above Aver- 

age,” Journal of Finance, 1998, 53 (6), 1887–1934. 

Rigobon, Roberto and Brian Sack, “Measuring the Reaction of Monetary Policy to the 

Stock Market,” Quarterly Journal of Economics, 2003, 118 (2), 639–669. 

Schmitt-Grohé, Stephanie and Martín Uribe, “Closing Small Open Economy Models,” 

Journal of International Economics, 2003, 61 (1), 163–185. 

Shiller, Robert J., “Do Stock Prices Move Too Much to be Justified by Subsequent Changes 

in Dividends?,” The American Economic Review, 1981, 71 (3), 421–436. 

40

, “From Efficient Markets to Behavioral Finance,” Journal of Economic Perspectives, 2003, 

17 (1), 83–104. 

Stein, Jeremy C., “Informational Externalities and Welfare-Reducing Speculation,” Journal 

of Political Economy, 1987, 95 (6), 1123–1145. 

Woodford, Michael, “Inflation Stabilization and Welfare,” Contributions to Macroeconomics, 

2002, 2 (1), Article 1. 

, “Robustly Optimal Monetary Policy with Near-Rational Expectations,” NBER Working 

Paper No. W11896, 2005. 

41

A First-order conditions 

Households trade contingent claims prior to the arrival of signals. They maximize unconditional 

expected utility subject to their budget constraint (3) with respect to contingent claims demand 

taking future demand of stocks and bonds as given. Maximizing ex-ante expected utility, i.e. 

prior to the arrival of information, 

max 

Q i t 

Et[ ˜ ∞ 

Et[ β t u(Ct)|st(i),Pt]] 

subject to budget constraint (3) yields the first order condition 

t=0 

u ′ (Ct(i))ω(zt+1,εt,θt(i))φθ(θt(i))φz(zt+1)φε(εt) = βEt[u ′ (Ct+1(i)]. 

φ· denotes the probability function corresponding to the normal distribution. 

Since agents maximize unconditional expectations of utility, they form undistorted beliefs 

about their future utility levels. Market clearing prices reflect true probabilities of receiving 

signals. Trading contingent claims completes markets and lets agents insure against risk due to 

idiosyncratic information they receive. It serves the purpose of eliminating the wealth distribu- 

tion from the state space. 

B Details on perturbation methods 

Before getting to the expansion of policy rules in state variables, I determine the steady-state 

value of capital and bonds. In the present model, I exploit the fact that deterministic steady- 

state levels for bonds are pinned down at B ∗ and for capital by 

R ∗ c = R ∗ b 

= 1 + r 

which allows to solve for steady-state consumption using the resource constraint. 

Having obtained the deterministic steady-state, we start exploiting the fact that the solution 

is analytic around the steady-state and expand optimal policies. We first have to expand in stock 

variables, here capital and bonds. We start out by totally differentiating equilibrium conditions 

(13) with respect to capital and bonds. We end up with a nonlinear system of equations with 

four variables and four unknowns (the first derivatives of the vector F(·, ·) with respect to K 

and B). Having solved the system and using its solution, we can recursively proceed to higher 

derivatives. The resulting system of equations is linear in all higher derivatives, starting with 

a second-order approximation. Having solved for all desired derivatives in capital and bond, 

42

we move on to the other known state, the log of productivity zt, where we can solve for higher 

derivatives and cross derivatives between z and K and B. It turns out that the second derivative 

with respect to z depends on the first mixed derivative of z with the previous states. The order 

is thus important. To generate a proposal, we expand with respect to the unknown variables εt 

and the news shocks zt+1. 

Having obtained an approximation for the deterministic system, we expand with respect to 

the standard deviation σ. All first and first mixed derivatives with respect to σ are zero. The 

second-order term, however, is important as it quantifies effects due to risk premia. 

B.1 Expansion for log returns 

In order to solve the signal extraction problem in closed form, we want all terms in the Euler 

equations to be lognormal. Therefore, we perform a nonlinear change of variables for the returns 

as well. This step is only important for the solution of the signal extraction problem. We start 

out by taking logs on both sides of the definition of returns (12) 

log(R bt 

c,t+1 ) = log(Pt+1 + Dt+1) − pt. 

The payoff of the asset is a nonlinear function in states. Therefore, we perform a nonlinear 

change of variables as we did with consumption and stock holdings to get: 

log(Pt+1 

+ 

Dt+1) = 

l∈S 

1 

|l|! 

∂ l log(Pt+1 + Dt+1) 

∂S l 

 

 

S=S ∗ 

 

i 

(Sli − S∗ li )li + rθθ 

Analogously to the computation of prices (17), we obtain the coefficients applying the chain rule 

such that we can utilize previously computed coefficients. Using this expression, we have all 

terms in the Euler equation in log-form. 

B.2 Orders of expansion 

Table 3 lists the order of expansions for each state variable I use to solve the model. In order 

to carry out welfare comparisons, I expand the value function in the same fashion as optimal 

policies with the same order. 

To see how good a given expansion approximates the true solution, we can compute the error 

in the optimality condition defined as 

 

 

 

 

F(s t ,s t+1 ) 

u ′ (Ct) 

 

 

 

= 

 

 

βRat 

1 

− 

u ′ (Ct) 

43 

b,t u′ (Ct+1) 

 

 

 

 

.

Expansion in 

Mixed with derivative k b zt εt zt+1 σ 

none 4 4 3 2 2 2 

1st order in k and b - - 2 2 2 2 

2nd order in k and b - - 2 2 2 - 

3rd order in k and b - - 1 - - - 

1st order in zt - - - 1 1 1 

1st order in εt - - - - 1 1 

Table 3: Order of approximation for different states. 

The higher the order, the smaller should be the error. In other words, we have an error estimation 

at hand that allows to check for convergence. Figure 11 shows the resulting Euler equation errors 

for equation (4). All variables except for capital are held at the deterministic steady-state. The 

figure shows how the error decreases with the order of approximation. 

logerror 

3 

4 

5 

6 

7 

8 

9 

3 4 5 

Figure 11: Euler equation error for Euler equation (4) plotted over capital. The error decreases 

with approximation order. The plot, here from order 1 to 4, displays convergence. 

C Details for signal extraction 

This section lays out how to derive expectations on the right-hand side of the Euler equations. 

Having solved for expectations, we impose market clearing to solve for all coefficients of the 

approximation. 

Let us begin with steps I and II of the approximation method using the expansions for 

optimal policies. We start out by taking logs on both sides of the Euler equations and use the 

44 

K

standard formula for the log of an expectation of a lognormal variable 6 to get 

−γct(i) = log(β) − ˜ E[γct+1(i) − r at 

c,t+1 |st(i),Pt] + γ2 

2 var[rat 

c,t+1 − ct+1(i)) st(i),Pt] (32) 

−γct(i) = log(β) + r at 

b,t − γ ˜ E[ct+1(i)|st(i),Pt] + γ2 

2 var[ct+1(i)) st(i),Pt] (33) 

Note that while expectations are subject to a belief shock, agent’s perception of all higher 

moments coincides with those of the optimal forecast. The form of Euler equations gives us a 

natural guess for the form of consumption 

ct(i) = cS(S k t ) + czt(S k t )zt + cε(S k t )µεt + czt+1 (Sk t )zt+1 + cθ(S k t )θt(i) 

and prices (see equation (22)). We guess the functional forms here and verify them subsequently. 

After plugging in for consumption and returns on the right-hand side of expectations, we 

see that we only need to forecast log productivity zt+1. The coefficients depend on Kt+1 and Bt 

which are common knowledge at time t: 

where 

−γct = κb,t − γczt(S k t ) ˜ E[zt+1|st(i),Pt]. (34) 

κb,t = log(β) + r at γ2 

b,t + 

2 var[ct+1(i)) st(i),Pt] − γcS(S k t ) + γcε(S k t )µε∗ + γczt+1 (Sk t )z∗ 

collects all known terms. Using the expansion for the logarithm of the payoff for the asset 

 

log(Pt+1 + Dt+1) = rS(S k t ) + rzt(S k t )zt + rε(S k t )µεt + rzt+1 (Sk t )zt+1 

we can analogously rewrite the Euler equation for capital as 

−γct + pt = κc,t + (rzt(S k t ) − γczt(S k t )) ˜ E[zt+1|st(i),Pt]. (35) 

Again, we collected constant terms in κc,t. 

We guess and verify later that expectations of future log productivity are a linear function 

of the private and public signal (as written in (23)). 

6 For a lognormally distributed random variable X, we can write the logarithm of its expectation as 

log E[X] = E[log X] + 1 

var[log X]. 

2 

45

In the last step, we subtract equation (32) from (33) to get 

pt = κc,t − κb,t + rzt(S k t ) ˜ E[zt+1|st(i),Pt]. 

Having derived the variable that we need to forecast, we can move on to steps three and 

five of the solution method. We now get to the point where the distortion in beliefs enters the 

equations. In order to simplify further proofs, I introduce a rescaled version of the belief shock. 

It will prove to be useful to change the scale of the belief shock that enters the system as esrztµε. 

It allows us to match coefficients in equation (34) more easily which give us the identities 

and in equation (35) to get 

czt(S k t ) = czt(Kt+1,Bt)(es + eppzt+1 ) 

cε(S k t ) = czt(Kt+1,Bt)(eppε + rzt 

cθ(S k t ) = czt(Kt+1,Bt)es 

czt 

µes) 

pzt+1 = rzt(es + eppzt+1 ) (36) 

pε = rzt(eppε + µes) (37) 

where I suppressed arguments. Using the identities, we can apply the projection theorem to get 

closed-form expressions for all coefficients. 

With the market clearing conditions at hand, we can 

⎡ 

⎢ 

⎣ 

where Λ = (z ∗ ,z ∗ ,pεµε ∗ + pzt+1 z∗ ) ′ and 

Σ = 

⎡ 

⎢ 

⎣ 

σ 2 z 

σ 2 z 

rz t es 

1−rz t ep σ2 z 

zt+1 

st(i) 

pt 

⎤ 

⎥ 

⎦ ∼ N (Λ,Σ) (38) 

σ2 rztes z 1−rztep σ2 z 

σ2 z + σ2 rztes θ 1−rztep σ2 z 

rztes 1−rztep σ2 z ( rz tes 1−rztep )2σ2 z + µ 2σ2 ε 

Applying the projection theorem lets us solve for the coefficients of the linear form of expec- 

tations. The conditional variable (zt+1|st(i),Pt) is itself normally distributed according to 

(zt+1|st,Pt) ∼ N 

 

Λ1 + Σ (1,2:3)Σ −1 

(2:3,2:3) 

 

st 

pt 

46 

 

− Λ2:3 

 

⎤ 

⎥ 

⎦ 

,Σ (1,1) − Σ (1,2:3)Σ −1 

(2:3,2:3) Σ (2:3,1)

Solving for coefficients of expectations yields a unique solution for the coefficients of expectations 

and 

es = 

r 2 zt µ2 σ 2 εσ 2 z 

r 2 zt µ2 σ 2 ε σ2 z + r2 zt σ2 θ σ2 z + r2 zt µ2 σ 2 ε σ2 θ 

ep = 

rztσ 2 θ 

r 2 zt µ2 σ 2 ε + r2 zt σ2 θ 

Recursively plugging the values into equations above, we can verify the loglinear guesses and 

compute coefficients. In particular, we get the following lemma: 

Lemma C.1 

Noise in prices is a multiple of distortions in beliefs. 

Proof: The verification of the guess delivers the proof. 

In order to finalize the procedure, we choose µ to match the coefficients of the expansion: 

D Proof of proposition 4.1 

. 

(39) 

µ = 1 

. (40) 

rztes 

The proof of the proposition involves taking comparative statics of the solution to competitive 

equilibrium with respect to dispersion of information. Lemma C.1 in appendix C shows that 

noise in prices is a multiple of belief shocks. The question now concerns the size of the factor. 

The solution to the signal extraction problem in appendix C delivers a closed-form solution 

for coefficients in expectations. From equation (36), we see that the factor µ that links distortions 

in beliefs with aggregate noise can be computed as in (40). It suffices to show that es → 0 as 

information gets more and more dispersed. Taking limits of equation (39) delivers the result. 

At the same time, however, coefficients in prices converge to a constant 

rztes 

1 − rztep 

E Proof of proposition 5.3 

→ rztσ 2 θ 

µ 2 σ 2 ε + σ 2 θ 

for σθ → ∞. 

The solution method shows that tax interventions of the form (26) do not change the response 

of the economy to known state variables such as capital stock Kt, aggregate bond holdings Bt−1, 

and total factor productivity zt. Hence, the response of stock returns to changes in total factor 

productivity rzt is unaltered. Prices are known at the time of formulating expectations about 

47

next period’s payoff. The reaction to news and belief shocks is lower for the case of stabilization 

policies but when forming expectations, agents take that into account. The signal extraction 

problem then makes up for the dampened reaction in prices such that expectations are unaltered 

after policy intervention. 

Equations (36) turn into 

(1 − t g )pzt+1 = rzt(es + eppzt+1 ) 

(1 − t g )pε = rzt(eppε + µes). 

Carrying out the signal extraction problem as in (38) yields the same weight on private signals. 

The weight on the public signal in expectations however is 

and hence expectations do not change. 

F Proof of lemma 5.4 

e t p 

1 

= ep 

1 − tg Having a backward-looking policy in place requires solving a different signal extraction problem. 

Payoffs to stocks no longer only depend on total factor productivity and unforecastable shocks 

but also previous period’s misjudgments. Agents thus forecast 

where 

and 

zt+1−t e µεt|st,Pt 

∼ N 

 

Λ1 + Σ (1,2:3)Σ −1 

(2:3,2:3) 

⎛ 

Σ e ⎜ 

= ⎜ 

⎝ 

t e2 µ 2 σ 2 ε + σ2 z 

σ 2 z 

 

st 

pt 

 

− Λ2:3 

 

Λ e = (z ∗ ,z ∗ ,pεµε ∗ + pzt+1 z∗ ) ′ 

rz t esσ 2 z 

1−rz t ep − te rz t esµ 2 σ 2 ε 

1−rz t ep 

σ 2 z 

σ 2 z + σ 2 θ 

rz t esσ 2 z 

1−rz t ep 

,Σ (1,1) − Σ (1,2:3)Σ −1 

(2:3,2:3) Σ (2:3,1) 

rztesσ2 z 

1−rztep − terztesµ 2σ2 ε 

1−rztep rztesσ2 z 

1−rztep r2 z e 

t 2 sµ 2σ2 ε 

(1−rztep) 2 + r2 z e 

t 2 sσ2 z 

(1−rztep) 2 

Applying the projection theorem to this problem solves the signal extraction problem. The 

weight on the private signal the becomes 

(t e + 1)µ 2 σ 2 εσ 2 z 

e e s = 

µ 2σ2 εσ2 z + σ2 θσ2 z + µ2σ2 εσ2 θ 

48 

⎞ 

⎟ 

⎠ 

 

(41)

and the weight households put on the public signal 

e e p = −te µ 2σ2 εσ2 z + σ2 θσ2 z − te µ 2σ2 εσ2 θ 

rzt µ 2σ2 εσ2 z + σ2 θσ2 z − te µ 2σ2 εσ2 

θ 

Both es and ep change with policy rules that condition on previous period’s misjudgment. 

G Proof of proposition 5.5 

The previous appendix demonstrates the signal extraction problem that agents solve when a 

policy of the form (27) is in place. The multiplier µ that links distortions in beliefs to aggregate 

noise in stock prices is defined as in (40). Hence, we have to look for the effects of policy on es 

and rzt. Using equation (41), we get 

∂rztes 

> 0. 

∂te The change of the multiplier µ thus takes the opposite sign which proves the proposition. 

49

Excessively Volatile Stock Markets - NYU Stern School of Business

Create successful ePaper yourself

Delete template?

Save as template?