The Role of Learning for Asset Prices

and Business Cycles

Fabian Winkler∗

1 September 2016

Abstract

I examine the implications of learning-based asset pricing in a model in which firmsface credit constraints that depend partly on their market value. Agents are learn-ing about stock prices, but have conditionally model-consistent expectations otherwise.The model jointly matches key asset price and business cycle statistics, while the com-bination of financial frictions and learning produces powerful feedback between assetprices and real activity, adding substantial amplification of business cycle shocks. Thepredictability in agents’ subjective forecast errors closely match that found in surveydata across a range of forecast variables. A reaction of the monetary policy rule to assetprice growth is beneficial under learning but not under rational expectations.

JEL: D83, E32, E44, G12

Keywords: Learning, Expectations, Financial Frictions, Asset Pricing, Survey Data

1 Introduction

Financial frictions are seen as a central mechanism by which asset prices interact withmacroeconomic dynamics. Yet our understanding of this interaction remains incomplete,in part due to the inherent difficulty of modeling asset prices. Business cycle models withfinancial frictions should ideally be based on an asset pricing theory that is able to captureat least some of the well-documented empirical regularities of asset price dynamics, such asvolatility and predictability of returns.

∗Board of Governors of the Federal Reserve System, 20th St and Constitution Ave NW, WashingtonDC 20551, fabian.winkler@frb.gov. I thank my former advisors Wouter den Haan and Albert Marcet, mydiscussant at the NBER Summer Institute Martin Schneider, as well as Klaus Adam, Andrea Ajello, JohannesBoehm, Peter Karadi, Albert Marcet, Stephane Moyen, Rachel Ngai, Markus Riegler, and participantsat the NBER Summer Institute 2016, Bank of Spain, Econometric Society World Congress in Montreal,Econometric Society Winter Meeting in Madrid, EEA Annual Meeting in Toulouse, ECB-CFS-Bundesbanklunchtime seminar, and Singapore Management University, for comments and suggestions. The views hereinare those of the author and do not represent the views of the Board of Governors of the Federal ReserveSystem or the Federal Reserve System.

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Figure I: Return expectations and expected returns.

24

68

101-

year

ret

urn

fore

cast

3.5 4 4.5log P/D ratio

-60

-40

-20

020

401-

year

rea

lised

ret

urn

3.5 4 4.5log P/D ratio

Expected nominal returns (left) are the mean response in the Graham-Harvey survey, realized nominalreturns (right) and P/D ratio are from the S&P 500. Data period 2000Q3–2012Q4. Correlation coefficientfor return forecasts ρ = .54, for realized returns ρ = −.44.

At the same time, there is evidence that expectations do not conform to the rational expec-tations hypothesis even in financial markets. The rational expectations hypothesis implies,for example, that investors are fully aware of return predictability in the stock market,expecting lower returns when prices are high and vice versa. Instead, measured expecta-tions imply they expect higher returns. This pattern has been documented extensively byGreenwood and Shleifer (2014) and is illustrated in Figure I. The left panel plots the mean12-month return expectation of the S&P500, as measured in the Graham-Harvey surveyof American CFOs, against the value of the P/D ratio in the month preceding the survey.The correlation is strongly positive: Return expectations are more optimistic when stockvaluations are high. This contrasts sharply with the actual return predictability in the rightpanel of the figure, where the correlation is strongly negative. Such a pattern can hardlybe reconciled with rational expectations.

These observations have sparked interest in asset pricing theories based on extrapolativeexpectations or learning. Adam, Marcet and Nicolini (2015) develop a learning-based assetpricing theory in which equilibrium asset prices are driven by fluctuations in subjective ex-pectations instead of discount factors. They show that in an endowment economy, learningis able to explain many so-called asset price puzzles.

In this paper, I examine the implications of a learning-based asset pricing theory for thebusiness cycle. I construct a model of firm credit frictions in which firms are subject to acredit constraint, the tightness of which depends on their market value. This dependencyimplies that stock market valuations affect access to credit and therefore investment. At thesame time, agents do not have rational expectations but are instead learning about pricegrowth in the stock market.

Deviating from the rational expectations hypothesis is a tricky business. One needs to ex-

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plicitly spell out the entire belief formation process, specifying how agents form expectationsabout future income, interest rates and so on. I develop a restriction on expectations whichessentially requires that agents’ expectations remain consistent with equilibrium conditionsother than stock market clearing. This implies that agents make the best possible forecastscompatible with their subjective beliefs about stock prices. These “conditionally model-consistent expectations” allow me to study in isolation the effects of asset price learningwhile retaining much of the familiar logic of rational expectations.

The model with learning jointly matches key business cycle and asset price moments, includ-ing the volatility and predictability of returns, negative skewness and heavy tails. What’smore, learning considerably magnifies the strength of the financial accelerator. The modeltherefore addresses the Kocherlakota (2000) critique that amplification of shocks throughfinancial frictions is usually quite weak. This weakness is due in part to the low endogenousasset price volatility in standard models (as previously observed by Quadrini 2011). Underlearning, a positive feedback loop emerges between asset prices and the production sideof the economy. When beliefs of learning investors are more optimistic, their demand forstocks increases. This raises firm valuations and relaxes credit constraints, in turn allowingfirms to move closer to their profit optimum. Firms are able to pay higher dividends to theirshareholders, raising stock prices further and propagating investor optimism. Endogenousasset price volatility and business cycle amplification become two sides of the same coin.

I also find that the model is able to replicate remarkably well the predictability of forecasterrors commonly found in survey data on expectations. This holds for stock returns as wellas for a wide range of macroeconomic variables. In the model, agents learn only aboutstock prices, yet their expectational errors spill over into forecasts of other variables. Forexample, when agents are too optimistic about asset prices, they also become too optimisticabout the tightness of credit constraints and therefore over-predict future real investmentand GDP, which is consistent with the patterns found in survey data. This finding suggeststhat the expectation formation process developed in this paper is quite useful even whenagents learn only about stock prices.

Finally, I also offer some implications for monetary policy. A positive response to stock pricegrowth in the interest rate rule – a form of “leaning against the wind” – is welfare-increasingin the learning model because it helps to stabilize expectations in financial markets and tomitigate asset price fluctuations. In the rational expectations version of the model, such areaction is not beneficial.

The remainder of the paper is structured as follows. Section 2 briefly discusses the relatedliterature. Section 3 presents the model and discusses the setup of expectation formationunder learning. To gain some understanding of the interaction between asset price learningand financial frictions, Section 4 first discusses a special case of the model that shuts offauxiliary frictions such as adjustment costs and nominal rigidities and permits a closed-form solution. Section 5 presents the main results using the full model. Section 6 containssensitivity checks, while Section 7 discusses implications for monetary policy. Section 8concludes.

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2 Related literature

The early literature on financial frictions emphasized their role for amplifying business cycleshocks (Kiyotaki and Moore, 1997; Bernanke and Gertler, 2001), but the quantitative im-portance of the “financial accelerator” mechanism is often found to be small (Kocherlakota,2000; Cordoba and Ripoll, 2004; Quadrini, 2011). The more recent literature on financialfrictions instead emphasizes shocks to borrowing constraints as independent drivers of thebusiness cycle (e.g. Jermann and Quadrini, 2012), or alternatively introduces direct shocksto asset prices. Miao, Wang and Xu (2015) develop a model with heterogeneous firms inwhich borrowing limits depend on stock market valuations, as in this paper. They provethe existence of rational liquidity bubbles and analyze a sunspot shock that governs the sizeof this bubble. Liu, Wang and Zha (2013) use a similar framework with land prices insteadof stock prices. In principle, such bubbly sunspot equilibria can arise in my model as well,but they do not exist in the range of parameter values I consider. Instead, this paper takesa different approach by going back to the question of financial frictions as an amplificationmechanism. Learning endogenously generates volatility in asset prices and interacts withfinancial frictions to form a feedback loop that amplifies standard business cycle shocks.

Besides learning, there exist other popular asset pricing theories that address asset pricingpuzzles, including long-run risk, habit, and disaster risk. Each of them has been shownto be compatible with standard business cycle moments in production economies (Boldrin,Christiano and Fisher, 2001, for habit; Tallarini Jr., 2000, for long-run risk; and Gourio,2012, for disaster risk). But each of them also has its problems (e.g. Lettau and Uhlig2000; Epstein, Farhi and Strzalecki 2013), and none of them is able to explain the disconnectbetween statistically predicted returns and expectations of returns in surveys. While Adam,Marcet and Nicolini (2015) and Barberis et al. (2015) have extensively documented theability of learning models to address many asset price regularities, this paper is the first toparsimoniously embed asset price learning into a fully fledged business cycle model and toexamine in detail the interaction of learning-based asset pricing and financial frictions.

There are a number of papers in the adaptive learning literature that are concerned withthe linkages between asset prices and the real economy, e.g. Milani (2008, 2011), Caputo,Medina and Soto (2010), Gelain, Lansing and Mendicino (2013). A finding that oftenemerges is that a policy that stabilizes expectations is welfare-increasing, and this is alsothe case here. I differ from the existing learning literature, where agents typically learnabout all forward-looking variables simultaneously, in being able to isolate the effects ofasset price learning. I do so by requiring expectations to be model-consistent conditionalon beliefs about stock prices. This allows me to incorporate asset price learning into abusiness cylce model in a parsimonious way. At the same time, it allows for expectational“spillovers” of forecast errors on asset prices into forecasts of other parts of the economy.

This paper also relates to a number of papers that try to explain asset price fluctuationswith learning about fundamentals instead of prices. This idea goes back to Timmermann(1996) but has recently become more prominent with very good applications by Fuster,Hebert and Laibson (2012), Hirshleifer, Li and Yu (2015), and Collin-Dufresne, Johannesand Lochstoer (2016). There, agents have extrapolative expectations about the exogenousdriving process of the economy (dividends or technology) but fully understand the resulting

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equilibrium pricing function, while in this paper this assumption of knowledge is reversed:Agents know the process for productivity and other fundamentals but are unsure abouthow the equilibrium price is formed. Because the price itself depends on beliefs, this allowsfor two-sided feedback between learning and the model outcomes, considerably magnifyingvolatility.

3 The model

3.1 Model setup

The economy is closed and operates in discrete time. It is populated by two types ofhouseholds. Lending households consume final goods and supply labor. They are risk-averse, trade debt claims on intermediate goods producers and receive interest from them.Firm owners only consume final goods. They are risk-neutral and trade equity claims onfirms and receive dividends from them. The intermediate goods producers, or simply firms,are at the heart of the model. They combine capital and labor into intermediate goods andare financially constrained.

In addition, the model has nominal price and wage rigidities and investment adjustmentcosts. These are introduced by way of adding three other types of firms into the model(which are all owned by the lending households): Final good producers transform interme-diate goods into differentiated final goods and set prices; labor agencies transform homo-geneous household labor into differentiated labor services and set wages; and capital goodsproducers produce new capital goods from final consumption goods. Finally, there is a fiscalauthority setting tax rates to offset steady-state distortions from monopolistic competition,and a central bank setting nominal interest rates. These additional elements of the modelare standard and their description is relegated to the appendix.

3.1.1 Households

Households with time-separable preferences maximizes utility as follows:

max(Ct,Lt,Bjt,Bgt )

∞t=0

EP∞∑t=0

βtC1−θt

1− θ − ηL1+φt

1 + φ

s.t. Ct = wtLt +Bgt − (1 + it−1)

pt−1

ptBgt−1 +

∫ 1

0(Bjt −Rjt−1Bjt−1) dj + Πt

Here, wt is the real wage received by the household and Lt is the amount of labor supplied.Consumption Ct is a standard Dixit-Stiglitz aggregator of differentiated consumption va-rieties with elasticity of substitution σp. Bg

t are real quantities of nominal one-periodgovernment bonds (in zero net supply) that pay a nominal interest rate it and pt is theprice level. Households also lend funds Bjt to intermediate goods producers indexed by

5

j ∈ [0, 1] at the real interest rate Rjt. These loans are the outcome of a contracting problemdescribed later on. Πt represents lump-sum profits and taxes.

The first-order conditions are standard. In what follows I define the stochastic discountfactor of the household as Λt+1 = βCt/Ct+1.

3.1.2 Intermediate good producers (firms)

The production of intermediate goods is carried out by a continuum of firms, indexedj ∈ [0, 1]. Firm j enters period t with capital Kjt−1 and a stock of debt Bjt−1 which needsto be repaid at the gross real interest rate Rjt−1. First, capital is combined with labor Ljtto produce output:

Yjt = (Kjt−1)α (AtLjt)1−α , (1)

where At is aggregate productivity. Labor is a CES combination of differentiated laborservices with elasticity of substitution σw, but the firm’s problem can be treated as if thelabor index was acquired in a competitive market at the real wage index wt. Output is soldcompetitively to final good producers at price qt. The capital stock depreciates at rate δ.This depreciated capital can be traded by the firm at the price Qt.

The firm’s net worth is the difference between the value of its assets and its outstandingdebt:

Njt = qtYjt − wtLjt +Qt (1− δ)Kjt−1 −Rjt−1Bjt−1. (2)

I assume that the firm exits with probability γ. This probability is exogenous and inde-pendent across time and firms. As in Bernanke, Gertler and Gilchrist (1999), exit preventsfirms from becoming financially unconstrained. If a firm does not exit, it needs to pay outa fraction ζ ∈ (0, 1) of its earnings as a regular dividend (where earnings Ejt are givenby Njt − QtKjt−1 + Bjt−1). The number ζ therefore represents the dividend payout ratiofor continuing firms.1 The firm then decides on the new stock of debt Bjt and the newcapital stock Kjt, maximizing the present discounted value of dividend payments using thediscount factor of its owners. Its balance sheet must satisfy:

QtKjt = Bjt +Njt − ζEjt (3)

If a firm does exit, it pays out its entire net worth as a terminal dividend.

3.1.3 Firm owners

Firm owners differ from households in their capacity to own intermediate firms. They arerisk-neutral and less patient than households. They can buy shares in firms indexed byj ∈ [0, 1]. As described above, if a firm exits, it pays out its net worth Njt as a terminal

1The optimal dividend payout ratio in this model would be ζ = 0, as firms would always prefer to buildup net worth to escape the borrowing constraint over paying out dividends. However, this would implythat aggregate dividends would be proportional to aggregate net worth, which is rather slow-moving. Theresulting dividend process would not be nearly as volatile as in the data. Imposing ζ > 0 allows to bettermatch the volatility of dividends and therefore obtain better asset price properties.

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dividend. Otherwise it pays a regular dividend of ζEjt. Denote the subset of firms alive atthe end of period t by Γt. Then, a firm owner’s utility maximization problem is given by:

max(Cft ,Sjt

)∞t=0

EP∞∑t=0

βtCft

s.t. Cft +

∫j∈Γt

SjtPjtdj =

∫j∈Γt

Sjt−1 (Pjt + ζEjt) dj (4)

+

∫j∈Γt−1\Γt

Sjt−1Njtdj (5)

Sjt ∈[0, S

](6)

where β ≤ β and S > 1. Consumption Cft is the same Dixit-Stiglitz aggregator of con-sumption varieties as for households.

The first term on the right-hand side of the budget constraint deals with continuing firmswhile the second term deals with exiting firms. In addition, firm owners face upper andlower bounds on traded stock holdings.2 The first-order condition of the firm owner is:

Sjt = 0 if Pjt >Sjt ∈

[0, S

]if Pjt =

Sjt = S if Pjt <

βEPt[Njt+11{j /∈Γt+1} + (ζEjt+1 + Pjt+1)1{j∈Γt+1}

]∀j ∈ Γt.

(7)

3.1.4 Borrowing constraint

In choosing their debt holdings, firms are subject to a borrowing constraint. The constraintis the solution to a particular limited commitment problem in which the outside option forthe lender in the event of default depends on the market value of the firm. Effectively, theborrowing constraint introduces a link between stock market valuations and investment,which has been empirically documented (e.g. Baker, Stein and Wurgler, 2003 and morerecently Hau and Lai, 2013).

Each period, lenders (households) and borrowers (firms) meet to decide on the lending offunds. Pairings are anonymous. Contracts are incomplete because the repayment of loanscannot be made contingent. Only the size Bjt and the interest rate Rjt of the loan canbe contracted in period t. Both the lender (a household) and the firm have to agree on acontract (Bjt, Rjt). Moreover, there is limited commitment in the sense that at the end ofthe period, but before the realization of next period’s shocks, firm j can always choose toenter a state of default. In this case, the value of the debt repayment must be renegotiated.If the negotiations are successful, then wealth is effectively shifted from creditors to debtors.The outside option of this renegotiation process is bankruptcy of the firm and seizure by the

2This renders demand for stocks finite under arbitrary beliefs. In equilibrium, the bounds are neverbinding.

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lender. Bankruptcy carries a cost of a fraction 1 − ξ of the firm’s capital being destroyed.The lender, a household, does not have the ability to operate the firm. It can liquidate thefirm’s assets, selling the remaining capital in the next period. This results in a recoveryvalue of ξQt+1Kjt. With some probability x (independent across time and firms), the lenderreceives the opportunity to “restructure” the firm if it wants. Restructuring means that,similar to Chapter 11 bankruptcy proceedings, the firm gets partial debt relief but remainsoperational. I assume that the lender has to sell the firm to another firm owner, retaininga fraction ξ of the initial debt. In equilibrium, the recovery value in this case will beξ (Pjt +Bjt) and this will always be higher than the recovery value after liquidation.

The optimal debt contract then takes the form of a leverage constraint with a weightedaverage of liquidation and market value of the firm:

Bjt ≤ (1− x)EPt Λt+1Qt+1ξKjt︸ ︷︷ ︸liquidation value

+x ξ (Pjt +Bjt)︸ ︷︷ ︸market value

(8)

The appendix formally derives this borrowing limit as the outcome of the limited commit-ment problem described above. It is worth noting that even under rational expectations,the market value of the firm Pjt +Bjt will generally be higher than the value of its capitalstock. This is because financial frictions prevent firms from investing to the point wherethe marginal return on capital is at its efficient level, so that capital already owned by afirm is more valuable than newly produced capital.

The borrowing constraint is similar to that in Miao and Wang (2011) who essentially lookat the case x = 1 and show that this type of borrowing constraint can lead to sunspotequilibria. The appendix shows that such multiple equilibria do not arise for the parametervalues considered in this paper.

3.1.5 Further model elements and shocks

Investment is subject to quadratic adjustment costs that move the price for capital goods:

Qt = 1 + ψ

(ItIt−1

− 1

)(9)

Further, the prices for final goods and labor are subject to Calvo rigidities, with probabilitiesof non-adjustment κ and κw and elasticities of substitution θ and θw, respectively. The pricefor intermediate goods qt equals the inverse of the gross markup of final goods producers.The monetary authority sets the nominal interest rate according to a Taylor-type interestrate rule:

it = ρiit−1 + (1− ρi)(β−1 + φππt + εit

), (10)

where φπ is the reaction coefficient on consumer price inflation, ρi is the degree of interestrate smoothing, and εit is an interest rate shock. Details on the setup of nominal rigiditiesand adjustment costs are provided in the appendix.

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Productivity evolves as an AR(1) process and there are two exogenous shocks (productivityand nominal interest rate).

logAt = (1− ρ) log A+ ρ logAt−1 + log εAt (11)

εAt ∼ iidN(0, σ2

A

)(12)

εit ∼ iidN(0, σ2

i

)(13)

3.2 Rational expectations equilibrium

I first describe the equilibrium under rational expectations. A rational expectations equi-librium is a set of stochastic processes for prices and allocations, a set of strategies in thelimited commitment game, and an expectation measure P such that the following holds forall states and time periods: Markets clear; allocations solve the optimization programs ofall agents given prices and expectations P; the strategies in the limited commitment gameare a subgame-perfect Nash equilibrium for all lender-borrower pairs; and the measure Pcoincides with the actual distribution of equilibrium outcomes.

Under a mild restriction on the exit probability γ, there exists a rational expectationsequilibrium characterized by the following properties.

1. All firms choose the same capital-labor ratio Kjt/Ljt. This allows one to define anaggregate production function and an internal rate of return on capital:

Yt = αKαt−1

(AtLt

)1−α(14)

Rkt = qtαYtKt−1

+Qt (1− δ)Kt−1 (15)

2. The expected return on capital is higher than the internal return on debt: EtRkt+1 >Rjt.

3. At any time t, the stock market valuation Pjt of a firm j is a linear function of itspost-dividend net worth QtKt − Bt. This permits one to write an aggregate stockmarket index as

Pt =

∫ 1

0Pjt = βEt [Dt+1 + Pt+1] . (16)

where aggregate dividends are given by Dt = γNt + ζEt.

4. Borrowers never default on the equilibrium path and borrow at the risk-free rate

Rjt = Rt =1

EtΛt+1. (17)

The lender only accepts debt payments up to the limit given by (8), which is linearin the firm’s net worth, and the firm always exhausts this limit.

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5. As a consequence of the previous properties of the equilibrium, all firms can be ag-gregated. Aggregate debt, capital, and net worth are sufficient to describe the inter-mediate goods sector:

Nt = RktKt−1 −Rt−1Bt−1 (18)

QtKt = (1− γ) ((1− ζ)Nt + ζ (Bt−1 −QtKt−1)) +Bt (19)

Bt = xEtΛt+1Qt+1ξKt + (1− x) ξ (Pt +Bt) . (20)

Proofs are relegated to the appendix.

3.3 Learning equilibrium

I require that the equilibrium under learning satisfies internal rationality (Adam and Marcet,2011). Specifically, given a subjective belief measure P of the distribution of model out-comes, an internally rational equilibrium is a set of stochastic processes for prices andallocations, and a set of strategies in the limited commitment game such that the follow-ing holds for all states and time periods: Markets clear; allocations solve the optimizationprograms of all agents given prices and expectations P; and the strategies in the limitedcommitment game are a subgame-perfect Nash equilibrium for all lender-borrower pairs.The only difference from a rational expectations equilibrium is that the measure P un-der which agents evaluate expectations can differ from the actual distribution of modeloutcomes.3

How, then, should one choose the subjective beliefs P? In a forward-looking business cyclemodel, there are many ways in which expectations affect equilibrium. Households need toform expectations about future interest rates and wages to determine their savings, investorsneed to form expectations about stock prices, firms need to forecast future productivity andwages in order to decide their demand for capital and so forth. This leaves many degrees offreedom to be filled. My focus in this paper is to concentrate on the effects of stock pricelearning, while keeping the model dynamics as close as possible to a rational expectationsequilibrium otherwise.

To this end, I construct the belief P in two steps: First, I specify the belief about stockprices that give rise to a learning problem; second, I impose what I call conditionally model-consistent expectations with respect to stock prices.

Under the subjective belief measure P, agents are not endowed with the knowledge of theequilibrium pricing function for stocks, i.e. the mapping of state variables and shocks toprices Pjt. To keep things simple and retain the linear aggregation property, I assume thatagents do believe that the value of an individual firm is proportional to its net worth, asunder rational expectations:

Pjt =EPt Njt+1

EPt Nt+1Pt. (21)

3It then becomes crucical to distinguish the subjective expectations EP [·] taken under P from the ex-pectation E [·] taken under the actual distribution of model outcomes.

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However, they are uncertain about the evolution of the aggregate market value Pt. Theyemploy a simple subjective model to forecast the market value, according to which Pt evolvesas a random walk with a small unobservable drift:

logPt = logPt−1 + µt + ηt, ηt ∼ N(0, σ2

η

)µt = µt−1 + νt, νt ∼ N

(0, σ2

ν

)where the innovations ηt and νt are independent of each other and across time, and alsoindependent of the other exogenous shocks in the model.4

Importantly, the innovations are unobservable, as is the mean growth rate µt. The onlyobservable in the system above is the price Pt. Bayesian updating of this belief systemamounts to a simple Kalman filtering problem. With an appropriate prior, the systemabove can be rewritten in terms of observables only:

logPt = logPt−1 + µt−1 + zt (22)

µt = µt−1 + gzt, (23)

where µt = EPt µt is the mean belief about the trend in stock price growth, and zt is thesubjective forecast error. Under P, zt is normally distributed white noise with variance σ2

z ,independent of the structural shocks εAt and εit. The belief µt is updated in the directionof the last forecast error: When agents see stock prices rising faster than they expected,they will also expect them to rise by more in the future. The parameter g is the learninggain which governs the speed of adjustment of price growth expectations.56

While agents think that Pt is a random walk with a drift, in equilibrium this will generallynot be the case. The equilibrium stock price must be such that the stock market clears.This implies that the Euler equation (7) has to hold with equality in the aggregate:

Pt = βEPt [Pt+1 +Dt+1]

= βEPt [Pt exp (µt + zt+1)] + βEPt [Dt+1]

=βEPt [Dt+1]

1− β exp(µt + 1

2σ2z

) . (24)

The value for the forecast error zt that equates (22) and (24) is not white noise but dependson the belief about future dividends and price growth.7 The expression also shows that,

4This implies that agents think that prices can diverge forever from economic fundamentals. In equilib-rium however, prices and dividends (and other fundamentals) will turn out to be cointegrated, contrary toagents’ expectations.

5The parameters of the state-space system and the observable system map into each other through therelations σ2

η = σ2p (1− g) and σ2

ν = σ2pg

2. Intuitively, a small learning gain g implies that agents believethere is little predictability in stock prices.

6I also impose that the belief about price growth µt is updated only at the end of the period, so thatagents make their forecasts in period t using µt−1 only. This “lagged belief updating” is common in thelearning literature (see Adam, Beutel and Marcet 2014 for a discussion) and avoids having to simulatenouslydetermine beliefs about the price and its realization. The appendix describes its implementation.

7The corresponding equilibrium values for the subjective forecasting error zt and the belief µt are zt =∆ logPt − µt−1 and µt = (1− g) µt−1 + g∆ logPt−1.

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since the denominator is close to zero, small changes in the belief µt can induce large swingsin equilibrium asset prices.

The specification of the subjective probability measure P is not complete yet. It does notyet specify how to calculate the expectation of future dividends EPt [Dt+1] above, or indeedany expectation of other model variables. The subjective probability measure P has tofully specify expectations about all variables in the model in order for an equilibrium to bewell-defined. In principle, there is an enormous amount of degrees of freedom still left tospecify P: Once we move away from rational expectations, agents could entertain arbitrarybeliefs about dividends, inflation, interest rates, wages and so on. Here is where I introducethe concept of conditionally model-consistent expectations in order to impose discipline onthe expectation formation process.

Definition. Consider an internally rational equilibrium with beliefs P. Agents’ subjectiveexpectations P are conditionally model-consistent if:

1. the distribution of the exogenous shocks ut under P coincides with their actualdistribution, and

2. for any endogenous model variable yt and t ≥ 0:

EPt [yt+1 | ut+1, Pt+1, µt+1] = yt+1 (25)

almost everywhere in equilibrium.

Conditional model consistency restricts the subjective belief P to have the maximum degreeof consistency with the model given agents’ misspecified belief about stock prices. Agentsendowed with such expectations may not know the equilibrium pricing function, but theymake the smallest possible expectational errors consistent with their subjective view aboutthe evolution of stock prices. Conditionally model-consistent expectations are close tobut different from rational expectations, for which condition (25) can be strengthened toEPt [yt+1 | ut+1] = yt+1.

I construct expectations satisfying conditional model-consistency as follows. I solve forexpectations that would be model-consistent in a model in which the stock price reallyevolved according to agents’ subjective beliefs. I take the set of equilibrium conditionsand remove the market clearing conditions for stocks and intermediate goods,8 replacingit with the subjective belief system (22)–(23). I then solve the model as under rationalexpectations, which leads to a subjective policy function yt = h (yt−1, ut, zt) where yt is thecollection of all endogenous model variables, ut is the collection of exogenous shocks, andzt is the subjective forecast error on stock prices. The policy function h together with thejoint distribution of (ut, zt)defines the subjective probability measure P. In order to solvefor the equilibrium, I then impose the market clearing condition in the stock market. Thisleads to a solution for the equilibrium stock price and, through Equation (22), a solutionfor the equilibrium subjective forecast error zt = r (yt−1, ut). Importantly, under P agents

8It is necessary to remove one more market clearing condition in addition to that for the stock market.Otherwise the rational expectations equilibrium would re-emerge owing to Walras’ law.

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think that zt is an unpredictable white noise process, while in equilibrium it is a functionof the states and the exogenous shocks. Finally, the actual policy function describing theequilibrium is computed as yt = g (yt−1, ut) = h (yt−1, ut, r (yt−1, ut)).

The appendix spells out this procedure in detail. I approximate the equilibrium both underlearning and under rational expectations using a second order perturbation method.

4 Inspecting the mechanism

In this section, I examine a special case of the model which does away with nominal rigidities,adjustment costs, net worth dynamics and risk aversion. This special case reduces thenumber of state variables and allows me to solve the model in closed form.

One insight is that neither learning nor financial frictions alone generate sizable amplificationof business cycle shocks or asset price volatility in a production economy. It is the interactionof these two features that leads to endogenous amplification.

4.1 Simplifying the model

First, I render the nominal rigidities redundant by setting κ = κw = 0 and σ = σw = 0.Investment adjustment costs are eliminated by setting ψ = 0.

Next, I simplify the financial structure of the firm. I set the exit rate of firms to γ = 0, sothat firms are infinitely-lived. To ensure that they do not escape the borrowing constraint,I then require them to pay out their earnings entirely every period, by setting ζ = 1. Theequations (18)–(20) describing aggregate firm dynamics now simplify to:

Nt = RktKt−1 −Rt−1Bt−1 (26)

Kt = Bt (27)

Bt = ξxKt + ξ (1− x) (Pt +Bt) . (28)

I further simplify preferences by setting the household coefficient of relative risk aversion toγ = 0 and the inverse elasticity of labor supply to φ→∞, implying inelastic labor supplyLt = L. For the firm owners I impose β = β so that one ends up with a representativehousehold. I also set the autoregressive coefficient for the productivity process to ρ = 1,effectively making all productivity innovations permanent.

4.2 Rational expectations equilibrium

The presence of financial frictions does not necessarily imply strong amplification, and thisis true in particular in this model under rational expectations. Start first with the caseξ = 1, where the borrowing constraint (28) is never binding. In this case, the model isjust a particular variant of the RBC model. Equilibrium in the labor market requires

13

equalization of the marginal rate of substitution and the marginal rate of transformationbetween consumption and labor:

wt = (1− α)A1−αt

(Kt

L

)α(29)

which implies that the expected marginal return on investment can be written as:

Et[Rkt+1

]= Rk (Kt, At) = α

(AtLe

ασ2A2

)αK−αt + 1− δ

The firm equates this expected return with the interest rate, which is constant at R = 1/β.Capital is simply proportional to productivity: Kt/At = K∗ for some fixed value K∗.

Once we introduce financial frictions by setting ξ < 1, how much amplification do weget? The answer: none. For all values of ξ strictly below one, the borrowing constraint isalways binding, but the capital stock and the stock price of the firm are still proportionalto productivity. The equilibrium is characterized by the following two equations:

Pt = AtP = Atβ(Rk(K, 1

)−R

)K

1− β (30)

Kt = AtK =ξx

1− ξPt (31)

The first equation pins down the stock market value of the firm, which depends on thecapital stock through expected dividends in the enumerator. These dividends depend oncapital through the size of the firm and the rate of return on capital. The second equationdetermines the capital stock that can be reached by exhausting the borrowing constraintthat depends on the stock market value. In the unique equilibrium, the capital stock isproportional to productivity, just as was the case when ξ = 1.

Financial frictions do not lead to any amplification or propagation of shocks in the rationalexpectations equilibrium. They have a level effect on output, capital, etc., but the dynamicsof the model are identical for any value of ξ (or x). Similarly, the behavior of asset pricesis entirely independent of financial frictions. The stock price evolves simply as:

logPt = logPt−1 + εt. (32)

Intuitively, with financial frictions, a shock to productivity raises asset prices just as muchas to allow the firm to instantly adjust the capital stock proportionately. At the same time,stock returns are not volatile and unpredictable at all horizons.

The complete irrelevance of financial frictions for the model dynamics is particular to theassumptions in this section, which make prices in the collateral constraint move exactly inlockstep with the marginal product of capital. However, the problem that low endogenousasset price volatility leads to low amplification through financial frictions is illustrative ofmost models with financial frictions (Quadrini, 2011).

14

4.3 Learning equilibrium

I now describe the equilibrium under learning. The first-order conditions of the householdimply that the interest rate still has to equal R = 1/β regardless of expectations. Also, thestatic labor demand equation of the firm is unchanged, so that labor market equilibriumimplies Equation (29) has to hold. The firm’s investment decision depends on the expectedreturn on investment. Here, I make use of conditionally model-consistent expectations:Expectations under the subjective measure P can be calculated using the equilibrium con-ditions of the model except for the market clearing condition for stocks and consumptiongoods. In this setting, we then have EPt

[Rkt+1

]= Rk (Kt, At), as under rational expec-

tations. Agents in the learning equilibrium are able to accurately infer the equilibriumwage, and therefore the optimal choice of labor and the return on capital, given today’sproductivity and capital stock.

By the same logic, they can also accurately predict next period’s dividend.9 They do notaccurately predict next period’s stock price, instead relying on the subjective law of motion(22)–(23). Applying the expression for the equilibrium stock price (24), here we have:

Pt =β(Rk (Kt, At)−R

)Kt

1− β exp(µt + 1

2σ2z

) .

We can infer the equilibrium realization of the subjective forecast error as zt = ∆Pt−1− µt.So while under P, agents think that zt is random white noise, in equilibrium it is a functionof the state variables and fundamental shocks of the model. This is precisely where rationalexpectations break under learning.

The learning equilibrium is summarized by the following three equations:

Pt =β(Rk (Kt, At)−R

)Kt

1− β exp(µt + 1

2σ2z

) (33)

Kt =ξx

1− ξPt (34)

µt+1 = µt −σ2ν

2+ g (∆ logPt − µt) (35)

The first equation is the determination of the equilibrium stock price as a function of thecapital stock and the belief of stock price growth. The second equation is the borrowingconstraint, which is always binding in equilibrium for any ξ < 110 The third equation is theupdating equation of the Kalman filtering problem. Here, the forecast error zt that agentsperceive as random noise has been substituted out by its equilibrium value.

Figure II depicts the dynamics of stock prices after a positive productivity innovation ε1 >0. The initial shock at t = 1 raises stock prices and the capital stock proportionally toproductivity through Equations (33) and (34), just as under rational expectations. Learning

9They do not accurately predict dividends more than one period ahead, as those depend on (misperceived)future stock prices.

10To see this, assume that Pt were sufficiently high so that the borrowing constraint didn’t bind. ThenRk (Kt, At) = R and therefore Pt = 0, a contradiction.

15

Figure II: Stock price dynamics under learning.

t

logPt − logP0

∆ logPt

µt

0 1 2 3 4 5 6 7

investors observe the rise in P1 and are unsure whether their positive forecast error z1 itis due to a transitive shock (η1 > 0) or a permanent increase in the growth rate of stockprices (ν1 > 0). They therefore revise their beliefs µ2 upward in Equation (35). In thenext period t = 2, the more optimistic beliefs increase the demand for stocks, and themarket clearing price in Equation (33) has to be higher, in turn relaxing credit constraintsand fueling investment. Beliefs continue to rise in subsequent periods as long as observedasset price growth (dashed black line in Figure II) is higher than the current belief µt (solidred line). The differences between observed and expected price growth are the subjectiveforecast errors (dotted red lines). In the figure, the increase in prices and beliefs ends att = 3, when the forecast error is zero. There is no need for a further belief revision. But inthe absence of subsequent shocks, no change in µt implies no change in the price Pt, so thatrealized asset price growth is zero at t = 4, at a time when agents expect strongly positiveprice growth. This triggers a downward revision in beliefs and an endogenous reversalin prices. Ultimately prices return to their steady-state levels. These learning dynamicsproduce return volatility and predictability.

Asset price learning affects economic activity because it influences current stock prices andtherefore the tightness of the borrowing constraint. But the feedback in this model is two-sided, as real activity affects stock prices through dividend payments. The strength ofthis channel depends on general equilibrium effects. Equation (33) shows that Kt entersthe expression for expected dividends twice. The multiplying factor Kt captures a partialequilibrium effect that is internalized by the firm. The internal rate of return on capitalis higher than the cost of debt and the firm therefore wants to increase its capital stockuntil it exhausts the borrowing constraint, increasing expected dividends. At the same timethough, higher levels of capital lower its marginal return Rk (Kt, At) because of decreasingreturns to scale at the aggregate level, increasing wages and decreasing expected dividends.

When financial frictions are severe enough (ξ is low), the partial equilibrium effect domi-nates. This case is depicted in Panel (a) of Figure III. The figure plots the stock pricingequation (33) and the credit constraint (34). When the degree of financial frictions is high,

16

Figure III: Endogenous response of dividends.

Kt

1−ξξ

Kt

Rk(Kt;At)Kt

R−exp(µt)

µ1

µ2

µ3

K∗

P1

P2

P3

Pt

(a) Amplification.

Kt

1−ξξ

Kt

Rk(Kt;At)Kt

R−exp(µt)

µ1

µ2

µ3

K∗

P1

P2

P3

Pt

(b) Dampening.

the credit constraint line is steep. Consider the effect of a positive productivity shock att = 1 as before, when the initial equilibrium is at P1 and µ1. The immediate effect will be aproportionate rise in stock prices and capital, together with a rise in beliefs from µ1 to µ2.This leads to higher stock prices at t = 2 and allows the firm to invest more and increase itsexpected profits—the partial equilibrium effect dominates. This adds to the rise in realizedstock prices, further relaxing the borrowing constraint and increasing next period’s beliefs.Stock prices, investment, and output all rise more than proportionally to productivity.

In Panel (b), credit constraints are not very severe (ξ is large) and the credit constraint lineis relatively flat. A relaxation of the borrowing constraint due to a rise in µ2 still allowsthe firm to invest and produce more, but in general equilibrium wages rise by so much thatdividends fall. This response of dividends dampens rather than amplifies the dynamics ofinvestment and asset prices.

This dampening effect can be so strong as to eliminate the effects of learning altogether.The appendix shows that in the limit of vanishing financial frictions, stock prices become apure random walk again:

logPt − logPt−1ξ→1−→ εt. (36)

As a consequence, the entire dynamics of the model become identical to those under rationalexpectations. As financial frictions disappear, the general equilibrium effects offset anydynamics from stock price learning. This shows that sizable amplification arises neitherfrom learning nor from credit frictions alone, but only from their interaction.

17

5 Quantitative results

I now go back to the general model, where I approximate equilibrium by a second-orderapproximation around the non-stochastic steady state (outlined in the appendix). First, Idiscuss the parameterization of the general model. I then review standard business cyclestatistics. Learning and asset price volatility account for a third of the volatility of output,pointing to the strength of the endogenous amplification mechanism. Next, I look at assetpricing moments and find that the model with learning closely matches the volatility ofstock prices (which is targeted by the estimation), but also the predictability of stockreturns, skewness and kurtosis. Impulse response functions confirm the presence of a strongamplification mechanism. The main channel is the endogenous volatility of asset pricesinduced by learning. Finally, I compare forecast errors made by agents in the model withthose observed in survey data. The patterns of predictability are remarkably similar, lendingcredibility to the assumed expectation formation process.

5.1 Choice of parameters

I partition the set of parameters into two groups. The first set of parameters is calibratedto first-order moments, and the second set is estimated by simulated method of moments(SMM) on second-order moments of US quarterly data (1962Q1–2012Q4).

5.1.1 Calibration

The capital share in production is set to α = 0.33. The depreciation rate δ = 0.025corresponds to ten percent annual depreciation. The trend growth rate of productivity isset to 1.5 percent annually so that g = 0.0037, and the persistence of productivity shocksis set to ρ = 0.95.11

The household discount factor is set such that the steady-state real interest rate equalstwo percent per year, implying a discount factor β = 0.9951. The elasticity of substitutionbetween varieties of the final consumption good, as well as that among varieties of laborused in production, is set to σ = σw = 4. The Frisch elasticity of household labor supply isset to 3, implying φ = 0.33, and risk aversion is set to θ = 1 (log utility in consumption).Finally, the firm owners’ discount factor is set to β = 0.9850 in order to produce an averagestock return of 6.65 percent per year.

The strength of the monetary policy reaction to inflation is set to φπ = 1.5, and the degreeof nominal rate smoothing is set to ρi = 0.85.

Three parameters describe the structure of financial constraints: x, the probability of re-structuring after default; ξ, the tightness of the borrowing constraint; and γ, the rate offirm exit. I calibrate the restructuring rate to x = 0.093. This is the fraction of US busi-ness bankruptcy filings in 2006 that filed for Chapter 11 instead of Chapter 7, and that

11The results are similar if one assumes permanent shocks to productivity, ρ = 1, which is more commonin the asset pricing literature.

18

Table I: Estimated parameters.

param. σa σi κ κw ψ ζ g

learning .00942 .000101 .597 .937 10.63 .684 .0131

(.00147) (.00277) (.093) (.215) (7.12) (.021) (.0013)

RE .0108 .000496 .729 .971 .110 .402 -

(.00153) (.000244) (.025) (.010) (.246) (.263)Parameters as estimated by simulated method of moments. Asymptotic standard errors in parentheses.

subsequently emerged from bankruptcy with an approved restructuring plan (a sensitivitycheck is included in Section 6.3).12 The remaining two parameters are chosen such that thenon-stochastic steady state of the model jointly matches the US average investment sharein output of 18 percent and an average ratio of debt to assets of one (the sample average inthe Fed flow of funds). The corresponding parameter values are γ = 0.0114 and ξ = 0.4712.

5.1.2 Estimation

The remaining parameters are the standard deviations of the technology and monetaryshocks (σA, σi), the degree of nominal price and wage rigidities (κ, κw), the size of investmentadjustment costs (ψ), the fraction of dividends paid out as earnings by continuing firms (ζ),and the learning gain (g). I estimate these six parameters to minimize the distance to a setof eight moments pertaining to both business cycle and asset price statistics: The standarddeviation of output; the standard deviations of consumption, investment hours worked, anddividends relative to output; and the standard deviations of inflation, the nominal interestrate, and stock returns (see also Tables II and III; all variables are in logs and all variablesexcept stock returns are HP-filtered). The set of estimated parameters θ solves

minϑ∈A

(m (θ)− m)′W (m (θ)− m)′ ,

where m (θ) are moments obtained from model simulation paths with 50,000 periods, m arethe estimated moments in the data, and W is a weighting matrix.13 I also impose that θhas to lie in a subset A of the parameter space which rules out deterministic oscillationsof stock prices.14 Such oscillations are not observed in the data, but can emerge in thelearning equilibrium when the learning gain is large. The restriction effectively constrainsthe degree of departure of subjective beliefs from rational expectations.

Table I summarizes the SMM estimates for both the learning and rational expectationsversion of the model. The first row presents the results under learning. Exogenous shocks

122006 is the only year for which this number can be constructed from publicly available data. Data onbankruptcies by chapter are available at http://www.uscourts.gov/Statistics/BankruptcyStatistics.

aspx. Data on Chapter 11 outcomes are analyzed in various samples by Flynn and Crewson (2009), Warrenand Westbrook (2009), Lawton (2012), and Altman (2014).

13I choose W = diag(

Σ)−1

where Σ is the covariance matrix of the data moments, estimated using a

Newey-West kernel with optimal lag order. This choice of W leads to a consistent estimator that placesmore weight on moments which are more precisely estimated in the data.

14θ /∈ A if there exists an impulse response of stock prices with positive peak value also having a negativevalue of more than 20% of the peak value.

19

come mainly from productivity shocks, since σi is estimated to be relatively small. TheCalvo price adjustment parameter κ implies retailers adjust their prices about every 5months on average.

The SMM procedure selects a high degree of nominal wage rigidities κw and of adjustmentcosts ψ. Even though these values are imprecisely estimated, they are substantially largerthan what is commonly found in the literature. High adjustment costs are needed becausethe high asset price volatility would otherwise translate into counterfactually large fluctu-ations in investment relative to output. In a sense, the amplification through asset pricelearning is too powerful and needs to be dampened again to fit the data. A high degree ofwage rigidity is needed in order to match the rather high volatility of employment in thedata, as it suppresses the wealth effect on labor supply.

The fraction of earnings paid out as dividends ζ is fitted to 68 percent, which is somewhathigher than the historical average for the S&P500 (about 50 percent). Finally, the learninggain g is within the range of values typically used in the learning literature. It implies thatagents believe the degree of predictability in stock prices to be relatively small.

The second row contains the parameters estimated under rational expectations. As underlearning, there are substantial standard errors around the point estimates. Nevertheless, atthese point estimates, the size of the shocks σa and σi is substantially larger under learning,implying that learning about stock prices leads to substantial amplification of shocks. Sincethe tightness of the borrowing constraint is much less volatile under rational expectations,the degree of investment adjustment costs required to fit the data is much smaller.

5.2 Business cycle and asset price moments

I now review the key business cycle and asset pricing moments in the data and acrossmodel specifications. Table II starts with business cycle statistics. Moments in the data areshown in Column (1). Moments for the estimated learning model are shown in Column (2),while Columns (3) and (4) contain the corresponding moments for the model under rationalexpectations and a frictionless comparison model in which financial frictions are shut off.15

The model parameters are held constant at the estimated values for the learning model inColumns (2) to through (4). Column (5) presents the moments under rational expectationswhen the parameters are re-estimated to fit the data.

The first row reports the standard deviation of detrended output. By construction, it ismatched well by the learning model in Column (2). When learning is shut off in Column (3),it drops one-third. This shows the great degree of amplification that learning adds to themodel. The standard (rational expectations) financial accelerator mechanism is present inthe model as well, since the volatility of output drops further in Column (4) when financialfrictions are shut off. But it is not as powerful as when it is combined with asset pricelearning. Of course, it is also possible to match output volatility with rational expectations,using larger shock sizes, as in Column (5).

15In the frictionless model, intermediate goods producers are owned by the household and face no financialconstraint. The model then reduces to a standard New-Keynesian model with adjustment costs and priceand wage rigidities.

20

Table II: Business cycle statistics in the data and across model specifications.

(1) (2) (3) (4) (5)

moment data learning RE fric.less RE re-estimated

output

volatility

σhp (Yt) 1.43%

(0.14%)

1.53%* 1.00 0.78 1.66%*

volatility rel.

to output

σhp (It) /σhp (Yt) 2.90

(.12)

2.83* 0.54 0.40 2.71*

σhp (Ct) /σhp (Yt) .60

(.035)

.61* 0.96 1.35 .60*

σhp (Lt) /σhp (Yt) 1.13

(.061)

1.20* 0.88 0.28 1.24*

σhp (Dt) /σhp (Yt) 3.00

(.489)

2.84* 3.35 - 1.41*

correlation

with output

ρhp (It, Yt) .95

(.0087)

.88 0.86 0.27 0.96

ρhp (Ct, Yt) .94

(.0087)

.47 0.82 0.99 0.80

ρhp (Lt, Yt) .85

(.035)

.86 0.58 -0.11 0.84

ρhp (Dt, Yt) .56

(.080)

.24 0.30 - 0.45

inflation σhp (πt) .27%

(.047%)

.31%* 0.27% 0.27% .21%*

nominal rate σhp (it) .37%

(.046%)

.11%* 0.10% 0.12% .09%*

Quarterly U.S. data 1962Q1–2012Q4. Standard errors in parentheses. πt is quarterly CPI inflation. it isthe federal funds rate. All following variables are in logarithms. Lt is total non-farm payroll employment.Consumption Ct consists of services and non-durable private consumption. Investment It consists of privatenon-residential fixed investment and durable consumption. Output Yt is the sum of consumption andinvestment. Dividends Dt are four-quarter moving averages of S&P 500 dividends. σhp (·) is the standarddeviation and ρhp (·, ·) is the correlation coefficient of HP-filtered data (smoothing coefficient 1600). Momentsused in the SMM estimation are marked with an asterisk.

The next set of rows report the standard deviation of consumption, investment, hoursworked and dividends relative to output. Moving from Column (2) to (3), the removalof learning leads to a large drop in the relative volatility of investment and, to a lesserextent, hours worked. This is because the estimated learning model features a high levelof investment adjustment costs to match investment volatility. Without large asset pricefluctuations generated by learning, investment becomes too smooth, as does the marginalproduct of capital and hence labor demand. The volatility of dividends is well matched bythe learning model, which is a prerequisite for stock prices in the model and in the data tobe comparable.

Next, I report the contemporaneous correlations of consumption, investment, hours workedand dividends with output. The values in the model with learning are in line with the data,but the correlation of consumption and output is quite low in the model. The reason is

21

Table III: Asset price statistics in the data and across model specifications.

(1) (2) (3) (5)

moment data learning RE RE re-estimated

excess

volatility

σ(Ret,t+1

)32.56%

(2.44%)

33.91%* 5.31% 4.86%*

σ(Pt

Dt

)41.08%

(6.11%)

30.58% 4.07% 3.88%

return

predictability

ρ(Pt

Dt, Ret,t+4

)-.297

(.092)

-.417 -.07 -.07

ρ(Pt

Dt, Ret,t+20

)-.585

(.132)

-.754 -.05 -.05

ρ(Pt

Dt, Pt+4

Dt+4

).904

(.056)

.54 .19 .879

negative

skewness

skew(Ret,t+1

)-.897

(.154)

-.560 .001 .095

heavy tails kurt(Ret,t+1

)1.57

(.62)

1.89 .06 .01

equity

premium

E(Ret,t+1

)4.06%

(1.93%)

4.06% 4.06% 4.06%

risk-free rate E(Rft

)1.99%

(.61%)

1.99% 1.99% 1.99%

σ(Rft

)2.34%

(.29%)

3.46% 2.42% 2.65%

Quarterly U.S. data 1962Q1–2012Q4. Standard errors in parentheses. Dividends Dt are four-quarter movingaverages of S&P 500 dividends. The stock price index Pt is the S&P 500. Excess returns Ret are annualizedquarterly excess returns of the S&P 500 over 3-month Treasury yields. σ (·) is the standard deviation; σhp (·)is the standard deviation of HP-filtered data (smoothing coefficient 1600); ρ (·, ·) is the correlation coefficient;skew (·) is skewness;kurt (·) is excess kurtosis. Moments used in the SMM estimation are marked with anasterisk.

that large stock price movements move investment at the expense of consumption, so thatoutput and consumption become less correlated. It is possible to improve this particularmoment by including it in the SMM estimation, but at the expense of matching asset pricingmoments.

The last two rows report the volatility of inflation and the nominal interest rate. Inflationvolatility is roughly in line with the data, but the nominal interest rate is less volatile acrossall model specifications.

Next, I present asset price statistics in Table III. The statistics correspond to some well-known asset price puzzles. The learning model fits them remarkably well, despite the factthat stocks are being priced by risk-neutral investors. Starting with excess volatility inColumn (2), the model with learning reproduces the standard deviation of excess returns in

22

the data and also comes close to the volatility of the P/D ratio.16 By contrast, the modelwith rational expectations in Column (5) cannot produce a similar amount of volatility,despite the fact that return volatility is explicitly targeted by the estimation.

Stock returns also exhibit considerable predictability by the P/D ratio at business-cyclefrequency. The same is true in the model with learning. Predictability is not targeted bythe estimation, and in fact it is somewhat stronger than in the data, reflected in a persistenceof the P/D ratio that is somewhat lower than in the data. Again, the rational expectationsmodel is not able to produce sizable return predictability.

The learning model also produces a distribution of returns that is negatively skewed andheavy-tailed to a similar degree as in the data, underlining the non-linearities in the assetprice dynamics that arise from the learning mechanism. Again, these features are absent inthe model under rational expectations.

The last part of the table displays the average equity premium and risk-free rate. Themodel is able to match the average equity premium, but this is a mechanical exercise: Inthe model, bonds are priced by households while stocks are priced by firm owners. Dueto this market segmentation, it is possible to obtain the desired equity premium simply bymaking firm owners less patient than households (β < β), as is standard in the financialfrictions literature. The average risk-free rate is likewise calibrated, and its volatility is low,as in the data.

5.3 Impulse response functions

Impulse response functions reveal the amplification mechanism at play. Figure IV plotsthe impulse responses to a persistent productivity shock. Red solid lines represent thelearning equilibrium, blue dashed lines represent the rational expectations version, andblack thin lines represent the comparison model without financial frictions. The impulseresponses are averaged across states and therefore mask the non-linearities present withlearning, but they are nevertheless instructive. Looking at the first row of impulse responses,output rises persistently after the shock due to both the increased productivity and therelaxation of credit constraints from higher asset prices. The increase in output is largerunder rational expectations than under the frictionless comparison; this is the standardfinancial accelerator effect. When learning is introduced, the response to the shock isamplified further. This also translates into amplification of the responses of investment,consumption, and employment. The amplification is due to two channels: First, learningleads to higher stock prices. The increase in firms’ market value allows them to borrowmore and invest and produce more. Second, agents under learning are not aware of themean reversion in stock prices and predict the stock price boom to last for a long time.Consequently, they overestimate the availability of credit and therefore production in thefuture, leading to an aggregate demand effect that increases output today (see also 6.1). The

16The P/D ratio is not as high as in the data is because the P/D ratio correlates somewhat too stronglywith dividends. The Campbell-Cochrane decomposition rt ≈ κ0 + κ1pdt− pdt−1 + ∆dt implies that V [rt] ≈(1− 2κ1ρ (pdt, pdt−1) + κ2

1

)V [pdt]+V [∆dt]+2Cov (∆dt,∆pdt). Since the learning model matches both the

volatility of dividends and returns, and has an autocorrelation of pdt that is low relative to the data, thelow volatility of pdt results from the covariance term of price and dividend growth.

23

Figure IV: Impulse responses to a persistent productivity shock.

0 8 16 24 32 400

0.5

1

1.5Output

0 8 16 24 32 40-2

0

2

4Investment

0 8 16 24 32 400

0.2

0.4

0.6

0.8

1Consumption

0 8 16 24 32 40-1

-0.5

0

0.5

1Employment

0 8 16 24 32 40-10

-5

0

5

10

15

20Stock prices

0 8 16 24 32 40-5

0

5Exp. dividends

0 8 16 24 32 40-0.4

-0.3

-0.2

-0.1

0

0.1

0.2Inflation

0 8 16 24 32 40-0.1

-0.08

-0.06

-0.04

-0.02

0Nominal rate

learning

RE

fric.less

Impulse responses to a one-standard deviation innovation in εAt, averaged over 5,000 random shock pathswith a burn-in of 1,000 periods. Stock prices, dividends, output, investment, consumption, and employmentare in 100*log deviations. Stock returns and the nominal interest rate are in percentage point deviations.

rise in stock prices in the second row of Figure IV is large under learning and accompaniedby an initial spike in dividend payments, although dividends subsequently fall below theircounterpart under rational expectations. The nominal interest rate falls less under learningas the monetary authority reacts to the inflationary pressures stemming from the relaxationin credit constraints.

Figure V plots the response to a temporary reduction in the nominal interest rate. Again, allmacroeconomic aggregates rise substantially more under learning than under both rationalexpectations and the frictionless benchmark. The monetary stimulus increases stock pricesand thus relaxes credit constraints. The consequent increase in aggregate demand raisesinflationary pressure, so that the systematic reaction of the interest rate rule raises theinterest rate sharply again after the shock.

5.4 Comparison to survey data on expectations

The rational expectations hypothesis asserts that “outcomes do not differ systematically[...] from what people expect them to be” (Sargent, 2008). Put differently, a forecast errorshould not be systematically predictable by information available at the time of the forecast.The absence of predictability is almost always rejected in the data.

Similarly, agents in the model under learning also make systematic, predictable forecasterrors. This holds not only for stock prices but also other endogenous model variables,

24

Figure V: Impulse responses to a monetary shock.

0 8 16 24 32 40-0.05

0

0.05

0.1

0.15Output

0 8 16 24 32 40-0.1

0

0.1

0.2

0.3Investment

0 8 16 24 32 400

0.02

0.04

0.06Consumption

0 8 16 24 32 40-0.1

0

0.1

0.2

0.3Employment

0 8 16 24 32 40-0.5

0

0.5

1

1.5Stock prices

0 8 16 24 32 40-0.5

0

0.5

1Exp. dividends

0 8 16 24 32 40-0.01

0

0.01

0.02

0.03Inflation

0 8 16 24 32 40-10

-5

0

5×10-3Nominal rate

learning

RE

fric.less

Impulse responses to a innovation in εmt, averaged over 5,000 random shock paths with a burn-in of 1,000periods. The size of the innovation is chosen to produce a 10 basis point fall in the equilibrium nominalrate. Stock prices, dividends, output, investment, consumption and employment are in 100*log deviations.Stock returns and the nominal interest rate are in percentage point deviations.

despite the fact that, conditional on stock prices, agents’ beliefs are model-consistent. Asystematic mistake in predicting stock prices will still spill over into a corresponding mistakein predicting the tightness of credit constraints, and hence investment, output, and so forth.Owing to the internal consistency of beliefs, I can compute well-defined forecast errors madeby agents in the model at any horizon and for any model variable.

Figure VI repeats the scatter plot of the introduction, contrasting expected and realized oneyear-ahead returns in a model simulation. The same pattern as in the data emerges: Whenthe P/D ratio is high, return expectations are most optimistic. In the learning model,this has a causal interpretation: High return expectations drive up stock prices. At thesame time, realized future returns are, on average, low when the P/D ratio is high. This isbecause the P/D ratio is mean-reverting (which agents do not realize, instead extrapolatingpast price growth into the future): At the peak of investor optimism, realized price growthis already reversing and expectations are due to be revised downward, pushing down pricestoward their long-run mean.

Table IV describes tests using the Federal Reserve’s Survey of Professional Forecasters(SPF) as well as the CFO survey data and compares the statistics to those obtained fromsimulated model data. Each entry corresponds to a correlation of the error of the meansurvey forecast with a variable that is observable by respondents at the time of the survey.Under the null of rational expectations, all correlation coefficients are zero.

25

Figure VI: Return expectations and expected returns in a model simulation.

3.5

44.5

51−

year

retu

rn fore

cast

2 2.5 3 3.5 4 4.5log P/D ratio

−100

−50

050

100

1−

year

realis

ed r

etu

rn

2 2.5 3 3.5 4 4.5log P/D ratio

Expected and realized nominal returns along a simulated path of model with learning. Simulation length200 periods. Theoretical correlation coefficient for subjective expected returns ρ = .47, for future realizedreturns ρ = −.38.

Columns (1) and (2) contain the correlations of future forecast errors with the P/D ratio.When stock prices are high, people systematically over-predict future stock returns andeconomic activity. For stock returns the model reproduces this pattern, as was alreadyshown in the scatter plot above. But the model also reproduces this pattern for output,investment, consumption and unemployment (proxied for by one minus employment in themodel). Where the model fails to replicate the data is for the nominal variables at thebottom of the table: When stock prices are high, forecasters under-predict future inflationand over-predict future nominal interest rates, while in the model the opposite result obtains.This discrepancy arises essentially because stock market booms are inflationary in the model,which seems not to be true in the data (Christiano et al., 2010).

Column (3) repeats the exercise for the growth rate of the P/D ratio, and here the modelreplicates the patterns in the data across all variables. Stock price growth positively predictsforecast errors, suggesting that agents’ expectations adjust slowly: They under-predict anexpansion in its beginning but then overshoot and over-predict it when it is about to end.In the model (Column 4), this pattern also emerges because expectations about asset pricesadjust slowly.

Column (5) reports the results of a particular test of rational expectations devised byCoibion and Gorodnichenko (2015). Since for any variable xt, the SPF asks for forecasts atone- through four-quarter horizons, it is possible to construct a measure of agents’ revision ofthe change in xt as Et [xt+3 − xt]− Et−1 [xt+3 − xt]. Forecast errors are positively predictedby this revision measure across all variables. Coibion and Gorodnichenko take this asevidence for sticky information models in which information sets are gradually updatedover time. But it is also consistent with the learning model: The correlation coefficients inColumn (6) are very similar to those in the data for aggregate activity, inflation and interest

26

Table IV: Forecast errors under learning and in the data.

(1) (2) (3) (4) (5) (6)

logPDt ∆ logPDt forecast revision

forecast variable data model data model data model

Rstockt,t+4 -.44 -.42 .06 .48 - -.25

(-3.42) (.41)

Yt,t+3 -.21 -.12 .22 .37 .29 .37

(-1.78) (2.42) (3.83)

It,t+3 -.20 -.41 .25 .44 .31 .48

(-1.74) (2.88) (3.79)

Ct,t+3 -.19 -.04 .21 .01 .23 .01

(-1.85) (2.37) (2.67)

ut,t+3 .05 .15 -.27 -.47 .43 .59

(.12) (-3.07) (6.07)

πt,t+3 .37*** -.09 .04 .31 .04 .10

(4.36) (0.43) (0.43)

it+3 − it -.22** 0.12 .05 .10 .24*** .38

(-2.48) (0.80) (2.66)

Correlation coefficients for mean forecast errors on one year-ahead nominal stock returns (Graham-Harveysurvey) and three quarters-ahead real output growth, investment growth, consumption growth, unemploy-ment rate, CPI inflation and 3-month treasury bill (SPF). t-statistics in parentheses. Regressors: Column(1) is the S&P 500 P/D ratio and Column (2) is its first difference. Column (3) is the forecast revision,as in Coibion and Gorodnichenko (2015). Data from Graham-Harvey covers 2000Q3–2012Q4. Data for theSPF covers 1981Q1–2012Q4. For the model, correlations are computed using a simulation of length 50,000,where subjective forecasts are computed using a second-order approximation to the subjective belief systemon a path in which no more future shocks occur, starting at the current state in each period. Unemploymentin the model is taken to be ut = 1− Lt. Stock returns in the model Rstockt,t+4 are quarterly nominal aggregatemarket returns.

rates.17

6 Sensitivity checks

6.1 Does learning matter?

In this model, large swings in asset prices lead to large swings in real activity throughtheir effect on credit constraints. But is learning necessary for this story at all? Maybe allthat matters for amplification is asset price volatility. In this section, I show that learninghas an effect on amplification over and above its effect on asset prices. Extrapolativeexpectations about asset prices also cause procyclical movements in aggregate demand,leading to additional amplification in the presence of nominal rigidities.

17The model predicts a negative correlation of forecast errors on stock returns with their forecast revisions.The CFO survey does not allow for the construction of the corresponding statistic in the data.

27

Figure VII: Does learning matter?

0 10 20 30 40−1

0

1

2

3

4Stock prices P

t / V

t

0 10 20 30 40−0.2

0

0.2

0.4

0.6

0.8Output Y

t

0 10 20 30 40−0.5

0

0.5

1

1.5

2Investment I

t

0 10 20 30 400

0.2

0.4

0.6

0.8Consumption C

t

counterfactual

learning

Solid red line: Impulse response to a one standard deviation positive productivity shock under learning.Black dash dotted line: Impulse response to a hypothetical rational expectations model with stock pricedynamics identical to those under learning (see text). The impulse responses in the figure are producedusing a first-order approximation to the model equations.

I replace the stock market value Pt in the borrowing constraint (20) with an exogenousprocess Vt that has the same law of motion as the stock price under learning. More precisely,I fit an ARMA(10,5) process for Vt such that its impulse responses are as close as possible tothose of Pt under learning (the exogenous shock in the ARMA process are the productivityand monetary shocks). I then solve this model, but with rational expectations. If learningonly matters because it affects stock price dynamics, then this hypothetical model shouldhave exactly identical dynamics to the model under learning.18

Figure VII shows that this is not the case. The ARMA process fits stock prices well: Theimpulse response of Pt under learning and Vt in the counterfactual experiment are indistin-guishable. But after a positive productivity shock, output, investment, and consumptionrise more under learning, even though the counterfactual model has the same stock pricedynamics by construction. The reason is that expectations of future asset prices matterbeyond their direct impact on current prices. Under learning, agents do not fully internal-ize mean reversion in stock prices and therefore predict that credit constraints are loosefor longer than they turn out to be. This leads to a wealth effect on households that in-creases their consumption, raising aggregate demand, and it leads to higher future expectedprices of capital goods EtQt+1, which enters the liquidation value of firms and hence relaxesborrowing constraints, even if stock prices are the same as under rational expectations.

18For this exercise I only compute a first-order approximation to the model equations.

28

These effects are powerful enough to create significant endogenous amplification throughthe departure of subjective beliefs from rational expectations.

6.2 Nominal rigidities

The model includes nominal rigidities. These allow for an investigation of the role of mone-tary policy in the next section, but they are also crucial for the quantitative fit of the model.Without price and wage rigidities, it is difficult to obtain comovement of consumption, in-vestment and employment in response to changes in investment demand that are unrelatedto TFP. This point has been made previously by Kobayashi and Nutahara (2010) in thecontext of news shocks and Ajello (2016) in the context of credit shocks, but applies justas well to changes in stock prices in this model.

An increase in stock prices relaxes credit constraints and allows firms to invest more andpay out higher dividends. But general equilibrium effects can dampen or even overturnthese effects, as has already been discussed in the simplified model of Section 4. Higherdemand for investment implies higher labor demand and therefore an increase in the realwage. It also tends to crowd out consumption today, implying higher real interest ratesthrough the consumption Euler equation. Both channels dampen the rise in firm profitsand investment.

Wage rigidities will counteract the dampening effects of real wage responses to changesin the tightness of credit constraints, allowing for a larger response of employment anddividends. Price rigidities, together with a relatively loose monetary policy rule, render theprice of intermediates qt pro-cyclical, implying larger dividend responses. They also dampenreal interest rate movements in response to changes in investment demand, mirrored byless crowding out of consumption. Together, nominal rigidities help the comovement ofinvestment, consumption and employment, and they also amplify the response of dividendsto changes in borrowing constraints, reinforcing the learning dynamics in the stock market.

To illustrate this point, I re-compute impulse responses of the model with learning, butwithout nominal rigidities (setting κ = κw = 0). I also reduce the size of investmentadjustment costs to ψ = 0.05. With the high degree of adjustment costs in the baselineversion, the model would include an explosive two-period oscillation and reducing ψ ensuresstability. Lowering adjustment costs also gives the real version a better chance at deliveringstrong impulse responses. Even then, the nominal version delivers far greater amplification.Figure VIII plots impulse responses to a positive productivity shock for both the nominaland real version of the model. Owing to lower adjustment costs, the initial response ofinvestment is expectedly stronger in the real version. However, the real wage wt rises moreand dividends rise less. This considerably dampens the learning dynamics and also mutesthe response of stock prices.

6.3 Borrowing constraint parameters

I now discuss the sensitivity of the results to the two main parameters affecting the borrow-ing constraint (20): The probability x that a firm can be sold off at market price after filing

29

Figure VIII: Role of nominal rigidities.

0 8 16 24 32 400

0.5

1

1.5Output

0 8 16 24 32 40−1

0

1

2

3Investment

0 8 16 24 32 400

0.2

0.4

0.6

0.8Consumption

0 8 16 24 32 40−5

0

5

10

15Stock prices

0 8 16 24 32 40−2

0

2

4Exp. dividends

0 8 16 24 32 400

0.2

0.4

0.6

0.8Real wage

learningκ=κ

w=0

Solid red line: Impulse response to a one-standard deviation positive productivity shock for the model withlearning and price and wage rigidities (“nominal” baseline). Black dash-dotted line: Impulse response to aproductivity shock for the model with learning but without nominal rigidities, re-estimated as in Section5.1.2 to fit the data (“real” comparison). The size of the shock shown is the same as in the nominal model.

for bankruptcy, governing the dependency of the constraint on stock prices; and the frac-tion of assets ξ preserved in bankruptcy, governing the overall tightness of the constraint.Figure IX plots the standard deviation of output and stock prices as a function of these twoparameters, respectively.

Panel (a) shows the role of the x parameter, which crucially affects the amplification mech-anism. The value x = 0 is a special case. At this point, stock prices do not enter theborrowing constraint and serve no role for allocations in the economy. Allocations underlearning and rational expectations coincide perfectly, even though stock price dynamics arestill amplified under learning. As x increases, the higher volatility of stock prices underlearning translates into higher volatility in real activity as well. Since swings in real activ-ity feed back into asset prices through their effect on dividends, the amplification becomesvery strong for high values of x until the dynamics become explosive. Beyond a value of xof about 0.2, no stable learning equilibrium exists. By contrast, the rational expectationsequilibrium barely depends on the parameter x.

Panel (b) shows the role of the ξ parameter. Amplification is hump-shaped with respect toξ. At ξ = 0, no collateral is pledgeable and firms have to finance their capital stock entirelyout of equity. In this case, fluctuations in stock prices again do not impact the economyand allocations coincide under learning and rational expectations; there is no amplificationfrom learning. However, as pledgeability increases to its maximum value (beyond which asteady state with permanently binding borrowing constraint does not exist), amplificationalso disappears. This mirrors the analysis of the simplified model in Section 4.

30

Figure IX: Sensitivity to borrowing constraint parameters.

0 0.05 0.1 0.150

1

2

3

sd. %

Output

0 0.05 0.1 0.150

10

20

30

sd. %

Stock prices

x

(a) Market value dependency x.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

1

2

sd. %

Output

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

5

10

15

20

sd. %

Stock prices

ξ

(b) Recovery rate ξ.Simulated model data HP-filtered with smoothing parameter 1600, sample length 50,000 periods. Thedashed black lines indicate parameter value in the estimated learning model.

7 Implications for monetary policy

Finally, I discuss how the interest rate rule followed by the monetary authority affects theequilibrium under learning, and whether it is beneficial to “lean against the wind”, i.e. toraise interest rates in response to asset price increases.

Consider extending the interest rate rule (10) as follows:

it = ρiit−1 + (1− ρi) (1/β + φππt + φY ∆ log Yt + φP∆ logPt) (37)

In addition to raising interest rates when inflation is above its target level (taken to bezero), the monetary authority can raise interest rates φ∆Y percentage points when real GDPgrowth increases one percentage point and φ∆P percentage points when stock price growthincreases one percentage point.19 I consider several values for the parameters φπ, φ∆Y , φ∆P .Table V summarizes how they affect the model under learning. In addition to the standarddeviations of output, stock prices, inflation and nominal interest rates, the table also reportsa measure of amplification by the learning dynamics and a measure of the welfare cost ofbusiness cycles.20

19I deliberately exclude levels or gap measures of output or asset prices from the interest rate rule. Doingso would imply that the monetary authority has more knowledge than the private sector under learning, asthe perceived equilibrium level of asset prices and the output gap depends on agents’ subjective beliefs, andat the same time communicates that knowledge through the rule.

20The amplification measure is the ratio of the standard deviation of output under learning over thatunder rational expectations at the same parameter values. The welfare cost of business cycles χ is definedas the fraction of steady-state consumption the household would need to give up in order to have its periodutility at the same level as the average stochastic period utility, in a steady state in which consumption andlabor are constant and equal to their average stochastic level, and price and wage dispersion is nil:

u(

(1− χ)E[Ct],E[Lt])

= E [u (Ct, Lt)] .

I simulate a model time series for (Ct, Lt, πt, πwt ) of 10,000 periods using the second-order approximation

31

Table V: Alternative monetary policy rules.

(1) (2) (3) (4) (5) (6)

baseline alternative rules optimized rules

φπ 1.50 3.00 1.50 1.50 1.42 1.27

φY 0.50 .61 .76

φP 0.10 .06

σ (Y ) 3.11% 3.93% 2.74% 2.57% 2.75% 2.26%

σ (P ) 26.11% 31.76% 19.00% 16.40% 17.87% 12.34%

σ (π) 0.32% 0.20% 0.34% 0.31% 0.36% 0.34%

σ (i) 0.18% 0.19% 0.15% 0.18% 0.18% 0.14%

welfare cost χ 0.0367% 0.0433% 0.0301% 0.0293% .0296% .0256%

Standard deviations of output, stock prices, inflation, and interest rates (unfiltered) under learning in percent.See Footnote 20 for the definition of the welfare cost χ. The interest rate smoothing coefficient is kept atρi = 0.85 for all rules considered.

Column (1) shows the baseline calibration under learning. The welfare cost of businesscycles is 0.0367 percent of average consumption. In Column (2), the coefficient on inflationφπ is doubled. This rule achieves a marked reduction in the volatility of inflation, but atthe cost of higher output volatility. Column (3) augments the baseline rule by a reactionto output growth. This rule reduces the volatility of output while keeping the volatilityof inflation unchanged. Column (4) considers a reaction of interest rates to stock pricegrowth, where the target rate is increased by 10 basis points when growth of stock pricesincreases by 1 percentage point (the immediate change in the nominal rate is only 1.5 basispoints due to interest rate smoothing). Such a reaction is quite effective in stabilizing theeconomy under learning. The volatility of output, stock prices and inflation is lower than inall previous rules. The welfare cost of business cycles is also lower than in Columns (1)–(3).

Because the previous rules are somewhat arbitrary choices, I also compute the optimalcoefficient values that minimize the welfare cost of business cycles.21 Column (5) displaysoptimized coefficients φπ and φY without a reaction to stock prices. The coefficients turn outto be quite close to the Taylor-type rule in Column (3). The resulting rule reduces welfarecosts to 0.0296 percent. Column (6) additionally allows for a reaction to stock prices prices.The optimal coefficient on stock prices is computed as φP = 0.06 while the coefficient oninflation is reduced. Allowing for a reaction to stock prices reduces the welfare cost furtherto 0.0256 percent.22 These results suggests that a positive interest rate reaction to stockprice growth in this model provides stabilizing effects that cannot be achieved by reacting to

method above, and I compute series for Ct and Lt using the exact formulae given in the appendix. I thenevaluate period utility using its exact formula as well to calculate the welfare loss. The expectation is thencomputed using averages over time.

21Note that this criterion is paternalistic because it minimizes the welfare cost of business cycles over time,rather than the private sector’s subjectively expected welfare cost.

22It is worth noting that despite the reaction stock prices in Column (6), the volatility of the nominal

32

output and inflation alone. Intuitively, raising rates when asset prices are rising (and vice-versa) acts to stabilize expectations in financial markets. In the model, this stabilizationworks mainly through changes in firms’ borrowing costs and dividend payouts, offsettingthe self-amplifying learning dynamics in the stock market. By contrast, under rationalexpectations this benefit of stabilizing financial market expectations is absent. Indeed, withthe rational expectations version of the model I find that if anything, the coefficient φP isnegative in the welfare-optimal rule (see the appendix). This is consistent with previousresults in the literature that rely on rational expectations (e.g. Faia and Monacelli, 2007;Gali, 2014).

8 Conclusion

In this paper, I have analyzed the implications of a learning-based asset pricing theory ina business cycle model with financial frictions. When firms’ borrowing constraints dependon their market value, learning in the stock market interacts with credit frictions to forma two-sided feedback loop between stock prices and firm profits that amplifies the learningdynamics. At the same time, it makes the financial accelerator mechanism more powerful,amplifying both supply and demand shocks. The model jointly matches standard businesscycle and asset pricing moments.

Despite the fact that beliefs are close to rational expectations, agents’ forecast errors onstock prices still spill over into their other forecasts. The resulting forecast error predictabil-ity closely match survey data on expectations on a range of variables.

An important innovation in developing the model was to introduce a belief system thatcombines learning about stock prices with a high degree of rationality and internal consis-tency. Beliefs are restricted in a way such that forecast errors conditional on future pricesand fundamentals are zero. This differs from most of the existing adaptive learning liter-ature where every forward-looking equation is parameterized separately. The method canbe widely applied in other models of the business cycle.

An examination of the sensitivity of the amplification mechanism to the monetary policy rulerevealed that a reaction of interest rates to stock price growth is beneficial under learning.This is because such a reaction effectively stabilizes expectations in financial markets. Thesame is not true in a rational expectations framework, illustrating that the choice of an assetprice theory can have important normative implications. Further research could examine indetail the policy implications of learning-based asset pricing and establish whether a policyof “leaning against the wind” is desirable in more general settings with asset price learning.

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36

Appendix

A Details on the model

A.1 Setup of adjustment costs and nominal rigidities

In the full model, final good producers (indexed by i ∈ [0, 1]) transform a homogeneous in-termediate good into differentiated final consumption goods using a one-for-one technology.The intermediate good trades in a competitive market at the real price qt (expressed inunits of the composite final good). Each retailer enjoys market power in her output market,and sets a nominal price pit for its production. A standard price adjustment friction a laCalvo means that a retailer cannot adjust her price with probability κ. Hence, the retailersolves the following optimization:

maxPit

∞∑s=0

(s∏

τ=1

κΛt+τ

)((1 + τ) pit − qt+spt+s)Yit+s

s.t. Yit+s =

(pitpt+s

)−σYt+s,

where Yt is aggregate demand for the composite final good. Since all retailers that canre-optimize at t are identical, they all choose the same price pit = p∗t . The derivation of thenon-linear aggregate law of motion for the retail sector is standard and the final equationsare:

p∗tpt

=1

1 + τ

σ

σ − 1

Γ1t

Γ2t

Γ1t = qt + κEPt Λt+1Yt+1

Ytπσt+1

Γ2t = 1 + κEPt Λt+1Yt+1

Ytπσ−1t+1 .

I assume that the government sets subsidies such that τ = 1/(σ−1) so that the steady-statemarkup over marginal cost is zero. Inflation πt = pt/pt−1 and the reset price are linkedthrough the price aggregation equation which can be written as

1 = (1− κ)

(p∗tpt

)1−σ+ κπσ−1

t

and the Tak-Yun distortion term is

∆t = (1− κ)

(Γ1t

Γ2t

)−σ+ κπσt ∆t−1.

This term ∆t ≥ 1 is the wedge due to price distortions between the amount of intermediategoods produced and the amount of the final good consumed. The amount of final goods

37

available for consumption and investment is Yt = Yt/∆t. Similarly, one can define Ct =Ct/∆t as the level of consumption the household could obtain if price distortions were zero.

Similarly to retailers, labor agencies transform the homogeneous household labor input intodifferentiated labor goods at the nominal price wtpt and sell them to intermediate firms atthe price wht, which cannot be adjusted with probability κw. Labor agency h solves thefollowing optimization:

maxwht

EPt∞∑s=0

(s∏

τ=1

κwΛt+τ

)((1 + τw)wht − wt+spt+s)Lht+s

s.t. Lht =

(whtwt

)−σwLt.

Since all labor agencies that can re-optimize at t are identical, they all choose the sameprice wht = w∗t . The first-order conditions are analogous to those for retailers. Again, Iassume that the government sets taxes such that τ = 1/(σw − 1) so that the steady-statemarkup over marginal cost is zero. Wage inflation πwt and the Tak-Yun distortion ∆wt aredefined analogously to final good producers.

Capital good producers operate competitively in input and output markets, producing newcapital goods using old final consumption goods. For the latter, they have a CES aggregatorjust like households. Their maximization program is entirely intratemporal:

maxIt

QtIt −(It +

ψ

2

(ItIt−1

− 1

)2)

In particular, they take past investment levels It−1 as given when choosing current invest-ment output. Their first-order condition defines the price for capital goods.

All of the profits made by the firms described above accrue to households. Similarly, allsubsidies by the government are financed by lump-sum taxes on households. The marketclearing conditions are summarized below. Supply stands on the left-hand side; demand onthe right-hand side.

Yt =

∫ 1

0Yjtdj =

∫ 1

0Yitdi

Yt =Yt∆t

= Ct + It +−ψ2

(ItIt−1

− 1

)2

+ Cet

Lt =

∫ 1

0Lhtdh

Lt =Lt

∆wt=

∫ 1

0Ljtdj

Kt =

∫ 1

0Kjtdj = (1− δ)Kt−1 + It

1 = Sjt, j ∈ [0, 1]

0 = Bgt

38

A.2 Properties of the rational expectations equilibrium

I consider a rational expectations equilibrium with the following properties that hold in aneighborhood of the non-stochastic steady-state.

1. The expected net discounted return on capital is strictly positive for both firm ownersand households: βEtRkt+1 > 1 and EtΛt+1R

kt+1 > 1.

2. At any time t, the stock market valuation Pjt of a firm j is linear in its net worth,with a slope that is strictly greater than one.

3. All firms choose the same capital-labor ratio Kjt/Ljt .

4. All firms can be aggregated. Aggregate debt, capital and net worth are sufficient todescribe the intermediate goods sector.

5. Borrowers never default on the equilibrium path and borrow at the risk-free rate, andthe lender only accepts debt payments up to a certain limit.

6. If the firm defaults and the lender seizes the firm, it always prefers restructuring toliquidation.

7. The firm always exhausts the borrowing limit.

Here, I derive restrictions on the parameters for existence of such an equilibrium. I firsttake the first two properties as given and show under which conditions the remaining oneshold, and then derive conditions for the first two properties be verified.

Value functions

An operating firm j enters the period with a predetermined stock of capital and debt. It isconvenient to decompose its value function into two stages. The first stage is given by:

Υ1 (Kjt−1, Bjt−1, st) = maxNjt,Ljt,Djt,Yjt

γNjt + (1− γ) (Djt + Υ2 (Njt −Djt, st))

s.t. Njt = qtYjt − wtLjt + (1− δ)QtKjt−1 −Rt−1Bjt

Yjt = Kαjt−1 (AtLjt)

1−α

Djt = ζ (Njt −QtKjt−1 +Bjt−1)

The aggregate state of the economy is denoted by st. In what follows, I will suppress thetime and firm indices for the sake of notation.

After production, the firm exits with probability γ and pays out all net worth as dividends.The second stage of the value function consists in choosing debt and capital levels as wellas a strategy in the default game:

Υ2

(N , s

)= max

K,B,strategy in default gameβE[Υ1

(K,B, s′

), no default

]+ βE

[Υ1

(K,B∗, s′

),debt renegotiated

]+ βE [0, lender seizes firm]

39

s.t. QK = N +B

Note that, since net worth N is non-negative around the steady state, the firm’s debt Bcannot exceed its capital stock K.

In the first stage, the first order condition with respect to L equalizes the wage with themarginal revenue. Since there is no firm heterogeneity apart from capital K and debt Band the production function has constant returns to scale, this already implies Property3 that all firms choose the same capital-labor ratio. Hence the internal rate of return oncapital is common across firms:

Rk = αq

((1− α)

qA

w

) 1−αα

+ (1− δ)Q

Taking Property 2 as given for now, Υ2 is a linear function

Υ2

(N , s

)= υsN

with slope νs > 1. Then Υ1 is homogeneous of degree one, and at the steady state (andtherefore in a neighborhood):

Υ1 (K,B, s) = N + (1− γ) (D −N + Υ2 (N −D, s))= N + (1− γ) (υs − 1) ((1− ζ)N + ζ (QK −B))

> N = RkK −RB.

Limited commitment problem

The second stage involves solving for the subgame-perfect equilibrium of the default gamebetween borrower and lender. Pairings are anonymous, so repeated interactions are ruledout. Also, only the size B and the interest rate R of the loan can be contracted (inequilibrium R = R but this is to be established first). The game is played sequentially:

1. The firm (F) proposes a borrowing contract(B, R

).

2. The lender (L) can accept or reject the contract.

• A rejection corresponds to setting the contract(B, R

)= (0, 0).

Payoff for L: 0. Payoff for F: βE[Υ1

(N , 0, s′

)].

3. F acquires capital and can then choose to default or not.

• If F does not default, it has to repay in the next period.

Payoff for L: EΛRB −B. Payoff for F: βE[Υ1

(K, RRB, s

′)]

.

40

4. If F defaults, the debt needs to be renegotiated. F makes an offer for a new debt levelB∗.23

5. L can accept or reject the offer.

• If L accepts, the new debt level replaces the old one.

Payoff for L: EΛRB∗ −B. Payoff for F: βE[Υ1

(K, RRB

∗, s′)]

.

6. If L rejects, then she seizes the firm. A fraction 1 − ξ of the firm’s capital is lost inthe process. Nature decides randomly whether the firm can be “restructured.”

• If the firm cannot be restructured, or it can but the lender chooses not to do so,then the lender has to liquidate the firm.Payoff for L: E [ΛQ′] ξK −B. Payoff for F: 0.

• If the firm can be restructured and the lender chooses to do so, she retains adebt claim of present value ξB and sells the residual equity claim in the firm toanother investor.Payoff for L: ξB + βE [Υ1 (ξK, ξB, s′)]−B. Payoff for F: 0.

Backward induction leads to the (unique) subgame-perfect equilibrium of this game. Startwith the possibility of restructuring. L prefers this to liquidation if

ξB + βE[Υ1

(ξK, ξB, s′

)]≥ EΛξQ′K.

This holds true at the steady state, as we have βRk > 1 (Property 1), Λ = β = 1/R, β ≤ βand Q′ = 1:

βΥ1 (ξK, ξB, s) > β(RkξK −RξB

)> ξ

(K − βRB

)> ξ (βK −B) .

Since the inequality is strict, it holds around the steady-state as well. This establishesProperty 6.

Next, L will accept an offer B∗ if it gives her a better expected payoff (lenders can diversifyamong borrowers so that their discount factor is invariant to the outcome of the game).The probability of restructuring is given by x. The condition for accepting B∗ is thereforethat

E [Λ] RB∗ ≥ x(ξB + βE

[Υ1

(ξK, ξB, s′

)])+ (1− x)E

[ΛQ′

]ξK.

Now turn to the firm F. Among the set of offers B∗ that are accepted by L, the firm willprefer the lowest one which satisfies the above restriction with equality. This follows fromΥ1 being a decreasing function of debt. This lowest offer will be made if it leads to a higher

23That the interest rate on the repayment is fixed is without loss of generality.

41

payoff than expropriation: βE[Υ1

(K, RRB, s

′)]≥ 0. Otherwise, F offers zero and L seizes

the firm.

Going one more step backwards, F has to decide whether to declare default or not. Itis preferable to do so if B∗ can be set smaller than B or if expropriation is better than

repaying, βE[Υ1

(K, RRB, s

′)]≥ 0.

What is the set of contracts that L accepts in the first place? From the perspective of L,there are two types of contracts: those that will not be defaulted on and those that will.If F does not default (B∗ ≥ B), L will accept the contract simply if it pays at least therisk-free rate, R ≥ R. If F does default (B∗ < B), then L accepts if the expected discountedrecovery value exceeds the size of the loan—i.e., E [Λ] RB∗ ≥ B.

Finally, let us consider the contract offer. F can offer a contract(B, R

)on which it will not

default. In this case, it is optimal to offer just the risk-free rate R = R. Also note that thepayoff from this strategy is strictly positive for any non-negative B that does not triggerdefault, since at the steady-state Rk > 1/β > 1/β > R and therefore

βE[Υ1

(K,B, s′

)]> βE

[RkK −RB

]= βE

[RkN +

(Rk −R

)B]

> 0.

F therefore prefers this contract to one that leads to default with expropriation. The payoffis increasing in the size of the loan B, since

∂

∂BE

[Υ1

(N +B

Q,B, s′

)]

=βE[Rk

Q−R+ (1− γ) (υs′ − 1)

((1− ζ)

(Rk

Q−R

)+ ζ

(Q′

Q− 1

))]>0.

Therefore, of all values for B that do not lead to default, F will want to choose the largestone, defined as:

B = max

B∣∣∣∣∣∣ x(ξB + βE

[Υ1

(ξ(N+BQ , s′

), ξB, s′

)])+ (1− x)E [ΛQ′] ξ N+B

Q −B≥ 0

.

In order for the borrowing constraint to be binding, it must be finite. Since the set abovecontains B = 0, this amounts to the condition that

x

(1 + β

∂

∂BE

[Υ1

(N +B

Q,B, s′

)])+ (1− x)E

[ΛQ′

]<

1

ξ(38)

which is satisfied for ξ small enough. Because Υ1 is homogenous of degree one, the borrowinglimit is linear in N and can be written as B = υB,sN .

42

F could also offer a contract(B, R

)that only leads to a default with debt renegotiation.

The optimal contract of this type is the solution to the following problem:

maxR,B,B∗

βE

[Υ1

(N +B,

RB∗

R, s′

)]

s.t.RB∗

R≥ B

RB∗

R= x

(ξB + βE

[Υ1

(ξN +B

Q, ξB, s′

)])

+ (1− x)E[ΛQ′

]ξN +B

Q

It is clear that the value of this problem is solved by setting R = R and B = B∗ = B, whichamounts to not defaulting. This establishes Properties 5 and 7.

Linearity of firm value

Since firms do not default and exhaust the borrowing limit B, the second-stage firm valueis

Υ2

(N)

= βE

[Υ1

(N + B

Q, B, s′

)]

= βE[Υ1

(1 + υB,s

QN, υB,sN , s

′)]

= βE[Υ1

(1 + υB,s

Q, υB,s, s

′)]

N .

We have therefore verified the linearity of Υ2. To establish Property 2, it remains to showthat the slope of Υ2 is greater than one. At the steady state:

υs = βΥ1 (1 + υB,s, υB,s, s)

= β

Rk + υB,s

(Rk −R

)︸ ︷︷ ︸

>0

1 + (1− γ)

>−1︷ ︸︸ ︷(υs − 1) (1− ζ)

︸ ︷︷ ︸

>0

+ (1− γ) (υs − 1) ζ

> βRk (1 + (1− γ) (υs − 1) (1− ζ)) + (1− γ) (υs − 1) ζ

> 1 + (1− γ) (υs − 1)

> 1.

Finally, the aggregated law of motion for capital and net worth needs to be established(Property 4). Denoting again by Γt ⊂ [0, 1] the indices of firms that are alive at the end of

43

period t, we have

QtKt = Qt

∫ 1

0Kjtdj =

∫j∈Γt

(Njt − ζEjt +Bjt) dj

= (1− γ) (Nt − ζEt) +Bt

Nt =

∫ 1

0Njtdj = RktKt−1 −Rt−1Bt−1

Bt =

∫ 1

0Bjtdj = xξ (Bt + Pt) + (1− x) ξEtΛt+1Qt+1Kt.

Return on capital

We can now establish a condition under which βRk > 1 holds (Property 1). From theaggregate equations above, and the definition of earnings E = N −QK +B, it follows thatin steady state:

Rk =RB

K+

1− ζ (1− γ)

(1− ζ) (1− γ)

(1− B

K

). (39)

Rearranging the above expression, one obtains that βRk > 1 holds at the steady state ifand only if:

γ > 1− β

ζβ + (1− ζ)1−βRB

K

1−BK

. (40)

A sufficient condition is that γ >(1−ζ)(1−β)1−ζ(1−β)

.

A.3 Conditions to rule out multiple equilibria

Collateral constraints often give rise to multiple equilibria due to their feedback effects: Lowasset prices reduce borrowing constraints and activity, which in depress asset prices and soon. This multiplicity appears even in the very early literature (Kiyotaki and Moore, 1997).More recently, Miao and Wang (2011) have shown that when firm borrowing constraintsdepend on equity value, multiple steady states are possible. In this section, I give conditionsunder which this type of multiplicity does not arise. These conditions are satisfied for theparameter values at which the model is simulated.

Miao and Wang look for an equilibrium in which firm value Υ2

(N , s

)is not linear but

affine in net worth N . Even a firm with zero net worth has positive value. This can be anequilibrium: The positive equity value enables the firm to borrow, acquire capital and paydividends from the returns; those expected dividends can justify the positive equity value.

Suppose that Υ2

(N , s

)= υsN + ϑs with ϑs ≥ 0 and υs > 1, and that βEtRkt+1 > 1.

Then the proof for the existence of an equilibrium satisfying properties 3–7 above still goes

44

through under the same conditions (38) and (40). The equation determining the coefficientϑs is:

υsN + ϑs = βE[κN,s′N + κB,s′B + (1− γ)ϑs′

]where

κN,s′ = (1 + (1− γ) (υs′ − 1) (1− ζ))Rk

Q− (1− γ) (υs′ − 1) ζ

Q′

Q

κB,s′ = (1 + (1− γ) (υs′ − 1) (1− ζ))

(Rk

Q−R

)− (1− γ) (υs′ − 1) ζ

(Q′

Q− 1

).

The borrowing limit B depends itself on equity value and therefore on the coefficients υsand ϑs:

B =E[ξxβ (1− γ)ϑs′ + ξN

((1− x) ΛQ′

Q + xβκN,s′)]

1− ξE[x+ (1− x) ΛQ′

Q + xβκB,s′] . (41)

Comparing coefficients, the equations determining υs and ϑs are:

υs = βE[κN,s′

]+ β

ξE[(1− x) ΛQ′

Q + xβκN,s′]

1− ξE[x+ (1− x) ΛQ′

Q + xβκB,s′]E [κB,s′]

ϑs = β (1− γ)

1 +ξxE

[κB,s′

]1− ξE

[x+ (1− x) ΛQ′

Q + xβκB,s′]E [ϑs′ ] .

Clearly, ϑs ≡ 0 is a solution to the second equation and corresponds to the equilibriumconsidered in this paper. It is the unique solution if the term multiplying E [ϑs′ ] is alwaysstrictly smaller than one. Around a steady state in which ϑs is zero, a sufficient conditionto guarantee uniqueness is that

β (1− γ)

(1 +

ξxβκB,s

1− ξx+ (1− x)β + xβκB,s

)< 1. (42)

This always holds for x small enough. It remains to establish conditions under which asteady state with ϑs > 0 can also be ruled out. Such a steady state would necessarily havethe term multiplying E [ϑs′ ] equal to one at the steady state. From this, it follows thatnecessarily

κB,s =1− β (1− γ)

ξβx(1− ξx− ξ (1− x)β)

υs =1− β (1− γ)

ξβx

1− ξx1− γ

Rk = R+κB,s

1 + (1− γ) (1− ζ) (υs − 1)(43)

45

holds at the steady state. Now, note that the values of κB,s and κN,s at the steady-stateare do not depend on ϑs, and that therefore the equilibrium borrowing limit B in (41) isincreasing in ϑs for any level of N . In particular then, equilibrium leverage B/K is also anincreasing in ϑs. Since the equilibrium return on capital is a decreasing function of leveragethrough Equation (39), the steady state with ϑs > 0 has a lower Rk than in the steady statewith ϑs = 0. A sufficient condition to guarantee that ϑs = 0 is the unique steady state istherefore that the corresponding steady-state value of Rk is higher than the one computedin (43).

A.4 Proof of the limit in (36)

Combine Equation (33) and (34) to obtain:

1− ξξ

(R− exp

(µt +

1

2σ2µ

))= α

(AtKt

)1−αEt[ε1−αt+1

]+ 1− δ −R.

It follows that:

∆ logPt = ∆ logKt

= ∆ logAt −1

1− α∆ log

(1− ξξ

(R− exp

(µt +

1

2σ2µ

))− 1 + δ +R

)ξ→1−→ ∆ logAt = εt.

B Solving for the learning equilibrium

This section describes how to construct and solve for the learning equilibrium with condi-tionally model-consistent expectations.

B.1 General formulation

To set the notation, I start with the solution of the standard rational expectations equilib-rium. Denote the n endogenous model variables by yt and the nu exogenous shocks by ut.The exogenous shocks are independent across time with joint distribution Fσ, mean zeroand variance σ2Σu. The solution of a (recursive) rational expectations equilibrium satisfiesthe equilibrium conditions:

Et [f−P (yt+1, yt, yt−1, ut)] = 0 (44)

Et [fP (yt+1, yt)] = 0 (45)

46

where fP denotes the stock market clearing condition and f−P collects the remaining n− 1equilibrium conditions.24 A recursive solution takes the form:

yt = gRE (yt−1, ut, σ) .

By the definition of rational expectations, the expectations in (44)–(44) are taken underthe probability measure induced by gRE and Fσ, so that the policy function gRE itself canbe found by solving:∫

f−P

(gRE (gRE (yt−1, ut, σ) , ut+1, σ) ,

gRE (yt−1, ut, σ) , yt−1, ut

)dFσ (ut+1) = 0 (46)∫

fP

(gRE (gRE (yt−1, ut, σ) , ut+1, σ) ,

gRE (yt−1, ut, σ)

)dFσ (ut+1) = 0. (47)

In the learning equilibrium, the probability measure P used by agents to form expectationsdoes not coincide with the actual probability measure describing the equilibrium outcomes.In particular, agents are not endowed with the knowledge that the stock price is determinedby the market clearing condition fP , and instead they form expectations about future pricesusing a subjective law of motion. This law of motion can be summarized in a function:

φ (yt, yt−1, zt) = 0

where yt = (yt, xt) incorporates the additional model variables xt introduced by the learningprocess, and zt is the subjective forecast error. In the mind of agents, this forecast error isan exogenous iid shock with distribution Gσ, mean zero and variance σ2Σz. I assume thatagents believe that z and u are mutually independent as well.

I impose discipline on the expectation formation process by requiring that agents have condi-tionally model-consistent expectations, as defined in the main text. I find such expectationsby computing a subjective policy function

yt = h (yt−1, ut, zt, σ)

which satisfies:∫f−P

(Ch (h (yt−1, ut, zt, σ) , ut+1, zt+1, σ) ,

Ch (yt−1, ut, zt, σ) , Cyt−1, ut

)dFσ (ut+1) dGσ (zt+1) = 0 (48)

φ (h (yt−1, ut, zt, σ) , yt−1, zt) = 0. (49)

Here, the matrix C just selects the original model variables yt from the augmented vectoryt (yt = Cyt). Solving for h effectively amounts to solving a different rational expectationsmodel in which the market clearing condition for the stock market is replaced by the sub-jective law of motion for stock prices. Once computed, the policy function h together with

24There are usually n + 1 equilibrium conditions in total, but one of the market clearing conditions isredundant due to Walras’ law. While under rational expectations, it is immaterial for the computation ofthe equilibrium which market clearing condition is left out, this choice can matter when constructing thelearning equilibrium with conditionally model-consistent expectations. Here I choose to omit the marketclearing condition for final consumption goods.

47

Fσ and Gσ defines a complete internally consistent probability measure P on all endogenousmodel variables. Under P, agents believe that the stock price follows the subjective law ofmotion φ, and P also satisfies the equilibrium conditions f−P :

EPt [f−P (yt+1, yt, yt−1, ut)] = 0.

This subjective belief is very close to rational expectations and preserves as much as possibleof its forward-looking, model-consistent logic while allowing for subjective expectationsabout stock prices.

Now, the subjective policy function h depends on the subjective forecast error zt, whichunder P is believed to be a white noise process. In equilibrium however, zt is insteaddetermined endogenously by the equilibrium stock price that clears the stock market. Thatis, the equilibrium value of the subjective forecast error is itself a function of the states andthe shocks:

zt = r (yt−1, ut, σ) (50)

The function r can be computed by imposing the remaining equilibrium in the stock market,represented by a functional equation EPt [ψ (yt+1, yt, yt−1, zt)] = 0. The function ψ includesthe stock market clearing condition fP and also an (optional) condition for lagged beliefupdating, explained below. Substituting the functional forms:∫

ψ

(h (h (yt−1, ut, r (yt−1, ut, σ) , σ) , ut+1, zt+1, σ) ,

h (yt−1, ut, r (yt−1, ut, σ) , yt−1, zt, σ)

)dFσ (ut+1) dGσ (zt+1) = 0. (51)

Note that while the current value of the forecast error zt was substituted out, the future valuezt+1 was not substituted out, as this value is still taken under the subjective expectation Pwhich treats it as an exogenous random disturbance.

The final equilibrium of the model is described by the objective policy function:

yt = g (yt−1, ut, σ) = h (yt−1, ut, r (yt−1, ut, σ) , σ) .

By construction, this policy function satisfies all equilibrium conditions of the model. Thisfunction g together with Fσ defines the equilibrium probability distribution of the modelvariables. It differs from the subjective distribution P only in that under P, zt is anunpredictable exogenous shock, whereas in equilibrium zt is a function of the state variablesand the structural shocks ut.

It is straightforward to see that the expectations thus constructed satisfy conditional model-consistency:

EPt [yt+1 | ut+1, Pt+1, µt+1] = h (yt, ut+1, zt+1, σ)

= h (yt, ut+1, r (yt, ut+1, σ) , σ)

= g (yt, ut+1, σ)

= yt+1.

48

B.2 Implementation

In this paper, I impose that beliefs about stock price growth are only updated at the endof every period, after the current stock price has been observed. To implement this “laggedbelief updating”, the subjective forecast error has to be two-dimensional, zt = (z1t, z2t).The subjective law of motion is:

φ (yt, yt−1, zt) =

∆ logPt − µt−1 − z1t

∆µt − gz2t

z1t − z1t−1

.

Two additional state variables are necessary to compute the subjective law of motion: thebelief µt and the lagged forecast error z1t, so that yt = (yt, µt, z1t).

To compute the equilibrium with lagged belief updating, I need to impose the stock marketclearing condition fP and also update the forecast error in the belief equation to be lastperiod’s forecast error in the price equation:

ψ (yt+1, yt, yt−1, zt) =

(Pt − β (Dt+1 + Pt+1)

z2t − z1t

).

The resulting expressions for the equilibrium price and beliefs are:

Pt =βEPt [Dt+1]

1− exp(µt + 1

2σ2z

)µt = (1− g) µt−1 − g∆ logPt−1.

B.3 Approximation with perturbation methods

I now describe how to compute a linear approximation of the objective policy functiong around the non-stochastic steady state y. The first step consists in deriving a first-order approximation of the subjective policy function h. This can be done using standardmethods, as the system of equations (48)–(49) can be solved as if it were a standard rationalexpectations model. The second step consists in finding the derivatives of the function r.Applying the implicit function theorem to Equation (51) leads to:

ry = −A−1

((∂ψ

∂yt+1hy +

∂ψ

∂yt

)hy +

∂ψ

∂yt−1

)ru = −A−1

(∂ψ

∂yt+1hy +

∂ψ

∂yt

)hu

rσ = −A−1

(∂ψ

∂yt+1hy +

∂ψ

∂yt

)hσ

where the matrix A is given by A =(

∂ψ∂yt+1

hy + ∂ψ∂yt

)hz + ∂ψ

∂zt. This matrix needs to be

invertible for the learning equilibrium to exist. The first-order derivatives of the actual

49

policy function g can be obtained by applying the chain rule:

g (yt−1, ut, σ) ≈ g (y, 0, 0) + gy (yt−1 − y) + guut + gσσ

gy = hy + hzry

gu = hu + hzru

gσ = hσ + hzrσ

The certainty-equivalence property holds for the subjective policy function h, hence hσ = 0.This implies that rσ = 0 and gσ = 0 as well, so certainty equivalence also holds underlearning.

Second- and higher-order perturbation approximations of g can be computed analogously.As in first order, only invertibility of the matrix A is required for a unique local solutionunder learning.

C Monetary policy rules under rational expectations

The table below mirrors Table V in the main text, but evaluated at the rational expectationsversion of the model, with parameters estimated to fit the same moments in the data as inthe version with learning.

Table VI: Alternative monetary policy rules under rational expectations.

(1) (2) (3) (4) (5) (6)

baseline alternative rules optimized rules

φπ 1.50 3.00 1.50 1.50 62.58 5.82

φY 0.50 107.44 21.27

φP 0.10 -7.50

σ (Y ) 3.71% 7.32% 3.03% 3.31% 1.74% 1.07%

σ (P ) 0.82% 2.42% 1.07% 0.79% 0.86% 0.59%

σ (π) 0.28% 0.09% 0.26% 0.26% 0.31% 0.34%

σ (i) 0.27% 0.23% 0.13% 0.25% 0.22% 0.26%

welfare cost χ 0.0727% 0.01583% 0.0463% 0.0625% .0305% .0303%

Standard deviations of output, stock prices, inflation, and interest rates (unfiltered) under rational expec-tations in percent. See Footnote 20 for the definition of the welfare cost χ. The interest rate smoothingcoefficient is kept at ρi = 0.85 for all rules considered.

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