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Essays in Macroeconomics Caines, Colin 2016

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Essays in MacroeconomicsbyColin CainesB.A. & Sc., McGill University, 2005M.A., The University of British Columbia, 2010A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Economics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)July 2016c© Colin Caines 2016AbstractThis thesis is composed of three chapters. The first chapter argues that boom-bust behavior inasset prices can be explained by a model in which boundedly-rational agents learn the process forprices. The key feature of the model is that learning operates in both the demand for assets and thesupply of credit. Interactions between agents on either side of the market create complementaritiesin their respective beliefs, providing an additional source of propagation. In contrast, the chaptershows why learning involving only one side on the market, the focus of most of the literature,cannot plausibly explain persistent and large price booms. Quantitatively, the model explainsrecent experiences in US housing markets. The full appreciation in US house prices in the 2000scan be generated from observed mortgage rate changes. The model also generates endogenousliberalizations in household lending conditions during price booms and replicates key volatilities ofhousing market variables at business cycle frequencies.The second chapter presents a learning model in which households are endowed with recursivepreferences. The chapter evaluates how the introduction of bounded rationality in beliefs effectsthe level of long run consumption risk in the economy. The chapter shows that structural learningframeworks currently found in the literature lead to a perception of low persistence in exogenousshocks, regardless of the underlying stochastic processes in the economy. Generating long run riskrequires a preference for late resolution of uncertainty.The third chapter provides an explanation for two features of the world saving distribution: (i)saving rates are significantly different across countries and they remain different for long periods oftime; and (ii) some countries and regions have shown very sharp changes in their average saving ratesover short periods of time. It formalizes a model of the world economy comprised of open economiesinhabited by heterogeneous agents endowed with recursive preferences. The model can generatethe time series behavior of saving observed in the data from measured productivity shocks. Themodel can also generate the sudden and long-lived increase in East Asian savings by incorporatingshocks to societal aspiration.iiPrefaceChapter 3 of this thesis is joint work with Prof. Amartya Lahiri at the Vancouver School ofEconomics. I was responsible for implementing the quantitative work in the chapter. The empiricaland theoretical components of the chapter are joint work. The chapter was first drafted by Prof.Lahiri.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Learning & House Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Background & Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 A Model With Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.1 Household Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.2 Lender Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3.3 Learning & Subjective Beliefs . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3.4 Equilibrium Under Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.4 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.5 Analytic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.6 Quantitative Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.6.1 Interest Rates & Boom-Bust in House Prices . . . . . . . . . . . . . . . . . . 251.6.2 Capturing Cyclical Variation in Housing Markets . . . . . . . . . . . . . . . 291.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 Learning & Recursive Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.2 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.3.1 Household Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.3.2 Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.3.3 Beliefs & Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.3.4 Equilibrium Under Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.4 Illustrating Long Run Risks Under Rational Expectations . . . . . . . . . . . . . . . 442.5 Quantifying the Learning Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47ivTable of Contents2.5.1 The Effects of Learning & Long Run Risks . . . . . . . . . . . . . . . . . . . 492.5.2 The Effect of the Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552.5.3 Internalizing Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 Explaining World Savings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.2 Two Facts on Cross-Country Saving . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.3 Recursive Preferences & Balanced Growth . . . . . . . . . . . . . . . . . . . . . . . 673.4 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.5 Quantifying the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.5.1 Adjustment Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783.5.2 A Five-Region Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803.6 Saving Miracles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86AppendicesA Chapter 1, Additional Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92B Chapter 1, Aggregation of Household Problem to Representative Agent . . . . 93C Chapter 1, Solving the Household’s Problem . . . . . . . . . . . . . . . . . . . . . 96D Chapter 2, Additional Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98E Chapter 2, Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101F Chapter 2, Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102G Chapter 3, Data & Calibration Notes . . . . . . . . . . . . . . . . . . . . . . . . . . 105G.1 Countries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105G.2 Calibration & Data Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107H Data Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108vList of Tables1.1 Calibrated Parameter Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.2 Business Cycle Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.3 Business Cycle Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.4 Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.1 Business Cycle Moments Under Rational Expectations . . . . . . . . . . . . . . . . . 482.2 Business Cycle Moments Under Learning (g = 0.002, Learning not Internalized) . . . 532.3 Business Cycle Moments Under Learning (g = 0.002, Learning not Internalized) . . . 542.4 Business Cycle Moments Under Learning: Effect of g . . . . . . . . . . . . . . . . . . 572.5 Business Cycle Moments Under Learning: Internalizing Learning (γ = 5, ψ = 1/1.5) 592.6 Business Cycle Moments Under Learning: Internalizing Learning (γ = 5, ψ = 6) . . 603.1 Parameterization of Baseline Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.2 Comparing Savings in the Model and the Data . . . . . . . . . . . . . . . . . . . . . 783.3 Parameterization of Model With Adjustment Costs . . . . . . . . . . . . . . . . . . 793.4 Parameterization of 5-Region Model With Adjustment Costs . . . . . . . . . . . . . 803.5 Simulated Moments: 5-Region Model With Adjustment Costs . . . . . . . . . . . . 81G.1 List of Countries in Regions, 3-Region Model . . . . . . . . . . . . . . . . . . . . . . 105G.2 List of Countries in Regions, 5-Region Model . . . . . . . . . . . . . . . . . . . . . . 106viList of Figures1.1 Home Price Index Futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 House Prices & Household Borrowing . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Response to Endowment Shock, Demand-Side Learning . . . . . . . . . . . . . . . . 231.4 Response of Log House Prices to Interest Rate Drop . . . . . . . . . . . . . . . . . . 251.5 Response of Log House Prices and Credit to Interest Rate Drop . . . . . . . . . . . . 271.6 Standard Deviation of q and fe(q/q−1) Relative to y . . . . . . . . . . . . . . . . . 321.7 Periodogram of Log House Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.1 Transitory & Permanent Shocks in Rational Expectations Model (γ = 5, ψ = 1/1.5) 442.2 Transitory & Permanent Shocks in Rational Expectations Model (γ = 5, ψ = 6) . . . 452.3 Influence of Learning (γ = 5, ψ = 6) . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.4 Effect of Preference for Timing of Risk in Learning Model . . . . . . . . . . . . . . . 522.5 Influence of g: Transitory & Permanent Shocks in Learning Model . . . . . . . . . . 563.1 Differences in Saving Rates Across Regions . . . . . . . . . . . . . . . . . . . . . . . 663.2 Saving Rates in the Asian Tigers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.3 Balanced Growth Evidence: Consumption-Output Ratios . . . . . . . . . . . . . . . 683.4 Impulse Response of Savings to Productivity . . . . . . . . . . . . . . . . . . . . . . 763.5 Saving Rates: Data & Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.6 Saving Rates Under Adjustment Costs: Data & Model . . . . . . . . . . . . . . . . 793.7 Response of Savings to Regime Switch . . . . . . . . . . . . . . . . . . . . . . . . . . 833.8 Saving Miracles: Data & Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84A.1 Maximum Log Price Deviation and Length of Propagation Following R Drop . . . . 92D.1 Effect of Internalizing Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98D.2 Effect of Preference for Timing of Risk in Learning Model . . . . . . . . . . . . . . . 99D.3 Influence of g: Transitory & Permanent Shocks in Learning Model . . . . . . . . . . 100viiAcknowledgementsI would like to thank my advisor Paul Beaudry and committee members Amartya Lahiri and JessePerla for their guidance during this work. I am also indebted to Yaniv Yedid-Levi and Henry Siufor helpful advice, as well as Florian Hoffmann for assistance on technical points. I would alsolike to thank the faculty and students in the economics department generally for comments andsuggestions during seminars.Finally, I would like to thank the Social Sciences and Humanities Research Council of Canadaand the University of British Columbia for generous support during this project.viiiDedicationTo Elyse (& Rosie)ixChapter 1Can Learning Explain Boom-BustCycles in Asset Prices? AnApplication to the US Housing Boom1.1 IntroductionFollowing the financial crisis of 2008 there has been an intense focus on the tendency of markets togenerate boom-bust patterns in asset prices. Explaining these episodes poses a difficult questionfor researchers: how can large and persistent price growth be explained in the absence of large andpersistent exogenous variation? Across a wide range of settings it has proven difficult to identifymarket fundamentals or frictions that can explain price booms as well as asset price volatilitymore generally. This chapter argues that boom-bust behavior in asset prices can be explained bya model in which boundedly rational agents learn the process for prices. The key feature of themodel is that, in contrast with the literature, learning operates in both the demand for assets andthe supply of credit. Propagation comes from the interaction between the two sets of agents in themodel, which creates complementarities in their respective beliefs. The quantitative performanceof the model is evaluated in the context of recent experiences in US housing markets. A singleunanticipated mortgage rate drop, consistent with that observed in the early 2000s, generates 20quarters of house price growth whilst capturing the full appreciation in US housing in the early2000s.The novel feature of this work is that it allows for learning in the credit supply problem facinglenders. This is in contrast to canonical asset pricing models with learning that restrict subjectivebeliefs to the demand side of the market. Models of bounded rationality allow for the possibilityof feedback loops to exist between subjective beliefs and observed outcomes. In order for suchenvironments to generate large and persistent asset price growth in response to a small set ofshocks, beliefs need to exhibit two properties. First, subjective beliefs must be highly responsive to11.1. Introductionobserved shocks. The response of outcomes to shifts in beliefs must be of sufficient size to generatesubsequent belief shifts. Second, the belief process itself must be sufficiently persistent to preventthe episode from dying out quickly. A contribution of this chapter is to show that in modelswith only demand side learning, there exists a trade-off between these two properties. In otherwords, increasing the elasticity of beliefs with respect to shocks comes at the cost of decreasing thepersistence of these beliefs. As a result of this trade-off such models struggle to explain asset pricebooms.Next, the chapter shows that this trade-off in the formation of the belief process can be brokenby extending bounded rationality to credit suppliers. When learning about prices operates in boththe demand side and credit supply side of the market there are complementarities in the beliefs ofbuyers and lenders. An increase in buyers’ price forecasts increases the capital gains they expectto receive on their assets, driving up demand. An increase in lenders’ price forecasts decreases thedefault rate they expect to face on their assets, leading to relaxed lending conditions. Each of theseactions drive up prices, and through the learning mechanism further increase the price forecastsof each type of agent. The chapter shows that this complementarity loosens the trade-off betweengenerating beliefs that are both persistent and responsive.Finally, the chapter shows that this mechanism can quantitatively capture many properties ofUS housing markets. The full appreciation in US housing seen in the early 2000s can be explainedwith observed mortgage rate movements. The calibrated model is also shown to replicate keyvolatilities of housing market variables at business cycle frequencies. Furthermore, the chapterexplains observed comovements in house prices and household leverage. The model developed hereis able to endogenously generate substantial liberalizations in households’ borrowing environmentconcurrent with periods of prolonged price growth.The chapter is structured as follows. Section 1.2 provides an overview of the literature inwhich this work is placed as well as a discussion of the recent experience in US housing markets.Section 1.3 presents the main model, outlining the microfoundations of agents’ beliefs, discussingthe decision problem and optimality conditions of households and lenders, and finally providing anequilibrium for the model under learning. A discussion of the model’s calibration is to be found insection 1.4. Section 1.5 presents the analytical findings of the chapter and demonstrates how thepresence of bounded rationality in both the demand for housing and the supply of credit breaks21.2. Background & Related Literaturea trade-off between the persistence of beliefs and the sensitivity of beliefs to shocks that existsin traditional learning models. Empirical findings are discussed in section 1.6; 1.6.1 examines theeffect of observed mortgage rate drops and highlights the model’s ability to capture much of theobserved experience in the US housing market post-2000, while 1.6.2 shows the model’s performancein attempting to match business cycle moments of the US housing market.1.2 Background & Related Literature1.2.1 BackgroundAcross almost all of the major urban centers in the United States price-rent ratios rose between20 and 70 percent in the 10 years leading up to the market crash. The experience in US housingis widely seen as being central to the subsequent crisis in global financial markets and the ensuingrecession. Housing wealth, about 80 percent of which is encompassed by the stock of owner-occupied homes, accounts for half of household net worth in the United States (Iacoviello (2011))and residential investment has been a relatively large component of US GDP growth over thepast 30 years (Wheaton and Nechayev (2010)). In spite of the prevalence and importance ofhousing booms, data on housing market fundamentals typically doesn’t display large volatilities.Consequently, standard frameworks struggle to explain large and persistent movements in houseprices1, suggesting a role for an expectations-based explanation. The overweighting of recentlyobserved information in the formation of beliefs, particularly beliefs about long horizon events, isdocumented in many settings2. An implication of such extrapolative behavior when applied tobeliefs about price movements is that agents tend to underpredict price growth during inflationaryperiods and overpredict growth during deflations.Figure 1.1 plots the 10-city composite of the Case-Shiller Home Price Index together with theprices of futures contracts that trade on this index on the Chicago Mercantile Exchange3. The1Literature has arisen as a result that attempts to explain housing booms as arising from a variety of differentand often conflicting mechanisms, including: varying supply-side constraints (see Gleaser et al. (2008), Bulusu,Duarte, and Vergaga-Alert (2013), Kiyotaki, Michaelides, and NIkolov (2011)), demand-side factors such as changesin credit conditions (Chu (2014), Favilukis, Ludvigson, and Nieuwerburgh (2010)), income instability (Nakajima(2011), Pastor and Veronesi (2006)), and social interaction mechanisms (Burnside, Eichenbaum, and Rebelo (2011)).Also see Justiniano, Primiceri, and Tambalotti (2015) for further discussion2See Adam and Marcet (2011), Eusepi and Preston (2011), Glaeser, Gyourko, and Saiz (2008) for discussion.3Note a version of this figure appears in Gelain and Lansing (2013).31.2. Background & Related LiteratureFigure 1.1: Home Price Index Futures1201401601802002202402006 2008 2010 2012 2014 2016price indexyearCase-Shiller Index and CME Futures Pricesfutures contracts provide a proxy for market expectations about future house prices, and they bearthe evidence of extrapolative expectations. Following the crest in US house prices in 2006, forecastssubstantially overpredicted the realized path of house prices for the best part of two years. Afterbeliefs about future growth adjusted in late 2007, the forecasted series missed the turning point atthe bottom of the market in 2009 and persistently underpredicted prices as inflation began in 2012.Similar evidence is documented in Case and Shiller (2003) and Piazzesi and Schneider (2009), whoestimate household beliefs about price increases and find increasing levels of optimism throughoutthe run-up in house prices in the mid 2000s.4This evidence suggests a potential mechanism for resolving the observed boom in prices with therelative lack of variation in market fundamentals. If agents’ beliefs about future events, particularlyin the long-run, are sufficiently responsive to shocks, booms may arise as self-confirming events:shocks to prices may shift the distribution of expected prices – and therefore expected capitalgains, expected default rates, etc... – enough to generate large increases in demand and thereforesubsequent price growth. In formulating a learning model in which boundedly rational agentsrecursively update beliefs about the process for house prices this chapter captures precisely thesekind of dynamics.4Gelain and Lansing (2013)41.2. Background & Related LiteratureFigure 1.2: House Prices & Household Borrowing1975 1980 1985 1990 1995 2000 2005 2010 20150.20.320.440.560.680.8net liability / GDPyear  200240280320360400pricenet credit mkt liabilities / gdpnet mortgage mkt liabilities / gdptotal mortgage mkt liabilities / gdpprice(a) US House Prices & Credit Market Liabilities1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014−0.01−0.00500.0050.010.015Net New Borrowing / Mkt Value of Housingyear  Mkt Value of Housingnet new mortgage liabilities / value of housingnew mortgage liabilities / value of housingMkt Value of Housing(b) Value of Housing & Mortgage LiabilitiesA key feature of the US house price boom was the large increase in credit provided to households.The left hand panel in figure 1.2 plots the All-Transaction House Price index from the FederalHousing Finance Agency together which three measures of net household liabilities as a fractionof GDP. The right hand panel shows net new borrowing by US households as a fraction of themarket value of housing stock. As can be seen, the data reveals close comovement between pricesand household credit during the late 1990s and early-mid 2000s. It seems reasonable, then, that astory of price booms in US housing markets should speak to this phenomenon. Under the learningframework developed in this chapter, a simple contracting mechanism between households andlenders endogenously generates substantial comovement in house prices and household leverage5.This is a result of the sensitivity of credit supply to subjective beliefs. Shifts in the distributionof expected price growth cause lenders to significantly liberalize lending conditions to households,driving inflation in house prices and subsequent shifts in lenders’ beliefs.5Box & Mendoza (2014) study the effect of credit market liberalizations in a learning framework. The authors areable to capture significant growth in land prices, however credit market liberalizations are exogenously imposed andthe model does not allow for feedback between agents’ exogenous variables and beliefs.51.2. Background & Related Literature1.2.2 Related LiteratureThis work follows from a wide literature that models learning behavior of economic agents so asto amplify and propagate shocks in macro models. In such frameworks agents are assumed to beuninformed about some process or set of processes and hence hold subjective beliefs about their truelaw of motion. These beliefs are updated over time to account for new information via a learningrule such as Bayesian updating, least squares updating, or constant gain learning (under such anupdating rule agents place a constant weight, called a gain, on new information). Typically, agentsperceive that temporary shocks have long run effects through their influence on learned beliefs.Much of the initial interest in this work arose in the monetary policy and business cycle literature.The ability of learning mechanisms to generate improved volatilities and comovements in macroe-conomic environments has been mixed 6. The range of results found in this literature can in part beaccounted for by the effect different learning mechanisms have on the microfoundations of agents’decision problems. Preston (2005) argued that researchers implicitly impose an inconsistency uponagents’ beliefs when assuming them to be uninformed about the future evolution of their own choicevariables, an assumption that is common in the learning literature. When agents are uninformedabout the true law of motion of some variable(s) in learning models, they forecast future valuesof these variables using a perceived law of motion that they have estimated from previously ob-served data. Hence, if uninformed about the law of motion of one of their own control variables,the agent’s forecasts of this object are not constrained to be consistent with what optimal choiceswould be given its beliefs about the evolution of other variables in the model. Suppose, for example,that an agent is uninformed about the true process governing investment. In a learning framework,forecasted investment would then be given by the expected value of an estimated process for invest-ment and not by the expected optimal response of investment to prices and exogenous shocks. Suchmodels therefore implicitly assume that agents are either uninformed about their decision problemin the future or that agents predict they will be making suboptimal decisions in the future.7This work is extended by Adam and Marcet (2011) who formalize the concept of internal ratio-nality. This stipulates that agents with subjective beliefs should make choices that are everywhere6See Williams (2003), Carceles-Poveda and Giannitsarou (2008), Huang, Liu, and Zha (2009), and Milani (2007).7Marshall and Shea (2013) provide an example of a learning model in housing where households are assumed tobe uninformed about the evolution of their own consumption.61.2. Background & Related Literatureoptimal conditional on these beliefs (ie. on and off the equilibrium path). In restricting subjectivebeliefs to the space of prices observed in the housing market the model formulated in this chapterbuilds upon this work. There is growing evidence that such frameworks can improve the inter-nal propagation mechanisms of models. Eusepi and Preston (2011) consider a real business cycleenvironment where agents are restricted to learning the parameters of wage and capital returnfunctions via constant gain learning. The specification allows the consumption-saving decisionsof households to be a discounted sum of subjective wage and rental rate forecasts. Relative torational expectations the persistence of shocks and overall volatilities are substantially increased.Similar results are found in Branch and McGough (2011), while Sinha (2011) suggests that suchlearning specifications can improve the performance of business cycle models in matching financialmoments8.9Asset pricing models with internally rational learning have achieved some success in explainingthe observed volatility and persistence in asset prices (most notably stock prices and house prices)over the business cycle, however this research has thus far failed to provide a convincing explanationof price booms. Adam, Beutel, and Marcet (2014) propose a learning framework in which boundedlyrational agents believe stock price growth to be governed by a simple linear hidden Markov model.A similar model is examined by Adam and Marcet (2010) with agents that are assumed to beuninformed about the evolution of stock returns instead of price growth. Both frameworks struggleto endogenously generate sequences of beliefs sufficient to yield price booms in general equilibriumsettings.Adam, Kuang, and Marcet (2011) implement a learning model, similar in spirit to that foundin Adam, Beutel, and Marcet (2014), to try to explain joint house price-currentaccount movementsin the G7 over the 2000s. The authors find that reductions in interest rates in the early 2000s cangenerate substantial increases in house prices as households become increasingly optimistic aboutprice appreciation, however the result is highly sensitive to the initial conditions of beliefs and pricegrowth at the time shocks hit. Similar models are formulated by Gelain and Lansing (2013) andGranziera and Kozicki (2012). This chapter is also closely related to Kuang (2014) who considers a8Similar frameworks can be found in Adam, Marcet, and Nicolini (2013), Branch (2014), and Williams (2003).9Note that a question that emerges from the initial learning literature in business cycle models is whether canonicalmacro models converge to rational expectations dynamics under learning. For a discussion of these issues see Evansand Honkapohja (2001), as well as Cellarier (2006), Cellarier (2008), Cho, Williams, and Sargent (2002), Evans andHonkapohja (2003), Ellison & Pearlman (2011), McCallum (2007), Williams (2003), and Zhang (2012).71.3. A Model With LearningKiyotaki and More (1997) environment with learning. Internally-rational agents in the model formbeliefs about the joint law of motion for collateral prices and collateral stock. The author showsthat the specified process of expectation formation can generate boom-bust cycles in prices andcredit. This chapter follows from Kuang (2014) in considering credit constraints to be an importantchannel through which learning can operate. This chapter goes further, however, in considering arole for internal rationality in the determination of loan-to-value ratios. It shows that under suchconditions boom-bust cycles can be obtained even with the simple models of expectation formationfound in Adam, Kuang, and Marcet (2011), Gelain and Lansing (2013), and Granziera and Kozicki(2012). The environment specified here also allows for belief heterogeneity between borrowers andlenders.Boz and Mendoza (2014) argue that credit market liberalizations in the late 1990s lie at theroot of the US housing boom. The authors propose a learning framework in which agents learn atwo-state Markov-switching model for collateral constraints, however the model cannot explain themajority of the growth in US house prices.10This chapter also contributes to a literature which considers the role of bounded rationalityin driving the credit supply choices of financial institutions. Luzzetti and Neumuller (2015) arguethat the dynamics of household debt and bankruptcy can in part be explained when lenders learnthe riskiness of the financial environment in which they operate. Similarly, Pancrazi and Pietrunti(2014) consider the role that boundedly-rational beliefs about prices on the part of lenders can playin determining debt and home equity extraction.1.3 A Model With LearningThe housing market is modelled as an open economy environment. The key interactions in themodel involve households who purchase and consume housing stock, and mortgage lenders whosupply households with credit in return for claims on their housing. Housing stock is treated as10Many of the papers in the learning literature make stronger assumptions about agents’ information structure.Gelain and Lansing (2013) study the behavior of US house prices under a form of learning where households makeforecasts of a composite variable which is composed of price-rent ratios and consumption growth. Williams (2012)considers a model where agents learn the mean and standard deviation of stock returns. The model includes anoccasionally-binding borrowing constraint about which the agent is uninformed and which it does not internalize inits decision problem. Branch, Petrosky-Nadeau, and Rocheteau (2014) examine US housing using a learning modelwith search and matching frictions in employment. Agents in the model believe that price growth will always exceedthe level of growth extrapolated off of recent data.81.3. A Model With Learningan asset which, in addition to providing households with a source of capital gains, yields a flow ofhousing services and can be posted as collateral when borrowing. Household borrowing is subject todefault. The chapter abstracts from modelling strategic behavior in the household’s default choice.While there is a rich literature that seeks to model the decision problem facing households whendefaulting on residential mortgages, these issues are not central to the mechanism considered in thechapter. The default choice is modelled to reflect observed default patterns in the data. Lendersare assumed to have access to an outside source of funds (ie. international financial markets) andare owned by agents outside the housing market.In modelling the lender’s problem the chapter does not seek to provide a complete descriptionof mortgage financing. Instead, the model provides a stylized contracting problem between lendersand households in order to capture observed properties of mortgage default as well as measuredcorrelations between default rates, price growth, and household leverage. Lenders set their supplyof credit in response to expected default rates and the value of their collateral claims in the eventof default. Household demand is driven by expected capital gains, expected default rates, and theircredit constraint.All agents are assumed to be uninformed about the determination of house prices, and thereforehold subjective beliefs about the evolution of these prices in the economy. Beliefs are homogeneousacross lenders and households, and are updated continuously to account for new information. Sta-tionary Bayesian updating of beliefs implies that learning follows the well-known constant gainalgorithm. Expectations about price growth at long horizons heavily weight recent price obser-vations. Importantly, the information structure respects the international rationality of agents’decision problem. Household and lender choices are optimal responses to their subjective beliefs.1.3.1 Household ProblemAn individual household has preferences over consumption and housing services given byEP0∞∑t=0βtu(ct, ht), where u(ct, ht) = cγct h1−γct (1.1)where P denotes the household’s subjective beliefs. The household receives an exogenous endow-ment yt each period out of which it can accumulate housing stock ht and purchase consumption91.3. A Model With Learninggoods (the numeraire in the economy). Housing stock has a price qt and depreciates at the rateδh. The household also has access to a technology that allows it to convert consumption goodsinto non-housing capital one-for-one. Capital can be rented to builders, who are owned by thehousehold, at a price pt and depreciates fully after one period. The household can also borrow froma credit supplier at a risk-free rate Rt by posting its housing stock as collateral. All debt contractsare one period in length. The household’s borrowing is limited by a collateral constraintRtbt ≤ θtEPt qt+1ht (1.2)which is assumed to bind each period. The flow budget constraint is given byct + qt (ht − (1− δh)ht−1) +Rt−1bt−1 + kt = yt + bt + ptkt−1 + Πt (1.3)Where Πt denotes profits earned by builders. At the beginning of each period the household isable to decide whether or not it wants to default on its debt repayment Rt−1bt−1. If it choosesnot to repay then the lender confiscates the household’s housing stock (1 − δh)ht−1. The modelabstracts from considerations of strategic default. Instead, households default so as to minimizetheir per-period repayment. However this process is subject to an idiosyncratic shock ζ, such thatthe household chooses to default if(1− δh)qtht−1 ≤ ζtRt−1bt−1, log ζt ∼ N(log ζ¯, σ2ζ ) (1.4)The shock ζ can be thought of as capturing differences in liquidity across households. This formu-lation allows the model to reflect the observed fact that only a subset of households who go intonegative equity end up having their mortgages foreclosed. The presence of default in the model im-plies that housing stock will be heterogeneous across households. Owing to the fact that per periodutility in (1.1) is Cobb-Douglas, the model nevertheless aggregates to a representative householdstructure with stochastic default (the representative agent loses only a fraction of his/her housingstock each period as a result of default). The formulation of the default problem in (1.4) is neces-sary to smooth the default choice after aggregation to a representative consumer. The aggregationis shown in appendix B.101.3. A Model With LearningThe household side of the model is completed by outlining the supply of new housing stock.Builders construct new housing stock from the physical capital they rent from households, accord-ing to a production function hs = f(k) = Akα. Profits earned by builders are returned to thehouseholds. The builders’ profit maximization problem is given bymaxk˜{Π(qt,pt, k˜) = qtAk˜α − ptk˜}=⇒ pt = αqtAk˜α−1t (1.5)1.3.2 Lender ProblemThe representative lender chooses the amount of credit to supply to the household at the noterate Rt. The lender is assumed to have access to an outside source of funds (ie. internationalfinancial markets) from which it can borrow. In the event of default, a household’s stock of housingis transferred to the lender. The lender can recoup the value of the collateral in the housingmarket, however it faces delay in doing so. In particular, the lender can only sell a unit of theforeclosed housing stock in its possession with probability µ each period. This assumption capturesthe presence of delay in the foreclosure process and allows the lender’s choice to depend on itsbeliefs about long-horizon events. The lender’s valuation of housing is also subject to a stochasticmarkdown, φt11. φt is assumed to follow a random walk in logs (which bounds φt > 0)log φt = log φt−1 + dt (1.6)which is known to the lender. The calibration of this process allows the model to match observedcorrelations between lagged house prices and household leverage. The lender’s profits are given byΠL|ht,P = −b˜t + β (1− Pr(default|P)) ·Rtb˜t + ... (1.7)Pr(default|P) · ∞∑j=1βj(1− δh)jµ(1− µ)j−1EPt (φt+jqt+j)ht11φt can be thought of as capturing non-monetary costs/benefits to liquidating foreclosed housing in a given period(ie. a liquidity value).111.3. A Model With Learningwhere P denotes the lender’s beliefs about future prices. Normalizing ΠL by the value of household’shousing stock, the profit maximization problem can be writtenmaxθ −θREPt(qt+1qt)+ β (1− Pr(default|P)) · θEPt(qt+1qt)+ ...Pr(default|P) ·[∑∞j=1 βj(1− δh)jµ(1− µ)j−1EPt(φt+jqt+jqt)] (1.8)=⇒ θ∗ = θ(mt) (1.9)Where θ is the value of the debt repayment Rtbt as a fraction of the expected value of the household’shousing stock, EP(qt+1ht). In other words, conditional on its observation of (qt, ht) and its beliefsP, the lender’s choice of credit supply b˜ is equivalent to determining the credit constraint that thehousehold faces in (1.2)12. The lender’s choice of θt implies that the perceived probability of defaultis given byPr(default|P) = Pr[log(qt+1qt)− log(ζt+1) ≤ log(θt1− δh)+ log(EPt(qt+1qt))](1.10)The lender’s credit supply choice is determined by the probability of default and the value ofits claims to the collateral in the event of default. As can be seen in (1.8), delay in the sale offoreclosure inventory implies that the value of the lender’s claim to the housing stock depends onlong-run forecasts of price growth13. This introduces a degree of convexity in the supply of creditwith respect to the belief mt.1.3.3 Learning & Subjective BeliefsSubjective beliefs are homogeneous across all agents in the economy. In order to impose discipline inlearning, beliefs are specified so as to respect the internal consistency of agents’ decision problems.More precisely, an agent i’s decision rule must be an optimal response to their beliefs P i everywhereon and off the equilibrium path. This restricts the range of variables over which agents can makesubjective forecasts. When agents make forecasts of their choice variables using a perceived law12As the household’s problem is calibrated to ensure that (1.2) binds in each period, the household will always bewilling to accept this contract.13Alternatively, this could be achieved through the presence of a foreclosure cost, however the assumed cost functionwould need to be a function of expected forward prices, which is not intuitive.121.3. A Model With Learningof motion which they estimate from past data (as is the case in learning models where agents areuninformed about the process governing the path of their own choice variables in the future, anassumption that is present in much of the literature), the forecasted values will not in general beoptimal responses to forecasted values of the other variables in the system. In such a case the agentis implicitly assumed to either be uninformed about their own decision problem or to be forecastingthat they will make suboptimal decisions at some future history14. Internal consistency of this kindrequires that agents only hold subjective beliefs about variables outside of their decision set. In themodel specified here agents are assumed to be uninformed about the true process governing houseprices in the economy. Agents meet the standard of Internal Rationality (Adam and Marcet (2011))in that their decision rules will be optimal conditional on their subjective beliefs. Furthermore, asprices are an equilibrium object, this formulation allows for feedback between subjective beliefs,which will be extrapolated from price data as a result of learning, and realized prices.As has been widely studied in the learning literature, simple rule-of-thumb updating rules cancapture the property that long-horizon price forecasts heavily weight recent price data, consistentwith the evidence discussed in sections 1.1 and 1.2. Such updating rules arise endogenously fromBayesian learning of parsimonious hidden state models. Agents in the model perceive that pricesare generated by the following data generating process (DGP)ln qtqt−1 = lnωt + qtlnωt = lnωt−1 + ωt qtωt iid∼ N 00, σ2q 00 σ2ω (1.11)where the persistent component of price growth, logωt, is a hidden state variable. Agents observethe realization of ln qtqt−1 and learn by updating beliefs about the distribution of lnωt. The choice ofperceived DGP follows a number of papers in the asset pricing learning literature, including Adam,Kuang, and Marcet (2011) and Adam, Beutel, and Marcet (2014). Optimal updating of 1.11 impliespatterns of forecast errors in prices consistent the evidence discussed in section Ch1-sec:Background.Under stationary Bayesian learning, the household’s posterior beliefs about lnωt are given bylnωt ∼ N(lnmt, σ0(σq, σω)2)(1.12)14See Preston (2005) or Eusepi and Preston (2011) for discussion.131.3. A Model With Learningσ20 =−σ2ω +√σ4ω + 4σ2qσ2ω2(1.13)The stationary Kalman filtering equations imply the constant gain algorithm for updating beliefsabout the posterior mean, lnmtlnmt = lnmt−1 + g(σq, σω) ·(lnqtqt−1− lnmt−1)(1.14)g(σq, σω) =σ0(σq, σω)2σ2q(1.15)Under constant gain updating (1.14), posterior beliefs will be a weighted average of past pricegrowth observations. The gain parameter g, which controls the weight agents place on new pricedata when forming beliefs, is equivalent to the inverse of the rate at which old observations arediscounted over time15.Given this belief structure, EPt (qt+j/qt) in (1.8) can be written asEPt(qt+jqt)= exp(j log(mt) +12j2σ20)· exp(12jσ2q)exp(12σ2ωj∑s=1s2)(1.16)A shift in beliefs mt influences the lenders’ supply of credit via its effect on perceived defaultprobabilities and the expected values of the lenders’ claims on housingmt ↑=⇒PPt (Default) ↓EPt(ΠL, defaultt+1)↑ =⇒ b˜t, θt ↑The household’s Euler equation for housing is given byqt =uh(t)uc(t)+ EPt[(β(1− δ) · λit+1λit· Γt+1 + θtR)· qt+1 − βθt(EPt qt+1) λit+1λitΓt+1](1.17)15The constant gain makes this a model of perpetual learning. Even when the DGP (1.11) is correctly specified,discounting of past data implies that mt will not converge in levels. In this case, however, beliefs should be ergodicallydistributed around the rational expectations equilibrium when the gain is small (see Evans and Honkapohja (2001)).141.3. A Model With Learningwhere Γt+1 = 1 − Pr(default|P) in (1.10). Increases in the economy-wide posterior mean of thepermanent component of price growth, logmt, directly influence the household’s housing demandvia three complementary channels: (i) increasing expected capital gains on housing, (ii) increasingthe supply of credit available to the household, and (iii) decreasing the household’s expected defaultprobability. As neither the lender nor the household understand the correct mapping betweenfundamentals and prices, it is assumed that neither agent is able to account for the effect of theiractions on future beliefs. In other words, when forecasting future prices the agents also do notinternalize the effect of future price movements on mt16.A few comments should be made at this point about the assumption that beliefs in the modelare homogeneous across all agents. Heterogeneity could be incorporated into the model in a numberof different ways: (i) households and lenders could be assumed to have the same perceived datagenerating process for prices (1.11) but differ with respect to rate at which they discount the past(ie. g varies between households and lenders), (ii) the perceived data generating process for pricescould differ between households and lenders, or (iii) heterogeneity could be assumed in the beliefsamongst households and lenders. In the case of (i), the dynamics of the model would be equivalentto formulation considered here with a different overall gain. In the case of (ii), a general commentabout the effect on the model’s dynamics cannot be made, however the available evidence doesnot support the conclusion that financial institutions and households hold structurally differentbeliefs about house prices 17. Finally, in the case of (iii), the model considered here abstracts froma transaction margin in housing and features an intentionally sparse contracting problem. This isdone so as to retain a focus on aggregate price movements. As a result, it is not a good frameworkfor considering heterogeneity in beliefs between households and lenders18.1.3.4 Equilibrium Under LearningThe equilibrium concept of the model is an Internally Rational Expectations Equilibrium (IREE),formalized by Adam and Marcet (2011). An Internally Rational Expectations Equilibrium for this16This is commonly referred to as the anticipated utility assumption (see Kreps (1998) and Sargent (1999)). Inpractice this assumption does not have a significant effect on the results presented in section 1.6 and is made forcomputational ease.17See Cheng, Raina, and Xiong (2012)18Note, Burnside, Eichenbaum, and Rebelo (2011) consider a housing model with heterogeneity between households.One difficulty facing frameworks such as this is calibrating the distribution of prior beliefs across agents.151.3. A Model With Learningeconomy is characteized by:1. A probability measure P i representing an agent’s beliefs over Ωs, where Ωs denotes the spaceof realizations of variables exogenous to an agent.2. A sequence of equilibrium prices {p∗t , q∗t }∞t=0 where p∗t , q∗t : ΩtS → RN+ ∀ t. Markets clear forall t, all realizations in ΩS almost surely in P i.3. A sequence of choice functions {c∗it, h∗it, k∗it, b∗it, k˜∗it, b˜∗it, h˜∗it}∞t=0 that maximize agent i’s objectivefunction conditional on P i. All agents i = 1, ..., I are internally rational.The IREE is closely related to the Self-Confirming Equilibrium concept in Fudenberg and Levine(1998). The key difference is that in a self-confirming equilibrium an agent’s beliefs only need tobe consistent with observations on the equilibrium path (ie. individuals do not observe behaviorthat contradicts their subjective beliefs). An IREE can also coincide with a rational expectationsequilibrium. In particular, a rational expectations equilibrium is an IREE in which subjectivebeliefs P i coincide with the objective probability distribution. Under rational expectations agentsinfer the correct process for prices from their knowledge of the system.Trivially, in order for the model to sustain an IREE in which P i deviates from the objectivedistribution, agents must not have access to the full information set available to agents in a rationalexpectations environment. In practice this implies that agents must be unaware of some equation(s)or identity(ies) in the rational expectations equilibrium. Agents in the learning model are assumedto be uninformed about the preferences and beliefs of other agents. Hence, they are uninformedabout the mechanism that links prices to state variables (computationally, this is equivalent to arepresentative agent who does not have the market clearing condition for housing in their informa-tion set). When households and lenders enter the marketplace they are unaware of how the pricesthey observe relate to the fundamentals of the housing market. As a result agents hold a subjectivebelief about the evolution of house prices and make decisions taking as given the price prevailingin the market and their own beliefs. It is important to emphasize, however, that while agents areunaware of the market clearing condition for housing, the house price that realizes is the price thatclears the housing market given agents’ beliefs and choices.The model is closed by specifying the market clearing condition for housing. The solution tothe household’s problem yields a housing demand equation hd(hdt−1, kt−1, bt−1, qt,mt, θt, yt|P). The161.4. Calibrationmarket clearing price q∗t is determined by the identityhd(hdt−1, kt−1, bt−1, q∗t ,mt, θt, yt|P)− (1− δh)(1−Dt)hdt−1 = Akαt−1 + µ(1− δh) ·(Dthdt−1 + Ft−1)(1.18)where Dt is the proportion of households who default in period t and Ft−1 is the inventory offoreclosed housing that the lender holds at the end of period t − 1. The left hand size of (1.18)denotes new purchases of housing after default choices are made. The amount of housing madeavailable for sale is given by the sum of newly constructed housing and the proportion µ of theforeclosure stock the lender is able to liquidate. The law of motion for F is given byFt = (1− µ)(1− δh) (Ft−1 +Dtht−1) (1.19)The household’s problem is solved via a form of parameterized expectations using spectralmethods. The details of the solution method can be found in appendix C. The lender’s decision ruleis approximated by a simple interpolation of the solution to (1.8). Given these two approximationsthe market clearing prices can be solved for any state vector via (1.18).1.4 CalibrationThe complete set of calibration results can be found in table 1.119. Exogenous variation in themodel comes from the endowment process yt and mortgage rate Rt. The endowment is estimatedas a log AR(1) process using detrended wages and salaries compensation data from the Bureau ofEconomic Analysis (BEA)2021. The mortgage rate series is taken from Freddie Mac’s 30-year fixedmortgage average for the United States. An Augmented Dickey-Fuller test on the series does notreject the hypothesis of a unit root, and the mortgage rate process is estimated as being a randomwalk in logs.22The delay in liquidating foreclosed housing, µ, is set so that foreclosure stock as a fraction of total19Appendix H lists the data sources used for this chapter.20The wages and salaries data is detrended using a bandpass filter. An AR(1) process is specified to limit thenumber of state variables in the model for computational ease.21The model presented in section 1.3 is a zero trend growth environment with a well-defined stationary steadystate, hence the model is simulated with shocks to detrended incomes.22When simulating the model, shocks to y and R are correlated. The correlation is estimated from measured shocksin the data.171.4. Calibrationhousing in steady state equals its 1996 value in the National Delinquency Survey of the MortgageBankers Association. The determination of credit supply in the model implies a correlation betweena weighted average of past prices (mt) and household leverage. The length and volatility of thelenders’ markdown shock {ds}Ts=1 are chosen so as to match this correlation as measured from thedata, as well as the level of loan-to-value (LTV) ratios on US mortgages. Using the Federal HousingFinance Agency’s (FHFA) All-Transaction House Prices Series for the United States, a sequencemˆt(g) is estimated using (1.14). Household leverage is measured using net mortgage liabilities fromthe Federal Reserve Financial Accounts as a fraction of the aggregate market value of non-farmresidential homes, taken from Heathcote and Davis (2007). The level of the series is adjusted tomatch the mean LTV ratio on US mortgages in 1996 Q1, measured in the American Housing Survey(AHS). The parameters are set so that (i) the correlation between these two series matches theimplied correlation of the lender’s choice of θt with mt in the model, and (ii) steady state θ matchesthe LTV ratio in the US in 1996 Q1.The parameters governing the liquidity shock, ζ¯ and σζ , in part control the level of default inthe model as well as its volatility. The pair (ζ¯, σζ) are set so that (i) the level of default in steadystate matches the aggregate delinquency rate on single-family residential mortgages in the US in1996 Q1 (measured by the St. Louis Federal Reserve), and (ii) the elasticity of default with respectto θ matches an estimated elasticity of the delinquency rate with respect to household leveragefrom 1992 Q1 to 2014 Q1.The gain parameter determines both the size and persistence of the response of beliefs to changesin prices. It is therefore key in governing the dynamics of the model. The beliefs are calibratedso as to match forecast errors taken from the data as follows. The perceived DGP for house pricegrowth (1.11) implies the relationshipV ar(logqtqt−1− log qt−1qt−2)= f (σq, σω) (1.20)This identity, together with the identities (1.13) and (1.15) implies a relationshipσ(g) = (σ0(g), σq(g), σω(g)) (1.21)181.4. CalibrationTherefore, the choice of g together with the variance in (1.20) implies the values of the priorvariances σq and σω. In order to choose g the left hand side of (1.20) is measured from the FHFAhouse price series23 and the model is simulated over a grid of g values (where for each g the priorsare set according to σ(g)) using shocks to yt and Rt measured in the data. The chosen gain isthat which minimizes the sum of squared errors between the vector of model-implied one-quarter-ahead forecast errors log qtqt−1 −EPt−1 logqtqt−1 and a data analog of this series. Forecast error data isconstructed using the prices of futures contracts on the S&P Case-Shiller home price index, whichtrade on the Chicago Mercantile Exchange. The futures prices can be thought of as a measure ofthe market’s expectations about house prices 24. The calibrated gain is 0.014. This value is slightlysmaller than the quarterly-implied gain parameter estimated in Adam, Kuang, and Marcet (2011)and sits within the range of values typically found in the learning literature25.The parameter γc in the utility function is equal to the consumption share of disposable income.This is set to 0.558 using BEA data on personal consumption expenditures26. The elasticity ofhousing supply in the model is given by α/(1− α). The parameter α is set using Saiz (2010), whoprovides estimates of local-level supply elasticities computed using data on land availability at theMSA level. The discount factor β is set to 0.96 so that the borrowing constraint binds in eachperiod. In order to ensure a stable solution to the household’s problem a compromise has to bemade in the calibration of the depreciation rate, δh, which is set to the relatively high value of 0.06.It should be noted, however, that this compromise serves to dampen rather than accentuate pricevolatility during model simulations.23The level of the house price series is set using the Census Bureau’s 2005 American Community Survey.24The futures contracts use price movements in the 10-city composite of the S&P Case-Shiller index, which coversthe following housing markets in the United States: Boston, Chicago, Denver, Las Vegas, Los Angeles, Miami, NewYork, San Diego, San Francisco, and Washington, DC. The data series for forecast errors begins in 2007 Q1.25See Adam, Kuang, and Marcet (2011). It should be noted that the size of gains found in the learning literaturevary with respect to the setting concerned. One should not necessarily expect estimated/calibrated gains to be thesame in models where agents learn about asset prices as in models where they learn about economy wide output orwages for example, as it is reasonable to assume that agents’ information about these objects differ.26γc is set equal to the 1999-2012 mean of the sum of the GDP shares for personal consumption expenditures ondurables, nondurables, and services, minus the GDP share of personal consumption expenditures on housing services(imputed rental of owner-occupied non-farm housing).191.5. Analytic ResultsTable 1.1: Calibrated Parameter ValuesParameter Valueβ discount factor 0.96δh depreciation of housing 0.06γc consumption share of income 0.558α curvature on housing production 0.0172θss steady state θ 0.834µ prob. of lender selling housing unit 0.69ρy persistence in yt 0.948defss steady state default rate 2.25%ζ¯ mean liquidity shock 0.124σζ std. dev of log liquidity shock 1.155σ0 posterior std. dev 7.957× 10−4σw priors std. dev. of ωt 9.482× 10−4σq priors std. dev. of qt 6.725× 10−2σd std. dev. of dt 0.026g gain parameter 0.0141.5 Analytic ResultsWhen both households and the suppliers of credit use subjective beliefs to forecast price movements,learning creates complementarities between the two sides of the market. This departure fromstandard demand-side learning frameworks bolsters the internal propagation mechanisms of themodel. In order to demonstrate the difficulty in generating price booms with demand-side learning,consider the model outlined in Section 1.3 without the lender’s problem discussed in 1.3.2. Such amodel is similar to the demand-side learning models of Adam, Kuang, and Marcet (2011); Adam,Beutel, and Marcet (2014); and Gelain and Lansing (2013)27. Self-confirming deviations in pricesoccur through a simple feedback mechanism28qt ↑=⇒ mt ↑=⇒ EPt [Capital Gain] =⇒ qt+1 ↑=⇒ mt+1 ↑27Such a setting can also be related to learning frameworks used to explain stock price volatilities. Winkler (2015)considers an environment in which both investors and firms learn the stock price of the firm. Investors are concernedabout capital gains on their investments while firms have a debt financing constraint that depends upon their marketvalue. This can be considered a decentralization of the setting considered here.28Note that shifts in expected capital gains operate on the household choice through two channels: (i) throughchanges in the expected resale value of housing stock, and (ii) through their effect on the household’s credit constraint.201.5. Analytic ResultsIn order to generate large and persistent price growth without relying upon a rich set of shocksto fundamentals, such a mechanism requires subjective beliefs to exhibit two properties. First,beliefs must be sufficiently responsive to price changes that the resulting response of demand drivessubsequent price increases. Second, the belief process mt itself must be highly persistent. Thetrade-off between the two can be illustrated by deriving a law of motion for mt. Writing thehousehold’s Euler equation (1.17) in simplified formqt = Θt + qtmtEPt [ρt+1] 29 (1.22)This impliesqt =Θt1−mtEPt [ρt+1](1.23)=⇒ log(qtqt−1)= log(ΘtΘt−1)+ log(1−mt−1EPt−1[ρt]1−mtEPt [ρt+1])(1.24)Substituting (1.24) into (1.14) yieldslogmt = (1− g) logmt−1 + g log(ΘtΘt−1)+ g log(1−mt−1EPt−1[ρt]1−mtEPt [ρt+1])(1.25)linearizing this equation yields:lnmt ≈ g1− ρ¯− gρ¯(1− ρ¯) ln(ΘtΘt−1)︸ ︷︷ ︸value of housing+ ρ¯Et ln(ρt+1ρt)︸ ︷︷ ︸price of consumption+≡P︷ ︸︸ ︷1− ρ¯− g1− ρ¯− gρ¯ · lnmt−1︸ ︷︷ ︸exp. capital gains + updating(1.26)29WhereΘt = uh(t)/uc(t)ρt+1 =(β(1− δ) · λit+1λit· Γt+1 + θtR)· Σt+1 − βθt(EPt Σt+1) λit+1λitΓt+1Σt+1 = exp(σ0t+1 + qt+1 + ωt+1)t ∼ N(0, 1)211.5. Analytic ResultsNote that the persistence parameter P is downward sloping with respect to gdPdg=−ρ¯2 + 2ρ¯− 1(1− ρ¯− gρ¯)2< 0 if ρ¯ 6= 1= 0 else(1.27)The law of motion (1.26) makes clear that in the absence of highly persistent shocks or stronginternal propagation mechanisms in the model, persistent growth in mt requires a relatively largevalue for P . Given (1.27), this can be achieved be lowering the value of the gain parameter.The gain parameter, however, determines the weight that agents place on new information whenupdating beliefs. Hence, a reduction in g diminishes the responsiveness of beliefs to price changes.The trade-off is illustrated in figure 1.3, which shows the response of log prices to a wage shock inthe demand-side learning environment. When the gain is low the shift in beliefs after the shock hitsis insufficient for the effect of higher expected capital gains to outweigh the effect of the wage shockdying out. As a result, the shock does not propagate. In contrast, under high gain calibrations theshock is propagated through prices. However, owing to household beliefs placing a relatively largerweight on current shocks (represented through the Θt and ρt terms in (1.26)), once the wage shockdies off mt is quick to readjust to fundamentals and qt returns to steady state faster than was thewas the case under a low gain. This tight trade-off between the persistence and responsiveness ofbeliefs under demand-side learning implies that such frameworks can struggle to generate the kindof sustained price growth seen in the data without a similarly persistent set of shocks.The credit supply problem in 1.3.2 introduces a complementary learning mechanism into themodel. Shifts in expected log price growth, logmt, lead the lender to increase credit supply dueto lower perceived default risk and higher expected payoffs in the event of default. As householdsare constrained this increases their demand for housing. As before, the shift in mt also pushes upthe demand for housing through its effect on expected capital gains. Hence, shifts in subjectivebeliefs give rise to credit supply changes that complement the effect of demand-side learning onprices. The parallel channels through which learning operates in the full model can be illustrated221.5. Analytic ResultsFigure 1.3: Response to Endowment Shock, Demand-Side Learning0 10 20 30 40 50g = 0.002g = 0.016g = 0.03as follows:q ↑=⇒ mt ↑=⇒Household:Lender:EPt [Capital Gain] ↑ PrP(Default) ↓EPt (Default Value) ↑ =⇒ θt ↑=⇒ Demand ↑=⇒ q ⇑Intuitively, the introduction of learning on the credit supply side should increase the persistencein the belief process. Following the an initial shock to prices and a shift in mt, the subsequent changein prices should be greater as there is both a demand and credit supply response to the belief shift.This will imply that the change in mt in the second period following the shock will be greater thanunder demand-side learning and so on. As a result, for any level of responsiveness of beliefs toshocks the propagation cycle should be longer.In order to explicitly show the influence of this parallel mechanism on the dynamics of householdbeliefs an approximate law of motion for mt is derived by combining (1.25) with the lender’s decision231.6. Quantitative Resultsrule θ(mt). Taking a linear approximation yieldslogmt ≈≡P ′︷ ︸︸ ︷{1− g(1 + µ1 + µ2∂¯θ∂m)}{1− g(µ1 + µ2∂¯θ∂m)} · logmt−1︸ ︷︷ ︸exp. capital gains+updating+credit supply+EtD(g, θ¯, R, ρ¯, γc) · log htht−1logρt+1ρt (1.28)where µ1, µ2 < 030. The autoregressive coefficient P ′ now has the following propertiesdP ′d∂θ¯/∂m> 0 ;d (abs(dP ′/dg))d∂θ¯/∂m< 0 (1.29)While the autoregressive coefficient is decreasing in the gain as before, the effect of learning incredit supply on beliefs is clear. First, increasing the elasticity of credit supply with respect tomt increases the persistence in mt conditional on the gain. Second, as the elasticity of creditsupply with respect to mt increases the influence of the gain on persistence P′ decreases. This isthe key influence of the complementarity induced by two-sided learning. If lenders are sufficientlyresponsive to their beliefs in the full model, the trade-off in trying to generate beliefs that are bothpersistent and sensitive to price changes that exists in demand-side learning frameworks can bebroken. Section 1.6 tests whether this is indeed the case in the calibrated model.1.6 Quantitative ResultsFollowing a number of papers in the literature the constant gain algorithm is modified for modelsimulations as follows:lnmt = lnmt−1 + g(σq, σω) ·(lnqt−1qt−2− lnmt−1)(1.30)30Combing the (1.24) with (1.3) and (1.2) yieldslogqtqt−1= µ1 logmtmt−1+ µ2θ¯ logθtθt−1+ µ3 loghtht−1+ µ4 logΓt+1Γt+ µ5 logΣt+1Σtµ0 = 1− ρ−(1− γcγc)·(θ¯R− δ); µ1 =1µ0·(ρ+θ¯R·(1− γcγc))µ2 =1µ0· 1R·(1− γcγc)µ3 = − 1µ0·(1− ρ+(1− θ¯R)(1− γcγc))µ4 =1µ0·(ρ− θ¯R)µ5 =ρµ0241.6. Quantitative ResultsThis assumption avoids simultaneity between the determination of prices and beliefs, and signif-icantly speeds up the computation31. Furthermore, in order to guarantee stability a constraintis imposed on the lenders credit supply choice when simulating the model: θt ≤ θ¯. This can beconceptualized as a regulatory constraint on LTV ratios. In practice, the model does not hit theconstraint when simulating at business cycle frequencies. θ¯ is set equal to Interest Rates & Boom-Bust in House PricesFigure 1.4: Response of Log House Prices to Interest Rate Drop-0.0500. 2001 2002 2003 2004 2005 2006 2007 2008TimeFull ModelDemand-Side LearningDataThe chapter now considers the model’s ability to endogenously generate persistent growth inhouse prices consistent with the observed boom in US housing markets in the mid-2000s. Giventhe paucity of empirical evidence that identifies significant trends in housing market fundamentalsduring this period, it is desirable that models should be able to generate persistent price growthfollowing a small set of shocks. This section evaluates the potential for interest rate movements inparticular to generate boom-bust periods in the calibrated model.In recent years there has been wide discussion about the extent to which monetary policycontributed to the 2008 financial crisis in general and to the house price boom more particularly.An argument commonly advanced in both the popular and academic literature is that persistently31See Adam, Kuang, and Marcet (2011) and Eusepi and Preston (2011).251.6. Quantitative Resultslow interest rates encouraged excessive borrowing through the early 2000s. The ensuing effect ofthe credit expansion on demand for housing may have in turn generated appreciation in houseprices over a near-10-year period. A growing empirical literature links interest rate movementswith periods of financial instability. Hott and Jokipii (2012) show that over the past 30 years,across a sample of 14 OECD countries, periods of low interest rates Granger-cause deviations ofhouse prices from fundamentals-implied levels (the authors’ characterization of a bubble). In asimilar vein, Ahearne et al. (2005) show that across advanced economies house price bubbles tendto be preceded by a period of loosening monetary policy.The early 2000s saw a period of abrupt decreases in mortgage rates across the US economy.Beginning in late 2000 the 30-year conventional mortgage rate in the US began a 3% drop, andthereafter remained relatively low until 2006. The rate drop coincided with an acceleration inthe aggregate house price index for the US. In the environment presented in section 1.3.1, sucha rate decrease not only drops the borrowing costs that households face, but also serves to relaxtheir credit constraints. As discussed in section 1.5, the learning framework considered here givesthe model strong internal propagation mechanisms by allowing for potentially large persistence insubjective beliefs without sacrificing the responsiveness of these beliefs to new information.In order to investigate whether the calibrated model can explain the pattern of house pricesin the 2000s the effect of an unanticipated drop in R is considered. The model is simulated fromsteady state with an initial interest rate set equal to the mean 30-year conventional mortgage ratein the US from 1996 Q1 to 2000 Q4. The drop in R is calibrated to match the mean US mortgagerate from 2001 Q1 to 2006 Q432. Figure 1.4 plots the response of log prices to the unanticipatedrate drop, together with both the response of prices when learning is restricted to the demandside of the housing market33and the actual path of the FHFA All-Transaction House Price Index.Under demand-side learning the shift in the distribution of expected capital gains is insufficient tosubsequently generate large shifts in housing demand. As a result the propagation of the interestrate shock in house prices is negligible. In the full model the shift in beliefs mt following the initialincrease in prices generates an increase in credit supply relative to the market value of housing. As32This follows from an exercise carried out in Adam, Kuang, and Marcet (2011). A similar exercise can be foundin Kuang (2014).33In the “demand-side learning model” considered from here on, only households are assumed to have the beliefsspecified in section Quantitative Resultsis clear in figure 1.4, this additional mechanism has a dramatic influence on the evolution of pricesfollowing the shock. The model can account for the full appreciation in US house prices in theearly 2000s with total growth between 2001 Q1 and 2006 Q4 slightly overstating the level observedin the data. Furthermore, the model can explain much of the persistence in prices following 2001.Following 2001 Q1, price growth persists for 20 quarters in the model, compared with 22 quartersin the data series. Importantly, the model is also capable of capturing asymmetry in boom-bustcycles. Following the peak in the simulated price series in figure 1.4, prices collapse to the steadystate level within 12 quarters.Figure 1.5: Response of Log House Prices and Credit to Interest Rate Drop1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009priceTime  abs(FE)priceabs(FE)(a) House Price and Absolute Value of the 1-Quarter-Ahead Forecast Error0.0040.0050.0060.0070.0080.0090.011998 2000 2002 2004 2006 2008Net New Borrowing / Market Value of HousingTimeboth sidesdemand sidedata(b) LeverageTwo comments should be made about this result. First, the literature on learning has notyet reached a settled view on how to appropriately calibrate gain parameters in constant gainlearning. As a result a wide range of values are found in the literature and the estimated gain inthis chapter sits in the high end of this spectrum. Second, as noted in section 1.4, the calibrationof the model suffers from the necessity of imposing a relatively high value for δh. In order to gaugethe sensitivity of the price response to these assumptions, the previous exercise was carried outover a grid of g and δh34. Figure A.1 in appendix A plots the maximum deviation of log prices34Note that for the different values of g the priors and parameters of the lender’s problem were recalculated as insection Quantitative Resultsfrom steady state following the R drop as well as the length of the propagation period (the numberof quarters of positive price growth following the shock) over (g, δh). Conditional on a given δh,decreasing the gain parameter actually increases the size of price growth following the interest rateshock. Similarly, increasing the depreciation rate on housing serves to augment rather than dampenprice growth in the model. In both cases, changes in δh or g from the values listed in table 1.1have an ambiguous effect on the length of the propagation period. However, the finding that pricegrowth persists for many quarters following the interest rate shock is robust to perturbations ofeither parameter in the neighborhood of the calibrated values.As discussed in section 1.1, the available data on price forecasts suggest that agents’ forecasterrors tend to be largest following turning points in the price series. This is consistent with themodel’s response to the R shock. The left hand panel of figure 1.5 plots the absolute value ofthe forecast error of log price growth in the model following the unanticipated rate drop. Thelarge spike in forecast errors at the point of the interest rate drop drives the initial appreciationin expected price growth. As beliefs adjust to higher prices the magnitude of the forecast errorsdecreases. This slows the growth in mt, and as a result growth in qt, until forecast errors go to zeroand the model hits a turning point. The ensuing collapse in the simulated prices and beliefs mt isdriven by a series of large negative forecast errors. The model also captures comovements in houseprices and household leverage.The right hand panel in figure 1.5 plots a simulated series of household leverage measures.Because debt contracts are one period in length and the framework does not feature an occasionallybinding constraint, the model abstracts from some dynamics of household debt. In order to makethe data and model series comparable, the right hand panel in figure 1.5 plots net new mortgageborrowing as a fraction of the market value of housing stock in both the data and the model35.Consistent with evidence provided in figure 1.2, the model generates increases in credit supplyand household leverage concurrent with the takeoff in house prices. Under demand-side learningthe growth in leverage is negligible. By contrast the full model can generate almost a third ofthe growth in the leverage measure seen in the data. Because the lender’s problem is highlystylized, the leverage series tracks house prices closely following the shock. This is the result of thelenders’ choice being predominantly a function of mt through the lender’s credit supply problem35Note, the series are normalized to 2001 Q1.281.6. Quantitative Resultsoutlined in section 1.3.2. Nevertheless, the framework suggests a potentially powerful mechanismfor endogenously generating liberalizations in credit markets during price booms.Table 1.2: Business Cycle MomentsData Learning Rational1978Q1 - 1978Q1 - Demand-Side Full Learning Expectations2014Q1 1990Q1 Learning Modelstd(·)/std(y)q 7.686 2.488 1.715 2.452 0.598q · h∗ 9.824 2.968 1.770 2.480 0.607fe(q/q−1) 1.367 1.367 0.556 0.652 0.334F · · 0.849 0.506θ 15.560 7.006 · 1.943 9× 10−4skewness(·)q 1.118 0.381 0.016 0.195 -0.102q · h 1.345 0.054 0.017 0.196 -0.102fe(q/q−1) -0.074 -0.074 0.016 0.065 0.016F · · -0.028 0.152 -0.011θ -0.021 0.208 · 0.178 0.086Reported moments are for log values. Simulation moments are taken from simulated sampleof size 50000 quarters, with the first 1000 quarters dropped as burn-in. Reported data momentsof one-quarter-ahead forecast errors are from CME data sample running from 2007Q1 to2011Q1. Wage data, y, is detrended using a bandpass filter with frequency range 1/32to 1/8 cycles per quarter.∗ data for market value of housing stock taken from Heathcote and Davis (2007), seesection Capturing Cyclical Variation in Housing MarketsIn order to test whether the results in section 1.6.1 come at the expense of the the model’s abilityto capture normal cyclical variation in prices and forecast errors, several large-sample simulationsare carried out. Table 1.2 shows moments for simulations of the full model as well as for the modelwhen learning only takes place on the demand side of the housing market, and the model underrational expectations. Note that the presence of the mid-2000s price boom has a large effect on291.6. Quantitative Resultsmeasured volatilities. Table 1.2 therefore includes data moments for a pre-boom sample covering1978Q1 - 1990Q1 in order to gauge moments during ‘normal’ periods. Under rational expectationsneither prices nor the market value of housing display anything like the volatility seen in the data.The standard deviation of each of these series, relative to wages, is only about 20-25% of that seenin the pre-boom sample. Furthermore, under rational expectations the relative standard deviationof the one-quarter-ahead forecast error of house price growth is a quarter of that measured in thedata.Table 1.3: Business Cycle CorrelationsData Learning RationalDemand-Side Full Learning ExpectationsLearning Modelcorr(·, y)q 0.234 0.438 0.362 0.964q · h 0.268 0.437 0.362 0.964fe(q/q−1) -0.613 0.054 0.060 0.319F · 0.062 0.260 -0.999θ -0.143 · 0.315 ≈ 0corr(·, R)q -0.586 -0.770 -0.766 0.219q · h -0.492 -0.772 -0.769 0.218fe(q/q−1) -0.433 -0.194 -0.237 0.113F · 0.225 -0.506 -0.142θ -0.869 · -0.694 -0.215Reported moments are for log values. Simulation moments are taken from simulatedsample of size 50000 quarters, with the first 1000 quarters dropped as burn-in. Reporteddata moments of one-quarter-ahead forecast errors are from CME data samplerunning from 2007Q1 to 2011Q1. Wage data, y, is detrended using a bandpass filterwith frequency range 1/32 to 1/8 cycles per quarter.The introduction of bounded rationality on the demand side of the market significantly increasesvolatilities in the model, however the relative standard deviation of prices and the market value ofhousing remain 31% and 40% below their pre-boom data values. By contrast, the full two-sided301.6. Quantitative Resultslearning model comes close to matching both of these moments. The model is able to capturealmost all the observed volatility in prices and the vast majority the relative standard deviationin the market value of housing. The volatility of the housing stock is relatively low in the model.This is a result of the very simple framework governing the construction of new housing stock.As a result the model performs better in explaining price variation than it does variation in themarket value of total housing stock. Introducing interaction between near-rational households andnear-rational lenders similarly increases the volatility of forecast errors, with the relative standarddeviation 17% higher under two-sided learning. While the model fails to capture the volatility inhousehold leverage, the introduction of bounded rationality amongst lenders nevertheless producesan extreme increase in the volatility of θ relative to the rational expectations model (as a resultthere is also a large increase in the volatility of the foreclosure inventory).In order to gauge the robustness of these findings, figure 1.6 plots the relative standard devi-ation of simulated prices and forecast errors in the model for different values of g. For each g,{σ0, σq, σω, µ,K} is recalibrated in line with section 1.4. As can be seen, in the region of the valueof g found in section 1.4 the volatilities displayed by the model remain close to the data valueslisted in table 1.2. As is also clear in table 1.2, the model provides a closer fit to the third momentsmeasured in the data. Neither the rational expectations model nor the demand-side learning modelcan produce the positive skew observed in house prices and household leverage. The model alsomatches the observed skewness in the forecast error series.Table 1.3 shows correlations between the simulated data and the exogenous processes y and R.Given the literature indicating that interest rates have been a key driver of house prices over the past30 years (see Hott and Jokipii (2012)) the correlations of simulated prices and forecast errors with Rare of particular interest. Both the rational expectations model and the model with only demand-side learning overstate the negative correlation between prices (and the market value of housingstock) and mortgage rates relative to the data. The introduction of learning amongst lenders haslittle effect on the measured value of this correlation. The full learning model also captures 55% ofthe observed negative correlation between R and forecast errors, an improvement upon the demand-side learning model, and comes close to matching the observed correlation between θ and R. Ascan also be seen in table 1.3, the full model can also capture the observed correlation between houseprices (and the market value of housing stock) and wages.311.7. ConclusionFigure 1.6: Standard Deviation of q and fe(q/q−1) Relative to y0.005 0.01 0.015 0.02 0.025 0.03 0.0351.522.53Gainrel. std(price) std(FE)rel. std(price)rel. std(FE)Turning to the time-series properties of the model, figure 1.7 plots the periodogram of pricesfor the three models listed in tables 1.2 and 1.3. Under rational expectations house prices fail todisplay the level of low-frequency variation seen in the data. This problem is alleviated throughthe introduction of bounded rationality in the model. Both the full model as well as the modelwith only demand-side learning can capture the bulk of the low frequency (ie. less than 0.2 cyclesper quarter) variation in the data, however the full learning model provides a marginally betterfit to the data spectrum. Autocorrelations for forecast errors and θ are shown in table 1.4. Thefull learning model matches the first order autocorrelation in forecast errors, however it overstatespersistence in the series at higher lags. The learning model also overstates persistence in householdleverage, however it significantly outperforms the rational expectations model in this regard.In sum, the model developed here weakly dominates the demand-side learning model alongthe dimensions discussed. The model captures almost all of the volatility in house prices over thebusiness cycle whilst providing a closer fit to observed correlations with market fundamentals.1.7 ConclusionThis chapter provides a framework for explaining asset price booms. A general equilibrium modelwith learning can quantitatively explain US house price growth in the 2000s and account for volatili-321.7. ConclusionTable 1.4: AutocorrelationData∗ Learning RationalDemand-Side Full Learning ExpectationsLearning Modelψt (fe(q/q−1)1 0.412 0.154 0.438 −4.27× 10−42 -0.091 -0.037 0.156 −7.19× 10−45 -0.068 -0.042 -0.054 −5.10× 10−310 -0.282 -0.034 -0.081 −1.08× 10−2ψt (θ)1 0.992 · 0.998 0.9992 0.984 · 0.995 0.9995 0.959 · 0.985 0.99810 0.862 · 0.971 0.998∗ autocorrelations for θ are taken on sample from 1978Q1-2014Q1Reported moments are for log values. Simulation moments are taken from simulated sampleof size 50000 quarters, with the first 1000 quarters dropped as burn-in. Reported datamoments of one-quarter-ahead forecast errors are from CME data sample running from2007Q1 to 2011Q1. Wage data, y, is detrended using a bandpass filter with frequencyrange 1/32 to 1/8 cycles per quarter.ties in house prices and forecast errors when agents learn subjective beliefs about prices. The modeldeparts from established frameworks in allowing for bounded rationality amongst both householdsand the suppliers of credit. In demand-side learning models there exists a trade-off between gener-ating subjective beliefs that are persistent and at the same time highly responsive to innovations.The interaction between the demand and credit supply sides of the market under learning boostthe persistence in beliefs, thereby breaking this trade-off. A single permanent and unanticipateddecrease in mortgage rates, consistent with the observed drop in US interest rates in the early2000s, produces 20 quarters of price growth whilst capturing the total growth in prices observedacross the US. The model also outperforms both demand-side learning and rational expectationsmodels in capturing key business cycle moments.331.7. ConclusionFigure 1.7: Periodogram of Log House Prices0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−55−50−45−40−35−30−25−20−15−10FrequencyPower/Frequency  Full ModelDemand−side LearningREEDataThere are several further issues that can explored using the framework developed here. Inparticular, the chapter has assumed homogeneous beliefs between households and credit suppliers,however this can be easily relaxed. The open economy assumption implies that interest rate move-ments in the model are exogenous. The model can be augmented to consider endogenous mortgagerates and the influence of movements in mortgage rate expectations amongst households. Similarly,a natural line of inquiry would be to augment the funding side of the lenders problem and considerthe effects of uncertainty about financial risk.34Chapter 2Learning in a Business Cycle Modelwith Recursive Preferences2.1 IntroductionLearning and adaptive expectations has been increasingly employed in macroeconomics in orderto address the failure of canonical models to generate rich amplification or propagation whilecapturing commonly-observed properties of household and investor expectations. While the processof updating beliefs can be an important source of propagation, the assumption that agents holdsubjective beliefs alters their understanding of how observed variation relates to long run outcomes.A large body of work now exists that uses variation in expected long run consumption growth toexplain key asset pricing moments. In canonical long run risk models, persistent and forecastablecomponents in the consumption growth process generate large variation in discount factors undercertain recursive preference structures. This chapter demonstrates that the introduction of adaptivelearning can have a substantial effect on long run consumption risks generated in otherwise standardframeworks.This chapter formulates a model in which agents learn by updating their beliefs about thestructural parameters of the economy. As has been studied elsewhere in the learning literature,agents in the model are uninformed about the technology that generates rental rates of capital andwages. The learning framework respects the internal rationality of agent choice while allowing forfeedback between boundedly-rational beliefs and equilibrium outcomes. Agents attempt to learnthe process governing wages and rental rates, which are themselves functions of investment choicesand therefore beliefs. In order for long run consumption risks to be priced, households in the modelare also endowed with Epstein-Zin preferences.Epstein-Zin preferences have become widely used in both macroeconomics and finance as theyallow the rate at which households discount consumption across periods to explicitly be a function of352.1. Introductionexpected long run consumption growth. As has been extensively studied in the literature, Epstein-Zin utility specifies preferences over the timing of the resolution of uncertainty. The combination ofthis preference with beliefs about the underlying stochastic processes in the economy generates longrun consumption risk36. When agents have a preference for the early resolution of uncertainty theyhave an aversion to variation in long run consumption growth. Environments in which shocks arepermanent are therefore riskier as variation in current consumption growth is positively correlatedwith variation in growth over long horizons. Conversely, a preference for the late resolution ofuncertainty generates more long run risk when shocks are temporary.The chapter shows that a key implication of the class of learning models considered here istheir effect on the perceived persistence of shocks. The propagation induced by feedback betweenboundedly-rational beliefs and outcomes increases overall volatility and variation in investment.However, unanticipated variation in wages and the rental rate of capital is perceived as being theresult of shocks with extremely low persistence, regardless of whether the underlying shocks thatcause it are themselves temporary or permanent. As a result, the generation of long run risks in thelearning model requires a preference for the late resolution of uncertainty. The is in contrast to muchof the macro finance literature studying long run risks, in which a preference for early resolution ofuncertainty is assumed. Importantly, this result is shown to be true regardless of whether agents areable to anticipate the evolution of their own beliefs. Agents who internalize the effect of observedoutcomes on the future evolution of beliefs can anticipate the long-run consequences of shocks.However, even when agents place a large weight on recent observations when updating beliefs, theperceived persistence in shocks is insufficient to generate long run risks with a preference for theearly resolution of uncertainty.The chapter is structured as follows. Section 2.2 discusses the literature in which this work isplaced. The model together with the learning framework and equilibrium concept are describedin section 2.3. Section 2.4 analyzes the model under rational expectations in order illustrate therole the preferences play in generating long run risks. The quantitative analysis of the full modelis presented in section 2.5. The chapter concludes in section 2.6.36See Lochstoer and Kaltenbrunner (2010)362.2. Related Literature2.2 Related LiteratureThe use of learning to generate amplification and persistence in macro models has met with mixedsuccess. Much of this literature has studied learning mechanisms in which households forecast futurevalues of their own choice variables. Bullard and Duffy (2002), Carceles-Poveda and Giannitsarou(2008), Ellison and Perlman (2011), Evans and Honkapohja (2001), Williams (2003), and Zhang(2012) all consider variants of the so-called Euler equation or reduced form approach to learning37. Under this framework households make forecasts using estimates of the law of motion forthe complete vector of endogenous variables in the economy, typically under the assumption thathouseholds know the functional form of the solution to the rational expectations equilibrium.These frameworks have been subject to two primary criticisms. First, in these models individu-als are limited to only caring about one-period-ahead forecasts of the economy. In standard settingsone-period-ahead forecasts of consumption, labour supply and investment will completely deter-mine decisions when combined with the optimality conditions of the household’s decision problem.This fundamentally changes the nature of the household’s decision problem relative to rationalexpectations. Under rational expectations, households make consumption-savings decisions con-tingent on a forecasted sequence of prices. Branch and McGough (2011) show that lengtheningthe forecasting horizon of boundedly-rational households can increase the propagation of shocks inmacro models. An additional criticism of the reduced form approach to learning is that it removesany conceptual distinction between the household’s beliefs and its decisions. Not only does thismake models difficult to interpret, but it also yields consumption-savings decisions that are notinternally consistent. Boundedly-rational households compute forecasts of future choice variablesusing their forecasting functions, however these forecasts are not restricted to be consistent withwhat optimal decisions will be given current beliefs about future prices and allocations. Agents areimplicitly assumed to not use part of their own information set when making choices38. 39This chapter contributes to an area of the learning literature in which households are consideredto be uninformed about some aspects of the underlying structure of the economy. This structural37Branch and McGough (2011)38Preston (2005)39For more discussion see Branch and McGough (2011), Bullard and Duffy (2002), Carceles-Poveda and Gian-nitsarou (2008), Ellison and Perlman (2011), Evans and Honkapohja (2001), McCallum (1999), McCallum (2007),Preston (2005), Williams (2003), or Zhang (2012).372.2. Related Literatureapproach to learning separates household decisions and beliefs, avoiding the consistency problemsinherent in the reduced form approach, while also making the household decision problem dependupon long-run forecasts. Preston (2005) and Eusepi and Preston (2011) show that learning frame-works in which households forecast only the paths of future prices can substantially improve theinternal propagation of shocks in standard business cycle models. The model presented in thischapter considers households who are uninformed about the formation of prices in the economy.Households attempt to learn the parameterization of the economy by estimating the relationshipbetween prices and the system’s state variables in each period. The nature of the household’s deci-sion problem is similar to rational expectations, the difference being that the perceived parametersunderlying the economy differ from their true values. The household is therefore not limited toconsidering only one-period-ahead forecasts in this framework, and both decisions and the expectedvalues of future choice variables are consistent with optimizing behavior given maintained beliefs.40This chapter also contributes to a literature which studies the effect of recursive preferences inmacro models. Following the introduction of the long-run risk model of Bansal and Yaron (2004),recursive preferences have become popular in macroeconomics. The underlying appeal is twofold.First, they provide greater flexibility in that they separate risk aversion and the intertemporal elas-ticity of substitution41. That is, separate parameters control how the agents treat uncertainty withinand between periods, implying a preference for either early or late resolution of uncertainty. Sec-ond, under this class of preferences the household’s discount factor becomes a function of expectedlong-run consumption growth. Recursive preferences can therefore bolster propagation mechanismsin models where agents infer information about future consumption from current shocks or in whichuncertainty varies over time42. For example, Malkhozov and Shamloo (2010) show that a modelwith recursive preferences and news shocks replicates key macroeconomic data, Gourio (2013) usesvariation in the probability of economic disaster together with recursive preferences to explain assetpricing moments, while Kuehn, Petrosky-Nadeau, and Zhang (2012) introduce recursive utility toa model with labour market frictions. Thus far, there has been limited study of whether modelswith non-rational expectations can generate the type of behavior proposed by Bansal and Yaron40Eusepi and Preston (2011) Preston (2005)41See Caldara et al. (2009) for discussion.42See Bansal and Yaron (2004) and Lochstoer and Kaltenbrunner (2010) for discussion.382.3. Model(2004) in production economies. Hirshleifer, Li, and Yu (2015) considers subjective expectations ina model with recursive preferences by introducing households that extrapolate future productivitygrowth from recent levels, matching key macroeconomic and financial moments in the process. Thischapter in contrast considers an environment that allows for feedback between agents’ beliefs andobserved outcomes, as they form beliefs about variables that are equilibrium objects. As discussed,this is in line with much of the recent learning literature.2.3 ModelThis section outlines the baseline model of the chapter. Households are endowed with Epstein-Zin preferences. The information structure is similar to the structural learning environments ofWilliams (2003) and Eusepi and Preston (2011). Individuals are uninformed about the formationof prices in the economy and hold boundedly-rational beliefs about their evolution over time. Inkeeping with much of the learning literature, beliefs are updated using a form of constant gainlearning.2.3.1 Household ProblemHouseholds hold Epstein-Zin preferences43 over consumption ctVt = max{c}{(1− β)c1−ψt + β(EPt[V 1−γt+1])1−γ} 11−ψ(2.1)where EPt [·] denotes the household’s expectations over its beliefs P at time t. Preferences of thistype separate the agent’s preference for risk across time periods and preference for risk across stateswithin a period. The parameters γ and ψ denote, respectively, the household’s coefficient of relativerisk aversion and the inverse of the elasticity of intertemporal substitution (EIS). Together thesetwo parameters govern the agent’s preference for the timing of the resolution of uncertainty. Whenγ > ψ (γ < ψ) the agent prefers the early (late) resolution of uncertainty.The rest of the household problem follows a standard real business cycle formulation. Agentsaccumulate capital kt, which they rent to firms at the rate rkt and inelastically supply 1 unit of43See Epstein and Zin (1989)392.3. Modellabour to firms at wage wt. The agent chooses both consumption, ct, and investment, it, and canalso borrow at+1 at the risk-free rate Rft . The flow budget constraint of the household is given byct + it + at+1 = rkt kt + wt +Rft−1at (2.2)The accumulation of capital is subject to convex adjustment costs, in particularkt+1 = (1− δ)kt + φ(itkt)kt (2.3)where φ(·) is given byφ(X) = a1 +a21− 1/ζX1−1/ζ (2.4)The magnitude of the capital adjustment costs is decreasing in ζ. Following Lochstoer andKaltenbrunner (2010) and Boldrin, Christiano, and Fisher (2001) the coefficients in the adjust-ment cost function a1 & a2 are set so that the steady state of the model coincides with the casewhere no adjustment costs are present.44The optimality conditions for the household are given byEPt [Mt+1 ·Rit+1] = 1 (2.5)EPt [Mt+1 ·Rft ] = 1 (2.6)where Mt+1 denotes the stochastic discount factor and is given byMt+1 = β(ct+1ct)−ψ ( Vt+1EPt (V1−γt+1 )1/1−γ)ψ−γ(2.7)The return on capital investment Rit+1 in equation 2.5 is given byRit+1 = φ′(itkt)·{rkt+1 +1− δ + φ(it+1/kt+1)φ′(it+1/kt+1)− it+1kt+1}(2.8)44a1 & a2 are set so that φ(iss/kss) = iss/kss and φ′(iss/kss) = 1. This implies a1 = 11−ζ (exp(µ) + δ − 1) anda2 = (exp(µ) + δ − 1)(1/ζ).402.3. Model2.3.2 TechnologyThe production technology is given byyt = exp(Zt)1−αkαt (2.9)where Zt is an exogenous TFP process. The technology impliesrkt = α · yt (2.10)wt = (1− α) · yt (2.11)Following Lochstoer and Kaltenbrunner (2010), the TFP process is given byZt = exp(µt+ z˜t) (2.12)z˜t = ρz˜t−1 + zt (2.13)where zt ∼ N(0, σ2z) and ρ ≤ 0. As has been studied extensively in the literature, the interactionbetween an agent’s expectation as to whether shocks are transitory or permanent (ie. whether ρ < 1or ρ = 1 in a rational expectations setting) and their preference over the timing of the resolutionof uncertainty is key in generating long run risks. In the exposition x˜t ≡ xt/ exp(µt) denotes avariable normalized to account for the effect of trend growth.2.3.3 Beliefs & LearningBeliefs are assumed to be homogeneous across all agents in the economy. Similar to chapter 1beliefs are specified so as to respect the internal consistency of the agent’s decision problem. Statedonce again, this implies that an agent’s decision rule must everywhere be an optimal response toits beliefs P. As a result only variables outside the agent’s decision set can have their true datagenerating process excluded from the agent’s information set. As with the learning environmentconsidered in Chapter 1 the model will feature an Internally Rational Expectations Equilibrium.The assumed information structure is similar to the structural learning frameworks studied inWilliams (2003) and Eusepi and Preston (2011). Agents are uninformed about the true process412.3. Modelgoverning prices rt and wt. In particular, the technology (2.9) and the associated pricing functions(2.10) & 2.11) are assumed to be outside the information set of the an agent. In order to makeinternally consistent choices agents must hold a belief about the behavior of the prices. Thetechnology (2.9) implies that rt and wt are log-linear with respect to capital. In order to minimizethe misspecification in beliefs, agents are also assumed to perceive this to be the case. Agentsbelieve that rt and wt are governed by the model log rtlog w˜t = ωr0ωw0+ ωr1ωw1 · log k˜t + rtwt (2.14)where [rt , wt ] ∼ N(0,Σ). Note, that (2.14) implicitly assumes that agents are aware of the trendrate of growth in economy, µ. Agents update their beliefs about the coefficients ω = [ωr0, ωw0 , ωr1, ωw1 ]′in (2.14) each period to account for new information45. The timing assumptions here are important.The agent enters period t having a belief about ω, denoted ωˆt−1 = [ωˆr0,t−1, ωˆw0,t−1, ωˆr1,t−1, ωˆw1,t−1]′. Itmakes its consumption-investment decision in t taking these values as given in 2.14. At the end ofthe period the household then updates its beliefs. 46As in chapter 1, optimal Bayesian updating of the coefficients in 2.14 can be shown to convergeto a simple constant gain algorithm when agents take ωt to be an unobserved state vector followinga random walk. That is, when agent’s believe log rtlog w˜t = ωr0,tωw0,t+ ωr1,tωw1,t · log k˜t + rtwt (2.15)ωt = ωt−1 + ωt (2.16)where ωt ∼ N(0, σω), then EPt [ωt] = ωˆt−1 evolves according toωˆt = ωˆt−1 + g · q′t(qtq′t)−1(xt − qtωˆt−1) (2.17)45Beliefs about the intercepts ωr0 and ωw0 can be thought of as reflecting uncertainty about long-run trends intechnology. Beliefs about the slope coefficients ωr1 and ωw1 can be thought of as reflecting uncertainty about howobserved prices correlate with capital.46The assumed belief structure in (2.14) departs from Eusepi and Preston (2011) in that it does not include aperceived law of motion for capital. Agents here understand that capital will evolve in line with optimal investmentresponses to observed prices and beliefs.422.3. Modelwhere xt = [log rt, log w˜t]′ andqt = 1 0 log k˜t 00 1 0 log k˜t (2.18)The details are shown in Appendix E. Three furthers points need to be be made at this juncture.First, the assumption that the agent updates its beliefs about ω after the the observation of rtand wt, and after having made its period t choices, is made so as to avoid simultaneity in thedetermination of ωˆt and xt. This assumption is common in the learning literature.47Second, asthe prices rt and wt are equilibrium objects they are themselves functions of ωˆt. Thus the modelallows for additional propagation mechanisms through the feedback between beliefs and outcomes.Finally, because ωˆt is a function of agent choices the model allows for the possibility that agentsinternalize the effect of their investment on future beliefs. In the quantitative evaluation that followsthe model is analyzed both under the assumption that agents internalize the impact of their choiceon their learning and that they do not.2.3.4 Equilibrium Under LearningThe equilibrium of the model under learning is an Internally Rational Expectations Equilibriumcharacterized by1. A probability measure P representing the homogeneous beliefs of agents over Ωs, where Ωsdenotes the space of realizations of variables exogenous to an agent.2. A sequence of equilibrium prices {r∗t , w∗t , Rf∗t } where r∗t , w∗t , Rf∗t : ΩtS → RN+ ∀ t. Marketsclear for all t, all realizations in ΩS almost surely in P.3. A sequence of choice functions {c∗t , i∗t , a∗t }∞t=0 that maximize the objective function of theagents conditional on P. All agents are internally rational.In order for the model to sustain an IREE where beliefs differ from those in a rational expectationsequilibrium, the agents cannot have access to the full information set available to agents underrational expectations. As outlined in section 2.3.3, this is achieved by removing the technology(2.9) as well as the functions (2.10) and (2.11) from the information set of agents in the learningmodel.47See Eusepi and Preston (2011) and Adam, Kuang, and Marcet (2011) for similar examples.432.4. Illustrating Long Run Risks Under Rational ExpectationsFigure 2.1: Transitory & Permanent Shocks in Rational Expectations Model (γ = 5, ψ = 1/1.5)00.0050.010.0150.02-20 0 20 40 60 80 100relative responseperiodsConsumptionTFP(a) Transitory Shock00.0050.010.0150.020.0250 20 40 60 80 100relative responseperiodsConsumptionTFP(b) Permanent ShockThe model is solved via a form of parameterized expectations with the assumption of a rep-resentative agent. Both the value function of the representative agent and the euler equation forcapital investment are approximated using the Chebyshev polynomials. The details of the solutionmethod can be found in appendix F.2.4 Illustrating Long Run Risks Under Rational ExpectationsBefore quantifying the learning model outlined in section 2.3 it is useful to make clear what is meantby ‘long run risk’ and to illustrate the use of Epstein-Zin preferences in generating them48. The formof the stochastic discount factor (2.7) indicates how the use of Epstein-Zin preferences changes theconsumption-saving decision relative to standard CRRA utility. The discount factor is composed oftwo components. The first is a function of one-period-ahead consumption growth and is identicalto the discount factor under CRRA utility. The second is a function of unanticipated variationin the continuation utility. The value of Epstein-Zin preferences is precisely that they allow forlong-horizon expectations about consumption growth to influence the pricing kernel separately fromtheir effect on one-period consumption growth. As has been studied extensively in the literature4948Much of the analysis in this section follows that found in Lochstoer and Kaltenbrunner (2010).49See Bansal and Yaron (2004) or Lochstoer and Kaltenbrunner (2010) for further discussion.442.4. Illustrating Long Run Risks Under Rational ExpectationsFigure 2.2: Transitory & Permanent Shocks in Rational Expectations Model (γ = 5, ψ = 6)00.0050.010.0150.02-20 0 20 40 60 80 100relative responseperiodsConsumptionTFP(a) Transitory Shock00.0050.010.0150.020.0250 20 40 60 80 100relative responseperiodsConsumptionTFP(b) Permanent Shockthe log of the discount factor can be writtenmt − EPt−1[mt] = −γ(∆ log ct − EPt−1[∆ log ct])− (γ − ψ) (∆vct − EPt−1[∆vct]) (2.19)where mt ≡ logMt and vc ≡ log VC . Unanticipated variation in the discount factor is decomposedinto two sources: unanticipated consumption growth and unanticipated changes in the relative valuefunction (which represents long run expected consumption growth). The variance of mt−EPt−1[mt]is commonly taken to be a measure of the level of overall perceived risk in the economy. Takingthe second moment of (2.19) yieldsvar(mˆt) = γ2var(∆ ˆlog ct)︸ ︷︷ ︸short run risk+ (γ − ψ)2var(∆vˆct) + 2γ(γ − ψ)cov(∆ ˆlog ct,∆vˆct)︸ ︷︷ ︸long run risk(2.20)where xˆ ≡ xt−EPt−1[xt]. Overall risk is composed of short run risks, which are a function of realizedconsumption growth, and long run risks, which are a function expected long run consumptiongrowth. Note that these long run consumption risks disappear when γ = ψ, as is the case withstandard power utility (indeed, CRRA preferences can be viewed as a special case of Epstein-Zinpreferences).Long run risk in (2.20) is a function of agents’ beliefs about the underlying structure of the452.4. Illustrating Long Run Risks Under Rational Expectationseconomy (which determine the variance and covariance terms) as well as their preferences. Theparameters γ and ψ specify a preference for the timing of the resolution of uncertainty. When thecoefficient of risk aversion exceeds the inverse of the elasticity of intertemporal substitution (whenγ > ψ) agents are said to have a preference for the early resolution of uncertainty. Given thechoice between two expected payoff-equivalent lotteries, one that offers a risky payoff for a certainnumber of periods and an certain payoff thereafter, and another that offers a certain payoff for theinitial periods and a risky payoff thereafter, agents would prefer the former. Seen another way,when cov(∆ ˆlog ct,∆vˆct) > 0 and shocks to realized consumption are correlated with variation inexpected consumption growth, then the environment is riskier when γ > ψ and agents experiencegreater variation in how they value consumption across states. Conversely agents have a preferencefor late resolution of uncertainty when γ < ψ for the opposite reasons.In order to make this distinction clear the model is solved and simulated under the assumptionof rational expectations (ie. agents are informed about the true mapping between {rt, wt} and themodel’s state variables). Table 2.1 shows moments for two calibrations of the model’s preferences. Inthe first case ({γ, ψ} = {5, 1/1.5}) the agent has a preference for the early resolution of uncertaintyand in the second ({γ, ψ} = {5, 6}) the agent has a preference for the late resolution of uncertainty50.For each case the model is simulated using transitory shocks (ρ = 0.95) and then using permanentshocks (ρ = 1).As can be seen in table 2.1, when γ > ψ the level of risk σ(m) is an order of magnitude largerwhen shocks are permanent instead of temporary51. With permanent shocks, realized consumptiongrowth is positively correlated with variation in expected long-run consumption growth (V/C).From (2.20), when γ > ψ this positive correlation pushes up the level of risk. When agents havea preference for early resolution of uncertainty they consider the permanent shock environment tobe riskier as they have an aversion to variation in expected consumption growth at long horizonsrelative to variation in realized consumption growth. Conversely, table 2.1 indicates that whenagents have a preference for late resolution of uncertainty they find the transitory shock environmentto be riskier for analogous reasons. When ρ < 1 shocks to realized consumption growth are50the parameterization of the model is largely taken from standard values found in the literature, see Lochstoerand Kaltenbrunner (2010) or Aldrich and Kung (2011).51Note that in 2.1, the level of short run risk, SRR, is taken to be γ · σ(∆ log c) in following Lochstoer andKaltenbrunner (2010). The level of long run risk is measured as the different between the total risk in the economyand SRR.462.5. Quantifying the Learning Modelnegatively correlated with expected long-run consumption growth. With γ < ψ agents have apreference for variation in expected long-run consumption growth all things being equal, and sothey find the transitory shock environment to be riskier. Mechanically, cov(∆ ˆlog ct,∆vˆct) < 0pushes up the level of risk when γ < ψ. Note that results are shown for only one value of ρ < 1.Decreasing ρ will have a monotonic effect on cov(∆ ˆlog ct,∆vˆct), so agents with a preference for lateresolution of uncertainty will find environments with lower values of ρ to be riskier.It should be noted that the riskiness of the two environments does not appear to have a qualita-tive effect on the financial moments in the model. From table 2.1, the Sharpe ratio of the realizedrisk premium (Ri−Rf ) is approximately an order of magnitude larger with permanents shocks foreach of the two sets of utility parameters. Similarly, for each set of utility parameters the Sharperatio of the expected risk premium (E[Ri − Rf ]) is approximately an order of magnitude largerwhen shocks are temporary.Figures 2.1 and 2.2 show the response of consumption to shocks for each of the two parameter-izations of the model under rational expectations. When ρ = 1, TFP shocks create an expectationof permanently higher consumption, hence for each of the two parameterizations a larger initialconsumption response is seen in the permanent shock case. Similarly, table 2.1 shows that for eachset of utility parameters, the use of permanent shocks increases the relative standard deviationof consumption growth and decreases the relative standard deviation of investment growth. In-creasing ψ decreases the intertemporal elasticity of substitution and hence the desire to smoothconsumption over time goes up. For each the two technology environments, figures 2.1 and 2.2 showa larger initial consumption when ψ is high, as well as a greater degree of consumption smoothingsubsequently.2.5 Quantifying the Learning ModelThis section provides a quantitative evaluation of the model under learning. The goal of the sectionis to clarify the role of the learning mechanism, hence the parameters are not estimated. For themost part the parameterization is standard. The discount factor (β) is set equal to 0.998, therate of trend growth (µ) is set equal to 0.004, the depreciation rate of capital (δ) is set equal to0.021, and the capital share of output (α) is set equal to 0.36. When simulating the model with472.5. Quantifying the Learning ModelTable 2.1: Business Cycle Moments Under Rational Expectations(i): γ = 5, ψ = 1/1.5 (ii): γ = 5, ψ = 6Transitory Permanent Transitory PermanentShocks Shocks Shocks Shocksσ(∆ log c)/σ(∆ log y) 0.347 0.546 0.463 1.055σ(∆ log i)/σ(∆ log y) 2.237 1.846 3.301 0.789µ(Rf ) 1.019 1.016 1.107 1.099σ(Rf ) 2.15× 10−3 5.65× 10−3 0.012 0.034µ(Ri −Rf ) 6.51× 10−5 5.62× 10−4 4.94× 10−4 3.25× 10−3σ(Ri −Rf ) 7.59× 10−3 7.64× 10−3 0.013 0.031SR(Ri −Rf ) 8.38× 10−3 0.074 0.039 0.104µ(E[Ri −Rf ]) 4.18× 10−5 5.44× 10−4 4.36× 10−4 3.24× 10−3σ(E[Ri −Rf ]) 6.15× 10−6 3.78× 10−3 2.53× 10−4 0.031SR(E[Ri −Rf ]) 6.791 0.144 1.719 0.105σ(m) 0.916 12.734 5.191 4.829SRR 0.022 0.035 0.030 0.068LRR(/ ∆c) −1.83× 10−2 5.42× 10−2 1.14× 10−2 −2.33× 10−2SRR(/risk) 5.461 0.393 0.963 1.036LRR(/risk) -4.461 0.607 0.037 -0.036σ(log V/C) 0.039 0.491 0.016 0.011corr(log(V/C),∆ log c) -0.052 0.485 −9.13× 10−4 0.273corr(∆ log(V/C),∆ log c) -0.999 0.895 -0.983 0.949In each case in the above table the model is parameterized as follows: β = 0.998, α = 0.36, δ = 0.021, µ = 0.004, ζ = 18,σz = 0.02. When simulating the model with transitory shocks ρ is set equal to 0.95. Interest rates are reported at annualizedvalues, all other moments are computed for variables at quarterly frequency. In order to control for differences in consumptionvolatility across the models, the price of risk σ(m) is normalized by σ(∆ log c). The level of short run risk, SRR, is taken tobe γ · σ(∆ log c) in keeping with Lochstoer and Kaltenbrunner (2010). The sample size is 110 000 periods with the first 10 000periods dropped as a burn-in482.5. Quantifying the Learning Modeltransitory shocks ρ is set equal to 0.95. Capital adjustment costs are held low, with ζ = 18. Thestandard deviation of the technology shocks is set equal to 2% across all of the specifications. As abaseline, the household’s prior belief about the shocks r and w is that they are uncorrelated witha standard deviation of 1% each. The Effects of Learning & Long Run RisksFigure 2.3 shows the response of consumption to both transitory and permanent technology shocksfor the learning model as well as the model under the assumption of rational expectations. Thegain parameter is set equal to 0.002 as in Eusepi and Preston (2011) 53. It is also assumed thatagents do not internalize learning in their decision problem. It is immediately apparent that theimpact effects in figure 2.3 are substantially different across the rational expectations and learningmodel. In particular, the initial response of consumption is considerably smaller in the learningmodel than under rational expectations. This is a result of the specification of beliefs in (2.14).While agents in the learning model understand that rt and w˜t are log-linear with respect to capitalthey do not observe the technology shocks z˜t54. From the standpoint of the agent, shocks to z˜tare perceived as being shocks to rt and wt . As a result, both permanent as well as persistent buttransitory shocks are perceived as being one-time shocks with no persistence. The income effect istherefore small and the agent invests almost all of the shock.The second factor which is apparent in figure 2.3 is that learning appears to induce more prop-agation in the model. Agents in the model are updating their beliefs about the law of motion ofequilibrium objects. As a result the model allows for feedback between {rt, wt} and ωt. Followingthe initial realization of the shock agents update their beliefs about ω, causing them to believe thatthere will be permanently higher prices {rt, wt}. As a result, income effects push up both consump-tion as well as investment (as does continued perceived shocks {r, w}). Investment continues todrive changes in rental rates and wages and the feedback between ω and {rt, wt} persists.52A similar parameterization is used by Lochstoer and Kaltenbrunner (2010) in a rational expectations model withrecursive preferences.53Note that the gain parameter chosen here is an order of magnitude smaller than that estimated in chapter 1.One should not expect these to be similar. Chapter 1 considers agents how are forming beliefs about house pricegrowth while this chapter considers agents forming beliefs about wages and rental rates. The assumption that agentsdiscount the past at different rates in these two settings simply implies that they differently informed about thesetwo sets of objects.54Agent beliefs in the learning model are therefore misspecified.492.5. Quantifying the Learning ModelThe impulse responses in figure 2.3 indicate the effect that learning should have on the model’sability to generate long-run risks. Under the structural learning framework considered here, unan-ticipated variation in rt and wt is perceived as being the result of extremely short lived shocks,regardless of whether the shocks that actually give rise to it are themselves permanent or transitory.As a result, agents who have a preference for the early resolution of uncertainty will not find thisbe a risky environment and the model should not generate positive long run risks.Table 2.2 shows moments from simulations of the learning model. Once again the gain isset equal to 0.002 and agents do not internalize learning in their decision problem. As in table2.1 the model is simulated when agents have a preference for the early resolution of uncertainty(γ = 5, ψ = 1/1.5) and when agents have a preference for the late resolution of uncertainty(γ = 5, ψ = 6). The effect of learning on how agents perceive shocks can be seen by examining themoments of the value function. For both parameterizations of the preferences learning induces alarge negative correlation between realized consumption growth and V/C. As a result, agents whohave a preference for the early resolution of uncertainty perceive the learning environment to beless risky. This is true when shocks are permanent or temporary as can be seen by comparing thenormalized variances of the pricing kernel M to table 2.1. The large negative correlation betweencurrent consumption growth and expected consumption growth that is induced by learning impliesthat the model doesn’t generate positive long run risks when agents have a preference for earlyresolution of uncertainty.As should now be expected the results are reversed when agents have a preference for lateresolution of uncertainty. In this case agents like variation in long run consumption growth andhence an environment in which shocks are perceived as being highly transitory is relatively morerisky. As a result the normalized variation in M is higher when γ = 5 and ψ = 6 than underrational expectations (see table 2.1), for both specifications of z˜t. The learning now does generatelong run risk. In table 2.2 about 8% of the perceived risk in the economy is accounted for by longrun risk when agents have a preference for late resolution of uncertainty.For both parameterizations of the preferences learning implies higher volatility of the financialvariables. The standard deviation of the risk-free rate as well as both the realized and expectedrisk premium are higher under learning, regardless of whether agents have a preference for earlyor late resolution of uncertainty. This is the result of the higher overall volatility that results from502.5. Quantifying the Learning Modellearning. Similarly, the added propagation mechanisms that result from learning imply that therelative standard deviation of both consumption and investment growth are higher under learningfor both parameterizations of the preferences. It is interesting to compare the dynamics of learningunder the different risk preferences. Figure 2.4 plots the response of consumption to permanent andtransitory shocks for each of the two parameterizations considered in table 2.255. Unsurprisingly,a lower intertemporal elasticity of substitution implies that the initial response of consumptionto the shock is significantly lower when ψ equals 6 instead of 1/1.5. It is interesting to see,however, that under learning the propagation of the shock and variation in consumption over timeis high with low elasticity of substitution. Because agents view deviations of rt and wt from theexpected values implied by (2.14) as the result of shocks with zero persistence, a desire for moreconsumption smoothing increases investment in response to observed shocks to rt and wt (this isreflected in the relative variances of consumption and investment growth in table 2.2 for the twoparameterization). This investment implies subsequent growth in both rt and wt, and throughupdating of beliefs increases in ωji,t, yielding further investment and so on. A lower intertemporalelasticity of substitution therefore bolsters the propagation mechanisms created by learning.A preference for late resolution of uncertainty requires that γ < ψ. Hence, this can be obtainedeither through a decrease in the intertemporal elasticity of substitution (an increase in ψ) or througha reduction in the coefficient of risk aversion. Table 2.3 shows simulation moments for two differentspecifications of the model with a preference for late resolution of uncertainty: (i) (γ = 5, ψ = 6)and (ii) (γ = 1/5, ψ = 1/1.5). The higher desire for consumption smoothing induced by a largervalue for ψ implies that the relative volatility of consumption growth is lower and that of investmentgrowth is higher in (i) than it is in (ii) (for both permanent and transitory shocks to z˜). Differencesin the volatility of the financial variables between the two parameterizations reflect differencesin overall volatility. Examining the risk variables, the large desire for consumption smoothingwhen ψ = 6 implies a significantly higher normalized volatility of M . However, the relatively lowcoefficient of risk aversion in parameterization (ii) implies a lower price of short run risk. As aresult, the proportion of overall risk accounted for by long run risks is an order of magnitude largerunder parameterization (ii).55The corresponding impulse responses for the beliefs, ωji,t, can be found in appendix D, figure D.2.512.5. Quantifying the Learning ModelFigure 2.3: Influence of Learning (γ = 5, ψ = 6)11.0011.0021.0031.0041.0051.0061.0071.008-20 0 20 40 60 80 100relative responseperiodsLearningREE(a) Consumption: Transitory Shock11.0051.011.0151.02-20 0 20 40 60 80 100relative responseperiodsLearningREE(b) Consumption: Permanent ShockFigure 2.4: Effect of Preference for Timing of Risk in Learning Model11.0011.0021.0031.0041.0051.0061.0071.008-20 0 20 40 60 80 100relative responseperiodsγ = 5 ; ψ = 6γ = 5 ; ψ = 1/1.5(a) Consumption: Transitory Shock11.0051.011.0151.021.025-20 0 20 40 60 80 100relative responseperiodsγ = 5 ; ψ = 6γ = 5 ; ψ = 1/1.5(b) Consumption: Permanent Shock522.5. Quantifying the Learning ModelTable 2.2: Business Cycle Moments Under Learning (g = 0.002, Learning not Internalized)(i): γ = 5, ψ = 1/1.5 (ii): γ = 5, ψ = 6Transitory Permanent Transitory PermanentShocks Shocks Shocks Shocksσ(∆c)/σ(∆y) 0.271 0.287 0.153 0.193σ(∆i)/σ(∆y) 2.359 2.357 4.4897 4.784µ(Rf ) 1.019 1.018 1.111 1.087σ(Rf ) 2.59×10−3 4.84×10−3 0.068 0.143µ(Ri −Rf ) -1.47×10−3 -4.06×10−4 -0.023 -9.16×10−3σ(Ri −Rf ) 0.028 0.049 0.065 0.127SR(Ri −Rf ) -0.053 -8.23×10−3 -0.352 -0.072µ(E[Ri −Rf ]) -6.03×10−4 5.09×10−4 -0.021 -7.29×10−4σ(E[Ri −Rf ]) 2.41×10−3 4.04×10−3 0.020 0.049SR(E[Ri −Rf ]) -0.250 0.126 -1.018 -0.015σ(m) 0.751 0.858 5.46 5.42SRR 0.017 0.018 9.87×10−3 0.012LRR(/ ∆c) -4.25 -4.14 0.463 0.424SRR(/risk) 6.66 5.83 0.915 0.922LRR(/risk) -5.66 -4.83 0.085 0.078σ(log V/C) 0.035 0.084 0.018 0.043corr(log(V/C),∆c) -0.081 -0.083 -0.105 -0.105cov(∆ log(V/C),∆c) -1.20×10−5 -1.22×10−5 -1.79×10−6 -2.36×10−6corr(∆ log(V/C),∆c) -0.999 -0.997 -0.941 -0.943In each case in the above table the model is parameterized as follows: β = 0.998, α = 0.36, δ = 0.021, µ = 0.004, ζ = 18,σz = 0.02. When simulating the model with transitory shocks ρ is set equal to 0.95. Interest rates are reported at annualizedvalues, all other moments are computed for variables at quarterly frequency. In order to control for differences in consumptionvolatility across the models, the price of risk σ(m) is normalized by σ(∆ log c). The level of short run risk, SRR, is taken tobe γ · σ(∆ log c) in keeping with Lochstoer and Kaltenbrunner (2010). The sample size is 110 000 periods with the first 10 000periods dropped as a burn-in532.5. Quantifying the Learning ModelTable 2.3: Business Cycle Moments Under Learning (g = 0.002, Learning not Internalized)(i): γ = 5, ψ = 6 (ii): γ = 1/5, ψ = 1/1.5Transitory Permanent Transitory PermanentShocks Shocks Shocks Shocksσ(∆c)/σ(∆y) 0.153 0.193 0.271 0.287σ(∆i)/σ(∆y) 4.4897 4.784 2.36 2.36µ(Rf ) 1.111 1.087 1.019 1.018σ(Rf ) 0.068 0.143 0.026 0.048µ(Ri −Rf ) -0.023 -9.16×10−3 -1.49×10−3 -4.25×10−4σ(Ri −Rf ) 0.065 0.127 0.028 0.049SR(Ri −Rf ) -0.352 -0.072 -0.054 -8.62×10−3µ(E[Ri −Rf ]) -0.021 -7.29×10−4 -6.22×10−4 4.93×10−4σ(E[Ri −Rf ]) 0.020 0.049 2.41×10−3 4.04×10−3SR(E[Ri −Rf ]) -1.018 -0.015 -0.258 0.121σ(m) 5.46 5.42 0.661 0.657SRR 9.87×10−3 0.012 6.98×10−4 7.07×10−4LRR(/ ∆c) 0.463 0.424 0.461 0.457SRR(/risk) 0.915 0.922 0.302 0.304LRR(/risk) 0.085 0.078 0.698 0.696σ(log V/C) 0.018 0.043 0.035 0.084corr(log(V/C),∆c) -0.105 -0.105 -0.081 -0.084cov(∆ log(V/C),∆c) -1.79×10−6 -2.36×10−6 -1.20×10−5 -1.22×10−5corr(∆ log(V/C),∆c) -0.941 -0.943 -0.999 -0.997In each case in the above table the model is parameterized as follows: β = 0.998, α = 0.36, δ = 0.021, µ = 0.004, ζ = 18,σz = 0.02. When simulating the model with transitory shocks ρ is set equal to 0.95. Interest rates are reported at annualizedvalues, all other moments are computed for variables at quarterly frequency. In order to control for differences in consumptionvolatility across the models, the price of risk σ(m) is normalized by σ(∆ log c). The level of short run risk, SRR, is taken tobe γ · σ(∆ log c) in keeping with Lochstoer and Kaltenbrunner (2010). The sample size is 110 000 periods with the first 10 000periods dropped as a burn-in542.5. Quantifying the Learning Model2.5.2 The Effect of the GainIn order to evaluate how the parameterization of the learning process effects the results in section2.5.1, the model is simulated over a range of values for the gain 56. Figure 2.5 plots the response ofconsumption to transitory and permanent shocks with g = 0.002 and g = 0.01. The correspondingimpulse responses for the beliefs ωji,t can be found in appendix D figure D.3. The influence of thegain on the dynamics is familiar from chapter 1. An increased gain causes a larger revision to beliefsfollowing the initial shock, causing a larger subsequent investment response. The feedback betweenbeliefs and {rt, wt} is augmented and hence there is more propagation of the shock with a highergain. The increased elasticity of beliefs with respect to {rt, wt} under the high gain specificationalso serves to decrease the persistence in the belief process and so as g increases so too does thespeed with which consumption returns to its long run path.It may seem odd that an order of magnitude change in the gain has a much less pronouncedeffect on real dynamics than would be the case in the environment studied in chapter 1. There area number of reasons for this. First, belief updating in (2.17) is a function of variation in the capitalstock, which is relatively low. Second, the presence of adjustment costs dampens the response ofinvestment to shifts in beliefs. Finally, the learning framework here is conceptually different asagents are updating beliefs about a set of four parameters and are essentially learning about boththe means of rt and wt as well as their slopes.Table 2.4 shows moments from simulations of the model when g = 0.002 and when g = 0.01.The table includes simulations for the two sets of utility parameters considered in table 2.2. As canbe seen, the additional propagation provided by the increased gain is too small to have a significanteffect on the relative volatilities of consumption and investment, or on the moments of the risk freerate and the risk premium.Crucially, the change in the gain has little noticeable effect on the amount of long-run riskgenerated by the model. The intuition for this result follows from the discussion in 2.5.1. Theagent does not anticipate the fact that it will update its beliefs in the future and so the changein the gain has no effect on the perceived relationship between realized consumption growth andexpected long-run consumption growth57. While variation in the gain parameter may change the56The prior variances about the errors in 2.15 are held constant.57Note, there is a slight decrease in the sample correlation between ∆c and log V/C owing the revisions in beliefs552.5. Quantifying the Learning ModelFigure 2.5: Influence of g: Transitory & Permanent Shocks in Learning Model11.0011.0021.0031.0041.0051.0061.0071.008-20 0 20 40 60 80 100relative responseperiodsg = 0.002g = 0.01(a) Consumption: Transitory Shock11.0051.011.0151.021.025-20 0 20 40 60 80 100relative responseperiodsg = 0.002g = 0.01(b) Consumption: Permanent Shockoverall propagation of shocks, the assumption that agents do not internalize learning when makingtheir decision implies that from their perspective changes in the gain have no bearing on the futurecourse of consumption. Thus the standard insight about long-run risk in rational expectations willnot be obtained through a particular choice of g.2.5.3 Internalizing LearningA key effect of the kind of learning framework considered here is that in addition to creatingadditional propagation mechanisms in the model, it introduces strong assumptions about howagents perceive shocks on impact. Agents perceive unanticipated variation in wages and the rentalrate of capital to be the result of shocks with zero persistence. As a result, recovering long runrisks requires a strong preference for the late resolution of uncertainty. The chapter now relaxes theassumption made thus far that agents do not internalize their learning when making their choices.Agents are now assumed to take account of future belief updating when making their consumption-saving decision. When the representative household observes unanticipated changes to the rentalrate of capital or to wages it anticipates that these will result in persistent changes in its beliefsabout ωji,t and hence that there should long run effects on consumption.The moments for the simulation are listed in tables 2.5 and 2.6. Table 2.5 reports moments forand therefore expected consumption growth at the end of the period.562.5. Quantifying the Learning ModelTable 2.4: Business Cycle Moments Under Learning: Effect of gγ = 5, ψ = 1/1.5 γ = 5, ψ = 6g = 0.002 g = 0.01 g = 0.002 g = .01σ(∆c)/σ(∆y) 0.271 0.274 0.153 0.170σ(∆i)/σ(∆y) 2.36 2.36 4.4897 4.483µ(Rf ) 1.019 1.019 1.111 1.111σ(Rf ) 2.59×10−3 0.024 0.068 0.059µ(Ri −Rf ) -1.47×10−3 -1.53×10−3 -0.023 -0.023σ(Ri −Rf ) 0.028 0.026 0.065 0.058SR(Ri −Rf ) -0.053 -0.059 -0.352 -0.402µ(E[Ri −Rf ]) -6.03×10−4 -6.34×10−4 -0.021 -0.021σ(E[Ri −Rf ]) 2.41×10−3 2.33×10−3 0.020 0.017SR(E[Ri −Rf ]) -0.250 -0.273 -1.018 -1.279σ(m) 0.751 1.190 5.46 5.36SRR 0.017 0.018 9.87×10−3 0.011LRR(/ ∆c) -4.25 -3.810 0.463 0.358SRR(/risk) 6.66 4.199 0.915 0.933LRR(/risk) -5.66 -3.199 0.085 0.067σ(log V/C) 0.035 0.025 0.018 0.014corr(log(V/C),∆c) -0.081 -0.211 -0.105 -0.230cov(∆ log(V/C),∆c) -1.20×10−5 -1.18×10−5 -1.79×10−6 -1.80×10−6corr(∆ log(V/C),∆c) -0.999 -0.982 -0.941 -0.881Agents do not internalize learning in any of the models. In each case in the above table the model is parameterized as follows:β = 0.998, α = 0.36, δ = 0.021, µ = 0.004, ζ = 18, σz = 0.02. Shocks are transitory with ρ set equal to 0.95. Interest ratesare reported at annualized values, all other moments are computed for variables at quarterly frequency. In order to control fordifferences in consumption volatility across the models, the price of risk σ(m) is normalized by σ(∆ log c). The level of shortrun risk, SRR, is taken to be γ · σ(∆ log c) in keeping with Lochstoer and Kaltenbrunner (2010). The sample size is 110 000periods with the first 10 000 periods dropped as a burn-in572.6. Conclusionthe model under the assumption that agents have a preference for early resolution of uncertainty(γ = 5, ψ = 1/1.5) while table 2.6 reports moments for the model under the assumption thatagents have a preference for late resolution of uncertainty (γ = 5, ψ = 6). It is clear that evenallowing for a large gain, the expected long run consumption growth following shocks that mightresult from internalizing learning is insufficient to change the qualitative properties of the model.Allowing agents to internalize learning slightly increases (decreases) the normalized volatility of Mwhen agents have a preference for early (late) resolution of uncertainty. This is in line with theaforementioned intuition. When agents have a preference for early (late) resolution of uncertaintythe environment becomes riskier (less risky) for them as they anticipate that observed shocks willresult in variation in long-run consumption growth. Similarly, the measured level of long run riskrelative to the overall variation in M is slightly increased (decreased) when agents have a preferencefor early (late) resolution of uncertainty.Nevertheless, quantitatively these changes are small. Assuming that agents internalize updatingof beliefs in their decision problem allows for a channel through which observed exogenous variationcan have long-run effects on consumption. Quantitative evaluation suggests, however, that theseanticipated long-run consumption effects are insufficient to yield positive long-run risks under thestandard assumption that agents prefer early resolution of uncertainty.2.6 ConclusionThis chapter considered the implications of structural learning for generating long run consumptionrisk. As has been studied elsewhere in the literature on bounded rationality, learning mechanismscan provide substantial amplification and propagation mechanisms in otherwise standard frame-works. This chapter demonstrates that a consequence of internal rationality is that it can changethe required preference for the timing of the resolution of uncertainty needed to generate long-runrisk. Agents in the model attribute observed variation to extremely short-lived shocks. This istrue even if agents can anticipate the effect that their decisions have on the path of future beliefs.Even when the underlying stochastic processes in the economy are subject to permanent shocks,boundedly-rational beliefs require a preference for the late resolution of uncertainty in order togenerate long run risks. A key assumption in the learning model is that agents do not have infor-582.6. ConclusionTable 2.5: Business Cycle Moments Under Learning: Internalizing Learning (γ = 5, ψ = 1/1.5)g = 0.002 g = 0.01Don’t Don’tInternalize Internalize Internalize Internalizeσ(∆c)/σ(∆y) 0.271 0.272 0.274 0.279σ(∆i)/σ(∆y) 2.359 2.358 2.36 2.36µ(Rf ) 1.019 0.019 1.019 1.019σ(Rf ) 2.59×10−3 0.026 0.024 0.024µ(Ri −Rf ) -1.47×10−3 -1.35×10−3 -1.53×10−3 -9.64×10−4σ(Ri −Rf ) 0.028 0.028 0.026 0.026SR(Ri −Rf ) -0.053 -0.049 -0.059 -0.037µ(E[Ri −Rf ]) -6.03×10−4 -4.86×10−4 -6.34×10−4 -5.03×10−5σ(E[Ri −Rf ]) 2.41×10−3 2.40×10−3 2.33×10−3 2.30×10−3SR(E[Ri −Rf ]) -0.250 -0.203 -0.273 -0.022σ(m) 0.751 0.795 1.190 1.310SRR 0.017 0.018 0.018 0.018LRR(/ ∆c) -4.25 -4.21 -3.810 -3.690SRR(/risk) 6.66 6.29 4.199 3.825LRR(/risk) -5.66 -5.29 -3.199 -2.825σ(log V/C) 0.035 0.035 0.025 0.025corr(log(V/C),∆c) -0.081 -0.080 -0.211 -0.200cov(∆ log(V/C),∆c) -1.20×10−5 -1.20×10−5 -0.982 -0.986corr(∆ log(V/C),∆c) -0.999 -0.999 -1.18×10−5 -1.16×10−5Agents have a preference for the early resolution of uncertainty in the above models (γ = 5, ψ = 1/1.5). In each case in theabove table the model is parameterized as follows: β = 0.998, α = 0.36, δ = 0.021, µ = 0.004, ζ = 18, σz = 0.02. Shocks arepermanent (ρ = 1). Interest rates are reported at annualized values, all other moments are computed for variables at quarterlyfrequency. In order to control for differences in consumption volatility across the models, the price of risk σ(m) is normalized byσ(∆ log c). The level of short run risk, SRR, is taken to be γ · σ(∆ log c) in keeping with Lochstoer and Kaltenbrunner (2010).The sample size is 110 000 periods with the first 10 000 periods dropped as a burn-in592.6. ConclusionTable 2.6: Business Cycle Moments Under Learning: Internalizing Learning (γ = 5, ψ = 6)g = 0.002 g = 0.01Don’t Don’tInternalize Internalize Internalize Internalizeσ(∆c)/σ(∆y) 0.153 0.193 0.170 0.173σ(∆i)/σ(∆y) 4.4897 0.478 4.483 0.445µ(Rf ) 1.111 1.109 1.111 1.111σ(Rf ) 0.068 0.142 0.059 0.058µ(Ri −Rf ) -0.023 -9.16×10−3 -0.023 -0.023σ(Ri −Rf ) 0.065 0.127 0.058 0.057SR(Ri −Rf ) -0.352 -0.072 -0.402 -0.403µ(E[Ri −Rf ]) -0.021 -7.78×10−4 -0.021 -0.021σ(E[Ri −Rf ]) 0.020 0.049 0.017 0.017SR(E[Ri −Rf ]) -1.018 -0.016 -1.279 -1.282σ(m) 5.46 5.42 5.36 5.35SRR 9.87×10−3 0.012 0.011 0.011LRR(/ ∆c) 0.463 0.423 0.358 -0.353SRR(/risk) 0.915 0.916 0.933 0.934LRR(/risk) 0.085 0.084 0.067 0.066σ(log V/C) 0.018 0.043 0.014 0.013corr(log(V/C),∆c) -0.105 -0.104 -0.230 -0.216cov(∆ log(V/C),∆c) -1.79×10−6 -2.37×10−6 -1.80×10−6 -1.84×10−6corr(∆ log(V/C),∆c) -0.941 -0.947 -0.881 -0.911Agents have a preference for the late resolution of uncertainty in the above models (γ = 5, ψ = 6). In each case in the abovetable the model is parameterized as follows: β = 0.998, α = 0.36, δ = 0.021, µ = 0.004, ζ = 18, σz = 0.02. Shocks arepermanent (ρ = 1). Interest rates are reported at annualized values, all other moments are computed for variables at quarterlyfrequency. In order to control for differences in consumption volatility across the models, the price of risk σ(m) is normalized byσ(∆ log c). The level of short run risk, SRR, is taken to be γ · σ(∆ log c) in keeping with Lochstoer and Kaltenbrunner (2010).The sample size is 110 000 periods with the first 10 000 periods dropped as a burn-in602.6. Conclusionmation about the underlying shocks in the economy. The results indicate that in order to maintainpreference for the early resolution of uncertainty in learning models of long-run risk, researchersmay wish to confine themselves to environments where agents are able to observe and form beliefsabout the sources of exogenous variation in the economy.61Chapter 3Explaining World Savings3.1 IntroductionData on the world saving distribution reveals two key features: (i) cross-country differences in sav-ing rates are significant and persistent; and (ii) some countries and regions have shown very sharpchanges in their average saving rates over short periods of time. These facts are problematic for thestandard model with time-additive preferences. Without equal rates of time preference, the asymp-totic distribution of world wealth is typically degenerate under additively separable preferences.While models with equal rates of time preference but cross-country differences in demographicsand productivity have had some success in accounting for part of the cross-country dispersion insaving rates, a substantial amount of variation still remains unexplained in these models.This chapter provides an alternative explanation for the observed saving patterns. It formalizesa model of the world economy that is comprised of open economies inhabited by infinitely-livedagents. The main point of departure from the standard neoclassical exogenous growth modelis that it endows agents with recursive preferences.58 Specifically, it follows Farmer and Lahiri(2005) and uses a modified version of recursive preferences. The key implication of the Farmer-Lahiri specification is that it generates a determinate steady state wealth distribution within agrowing world economy, a feature that typical models with recursive preferences cannot generate.The problem with the standard recursive preference specification is that it is inconsistent withbalanced growth. The existence of balanced growth requires homothetic preferences. However,58The dynamic properties of models with recursive preferences and multiple agents were analyzed in a celebratedpaper by Lucas and Stokey (1984). Assuming bounded utility, Lucas-Stokey studied an endowment economy. Conse-quently, the results of Lucas-Stokey cannot be directly applied to growing economies. There is a small literature onrecursive preferences with unbounded aggregators. Boyd (1990) has developed a version of the contraction mappingtheorem that can be used to generalize Lucas-Stokey’s proof of existence of a utility function to the unbounded case.If preferences are time-separable, King, Plosser, and Rebelo (1988) showed that the period utility function must behomogenous in consumption and Dolmas (1996) has generalized their result to the case of recursive utility. Farmerand Lahiri (2005) have applied the Dolmas result to a multi-agent economy and have established that homogeneityrules out the assumption of increasing marginal impatience. Hence, the existence of an endogenous stationary wealthdistribution is inconsistent with balanced growth.623.1. Introductionunder homothetic preferences, the asymptotic wealth distribution in multi-agent environments iseither degenerate or reflects the initial wealth distribution.One might of course consider the issue of balanced growth to be irrelevant to understanding sav-ings behavior. However, the inconsistency of standard recursive preferences with balanced growthis problematic if one’s goal is to explain saving rates. In particular, given the constancy of bothlong run average growth rates and saving rates for most groups of countries (regions or continents),understanding long run patterns of savings would appear to be intrinsically linked to long runsteady state dynamics. As is well known, the Kaldor growth facts are quite stark in suggestingbalanced growth to be a robust feature of long run growth. Hence, the chapter argues that anymodel attempting an explanation of dispersions in long run saving rates across countries should beconsistent with balanced growth.To overcome the problems with the standard recursive preference specification, the chapterfollows Farmer and Lahiri (2005) and constructs a model of recursive utility in which agents careabout relative consumption. It is assumed that preferences are described by an aggregator thatcontains current consumption, future utility, and a time-varying factor that is external to the agentbut grows at the common growth rate in a balanced growth equilibrium. In assuming that agentsderive utility from their place in the consumption distribution, the preference structure will beconsistent with the well-known Easterlin Paradox. Following Easterlin (1974) a number of studieshave shown empirical measures of utility and satisfaction to be increasing in relative rather thanabsolute wealth. Such evidence can be found in Easterlin (1995), Helliwell (2003), and Di Tella,MacCulloch, and Oswald (2003) 59.The assumed time dependence allows for preferences to exhibit increasing marginal impatience,which is a necessary condition for a non-degenerate asymptotic wealth distribution. A positiveproductivity shock in the model induces a rise in saving which ultimately reverts back to its priorlevel due to the increasing marginal impatience of agents as their wealth rises relative to worldwealth, thereby preserving a determinate asymptotic wealth distribution. Equally important, thisspecification implies that different preferences induce different consumption to wealth ratios ofdifferent agents which, in turn, map into different steady state saving rates of different countries.The chapter starts by demonstrating that the modified preferences of Farmer and Lahiri (2005)59see Clark, Frijters, and Shields (2008) for futher discussion.633.1. Introductioncan account for not only the observed differences in average long run saving rates between regions forthe period 1970-2010 but also the time series behavior of region-specific saving rates. Strikingly, themodel achieves these two goals without appealing to either different externality factors or frictionsin factor and goods flows. This feature of the model is first established by examining a two-region,heterogenous agent world economy without any impediments in factor or goods mobility acrossregions. Importantly, this version of the model assumes that the external factor in preferencesis indexed to the common world per capita consumption level. Hence, all agents have the sameexternal factor. A calibrated version of this two-region model is shown to quantitatively match boththe average region-specific levels of savings between 1970-2010 as well as the time series movementsin saving rates in the two regions during this period. Encouragingly, the model can match thesefacts despite allowing only two factors to vary across countries (one preference parameter and laborproductivity) and using a common world productivity process as the only exogenous driver of timeseries fluctuations around the balanced growth path. The chapter then establishes the same resultsfor a richer five-region world economy.Lastly, the chapter turns to saving miracles. In order to generate sudden switches in savingrates, an additional modification is proposed. Specifically, the world is divided into three regions:the G7, the Asian tigers, and Emerging countries. Crucially, the external factor is now allowedto be different in levels for the three groups even though it is still constrained to have the samegrowth rate. The basic idea behind this is that all societies have role models/peer groups that theywant to keep up with or imitate. This approach to explaining miracles amounts to a hypothesisthat these sudden transformations of economies occur due to changes in their aspirations. Moreparticularly, under the formalization considered here the steady state saving rates are functionsof the external factor in preferences which describes the benchmark relative to whom the countryevaluates itself. The model can generate the observed saving miracle of the Asian Tigers if theirbenchmark external factor is switched in 1970 from the average consumption level of the developingworld to the G7 consumption level. The model predicts that saving rates rise towards the observedlevels in the data as the economy starts building its consumption towards its new desired level.As consumption rises however, increasing marginal impatience starts to become stronger over timewhich eventually induces the saving rates to come back down. The model predictions are shown tomatch the facts quantitatively as well as qualitatively for the Asian Tigers.643.2. Two Facts on Cross-Country SavingThis work is related to two different strands of literature. The first is the relatively large bodyof research focused on explaining the dispersion of saving rates across countries. Explanations forthe observed variation in cross-country savings have typically focused on variations in per capitaincomes, productivity growth, fertility rates or the age distribution of the population. Contributionsalong these lines can be found in Mankiw, Romer, and Weil (1992), Christiano (1989), Chen,Imrohoroglu, and Imrohoroglu (2006), Horioka and Terada-Hagiwara (2012), Loayza, Schmidt-Hebbel, and Serven (2000) and Tobing (2012). These papers typically find significant explanatorypower for demographics and some explanatory power for per capita income (though the directionof causality there is somewhat unclear). However, a significant part of the saving variability in thedata continues to remain unaccounted for.This chapter is also related to the work on recursive preferences and stationary wealth distri-butions that goes back in its modern form to Lucas and Stokey (1984) and Epstein and Hynes(1983). Of particular relevance to this work are the contributions of Boyd (1990), Dolmas (1996)and Ben-Gad (1998) who focused on characterizing the stationary wealth distribution in growingeconomies. A second line of research has examined the implications of recursive preferences forstationary wealth distrbitution in growing economies. Also relevant to this work are the papersby Uzawa (1969), Mendoza (1991) and others who examined the effects of endogenously varyingdiscount rates on the equilibrium dynamics of the neoclassical growth model.The next section describes some of the key data features that motivate this study. Section3.3 quickly reviews the key issues associated with recursive preferences under balanced growth aswell as the “fix” to the problem suggested by Farmer and Lahiri (2005). Section 3.4 presents anddevelops the model. Section 3.5 discusses the calibration of the model and examines its quantitativefit to average saving rates in a two-region economy. Section 3.6 discusses miracles while the lastsection concludes.3.2 Two Facts on Cross-Country SavingThere are two features of the data that should be highlighted. First, is the sustained differences insaving rates across groups of countries. To do this the countries are collected into three groups: the653.2. Two Facts on Cross-Country SavingG7, Emerging, and Sub-Saharan Africa.60 Figure 3.1 plots the savings rates of these three groupsof countries between 1970 and 2010. The figure illustrates another key feature of the data. Savingsrates are different for different countries for long periods of time. Further, they show little or noevidence of convergence.Figure 3.1: Differences in Saving Rates Across Regions510152025national saving1970 1980 1990 2000 2010yearG7 Emerging MktSub−SaharaNational Saving 1970−2010While the overall pattern suggests that saving rates are persistent, the data does have anotherimportant aspect: in some countries saving rates show sudden and sharp swings over relativelyshort periods of time. Figure 3.2 highlights this by plotting the saving rates in the Asian Tigersbetween 1960 and 2010. Clearly, saving rates in the Asian economies increased very sharply from1960 onward. In a short time period of 20 years starting in 1970, their saving rates rose by almost15 percentage points. Since the mid-1980s, the average saving rate in the Asian tigers has exceededthe average saving rate in the G7 countries by over 10 percentage points.This chapter argues that this data can be explained by allowing the rate of time-preference tovary across countries using a modified version of recursive preferences. In the standard model ofrecursive preferences studied by Lucas and Stokey (1984) and Epstein and Hynes (1983), agents60The list of countries in each group is in the Appendix.663.3. Recursive Preferences & Balanced GrowthFigure 3.2: Saving Rates in the Asian Tigers152025303540national saving1960 1970 1980 1990 2000 2010yearG7 Asian TigersNational Saving 1960−2010become less patient as they become richer in an absolute sense. This idea is adapted to the case ofa growing economy with the assumption that agents become less patient as they become richer ina relative sense.3.3 Recursive Preferences & Balanced GrowthAs discussed in Section 3.1, a key goal of the work is to examine the ability of a modified versionof recursive preferences to rationalize the cross-country saving facts. Before presenting the modelit is worthwhile to review why a modification is needed at all. In a nutshell, the need to modifythe standard recursive specification arises because of an interest in analyzing environments withsteady state balanced growth. Balanced growth is not only one of the celebrated Kaldor facts butalso happens to characterize the modern data aggregated by regions. This can be seen from Figure3.3 below which shows the time series behavior of the consumption-to-output ratio for G7 anddeveloping countries separately. The figure makes clear that on average, consumption and outputtend to grow at similar rates over long periods of time. Hence, it should be desirable to retainbalanced growth as a key feature in any model of world saving.The baseline recursive preference structure, however, is not consistent with balanced growth.6161It should be noted that the celebrated Lucas and Stokey (1984) paper that studied recursive preferences in aheterogenous agent economy did not have long run growth. Hence, this inconsistency was not germane to their work.673.3. Recursive Preferences & Balanced GrowthFigure 3.3: Balanced Growth Evidence: Consumption-Output Ratios6065707580C / Y 1970 1980 1990 2000 2010yearG7 Emerging Mkt & SubSaharaConsumption−Output RatioTo see this, consider the following recursive aggregator of preferencesut = W (ct, ut+1)With heterogenous agents, there exists an asymptotic stationary wealth distribution if along asteady state balanced growth path all agents equateW iu(ci, ui)= W ju(cj , uj)= 1/R for all i, j (3.1)where c is consumption, u is utility and R is the steady state interest factor common to all agents.Dolmas (1996) showed that this can only occur if the aggregator W is homogenous of the formW (λx, λγy) = λγW (x, y) (3.2)When this homogeneity condition is satisfied Wu becomes a constant along a balanced growth paththereby making it possible for the endogenous rate of time preference to remain constant and equalto the constant interest rate along a balanced growth path. More fundamentally, Wu becomes anumber that is independent of c and u in steady state.While the condition above is intuitively obvious, Farmer and Lahiri (2005) showed that thishomogeneity condition also implies that the stationary wealth distribution in a heterogenous agent683.3. Recursive Preferences & Balanced Growtheconomy is generically degenerate. It leads to a non-degenerate steady state wealth distributiononly in the knife-edged case of W i = W j . But those are exactly the implications in the case ofadditively separable preferences. In other words, the key Farmer-Lahiri result is that recursivepreferences do not add anything to our understanding of stationary wealth distributions beyondwhat we already know from additively separable preferences.In the context of recursive preferences in environments without steady state growth, Lucasand Stokey (1984) proved the existence of a stationary wealth distribution as long as preferencesexhibited increasing marginal impatience, i.e., agents became more impatient as they grew wealth-ier. Intuitively, rising impatience bounds the desire to accumulate assets by raising the desire toconsume. The problem of the specification in equation (3.2) is that there is no force akin to theincreasing marginal impatience of Lucas-Stokey that can endogenously equate it across agents.Consequently, the equilibrium has a knife-edge property to it.Farmer and Lahiri (2005) showed however that the introduction of an externality into preferencescould fix this problem. In particular, they considered preferences of the formui = W i(cit, uit+1, at)where at is a factor that is external to the individual. This could stand in for habits, the av-erage consumption level of the economy, or any other factor provided it grows at the rate ofsteady state growth and, as formalized here, is external to the individual household. Farmer-Lahiri showed that as long as the aggregator W was homogenous in all three arguments so thatW (λx, λγy, λz) = λγW , an economy with heterogenous agents would give rise to an endogenouslydetermined stationary wealth distribution with different agents choosing different saving levels inorder to equate W iu(c˜i, gγi u˜i, 1)= W ju(c˜j , gγj u˜j , 1)where x˜ = xa and g denotes the steady stategrowth. Moreover, the homogeneity property also ensures that Wu would be constant in steadystate. Hence, this specification can generate steady state differences in saving rates across agentsfacing a common vector of prices. The rest of the chapter will examine the potential of thesepreferences to account for the disparity in saving rates across the world.693.4. The Model3.4 The ModelThe environment is a world economy consisting of N small economies. Each country i is populatedby li agents and this measure remains constant over time. Introducing population growth into themodel is a straightforward extension that does not change any fundamental result. With no lossof generality the world population is normalized to unity so that∑i li = 1. It is assumed that theworld economy has an integrated market for goods and factors. Hence, there are no impedimentsto goods or factor flows throughout the world. This assumption makes the role of the preferencestrucutre in accounting for saving disparities particularly stark.Agents globally are endowed with one unit of labor time which they supply inelastically tothe market. It is assumed that all agents within a country have identical preferences, but thatpreferences of agents across countries maybe different. The preferences of the representative agentin country i are described by the recursive representationuit =cθiit ζ¯1−θiitθi+ Et[βiuδiit+1ζ¯1−δiitδi](3.3)where c denotes consumption and u denotes utility. This recursive preference specification is stan-dard except for the argument ζ¯i which stands for an externality in preferences. It is external to theindividual but is indexed by i since this externality parameter is allowed to vary across countries.This externality could represent a number of different things including external habits, relativeconsumption (“keeping up with the Jones’s”) etc. Allowing it to vary across agents implies, forexample, that the relative consumption targets could vary across countries. Note that these prefer-ences reduce to the standard additively separable across time specification in the special case wherewhen δ = 1. Ceteris paribus, a higher δ makes agents more patient by raising the discount factor.It is easy to check that this aggregator is linearly homogenous, thereby satisfying the homogeneityand regularity conditions needed for the existence of a Balanced Growth Path (BGP) as shown inFarmer and Lahiri (2005).Agents have two sources of income: wage income from working and capital income earned byrenting out their capital to firms. Households save by accumulating capital. Income can be used703.4. The Modelfor either consumption or saving. The budget constraint for households is thus given bycit + ιit = ritkit + wit (3.4)where k is the capital stock of household i at the beginning of period t while ι denotes saving. ris the rental rate on capital from country i while wi is the wage rate of labor from country i. Thecapital stock of the household evolves according to the accumulation equationkit+1 = (1− d) kit + ιit , ki0 given for i = 1, ..., N (3.5)where d is the depreciation rate.Agents maximize utility subject to equations (3.4) and (3.5). The first order condition describingthe optimal consumption-saving plan is(citζ¯it)θi−1= βiEt[(cit+1ζ¯it+1)θi−1(uit+1ζ¯it+1)δi−1(1 + rit+1 − d)]There is a common world production technology which produces output according toYt = AtKαt[N∑i=1(γili)η] 1−αηwhere K =∑Ni=1 kili is capital and li denotes labor from country i. γi is the the labor productivityof workers from country i. This is allowed to vary across countries but not across time. η controlsthe elasticity of substitution across the different types of labor.A is the productivity of the technology and is given byAt = ezta1−αtwhere a and z are productivity processes described byat = gat−1 (3.6)713.4. The Modelzt = ρzt−1 + σεt (3.7)Thus, a is the long run trend in TFP with g being the trend growth of productivity while zrepresents TFP fluctuations around the trend.Optimality in factor markets dictates that factor prices must be given by:rt = αytkt(3.8)wit = (1− α) ytli(γili)η∑Ni=1 (γili)η(3.9)where y and k denote per capita world output and capital, respectively. Note that since the worldpopulation has be normalized to unity, aggregate world output and capital are also per capitaworld output and capital: Y = y, K = k. The wage rate is country-specific as labor productivityis different and different types of labor are not perfect substitutes. On the other hand, the rentalrate is identical for the different types of capital since they are perfect substitutes in productionand they are all equally productive.Any world equilibrium in this economy must clear goods and factor markets:N∑i=1li (cit + ιit) = yt (3.10)N∑i=1likit = kt (3.11)N∑i=1li = 1 (3.12)Equation (3.10) is the goods market clearing condition which dictates that the total demand forconsumption and investment by the world must equal the world GDP. Equation (3.11) says thattotal world capital supply must equal capital demand while equation (3.12) is the correspondingworld labor market clearing condition.723.4. The ModelDefinition 3.4.1 An equilibrium in this economy is a set of allocations {cit, kit, ιit, yt} and prices{wit, rt} such that at each t (a) all households solve their optimization problem given prices; (b)firms maximize profits given prices; and (c) the allocations clear all markets.The model presented here departs from a standard neoclassical framework along two dimensions.First, it allows for what is essentially a form of external consumption habits. Second, it allows forpreference heterogeneity. Heterogeneity in the parameters in (3.3) is essential for generating adeterminate asymptotic saving distribution. In order to make this clear it is worth sketching outa brief description of how the recursive specification works in steady state. Let x˜ = xa . In steadystate, the rate of time preference for agent i is given byW iu = βi(gu˜i)δi−1while steady state normalized utility isu˜i =(Rβi)11−δigwhere R = 1+r−d. Using the definition of the aggregator, a steady state expression for normalizedconsumption follows:c˜i =[θiu˜i − θiβiδi(u˜ig)δi] 1θiHence, each u˜ maps into a different c˜. In this set-up, different δ′s and β′s imply different steadystate u˜′s. The rate of time preference Wu is equated across agents by different steady state c˜′sand u˜′s. Hence, different δ′s and β′s across agents induce a dispersion in steady state saving ratesacross agents. 62For a growing economy characterized by agents with such heterogenous preferences, Farmerand Lahiri (2005) used the results of Lucas and Stokey (1984) to prove that there exists a uniqueconvergent path to a unique steady state with a stationary distribution of saving rates provided cand u are both “non-inferior”63, and preferences display increasing marginal impatience, i.e., Wu is62In the absence of heterogeneity in the preference parameters savings would be asymptotically equated acrossagents. The effect of reference-dependence in preferences in this case would be seen only in the dynamics of savingsbehavior.63c and u are non-inferior if c < c′ and u > u′ =⇒ W˜ ic (c,u)W˜ iu(c,u)>W˜ ic(c′,u′)W˜ iu(c′,u′) , i = 1, 2.733.5. Quantifying the Modeldecreasing in c. The specification considered here satisfies all the conditions of Farmer and Lahiri(2005). Hence, their results apply to this model as well.3.5 Quantifying the ModelThe model is now taken to the data. Before proceeding to the quantification there are somemeasurement issues that need to be addressed. Specifically, measures for saving, the capital stockand the share of income accruing to capital by region are needed. The savings rate for each regioni is constructed from the relationSiY i= 1− CiY i− IiY iwhere Ci, Ii and Y i denote aggregate consumption, investment and output of region i. For eachregion their aggregate output, consumption, capital stock, workers and investment are computedby summing across all countries in the region.Capital stocks for each country in the sample are constructed using the perpetual inventorymethod. For the capital share numbers a number of alternative approaches were tried ranging froma constant 1/3 share of capital for all countries to the numbers computed by Caselli and Feyrer(2007) as well as those from Bernanke and Gurkaynak (2002). Since the results are robust to thesealternative approaches, in the following the capital share is set to a common 1/3 for all countries.To compute productivity in each region the production function is used to get:Productivity t =ytkαt(N∑i=1(γilit)η) 1−αηThe series is detrended using a linear trend and the linear trend is set equal to a1−αt . The detrendedseries is set equal to z, which in turn is used to estimatezt = ρˆzt−1 + εˆtThe computed residuals εˆt are used as shocks to the model.The first important choice to make is the choice of the externality parameter. For the baseline743.5. Quantifying the Modelcase this is set toζ¯it = c¯t for all i and twhere c¯t denotes the average per capita world consumption level. This is the most neutral startingpoint where there is a common reference consumption target for all countries. Later this assumptionwill be relaxed to explore its potential in explaining saving miracles.The world is divided into three groups: G7 economies, Asian tigers, and Emerging economies.The Emerging economies are deliberately separated from the Asian Tigers since the Tigers are ana-lyzed separately. Details regarding the countries in each region, the data and the series constructionare contained in Appendix G.A two-country version of the model presented above is first examined. To align the two-countrymodel to the data the world is normalized to only comprise the G7 and Emerging economies.64 Wecalibrate the model to match data on two regions (the G7 and Emerging economies) in 1970.The calibration strategy is to first set the vector (βG7, βEmg, θG7, θEmg, η) exogenously. For thebaseline parameterization these parameters are set to be identical across countries so as to retainthe focus of the analysis on recursive preferences and the key aspect of heterogeneity in preferencesemanating from the non-additively separable component of preferences. Accordingly, wset βG7 =βEmg = 0.97 and θG7 = θEmg = 0.8. Lastly, η is set equal to 1, which implies that efficiencyadjusted labor from different countries are perfect substitutes. β = 0.97 is relatively standardnumber for the discount factor in cross-country studies with additively separable preferences.The regional l′is are chosen to match the relative labour shares in 1970 which were 0.205 and0.795 for the G7 and the emerging economies, respectively. The vector (δG7, δEmg, γG7, γEmg) isthen set so that the steady state values of the regional saving rates and relative world capital sharesare equal to those observed in 1970. Note that the procedure targets the regional saving rates andworld capital shares. Hence, four parameters are chosen. The parameter choices and the datatargets for 1970 are summarized in Table 3.1 below:To illustrate the mechanics of the model the response of the saving rates of the two regions toa one standard deviation positive shock to the common productivity process z is plotted. Figure3.4 shows that the two regions respond differently to the same productivity shock. Specifically, the64Below a richer five-country world is calibrated by splitting the Emerging economies into four groups. The two-country specification is first presented in order to build intuition.753.5. Quantifying the ModelTable 3.1: Parameterization of Baseline ModelData moments (1970) Parameterssaving rate world capital share δi γiG7 0.247 0.698 0.978 2.803Emerging economies 0.171 0.302 0.969 0.535(βG7, βEmg, θG7, θEmg, η) = (0.98, 0.98, 0.8, 0.8, 1)This table summarizes the regional data targets and the corresponding parameter choicesto enable the baseline model to match those moments.saving rate in the G7 rises less and declines faster than in the emerging economies. The reason forthis is the increasing marginal impatience that is built into these preferences. The high consumptionregion (the G7) is also more impatient and hence responds less and and adjusts downwards fasterin response to the same increase in the real interest rate relative to the emerging economies. Thestationary world wealth distribution is non-degenerate precisely due to this feature of preferences.Figure 3.4: Impulse Response of Savings to Productivity00.050.10.150 5 10 15 20G7EmergingNote: The graph depicts the response of saving rates in the two regions to a positive one standarddeviation shock to the common productivity process zt.So, how well does the model explain world saving behavior? This is examined next by generatingthe saving from the model in response to the measured productivity shocks in the data between 1970and 2010. Recall that the model was parameterized to mimic data in 1970. All those parametersare kept fixed across time for these simulations. Figure 3.5 shows the model and data generatedsaving rates for the G7 and the emerging economies. Panel (a) shows the time series of savings forthe two regions while panel (b) of Figure 3.5 plots the difference between saving rates of the two763.5. Quantifying the Modelregions in the model and the data. The figures make clear that the model tracks the trend behaviorof saving rates in the two regions quite well. This shows up clearly in panel (b) where the differencesin saving rates between the regions in the model closely tracks the data. The noteworthy aspectof this is that, despite the fact that the model has not been calibrated to match the time seriesbehavior of the regional saving rates, it tracks turning points in the saving rates of both regionsquite closely with just one common exogenous productivity process.Figure 3.5: Saving Rates: Data & Model0. 1975 1980 1985 1990 1995 2000 2005 2010G7, dataEmerging, dataG7, modelEmerging, model0. 1975 1980 1985 1990 1995 2000 2005 2010datamodel(a) time series of saving rates (b) difference in saving ratesNotes: Panel (a) shows the saving rates in the G7 and emerging economies in the model and the databetween 1970 and 2010. Panel (b) shows the difference between the saving rates of the G7 and emergingeconomies in the model and the data during this period.Table 3.2 also shows that the baseline model performs quite well in reproducing the correlation ofsaving rates between the two regions. Additionally, the model generated saving rates also correlatequite strongly with the actual saving rates for each region individually. How sensitive are thesimulation results to the assumed parameter values for θ, a parameter for which direct estimatesdon’t exist? Table 3.2 sugests that the results of the baseline model are quite robust to changingθ. Neither the volatility nor the correlation statistics change too much in response to changing thebaseline symmetric specification (θG7, θDev) = ( to an asymmetric one.Overall, these results suggest that the model does a good job of accounting for saving behaviorover time both within regions as well as differences across regions of the world.773.5. Quantifying the ModelTable 3.2: Comparing Savings in the Model and the DataData Model(θG7, θDev)(0.8, 0.8) (0.9, 0.8) (0.8, 0.9)std dev(sG7) 0.981 2.694 2.732 2.641std dev(sDev) 1.099 2.940 2.992 2.919corr(sG7, sDev) 0.662 0.973 0.965 0.950corr(smodelG7 , sdataG7 ) 0.655 0.653 0.665corr(smodelDev , sdataDev ) 0.572 0.568 0.572Note: The table reports moments of saving rates in the data andthe model for the period 1970-2010.3.5.1 Adjustment CostsThe feature of the data that the model does not reproduce as well is the volatility of saving. Inthe data saving rates are much smoother than in the model. Table 3.2 summarizes the data andmodel moments of saving rates in the two regions. The table makes clear that under the baselineparameterization, the volatility of saving rates in the model is three-fold larger than in the data.This is mainly due to the fact that investment is too volatile in the model relative to the data. Ashas been noted by countless authors, this problem is easy to address by introducing adjustmentcosts on investment. Adjustment costs are introduced into the model by modifying the capitalaccumulation equation in each country askit+1 = (1− d) kit + ιit − kitφ(ιitkit), ki0 given for i = 1, ..., Nwhere φ′ > 0, φ′′ > 0. In calibration considered here the adjustment cost function is given by thequadratic form:φ( ιk)=b2( ιk+ 1− d− g)2This specification implies that adjustment costs are zero along any deterministic steady state growthpath. Moreover, the model reduces to one with no adjustment costs when b = 0. A crucialassumption underlying this formulation is that the adjustment cost function is common acrosscountries, i.e., there is a common b. Allowing b to be country-specific would simply improve thepotential fit of the model.783.5. Quantifying the ModelThe parameter b is calibrated to match the standard deviation of∑i ωitsit, where ωit =litkit∑i litkitis the world capital share of country i. Hence, b is picked to match the volatility of world savings. Inthe two-country world analyzed above, the implied b is 0.47. The rest of the parameter configurationfor the model under adjustment costs is summarized in Table 3.3 below:Table 3.3: Parameterization of Model With Adjustment CostsParametersδi γi bG7 0.978 2.803 0.47Emerging economies 0.969 0.535 0.47(βG7, βEmg, θG7, θEmg, η) = (0.98, 0.98, 0.8, 0.8, 1)This table summarizes the parameter choices to enable themodel with adjustment costs to match the targeted datamoments.How does the model’s predictions for the regional saving rates change with adjustment costs?Figure 3.6 shows the predicted saving rates for the two economies:Figure 3.6: Saving Rates Under Adjustment Costs: Data & Model0. 1975 1980 1985 1990 1995 2000 2005 2010G7, dataEmerging, dataG7, modelEmerging, modelNotes: The figure shows the saving rates in the G7 and emerging economies inthe model with adjustment costs and the data between 1970 and 2010.Figure 3.6 shows that the introduction of adjustment costs allows the model to match thevolatility of country-specific saving rates without changing the fit of the average level of saving.What is possibly somewhat surprising is that the model is able to match the individual saving793.5. Quantifying the Modelvolatilities even with a common adjustment cost function. This reflects the fact that the primarydifference in saving behavior across countries is the level of savings rather than their volatility whichtends to be relatively similar.3.5.2 A Five-Region ExtensionThe model presented above aggregated the world economy into two broad regions – the G7 andEmerging economies. However, there is a lot of dispersion in saving behavior across the Emerginggroup which comprises countries from sub-Saharan Africa, south Asia as well as Latin America.Can the model capture the saving heterogeneity across such a wide spectrum of countries? This isanswered this by expanding the two-region model to a five-region economy. This done by breakingup the Emerging economy group into four groups of countries: sub-Saharan Africa, Latin America,Middle East and North Africa, and Developing Asia. The data appendix provides the list ofcountries in each group.As before δ and γ are allowed to vary across regions. These parameters are calibrated to matchthe saving rates in each region in 1970. The resulting parameter configuration for the 5-countryworld economy is given in Table 3.4. The adjustment cost parameter b is set so that the standarddeviation of capital share-weighted world saving in the model matches the data value. The modelis solved to a 2nd order approximation.Table 3.4: Parameterization of 5-Region Model With Adjustment Costs1970 data targets ParametersSaving rate Capital share δi γiG7 0.247 0.715 0.978 2.962Sub Saharan Africa 0.126 0.034 0.969 0.787Latin America 0.147 0.085 0.975 2.169Middle East and North Africa 0.317 0.043 0.970 0.790Developing Asia 0.165 0.123 0.966 0.293(β, θ, η, b) = (0.98, 0.87, 1, 0.39)This table summarizes the parameter choices to enable the five region model withadjustment costs to match the targeted data moments.How well does the model perform in matching the time series behavior of regional saving ratesin the data? The particular interest of this work lies in examining the ability of the model to matchthe variation of saving rates across regions. Table 3.5 compares the key moments of interest in803.6. Saving Miraclesthe data and the model. In terms of the time-series variability of saving rates within each region,the model does very well in matching the G7 but tends to under-predict the variability of savingrates in the other four regions, especially the Middle east and north Africa. This though is nottoo surprising given the special dependence of this region on volatile oil revenues and the one-shock specification. More interestingly, the table 3.5 also shows that the model does rather wellin matching the dispersion of saving rates across regions. Thus, the average standard deviationof saving between regions is 6.38 in the data and 7.98 in the model. Given the previously notedinability to match the volatility of saving in the middle east, this is considered to be a success ofthe model in matching the cross-sectional dispersion of world savings.Table 3.5: Simulated Moments: 5-Region Model With Adjustment CostsData Modelstd dev (SG7) 0.981 0.914std dev (SSubSahara) 1.835 0.753std dev (SLatinAmerica) 1.163 0.762std dev (SMidEast+NAfrica) 4.478 1.011std dev (SDevAsia) 1.693 0.894mean σbtwt 6.376 7.984corr(σbtwt,data, σbtwt,model)0.419Note: The table reports moments of saving rates in the data and themodel for the period 1970-2010. σbtwt denotes the between-region stan-dard deviation of saving at time t.3.6 Saving MiraclesThe second motivation of this work was the observation that some countries have shown sudden anddramatic increases in their average saving rates. We showed an example of this in Figure 3.2 whichdepicted the sharp pickup in the average saving rate of the Asian Tigers. The chapter now turnsto demonstrating how the model can accommodate these dramatic saving miracles. The principalidea is that saving behavior is dictated by our goals which, in turn, is often determined by ourposition relative to a comparison group. If a society begins to aspire to have the wealth levels of amuch richer comparison group then its saving levels have to respond to achieve that new goal. Akey feature of the recursive preference structure that has been formalized here is the presence of813.6. Saving Miraclesrelative consumption. In the model presented in the previous section the relative consumption levelin preferences was just the world average per capita consumption. In this section the consequencesof a country changing its relative comparison group from the average world level to a richer cohortwill be examined. Could such a change generate an increase in saving rates similar in magnitude tothe rise in Asian savings we saw in Figure 3.2? It should be clarified at the outset that this chapteris not building a theory of aspirations. Rather, it is quantitatively exploring the dynamic generalequilibrium consequences of a change in aspirations.Consider a world economy comprising three regions. Let the two regions now be the G7,Emerging economies, and the Asian Tigers. In other words, the world economy is now beingexplained from the two-region perspective of the previous section, with the Asian Tigers alsoincluded. Recall that at each date t the externality in preferences for each i is denoted by ζi.Let average per capita world consumption beC = lG7cG7 + lEmgcEmg + lT igerscT igerswhere ci is the per capita consumption of region i. Consider two regimes:Regime 1: ζG7 = ζEmg = ζT igersRegime 2: ζG7 = ζEmg = C ; ζT igers = ζG7Under Regime 1 all three regions value their own consumption relative to the world per capitaconsumption level. Under Regime 2 however, the Asian Tigers switch their comparison group tothe G7 while the other two regions continue to use the world average as the relevant consumptioncomparison group. We consider an environment where at some date t∗, the regime switches fromRegime 1 to Regime 2. Given that per capita consumption is higher in the G7, such a regime switchrepresents a switch to a higher aspiration level for the Asian Tigers.In the context of the model, could such a regime switch account for the almost 15 percentagepoint increase in the saving rate of the Asian Tigers since 1970? To answer this question themodel is calibrated by choosing parameter values such that the model reproduces the steady statesaving rates, world capital shares and world labor shares of the three regions in 1960 under the823.6. Saving Miraclesmaintained assumption of the world being in Regime 1. The model is then perturbed with a regimeswitch to Regime 2 in 1970, where the reference consumption level for the Tigers increases to the percapita consumption of the G7 economies. Keeping the parameters underlying the initial calibrationunchanged, the estimated productivity process from 1960 to 2010 is then fed into the model andsimulate the equilibrium paths for the three economies. The model generated saving rates are thencomared with their data counterparts.To clarify the role of the regime switch in generating the saving increase, Figure 3.7 shows thesaving rates of the G7 and the Tigers when all productivity movements are shut down and only theregime switch in 1970 is allowed. Since the world economy is in steady state in 1960, saving ratesare constant till 1970 when the regime switch occurs. From that date onward savings of the Tigersrises while the G7 saving rate initially declines before recovering towards its original level.65Figure 3.7: Response of Savings to Regime Switch00. 1970 1980 1990 2000 2010G7, modelTigers, modelNote: The graph depicts the response of saving rates in the two regions when the only shock is a changein the reference consumption level of the Asian Tigers to per capita consumption of the G7 in 1970.What are the predicted saving rates of the model when the measured productivity shocks areincorporated? Panel (a) of Figure 3.8 shows the simulated path of savings under the estimatedproductivity process between 1960 and 2010 with a regime switch from Regime 1 to Regime 2 in1970. As a point of contrast, panel (b) of Figure 3.8 plots the saving rates under the assumption65The decline in the G7 saving on impact occurs due to the fall in the interest rate that is induced by the rise inthe desired savings of the Tigers. Note that the saving rate of the Emerging economies is not plotted here in orderto keep the graph uncluttered.833.7. Conclusionof no change in regime. The fit in Panel (a) is considered to be quite remarkable in terms of howwell the model-generated saving rates track the actual saving rates.Figure 3.8: Saving Miracles: Data & Model1960 1970 1980 1990 2000 2010G7, modelTigers, modelG7, dataTigers, data1960 1970 1980 1990 2000 2010G7, modelTigers, modelG7, dataTigers, data(a) saving rates under regime switch (b) saving rates without regime switchNotes: Panel (a) shows the saving rates in the G7 and the Asian Tiger economies in the model under aswitch in the comparison group for the Tigers with the data between 1970 and 2010. Panel (b) showsthe saving rates of the G7 and the Asian Tigers in the model without any switch in regime.In summary, the switch to a higher aspiration is key for the model to reproduce the sharpincrease in the saving rates of the Asian Tigers. This result is indicative of the power of theaspiration mechanism to explain the rapid growth of savings in Asia.3.7 ConclusionThe variation in saving behavior across countries has long been a puzzle and a challenge to explainfor standard neoclassical models. This chapter has explored the explanatory potential of recursivepreferences and preference heterogeneity in jointly accounting for the cross-country saving data.The model presented here used a preference specification that displays a form of relative consump-tion. Specifically, agents of a country derive utility from consumption relative to the consumptionof a reference group. The specification implies that when countries are poor they display high pa-tience and high saving rates. As their consumption gets closer to the levels of their reference grouphowever they become more impatient, a property that Lucas and Stokey (1984) called “increasingmarginal impatience”. This feature of preferences keeps the wealth distribution from becomingdegenerate even when preferences are heterogenous across countries.843.7. ConclusionThese preferences were applied to a multi-country world economy model with free capital flowsacross countries and the model was calibrated to match the long-run differences in saving ratesacross countries. Using only productivity shocks to a common world production technology as anexogenous driver, it was then shown that that the calibrated model can also account for the shortrun differences in saving rates across countries.In addition, it was also shown that a change in the aspirations of societies, as captured by achange in the reference consumption basket they use to value their own utility, can account forsudden and sharp changes in saving rates. Thus, the model can account for the rapid increase inAsian saving rates and its overall behavior between 1960 and 2010 by allowing for a change in thereference basket being used by the Asian economies from the average world consumption level tothe G7 consumption level in 1970. Intuitively, a higher reference consumption level induces greatersaving as accumulating greater wealth is the only way to achieve a higher steady state consumption.The results in this chapter indicate that this class of models has great potential in also helpingus understand changes in the wealth distribution within countries over time. Wealth evolves asa function of saving. Accounting for differential saving rates is thus key to explaining wealthdistributions and changes therein. 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Assume that all notation is in terms of individual household i in a heterogeneous agentseconomy. All agents are assumed to have the same information, preferences, credit constraint,etc... Assume that the households have access to a complete set of Arrow-Debreu securities inconsumption goods. Also assume that the household has access to a security that is indexed tohouse prices (in other words, a security that delivers in units of housing). Assume that for bothof these securities the household is exempt from obligations/payments if it defaults. For ease ofnotation, physical capital kt is omitted from the exposition, but its presence does not alter thesteps outlined below.• The budget constraint and credit constraint of of household i arecit + qthit +Rbit−1 + pcteict + pht eiht = yit + (1− δ)hit−1 + bit + eict−1 + qteiht−1 (B.1)bit ≤(θR)E˜t[qt+1]hit ≡ Q˜thit (B.2)these implycit + (qt − Q˜t)hit = y˜it +{(1− δ)qt −RQ˜t−1}hit−1 (B.3)wherey˜it = yit − pcteict − pht eiht + qteiht−1 + eict−1 (B.4)• Let1− Γt = Pr(qtqt−1< ζitθ1− δ)= Pr(Default| qtqt−1)(B.5)• If the household defaults, it loses its stock of housing and no longer has to repay its debt bt−1• The household’s maximization problem can be written as followsmaxEPt∑tβtu(cit, hit) + λit(yit + (1− δ)qtΓthit−1 + bit + Γteict−1 + qtΓteiht−1...−cit − qthit −RΓtbit−1 − pcteict − pht eiht)+µit(θEPt {qt+1}hit −Rbit) (B.6)93Appendix B. Chapter 1, Aggregation of Household Problem to Representative Agent• The first order conditions for this problem arepct = β · EPt[λit+1λitΓt+1](B.7)pht = β · EPt[λit+1λitΓt+1qt+1](B.8)λit = uc(cit, hit) ≡ uic(t) (B.9)µit =λitR− βλit+1Γt+1 (B.10)qtλit = uih(t) + µitθqt+1 + β(1− δ)λit+1qt+1Γt+1 (B.11)=⇒ qt = uih(t)uic(t)+ EPt[(β(1− δ) · λit+1λit· Γt+1 + θR)· qt+1 − βθ(EPt qt+1) λit+1λitΓt+1](B.12)• From the securities market:pct = β · EPt[λit+1λitΓt+1]= β · EPt[λjt+1λjtΓt+1](B.13)pht = β · EPt[λit+1λitΓt+1qt+1]= β · EPt[λjt+1λjtΓt+1qt+1](B.14)– So these markets imply that the price of consumption and default-weighted housing isthe same for all agents• Rewrite the budget constraint, and combine with the assumption that the borrowing con-straint bindscit + qthit +RQ˜t−1hit−1 = y˜it + (1− δ)qthit−1 + Q˜thit (B.15)=⇒ cit + (qt − Q˜t)hit = y˜it +((1− δ)qt −RQ˜t−1)hit−1 (B.16)94Appendix B. Chapter 1, Aggregation of Household Problem to Representative Agent• From the household Eulerqt =1− γcγc· cithit+ β(1− δ) EPt[λit+1λitΓt+1qt+1]︸ ︷︷ ︸common across agents−βθ (EPt qt+1) EPt [λit+1λit Γt+1]︸ ︷︷ ︸comon across agents...+θREPt [qt+1]︸ ︷︷ ︸Q˜t=⇒ cithit=1− γcγc{qt − Q˜t − β(1− δ)EPt[λit+1λitΓt+1qt+1]+ βθ(EPt qt+1)EPt[λit+1λitΓt+1]}︸ ︷︷ ︸≡Bt, common across agents=⇒ cit = Bthit (B.17)• Substitute this into the modified budget constraint (B.16)cit + (qt − Q˜t)hit = y˜it +((1− δ)qt −RQ˜t−1)hit−1=⇒ cit + (qt − Q˜t)B−1t cit = y˜it +((1− δ)qt −RQ˜t−1)hit−1=⇒(1 + (qt − Q˜t)B−1t)cit = y˜it +((1− δ)qt −RQ˜t−1)hit−1=⇒(1 + (qt − Q˜t)B−1t)∑icit =∑iy˜it +((1− δ)qt −RQ˜t−1)∑ihit−1 (B.18)– Note that ∑icit = Ct∑ihit = Ht∑iy˜it =∑i(yit − pcteict − pht eiht + qteiht−1 + eict−1)= Yt − pct∑ieict︸ ︷︷ ︸=0−pht∑ieiht︸ ︷︷ ︸=0+qt∑ieiht−1︸ ︷︷ ︸=0+∑ieict−1︸ ︷︷ ︸=0where the zeros are the result of market clearing in securities=⇒(1 + (qt − Q˜t)B−1t)Ct = Yt +((1− δ)qt −RQ˜t−1)Ht−1 (B.19)which is equivalent to a representative agents’ choice95Appendix CChapter 1, Solving the Household’sProblemThe solution to the household problem is computed using a parameterized expectations algorithm(PEA). Rewriting the household Euler (1.17) in simplified notationqt − uh(t)uc(t)= EPt [Ω(qt+1, ct+1, ht+1)|ct, ht, bt, qt,mt, θt, yt, Rt] (C.1)The expectation on the right hand side is approximated asexp (e(Xt, γ)) (C.2)The basis functions for e(·) are the Chebyshev polynomials. The vector of state variables for thehousehold, Xt, is (ht−1, kt−1, qt,mt, θt, yt, Rt)66. Define γ to be a coefficient vector. The order ofthe approximation along each dimension is given by (rh, rk, rq, rm, rθ, ry, rθ), hence γ is an rh ×rk × rq × rm× rθ × ry × rθ vector. Let ψi(x) denote the Chebyshev polynomial of order i evaluatedat x. Because the Chebyshev polynomials are defined on [−1, 1] the state variables are rescaledaccording tox˜ = 2x− x¯x¯− x − 1 ∈ [−1, 1]The approximating function e(Xt, γ) can be writtene ((h, k, q,m, θ, y, R), γ) = T (h, k, q,m, θ, y, R)γ (C.3)= (Th(h)⊗ Tk(k)⊗ Tq(q)⊗ Tm(m)⊗ Tθ(θ)⊗ Ty(y)⊗ TR(R)) γwhere Tx(x) = (ψ0(x˜), ψ1(x˜), ..., ψrx(x˜))The coefficient vector γ is solved by appealing to the Chebyshev interpolation theorem andsetting the approximating function e(·) to be exact at the Chebyshev nodes. γ therefore solves thesystemexp(EPt [Ω(·)])− e ((h, k, q,m, θ, y, R), γ) = 0 ∀ vectors of nodes (h, k, q,m, θ, y, R) (C.4)which is an rh× rk× rq× rm× rθ× ry× rθ system of equations with rh× rk× rq× rm× rθ× ry× rθunknowns. (C.4) is solved by an iterative procedure. Let xˆ denote the rx-vector of Chebyshev66In order to limit the size of the space the following approximation is applied bt ≈ θtRt qtmtht.96Appendix C. Chapter 1, Solving the Household’s Problemnodes along dimension x and Xˆ = hˆ× kˆ × qˆ × mˆ× θˆ × yˆ × Rˆ1. Start from an initial guess γ02. Compute the vector e (·, γ0) for all Chebyshev nodes hˆ× kˆ × qˆ × mˆ× θˆ × yˆ × Rˆ using (C.3)e(hˆ× kˆ × qˆ × mˆ× θˆ × yˆ × Rˆ, γ0)= T (hˆ, kˆ, qˆ, mˆ, θˆ, yˆ, Rˆ) · γ0≡ B · γ03. Given e(X, γ0) one can back out a corresponding vector of choice variables (ct, ht, bt, kt) fromthe model equations outlined in section 1.3.1, call this υ(X, γ0). Compute the expectationEPt [Ω(·)] at all the Chebyshev nodes using a Gaussian quadrature, call this vector Y (Xˆ, γ0),Y (Xˆ, γ0) =ˆΩ(qt+1, ct+1(υ(Xˆ, γ0)), ht+1(υ(Xˆ, γ0))|Xˆ, υ(Xˆ, γ0))dF (yt+1, qt+1|qt, yt,P)4. Computeγ∗ =(B′B)−1B′ log Y (Xˆ, γ0)5. Update the guessγ1 = ωγ∗ + (1− ω)γ0for ω ∈ (0, 1)676. Repeat 1-5 until convergence6867The procedure fails to converge when γ1 = γ∗, necessitating this step.68Further details on such methods can be found in Christiano and Fisher (2000).97Appendix DChapter 2, Additional FiguresFigure D.1: Effect of Internalizing Learning11.0011.0021.0031.0041.0051.0061.0071.008-20 0 20 40 60 80 100relative responseperiodsnot internalizedinternalizedpermanent shock to c: γ = 5, ψ = 6, g = 0.0198Appendix D. Chapter 2, Additional FiguresFigure D.2: Effect of Preference for Timing of Risk in Learning Model00.00020.00040.00060.00080.0010.00120.0014-100 0 100 200 300 400 500relative responseperiodsωr0ωr1ωw0ωw1(a) ω reponse to transitory shock, γ = 5, ψ = 1/1.500.00020.00040.00060.00080.0010.0012-100 0 100 200 300 400 500relative responseperiodsωr0ωr1ωw0ωw1(b) ω reponse to transitory shock, γ = 5, ψ = 600.0020.0040.0060.0080.01-100 0 100 200 300 400 500relative responseperiodsωr0ωr1ωw0ωw1(c) ω reponse to permanent shock, γ = 5, ψ = 1/1.500.0020.0040.0060.0080.01-100 0 100 200 300 400 500relative responseperiodsωr0ωr1ωw0ωw1(d) ω reponse to permanent shock, γ = 5, ψ = 699Appendix D. Chapter 2, Additional FiguresFigure D.3: Influence of g: Transitory & Permanent Shocks in Learning Model05e-050.00010.000150.00020.000250.0003-20 0 20 40 60 80 100relative responseperiodsωr0ωr1ωw0ωw1(a) ω : g = .002, transitory shocks00.00020.00040.00060.00080.0010.0012-20 0 20 40 60 80 100relative responseperiodsωr0ωr1ωw0ωw1(b) ω : g = .01, transitory shocks00.0010.0020.0030.0040.0050.006-100 0 100 200 300 400 500relative responseperiodsωr0ωr1ωw0ωw1(c) ω : g = .002, permanent shocks00.0020.0040.0060.0080.01-100 0 100 200 300 400 500relative responseperiodsωr0ωr1ωw0ωw1(d) ω : g = .01, permanent shocks100Appendix EChapter 2, LearningThis appendix outlines the derivation of the updating rule for beliefs (2.17). Write the household’sbelief as a linear hidden state model(log rtlog w˜t)=(ωr0,tωw0,t)+(ωr1,tωw1,t)· log k˜t + zt (E.1)ωt = ωt−1 + ωt (E.2)where iid∼ N(0,Σz) and w iid∼ N(0,Σω). Equation (E.1) is the observation equation and (E.2) thestate equation. As in section 2.3.3 Write xt = [log rt, log w˜t]′ andqt =(1 0 log k˜t 00 1 0 log k˜t)Then optimum Bayesian updating is given by the Kalman filtering equationsωˆt = ωˆt−1 +Kt(zt − qtωˆt−1) (E.3)Kt = Ptq′t(qtPtq′t + Σz)−1 (E.4)Pt+1 = Pt − Ptq′t(qtPtq′t + Σz)−1qtPt + Σω (E.5)Assume that as t→∞qtPtq′t + Σz ≈ Σz (E.6)q′t(Σz)−1qt → Γ (E.7)Then, assuming Σω = g˜2Γ−1 and Pt → P as t → ∞, where P = g˜ · Γ−1, then (E.3) and (E.4) canbe rewrittenωˆt = ωˆt−1 + g˜ · q′(qq′)−1(Σz)−1(xt − qtωˆt−1) (E.8)If rt and wt are uncorrelated then (E.8) can be rewrittenωˆt = ωˆt−1 + g · q′(qq′)−1(xt − qtωˆt−1) (E.9)101Appendix FChapter 2, Solution MethodThe solution to the household problem is computed using a parameterized expectations algorithm(PEA). Rewriting the household Euler (2.5) and value function (2.1) in simplified notationc−ψt = EPt[Ωe(ct+1, Vt+1)|kt, rt , wt , ωr0,t−1, ωr1,t−1, ωw0,t−1, ωw1,t−1](F.1)Vt = Ωv(kt, rt , wt , ωr0,t−1, ωr1,t−1, ωw0,t−1, ωw1,t−1) (F.2)The expectation on the right hand side is approximated asexp (e(Xt, γe)) (F.3)The basis functions for e(·) are the Chebyshev polynomials. Similarly approximate the valuefunction asv(Xt, γv) (F.4)The vector of state variables for the household, Xt, is (kt, rt , wt , ωji,t−1). Define γe and γv to becoefficient vectors. The order of the approximation along each dimension is given by(rk, r,r, r,w, rω,r0, rω,r1, rω,w0, rω,w1), hence γj is an (rk×r,r×r,w×rω,r0×rω,r1×rω,w0×rω,w1) vec-tor. Let ψi(x) denote the Chebyshev polynomial of order i evaluated at x. Because the Chebyshevpolynomials are defined on [−1, 1] the state variables are rescaled according tox˜ = 2x− x¯x¯− x − 1 ∈ [−1, 1]The approximating function e(Xt, γ) can be writtene ((k, r, w, ωr0, ωr1, ωw0 , ωw1 ), γe) = T (k, r, w, ωr0, ωr1, ωw0 , ωw1 )γe (F.5)= (Tk(k)⊗ Tr(r)⊗ ...⊗ Tω,w1(ωw1 )) γev ((k, r, w, ωr0, ωr1, ωw0 , ωw1 ), γv) = T (k, r, w, ωr0, ωr1, ωw0 , ωw1 )γv (F.6)where Tx(x) = (ψ0(x˜), ψ1(x˜), ..., ψrx(x˜))The coefficient vectors γe and γv are solved by appealing to the Chebyshev interpolation theoremand setting the approximating functions e(·) v(·) to be exact at the Chebyshev nodes. γe and γv102Appendix F. Chapter 2, Solution Methodtherefore solve the systemexp(EPt [Ω(·)])− e ((k, r, w, ωr0, ωr1, ωw0 , ωw1 ), γe) = 0 (F.7)Ωv(·)− v ((k, r, w, ωr0, ωr1, ωw0 , ωw1 ), γv) = 0which is a 2× rk × r,r × r,w × rω,r0× rω,r1× rω,w0× rω,w1 system of equations with 2× rk × r,r ×r,w×rω,r0×rω,r1×rω,w0×rω,w1 unknowns. (F.7) is solved by an iterative procedure. Let xˆ denotethe rx-vector of Chebyshev nodes along dimension x and Xˆ = kˆ × ˆr × ˆw × ωˆr0 × ωˆr1 × ωˆw0 × ωˆw11. Start from an initial guess {γe,0, γv,0}2. Compute the vectors e (·, γe,0) and v (·, γv,0) for all Chebyshev nodes Xˆ using (F.5)e(Xˆ, γe,0)= T (Xˆ) · γe,0≡ B · γe,0v(Xˆ, γv,0)= T (Xˆ) · γv,0≡ B · γv,03. Given e(X, γe,0) & v(X, γv,0) one can back out the corresponding choice ct from the modelequations outlined in section 2.3.1, call this c(X, γ0). Compute the expectations EPt [Ωe(·)]and EPt[Ωv(·)1−γ]at all the Chebyshev nodes using a Gaussian quadrature, call these vectorsY (Xˆ, γ0) and RV (Xˆ, γ0)Y (Xˆ, γ0) =ˆΩe(c(Xˆ ′, γ0), v(Xˆ ′, γ0)|Xˆ ′, γ0)· dF (Xˆ ′|Xˆ, c(Xˆ, γ0),P)RV (Xˆ, γ0) =ˆΩv(c(Xˆ ′, γ0), v(Xˆ ′, γ0)|Xˆ ′, γ0)1−γ · dF (Xˆ ′|Xˆ, c(Xˆ, γ0),P)4. Computeγ∗e =(B′B)−1B′ log Y (Xˆ, γ0)γ∗v =(B′B)−1B′((1− β)c(Xˆ, γ0)1−ψ + βRV 1−γ) 11−ψ5. Update the guessγe,1 = ωγ∗e + (1− ω)γe,0γv,1 = ωγ∗v + (1− ω)γv,0for ω ∈ (0, 1]6969The procedure can fail to converge when γ1 = γ∗, necessitating this step.103Appendix F. Chapter 2, Solution Method6. Repeat 1-5 until convergence7070Further details on such methods can be found in Christiano and Fisher (2000).104Appendix GChapter 3, Data & Calibration NotesG.1 CountriesTable G.1: List of Countries in Regions, 3-Region ModelRegion CountriesAsian Tigers Hong Kong, Indonesia, Korea, Malaysia, Philippines, Singapore, Taiwan, Thailand, VietnamEmerging Afghanistan, Angola, Albania, Algeria, Argentina, Armenia, Bahamas, Bahrain,Bangladesh, Barbados, Belize, Benin, Bhutan, Bolivia, Botswana, Brazil, Brunei, Bulgaria,Burkina Faso, Burundi, Cambodia, Cameroon, Cape Verde, Central African Republic,Chad, Chile, China, Colombia, Comoros, Democratic Republic of Congo, Republic of Congo,Costa Rica, Cote d’Ivoire, Dominican Republic, Ecuador, Egypt, El Salvador, Equatorial Guinea,Ethiopia, Fiji, Gabon, Gambia, Ghana, Guatemala, Guinea, Guinea-Bissau, Guyana, Haiti,Honduras, Hungary, India, Iran, Iraq, Jamaica, Kenya, Laos, Lebanon, Lesotho, Liberia,Madagascar, Malawi, Maldives, Mali, Maritius, Mauritania, Mexico, Mongolia, Morocco,Mozambique, Namibia, Nepal, Nicaragua, Niger, Nigeria, Oman, Pakistan, Panama,Papua New Guinea, Paraguay, Peru, Poland, Romania, Rwanda, Senegal, Sierra Leone,Solomon Islands, South Africa, Sri Lanka, Sudan, Suriname, Swaziland, Syria,Tanzania, Trinidad Tobago, Togo, Tunisia, Turkey, Uganda, Uruguay, Venezuela, Zambia,ZimbabweG7 Canada, France, Germany, Italy, Japan, United Kingdom, United States105G.1. CountriesTable G.2: List of Countries in Regions, 5-Region ModelRegion CountriesDeveloping Asia Afghanistan, Bangladesh, Bhutan, Brunei, Cambodia, China, Fiji, IndiaIndoneisa, Malaysia, Maldives, Nepal, Pakistan, Papua New Guinea,Philippines, Solomon Islands, Sri Lanka, Thailand, VietnamG7 Canada, France, Germany, Italy, Japan, United Kingdom, United StatesLatin America & Carribbean Argentina, Bahamas, Barbados, Belize, Bolivia, Brazil, Chile,Colombia, Costa Rica, Dominican Republic, Ecuador, El Salvador,Guatemala, Guyana, Haiti, Honduras, Jamaica, Mexico, Nicaragua,Panama, Paraguay, Peru, Suriname, Trinidad & Tobago,Uruguay, VenezualaMiddle East & Northern Africa Algeria, Bahrain, Egypt, Iran, Iraq, Jordan, Lebanon, Mauritania,Morocco, Oman, Sudan, Syria, TunisiaSub Sahara Angola, Benin, Botswana, Burkina Faso, CameroonCentral Africa Republic, Comoros, Democratic Republic of Congo,Republic of Congo, Cote d’Ivoire, Equatorial Guinea,Ethiopia, Gabon, Gambia, Ghana, Guinea, Guinea-Bissau, KenyaLesotho, Liberia, Madagascar, Malawi, Mali, Maritius, MozambiqueNamibia, Niger, Nigeria, Rwanda, Sao Tome Principe, Senegal,Seychelles, Sierra Leone, South Africa, Sudan, Swaziland, Tanzania,Togo, Uganda, Zambia, Zimbabwe106G.2. Calibration & Data ComputationG.2 Calibration & Data ComputationThe model is calibrated to match data on two regions over the period 1970. The vector (β1, β2, θ2, θ2, η)is set exogenously, while li are set to match relative labour shares in at the start of the sample.(δ1, δ2, γ1, γ2) is then set so that the steady state values of the regional saving rates and relativecapital shares are equal to those observed in the data at the start of the sample. We set the capitalshare of output for each country as αi = 0.33.The saving rate of country is computed as1− consumption share of rgdp per capita− investment share of rgdp per capitaThe savings rates of each region are averaged across regions to obtain regional saving rates (averagesare unweighted)The capital stock of each country in the sample is constructed using the perpetual inventorymethod. Assuming a depreciation rate of 0.06 for each country and using the growth rate of realgdp per capita the steady state relationship is given byki1970 = ii1970/(dep+ girgdp)In computing the capital stock numbers it is assumed that depreciation (dep) is 0.06 for all countries.The growth rate grgdp is measured as the average growth rate of rgdp per worker in the first tenperiods of the sample.Labour supply as measured by the number of workers, is computed aslab =real gdp per capita × populationreal gdp per workerThe output and labour supply of each country are summed across regions to obtain output perworker and capital per worker for each region.TFP shocks are measures from the data. For each region, prodt =ytkαtwas computed and theseries was detrended using a linear trend. The linear trend was set equal to at. The detrendedseries was then set equal to zt and estimate zt = ρˆzt−1 + ˆt. The residuals ˆt are used as shocks inthe simulated model.107Appendix HData AppendixUS. Bureau of Economic Analysis, Compensation of employees: Wages and salaries [A576RC1Q027SBEA],retrieved from FRED, Federal Reserve Bank of St. Louis,https://research.stlouisfed.org/fred2/series/A576RC1Q027SBEA/US. Bureau of Labor Statistics, Consumer Price Index for All Urban Consumers: All Items LessFood & Energy [CPILFENS], retrieved from FRED, Federal Reserve Bank of St. Louis,https://research.stlouisfed.org/fred2/series/CPILFENS/, August 1, 2015.US. Bureau of Economic Analysis, Personal Consumption Expenditures [PCECA], retrieved fromFRED, Federal Reserve Bank of St. Louis,https://research.stlouisfed.org/fred2/series/PCECA/, August 1, 2015.US. Bureau of Economic Analysis, Personal consumption expenditures: Services: Housing: Imputedrental of owner-occupied nonfarm housing [DOWNRC1A027NBEA], retrieved from FRED, Fed-eral Reserve Bank of St. Louis,https://research.stlouisfed.org/fred2/series/DOWNRC1A027NBEA/, August 1, 2015.US. Bureau of Economic Analysis, Shares of gross domestic product: Personal consumption expen-ditures: Durable goods [DDURRE1A156NBEA], retrieved from FRED, Federal Reserve Bankof St. Louis,https://research.stlouisfed.org/fred2/series/DDURRE1A156NBEA/, August 1, 2015.US. Bureau of Economic Analysis, Shares of gross domestic product: Personal consumption ex-penditures: Nondurable goods [DNDGRE1A156NBEA], retrieved from FRED, Federal ReserveBank of St. Louis,https://research.stlouisfed.org/fred2/series/DNDGRE1A156NBEA/, August 1, 2015.US. Bureau of Economic Analysis, Shares of gross domestic product: Personal consumption expen-ditures: Services [DSERRE1A156NBEA], retrieved from FRED, Federal Reserve Bank of St.Louis,https://research.stlouisfed.org/fred2/series/DSERRE1A156NBEA/, August 1, 2015.Freddie Mac, 30-Year Fixed Rate Mortgage Average in the United States [MORTGAGE30US],retrieved from FRED, Federal Reserve Bank of St. Louis,https://research.stlouisfed.org/fred2/series/MORTGAGE30US/, August 1, 2015US. Federal Housing Finance Agency, All-Transactions House Price Index for the United States[USSTHPI], retrieved from FRED, Federal Reserve Bank of St. Louis,108Appendix H. Data Appendixhttps://research.stlouisfed.org/fred2/series/USSTHPI/, August 1, 2015.Board of Governors of the Federal Reserve System (US), Households and Nonprofit Organizations;Credit Market Instruments; Asset, Level [HSTCMAHDNS], retrieved from FRED, Federal Re-serve Bank of St. Louis,https://research.stlouisfed.org/fred2/series/HSTCMAHDNS/, August 1, 2015.Board of Governors of the Federal Reserve System (US), Households and Nonprofit Organizations;Credit Market Instruments; Liability, Level [CMDEBT], retrieved from FRED, Federal ReserveBank of St. Louis,https://research.stlouisfed.org/fred2/series/CMDEBT/, August 1, 2015.Board of Governors of the Federal Reserve System (US), Real Gross Domestic Product [GDPC1],retrieved from FRED, Federal Reserve Bank of St. Louis,https://research.stlouisfed.org/fred2/series/GDPC1/, August 1, 2015.Board of Governors of the Federal Reserve System (US), Households and Nonprofit Organizations;total mortgages; asset, Level [HNOTMAQ027S], retrieved from FRED, Federal Reserve Bankof St. Louis,https://research.stlouisfed.org/fred2/series/HNOTMAQ027S/, August 1, 2015.Board of Governors of the Federal Reserve System (US), Households and Nonprofit Organizations;total mortgages; liability [HNOTMLQ027S], retrieved from FRED, Federal Reserve Bank of St.Louis,https://research.stlouisfed.org/fred2/series/HNOTMLQ027S/, August 1, 2015.Board of Governors of the Federal Reserve System (US), Households and Nonprofit Organizations;Credit Market Instrutments; Asset, Level [CMIABSHNO], retrieved from FRED, Federal Re-serve Bank of St. Louis,https://research.stlouisfed.org/fred2/series/CMIABSHNO/, August 1, 2015.Board of Governors of the Federal Reserve System (US), Households and Nonprofit Organizations;Home Mortgages; Liability [HMLBSHNO], retrieved from FRED, Federal Reserve Bank of St.Louis,https://research.stlouisfed.org/fred2/series/HMLBSHNO/, August 1, 2015.Board of Governors of the Federal Reserve System (US), Households; Owner’s Equity in RealEstate as a Percentage of Household Real Estate, Level [HOEREPHRE], retrieved from FRED,Federal Reserve Bank of St. Louis,https://research.stlouisfed.org/fred2/series/HOEREPHRE/, August 1, 2015.Board of Governors of the Federal Reserve System (US), Households and Nonprofit Organizations;Credit Market Instruments; Liability, level [HSTCMDODNS], retrieved from FRED, FederalReserve Bank of St. Louis,https://research.stlouisfed.org/fred2/series/HSTCMDODNS/, August 1, 2015.Board of Governors of the Federal Reserve System (US), Households; Owner’s Equity in RealEstate, Level [OEHRENWBSHNO], retrieved from FRED, Federal Reserve Bank of St. Louis,109Appendix H. Data Appendixhttps://research.stlouisfed.org/fred2/series/OEHRENWBSHNO/, August 1, 2015.Board of Governors of the Federal Reserve System (US), Delinquency Rate On Single-Family Res-idential Mortgages, Booked In Domestic Offices, All Commercial Banks [DRSFRMACBS], re-trieved from FRED, Federal Reserve Bank of St. Louis,https://research.stlouisfed.org/fred2/series/DRSFRMACBS/, August 1, 2015.S&P Dow Jones Indices LLC, S&P/Case-Shiller 10-City Composite Home Price Index [SPCS10RNSA],retrieved from FRED, Federal Reserve Bank of St. Louis,https://research.stlouisfed.org/fred2/series/SPCS10RNSA/, August 1, 2015.Davis, Morris A. and Jonathan Heathcote, 2007, “The Price and Quantity of Residential Land in theUnited States,” Journal of Monetary Economics, vol. 54 (8), p. 2595-2620; data located at Landand Property Values in the U.S., Lincoln Institute of Land Policy http://www.lincolninst.edu/resources/Saiz, Albert. 2010. “The Geographic Determinants of Housing Supply,” The Quarterly Journal ofEconomics 125: 1253-1296.U.S. Dept. of Commerce, Bureau of the Census. AMERICAN HOUSING SURVEY, 2005: NA-TIONAL MICRODATA. ICPSR04593-v1. Washington, DC: U.S. Dept. of Commerce, Bureauof the Census [producer], 2006. Ann Arbor, MI: Inter-university Consortium for Political andSocial Research [distributor], 2007-06-13. http://doi.org/10.3886/ICPSR04593.v1Bloomberg L.P. CME Case Shiller Home Price Futures 1/5/2007 to 1/2/2015. Retrieved May 20,2015 from Bloomberg database.Heston, Alan, Robert Summers and Bettina Aten (2012). Penn World Table Version 7.1. Center forInternational Comparisons of Production, Income and Prices at the University of PennsylvaniaURL http://pwt.econ.upenn.edu/.110


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