The review of economic studies , tập 77, số 4, 2010

THE REVIEW OF Vol 77(4) No 273 ECONOMIC STUDIES Estimating Intertemporal Allocation Parameters using Synthetic Residual Estimation Sule Alan and Martin Browning 1231 Missing Women: Age and Disease Siwan Anderson and Debraj Ray 1262 Can Gender Parity Break the Glass Ceiling? Evidence from a Repeated Randomized Experiment Manuel F Bagues and Berta Esteve-Volart 1301 Political Competition, Policy and Growth: Theory and Evidence from the US Timothy Besley, Torsten Persson and Daniel M Sturm 1329 Modelling Income Processes with Lots of Heterogeneity Martin Browning, Mette Ejrnæs and Javier Alvarez 1353 Habits Revealed Ian Crawford 1382 Cary Deck and Harris Schlesinger 1403 How Important Is Human Capital? A Quantitative Theory Assessment of World Income Inequality Andrés Erosa, Tatyana Koreshkova and Diego Restuccia 1421 Exploring Higher Order Risk Effects The Long and Short (of) Quality Ladders Amit Khandelwal 1450 Labour-Market Matching with Precautionary Savings and Aggregate Fluctuations Per Krusell, Toshihiko Mukoyama and Ays¸egül S¸ahin 1477 October 2010 Efficient Estimation of the Parameter Path in Unstable Time Series Models Ulrich K Müller and Philippe-Emmanuel Petalas 1508 Choosing the Carrot or the Stick? Endogenous Institutional Choice in Social Dilemma Situations Matthias Sutter, Stefan Haigner and Martin G Kocher 1540 Why Has House Price Dispersion Gone Up? Stijn Van Nieuwerburgh and Pierre-Olivier Weill 1567 THE REVIEW OF ECONOMIC STUDIES The Review was started in 1933 by a group of young British and American Economists It is published by The Review of Economic Studies Ltd, whose object is to encourage research in theoretical and applied economics, especially by young economists, and to publish the results in The Review of Economic Studies EDITORIAL COMMITTEE Joint Managing Editors BRUNO BIAIS, University of Toulouse MARCO OTTAVIANI, Northwestern University IMRAN RASUL, University College London ENRIQUE SENTANA, CEMFI KJETIL STORESLETTEN, University of Oslo Editorial Office Manager ANNIKA ANDREASSON, IIES, Stockholm University Secretary and Business Manager CHRISTOPHER WALLACE, Trinity College, Oxford Foreign Editors DONALD ANDREWS, Yale University MARCO BATTAGLINI, Princeton University DIRK BERGEMANN, Yale University MATTHIAS DOEPKE, Northwestern University RAY FISMAN, Columbia University BURTON HOLLIFIELD, Carnegie Mellon University 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1467-937X (Online) Review of Economic Studies (2010) 77, 1231–1261 © 2010 The Review of Economic Studies Limited 0034-6527/10/00411011$02.00 doi: 10.1111/j.1467-937X.2010.00607.x Estimating Intertemporal Allocation Parameters using Synthetic Residual Estimation and MARTIN BROWNING University of Oxford First version received September 2004; final version accepted September 2009 (Eds.) We present a novel structural estimation procedure for models of intertemporal allocation This is based on modelling expectations errors directly; we refer to it as synthetic residual estimation (SRE) The flexibility of SRE allows us to account for measurement error in consumption and for heterogeneity in intertemporal allocation parameters An investigation of the small sample properties of the SRE estimator indicates that it dominates generalized method of moments (GMM) estimation of both exact and approximate Euler equations in the case when we have short panels and noisy consumption data We apply SRE to two panels drawn from the Panel Study of Income Dynamics (PSID) and estimate the joint distribution of the discount factor and the elasticity of intertemporal substitution We reject strongly homogeneity of the discount factor and the elasticity of intertemporal substitution We find that, on average, the more educated are more patient and less willing to substitute intertemporally than the less educated Within education strata, patience and willingness to substitute are positively correlated INTRODUCTION We consider the familiar intertemporal allocation model with iso-elastic preferences If we have exponential discounting and there are no liquidity constraints, the resulting exact Euler equation for consumption growth is: Ct+1 Ct −γ (1 + rt+1 ) β = εt+1 (1) where Ct is consumption in period t, γ is the coefficient of relative risk aversion, β is the discount factor, and rt+1 is the real rate of interest between periods t and t + In this framework, the elasticity of intertemporal substitution (eis) is the reciprocal of the coefficient of relative risk aversion Therefore we use these two terms interchangeably throughout the text The term εt+1 is a “surprise” term which satisfies the orthogonality condition: Et (εt+1 ) = (2) where Et (.) denotes the expectation operator conditional on information available at time t 1231 Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 SULE ALAN University of Cambridge 1232 REVIEW OF ECONOMIC STUDIES The other widely used data resource for consumption studies are quasi-panels These are constructed from cross-section expenditure survey information by taking within-period means following the same population (for example, means over all the 25 year olds in one year and all the 26 year olds in the next year) Although this averaging reduces the effect of measurement error, the construction of quasi-panels from samples that change over time induces sampling error, which is very much like measurement error In the wider measurement error literature, resolutions of the problem for non-linear estimators have only been possible in particular circumstances; see Hausman (2001), Schennach (2004), Wansbeek (2001), and Hong and Tamer (2003) If the measurement is classical in the sense of being multiplicative and independent of everything else © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 Over the past quarter century, this theoretical framework has been the principal vehicle for estimating preference parameters such as β and γ , and for testing for the validity of the standard orthogonality assumptions in general and for the “excess sensitivity” of consumption to predictable income growth in particular Generalized method of moments (GMM) estimation is based on the orthogonality condition (2) using instruments dated t or before such as lagged consumption, interest rate, and income variables The attraction of estimation based on equation (1) is that one can estimate the preference parameters without explicitly parameterizing the stochastic environment that agents face Browning and Lusardi (1996) discuss the results of 25 studies using Euler equation methods on micro data and conclude that the results are disappointing A number of subsequent Monte Carlo-based papers have investigated why we experience this failure (Carroll, 2001; Ludvigson and Paxson, 2001; Attanasio and Low, 2004) The problems identified are manifold, but the most important seems to be the paucity of appropriate data (long panels on demands or consumption) and the substantial measurement error in consumption (see Shapiro, 1984; Altonji and Siow, 1987; Runkle, 1991) Regarding the latter, Runkle (1991), for example, estimates that 76% of the variation in the growth rate of food consumption in the Panel Study of Income Dynamics (PSID) is noise.1 Measurement error of this magnitude means that we cannot use the exact Euler equation for estimation since the equation is non-linear in parameters (a point first made in the general context of non-linear GMM by Amemiya (1985)) Generally, the presence of measurement error when estimating non-linear equations raises serious and difficult problems.2 Alan, Attanasio, and Browning (2009) present two estimation strategies that allow for “classical” measurement error in exact Euler equations, but these require moderate length panels and cannot be extended to allow for heterogeneity in preference parameters An early reaction to these problems was to linearize equation (1) by taking logs and approximating ln εt+1 in some way The use of linearized Euler equations (whether first or second order) solves the measurement error problem3 but the transformation ln εt+1 introduces latent variables that lead to violations of the orthogonality conditions exploited by GMM methods Given these problems, Carroll (2001) concludes that “empirical estimation of consumption Euler equations should be abandoned” On the other hand, Attanasio and Low (2004) present results based on simulation data that suggest that Carroll’s conclusion is overly pessimistic if we have long panels (40 periods, say) and time series variation in real rates We not find this conclusion too comforting for empirical work since we not have long consumption panels Thus the emerging consensus seems to be that we must give up on empirical Euler equations and resort to estimating (or calibrating) structural dynamic programming models based on specifying the environment agents face (see Hubbard, Skinner, and Zeldes, 1995; Carroll and Samwick, 1997; Gourinchas and Parker, 2002) The main problem with this approach is that because of computational complexities it can only accommodate very limited sources of uncertainty and heterogeneity In this paper, we propose an alternative approach to estimating the parameters of intertemporal allocation The key to our approach is that, associated with every structural model, there ALAN & BROWNING SYNTHETIC RESIDUAL ESTIMATION 1233 The class of SMD estimators (see Hall and Rust, 2002) includes the efficient method of moments procedure of Gallant and Tauchen (1996) and the indirect inference method of Gouri´eroux, Monfort and Renault (1993) Current GMM methods not provide any way to allow for heterogeneity See Attanasio, Banks and Tanner (2002), Vissing-Jorgensen (2002), and Guvenen (2006) for evidence based on stock market participation, and Dohmen et al (2005) and Guiso and Piaella (2001) for evidence based on survey responses to risk attitude questions © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 is a distribution for the expectations errors (the εt+1 ’s in equation 1) If we knew this distribution, then we could identify preference parameters from a path of consumption and interest rates The problem is that the distribution of expectations errors is not known and may depend on preference parameters in an unknown way In Section we show that for a wide class of models with heterogeneous agents, the distribution of (pooled) expectations errors can be well approximated by a mixture of two lognormals which is independent of preference parameters Using this result, we show that from equation (1) we can jointly identify the preference parameters and the parameters of the mixture distribution We term our new procedure synthetic residual estimation (SRE) to reflect that it relies on generating synthetic expectations errors, εt+1 There are a variety of possible estimation methods that could be used; we use simulated minimum distance (SMD)4 since it allows us to adopt heterogeneity schemes for which it is very difficult to write down the likelihood function We lay out the details of our simulation-based estimation procedure in Section In Section 4, we compare the small sample properties of SRE with linearized and exact GMM when we not have any heterogeneity.5 These Monte Carlo results suggest that even when there is considerable measurement error (e.g half the observed consumption growth variance is due to noise), SRE works well both in absolute terms and relative to GMM, even for moderately short panels In the second half of the paper we present an empirical application of SRE The major innovation in our modelling is that we allow heterogeneity in the discount factor and the coefficient of relative risk aversion A number of regularities observed in consumption and wealth data can be rationalized by allowing for heterogeneity in the discount factor and/or in risk aversion The most important of these is the heterogeneity in lifetime wealth accumulation by households with similar earnings profiles This requires heterogeneity in the discount factor (see Samwick, 1998; Krusell and Smith, 1998; Hendricks, 2007) The only estimates of the distribution of discount factors within the context of consumption life-cycle models are due to Lawrance (1991), Samwick (1998), and Cagetti (2003) Heterogeneity in risk aversion (or eis) also has great potential for explaining some regularities, particularly for household portfolio allocations To our knowledge, there are no estimates of the distribution of the eis in the consumption literature There are, however, several papers in the literature that indicate that the eis is very likely to be heterogeneous.6 In the empirical application we consider two samples of households drawn from the PSID from 1974 to 1987, based on their broadly defined education group membership In Section 5, we present our sample selection, variable definitions, and some of the features of our two samples In particular, we show that even within education strata there is considerable variation across households in the mean and standard deviation of consumption growth We use this variation to identify the joint distribution of the discount factor and the coefficient of relative risk aversion We present our results and their implications in Section In line with other studies based on consumption and wealth data, we find that the more educated are more patient than the less educated The median discount factors are 0.93 and 0.96 for the less educated and the more educated, respectively There is also considerable heterogeneity within education strata, with a significant fraction of each stratum having a discount factor below 0.9 and a high proportion of the educated having a value close to unity We discuss how these 1234 REVIEW OF ECONOMIC STUDIES THE DISTRIBUTION OF EXPECTATIONS ERRORS Our estimator is based on sampling from the conditional distribution of the expectations errors (the εt+1 ’s in the Euler equation (1)) Our motivation for this is that we found that this distribution displays some strong regularities across many of the simulation models considered in the literature We illustrate this in this section The data generating process we use is very standard; details are given in Appendix A Beginning-of-period assets or debts, At , plus within-period income, Yt , minus consumption, Ct , are carried forward from period t to t + at a real interest rate of rt+1 : At+1 = (1 + rt+1 )(At + Yt − Ct ) (3) Assuming exponential discounting and an iso-elastic felicity function, this gives the Euler equation (1) We present simulation results for 17 variants of the standard model These differ in the curvature of the felicity function (γ ); the time discount rate (δ = (1 − β) /β); the income process parameters; whether the interest rate is stochastic; the presence of liquidity constraints; and the degree of measurement error Our environment has agents with a finite lifetime of 80 periods, with no bequest motive and no initial assets Agents face two types of income shocks, permanent and transitory For agent h, the assumed income process is: Yt = Pt ut (4) where ut is an independent and identically distributed (iid) lognormal shock to transitory income with unit mean and a constant variance exp σu2 − Pt is permanent income which follows a log random-walk process: Pt = Pt−1 zt (5) where zt is an iid lognormal shock to permanent income with unit mean and a constant variance exp σz2 − In our simulations we set σu = σz = 0.1, and also experiment with σz = 0.15 Values such as these are conventional in the consumption and income literature; see Gourinchas and Parker (2002) and Low, Meghir and Pistaferri (2008) We assume that the innovations to income are independent over time and across individuals so that we assume away aggregate shocks to income The real interest rate has a mean of 0.03 and is assumed to be the same for everyone between any two periods For the variants that have stochastic interest rates, the process is an AR (1) with a mean of 0.03, an AR parameter of 0.6, and an error with a standard deviation of 0.025 © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 estimates should be interpreted in Section as our sample selection procedure excludes all liquidity-constrained and potentially high-discount-rate households For the coefficient of relative risk aversion, we find that the less educated households are less risk averse than the more educated households The medians of the two distributions are 6.2 and 8.4, respectively These values are higher than those estimated in consumption-based studies but closely in line with wealth- and portfolio choice-based studies The finding that the less educated have a higher discount rate and a lower coefficient of relative risk aversion than the more educated implies that patience and risk aversion are positively correlated across the two education strata Within strata, however, we find the opposite result of a negative correlation between patience and risk aversion; this is consistent with experimental evidence, which uses subjects who have the same education level ALAN & BROWNING SYNTHETIC RESIDUAL ESTIMATION 1235 TABLE Simulated models Model Coeff RRA γ 10 11 12 13 14 15 16 17 0.05 No 0.05 No 0.15 No 0.05 No 0.05 No 0.15 No 0.05 No 0.15 No 0.05 Yes 0.15 Yes 0.15 Yes 4/2 0.05 No 0.05/0.15 No 0.05 No 4/2 0.05/0.15 No Model with moderate measurement error (30% noise) Model with high measurement error (85% noise) (1&2) (1&3) (1&4) (1&2&3&4) Discount rate δ Real interest rate stochastic Income process, σz Liquidity constraint 0.1 0.1 0.1 0.15 Carroll Carroll 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1/0.15 0.1/0.15 No No No No Implicit Implicit Yes Yes No No Yes No No No No Notes: The interest rate is 0.03 for constant interest rate models In models with stochastic interest rates (models 9–11), the interest rate is assumed to have a mean of 0.03, a standard deviation of 0.025 and an AR(1) coefficient of 0.6 The standard deviation of (the logarithm of) shocks to transitory income, σu , is set to 0.1 for all models, and the standard deviation of (the logarithm of) shocks to permanent income is denoted as σz “Carroll” income process refers to the assumption that transitory shocks take the value zero with a 1% probability All models are solved for T = 80 periods and simulated for N = 10, 000 agents © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 For each variant, we first solve the dynamic program and generate a decision rule for each period Using the decision rules, we simulate 80-period consumption paths for each of 10,000 simulated individuals We then remove the first 20 and the last 20 periods for each agent to minimize starting and end effects and obtain 40 periods For each pair of adjacent simulated periods, we construct the expectation error (εt+1 ) according to equation (1) This gives 39 expectations errors for each of our agents However, we lose one more period (giving a total of 38 periods), as we want to assess the dependence of the variance of εt+1 on εt Table presents the features of all 17 variants we consider The second to fourth columns report the coefficient of relative risk aversion, the discount rate, and whether the interest rate is stochastic, respectively The standard deviation of the logarithm of permanent income shocks (σz ) is presented in the fifth column The last column indicates whether we impose a liquidity constraint or not We take model as our benchmark variant and make changes one at a time Model lowers the coefficient of relative risk aversion from to 2; model increases the discount rate from 0.05 to 0.15; and model increases the standard deviation of the logarithm of permanent income shocks from 0.1 to 0.15 Models and impose an implicit liquidity constraint; this process is examined under two different impatience levels For models with an explicit liquidity constraint (models 7, 8, and 11) we have to take account of the fact that the Euler equation does not hold for all periods To this, we remove shocks if the end of period assets in the previous period are zero; that is, if the agent does not carry forward assets between t and t + (At = 0), then εt+1 for that agent is dropped We experiment with stochastic interest rates in models to 11 These models are also examined with and without liquidity constraints 1236 REVIEW OF ECONOMIC STUDIES TABLE Tests for expectations errors Test for equality of distributions Model L-test M -test 10 11 12 13 14 15 16 17 17.4 45.5 0.01 0.00 15.1 0.00 18.9 41.6 14.1 0.01 68.2 0.00 0.00 0.00 0.00 51.9 4.03 53.8 20.6 78.6 13.1 60.6 28.8 52.2 42.0 54.7 48.9 48.9 65.8 39.1 15.8 79.4 50.5 33.8 Heteroskedasticity Coefficient (ω) ˆ −0.010 −0.011 0.079 −0.195 −0.011 3.05 −0.010 0.003 −0.013 0.053 −0.004 −0.005 049 −0.086 −0.015 −0.536 −2.92 t-ratio −2.7 −0.3 5.4 −13 −3.0 39.7 −2.6 0.7 −3.1 2.5 −0.74 −1.5 4.4 −6.7 −2.2 −12.2 −3.35 Notes: L-test refers to p -values obtained from Kolmogorov–Smirnov equality-of-distributions test under the lognormality assumption M -test refers to p -values obtained from the same test under mixture of two lognormals The last two columns give the estimated slope parameter (ωˆ ) and associated t -value from the regression of (εh,t − 1)2 on (εh,t −1 − 1) to assess the conditional heteroskedasticity in expectations errors When performing this test, we not allow the parameters of the lognormal distribution to be estimated With such a large sample size, this should be largely irrelevant To make the estimation tractable, we only consider 500 households (19, 000 observations) for each variant © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 Note that models 6, 8, and 11 simulate “buffer stock” savers, as they generate very little assets due to high impatience and liquidity constraints To capture the effect of heterogeneity, we also experiment with some mixed models Model 12 is generated by mixing simulated paths of models and with equal probability (heterogeneity in the coefficient of relative risk aversion); model 13 is the mixture of models and (heterogeneity in the discount rate); model 14 is a mixture of models and (heterogeneity in the variance of the income process); and model 15 is a mixture of models 1–4 Finally, models 16 and 17 add noise to the consumption paths obtained from the baseline model (model 1) In model 16 (respectively, model 17) we introduce moderate (respectively, high) noise so that 30% (respectively, 85%) of the variance of consumption growth is due to measurement error The unconditional mean of the expectations errors is unity for all models except for those with measurement error To test for the functional form of the distribution of errors, we first estimate the parameters of a lognormal and then of a mixture of two lognormals for the expectations errors generated by each of the 17 models We then perform a Kolmogorov–Smirnov goodness-of-fit test for the error distribution against these estimated parametric distributions.7 Columns and of Table present the p-values of the test statistics for each model These indicate that we cannot always fit a lognormal but a mixture of two lognormals always fits well, even for models with heavily skewed distributions and thick tails such as those with implicit liquidity constraints (models and 6) or when we mix homogeneous models ALAN & BROWNING SYNTHETIC RESIDUAL ESTIMATION 1237 (models 12–15).8 It is this regularity that underpins our estimation procedure The final two columns give the slope parameter and associated t-value from the regression of the square of the current expectations error (minus the mean of unity) on the lagged expectations error: (εh,t − 1)2 = φε + ω(εh,t−1 − 1) + h,t (6) SYNTHETIC RESIDUAL ESTIMATION (SRE) 3.1 Overview Our estimation procedure is a variant of SMD, which involves matching statistics from the data with statistics from a simulated model.9 We define a J -vector of statistics (auxiliary parameters) and calculate them from the data, λD We simulate the model using parameters θ and calculate the auxiliary parameters for the simulated data, λS (θ) The final step is to choose parameters that minimize the weighted distance between the sample and simulated auxiliary parameters To this, we take a J × J positive definite, data-dependent weighting matrix, W , and define the SMD estimator: θ SMD = arg λS (θ) − λD W λS (θ ) − λD θ (7) Asymptotic properties of this estimator are given in Gourieroux, Monfort and Renault (1993) The novelty of our approach is that, rather than simulating the full model, we simulate the expectations errors and use these to construct consumption paths For the exposition here, we consider a balanced panel with h = 1, , H households and t = 1, , T periods In the empirical section, we discuss how to deal with the unbalanced panel that we actually use.10 We allow the discount factor, β, and the coefficient of relative risk aversion, γ , to be heterogeneous with some stochastic dependence between the two distributions and the initial values of consumption There are four steps for the simulation procedure In the first step, we simulate expectations errors that have the properties identified in the previous section Thus we simulate mixtures of two unit mean lognormals, allowing for conditional heteroskedasticity In the second step, we simulate values for initial values and preference parameters In the third step, we take the simulated expectations errors, the initial values, and the simulated preference parameters and generate consumption paths using equation (1) Finally, we add measurement error The simulation procedure takes a set of 15 model parameters We present a sketch of the parameters here using the notation μ for a location parameter, φ for a dispersion parameter, We also calculated Kruskal–Wallis and Wilcoxon signed-rank test statistics The results are similar; that is, we not reject the mixture of lognormals for any of the models A detailed description of the general SMD procedure we use is given in Appendix B 10 We also postpone to the empirical section any discussion of how to allow for time-varying observable factors such as household composition © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 We run this regression to assess the degree of conditional heteroskedasticity in expectations errors The t-values in Table indicate strong conditional heteroskedasticity for most models We have not been able to establish theoretically the sign of the dependence between past shocks and the subsequent variance It depends on the level of accumulated assets and the marginal propensity to consume out of income Nevertheless, the simulations suggest that it is important to account for such a dependence Therefore, in our estimation procedure, we shall allow for this conditional heteroskedasticity and estimate ω 1238 REVIEW OF ECONOMIC STUDIES θ = φε1 , φε2 , ω, π , μ1 , φ1 , μβ , φβ , ωβ1 , μγ , ωβγ , φγ , ωγ , μm , φm (8) The simulation procedure takes θ and returns consumption paths for each household for t = 1, , T These parameters are the input for the optimization routine 3.2 Simulating expectations errors To simulate expectations errors, we draw four sets of mutually independent pseudo-random numbers: ν1h,t , ν2h,t , ν3h,t , ν4h,t for all h and t = 0, 1, , T 11 The variables ν1h,t , ν2h,t , and ν4h,t are standard normal variables and ν3h,t is a uniform on [0, 1] The expectations errors, εh,t ’s, are simulated recursively We define two variances by: σk2 = exp(φεk ), k = 1, (9) where the exponential is taken to ensure that the variance is positive Then we define two initial heterogeneous error terms by: εkh,0 = exp − ln + σk2 + ln + σk2 νkh,0 , k = 1, (10) By construction, each of these terms has a unit mean We then mix these distributions with a mixing parameter given by: dh,0 = 50 ∗ ν3h,0 − π where (.) is the standard normal cumulative distribution function (cdf) This is a “smoothed” indicator function which takes or for values of the uniformly distributed random draws v3h,0 that are not very close to π Such smoothed indicators are routinely used to facilitate derivative-based optimization These values control whether household h draws from the first simulated residual distribution or the second, so that: εh,0 = dh,0 ε1h,0 + (1 − dh,0 )ε2h,0 (11) 11 The start at t = is to give a first draw (νkh,0 ) which is used in generating the conditional heteroskedasticity for the t = observation © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 and ω for a parameter controlling the dependence between parameters The parameters for the approximated expectations errors distribution are denoted as (φε1 , φε2 , ω, π ) The parameters φε1 and φε2 are for the dispersions of the two components of the mixture of lognormals (the means are fixed at unity) The parameter ω controls the extent of conditional heteroskedasticity and π controls the mixing probabilities For the distribution of the initial level of consumption, we have the location and dispersion parameters (μ1 , φ1 ) For the discount factor we have three parameters: μβ , φβ , ωβ1 These are, respectively, related to the discount factor location, dispersion, and the dependence between the discount factor and initial consumption The parameters for the coefficient of relative risk aversion are denoted as μγ , φγ , ωβγ , ωγ These are, respectively, related to the location, dispersion, the dependence between the discount factor and coefficient of relative risk aversion, and the dependence between the coefficient of relative risk aversion and initial consumption The final pair of parameters (μm , φm ) are the location and dispersion parameters for the measurement error We denote the vector of model parameters as: VAN NIEUWERBURGH & WEILL HOUSE PRICE DISPERSION 1593 pw-bp 12 10 data model 1975 1980 1985 1990 time 1995 2000 2005 Figure Price–wage sensitivity Notes: Each period we run a cross-sectional regression of real house prices on real wages The slope coefficient bp is computed as the population-weighted covariance of prices and wages, divided by the population-weighted variance of wages The dashed line with circles denotes the time series for bp in the data, while the solid line (no markers) denotes the same slope in the model The model’s slope is computed by feeding in the observed wage data, and evaluating them at the equilibrium price function 46.95% in the first period of the transition It then rises gradually to 55.18% over the next 32 periods Although the initial drop is counter-factual (we explain why it occurs below and propose a resolution in Section 4.6.2), the 8.2% increase between 1976 and 2007 is similar (iden© 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 1594 REVIEW OF ECONOMIC STUDIES Population Distribution Across Wage Quintiles 0.9 0.8 1975 2007 final steady state 0.7 Fraction 0.6 0.5 0.4 0.3 0.2 0.1 Wage Quintile Figure Population distribution Notes: This figure plots the population distribution by wage quintile in the benchmark model We use the model with an increasing productivity dispersion to generate a population time series for each MSA We sort the MSAs into five equally sized wage bins and calculate the ratio of the number of people in each quintile to the number of people in the economy (normalized to 1) The graph shows the distribution in the initial steady state (1975, left bars), after 32 years (2007, middle bars), and in the final steady state (right bars) © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 tical) to the 9.4% (8.2%) increase in the data between 1976 (1975) and 2007 The increase in population concentration after 1976 is made possible by increased construction in highproductivity regions In contrast, low-productivity regions see no construction and a declining housing stock because of depreciation, and they lose population This construction pattern facilitates population concentration in the highest wage quintile Figure shows that the population further concentrates towards the highest productivity regions as the economy moves towards the final steady state, at which point Q5 is 80.50% Even though there are no more exogenous changes to the productivity process after 2007 and the construction threshold has reached its steady-state value, the housing distribution continues to adjust towards its steady state This continued population concentration towards high-wage regions explains why the populationweighted average and CV of wages and house prices in Figures and 5, respectively, continue to increase in the model after 2007 As mentioned above, one problem with the benchmark calibration is the large initial drop in Q5 between the initial steady state and the first period of the transition This drop is an artefact of the specific mechanics driving the increase in cross-sectional wage dispersion Indeed, in order to obtain a gradual, linear increase in the dispersion of log productivity, plotted in the left panel of Figure 4, we need a large jump in the innovation variance of log productivity in period 1, σa1 This high innovation variance acts as a big shock to the wage distribution between the initial steady state and period Previously high-productivity, high-population regions may draw a very negative productivity innovation which puts them in a lower wage quintile and vice versa This breaks the strong association between population and wages, VAN NIEUWERBURGH & WEILL HOUSE PRICE DISPERSION 1595 resulting in the initial drop in Q5.29 One consequence of these productivity dynamics is that the rank correlation of wages between adjacent years is counter-factually low For instance, the rank correlation between the initial steady state and the first period of the transition is 68.8% compared to a rank correlation of 96.8% in the 1975–1976 wage data This rank correlation gradually increases along the transition path and exceeds 95% only 20 years into the transition In the data it is above 95% throughout In Section 4.6.2 below, we consider an alternative calibration that increases productivity dispersion in a rank-preserving way Its prediction for the fraction of jobs in the highest wage quintile matches the data The increasing dispersion in productivity causes migration from previously productive to newly productive regions The migration pattern and the magnitude of the migration rate predicted are similar in model and data To measure migration in the data, we use US Census data for the in-migration and outmigration between 1995 and 2000, available for 271 MSAs Net migration is defined as the difference between in- and out-migration We focus on the sub-population of young (25–39), single, college-educated because this group is more likely to move for productive reasons and, therefore, to more closely approximate the agents in our model We sort regions into 25 wageper-job bins and compute net migration rates for each bin We adjust for population growth by scaling the population in 2000 so that it is the same as in 1995 We measure net migration in the same way in the model; Appendix C.2 (Supporting Information) contains the details When we compare model to data, we find a similar pattern: out-migration from low-wage areas and in-migration into high-wage areas The top 10% lowest wage regions see an outmigration of about 7% in both model and data The top 10% highest wage regions see an in-migration of about 1.5% in both model and data Figure shows the annual net migration rate in the benchmark model and in the data These results suggest that the assumption of frictionless reallocation does not lead to excessive migration This is due to the high persistence of wages Indeed, consider the equilibrium of a version of our model with constant wages: in that extreme case, nobody would find it optimal to move despite perfect mobility 4.6 Robustness In this section, we discuss two alternative calibrations and their implications for wages and house prices 4.6.1 Alternative 1: calibrating to initial CV of house prices First, we explore a calibration in which the initial steady-state CV of house prices matches the 1975 value of 0.154 in the data; see Table 2, Panel C This is an alternative to matching the 1975 sensitivity of prices to wages bp in Panel B The calibration, which continues to match the level and CV of wages in 1975 and 2007, features less ability dispersion and more regional productivity dispersion than the benchmark In particular, the cross-sectional standard deviation of ability 29 Since we assume that the innovation variance reaches its steady-state value from period 33 onwards, the linear increase in the standard deviation of log productivity also requires that the innovation variance increase between period and period 32, and that it overshoot its final steady-state value Precisely, the time path for the innovation standard deviation of log productivity in our benchmark calibration is as follows: 0.0004 (initial steady state), 0.0068 (period 1), 0.008 (period 2), gradually increasing to 0.0263 (period 32), then constant at 0.0103 (from period 33 until final steady state) © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 4.5 Mobility 1596 REVIEW OF ECONOMIC STUDIES model data 2 10 10 15 20 25 real wage bin Figure Net migration rates Notes: The model plots net migration rates in the benchmark model (left bars) and in the data (right bars) Each pair of bars represents the net migration (in-migration minus out-migration) between 1995 and 2000 of a group of regions with similar real wages In particular, we form 25 groups of regions, sorted by their real wage from lowest to highest The data is from the US Census for young, single, college-educated persons The model computes migration rates in the same way as in the data is 0.056 compared to 0.079 in the benchmark calibration (ke is 25.28 compared to 17.89), the initial productivity dispersion is higher (0.039 compared to 0.006), and it rises to a higher value in the final steady state (0.135 vs 0.103) This calibration predicts a rise in the populationweighted CV of house prices of 53 points, similar to the 51 point increase in the benchmark It generates a substantially larger increase in house price levels: a 23% increase compared to an 11% increase in the benchmark and a 33% increase in the data The downside to matching the initial CV of house prices is that we overstate the initial sensitivity coefficient: bp0 = 5.37 versus 0.81 in the 1975 data Both models feature a similar increase in this sensitivity coefficient Hence, this alternative calibration introduces excess sensitivity of house prices to wages: a year-by-year cross-sectional regression of house prices on wages delivers a slope coefficient of 8.2 (averaged over time) inside the model and 3.2 in the data The excess sensitivity problem is still less pronounced than in the model without ability dispersion though The R for house prices in equation (26) is 19% in this model compared to 26.5% in the data and 31% in the benchmark model © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 percent per year VAN NIEUWERBURGH & WEILL HOUSE PRICE DISPERSION 1597 at = avg(at−1 ) + ρt (at−1 − avg(at−1 )) + σat εt , where avg(at−1 ) denotes the cross-sectional average productivity Setting ρt > allows us to increase dispersion in a rank-preserving way; Appendix C.1 (Supporting Information) explains the mechanics in detail As in the benchmark model, we match the 1975 sensitivity coefficient bp0 The calibration is essentially identical to the benchmark model The main moments of interest are discussed in Panel D of Table First, the model continues to generate a large increase in the CV of house prices for a given increase in the CV of wages The increase is 60 points compared to 51 points in the benchmark Second, it generates a larger increase in house prices: 19% increase compared to 11% in the benchmark Third, the sensitivity coefficient increases strongly from 0.81 in 1975 to 11.95 in 2007 Fourth, and most significantly, this calibration generates population dynamics close to those observed in the data The fraction of people working and living in the top 20% of regions in terms of wage is 64.88% in the initial steady state and rises to 74.77% after 32 periods This is close to the 73.09% in the 2007 data This calibration avoids the steep drop in Q5 in the first period of the transition, which we noted for the benchmark model Instead, the 1976 value for Q5 in the model is 64.62%, close to the initial steady-state value The population then gradually relocates towards the newly productive regions In the final steady state, Q5 reaches a value of 80.37%, similar to the benchmark model The key difference with the benchmark model, therefore, is the transition path of Q5 Because it avoids the initial drop in population, this model generates a higher increase in population-weighted house prices and a higher population-weighted sensitivity of prices to wages By the same token, this version of our model matches the rank correlation of wages between adjacent years It is 99.55% on average in the model and 99.22% on average in the data Finally, the model generates an R statistic of 25.1%, close to the 26.5% number in the data 4.7 Increase in regulation In this section, we use our model to pursue an alternative explanation for the increase in the level and dispersion of house prices: housing supply regulation has become tighter over time and has gradually spread geographically (from the coastal areas inland) To keep matters as simple as possible, we hold all parameters of the benchmark calibration fixed with one © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 4.6.2 Alternative 2: rank-preserving increase in wage dispersion Second, we explore a calibration in which the increase in the standard deviation of productivity dispersion is engineered in a different way As explained above, in order to generate a linear increase in dispersion, we need that the innovation variance jumps from the initial steady state to the first period of the transition, then gradually rises until period 33, and then jumps back down to the final steady-state value The initial jump in innovation variance introduces too much “mixing” in the cross-sectional productivity distribution and leads to a counter-factually low rank correlation between region-specific wages in adjacent periods As an alternative, we consider a lower time path for the innovation standard deviation of log productivity: a linear rise from its initial steady state to its final steady-state value over 33 periods Because this lower time path of productivity shocks, by itself, generates a smaller increase in productivity dispersion than in our benchmark, we add a second engine of productivity dispersion We deterministically increase the productivity of regions above the average and, vice versa, decrease the productivity of regions below the average Importantly, unlike random productivity shocks, these deterministic changes preserve the rank of regions in the productivity distribution Formally, the law of motion for log productivity becomes: 1598 REVIEW OF ECONOMIC STUDIES exception Instead of increasing the dispersion of productivity over time, we hold it fixed at its initial steady-state level The resulting initial steady-state wage distribution is the same as in our benchmark model and matches the observed 1975 wage distribution as before Instead, we tighten regulation We let the number of permits at time t in a region with productivity A be t (A) = πa A Amin φt 0.305 0.3 permits 0.295 0.29 initial ss final ss 0.285 16.6 16.8 17 17.2 17.4 17.6 17.8 productivity 18 18.2 18.4 18.6 Figure Tightening housing supply regulation Notes: This figure plots the permit function (A) = πa (A/Amin )φ The top line denotes the situation in 1975 when πa = 0.30026 and φ = The bottom line denotes the situation in 2007 and beyond when πa = 0.30026 and φ = −0.5 In the years between 1975 and 2007, φ decreases linearly from to −0.5, so that the permit function gradually rotates from the top line to the bottom line © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 The elasticity parameter φt = in 1975 as in our benchmark model We let it decrease from to φt = −0.5 between periods and 32 of the transition We keep it constant at −0.5 after 2007 (periods 33 and later) Figure illustrates how lowering the supply elasticity parameter φ reduces the number of permits, and more so in highly productive regions This rotation captures the stylized fact that supply regulation gradually tightened over the last three decades, especially among more productive regions such as the coastal metropolitan areas Panel E of Table shows the results of the regulatory tightening exercise We find that decreasing building permits has quantitatively minor effects on average house prices and on the dispersion of house prices While both increase, the increases are quantitatively small compared VAN NIEUWERBURGH & WEILL HOUSE PRICE DISPERSION 1599 4.8 Rental prices In the model, the price of a house equals the present discounted value of the rents An alternative to testing the model’s implications for house prices would be to test its implications for rents After all, the spatial equilibrium model also predicts a relationship between rents and wages We collected nominal rent data from the Fair Market Rents database, as detailed in Appendix D.3 (Supporting Information) As we did with nominal house prices, we deflate them by the regional non-housing consumer price index (CPI) as well as by the trend in house size Census data suggest that the size of multi-family homes, which is likely to be rental housing, grew at the same rate as single-family housing, which is likely to be owner-occupied The rental data are only available in 1982 and from 1984 to 2007 As the last two rows of Table 2, panel A, show, de-trended real rents seem to have fallen from $4220 per year in 1982 to $3420 in 2007 The CV increased only moderately, from 0.153 in 1982 to 0.190 in 2007 Our model generates a moderate increase in average rents and a large increase in the CV of rents, just as with house prices Therefore, while the model can account for the observed increase in house price dispersion, it produces an increase in rent dispersion that is too large relative to the data This is akin to the excess volatility puzzle, according to which equity prices are too volatile relative to their underlying dividends A potential explanation for this divergence is that the cash flows entering the present-value formula for house prices are unlikely to be the rents we measure in the data Glaeser and Gyourko (2009) make several compelling empirical observations suggesting that house price and rent series can be best understood as the costs of two different types of housing, reflecting different demands on two related, but not directly comparable, markets.30 This market segmentation causes a (severe) selection problem when comparing the present value of observed rents to observed house prices This selection problem could help explain our empirical observation 30 Rents in our model are then to be interpreted as the per-period user cost of owner-occupied housing Since there are no regional data on the user cost, and since single-family ownership price data are of high quality, it seems natural to test the model using house price data instead Like us, the bulk of the spatial location literature derives implications for (implicit) rents, but almost always tests them on owner-occupied house price data (recent examples are Gyourko, Mayer, and Sinai, 2006; Glaeser and Gyourko, 2006) © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 to those we found in our benchmark exercise The same is true for the price–wage sensitivity which increases only slightly over time The population in Q5 is also almost constant Finally, our R goodness-of-fit metric for house prices is only 5%, one-fifth of its value in the data and one-sixth of its value in the benchmark model The intuition for the small impact of regulation is simple While tighter regulation reduces the supply of houses in high-wage metropolitan areas, the equilibrium response of labour is to move out, thereby effectively reducing the housing demand in those same areas The net effect is a tiny increase in price A similar intuition is at work in the closed city model of Arnott and MacKinnon (1977) and the open city model of Aura and Davidoff (2008) We have explored alternative values for the supply elasticity parameter φ (−3, −1, −0.1, and even +1) The results were quantitatively similar across cases because of the endogenous response of mobility to the various regulatory changes In the same vein, tightening regulation alongside an increase in wage dispersion delivers the same quantitative results as in our benchmark calibration with constant regulation Impediments to labour mobility, absent from the model, may slow down the reduction in housing demand, but are unlikely to reverse it These results suggest that an increase in wage dispersion is an important ingredient to generate a quantitatively meaningful increase in house price level and dispersion 1600 REVIEW OF ECONOMIC STUDIES that owner-occupied house price dispersion increased much more than rent dispersion: indeed, the increase in income inequality, the key driving force of our model, was most pronounced in the top half of the income distribution, a group that is more likely to be composed of homeowners One way to address selection would be to study housing units that are both for rent and for sale: unfortunately, there exists no such regional panel dataset for the United States, but other countries or certain regions within the US may have such data Another way to go would be to develop a model where agents choose to self-select into the rental or the ownership market This extension is left for future research Our paper provides a new general equilibrium framework for analysing the joint dynamics of regional income, house prices, and housing quantities It extends the Rosen–Roback spatial equilibrium model along several dimensions in order to establish closer contact with the data We use our framework to study the quantitative effect of wage dispersion and housing supply regulation for the regional house price level and its dispersion The model accounts for several features of the joint price–wage distribution Faced with an increase in the productivity dispersion across metropolitan areas, households choose to reallocate from lower towards higher productivity metropolitan areas This pushes up house prices in high-wage areas The observed increase in wage dispersion is sufficient to generate the observed increase in the house price dispersion across metropolitan areas The same 30 years since 1975 also saw a tightening of housing supply regulation, especially in the coastal areas One might think that the supply effect induced by this regulatory tightening could, in and of itself, account for the increase in house price level and dispersion However, because of the equilibrium response of households to move out of the more tightly regulated areas, the house price effects of tightening supply restrictions are small So, while supply constraints are important, the increase in wage dispersion is an essential part of the explanation The model’s prediction of an increasingly strong cross-sectional sensitivity of house prices to wages is consistent with the data It suggests that increasing dispersion of regional productivity, as opposed to an increasing dispersion in the ability of households, underlies the changes in spatial location, wage, and house price patterns we have observed over the last three decades APPENDIX A PROOFS31 A.1 Preliminary results The following Lemma compiles technical results which are used in the following subsection Lemma Consider some strictly increasing strictly concave, and twice continuously differentiable function v : (0, ∞) → R Suppose that v (h) goes to minus infinity as h goes to zero, and that v (h) goes to zero as h goes to infinity Then The derivative v (h) goes to infinity as h goes to zero, and goes to zero as h goes to infinity The function hv (h) goes to zero as h goes to infinity The function w (h) ≡ hv (h) − v (h) is continuous and strictly decreasing, goes to zero as h goes to infinity, and goes to infinity as h goes to zero The function (x ) = 1/w −1 (x ) is continuous and strictly increasing It can be extended by continuity at zero with (0) = It goes to infinity as x goes to infinity 31 Appendices B–E are available as Supporting Information online (www.restud.org.uk) © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 CONCLUSION VAN NIEUWERBURGH & WEILL HOUSE PRICE DISPERSION 1601 The function R(x ) ≡ v ◦w −1 (x ) is increasing, convex and continuous, goes to zero as x goes to zero, and goes to infinity as x goes to infinity Consider any density g(A) such that, for all x ∈ R, Amax G(x ) = (max{A − x , 0}) g(A) dA < ∞ Amin Then, the function G(x ) is continuous Proof v (h1 ) + h2 v (h1 ) − v (h2 ) ≥ h1 v (h1 ) ≥ Letting h1 go to infinity shows that −v (h2 ) ≥ lim suph→∞ hv (h) ≥ for all h2 Letting h2 go to infinity shows that hv (h) also goes to zero as h goes to infinity Consider the function w (h) ≡ hv (h) − v (h) The above results show that w (h) goes to zero as h goes to infinity Because w (h) = hv (h) < 0, it follows that w (h) ≥ Lastly, since w (h) ≥ −v (h), letting h go to zero shows that w (h) goes to infinity as h goes to zero The previous paragraph implies that the function (x ) = 1/w −1 (x ) is well defined It is continuous and increasing, and goes to zero as x goes to zero and to infinity as x goes to infinity Lastly, consider the function R(x ) is increasing because both v (x ) and w −1 (x ) are decreasing Points and of the Lemma imply that it goes to zero as x goes to zero, and to infinity as x goes to infinity In order to prove that R(x ) is convex, note that R (x ) = v ◦w −1 (x ) w ◦w −1 (x ) = v ◦w −1 (x ) w −1 (x ) × v ◦w −1 (x ) = , w −1 (x ) where the second line follows from the fact that w (h) = hv (h) Since w −1 (x ) is decreasing, it follows that R (x ) is increasing, which establishes convexity Pick any x ∈ R and some η > Then, for all y ∈ [x − η, x + η] α |G(x ) − G(y)| ≤ (max{A − x , 0}) − (max{A − y, 0}) g(A) dA Amin Amax × (max{A − x , 0}) − (max{A − y, 0}) g(A) dA α ≤ α (max{A − x , 0}) − (max{A − y, 0}) g(A) dA Amin Amax ×2 (max{A − x + η, 0}) g(A) dA, α where the second inequality follows because (x ) is decreasing Now, because G(x − η) < ∞, it follows that for all ε > there exists some α > such that the second integral on the right-hand side is less than ε/2 Since the function (max{z , 0}) is uniformly continuous over the compact [0, α − x + η], there exists some η < η, such that |x − y| < η implies that | (max{A − x , 0}) − (max{A − y, 0}) | < ε/2 Plugging this back into the first integral on the right-hand side shows that |x − y| < η implies that |G(x ) − G(y)| < ε || © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 For any h1 > h2 , concavity implies that v (h2 )(h1 − h2 ) ≥ v (h1 ) − v (h2 ) Therefore, v (h2 )h1 ≥ v (h2 )h2 + v (h1 ) − v (h2 ) ≥ v (h1 ) − v (h2 ) Letting h2 go to zero in the inequality implies that v (h2 ) goes to infinity as h2 goes to zero Second, since v (h) is positive and decreasing, it has some positive limit v as h goes to infinity Since v (h) is concave, then for all h1 > h2 , ≥ v (h1 ) ≥ v (h2 ) + v (h1 )(h1 − h2 ) ≥ v (h2 ) + v (h1 − h2 ) Letting h1 go to infinity shows that v = Therefore, v (h) goes to zero as h goes to infinity Rearranging the previous inequality implies that 1602 REVIEW OF ECONOMIC STUDIES A.2 Proofs of the results in the text A.2.1 Proof of Proposition We proceed in five steps First, we let εit ∈ [e, e] be the set of households who find it optimal to locate in an island with current productivity Ait Then, we have Result If, for some j , F (εjt ) = 0, then εit = ∅ for all i < j To prove the first statement, consider an island Ait with a measure F (εit ) = of households Then the rent must be Rit = For all j < i , we then have that Rjt ≥ = Rit and Ajt < Ait Thus, island j is strictly less attractive to household than island i and, consequently, is not populated, i.e εjt = ∅ We then show that Indeed, household e ∈ εit ∩ εjt is indifferent between island i and j if and only if eAit + v (hit ) − Rit hit = eAjt + v (hjt ) − Rjt hjt Since there is a unique e solving this equation, the result follows We then have Result If, for some i , F (εit ) > 0, then F (εjt ) > for all j > i Note that, because of Result 2, all households e in the set εit live in islands of type i Thus, islands of type i must be inhabited by a positive measure of households, so housing demand is non-zero and Rit > Now, if an island j > i were populated by a measure zero of households, then Rjt = and households e ∈ εit would strictly prefer it over island i because of its lower rent and strictly higher wage, which would contradict optimality Let p be the smallest integer such that F (εit ) > and let eit be the infimum of εit for all i ≥ p This infimum is well defined since, by the previous result, then εit is not empty for all i ≥ p For i = N + 1, we define eN +1t ≡ e Next, we show Result Suppose Ait < Ajt If household e ∈ [e, e] weakly prefers island j to i , then all households e > e strictly prefer j to i This result follows because the household’s objective function is super-modular Since eAjt + v (hjt ) − Rjt hjt > eAit + v (hit ) − Rit hit ≥ ⇔ e(Ajt − Ait ) ≥ v (hit ) − v (hjt ) − Rit hit + Rjt hjt , (A1) and Ajt > Ait , then the inequality is strict for all e > e Equipped with this result, we obtain Result The lowest ability cutoff is ept = e and, for all i ≥ p, εit = [eit , ei +1t ] First note that the sets εit are increasing Otherwise, suppose we had j > i , e ∈ εit , e ∈ εjt and e < e Then e weakly prefers j to i and so, by result 4, e would strictly prefer j to i , a contradiction It then follows that the sequence eit is weakly increasing Now all e ∈ (eit , ei +1t ) must belong to εit Otherwise, if some e ∈ (eit , ei +1t ) belonged to εjt for some j < i , then we could find some e ∈ εit < e By Result 4, e would strictly prefer i to j , which is a contradiction Also, if e belonged to εjt for j > i , then ejt ≥ ei +1t > e, which contradicts the fact that ejt is the infimum of εjt By a similar line of argument, we can show that any e ∈ / (eit , ei +1t ) cannot belong to εit This shows that (eit , ei +1t ) ⊆ εit ⊆ [eit , ei +1t ] It then follows that eit < ei +1t because otherwise εit would have measure zero Since, in an equilibrium, all household must live in some location, we must also have that εpt = e Lastly, letting e → ei +1t from the left and from the right in equation (A1) for i and i + 1, we obtain that household ei +1t is indifferent between island i and island i + 1, so both eit and ei +1t belong to the set εit A.2.2 Proof of Proposition In this proof, we suppress time subscripts to simplify notation We proceed in two steps First, we show that the system of difference equations (17) and (18) has a unique solution with e1t = e and eN +1t = e Then, we show that the unique solution of the difference equation is indeed the basis of an equilibrium assignment of households to islands © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 Result Consider i = j Then εit ∩ εjt is either empty or is a singleton VAN NIEUWERBURGH & WEILL HOUSE PRICE DISPERSION 1603 Step To solve the system of difference equations, it is useful to consider the following change of variables: e = F −1 (min{1 − q, 1}) ≡ ψ(q), (A2) where qi ∈ (−∞, 1] The variable q is a “generalized percentile”, such that when q ≤ 0, then e = e and when q = 1, e = e In terms of qi , the system of difference equation becomes qi − qi +1 = μi Hi (max{ψ(qi )Ai − Ui , 0}) , Ui +1 = (ψ(qi +1 ) − ψ(qi )) Ai + Ui (A3) (A4) qi +1 = qi − μi Hi (max{ψ(qi )Ai − Ui , 0}) Then there are three cases to consider If qi < 1, then qi +1 is strictly increasing in U1 given that qi and Ui − ψ(qi )Ai are strictly increasing in U1 , and ( · ) is an increasing function The second case is if qi = and qi +1 < 1; then ψ(qi )Ai − Ui > and so qi +1 is strictly increasing in U1 Lastly, if qi = and qi +1 = 1, then ψ(qi )Ai − Ui ≤ 0, and we obtain that qi +1 is increasing in U1 Now turn to: Ui +1 − ψ(qi +1 )Ai +1 = Ui − ψ(qi )Ai + (Ai − Ai +1 )ψ(qi +1 ) (A5) The result follows by the induction hypothesis and because Ai − Ai +1 < and ψ(q) is decreasing We thus have that qN +1 is increasing, and strictly increasing if qN +1 < Moreover, for all U1 > eAN , Ui ≥ U1 > eAN ≥ ei Ai for all i , so qi = for all i On the other hand, qi +1 − qi < − {μj Hj } × j ∈{1, ,N } (max{ψ(q1 )A1 − U1 , 0}) , because, from equation (A5), the sequence ψ(qi )Ai − Ui is increasing in i Since ψ(q1 ) = e, it follows that, as U1 goes to minus infinity, qi +1 − qi goes to minus infinity, and so does qN +1 - Taken together, these properties show that there exists a unique U1 such that qN +1 = Step We now verify that the solution we constructed is indeed the basis of an equilibrium That is, given +1 {Ui }Ni=1 and {ei }Ni =1 , we let: p = min{i ∈ {1, , N } : ei Ai − Ui > 0}, ni = Hi (max{ei Ai − Ui , 0}) , hi = Hi /ni , Ri = v (hi ) The labour market clears by construction of ni and the housing market clears by construction of hi The rent Ri makes it optimal for a household in island i to consume hi All we need to verify is that, for all i ≥ p, households e ∈ [ei , ei +1 ] find it indeed optimal to live in island i We start by noting that: Ui (e) ≡ eAi + v (hi ) − Ri hi = eAi + v (hi ) − v (hi )hi = eAi − w (hi ) = eAi − max{ei Ai − Ui , 0} = (e − ei )Ai + min{ei Ai , Ui } For e = ep , we have that, for i < p, Ui (e) = + eAi , since Ui = Up ≥ eAp > eAi For i = p, then Up (e) ≥ eAp by definition of p Thus, e prefers island p to any island i < p, and so does any household e ≥ e because location choices are monotonic in ability Thus, islands i < p are not populated Now, for any i ≥ p, we have that Ui ≤ ei Ai because of equation (17) and the fact that ei +1 > ei Thus: Ui +1 (e) = (e − ei +1 )Ai +1 + Ui +1 = (e − ei +1 )Ai +1 + Ui + (ei +1 − ei )Ai i = Ui (e) + (e − ei +1 )(Ai +1 − Ai ) = Up (e) + (e − ej +1 )(Aj +1 − Aj ), j =p © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 +1 One first sees that a pair of sequences {Ui }Ni=1 and {ei }Ni =1 solves equations (17) and (18) with e1 = e and eN +1 = e +1 if and only if the pair of sequences {Ui }Ni=1 , qi = {1 − F (ei )}Ni =1 solves equations (A3) and (A4) with q1 = and qN +1 = Now, given an initial condition U1 and q1 = 1, we show that Ui − ψ(qi )Ai is strictly increasing in the initial condition U1 , qi is increasing in the initial condition U1 , and strictly increasing if qi < We proceed by induction Clearly, the property is true for i = Now, suppose it is true for all j ≤ i We have that 1604 REVIEW OF ECONOMIC STUDIES where the second equality follows by equation (18), the third equality by definition of Ui (e), and the last equality by iterating backward until j = p The terms of the sum are positive if and only if e ≥ ej +1 It thus follows that a household finds it optimal to locate in the largest j such that e ≥ ej In other words, households in [ei , ei +1 ] find it optimal to locate in islands of type i A.2.4 Proof of Proposition We start from the definition Uit = eit Ait + max {v (h) − Ri h} h≥0 ≡ eit Ait − θ(Ri ), (A6) where θ(R) ≡ minh≥0 {Rh − v (h)} Note that, for each h, the function h → Rh − v (h) is positive, increasing, and affine Being the upper envelope of such a family of functions, θ(R) is increasing and concave We then write θ(Ri +1t ) − θ(Rit ) Ri +1t − Rit Ri +1t − Rit × = Ai +1t − Ait θ(Ri +1t ) − θ(Rit ) Ai +1t − Ait (A7) The first term is increasing in i because, as argued above, the function θ(R) is increasing and concave The second term is also positive and increasing Indeed, using equation (A6), we have that θ(Rit ) = Uit − eit Ait Therefore: θ(Ri +1t ) − θ(Rit ) = ei +1t Ai +1t − Ui +1t + Uit − eit Ait = ei +1t Ai +1t − Uit − (ei +1t − eit )Ait + Uit − eit Ait = ei +1t (Ai +1t − Ait ), where the second line follows from the indifference equation (18) Thus, the second term of equation (A7) is simply equal to ei +1t , which is increasing because of assortative matching of ability with productivity Acknowledgements We thank Fernando Alvarez, Yakov Amihud, Andy Atkeson, David Backus, Markus Brunnermeier, V.V Chari, Morris Davis, Matthias Doepke, Xavier Gabaix, Dirk Krueger, Lars Hansen, Jonathan Heathcote, Christian Hellwig, Hugo Hopenhayn, Narayana Kocherlakota, Samuel Kortum, Ricardo Lagos, Robert Lucas, Hanno Lustig, Erzo G.J Luttmer, Franc¸ois Ortalo-Magn´e, Christopher Mayer, Ellen McGrattan, Holger Mueller, Torsten Persson, Florian Pelgrin, Andrea Prat, Enrichetta Ravina, Victor Rios-Rull, Jean-Charles Rochet, Esteban Rossi-Hansberg, Robert Shimer, Thomas Sargent, Kjetil Storesletten, Laura Veldkamp, Gianluca Violante, Mark Wright, Randall Wright, and the seminar participants at NYU, IIES in Stockholm, University of Oslo, LSE, HEC Paris, UCLA, Yale University, University of Minnesota, the NBER asset pricing meetings, the University of Chicago, the AEA meetings in Chicago, MIT, the University of Pennsylvania, HEC Lausanne, the Chicago Fed, the NBER public economics and real estate meetings, Minnesota Macro, and Stanford University for comments We thank the Editor and two anonymous referees for their comments We acknowledge financial support from the Richard S Ziman Center for Real Estate at UCLA REFERENCES ALVAREZ, F and VERACIERTO, M (2000), “Labor-Market Policies in an Equilibrium Search Model”, NBER Macroeconomic Annual 1999, Vol 14 (Cambridge and London: MIT Press) 265–304 ALVAREZ, F and VERACIERTO, M (2006), “Fixed-term Employment Contracts in an Equilibrium Search Model”, (NBER Working Paper No 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