House Price Momentum and Strategic Complementarity Part 1

House Price Momentum and Strategic Complementarity Abstract House price changes are positively autocorrelated over two to three years, a phenomenon known as momentum. While several frictions that cause momentum have been identified, existing explanations cannot quantitatively explain the magnitude of momentum found in the data. This paper introduces, empirically grounds, and quantitatively analyzes an amplification mechanism that can generate substantial momentum from a broad class of small frictions. The amplification is due to a concave demand curve in relative price, which implies that increasing the quality-adjusted list price of a house priced above the market average rapidly reduces its probability of sale, but cutting the price of a below-average priced home only slightly improves its chance of selling. This creates a strategic complementarity that incentivizes sellers to set their list price close to others’. Consequently, frictions that cause slight insensitivities to changes in fundamentals lead to prolonged adjustments because sellers gradually adjust their price to stay near the average. I provide new micro empirical evidence for the concavity of demand—which is often used in macro models with strategic complementarities—by instrumenting a house’s relative list price with a proxy for the seller’s equity and nominal loss. I find significant concavity, which I embed in an equilibrium housing search model. Quantitatively, the calibrated model amplifies frictions that cause momentum by a factor of two to three. 1 Introduction A puzzling and prominent feature of housing markets is that aggregate price changes are highly positively autocorrelated, with a one percent annual price change correlated with a 0.30 to 0.75 percent change in the subsequent year (Case and Shiller, 1989).1 This price momentum lasts for two to three years before prices mean revert, a time horizon far greater than most other asset markets. Substantial momentum is surprising both because most pricing frictions dissipate quickly and because predictable price changes should be arbitraged away by households, either by altering their bidding and bargaining or by re-timing their purchase or sale. Indeed, while several papers have explained momentum by introducing frictions into models of housing markets, these frictions need to be unpalatably large to quantitatively explain house price momentum. This paper introduces, empirically grounds, and quantitatively analyzes an amplification mechanism for a variety of underlying frictions that can generate substantial momentum without the need for implausibly-large frictions. The mechanism relies on a strategic complementarity among list-price-setting sellers that makes the optimal list price for a house depend positively on the prices set by others. Strategic complementarities of this sort are frequently used in macroeconomic models, but there is limited empirical evidence of their importance or strength. In analyzing momentum in the housing market, I provide direct micro empirical evidence for a prominent strategic complementarity in the macroeconomics literature and, using a calibrated equilibrium search model, demonstrate that its ability to amplify underlying frictions is quantitatively significant. The propagation mechanism I introduce relies on two components: costly search and a demand curve that is concave in relative price. Search is inherent to housing because no two houses are alike and idiosyncratic taste can only be learned through costly inspection. Concave demand in relative price implies that the probability a house sells is more sensitive to list price for houses priced above the market average than below the market average. While concave demand may arise in housing markets for several reasons, I focus on the manner in which asking prices direct buyer search. The intuition is summarized by an advice column for sellers: “Put yourself in the shoes of buyers who are scanning the real estate ads...trying to decide which houses to visit in person. If your house is overpriced, that will be an immediate turnoff. The buyer will probably clue in pretty quickly to the fact that other houses look like better bargains and move on.” 2 In other words, the probability that a house is visited by buyers decreases rapidly as a home’s list price rises relative to the market average. This generates a concave demand curve in relative price because at high relative prices buyers are on the margin of looking and purchasing, while at low relative prices they are mostly on the margin of purchasing. Concave demand incentivizes list-price-setting sellers—who have market power due to search frictions—to set their list prices close to the mean list price. Intuitively, raising a house’s relative list price reduces the probability of sale and profit dramatically, while lowering its relative price increases 1 See also Cutler et al. (1991), Head et al. (2014), and Glaeser et al. (2014). “Settling On The Right List Price for Your House,” Ilona Bray, http://www.nolo.com/legal-encyclopedia/listinghouse-what-price-should-set-32336-2.html. 2 1 the probability of sale slightly and leaves money on the table. In a rational expectations equilibrium with identical sellers, all sellers change their prices simultaneously and no gradual adjustment arises from the incentive to be close to the mean. However, modest frictions that generate differential insensitivities of prices to changes in fundamentals cause protracted price adjustments because sellers optimally adjust their list price gradually so their price does not stray too far from the market average. Importantly, this strategic complementarity amplifies any friction that generates heterogeneity in the speed of adjustment, a class that includes most of the frictions proposed by the literature, such as staggered pricing, backward-looking expectations, learning, and the gradual spread of sentiment. This paper is focused on the degree of amplification, and is agnostic as to which particular friction is at work in practice. To evaluate the concavity of the effect of unilaterally changing a house’s relative quality-adjusted price on its probability of sale, I turn to micro data on listings for the San Francisco Bay, Los Angeles, and San Diego metropolitan areas from 2008 to 2013. I address bias caused by unobserved quality by instrumenting a house’s relative list price with the amount of local price appreciation since the seller purchased. Sellers with less appreciation since purchase set a higher price for two reasons. First, sellers often use the equity they extract from their current home to make a down payment on their next home (Stein, 1995; Genesove and Mayer, 1997). When the amount of extracted equity is low, sellers are on a down payment constraint, each dollar of equity extracted can be leveraged heavily, and the seller’s marginal utility of cash on hand is high. Conversely, with higher equity extraction, the down payment constraint is less likely to be binding, each dollar extracted is not leveraged to the same extent, and the seller’s marginal utility of cash on hand is lower. Given their higher marginal utility of cash on hand, sellers extracting less equity set a higher list price. Second, home sellers exhibit nominal loss aversion (Genesove and Mayer, 2001). Local appreciation since purchase is a noisy proxy for the exact appreciation of any given house, so there is a negative relationship between appreciation since purchase and both the probability and size of a nominal loss and hence the list price a seller sets. Because I compare listings within a ZIP code and quarter, the supply-side variation based on appreciation since purchased identifies the curvature of demand if unobserved quality is independent of when a seller purchased their home. The instrumental variable estimates reveal a statistically and economically significant concave demand curve that is highly robust to various controls, sample restrictions, and proxies for unobserved quality and date of purchase. My findings about the concavity of demand are also robust to other sources of relative price variation that are independent of appreciation since purchase. To assess the strength of this propagation mechanism, I embed concave demand in a DiamondMortensen-Pissarides equilibrium search model and consider two separate frictions cited by the literature to assess the amplification mechanism’s ability magnify frictions. First, I consider staggered pricing whereby overlapping groups of sellers set prices that are fixed for multiple periods (Taylor, 1980). Concave demand induces sellers to only partially adjust their prices when they have the opportunity to do so, and repeated partial adjustment manifests itself as momentum. Second, I introduce a small fraction of backward-looking rule-of-thumb sellers as in Campbell and 2 Mankiw (1989) and Gali and Gertler (1999). Backward-looking expectations are frequently discussed as a potential cause of momentum (e.g., Case and Shiller, 1987; Case et al. 2012), but some observers have voiced skepticism about widespread non-rationality in housing markets given the financial importance of housing transactions for most households. With a strategic complementarity, far fewer backward-looking sellers are needed to explain momentum because the majority of forward-looking sellers adjust their prices gradually so they do not deviate too much from the backward-looking sellers (Haltiwanger and Waldman, 1989; Fehr and Tyran, 2005). This, in turn, causes the backward-looking sellers to observe more gradual price growth and change their price by less, creating a two-way feedback that amplifies momentum. I calibrate the parameters of the model that control the shape of the demand curve to match the micro empirical estimates and the remainder of the model to match steady state and time series moments. I find that a two-month staggered pricing friction without concavity is amplified into four to six months of gradual adjustment, and between a half and a third as many backward-looking sellers are needed to explain the amount of momentum in the data with concavity relative to without concavity. I conclude that concave demand amplifies underlying frictions by a factor of between two and three. The amplification mechanism adapts two ideas from the macro literature on goods price stickiness to frictional asset search. First, the concave demand curve is similar to “kinked” demand curves (Stiglitz, 1979) which, since the pioneering work of Ball and Romer (1990) has been frequently cited as a potential source of real rigidities. In particular, a “smoothed-out kink” extension of Dixit-Stiglitz preferences proposed by Kimball (1995) is frequently used to tractably introduce real rigidities through strategic complementarity in price setting. Second, the repeated partial price adjustment caused by the strategic complementarity is akin to Taylor’s (1980) “contract multiplier.” A lively literature has debated the importance of strategic complementarities and kinked demand in particular for propagating goods price stickiness by analyzing calibrated models, by assessing whether the ramifications of strategic complementarities are borne out in micro data (Klenow and Willis, 2006), by examining exchange-rate pass through for imported goods (e.g., Gopinath and Itshoki, 2010; Nakamura and Zerom, 2010), and by estimating the response of firms’ pricing to their own costs and the prices of their competitors, instrumenting costs prices with exchange rates (Amiti et al., 2016). My analysis of housing markets adds to this literature by directly estimating a concave demand curve and assessing its ability to amplify frictions in a calibrated model. The remainder of the paper proceeds as follows. Section 2 provides facts about momentum and describes why existing explanations cannot quantitatively explain momentum. Section 3 analyzes micro data to assess whether housing demand curves are concave. Section 4 presents the model. Section 5 calibrates the model to the micro estimates and assesses the degree to which strategic complementarity amplifies momentum. Section 6 concludes. 3 2 Momentum: Facts and Explanations Since the pioneering work of Case and Shiller (1989), price momentum has been considered one of the most puzzling features of housing markets. While other financial markets exhibit momentum, the housing market is unusual for the 8 to 14 quarter horizon over which it persists.3 House price momentum has consistently been found across cities, countries, time periods, and price index measurement methodologies (Cho, 1996; Titman et al., 2014). Figure 1 shows two nation-wide measures of momentum for the CoreLogic repeat-sales house price index for 1976 to 2013 and a third measure for the same index across 103 cities.4 Panel A shows that autocorrelations are positive for 11 quarterly lags of the quarterly change in the price index adjusted for inflation and seasonality. Panel B shows an impulse response in log levels to an initial one percent price shock estimated from an AR(5). In response to the shock, prices gradually rise for two to three years before mean reverting. Finally, panel C shows a histogram of AR(1) coefficients estimated separately for 103 metropolitan area repeat-sales house price indices from CoreLogic using a regression of the annual change in log price on a one-year lag of itself as in Case and Shiller (1989): t,t 4 ln p 1 = 0 + 1 t 4,t 8 ln p + ". (1) is positive for all 103 cities, and the median city has an annual AR1 coefficient of 0.60.5 Appendix B replicates these facts for a number of countries, price series, and measures of autocorrelation and consistently finds two to three years of momentum. Momentum is a puzzle because forward-looking models have a strong arbitrage force that eliminates momentum, even with short sale constraints. In Walrasian models, this works through potential buyers and sellers re-timing their transactions, while in search models it works through the outside options of buyers and sellers. Construction also serves as an arbitrage force. Furthermore, momentum cannot be empirically explained by serially correlated changes in fundamentals. Case and Shiller (1989) argue that momentum cannot be explained by autocorrelation in interest rates, rents, or taxes, and Capozza et al. (2004) find significant momentum after accounting for six comprehensive measures of fundamentals in a vector error correction model. Pulling together the empirical and theoretical arguments, Glaeser et al. (2014) estimate a dynamic spatial equilibrium model and find that “there is no reasonable parameter set” consistent with short-run momentum. In recent years, a number of frictions have been added to models of the housing market to explain momentum. These frictions fall into four main categories. First, gradual learning about market conditions by sellers can create momentum. Anenberg (2014) structurally estimates a partial equilibrium model with learning and finds an annual AR(1) coefficient of 0.165, which is about a quarter of the median city in the CoreLogic data. The inability 3 See Cutler et al. (1991) and Moskowitz et al. (2012) for analyses of time series momentum for different assets. As discussed in Appendix B, price indices that measure the median price of transacted homes display momentum over roughly two years as opposed to three years for repeat-sales indices. Appendix B also shows that there is no evidence of an asymmetry in falling markets relative to rising markets, although the test for asymmetry has limited statistical power. 5 This paper does not attempt to explain the variation in momentum across MSAs. 4 4 Figure 1: Momentum in Housing Prices B. Impulse Response of Seas Adj. Log Real Price Levels, AR(5) -0.50 Autocorrelation 0.00 0.50 Log Price Impulse Response to Shock 1.00 A. Autocorrelations of Seas Adj. Quarterly Real Price Changes .06 .04 .02 0 0 5 10 Quarterly Lag 15 0 20 5 10 Quarters After Shock 15 20 0 1 Density 2 3 4 C. Histogram of Annual House Price Change AR(1) Coefficients for 103 Cities 0 .2 .4 .6 Autoregressive Coefficient on Log Annual Change in HPI .8 Notes: Panel A and B show the autocorrelation function for quarterly real price changes and an impulse response of log real price levels estimated from an AR(5) model, respectively. The IRF has 95% confidence intervals shown in grey. An AR(5) was chosen using a number of lag selection criteria, and the results are robust to altering the number of lags. Both are estimated using the CoreLogic national repeat-sales house price index from 1976-2013 collapsed to a quarterly level, adjusted for inflation using the CPI, and seasonally adjusted. Panel C shows a histogram of annual AR(1) coefficients of annual house price changes as in regression (1) estimated separately on 103 CBSA division repeat-sales house price indices provided by CoreLogic. The local HPIs are adjusted for inflation using the CPI. The 103 CBSAs and their time coverage, which ranges from 1976-2013 to 1995-2013, are listed in Appendix A. of learning to explain the amount of momentum in the data reflects the fact that Bayesian learning about buyer valuations happens relatively quickly. One would need to dramatically slow down learning with an extreme parameterization to generate three years of momentum. Second, Head et al. (2014) demonstrate that search frictions can generate momentum by making the adjustment of market tightness gradual in response to shocks to fundamentals. In a boom in their model, most sellers sell but only some buyers buy, so the number of searching buyers “builds up” as buyers persistently enter, which causes market tightness to rise over time as more houses are rented out to accommodate the growing mass of buyers and fewer houses are listed for sale. Head et al. calibrate their model and obtain only 40 percent of the autocorrelation in the data over one year and no momentum over two years. Over 85 percent of the price impulse response occurs in the first quarter after a shock in contrast to the impulse response in Figure 1, in which under 25 percent occurs in the first quarter. Third, a behavioral finance literature hypothesizes that investors initially under-react to news due to behavioral biases (Barberis et al., 1998, Hong and Stein, 1999) or loss aversion (Frazzini, 5 2006) and then “chase returns” due to extrapolative expectations about price appreciation. Both extrapolative expectations and loss aversion are considered to be important forces in the housing market (Case and Shiller, 1987; Genesove and Mayer, 2001). Recently, Glaeser and Nathanson (2016) have provided a behavioral theory of momentum in which agents neglect to account for the fact that previous buyers were learning from prices and instead take past prices as direct measures of demand. Their calibrated model can match quantitatively explain momentum when all agents are behavioral and observe prices with a somewhat-long six-month lag. Momentum is dampened to a third of the observed amount when half of sellers are fully rational, and recent survey and experimental evidence suggest that under half of agents expectations appear to be substantially extrapolative (Kuchler and Zafar, 2016; Armona et al. 2016). Fourth, Burnside et al. (2015) show that in a model with belief heterogeneity, momentum could result from optimism and pessimism that gradually spreads through social interactions, akin to epidemiological models of the spread of disease. In their model, agents with tighter priors are more likely to convert others to their beliefs, and in order to generate sufficient momentum, the relative confidence of optimists and pessimists must be extremely close so that the relative number of optimists grows gradually over the course of a decade instead of suddenly in an “epidemic.” This tight parameter restriction could be relaxed with the amplification mechanism presented here.6 For all four types of frictions, the degree of friction necessary to match the amount of momentum in the data is unpalatably large. This paper introduces a strategic complementarity in pricing that amplifies the effect of any friction that generates heterogeneity in beliefs, a class that includes learning, backward-looking sellers, and gradually spreading beliefs, among others. With the additional momentum provided by the strategic complementarity, the frictions identified by the literature are able to quantitatively explain momentum with substantially more reasonable parameterizations. 3 The Concavity of Demand in the Housing Market I propose an amplification channel for momentum based on search and a concave demand curve in relative price. Search is a natural assumption for housing markets, but the relevance of concave demand requires further explanation. A literature in macroeconomics argues that strategic complementarities among goods producers can amplify small pricing frictions into substantial price sluggishness by incentivizing firms to set prices close to one another. Because momentum is similar to price stickiness in goods markets, I hypothesize that a similar strategic complementarity may amplify house price momentum. There are several reasons why concave demand may arise in housing markets. First, buyers may avoid visiting homes that appear to be overpriced. Second, buyers may infer that underpriced homes are lemons. Third, a house’s relative list price may be a signal of seller type, such as an unwillingness 6 Burnside et al. obtain such a parameter restriction through a simulated method of moments procedure. Their data implies that the change in the number of optimists is growing at a decreasing rate between 7 and 10 years after the start of the boom (assumed to be 1996), which implies a very gradual spread of optimism. This estimation strategy does not provide any direct evidence inconsistent with the strategic complementarity presented here because their model with a strategic complementarity in price setting could fit the same data with a faster spread of optimism. 6 to negotiate (Albrecht et al., 2016). Fourth, homes with high list prices may be less likely to sell quickly and may consequently be more exposed to the tail risk of becoming a “stale” listing that sits on the market without selling (Taylor, 1999). Fifth, buyers may infer that underpriced homes have a higher effective price than their list price because their price is likely to be increased in a bidding war (Han and Strange, 2016). Sixth, the law of one price—which would create a step-function demand curve—may be smoothed into a concave demand curve by uncertainty about what a house is worth. Nonetheless, concrete evidence is needed for the existence of concave demand in housing markets before it is adopted as an explanation for momentum. Consequently, this section assesses whether demand is concave by analyzing micro data on listings matched to sales outcomes for the San Francisco Bay, Los Angeles, and San Diego metropolitan areas from April 2008 to February 2013.7 3.1 3.1.1 Empirical Approach Econometric Model To keep the analysis transparent, the unit of observation is a listing associated with an initial log list price, p, and the outcome of interest is a summary statistic of the time to sale distribution, d. In the main text, d is an indicator for whether the house sells within 13 weeks, with a withdrawal counting as a non-sale, and I vary the horizon and use time to sale for d in robustness checks. The data consist of homes, denoted with a subscript h, from markets defined by a location ` (a ZIP code in the data) and time period t (a quarter in the data). The relevant demand curve for list-price-setting sellers is the effect of unilaterally changing a house’s quality-adjusted list price relative to the average list price in the area on its probability of sale. To simplify notation, I combine the average list price in location ` at time t and the qualityadjustment for house h into a single “target” list price for house h, p̃h`t . p̃h`t has two additive components: the average log list price in location ` at time t, represented by a fixed effect ⇠`t , and quality qh`t that is only partially observable to the econometrician: p̃h`t = ⇠`t + qh`t . (2) I model the probability of sale dh`t as being a potentially nonlinear function of ph`t p̃h`t net of a market-specific effect and an error term: dh`t = g (ph`t p̃h`t ) + `t + "h`t . (3) The demand curve in relative price g (·) is assumed to be invariant across markets defined by a location and time net of an additive fixed effect `t that represents local market conditions. "h`t is an error term that represents luck in finding a buyer and is assumed to be independent of ph`t p̃h`t .8 7 These areas were selected because both the listings and transactions data providers are based in California, so the matched dataset for these areas is of high quality and spans a longer time period. 8 Traditionally, endogeneity concerns stem from the correlation of the error term "h`t with price. This source of 7 I call ph`t p̃h`t the seller’s log relative markup, as it represents the “markup” a seller is asking over the quality-adjusted average list price for a house in location ` at time t. If qh`t were observable, one could directly estimate (3), but quality qh`t almost certainly has an important unobserved component that is likely positively correlated with price, leading to an estimated demand curve that is less elastic than the true demand curve. To surmount this challenge, I use a non-linear instrumental variable approach that traces out the demand curve using plausibly exogenous supply-side variation in seller pricing behavior. I also use an “OLS” approach that assumes away unobserved quality to assess how the instrument affects the measured shape of g (·). Formally, I model quality as a linear function of observed measures of quality Xh`t and quality unobserved by the econometrician uh`t : qh`t = Xh`t + uh`t . (4) Combining (2) and (4), the reference price p̃h`t can be written as: p̃h`t = ⇠`t + Xh`t + uh`t . (5) I use two observed measures of quality Xh`t in my baseline specification and introduce other measures and more flexible functional ⇣ ⌘forms for Xh`t in robustness tests. First, I use a repeat-sales predicted repeat `t price p̂h`t = log Ph`⌧ `⌧ , where Ph`⌧ is the price at the previous sale date ⌧ and `t is a ZIP code-level repeat-sales house price index index. Second, I use a hedonic predicted price p̂hedonic , t which is the sum of the static value of a house’s hedonic characteristics and a date of sale fixed effect, both estimated within a ZIP code. The construction of both indices follows standard practice and is detailed in Appendix A. Unobserved quality uh`t introduces two difficulties in identifying g (·). First, there is an endogeneity problem as unobserved quality and price are likely positively correlated. Second, unobserved quality creates a measurement error problem because the true p̃h`t is not observable. Both of these problems may lead to bias in the estimated g (·). I identify g (·) in the presence of unobserved quality by introducing an instrumental variable zh`t which is described in the next section that generates supply-side variation in ph`t p̃h`t and that is independent of unobserved quality. To allow for nonlinearity in the first stage, I let zh`t affect price through a flexible function f (·). Then g (·) is identified if: Condition 1. zh`t ? (uh`t , "h`t ) and ph`t = f (zh`t ) + p̃h`t = f (zh`t ) + ⇠`t + Xh`t + uh`t . (6) bias is absent here as the effect of demand shocks on average price levels is absorbed into ⇠`t and the effect of prices on aggregate demand is absorbed into `t . 8 The first half of Condition 1 is an exclusion restriction that requires zh`t have no direct effect on the outcome, either through fortune in finding a buyer "h`t or through unobserved quality uh`t . I discuss this assumption when I introduce the instrument in subsection 3.1.2. The second part of Condition 1 requires that zh`t is the only reason for variation in ph`t p̃h`t , which is effectively a no measurement error assumption. I discuss this assumption and the robustness of my results to measurement error in subsection 3.1.3. Under Condition 1, ph`t p̃h`t = f (zh`t ), and g (·) can be estimated by a two-step procedure that estimates equation (6) by OLS as a first stage and then uses the predicted f (zh`t ) as ph`t p̃h`t to estimate equation (3) as a second stage by OLS. Both equations weighted by the inverse standard deviation of the error in the repeat-sales index to account for the reduced precision of the predicted prices in areas with fewer transactions, and I use a quintic polynomial for f (·). I show that neither of these assumptions matters for my results in robustness tests. I assess the degree of concavity in two ways. First, I use a quadratic g (·) and test whether the quadratic term is statistically distinguishable from zero. To account for spatial correlation, I calculate standard errors by block bootstrapping the entire procedure and clustering on 35 units defined by the first three digits of the ZIP code (ZIP-3). The bootstrapped 95 percent confidence interval is my preferred test for concavity. Second, to visualize the data, I construct a binned scatter plot, which approximates g (·) using indicator variables for the 25 equally-sized bins of ph`t p̃h`t , as detailed in Appendix C. I also overlay a third-order polynomial fit with pointwise 95 percent confidence bands. 3.1.2 Instrument Due to search frictions, home sellers face a trade-off between selling at a higher price and selling faster. Sellers with a higher marginal utility of cash on hand will choose a higher list price and longer time on the market. My instrument takes advantage of two sources of variation in the marginal utility of cash on hand that are plausibly independent of unobserved quality. The first source of variation is that sellers who extract less equity upon sale on average have a higher marginal utility of cash on hand. This is the case because many sellers many sellers use the equity they extract from sale for the down payment on their next home (Stein, 1995). At low levels of equity extraction, households are on a down payment constraint, and each dollar of equity extracted is used towards the next down payment and leveraged, making the marginal utility of cash on hand is high. By contrast, sellers extracting significant equity have enough cash on hand that they no longer face a binding down payment constraint. Because they do not leverage up each additional dollar of equity they extract, their marginal utility of cash on hand is lower. The presence of a binding down payment constraint depends on an individual seller’s liquid asset position and their access to credit, but on average sellers that extract more equity have a less binding down payment constraint, lower marginal utility of cash on hand, and set lower list prices. The second source of variation is loss aversion. Using data on condominiums in Boston in the 1990s, Genesove and Mayer (2001) show that sellers experiencing a nominal loss set higher list 9 prices, attain a higher selling price, and take longer to sell. I use one instrument to capture the effect of both of these sources of variation: ⇣ ⌘ the log of `t appreciation in the ZIP repeat-sales house price index since purchase zh`t = log `⌧ , where is the repeat-sales house price index, t is the period of listing, and ⌧ is the period of previous sale. For loss aversion, zh`t is a proxy for whether the seller is facing a nominal loss and how large the loss will be because it measures average appreciation in the local area rather than the appreciation of any particular house. For equity extraction, financing and refinancing decisions make the equity of sellers endogenous. zh`t is isomorphic to equity if all homeowners take out an identical mortgage and do not refinance and is thus an instrument for the exogenous component of equity. I am agnostic as to the importance of each source of variation in the first stage relationship. For the instrument to be relevant, f (zh`t ) must be have a significant effect on ph`t in the first stage equation (6). I show below that the first stage is strong and has the predicted effect of lowering the list price when appreciation since purchase is high.9 Importantly, it is smooth and monotonic, so nonlinearity in f (zh`t ) does not drive the results on g (·). The exclusion restriction for the instrument to be valid is zh`t ? (uh`t , "h`t ) in Condition 1, which requires that appreciation since purchase zh`t have no direct effect on the probability of sale dh`t , either through the error term "h`t or through unobserved quality uh`t . If this is the case, zh`t only affects probability of sale through the relative markup ph`t p̃h`t . Because I use ZIP ⇥ quarter of listing fixed effects ⇠`t , the variation in zh`t comes from sellers who sell at the same time in the same market but purchase at different points in the cycle. Condition 1 can thus be interpreted as requiring that unobserved quality be independent of when the seller purchased. This assumption is difficult to test because I only have a few years of listings data, so flexibly controlling for when a seller bought weakens the effect of the instrument on price in equation (6) and widens the confidence intervals to the point that any curvature is not statistically significant. Nonetheless, I evaluate the identification assumption in five ways in robustness tests in Appendix C. First, I include observable measures of quality in Xh`t . Second, I show that the observable measures of quality are either uncorrelated with the date of purchase (bedrooms and bathrooms) or roughly linear in date of purchase (age, rooms, lot size) and do not appear to vary systematically with the housing cycle. This implies that any unobservables sorted in the same way as these observables would be captured by a linear time trend. This motivates my third test, which includes a linear time trend in date of purchase or time since purchase. Fourth, I limit the sample to sellers who purchased prior to 2004 and again include a linear time trend, eliminating variation from sellers who purchased near the peak of the bubble or during the bust. In all of these tests, the results remain robust, although standard errors widen with smaller sample sizes. Fifth, I show that the shape of ⇣ ⌘ `t In the first stage regression, appreciation since purchase enters both through zh`t = log `⌧ and through Xh`t , ⇣ ⌘ `t which includes p̂repeat = log Ph`⌧ `⌧ = log (Ph`⌧ ) + zh`t . The coefficient on log (Ph`⌧ ) and zh`⌧ entering through h`t 9 p̂repeat are restricted to be the same. f (zh`t ) is identified in the first stage through the differential coefficient on zh`t h`t and log (Ph`⌧ ). I address concerns that the identification of f (zh`t ) is coming from introducing p̂repeat linearly but h`t zh`t nonlinearly in the first stage by introducing p̂repeat with a polynomial of the same order as f (·) in robustness h`t tests, and the results are virtually unchanged. 10 the estimated demand curve is similar for IV and OLS, which assumes away unobserved quality, although OLS results in a far more inelastic demand curve due to bias created by the positive correlation of price with unobserved quality. If in spite of these robustness tests, homes with very low appreciation since purchase are of substantially lower unobserved quality despite their higher average list price, my identification strategy would overestimate the amount of curvature in the data.10 To make the exclusion restriction more plausible, I focus on sellers for whom the exogenous variation is cleanest and consequently exclude four groups from the main analysis sample. I relax these exclusions in robustness tests. First, I drop houses sold by banks after a foreclosure (often called REO sales), as the equity of the foreclosed-upon homeowner should not affect the bank’s list price. Second, many individuals who have had negative appreciation since purchase are not the claimant on the residual equity in their homes—their mortgage lender is. For these individuals, appreciation since purchase is directly related to their degree of negative equity, which in turn affects the foreclosure and short sale processes of the mortgage lender or servicer. Because I am interested in time to sale as determined by the market rather than how long a mortgage servicer takes to approve a sale, I exclude these individuals based on two proxies for equity detailed when I describe the data in the next subsection. Second, investors who purchase, improve, and flip homes typically have a low appreciation in their ZIP code since purchase but improve the quality of the house in unobservable ways, violating the exclusion restriction. To minimize the effect of investors, I exclude sellers who previously purchased with all cash, a hallmark of investors. Third, I drop sellers who experience extreme price drops since purchase (over 20 percent) as these are unusual sellers who only have positive equity if they took out a very unusual initial mortgage. 3.1.3 Measurement Error The second part of Condition 1 requires that zh`t is the only reason for variation in ph`t p̃h`t . This is a strong assumption because there may be components of liquidity that are unobserved or other reasons that homeowners list their house at a price different from p̃h`t , such as heterogeneity in discount rates. If the second part of Condition 1 did not hold, the estimates would be biased because the true ph`t p̃h`t would equal f (zh`t ) + ⇣h`t , and the unobserved measurement error ⇣h`t enters g (·) nonlinearly. This is an issue because the measurement error induced by unobserved quality is non-classical. In (6), unobserved quality is a residual that is independent of observed quality Xh`t , and so the measurement error induced by uh`t is Berkson measurement error, in which the measurement error is independent of the observed component, rather than classical measurement error, in which the measurement error is independent of the truth p̃h`t . An instrument such as zh`t can address classical measurement error in a non-linear setting, but it cannot address Berkson measurement error, which is why an additional assumption is necessary. 10 One potential concern is that sellers with higher appreciation since purchase improve their house in unobservable ways with their home equity. However, this would create a positive first stage relationship between price and appreciation since purchase while I find a strong negative relationship. 11 I use two strategies to show that the bias created by measurement error does not cause significant spurious concavity. First, I prove that if the measurement error created by other sources of variation in the relative markup ph`t p̃h`t is independent of the variation induced by the instrument, the measurement error would not cause spurious concavity. Intuitively, noise in ph`t the observed probability of sale at each observed ph`t of sale at true ph`t p̃h`t would cause p̃h`t to be an average of the probabilities p̃h`t s that are on average evenly scrambled. Consequently, the curvature of a monotonically-decreasing demand curve is preserved. An analytical result can be obtained if the true g (·) is a polynomial regression function as in Hausman et al. (1991): Lemma 1. Consider the econometric model described by (3) and (5) and suppose that: zh`t ? (uh`t , "h`t ) , ph`t = f (zh`t ) + ⇣h`t + p̃h`t , (7) (8) ⇣h`t ? f (zh`t ), and the true regression function g (·) is a third-order polynomial. Then estimating g (·) assuming that ph`t = f (zh`t ) + p̃h`t yields the true coefficients of the second- and third-order terms in g (·). If g (·) is a second-order polynomial, the same procedure yields the true coefficients of the first- and second-order terms. Proof. See Appendix C. While a special case, Lemma 1 makes clear that the bias in the estimated concavity is minimal if ⇣h`t ? f (zh`t ). Second, while spurious concavity is a possibility if the measurement error created by other sources of variation in the relative markup were correlated with the instrument, the amount of concavity generated would be far smaller than the concavity I observe in the data. Appendix C presents Monte Carlo simulations that show that if the instrument captures most of the variation in the relative markup ph`t p̃h`t at low levels of appreciation since purchase but very little of the variation at high levels of appreciation since purchase, spurious concavity arises because the slope of g (·) is attenuated for low relative markups but not high relative markups. However, to spuriously generate a statistically-significant amount of concavity, one would need a perfect instrument at low levels of appreciation since purchase and all of the variation in price at high levels of appreciation since purchase to be measurement error. Because this is implausible, I conclude that spurious concavity due to measurement error is not driving my findings. 3.2 Data I combine data on listings with data on housing characteristics and transactions. The details of data construction can be found in Appendix A. The listings data come from Altos Research, which every Friday records a snapshot of homes listed for sale on multiple listing services (MLS) from several publicly available web sites and records the address, MLS identifier, and list price. The housing characteristics and transactions data come from DataQuick, which collects and digitizes 12 public records from county register of deeds and assessor offices. This data provides a rich onetime snapshot of housing characteristics from 2013 along with a detailed transaction history of each property from 1988 to August 2013 that includes transaction prices, loans, buyer and seller names and characteristics, and seller distress. I limit my analysis to non-partial transactions of single-family existing homes as categorized by DataQuick. I match the listings data to a unique DataQuick property ID. To account for homes being delisted and re-listed, listings are counted as contiguous if the same house is re-listed within 90 days and there is not an intervening foreclosure. If a matched home sells within 12 months of the final listing date, it is counted as a sale, and otherwise it is a withdrawal. The matched data includes 78 percent of single-family transactions in the Los Angeles area and 68 percent in the San Diego and San Francisco Bay areas. It does not account for all transactions due to three factors: a small fraction of homes (under 10%) are not listed on the MLS, some homes that are listed in the MLS contain typos or incomplete addresses that preclude matching to the transactions data, and Altos Research’s coverage is incomplete in a few peripheral parts of each metropolitan area. I limit the data to homes listed between April 2008 and February 2013.11 I drop cases in which a home has been rebuilt or significantly improved since the transaction, the transaction price is below $10,000, or a previous sale occurred within 90 days. I exclude ZIP codes with fewer than 500 repeat sales between 1988 and 2013 because my empirical approach requires that I estimate a local house price index. These restrictions eliminate approximately five percent of listings. As discussed above, the instrumental variables analysis omits REO sales, investors who previously purchased with all cash, individuals who experience extreme price declines, and individuals with substantial negative equity. I create two different IV samples that identify listings in which the seller has substantial negative equity in two different ways. First, I create a proxy for the equity of all sellers at listing using DataQuick data on the history of mortgage liens against each property along with the loan amount, loan type (fixed or adjustable rate), and an estimate of the interest rate based on loan and property characteristics. The data does not indicate when mortgages are prepaid and omits several features the payment schedule. Consequently, I follow DeFusco (2015) and create a “debt history” for each listed property that estimates the outstanding mortgage debt by making assumptions about unobserved mortgage characteristics and when liens are paid off as detailed in Appendix A. Because the resulting proxy for seller equity is noisy, in the first IV sample I exclude sellers with less than -10 percent estimated equity, and I vary this cutoff in robustness tests. I create a second IV sample by excluding listings in which DataQuick has flagged the sale as a likely short sale or withdrawn listings that are subsequently foreclosed upon in the next two years. 11 The Altos data begins in October 2007 and ends in May 2013. I allow a six month burn-in so I can properly identify new listings, although the results are not substantially changed by including October 2007 to March 2008 listings. I drop listings that are still active on May 17, 2013, the last day for which I have data. I also drop listings that begin less than 90 days before the listing data ends so I can properly identify whether a home is re-listed within 90 days and whether a home is sold within six months. The Altos data for San Diego is missing addresses until August 2008, so listings that begin prior to that date are dropped. The match rate for the San Francisco Bay area falls substantially beginning in June 2012, so I drop Bay area listings that begin subsequent to that point. 13 Table 1: Summary Statistics For Listings Micro Data Sample All Prior Trans IV IV2 All Prior Trans IV IV2 Only houses that sold? No No No No Yes Yes Yes Yes Transaction 72.30% 74.60% 68.70% 62.70% 100% 100% 100% 100% Prior Transaction 62.70% 100% 100% 100% 64.70% 100% 100% 100% REO 22.40% 27.80% 0% 0% 28.70% 35.60% 0% 0% Short Sale or 19.40% 23.60% 13.50% 0% 19.80% 24.30% 15.90% 0.00% Estimated Equity < -10% 30.90% 49.30% 100% 88.40% 31% 47.90% 100.00% 94.20% Initial List Price $644,556 $595,137 $859,648 $861,254 Subsequent Foreclosure $580,150 $548,132 $824,239 $845,253 Transaction Price $532,838 $501,588 $758,803 $781,091 Weeks on Market 15.32 16.46 14.01 12.86 61.20% 57.50% 0.652 0.685 Sold Within 13 Wks 44.30% 42.90% 44.80% 42.90% Beds 3.283 3.239 3.337 3.328 3.264 3.229 3.327 3.322 Baths 2.185 2.126 2.331 2.316 2.141 2.102 2.31 2.308 Square Feet 1,811 1,732 1,969 1,953 1,759 1,702 1943.6 1944.1 N 663,976 416,373 140,344 137,238 480,258 310,758 96,400 86,033 Notes: Each column shows summary statistics for a different sample of listings. The four samples used are the full sample of listings matched to a transaction, houses with an observed prior transaction (or if the observed prior transaction is not counted as a sales pair, for instance because there is evidence the house was substantially renovated), and the first and second IV samples. The first set of four columns provides summary statistics for all listed homes regardless of whether they sell The second four columns limits the summary statistics to houses that sell. The data covers listings between April 2008 and February 2013 in the San Francisco Bay, Los Angeles, and San Diego areas as described in Appendix A. REOs are sales of foreclosed homes and foreclosure auctions. Short sales and subsequent foreclosures include cases in which the transaction price is less than the amount outstanding on the loan and withdrawals that are subsequently foreclosed on in the next two years. The estimation procedure for equity is described in Appendix A, and a -10 percent threshold is chosen because the equity measure is somewhat noisy. The data set consists of 663,976 listings leading to 480,258 transactions. I focus on the 416,373 listings leading to 310,758 transactions with an observed prior transaction in the DataQuick property history going back to 1988. The two IV samples are substantially smaller because of the large number of REO sales and listings by negative equity sellers from 2007 to 2013. Roughly 28 percent of listings with an observed prior transaction sample are REO sales, and dropping these leads to approximately 300,000 listings. Another 34,000 sales by likely investors who initially purchased with all cash. Of the remaining 266,000 listings, roughly 126,000 have estimated equity at listing under -10% or appreciation since purchase under -20%, so the first IV sample consists of 140,344 listings leading to 96,400 transactions. Roughly 129,000 of the 266,000 listings are flagged as a short sale by DataQuick, subsequently foreclosed upon, or have appreciation since purchase under -20%, leading to a second IV sample of 137,238 listings leading to 86,033 transactions. Table 1 provides summary statistics for the subsamples that I use in the analysis. 14 3.3 Results Figure 2 shows first and second stage binned scatter plots for both IV samples: IV sample one which excludes extreme negative equity sellers, and IV sample two which excludes short sales and withdrawals that are subsequently foreclosed upon. The results are similar for both samples. As shown in panel A, the first stage is strong, smooth, and monotonic. This is the variation that identifies the shape of demand and is the x-axis of panel B. The first stage is strong with a joint F statistic for the third order polynomial of the instrument in (6) of 206 in the baseline specification for IV sample one. Panel B shows a clear concave relationship in the second stage, with very inelastic demand for relatively low priced homes and elastic demand for relatively high priced homes. This curvature is also visible in the cubic polynomial fit. Table 2 shows regression results when g (·) is approximated by quadratic polynomial. Columns 3 and 5 show the IV results for each IV sample. In both cases, there is clear concavity: the quadratic term is negative and highly significant, and the bootstrapped 95 percent confidence interval for the quadratic term is bounded well away from zero. As a point of comparison, Column 1 shows the OLS results that assume away unobserved quality for the full set of sales with a observed prior transaction. The fixed effects are at the ZIP ⇥ quarter ⇥ REO seller ⇥ short seller level to prevent distressed sales from biasing the results. Columns 2 and 4 show similar OLS results for IV sample one and IV sample two. Reassuringly, concavity is not unique to the IV specification. In all three samples, OLS displays clear and significant concavity, and binned scatter plots in Appendix C show that most of the difference across samples is from extreme quantiles that do not drive concavity in the IV specification. The expected bias is also present: the demand curve is flatter because p̃h`t is positively correlated with ph`t due to omitted unobserved quality. Finally, while there is slightly more concavity in the IV samples, the 95 percent confidence intervals on the quadratic terms overlap. This suggests that the concavity found in the IV samples is not due to sample selection. At the mean price, the sample one estimates imply that raising one’s price by one percent reduces the probability of sale within 13 weeks by approximately 2.7 percentage points on a base of 48 percentage points, a reduction of 5.6 percent. This corresponds to a one percent price hike increasing the time to sale by five to six days. By contrast, increasing the list price by five percent reduces the probability of sale within 13 weeks by 21.5 percentage points, a reduction of 45 percent. These figures are slightly smaller than those found by Carrillo (2012), who estimates a structural search model of the steady state of the housing market with multiple dimensions of heterogeneity using data from Charlottesville, Virginia from 2000 to 2002. Although we use very different empirical approaches, in a counterfactual simulation, he finds that a one percent list price increase increases time on the market by a week, while a five percent list price increase increases time on the market by nearly a year. Carrillo also finds small reductions in time on the market from underpricing, consistent with the nonlinear relationship found here. Appendix C shows the finding of concavity is highly robust. Across both samples, the results are robust across geographies, time periods, and specifications, although in a handful of cases restricting 15 Figure 2: Instrumental Variable Estimates of the Effect of List Price on Probability of Sale IV Sample 1: Excluding Low Estimated Equity B. Second Stage .34 −.04 .36 .38 .4 Prob Sell in 13 Weeks .42 .44 .46 .48 Log Initial List Price Relative to Mean −.02 0 .02 .5 .52 .04 .54 A. First Stage −.5 0 .5 Log Appreciation Since Purchase 1 −.04 −.02 0 Log Relative Markup .02 .04 IV Sample 2: Excluding Short Sales and Subsequent Foreclosures B. Second Stage .32 −.04 .34 .36 .38 Prob Sell in 13 Weeks .4 .42 .44 .46 Log Initial List Price Relative to Mean −.02 0 .02 .48 .5 .04 .52 A. First Stage −.5 0 .5 Log Appreciation Since Purchase 1 −.04 −.02 0 Log Relative Markup .02 .04 Notes: For both samples, Panel B shows a binned scatter plot of the probability of sale within 13 weeks net of ZIP ⇥ first quarter of listing fixed effects (with the average probability of sale within 13 weeks added in) against the estimated log relative markup p p̃. It also shows an overlaid cubic fit of the relationship, as in equation (3). To create the figure, a first stage regression of the log list price on a fifth-order polynomial in the instrument, fixed effects at the ZIP x first quarter of listing level, and repeat sales and hedonic log predicted prices, as in (6), is estimated by OLS. The predicted value of the polynomial of the instrument is used as the relative markup. The figure splits the data into 25 equally-sized bins of this estimated relative markup and plots the mean of the estimated relative markup against the mean of the probability of sale within 13 weeks net of fixed effects for each bin, as detailed in Appendix C. Before binning, the top and bottom 0.5 percent of the log sale price residual and any observations fully absorbed by fixed effects are dropped. The entire procedure is weighted by the reciprocal of the standard deviation of the prediction error in the repeat-sales house price index in the observation’s ZIP code from 1988 to 2013. IV sample 1 drops sales of foreclosures, sales of homes with less than negative 20 percent appreciation since purchase, sales by investors who previously purchased with all cash, and homes with under -10 percent estimated equity. IV sample 2 does away with the estimated equity requirement and instead drops DataQuick determined short sales and withdrawn listings that are foreclosed upon in the subsequent year. The grey bands indicate a pointwise 95-percent confidence interval for the cubic fit created by block bootstrapping the entire procedure on 35 ZIP-3 clusters. Panel A shows the first stage relationship between the instrument and log initial list price in equation (6) by residualizing the instrument and the log initial list price against the two predicted prices and fixed effects, binning the data into 25 equally-sized bins of the instrument residual, and plotting the mean of the instrument residual against the mean of the log initial list price residual for each bin. The first-stage fit is overlaid. N = 140,344 observations for IV sample 1 and 137,238 observations for IV sample 2 prior to dropping unique zip-quarter cells and winsorizing. 16 Table 2: The Effect of List Price on Probability of Sale: Regression Results Estimator Sample Controls Constant Linear Quadratic Quadratic Bootstrapped 95% CI N Dependent Variable: Sell Within 13 Weeks (1) (2) (3) OLS OLS IV IV Sample 1: All With Prior Obs Excluding Prior Obs REO, Investors, < -10 % Estimated Equity > 20% Depreciation ZIP⇥ Quarter FE ZIP X Quarter FE Repeat and Hedonic Repeat and Hedonic Predicted Price, Predicted Price Distress FE 0.461*** 0.496*** 0.480*** (0.005) (0.009) (0.008) -0.216*** -0.293*** -2.259*** (0.016) (0.021) (0.346) -0.634*** -1.062*** -40.955*** (0.088) (0.188) (10.271) [-0.832, [-1.518, [-69.168, -0.498] -0.825] -28.758] 416,373 140,344 140,344 (4) (5) OLS IV IV Sample 2: Prior Obs Excluding REO, Investors, Short Sales > 20% Depreciation ZIP X Quarter FE Repeat and Hedonic Predicted Price 0.475*** (0.011) -0.295*** (0.034) -0.802*** (0.227) [-1.319, -0.490] 137,238 0.461*** (0.009) -1.932*** (0.291) -29.208*** (7.206) [-48.241, -20.496] 137,238 Notes: * p <0.05, ** p<0.01, *** p<0.001. Each row shows regression coefficients when g(·) in equation (3) is approximated using a quadratic polynomial. This relationship represents the effect of the log relative markup on the probability of sale within 13 weeks. For IV, a first stage regression of log list price on a fifth-order polynomial in the instrument, fixed effects at the ZIP x first quarter of listing level, and log predicted price using both a repeat-sales and a hedonic methodology, as in (6), is estimated by OLS. The predicted value of the polynomial of the instrument is used as the relative markup in equation (3), which is estimated by OLS. For OLS, quality is assumed to be perfectly measured by the hedonic and repeat-sales predicted prices and have no unobserved component. Consequently, the log list price is regressed on fixed effects and the predicted prices and uses the residual as the estimated relative markup into equation (3), as described in Appendix C. Both procedures are weighted by the reciprocal of the standard deviation of the prediction error in the repeat-sales house price index in the observation’s ZIP code from 1988 to 2013. Before running the second-stage regression, 0.5 percent of the sample is winsorized on each end of the distribution of the relative markup, and any observations fully absorbed by fixed effects are dropped. Standard errors and the 95 percent confidence interval for the quadratic term are computed by block bootstrapping the entire procedure on 35 ZIP-3 clusters. For column 1, the full set of listings with a previous observed transaction are used. To prevent distressed sales from biasing the results, the fixed effects are at the quarter of initial listing x ZIP x distress status level. Distress status corresponds to three groups: normal sales, REOs (sales of foreclosed homes and foreclosure auctions), and short sales (cases where the transaction price is less than the amount outstanding on the loan and withdrawals that are subsequently foreclosed on in the next two years). IV sample 1 drops sales of foreclosures, sales of homes with more than a 20 negative appreciation since purchase, sales by investors who previously purchased with all cash, and homes with under -10 percent estimated equity. IV sample 2 does away with the estimated equity requirement and instead drops DataQuick determined short sales and withdrawn listings that are foreclosed upon in the subsequent year. The number of observations listed is prior to dropping observations that are unique to a ZIP-quarter cell and winsorizing. 17 to a smaller sample leads to insignificant results. The results are also robust to time trends in date of purchase, limiting the sample to sellers who purchased before the bubble, controlling for other measures of quality and nearby foreclosures, using different functional forms for f (·) and for the observables Xh`t , allowing for differential sorting by letting vary across time and space, accounting for the uniqueness of a house in its neighborhood, and accounting for different price tiers within ZIP codes, and using alternate measures of whether a house sells quickly. Finally, the results are robust to changing the criteria for inclusion in each IV sample and to using transaction prices rather than list prices, which assuages concerns that bargaining or price wars undo the concavity in list price. Appendix C also shows that concavity is clearly visible in the reduced-form relationship between the instrument and probability of sale, providing further reassurance that the concavity is not being driven by the first stage. The instrumental variable results thus provide evidence of demand concave in relative price for these three MSAs from 2008 to 2013.12 4 A Search Model of House Price Momentum This section introduces an equilibrium search model with concave demand which I calibrate to my micro evidence to quantitatively assess how much concave demand can amplify small frictions that create momentum. The model builds on search models of the housing market, such as Wheaton (1990), Krainer (2001), Novy-Marx (2009), Piazzesi and Schneider (2009), Genesove and Han (2012), Ngai and Tenreyro (2013), and Head et al. (2014). I first introduce a model of a metropolitan area with a fixed population and housing stock, describe the housing market, and show how sellers set list prices. I then add two frictions that create some initial momentum, staggered price setting and backward-looking rule-of-thumb sellers, and define equilibrium. Table 3 summarizes the notation. 4.1 Setting Time is discrete and all agents are risk neutral. Agents have a discount factor of . There is a fixed housing stock of mass one, no construction, and a fixed population of size N .13 There are four types of homogenous agents: a mass Bt of buyers, St of sellers, Ht of homeowners, and Rt of renters. Buyers, sellers, and homeowners have flow utilities (inclusive of search costs) b, s, and h and value functions Vtb , Vts , and Vth , respectively. Buyers and sellers are active in the housing market, which is described in the next section. The rental market, which serves as a reservoir of potential buyers, is unmodeled. Each agent can own only one home, which precludes short sales and investor-owners. Sellers and buyers are homogenous but sellers may differ in their list prices. Each period with probability h and r, t respectively, homeowners and renters receive shocks that cause them to separate from their current house or apartment, as in Wheaton (1990). A 12 Aside from the tail end of my sample, this period was a depressed market. The similarity between my results and Carrillo’s provide some reassurance that the results I find are not specific to the time period, but I cannot rule out that the nonlinearity would look different in a booming market. 13 Construction is omitted for parsimony, as it would work against momentum with or without concavity, leaving the relative amount of amplification roughly unchanged. See Head et al. (2014) for a model with construction. 18 Table 3: Notation in the Model Variable Description Note Masses N Total Population (Housing Stock Mass One) Bt Endogenous Mass of Buyers Value Fn Vtb St Endogenous Mass of Sellers Value Fn Vts Rt Endogenous Mass of Renters Ht Endogenous Mass of Homeowners Value Fn Vth Flow Utilities b Flow Utility of Buyer (Includes search cost) s Flow Utility of Seller (includes search cost) h Flow Utility of Homeowner Moving Shock Probabilities h Prob Homeowner Gets Shock r t Prob Renter Gets Shock Exogenous Driving Process Other Fixed Parameters Discount Factor L Probability Seller Leaves Metro Area V0 PDV of Exiting Metro Area Utilities " Match-Specific One-Time Utility Benefit h Permanent House Quality ⇠ F (") Mean Zero Stochastic Draws ⌘h,t Noise in h in Binary Signal About Quality, IID Common to All Buyers in Period t ⇠ G (⌘) Parameters / Values ✓tm ✓˜t qm (✓tm ) ⇠m Market Tightness = B/S in Submarket m = {f, d} Endogenous Vector of ✓ m s Endogenous Prob. Seller Meets Buyer in m (Matching Fn) Endogenous Constant in Matching Function in m Matching Function Elasticity in all ms pt List Price Adjusted for Permanent Quality Endogenous ⌦t Distribution of Prices Endogenous "⇤t Threshold " for Purchase Endogenous µ Threshold for Binary Signal Distribution Parameters ", "¯ Bounds of Dist of F ("), Uniform With Mass Point Size of Mass Point of Uniform Dist at "¯ Generalized Normal Variance Parameter ⇣ Generalized Normal Shape Parameter 19 renter who gets a shock enters the market as a homogenous buyer. A homeowner who gets a shock leaves the MSA with probability L, in which case they become a seller and receive a net present value of V 0 for leaving, and remains in the city with probability 1 L. If they remain in the city, they simultaneously become a buyer and a homogenous seller. These two roles are assumed to be quasi-independent so that the value functions do not interact and no structure is put on whether agents buy or sell first, as in Ngai and Tenreyro (2013) and Guren and McQuade (2015). Because the population is constant, every time a seller leaves the city they are replaced by a new renter. I defer the laws of motion that formalize the system until after I have defined the probabilities of purchase and sale. The value function of the homeowner is: Vth = h + Et 4.2 h h h s Vt+1 + LV 0 + (1 i ⇣ b L) Vt+1 + 1 h ⌘ i h Vt+1 . (9) The Housing Market The search process occurs at the beginning of each period and unfolds in three stages. First, sellers post list prices p̂t . Second, buyers observe a noisy binary signal about each house’s quality relative to its price. Buyers direct their search either towards houses that appear reasonably-priced for their quality or towards houses that appear to be overpriced for their quality, which defines two sub-markets: follow the signal (submarket f ) or do not follow the signal (submarket d). After choosing a submarket, buyers search randomly within the submarket an stochastically find a house to inspect. Third, matched buyers inspect the house and decide whether to purchase it. At the inspection stage, buyers observe their idiosyncratic valuation for the house ", which is match-specific, drawn from F (") at inspection, and realized as utility at purchase. They also observe the house’s permanent quality h, which is common to all buyers, mean-zero, gained by a buyer at purchase, and lost by a seller at sale. I assume all sales occur at list price, or equivalently that risk neutral buyers and sellers expect that the average sale price will be an affine function of the list price. Letting pt ⌘ p̂t her surplus from doing so Vth + " b " > pt + b + Vt+1 Vth ⌘ "⇤t (pt ) h be the quality-adjusted list price, the buyer purchases if his or pt b Vtb is positive. This leads to a threshold rule to buy if and a probability of purchase given inspection of 1 The assumption that houses sell at list price is made for tractability.14 15 F ("⇤t ). While clearly a simplification, it is a reasonable approximation in two ways. First, for average and median prices it is realistic: Appendix D shows that in the merged Altos-DataQuick micro data, the modal transaction price is the list price, and the average and median differences between the list and transaction price are less than 0.03 log points and do not vary much across years. Second, conditional on unobserved quality, the difference between initial list and transaction prices does not vary with the initial list 14 This assumption restricts what can occur in bargaining or a price war. Several papers have considered the role of various types of bargaining in a framework with a list price in a steady state search model, including cases in which the list price is a price ceiling (Chen and Rosenthal, 1996; Haurin et al., 2010), price wars are possible (Han and Strange, 2016), and list price can signal seller type (Albrecht et al., 2016). 15 An important feature of the housing market is that most price changes are decreases. Consequently, the difference between the initial list price and the sale price fluctuates substantially over the cycle as homes that do not sell cut their list price. I abstract from such duration dependence to maintain a tractable state space. 20 price. To show this, Appendix D uses the IV procedure in Section 3 replacing dh`t with the difference between list and transaction prices. There is no strong relationship between relative quality-adjusted list prices and transaction prices, and if anything setting an average price results in a slightly higher transaction price. Consequently, the list price is the best predictor of the transaction price from the perspective of a list-price-setting seller. Since utility is linear, adding adding empirically-realistic ex-post bargaining that maintains the mean transaction price as the list price would not alter the seller’s list price setting incentives as shown in Appendix D. At the signal stage, buyers observe a binary signal from their real estate agent or from advertisements that reveals whether each house’s quality-adjusted price relative to the average qualityadjusted price is above a threshold µ. However, quality h (or equivalently the observation of the average price) is subject to mean zero noise ⌘h,t ⇠ G (·), where G (·) is assumed to be a fixed distribution.16 This noise, which represents how well a house is marketed in a given period, is common to all buyers but independent and identically distributed across periods. The signal thus indicates a house is reasonably priced if, pt E⌦t [pt ] ⌘h,t  µ, where ⌦t is the cumulative distribution function of list prices and the notation E⌦t [·] represents an expectation with respect to the distribution of prices ⌦t rather than an intertemporal expectation. Consequently, a house with quality-adjusted price pt appears reasonably priced and is searched by buyers in submarket f with probability 1 G (pt submarket d with probability G (pt µ). I assume that search is more efficient if buyers E⌦t [pt ] E⌦t [pt ] µ) and is searched by buyers in follow the signal than if they do not because they have the help of a realtor or are looking at bettermarketed homes. In equilibrium, buyers follow a mixed strategy and randomize whether they search submarket f or d so that the value of following the signal is equal to the value of not following it. I consider an equilibrium in which all buyers choose the same symmetric strategy with a probability of following the signal of t. After choosing a submarket, buyers search randomly within that sub-market and cannot direct their search to any particular type of home within that market. The probability a house in submarket m meets a buyer is determined according to a constant returns to scale matching function, q m (✓tm ), where ✓tm is the ratio of buyers to sellers in submarket m = {f, d}. The probability a buyer meets a seller is then q m (✓tm ) /✓tm . The matching function captures frictions in the search process that prevent all reasonably-priced homes and all buyers from having an inspection each period. For instance, buyers randomly allocating themselves across houses may miss a few houses, or there may not be a mutually-agreeable time for a buyer to visit a house in a given period. The mass of sellers in the f submarket is St times the weighted average probability that any given seller is in the f submarket E⌦ [1 G (·)], and the mass of sellers in the d submarket is 16 Because the signal reveals no information about the house’s permanent quality quality "m , the search and inspection stages are independent. 21 h, posted price p̂t , or match similarly St E⌦ [G (·)]. Consequently, the market market tightness in the f and d submarkets is: ✓tf = ✓td = Btf ollow Bt t = f St E⌦t [1 G (pt E⌦t [pt ] St Btdo not f ollow Bt (1 t) = d St E⌦t [G (pt E⌦t [pt ] St µ)] µ)] . The probability a buyer who follows the signal buys a house is then: P r [Buy|Follow] = = = ⇣ ⌘ q f ✓tf Z ✓tf ⇣ ⌘ q f ✓tf Z 1 G (pt E⌦t [pt ] µ) (1 E⌦t [1 G (pt E⌦t [pt ] µ)] (1 G (pt E⌦t [pt ] h ⇣ ⌘i 1 E⌦t df pt, ⌦t , ✓˜t , t ✓t F ("⇤t (pt ))) d⌦t (pt ) µ)) (1 t ✓t F ("⇤t (pt ))) d⌦t (pt ) where ✓t = Bt /St is the aggregate market tightness, and ⇣ ⌘ ⇣ ⌘ df pt , ⌦t , ✓˜t = q f qtf (1 G (pt E⌦t [pt ] µ)) (1 F ("⇤t (pt ))) , (10) is the demand curve faced by a seller in the f submarket. Similarly, the probability a buyer buys if they do not follow the signal is: P r [Buy|Do Not Follow] = where, 1 (1 ⇣ ⌘ ⇣ ⌘ dd pt , ⌦t , ✓˜t = q d ✓td G (pt t ) ✓t h E⌦t dd (pt, ⌦t , ✓t , E⌦t [pt ] µ) (1 t) i , F ("⇤t (pt ))) , (11) is the demand curve faced by a seller in the d submarket. Note that the demand curve faced by sellers, which is the ex-ante probability of sale for a house with a list price pt , can be written as: ⇣ ⌘ d pt , ⌦t , ✓˜t = P r [Good Signal] P r [Sell|Good Signal] + P r [Bad Signal] P r [Sell|Bad Signal] ⇣ ⌘ ⇣ ⌘ = df pt , ⌦t , ✓˜t + dd pt , ⌦t , ✓˜t . (12) Similarly, the total probability a buyer buys given the P r [Buy] = = h ⇣ ⌘i 1 E⌦t df pt, ⌦t , ✓˜t + (1 t ✓t h ⇣ ⌘i 1 E⌦t d pt , ⌦t , ✓˜t . ✓t t t randomization strategy is: t) 1 (1 t ) ✓t h E⌦t dd (pt, ⌦t , ✓t , t) i Given these probabilities of purchase and sale, the stock of buyers is equal to the stock of buyers 22 who failed to buy last period plus the stock of renters and flow of new entrants who decide to buy, while the stock of sellers is equal to those sellers who failed to sell last period plus homeowners who put their house up for sale. These are formalized by: Bt = ✓ 1 ⇣ St = 1 h ⇣ ⌘i◆ ˜ E⌦ d pt 1 , ⌦ t 1 , ✓ t 1 Bt 1 + ✓t 1 t 1 h ⇣ ⌘i⌘ E⌦t 1 d pt 1 , ⌦t 1 , ✓˜t 1 St 1 + h H t 1 r t 1 Rt 1 + (1 L) h Ht 1 (13) (14) 1. Because there are mass one of homes that can either be owned by a homeowner or up for sale and mass N of agents who can either be renters, homeowners, or buyers, (15) 1 = Ht + St N (16) = Rt + B t + Ht . These equations together with (13) and (14) implicitly define laws of motion for Ht and Rt . When a buyer buys, they receive the expected surplus E [" of pt . The value of a buyer who follows, Vtb,f , "⇤t |" > "⇤t ] where "⇤t is a function a buyer who does not follow Vtb,d , and a buyer prior to choosing a submarket Vtb , are: Vtb,f Vtb,d Vtb h ⇣ ⌘ i 1 E⌦t df pt, ⌦t , ✓˜t E [" "⇤t |" > "⇤t ] t ✓t h ⇣ ⌘ i 1 b = b + Et Vt+1 + E⌦t dd pt , ⌦t , ✓˜t E [" "⇤t |" > "⇤t ] (1 t ) ✓t n o b,f b,d = max Vt , Vt . b = b + Et Vt+1 + (17) (18) In equilibrium, buyers are indifferent between the two markets, so Vtb,f = Vtb,d or: h ⇣ ⌘ E⌦t df pt , ⌦t , ✓˜t E [" h ⇣ ⌘ d ˜ E⌦t d pt , ⌦t , ✓t E [" i "⇤t |" > "⇤t ] i = 1 ⇤ ⇤ "t |" > "t ] This pins down the fraction of buyers who go to submarket f , b Vtb = b + Et Vt+1 + t. h ⇣ ⌘ 1 E⌦t df pt, ⌦t , ✓˜t E [" t ✓t t (19) . t Vtb can then be rewritten as: i "⇤t |" > "⇤t ] . (20) Sellers have rational expectations but set their list price before ⌘h,t is realized and without knowing the valuation of the particular buyer who visits their house. The demand curve they face ⇣ ⌘ ˜ is d pt , ⌦t , ✓t , so the seller value function is: n ⇣ ⌘⇥ s Vts = s + Et Vt+1 + max d pt , ⌦t , ✓˜t pt pt I solve for the seller’s optimal price in the next subsection. 23 s s Et Vt+1 ⇤o . (21) I parameterize the model by assuming distributions for F (·), the distribution of idiosyncratic match quality, and G (·), the noise in the signal. I assume that F ("m ) is a uniform distribution on [", "¯] with a mass point of mass at "¯. In Figure 2, the demand curve for below average priced homes is very flat, which implies a very low density of F (·) at the margin. If there were not a mass point at the top of F (·), the low density would imply a very large upper tail conditional expectation E [" "⇤ |" > "⇤ ], which in turn implies a very high value of future search to buyers. Adding a mass point allows me to control the value of future search and target a realistic buyer search cost. In practice, this means that there are many buyers who like the property, many who do not, and a few in between. For G (·) I use a type 1 generalized normal distribution with a PDF of g (x) = 2 ⇣ (|x µ|/ )⇣ . (1/⇣) e This is the same as a normal if the shape parameter ⇣ is equal to two. When ⇣ > 2, the PDF is more flat topped” which results in a CDF that is more kinked rather than smoothly s-shaped. This allows me to capture a feature of Figure 2: the demand curve, which inherits the properties of the CDF G (·), is somewhat kinked. These two distributions provide a close fit to the microdata. I also assume that the matching function is Cobb-Douglas q m (✓) = ⇠ m ✓ , as is standard in the housing search literature, with ⇠ f > ⇠ d . While these functional form assumptions matter somewhat for the precise quantitative predictions of the model, they are not necessary for the intuitions it illustrates. This setup leads to a locally concave demand curve with considerable curvature in the neighborhood of the average price At below average prices, the house receives a good signal with near certainty and the demand curve is dominated by the trade-off between idiosyncratic match quality "m and price, so demand is less elastic. At above average prices, the demand curve is dominated by the fact that perturbing the price affects whether the house gets a good signal or a bad signal, in which case it ends up in a market with few buyers who match with less efficiency. To illustrate how the demand curve is built ⇣ ⌘up from its various components, Figure 3 shows the shapes of the probability of inspection q ✓tf (1 G (pt E⌦t [pt ] µ)) + q ✓td G (pt E⌦t [pt ] µ), the probability of purchase conditional on inspection 1 F ("⇤ (pt )), and the overall demand curve faced by sellers d (pt , ⌦t , ✓t ), equal to the product of the first two panels. The probability of inspection and overall demand curve are split into the f and d components, revealing that the d submarket has negligible effect on the overall demand curve because it has few buyers. Note that the axes are switched from a standard Marshallian supply and demand diagram to be consistent with the empirical estimates. 4.3 Price Setting The strategic complementarity enters through the seller’s optimal price. Sellers have monopoly power due to costly search, and the optimal pricing problem they solve in equation (21) is the same as that of a monopolist facing the demand curve d except that the marginal cost is replaced by the s . The optimal pricing strategy is seller’s outside option of searching again next period s + Et Vt+1 s , and the markup varies inversely with relative price, a markup over this outside option s + Vt+1 creating a strategic complementarity. With an initial friction that generates some heterogeneity in the speed of adjustment, the strategic complementarity causes quick adjusters to adjust their price 24 Figure 3: The Concave Demand Curve in the Model Prob Inspect Prob Buy | Inspect Demand Curve Total Demand Curve Follow Demand Curve Do Not Follow Demand Curve Prob of Sale Prob Inspect Prob Buy | Inspect Total Follow Do Not Follow 1 Relative Price 1 Relative Price 1 Relative Price Notes: The figures are generated using calibration described in Section 5. All probabilities are calculated assuming all other sellers are setting the steady state price and considering the effect of a unilateral deviation. The first panel shows the total probability of inspection and the components coming form the f and d submarkets, with the f submarket almost entirely overlapping the total and the d submarket essentially zero because few buyers are in the d submarket. The second panel shows the probability of purchasing given inspection. The third panel shows the demand curve, which is the product of the two. Again, the f submarket demand curve df is essentially on top of the total demand curve d and d submarket demand curve dd is near zero. Note that the axes are swapped from the traditional Marshallian supply and demand diagram in order to be consistent with the empirical analysis in Section 3. s jumps upon the change in fundamentals, but more gradually when fundamentals change: s + Vt+1 raising the list price above the market average erodes the markup, so the optimal price does not change much on impact. To formalize these intuitions in my model, I focus on a symmetric equilibrium.17 Sellers do not internalize that their choice of pt affects the average price, which they treat as given. Seller optimization implies: Lemma 2. The seller’s optimal list price at the interior optimum is: ⇣ ⌘3 d pt , ⌦t, ✓˜t s 5 pt = s + Et Vt+1 + Et 4 @d(pt ,⌦t ,✓˜t ) 2 @pt s = s + Et Vt+1 + 1 f ("⇤t ) F ("⇤t ) + g (pt E⌦t [pt ] µ) 1 G(pt E⌦t [pt ] µ) ✓ 1 1 dd (pt ,⌦t ,✓˜t ) E⌦t [pt ] µ) d(pt ,⌦t ,✓˜t ) 1 G( pt ◆, (22) where dd is defined by (11) and d is defined by (12). 17 Although the seller’s problem may not be globally concave, I focus on an interior optimum and later check that the interior optimum is the global optimum by simulation. This turns out not to be a concern in the baseline calibration in Figure 3 as the mass point in the idiosyncratic taste distribution occurs before the probability of inspection curve flattens. In some alternate calibrations, the mass point is further out, and the demand curve is non-concave. 25 Proof. See Appendix E. In equation (22), the seller markup is written as an additive markup equal to the reciprocal of the semi-elasticity of demand, d(pt ,⌦t, ✓t ) @d(pt ,⌦t ,✓t ) @pt . The semi-elasticity, in turn, is equal to the sum of the hazard rates of the idiosyncratic preference distribution F (·) and the distribution of signal noise G (·) adjusted for the share of sales that occur in the d submarket, the term in parenthesis. This creates a local strategic complementarity because the elasticity of demand rises as relative price increases, causing the optimal additive markup to fall and pushing sellers to set prices close to those of others. Mathematically, this works through relative price pt E⌦t [pt ] entering the hazard rate of the signal G (·), which is rising in relative price so the additive markup is falling in relative price.18 However, in a rational expectations equilibrium in which all sellers can set their price flexibly, all sellers choose the same list price and pt = E⌦t [pt ], so there is no change in relative prices to affect the markup. A shock to home values thus causes list price to jump proportionally to the seller’s outside option, and there is no momentum. In the terminology of Ball and Romer (1990), concave demand is a real rigidity that only amplifies nominal rigidities. Consequently, I introduce frictions that generate some heterogeneity in the insensitivity of prices to fundamentals. As discussed in Section 2, there are several frictions that the literature has identified that act in such a manner, and I am agnostic as to which underlying frictions are at work in practice. In this paper I separately introduce two tractable, transparent, and well-understood frictions, a small fraction of rule-of-thumb sellers and staggered pricing, and demonstrate concave demand’s ability to amplify each of them. I call these the “rule of thumb model” and the “staggered pricing model” and define their equilibria separately. I leave a formal analysis of other frictions that may interact with concave demand to future work. 4.4 Source of Insensitivity 1: A Small Fraction of Rule-of-Thumb Sellers Since Case and Shiller (1987), sellers with backward-looking expectations have been thought to play an important role in housing markets. Previous models assume that all agents have backwardlooking beliefs (e.g., Berkovec and Goodman, 1996), but some observers have found the notion that the majority of sellers are non-rational unpalatable given the financial importance of housing transactions for many households. Indeed, recent evidence from surveys and experiments finds significant heterogeneity in backward-looking expectations: Kuchler and Zafar (2016) analyze survey expectations and find significant extrapolation by lower-educated households that make up 44 percent of the sample but minimial extrapolation by higher-educated households, and Armona et al. (2016) estimate that 41 percent of households are extrapolators using an experimental intervention. Consequently, I introduce a small number of rule-of-thumb sellers, as in Campbell and Mankiw (1989), and assess quantitatively what fraction of sellers is needed to be non-rational to explain the momentum in data, similar to Gali and Gertler (1999). 18 Because few buyers are in the d submarket, the term in parenthesis is near unity. 26 I assume that at all times a fraction 1 ↵ of sellers are of type R (rational) and set their list price pR t rationally according to Lemma 2 and (22) but a fraction ↵ of sellers are of type E (extrapolator) and use a backward-looking rule of thumb to set their list price pE t . Specifically, they set their price equal to the most recently observed price plus a fraction of the most recently observed inflation: pE t = pt 2 + pt 3 + pt 3 4 + ✓ pt 2 + pt 3 + pt 3 4 pt 5 + pt 6 + p t 3 7 ◆ , (23) where pt is the transaction-weighted average price at time t: pt = E ↵dE t pt + (1 ↵dE t + (1 R ↵) dR t pt . ↵) dR t (24) Such a rule is a common assumption in models with backward-looking expectations and in the New Keynesian literature and is frequently motivated by limited knowledge, information costs, and extrapolative biases (e.g., Hong and Stein, 1999; Fuster et al. 2010).19 For tractability and parsimony, I assume that regardless of whether rational or backward-looking sellers sell faster, inflows adjust so that ↵ of the active listings are houses owned by backward-looking sellers at all times. To conserve notation, in the model with rule of thumb sellers, St refers to the total number of sellers, but Vts is now the value function for the rational sellers. Vts remains as in equation (21), and the value function of a buyer remains (20), but now there are two prices pE t and pR t in the market. 4.5 Source of Insensitivity 2: Staggered Price Setting Prices in housing markets are adjusted only infrequently, with the median price lasting two months as shown in the Altos listings data in Appendix D. This is the case because it takes time to market a house and collect offers, and lowering the price frequently can signal that the house is of poor quality. While likely not the most important pricing friction in housing markets, infrequent price adjustment has the virtue of being familiar and tractable. I introduce it into the baseline model by assuming that groups of sellers set prices in a staggered fashion as in Taylor (1980). In particular, I assume there are N groups of sellers that set prices every N periods, typically using N = 2 in monthly simulations. Denote the prices pt , value functions Vts , seller masses St , and purchase thresholds "t of a vintage of sellers that set prices ⌧ = {0, ..., N 1} periods ago by ⌧ superscripts. Buyers receive the same signals buyer’s problem and value function remain the same, s while the seller’s value function is as in (21), except the value function for Vts,⌧ has Vt+1 terms s,⌧ +1 replaced by Vt+1 for ⌧ = {0, N s,0 2} and by Vt+1 for ⌧ = N 1. Seller optimization implies an optimal list price that is reminiscent of Taylor or Calvo pricing: 19 I use three-month lag to match the lag with which house price indices are released. I use three-month averages to correspond to how major house price indices are constructed and to smooth out saw-tooth patterns that emerge with non-averaged multi-period lags. I microfound such a rule using a model of limited information and near rationality in Appendix E 27 Lemma 3. If posted prices last N periods, the seller’s optimal reset price p0t is: p0t where Dtj (p) = Et hQ j 1 ⌧ =0 ⇣ 1 PN 1 ⌧ D ⌧ p0 t t 1 ⌧ ⌧ 0 D t pt ⌧ =0 ⌧ '⌧ t t ⌧ t ⌧ =0 = P N (25) ⇣ ⌘⌘i ⇣ ⌘ d⌧ p, ⌦t+⌧ , ✓˜t+⌧ d p, ⌦t+j + ✓˜t+j is the expected probability the " @d p ,⌦ ,✓˜ # ( house is sold j periods after the price is set, ⌧ t = Et t t+⌧ ) t+⌧ @pt d(pt ,⌦t+⌧ ,✓˜t+⌧ ) is the expected semi-elasticity s,⌧ +1 of demand with respect to price after ⌧ periods from Lemma 2, '⌧t = s+Et Vt+⌧ +1 + optimal flexible reset price ⌧ periods after the price is set, and s,N Vt+N = s,0 Vt+N . 1 ⌧ t is the expected Proof. See Appendix E. As is standard in staggered pricing models, the optimal price is a weighted average of the optimal flexible prices that are expected to prevail on the equilibrium path until the seller can reset his or her price. The weight put on the optimal flexible price in period t + ⌧ is equal to the discounted probability of sale in period t + ⌧ times the semi-elasticity of demand in period t + ⌧ . Intuitively, the seller cares more about periods in which probability of sale is higher but also about periods in which demand is more elastic because perturbing price has a larger effect on profit. In addition to the new price setting rule, the law of motion for sellers (14) needs to be altered to N different laws of motion for each of the the N groups of sellers each with separate prices. These modifications are relegated to Appendix E. 4.6 Equilibrium I add a stochastic shock process to the model to examine its dynamic implications. The results do not depend qualitatively or qualitatively depend on the particular shock used, but the positive correlation between price and volume in the data implies that demand-side shocks dominate. For simplicity, I use a shock to the entry rate of renters r t so that a positive shock causes the entry of potential buyers into the market, increasing demand and pushing up prices.20 An example of such a shock would be a change in credit standards for new homeowners or changes in the price expectations of renters. I implement the shock by assuming that r t follows a mean-reverting AR(1) process: r t = ¯r + ⇢ r t 1 ¯ r + ut with ut ⇠ N 0, 2 ⌘ and iid. (26) An equilibrium with a fraction ↵ of backward-looking sellers is defined as:21 20 See Guren (2015), a previous working paper version of this paper, for a model with an endogenous entry decision whereby agents who receive shocks have the opportunity to pay a randomly-drawn fixed cost to avoid entry. The working paper shows how momentum interacts with endogenous entry to explain the relationships between price, volume, and inventory in the data, in particular the strong “housing Phillips curve” relationship between price changes and inventory levels. 21 An analogous equilibrium with N groups of backward-looking sellers is defined in Appendix E. Aside from switching i 2 {E, R} for ⌧ = {0, ..., N 1}, it differs from the above definition of an equilibrium with backwardlooking sellers in three key ways. First, prices are (25) for sellers that can reset their prices and fixed for sellers that 28 i Definition ⇣ ⌘1. Equilibrium with a fraction ↵ of backward-looking sellers is a set of prices pt , demands d pi , ⌦t , ✓˜t , and purchase cutoffs "⇤,i for each type of seller i 2 {E, R}, a transaction-weighted t t average price pt , rational seller, buyer, homeowner, and renter value functions Vts , Vtb , Vth , a probability that buyers follow the signal t, stocks of each type of agent Bt , St , Ht , and Rt , and a shock to the flow utility of renting xt satisfying: 1. Optimal pricing for rational sellers (22) and the pricing rule (23) for backward-looking sellers, which depends on lagged transaction-weighted average prices (24); i b 2. Optimal purchasing decisions by buyers: "⇤,i t = pt + b + Vt+1 Vth ; 3. The demand curve for each type of seller i 2 {E, R} in the f submarket (10), the d submarket, (11), and the aggregate (12), all of which result from buyer search behavior; 4. The value functions for buyers (20), rational sellers (21), homeowners (9); 5. The laws of motion for buyers (13) and sellers (14) and the closed system conditions for homes (15) and people (G) that implicitly define the laws of motion for homeowners and renters; 6. Buyers are indifferent across markets (19); 7. All agents have rational expectations that r t evolves according the AR(1) process (26). The model cannot be solved analytically, so I simulate it numerically using a log-quadratic approximation pruning higher-order terms as in Kim et al. (2008) around a steady state described in Appendix E in which ut = 0 8 t implemented in Dynare (Adjemian et al., 2013).22 Appendix G shows that the impulse responses are almost identical in an exactly-solved deterministic model with an unexpected permanent shock, so approximation error is minimal. 5 Amplification of Momentum in the Calibrated Model To quantitatively assess the degree to which concave demand curves amplify house price momentum, this section calibrates the model to the empirical findings presented in Section 3 and a number of aggregate moments. 5.1 Calibration In order to simulate rule of thumb model, 22 parameters listed in Table 5 must be set. The staggered pricing model requires two fewer parameters. This section describes the calibration procedure and targets, with details in Appendix F. Because a few parameters are based on limited data cannot. Second, the laws of motion for each vintage of sellers in the Appendix replace the laws of motion in the text. Third, the value functions for each vintage of sellers are similarly altered. 22 Because of the mass point in the F (·) distribution, the model is not smooth. However, a perturbation approach is appropriate because the mass point at "¯ is virtually never reached (less than 0.1 percent of the time in simulations). 29 and subject to some uncertainty, I use the baseline calibration for exposition and use 14 different parameterizations to determine a plausible range for the amplification of each underlying friction. Three components of the calibration control the shape of the demand curve and thus have a firstorder impact on momentum: the local density of the idiosyncratic quality distribution F (·) controls the elasticity of demand for low-priced homes that are certain to be visited; and ⇣, the variance and shape parameters of the signal distribution, control how much the elasticity of demand changes as relative price increases; and µ, the threshold for being overpriced, controls where on the curve the average price lies. The other parameters have a second order effect on momentum. Consequently, the first step of the calibration sets these three components to match the instrumental variable binned scatter plot from Section 3. The second step calibrates the rest of the model to match steady state and time series moments. For the first step, I approximate the probability of sale as a function of a few key parameters, the relative list price I observe in the data, and a fixed effect that absorbs the aggregate market conditions so that the model can be directly compared to my empirical specification. This allows me to approximate the model demand curve out of steady state with the heterogeneity in the data for the purposes of calibration and then conduct dynamic simulations with the heterogeneity suppressed to maintain a tractable state space. Specifically, Appendix E shows that the probability of sale at the time the list price is posted can be approximated as: d (pt E⌦t [pt ]) ⇡ t (1 2 ✓ F ("⇤mean + pt E⌦t [pt ])) ⇥ ◆ mean [1 G (pt E⌦t [pt ] µ)] E⌦t [1 G(pt E⌦t [pt ] µ)] ✓ ◆ (1 mean ) ⇠d + ⇠f E G p E [p ] µ G (pt E⌦t [pt ] µ) )] ⌦t [ ( t ⌦t t 6 6 4 3 (27) 7 7, 5 where t is a fixed effect that summarizes the state of the market at time t, and "⇤mean and mean are the mean values of these variables over the cycle. Appendix E also explains why the approximation error is small. I solve for mean by approximating it by its steady state value using a steady-state version of (19). (27) is then used to simulate the probability of sale in three months for each E⌦ [pt ], and the density of F (·), µ, and p are chosen to minimize the mean squared error between the simulated demand curve and the IV binned-scatter plot with t chosen match the average probability of sale.23 24 25 Evaluating (27) requires values for ⇠d , ⇠f "⇤mean , and a parametrization of F (·) given its density. 23 To minimize the importance of outliers, 2.5 percent of the data is Winsorized from each end rather than 0.5 percent in Section 3. 24 Setting ⇣ by minimizing the distance between the model and data leads to a very large ⇣ that leads to numerical error when the dynamic model is solved using perturbation methods. Consequently, I choose ⇣ = 8, which gets nearly all the way to the improvement in mean squared error from increasing ⇣ above two while reducing numerical error. The results are not sensitive to this choice of ⇣. 25 Through equations (21) and (22), the seller search cost is determined by the elasticity of demand given the steady state price and probability of sale. In Figure 2, the zero point is just on the inelastic side of the demand curve, yielding an extremely high seller search cost. Because the zero point corresponding to the average price is not precisely estimated and depends on the deadline used for a listing to count as a sale, I use a zero point within one percent of the estimated zero point that gives a more plausible demand elasticity and seller search cost. 30 Figure 4: Model Fit Relative to Instrumental Variable Estimates 0.52 Model Data 0.5 Prob of Sale in 13 Weeks 0.48 0.46 0.44 0.42 0.4 0.38 0.36 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 Log Relative Markup Notes: The Xs are the binned scatter plot from the IV specification with 2.5 percent of the data from each end Winsorized to reduce the effects of outliers. The dots are the simulated probabilities of sale in three months at each price calculated using (27) and approximating t by its steady state value using (19) as described in the text. The baseline values are described here, and the parameters are permuted below. I assume "⇤mean = $100k, which I show in robustness tests is a normalization that has no impact on the economics of the model. I also assume a value of ⇠ d /⇠ f = 1/2. This has no analog in the data, and I show in robustness checks that it is of minimal importance for my quantitative results.26 To fully parameterize F (·), I introduce two additional moments that along with the assumed "⇤mean and density pin down ", "¯, and 1 F ("⇤mean ) : the average fraction of home inspections that lead to a purchase and the average mean excess function E [" "⇤mean |" > "⇤mean ]. The average fraction of home inspections that lead to a purchase is set to 1/10 to match buyer surveys from National Association of Realtors surveys analyzed by Genesove and Han (2012). The mean excess function is selected to match a reasonable target for the buyer flow cost of 0.75 percent of the purchase cost of the house per month, so that the average seller incurs 3 percent of the purchase cost as search costs, and this parameter is varied in robustness tests. Figure 4 shows the IV binned scatter plot for the first IV sample alongside the model’s predicted three-month probability of sale for the (F density, , µ) that minimize the distance between the model and the data. The demand curve in the calibrated model captures the curvature in the data well. The second step in the calibration sets the remaining parameters to match steady state moments listed in the first three panels of Table 4 as detailed in Appendix F. These targets are either from 26 I choose a value of 1/2 to limit the degree of non-convexity in the model. If ⇠ d is high enough relative to ⇠ f , sellers may have an incentive to deviate and set a price of "¯ when prices are rising and buyers visit rarely. Appendix E shows numerically that with ⇠ d /⇠ f = 1/2 this never happens. Empirically, I rule out a substantially higher ⇠ d because the stronger incentive to deviate would generate asymmetries in house prices not present in the data. 31 Table 4: Calibration Targets For Baseline Calibration Steady State Parameter or Moment Value Source / Justification Parameters (Matching Function Elasticity) L (Prob. Stay in MSA) 0.8 Genesove and Han (2012) 0.7 Anenberg and Bayer (2015) Aggregate Targets Annual Discount Rate 5% Time on Market for Sellers 4 Months Approx average parameter value in literature Time on Market for Buyers 4 Months ⇡ Time to sell (Genesove and Han, 2012) Homeownership Rate 65% Time in House For Owner Occupants 9 Years American Housing Survey, 1997-2005 Long run average, 1970s-1990s Prob Purchase | Inspect 0.1 Mean buyer visits 10 homes (Genesove and Han, 2012) Assumed Values Steady State Price $760k Average transaction price in IV sample h (Flow Utility of Homeowner) $6.78k 2/3 of House Falue From Flow Util (Normalization) 0.5 Limited Incentive to “Fish” ⇠ d /⇠ f "⇤ in steady state b/P (Flow Utility of Buyer Rel To Price) $100k Normalization .75% of Price Average total buyer search costs 3% of price 0.4 Based on Case et al. (2012) Time Series Moments SD of Annual Log Price Changes 0.065 CoreLogic national HPI adjusted for CPI, 1976-2013 ⇢ (Monthly Persistence of AR1 Shock) 0.950 Head et al. (2014, 2016) Autocorr of Local Pop Growth Table 5: Calibrated Parameter Values for Baseline Rule of Thumb Model Parameter ⇠f Interpretation Value Parameter Interpretation Value Monthly Discount Factor 0.996 -b/P Flow Util of B (search cost)/Price 0.75% Matching Fn Elasticity 0.8 -s/P Flow Util of S (search cost)/Price 2.1% Matching Fn Efficiency, Follow 2.540 "¯ Upper Bound of F Dist $ 161 k " Lower Bound of F Dist -$5,516 k Matching Fn Efficiency, Defy 1.270 h Monthly Prob H Moving Shock 0.009 Weight of Mass Point at "¯ 0.090 ¯r Ave Monthly Prob R Moving Shock 0.013 Variance Param in G (·) 39.676 N Population 1.484 ⇣ Shape Param in G (·) 8 L Prob Leave MSA 0.7 µ Threshold for Signal 39.676 SD of Innovations to AR(1) shock 0.00004 ⇠d AR(1) Param in Rule of Thumb 0.4 V0 NPV of Leaving MSA $2,776 k ⇢ Persistence of AR(1) shock 0.950 h Flow Util of H $6.783 k ↵ Fraction Backward Looking 0.299 ⌘ Notes: These parameters are for the baseline calibration. The calibration is monthly. 32 other papers or are long-run averages for the U.S. housing market, such as the homeownership rate, the average amount of time between moves for buyers and renters, and the average time on the market for buyers and sellers. The more speculative targets are varied in robustness tests. To calibrate the driving shock process which controls the rate at which renters become buyers, I match two time series moments as indicated by the bottom panel of Table 4. The monthly persistence of the shock is set to match the persistence of local income shocks as in Glaeser et al. (2014). The variance of the iid shock is set to match the standard deviation of annual log price changes in stochastic simulations. For the backward-looking model, I adjust ↵ and recalibrate the model until the impulse response to the renter entry shock matches the matches the 36 months of positively autocorrelated price changes in the AR(5) impulse response estimated on the CoreLogic national house price index in Section 2. Table 5 summarizes the baseline calibrated parameter values for the backwards-looking model. For the staggered pricing model, I use N = 2 groups in a monthly calibration to match the median time to adjust price in the data described in Appendix D. Because I cannot generate enough momentum to match the data using the staggered pricing model, I use the backwardlooking calibration procedure and then report results that take out backward-looking sellers and add staggering. 5.2 Quantitative Results on Amplification To assess the degree of momentum in the data and in the model, I compare the calibrated rule of thumb and staggered pricing models to a frictionless model and to calibrated models with either backward-looking sellers and staggered pricing but no concavity. The simulations without concavity use a demand curve with the same steady state probability of sale and additive markup for the average house as the concave model, but the markup is constant regardless of the relative price as detailed in Appendix E. Figure 5 shows the impulse responses graphically by plotting impulse responses under the baseline calibration. The impulse response is computed as the average difference between two sets of simulations that use the same sequence of random shocks except for one period in which an additional standard deviation shock is added. Impulse responses for downward shocks are approximately symmetric and shown in Appendix G. Panel A shows these impulse responses for rule of thumb model. The solid line shows the model impulse response that attains a 36 month impulse response with 30 percent backward-looking sellers, while the dotted line shows the estimated AR(5) impulse response from the CoreLogic data with the 95 percent confidence interval shown as thin grey lines. The two impulse responses nearly identical: they both jump very little on impact and rise smoothly before flattening out at 36 months. The dashed line shows the model without backward-looking sellers but with concave demand. In this case, the optimal price jumps nearly all the way to its peak level immediately. There is essentially no momentum because there is no initial stickiness for the strategic complementarity to amplify. The dash-dotted line shows the non-concave demand impulse response. Prices jump over half of 33 Figure 5: Price Impulse Response Functions: Model and Data A: Rule of Thumb Model B: Staggered Pricing Model 0.01 0.06 0.009 0.04 0.015 0.01 0.02 Log Impulse Response to 1 SD Shock 0.02 Log AR(5) Impulse Response in Data Log Impulse Response to 1 SD Shock 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.005 0 29.88% Backward Looking, Concave 29.88% Backward Looking, Non−Concave No Backward Looking, Concave CoreLogic HPI 0 6 12 18 Months 24 30 36 Staggered and Concave Staggered, Nonconcave Flexible Prices 0.001 0 0 0 2 4 6 Months 8 10 12 Notes: Panel A figure shows impulse responses to a one standard deviation shock to the renter probability of becoming a buyer in the rule of thumb model with and without concavity as well as a fully-flexible model (↵ = 0). The calibration removing concave demand maintains the steady state markup and probability of sale as described in Appendix E.. Also shown on the right vertical axis in the figure in the dotted black line and with grey 95% confidence intervals is the impulse response to a one standard deviation price shock estimated from a quarterly AR(5) for the seasonally and CPI adjusted CoreLogic national house price index for 1976-2013, as in Figure 1. The vertical axes of the model and AR(5) are different because the AR(5) is a quarterly shock, while the model is a monthly shock. Panel B shows impulse responses to a one standard deviation shock to the renter probability of becoming a buyer in the staggered pricing model as well as a model with no staggering and a model with no concavity. Simulated impulse responses are calculated by differencing two simulations of the model from periods 100 to 150, both of which use identical random shocks except in period 101 in which a one standard deviation negative draw is added to the random sequence, and then computing the average difference over 100 simulations. the way to their peak value upon the impact of the shock and continue to rise for seven months, at which point they begin to mean revert. One can generate a 36 month impulse response without concave demand, but this requires 74 percent of sellers to be backward looking.27 28 Far fewer backward-looking sellers are needed to match the data with concave demand in the rule-of-thumb model because the strategic complementarity creates a two-way feedback. When a shock occurs, the backward-looking sellers are not aware of it for several months, and the rational sellers only slightly increase their prices so that they do not dramatically reduce their chances of attracting a buyer. When the backward-looking sellers do observe increasing prices, they observe a much small increase than in the non-concave case and gradually adjust their price according to their AR(1) rule, reinforcing the incentives of the rational sellers not to raise their prices too quickly. Panel B shows similar impulse responses for the staggered model. Again, the solid line shows the 27 The 30 percent backward-looking sellers I find are necessary to explain the data is slightly below the 40 to 50 percent of the population that is significantly extrapolative found by Kuchler and Zafar (2016) and Armona et al. (2016). This is likely the case because the model excludes several strong arbitrage forces that work against momentum such as construction and endogenous entry, which would increase the necessary fraction of backward-looking sellers. 30 percent is also of the same order of magnitude as Gali and Gertler (1999), who find find that 26 percent of firms are backward looking in a structurally estimated New Keynesian model. 28 Because rational sellers are incentivized to “imitate” the backward-looking sellers, the loss from failing to optimize for the backward-looking sellers is relatively small, on average under half a percent of the sales price. 34 model with concave demand, the dashed line shows the model with flexible prices, and the dashdotted line shows the model without concavity. Without both concave demand and staggering, reset prices jump on impact and reach a convergent path to the stochastic steady state as soon as all sellers have reset their prices, as indicated by the dotted red line and the dashed green line. In combination, however, the two-month staggered pricing friction is amplified into seven months of autocorrelated price changes. While this is only one sixth of the momentum observed in the data, this experiment reveals concave demand to be a powerful amplification mechanism. The gradual impulse response results from sellers only partially adjusting their list prices when they have the opportunity to do so in order to not ruin their chances of attracting a buyer by being substantially overpriced. Repeated partial adjustment results in serially correlated price changes that last far beyond the point that all sellers have reset their price.29 Table 6 summarizes the amount of amplification generated by concave demand in the calibrated model by showing summary statistics for the amplification of momentum that compare the concave model impulse response to the nonconcave model impulse response under 14 different calibrations. For the rule of thumb model, amplification is measured as the ratio of the fraction of backwardlooking sellers ↵ without concavity to ↵ with concavity. For the staggered model, amplification is measured as the ratio of the maximum period of the impulse response with concavity to the maximum period without concavity. In the model without concavity, this is always two periods because prices fully adjust once all sellers have had the opportunity to change their price. To capture the number of periods with nontrivial autocorrelation, my preferred measure of amplification replaces the number of periods to reach the maximum with number of periods to reach 99 percent of the maximum. For the baseline calibration, both the rule of thumb and staggered pricing models amplify both frictions by a factor of 2.5. Table 6 reveals that this is a robust finding, as across a broad range of calibrations and for both frictions, concave demand robustly amplifies momentum by a factor of two to three. The parameter that most affects the degree of amplification is the seller’s search cost. This is the case because search costs create market power for list-price-setting sellers. As search costs fall, the market power of these sellers is eroded, and the incentive to set once’s price close to the market average, which enters as an additive markup in equation (22), shrinks relative to changes in market conditions, which enters through the seller’s outside option of searching again next period. The seller search cost is pinned down by the elasticity of demand at the average price and is 2.1 percent of the average sale price per month. The average seller, who is on the market for four months, thus incurs search costs equal to 8.4 percent of the transaction price. This is a plausible figure given six-percent realtor fees, maintenance and staging costs to get the house into condition for listing, the nuisance of listing ones house, and the fact that many sellers need to sell quickly due to the high costs of holding multiple houses. Nonetheless, because a 2.1 percent monthly seller 29 With staggered pricing there are further dynamic incentives because price resetters leapfrog sellers with fixed prices and are subsequently leapfrogged themselves. The interested reader is referred to Appendix G.3 for a detailed discussion of the dynamic intuition with staggered pricing. 35 Table 6: Summary Statistics for Amplification of Momentum Across Different Calibrations Rule of Thumb Model IV Sample 1 (Baseline) Staggered Pricing Model ↵ ↵ Ampli- Max Ampli- 99% of Max Ampli- Concave Nonconcave fication Period fication Period fication 0.2988 0.7406 2.479 7 3.5 5 2.5 IV Sample 2 0.2637 0.7188 2.726 8 4 6 3 -b/P of 0.25% 0.2988 0.7422 2.484 7 3.5 5 2.5 -b/P of 1.25% 0.2988 0.7375 2.468 7 3.5 5 2.5 -s/P of 1.0% 0.4062 0.7664 1.887 6 3 4 2 -s/P of 1.5% 0.3496 0.757 2.165 6 3 5 2.5 -s/P of 2.5% 0.2715 0.7188 2.648 7 3.5 6 3 "⇤ = $50k 0.2988 0.7375 2.468 7 3.5 5 2.5 "⇤ = $150k 0.2988 0.7375 2.468 7 3.5 5 2.5 h 1/2 of Flow Util 0.2988 0.7406 2.479 7 3.5 5 2.5 h 4/5 of Flow Util 0.2988 0.7375 2.468 7 3.5 5 2.5 f 0.3145 0.7453 2.370 7 3.5 5 2.5 = 0.3 0.2617 0.7008 2.678 N/A N/A N/A N/A = 0.5 0.3457 0.7852 2.271 N/A N/A N/A N/A d ⇠ /⇠ = 0.75 Notes: Each row shows a different robustness check. The entire model is recalibrated given the indicated parameter change. The first column shows ↵, the share of backward-looking sellers necessary for a 36-month price impulse response in the rule of thumb model, and the second column shows the required ↵ in a model with no concavity. For the no concavity case, the demand cure is calibrated to maintain the steady state markup and probability of sale as described in Appendix E. The amplification column shows the ratio of the first to the second column. The third and fifth columns show the period in which the maximum price and 99% of the maximum price are reached in the staggered pricing model with the same calibration. The fourth and sixth columns report the ratio of the maximum and 99% of the maximum period in the staggered model to the model without concavity, which results in a two-month impulse response as prices fully adjust once all sellers have the opportunity to do so. Because the seller search cost is pinned down by the elasticity of demand along with the probability of sale and price, the robustness checks that vary s/p alter the binned scatter plot that is used for the calibration by slightly scaling the probability of sale relative to its median value to obtain a more or less elastic demand curve, as described in Appendix F. This approach was chosen because it errs on the side of reducing the amount of concavity. search cost may be considered high, I vary it in robustness tests.30 When I do so, I find that the factor of amplification falls to 1.89 (2.0 for staggered) for a monthly search cost of 1.0 percent of the purchase price and rises to 2.65 (3.0 for staggered) when I raise the seller search cost to 2.5 percent of the purchase price. The other parameters that are calibrated to somewhat speculative targets and consequently altered in Table 6 have a smaller effect on the factor of amplification.31 I thus conclude that concave demand amplifies both frictions by a factor of two to three. 30 Because the seller search cost is pinned down by the elasticity of demand, in order to vary it I must alter the binned-scatter plot to which I calibrate to obtain a more elastic demand curve. To do so, I stretch the probability of sale around its median value. I take this approach as opposed to compressing the relative list price because it reduces the concavity slightly while compressing the relative list price increases concavity, and I do not want to artificially increase concavity. 31 Calibrating to IV sample two yields a slightly higher degree of amplification. Although the demand curve is slightly less concave in IV sample two, it is more inelastic, yielding a larger seller search cost. With a seller search cost equal to what I use for IV sample one, I would obtain slightly less amplification than for IV sample one. 36 5.3 Amplification of Other Frictions Because concave demand induces sellers to set their list price close to the market average, it amplifies any friction that creates heterogeneity in the speed of adjustment by list-price-setting sellers. Beyond the two frictions I incorporate in my model to quantitatively assess the amplification of momentum created by list-price-setting sellers, concave demand would amplify momentum created by learning in a model with incomplete information or by the gradual spread of sentiment in a model with belief disagreement. Although formally modeling these channels is beyond the scope of this paper, it is worth briefly discussing why each of these frictions would be amplified. In Burnside et al.’s (2015) model of gradually-spreading sentiment, momentum arrises from the gradual conversion of pessimists to optimists through social interactions. With a strategic complementarity, the optimistic agents would not want to raise their price too much relative to pessimists, creating the same amount of momentum with more rapid social dynamics. With incomplete information and learning, strategic complementarities can cause very gradual price adjustment even if first-order learning occurs somewhat rapidly as in Anenberg (2014) because the motive to price close to others makes higher order beliefs matter. Learning about higher order beliefs is more gradual—which in turn makes price adjustment more gradual—because agents must learn not only about fundamentals but also about what everyone else has learned as in a Keynesian beauty contest. Such a model has the potential to quantitatively explain momentum without appealing to non-rationality. For instance, Anenberg (2014) obtains about a quarter of the momentum in the data from learning. If the degree of amplification of learning is similar to what I find for staggered pricing and backward-looking sellers, this would come close to explaining the amount of momentum observed in the data. 6 Conclusion The degree and persistence of autocorrelation in house price changes is one of the housing market’s most distinctive features and greatest puzzles, and existing explanations rely on unpalatably large frictions to quantitatively explain momentum. This paper introduces a mechanism that amplifies many proposed frictions into substantial momentum, allowing the needed frictions to be of a plausible magnitude. Search and concave demand in relative price together imply that increasing one’s list price above the market average is costly, while lowering one’s list price below the market average has little benefit. This strategic complementarity induces sellers to set their list prices close to the market average. Consequently, frictions that cause heterogenous insensitivity to changes in fundamentals can lead to prolonged autocorrelated price changes as sellers slowly adjust their list price to remain close to the mean. I provide evidence for concave demand in micro data and introduce an equilibrium search model with concave demand that is calibrated to match the amount of concavity in the micro data. Quantitatively, concave demand amplifies momentum created by staggered pricing and a fraction of backward-looking rule of thumb sellers by a factor of two to three. Importantly, concave demand 37 amplifies any other pricing frictions that creates heterogeneity in the speed of price adjustment because the incentive to price close to the average makes sellers who would change their price quickly instead respond sluggishly. Assessing which frictions are relevant is an important path for future research. Beyond the housing market, this paper shows how a central idea in macroeconomics—that strategic complementarities can greatly amplify modest frictions—can be applied in new contexts. These contexts can, in turn, serve as empirical laboratories to study macroeconomic phenomena for which micro evidence has proven elusive. In particular, many models with real rigidities (Ball and Romer, 1990) use a concave demand curve. This paper provides new evidence that a concave demand curve in relative price is not merely a theoretical construct and can have a significant effect on market dynamics. 38