Arrested Development: Theory and Evidence of Supply-Side Speculation in the Housing Market

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1 Arrested Development: Theory and Evidence of Supply-Side Speculation in the Housing Market Charles G. Nathanson Kellogg School of Management Northwestern University Eric Zwick Booth School of Business University of Chicago and NBER September 2015 Abstract This paper studies the role of speculation in amplifying housing cycles. Speculation is easier in the land market than in the housing market due to frictions that make renting less efficient than owner-occupancy. As a result, undeveloped land both facilitates construction and intensifies the speculation that causes booms and busts in house prices. This observation reverses the standard intuition that cities where construction is easier experience smaller house price booms. It also explains why the largest house price booms in the United States between 2000 and 2006 occurred in areas with elastic housing supply. JEL Codes: D84, G12, G14, R31 We thank John Campbell, Edward Glaeser, David Laibson, and Andrei Shleifer for outstanding advice and Tom Davidoff, Morris Davis, Robin Greenwood, Sam Hanson, Chris Mayer, Alp Simsek, Amir Sufi, Adi Sunderam, Jeremy Stein, Stijn Van Nieuwerburgh, and Paul Willen for helpful comments. We also thank Harry Lourimore, Joe Restrepo, Hubble Smith, Jon Wardlaw, Anna Wharton, and CoStar employees for enlightening conversations and data. Prab Upadrashta provided excellent research assistance. Nathanson thanks the NSF Graduate Research Fellowship Program, the Bradley Foundation, the Becker Friedman Institute at the University of Chicago, and the Guthrie Center for Real Estate Research for financial support. Zwick thanks the University of Chicago Booth School of Business, the Neubauer Family Foundation, and the Harvard Business School Doctoral Office for financial support.

2 Asset prices go through periods of sustained price increases, followed by busts. To explain these episodes, economists have developed theories based on disagreement, speculation, and strategic trading. This literature focuses on the behavior of asset prices in stock markets, but it is natural to ask whether these ideas can explain housing markets as well. Like any other financial asset, housing is a traded, durable claim on uncertain cash flows. An enduring feature of housing markets is booms and busts in prices that coincide with widespread disagreement about fundamentals (Shiller, 2005), and there is a long history of investors using real estate to speculate about the economy (Kindleberger, 1978; Glaeser, 2013). Yet housing differs in a fundamental way from the typical asset studied in finance. The typical financial asset is in fixed supply, and its dividends are worth the same to all buyers. Housing is a good its value derives from the utility flows it delivers to end users. dividends from housing have different values for different people. And because firms can respond to high prices with new construction, housing supply is not fixed. In this paper, we incorporate speculation into a neoclassical, price-theoretic model of housing to examine whether the finance view of price booms generalizes to non-financial markets. 1 The translation is not seamless: speculation affects house prices only under certain conditions. In particular, we find a non-monotonic relationship between the elasticity of housing supply and the tendency of speculation to raise house prices. The A simple formula maps investor beliefs to house prices using intuitive, measurable objects like the demand and supply elasticities of housing. We apply this formula to understand the variation across cities during the housing boom of 2000 to In particular, we explain why the strongest house price growth occurred in cities, like Las Vegas, where housing supply before 2000 seemed able to absorb rising demand and prevent price growth. The extreme price growth in these anomalous elastic cities is a well-known puzzle for existing theories of housing markets, in which rapid construction holds down prices (e.g., Glaeser, Gyourko and Saiz, 2008). Our model combines finance and price theory using three ingredients. First, as in Miller (1977), Harrison and Kreps (1978), Diether, Malloy and Scherbina (2002), Scheinkman and Xiong (2003), and Simsek (2013), market participants agree to disagree about the future level of asset prices. In our setting, disagreement about prices arises in response to an unexpected housing demand shock. Some people believe the shock will persist; others believe it will dissipate. This disagreement generates trade because people hold heterogeneous priors and do not update their own beliefs upon learning the beliefs of others. Morris (1996) argues that such disagreement best fits unprecedented situations like unanticipated secular 1 Other papers have applied speculative finance models to housing. Piazzesi and Schneider (2009) and Burnside, Eichenbaum and Rebelo (2014) incorporate optimism and non-standard learning into search models of the housing market, Favara and Song (2014) incorporate a rental margin into a model of disagreement with short-selling constraints, and Giglio, Maggiori and Stroebel (2014) empirically evaluate whether rational bubbles exist in the housing market. Unlike those papers, our work focuses on housing supply; we explore how a realistic model of housing supply alters the predictions of finance models of speculation. 2

3 shifts in housing demand in which people have not yet had a chance to engage in rational learning. As Glaeser (2013) documents, housing booms have historically been accompanied by unanticipated events like the settlement of new cities or the discovery of new resources. Second, we permit housing to have flexible supply that responds to current demand as well as expectations about future demand. Housing is supplied by a competitive market of developers, who buy land at market prices and turn it into housing. As in Saiz (2010), the amount of developable land is fixed in the long run by geography and regulation, leading the marginal cost of housing supply to rise as the city grows. Free entry in construction links land and house prices, so investors who are optimistic about housing demand can speculate in both land and housing markets. Following models of speculation in stock markets, we rule out short-selling in land and housing. The case for short-sale constraints is even stronger in real estate, where a lack of asset interchangeability makes it impossible to cover a short. Third, residents receive heterogeneous utility from living in housing, and some of this utility accrues only when they own their houses. This non-transferable ownership utility captures the inefficiencies arising from the separation of ownership and control. Such moral hazard inefficiencies have long been recognized in corporate finance (Shleifer and Vishny, 1997), and Henderson and Ioannides (1983) use them to explain why some residents choose to own rather than rent. The equilibrium result of ownership utility is that homeownership is dispersed among individual residents rather than concentrated among a few landlords who rent out the housing stock. Dispersed ownership is one of the most salient aspects of the housing market, with over 60% of the housing stock owner-occupied. Our analysis examines how an identical shock affects house prices in cities at different stages of development. We first characterize how house prices aggregate disparate beliefs about future demand. Only two statistics from the belief distribution matter the average belief and the most optimistic belief and the belief implied by market prices is a weighted average of these two. The weight on the most optimistic belief increases with the availability of land and decreases with the share of housing that is owner-occupied. A sufficient statistic for the influence of optimism is the short-run elasticity of housing supply, which reflects how easily speculators can participate in the market. When a city consists solely of owneroccupied housing, prices reflect the average belief and are not biased toward optimism. This result holds even though short-selling housing is impossible. Speculation raises prices only when optimists can take concentrated positions. Because owner-occupied housing is dispersed among residents with varying beliefs, owner-occupied housing subdues speculation. We explore how belief aggregation combines with classical supply and demand forces to influence prices in response to the shock. A given realization of future demand affects house prices less when building houses in the future is easy, that is, when the long-run supply elasticity is high. This classical effect weighs against the result on belief aggregation, which 3

4 holds that house prices look more optimistic when the short-run supply elasticity is high. And when neither the short-run nor long-run elasticities are high, the dispersion economies of owner-occupied housing dominate. Thus speculation amplifies house prices most in the middle: a city with short-run elastic, but long-run inelastic supply. In such a city, developers can build housing easily today, but anticipate running out of available land in the near future. This theoretical condition likely characterized the anomalous elastic cities at the start of the boom in Several of these cities face long-run limits to their growth, but little regulation of current construction. For instance, Las Vegas is surrounded by land owned by the federal government, and Congress passed a law in 1998 prohibiting the sale of land outside a development ring depicted in Figure 1. 2 During the boom, land investors acted as if they expected these governments to stop selling land and restrict future development. We show in Section 4 that land prices rose strongly in these cities, which would not have occurred had investors anticipated unlimited land in the long run. We present evidence of land market speculation between 2000 and 2006 from U.S. public homebuilders behavior we term supply-side speculation. These firms tripled their land holdings during this time, while land prices rose significantly across the country. Statements by these firms in their financial reports confirm that perceived land supply constraints drove this behavior. At the same time, the homebuilding industry saw its stocks short-sold more frequently than 95% of the industries in the United States. Our model also offers new predictions for the variation in house price booms within a city. Optimistic speculators hold rental housing, just as they hold land. All else equal, prices appear more optimistic and hence house price booms are larger in types of housing that are easier to rent out, such as condos and multifamily units. Similarly, neighborhoods where a greater share of housing is rented witness stronger price increases. This prediction matches the data: house prices increased more from 2000 to 2006 in neighborhoods where the share of rental housing in 2000 was higher. 2 Las Vegas provides a stark illustration of our model. The ample raw land available in the short run allowed Las Vegas to build more houses per capita than any other large city in the U.S during the boom. At the same time, speculation in the land markets caused land prices to quadruple between 2000 and 2006, rising from $150,000 per acre to $650,000 per acre, and then lose those gains. This in turn led to a boom and bust in house prices. The high price of $150,000 for desert land before the boom and after the bust demonstrates the binding nature of the city s long-run development constraint. A New York Times article published in 2007 cites investors who believed the remaining land would be fully developed by 2017 (McKinley and Palmer, 2007). The dramatic rise in land prices during the boom resulted from optimistic developers taking large positions in the land market. In a striking example of supply-side speculation, a single land development fund, Focus Property Group, outbid all other firms in every large parcel land auction between 2001 and 2005 conducted by the federal government in Las Vegas, obtaining a 5% stake in the undeveloped land within the barrier. Focus Property Group declared bankruptcy in

5 FIGURE 1 Long-Run Development Constraints in Las Vegas Figure 2-9: Las Vegas Valley Development: Notes: This figure comes from page 51 of the Regional Transportation Commission of Southern Nevada s Regional Transportation Plan (RTCSNV, 2012). The first three pictures display the Las Vegas metropolitan area in 1980, 1990, and The final picture represents the Regional Transportation Commission s forecast for The boundary is the development barrier stipulated by the Southern Nevada Public Land Management Act. The shaded gray region denotes developed land Regional Transportation Plan,

6 1 A Housing Market with Disagreement Our housing market model is set in discrete time with an infinite horizon. The infinite horizon allows us to compare the effect of an identical demand shock in cities at different stages of development. Inside this infinite horizon framework, we embed a two-period model of disagreement. A two-period disagreement model inside an infinite horizon housing model provides a simple setting for studying the static interaction between disagreement and house prices. Housing Supply. The city we study has a fixed amount of space S. This space either is used for housing or remains as undeveloped land. The total housing stock in the city at time t is H t and the remaining undeveloped land is L t, so S = H t + L t for all t. A continuum of real estate developers invest in land and construct housing from the land at a cost of K per unit of housing. The aggregate supply of new housing is H t. Construction is instantaneous, and housing does not depreciate: H t = H t + H t 1. Construction is also irreversible: H t 0. Both housing and land are continuous variables, and one unit of housing requires one unit of land. Undeveloped land possesses some use other than residential housing. The role of this alternate use in the model is to smooth out the city s development as land is depleted, leading the housing supply elasticity to decrease continuously with development. Without this alternate use, short-run housing supply would be perfectly elastic up to the point of land exhaustion, after which it would become perfectly inelastic. This crude structure would not affect our results; in fact, the results are much easier to demonstrate in this case. We introduce the alternate use to show that our results are robust to a continuous supply elasticity, and to match the empirical likelihood that this elasticity does decline continuously with development. 3 To this end, the developers rent out land on spot markets at a price of r l t. Rental demand for undeveloped land comes from firms, such as farms, that use the city s land as an input. These firms buy their inputs and sell their products on the global market. Therefore, their aggregate demand for land depends only on r l t and not on any other local market conditions. This aggregate rental demand curve is D l (r l t), where D l ( ) is smooth, decreasing, and positive function such that D l (0) S. 3 Other papers have produced supply elasticities that decline smoothly using different model specifications. Saiz (2010) does so by assuming homeowners must commute to a central business district, and Paciorek (2013) models construction costs as increasing in the share of available land that has been developed. As we discuss in Section 4, an empirical literature surveyed by Gyourko (2009) indicates that housing supply elasticities declined tremendously between 1970 and 2000 in certain cities in the United States. 6

7 The profit flow of a developer j at time t is π j,t = rtl l j,t + p l t(l j,t 1 L j,t ) + (p h t p l t K) H } {{ } j,t, (1) } {{ } development profit homebuilding profit where p h t is the price of housing and p l t is the price of land. The real estate development industry faces no entry costs, so the industry is perfectly competitive. Because homebuilding is instantaneous and does not depend on prior land investments, profits from this line of business must be zero due to perfect competition. 4 We denote the aggregate homebuilding profit by π hb t = (p h t p l t K) H t. Each developer begins with a land endowment and issues equity to finance its land investments. It maximizes its expected net present value of profits E j t=0 βt π j,t. The operator E j reflects firm j s expectation of future land prices. Firm-specific beliefs represent the beliefs of the firm s CEO, who owns equity, cannot be fired, and decides the firm s land investments. The number of each developer s equity shares equals the amount of land it holds, and each developer pays out its land rents as dividends. developer equity therefore equals the market price p l t of land. Individual Housing Demand. The market price of A population of residents live in the city and hold its housing. These residents receive direct utility from consuming housing. Lower-case h denotes the flow consumption of housing, whereas upper-case H denotes the asset holding. Flow utility from housing depends on whether housing is consumed through owner-occupancy or under a rental contract. Residents also derive utility from non-housing consumption c. Each resident i maximizes the expected present value of utility, given by E i t=0 β t u i (c t, h own t, h rent t ), where β is the common discount factor. Flow utility u i (,, ) has three properties. First, it is separable and linear in c. This quasilinearity eliminates risk aversion and hedging motives. Second, owner-occupied and rented housing are substitutes, and residents vary in which type of contract they prefer and to what degree. Substitutability of owner-occupied and rented housing fully sorts residents between the two types of contracts; no resident consumes both types of housing simultaneously. Finally, residents face diminishing marginal utility of owner-occupied housing. This property leads homeownership to be dispersed among residents in equilibrium. 4 We discuss evidence supporting the perfect competition assumption in Section 4. 7

8 The utility specification we adopt that features these three properties is u i (c, h own, h rent ) = c + v(a i h own + h rent ), (2) where a i > 0 is resident i s preference for owner-occupancy, and v( ) is an increasing, concave function for which lim h 0 v (h) =. The distribution of the owner-occupancy preference parameter a i across residents is given by a continuously differentiable cumulative distribution function F a, which is stable over time. Owner-occupancy utility is unbounded: df a has full support on R +. This utility arises from an agency problem in which a resident would like to maintain the house in some idiosyncratic way, but doing so is costly for the owner and is not contractible in detail. The Appendix derives the functional form in equation (2) as the equilibrium outcome of such an agency problem. Resident Optimization. Residents hold three assets classes: bonds B, housing H, and developer equity Q. Global capital markets external to the city determine the gross interest rate on bonds, which is R t = 1/β, where β is the common discount factor. Residents may borrow or lend at this rate by buying or selling these bonds in unlimited quantities. In contrast, housing and developer equity are traded within the city, and equilibrium conditions determine their prices p l t and p h t. Homeowners earn income by renting out the housing they own in excess of what they consume. The spot rental price for housing is r h t ; landlord revenue is therefore rt h (H i,t h own i,t h rent i,t ). Shorting housing is impossible, but residents can short developer equity. Doing so is costly. Residents incur a convex cost k s (Q) to short Q units of developer stock, where k s (0) = 0 and k s, k s > 0. As in Duffie, Garleanu and Pedersen (2002), this convex cost is a reduced form capturing features like risk and credit that are omitted from the model but limit shorting empirically. 5 Short-sale constraints in the housing market result from a lack of asset interchangeability. Although housing is homogeneous in the model, empirical housing markets involve large variation in characteristics across houses. This variation in characteristics makes it essentially impossible to cover a short. Unlike in the housing market, asset interchangeability holds in the equity market, where all of a firm s shares are equivalent. The Bellman equation representing the resident optimization problem is V (B i,t 1, H i,t 1, Q i,t 1 ) = max B i,t,h i,t,q i,t c i,t,h own i,t,h rent i,t c i,t + v(a i h own i,t + h rent i,t ) } {{ } flow utility + βe i,t V (B i,t, H i,t, Q i,t ), (3) } {{ } continuation value 5 In Duffie, Garleanu and Pedersen (2002), each investor can short only 0 or 1 units of stock. This specification provides the simplest form of a convex cost: the marginal cost of shorting beyond 1 unit is infinite, whereas the marginal cost of shorting beyond 0 units is 0. As in the current paper, Duffie, Garleanu and Pedersen (2002) study short-selling when investors hold heterogeneous beliefs and agree to disagree, and make a reduced form convex cost assumption to enable the study of short-selling in a Walrasian market. 8

9 where the maximization is subject to the short-sale constraint 0 H i,t, the ownership constraint h own i,t H i,t, and the budget constraint R t B i,t 1 B } {{ i,t + } c }{{} i,t borrowing costs consumption p h t (H i,t 1 H i,t ) } {{ } housing returns + p l t(q i,t 1 Q i,t ) } {{ } equity returns + rt h (H i,t h own i,t h rent i,t ) } {{ } housing rental income + rtq l } {{ i,t } max(0, k s ( Q i,t )). } {{ } dividends shorting costs Aggregate Demand and Beliefs. Aggregate demand to live in the city equals the number of residents N t. This aggregate demand consists of a shock and a trend: log N } {{ } t = z t + log N }{{} t. } {{ } demand shock trend The trend component grows at a constant positive rate g: for all t > 0, log N t = g + log N t 1. The shocks z t have a common factor x. The dependence of the time-t shock on the common factor x is µ t, so that z t = µ t x. Without loss of generality, µ 0 = 1: the time 0 shock z 0 equals the common factor x. We denote µ = {µ t } t 0. At time 0, residents observe the following information: the current and future values of trend demand N t, the trend growth rate g, the current demand N 0, the current shock z 0, and the common factor x of the future shocks. They do not observe µ, the data needed to extrapolate the factor x to future shocks. Residents learn the true value of the entire vector µ at time t = 1. The resolution of uncertainty at time t = 1 is common knowledge at t = 0. Residents agree to disagree about the true value of µ. At time 0, resident i s subjective prior of µ is given by F i, an integrable probability measure on the compact space M of all possible values of µ. These priors vary across residents, and knowing the priors of other residents does not lead to any Bayesian updating. This agree-to-disagree assumption rules 9

10 out any inference from prices at t = 0. Therefore, for instance, an investor who ends up holding the land can realize he must be the most optimistic investor, but this realization fails to move his posterior away from his prior. As argued by Morris (1996), this heterogeneous prior assumption is most appropriate when investors face an unusual, unexpected situation like the arrival of the shock we are studying. Examples in the housing market include the settlement of new cities (like Chicago in the 1830s) or the discovery of new resources (like the Texas oil boom of the 1970s). In the case of the U.S. housing boom between 2000 and 2006, we follow Mian and Sufi (2009) in thinking of the shock as the arrival of new securitization technologies that expanded credit to low-income borrowers, although an equally valid interpretation would be demographic shifts leading to a secular increase in housing demand. The shock to housing demand between 2000 and 2006 is x, and µ represents the degree to which this shock persists after Even economists disagreed about µ during the boom, as Gerardi, Foote and Willen (2010) demonstrate. The resulting subjective expected value of each µ t is µ i,t = µ M tdf i, and the vector of resident i s subjective expected values of each µ t is µ i = {µ i,t } t 0. The subjective expected value µ i uniquely determines the prior F i. The distribution of µ i itself across residents admits an integrable probability distribution F µ on M, which is independent from the distribution F a of owner-occupancy preferences. The CEOs of the development firms are city residents, so their beliefs are drawn from the same distribution F µ. Equilibrium. Equilibrium consists of time-series vectors of prices p L (µ), p H (µ), r l (µ), r h (µ) and quantities L(µ), H(µ) that depend on the realized value of µ. These pricing and quantity functions constitute an equilibrium when housing, land, and equity markets clear while residents and developers maximize utility and profits: Consider pricing functions p h (µ), p l (µ), r h (µ), r l (µ) and quantity functions H(µ), L(µ). Let Hi,t, Q i,t, (h own i,t ), and (h rent i,t ) be resident i s solutions to the Bellman equation (3) given his owner-occupancy preference a i, his beliefs µ i, and these pricing functions. Let L j,t be developer j s land holdings that maximize expected net present value of profits in equation (1), given the pricing functions; L t is the sum of these land holdings across developers. The pricing and quantity functions constitute a recursive competitive equilibrium if at each t: 1. The sum of undeveloped land and housing equals the city s endowment of open space: S = L t (µ) + H t (µ). 2. Flow demand for land equals investment demand from developers, which equals the 10

11 resident demand for their equity: L t (µ) = L t = D l (r l t(µ)) = 3. Resident stock and flow demand for housing clear: H t (µ) = N t (µ) 0 M H i,tdf µ df a = N t (µ) 4. Construction maximizes developer profits: 5. Developer profit from homebuilding is zero: Elasticity of Housing Supply. 0 H t (µ) H t 1 (µ) arg max πt hb. H t 0 max πt hb = 0. H t M M Q i,tdf µ df a. ((h own i,t ) + (h rent i,t ) )df µ df a. The housing supply curve is the city s open space S less the rental demand for land D l (r l t). We denote the elasticity of this supply curve with respect to housing rents rt h by ɛ S t. The supply elasticity determines the construction response to the shocks {z t }. It will also serve as a sufficient statistic for the extent to which land speculation affects house prices. This section describes the supply elasticity ɛ S t growth path, which occurs when x = 0. along the city s trend The relationship between land rents rt l and house rents rt h allows us to calculate this elasticity. Because trend growth g > 0, new residents perpetually enter the city, and developers build new houses each period. Perpetual construction ties together land and house prices. In particular, as developers must be indifferent between building today or tomorrow, house rents equal land rents plus flow construction costs: r h t = r l t + (1 β)k. The supply of housing is open space net of flow land demand: S D l (r h t (1 β)k). The elasticity of housing supply is thus ɛ S t rt h (D l ) /(S D l ). When the flow land demand D l features a constant elasticity ɛ l, the elasticity of housing supply takes on the simple form ɛ S t = ( ) rt h S 1 ɛ l, (4) rt h (1 β)k H t where H t is the housing stock at time t. The arrival of new residents increases both rents r h t 11

12 and the level of development H t /S. The supply elasticity given in equation (4) unambiguously falls (see Appendix for proof): Lemma 1. Define housing supply to be the residual of the city s open space S minus the flow demand for land: S D l. The elasticity ɛ S t of housing supply with respect to housing rents rt h decreases with the level of city development H t /S along the city s trend growth path. 2 Supply-Side Speculation At time 0, residents disagree about the future path of housing demand. Speculative trading behavior results from this disagreement. This section describes how owner-occupancy frictions crowd speculators out of owner-occupied housing and into rental housing and land. Demand and supply elasticities determine how prices aggregate the beliefs of owner-occupants and of optimistic speculators. 2.1 Land Speculation and Dispersed Homeownership We first consider the developer decision to hold land at time 0. Developer j s first-order condition on its land-holding L j,0 is 1/β E }{{} j p l 1/(p l 0 r0) l, } {{ } risk-free rate expected land return with equality if and only if L j,0 > 0. A developer invests in land if and only if it expects land to return the risk-free rate. At time 0, developers disagree about this expected return on land because they disagree about the future path of housing demand. The developers that expect the highest returns invest in land, while all other developers sell to these optimistic firms and exit the market. We denote the optimistic belief of the developers who invest in land by Ẽpl 1 max µj E(p l 1 µ j ). Optimistic residents finance developer investments in land through purchasing their equity. Less optimistic residents choose to short-sell developer stock. Developer stock allows residents to hold land indirectly: its price is p l 0 and it pays a dividend of r l 0. Resident i holds this equity only if he agrees with the land valuation of the optimistic developers, in which case E i p l 1 = Ẽpl 1. Otherwise, he shorts the equity, and his first-order condition is k s( Q i,0) = β(ẽpl 1 E i p l 1). Disagreement increases the short interest in the equity of the developers holding the land. Without disagreement, Ẽpl 1 = E i p l 1 for all residents, so no one shorts. 12

13 Only the most optimistic residents hold housing as landlords. A resident is a landlord if he owns more housing than he consumes through owner-occupancy: H i > h own i. The firstorder condition of the Bellman equation (3) with respect to H i,0 when it is in excess of h own i,0 is 1/β E }{{} i p h 1/(p h 0 r0) h, (5) } {{ } risk-free rate expected housing return with equality if and only if H i,0 > h own i,0. Only the most optimistic residents invest in rental housing, just as only the most optimistic developers invest in land. Land and rental housing share this fundamental property. During periods of uncertainty, the most optimistic investors are the sole holders of these asset classes. During the housing boom of the early 2000s, landlord investment constituted a significant component of real estate purchases. Haughwout et al. (2011) show that the fraction of new mortgages to borrowers with at least two rose from 20% to 35% between 2000 and 2006, and Chinco and Mayer (2014) show that 18.4% of single-family home purchases between 2000 and 2007 were to buyers whose property tax bill mailing address differed from the property address. Owner-occupancy utility crowds these optimistic investors out of owner-occupied housing, which remains dispersed among residents of all beliefs. The decision to own or rent emerges from the first-order conditions of the Bellman equation (3) with respect to h own i,0 and h rent i,0. We express these equations jointly as v (a i (h own i,0 ) + (h rent i,0 ) ) = min a 1 i (p h 0 βe i p h } {{ } 1), r0 h } {{ } marginal utility of housing owning }{{} renting. (6) The left term in the parentheses denotes the expected flow price of marginal utility v from owning a house; the right term denotes the flow price of renting. A resident owns when the owner-occupancy price is less than the rental price. As long as the owner-occupancy preference a i is large enough, resident i decides to own even if his belief E i p h 1 is quite pessimistic. Homeownership remains dispersed among residents of all beliefs. This point relates to the work of Cheng, Raina and Xiong (2014), who find that securitized finance managers did not sell off their personal housing assets during the boom. They interpret this result as evidence that these managers had the same beliefs as the rest of the market about future house prices. An alternative interpretation, made clear by equation (6), is that the managers did doubt the market valuations, but continued to own housing due to the ownership utility a i. We gain additional intuition about the own-rent margin by substituting equation (5) into 13

14 equation (6). We denote the most optimistic belief about future house prices, the one held by landlords investing in rental housing, by Ẽph 1 max µi E(p h 1 µ i ). The decision to own rather than rent reduces to a i 1 + β(ẽph 1 E i p h 1). (7) r0 h Without disagreement, a resident owns exactly when he intrinsically prefers owning to renting, so that a i 1. Disagreement sets the bar higher. Some pessimists for whom a i 1 choose to rent because they expect capital losses on owning a home. Other pessimists continue to own because their owner-occupancy utility is high enough to offset the fear of capital losses. Proposition 1 summarizes these results. Proposition 1. Owner-occupancy utility crowds speculators out of the owner-occupied housing market and into the land and rental markets. The most optimistic residents those holding the highest value of E i p h 1 buy up all rental housing and finance optimistic developers who purchase all the land. In contrast, owner-occupied housing remains dispersed among residents of all beliefs. Proposition 1 yields two empirical predictions about developer behavior during a boom. The most optimistic developers buy up all the land. Unless they start out owning all the land, these optimistic developers increase their land positions following the demand shock. They hold this land as an investment rather than for immediate construction. The second prediction concerns short-selling. Residents who disagree with the optimistic valuations of developers short their equity. Prediction 1. Optimistic developers accumulate land in excess of their immediate construction needs. Prediction 2. Disagreement increases the short interest of developer equity. The idea that real estate speculation transpires primarily in land departs from the literature, which has focused mostly on investors in houses. 6 In Section 4, we test these predictions by examining the balance sheets and market equity of U.S. public homebuilders during the boom and bust of the early 2000s. 2.2 Belief Aggregation Prices aggregate the heterogeneous beliefs of residents and developers holding housing and land. The real estate market consists of three components: land, rental housing, and owneroccupied housing. The most optimistic residents hold the first two, while the third remains 6 See, for example, Barlevy and Fisher (2011), Haughwout et al. (2011), Chinco and Mayer (2014), and Bayer et al. (2015). 14

15 dispersed among owner-occupants. House prices reflect a weighted average of the optimistic belief and the average belief of all owner-occupants. The weight on the optimistic belief is the share of the real estate market consisting of land and rental housing; the weight on the average owner-occupant belief is owner-occupied housing s share of the market. To derive these results, we take a comparative static of the form p h 0/ x. The shock z = µx scales with the common factor x. We differentiate with respect to x at x = 0 to explore how prices change as the shocks, and hence the ensuing disagreement, increase. Our partial derivative holds current demand N 0 constant to isolate the aggregation of future beliefs. There are both costs and benefits to this first-order approach. The benefit is it allows us to derive simple, intuitive formulas in terms of supply and demand elasticities. A long literature in public economics and price theory uses perturbation arguments for this reason (Chetty, 2009). The cost is that we abstract away from the effects of a large shock on the housing market. For instance, a large enough shock would lead optimistic investors to displace nearly all owner-occupants from their homes. To us, the clarity of the first-order approach outweighs what is lost by not considering the effects of a large shock more explicitly. We first use equation (5) to write p h 0 = r h 0 + βẽph 1. The shock increases the optimistic belief βẽph 1, directly increasing prices. It also changes the market rent r h 0. This rent is determined by the intersection of housing supply and housing demand: S D ( l r0 h (1 β)k ) = D } {{ } 0(r h 0) h, (8) } {{ } housing supply housing demand where D h 0(r h 0) = N 0 1+β( Ẽp h 1 E ip h 1 )/rh 0 (v ) 1 (r h 0)dF a df µ M 0 } {{ } rental housing ( ) + N 0 a 1 M 1+β(Ẽph 1 E i (v ) 1 a 1 i (r0 h + β(ẽph 1 E i p h 1)) df a df µ. ip h 1 )/rh 0 } {{ } owner-occupied housing (9) The housing demand equation follows from equation (6) and equation (7). We determine the shock s effect on rents by totally differentiating equation (8) with respect to x at x = 0, keeping current demand N 0 constant. When the elasticity of housing demand ɛ D is constant, the resulting comparative static p h 0/ x adopts the simple form given in the following proposition, which we prove in the Appendix. Proposition 2. Consider the partial effect of the shock in which current demand N 0 stays 15

16 constant but future house price expectations E i p h 1 change. The change in house prices averages the changes in the optimistic resident belief and the average belief: p h 0 x = ɛs 0 + (1 χ)ɛ D βẽph 1 ɛ S 0 + ɛ D x + χɛd ɛ S 0 + ɛ D βep h 1 x, (10) where Ẽph 1 = max i E i p h 1 is the most optimistic belief, Ep h 1 = M E ip h 1dF µ is the average belief, ɛ S 0 is the elasticity of housing supply at time 0, ɛ D is the elasticity of housing demand, and χ = 0 (hown i,0 ) df a /H 0 is the share of housing that is owner-occupied when x = 0. The weight on the optimistic belief in Proposition 2 represents the share, on the margin, of the real estate market owned by speculators. The supply elasticity ɛ S 0 represents land, and (1 χ)ɛ D represents rental housing. The remaining χɛ D represents owner-occupied housing and is the weight on the average owner-occupant belief. The average owner-occupant belief coincides with the unconditional average belief because at x = 0, beliefs and tenure choice are independent. Proposition 2 yields four corollaries on the difference in belief aggregation across cities and neighborhoods. Prices look more optimistic when the weight (ɛ S 0 + (1 χ)ɛ D )/(ɛ S 0 + ɛ D ) is higher. This ratio is greater when the supply elasticity ɛ S 0 is higher: Corollary 2.1. Prices look more optimistic when the housing supply elasticity is higher, i.e. in less developed cities. Disagreement reverses the common intuition relating housing supply elasticity and movements in house prices. Elastic supply keeps prices low by allowing construction to respond to demand shocks. But land constitutes a larger share of the real estate market when supply is elastic. Speculators are drawn to the land markets, so elastic supply amplifies the role of speculators in determining prices during periods of disagreement. When supply is perfectly elastic, ɛ S 0 = and prices reflect only the beliefs of these optimistic speculators: Corollary 2.2. When housing supply is perfectly elastic, house prices incorporate only the most optimistic beliefs; they reflect the beliefs of developers and not of owner-occupants. Recent research has measured owner-occupant beliefs about the future evolution of house prices. 7 In cities with elastic housing supply, such as the cities motivating this paper, developer rather than owner-occupant beliefs determine prices. Data on the expectations of homebuilders would supplement the research on owner-occupant beliefs to explain prices in these elastic areas. 7 See Landvoigt (2014), Case, Shiller and Thompson (2012), Burnside, Eichenbaum and Rebelo (2014), Soo (2013), Suher (2014), and Cheng, Raina and Xiong (2014). 16

17 Prices aggregate beliefs much better when housing supply is perfectly inelastic (ɛ S 0 = 0) and all housing is owner-occupied (χ = 1). In this case, the price change depends only on the average belief Ep h 1: Corollary 2.3. When the housing stock is fixed and all housing is owner-occupied, prices reflect the average belief about long-run growth. In many settings, such as when investor information equals a signal plus mean zero noise, prices reflect all information when they incorporate the average private belief of all investors. Owner-occupied housing markets with a fixed housing stock display this property, even though short-selling is impossible and residents persistently disagree. These frictions fail to bias prices because homeownership remains dispersed among residents of all beliefs, due to the utility flows that residents derive from housing. The weight (ɛ S 0 + (1 χ)ɛ D )/(ɛ S 0 + ɛ D ) on optimistic beliefs is also higher when χ is lower: Corollary 2.4. Prices look more optimistic when a greater share of housing is rented. Speculators own a greater share of the real estate market when the rental share 1 χ is higher. Prices bias towards optimistic beliefs in market segments where more of the housing stock is rented. 3 The Cross-Section of City Experiences During the Boom We derive a formula for the total effect of the shock z on house prices. This formula expresses the house price boom as a function of the city s level of development when the shock occurs. Our analysis up to this point has explored the partial effect of how prices aggregate beliefs E i p h 1, without specifying how these beliefs are formed. To derive the total effect of the shock, we express the changes in these beliefs in terms of city characteristics and the exogenous demand process. Specifically, we calculate the partial derivative log p h 0/ x holding all beliefs fixed at µ i = µ, and then use Proposition 2 to derive the total effect of the shock x on house prices. As before, we evaluate derivatives at x = 0. 8 At time 0, each resident expects the shock z t to raise log-demand at time t by µ t x. The resulting expected change in rents rt h depends on the elasticities of supply and demand at time t: log E 0 rt h = µ t x ɛ S t + ɛ. D 8 Evaluating derivatives at x = 0 describes the model when construction occurs in each period. When x is large enough and the shock z might mean-revert, a construction stop at t = 1 is possible and anticipated by residents at t = 0. This feature of housing cycles distracts from our focus on housing booms and how they vary across cities. 17

18 This equation follows from price theory. When a demand curve shifts up, a good s price increases by the inverse of the total elasticity of supply and demand. The total effects of the shocks {z t } on the current house price p h 0 follows from aggregating the above equation across all time periods, using the relation p 0 = E 0 t=0 βt rt h : log p h 0 x = µ ɛ S + ɛ D. (11) The mean persistence of the shock is µ = t=0 µ tβ t r h t (ɛ S t +ɛ D ) 1 / t=0 βt r h t (ɛ S t +ɛ D ) 1, and ɛ S is the long-run supply elasticity given by the weighted harmonic mean of future supply elasticities in the city: ɛ S ɛ D + t=0 β tr h t t=0 βt r h t (ɛ S t + ɛ D ) 1. The higher this long-run supply elasticity, the smaller the shock s impact on current house prices, holding µ fixed. We now put together the two channels through which the shock changes prices. Equation (11) expresses the price change that results when µ is known, and equation (10) describes how prices aggregate residents heterogeneous beliefs about µ. Proposition 3 states the total effect d log p h 0/dx, which we formally calculate in the Appendix. Proposition 3. The total effect of the shock x on current house prices is d log p h 0 dx ( ) ɛ S = 0 + (1 χ)ɛ D µ + χɛd ɛ S 0 + ɛ D ɛ S 0 + ɛ µ } {{ D } aggregate belief 1 } ɛ S {{ + ɛ D } pass-through, (12) where ɛ S 0 is the current elasticity of housing supply, ɛ S is the long-run supply elasticity, ɛ D is the elasticity of housing demand, χ is the share of housing that is owner-occupied, µ is the mean persistence of the most optimistic belief about µ, and µ is the mean persistence of the average belief. Equation (12) is the key result of this paper. This equation expresses the size of a house price boom in terms of observable city characteristics like supply and demand elasticities for housing. By using different values for the parameters, we now predict which types of cities experience the sharpest house price increases after a demand shock. We use these predictions in Section 4 to explain the variation in price growth across cities between 2000 and To start, equation (12) demonstrates how a city with perfectly elastic housing supply can experience a house price boom. Housing supply is perfectly elastic when ɛ S 0 =. In this case, the house price boom is µx/( ɛ S + ɛ D ). This price increase is positive as long as the long-run supply elasticity ɛ S is not also infinite. 18

19 Prediction 3. A house price boom occurs in a city where current housing supply is completely elastic, construction costs are constant, and construction is instantaneous. Supply must be inelastic in the future for such a price boom to occur. In the Appendix, we prove that a limiting case exists in which ɛ S 0 = while ɛ S <. A house price boom results from a shock to current demand accompanied by news of future shocks. When supply is inelastic in the long run, these future shocks raise future rents, and prices rise today to reflect this fact. This price change occurs even if supply is perfectly elastic today, because residents anticipate the near future in which supply will not be able to adjust as easily. This supply condition elastic short-run supply, inelastic long-run supply occurs in cities at an intermediate level of development. Figure 2(a) demonstrates the possible combinations of short-run and long-run supply elasticities in a city. We plot the pass-through 1/(ɛ S + ɛ D ); a higher pass-through corresponds to a lower elasticity. Lightly developed cities have highly elastic short-run and long-run supply, and heavily developed cities have inelastic short-run and long-run supply. In the intermediate case, current supply is elastic while long-run supply is inelastic, reflecting the near future of constrained supply. Disagreement amplifies the house price boom the most in exactly these nearly developed elastic cities. The amplification effect of disagreement equals the extent to which optimists bias the price increase given in equation (12). When owner-occupancy frictions are present (χ = 1), the difference between the price boom under disagreement and under the counterfactual in which all residents hold the average belief µ is ɛ S 0 µ µ ɛ S 0 + ɛ D ɛ S + ɛ. D This amplification is largest in nearly developed elastic cities, where ɛ S 0 is large and ɛ S is small. Because this amplification increases in ɛ S 0 and decreases in ɛ S, nearly developed elastic cities provide the ideal condition for disagreement to amplify a house price boom. Figure 2(b) plots the house price boom given by equation (12) across different levels of city development, for both the case of disagreement and the case in which all residents hold the average belief. The amplification effect of disagreement is the difference between the two curves. Optimistic speculators amplify the price boom the most in the intermediate city. Highly elastic short-run supply facilitates speculation in land markets, biasing prices towards their optimistic belief. This bias significantly increases house prices because housing supply is constrained in the near future. The optimism bias is smaller in the highly developed city. As a result, price increases in intermediate cities are as large as the price boom in the highly developed areas. In fact, the price boom in some intermediate cities can exceed that in the highly developed 19

20 FIGURE 2 Comparative Statics with Respect to Level of Development a) Supply Elasticity 1.8 Short Run Long Run Pass Through of Demand Shocks to Prices 0 0 Low Intermediate High 125 Developed Years of Development Developed Developed b) Price Increase 7% With Disagreement Without Disagreement Annualized Price Increase 4% 0% 0 Low Intermediate High 125 Years of Development Developed Developed Developed Notes: The parameters we use are µ = 1, µ = 0.2, x = 0.06, g = 0.013, ɛ D = 1, β = , and ɛ l = 1. We hold the amount of space S fixed and vary the initial trend demand N 0. The x-axis reports annualized trend demand given by log N 0 /g. (a) Short-run pass-through is 1/(ɛ S 0 + ɛ D ); long-run pass-through is 1/( ɛ S + ɛ D ). We calculate the rent and housing stock at each level of development using equation (A1) in the Appendix, and then calculate the supply elasticities using equation (4). (b) Each curve reports the derivative in equation (12) times x, which we calculate using the elasticities shown in Panel (a). The without disagreement counterfactual uses µ = µ = 0.2 instead of µ = 1 > µ =

21 cities. The parameters we use in Figure 2(b) generate an example of this phenomenon. This surprising result reverses the conclusion of standard models of housing cycles, in which the most constrained areas always experience the largest price increases. This reversal occurs as long as owner-occupancy frictions are high and the extent of disagreement is sufficiently large: Prediction 4. If disagreement and owner-occupancy frictions are large enough, then the largest house price boom occurs in a city at an intermediate level of development. There exists χ < 1 and δ > 0 such that for χ χ 1 and µ µ δ, the price boom d log p h 0/dx is strictly largest at an intermediate level of development N 0 <. The next prediction of equation (12) concerns why house price booms occur in some elastic cities but not in others. Elastic cities are those for which ɛ S 0. As shown in Figure 2(a), these cities differ in their long-run supply elasticity ɛ S. When ɛ S =, the house price boom d log p h 0/dx = 0. Prices remain flat because construction can freely respond to demand shocks now and for the foreseeable future. House prices increase in elastic cities only when the development constraint will make construction difficult in the near future: Prediction 5. Consider two cities that experience the same demand shock and in which current housing supply is perfectly elastic (ɛ S 0 = ). House prices rise more in the city in which the long-run supply elasticity ɛ S is lower. Empirically, elastic cities could be distinguished based on the level of house prices and the extent of development before the shock. Because house prices increase with development, elastic cities nearing their development constraints should have higher house prices than other elastic cities. Finally, equation (12) predicts variation in house price increases within a city. A city s housing market consists of a number of market segments, which are subsets of the housing market that attract distinct populations of residents. Because they attract distinct populations, we can analyze them using equation (12), which was formulated at the city-level. All else equal, housing submarkets in which χ is higher experience smaller house price booms. Recall that χ is the share of the housing stock that is owner-occupied rather than rented when x = 0. It is a sufficient statistic for the distribution F a of owner-occupancy utility. When χ is larger, the price increase d log p h 0/dx is smaller: d log p h 0 χ dx = ɛd ɛ S 0 + ɛ D µ µ ɛ S + ɛ D < 0. This derivative is negative because the optimistic belief µ exceeds the average belief µ. We summarize this result in our final prediction: 21

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