Essay

Risk in Housing Markets: An Equilibrium Approach⇤
Aurel Hizmo† NYU Stern January 30, 2012

Abstract Homeowners are overexposed to city-speciﬁc house price risk and income risks, which may be very di cult to insure against using standard ﬁnancial instruments. This paper develops a micro-founded equilibrium model that transparently shows how this local uninsurable risk a↵ects individual location decisions and portfolio choices, and ultimately how it a↵ects prices in equilibrium. I estimate a version of this model using house price and wage data and provide estimates for risk premia for di↵erent cities, which imply that homes are on average about $20000 cheaper than they would be if owners were risk-neutral. This estimate is over $100000 for volatile coastal cities. Next, I simulate the model to study the e↵ects of ﬁnancial innovation on equilibrium outcomes. Creating assets that hedge city-speciﬁc risks increases house prices by about 20% and productivity by about 10%. The average willingness to pay for completing the market per homeowner is between $10000 and $20000. Welfare gains come both from better risk-sharing and from more e cient sorting of households across cities.

⇤ I am deeply grateful to Patrick Bayer, Andrew Patton, and Peter Arcidiacono for their encouragement and support. I also thank Robert McMillan, Tim Bollerslev, Vish Viswanathan, Chris Timmins, Jimmy Roberts and the seminar participants at Duke Finance, the ERID Conference at Duke, Fed Board, NYU Stern, Penn State, Philadelphia Fed, University of Chicago, UPenn Wharton, USC Marshall, and Washington University in St. Louis for their helpful comments. † Department of Finance, Stern School of Business, New York University, 44 W. 4th Street, Suite 9-74, New York, NY 10012. Email: ahizmo@stern.nyu.edu.

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Introduction

Throughout their lifetimes, households face many long-term economic risks, most of which may be di cult to insure against.1 Social safety nets and insurance mechanisms are particularly limited for long-term income and house price risk. The inability to e ciently spread risk across many households has direct adverse welfare e↵ects since individual households may bear most of the burden of negative income and housing wealth shocks. Also, because of risk-aversion, many talented individuals may forgo brilliant careers and investment opportunities that are deemed too risky to undertake. E↵ective risk sharing can therefore limit downside risk and also enable individuals to make better choices, which leads to higher productivity and welfare in the economy. Such considerations have motivated Shiller (1993, 2003) and others to call for the creation of ﬁnancial instruments to enable widespread sharing of risks that are not directly traded in equity markets. Risks management considerations are especially important when a young household chooses a combined labor and housing market. Since most homeowners live near their workplace, they are exposed to a considerable amount of location-speciﬁc income and house price risk. From a portfolio management point of view, households are very much concerned about location-speciﬁc risk since a disproportionate share of their wealth is invested in one particular house, and they cannot reoptimize very frequently due to large reallocation costs.2 To illustrate the house price volatility homeowners face Figure 1 shows the real house price index for a set of 20 major US cities. Although the US national house prices have historically been fairly steady, this is not the case for many cities in the US: individual cities are very volatile.3 Since much of the cross-sectional volatility is idiosyncratic, standard portfolio theory implies that households would ideally hold a diversiﬁed portfolio of homes rather than only owning in one particular city. The under-diversiﬁcation problem is exacerbated even more by the fact that individual homeowners cannot easily hedge local income and house price by using standard ﬁnancial instruments.4 The riskiness of a particular location can
1 Many studies test and strongly reject the hypothesis that people share risks e↵ectively. Some examples include Zeldes (1989), Cochrane (1991), Hayashi, Altonji and Kotliko↵ (1996), Athanasoulis and van Wincoop (2000). 2 Tracy, Schneider, and Chan (1999) report housing wealth comprises about two-thirds of the typical households portfolio. 3 While there is little short run volatility in housing markets, the focus here is on long run volatility since homes are traded infrequently. 4 The correlation between the stock market and average labor income is usually estimated to be close to zero. See for example Cocco, Gomes, and Maenhout (2005). Campbell and Viceira (2002) do ﬁnd a positive correlation of income with lagged stock returns that is as high as .5 for college graduates. The correlation between housing returns and the aggregate stock market or REITs is also found to be very low (see Flavin and Yamashita (2002), and Hinkelmann and Swindler (2006)). Hinkelmann and Swindler (2006) also ﬁnd very little correlation between prices of futures contracts (not including the new housing futures) and house price returns, stressing that the creation of housing futures should be very beneﬁcial for hedging purposes. Constructing local stock price indexes for largest

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FHFA House Price Index (1995=100) 260 240 220 200 House Price Index 180 160 140 120 100 80 60 1975 1980 1985 1990 Year 1995 2000 2005 2010
USA Atlanta Baltimore Chicago Cleveland Dallas Denver Detroit Houston LosAngeles Miami Minneapolis NewYork Oakland Philadelphia Phoenix Pittsburgh SanDiego Seattle StLouis Tampa Washington

Figure 1: The Real House Price Index for Major US Cities therefore simultaneously a↵ect households’ location and portfolio choice decisions and ultimately housing prices. In turn, ﬁnancial innovation that allows households to better control their exposure to these risks may have a major impact on labor market choice, home prices and welfare. The goal of this paper is to show both theoretically and empirically how exposure to uninsurable city-speciﬁc risks a↵ects house prices, as well as household location and portfolio decisions. Understanding the nature and magnitude of these e↵ects is crucial in evaluating the beneﬁts of creating new ﬁnancial instruments that facilitate risk sharing, such as those proposed by Shiller (1993, 2003) . To that end, I ﬁrst develop an equilibrium theory that shows in a transparent way how location-speciﬁc risk is capitalized into house prices and how it a↵ects productivity and welfare in the economy. The model is estimated using wage and price data for individual metropolitan areas in the US, providing empirical support for the idea that risk is priced in housing markets. I then simulate the model to study the beneﬁts from creating ﬁnancial assets that correlate with city-speciﬁc house prices and income. The backbone of the paper consists of a novel micro-founded dynamic equilibrium model that merges standard methodologies used in urban economics, which study sorting and spatial properties employers from 16 cities in California, Hizmo (2010) reports a correlation of .36 between the local house price returns and the local stock price index.

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of the problem, with models from continuous-time ﬁnance that study ﬁnancial assets. To my knowledge, this is the ﬁrst ﬂexibly estimable model that simultaneously considers risk-aversion, multiple sources of uncertainty, rich agent heterogeneity, sorting, portfolio choice and asset prices in one uniﬁed framework. The main features in the model that drive most of the interesting results are that markets are incomplete, in that there is no perfect hedging of risks, houses are indivisible, and agents are heterogeneous in terms of their productivity and risk preferences. The general setting of the model is very similar to that in Ortalo-Magne and Prat (2010) (OMP hereafter).5 Each instant, a generation of households are born and live for T periods. Households initially choose a single labor and housing market from a system of cities. Once households choose a city to live in, they are assumed to live there until the end of their life. Unlike the main speciﬁcations of OMP, I model houses as indivisible assets. Each household must buy one house in order to live in a city, which gives them access to local wages and amenities.6 Households also have access to a risk-free asset and the stock market, and can adjust their portfolios continuously. At the end of their lives, households sell their homes and their ﬁnancial assets, and consume their terminal wealth. Because they cannot hold shares of homes in di↵erent cities, households have to resort to using stocks to insure against their income and house price risks.7 Because stocks may not be correlated with every dimension of risk that households face, households may face city-speciﬁc uninsurable risks. The solution to the household problem involves two steps. Conditional on location choice, households decide how to invest their wealth in the ﬁnancial market given their local income and house price process. Taking into account the utility derived from these optimal portfolio decisions, households choose the city that provides them with the highest expected utility. The portfolio choice
OMP is the ﬁrst paper to theoretically show the links between location and portfolio choice in an equilibrium model. Building on their elegant work, I extend the theory in number of important dimensions. Since the main focus in OMP is not empirical, income in di↵erent cities follows a random walk (in discrete time) with no drift and so do stock prices. I consider city-speciﬁc income processes that are richer and match empirical patterns of wage processes. Here, the growth in wages is modeled as a city-speciﬁc mean reverting process. Also, because of the continuous time setting, this model can handle stock prices that follow geometric Brownian motions with drift, which is in line with the modern continuous-time ﬁnance literature. In addition, this model also includes amenities, heterogeneity in preferences for local amenities, and heterogeneity in risk-aversion. In terms of the equilibrium, OMP conjecture a price process and show the existence of an equilibrium, while saying relatively little about about uniqueness. This paper proves that a linear stationary equilibrium is unique in its class, and does not rely on the existence of “hyper-marginal” households. 6 The assumption that everyone owns is made for tractability purposes. Focusing on the owner-occupied rather than the rental market and could be justiﬁed by the observation that the homeownership rate in the US is around 70%. 7 Since agents do not migrate to di↵erent cities every period they are concerned only about long-term house price risk, or the change in house prices from the time when they bought the home to the time they will sell it before they die.
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problem with exogenous income that households face in this model is studied extensively in the literature. Generally the solutions are either found numerically and/or under the assumption that markets are complete.8 Using transformations of the Hamilton-Jacobi-Bellman equation similar to Henderson (2005), I derive a closed-form solution of the portfolio choice with housing, exogenous income and incomplete markets. The explicit solution is crucial here since it is used to solve for the equilibrium allocation of individuals across space. Given optimal portfolio decisions, I then solve for the spatial allocation of households. The equilibrium concept used here follows in the tradition of classic urban models of Rosen (1979) and Roback (1982) where spatial equilibrium is found by equalizing utility di↵erences across cities. The more modern version of these models is the static horizontal sorting model of Bayer, McMillan, and Rueben (2005) or Bayer and Timmins (2005). Under a static setting, these models prove existence and uniqueness of a spatial equilibrium under very general assumptions about household heterogeneity. I make direct use of these horizontal sorting results in my model.9 Merging closed-form results from the optimal portfolio choice problem with a horizontal sorting model of individuals across space, I prove the existence and uniqueness of a linear stationary equilibrium. One key theoretical result is that home prices are derived to be a closed-form function of the underlying productivity of the economic base of a city minus a city-speciﬁc premium, which is a function of agent heterogeneity, sorting and risks in the particular city. Home prices can also be interpreted as the expected discounted sum of future dividends of the marginal household in that city plus a risk premium. In equilibrium, risk premia of homes takes a linear factor structure where the tradable part of risk is priced at the market price of risk. The non-insurable part of the local risk is also priced in equilibrium but its price is the risk-aversion parameter of the marginal person who lives there. In general, cities exposed to higher amounts of non-insurable risks have lower home prices in order to compensate residents for the extra risk they are taking. Financial asset portfolio decisions are also found to be a generalized version of classic results in portfolio choice in ﬁnance. Stock holdings consist of a classic myopic term as found in Merton (1969), and a hedging demand term that depends on the correlation of the particular stock with
Examples that use numerical methods to solve the portfolio choice problem with housing and exogenous income include Cocco (2004), Yao and Zhang (2005), and Van Hemert (2009). Because these models focus on numerical solutions to single agent problems, they allow for general income processes and utility function, as well as own/rent decisions. Kraft and Munk (2010) solve the optimal portfolio choice in closed-form but they assume that all individuals are renters and markets are complete. 9 In terms of dynamic settings, the closest models are Glaeser and Gyourko (2010) and Van Nieuwerburgh and Weill (2010). These models are designed to study cross-sectional and time-series properties of house prices and wages. This paper di↵ers from these models as it includes risk-aversion, portfolio choice and asset markets in the typical location choice equilibrium model.
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income in the city a household lives in. The equilibrium rate of return for a particular stock also depends on its correlation with income in all of the cities and the distribution of the risk-aversion parameter over the population.10 Stocks that are negatively correlated with income and house prices are in large demand for hedging purposes. In order for the stock market to clear the returns for these stocks are lower in equilibrium. An interesting insight to emerge is that households sort not only on income and amenities, but also on the uninsurable local risk. Households that are more risk-averse are more likely to locate in cities with less uninsurable risk. Sorting on risk leads in turn to misallocation of human capital. Risk-averse households do not move to the labor market where they are most productive because they may ﬁnd that city too risky. Instead, some other less risk-averse and less productive household will move to the risky city in question. The creation of ﬁnancial instruments that can be used as a hedge against noninsurable local risk can therefore reduce the incentives to sort on risk and increase productivity and welfare in the economy. In the limiting case where all risk is tradable, households do not sort on risk and they locate in the labor market where their productivity is highest. I estimate a version of the model using wage and house price data for 216 metropolitan areas in the US under the assumption that agents are homogeneous.11 Instead of looking for all the underlying factors that drive income and house prices in the US I focus on three factors that capture a large share of the common variation across metropolitan areas. These factors are constructed in the spirit of the three Fama-French factors. The ﬁrst factor (HMKT) is the average price growth across metropolitan areas. The second (SMBH) is price growth in high-price metropolitan areas minus growth in low-price ones. The third (HMLH) is growth in metropolitan areas with high price-to-wage ratios minus growth in ones with low price-to-wage ratios. The national HMKT factor turns out to have a correlation of .6 with aggregate REIT returns, while the other two factors are uncorrelated with REITs or the three Fama-French factors. This suggests that the national factor may be spanned by traded assets while the other two most likely are not. I estimate the model under di↵erent assumptions about the factors’ tradability and I ﬁnd that areas with higher nontradable variance demand a house price risk premium, which is reﬂected in lower house prices.
Other studies have also found that the presence of nontradable income and housing wealth in the household’s portfolio can a↵ect asset prices. Examples that study the impact of housing decisions and prices on ﬁnancial asset prices include OMP, Piazzesi, Schneider, and Tuzel (2007), Lustig and van Nieuwerburgh (2005), and Yogo (2006). The interaction of labor income risk and asset prices is studied in Constantinides, Donaldson, and Mehra (2002), Santos and Veronesi (2006), and Storesletten, Telmer, and Yaron (2007). 11 The model could be fully estimated using individual migrations decisions data available from the Decennial Census or the PSID. Using individual migration decisions allows for direct estimation of heterogeneity in preferences and in productivity. In a companion paper, I estimate the full structural model using a two-stage procedure closely related to the popular random coe cient logit model of Berry, Levinsohn and Pakes (1995).
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The estimated risk premia imply that on average homes are about $20000 cheaper than they would be if owners were risk-neutral, although these estimates vary from $135000 for San Francisco, CA to -$20000 for Denver, CO. To my knowledge no other estimates of housing risk premia exist in the previous literature. Using the estimates for 216 US metropolitan areas, I then simulate the model to study the e↵ect of ﬁnancial innovation on house prices, household sorting across space and on overall productivity and welfare in the economy. I start from the baseline where only the national HMKT factor is spanned by traded stocks. I then consider the cases where new ﬁnancial instruments are created that correlate with the two other factors proposed above, and the case when the ﬁnancial instrument allows for perfect insurance. The creation of tradable ﬁnancial instruments that correlate with housing and income risk improves households’ ability to hedge risk and consequently lowers housing risk premia. Due to heterogeneity in risk preferences, this leads to a di↵erent sorting of households across space. In the new sorting equilibrium, human capital is allocated more e ciently, leading to higher overall productivity welfare. For a reasonable range of risk-aversion parameters, I ﬁnd that completing markets can increase home prices by about 20 percent, increase productivity in the economy by 10 percent, and significantly improve welfare.12 The average willingness to pay for access to ﬁnancial instruments that correlate with all of the sources of risk in the economy is between $10000 and $20000 per homeowner.13 The willingness to pay is, however, much higher for households that are very risk-averse, or households who live in locations with high noninsurable volatility. Overall, these ﬁndings do depend on the distribution of the risk-aversion parameter over the population. When heterogeneity in risk-aversion is large, productivity increases, but prices and welfare respond less to completing markets because of sorting e↵ects. When households are homogeneous, all of the beneﬁts of completing markets are directly reﬂected into higher house prices, and there is no gains in productivity. The rest of the paper is organized as follows: Section 2 develops a joint equilibrium theory of location and portfolio choice. In Section 3, I describe the estimation procedure and ﬁt the model to annual wage and house price data for 216 metropolitan areas in the US for the last 25 years.
Under the popular “standard incomplete markets” framework in macroeconomics, Heathcote, Storesletten and Violante (2008) also ﬁnd that insuring wage risk has large welfare and productivity implications. The gain in productivity is through a di↵erent channel in their model. Under market incompleteness the less productive workers work too much while high productivity agents work too little. Similar results are also found in Pijoan-Mas (2006). My paper proposes a di↵erent channel through which market completeness a↵ects productivity. Completing markets increases productivity because workers are matched with jobs more e ciently because of sorting e↵ects. 13 To my knowledge there exist no previous estimates in the literature for the willingness to pay for access to these new ﬁnancial instruments.
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Section 4, uses the model to conduct a series of general equilibrium counter-factual simulations designed to study the impact of creating new ﬁnancial instruments that allow individuals to hedge housing risk; and Section 5 concludes.

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The Model

The classical Capital Asset Pricing Model (CAPM) states that only variation related to the market factor should be priced in equilibrium. In contradiction with CAPM, Hizmo (2010) and Case et al. (2010) ﬁnd that factors not related to market returns and idiosyncratic risks are priced in housing market returns. The likely reason why CAPM fails to describe housing returns fully is that housing markets are characterized by many frictions that limit arbitrage. Because houses are large and indivisible, most households only own a home in one city, which exposes them to a large amount of local risk. This, combined with households’ inability to perfectly hedge income and house price risks, gives rise to a unique problem that typical asset pricing models are not well-suited to tackle. This paper develops a micro-founded dynamic equilibrium model that accounts for all of the frictions mentioned above and ﬁts the deviations from the CAPM. This ﬂexible model simultaneously considers risk-aversion, multiple sources of uncertainty, rich household heterogeneity, sorting, portfolio choice and asset prices in one uniﬁed framework. In equilibrium, risk premia of homes turns out to take a linear factor structure where the non-tradable part of the local risk is priced. Interestingly, equilibrium ﬁnancial asset returns are also a↵ected by the frictions in housing markets. Because of the explicit nature of the solution, the main equations of the model can be directly estimated using house price and income data. The model is also well suited for counter-factual simulations since the derived equilibrium is unique for any given set of parameters. In the rest of this section, I describe the setting of the model, prove existence and uniqueness of the equilibrium, and discuss the theoretical results.

2.1
2.1.1

The Setting
Cities and population

There is a measure one of households born at time t and they live until t + T .14 I consider an overlapping generations model where a new cohort of people is born in every instant. Households
Allowing for random life spans would complicate the model since households would want to hedge their mortality risk on top of their income and house price risk. We abstract from these complications in the current setting since our main goal is to understand the e↵ects of wage and house price risk.
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must buy only one house to live either in the countryside or in any city. They choose a location after they are born and live there until the end of their lives, at which point they sell their house to the new incoming generation. Such transactions occur continuously since at every instant in time a generation is born and a generation dies. Suppose there are L cities denoted l = 1, ..., L with countryside l = 0. Each city l has nl houses available for people to move into at any given period of time and cities are ﬁxed in size. The total ´T supply of a city is 0 nl ds = T nl . Assume that one household must occupy exactly one house.

Not all of the households from the same cohort can locate in cities since housing is scarce there. This means that a share of the population must locate in the countryside. The reason why the countryside is important here is that it serves as the outside choice that can be used the set the level of utility and prices in the spatial equilibrium. This will become obvious when we discuss uniqueness of the equilibrium. 2.1.2 Financial asset market
0

1 m Suppose that the whole economy is driven by m independent Brownian motions Bt = Bt ...Bt .

Also suppose that people can invest in n risky ﬁnancial assets and one risk-less asset.15 The risky assets do not pay any dividends.16 Households can hold any amount of these assets and there are no transaction costs or other frictions. The evolution of the risky asset prices is given by: dPti /Pti = µi dt + t i i1 dBt

+

i i2 dBt

+ ... +

i im dBt

i = 1, 2, ...n

For compactness, we write this in matrix form as: DPt1 dPt = µt dt + ⌃P dBt ⇥ ⇤ ⇥ ⇤ where Pt = Pt1 , ..., Ptm , DPt is a n ⇥ n matrix with diagonal equal to Pt , µt = µ1 , ..., µm , ⌃P is t t the n ⇥ m matrix whose rows are the volatilities of Pti .
15

Although we could potentially deal with a more general case, for simplicity assume that the
Notice that depending on m and n the stock market could be complete or incomplete. We will have m > n in our model which will imply that markets are not complete. 16 The assumption that stocks do not pay any dividends is made for analytical simplicity and is standard in the ﬁnance literature. For the purpose of our model it doesn’t really matter if returns from the ﬁnancial assets are coming from dividends or from price appreciation. All we are interested in is what returns are and how they are correlated with other assets in the economy.

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coe cients µi = µi and t

ijt

=

ij

are constant over time so we have: DPt1 dPt = µdt + ⌃P dBt

The matrix ⌃P , which captures the exposure of stocks to the underlying risk factors Bt , is given exogenously. In contrast, the expected return vector µ is an equilibrium object. Given the volatility matrix ⌃P , I solve for the equilibrium rate of return vector µ that clears the ﬁnancial asset market. For now, we do not consider other kinds of assets or derivatives in this economy. The inclusion and pricing of other sorts of assets or derivatives could also be easily handed in this framework because of the standard normality assumptions that we are making as to how shocks evolve in the economy. Therefore, the set of stock used here can be thought of as the basis that spans all the other ﬁnancial assets in the economy that we do not consider. 2.1.3 The housing dividend

Households must buy only one house in order to live either in the countryside or in any city. As mentioned, they decide where to live when they are born and live in the same place until the end of their lives. The countryside does not pay any wages and does not o↵er any amenities.17 When a household moves to a city, it gets utility from the local amenities as well as wages from local ﬁrms. The per period housing dividend a household living in city l receives in terms of dollars is: l l Dit = wit + iM l

where wit is the wage household i receives at time t and in city l.18 The taste parameter i iM

l

is the value it receives from amenities

capture heterogeneity in the population of preferences for city-

speciﬁc amenities, such as school quality, crime or weather. The wages household i receives are given by: l l wit = yt + ⇠i + "l i

This assumption is not as stringent as it may seem. House prices and utility in equilibrium will be relative to the outside choice i.e. the countryside. The utility and prices of the outside choice are set to zero for simplicity. In principle, the outside utility could be set to any level and this does not a↵ect the equilibrium allocation of households or asset prices. Changing the level of the outside choice utility only increases the price level in every city by a constant. This is a standard property of standard discrete choice spatial models. 18 Note that this should be interpreted as the net dividend from living in one house, which is the part of wages and amenities left over after households pay taxes and other costs to live in the city.

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l where yt is time-varying city productivity, ⇠i is a worker-speciﬁc ﬁxed e↵ect that does not vary

across cities and "l is a city-worker match ﬁxed e↵ect. We can think of yt as the part of the cityi speciﬁc productivity that does not depend on who works there. The ﬁxed e↵ect ⇠i can be thought of as the e↵ect of an individual’s general education or expertise on wages, which does not depend on where the household locates. The last term "l can be interpreted as the e↵ect of industry or i ﬁrm-speciﬁc human capital on wages. Workers specialized in the auto industry will have a higher "l in Detroit, MI, and workers who specialize in the high tech industry will have a higher "l in San i i Jose, CA. The only part of wages that is stochastic from the household’s point of view is yt . Suppose that l the productivity of city l at time t denoted by yt evolves according to:19

l dyt = sl dt t ⇣ l dsl = ml t

⌘ sl dt + ⌃l dBt t s

This process means that the city-speciﬁc productivity yt grows with a stochastic drift st . This drift follows a mean-reverting stochastic process also known as the Ornstein-Uhlenbeck process, which is the continuous-time analog of a discrete time AR(1) process. The stochastic shocks that govern this process are the underlying sources of risk in the economy Bt . The vector ⌃l , which is m ⇥ m, s governs the exposure of city l to these underlying sources of risk. The city-speciﬁc parameter ml is the long-run average of the drift st , while the parameter reverts to the mean ml . High values of l l

governs the speed at which the process

imply that the process reverts to the long run mean

quickly, and low values imply that the process is more likely to wander o↵ far from the mean for extended periods of time. This process implies that income will on average increase by the average amount ml , but there will be “business cycles” where income increases faster or slower than the long run average ml . 2.1.4 The utility speciﬁcation

A household is born at some time t, buys a house in a city, starts working and receives wages, and continuously invests his wealth in ﬁnancial assets. At the end of his lifetime, the household sells his home and all of his assets and consumes all of his accumulated wealth. It is assumed that there
This exact form of the evolution of the city-speciﬁc productivity is not essential for the model. The choice of this particular stochastic process is motivated by empirical patterns of the metropolitan area-speciﬁc average income series, which are used to estimate of the model in the next section.
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is no intermediate consumption; all of the wealth is consumed at the end of his life. Allowing for intermediate consumption complicates the solution of our problem to the point where analytical solutions are not possible. In fact, there are no known analytical solutions in the literature to the portfolio choice problem with exogenous income and incomplete markets, unless the income process is extremely simplistic.20 At birth time t0 , households maximize their lifetime utility given by: V (Xt0 , wt0 , st0 , l) = sup
✓t ,l

Et 0 e

i

h

Xt

0 +T

+M l

1 ir

⇣

erT

1

⌘i

(1)

where Xt0 +T is the ﬁnancial wealth accumulated through the lifetime and the second term is the utility accumulated from access to the local amenities. Each household chooses the optimal location l in which to live and each period chooses the optimal ✓, which is the dollar amount invested in stocks. The above maximization problem is constrained by the wealth evolution equation, which for t 2 (t0 , t0 + T ) is: dXt = ✓it dPt + r (Xit Pt ✓it ) dt + wit dt

where Xt is total wealth and ✓t is amount of money invested in stocks. Suppose an individual will die at time T , which means he is born at t0 = T T . Using the constraint, we can solve for the

terminal wealth and write the optimization problem for any t 2 (t0 , T ) as: Vtl = sup
✓t ,l

Et e

i

(er(T

t) X

´T t+ t

er(T

u)

1 0 (✓u DPu dPu

1 10 ✓u rdu+wu du)+ r (er(T

t)

1) i M l +pl ) T

(2)

Households therefore choose a city l to work and live in, and a series of stock holdings ✓t that maximize their lifetime utility, which is deﬁned over their terminal wealth. This wealth is composed of their capital gains on the home they bought, the wealth accumulated from wages they received, the wealth accumulated from investing in the stock market, and the dollar value of the utility accumulated from having access to local amenities.
Merton (1971) solves this problem by assuming deterministic income. Svensson and Werner (1993) provide solutions in the case of inﬁnitely lived agents with intermediate consumption in the case where income is drawn from an iid normal distribution, meaning that all of the shocks to income are temporary. See Henderson (2005) for a more detailed discussion.
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2.2

Equilibrium

An equilibrium in this economy is a set of home prices, asset prices and portfolio and location decisions such that households maximize utility and the asset markets and the housing markets clear. I consider a stationary equilibrium where the mass of houses available to a new cohort nl is constant over time.21 This means that there are nl households of every possible age in one city. Because of the complexity of the problem considered, I focus only on equilibrium prices that are linear in the states: l p l = Al y t + B l s l + C l t t

(3)

where Al , B l , and C l are city-speciﬁc constants to be determined in equilibrium. Starting with prices of this form, I show that there exists a unique equilibrium and that the values for these three vectors A = [A1 , A2 , ..., AL ], B = [B 1 , B 2 , ..., B L ] and C = [C 1 , C 2 , ..., C L ] are determined uniquely in equilibrium. The price for the countryside is normalized to zero.22 Several steps are needed to prove that there exists a unique equilibrium . First, conditional on a household having located in a speciﬁc city, I solve for optimal value function and for optimal portfolio choice. Next, I show that given vectors A and B, the equilibrium stock returns are uniquely determined by asset market clearing. Then, I turn to the location decision which gives unique values for A, B and C and a unique sorting equilibrium. These vectors A, B and C ensure equilibrium both in the housing and the asset market. 2.2.1 Portfolio choice and stock market equilibrium

Conditional on living in some location l, each household continuously chooses where to allocate his money. The problem given in equation (2), conditional on a choice of l, is the optimal portfolio choice problem for an investor facing imperfectly hedgeable stochastic income. The returns on income and stocks are imperfectly correlated, so the market is incomplete. This is a complicated class of problems to solve and very few instances in the ﬁnance literature have been successful in ﬁnding a closed-form solution.23
The assumption of ﬁxed supply is not empirically realistic for many metropolitan areas, although it may ﬁt the most volatile and supply constrained coastal metropolitan areas. Allowing for elastic supply is ﬁrst on the list of many extensions to the current model. 22 The assumption on the linearity of equilibrium prices is not as limiting as it seems. The equilibrium prices found from this assumption turn out to be the expected discounted sum of future dividends of the marginal person that lives in the city plus a city-speciﬁc risk premium as shown later in a proposition. 23 Henderson (2005) is the closest example to this paper in providing closed-form results. Under the special case of iid and normally distributed income, Svensson and Werner (1993) obtain explicit results in an inﬁnite horizon problem. Similar results are also found by Du e and Jackson (1990) and Tepla (2000) in a ﬁnite horizon problem.
21

13

To ﬁnd the value function, I follow the logic laid out by Henderson (2005).24 First, I take the ﬁrst order conditions of the Hamilton-Jacobi-Bellman (HJB hereafter) and substitute out ✓t and Xt . This gives rise to a nonlinear partial di↵erential equation we need to solve. The nonlinear HJB equation is then transformed to a linear partial di↵erential equation. At this point, we do a change of measure and use the Feynman-Kac theorem to get the value function with the portfolio choice substituted out. Taking the expectation, we achieve the result in the following proposition, which is proved in the appendix. Proposition 1. (Value Function) The value function of an household at some time t 2 (t0 , T ) conditional on being in city l is: V (t, y, s, l) = with: Utl =  er(T ⇣
t)

eUt

l

(4)

i

Xt +

1 + 2

1 ⇣ r(T e r

t)

2 i

⌃l ss

1 ⌃l ⌃P P ⌃l s sP P

⌘ˆ

T

ˆ ⌘⇣ ⌘ ˆl l ˆl ˆl 1 M l + ⇠i + "l + kt1 yt + kt2 sl + kt3 + i i t ⇣ ˆl ku2 ⌘2 du 1 0 µ 2 t
1 10 r ⌃P P (µt

T

t

ˆ ˆl k4 ku2

q ⌃l du ss

1r) (T

t)

t

ˆl ˆ where kt1 ...k4 are deﬁned as ˆl kt1 = Al (

1

er(T r
t)

t)

ˆl kt2

=

Al 1

1

e l l (T

+ Ble l (T

l (T

t)

er(T lr

t)

+

e

er(T l (r + l )

t)

t)

)

Other cases that study the portfolio choice problem with exogenous income either solve the model numerically or make the assumption that markets are complete. Examples that use numerical methods include Cocco (2004), Yao and Zhang (2005), and Van Hemert (2009). Kraft and Munk (2010) solve the optimal portfolio choice in closed-form but they assume that markets are complete. 24 The problem is di↵erent from that in Henderson (2005) since here risks and stocks are multidimensional, the income process is governed by a state variable that does not depend on the level of income, and agents hold a risky house in their portfolio.

14

ˆl kt3

= m

l

1 ⇣ rT + 1 r2 ˆl k4 =

(

ˆl kt2 + Al (T er(T

t) + B l +
t)

t 1 ⌘

er(T r + Cl

t)

(rt + 1) 1r)

and Al , B l , C l , ml , l ,"l and ⌃l are city-speciﬁc constants and ⌃KL = ⌃K ⌃0 . s i L Notice that the value function above is the exponential of a linear function in states. All of the ˆl ˆl functions k1t ...k4 do not depend on the states or individual heterogeneity parameters. This simple structure is very helpful in solving the portfolio and location decisions given in the next proposition and proven in the appendix. Proposition 2. (Portfolio Choice) A household of type i living in city l at time t 2 (t0 , T ) holds stocks in the dollar amount given by the vector:
1

1 ⌃l ⌃P P (µ sP p ⌃l ss

✓ilt = (⌃P P )

"

ˆl (µ 1r) kt2 + r(T t) ⌃l s P r(T t) e ie

#

(5)

ˆl where kt2 is a city-speciﬁc nonrandom function of time that depends on vectors A and B. The optimal portfolio in the stock, ✓ilt given in (5) is comprised of two components. The ﬁrst is the same term that Merton (1969) ﬁnds as an optimal strategy in the absence of stochastic income. This strategy is myopic as it is the portfolio choice for an investor who ignores income risk and only looks one period ahead. Since income is risky and is correlated with stock returns, the optimal portfolio also includes a hedging component. This second term hedges the change of stochastic income and can be interpreted as the inter-temporal hedging demand as in Merton (1971).25 Notice that the second term does not depend on the risk-aversion parameter. This is because income is normally distributed and it is instantaneously riskless, which means that investors do not hedge income but the change in income st . Now we turn to equilibrium in the stock market conditional on values of A and B. There is a ﬁxed supply of measure one of stocks and in equilibrium this should be equal to the aggregate demand for stocks. The aggregate demand for stocks of every household type i at age t in city l
25

A similar hedging demand is also found in Henderson (2005), Du e and Jackson (1990) and Tepla (2000).

15

should equal the total stock supply: ( L X ˆ l=1 i :V

l

V k ,8k

ˆ

T

0

1 ⌃P P

"

# ) ˆl (µ 1r) kt2 l + r(T t) ⌃P s dtd ( i ) = S r(T t) e ie i, where ( i ) is the cumulative density function for

nl is the number of homes available in city l,

and S is the net supply of assets, which could be zero. Simplifying and solving for µ, we get the equilibrium market returns. Proposition 3. (Stock Market Equilibrium) Given a set vectors A and B the stock returns are uniquely determined in equilibrium: r⌃P P S + r 1 PL ⇣ l=1 r(T

µ=

e

⌘ ´ T k2t ˆl nl ⌃l s 0 er(T t) dt P ´ 1 + 1r t) d ( i) i i

(6)

ˆl where k2t is a city-speciﬁc constant that depends on vectors A and B and the age t of the individual. The equilibrium rates of returns for stocks in this case do depend on the distribution of the riskaversion parameter over the population but do not depend on the location decisions of individuals.26 Equilibrium returns do not only depend on the variance-covariance matrix of the stock but also on the covariance of that particular stock with the local shocks in each city and the size of those cities. This ﬁnding is a deviation from standard equilibrium models where local nondiversiﬁable risk is ignored. If stocks are positively correlated with local shocks, they will have a higher rate of return than otherwise in order for the market to clear. Further interpretation of this result is ˆl postponed until after we solve for the equilibrium value of k2t , which makes this equation much more transparent. 2.2.2 Housing market equilibrium

We turn now to equilibrium in housing markets. At birth time t0 , households maximize the value l function in equation (4) with initial wealth Xt0 = X0i

pl 0 = X0i t

l Al y t 0

B l sl 0 t

C l . Given their

optimal portfolio decisions given by Proposition 2, at time t0 they choose the optimal location l that maximizes their lifetime utility in equation (4). Any monotonic transformations of the value
The reason that equilibrium rates of return do not depend on the location decisions of individuals is that, once the move to a city, all individuals face the same exposure to the city’s risk factors given by ⌃l . If I were to allow the s variance of wages to be individual speciﬁc then the location decisions of individuals would a↵ect asset prices directly. I have solved the model in that case but do not present it here since allowing for individual heterogeneity in exposure to factors makes the solution much clumsier and harder to interpret.
26

16

function do not change the maximization problem a household faces. After taking the log of the value function and dropping any terms that are the same across choices, the maximization problem for a household born at time t0 the log value function takes a very simple form. Proposition 4. (Optimal Location) A household i born at time t0 chooses his/her optimal location by solving for the maximum log value function:

maxl Utl0 i =

iM

l

l l l l + k 1 yt 0 + k 2 s l 0 + k 3 t

rC l

l i k4

+ "l i

(7)

l l where k1 ...k4 are location-speciﬁc constants that do not depend on states, individual heterogeneity,

or the vector C: l k1 = 1

rAl ⇣ + Bl e ⌘

l k2

=

⇣

r erT 1

erT e T erT + l lr (r + l )

⌘ 1

(

A

l1

e l lT

lT

erT

l

)

l k3

=

⇣

r erT l ml k2

⌘ Al ml T 1
1 ⌃sP ⌃P P



ml

1 ⇣ rT + 1 r2 ˆ
T 0 l k5 du

erT

⌘

(µ ⌘ˆ

1r)
T

1⇣ l k4 = ⌃ss 2 l k5u

1 ⌃ l ⌃P P ⌃ l s sP P

0

⇣

l k5u + Beru

⌘2

du

=

r (eru 1 1)

(

A

l1

e l l

lu

eru e u eru + l lr (r + l )

)

⇣ + Bl e

lu

eru

⌘

l l The constant k1 ...k4 are found by setting T

ˆl ˆl t0 = T in the constantsk1t ...k4 in Proposition 1. The

ˆ kit in Proposition 1 depend on time t because they are functions of the number of years left until death T l l t. Since at birth there are T years left until death, the constants k1 ...k4 do not depend

on time but only on total lifespan T . 17

Given the maximization problem (7), household i chooses location l if the utility it receives from this choice exceeds the utility it receives from all other choices, or when: Uil ¯ where Uil = Uil Uik ) ¯ Uil + "l i ¯ Uik + "k i ) "l i "k i ¯ Uik ¯ Uil 8l, k

"l . The location-worker speciﬁc ﬁxed e↵ect "l is drawn from some probability i i

distribution for each household. Because we have a continuum of households, every possible value of "l is drawn in each generation. Therefore we can interpret the mass of individuals with "l in some i i given range as the probability of drawing "l from that range. If we interpret "l as a random variable, i i the probability that a household chooses location l can be written as a function of everything that enters in Uil :27 l P ri = fl ( i , i , A, B, C, K)

(8)

Aggregating these probabilities in equation (8) for all households gives the demand for each city: l Dt

=

ˆ

fl ( i , i , A, B, C, K) dF ( i , i ) i, i

where F ( i , i ) is the joint distribution of the

i

and

i

parameters over the population. In equi-

librium, demand for each city has to equal supply in each city: l D t = nl

8l

)

ˆ

fl ( i , i , A, B, C, K) dF ( i , i ) = nl , i, i

8l

(9)

The sorting problem described here is very similar to horizontal sorting models in urban economics. The closest models in structure are those developed by Bayer, McMillan and Rueben (2005) (BMR) and Bayer and Timmins (2005). At birth, which is when households choose where l l to locate, k1 ...k4 are city-speciﬁc characteristics. This means that the current framework maps

directly to the BMR sorting model with the vector C serving as a price that clears the market. If demand exceeds supply, C l is bid up until demand equals supply, and vice versa. Just as in the BMR model, given ﬁxed city characteristics, there is a unique “price” vector C that leads to a unique sorting equilibrium. Proposition 5. (Sorting) Given a set of vectors A and B and a set of city characteristics l l k1 ...k4 , if "l is drawn from a continuous distribution, a unique vector C solves the system of equai

Again the probability that a household of type i chooses location l is the same as the mass of households of type i that choose location l. I interpret this as a probability only to relate these results to the urban horizontal sorting models as shown below.

27

18

tions given in (9). Moreover, the equilibrium spatial allocation of households is unique (within the class of linear stationary equilibria). The ﬁrst part of the Proposition 5 is proved in Proposition 1 of BMR and the second part in Proposition 2 of Bayer and Timmins (2005). Their proofs will not be reproduced here. Now we turn to determine the unique values of the vectors A and B. Note that the distribution F ( i , i , "i ) is not time-dependent. The quantity of available houses in each neighborhood nl is also constant across time by assumption. That means that the same set of heterogeneous individuals will be drawn each period and they will sort across the same set of homes. The sorting problem will be identical in each period except that the states yt and st may change. Because A and B are constant across time, in order for the constant vector C to clear the markets each period, we need the utility from living in a city not to depend on states yt and st . If the utility function given in l l (7) depended on the states, the vectors A, B and C could not be constant. Setting k1 = k2 = 0, 8l

gives the unique values for A and B. The result is given in the next proposition which is proved in the appendix. Proposition 6. (Equilibrium Prices) Given a time invariant distribution of household

characteristics F ( i , i , "i ), the equilibrium house prices are given by: 1 l 1 pl = y t + t r r (r + where ⇡ l = C l l ml . r2 (r+ l )

l)

sl + t

l ml

r2 (r +

l)

+ ⇡l

(10)

This price function gives a unique sorting equilibrium in housing markets

as shown in Proposition 5 and gives unique equilibrium rates of return for stocks as shown in Proposition 3. Proposition 6 shows that the price function given in (10) ensures market clearing in both the stock and the housing market. Not only is this equilibrium price unique among the ’linear-inthe-states’ class of functions, but it also results in a unique equilibrium in the housing and asset markets.

2.3
2.3.1

Analysis of the Equilibrium
Equilibrium house prices

The linear structure of equilibrium house prices is not as limiting as it may ﬁrst seem. The price function given in (10) has also the appealing feature that it can be expressed as the expected discounted sum of productivity in one city plus a city-speciﬁc constant, ⇡ l , which captures risk 19

premia and heterogeneity in preferences in each city. Proposition 7. Under the assumed processes for the evolution of yt and st , the equilibrium price function (10) is equivalent to the price function: pt = Et ✓ˆ
1

e

t

r(u t) l yt dt

◆

+ ⇡l

where ⇡ l is a city-speciﬁc constant determined in equilibrium that captures housing risk premia and the e↵ect of individual heterogeneity in prices. l In order to prove the proposition above, I plug in the assumed process for yt and solve the

double integral that discounts the cash ﬂows to the present and takes the expectation. Given the l properties of the yt process, we can switch the order of integration and solve for the double inﬁnite

integral to get equation (10). The details of the proof can be found in the appendix. 2.3.2 Sorting and the factor structure for housing risk premia

The analysis so far has been conducted under a general form for ⌃P , which is the co-movement of stock returns in relation to the underlying sources of uncertainty. Without loss of generality, consider the case where stock i’s returns are driven by only one underlying Brownian motion. i Suppose there are n such stocks and that stock i is only driven by Brownian motion Bt . This

means that all the o↵-diagonal elements of ⌃P are zero. There are a total m Brownian motions that drive the economy so if m > n, perfect hedging cannot be achieved. The reason for this transformation of the model is that it greatly simpliﬁes the interpretation of the results, and it delivers equations that I can easily estimate using house price and wage data. Proposition 8. If the n stocks are driven by the ﬁrst n sources of risk B 1 ... B n and all the o↵-diagonal elements of ⌃P are zeros, then the location-speciﬁc value function is: n X j=1

maxl Uil =

iM

l

r⇡ l

lj j s

1 r (r +

l)

i

j=n+1

m X ⇣

lj s

⌘2

4r2 (r

⇣

erT +

⌘ 1

l )2

+ "l i

(11)

where

j

is the Sharpe’s ratio of stock j given in equilibrium by: r == j j PS

j

=

µj j P

r

+T e

⇣

1

rT

PL

l=1

⌘´

✓ i lj nl s r+ (

l

)

1

◆

(12)

d ( i) i

20

and

j P

and

lj s

are the (j,j)th and jth element of ⌃p and ⌃l respectively. s

Proposition 8 can be derived by plugging the equilibrium house price function in equations (6) and (7). First note how exposure to a source of risk a↵ects utility and in turn prices. A city can be exposed to an underlying source of risk that is spanned by traded stocks or to one that is not hedgeable by traded assets. If the underlying source of uncertainty j is spanned, the standard deviation lj s

of the city’s exposure to it enters utility in a linear form and is multiplied by the price

of risk (Sharpe’s ratio) for that source of uncertainty. Otherwise, the unspanned risk factors enter jointly in the term that is multiplied by the risk-aversion parameter
i.

Notice that if all of the

sources of risk were spanned, the latter term would drop out of the utility function and not a↵ect prices in equilibrium. If this is not the case, however, idiosyncratic risk speciﬁc to a city that is not spanned by the traded assets will a↵ect utility and it will be priced in equilibrium. The equilibrium market prices of risk rates and the volatility j P, lj s , j

for stock j in this case depend not only on the interest

but also on its correlation with the income and house price processes in which is weighted by the size of the city nl . If stock j is positively

all of the cities captured by

correlated with local shocks for a large share of the homes in the economy, they will have a higher rate of return and a higher market price of risk than otherwise. This is because most people will want to short the stock in question for hedging purposes. For every person who shortsells the stock, there must be someone who holds it. Therefore, in order for it to be worth it for people to hold this stock, the rate of return and the market price of risk need to be higher. The market price of risk does not depend on the location decisions of individuals since lj s

is not individual speciﬁc. If it

was, the allocation of individuals across space would directly a↵ect equilibrium asset returns. The distribution of the risk-aversion heterogeneity also a↵ects the market prices of risk as can be seen from the integral in the denominator. 2.3.3 Talent allocation

In this equilibrium, individuals do not always choose the location where they are most productive or where their "l is the highest. As a result, productivity in the economy is not maximized and i there are two reasons for this. Firstly, workers do not go where they are most productive because of amenities that di↵erent places possess. A worker who is very productive in low amenity city A but not so productive in a high amenity city B may decide to locate in B if he places a lot of weight on amenities. The same kind of productivity ine ciencies also arise when cities have di↵erent levels of non21

diversiﬁable risk. An individual who is very risk-averse may not decide to locate in a city with a high level of nondiversiﬁable risk. If all of the sources of uncertainty were hedgeable, however, this kind of sorting ine ciency would not be present. In our counterfactual simulations in the section 4, we look at how much more productive the US would be as a result of creating assets that span presently hedgeable sources of risk.

3

Estimation

The equilibrium model developed in the previous section can be fully estimated by combining individual migration data from the decennial Census, housing price data from FHFA, wage data available from the Bureau of Economic Analysis, and amenity data such as crime, weather and quality of schooling from other sources. Instead of using migration data, I take a simple approach in this paper and estimate the model assuming homogeneity in preferences. I later use the estimated parameters of the model in a series of counterfactual simulations in which I allow for various degrees of preference heterogeneity. I estimate the model using house price and wage data at the metropolitan area level for the US. Home prices are constructed by using the annual FHFA home price indices at the metropolitan area level, which are scaled up by using median home price levels from the 2000 Census. The wage data used consist of the personal income per capita measure made available by the Bureau of Economic Analysis. I also construct several proxies for amenities, which are: crime levels from the FBI’s publication “Crime in the United States”, population, population density, and house density from the decennial Census, and a series of weather variables from the Area Resource File maintained by the Quality Resource Systems. The ﬁnal sample analyzed here consists of annual data for the period from 1985 to 2008 for 216 metropolitan areas (MSAs hereafter). Under the assumption of homogeneity in preferences for amenities and for risk, the price equation becomes linear in risk factors. The main estimating equation therefore is: 1 l 1 pl = y t + t r r (r + where 1 ⇡ = M r l n X i=1

l)

sl + t

l ml

r (r +

l)

+ ⇡l

(13)

i 2 r (r

il s

+

l)

n X i=1

il 2 erT s 4r3 (r +

1
l)

+"

(14)

l Empirically, pl is the price level in any particular metropolitan area l, yt is the net income received,

22

st is the instantaneous growth in income, and ml is the expected growth in income. The price premium ⇡ l is composed of the e↵ect of amenities M , the e↵ect of the exposure to traded sources of risk, and the exposure to local noninsurable sources of risk. Our goal is to estimate all of the parameters of the price equation in order to get estimates for risk premia. Estimating the price equation is achieved in several steps. First, I use time-series regressions to estimate , as well as to identify common risk factors Bt that drive the economy. Using these factors, I can estimate the factor loadings il s

for each metropolitan area. In the second step, I use cross-sectional regressions i to estimate equation (14) and get estimates of the market prices of risk

and the risk-aversion

parameter . Using these parameters, I then calculate the implied risk premia and amenities for each of the MSAs.

3.1

Time-Series Factor Decomposition

In order to estimate the model, we would ﬁrst need estimates of the underlying factors that drive the economy in each metropolitan area, which are given by the Brownian motions Bt in the model. Using price and wage data, we can only identify the total e↵ect that these factors from time t to time t as given by Ftl = = ˆ t 1

t 1 1

⌃l dBt s t 1 dBt

ˆ

+

2

t 1

ˆ

t

t 1

2 dBt

+ ... +

m

ˆ

t

t 1

m dBt

(15)

Breaking this total e↵ect up into the actual fundamental factors, as in the second line of the above equation, would require a deeper analysis into the forces that drive the local industry of each metropolitan area, which is beyond the scope of this paper. If we have an estimate of Ftl we could ´t i use any unobserved factor model, such as principal factor analysis, to estimate the factors t 1 dBt and the factor loadings
i.

While this procedure would be rigorous, the estimated unobserved

factors would be very hard to interpret. To avoid interpretation issues, here I ﬁrst estimate the total e↵ects of these underlying factors Ftl for each metropolitan area, and then I decompose the total e↵ects into common empirical factors that are constructed in the spirit of the Fama-French factors for the stock market.

23

3.1.1

Total E↵ect of Factors

First, I estimate the total e↵ect of the underlying risk factors on wages and prices Ftl . As in the model I assume that the growth in average income in each metropolitan area evolves according to: l dyt = sl dt t ⇣ dsl = ml t

⌘ sl dt + ⌃l dBt t s

l which means that the change in income dyt has a mean reverting drift sl . The instantaneous t

income drift sl is driven by the exposure of the metropolitan area to the underlying factors Bt . The t parameters of this model cannot be identiﬁed by using only income data unless income is observed continuously over time. Using equation (13), however, we can estimate everything by exploiting both wage and house price data even though they are observed over discrete time intervals. The estimation procedure is straight-forward and follows a series of linear regressions. Note that rearranging the equation (13), we can deﬁne xt as the gap between prices and wages, which is the observable part of our model: x l = pl t t 1 l 1 yt = r r (r + sl + t l ml

l)

r2 (r +

l)

+ ⇡l

The variables on the left hand side are the observable annual prices and wages in city l. All of the variables in the right hand side are unobservable and are to be estimated. Since sl is assumed t to follow an Ornstein-Uhlenbeck process, we know that the observable xl should also follow an t Ornstein-Uhlenbeck process. The evolution of xl can be described by the equation: t ⇣ ⌘ 1 xt + r (r + ) ˆ t+1 (t+1 u)

xt+1 = x 1 ¯ where x = ¯
1 r(r+ ) m ml r2 (r+ )

e

+e

e

t

⌃l dBt s

(16)

⇡ l . The equation above looks like a simple AR(1) process. We can

estimate the equation (16) by running the following regressing for each MSA: x l = a l + bl x l + u l t+1 t t We can estimate the and by setting the parameters: ˆ= ln(ˆ b) (17)

24

ˆ Notice also that we can get an estimate of Ftl =

that drive the economy from t to t + 1, from the residual of the regression above. The estimate is: ⇣ v ⌘2 u u ln(ˆ b) u t 2ln(ˆl ) 1 l b l l ˆ ˆ ⇣ ⌘2 ˆl ut = c ut ˆl b 1 b

⇣´ \ ⌘ t+1 l ⌃s dBt , which is the total sum of factors t

ˆ Ftl = r2 r

Using annual house price and income data for 216 metropolitan areas in the US for the period of 1985-2008, I estimate the parameters , l and a vector Ftl for each metropolitan area. I deﬂate prices and wages by the CPI with base year 2008 in order to eliminate any e↵ects of inﬂation. Although inﬂation risk can signiﬁcantly a↵ect lifetime wealth, I abstract from it in the model and estimate the equations of interest using real variables. Also, since the model abstracts from time-varying interest rates, I set interest rate to r = 0.04. The ﬁrst regression that I estimate separately for each metropolitan area is equation (17), which substituting for what xt is in terms of data reads: Hpricel t ⇣ ↵Incomel = al + bl Hpricet t ↵Incomel t ⌘ + cl Ftl (18)

1

1

where Hpricel is the median house price level in MSA l and Incomet is the per capita income. In the model ↵ would be equal 1/r if all of the income that an individual receives can be saved or consumed. In other words the model states that the coe cient on net income should be 1/r. In reality only a portion of the labor income can be considered as net beneﬁt since individuals pay taxes and also incur utility losses from sacriﬁcing leisure in order to receive that income. Estimating the parameter ↵ directly can be di cult since wages are correlated with unobserved amenities in each MSA leading to serious endogeneity problems. Because nicer places generally o↵er higher wages the estimated parameter will be biased upwards. For my sample, the estimated parameter is 3.7, which is very high, implying very negative amenities for many metropolitan areas that have relatively low price to wage ratios. In this analysis I set ↵ to 2.5, which implies reasonable levels and ranking of amenities across metropolitan areas as it will be shown later. The qualitative nature of the results does not depend on the exact value of this parameter. Setting ↵ between 2 and 3 yields very similar results. In principle, could be estimated separately for each MSA. Here, I ﬁx to be the same across

metropolitan areas in order to keep the interpretation of the other parameters of the model simple.

25

The estimate parameter here is ˆ = .0315 and is signiﬁcantly di↵erent from zero at the 99% level. The interpretation of this parameter is that xt , which is the di↵erence between prices and wages, does not revert very quickly to the mean. If the di↵erence between prices and wages is too high, we would expect it to remain high for quite some time before it eventually reverts back to the normal level. The parameter b from the discrete time AR(1) regression (18) is .9689 (.0041) meaning that xt is close to being non-stationary. The total variance parameter  varies widely across metropolitan areas. The most volatile MSAs are San Francisco-San Mateo-Redwood City, CA (105.35), Santa Ana-Anaheim-Irvine, CA (97.88) and Honolulu, HI (97.21). The least volatile MSAs are South Bend-Mishawaka, IN-MI (3.98), Huntington-Ashland, WV-KY-OH (4.47) and McAllen-EdinburgMission, TX (5.26). I also estimate Ftl for each metropolitan area from the residual of the regression (18). While there is no particular interest in the values of Ftl , decomposing it into di↵erent factors is crucial for the estimation of the model. 3.1.2 Three Common Factors for Housing Markets l i

In order to estimate equation (14), we need estimates for

which are the MSA’s exposure to the

underlying factors that drive the economy. This means that we need to decompose Ftl in several factors and estimate the factor loadings l i

for each metropolitan area as in equation (15). As it is

very di cult to uncover all of the factors that drive the local economy for each city, I decompose Ftl into a few factors that capture much of the variation in the data.28 These factors are constructed very similarly to the popular factors that Fama and French (1993) use to explain common variation in the ﬁnancial asset markets.29 The ﬁrst factor, denoted housing market (HMKT hereafter), is the annual house price returns for the whole US housing market. The second factor, denoted housing small-minus-big (SMBH), is deﬁned as the average returns in MSAs in the bottom half of the price level distribution in a given year minus and the average return in MSAs in the top half. This factor replicates a self-ﬁnancing diversiﬁed portfolio that holds houses in low priced metropolitan areas and shorts houses in high
28 Hizmo (2010, 2) shows that the same three factors considered here explain about 90 percent of the time-series and cross-sectional variation in the house price returns for twenty ﬁve diversiﬁed housing portfolios constructed by sorting metropolitan areas on price level and price over wage ratios. For individual metropolitan areas these three factors explain about 50 percent of the variation. 29 The Fama/French factors are constructed using 6 portfolios formed on size (market capitalization) and book to market value ratios. The ﬁrst factor is the excess return on the stock market. The second factor, small minus big, is the average return on the three small portfolios minus the average return on the three big portfolios. The third factor, high minus low, is the average return on the two value portfolios minus the average return on the two growth portfolios. These three factors combined can explain over 90% of the time-series and cross-sectional variation of any well diversiﬁed stock portfolio.

26

The Thre e Housi ng Marke t Fac tors 130

120 Fac tor I nde x (1995=100)

110

100

90

80

HMKT HMLH SMBH 1985 1990 1995 Ye ar 2000 2005

70

Figure 2: The Factor Decomposition of Local Risk price ones. The third factor, denoted housing high-minus-low (HMLH), is deﬁned as the average returns in MSAs in the bottom 30 percent of the price level distribution in a given year minus the average return in MSAs in the top 30 percent. This factor replicates a self-ﬁnancing diversiﬁed portfolio that holds houses in metropolitan areas with high price-to-wage ratios and shorts houses in areas with low price to wage ratio. Similar to the small-minus-big Fama-French factors, SMBH intends to capture size e↵ects in MSA price returns. On the other hand, HMLH is intended to capture growth e↵ects since a high price to wage ratio can be an indicator of high expected growth. For ease of interpretation I orthogonalize the three factors through a series of regressions. First, I regress HMKT on HMLH and SMBH and redeﬁne HMKT as the regression residual. Then I regress HMLH on SMBH and redeﬁne HMLH as the regression residual. This procedure gives three factors that are orthogonal to each other. The qualitative nature of the results is not a↵ected by this orthogonalization. The three factors are displayed in Figure 2. I decompose the local growth term Ftl down into the e↵ect of three factors and a residual term 27

". The estimated regression for each metro area is: Ftl = ↵l + l HM KT HM KTt

+

l SM BH

· SM BHt +

l HM LH

· HM LHt + "i t

(19)

The results are summarized in Table 1. The coe cients in this table are presented at the mean across metropolitan areas. In parentheses I show the number of times a coe cient is found to be signiﬁcantly di↵erent from zero at the 90% level. The distribution of the estimated R-squared is summarized by its mean, minimum and maximum value. While the magnitude of these coe cients is hard to interpret, their sign is straightforward. On average, when the housing market factor HMKT increases, so do wages and house prices. The higher the increase in this factor the more prices will deviate from wages. A similar result holds for the HMLH factor. If the growth in high price to wage metropolitan areas is higher than in low price to wage ones, then on average prices and wages will increase, and the price-wage gap will increase. The opposite is found when low priced MSA appreciate faster than high priced MSAs as it can be seen by the average coe cient on the SMBH factor. The HMKT factor alone on average explains about 13% of the time-series variation, the SMB factor about 22%, and the HMLH factor about 16%. The three factors combined explain about 50% of the time-series variation on average, although the R-squared for particular metropolitan areas ranges from .02 to .90. Table 7 in the Appendix displays all of the factor loadings for the three factors and the estimated R-squared coe cients for all of the 216 metropolitan areas. The MSAs with high R-squared coe cients are generally metropolitan areas with high volatility, growth, price levels, population and density. The opposite is true for MSAs with low R-squared coe cients. The top MSA’s in terms of the R2 are Washington-Arlington-Alexandria, DC-VA-MD-WV (.90), Bakersﬁeld, CA (.87) Riverside-San Bernardino-Ontario, CA (.86). The bottom MSA’s are South Bend-Mishawaka, IN-MI (.026), Charlotte-Gastonia-Concord, NC-SC (.027) and Elkhart-Goshen, IN (.047). In terms of factor loadings, cities like New York, Boston, Washington DC, San Francisco have large positive loadings on the HMKT factor, and large negative loadings on SMBH. MSA’s near California or Florida generally have high loadings on the HMLH factor, while other large MSAs in Massasuchets, Connecticut or New York have negative loadings. Overall there is a large amount heterogeneity in the exposure of particular metropolitan areas to the three factors, which ﬁts the general observation that local housing markets are very di↵erent from each-other, and do not usually follow the national market.

28

Table 1: Time-Series Regressions of the Local Economic Base on Three Risk Factors (1) HMKT 7.8257 (102) SMBH -11.7965 (107) HMLH 7.7780 (98) R2 distribution Mean Min Max .1290 .0001 .4810 .2172 .0001 .7466 .1583 .0001 .7392 .5049 .0264 .9036 (2) (3) (4) 7.8363 (147) -11.7404 (121) 7.7919 (115)

Groups

216

216

216

216

Note - The dependent variable in all of the regressions is the estimated process F that drives wages and prices in each metro area . Speciﬁcations (1)-(4) show the mean coe cients from time-series regressions that are run separately for each MSA. In parentheses is shown the number of times a coe cient is found to be signiﬁcantly di↵erent from zero at the 90% level. The sample consists of annual data from 1985 to 2008.

29

3.1.3

Are the Factors Spanned by Traded Assets?

While the three proposed factors do explain a large share of the variance in the underlying local growth term Ftl , it may not be possible for homeowners to use them to hedge house price and income risk. First of all, it is not feasible for any household to hold multiple homes in several metropolitan areas to replicate any of the three factors. Perhaps, the only feasible way to hedge against these sources of risk would be to invest in ﬁnancial assets that correlate with the three factors. In order to get an idea about how much of the variance of these three factors can be spanned by using stock returns I regress each of the three factors on the three original Fama-French factors and on REIT returns. The REIT returns used here come from the aggregate NAREIT index for REITs that invest in mortgage issued on real estate and construction. The results are almost identical if we use Equity or Hybrid REIT indexes. The Fama-French factors are three portfolios constructed using 6 portfolios formed on market capitalization (size) and book-to-market value ratios. The ﬁrst factor, Mkt-Rf is the excess return on the stock market. The second factor, small minus big (SMB), is the average return on the three small portfolios minus the average return on the three big portfolios. The third factor, high minus low (HML), is the average return on the two high book-to-market ratio portfolios minus the average return on the two low book-to-market ratio portfolios. These three factors combined can explain over 90% of the time-series and cross-sectional variation of any well diversiﬁed stock portfolio. The results are presented in Table (2). The housing market factor HMKT is the only one that seems to be very correlated with traded assets. In the ﬁrst speciﬁcation, HMKT is regressed on REIT returns. The coe cient on REIT is positive and statistically signiﬁcant with the magnitude and standard error of .0396 (.0116). The correlation coe cient between the HMKT and the REIT returns is about 0.6. As it can be seen in the second speciﬁcation, adding the Fama-French factors does not increase the R-squared and all of the coe cients on these factors are not statistically signiﬁcant. In both speciﬁcations, the coe cient on REIT is positive and statistically di↵erent from zero. When the same regressions are estimated for the SMBH and HMLH factor, all of the coe cients are found to be small in magnitude and not statistically signiﬁcant from zero. The R-squared coe cients from these regressions are also very low. I interpret these results as evidence that homeowners are able to hedge most the HMKT factor risk by using tradable assets. On the other hand, tradable assets do not seem to be helpful at all at hedging the risk that is due to the

30

Table 2: Predicting Housing Factors with Stock Returns HMKT REIT .0396⇤⇤ (.0116) Mkt-Rf .0488⇤⇤ (.0130) -.0111 (.0213) SMB .0028 (.0332) HML -.0081 (.0276) .0033 (.0191) SMBH .0097 (.0206) -.0174 (.0338) .0150 (.0526) -.0565 (.0438) HMLH .0028 (.0140) .0059 (.0155) .0039 (.0253) -.0091 (.0394) -.0242 (.0328)

R2 Years

0.345 24

0.355 24

0.001 24

0.084 24

0.002 24

0.042 24

Note - The dependent variables are the three housing factors. The independent variables are aggregate mortgage REIT returns from the FTSE NAREIT series, and the three Fama-French factors downloaded from Kenneth French’s website. The sample consists of annual data from 1985 to 2008. The standard errors are given in parentheses.
⇤⇤

statistical signiﬁcance at the 95% level

SMBH and the HMLH factors.

3.2

Market Prices of Risk and Risk Premia

After estimating the factor loadings in the previous section, we have all of the ingredients to estimate the cross-sectional equation (14). For this we need an estimate of ⇡ l , which can be given by: ✓ ⇡ = E pl t ⇣ = E pl t = E l ml 1 sl + 2 t r (r + l ) r (r + l ) ⌘ ml l ↵wt r2 ! pl pl 1 t t l ↵wt r l ↵wt

◆

pl t

where the second equality uses the fact that E(sl ) = ml and the third equality uses the fact that t E pl t pl t
1

= ml /r. Instead of ﬁrst estimating ⇡ and then estimating equation (14), I do it all

in one step by using a between-e↵ects panel data estimator that only uses cross-sectional variation in the data. The estimating equation is: 31

l ⇡t =

1 Ml r

l 1 ˆHM KT

+

l 2 ˆSM BH

r2

where

⇣

⌘ r+ ˆ l ↵wt

+

l 3 ˆHM LH

l ˆerr ⇣ ⌘ + el t 3 r+ ˆ 4r

2

(20)

l ⇡t = p l t

pl t

pl t r

1

The parameters ˆ are the factor loadings from the previous section and

err

is the standard deviation

of the residual from regression (19). The variable M captures amenities, which in this case are average temperature in January and July, average hours of sun in January, average humidity in July, a crime index that weights violent crimes ten times more than non-violent ones, population size, population density, housing density and percent are of water of MSA. The parameters to be estimated are , and the risk-aversion parameter .

I estimate equation (20) to get estimates for the market prices of risk for the three proposed factors and for the parameter of risk aversion. The results are presented in Table 3. Speciﬁcation l (1) assumes that all the three factors are spanned and the residual variance ˆerr 2

is not. The

estimated risk prices and the risk-aversion parameter are all statistically signiﬁcant at the 95% percent level. As it can be seen from the estimate of the risk-aversion parameter, higher idiosyncratic variance leads to cheaper house prices. Households need to be compensated by one dollar decrease in current house prices for every increase of $300000 in the variance to their lifetime wealth. The Rsquared coe cient is fairly high at .6859 meaning that the three factors explain a large share of the cross-sectional variation in prices and wages. Even if we didn’t control for amenities, the R-squared would be rather high at .5165. In the next three columns, I repeat the same procedure under di↵erent assumptions about which factors are spanned and which aren’t. Speciﬁcation (2) assumes that both HMKT and SMB are spanned by traded assets while HMLH isn’t. In speciﬁcation (3) only the HMKT factor is spanned, and in speciﬁcation (4) all of the volatility in each metropolitan areas is not spanned by any traded asset. The results are very similar under these di↵erent assumptions both in terms of magnitudes and in terms of their signiﬁcance. The estimated absolute risk-aversion parameter is around 3 · 10 net wages. Using the estimates in Table 3 together with the factors loadings for each metropolitan area, we can estimate the MSA speciﬁc risk premia and the implied amenities from equation (20). Using risk prices from speciﬁcation (1) of Table3, the appendix Table 8 shows the estimated risk premia 32
6,

which implies a relative risk-aversion parameter of 1 on average if we

were to assume that a household lives in the median city in terms of volatility and earns $20000 in

Table 3: Cross-Sectional Estimates of Risk Prices for Traded Factors (1) ˆ HM KT ˆ SM BH ˆ HHM L ˆ /105 -.1166⇤⇤ (.0591) -.1285⇤⇤ (.0241) .1649⇤⇤ (.0302) -.3462⇤ .1944 -.5932⇤⇤ (.1973) -.3163⇤⇤ (.0735) -.2618⇤⇤ (.0575) (2) -.2093⇤⇤ (.0629) -.1223⇤⇤ (.0263) (3) -.0473 (.0648) (4)

Amenities

Yes

Yes

Yes

Yes

R2 Groups N. Obs.

0.6859 177 4230

0.6471 177 4230

0.5596 177 4230

0.5559 177 4230

Note - The estimated coe cients represent the market price of risk for each of the factors if they are traded. The dependent variable is the part of house prices that is not due to wages and expected growth. The independent variables are the factor loadings estimated from time-series regressions multiplied by 1/ r2 (r + ) . The absolute risk-aversion parameter ˆ is the coe cient on the variable ˆERR , which is the standard deviation of the residual from each time-series regression multiplied by e30r / 4 ⇤ r3 (r + )2 as the model predicts. The coe cients in this table are estimated by a panel data between estimator that only uses cross-sectional variation. The amenities included are weather, crime, population, density, and percent water area in MSA. The standard errors are shown in parentheses.
⇤

statistical signiﬁcance at the 90% level statistical signiﬁcance at the 95% level

⇤⇤

33

and the implied amenity value for all the metropolitan areas in alphabetical order. A sample of the ﬁfty most populated MSAs sorted by risk premia is displayed in Table 4. Negative estimates for risk premia are interpreted as the dollar amount a homeowner needs to be compensated for living in a risky MSA. Santa Ana-Anaheim-Irvine, CA is the MSA with the highest risk premium of -$140048. This means that prices there are $140048 cheaper than they would be if all homeowners were risk-neutral. Since homeowners are risk-averse, they are compensated by cheaper home prices for taking the risk of owning. On the bottom of the table we can see that Denver-Aurora-Broomeld, CO experiences home prices than are about $20000 higher than they would be if homeowners were risk-neutral. This is because Denver has almost opposite factor loadings to Santa Anna or San Francisco and lower overall volatility. Overall, coastal cities have large risk premia while the Midwest and the southern metropolitan areas have the lowest. The part of home prices that is not due to wages or risk premia is used as an estimate for amenities in that MSA. For example, after taking out the e↵ects of wages, expected growth and risk premia from home prices in Santa Ana, the implied value of amenities is about $300000. Out of the ﬁfty largest cities in Table 4, the worst MSA in terms of amenities is Detroit-Livonia-Dearborn, MI with a value of about -$32000 and the best metropolitan area is San Francisco-San Mateo-Redwood City, CA with a value of amenities of $316297.

4

Simulations

Using estimates for 216 US metropolitan areas I simulate the model to study the e↵ect of ﬁnancial innovation on house prices, household sorting across space and on overall productivity in the economy. In order to simulate the model I need estimates of the individual-MSA productivity match "l . I take estimates of this distribution from the previous literature.

4.1

The wage equation parameters

Using very detailed conﬁdential migration data on a large set of individuals, Bishop (2008) estimates a wage regression: l l wit = f (agei ) + !i + µl + ✓i + ⌘i + et t

where wages of individual i who works in city l at time t are regressed on age dummies and individual speciﬁc characteristics !i and city-by-year ﬁxed e↵ects µl . The parameters of interest that are used t l here are the standard deviations for the individual-city match ✓i and the individual ﬁxed e↵ect ⌘i .

34

Table 4: The Fifty Most Populated MSAs Sorted on Risk Premia SantaAna-Anaheim-Irvine,CA SanFrancisco-SanMateo-RedwoodCity,CA SanJose-Sunnyvale-SantaClara,CA LosAngeles-LongBeach-Glendale,CA SanDiego-Carlsbad-SanMarcos,CA Oakland-Fremont-Hayward,CA Sacramento-Arden-Arcade-Roseville,CA Riverside-SanBernardino-Ontario,CA WestPalmBeach-BocaRaton-BoyntonBeach,FL FortLauderdale-PompanoBeach-DeerﬁeldBeach,FL(MSAD) Miami-MiamiBeach-Kendall,FL LasVegas-Paradise,NV Baltimore-Towson,MD Orlando-Kissimmee,FL Phoenix-Mesa-Scottsdale,AZ Edison-NewBrunswick,NJ Wilmington,DE-MD-NJ Newark-Union,NJ-PA NewYork-WhitePlains-Wayne,NY-NJ Nassau-Su↵olk,NY Portland-Vancouver-Beaverton,OR-WA Camden,NJ Providence-NewBedford-FallRiver,RI-MA Philadelphia,PA Chicago-Naperville-Joliet,IL Milwaukee-Waukesha-WestAllis,WI LakeCounty-KenoshaCounty,IL-WI Boston-Quincy,MA RockinghamCounty-Stra↵ordCounty,NH Minneapolis-St.Paul-Bloomington,MN-WI Cambridge-Newton-Framingham,MA Peabody,MA Gary,IN St.Louis,MO-IL Nashville-Davidson-Murfreesboro-Franklin,TN Warren-Troy-FarmingtonHills,MI Detroit-Livonia-Dearborn,MI SanAntonio,TX Columbus,OH Charlotte-Gastonia-Concord,NC-SC Indianapolis-Carmel,IN Cincinnati-Middletown,OH-KY-IN KansasCity,MO-KS Pittsburgh,PA Cleveland-Elyria-Mentor,OH Atlanta-SandySprings-Marietta,GA Houston-SugarLand-Baytown,TX Dallas-Plano-Irving,TX NewOrleans-Metairie-Kenner,LA Denver-Aurora-Broomﬁeld,CO Price 482828 714716 520378 340842 361444 351486 235755 191675 243844 239411 251186 178390 267441 192993 156698 330532 244404 390079 402722 411170 276563 195479 227287 190433 248207 230988 218836 336747 239595 186015 387527 318301 117579 137199 167671 123421 58617 151913 165190 197281 122092 166943 144052 126616 144077 191194 159217 149710 163257 258493 Wages 51894 76042 58531 42265 46649 53093 41119 30634 58358 41974 35887 39920 47881 35717 36156 51865 43643 56655 54540 57617 39942 42626 40887 47361 45510 42824 51782 55220 45231 47653 60093 50895 35922 41823 39768 44488 32094 34937 38741 39621 39297 39066 40396 42104 40118 38336 45835 43458 41740 48010 Amenities 316297 382048 304677 200361 208206 177091 137359 115268 49911 94247 89587 60215 78048 70500 37786 112694 86808 140479 140388 132911 85134 44754 69347 17739 70747 67918 33225 92699 57195 7047 127056 107415 330 -2587 16522 -13735 -32071 33157 37593 41230 -1721 37096 8017 -15350 21777 47676 1872 19554 -5321 51834 Risk Premia -140048 -135615 -130572 -97654 -90688 -88480 -77007 -62994 -59038 -57586 -49034 -44996 -43632 -42265 -34795 -34683 -32946 -31607 -31317 -30188 -29251 -27447 -22110 -21036 -15429 -11209 -10426 -7879 -6151 -5723 -4981 -4670 -3063 -2983 -1207 314 533 1326 2023 2221 2270 2389 2919 3052 5183 6041 6079 7398 13291 20281

Note - All the variables are for year 2008 and given in 2008 dollars. The risk premia is estimated by the exposure of a metropolitan area to the three factors and to the idiosyncratic variance. A negative risk premium should be interpreted as a cheaper house price due to the exposure of the MSA to risk. The amenities are calculated as the amount left over from prices after removing the e↵ect of wages, expected growth, and risk premia.

35

In the model presented here these variables correspond to "t and ⇠i respectively. The estimated i standard deviations that are used in my model are ˆ" = $15499.21 and ˆ⇠ = $7073.40 in year 2000 dollars.

4.2

Risk-Aversion Parameters

While there is some consensus in the literature about the magnitude of the relative risk-aversion (RRA) parameter at least for Constant Relative Risk Aversion utility functions, there is no agreement on the absolute risk-aversion (ARA) coe cient.30 For these simulations I experiment with a range of absolute risk-aversion parameters anchored to be not to far from the estimates of Table 3. I simulate the model for ARA coe cients from 3 · 10
6

to 3 · 10

5,

which in the simulations imply

average RRA coe cients from .41 to 4.15. These values for the RRA are in line with estimates of the previous literature that ﬁnd that the RRA coe cient is between 1 and 5. In the case where I allow for heterogeneity in risk-aversion I draw these parameters from a uniform distribution. The average ARA and RRA parameters are the same as above except for that I allow for bounds around the values given.

4.3

Results

I ﬁrst simulate the model under the assumption that there is no heterogeneity in risk-aversion. All the results in the homogeneous agents case will be driven by the fact that completing the market will lead to better risk-sharing. I start from the baseline where there is no correlation between stocks and and factors that drive the local economy. I then consider the cases when new ﬁnancial instruments are created that correlate with three factors proposed above and the case when the ﬁnancial instrument allows for perfect insurance. The results from a series of simulations are presented in Table 5. In the ﬁrst panel I study the e↵ects of market completeness on house prices. In the ﬁrst row the model is simulated for ARA of .3 · 10
5,

which on average implies a RRA

of .41 for the simulated wealth levels. Starting from the case where all variance is noninsurable, if we create an asset that spans the HMKT factor, prices will increase by 2.4%.31 If in addition we create another asset that also spans the HMLH factor, prices will increase by a total of 3.14%
See for example Vigna (2009) We have reason to believe that households can hedge against the HMKT risk factor by using existing ﬁnancial assets as shown from the high R-squared estimates when we regress HMKT on traded ﬁnancial assets in the previous section. If we agree that the HMKT factor is already spanned, we can alternatively interpret the negative of the magnitudes from these simulations as the e↵ect of prohibiting the households from using ﬁnancial assets to hedge the HMKT risk.
31 30

36

in relation to what they where when no factors were spanned. If we span all the three factors, prices will go up by 4.24%. If we create assets that span not only the three factors but all of the remaining variation in prices and wages, prices would go up by about 6%. In the next few rows I simulate the model again with higher ARA and RRA parameters. Price changes are very sensitive to the risk-aversion parameter. In the extreme case of a RRA of 4.15 prices increase by 40% when we span all sources of variance. The second panel in Table 5 shows welfare e↵ects of completing the market in terms of the compensating variation. The compensating variation is the amount of dollars a household should be compensated after a policy change in order to reach the initial utility level. Here I interpret the compensating variation as the maximum payment a homeowner is willing to make in order to have access to a ﬁnancial asset that spans a particular factor. In the ﬁrst row the model is simulated for ARA of .3 · 10
5,

which on average implies a RRA of .41 for the simulated wealth levels. Starting

form the case where all variance is noninsurable, on average homeowners would be willing to pay $858 dollars in order to gain access to a ﬁnancial asset that spans the HMKT factors, $2680 for a ﬁnancial asset that spans all three factors, and $3600 for one that spans all of the volatility in the market. The willingness to pay is again closely tied to the risk-aversion parameter. Homeowners are willing to pay $7760, $15097 and $22125 if their RRA parameters were 1.38, 2.77 and 4.15 respectively. The willingness to pay is as high as $31342 for an asset that spans all sources of volatility for the case of high RRA of 4.15. Next I turn to simulating the model by allowing heterogeneity in risk-aversion over the population. The results from a series of simulations are displayed in Table 6. In each row the mean of the risk-aversion parameters is set to be the same as in Table 5. The ﬁrst panel shows e↵ect of completing markets on prices. In the ﬁrst row for example, I draw the absolute risk-aversion parameter from a continuous uniform distribution U (1, 59) · 10
7,

which for the average wealth implies RRA

coe cients as if they were drawn from the uniform distribution U (.01, .8). The e↵ects on prices are similar to those previous table. The range of price increases goes from 2.67% when homeowners are given access to assets that span the HMKT factor to 7.12% when all possible sources of risk are spanned. Prices can jump up by 43% when all variance is spanned if the RRA is drawn from U (.02, 12). For a more reasonable range of RRA parameters drawn from U (.02, 7), prices increase from 5.5% when only the HMKT is spanned to 30% when all the sources of volatility are spanned. The second panel in Table 6 simulates the e↵ects of market completeness on productivity increases. In general the creation of tradable ﬁnancial instruments that correlate with housing and 37

Table 5: The E↵ects of Completing Markets on Prices and Welfare when Households have Identical Risk Preferences HMKT+ HMKT Price % ARA .3 · 10 1 · 10 2 · 10 3 · 10 ARA .33 · 10 1 · 10 2 · 10 3 · 10
5 5 5 5 5 5 5 5

HMKT+ SMBH

HMKT+ HMLH+SMB

All Variance

HMLH

RRA 0.41 1.38 2.77 4.15 2.64 4.08 6.15 8.21 3.14 5.75 9.48 13.21 3.74 7.77 13.52 19.27 4.24 9.43 16.85 24.26 5.89 14.92 27.83 40.73

Willingness to Pay $ RRA 0.41 1.38 2.77 4.15 856 1622 2775 3901 1243 2915 5411 7771 2292 6467 12461 18255 2680 7760 15097 22125 3600 10895 21339 31342

Note - The model is simulated for 216 MSAs using parameters from estimated house price and wage processes. Each column displays the e↵ect of creating ﬁnancial instruments that correlate with the given housing factors. The last column displays the e↵ects of creating ﬁnancial instruments that span all of the variance in wages and prices. The ﬁrst panel shows percent changes in average house prices, and the second panel shows the average compensating variation in dollars. The willingness to pay is the average amount a household would be willing to pay for the creation of the ﬁnancial instrument. ARA stands for the coe cient of absolute risk-aversion. RRA is the coe cient of relative risk-aversion implied by the ARA for the average household.

38

Table 6: Then E↵ects of Completing Markets on Prices, Productivity and Welfare with Heterogeneous Risk Preferences HMKT+ HMKT Price % ARA U (1, 59) · 10 U (1, 199) · 10 U (1, 399) · 10 U (1, 599) · 10
7 7 7 7

HMKT+ SMBH

HMKT+ HMLH+SMB

All Variance

HMLH

RRA U (.01, .8) U (.01, 3) U (.02, 7) U (.02, 12) 2.67 3.90 5.50 7.27 3.40 6.15 9.74 13.65 4.60 9.14 14.95 21.51 5.36 11.54 19.38 28.15 7.12 16.87 29.15 42.84

Productivity % ARA U (1, 59) · 10 U (1, 199) · 10 U (1, 399) · 10 U (1, 599) · 10
7 7 7 7

RRA U (.01, .8) U (.01, 3) U (.02, 7) U (.02, 12) 0.29 0.78 0.91 1.01 0.74 1.84 2.29 2.67 2.20 6.69 8.39 9.16 2.43 8.04 10.86 12.47 2.62 9.78 14.59 18.58

Willingness to Pay $ ARA U (1, 59) · 10 U (1, 199) · 10 U (1, 399) · 10 U (1, 599) · 10
7 7 7 7

RRA U (.01, .8) U (.01, 3) U (.02, 7) U (.02, 12) 894 1344 2009 2782 1280 2388 3910 5617 2260 4561 7469 10953 2665 5665 9431 13869 3554 8119 13758 20257

Note - The model is simulated for 216 MSAs using parameters from estimated house price and wage processes. Each column displays the e↵ect of creating ﬁnancial instruments that correlates with the given housing factors. The last column displays the e↵ects of creating ﬁnancial instruments that span all of the variance in wages and prices. The ﬁrst panel shows percent changes in average house prices, the second shows percent changes in average wages, and the third panel shows the average compensating variation in dollars. The willingness to pay is the average amount a household would be willing to pay for the creation of the ﬁnancial instrument. ARA stands for the coe cient of absolute risk-aversion. RRA is the coe cient of relative risk-aversion implied by the ARA for the average household.

39

income risk improves the households’ ability to hedge risk and consequently lowers housing risk premia. In addition when there is heterogeneity in risk preferences, lowering the amount of noninsurable volatility leads to a di↵erent sorting of households across space. In the new sorting equilibrium, human capital is allocated more e ciently leading to higher overall productivity or wage level in the economy. The simulated e↵ects on productivity are sensitive to the assumption on the distribution of the risk-aversion parameters. For a reasonable range, when the RRA parameters are distributed according to the uniform distribution U (.02, 7), wages increase by 8.39% when all of the three factors are spanned, and they increase by 10.86% when all of the sources of volatility are spanned. The last panel of Table 6 shows welfare e↵ects of completing the market in terms of willingness to pay in dollars. The average willingness to pay for access to a tradable asset varies widely with the risk-aversion parameters as well. For the range of RRA parameters that are drawn from the uniform distribution U (.02, 7) on average homeowners would be willing to pay $2009 dollars in order to gain access to a ﬁnancial asset that spans the HMKT factors, $9431 for a ﬁnancial asset that spans all three factors, and $13758 for one that spans all of the volatility in the market. Taken together, results from these simulations can be taken as support to the idea that creating ﬁnancial instruments that improve the households ability to manage risk is very beneﬁcial in many levels. Particularly, under di↵erent assumptions about parameters values, simulations show that on average creating assets correlate with all of the sources of volatility signiﬁcantly increase productivity, wages and welfare.

5

Conclusion

In order to understand the links between underlying risk factors, house prices and household location decisions I develop a micro-founded equilibrium model. The approach used is unique in that it merges standard methodologies used in urban economics, which study sorting and spatial properties of the problem, with models from continuous-time ﬁnance that study ﬁnancial assets. This ﬂexible and estimable model simultaneously considers risk-aversion, multiple sources of uncertainty, rich agent heterogeneity, sorting, portfolio choice and asset prices in one uniﬁed model. One key theoretical result is that home prices are derived to be a closed-form function of the underlying productivity of the economic base of a city minus a city-speciﬁc risk premium, which is a function of agent heterogeneity, sorting and risks in the economy. The problem of household sorting across

40

space turns out to be very similar to that studied by Bayer et. al. (2005). Asset portfolio decisions are also found to be a generalized version of classic results in portfolio choice in ﬁnance. The model is then estimated using US price and wage data and is simulated to study the e↵ect of completing markets on house prices, household sorting across space and on overall productivity in the economy. The estimated risk premia imply that on average homes are about $20000 cheaper than they would be if owners were risk-neutral, although there is large heterogeneity across metropolitan areas. Creating ﬁnancial instruments that can be used for hedging purposes, lowers housing risk premia, increases welfare and increases productivity in the economy through sorting e↵ects. For a reasonable range of risk-aversion parameters, I ﬁnd that completing markets can increase home prices by about 20 percent, increase productivity in the economy by 10 percent and signiﬁcantly improve welfare. The average willingness to pay for access to ﬁnancial instruments that correlate with all of the sources of risk in the economy is between $10000 to $20000 per homeowner, depending on the assumed risk-aversion. This willingness to pay also varies widely across individuals and the sources of risk they are exposed to: more risk averse agents and agents that locate in more volatile cities are willing to pay much more than the average to gain access to complete markets. Taken together, these ﬁndings draw attention to the potential beneﬁts for creating ﬁnancial instruments that correlate with house prices and income in every metropolitan area. Although these beneﬁts are large, comparably large implementation di culties may exist in creating a market for instruments to manage housing risk. As in any new market, some di culties include marketing to and educating homeowners about these products, pricing them correctly, as well as creating and maintaining enough liquidity in these markets. The results in this paper suggest that perhaps the lack of a well-functioning market for hedging housing risk is not because this risk is trivial, but rather because we haven’t found a way to implement the idea successfully yet.

References
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41

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117 (518), 353–374. Bayer, Patrick Robert McMillan and Kim Rueben, “An Equilibrium Model of Sorting in an Urban Housing Market,” NBER Working Paper 10865, 2009. Berry, Steven, James Levinsohn, and Ariel Pakes, “Automobile Prices in Market Equilibrium,” Econometrica, July 1995, 63 (4), 841–90. Bishop, K., “A dynamic model of location choice and hedonic valuation,” Unpublished, Washington University in St. Louis, 2008. Campbell, J.Y. and L.M. Viceira, Strategic asset allocation: portfolio choice for long-term investors, Oxford University Press, USA, 2002. Case, Karl E., John Cotter, and Stuart A. Gabriel, “Housing Risk and Return: Evidence From a Housing Asset-Pricing Model,” Working Papers 201005, Geary Institute, University College Dublin 2010. Cocco, J.F., “Portfolio choice in the presence of housing,” Review of Financial studies, 2005, 18 (2), 535. , F.J. Gomes, and P.J. Maenhout, “Consumption and portfolio choice over the life cycle,” Review of ﬁnancial Studies, 2005, 18 (2), 491. Cochrane, J.H., “A simple test of consumption insurance,” Journal of Political Economy, 1991, 99 (5), 957–976. Coulson, Edward Crocker Liu and Sriram Villupuram, “Urban Economic Base as a Catalyst for Movements in Real Estate Prices,” Working Paper, 2008. Du e, D., Dynamic Asset Pricing Theory, Princeton University Press, 1996. Du e, Darrell and Matthew O. Jackson, “Optimal hedging and equilibrium in a dynamic futures market,” Journal of Economic Dynamics and Control, 1990, 14 (1), 21 – 33. Flavin, M. and S. Nakagawa, “A model of housing in the presence of adjustment costs: A structural interpretation of habit persistence,” American Economic Review, 2008, 98 (1), 474– 495. 42

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Ortalo-Magne, Francois and Andrea Prat, “Spatial Asset Pricing: A First Step,” Working Paper, 2010. Piazzesi, M., M. Schneider, and S. Tuzel, “Housing, consumption and asset pricing,” Journal of Financial Economics, 2007, 83 (3), 531–569. Pijoan-Mas, J., “Precautionary savings or working longer hours?,” Review of Economic Dynamics, 2006, 9 (2), 326–352. Roback, J., “Wages, rents, and the quality of life,” The Journal of Political Economy, 1982, 90 (6), 1257. Rosen, S., “Wage-based indexes of urban quality of life,” Current issues in urban economics, 1979, 3. Shiller, R.J., Macro markets, Clarendon Press, 1993. , “The new ﬁnancial order: Risk in the 21st century,” Cato Journal, 2003, 23 (2). Shreve, Steven E., Stochastic calculus for ﬁnance. II Springer Finance, New York: SpringerVerlag, 2004. Tepla, L., “Optimal hedging and valuation of nontraded assets,” European Finance Review, 2000, 4 (3), 231. Tracy, J.S., H.S. Schneider, and S. Chan, “Are stocks overtaking real estate in household portfolios?,” Current Issues in Economics and Finance, 1999, 5 (5). Vigna, Elena, “Mean-variance ine ciency of CRRA and CARA utility functions for portfolio selection in deﬁned contribution pension schemes,” CeRP Working Papers, September 2009, (89). Yao, R. and H.H. Zhang, “Optimal consumption and portfolio choices with risky housing and borrowing constraints,” Review of Financial Studies, 2005, 18 (1), 197. Zeldes, S.P., “Consumption and liquidity constraints: an empirical investigation,” The Journal of Political Economy, 1989, 97 (2), 305–346.

44

Appendix
A
A.1

Proofs
Proof of Proposition 1

The problem at some time t faced by each household who dies at time T is: maxV (Xt , wt , xt ) = sup
✓t ,l

Et e

i

1 (XT + r (er(T

t)

1) i A l )

(21)

For notational simplicity we drop the i and l subscript from now on. Conditional on being in location l we solve for the value function and for the optimal portfolio decisions. The maximization problem is subject to the wealth evolution equation:
0 dXt = ✓t DPt1 dPt + Xt

10 ✓t rdt + wt dt 10 ✓t rdt + wt dt

0 = ✓t (µdt + ⌃P dBt ) + Xt

where ✓t is nx1. The Hamilton-Jacobi-Bellman equation associated with this problem is ˙ 0 = V + VX Xt
0 0 ✓t 1 r + wt + ✓t µt + Vy st + Vs (m 32 :

1 1 0 st ) + VXX ✓t ⌃P ⌃0 ✓ + Vss ⌃s ⌃0 P s 2 2

0 +VXs ✓t ⌃P ⌃0 x

Now we take FOC and solve for ✓t : ✓t = 1 VXX
1 ⌃P P [VX (µt

1r) + VXs ⌃P s ]

where for simplicity we we have deﬁned ⌃Y Z = ⌃Y ⌃0 . Plugging back in and simplifying we get: Z ˙ 0 = V + VX (Xr + wt ) + Vy st + Vs (m 1 st ) + Vss ⌃ss 2 ⇤

1 2 1 VX (µ0 10 r) ⌃P P (µt 1r) t 2 V ⇥ 0 XX 1 1 1 VX VXs (µt 10 r) ⌃P P ⌃P s + ⌃sP ⌃P P (µt 2 VXX 2 ⌃ ⌃ 1⌃ 1 VXs sP P P P s 2 VXX
32

1r)

Svensson and Werner (1993) derive a similar HJB for a related problem

45

Substitute initial wealth out of the value function. V (t, Xt , wt , st ) = sup✓s = e i Et e [Xt er(T

i

(er(T

t) X

´T t+ t

er(T

u)

1 0 (✓u DPu dPu

10 ✓u rdu)+

´T t er(T

u) w

1 u du+ r

(er(T

t)

1) i A l )

t) + 1 r

(er(T

t)

1)( i Al +⇠i +"l )] i

g(T

t, yt , st )

with terminal condition g(0, w, s) = e 0 = g ˙ +g gs ✓ ✓ (m
t)

[pT ] .

Finally in the ﬁnal simpliﬁed form: 1r
0 1 ⌃P P ⌃sP

st ) yt +

er(T

1 0 µ 2 t

1⇥ 0 µt 2

1 10 r ⌃P P (µt

1 gss ⌃ss gy st 2 1 1 g 2 ⌃P s ⌃P P ⌃sp + s 2 g

+ ◆ 1r)

1 ⌃ P s ⌃P P

(µt

1r)

⇤

◆

Since this is nonlinear we need to do a change of variables to get it to be linear. Making the substitution: g (T we get: 0 =
1 ⌃ss ⌃P s ⌃P P ⌃sP ˙ + er(T t) yt ⌃ss ✓ 1⇥ 0 1 1 (m st ) µt 10 r ⌃P P ⌃P s + ⌃sP ⌃P P (µt s 2 1 ss ⌃ss y st 2

t, y, s) =

(T

t, y, s) ⌃ss

⌃ss ⌃P s ⌃ 1 ⌃sP PP

e

1 (µ0 t 2

1 10 r)⌃P P (µt 1r)(T t)

1r)

⇤

◆

Making a last substitution G (t, y, s) = ˙ 0 = G er(T
t)

(T

t, y, s) we get the linear pde: (m st )
1 ⌃P s ⌃P P (µt

yt

⌃ss

1 ⌃sP ⌃P P ⌃P s G+ ⌃ss

1r) Gs

1 + ⌃s ⌃T Gss + Gy st s 2 with terminal condition: G(T, y, s) = e 2 (µt
1 0 1 10 r)⌃P P (µt 1r) ⌃ss ⌃P s ⌃ 1 ⌃sP PP ⌃ss

(T t)

⌃ss

⌃P s ⌃ 1 ⌃sP PP ⌃ss

pT

46

Now we do a change of measure by multiplying the probability density function by: " 1 1 ⌃sP ⌃P P (µt p ⇠T = exp 4 2 ⌃ss 2 2 1r) #2 "
1 ⌃sP ⌃P P (µt p ⌃ss

(T

t)

1r)

#

⌃ p s (BT ⌃ss

Bt ) 5

3

Then we can use the Feynman Kac theorem to get:33 G (t, y, s) = E 4⇠T G(T, y, s)e variable ✓: V (t, y, s) = e [Xt er(T
t) + 1 r



⌃ss

⌃sP ⌃ 1 ⌃P s PP ⌃ss

´T t er(T

u) y

u du

3 5

Now we can ﬁnally write the value function in a form that does not involve the portfolio decision

(erT

t

1)( i Al +⇠i +"l )] i

1 (µ0 t 2

1 10 r)⌃P P (µt 1r)(T t)

[G (t, y, s)] ⌃ss

⌃ss ⌃sP ⌃ 1 ⌃P s PP

In order to save some notation: ⇤ ¯⇥ eF Et eFT ⌃ss
⌃ss ⌃sP ⌃ 1 ⌃P s PP

V (t, y, s) = where:

F

=

1 1 0 1 ⌃sP ⌃P P (µt 1r) 1 µt 10 r ⌃P P (µt 1r) (T t) (T 1 2 2 ⌃ss ⌃sP ⌃P P ⌃P s  ⇣ ⌘ 1 rT t l l Xt er(T t) + e 1 i A + ⇠ i + "i r

2

t)

FT

=

" i 1 ⌃sP ⌃P P (µt p ⌃ss

1r)

#

⌃ss

1 ⌃sP ⌃P P ⌃P s ⌃ss

⌃ p s (BT ⌃ss ✓ˆ T er(T t Bt )
u)

yu du + AyT + BsT + C

◆

Notice that F is just a constant while the term FT depends on the states. Using the assumed processes for yt and st we can write FT in terms of states at time t. First we make a transformation that transforms the m independent Brownian motions Bt to a single brownian motion Zt for each
33

See Du e (1996), Theorem and Condition 2 on p. 296

47

location. To do this we redeﬁne: ⌃s dZt = p dBt ⌃ss which means that now we can write the evolution of the drift st of the income process as: dst = = with  = p (m (m st ) dt + ⌃s dBt st ) dt + dZt

⌃ss . This means that st is just a simple one dimensional Ornstein-Uhlenbeck process.

We can use the solutions to the processes yt and st to write the components of FT as:
(T t)

sT = st e

+m 1

⇣

e

(T t)

⌘

+

ˆ

T

e

(u T )

dZu

t

yT

= yt +

ˆ

T

su du 1⇣ 1 e
(T t)

t

= yt + s t

⌘

✓ + m (T

t)

1⇣

1

e

(T t)

⌘◆

+

ˆ

T

t

⇣

1

e

(T s)

⌘

dZs

ˆ

T

e

r(T u)

t

✓ ⌘ ⌘ 1⇣ 1 1⇣ 1 ⇣ r(T t) yu du = yt 1 e + st 1 er(T t) + e (T r r r+ ⌘ ⌘ 1 ⇣ 1⇣ m 2 rT + 1 er(T t) (rt + 1) + mt 1 er(T t) r✓ r ⌘ ⌘◆ m 1⇣ 1 ⇣ r(T t) (T t) r(T t) 1 e + e e r r+ ˆ T  ⌘ ⌘  1⇣ 1 ⇣ 1 er(T s) + e (T s) er(T s) dZt r r+ t

t)

e

r(T t)

⌘◆

Rewriting FT in terms of states at time t we get: ⇣ ⌘ ˆ ˜ ˜ ˜ k1t yt + k2t st + k3t +
T

FT = p

i

t

⇣

˜ k4

where dZt = ⌃s / ⌃ss dBt and: ˜ k1t = ⌃ss
1 ⌃sP ⌃P P ⌃P s ⌃ss

⌘ p ˜ ⌃ss k2t dZu i !

A

1

er(T r

t)

48

˜ k2t =

⌃ss 1

1 ⌃sP ⌃P P ⌃P s ⌃ss

(

A

1

e

(T t)

+ Be
t)

(T t)

er(T r

t)

+

e

(T t)

er(T (r + )

)

˜ k3t =

˜ mk 2 + m

⌃ss

1 ⌃sP ⌃P P ⌃P s ⌃ss

1 ⇣ rT + 1 r2

er(T

t)

⌘ (rt + 1) + C

(

Am (T

t) + Bm +

mt 1

er(T r

t)

The only random part of FT mean and variance:34

deterministic function and thus a martingale. Therefore, the distribution of FT is a normal with ⇣ ˜ ˜ ˜ k1t yt + k2t st + k3t ˜ k4 i 1 ⌃sP ⌃P P (µ 1r) p ⌃ss ⇣ ⌘ ´T p ˜ ˜ now t k4 ⌃ss k2t dZt , which is an Ito integral over a i

˜ k4t =

Et (FT ) = V ar (FT ) = ˆ

i T

t

⇣

Using the fact that FT is normally distributed we can write the value function as:

⌘2 p ˜ ⌃ss k2t dt

⌘

V (t, y, s) =

e

F

h

e

E(FT )+ 1 V ar(FT ) 2

i

⌃ss

⌃ss ⌃sP ⌃ 1 ⌃P s PP

Simplifying the above equation and including the location superscript l, the value function is: V (t, y, s, l) = with: Utl =  er(T ⇣
t)

eUt

l

i

Xt +

1 + 2

1 ⇣ r(T e r

t)

2 i

⌃l ss

1 ⌃l ⌃P P ⌃l s sP P

⌘ˆ

T

ˆ ⌘⇣ ⌘ ˆl l ˆl ˆl 1 M l + ⇠ i + " l + k 1 yt + k 2 s l + k 3 + i i t ⇣ ˆl kt2 ⌘2 du 1 0 µ 2 t
1 10 r ⌃P P (µt

T

t

ˆl ˆl k4 k2 t)

q ⌃l du ss

1r) (T

t

ˆl where we deﬁne kit =
34

1 ⌃l ⌃l ⌃P P ⌃l s ˜ l ss sP P kit l ⌃ss

ˆl ˜l for i = 1...3 and k4 = k4 .

⌅

See Shreve (2004) chapter 4.

49

A.2

Proof of Proposition 2

For clarity we suppress the city-speciﬁc l superscript. In the proof of Proposition 1, I show that the optimal amount invested in stocks for an individual who is t l is: ✓ilt = 1 VXX (⌃P P )
1

t0 years old and lives in location i

h

VX (µ

1r) + VXs ⌃l s P

Taking the derivatives of the value function equation (4): l V (t, y, s, l) =

eUt

VX =

er(T

l t) Ut

e

VXX =

l 2 2r(T t) Ut

e

e

VXs =

l 2 ˆ l r(T t) Ut k2 e e

Substituting these derivatives in the equation for ✓ilt we get: ✓ilt = (⌃P P ) where: ˆ k2t = ( A 1 1 e er(T r
(T t) 1

"

1 er(T
t)

(µt

1r) +

ˆl k2t er(T
t)

⌃l s P

#

+ Be
t)

(T t)

+

e

(T t)

er(T (r + )

t)

) ⌅

A.3

Proof of Proposition 6

For a given A and B, in each period we look for a C l that clears the housing markets in some given period t. For the generation of agents born in some period t, the problem in equation 7 is identical to static horizontal sorting models studied in the urban economics literature. Bayer, McMillan and Rueben (2005) prove that under the assumption that "l has continuous support there exist a unique i 50

vector C = {C 1 , C 2 ...C L } that clears the market and gives the unique sorting of households across space. Their proof applies exactly to this problem so is not reproduced here. The reader is referred to the original article for details. Given that we now have unique values for C l , we turn to determining what A and B need to be in order for the markets to be in equilibrium across time. First note that a measure 1 of households is born and looking to buy a home each period. Because the joint distribution of heterogeneity in Pr i, l i , ⇠i , "i

is iid over time, the same set of households will be in the market in every time i, l i , ⇠i , "i

period. In other words in every period the full support of the joint distribution P r will be realized.

Consider the sorting problem in equation 7 at two di↵erent time periods t and u where yt 6= yu and st = su . If k1 6= 0 then the equilibrium C of the sorting equilibrium in period t will have to be di↵erent than that in period u. Because we are looking for a vector C that is constant across time we conclude that it must be the case that k1 = 0 for every location l. A similar argument leads to l the conclusion that in order to have A, B, and C to be constant across time it must be that k2 = 0 l l for every location l. Setting k1 = k2 = 0 we ﬁnd the unique solutions be:

Al = Bl =

1 r
l)

1 r (r +

Given the above solutions A, and B we can now get the unique equilibrium market returns for stocks in equation 6. We have therefore found the three vectors A, B, and C that give rise to equilibrium both in the asset market and the housing market. ⌅

A.4

Proof of Proposition 7

The location l superscript will be omitted for the rest of the proof for notational simplicity, and this analysis is the same for any location l. The idea here is to use the solution to the process for yt to solve for the expectation in closed-form. Recall that the processes that drive income are: dyt = st dt dst = (m st ) dt + dZt

51

p p where dZt = ⌃s / ⌃ss dBt and  = ⌃ss . This change of variables transforms the st process to a simple one dimensional Ornstein-Uhlenbeck process. The solution for this process for some T > t is: sT = st e
(T t)

⇣ +m 1

e

(T t)

⌘

+

ˆ

T

e

(v T )

dZv

t

For any T > t we can write yT as: yT = yt + = yt + = yt + ˆ ˆ ˆ
T

t T

t T

su du ✓ st e ⇣ st e 1 e

(u t)

+m 1 +m 1 ⇣

⇣

e

(u t)

(u t)

t

= yt + s t

1⇣

(T

e ✓ ⌘ t) + m (T

(u t)

⌘⌘

⌘

+

ˆ

u

e ˆ

(v u) T

dZv du
T

t

du + 1

t)

1⇣

t

✓ˆ

◆

v

e

(T t)

e ⌘◆

(v u)

du dZv ⇣ 1 e
(T v)

+

ˆ

T

◆

t

⌘

dZv

where in the second equality we substitute in for the solution to su , in the third equality we changed the order of integration for the last term, and in the last equality we solve the deterministic integrals. We can now substitute the solution to y directly in the price equation: ˆ 1 pt = Et e r(u t) yu du + ⇡ ˆ 1t = e r(u t) Et (yu ) du + ⇡ ✓ ˆt 1 1⇣ r(u t) = e yt + s t 1 e t (u t)

=

1 1 yt + st + 2 m+⇡ r r(r + ) r (r + )

⌘

+m u

✓

t

1⇣

1

e

(u t)

⌘◆◆

du + ⇡

In the second equality we interchange the integral with the expectation.35 In the third equality we ⇣´ ⌘ u substitute for the expected value of yu . Here we use the fact that E t  1 e (u v) dZv = 0 since Ito integrals of deterministic functions are martingales.
35

⌅

In order to interchange the integral and the expectation, ﬁrst notice that the expectation is an integral over some probability measure. We ⇣ then use Fubini’s theorem that says that the order of integration can be changed as long can ⇣´ ⌘ ⌘ ´1 u  as t e r(u t) Et 1 e (u v) dZv du < 1. This condition is satisﬁed for our problem. The intuition is t that the expectation of the absolute value of the martingale grows slower that the discounting term e speaking the expectation grows linearly over time while the discounting term grows exponentially. r(u t)

. Broadly

52

B

Data Appendix
Table 7: The Factor Loadings and the R-squared
Metropolitan Area Akron,OH Albany-Schenectady-Troy,NY Albuquerque,NM Allentown-Bethlehem-Easton,PA-NJ Amarillo,TX Anchorage,AK AnnArbor,MI Atlanta-SandySprings-Marietta,GA Augusta-RichmondCounty,GA-SC Austin-RoundRock,TX Bakersﬁeld,CA Baltimore-Towson,MD BarnstableTown,MA BatonRouge,LA Beaumont-PortArthur,TX Bellingham,WA Bethesda-Frederick-Rockville,MD Binghamton,NY Birmingham-Hoover,AL Bloomington-Normal,IL BoiseCity-Nampa,ID Boston-Quincy,MA Boulder,CO Bremerton-Silverdale,WA Bridgeport-Stamford-Norwalk,CT Bu↵alo-NiagaraFalls,NY Burlington-SouthBurlington,VT Cambridge-Newton-Framingham,MA Camden,NJ Canton-Massillon,OH CapeCoral-FortMyers,FL Casper,WY CedarRapids,IA Charleston-NorthCharleston-Summerville,SC Charlotte-Gastonia-Concord,NC-SC Charlottesville,VA Chattanooga,TN-GA Cheyenne,WY Chicago-Naperville-Joliet,IL Chico,CA HMKT 3.059 4.554 11.580 3.490 4.884 18.350 6.879 7.654 3.913 7.783 11.340 7.352 25.493 8.048 6.983 -1.084 11.745 4.753 4.780 4.266 3.750 23.981 14.178 4.590 34.996 0.267 5.099 25.008 4.685 3.639 7.485 4.515 3.024 10.308 0.862 6.853 4.609 6.160 5.863 12.545 SMBH -0.174 -18.886 -1.059 -24.722 2.958 3.117 -8.313 -4.361 -0.744 8.152 -21.111 -22.227 -41.569 6.369 6.474 -9.358 -45.196 -7.221 0.441 0.564 -0.033 -36.247 6.945 -8.663 -49.136 -3.779 -21.748 -37.554 -19.207 2.355 -15.013 9.585 1.403 -7.175 0.501 -15.302 -1.685 0.039 -12.193 -22.399 HMLH 0.341 2.885 16.847 0.453 0.661 10.947 -2.644 -2.651 3.810 3.007 22.663 16.914 -9.980 8.294 4.988 27.285 22.672 -1.851 2.174 2.004 14.016 -13.300 -2.888 22.584 -8.938 0.404 3.842 -18.296 6.015 -0.821 17.192 13.116 1.477 8.320 -1.556 14.728 1.836 4.876 4.882 25.952 R-squared 0.107 0.588 0.470 0.583 0.208 0.209 0.371 0.596 0.360 0.131 0.872 0.846 0.767 0.538 0.596 0.780 0.852 0.225 0.370 0.236 0.506 0.695 0.245 0.672 0.600 0.105 0.804 0.658 0.718 0.286 0.752 0.285 0.213 0.489 0.027 0.741 0.435 0.271 0.783 0.785

53

Table Continued: The Factor Loadings and the R-squared
Metropolitan Area Cincinnati-Middletown,OH-KY-IN Cleveland-Elyria-Mentor,OH CollegeStation-Bryan,TX ColoradoSprings,CO Columbia,SC Columbus,OH CorpusChristi,TX Corvallis,OR Dallas-Plano-Irving,TX Dalton,GA Davenport-Moline-RockIsland,IA-IL Dayton,OH Deltona-DaytonaBeach-OrmondBeach,FL Denver-Aurora-Broomﬁeld,CO DesMoines-WestDesMoines,IA Detroit-Livonia-Dearborn,MI Durham-ChapelHill,NC EauClaire,WI Edison-NewBrunswick,NJ ElPaso,TX Elkhart-Goshen,IN Erie,PA Eugene-Springﬁeld,OR Evansville,IN-KY Fayetteville-Springdale-Rogers,AR-MO Flint,MI FortCollins-Loveland,CO FortLauderdale-PompanoBeach-DeerﬁeldBeach,FL(MSAD) FortWayne,IN Fresno,CA Gary,IN GrandJunction,CO GrandRapids-Wyoming,MI Greeley,CO Greensboro-HighPoint,NC Harrisburg-Carlisle,PA Hartford-WestHartford-EastHartford,CT Holland-GrandHaven,MI Honolulu,HI Houston-SugarLand-Baytown,TX Huntsville,AL Indianapolis-Carmel,IN Jackson,MS HMKT 4.171 5.112 8.952 11.970 2.965 3.280 10.986 1.374 7.948 2.745 2.765 2.790 9.910 15.812 2.158 2.960 3.903 2.270 22.317 2.602 1.795 3.309 0.533 4.108 3.640 6.336 10.549 17.287 2.076 12.855 1.152 7.009 1.467 12.939 1.948 3.728 9.397 1.718 -18.653 8.153 4.965 1.691 4.559 SMBH -2.410 0.422 5.730 1.701 -0.983 -1.874 2.339 5.700 2.233 -1.786 3.424 -0.443 -16.977 4.771 0.827 -2.075 -2.138 -1.993 -43.143 0.270 0.807 -0.228 -2.695 1.509 -4.286 -3.552 2.618 -28.214 -1.141 -23.551 1.936 6.067 -1.924 1.786 -1.273 -3.242 -32.412 -2.755 -51.363 5.778 -0.057 -0.066 -0.186 HMLH -0.813 -0.207 4.084 6.523 0.568 -0.808 7.058 21.990 0.370 1.674 3.542 -0.885 17.566 -2.136 1.303 -0.316 -1.892 -0.148 0.507 5.954 1.221 0.566 16.770 0.606 6.666 -0.652 0.364 27.252 -0.763 25.696 4.186 13.060 -1.448 -0.790 -3.363 5.761 -13.102 -3.192 60.250 4.499 0.752 -0.672 2.523 R-squared 0.465 0.282 0.392 0.634 0.241 0.319 0.576 0.670 0.233 0.130 0.292 0.133 0.857 0.437 0.147 0.205 0.177 0.092 0.773 0.230 0.047 0.211 0.759 0.401 0.488 0.238 0.316 0.837 0.124 0.805 0.416 0.384 0.289 0.282 0.311 0.506 0.667 0.172 0.738 0.391 0.187 0.102 0.244

54

Table Continued: The Factor Loadings and the R-squared
Metropolitan Area Jacksonville,FL Janesville,WI Kalamazoo-Portage,MI KansasCity,MO-KS Kennewick-Pasco-Richland,WA Knoxville,TN LaCrosse,WI-MN Lafayette,LA LakeCounty-KenoshaCounty,IL-WI Lakeland-WinterHaven,FL Lancaster,PA Lansing-EastLansing,MI LasCruces,NM LasVegas-Paradise,NV Lexington-Fayette,KY Lima,OH Lincoln,NE LittleRock-NorthLittleRock-Conway,AR Longview,TX Longview,WA LosAngeles-LongBeach-Glendale,CA Louisville-Je↵ersonCounty,KY-IN Lubbock,TX Macon,GA Madera-Chowchilla,CA Madison,WI Manchester-Nashua,NH Mansﬁeld,OH Medford,OR Memphis,TN-MS-AR Merced,CA Miami-MiamiBeach-Kendall,FL Midland,TX Milwaukee-Waukesha-WestAllis,WI Minneapolis-St.Paul-Bloomington,MN-WI Mobile,AL Modesto,CA Monroe,LA Monroe,MI Napa,CA Naples-MarcoIsland,FL Nashville-Davidson-Murfreesboro-Franklin,TN Nassau-Su↵olk,NY HMKT 9.029 1.128 4.265 5.302 0.068 4.363 4.909 5.760 2.766 7.457 0.167 4.135 5.886 11.720 4.485 2.825 2.833 4.994 5.003 2.226 14.561 1.889 5.479 3.188 11.664 3.246 15.200 3.021 11.435 0.303 10.050 13.639 9.405 5.457 9.183 6.034 13.173 5.386 3.564 20.727 16.584 4.070 25.344 SMBH -11.038 1.005 0.214 -3.242 2.742 -0.468 0.216 9.518 -9.366 -9.498 -7.425 -3.783 -2.965 -20.360 -2.457 -1.018 0.498 0.834 3.210 4.943 -54.850 0.305 2.047 -1.321 -29.035 -2.106 -28.583 1.412 -19.949 -3.176 -27.243 -18.152 13.731 -5.180 -8.632 3.625 -34.490 3.338 -3.929 -52.064 -33.255 -1.678 -47.593 HMLH 11.407 4.419 1.585 -1.073 7.110 4.608 3.743 9.392 0.953 16.365 6.818 -0.958 8.585 22.298 0.756 0.448 1.926 1.884 4.121 16.713 29.291 0.683 2.525 0.838 35.194 5.315 -5.439 1.429 26.830 -2.724 21.825 26.455 8.258 6.919 2.726 4.347 24.795 2.515 0.131 29.391 32.508 1.498 -3.155 R-squared 0.698 0.146 0.204 0.548 0.199 0.515 0.375 0.690 0.458 0.564 0.549 0.338 0.641 0.865 0.343 0.140 0.260 0.388 0.539 0.660 0.848 0.187 0.396 0.227 0.813 0.210 0.665 0.171 0.849 0.196 0.737 0.765 0.433 0.558 0.575 0.185 0.804 0.328 0.164 0.738 0.760 0.164 0.830

55

Table Continued: The Factor Loadings and the R-squared
Metropolitan Area NewHaven-Milford,CT NewOrleans-Metairie-Kenner,LA NewYork-WhitePlains-Wayne,NY-NJ Newark-Union,NJ-PA Niles-BentonHarbor,MI Oakland-Fremont-Hayward,CA Odessa,TX Ogden-Clearﬁeld,UT OklahomaCity,OK Olympia,WA Omaha-CouncilBlu↵s,NE-IA Orlando-Kissimmee,FL Oxnard-ThousandOaks-Ventura,CA PalmBay-Melbourne-Titusville,FL Peabody,MA Pensacola-FerryPass-Brent,FL Peoria,IL Philadelphia,PA Phoenix-Mesa-Scottsdale,AZ Pittsburgh,PA PortSt.Lucie,FL Portland-SouthPortland-Biddeford,ME Portland-Vancouver-Beaverton,OR-WA Poughkeepsie-Newburgh-Middletown,NY Providence-NewBedford-FallRiver,RI-MA Provo-Orem,UT Pueblo,CO Racine,WI Raleigh-Cary,NC Reading,PA Redding,CA Reno-Sparks,NV Richmond,VA Riverside-SanBernardino-Ontario,CA Roanoke,VA Rochester,MN Rochester,NY Rockford,IL RockinghamCounty-Stra↵ordCounty,NH Sacramento-Arden-Arcade-Roseville,CA Saginaw-SaginawTownshipNorth,MI Salem,OR Salinas,CA HMKT 17.303 9.893 22.818 24.421 1.460 17.355 2.770 6.929 8.301 2.077 3.561 10.175 24.815 8.127 24.809 9.372 1.594 3.385 7.923 3.354 12.800 9.980 6.510 20.442 13.802 6.015 8.399 4.853 3.576 0.805 8.762 12.459 4.555 12.173 2.537 4.972 3.017 1.174 17.781 11.827 3.127 4.198 17.464 SMBH -31.329 6.286 -45.541 -44.044 -3.758 -53.726 6.388 11.549 7.904 -3.902 1.334 -17.405 -76.045 -13.584 -33.914 -11.313 2.570 -16.903 -14.578 0.944 -20.766 -21.924 -1.059 -39.836 -31.162 11.759 0.135 -6.015 0.891 -10.277 -21.875 -23.911 -9.201 -31.933 -2.519 -3.118 -4.286 -0.526 -27.745 -38.269 -1.026 3.304 -47.843 HMLH -5.593 1.545 -2.468 -2.095 0.369 23.873 4.490 8.165 8.067 22.664 0.542 21.053 22.819 14.058 -13.787 13.351 0.530 2.694 16.295 0.569 19.446 -3.937 22.181 -4.061 -1.481 12.309 1.133 6.549 0.982 4.342 23.831 23.915 9.631 25.135 4.227 -1.308 -3.217 4.274 -8.645 23.842 1.013 14.215 23.966 R-squared 0.657 0.507 0.815 0.750 0.310 0.796 0.372 0.512 0.721 0.741 0.394 0.786 0.819 0.864 0.665 0.795 0.149 0.690 0.680 0.197 0.844 0.810 0.700 0.747 0.806 0.439 0.377 0.606 0.070 0.514 0.734 0.774 0.644 0.865 0.403 0.238 0.282 0.394 0.712 0.688 0.264 0.697 0.783

56

Table Continued: The Factor Loadings and the R-squared
Metropolitan Area SaltLakeCity,UT SanAntonio,TX SanDiego-Carlsbad-SanMarcos,CA SanFrancisco-SanMateo-RedwoodCity,CA SanJose-Sunnyvale-SantaClara,CA SanLuisObispo-PasoRobles,CA SantaAna-Anaheim-Irvine,CA SantaBarbara-SantaMaria-Goleta,CA SantaCruz-Watsonville,CA SantaFe,NM SantaRosa-Petaluma,CA Savannah,GA Scranton-Wilkes-Barre,PA Seattle-Bellevue-Everett,WA Sebastian-VeroBeach,FL Shreveport-BossierCity,LA SouthBend-Mishawaka,IN-MI Spokane,WA Springﬁeld,IL Springﬁeld,MA Springﬁeld,MO Springﬁeld,OH St.Louis,MO-IL Stockton,CA Syracuse,NY Tacoma,WA Tallahassee,FL Tampa-St.Petersburg-Clearwater,FL Toledo,OH Topeka,KS Trenton-Ewing,NJ Tucson,AZ Tulsa,OK Tyler,TX Vallejo-Fairﬁeld,CA VirginiaBeach-Norfolk-NewportNews,VA-NC Visalia-Porterville,CA Warren-Troy-FarmingtonHills,MI Washington-Arlington-Alexandria,DC-VA-MD-WV Waterloo-CedarFalls,IA Wenatchee-EastWenatchee,WA WestPalmBeach-BocaRaton-BoyntonBeach,FL Wilmington,DE-MD-NJ HMKT 5.335 10.301 21.814 19.466 12.929 14.438 28.367 20.126 13.819 10.337 14.441 7.408 3.243 0.932 15.148 7.884 -0.108 3.372 0.849 7.316 3.873 2.514 4.136 16.603 2.235 5.135 4.799 11.015 3.403 4.245 15.577 8.111 8.037 5.397 17.277 5.923 7.391 5.288 16.022 2.769 -1.048 19.420 4.371 SMBH 10.671 5.155 -55.869 -90.385 -63.083 -56.354 -82.096 -50.303 -65.713 1.302 -51.070 -3.703 -1.967 -14.413 -17.709 2.726 -0.415 0.469 -0.990 -25.214 0.007 0.330 -5.360 -41.185 -8.397 -6.659 -5.175 -16.470 -1.024 -0.497 -33.589 -12.010 7.470 2.634 -37.947 -16.568 -22.658 -4.561 -47.490 3.148 6.139 -32.849 -20.335 HMLH 13.153 8.380 22.163 12.650 18.547 28.803 38.745 17.341 18.778 17.763 20.683 7.944 2.545 26.672 18.043 5.785 0.439 15.803 3.784 -4.894 3.867 -1.375 0.628 25.143 -1.345 19.555 11.619 16.055 -0.393 3.057 -2.730 15.598 1.141 0.398 23.233 14.717 24.832 -1.140 23.720 5.393 15.622 26.344 9.114 R-squared 0.530 0.456 0.727 0.744 0.536 0.709 0.837 0.748 0.687 0.606 0.693 0.648 0.106 0.566 0.737 0.382 0.026 0.639 0.168 0.772 0.394 0.157 0.666 0.792 0.392 0.628 0.714 0.853 0.239 0.477 0.714 0.709 0.700 0.132 0.780 0.788 0.825 0.272 0.904 0.181 0.327 0.865 0.810

57

Table Continued: The Factor Loadings and the R-squared
Metropolitan Area Wilmington,NC Winston-Salem,NC Worcester,MA York-Hanover,PA HMKT 6.809 4.749 17.265 2.545 SMBH -5.212 -0.863 -28.101 -9.258 HMLH 14.656 0.574 -7.750 8.270 R-squared 0.567 0.311 0.720 0.525

Table 8: The Estimated Risk Premia and Implied Amenities for 2008
Metropolitan Area Akron,OH Albany-Schenectady-Troy,NY Albuquerque,NM Allentown-Bethlehem-Easton,PA-NJ Amarillo,TX AnnArbor,MI Atlanta-SandySprings-Marietta,GA Augusta-RichmondCounty,GA-SC Austin-RoundRock,TX Bakersﬁeld,CA Baltimore-Towson,MD BarnstableTown,MA BatonRouge,LA Beaumont-PortArthur,TX Bellingham,WA Binghamton,NY Birmingham-Hoover,AL Bloomington-Normal,IL BoiseCity-Nampa,ID Boston-Quincy,MA Boulder,CO Bridgeport-Stamford-Norwalk,CT Bu↵alo-NiagaraFalls,NY Burlington-SouthBurlington,VT Cambridge-Newton-Framingham,MA Camden,NJ Canton-Massillon,OH CapeCoral-FortMyers,FL Casper,WY CedarRapids,IA Charleston-NorthCharleston-Summerville,SC Charlotte-Gastonia-Concord,NC-SC Charlottesville,VA Price 133978 191780 292153 229955 128082 165966 191194 120856 198370 143663 267441 317123 168206 140564 284456 120595 153568 152073 163398 336747 351691 436013 115580 251974 387527 195479 122679 118378 194152 142387 212297 197281 282797 Wage 37893 42523 35415 38208 34729 39107 38336 33056 37362 30047 47881 51194 36346 35507 35592 34367 39886 38865 35615 55220 50058 79108 37647 41139 60093 42626 32763 40898 52185 38811 35447 39621 43344 Amenities 17114 49034 144544 100250 18884 43073 47676 12285 49877 75981 78048 88890 31159 22388 127464 19784 5495 29517 24438 92699 96189 122523 10385 90307 127056 44754 16914 1526 -15784 18218 48345 41230 88028 Risk Premia 1488 -23928 -19299 -30157 5841 -1008 6041 -2980 1987 -46690 -43632 -15598 1614 6259 -53727 -3927 1669 1277 -18861 -7879 16956 -36123 -6018 -26295 -4981 -27447 6955 -36267 -12044 1944 -12361 2221 -33482

58

Table continued: The Estimated Risk Premia and Implied Amenities for 2008
Metropolitan Area Chattanooga,TN-GA Cheyenne,WY Chicago-Naperville-Joliet,IL Chico,CA Cincinnati-Middletown,OH-KY-IN Cleveland-Elyria-Mentor,OH CollegeStation-Bryan,TX ColoradoSprings,CO Columbia,SC Columbus,OH Corvallis,OR Dallas-Plano-Irving,TX Dalton,GA Davenport-Moline-RockIsland,IA-IL Dayton,OH Deltona-DaytonaBeach-OrmondBeach,FL Denver-Aurora-Broomﬁeld,CO DesMoines-WestDesMoines,IA Detroit-Livonia-Dearborn,MI Durham-ChapelHill,NC EauClaire,WI Edison-NewBrunswick,NJ ElPaso,TX Elkhart-Goshen,IN Erie,PA Eugene-Springﬁeld,OR Evansville,IN-KY Fayetteville-Springdale-Rogers,AR-MO Flint,MI FortCollins-Loveland,CO FortLauderdale-PompanoBeach-DeerﬁeldBeach,FL(MSAD) FortWayne,IN Fresno,CA Gary,IN GrandJunction,CO GrandRapids-Wyoming,MI Greeley,CO Greensboro-HighPoint,NC Harrisburg-Carlisle,PA Hartford-WestHartford-EastHartford,CT Holland-GrandHaven,MI Houston-SugarLand-Baytown,TX Huntsville,AL Price 126886 176181 248207 229208 166943 144077 139690 208343 139798 165190 277148 149710 118068 114984 122437 156630 258493 152250 58617 188378 153334 330532 135003 135147 95974 207351 110415 149133 113815 223706 239411 93630 184897 117579 231281 88277 195624 143837 167891 240972 160533 159217 157478 Wage 34784 44613 45510 32349 39066 40118 28176 38221 35328 38741 37755 43458 28675 38571 35526 32098 48010 42506 32094 40927 33193 51865 28071 32263 32294 33522 36329 32537 29488 38848 41974 34176 30997 35922 36665 33582 28402 35405 39106 50755 33009 45835 38259 Amenities 1780 17075 70747 119236 37096 21777 44535 60144 15821 37593 99086 19554 18833 -10157 18824 52917 51834 9953 -32071 35107 25137 112694 39736 35436 -3583 63215 -4238 39790 33532 58087 94247 -4478 99964 330 44430 -16069 88787 28144 31027 72715 50988 1872 31236 Risk Premia -265 -2685 -15429 -54105 2389 5183 7269 3406 721 2023 -26832 7398 -2710 634 2942 -35650 20281 764 533 3012 -784 -34683 -7314 -204 1804 -27738 4658 -11657 1381 10117 -57586 1411 -54344 -3063 -9774 1189 11137 4960 -8813 -15708 2065 6079 2638

59

Table continued: The Estimated Risk Premia and Implied Amenities for 2008
Metropolitan Area Indianapolis-Carmel,IN Jackson,MS Jacksonville,FL Janesville,WI Kalamazoo-Portage,MI KansasCity,MO-KS Kennewick-Pasco-Richland,WA Knoxville,TN LaCrosse,WI-MN Lafayette,LA LakeCounty-KenoshaCounty,IL-WI Lakeland-WinterHaven,FL Lansing-EastLansing,MI LasCruces,NM LasVegas-Paradise,NV Lexington-Fayette,KY Lima,OH LittleRock-NorthLittleRock-Conway,AR Longview,TX Longview,WA LosAngeles-LongBeach-Glendale,CA Louisville-Je↵ersonCounty,KY-IN Lubbock,TX Macon,GA Madera-Chowchilla,CA Madison,WI Manchester-Nashua,NH Mansﬁeld,OH Medford,OR Memphis,TN-MS-AR Merced,CA Miami-MiamiBeach-Kendall,FL Midland,TX Milwaukee-Waukesha-WestAllis,WI Minneapolis-St.Paul-Bloomington,MN-WI Mobile,AL Modesto,CA Monroe,LA Monroe,MI Napa,CA Naples-MarcoIsland,FL Nashville-Davidson-Murfreesboro-Franklin,TN Nassau-Su↵olk,NY Price 122092 140458 196673 144590 147363 144052 165756 149041 149199 135445 218836 156411 115284 150517 178390 144070 107168 134756 95354 203250 340842 133968 103485 123826 233514 216103 225464 110216 257085 118286 120258 251186 139240 230988 186015 130544 159318 118373 117573 393989 232223 167671 411170 Wage 39297 36054 40028 31826 33685 40396 33040 34696 35263 40182 51782 32572 33844 27855 39920 36413 30351 39012 36046 29703 42265 37995 32447 34147 26524 44172 45432 29719 34506 38577 27871 35887 53968 42824 47653 30567 31485 32204 33397 52169 62559 39768 57617 Amenities -1721 21424 36462 36699 33436 8017 66283 21846 20591 -2779 33225 44635 10074 52528 60215 22959 16977 5170 -20838 77843 200361 784 6290 13047 160921 44371 63246 24949 111794 -3265 79226 89587 -49126 67918 7047 13332 105736 5546 12661 201297 32878 16522 132911 Risk Premia 2270 -203 -21340 -5532 1337 2919 -9875 -3172 -861 1883 -10426 -30456 594 -10469 -44996 113 417 2762 2229 -18158 -97654 1069 3503 46 -77485 -8648 -15225 1903 -52034 -189 -57403 -49034 7712 -11209 -5723 371 -66758 4566 -2682 -96628 -76764 -1207 -30188

60

Table continued: The Estimated Risk Premia and Implied Amenities for 2008
Metropolitan Area NewHaven-Milford,CT NewOrleans-Metairie-Kenner,LA NewYork-WhitePlains-Wayne,NY-NJ Newark-Union,NJ-PA Niles-BentonHarbor,MI Oakland-Fremont-Hayward,CA Odessa,TX Ogden-Clearﬁeld,UT OklahomaCity,OK Olympia,WA Omaha-CouncilBlu↵s,NE-IA Orlando-Kissimmee,FL Oxnard-ThousandOaks-Ventura,CA PalmBay-Melbourne-Titusville,FL Peabody,MA Pensacola-FerryPass-Brent,FL Peoria,IL Philadelphia,PA Phoenix-Mesa-Scottsdale,AZ Pittsburgh,PA PortSt.Lucie,FL Portland-SouthPortland-Biddeford,ME Portland-Vancouver-Beaverton,OR-WA Poughkeepsie-Newburgh-Middletown,NY Providence-NewBedford-FallRiver,RI-MA Provo-Orem,UT Pueblo,CO Racine,WI Raleigh-Cary,NC Reading,PA Redding,CA Reno-Sparks,NV Richmond,VA Riverside-SanBernardino-Ontario,CA Roanoke,VA Rochester,NY Rockford,IL RockinghamCounty-Stra↵ordCounty,NH Sacramento-Arden-Arcade-Roseville,CA Saginaw-SaginawTownshipNorth,MI Salem,OR Salinas,CA SaltLakeCity,UT Price 229016 163257 402722 390079 144779 351486 84268 209648 128285 254566 135631 192993 405815 100877 318301 161147 126718 190433 156698 126616 149184 211700 276563 303410 227287 241458 133986 177849 213951 153884 219005 194248 214436 191675 169746 120462 114019 239595 235755 55675 194581 247201 221381 Wage 46918 41740 54540 56655 33669 53093 34622 32799 38882 39988 43012 35717 46787 37035 50895 33338 40787 47361 36156 42104 39777 41522 39942 40119 40887 23814 30564 37012 39602 36256 34527 46929 42309 30634 38727 39812 32955 45231 41119 30143 32016 42857 38237 Amenities 64867 -5321 140388 140479 19300 177091 -24432 60305 6263 82811 -7787 70500 259377 1245 107415 51831 -12718 17739 37786 -15350 32702 41508 85134 138304 69347 102756 25416 49031 65866 45164 105379 62649 52788 115268 24979 14085 17320 57195 137359 -27700 52384 133215 43451 Risk Premia -17634 13291 -31317 -31607 -3714 -88480 2141 5439 4799 -37145 4075 -42265 -111207 -28061 -4670 -23593 3105 -21036 -34795 3052 -40431 -10423 -29251 -26197 -22110 -3380 5662 -12039 921 -18381 -54805 -53131 -20864 -62994 -6874 1750 -5917 -6151 -77007 180 -13689 -80905 -4870

61

Table continued: The Estimated Risk Premia and Implied Amenities for 2008
Metropolitan Area SanAntonio,TX SanDiego-Carlsbad-SanMarcos,CA SanFrancisco-SanMateo-RedwoodCity,CA SanJose-Sunnyvale-SantaClara,CA SanLuisObispo-PasoRobles,CA SantaAna-Anaheim-Irvine,CA SantaBarbara-SantaMaria-Goleta,CA SantaCruz-Watsonville,CA SantaFe,NM SantaRosa-Petaluma,CA Savannah,GA St.Louis,MO-IL Warren-Troy-FarmingtonHills,MI WestPalmBeach-BocaRaton-BoyntonBeach,FL Wilmington,DE-MD-NJ Price 151913 361444 714716 520378 389682 482828 277409 495718 310957 354335 176647 137199 123421 243844 244404 Wage 34937 46649 76042 58531 40635 51894 47957 51140 44927 47755 39183 41823 44488 58358 43643 Amenities 33157 208206 382048 304677 256355 316297 141104 291999 102140 202088 19891 -2587 -13735 49911 86808 Risk Premia 1326 -90688 -135615 -130572 -110656 -140048 -73889 -113154 -16706 -89487 -8893 -2983 314 -59038 -32946

Note - All the variables are for year 2008. The risk premia is estimated by the exposure of a metropolitan area to the three factors and to the idiosyncratic variance. A negative risk premium should be interpreted as a cheaper house price due to the exposure of the MSA to risk. The amenities are calculated as the amount left over from prices after removing the e↵ect of wages, expected growth, and risk premia.

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