Hedonic Equilibrium in Metropolitan Housing Markets - PowerPoint Presentation

Hedonic Equilibrium in Metropolitan Housing Markets Dennis Epple Prepared for presentation at University College London June 2018

Very Excellent Coauthors This presentation is based on joint research with: Holger Sieg, University of Pennsylvania Luis Quintero, Johns HopkinsWorking paper:“A New Approach to Estimating Equilibrium Models for Metropolitan Housing Markets” 2

Why Another Model? Existing hedonic equilibrium models permit us to study housing markets and to evaluate policy impacts across locations within a metropolitan area. Our focus is on developing a hedonic framework to conduct analysis across metropolitan areas including:Comparison of housing prices as a function of quality across metropolitan areas.Calculating the compensating variation required to make a given household equally well off in two different metropolitan markets. This is of value for study of agglomeration economies. Calculating the stock of amenity-adjusted housing in different metropolitan markets—and comparing across markets. 3

Objectives of Our Research 1. Formulate and Estimate an Equilibrium Model of Metropolitan Housing Markets:Comprehensive housing coverage, both rental and owner-occupiedComprehensive measurement of housing quality and all associated amenities and disamenities 2. Estimate annualized rental price as a function of quality across the stock of houses in multiple metropolitan areas. 3. Applications of of this new approach include: We compare hedonic price functions and the distributions of qualities of housing stocks across metropolitan areas. Our application compares Chicago and New York metropolitan areas.We calculate the compensating variation, as a function of income and household type, that would be required for households in Chicago to be equally well off in New York. 4

Related Research There is a very large related literature admirably reviewed in: Kuminoff , Nicolai V., Christopher Timmins, and V. Kerry Smith (2013) “The New Economics of Equilibrium Sorting and Policy Evaluation Using Housing Markets,” Journal of Economic Literature, 51(4 ), 1007-1062 . Related work: Landvoigt , Tim, Martin Schneider, and Monika Piazzesi, “The Housing Market(s) of San Diego” American Economic Review, April 2015. 5

Features of Our Model Key assumptions: Housing quality is continuous and unidimensional.Other important model elements:Quality is unobserved.Quality is comprehensive, encompassing all features of a dwelling and its location. Thus, in addition to structural characteristics, quality incorporates the presence of a subway stop near a dwelling, an art museum in the metropolitan area in which the dwelling is located, balminess or harshness of climate , and so forth.Changing metropolitan population size and demographic composition, taken as exogenous, drive changes in the metropolitan housing market equilibrium from period to period.Households vary in income and demographic characteristics (e.g., age of head, children). 6

Data Requirements For empirical implementation we require data for at least two markets: A given metropolitan area for two or more points in time and/or multiple metropolitan areas. For each metropolitan area, we need data for a representative sample of dwellings:Property value or rental payment for each dwelling.Income and characteristics of occupants for each dwelling. 7

Application and Data The American Housing Survey is the most comprehensive survey of US metropolitan areas. For each survey year, a subset of metropolitan areas is chosen for extended survey. A sample of dwellings is selected in each metropolitan area that is surveyed. Occupant income and demographic characteristics (age of household head, size of household) are obtained.Rent is obtained for each surveyed renter-occupied dwelling.Value is estimated for each owner-occupied dwelling.The Chicago and New York Metropolitan Areas were both surveyed in both 1999 and 2003, providing data for our application. 8

House Values and Annual Rental Values For our analysis, we need annualized rental values for owner-occupied units, and we need property values for renter-occupied units. We assume that households of a given income, y, and demographic characteristics, x, are indifferent between owning and renting. For each metropolitan area in our sample and each date, we match renter-occupants who have income y and characteristics x to owner-occupants with the same income and characteristics. We assume the observed rent paid by the former equals implicit rent paid by the latter.We then estimate a non-parametric relationship, v t (V) that maps house value, V, to rent, v, across the range of house values. We thereby obtain implicit rents for all owner-occupied dwellings and can then combine the data for owner-occupied and rental units to calculate the cumulative distribution function, G t (v), for rents of all dwellings in a metropolitan area at date t.Proposition 2 shows that vt(V) and its inverse are non-parametrically identified . Hence, we can invert v t (V) to obtain estimated market values, V, of renter-occupied dwellings. User Cost 9

Implicit Rents In Chicago Metro Area 10 Implicit rents ranged between 6% and 8% of house values in Chicago in 1999 and 2003

Overview of Remainder of Presentation Model and derivation of equilibrium for households of only one-type—households varying only in income. Begin by considering a metropolitan area at two points in time. Then extend to multiple metropolitan areas.Then generalize to multiple observed household types based on age and familiar size. Then further generalize to multiple unobserved household types . Empirical results from joint estimation of model for Chicago and New York for each of the above.11

Why Spend Time on the One-Type Model Our approach is easiest to convey with the one-type model. Several key findings from the one-type model are remarkably similar to the results of the multiple type models. For cross-country applications (e.g., comparison of London and New York metro areas), data for estimating the one-type model are likely to be more readily available.12

Notation y household income F t(y) CDF of income in metro area at date tv house rent (implicit rent if owner-occupied)G t (v) CDF of house rents, at date t. h house quality R t (h) CDF of house qualities at date trt(h) density of house qualities at date tv t (h) hedonic price function U( h,b ) utility for housing and numeraire bundle c t (h) user cost factor for housing of quality h V t (h) market value of house of quality h Note: In general, t can denote different time periods in a given metro area or different metro areas, or both. 13

Derivation of Hedonic Equilibrium Price Function Let h be the quantity of services provided by a dwelling and its environment. We treat h as latent. Let vt(h) be the hedonic price function specifying annual rental value of a house providing services h.When households differ only by income, the following are sufficient to derive the hedonic price function for a metropolitan area a date t:The metropolitan distribution of income, F t (y), and the metropolitan distribution of house rent, G t (v), both of which are observed. A specification of the utility function, U( h,b), over h and the numeraire bundle b.14

Derivation of Hedonic Price Function Derivation of the hedonic price function v(h) for a given metropolitan area at a given date. (Subscript t is suppressed for simplicity). Step 1: A household with income y chooses housing h at date t to maximize: U(h,b) s.t. y=v(h)+b. Equivalently, the household maximizes: U(h, y-v(h)) Differentiating the above with respect to h yields the first-order condition: Rosen, JPE, 1974. 15

Step 1 16 Indifference curve •

Step 2: Derivation of Hedonic Price Function Step 2: With demand for quality increasing in income, and rent increasing in quality, the CDF for each percentile of y equals the CDF for the corresponding percentile of v: F(y)=G(v) This is a market-equilibrium relationship that provides a mapping from v to y: y=y(v) This tells us, for each v, the income, y, of the household that spends v on housing. This idea is from: Landvoigt, Schneider, and Piazzesi , AER, 2015. 17

Illustration of y(v) Mapping in Step 2 The 60 th percentile of housing expenditure ($13,700) is consumed by the household at 60th percentile of income ($59,000). 18

Step 3: Derivation of Hedonic Price Function From Step 1 :From Step 2: y=y(v)  y(h)=y(v(h)) Step 3: Substitute y(v(h)) into first-order condition from Step 1 to obtain the following differential equation for v(h): Given a specification of the utility function and given observed distributions of income and housing expenditure, the above differential equation can be solved to obtain v(h), the hedonic price function. In estimation, this differential equation is solved numerically. 19

Parameterization Of Utility We adopt the utility function: U= ln(1-(h+)  )+ ln(y-v(h)- ) , , , , and  are parameters. This odd-looking utility function is relatively flexible and proves well suited to our analysis.We retain this functional form throughout, allowing parameters to vary across type in the multiple-type versions of the model.   20

Parametric Example A parametric example that has closed-form solution for the hedonic equilibrium price function is useful for illustration.For our estimated models, we solve for equilibrium numerically and thus do not need to do not need to impose parametric forms on the distributions of income or house values.Example: Suppose the densities of income and house values in period t are lognormal: 21

Equilibrium In Parametric Example Implementing Steps 1 through 3, we obtain the hedonic price function: Parameters without subscripts ( , , ,  ) are preference parameters. Parameters (t,t ) are the mean and standard deviation of the distribution of ln(y). Parameters ( t , t ) are parameters of the distribution of ln(v). This example illustrates that the hedonic price function changes over time if the distributions of income and/or house values change over time. 22

Derivation of Equilibrium Step 4 Recall that we take the metropolitan income distributions, Ft(y), and populations to be exogenous. The distribution of house values, Gt(v) is endogenous. Hence, the hedonic price function obtained from Steps 1 through 3 can be thought of as the demand side of the model. The remaining equilibrium condition determining G t(v) comes from the supply side of the model.Let Rt(h) be the CDF of metropolitan housing supply, which is fixed within period but can vary between periods. The following market equilibrium condition must hold for all h: Step 4: G t ( v t (h)) = R t (h) 23

Parameterization of Housing Supply We take the distribution of house qualities in the initial period in each metropolitan area as given. r t(h) denotes the “quantity” (i.e., density) of housing of quality h.The change in the quantity of quality h from period to period in a metro area is given by the following function which assumes the elasticity of supply  is the same across the quality dimension: The constant A t normalizes the distribution to 1 .Note that we use market value, Vt(h), in the above supply function because housing producers care about the market value for which a unit can be sold. We observe V for owner-occupied units and, as noted earlier, we impute V for renter-occupied units. 24

Identification We assume the utility function is invariant across time and across markets . We show (Proposition 3) that, we can adopt an arbitrary scaling of h. This is intuitive; there is no inherent scaling of units of housing quality.Hence, we let house quality equal to house rent, h=v, for one metropolitan area (Chicago) at one point in time (1999). This scaling provides a cardinalization of h, and sets the hedonic price function v(h)=h in Chicago in 1999. With this scaling of h, the distribution of quality is equal to the distribution of rents for Chicago in 1999. For all other periods and other metro areas, the distributions of h and v, and the hedonic price functions, are estimated. In Appendix A we present a model with flexible parametric specifications of the distributions of y and v. This model has a closed-form solution for the hedonic price function. We prove that it is identified with data for two or more metropolitan markets. This can be a given metropolitan area at two or more points in time, two or more different metropolitan areas, or both. For the one-type model, we obtain virtually the same results when estimating with this parametric form as we do when estimating with the distributions of y and v being non-parametrically estimated. 25

Estimation Is By GMM For each metropolitan area, m (Chicago, NY), and each date, t (1999,2003), estimate the rent-to-value function; impute rents for owner-occupied housing; and impute market values for renter-occupied housing. This needs to be done only once.For one metro area at one date (Chicago, 1999), normalize house qualities to equal house rents. Discretize quality into J intervals, with qualities h j . This needs to be done only once. Outer Loop: Choose a vector of structural parameters denoted . (In the single-type model,  is the utility function parameters and the supply elasticity.) Inner Loop: Iteratively solve the model until the hedonic price functions in each market and associated endogenous distributions of house values equilibrate demand and supply for each quality level in each market. Calculate the predicted joint distribution of income and rents Form orthogonality conditions based on the difference between the observed joint distributions of income and rents , denoted , and their predicted counterparts . Update  until the GMM objective is minimized. More detail. 26

Application: Joint Estimation for 4 Markets back Two metropolitan markets for two time periods each: New York and Chicago in 1999 and 2003. Populations: CMA 9.8 NYMA 19.9 27 Some totally irrelevant New York trivia Some totally irrelevant Chicago trivia

Income Distributions for The Two Metro Areas 28 Chicago Income distribution stochastically dominates NY distribution i.e., NY metro area is poorer.

Rent Distributions for the Two Metro Areas 29 Rent distribution in NYMA is shifted to the right relative to CMA, i.e., NY is more expensive.

Preference and Supply Parameters The supply elasticity is for a period of four years, i.e. a 1% increase in price over four years for a give quality elicits a .055% increase in quantity of that quality. Demand implications of the four preference parameters are best illustrated by plots of price and income elasticities. In interest of time, I will defer those to the multiple-type results. 30

Fit to Income and Rent CDF’s NYMA 31

Fit to Income and Rent CDF’s Chicago MA 32

Fit of Supply-Demand Condition NY 2003 F 33

Hedonic Prices and Qualities NY and Chicago Metro Areas Points a, b, c, d show rents paid and qualities obtained in Chicago by households at the 20 th , 40 th , 60 th , and 80th percentiles of income in Chicago. Points A, B, C, and D show rents those same households would pay in New York, and the qualities they would obtain. 34 For each household, rent is higher in NY Metro area and quality is lower.

Agglomeration Economies How much must aggregate annual income in NYMA exceed income for comparable utility in CMA? Our model implies aggregate CV of NYMA relative to CMA to be $113 billion annually. This is the combined payment that New Yorkers make in excess of the expenditure that would give them the same utility in Chicago. Roughly $5,700 per capita.As a consistency check we calculate aggregate annual rental expenditures for New York two ways:Using New York Hedonic Price Function: $198 billion.Using Chicago Hedonic Price Function: $81 billion The difference, $117 billion, closely approximates the $113 billion from our calculation of aggregate CV. 35

Generalization to Multiple Observed Household Types We allow utility to depend on household type: i, i,  i ,  i , and  i are type-specific parameters.Because preference parameters differ across types, there are a total of 15 preference parameters in the NY- Chicago application that follows. 36

Choosing Household Types We used k-means clustering to choose the optimal number of clusters based on age of head, family size, and share spent on rent. The centroids of the estimated clusters identify 3 groups that behave distinctly in their housing consumption:1. Young households mostly without children. 2. Middle age households with children. 3. Older households with few having children living in the household . k-means clustering is a cluster-analysis method that chooses clusters to minimize the sum of squared distances of observations from cluster means. The number of clusters is determined by the “silhouette” method which chooses the number of clusters to achieve comparable dissimilarity within clusters. 37

Remarkable Demographic Similarities: NY and Chicago M.A.’s 38 More demographic detail

Income Distributions By Type NYMA and CMA 39 Older households have markedly lower incomes in both MA’s. Young and middle-aged households have comparable income distributions in NYMA. Middle-aged households have higher incomes than young households in Chicago.

Shares of Income Spent on Housing 40 Expenditure share declines as income rises for all hh types. Ranking by type is similar in Chicago and NY: Old spend highest share of income on housing, then young, then middle aged. Old poor spend a very high proportion on housing. A NYT article highlighted the plight of NY elderly:“Up in years and all but priced out of New York,” April 30, 2014.

Estimation Results: Supply and Preference Parameters 41 .0055 (.0013)

Implied Price and Income elasticities 42 Older households respond less to price changes and less to income changes than young and middle-aged.

Robustness: Hedonic Price Functions Hedonic price functions from one-type and three-type models are nearly identical. 43

Robustness: CV’s from One-Type & Three Type CV from one-type model is shown in solid black. Weighted CV from three-type virtually identical to one-type. CV as share of income is particularly high forold poor in New York. 44

Unobserved Types A weakness of the model with multiple observed types is that it assumes perfect correlation of income and housing consumption for each observed type. This does not accord with the data. Hence, we extend the model to unobserved types.The utility function of each unobserved type, i, is the same form we have used thus far, with parameters differing across type. The fraction of unobserved type i is determined by the following linear function of observed characteristics. 45

Unobserved Types Continued A model with five unobserved types accounts well for the correlations between income and observed type, as shown next slide. The estimated four-year supply elasticity (.072) is almost identical to that obtained with observed types.Compensating variation as a function of income to move from Chicago to New York is also very similar to results obtained with preceding models. 46

Predicted and Observed Correlations 47

Housing Quality Stocks: NY and Chicago Metro Areas 48 Relative to Chicago, New York has a much larger stock. (NYMA has much larger population than CMA.) NYMA stock is more heavily concentrated at lower qualities. This is intuitive: NY has higher prices at all qualities, and NY income distribution is shifted left relative to Chicago.

Concluding Commercial Message We see our new method as having a number of advantages for studying metropolitan-level housing markets.It does not require any a priori assumptions about the characteristics that determine house quality. The one-type is easily implementable using metropolitan-level data on the distribution of house values and rents and the distribution of household income. It provides a straightforward summary of the changes in prices and quantities across the house quality distribution. It is comprehensive in incorporating not only services provided by a dwelling but all location-specific amenities including metropolitan-level amenities (e.g., warm glow of rooting for Cubs).It provides a new tool for study of variation across metropolitan areas, including agglomeration economics 49

Prospective Applications Cross-country comparisons, e.g., New York and London metro areas. Robustness of results from one-type model are encouraging for this application because cross-country data for estimating the one-type model are available. Incorporate federal taxes to investigate how treatment of owner and renter housing affects prices and supply across the quality spectrum. Extend the model to incorporate tenure choice to investigate: How does tax federal policy affects tenure choice and prices across the quality spectrum? How do differences in demographics and income distributions give rise to differences in tenure choice? Chicago and NYC make for an interesting application because of large differences in ownership rates. Investigate effects of rent stabilization policies. Extend the model to incorporate state and local taxes to permit investigation of how alternative local tax policies impact the distribution of housing prices and quantities across the quality spectrum. 50

House Value and Rent: Poterba back User cost of a dwelling is the annualized cost per unit of housing: vt(h) = ct(h) V t (h) For example, assuming constant rates for all relevant elements of user cost ( Poterba , 1992) derives:ct(h)=(i+p+-) where i is nominal risk-adjusted interest rate for housing investments,  p is the property tax rate,  is the rate of maintenance and depreciation, and  is the rate of housing appreciation. We assume competitive pricing of housing and no transactions costs, so rent equals user cost. We estimate c t (h) non-parametrically, so do not need to make assumptions about rate of appreciation, etc. 51

Some Totally Irrelevant New York Trivia back 1. There is a birth in New York City every 4.4 minutes.2. There is a death in New York City every 9.1 minutes.3. On Nov. 28, 2012, not a single incident of violent crime in NYC was reported for an entire day. The first time in basically ever. 4. The Federal Reserve Bank on New York’s Wall Street contains vaults that are located 80 feet beneath the bank and hold about 25 percent of the world’s gold bullion . 5. Prior to 1957, a pneumatic mail tube system was used to connect 23 post offices across 27 miles. At one point, it moved 97,000 letters a day . 6. Prior to World War II, everyone in the entire city who was moving apartments had to move on May 1 . 52

Some Totally Irrelevant Chicago Trivia back Ten major events that occurred between the Chicago Cubs two successive wins (1908, 2016) of the World Series championship:1. Radio was invented; Cubs fans got to hear their team lose. 2 . TV was invented; Cubs fans got to see their team lose. 3 . Baseball added 14 teams; Cubs fans get to see and hear their team lose to more clubs.4. Haley's comet passed Earth twice.5. Famed Cubs announcer Harry Caray was born....and died. 6. The NBA, NHL and NFL were formed, and Chicago teams won championships in each league. 7. Sixteen U.S. presidents were elected. 8. There were 11 amendments added to the Constitution. 9. Prohibition was created and repealed. 10. Man landed on the moon, as did several home runs given up by Cubs pitchers. 53

Age and Children By Household Type back 54

Flexible Parameterization of Ft (y) and G t (v) backWe permit the density of income and house value to vary by period. We use the very flexible four-parameter generalization of the lognormal: Suppressing t, the GLN4 for income is as follows: This reduces to the lognormal when r=2 and =0. 55

Formal Details of Estimation Estimate the rent-to-value function; impute rents for owner-occupied housing; and impute values for renter-occupied housing. Estimate the remaining structural parameters by matching the observed joint distributions of income and rents to the predicted distributions of income and rents. More formally, we estimate the structural parameters of the model using the following estimation algorithm:Estimate for each t the rent-to-value function, v t ( V t ), and its inverse, using a non-parametric matching estimator. Use the rent-to-value function to impute rents for owner-occupied units, obtain the market distribution for rents, and compute the joint distributions for income and rents: . Discretize the rent distribution in period 1 into J intervals indexed by j , and normalize housing quality to obtain h j and r j 1 for all j. Choose a vector of structural parameters, denoted by  , which includes the parameters of the type distribution and the parameters of the income distributions for each unobserved type.   56

Formal Details of Estimation Continued back Solve for the implied equilibrium prices in all periods t> 1 :Guess values of vjt and impute implied V jt Calculate the highest income, jt, for which hj is the optimal choice. Calculate the demand for h j : H d jt ( v t ) = F ( jt )- F( j-1t ) Calculate supplies : Check whether equilibrium holds: H d jt ( v t ) = r jt ( v jt ) Repeat until equilibrium prices have been found for all periods. Calculate predicted joint distribution in equilibrium: Form orthogonality conditions based on difference between predicted joint distribution and the observed joint distribution Update  until GMM objective is minimized.   57