Land development and frictions to housing supply over the business cycle

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) Land development and frictions to housing supply over the business cycle Hyunseung Oh, Choongryul Yang, Chamna Yoon 2024-010 Please cite this paper as: Oh, Hyunseung, Choongryul Yang, and Chamna Yoon (2024). “Land development and frictions to housing supply over the business cycle,” Finance and Economics Discussion Series 2024-010. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2024.010. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Land development and frictions to housing supply over the business cycle∗ Hyunseung Oh† Choongryul Yang‡ Chamna Yoon§ February 2, 2024 Abstract Using a novel data set of U.S. residential land developments, we document that the average time to develop residential properties—which includes both the time spent preparing land infrastructures and construction—is about three years, consistent with sizable lags in housing investment projects. We show that the time to develop is highly dispersed across locations, a finding that helps quantify thehousingsupplyelasticitythatisrelevantforassessinglocalhousingvariations over the business cycle. We also show that incorporating long and dispersed time todevelopintoanotherwisestandardhousinginvestmentmodelhelpsrationalize some empirical facts on the housing market. Our model implies that policies to boosthousingsupplyarelesseffectiveinimmediatelystabilizinghousepricesfor regionswherelanddevelopmenttakesalongtime. Keywords: Housingsupply;housepricedynamics;residentialinvestment. JELClassificationNumbers: E22,E32,R31. ∗We thank Harun Alp, Etienne Gagnon, Daniel Garcia, Greg Howard, Matteo Iacoviello, Ryan Kim, Emi Nakamura, Joseph Nichols, Jón Steinsson, Luis Torres, and numerous seminar and conference participants for helpfulcommentsandsuggestions. Theviewsexpressedaresolelytheresponsibilityoftheauthorsandshould not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or of any otherpersonassociatedwiththeFederalReserveSystem. †FederalReserveBoard. E-mail: hyunseung.oh@frb.gov. ‡FederalReserveBoard. E-mail: choongryul.yang@frb.gov. §SeoulNationalUniversity. E-mail: chamna.yoon@snu.ac.kr. 1

1 Introduction Researchers have argued that significant adjustment costs are needed in standard investment models to account for the empirically slower response of investment to economic shocks. Because ships and factories cannot be built in a day, these adjustment costs are typically motivatedasstand-insforthetimeittakestoproducenewcapitalandthedifficultiesinadjusting investmentplansoncetheyareintrain. Astrandofliteratureonthehousingmarketalsohighlightsinelasticshort-runhousingsupplyasasensiblefeaturetoexplainthedynamicproperties of the housing market, such as the difference in the short- and long-run housing market reactionstoCOVID-19(Howard,LiebersohnandOzimek,2023).1 Quantitatively,littleisknown, however, about how long it takes to develop land and what that implies on the adjustment of housingsupplyovertime. In this paper, we document the time it takes to build a house from undeveloped piece of land across regions using a new data set and study its aggregate and cross-regional implications using a model of investment dynamics. A desirable feature of our data set is that we observethetimeittakesnotonlytobuildstructuresonadevelopedlotbutalsotodevelopland infrastructureonavacant,undevelopedpieceofland. Accordingly,weareabletoanalyzethe comprehensiveprocessofresidentialconstructionacrossmajorU.S.regions. Empirically, we document two stylized facts on land development. First, residential land developmentisindeedalengthyprocessthattakesmorethanthreeyears,onaverage,afterreceivingapreliminaryapprovalofthesiteplanfromthelocalgovernment,includingmorethan a year, on average, to develop raw land with a subdivision map into a lot on which structures could be built. Second, the time it takes to develop land is highly dispersed across locations, even after controlling for an extensive list of variables that are likely to affect local construction demand. In turn, we find that a county’s median time to develop (TTD) is associated not only with a measure of its long-run housing supply elasticity, but also with adverse local weatherconditionsthathinderconstructionactivityintheshortrun. Thisinformationsuggests thatlocalfactorsthatdetermineTTDarenotfullyalignedwiththelocalfactorsthatdetermine housingsupplyinthelongrun. We then use the local variations in TTD to quantify the local housing supply elasticity, a measure that has taken center stage in the macro-housing literature. As housing wealth is aboutone-thirdofthetotalnetworthofU.S.households,housepricechangeshavesignificant spillover effects to the broader economy and estimates of the housing supply elasticity are 1Manymodelsthataccountforresidentialinvestmentandhousepricedynamicsrelyontheassumptionof fixed land supply—for example, Davis and Heathcote (2005); Kiyotaki, Michaelides and Nikolov (2011); and Kaplan,MitmanandViolante(2020). 2

frequently used to decipher the causal effect of house price changes to economic activity.2 While the existing estimates of the housing supply elasticity mainly focus on the long-run determinantsofhousingsupply,weshowthatTTDhelpsquantifythehousingsupplyelasticity morerelevantfortheevolutionofhousingsupplywithinthenextfiveyears. Towardsthat,we elaborate a housing investment model with TTD and derive analytical expressions that relate the short-run (up to five years) housing supply elasticity to TTD and the long-run housing supply elasticity. Combining the model and the data, we find that short-run housing supply elasticities vary significantly across counties and are indeed smaller than, and distinct from, correspondinglong-runelasticities. WeshowthattheconsiderationofrelativelylonganddispersedTTDinanotherwisestandardlocalgeneralequilibriummodelofhousinginvestmenthelpsrationalizeanumberofempiricalfacts. First,whentheTTDfrictionisincluded,ourotherwisestandardmodelnolonger predicts a counterfactual, strongly negative correlation between a region’s relative price and relativequantityinresponsetoacommondemandshockacrossregions. Intuitively,aslengthy TTDlowerstheshort-runsupplyelasticityinallregions,developersintheregionwithahigher long-run supply elasticity are more likely to substitute short-run supply with long-run supply as they internalize the larger gap between the short- and long-run supply elasticities. Second, whentheautocorrelationofthecommondemandshockisnothighlypersistent,ourmodelpredicts that a region’s house price growth is better explained by the short-run supply elasticity than by the long-run supply elasticity. Using county-level data, we regress a county’s house price change (relative to the national house price change) on the short- to long-run housing supply elasticities. Since the early 2010s, we find that our short-run supply elasticity outperforms the long-run supply elasticity in accounting for the observed cross-county variation in houseprices. Finally, we draw a policy implication by conducting a counterfactual exercise where the government aims to stabilize house prices by a discretionary housing supply policy. When TTD is present, we find that government incentives to boost the housing supply affect house prices through the expectations channel of future housing supply. Therefore, the policy could be somewhat less effective in immediately stabilizing house prices for regions where land developmenttakesalongtime,butitcouldbemoreeffectiveinstabilizingmedium-runhouse pricesforthoseregions. 2According to the 2022 financial accounts of the United States from the Federal Reserve Board’s flow of funds statistics, the total real estate at market value for households and nonprofit organizations is 47.1 trillion dollarsandthenetworthofhouseholdsandnonprofitorganizationsis143.7trilliondollars. 3

Relatedliterature. Studiesonhousingsupplymainlyfocusonestimatingitslong-rundeterminants (Saiz, 2010; Baum-Snow and Han, 2022; Lutz and Sand, 2022), and these estimates are typically used to identify regional variations to economic shocks (Mian, Rao and Sufi, 2013; Mian and Sufi, 2014; Davis and Haltiwanger, 2019; Bhattarai, Schwartzman and Yang, 2021). This approach could be problematic when the economic shock of interest does not persist in the long run and when the short-run determinants of housing supply differ from the long-run estimates. For example, Guren, McKay, Nakamura and Steinsson (2020) suggest that a puzzling feature of the cross-regional housing price and quantity correlation discussed in Davidoff (2016) could be potentially resolved by assuming a lower short-run housing supply elasticity in all regions. Not much is known, however, about the determinants of housing supplyelasticityinabusinesscyclefrequency,withthenotableexceptionofTopelandRosen (1988), who estimate the short- and long-run elasticities of housing supply and find that most of the long-run response occurs within a year. We contribute to this literature by (i) quantifyingfrictionstohousingsupplyatabusinesscyclefrequencybasedontheobserveddurationof land development and (ii) studying the implications of our quantified frictions on the housing market through an equilibrium model of housing investment. By investigating the link between new housing supply data and the elasticity of housing supply, we also complement the literaturethatstudiesthesensitivityoflocaleconomicactivitytohouseprices(Guren,McKay, NakamuraandSteinsson,2021;GrahamandMakridis,2023). Researchinthistopicidentifies plausibly exogenous house price variations by focusing on variables of the local economy or existing housing characteristics; we argue that data on the timing of new housing supply also capture an important source of local variation in house prices that could be used to estimate thesensitivityoflocaleconomicactivitytohouseprices. In the literature on business cycles, time to build has been noted as a key friction to investment dynamics at least since Kydland and Prescott (1982). Subsequently, several papers document the duration of building capital using newly available data or study its implications on investment behavior (Lucca, 2007; Kalouptsidi, 2014; Sarte, Schwartzman and Lubik, 2015; Millar, Oliner and Sichel, 2016; Kydland, Rupert and Šustek, 2016; Oh and Yoon, 2020; Meier, 2020; Charoenwong, Kiruma, Kwan and Tan, 2024). Our paper contributes to thislineofwork,bothbyprovidingnewstylizedfactsonthecomprehensiveconstructionprocessfromundevelopedlandtothecompletionofnewstructuresandbydeliveringanumberof housingmarketimplicationsofthenewstylizedfacts. Relatedly,ourworkcontributestoexisting studies of housing investment (Mayer and Somerville, 2000; Green, Malpezzi and Mayo, 2005; Haughwout, Peach, Sporn and Tracy, 2013; Paciorek, 2013; Murphy, 2018; Nathanson and Zwick, 2018). More broadly, our findings could also shed light on the importance of 4

TTD frictions for nonresidential structures as both residential and nonresidential structures arelikelytosharesomecommonhurdlesfromthesitedevelopmentstage. As discussed earlier, the implications of these residential construction dynamics are less explored in the housing and macro literature. Most models that study the aggregate implications of the housing sector abstract from the dynamic aspect of land development (Davis and Heathcote, 2005; Iacoviello and Neri, 2010). Following the spirit of Glaeser, Gyourko andSaiz(2008),weexploretheaggregateandregionalimplicationsofhousingsupplywitha focusontheshort-rundynamics. Structure of the paper. In section 2, we present the land development data and estimate a TTDmeasurethatiscomparableacrossregionsbycontrollingforvariousfactors. Insection3, we develop a TTD model of housing investment and analytically derive housing supply elasticities in each horizon. In section 4, we quantify local housing supply elasticities in the short to medium run by using the model derivations and the empirical TTD estimates. In section 5, we study three implications of our model for house prices and housing quantity. Section 6 concludes. The Online Appendix provides additional details and sensitivity analyses of our theoreticalandempiricalresults. 2 Duration of land development across regions Inthissection,weuseauniquedatasetthattracksdevelopmentactivitiesforresidentialproperties in the U.S. to measure the duration of land development across regions. The desirable feature of our data set is that it includes the period of site development prior to building construction. Weshowthatthedurationoflanddevelopmentthatincludestheperiodofsitedevelopment islengthy and varies widelyacross regions. Thesevariations persist even aftercontrolling for anumberofobservableregionaldemandfactors. 2.1 Land development data and summary statistics Our data set comes from Zonda, which provides data and analysis to the national residential home-building industry. The data set is constructed from Zonda’s survey markets data, which covermanyofthemajormetroareaswithhighresidentialconstructionactivityintheU.S.The survey markets data put together a quarterly construction status survey in new home subdivisions, an area containing a large number of houses or apartments to be built close together at 5

Table1: Newhousingcompletionsbetween2003and2019 (unit: 1,000housing) Zonda Census Coverage Totalhousing 7,790 20,020 39% Single-familyhousing 5,939 15,314 39% the same time. Large subdivisions are often broken down to multiple sections, each of which is typically builtby a single-builder company. The data set displaysthe total number of housing units as well as other construction characteristics by sections. We have access to this data setfrom2000to2021. AsshowninTable1,ourdatasetincludesalargenumberofnewhousingsupplyacrossthe U.S.Between2003and2019,thedatasetcontains222,868developedsectionswithatotalof 7.8millionunitsofnewhousing. Forreference,theCensusBureaureportsasumof20million new housing completions in the same period, implying a 39 percent Census coverage ratio of our data set. Our data set is not biased toward multi-unit housing development, as the Census coverage of single-unit housing completion is also around 39 percent.3 These completions are distributed over 318 counties in 113 core-based statistical areas (CBSAs) that represent 48 percent of the U.S. population. The average population size of those CBSAs is 1,590,428, whichis4.7timeslargerthanthatoftheU.S.averageCBSA.4 Besides the high coverage ratio, a desirable feature of the data set is that it continuously tracks the construction status of subdivisions and sections. Land development is a lengthy process, starting from the acquisition of land by developers and the design of a development plan. Theplanisthensubmittedtotheappropriatemunicipalityforapreliminaryreview. The profile of the subdivision is first created and labeled as future lots in our data set when the municipalitygrantsapreliminaryapprovalasafirststepintheprocessor,iftheapprovaldate is not available, after Zonda reviews and verifies the site plan submitted to the municipality. During each quarterly survey, the lot remains as future lots while there is ongoing land development, such as the site having survey stakes or equipment. It follows that the final site plan is submitted and approved, and the necessary permits are processed. Thereafter, the infrastructure for the land is developed and the lot is now labeled as active. At this stage, separate homebuilders enter for construction projects in the active lots not pursued by the developer. 3In our data set, single-family housing units comprise 82.6 percent of the completions, followed by townhouses(10.2percent),condos(2.3percent),andduplexes(1.2percent).WeshowintheAppendixthattheCensus coveragedoesnotsignificantlyfluctuateacrossyears. 4Ofthetop20CBSAsrankedbythe2020Censuspopulation, only2CBSAs(BostonandSeattle)arenot includedinourdataset. 6

When there is excavation activity with a slab or basement on these vacant developed lots, the units are classified as under construction, consistent with the Census Bureau’s definition of housing starts. After the completion of home construction, each house is classified as either a finished vacant unit or an occupied unit, depending on its status of sales. Eventually, the subdivision is classified as built out, and it exits the data set when the number of occupied unitsequalsitstotalunits. Based on this classification, we define a TTD measure for each new development section. TTD is defined as the duration between the quarter when the municipality likely approves a preliminary site plan and the quarter when the number of finished units (vacant or occupied) reaches at least half of the total number of units. The unique feature of our data set is that it captures the earliest stage of a completed development with a plan that is as concrete as a preliminary map submitted to the municipality. Our definition for the beginning quarter of TTD fully takes this feature on board; in the Appendix, we present results using an alternative definition for the beginning quarter of TTD.5 The definition for the end quarter of TTD is consistent with the Census Bureau’s definition of the completion of a multi-unit building, as it classifies the construction of a multi-unit building as complete when half of the units are finished. It is worth noting that the Census Bureau tracks construction time per building, whereas we can only track construction time per section. Therefore, our measure of the section’s construction time could be longer than the construction time of an average building in that section if a developer decides to build structures sequentially rather than simultaneously. In the Appendix, we study the sensitivity of our empirical results when the end quarter of TTD is defined earlier than our baseline—that is, the date at which the number of finished unitsreachesaquarterofthetotalnumberofunitsinthesection. For the remaining analysis, we adopt the following sample selection criterion. Between 2003 and 2019, 222,868 sections were completed. We dropped 102,575 observations without informationonTTD(forexample,missingstartdates),resultingin120,293observations.6 We further dropped 16,097 observations without lot size information or demand controls, leaving uswith104,196observations. 5Ourbaselinedefinitionisalsodrivenbydataavailability. Inthedataset, thepreliminaryapprovaldateis missing for many sections, which limits our sample size substantially. In the Appendix, we provide details on howwemeasurethebeginningquarterofdevelopment. 6Specifically,wedefinethestartdateasthefirstquarterwhenthetotalnumberofplannedunitsisequalto thetotalnumberoffuturelots,basedonZonda’squarterlyreviewofnewlysubmittedmapsatthemunicipality. Wefindthatourdefinedstartdateisclosetothemunicipalityapprovaldateofthepreliminarysiteplan, when thelatterdateisavailableinthedataset. Wedroppedsectionswheretheirfirstobservationalreadyhadpositive activelots,aslanddevelopmentonthesesectionslikelystarted(accordingtoourdefinition)beforetheyentered thedataset. 7

Table2: SectionTTDstatistics (unit: days) SiteTTD BuildingTTD TotalTTD Mean 569 760 1,329 Std. dev. 760 792 1,077 IQR 458 548 1,006 P10 91 182 366 P25 181 275 638 P50 275 458 1,004 P75 639 823 1,644 P90 1,278 1,734 2,922 Observations 104,196 104,196 104,196 Note: Eachobservationisasubdivisionorasectionofasubdivisionwhentherearemultiplesectionsinasubdivision. IQRstandsfortheinterquartilerange(P75−P25). FivedifferentpercentilesofeachTTDdistributionare shown—forexample,P50referringtothemedian(50thpercentile)ofthedistribution. 2.2 Stylized facts on the duration of housing development The total time it takes for housing development (TTD) comprises two parts: time to develop infrastructure at the site (site TTD) and time spent on the construction of buildings (building TTD). Just by looking into the raw measures of TTD based on Table 2, we find two stylized factsonthedurationofhousingdevelopmentthatstandout. First, housing development is a lengthy process with significant time spent on land development. As shown in the first row of the table, housing development takes a total of 1,329 days on average. While less emphasized in the literature because of limited data availability, we find that the duration between the land development plan approval and the finishing of site development is substantially long, averaging 569 days. The mean construction time of buildingsonthesedevelopedsitesis760days. Second, there is substantial heterogeneity in the duration of housing development. The standarddeviationandtheinterquartilerangeoftotalTTDarebotharoundthreeyears(1,077 days and 1,006 days, respectively). The significant heterogeneity is also pronounced in site TTD,asitsstandarddeviationismorethantwoyears. Note that the distribution of TTD is skewed to the right, as the mean is larger than the median in all TTD measures. This finding is also evident from the lengthy TTD at the 90th percentile. 8

2.3 Controlled measures of TTD The lengthy and highly dispersed TTD across sections documented above could be driven by various factors. Our goal is to measure the developers’ TTD constraint for housing development that is comparable across locations. Toward that goal, we need to first control for differencesinconstructioncharacteristicsthatarelikelytoaffectTTD.Foreachdevelopment section, the data set includes some of that information, such as the number of housing units, the average lot size, the type of housing, and the builder(s) of each section. Indeed, these construction characteristics have substantial variations. For example, there are an average of 42housingunitspereachsection,andthestandarddeviationisalso42units. The first column of Table 3 shows the regression result when the log of TTD is regressed on several construction characteristics in our sample. Builder fixed effects are included for the top 15 builders in our sample. Completion year fixed effects are also included to abstract from time variations in TTD. We find that both the number of units and the (average) lot size per housing unit are positively associated with TTD. One percent increases in the number of units and in the lot size per unit imply 0.131 percent and 0.136 percent increases in TTD, respectively. These results are highly significant and consistent with the findings in Oh and Yoon (2020), where the square footage of a single-family house under construction is shown to be positively associated with its time to build. In terms of the type of housing, townhouses andcondostakealongertimetocompleterelativetosingle-familydevelopments. Even after we control for construction characteristics, TTD could also be driven by local economic factors that are linked to the developers’ incentives in those locations. Because our model does not feature the developer’s location choice, we would also need to control for these factors. The second column of Table 3 adds a number of local controls potentially relevant for housing supply—such as a Bartik-type variable that measures the local demand pressure, indicators of sand and coastal states, population shares of immigrants and collegeeducated adults, and population density—as suggested in Davidoff (2016). We also include the annual county-level GDP in the regression to control for any country-level time-varying economic factors. The regression results show that several of these local economic factors are associated with TTD in a statistically significant manner. Counties with higher Bartik demand pressure, immigrant share, and population density, or counties not in sand states and especially in coastal states, experience longer duration in development, as shown in the Appendix. Evenafterwecontrolfortheselocaleconomicfactors,however,theR-squaredshows limited improvement over the regression in the first column, and the regression coefficients for construction characteristics remain robust. These results suggest that local economic fac- 9

Table3: SectionTTDregressionresults Variables (1) (2) (3) Log(numberofunits) 0.131*** 0.140*** 0.138*** (0.00427) (0.00436) (0.00417) Log(lotsize) 0.136*** 0.141*** 0.140*** (0.00480) (0.00492) (0.00477) Singlefamily − − − Townhouse 0.208*** 0.198*** 0.187*** (0.0111) (0.0113) (0.0112) Condo 0.191*** 0.227*** 0.235*** (0.0417) (0.0414) (0.0384) Duplex 0.0436 0.0341 0.0396 (0.0309) (0.0310) (0.0293) Etc. 0.0233 0.0461* 0.0444* (0.0263) (0.0261) (0.0246) Builderfixedeffect (cid:88) (cid:88) (cid:88) Yearfixedeffect (cid:88) (cid:88) (cid:88) Localcontrols (cid:88) Localcontrols×Year (cid:88) Constant 4.505*** 4.432*** 5.284*** (0.0505) (0.0883) (0.184) Observations 104,196 104,196 104,196 R-squared 0.282 0.289 0.330 Note:Regressionwithlog(TTD)asthedependentvariable.LocalcontrolvariablesincludeBartik-typepredicted industryemploymentgrowth,indicatorsforsandstateandcoastalstate,populationshareofimmigrants,populationshareofcollegeeducated,populationdensity,andcountyrealGDP.Robuststandarderrorsarereportedin parentheses. ***p<0.01,**p<0.05,*p<0.1. tors might play a limited independent role after taking into account the developer’s choice of constructioncharacteristics. ThethirdcolumnofTable3allowsforadditionalflexibilityinthetime-varyingresponseof TTD to local economic factors by interacting the time-invariant local controls with year fixed effects. While the R-squared moderately improves to 33 percent, the regression coefficients onconstructioncharacteristicsremainrelativelyrobustacrossallthreespecifications. 10

Table4: County-levelTTDstatistics (unit: days) RawTTD Reg. (1) Reg. (2) Reg. (3) Mean 1,044 973 969 971 Std. dev. 426 317 317 287 IQR 365 396 371 319 P10 640 605 605 631 P25 822 764 784 803 P50 1,005 954 959 964 P75 1,187 1,160 1,155 1,123 P90 1,369 1,369 1,358 1,348 Observations 267 267 267 267 Note: Each observation is a county’s median TTD. We use counties with at least 10 completed sections observed. IQRstandsfortheinterquartilerange(P75−P25). FivedifferentpercentilesofeachTTDdistributionare shown—forexample,P50referringtothemedian(50thpercentile)ofthedistribution. 2.4 County-level TTD Using the regression results in Table 3, we now construct the county-level TTD for the representativehousing. Thatis,wenormalizeTTDforeachsectionbyassumingthatthecontrolled observables are at their national average values and add the fitted residuals. We then take the median of this value for each county of interest. Note that we use the median instead of the mean, as the distribution of TTD is skewed to the right. Moreover, the median TTD should berelativelyinsensitivetoanyremainingtime-varyingdemandfactorsofTTD,suchasthefat righttailinconstructiontimeduringtheGreatRecession(OhandYoon,2020). Table 4 presents some cross-county moments of each county’s measure of TTD. In the first column (“Raw TTD”), we observe that the average of the county-level TTD using raw section TTD data is almost the same as the average of the section TTD itself. The standard deviation and the interquartile range are sizable at 426 days and 365 days, respectively. The secondtofourthcolumnspresentthesamestatisticsusingthecontrolledTTDestimatesinTable 3. Controlling for construction characteristics and focusing on a nationally representative housingdevelopment,wefindthatthecross-countymeanTTDis973days,justlessthanthree years. The standard deviation and the interquartile range are also sizable at around one year. Results for the third and fourth columns are similar to those of the second column, consistent with the R-squared results in Table 3, suggesting that after one controls for construction characteristics,themarginalcontributionoflocaldemandfactorsislimited. 11

2.5 The geographic determinants of the land development process Land development is a major topic of interest in civil engineering, as each construction site poses unique engineering challenges based on soil characteristics, topography, weather, and otherphysicalfeatures(Kone,2006). Assuch,developerscreatenotonlyamasterplandesign that conceptualizes their new development at the location of interest, but also a site engineering plan that adapts the master plan design to the physical properties of the site. These site engineeringplansinclude(i)agradingplanthatshowstheelevationsofgroundsandbuildings, (ii)astormwatermanagementplanthatshowsthevolumeandrateofstormwaterrunoff,and (iii)anerosionandsedimentcontrolplanthatshowstheerosioncontrolbarriersandmaterials atthesite. Thelocalgovernmentregulationondevelopmentvariesbasedonitstransitoryand permanent environmental effects, and this factor also plays an important role in shaping the engineeringplansofeachsite. As shown in Table 4, our county-level TTD measures exhibit significant degrees of crossregional variation even after controlling for construction characteristics and local economic factors. Based on the described process of land development, we then ask whether any observable geographic differences in engineering challenges could account for that variation. Some of the geographic differences could be highly correlated with factors that determine long-run housing supply described in Saiz (2010), but other geographic differences that do not play a large role in accounting for the long-run housing supply could nevertheless matter for the duration of land development. For example, each location is exposed to different weatherconditionsthatmightnotmateriallyaffectthedecisiontodeveloplandbutmightstill matter for the cost and duration of land development. That is, locations with extreme storm and heat conditions could still be desirable for new construction, but severe weather will occasionally delay on-site construction activity and the developer’s building design to tolerate extremeweatherconditionsmightfurtherlengthenthedevelopmentprocess. Table5displaystheregressionresultusingourcontrolledTTDmeasuresandtheobserved geographicdeterminantsdescribedinthissection. Boththelong-runhousingsupplyelasticity and rainfall intensity affect TTD in a statistically significant sense. That is, the duration of land development is longer for locations where (i) the long-run housing supply is limited and (ii)rainfallsaremoreintense,morefrequent,orboth. HeatalsodelaysTTD,butitsstatistical significance is less pronounced. A careful study of these engineering challenges is beyond the scope of this paper, but we think that the significance of these geographic determinants in accounting for the variation in TTD across locations suggests the potential of our controlled TTDmeasuretoreflectplausiblyexogenousvariationstoeconomicshocks. 12

Table5: County-levelTTDregressionresults Variables Reg. (1) Reg. (2) Reg. (3) Saizelasticity −0.162*** −0.150*** −0.134*** (0.047) (0.048) (0.041) Rainfallintensity 0.125*** 0.099*** 0.099*** (0.022) (0.022) (0.021) Heat 0.030 0.049** 0.036* (0.022) (0.022) (0.021) Observations 222 222 222 R-squared 0.206 0.192 0.190 Note: Weusecountieswithatleast10completedsectionsobserved. “Rainfallintensity”measurestherainfall inches per hour on a storm of one-hour duration and a 100-year return period for each county (Data source: NationalOceanicandAtmosphericAdministration’sAtlas14PrecipitationFrequencyEstimates). “Heat”—that is,coolingdegreedays—isameasureoftheyear’stemperaturehotness,calculatedasthedifferencebetweenthe dailytemperaturemean(thesumofthehighandlowtemperaturesdividedbytwo)and65degreesFahrenheit, multipliedbythenumberofdayswithapositivevalueofthisdifferenceinagivenyear(Datasource: National Centers for Environmental Information’s Annual Climatological Data). Robust standard errors are reported in parentheses. ***p<0.01,**p<0.05,*p<0.1. 3 Time-to-develop model of housing investment This section outlays the theoretical framework that ties our TTD data to the housing supply elasticity, which is a key measure of interest in the housing literature. In the model, the developermakeshousinginvestmentdecisionsunderaTTDconstraint. Weanalyticallyderive and characterize the short- to long-run housing supply elasticities as a function of several parameters,includingTTD.Asourfocusistoanalyticallyformulatehousingsupplyelasticity based on TTD, it is important to note that our framework abstracts from the endogeneity of TTDorotherpotentialshort-rundeterminantsofhousingsupplystudiedintheliterature. 3.1 Model description Inperiodt,thedeveloperproduceshousingunits,I ,usinginputsbuiltincurrentandprevious t periods,{U } ,basedonthefollowingTTDconstructionfunction: t−p|t p=0,1,···,P (cid:32) (cid:33) θ P θ−1 (cid:88) θ−1 I = U θ , θ > 0. (3.1) t t−p|t p=0 13

The parameter P is the number of periods it takes to complete a project from the beginning, andθgovernsthesubstitutabilityofthedifferentstagesofconstruction. ThisgeneralizedTTD specification follows Sarte et al. (2015) and nests the investment assumption in Kydland and Prescott(1982)asaspecialcasewhenθ → 0.7 Inturn,thedeveloperbuildsconstructioninputsU atalotwherehousingcompletions t|t+p are scheduled in period t + p for each p ∈ {0,1,··· ,P}. These inputs are built based on a Cobb-Douglasproductionfunction: U = M1−α Nα , ∀p ∈ {0,1,··· ,P}, (3.2) t|t+p t+p−P|t+p t|t+p where M is the housing permit in the beginning period of development t + p − P t+p−P|t+p for the lot where new housing is scheduled to be completed in period t+p and N is the t|t+p amountofvariableconstructioninputatthatlot. In each period, the developer purchases building permits, M , from the local governt|t+P ment at a price, q , for a lot that is at the beginning stage of development. Moreover, the M,t developer hires variable construction inputs for each lot under development at a competitive cost, w . When a lot is fully developed, its completed new houses, I , are sold at a unit price, t t q . Thedeveloper’sprofitinperiodt,Φ ,is t t Φ = q I −q M −w N , t t t M,t t|t+P t t P (cid:88) where N = N . (3.3) t t|t+p p=0 The developer builds new houses at multiple lots by purchasing building permits and utilizingvariableconstructioninputstomaximizeitsdiscountedsumofprofits: ∞ (cid:88) max E Λ (q I −q M −w N ), 0 0|t t t M,t t|t+P t t {It,M t|t+P ,Nt,{N t|t+p ,U t|t+p }P p=0 } t=0 subjecttotheTTDequations(3.1),(3.2),and(3.3). ThevariableΛ isthestochasticdiscount 0|t factorbetweenperiods0andt,andE istheexpectationsoperatorconditionaloninformation t availableinperiodt. 7Consistent with our TTD construction function when θ → 1, we assume that I = (cid:81)P U when t p=0 t−p|t θ =1. Tosimplifytheanalysiswithoutlossofgenerality,weassumeθ (cid:54)=1inthissection. 14

Finally,permitsaresuppliedbythelocalgovernmentandareelastictothehouseprice: M = M ¯ qγ. (3.4) t|t+P t Thisassumptioniscommonintheliteratureandisconsistentwiththelong-runhousingsupply elasticityasafunctionofγ (Guren,McKay,NakamuraandSteinsson,2020). 3.2 Developer’s housing supply We denote µ as the period-t Lagrange multiplier of equation (3.2) for each p and express t|t+p the respective optimality conditions of (i) construction at each stage, (ii) variable inputs at eachstage,and(iii)buildingpermitsatthebeginningstageofdevelopmentasfollows: (cid:34) (cid:18) (cid:19)1(cid:35) I θ µ = E Λ q t+p forp = 0,1,··· ,P, t|t+p t t|t+p t+p U t|t+p w = αµ M1−α Nα−1 forp = 0,1,··· ,P t t|t+p t+p−P|t+p t|t+p (cid:34) (cid:35) P (cid:88) U q = (1−α)E Λ µ t+p|t+P . M,t t t|t+p t+p|t+P M t|t+P p=0 The first condition shows that construction at each stage is chosen such that its shadow value, µ , is equal to its marginal contribution to the expected discounted value of the completed t|t+p house in the future. The second and third conditions equate the costs of variable inputs and thebuildingpermittotherespectivemarginalproducts. Afterlog-linearizingthepreviousoptimalityconditionsaswellasequations(3.1)through (3.4),wederivethefollowinglemmathatsummarizesthemodel’shousingsupplyconditions. Lemma1(log-linearizeddynamichousingsupplyequilibrium) Let each hatted variable be the log deviation from its steady-state value. The following log-linearized equilibrium conditionssummarizetheTTDmodelofhousinginvestment: P 1 (cid:88)(cid:16) (cid:17)p I ˆ = β ˜α(θ−1)/θ U ˆ , t t−p|t B(P) p=0 (cid:18) (cid:19) 1−α 1 1 1−α + U ˆ = E I ˆ +E qˆ + M ˆ −wˆ +E (λ ˆ −λ ˆ ), t|t+p t t+p t t+p t+p−P|t+p t t t+p t α θ θ α ˆ ˆ ˆ U = (1−α)M +αN , t|t+p t+p−P|t+p t|t+p 15

(cid:32) (cid:33) ˜ P 1−β (cid:88) N ˆ = β ˜pN ˆ , t 1−β ˜1+P t|t+p p=0 ˆ M = γqˆ, t|t+P t where λ β ˜(α(θ−1)/θ)(1+t) −1 Λ = βp t+p , β ˜ = βθ+α( θ 1−θ), and B(t) = . t|t+p λ β ˜α(θ−1)/θ −1 t Note that in Lemma 1, we introduced the deterministic per-period discount factor parameter β < 1. As such, the stochastic discount factor between period t and t + p, Λ , is t|t+p decomposed into the deterministic discount factor βp and the net stochastic discount factor λ /λ . t+p t UsingLemma1,thefollowingpropositionderivesthedeveloper’speriod-by-periodhousingsupplychoiceasafunctionofhousepricesandothergeneralequilibriumforces. Proposition2(period-by-periodhousingsupplycurve) Basedonthelemmaandassuming a steady-state equilibrium before period 0, we derive the following period-by-period housing supplycurve:  Υ (P)qˆ +GE (P), ∀t ∈ [0,P), ˆ t t t I = t  α qˆ +γqˆ +G(cid:103)E (P), ∀t ∈ [P,∞), 1−α t t−P t where B(t) Υ (P) = , t (cid:0) 1−α + 1 (cid:1) B(P)− 1B(t) α θ θ t Υ (P) (cid:88) (cid:16) (cid:17) GE (P) = − t (β ˜α(θ−1)/θ)j wˆ +(λ ˆ −λ ˆ ) , t t−j t−j t B(t) j=0 P Υ (P) (cid:88) (cid:16) (cid:17) G(cid:103)E (P) = − P (β ˜α(θ−1)/θ)j wˆ +(λ ˆ −λ ˆ ) . t t−j t−j t B(P) j=0 All the proofs are available in the Appendix. Proposition 2 decomposes housing supply into its partial equilibrium and general equilibrium components. The general equilibrium component, denoted by GE (P) and G(cid:103)E (P), depends on the current and past histories of t t constructionwagesandthestochasticdiscountfactors. Ourobjectofinterestinthissectionis housing supply in partial equilibrium, and the general equilibrium forces that depend on the setup of the overall economy will be studied in section 5. The partial equilibrium component 16

Figure1: Theoreticalstatichousingsupplyelasticities consists of the current housing price before the TTD constraint (t < P) and the P-period lagged house price after the TTD constraint (t ≥ P). The partial equilibrium component has a well-defined static housing supply elasticity, which is Υ (P) when t < P and α/(1 − α) t whent ≥ P. Thefollowingcorollaryprovidesussomeusefulcomparativestaticswithregard tothederivedstatichousingsupplyelasticities. Corollary3(comparativestatics) The static housing supply elasticity when t < P, Υ (P), t has two properties. First, Υ (P) is positive and an increasing function of time, with an upper t boundofα/(1−α): α 0 < Υ (P) < Υ (P) < . t−1 t 1−α Second,Υ (P)islargerwhentheTTDconstraintP isshorter: t ˜ ˜ Υ (P) > Υ (P) whenP < P. t t Corollary3isvisualizedinFigure1. Asobserved,thestatichousingsupplyelasticityisan increasing function up to the TTD constraint. Afterward, the static housing supply elasticity is determined by the parameter α, which represents the production elasticity to variable construction inputs. Comparing the housing supply elasticity between two regions with different 17

˜ TTD constraints, P and P, we find that the region with a shorter TTD constraint has a higher static housing supply elasticity for two reasons. First, housing supply is more flexible during periods under construction, represented by area A in the figure. Second, housing supply determined at the beginning period is completed earlier, represented by area B in the figure. In turn,areaA+B isthecumulativehousingsupplyelasticitydifferenceinthetworegions. As notedearlier,ourmodelneststheTTDassumptioninKydlandandPrescott(1982)asaspecial case when θ → 0. In this case, the static housing supply elasticity becomes a step function: 0 before the TTD constraint is reached and α/(1−α) afterward. As such, the difference in the statichousingsupplyelasticityacrossregionsinKydlandandPrescott(1982)arisesonlyafter theTTDconstraintisreachedinthemoreflexibleregion,whichisareaB. 3.3 The short- and long-run empirical housing supply elasticities Using the proposition, we define the T-horizon housing supply elasticity that is consistent withexistingempiricalmeasuresofthehousingsupplyelasticity. Definition4(T-horizonhousingsupplyelasticity) The T-horizon housing supply elasticity is defined as the average of the theoretical partial equilibrium period-by-period housing supply elasticities between periods 0 and T. Using Proposition 2 and assuming Υ (P) = t α/(1 − α) when t ≥ P for simplicity of notation, we define the T-horizon housing supply elasticitywithP-periodTTD,E (P),as T E (P) ≡ ∆T t=0 I ˆ t ≡ 1 (cid:88) T (cid:2) Υ (P)+γ(T −P +1)×1 (cid:3) , (3.5) T ∆T qˆ T +1 t {T≥P} t=0 t t=0 where 1 is an indicator function equal to 1 when T ≥ P. Moreover, the long-run {T≥P} housingsupplyelasticity,E ,isdefinedas ∞ α E ≡ lim E (P) = +γ. ∞ T T→∞ 1−α By taking an average of the period-by-period housing supply elasticities, our T-horizon housing supply elasticity summarizes the evolution of housing supply over time based on the supply-sidebehavior. IntheshortrunwhenT < P,housingsupplyelasticityisnotafunction of γ but a function of TTD and other parameters of the housing construction function. In the longrun,housingsupplyelasticityispurelyafunctionofγ andα;TTDisnolongerrelevant. For P ≤ T < ∞, housing supply elasticity is a weighted average of TTD and the long-run elasticity,wherethelattermattersmoreasT → ∞. 18

Of note, this definition is not the only way to characterize the T-horizon housing supply elasticity. Depending on the endogenous forces that drive house prices, different weights on theperiod-by-periodhousingsupplyelasticitiesmightbettercharacterizetheaveragehousing supply elasticity over the horizon of interest. Indeed, our unweighted average of the periodby-period housing supply elasticities might be viewed as an agnostic measure to the various drivingforcesofthehousepricethroughoutthehorizon. 4 Quantifying housing supply elasticities in each horizon Using the duration of land development statistics in section 2 and our theoretical result in section 3, this section presents regional housing supply elasticities in each horizon. We ask whetherthesignificantTTDvariationswefindinthedatatranslatetosignificantvariationsin the T−horizon housing supply elasticities by comparing those with the counterpart long-run elasticities. In this section, our main focus is on quantifying housing supply elasticities at different horizons; their relevance in accounting for housing market dynamics will be studied insection5. 4.1 Parameterization Recallcountyi’sT-horizonhousingsupplyelasticityfromDefinition4. Inequation(3.5),the elasticityEi(P )requiresfiveinputs: P ,γ ,α,β,andθ.8 Webeginbydiscussingthecalibra- T i i i tionof thekey county-levelparametersP andγ . First,P is calibrated bythe countymedian i i i TTD estimate in the previous section. Note that the TTD estimate is the time span between the approval of the preliminary site plan and the completion of the project. As such, it is inclusive of the county’s average time span between submitting a project for final approval and receiving a decision, documented in Gyourko, Hartley and Krimmel (2021). Next, the permit supply elasticity parameter (γ ) is derived from Saiz’s long-run housing supply elasticity i togetherwithα describedbelow. For the remaining parameters, we set the time discount factor, β, at 0.99, as the model is calibrated at a quarterly frequency. The construction labor share, α, is set at 0.385 which implies that a county with the smallest Saiz’s supply elasticity has a permit elasticity γ at i its lower bound of zero. This value of α is consistent with our estimate of the construction laborincomeof37percentintheKLEMSaccountwhenweassumethatoverheadlaborcosts 8Duetodataavailability,weassumethatthetimediscountfactor(β),theconstructionlaborshare(α),and theelasticityofsubstitutionacrossconstructionstages(θ)areidenticalacrossallcounties. 19

are about 10 percent of the total labor cost.9 The elasticity of substitution across construction stages, θ, is set at 0.5 in our baseline. The calibration of θ is not straightforward, but we think a complementarity assumption (θ < 1) is intuitive, as most construction activities are conducted sequentially on site. Another piece of evidence that tells us that θ is small is the minor difference in our data between the number of housing units planned at the beginning of development and the number of housing units completed at the end of development. For example, the total number of housing units submitted at the beginning of development does not change at the end of development for about 60 percent of the sections in our sample. In the Appendix, we show that the regional variation of our supply elasticity measures is less sensitivetothisparameter. 4.2 The T−horizon housing supply elasticity InFigure2,wequantifythecounty-levelhousepriceelasticityforeachhorizon. Theleftpanel compares our measure of horizon-specific elasticities with the Saiz elasticity. The takeaway from this panel is that the shorter-run housing supply elasticities are not simply a monotonically smaller version of the long-run supply elasticities. While one-year (T = 4) housing supply elasticities are close to zero and show little variability, five-year (T = 20) housing supplyelasticitiesshowsignificantvariations. AsT increases,theT−horizonhousingsupply elasticitiesincreaseandconvergetothe45degreelinethatequatesthelong-runelasticity. The right panel shows the standard deviation of the T−horizon elasticities for each T normalized by the standard deviation of the long-run elasticities. The blue solid line plots the results when we use the county-level TTD measure used in the regression (3) column in Table 4. For example, when T = 9, our T−horizon supply elasticity is half as variable in standard deviation as the long-run elasticity. The red dashed line plots the results when we assume that all counties have the same TTD at the national median of T = 11. In this case, the cross-county variation in housing supply elasticity obviously disappears in the short run. Therefore, the difference between the blue solid line and the red dashed line quantifies the share of TTD variations in accounting for the cross-county variations of housing supply elasticities for each T. We find that the difference between the blue solid line and the red dashed line remains sizable especially for lower values of T, suggesting that TTD variations play a significant role in accounting for total variations in each of the T−horizon housing supply elasticity. For example, at T = 16, TTD contributes to about 50 percent of the total 9KLEMSstandsforK-capital,L-labor,E-energy,M-materials,andS-purchasedservices;thetermrefersto broadcategoriesofintermediateinputsconsumedbyindustriesintheirproductionofgoodsandservices. 20

5 4 3 2 1 0 ))P(i e( yticitsalE retrauQ-T i T Short- to Long-run Supply Elasticity 45o Line 0.8 T = 4 T = 20 T = 36 0.6 0.4 0.2 0.0 0 1 2 3 4 5 Saiz Long-Run Elasticity )yticitsalE ziaS(.dtS/))P(i e(.dtS i T TTD Contribution to Elasticity Variation Baeline Common TTD 0 4 8 12 16 20 24 28 32 36 40 Horizon (T) Figure2: Supplyelasticitiesineachhorizon Note: Theleft panelshowsthe scatterplotcomparing theSaizsupply elasticityandthe T−horizonsupply elasticitieswithT = 4,20,36quarters. Therightpanelshowsthecross-countyvariationoftheT−horizon supplyelasticitieswhenweuse(i)theTTDmeasureconstructedbyusingregression(3)inTable3(bluesolid line)and(ii)themediancountyTTDmeasureof11quarters(reddashedline). variationintheT−horizonhousingsupplyelasticity. Insum,we findthatthelargevariationsof TTDdocumentedinthedatatranslate toquantitatively sizable variations in the shorter-run housing supply elasticities. This suggests that our shorter-run housing supply elasticities could provide a new perspective compared to the long-runhousingsupplyelasticityinthestudyofhousingdynamics. Nevertheless, our focus so far on housing supply elasticities has limitations because the equilibrium housing market behavior is a combination of both housing supply and demand anditisunclearwhetherthemeasuredvariationsinsupplyelasticitieswouldimplysignificant variationsinhousingmarketvariablesthatweobservethroughthedata. Thatisthetopicofthe next section, where we study the equilibrium effects of short-run housing supply elasticities onhousepricesandhousinginvestment. 5 Theory and application of short- and long-run elasticities We documented that TTD is lengthy and dispersed across counties. Mapping the empirical TTD into our model’s analytical expression for housing supply elasticity, we find that the short-run housing supply elasticity is also highly dispersed across counties. As the housing supplyelasticityisderivedfromapartialequilibriummodelofhousingdevelopers,ourresults 21

are not specific to the nature of economic shocks or the endogenous response of housing demand. However,housingsupplyelasticitydoesnotdirectlyinformusaboutthedynamicsof house prices and quantities that are of interest in the literature. To understand its implications onhousepricesandquantities,oneneedstospecifythedemandsideofhousing. In this section, we study the theoretical implications of our short- and long-run housing supply elasticities on house prices and quantities using a local general equilibrium model of housinginvestment. Thatis,weextendthepartialequilibriummodelofdevelopersinsection3 byincorporatingthehouseholdandthenondurablegoods–producingsectors. Afterspecifying the local general equilibrium model, we hit the economy with a common housing demand shock and study the differential local house price and quantity responses. We show that our calibrated model predicts two salient housing market features that can be tested with house price and quantity data. We also investigate the effectiveness of a government’s discretionary housingsupplypolicywhenTTDistakenintoaccount. 5.1 Local general equilibrium TTD model The local economy consists of housing developers, households, nondurable goods producers, and a local government. Since the national central bank sets the interest rate, we assume that the local economy takes it as given. Therefore, the bond and nondurable goods markets do not clear locally, analogous to the assumptions in small open economy models in the international macro literature. Housing developers follow the same specification and notation as in section3. Below,wedescribethehouseholds,therestoftheeconomy,andtheequilibriumof themodel. 5.1.1 Households Therepresentativelocalhousehold’sexpectedlifetimeutilityis ∞ (cid:88) E βtU(C ,H ,N ,N ;ϕ ), (5.1) 0 t t n,t t t t=0 whereC isthehouseholdconsumptionofnondurablegoods,H istheserviceflowofhousing, t t N is the labor supply for the nondurable goods sector, and ϕ is an exogenous process for n,t t housing demand. The household’s one-period subjective discount factor, β, is consistent with thehousingdevelopers’deterministicdiscountfactorspecifiedinsection3. Thehousehold’sserviceflowofhousingisproportionaltoitshousingstock. Forsimplicity 22

ofnotation,thehousingstockisalsodenotedasH . Thehousingstockevolvesovertimeby t H = (1−δ)H +I , (5.2) t t−1 t whereδ isthedepreciationrateofthehousingstock. Thehouseholdflowbudgetconstraintisgivenby B ψ C +q I + t+1 + b B2 = w N +w N +B +Φ +T , (5.3) t t t R 2 t+1 n,t n,t t t t t t t whereB isthehousehold’sone-periodbondholdingsthatmatureinperiodt+1,R isthe t+1 t gross bond interest rate between periods t and t+1, w is the real wage for working in the n,t nondurable goods sector, Φ is the period-t profit of developers because households are the t final owners of the developers, and T is transfers from the local government. As is standard t insmallopeneconomymodels,thehouseholdissubjecttothebondportfolioadjustmentcost ψ B2 /2. The parameter ψ is calibrated to be positive for stability in solving the model but b t+1 b smallenoughtonotmateriallyaffectthemodeldynamics. 5.1.2 Therestoftheeconomy As we will discuss next, the rest of the economy consists of the nondurable goods producers, the local government, and the market-clearing conditions. We also specify the exogenous processforhousingdemandthatweuseforlaterapplications. Nondurablegoodsproducers. Therepresentativenondurablegoodsproduceroperateswith ¯ a linear production technology, Y = ZN , where Y is the output of the nondurable good t n,t t ¯ and Z captures its productivity. The profit of the producer is Y − w N , where both the t n,t n,t input and output markets are perfectly competitive. The nondurable goods are tradable to otherregions. Local government. As specified in equation (3.4), the supply of housing permits is determinedbyitslocalgovernment,whichinturniselastictotheregion’sequilibriumhouseprice. Foreachhousingpermit,thelocalgovernmentcollectsafee,q ,fromdevelopers. Thelocal M,t government also collects the bond portfolio adjustment cost from households. The local governmentfollowsabalancedbudgetbyrebatingbackitsrevenuetothehouseholdsintheform 23

oftransfersT : t ψ T = q M + b B2 . (5.4) t M,t t 2 t+1 Market clearing. The labor markets for the nondurable goods sector and the construction sector clear by equating the supply and demand of labor in each sector. The permit market clears by equating permit supply to permit demand. The market for new housing clears by equatingthesupplyanddemandofnewhousinginvestment. Thebondandnondurablegoods markets do not clear locally as we assume that the interest rate is exogenously determined by the national central bank. Finally, the following resource constraint of the local economy needstobesatisfied: B t+1 C + = w N +B . (5.5) t n,t n,t t R t Exogenous housing demand. The exogenous component of housing demand, ϕ , is att tached to the household’s preference over the housing stock H in the utility function (5.1) t andfollowsafirst-orderautoregressiveprocessinlogs: logϕ = (1−ρ )ϕ¯+ρ logϕ +(cid:15) , (5.6) t ϕ ϕ t−1 ϕ,t where (cid:15) is the exogenous housing demand shock and ρ is the persistence of exogenous ϕ,t ϕ housingdemandfromitsmeanϕ¯. 5.1.3 Equilibrium The local general equilibrium is a set of variables {U ,N ,µ }P , M , N , I , t|t+p t|t+p t|t+p p=0 t|t+P t t H , C , Y , N , B , w , w , q , R for t ≥ 0 such that taking as given the endogenous t t t n,t t+1 t n,t t t prices w , w , and q , the exogenous processes R and ϕ , and the initial conditions B and t n,t t t t 0 H ,thefollowingconditionshold: −1 1. Housingdevelopersmaximizetheirprofitsubjectto(3.1)through(3.3). 2. Householdsmaximizetheirlifetimeutility(5.1)subjectto(5.2)and(5.3). 3. Nondurablegoodsproducersmaximizetheirprofit. 4. Thelocalgovernmentsuppliespermitsandbalancesitsbudgetaccordingto(5.4). 24

Quantity ecirP Inelastic Supply C Elastic Supply B Demand (D(cid:48)) A Demand (D) Figure3: Housingsupplyanddemandcurves 5. Markets clear for nondurable goods labor, construction labor, permits, and housing investment,andtheresourceconstraint(5.5)issatisfied. We assume that the interest rate is fixed by the national central bank. The exogenous componentofhousingdemandfollows(5.6). 5.2 Relative price and relative quantity correlation in the housing market Consider two regions with different housing supply elasticities. In Figure 3, the housing supply curves of the two regions are denoted as inelastic supply and elastic supply. Assume that the initial housing market equilibrium for both regions is at A, where the demand curve is denoted as D. When there is positive housing demand that shifts the demand curve from D to D(cid:48), the equilibrium price and quantity responses are different for the two regions. In the inelastic supply region, the equilibrium is formed at C, where prices increase by a lot and quantitiesincreasebylittle. Bycontrast,intheelasticsupplyregion,theequilibriumisformed at B, where prices increase by little and quantities increase by a lot. This outcome implies that conditional on a demand shock, the cross-regional correlation between the relative price andtherelativequantityshouldbenegative. In the data, however, the cross-county correlation of the relative price and the relative 25

quantity is mildly positive. Using the county house price indexes from the Federal Housing Finance Agency and the annual estimates of the number of housing units from the Census Bureau, we find that the cross-county correlation of the annual growth rates of prices and quantities is positive in 15 years between 2001 and 2019 with an average correlation of 0.13. This discrepancy between theory and data poses a challenge to the view that housing supply elasticities play an important role in the cross-regional dynamics of the housing market (Davidoff,2016). We provide an explanation for this discrepancy through a calibrated local general equilibrium model, as specified above. When a positive demand shock hits the economy, the local governmentssupplynewpermits,andtheregionwithelasticlong-runhousingsupplyobserves alargernumberofnewpermitssuppliedbythelocalgovernmentcomparedtotheregionwith inelastic long-run housing supply, as shown in the left panel of Figure 4. This response is thestandardleveleffect ofthesupplyelasticity—thatis,housingsupplyishigherinlocations withhigherlong-runelasticity. WithTTD,however,ourmodelalsogeneratesaslopeeffectofthesupplyelasticityasTTD generates a positive slope between the short- and long-run supply elasticity.10 As developers ineachregiondealwithmultipleprojectsunderdifferentstagesofconstruction,theymakethe most out of this process by allocating more variable inputs to the project with a large number of permits at the expense of lowering inputs of some other projects with fewer permits. The middle panel of Figure 4 shows that this substitution channel is stronger in an elastic region relative to an inelastic region, because developers in regions with high long-run supply elasticityfaceasteeperslopeoftheelasticitypathandhenceastrongerintertemporalsubstitution motive. As a consequence, the right panel of Figure 4 shows that new housing quantities in the elastic region are lower than those in the inelastic region until the period when the TTD constraintismet. Quantifying the county-level TTD and long-run elasticity data, Figure 5 presents the conditional relative price and relative quantity rank correlations at each horizon under a common housing demand shock. In the baseline, the correlation turns positive in the short run, peaks ataroundsixquarters,andthengraduallytrendsdownuntilitmovesintonegativeterritoryas the TTD friction is lifted in many counties. We also plot the correlation in a counterfactual model in which TTD for all counties is set to be at the national median of 11 quarters. In this case, the only source of cross-county variation is the long-run elasticity. This model predicts that the conditional cross-county correlation is either 1 in the short run or −1 in the longer run when the TTD constraint is lifted. In both models, the long-run correlations are −1, as 10Forsimplicityofdiscussion,weassumethattheTTDconstraintisthesame11quartersforbothregions. 26

1.2 1.2 1.2 1 1 1 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 -0.2 -0.2 0 10 20 30 10 5 0 0 10 20 30 Figure4: Impulseresponsestoacommondemandshock Note: Thisfigureshowstheimpulseresponseofthehousingpermit(leftpanel),time-0responsesofeachpstageconstructionlabor(middlepanel),andtheimpulseresponseofnewhousingconstruction(rightpanel), toacommondemandshock. Bluesolidlinesshowtheresponsesintheregionwithinelastichousingsupply (γ = 0.9)andreddashedlinesshowtheresponsesintheregionwithelastichousingsupply(γ = 3.0). The commondemandshockisassumedtohaveapersistenceof0.95. housingsupplyinthelongrunisdeterminedonlybythelong-runelasticity. Wegaugetheempiricalrelevanceofthetheoreticalchannelbysimulatingourmodelwith common demand shocks and computing the unconditional cross-county correlations of the annual growth rates of prices and quantities in the same manner as our empirical estimates above.11 Weevaluatetheimportanceofthepositiveshort-runrelativepriceandrelativequantitycorrelationingeneratingtherelativepriceandrelativequantitycorrelationobservedinthe databycomparingourresultswithandwithoutTTD. Figure 6 indicates that the strong negative cross-county correlation between the annual growth rates of prices and quantities disappears as the short-run elasticity is assumed to be smaller than the long-run elasticity, although the correlation does not become mildly positive like in the data. This result suggests that while the discrepancy is much smaller than in the model without any TTD, other shocks are also needed to fully account for the correlation we observe in the data, such as the location-specific housing demand shocks studied extensively in Howard and Liebersohn (2021, 2023). The figure also illustrates that what matters for the model’s cross-county correlation is a lengthy median TTD rather than the variation of TTD across counties. That is, there is little difference in the correlation between the baseline case 11WepresentourmodelcalibrationintheAppendix. 27

1 0.5 0 -0.5 -1 0 5 10 15 20 25 30 Figure5: Therelativepriceandrelativequantitycorrelationunderacommondemandshock Note: This figure shows the Spearman’s rank correlation between (cumulative) house price response and (cumulative)newconstructionresponsetoacommondemandshock. Bluelineshowsthecorrelationinthe baseline model with cross-county variations in both TTD and the long-run elasticities. Red line shows the correlationinthemodelwithcommonTTDforallcounties. andthecaseinwhichcountieshaveacommonTTD. Overall, our argument supports the view that housing supply elasticities could play an importantroleinaccountingforhousingmarketdynamicsifshort-runhousingsupplyfrictions intheformofTTDarealsotakenintoaccount,relatedtoapointraisedinGurenetal.(2020). 5.3 House price dynamics in the short and long run Our local general equilibrium model predicts that house prices react more sensitively to a housing demand shock in locations with inelastic housing supply conditions. With TTD constraints, there is a wedge between the short- and long-run housing supply elasticities, and section4findsthatcross-countyvariationsintheshort-runhousingsupplyelasticityarelarge and distinct from the variations in the long-run elasticity. As such, we now study the quantitative importance of these measured short-run housing supply elasticities in accounting for housepricedynamicsbothinourmodelandinthedata. Model investigation. To investigate the degree to which our short-run housing supply elasticitiesshouldinfluencehousepricesintheory,wesolveourmodelandcomputeeachcounty’s housepriceresponsetoacommonhousingdemandshock. 28

0.2 0 -0.2 -0.4 -0.6 -0.8 0 4 8 11 12 16 20 Figure6: Unconditionalrelativepriceandrelativequantitycorrelation: Modelversusdata Note:ThisfigureshowstheaverageofSpearman’srankcorrelationbetweenhousepriceandnewconstruction inresponsetoacommondemandshock.Bluesolidlineshowstheaveragecorrelationofthesimulateddatain thecommonTTDmodelsasafunctionofTTD(P). Redcrossshowstheaveragecorrelationinthebaseline modelwherebothTTDandlong-runelasticityareheterogeneous(medianTTDis11quarters).Lastly,orange dashedlineshowstheaveragecorrelationinthe2001-2019UScountydata. In the left panel of Figure 7, we plot the rank correlation for each horizon T between a county’s T-horizon housing supply elasticity, E , and its impact house price response to T a common housing demand shock, q . For each persistence parameter of interest for the 0 housing demand shock, the rank correlation is negative as house prices tend to increase more in counties where housing supply elasticities are lower. The rank correlations are not exactly −1, however, because the county rankings of the T-horizon housing supply elasticity are not thesameacrossthehorizons,asdiscussedinsection4. Moreover,thenon-monotonerankings of the various horizons of housing supply elasticities suggest that the rank correlation should depend on the persistence of the housing demand shock. In the figure, we find that when the persistence is relatively high (ρ = 0.92), the rank correlation is close to −1 for housing ϕ supply elasticities at a horizon of 7 or more years. This near-perfect, negative alignment between the longer-run housing supply elasticities and the impact house price response no longerholdswhenthepersistenceisrelativelylow(ρ = 0.88),asthelowestrankcorrelation ϕ isaround−0.9atthefour-yearhousingsupplyelasticityandthecorrelationisaround−0.7at thelong-runhousingsupplyelasticity. Theseresultssuggestthatourshort-runhousingsupplyelasticitieshaveimplicationsonthe richerdynamicsofhousepricesnotcapturedbythelong-runhousingsupplyelasticity. Thatis, 29

-0.2 20 -0.4 15 -0.6 10 -0.8 5 -1 0 0 4 8 12 16 20 0.85 0.88 0.91 0.94 0.97 1 Figure7: Short-runhousingsupplyelasticitiesandhousepricesacrosscounties Note: The left panel plots the Spearman’s rank correlation coefficient (at each T) between a county’s Thorizonhousingsupplyelasticityandthesizeofitsimpacthousepriceresponse,wheneachcountyishitby acommonhousingdemandshocksubjecttothethreecalibratedpersistenceparameters. Thecirclemarkerof eachlineindicatesthelowestcorrelationcoefficientforthegivenpersistenceparameter.Therightpanelplots theT valuethatisconsistentwiththelowestcorrelationbetweentheT-horizonhousingsupplyelasticityand the size of the impact house price response for each persistence parameter ρ ∈ (0.85,1) of the common ϕ housingdemandshock. when the persistence of the common housing demand shock is relatively low, our shorter-run housing supply elasticities are more relevant in accounting for the sensitivity of house prices in terms of exhibiting the lowest rank correlation. In the right panel of Figure 7, we show the optimal horizon T∗ at which the Spearman’s rank correlation between the T-horizon housing supply elasticity and the impact house response is minimized, for each persistence parameter ρ ofthehousingdemandshock: ϕ T∗(ρ ) = arg min Corr(E ,q (ρ )). ϕ T 0 ϕ T∈[1,∞) We find that the most relevant housing supply elasticity depends on the persistence of the housing demand shock.12 For example, when the housing demand shock of interest is highly persistent at or above 0.98, the optimal T-horizon housing supply elasticity is 15 years or higher, suggesting that the Saiz long-run elasticity is a relatively good benchmark to study 12Weonlyplotthecasewherethepersistenceparameterimpliesthatthehalf-lifeoftheshockismorethana year. 30

housepriceresponses. However,whenthepersistenceofthehousingdemandshockofinterest is less than 0.90, the optimal T-horizon housing supply elasticity is 7 years or lower and the relevanceofthelong-runelasticityisdiminished. In sum, we find that when the housing demand shock of interest has a relatively low persistence, housing supply elasticities with short horizons are more relevant than the long-run elasticity in accounting for the relative house price dynamics across counties. The performance of the long-run elasticity is weak because shocks with lower persistence do not last long enough to affect housing supply in the long run. We also verify that when the housing demandshockofinterestishighlypersistent,thelong-runhousingsupplyelasticitydominates theshort-runelasticityasvariationsinTTDmatterslessinthiscase. These findings hold even when we allow for cross-regional spillovers. In the Appendix, westudyatwo-regiongeneralequilibriummodelwithasymmetrichousingsupplyconditions. Specifically,weassumethattheshort-runhousingsupplyelasticityislargerinregiononebut that the long-run housing supply elasticity is larger in region two. Conditional on a positive common housing demand shock, we find that the impact house price response is larger in region two, but the response reverses eventually, and the medium-run house price response is largerinregiontwo. Empirical exercise. To study the empirical relevance of our short-run housing supply elasticityinaccountingforhousepricedynamics,wefocusonfourepisodesoftherecenthousing cycle: (1) the 2000s housing boom (2002–06), (2) the 2000s housing bust (2006–09), (3) the 2010s housing recovery (2012–19), and (4) the COVID housing boom (2020–22). In each of these episodes, we estimate the following relative house price regression for each horizon T ∈ {1,2,3,···}: ∆log(P /P ) = κ E ˜i +ΩX +u . (5.7) i N T T i i The left hand side variable is the log change of county i’s house price index, P , relative to i the national house price index, P , between the years of interest. The variable E ˜i is county N T i’sT−horizonhousingsupplyelasticitystandardizedtohavethesamecross-countyvariation acrossT. Weallowforasetofcontrolvariables(includingaconstant),X ,aswellasaresidual i term, u , that captures other unmodeled drivers of the relative house price. Because a higher i value of the housing supply elasticity should imply a lower sensitivity of the relative house price to shocks, a simple test to validate the empirical relevance of the T−horizon housing supplyelasticityistoshowthattheestimatedcoefficientκ issignificantlynegative. T Figure 8 displays the estimation results of the relative house price regression in each of 31

0.00 -0.10 -0.20 -0.30 -0.40 -0.50 ) k( stneiciffeoC noissergeR T 2000s housing boom 2000s housing bust 0.00 -0.10 -0.20 -0.30 -0.40 -0.50 4 8 12 16 20 4 8 12 16 20 Supply Elasticity Horizon (T) Supply Elasticity Horizon (T) 0.10 0.05 0.00 -0.05 -0.10 ) k( stneiciffeoC noissergeR T 2010s housing recovery COVID housing boom 0.10 0.05 0.00 -0.05 -0.10 4 8 12 16 20 4 8 12 16 20 Supply Elasticity Horizon (T) Supply Elasticity Horizon (T) T-Year Elasticity – 1 S.D. – 2 S.D. Figure8: Relativehousepriceregressioncoefficients Note: Eachfigureshowstheestimatedcoefficientsforκ asafunctionofthestandardizedsupplyelasticity T horizonT inequation(5.7). Theconfidenceintervalsforthe1and2standarddeviationsoftheestimatesare shownasthedarkgrayandlightgrayareas,respectively. Thefourfigurespresenttheresultsusingdatafor 2002–06(topleftpanel),2006–09(toprightpanel),2012–19(bottomleftpanel),and2020–22(bottomright panel). Local control variables include Bartik-type predicted industry employment growth, indicators for sandstateandcoastalstate,populationshareofimmigrants,populationshareofcollegeeducated,population density,andcountyrealGDPgrowth. thefourepisodes. Duringthe2000shousingboomandbustperiods,wefindthatthelong-run elasticityismorerelevantthantheshort-runelasticityinaccountingforchangesinacounty’s relativehouseprice,inthesensethattheestimatedcoefficientsforthelong-runelasticitiesare more negative than the coefficients for the short-run elasticities. In both the 2010s housing recoveryandtheCOVIDhousingboomperiods,however,theshort-runelasticityoutperforms thelong-runelasticity. Theestimatedcoefficientsfortheelasticitiesbelowthe10-yearhorizon are negative and the two standard deviation confidence intervals around the 3- and 4-year elasticitiesremainbelowzero. Ontheotherhand,theestimatedcoefficientsforlongerhorizon elasticitiesarepositiveandnotdistinguishablefromzero. 32

Through the lens of our model, the results imply that both the positive housing demand shock in the 2000s boom and the negative housing demand shock in the 2000s bust were perceived as highly persistent, leading to the higher relevance of the long-run elasticity in accounting for house prices. After the 2000s housing cycle, however, agents might have expected the 2010s recovery to be less persistent, possibly reflecting on the recent housing boomexperience,whichthenmadetheshort-runelasticitymorerelevanttoaccountforhouse prices. That is, even in a location where the long-run elasticity is high and buildable land is plentiful, residential developers in the 2010s might have thought that if TTD is too lengthy, pursuingnewdevelopmentisnotasworthyasbeforeduetoconcernsthatthepositivedemand couldquicklyreversecourse. ThesamegoeswiththeCOVIDboom,wheredevelopersmight havecontinuedtobelievethatthehigherhousingdemandinducedbythegreaterflexibilityof work-from-homewouldnotlastoncetheviruswasundercontrol. Finally, we note some caveats to the empirical analysis. First, the estimated regression coefficients since the 2010s are smaller in absolute value compared to the 2000s. It is indeed likelythatnationalhousingshocksplayedalimitedrolesincethe2010samidlocation-specific shocks in the housing market. Second, TTD could have shifted especially during the COVID boom when there were known bottlenecks to construction, such as the shortage in lumber. While we think these bottlenecks were widespread across the country and did not meaningfullyaffecttherelativeTTDacrosslocations,ifTTDwasdisproportionatelyshiftedinseveral locations to the extent that the overall TTD rankings were significantly changed, then our resultsshouldbetakenwithmorecaution. Thehousingwealtheffect. Followingthediscussionthatourshort-runhousingsupplyelasticity is as relevant as the long-run housing supply elasticity in accounting for relative house price dynamics across locations, we now estimate the housing wealth effect implied by our elasticity. Following Guren et al. (2020), the housing wealth effect we estimate is the elasticity of retail employment per capita to real house prices at the CBSA level. We construct an instrument for the real house price change by interacting our 5-year housing supply elasticity with the national real house price change in the same period. We follow the analysis in Guren et al. (2020) and display the 10-year rolling window estimates of the housing wealth effect in Figure 9. For example, the point estimate for 1990q1 is based on the regression between1985q1and1995q1. Forcomparison,wealsopresentthesamesampleestimatesusing boththesensitivityinstrumentofGurenetal.(2020)andthehousingsupplyelasticityofSaiz (2010). We find that our results are consistent with those based on the other two instruments especiallyaftertheearly2000s. Assuch,weconfirmthatthemagnitudeofthehousingwealth 33

0.6 0.4 0.2 0.0 -0.2 secirP esuoH ot pmE liateR fo yticitsalE VI 5-Year Elasticity Sensitivity Instrument Saiz Elasticity 1990q1 1995q1 2000q1 2005q1 2010q1 Midpoint of 10 Year Window Figure9: Theelasticityofretailemploymentpercapitatohousepricesover10-yearwindows Note:ThefigureplotstheelasticityofretailemploymentpercapitatorealhousepricesattheCBSAlevelfor rolling10-yearsampleperiodsforthreedifferentmethods. Eachpointindicatestheelasticityfora10-year sample period with its mid-point in the quarter stated on the horizontal axis. The black solid line uses an instrument that interacts the 5-year housing supply elasticity derived from equation (3.5) with the national annual log change in house prices. The red dashed line uses the sensitivity instrumental variable estimator inGurenetal.(2020). Thegreendottedlineusesaninstrumentthatinteractstheestimatedhousingsupply elasticityfromSaiz(2010)withthenationalannuallogchangeinhouseprices. Allthreespecificationsuse the same controls and CBSA fixed effects as described in Guren et al. (2020). The sensitivity instrument alsoincludesregion-timefixedeffects,whilethe5-yearelasticityandtheSaizelasticityonlyusetimefixed effects. Thegrayareashows95%confidenceintervalsforthe5-yearelasticityregressionwherethestandard errors are constructed using two-way clustering by CBSA and time. All cases use the same sample of 79 CBSAswheretheTTDdataareavailable. effect discussed in Guren et al. (2020) is also sensible when using our elasticity that is based onadifferentsourceofvariationfromtheexistingapproaches. 5.4 The effectiveness of housing supply policy in stabilizing house prices Arapidincreaseinhousepricesraisesconcernsofpolicymakers,asthesedevelopmentscould subsequently lead to outsized drops in those prices that amplify stress in the financial system andthebroadereconomy. Assuch,stabilizinghousepricesisakeyobjectiveofpolicymakers, and various measures are discussed and implemented in practice. In this part, we study the effectivenessofthegovernment’sdiscretionaryhousingsupplypolicyasatoolforhouseprice stabilizationwhenTTDistakenintoaccount. 34

Before the analysis, we clarify what we mean by discretionary housing supply policy. As summarized in Glaeser and Gyourko (2008), new construction in the U.S. is regulated in terms of building codes and land-use rules. In particular, there are numerous examples of land-use regulations that directly limit housing supply across regions, such as minimum lot size requirements, height restrictions, or growth-control policies. The discretionary housing supplypolicywehaveinmindisatemporaryrelaxationoftheseexistingland-useregulations, asinpractice,newdevelopmentcouldreceiveawaivertosomeoftheregulations. To be specific, we modify the local government’s permit supply assumption in equation (3.4)tothefollowing: M = v M ¯ qγ, logv = ρ logv +εv, t|t+P t t t v t−1 t where the variable v indicates the government’s discretionary housing supply policy that folt lowsafirst-orderautoregressiveprocessinlogs. Figure10plotstheimpulseresponsefunctionsofthehouseprice,cumulativehousingconstruction (as a percentage of the initial housing stock), and housing permits conditional on a discretionarypermitsupplyshockthatincreasesthe10-year(40-quarter)cumulativeconstruction by 2 percent of the housing stock. Compared with the result when TTD is assumed to be zero, a positive discretionary housing supply policy is somewhat less effective in reducing housepricesintheshortrunwhenTTDissetatthenationalmedianof11quarters. Notethat thepeakdeclineinhousepricesoccursataroundtwotothreeyearswithTTD,comparedwith the peak declineat around one year without TTD.This difference implies that a discretionary housing supply policy with TTD could be an effective tool to stabilize house prices in the medium run through its effects on forming expectations about future supply conditions. As theliteraturefindsthathousepricestendtoshowmomentumintheshorttomediumrun,these discretionary supply policies could be effective in countering that momentum by controlling theexpectationsoffuturesupplyundertheTTDcommitment. Inconclusion,settingasidethepoliticalconstraintsinimplementingadiscretionaryhousing supply policy, we find that lengthy TTD might also somewhat limit its effectiveness in stabilizinghousepricesintheshortrun. WhileadiscretionaryhousingsupplypolicytostabilizehousepricesmightnothavebeenadiscussionatthenationallevelintheU.S.,thispolicy was implemented in Korea to tackle surging house prices in early 2021.13 Our analysis suggests the potential challenges of such a policy when TTD is lengthy. Of note, a nationwide 13See Cynthia Kim (2021), “S. Korea to Boost Seoul Housing Supply by 10% to Calm Buying Frenzy,” Reuters,February4,https://news.trust.org/item/20210204030650-r2wji. 35

0 2.5 50 -0.2 2 40 -0.4 -0.6 1.5 30 -0.8 1 20 -1 0.5 10 -1.2 -1.4 0 0 -1.6 -0.5 -10 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 Figure10: Modelresponsestoadiscretionarypermitsupplyshock Note:Thisfigureshowstheimpulseresponsesofhouseprice(leftpanel),cumulativehousingconstructionas apercentoftheinitialhousingstock(middlepanel),andpermitsupply(rightpanel),toadiscretionarypermit supplyshockwith0.9persistence.WecomparethemodelresponseswithoutTTD(bluesolidlines)andthose withthemedianTTDconstraint(reddashedline). Thesizeofthepermitsupplyshocksinbothmodelsare scaledtohavethesamecumulativeconstructionresponseat40quarters. housing supply policy is likely to interact with the interest rate, which is not allowed in the aboveexperimentusingalocalgeneralequilibriummodel. 6 Conclusion In this paper, we use a TTD model of housing investment to formulate a link between shortand long-run housing supply elasticities and analyze TTD for residential development across theU.S.usingauniquedataset. Wethenquantifyfrictionstohousingsupplyoverthebusiness cycleacrossmajorcountiesanddrawtheirimplicationsforhousingmarketdynamicsthrough alocalgeneralequilibriummodel. As we stated, a comprehensive process for land development takes about three years, on average, in the U.S. This feature alone introduces a large difference between the short- and long-run housing supply elasticities. In this paper, we adopt insights from the investment adjustment cost literature to shed light on the role that lengthy and dispersed TTD could play on housing market dynamics. Toward that objective, we abstract from several features of the data set that might be useful for future research. First, one could explore the time-varying 36

nature of TTD, especially during the recent periods. While Oh and Yoon (2020) study the cyclicalpatternoftime-to-buildinthecontextofthe2002–2011housingboom-bustcycle,its lower frequency trend could also be explored in the context of understanding the half century decline in construction-sector productivity (Goolsbee and Syverson, 2023). Second, our TTD regression results suggest that geographic determinants could play a key role in construction activity. As most construction activity is still conducted on site, climate change and environmentalregulationwouldalsohaveafirst-ordereffectontheconstructionsector. Wehopethat our modeling framework as well as our granular TTD data open up a new avenue of research alongtheselines. 37

References Baum-Snow, Nathaniel and Lu Han, “The Microgeography of Housing Supply,” mimeo, 2022. Bhattarai, Saroj, Felipe Schwartzman, and Choongryul Yang, “Local Scars of the US HousingCrisis,”JournalofMonetaryEconomics,2021,119,40–57. Charoenwong, Ben, Yosuke Kiruma, Alan Kwan, and Eugene Tan, “Capital Budgeting, Uncertainty,andMisallocation,”JournalofFinancialEconomics,2024,153,103779. Davidoff, Thomas, “Supply Constraints Are Not Valid Instrumental Variables for Home Prices Because They Are Correlated With Many Demand Factors,” Critical Finance Review,2016,5,177–206. Davis, Morris and Jonathan Heathcote, “Housing and the Business Cycle,” International EconomicReview,2005,46(3),751–784. Davis, Steven J and John C Haltiwanger, “Dynamism Diminished: The Role of Housing Markets and Credit Conditions,” Working Paper 25466, National Bureau of Economic ResearchJanuary2019. Glaeser, Edward L. and Joseph Gyourko, Rethinking Federal Housing Policy, The AEI Press,2008. Glaeser,EdwardL,JosephGyourko,andAlbertSaiz,“HousingSupplyandHousingBubbles,”JournalofUrbanEconomics,2008,64(2),198–217. Goolsbee, Austan and Chad Syverson, “The Strange and Awful Path of Productivity in the U.S.ConstructionSector,”NBERWorkingPaper,2023,30845. Graham, James and Christos A. Makridis, “House Prices and Consumption: A New Instrumental Variables Approach,” American Economic Journal: Macroeconomics, January 2023,15(1),411–43. Green, Richard K, Stephen Malpezzi, and Stephen K Mayo, “Metropolitan-Specific EstimatesofthePriceElasticityofSupplyofHousing,andTheirSources,”AmericanEconomic Review,2005,95(2),334–339. Guren, Adam, Alisdair McKay, Emi Nakamura, and Jon Steinsson, “What Do We Learn From Cross-Regional Empirical Estimates in Macroeconomics,” NBER Macroeconomics Annual,2020,pp.175–223. 38

, , , and , “Housing Wealth Effects: The Long View,” Review of Economic Studies, 2021,88(2),669–707. Gyourko, Joseph, Jonathan S. Hartley, and Jacob Krimmel, “The Local Residential Land UseRegulatoryEnvironmentacrossU.S.HousingMarkets: EvidencefromaNewWharton Index,”JournalofUrbanEconomics,July2021,124,103337. Haughwout, Andrew, Richard W. Peach, John Sporn, and Joseph Tracy, “The Supply Side of the Housing Boom and Bust of the 2000s,” in Edward L. Glaeser and Todd Sinai,eds.,HousingandtheFinancialCrisis,UniversityofChicagoPress,2013,chapter2, pp.69–104. Howard,GregandJackLiebersohn,“WhyistheRentsoDarnHigh? TheRoleofGrowing Demand to Live in Housing-Supply-Inelastic Cities,” Journal of Urban Economics, 2021, 124,103369. and , “Regional Divergence and House Prices,” Review of Economic Dynamics, 2023, 49,312–350. , , and Adam Ozimek, “The Short- and Long-Run Effects of Remote Work on U.S. HousingMarkets,”JournalofFinancialEconomics,2023,150(1),166–184. Iacoviello, Matteo and Stefano Neri, “Housing Market Spillovers: Evidence from an Estimated DSGE Model,” American Economic Journal: Macroeconomics, April 2010, 2 (2), 125–64. Kalouptsidi,Myrto,“TimetoBuildandFluctuationsinBulkShipping,”AmericanEconomic Review,2014,104(2),564–608. Kaplan, Greg, Kurt Mitman, and Giovanni L. Violante, “The Housing Boom and Bust: ModelMeetsEvidence,”JournalofPoliticalEconomy,2020,128(9),3285–3678. Kiyotaki, Nobuhiro, Alexander Michaelides, and Kalin Nikolov, “Winners and Losers in HousingMarkets,”JournalofMoney,CreditandBanking,2011,43(2–3),255–296. Kone,D.Linda,LandDevelopment,10ed.,BuilderBooks.com,2006. Kydland, Finn E. and Edward C. Prescott, “Time to Build and Aggregate Fluctuations,” Econometrica,1982,50(6),1345–1370. , Peter Rupert, and Roman Šustek, “Housing Dynamics over the Business Cycle,” InternationalEconomicReview,2016,57(4),1149–1177. Lucca,DavidO.,“ResuscitatingTime-to-Build,”mimeo,2007. 39

Lutz,ChandlerandBenSand,“HighlyDisaggregatedLandUnavailability,”mimeo,2022. Mayer, Christopher J. and C. Tsuriel Somerville, “Residential Construction: Using the Urban Growth Model to Estimate Housing Supply,” Journal of Urban Economics, 2000, 48,85–109. Meier, Matthias, “Supply Chain Disruptions, Time to Build, and the Business Cycle,” workingpaper,2020. Mian, Atif and Amir Sufi, “What Explains the 2007-2009 Drop in Employment?,” Econometrica,2014,82(6),2197–2223. ,KamaleshRao,andAmirSufi,“HouseholdBalanceSheets,Consumption,andtheEconomicSlump,”QuarterlyJournalofEconomics,2013,128(4),1687–1726. Millar, Jonathan N., Stephen D. Oliner, and Daniel E. Sichel, “Time-To-Plan Lags for Commercial Construction Projects,” Regional Science and Urban Economics, 2016, 59, 75–89. Murphy, Alvin, “A Dynamic Model of Housing Supply,” American Economic Journal: EconomicPolicy,November2018,10(4),243–267. Nathanson, Charles G. and Eric Zwick, “Arrested Development: Theory and Evidence of Supply-Side Speculation in the Housing Market,” Journal of Finance, December 2018, 73 (6),2587–2633. Oh, Hyunseung and Chamna Yoon, “Time to Build and the Real-Options Channel of ResidentialInvestment,”JournalofFinancialEconomics,2020,135(1),255–269. Paciorek, Andrew, “Supply Constraints and Housing Market Dynamics,” Journal of Urban Economics,2013,77,11–26. Saiz, Albert, “The Geographic Determinants of Housing Supply,” Quarterly Journal of Economics,2010,125(3),1253–1296. Sarte, Pierre-Daniel, Felipe Schwartzman, and Thomas A. Lubik, “What Inventory Behavior Tells Us About How Business Cycles Have Changed,” Journal of Monetary Economics,2015,76,264–283. Topel, Robert and Sherwin Rosen, “Housing Investment in the United States,” Journal of PoliticalEconomy,1988,96(4),718–740. 40