Nonlinear Effects of Loan-to-Value Constraints

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) Nonlinear Effects of Loan-to-Value Constraints C. Bora Durdu and Sergio Villalvazo 2024-081 Please cite this paper as: Durdu, C. Bora, and Sergio Villalvazo (2024). “Nonlinear Effects of Loan-to-Value Constraints,” Finance and Economics Discussion Series 2024-081. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2024.081. 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.

Nonlinear Effects of Loan-to-Value Constraints∗ C. Bora Durdu Sergio Villalvazo Federal Reserve Board September 2024 Abstract Thispaperinvestigatestheimpactofloan-to-value(LtV)borrowingconstraintsin models with occasionally binding credit constraints. These constraints give rise to a Fisherian debt-deflation mechanism, where exogenous shocks can trigger cascading effects resulting in significant declines in consumption, asset prices, and borrowing reversals—characteristicoffinancialcrises. However,recentliteraturechallengestraditional view by suggesting that collateral constraints may not always exacerbate financial disturbances but could instead foster dynamics leading to multiple equilibria. Buildingonthisdiscussion,thepaperexploresequilibriumassetpricingmodelswith LtV collateral constraints, identifying critical thresholds that govern asset price dynamics, consumption patterns, and current account behaviors. Our analysis uncovers that when the LtV limit is close to zero, tighter constraints induce smaller drops in consumption during crises. Conversely, when the LtV limit is close to one, we observe that tighter constraints induce larger drops in consumption during crises. The nonlinear relationship between the LtV ratio and adverse effects on macroeconomic outcomes aligns with cross-country evidence regarding the relationship between the leveloffinancialdevelopmentandtheseverityofconsumptiondeclinesduringcrises. J.E.L.classificationcodes: E31,E37,E52,F41,G01. Keywords: Financialcrises,Loan-to-valueconstraints,Debt-deflation. ∗WethankLucaGuerrieri,GastonNavarro,AlexVardoulakisandSarahZoiforhelpfuldiscussionsand comments,andJulianWangforexcellentresearchassistance. Wearealsogratefulforcommentsbyconference and seminar participants at the 2024 MEA and ITAM Alumni Meetings. The views expressed in this paperarethoseoftheauthorsandshouldnotbeattributedtotheBoardofGovernorsoftheFederalReserve Systemoritsstaff. 1

1 Introduction In the wake of the 1990s’ sudden stops experienced by numerous emerging economies and the economic turmoil of the 2007-2008 Global Financial Crisis, the quest for models capable of elucidating the dynamics of these crises has intensified. Pioneered by the work of Mendoza (2010), one prominent avenue has been the exploration of models featuring occasionally binding credit constraints, both for normative and positive analysis. Central to these models is the incorporation of loan-to-value (LtV) type borrowing constraints, whichcapborrowingatafractionofthemarketvalueofassetholdings. These models, enriched by the presence of occasionally binding constraints, unveil a Fisheriandebt-deflationmechanism. AsshownbySchmitt-GrohéandUribe(2017),invulnerable economic states, exogenous shocks can trigger the tightening of these constraints, precipitatingacascadeofeffects. Asassetpricesplummetinresponse,theconstrainttightens further, exacerbating deflationary pressures. This feedback loop often manifests in sharp declines in consumption, asset prices, and borrowing reversals—hallmarks of financialcrises. However,recentresearch,epitomizedbytheworkofSchmitt-GrohéandUribe(2021), challenges conventional wisdom regarding the implications of collateral constraints. Contrary to prior intuition, these authors argue that the effects of collateral constraints may not always amplify financial disturbances. Rather, they posit that such constraints could engender dynamics seemingly at odds with earlier literature, including the emergence of multipleequilibria. In particular, Schmitt-Grohé and Uribe (2021) highlight the nonlinearity of adjustment in economies subject to collateral constraints. These dynamics give rise to two distinct equilibria: one characterized by self-fulfilling financial crises driven by pessimistic views on collateral values, and another marked by underborrowing and excessive precautionary savingsasawaytoself-insure. This paper contributes to this evolving literature by rigorously characterizing the analyticalsolutionofequilibriumassetpricesinsmallopeneconomiesfeaturingloan-to-value collateral constraints. Specifically, we investigate the ramifications of varying collateral fractions on key macroeconomic and financial variables. Our analysis uncovers that when the LtV limit is close to zero, tighter constraints induce smaller drops in consumption during crises. Conversely, when the LtV limit is close to one, we observe that tighter constraintsinducelargerdropsinconsumptionduringcrises. 2

ThenonlinearimpactofLtVlimitsisinfluencedbyageneralequilibriumeffectcaused by asset prices. When the LtV limit rises, two conflicting outcomes occur. Firstly, increasedindebtednessmakestheeconomymorefragile,leadingtoagreaterdeclineinconsumption. Secondly,amorerelaxedcollateralconstraintlessenstheimpactonassetprices. In economies with low LtV, the first effect is more significant, whereas in economies with highLtV,thesecondeffectprevails. To assess the empirical validity of our theoretical findings, we examine the crosscountryevidenceonhowtheleveloffinancialdevelopment,whichproxiesthemodel’sLtV limits, is correlated with the severity of consumption declines during a financial crisis. To dothis,weconstructapaneldatabaseofSuddenStopepisodesfromBianchiandMendoza (2020), a financial development index developed by Svirydzenka (2016) and macroeconomic aggregates from The World Bank (2024). Predictive quadratic regressions validate our theoretical findings. In particular, we find a U-shaped relationship between consumptiongrowthandfinancialdevelopmentduringcrises. Our paper is related to several strands of the literature. The investigation of financial crises, particularly within the context of emerging economies, has been significantly advancedbymodelsincorporatingoccasionallybindingcreditconstraints. Thesemodels,pioneered by Mendoza (2010), have provided essential insights into the mechanisms driving sudden stops and economic downturns. Central to this approach is the loan-to-value (LtV) constraint,whichlimitsborrowingbasedonthemarketvalueofassets. Schmitt-Grohéand Uribe (2017) further developed these models by examining stock collateral constraints, revealing how exogenous shocks can tighten these constraints, leading to a Fisherian debtdeflationcyclethatexacerbatesfinancialinstability. Theliteraturehasalsoexploredthepossibilityofmultipleequilibriaarisingfromthese constraints. Schmitt-Grohé and Uribe (2021) analyzed flow collateral constraints, demonstrating that these models can result in nonlinear dynamics and the potential for multiple equilibria. Similarly, Jeanne and Korinek (2019) discussed the heuristic implications of stock collateral constraints, suggesting that under certain conditions, these constraints couldleadtoself-fulfillingfinancialcrisesorexcessiveprecautionarysavingsbehavior. Inadditiontothetheoreticalcontributions,variousstudieshaveemployedthesemodels to explore the broader implications of collateral constraints. For instance, Bianchi and Mendoza (2018) and Jeanne and Korinek (2019) applied similar models to investigate the role of pecuniary externalities in financial crises, while other scholars such as Lorenzoni (2008), Mendoza (2010), Bianchi (2011), and Benigno, Chen, Otrok, Rebucci, and Young 3

(2013, 2016), examined how these constraints influence overborrowing and the design of optimalmacroprudentialpolicies. Devereux,Young,andYu(2019)alsocontributedtothis discussionbyexploringthecapitalflowdynamicsunderdifferentregulatoryframeworks. Moreover,therelationshipbetweenfinancialdevelopmentandeconomicoutcomeshas been extensively studied. Levine (1997), Loayza and Ranciere (2006), and Bordo and Meissner (2015) explored how financial development affects economic growth, business cycles, and the severity of financial crises. These studies provide empirical support for the theoretical models, suggesting that the level of financial development can significantly influencetheimpactofcollateralconstraintsonmacroeconomicstability. The remainder of the paper is organized as follows. Section 2 describes the sudden stop model with perfect foresight and characterizes its analytical solution. In Section 3 we extendthemodeltoanenvironmentwithuncertainty. Wesolvethemodelnumericallyand show that our results are driven by a general equilibrium effect through the asset’s price. Section4describesthemodelvalidationexerciseandshowsthatourtheorypredictionsare inlinewiththedata. Finally,Section5concludes. 2 Model with Perfect Foresight The model is similar to Bianchi and Mendoza (2018) and Jeanne and Korinek (2019). The small open economy is populated by a continuum of homogeneous households. The representative household maximizes a CRRA utility function, u(c) and only derives utility from consumption, c. The household buys next period international bond holdings, b(cid:48), that payanexogenousinterestrateRandalsobuysnextperioddomesticassetholdings,a(cid:48),with an endogenous price q. The domestic asset net supply is fixed and normalized to 1. Total endowment output in the economy is given by y, whose share of labor income is given by (1 − α) and share to capital (dividends) income by α. Finally, the household’s credit is constrainedbyastandardloan-to-valuecollateralconstraintinwhichdebtcannotexceeda constant fraction κ of the market value of next period asset holdings.1 The problem of the householdinrecursiveformisgivenby: 1The timing of assets used as collateral on the right-hand side of the constraint relies on assumptions aboutthecharacteristicsofcreditcontractsandtheirenforcement. Bothtimingapproacheshavebeenwidely adopted. FormoredetailsseeBianchiandMendoza(2020) 4

V(a,b) = max u(c)+βV(a(cid:48),b(cid:48)) s.t. (1) {c,b(cid:48),a(cid:48)} c+R−1b(cid:48) +qa(cid:48) = (1−α)y +a(αy +q)+b R−1b(cid:48) ≥ −κqa(cid:48). Let λ > 0 and µ ≥ 0 be the multipliers on the budget constraint and collateral constraint, respectively, and define µˆ = µ/λ ≥ 0. Then, assuming the market clearing condition, a = a(cid:48) = 1,theequilibriumconditionsare: u (c) = Rβu (c(cid:48))+µ (2) c c u (c(cid:48))(αy(cid:48) +q(cid:48)) c q = β (3) u (c)(1−κµˆ) c 0 = µ(R−1b(cid:48) +κq) (4) c = y +b−R−1b(cid:48). (5) Where u (c) corresponds to the first derivative of the utility function with respect to conc sumption. Now assume b > −Rκq such that the economy is not constrained and Rβ = 1, to analyze a stationary equilibrium in which consumption is constant: c = c(cid:48). From Eq. 2-5 weobtain: c = c(cid:48) = y +(1−β)b (6) µ = µˆ = 0 (7) βαy q = . (8) 1−β The equilibrium characterized by Eq. 6-8 is a well known result. The household will keep a constant consumption equal to the total endowment, y, plus or minus the interest paid or earneddependingonthesignoftheinitialbondposition,b. Thecurrentaccountisconstant and zero, CA = b(cid:48) −b = 0, the collateral constraint is not binding by assumption and the t equilibriumexhibitsthestandardforwardlookingassetprice. Now assume that the household is marginally constraint, i.e., the debt is at its highest level and the collateral constraint multiplier is zero: b = −Rκq and µ = 0. Then the 5

equilibriumallocationsbecome: καy b = b(cid:48) = − (9) 1−β c = y(1−κα). (10) 2.1 Analytical Results The analytical results in this section provide a clear and concise derivation of the equilibrium allocations and asset prices in a simplified model setup with a logarithmic utility function(thecoefficientofriskaversionisequalto1). Byassuminganunexpectedwealthneutralshocktotheeconomyandasequentialframework(Eq. 11),wecanexplorehowthe LtVlimitinfluencestheoutcomesofthemodelunderdifferentscenarios. Theclosed-form solutions derived here help to uncover key insights into the dynamics of asset prices, consumption,anddebt,particularlywhentheeconomyfacesabindingcollateralconstraint.  y fort ≤ −1    y = γy fort = 0withγ < 1 (11) t    y˜= y(1+(R−1)(1−γ)) fort ≥ 1. Thesolutionofthemodelstartsbyconsideringtwoscenariosbasedontherelationship between the equilibrium asset price in period 0 and period 1. In period t = 0, the endowmentlevelintheeconomyislowerthaninthepreviousperiodandfromt = 1onward,the endowment is larger such that the net present value stays constant.2 It is straightforward to see that, due to the shock, the household would like to smooth her consumption by increasingherdebtinperiod0. However,notethatinthepreviousperiod,thehouseholdwas holdingthemaximumpossibledebt. Therefore,therearetwopossiblescenarios. Scenario 1: If the equilibrium price in period 0 is less than or equal to the unconstrained equilibrium price in period 1 (q ≤ q = βαy˜), then, the maximum debt in period 0 1 1−β 0 is supported in an unconstrained period 1 and, hence, the household can maintain a con- 2Notethatintheabsenceofthecollateralconstraint,suchwealthneutralshockwouldnothaveanyeffect on consumption. The economy would be able to perfectly smooth consumption by adjusting the current account. Duringperiod0,theeconomywouldborrowfromabroadandfromperiod1onward,theeconomy wouldpaytheinterestsofsuchborrowingwiththeadditionalendowment. 6

stant consumption from period 1 onward. The equilibrium allocations are such that the householdconsumesasmuchaspossibleinperiod0sincesheisatthedebtlimitand,from period 1 onward, the household consumes a constant level. Note that q = βαy < βαy˜, −1 1−β 1−β hence this case allows for some asset price inflation in period 0. Whenever there is asset priceinflation,thecurrentaccountwillbepro-cyclical. Scenario 2: If the equilibrium price in period 0 is greater than the unconstrained equilibriumpriceinperiod1(q > βαy˜),thenthemaximumdebtinperiod0isnotsupportedin 0 1−β an unconstrained period 1 and hence the household cannot maintain a constant consumption from period 1 onward. The equilibrium allocations are such that the household gets as much debt as possible in period 0 as long as c ≥ c and, from period 1 onward, the −1 0 householdstartsadeleveragingprocess. Inthisprocess,thehouseholdholdsthemaximum amount of debt possible in each period and slowly increases her consumption until she reaches the steady state in which the constraint is marginally binding and holds a constant consumptionlevel. Notethatinthiscase,theeffectsoftheshockarepersistent,thecurrent account is pro-cyclical, the asset price increases in period 0 and slowly decreases until it convergesto βαy˜,whichisacounter-intuitivedynamic. 1−β Now we turn to characterizing for which LtV limits we are in the first scenario. Let c−1 z(κ) = 1 ∈ (0,1]. Note that z(·) is an implicit function of κ since the equilibrium c−1 0 consumption choices depend on it and, z(·) ≤ 1 since c ≤ c , because if not then the 0 1 household could save in period 0 to smooth her consumption. From Eq. 3 we get the followingrelation: q ≤ q 0 1 ⇔ βz(κ)(αy˜+q ) βαy˜ 1 ≤ 1−κ(1−z(κ)) 1−β ⇔ z(κ)(1−κ) ≤ 1−κ. The last inequality is satisfied only for all 0 ≤ κ ≤ 1. Hence, for 0 ≤ κ ≤ 1 we are in scenario1andforκ > 1weareinscenario2. Assumption1. LettheLtVlimitbesuchthat0 ≤ κ ≤ 1. Under Assumption 1, the following equations summarize the equilibrium allocations 7

andassetprices: βαy q = (12) −1 1−β c = y(1−κα) (13) −1 καy b = − (14) 0 1−β (cid:18) (cid:19) γ(1−β)−κα c = γy +b −R−1b = y +κq (15) 0 0 1 0 1−β b = −Rκq = b forτ ≥ 2 (16) 1 0 τ 1−β c = y˜+b −R−1b = y˜− κq = c forτ ≥ 2 (17) 1 1 2 0 τ β βαy˜ q = = q forτ ≥ 2. (18) 1 τ 1−β Note that the only unknown is the current price q . Combining Eq. 2-3 and subtituing 0 Eq. 15-18,weobtainaquadraticpolynomial: c−1(αy˜+q ) q = β 1 1 0 c−1(1−κµˆ ) 0 0 ⇔ c (αy˜+q ) 0 1 q = β 0 κc +(1−κ)c 0 1 ⇔ (cid:18) (cid:19) βαy˜ c 0 q = 0 1−β κc +(1−κ)c 0 1 ⇔ (cid:16) (cid:17) (cid:18) (cid:19) y γ(1−β)−κα +κq βαy˜ 1−β 0 q = 0 (cid:16) (cid:16) (cid:17) (cid:17) (cid:16) (cid:17) 1−β κ y γ(1−β)−κα +κq +(1−κ) y˜− 1−βκq 1−β 0 β 0 ⇔ 0 = Aq2 +Bq +C. 0 0 8

Where: κ A = (κ+β −1) β (cid:18) (cid:19) (cid:18) (cid:19) κα βακ B = κyγ 1− +y˜ 1−κ− γ(1−β) 1−β (cid:18) (cid:19) βαy˜yγ κα C = −1 . 1−β γ(1−β) Whichcanbesolvedusingthegeneralquadraticformulasolution: √ √ −B + B2 −4AC −B − B2 −4AC qH = and qL = . (19) 0 2A 0 2A Equation 19 is the analytical solution for the asset price on the period when the unexpectedwealthneutralshockhitstheeconomy. Inthenextsubsectionweusethisanalytical solution to characterize the parameter regions where the equilibrium is unique, multiple or non-existent. 2.1.1 CharacterizationoftheEquilibriumAssetPriceandCases Inthissubsection,weobtainthegeneralcasesforuniqueness,multiplicityornon-existence oftheequilibrium. Assumption2. Thedropintheendowmentisnottoolarge: γ > α. Assumption 2 is plausible since γ corresponds to 1 minus the size of the shock so it is expected to be close to 1 and α corresponds to the capital income share which is expected tobelessthan 1. Nowweobtainthefollowingcases: 3 Cases:  A = 0 ⇔ (κ = 1−β orκ = 0)thenq = −C > 0sinceB > 0,C < 0.   0 B    A > 0andC < 0 ⇔ κ ∈ [1−β,min{1, (1−β)γ}]thenqH > 0. Unique α 0 A < 0 ⇔ κ < 1−β then       qH > 0, qL > 0butonlyqL > 0isconsistentwithc ≤ c . 0 0 0 0 −1 (cid:110) Multiple(pair) A > 0,C > 0andB2 −4AC > 0thenqH > 0, qL > 0. 0 0 (cid:110) Non-existence B2 −4AC < 0. 9

In summary, 0 ≤ κ < min{1, (1−β)γ} guarantees a unique equilibrium. Intuitively, α more volatile economies (lower γ) and economies with higher capital returns (higher α) require a lower LtV limit to guarantee a unique equilibrium. This result is important since itsaysthatauniqueequilibriumisguaranteedtoexistwhentheLtVlimitisclosetozero. Having a closed form solution for the equilibrium asset price allows us to get the followingresults. Proposition 1. If the economy is under a unique equilibrium (0 ≤ κ < (1 − β)γ ≤ 1), α thentheDebt-to-GDPratioislessthantheshocksize: −b < γ. y Toobtainthisresult,weusethebondexpressioninEq. 9dividedbytheendowmentand note that under a unique equilibrium, then α < (1−β)γ. Combining the two expressions κ deliversProposition1. Proposition2. Whendebtisnotpossible(κ = 0)then: • One-to-onemappingfromshocktodropinpriceandconsumption, q0 = c0 = γ. q−1 c−1 • CrisisamplificationordampeningfromincreasesintheLtVfromzerodependonthe capital share for the asset price and unambiguous crisis amplification for drop in consumption:  (cid:12) (cid:12) dq 0(cid:12) ≥ 0ifα ≤ (1−β)γ dc 0 /c −1(cid:12) (cid:12) = , (cid:12) < 0. dκ(cid:12) dκ (cid:12) κ=0 < 0ifα > (1−β)γ κ=0 The first part of this result is obtained by substituting κ = 0 in Eq. 12-15 and 19. The second part is obtained after differentiating Eq. 19 with respect to κ and evaluating it at κ = 0. Proposition3. WhentheLtVlimitisatthemaximumandtheequilibriumisunique(κ = 1 andα = (1−β)γ)then: • One-to-one mapping from shock to drop in consumption, c0 = γ, and there is a c−1 slightpriceinflation, q0 > 1. q−1 • Crisis amplification from decreases in the LtV from one in both the asset price and consumption: (cid:12) (cid:12) dq 0(cid:12) dc 0 /c −1(cid:12) (cid:12) > 0, (cid:12) > 0. dκ(cid:12) dκ (cid:12) κ=1 κ=1 10

The proof of this result is analogous to how we obtained Proposition 2 evaluating the derivativesatκ = 1andα = (1−β)γ. Proposition 4. When α = (1 − β)γ and hence the equilibrium is unique over all κ ∈ [0,1],thenpricedeflationhappenswhenκ < 1−γ(1−β) andinflationwhenκisabovesuch 1−γ(1−β)2 threshold. This result is obtained by substituting α = (1−β)γ and comparing q with q from −1 0 Eq. 19,thensinceq isincreasinginκ(Proposition2)wesolvefortheκsuchthat q0 = 1 0 q−1 toobtainthethreshold. The derived propositions further elucidate the implications of the model. For instance, the finding that the debt-to-GDP ratio is constrained by the shock size under a unique equilibrium underscores the importance of managing leverage in the economy. Moreover, the analysis shows that the effects of increasing the LtV limit are nuanced: while it can dampencrisesinsomecases,itmayamplifytheminothers,dependingonthecapitalshare andcurrentLtVlevel. Thisresultemphasizesthetrade-offsinherentinfinancialregulation andtheneedforcarefulcalibrationofLtVlimitstobalanceeconomicgrowthandfinancial stability. Overall, this section demonstrates the power of analytical modeling in uncovering the complex interplay between collateral constraints, asset prices, and macroeconomic outcomes. The insights gained from the closed-form solutions provide a solid foundation for understanding the broader dynamics explored in the more complex, stochastic version of themodel. 2.2 Numerical Exercise To further dissect the mechanics of our model, we analyze numerically a version of the modelwithoutuncertainty,allowingustoisolatethecoredynamicsatplay. Thisapproach helps in understanding the fundamental behavior of the model under different levels of the loan-to-value(LtV)limit,κ,particularlyfocusingontheexistenceandnatureofequilibria. The parameters used for the numerical exercise are shown in Table 1. These are standardvaluesintheliterature. The results presented in the following figures highlight three distinct scenarios: cases where the model yields a unique equilibrium, multiple equilibria, or no equilibrium at all. These scenarios are critical for understanding the potential stability or instability of the 11

Table1: ParameterValues Parameter Value β,(R = β−1) discountfactor 0.94 y totalendowment 1.0 γ dropendowment 0.95 α capitalshare ∈ {(1−β)λ = 0.057, 0.15, 0.20} κ LtVlimit ∈ [0, 1] economyundervaryingfinancialconstraints. UniqueEquilibriumCase Figure1specificallyillustratesthecasewherethecapitalshareα,isequalto(1−β)γ, themaximumvaluethatguaranteesauniqueequilibriumacrossallLtVlimitsrangingfrom 0 to 1. This particular setting allows us to observe how the economy responds when the LtV limit is varied in a controlled, predictable environment, free from the complexities introducedbyuncertainty. Panela)ofFigure1demonstratesamonotonicdampeninginthedeclineofassetprices as the LtV limit increases. This behavior aligns with the intuition that higher LtV limits, by loosening borrowing constraints, mitigate the downward pressure on asset prices. As the collateral constraint becomes less binding, the market is able to absorb shocks more efficiently,preventingasharpdropinassetvalues. Panelb)showsaU-shapedresponseintheconsumptiondropastheLtVlimitincreases. Thisnon-monotonicpatternisparticularlynoteworthyasitreflectsthedelicatebalancebetween increased borrowing capacity and heightened vulnerability. At lower LtV limits, the economy is less leveraged, and the impact of shocks on consumption is relatively contained. However, as the LtV limit increases, the increased leverage makes the economy more susceptible to shocks, leading to a more pronounced decline in consumption. Beyond a certain point, though, the loosening of constraints allows for greater consumption smoothing,hencetheU-shapedpattern. Panel c) presents the corresponding change in the current account to GDP ratio, which exhibits a mirror image of the U-shaped consumption response. This inverse relationship highlights the trade-offs faced by the economy as it adjusts to shocks under different LtV regimes. WhentheLtVlimitislow,thelimitedborrowingcapacityleadstosmallerswings in the current account. As the LtV increases, the economy finances consumption by bor- 12

rowing more and when the economy becomes overly leveraged, the need to correct this imbalance in the face of adverse shocks results in a sharp reversal, hence the mirrored U-shape. Thesefindingsfromtheuniqueequilibriumcasehighlightthecriticalrolethatcollateral constraints play in shaping the economy’s response to shocks. The existence of a unique equilibrium in the setting here indicates a predictable and stable economic environment where policy interventions can be more effectively tailored. The U-shaped consumption responseandthemirroredcurrentaccountbehaviorfurtheremphasizethenonlineareffects of varying LtV limits, suggesting that while increasing the LtV limit can offer short-term benefitsintermsofconsumptionsmoothing,italsointroduceslonger-termrisksassociated withhigherleverage. 1 0 2 0 -1 1.5 -1 -2 -2 -3 1 -3 -4 0.5 -4 -5 0 -5 -6 -6 -7 -0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (a)AssetPrice (b)Consumption (c)CurrentAccounttoGDPratio Figure1: Percentchangesduringcrisisforlowcapitalshare(α) We now explore the behavior of the model when the capital share (α) is set to medium and high levels, specifically 0.15 and 0.20, as illustrated in Figures 2 and 3, respectively. Thesescenariosallowustoexaminehowincreasingthecapitalshareinfluencesthestabilityoftheeconomy,particularlyintermsoftheexistenceandmultiplicityofequilibria. MultipleEquilibria In Figure 2, where α is set to 0.15, the model begins to exhibit more complex dynamics compared to the unique equilibrium case. At this medium capital share level, certain values of the LtV limit lead to multiple equilibria. The solid blue lines correspond to the equilibrium obtained with the “high” asset price while the dashed red lines correspond to the equilibrium with the “low” asset price from Eq. 19. Note that the dynamics of the “high” asset price are similar to the dynamics under a unique equilibrium and are more stablerelativetothe“low”assetprice,whichgeneratestotalassetpricecollapsesforsome 13

LtVlimits. This outcome reflects the increased role of capital in the production process, which amplifiesthefeedbackeffectsbetweenassetpricesandcollateralconstraints. Asthecapital share rises, the value of collateral becomes more sensitive to fluctuations in asset prices, makingtheeconomymorepronetononlinearandpotentiallyunstableresponsestoshocks. In this scenario, the economy can settle into different equilibrium states depending on initial conditions or exogenous shocks. For instance, one equilibrium might be characterized by relatively stable asset prices and consumption, while another might involve more pronounced declines in these variables. This multiplicity of equilibria introduces an element of unpredictability, as small perturbations can push the economy from one equilibriumtoanother,potentiallytriggeringafinancialcrisisorexacerbatingitseffects. 0 0 100 -20 -20 80 -40 -40 60 -60 -60 40 -80 -80 20 -100 -100 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (a)AssetPrice (b)Consumption (c)CurrentAccounttoGDPratio Figure2: Percentchangesduringcrisisformediumcapitalshare(α) Non-ExistentEquilibrium AtcertainLtVlevels,themodelfailstoconvergetoanequilibrium,indicatingabreakdown in the standard market-clearing mechanism. This situation could arise when the tightening of collateral constraints due to falling asset prices creates a feedback loop so severethattheeconomycannotstabilize,reflectinganextremeformoffinancialfragility. Figure 3 further explores this dynamic by increasing the capital share to 0.20. At this elevatedlevel,theeconomybecomesevenmoresensitivetochangesintheLtVlimit,with abroaderrangeofLtVvaluesleadingtomultipleequilibriaornon-existentequilibria. The high capital share scenario underscores the dangers of high leverage in an economy where capital plays a dominant role. As α increases, the productive capacity of the economy becomes more reliant on capital, and thus, more exposed to fluctuations in asset prices. The tightening of collateral constraints in response to declining asset values can quickly spiral out of control, leading to severe disruptions in consumption and external 14

balances. In these high α scenarios, the model demonstrates how elevated capital intensity can exacerbate financial instability, particularly in the presence of loose borrowing constraints (highLtVlimitswhereκ > (1−β)γ). Thecombinationofhighleverageandasignificant α capital share creates conditions where financial markets can no longer clear, resulting in a collapseofassetprices,asharpcontractioninconsumption,andpotentially,afailureofthe economytoreachanequilibrium. 0 0 100 -20 -20 80 -40 -40 60 -60 -60 40 -80 -80 20 -100 -100 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (a)AssetPrice (b)Consumption (c)CurrentAccounttoGDPratio Figure3: Percentchangesduringcrisisforhighcapitalshare(α) The analysis of the model under medium and high capital share levels provides critical insightsintotheconditionsunderwhichfinancialmarketsmaybecomeunstable. Thepresence of multiple equilibria and the potential for non-existent equilibria highlight the risks associatedwithhighleverageandsignificantcapitalintensityintheeconomy. Fromapolicyperspective,thesefindingssuggestthatregulatorsneedtobeparticularly cautious when allowing high LtV limits in economies with a high capital share. While higher LtV limits can facilitate investment and growth during stable periods, they can also leadtomultipleandpotentiallyunstableequilibria,increasingtheriskoffinancialcrises. In economies where capital plays a major role, policies that aim to moderate leverage andensuremorerobustcollateralframeworkscouldhelppreventtheonsetoftheseunstable equilibria. Additionally, understanding the conditions that lead to non-existent equilibria can inform the design of safety nets and intervention mechanisms that could prevent the economyfromspiralingintoastatewheremarketsfailtoclearaltogether. Overall, the findings from these scenarios underscore the importance of considering the interaction between capital intensity and financial constraints in macroeconomic models. By highlighting the potential for multiple or non-existent equilibria, this analysis adds an important dimension to our understanding of financial stability and the factors that can 15

precipitate or mitigate financial crises. This analysis also reinforces the importance of considering nonlinear dynamics in policy design. Policymakers need to be aware that increasingfinancialflexibilitythroughhigherLtVlimitsmayonlybebeneficialuptoapoint. Beyond that, the increased vulnerability can lead to more severe downturns in the event of adverse shocks. The insights gained from this simplified model without uncertainty provideafoundationalunderstandingthatcomplementsthemorecomplexdynamicsobserved inthestochasticversionofthemodel,whichwepresentinthenextsection. 3 Model with Uncertainty There is not much we can do analytically when there is uncertainty in the economy. However,inthissectionwecomputeanextensionofthemodelwheretheincomeendowment,y, isstochasticandfollowsanAR(1)Markovprocess. Specifically,logy = (1−ρ)log(y )+ t ss ρlog(y ) + σ (cid:15) , where (cid:15) ∼ N(0,1). The additional parameters will take the values t−1 y t t ρ = 0.70 and σ = 0.025, which are common values used in the literature. The rest of y the parameters stay the same as in Table 1 with the exception of the interest rate which is lowered such that Rβ < 1 to guarantee a non-degenerate limiting wealth distribution: R = 1.03.3 Therecursiverepresentationoftheproblemofthehouseholdbecomes: V(a,b,s) = max u(c)+βE [V(a(cid:48),b(cid:48),s(cid:48))] s.t. (20) s(cid:48)|s {c,b(cid:48),a(cid:48)≥0} c+R−1b(cid:48) +qa(cid:48) = (1−α)y(s)+a(αy(s)+q)+b R−1b(cid:48) ≥ −κqa(cid:48). 3.1 Quantitative Results In this section, we analyze the quantitative implications of varying the loan-to-value (LtV) limit, κ, in an economy where the households are exposed to an aggregate shock to their endowment. Therepresentativeagentframeworkallowsustomodeltheaggregateimplications of endowment shocks, providing a clear lens through which to observe the dynamics offinancialcrisesunderdifferentlevelsofcollateralconstraints. 3Toefficientlysolvethemodelwithuncertainty,werelyontheFiPItglobalalgorithmproposedbyMendoza and Villalvazo (2020), which converges for parameters close to the parameter regions that deliver a uniqueequilibriumderivedinSection2.1.1. Hence, toanalyzetheroleofaLtVlimitbetween0and1we usedacapitalincomesharebelowtheuniquenessthreshold,specificallyα=0.02. 16

To explore these dynamics, we solve, simulate, and derive impulse response functions (IRFs)startingfromthestochasticsteadystateforarangeofLtVlimits. Figure4illustrates theimpactofa2standarddeviationshockonkeymacroeconomicvariables—namelyasset prices, consumption, and the current account to GDP ratio—across different values of κ. Thesolidredlinesinthefiguredepictthechangesinthesevariablesonimpact. Our findings reveal a non-monotonic relationship between the LtV limit and the economy’sresponsetoshocks,consistentwiththetheoreticalpredictionsoutlinedintheearlier sections using the deterministic model. Specifically, as κ increases, we observe that the effectsoftheshockonassetprices,consumption,andthecurrentaccountvaryinanonlinearfashion. Thisnon-monotonicityunderscoresthecomplexrolethatfinancialconstraints play in amplifying or dampening the impact of shocks in the presence of aggregate uncertainty. A crucial aspect of these results is the role of the economy’s stochastic steady state, where βR < 1. This condition ensures that the economy operates in a constrained state, making the simulations directly comparable to the analytical results obtained earlier. The constrained state reflects the inherent vulnerability of the economy to shocks, particularly when borrowing limits are binding, and the LtV limit is a pivotal factor in determining the severityoftheeconomy’sresponse. To further understand the mechanisms driving these results, we contrast the general equilibrium effects with partial equilibrium responses. The blue dashed lines in Figure 4 representtheoutcomeswhentheassetpriceisheldfixedatitsstochasticsteadystatelevel, effectivelyisolatingtheroleofpecuniaryexternalities. Inthispartialequilibriumscenario, the decline in consumption is relatively flat across different LtV limits, indicating that without the feedback loop generated by fluctuating asset prices, the economy’s response toshocksismoremuted. Thiscomparisonhighlightsthecriticalimportanceofpecuniaryexternalitiesinshaping the overall dynamics. The non-monotonic consumption behavior observed in the general equilibrium setting is largely driven by the interaction between asset prices and collateral constraints. Specifically, when the LtV limit increases, two opposing forces come into play. On one hand, higher LtV limits allow for greater indebtedness, making the economy more vulnerable to shocks and amplifying the drop in consumption. On the other hand, a looser collateral constraint mitigates the downward pressure on asset prices, dampening theoverallimpactoftheshock. IneconomieswithlowLtVlimits,thevulnerabilityeffectdominates,leadingtoamore 17

pronounced decline in consumption. Conversely, in economies with high LtV limits, the mitigating effect on asset prices prevails, resulting in a less severe consumption decline. This interplay between vulnerability and mitigation is crucial for understanding the nonlinearresponsesofeconomiestofinancialshocksandunderscoresthenuancedroleofLtV limitsincrisisdynamics. Overall,thesequantitativeresultsnotonlyreinforcethetheoreticalpredictionsbutalso providearicherunderstandingofthemechanismsatplay. ByillustratinghowtheLtVlimit influences the economy’s response to shocks through both direct and indirect channels, this analysis contributes to a more comprehensive view of the factors that drive financial instability and offers insights into the potential policy implications of varying collateral constraints. 4 -2 1 -2.5 0.5 2 -3 0 0 -3.5 -0.5 -4 -1 -2 -4.5 -1.5 -5 -2 -4 -5.5 -2.5 -6 -6 -3 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 (a)AssetPrice (b)Consumption (c)CurrentAccounttoGDPratio Figure4: Changeonimpactaftera2s.d. shock 4 Model Validation To empirically validate the theoretical predictions discussed in this paper, we construct a comprehensivepaneldatabaseencompassingSuddenStopepisodesfromBianchiandMendoza (2020), alongside a financial development index developed by Svirydzenka (2016) and macroeconomic aggregates sourced from the World Bank’s World Development Indicators(WDI),TheWorldBank(2024). Thisdatasetenablesustoexaminetherelationship between financial development and key macroeconomic outcomes during financial crises, therebyprovidingareal-worldtestofourmodel’simplications. The validation exercise focuses on a descriptive regression analysis, where we investigate the impact of varying levels of financial development on asset prices, consumption growth, and the change in the current account to GDP ratio. Specifically, we estimate the 18

followingquadraticregressionmodel: Dependent % = β +β FD +β FD2 +(cid:15) . (21) i,t 0 1 i,t 2 i,t i,t Here, the dependent variables are the percentage changes in asset prices, consumption growth, and the current account to GDP ratio, while FD represents the financial develi,t opmentindex. Thisquadraticspecificationallowsustocapturepotentialnonlineareffects, consistentwiththetheory’spredictionthattheimpactoffinancialdevelopmentisnotlinear butratherexhibitsacomplex,non-monotonicrelationship. (a)AssetPrice (b)Consumption (c)CurrentAccounttoGDPratio Figure5: Regressionestimateswith90percentconfidencebands As illustrated in Figure 5, the empirical findings align closely with the theoretical predictions. The regression results show a monotonic increase in asset price changes with higher levels of financial development, which supports the model’s assertion that more developed financial systems are better equipped to absorb shocks without triggering severe asset price declines. Moreover, the data reveal a U-shaped relationship between consumption growth and financial development, echoing our theoretical findings that economies with intermediate levels of financial development are more vulnerable to consumption declines during crises. In contrast, economies at either low or high ends of the financial developmentspectrumexhibitlesssevereconsumptiondeclines,duetodifferingmechanisms offinancialconstrainttightening. Similarly, the change in the current account to GDP ratio displays a mirror image of the U-shaped relationship observed for consumption growth. This finding underscores the model’s implication that financial development influences the external balance adjustment processduringcrises,withintermediatelevelsofdevelopmentleadingtomorepronounced swingsinthecurrentaccount. 19

Theseresultsproviderobustempiricalsupportforthetheoreticalframeworkoutlinedin this paper. They highlight the importance of considering nonlinearities in the relationship between financial development and macroeconomic outcomes during crises, and reinforce theideathattheeffectivenessoffinancialdevelopmentinmitigatingcrisisimpactsishighly context-dependent. Byvalidatingourmodel’spredictionswithreal-worlddata,thissection bridges the gap between theory and practice, offering insights that are practically relevant forpolicymakers. 5 Conclusion This paper has delved into the intricate dynamics of financial crises, particularly focusing ontheroleofloan-to-valueborrowingconstraintsinshapingdynamicsofequilibriumasset pricing models with occasionally binding collateral constraints. By elucidating the Fisheriandebt-deflationmechanisminherentinthesemodels,wehaveexaminedtheimplications ofawiderangeofvaluesforloan-to-valuelimitsanduncoveredacriticalthresholdthatdeterminesassetpricemovements,consumptionpatterns,andcurrentaccountbehaviors. We found that increasing the LtV limit leads to nonlinear effects on key macroeconomic variables such as asset prices, consumption, and the current account. Specifically, a unique equilibrium exists when the capital share is set at the level that guarantees stability across allLtVlimits. Inthisscenario,higherLtVlimitsmitigatedeclinesinassetpricesandcreate a U-shaped consumption response, balancing the benefits of increased borrowing capacity withtherisksofheightenedleverage. However, when capital share increases, the model exhibits more complex dynamics, including the possibility of multiple or non-existent equilibria. This reflects the greater sensitivity of asset prices to shocks, especially in highly leveraged economies. These findings underscore the risks associated with high leverage and loose borrowing constraints, particularly in economies with significant capital intensity. The presence of multiple equilibria or the failure of markets to clear altogether suggests that policy measures should carefully moderate LtV limits, balancing the benefits of credit expansion with the need for financialstability. These theoretical findings are validated by cross-country relationship between level of financial development and severity of consumption decline during financial crises. In particular, a U-shaped relationship between consumption growth and financial development index emerges during crises. These results provide robust empirical support for the the- 20

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