Debt Deflation Effects of Monetary Policy

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Debt Deflation Effects of Monetary Policy Li Lin, Dimitrios P. Tsomocos, and Alexandros P. Vardoulakis 2014-37 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.

∗ Debt Deflation Effects of Monetary Policy † ‡ § Li Lin Dimitrios P. Tsomocos Alexandros P. Vardoulakis May 7, 2014 Abstract ThispaperassessestherolethatmonetarypolicyplaysinthedecisiontodefaultusingaGeneral Equilibrium model with collateralized loans, trade in fiat money and production. Longtermnominalloansarebackedbycollateral,thevalueofwhichdependsonmonetarypolicy. Thedecisiontodefaultisendogenousanddependsontherelativevalueofthecollateraltoface valueoftheloan. Defaultresultsinforeclosure,higherborrowingcosts,inefficientinvestment andadecreaseintotaloutput.Weshowthatpre-crisiscontractionarymonetarypolicyinteracts withFisheriandebt-deflationdynamicsandcanincreasetheprobabilitythatacrisisoccurs. Keywords: Default,Collateral,Debt-deflation JELClassification: E4;E5;G0;G1;G2 ∗WearegratefultotheseminarparticipantsfortheirhelpfulcommentsinBanquedeFrance,theBankof Korea,theMiniConferenceonEconomicTheoryintheUniversityofIllinois,the7thAnnualCowlesConferenceonGeneralEquilibriumanditsApplicationinYaleUniversity, the11thSAETConferenceinFaro, the43rdMMFConferenceinBirminghamBusinessSchool,the2013AnnualmeetingoftheAmericanEconomicAssociation,theNewSchoolofEconomicsinMoscow,theHigherSchoolofEconomicsinMoscow, PekingUniversity,andtoFernandoAlvarez,RegisBreton,JohnGeanakoplos,GaelGiraud,CharlesGoodhart, Christian Hellwig, Herakles Polemarchakis, David Rappoport and Skander Van den Heuvel for their helpfulcomment. Allremainingerrorsareours. Theviewsexpressedinthispaperarethoseoftheauthors anddonotnecessarilyrepresentthoseofFederalReserveBoardofGovernors,anyoneintheFederalReserve System,oranyoftheinstitutionswithwhichweareaffiliated. †InternationalMonetaryFund,UnitedStates;email: llin@imf.org ‡Said Business School and St. Edmund Hall, University of Oxford, United Kingdom; email: dimitrios.tsomocos@sbs.ox.ac.uk §Board of Governors of the Federal Reserve System, United States; email: alexandros.vardoulakis@frb.gov 1

1 Introduction Thefinancialcrisisof2007-2008hasrenewedtheinterestintheabilityofmonetarypolicy tomitigatetheadverseconsequencesthatfinancialfrictionscanhaveonrealeconomicactivity. Mishkin (2009) and Gertler and Karadi (2011) argue that accommodative monetary policyishelpfulduringfinancialcrisisepisodes. Thispapertakesastepbackandexamines whether pre-crisis contractionary monetary policy can increase the likelihood that a crisis occursinthefutureand,ifyes,whatareitsrealeffects. OurmodelcansuccinctlynestcompetingvisionsofthecausesoftheGreatDepression (and of similar episodes) where debt-deflation dynamics act as an amplification mechanism.1 On the one hand, Friedman and Schwartz (1963) find a high positive correlation betweenmoneysupplyandoutputandconcludethatthedeclineinthemoneystockbefore the Great Depression was a substantial factor for the subsequent deflation and decline in GDP. On the other hand, Bernanke (1983) establishes that the Great Depression can be better explained when one explicitly models the financial frictions, which can impede the supply of credit to the real economy and, thus, GDP growth. Our analysis suggests that monetary forces are capable of inducing debt-deflation dynamics, but only when they exacerbate the underlying financial frictions, which in our model lead to default. Thus, we proposea“debt-deflation"channelofmonetarypolicy. We examine the effects of monetary policy on total output within a framework of fully flexible prices. The underlying friction is that we allow agents to (endogenously) default ontheirlong-termloanobligations. Thus,thereisaneedforcollateraltobacktheseloans. In all other respects, we maintain all the structural characteristics of General Equilibrium analysis,i.e. optimizingbehavior,perfectlycompetitivemarketsandrationalexpectations. We show how an adverse monetary shock in the present can lead to over-in-debtness 1TheoriginofthisviewcanbetracedbacktoFisher(1933). Hisanalysisisbasedontwofundamental conditions, over-indebtedness and deflation. He argued that over-indebtedness can precipitate deflation in futureperiodsandsubsequentlyliquidationofcollateralizeddebtandbankruptcy,whichcanleadtofiresales suppressingthevalueofthecollateralevenfurther. Hence,theinitialdeflationarypressuresareexacerbated andtheyprecipatetoevenhigherdefault,and,ultimately,toloweroutput. 2

and future deflation that in some state of the world can result in default, debt liquidation, reallocation of capital and finally reduction in GDP. Market incompleteness is central to ouranalysis,sinceagentscannotwritecomprehensivecontractsandhedgethepossibilityof default. Weconsideratwoperiodeconomypopulatedbyentrepreneurs,whobothconsume and produce, and show under what conditions the system can move to a state which is characterized by defaults on collateralized loan obligations. Agents engage into long-term borrowingtobuytheproductiveassets,whichtheypledgeascollateraltosecuretheirloan. The decision to default is endogenous and depends on the difference between the value of the collateral and the loan as in Geanakoplos (2003). We introduce money to emphasize how a nominal shock, and not only a productivity shock, can lead to financial fragility and areductioninGDP. Our result can be summarized as follows. Consider the case where debt is fully collateralized and entrepreneurs never default, since the nominal value of their contractual obligation is less than the value of their pledged collateral. Assume also that there is an adverse future state of the economy whereby the value of their debt is equal to the value oftheircollateral,and,thus,theborrowerisindifferentbetweendefaultandfullyrepaying hisloan. Inotherwords,heison-the-vergeofdefaulting. Anyfurtheradverseshockinthe economy that reduces further the value of his collateral will inevitably provide him with an incentive to default, since the benefit from defaulting will exceed its cost. In such a situation,theimpactoftherealeconomybecomesevident. Whentheentrepreneurdefaults he loses the capital asset he has pledged as collateral and, therefore, his production will decrease. Subsequently, he needs to attract new capital in the market under more stringent financialconditions. Theupshotoftheargumentisthatthisprocessmayleadtoproductive inefficiency due to capital reallocation to firms that are not debt or liquidity constrained, yet their marginal product of capital is lower. We refer the reader to Gilchrist et al. (2013) foranempiricalassessmentofthemagnitudeofthelossinaggregateresourcesduetosuch misallocationandforareviewoftherelatedliterature. 3

Our work relates to the strand of literature that argues that the financial crisis and in particular defaults on financial contracts can lead to economic recessions. Bernanke and Gertler(1989)andBernankeetal.(1999)modelacreditconstraint,arisingfromcostlystate verification, whereby the firm is only able to obtain collateralized loans and the amount of credittothefirmshrinksinthepresenceofdeflationarypressuresonthepricesofitsassets. Thisintroducesanexternalfinancepremium,whichincreaseswithadecreaseintherelative price of capital. In turn, an increase in the cost of capital will result in a decrease in the marginal product and a reduction in GDP. They show that GDP and investment do not only depend on the fundamentals and productivity, but also on the soundness of the firms’ financial situation, which is an important source of financial instability. We argue in this paper that informational asymmetries are not the crucial element for the financial situation of firms to result in GDP contraction. Instead, the possibility of positive default and asset liquidationprovidethegenesisofachainreactionthatweakensthestabilityofthefinancial systemandresultsintolowerproduction. Our approach is also related to the work on the debt deflation theory of Sudden Stops (Mendoza (2006), Mendoza (2010), and Mendoza and Smith (2006)). These papers introduce collateral constraints similar to Kiyotaki and Moore (1997) in an RBC model of a Small Open Economy to show that when debt is sufficiently high, an adverse productivity shocktriggerstheconstraintsandresultsinafire-salesspiral,fallingpricesandareduction in output. Our results point to the same direction, though contrary to them we consider a monetary economy with nominal contracts and focus on monetary shocks, which have not been thoroughly studied in the literature. In addition, they do not allow for the possibility of default. The latter is crucial for our analysis, since it is the reason that capital gets reallocated to result in inefficient production. Due to fully flexible nominal prices, monetary policy only affects the price level in the final period and not the total output in the absence of default.2 However, default makes credit conditions more adverse and capital is 2EggertssonandKrugman(2012)showthatwhenpricesaresticky, deleveraginganddeflationwillstill affectoutputduetoareductioninaggregatedemand. 4

notallocatedefficiently. We contribute to the aforementioned papers by studying the effect of nominal loan contracts on the propagation of shocks and output. Importantly, Bernanke et al. (1999) focus on real contracts and argue that the modeling of nominal ones is an important step for future research. In our work, nominal long-term loans play a crucial role, since their face value is invariant to deflationary pressures, while the value of collateral that backs themisnot. Ourframeworkisrichenoughtoanalyzeproductivityshocksasthecauseofdebtdeflation. Anumberofpapersmodelfire-salesduetoadverseproductivityorfundingshocksto capture debt-deflationary effects on asset prices leading to loss spirals and financial instability.3 However,wechoosetofocusonthemonetarychannel,sinceitistheleastexplored in the literature. In our model, default on the collateralized loan due to a fall in the value of collateral, exacerbated the debt-deflation dynamics leading to further price declines. Agents do not face additional borrowing constraints and the drop in output is due to an inefficient reallocation of capital. The introduction of funding constraints as in the papers above,wouldexacerbatethechannelthatwedescribe. The rest of the paper proceeds as follows. Section 2 presents the model, while section 3 discusses the equilibrium. Section 4 describes how monetary policy can cause default and how the latter results in higher borrowing costs, capital reallocation and lower output. Section5concludes. TheproofsarerelegatedtotheAppendix. 2 The Model We build a general equilibrium model where two types of agents interact to produce a consumption good. Agents are considered to be entrepreneurs, who both produce and consume the same good. Production happens through the utilization of another capital 3Forexample,ShleiferandVishny(1992),GrombandVayanos(2002),FostelandGeanakoplos(2008), Lorenzoni(2008),BrunnermeierandPedersen(2009),AdrianandShin(2009),Korinek(2011). 5

good, from which agents derive no utility at any point in time. Nonetheless, its price will be always positive in the beginning of every period, since it is essential for the production of the consumption good from which agents derive utility. An important consequence of defaultisthereallocationofresources. Theagentthatdefaultslosesthepledgedcollateral, which is put for sale in the market. Heterogeneity is an important factor, since it is the reallocation of collateral that results in lower aggregate output. In a general equilibrium framework,themarketforthecapitalgoodclearsandallcapitalwillbeusedforproduction. Totaloutputdependsontheefficientuseofcapital,whichmeansanyadditionalunitshould end up to the agent with the higher marginal productivity. However, in the presence of financing frictions capital may end up with the least productive agent, since he may not facethesefrictions. We assume that production takes time and receipts from the sale of goods are not immediately available. This creates the need for a short term funding market, which bridges the gap between expenditures and receipts from sales. Implicitly, agents cannot directly tradethecapitalgoodagainstthesubsequentproductionoftheconsumptiongoodandthey cannot not write their own IOUs to facilitate these transactions. Instead, they need to hold money,whichismodeledascreditfromthecentralbank,totransactinthecapitalandconsumption goods’ markets. The transaction demand for money motive naturally emerges from the cash-in-advance constraint. Since capital is a durable good, in view of the inherent moral hazard problem of honoring long-term debt obligation, agents are required to pledge the capital they purchase as collateral. Finally, the introduction of uncertainty is crucial, since under certainty there would be no default. Without loss of generality, we allowfordefaultinonlysomerealizationsinthefuture. The possibility of default on the contractual obligations that an agent undertakes underscores the necessity for our cash-in-advance constraints. The interplay of liquidity and default justifies fiat money as the stipulated mean of exchange. Otherwise, the mere presence of a monetary sector without any possibility of endogenous default or any other fric- 6

tion in equilibrium would become a veil without affecting real trade and, eventually, final equilibrium allocation. Indeed, cash-in-advance constraints are the minimal institutional arrangement to capture a fundamental aspect of liquidity and how it interacts with default toaffecttherealeconomy. To sum up, our minimal modeling characteristics are agents’ heterogeneity, consumptionandadurablecapitalgood,acollateralizedlong-termloanandshort-termloanmarkets, flexible prices, a monetary economy, uncertainty and incomplete markets. Even though complexity increases with the introduction of these characteristics, we are able to solve the model in closed-form and derive analytical results for our thesis. We now describe the modelinamorerigorousmanner. 2.1 The Economy We consider a two-period monetary general equilibrium model with production, where agents know the present (t =0) but face an uncertain future (t =1), when nature chooses oneofthestatesoftheworlds∈S={1,2}withprobabilityπ . State1and2areotherwise s the same except that there is a lower short-term money supply by the central bank in state 2 than in state 14. Let S∗ = {0}∪S be the set of all states. There are two goods in the economy. Good1isacommodityandisperishable. Good2isacapitalgoodandisdurable. Two heterogeneous agents, a and b trade these two goods. Agent a has an endowment e ∈ R of the capital good at t = 0, while the poor agent b has zero endowment of the ++ capital good at every point in time. Agents are not endowed with the commodity good, but rather use capital to produce it. Agents obtain utility from consuming the commodity, while the capital has no consumption value and is only used for production. Let xh be the s∗ consumption of commodity in state s∗ by agent h∈H. For the purpose of finding a closed 4We can consider this as a monetary shock. This is the only source of uncertainty in the model. Alternatively, we could have distinguished the two states via a productivity shock. What matters is that there is somefundamentaluncertaintybetweenthetwostates,thusourresultswouldbequalitativelythesameunder aproductiveshockaswell. 7

form solution, we assume a logarithmic utility function υ(xh ) = ln(xh ) : R → R,∀s∗ ∈ s∗ s∗ + S∗,h∈H. Let yh be the capital good held by agent h at the end of state s∗. Both agents s∗2 have Cobb-Douglas production functions given by Fh(yh )=Ah (yh ) σ :R →R,∀s∗ ∈ s∗ s∗2 s∗ s∗2 + S∗,h∈H,whereAh isthetotalfactorproductivityandσistheoutputelasticityofcapital. s∗ Production takes place within each period. Without loss of generality, we let both states occur with equal probability (i.e. π = π = 1/2), and assume Ah = 1,∀s∗ ∈ S∗,h ∈ H, 1 2 s∗ σ=0.3ande=2. 2.2 Money, Short-term Money Markets, and Money Storage Moneyinourmodelisthestipulatedmeansofexchangeandastoreofvalue. Weintroduce it through cash-in-advance constraints, such that an agent can purchase either the capital or the commodity in the relevant markets only by paying in money. Although money is fiat and has no intrinsic (consumption) value, it has value because it is essential for the conduct of transactions in the goods’ markets. Agents cannot print their own money and they have to borrow it from the Central Bank, which intervenes directly in the short-term and long-term money markets. In particular, when the central bank purchases intra-period bonds within each state of the world, it injects a quantity of money M s∗ ,∀s∗ ∈S∗ into the system. Moreover, when the central bank extends a collateralized loan at t =0, it injects a quantity of money m¯ into the system5. Money exits the system when agents repay their short-term and long-term loan to the central bank. At the end of period 2 all money will exitthesystem,sinceithasnoconsumptionvalueforanyagent. Fors∗∈S∗,letµh betheamountoffiatmoneythatagenthchoosestooweintheshorts∗ term money market and r s∗ be the short-term interest rate. From market clearing, we have 5Collateralizedlong-termloanextensionisnotanunusualfunctionofmoderncentralbanksespeciallyin theaftermathofthe2007financialcrisis.Alternatively,onecouldthinkofgovernmentsponsoredinstitutions, which extend collateralized loans, e.g. Freddie Mac or Fannie Mae in the case of mortgages. Abstracting from a competitive optimizing banking sector allows us to focus on the effects of credit extension/money supplybythecentralbankondefaultandoutput. However,bydoingsowecannotderiveconclusionsabout financialfragilityandthepossibilityofcreditcrunches,whichissuesarekeptforfurtherresearch. 8

that 1+r s∗ = ∑ h∈H µh s∗ /M s∗. Thus, the ratio of nominal value of loans over the central bank’s credit extension determines the gross nominal interest rate. The amount of fiat money that each agent h borrows is µh s∗ /(1+r s∗ ). Agents do not default in the short-term moneymarkets. sincethereisnouncertaintyabouttheirproductionwithineachperiodand theirshort-termborrowingcanbefullycollateralized. The only way for agents to transfer money across periods is through a money storage technology,potentiallyofferedbythecentralbank. Agentamaystored amountofmoney in the beginning of t = 0 so that he will be able to use it at t = 1.6 We assume that the money storage technology is only available at the beginning of t = 0, not in the end of t =0. 2.3 Commodity and capital good markets Denote by p s∗1 the price of the commodity and p s∗2 the price of capital in s∗ ∈S∗. These are taken as given by both agents to maintain price-taking behavior. Let bh and bh , s∗1 s∗2 ∀h∈H,betheamountoffiatmoneyspentbyagenthtotradeinthecommodityandcapital goods’ markets in state s∗ ∈S∗. In addition, let qh and qh be the amount of commodity s∗1 s∗2 and capital offered for sale in state s∗ ∈S∗ by h. In equilibrium, at positive levels of trade, 0 < p s∗1 = ∑ h∈H bh s∗1 /∑ h∈H qh s∗1 < ∞, and 0 < p s∗2 = ∑ h∈H bh s∗2 /∑ h∈H qh s∗2 < ∞. Note thatagentscannotsellcommoditiesorcapitalgoodstheydonotown. The amount of capital good held by agent a at the end of t =0 is ya =e−qa , while 02 02 in state s it is ya =e−qa −qa .7 The amount of capital good held by agent b at the end s2 02 s2 of t = 0 is yb = bb /p , while in state s agent b(cid:48)s final holdings depend on whether he 02 02 02 defaultsonthecollateralizedloanornot,whichisdiscussedinthefollowingsection. 6In our model only agent a stores money intertemporally. However, the arrangement described here applies to agent b as well; the same goes for the next section where we only describe agent b taking out a collateralizedloan. 7Wehavemodeledagentasellingthecapitalgoodinbothperiods. Intheinitialperiod,thisisalwaystrue sinceheistheonlyoneendowedwithit. However,itmaywellbethecasethathebuysbacksomecapitalin thesecondperiod. Ifthiswasthecaseqa wouldbenegativeandthecash-in-advanceconstraintswouldneed s2 tobeadjustedaccordingly. Notethatthisdoesnotaffecttheresultsofourthesis. 9

As mentioned, all transactions are intermediated through the use of fiat money, i.e. the proceeds from commodity sales in state s∗ cannot be used to purchase the capital good directly,andviceversa. Thisinstitutionalarrangementisafundamentalfeatureofamodel that captures the importance of liquidity constraints and generates a transaction demand for fiat money. We have chosen to introduce money in our model through cash-in-advance constraints as it is methodologically convenient and captures the way goods prices are determinedthroughtheQuantityTheoryofMoney(QTM),wherebybothpricesandquantities are affected when monetary variables change.8 Cash-in-advance constraints should beviewedasliquidityconstraintsthatdistinguishgoodsfromliquidwealth. An alternative way to introduce a demand for money, is by incorporating money balances in the utility and production function. Stein (2012) considers such a model where banks engage in money creation and show that this can lead to financial instability due to fire-sales. When banks try to retain the riskless character of their IOUs, they will need to liquidate a part of their portfolio in bad realizations. Although in his model prices are flexible,monetarypolicycanplayarolethroughcontrollingmoneycreation. Thereasonis that money enters as an input in the objective function of both households and firms function. In our framework, the only role for money aggregates is to determine the price level of goods through the QTM. A change in the quantity of money will have no real effect on outputinthefinalperiodifagentschoosenottodefaultontheirlong-termobligations. The only effect would be an adjustment in prices, since prices are fully flexible. However, the moneystockintheinitialperiodaffectstheinvestmentdecisionbyagentb. Thisisanother financing friction due to the fact that the long-term loan needs to be backed by collateral. Giventhescarcityofcollateral,achangeinM willaffectinvestmentdecisions. Monetary 0 policy has real effects, when deflationary pressures due to a lower money supply induces 8The methodology is close to Dubey and Geanakoplos (2006), Tsomocos (2003) and Goodhart et al. (2006), whointroducecash-in-advanceconstraintstoexaminetheinteractionbetweenliquidityanddefault andanalyzefinancialstability. However,onlyGoodhartetal.(2010)examinetheinteractionbetweenmoney andcollateralvaluesinthecaseofmortgages. 10

agentstodefaultafteracertainpoint,whichresultsinareallocationofresources.9 Wecall this the debt deflation channel of monetary policy, which is described in detail in section 4. We discuss the endogenous decision of agents to default in the following section. Recapitulatively,ourdebtdeflationchannelisinitiatedviapositivedefault,thusemphasizingthe important interconnection of liquidity and default. Consequently, the externality induced bypositivedefaultleadstoinefficientcapitalallocationandinvestmentintheeconomy. 2.4 Default and Collateralized Loan Intheinitialperiod,agentbfinanceshisinvestmentinthecapitalgoodboththroughshorttermandcollateralizedborrowing. Whenheborrowsfromthecollateralizedloanmarket,10 he pledges the capital purchased as collateral. In the second period, the borrower either delivers in full the amount of the collateralized loan or defaults. In the case of default, the collateral pledged is foreclosed and is put for sale in the secondary capital market. The receipts are transferred to the central bank and determine the effective return on the collateralizedloan. Formally, at t = 0, agent b takes out a collateralized loan to finance the purchase of the capital good. The interest rate is r¯ and he promises to payback µ¯ in the next period. The collateralized loan extension is therefore m¯ = µ¯/(1+r¯), since the credit extension is m¯. He spends bb ≤ µ¯/(1+r¯)+µb/(1+r )11 amount of money to purchase bb /p 02 0 0 02 02 amount of the capital good, which he then pledges as collateral. We denote by C the 9An advantage of our model is that it yields closed-from results. Hence, we are able to identify clearly thepropagationmechanismandpresenttheunfoldingofevents, throughwhichmonetarypolicyaffectsthe decisiontodefaultandsubsequentlytheallocationofcapitalandtotaloutput.Tothatextentwedonotengage in a detailed discussion of optimal monetary policy, but rather propose default as an additional channel for affectingaggregateoutput. 10Asmentionedthepriceofthecapitalgoodwillbehigherthanthetheproceedsforgoodssales,sinceit isdurableandcanbeusedforproductioninthesecondperiodaswell.Thus,agentbwillpartiallyfinancehis capital good’s purchases through short-term borrowing or equivalently his income from goods sales within thesameperiod,andpartiallythroughalong-termloanagreement. µb/(1+r ) 11The ratio 0 0 determines the margin on the collateralized loan, i.e. how much individual rebb 02 sourcesagentbhastoutilizetopurchasethecapitalgood. Thelowerthemargin,theeasierfortheagentto purchasecapitalbyusingitascollateral. 11

amount of collateral pledged in terms of units of the capital good, i.e.C =bb /p . Thus 02 02 the collateralized loan is defined by both the interest rate and the collateral requirement. At t = 1, the agent will deliver min (µ¯,p C). If p C ≥ µ¯, then agent b does not default s2 s2 on the collateralized loan and delivers the full amount µ¯. This is not a naive assumption. Due to our General Equilibrium framework every contract is priced in equilibrium. When equilibrium prices are such that the value of the collateral in the future is less than the amounttheagenthastorepay,hewouldratherdefault,purchasethesameamountofcapital fromthesecondarymarketandbebetteroff.12 Defaultisanendogenousdecisionstemming from utility optimization. Only when equilibrium prices are such that the value of the collateral is higher than the nominal value of the loan will the agent repay fully. This is thedebtdeflationchannelthroughwhichmonetarypolicyandmoneysupplymatterforthe determination of asset prices and they affect the decision to default and aggregate output, whichisanalyzethoroughlyinsection4. Moreover, agent b spends an additional amount of money bb in the capital market at s2 t=1, which brings his final capital good’s holdings to yb = bb /p +bb /p .13 When s2 02 02 s2 s2 p C <µ¯, the borrower will give up the collateralC, which is then sold on the market for s2 p C. Hewillthenspendbb topurchasethecapitalgoodandhisholdingisyb =bb /p . s2 s2 s2 s2 s2 2.5 Time-structure of the markets At t =0, the short-term (intra-period) money and collateralized loan markets open. Then commodityandcapitalgoodmarketsmeet. Agentsproducewithintheperiod. Settlements of short-term loans occur at the end of each period. Finally, consumption takes place. The samemarketactivitiestakeplaceatt =1inallthestatesandinadditionagentbrepaysthe 12Animplicitassumptionisthattheagentisnotfurtherpenalizedfordefaultingapartfromlosingthecapitalgoodhisowns. Giventhatthereisadditionalpunishment,thewedgebetweentheloanandthecollateral value has to be higher for him to default. Such an assumption only adds complexity and does not alter the mechanismthroughwhichmoneysupplyaffectsthedecisiontodefault. 13Inprinciple,theagentmaychoosetosellsomeofthecapitalgoodheownsinthesecondperiod. Inthis case,bb isnegativeandthecash-in-advanceconstraintsneedtobeadjustedaccordingly. Againthisdoesnot s2 affecttheresultsofourthesis. 12

collateralizedloansoralternativelydefaultsandthepledgedcollateralisforeclosed. 2.6 Budget sets Denote the macro variables which are determined in equilibrium, and which every agent regards as fixed, by ηηη = (ppp,rrr,r¯) ∈ R2S∗ ×RS∗ ×R . Denote σσσa ∈ ∑ (ηηη), where σσσa = + + + a (bbba,qqqa,µµµa,d)∈RS∗ ×RS∗ ×RS∗ ×R andσσσb ∈∑ (ηηη),whereσσσb =(bbbb,qqqb,µµµb,µ¯)∈RS∗ × + + + + b + RS∗ ×RS∗ ×R the vectors of agent a and b’s market decisions. Agent a’s optimization + + + problemisasfollows max Πa =ln(xa)+∑π ln(xa) 0 s s σσσa∈∑∑∑a s∈S s.t.Ba(ηηη)={σσσa ∈∑ (ηηη):(01a)−(s2a)} a where: µa (01a) ba +d ≤ 0 01 1+r 0 (02a) µa ≤ p qa 0 02 02 µa (s1a) ba ≤ s +d s1 1+r s (s2a) µa ≤ p qa s s2 s2 (01a) says that in the beginning oft =0, agent a borrows short-term to purchase commodities and deposits the rest. (02a) says that in the end of t = 0, agent a repays the short-term loan using the proceeds of capital sales. (s1a) says that in the beginning of each state s∈S, agent a uses the deposits and short-term borrowing to purchase the commodity. (s2a) says that in the end of each state s ∈ S, agent a repays the short-term loan using the proceeds of capital sales. The capital owned by agent a in the end of each period is ya =e−qa and ya =e−qa −qa respectively, as discussed in section 2.3. Note 02 02 s2 02 s2 13

that agent a cannot sell more of the capital good than what he initially owns, i.e.qa <e, 02 qa s2 < ya 02 . x s a ∗ = (ya s∗2 )σ+ba s∗ /p s∗ is agent a’s consumption, which is equal to what he producesplusthe(net)purchasesofthecommodity. Agentb’soptimizationproblemisasfollows: max Πb =ln(xb)+∑π ln(xb) 0 s s σσσb∈∑∑∑b s∈S s.t.Bb(ηηη)={σσσb ∈∑ (ηηη):(01b)−(s2b) b where: µb µ¯ (01b) bb ≤ 0 + 02 1+r 1+r¯ 0 (02b) µb ≤ p qb 0 01 01 bb (03b) C= 02 p 02 µb (s1b) µ¯+bb ≤ s ifbdoesnotdefaultinstates s2 1+r s µb (s1b) bb ≤ s ifbdefaultsinstates s2 1+r s (s2b) µb ≤ p qb s s1 s1 (01b) says that in the beginning of t = 0, agent b enters both a short-term and a collateralized loan to purchase the capital good. (02b) says that in the end of t = 0, agent b repays the short-term loan using the proceeds of commodity sales. (03b) says that agent b puts all the capital good it bought as collateral for the intertemporal loan. (s1b) says that in the beginning of each state s ∈ S, agent b borrows short-term to purchase more of the capital good and also to repay the collateralized loan if he chooses not to default. If he chooses to default, he does not repay the collateralized loan and uses the money borrowed short-term only to purchase capital, since the capital he owned has been seized and put for sale. (s2b) says that in the end of each state s∈S, agent b repays the short-term loan using 14

the proceeds of the commodity sales. xb =(yb ) σ −qb is agent b’s consumption, which s∗ s∗2 s∗ is equal to the amount of the commodity he produces minus what he sells to repay his short-termloan. Thecapitalownedbyagentbintheendofeachperiodisyb =bb /p at 02 02 02 t =0, yb =bb /p +bb /p in state s if b does not default and yb =bb /p in state s if s2 02 02 s2 s2 s2 s2 s2 bchoosestodefault. 3 Equilibrium Wesaythat(ηηη,(σσσh) )isaMonetaryCollateralEquilibrium(MCE)fortheeconomy h∈H E{υ,e,F;MMM,m¯},iff: ba (i) p = s∗1, ∀s∗ ∈S∗ s∗1 qb s∗1 bb (ii) p = 02 02 qa 02 bb (ii(cid:48)) p = s2 ifbdoesnotdefaultinstates∈S s2 qa s2 bb (ii(cid:48)(cid:48)) p = s2 ifbdefaultsinstates∈S s2 qa +C s2 µ¯ (iii) 1+r¯= m¯ ∑ µh (iv) 1+r s∗ = h∈H s∗ ∀s∗ ∈S∗ M s∗ (v) σσσh ∈ argmaxΠh σσσh∈Bh(ηηη) Condition (i) says that the commodity market clears. Conditions (ii), (ii(cid:48)) and (ii(cid:48)(cid:48)) say thatthecapitalgoodmarketsclearforalls∗∈S∗. Condition(iii)saysthatthecollateralized loanmarketclears. Condition(iv)saysthattheshort-termmoneymarketsclear. Condition (v) says that both agents optimize. In sum, all markets clear, expectations are rational, i.e. future prices and interest rates are correctly anticipated, and agents optimize given their budgetsets. 15

Since agent b is not endowed with capital and agent a has a decreasing returns to scale production function, there will always exist gains from trade. Thus, agent a will sell part of his capital endowment to b and subsequently buy back some of b’s output. We refer the interested reader to Geanakoplos and Zame (2013) for details about the existence of equilibrium. Hereafter, we analyze two types of equilibria: an equilibrium where there is no default inanystate,andanequilibriumwherethereisdefaultonthecollateralizedloaninonestate in the second period. We show that there is a threshold such that the former equilibrium obtains if the money supply at t=0 is higher than the threshold, while the latter obtains otherwise. Weconsiderthatthebudgetconstraintsofagentsarealwaysbindinginequilibrium, i.e., agents do not hold idle cash within each period. To guarantee this, we postulate that there is an infinitely small cost when agents hold money within each period, which could be rationalized as the fee of maintaining checking accounts. Hence, agents will use all the borrowed funds to trade in the goods’ markets and the budget constraints Bh(ηηη) are alwaysbinding. 4 Debt Deflation Channel of Monetary Policy Themainobjectiveofthispaperistocharacterizethedebtdeflationchannelofmonetary policy. Asalreadymentioned,wewanttoexaminethewaythatthemoneystockmattersfor theaggregateoutputlevel. Giventhatwehaveabstractedfromanyotherfinancialfrictions apartfromdefaultandsincepricesareflexible,forthemostpartmonetarypolicywillonly affect the general price level, while production will be efficient if there is no default. The money supply in the initial period affects the allocation of capital and total output at t=0, while the latter is maximized at t=1 given that no default occurs in any state of the world. Thisistheconclusionofproposition4.5. Toreachthisconclusionwesolvefortheagents’ optimal production decision in proposition 4.3 and show that there is a wedge between the 16

marginal productivity of the two agents, which is a function of the short-term interest rate. Proposition 4.1 models the term-structure of interest rates and proposition 4.4 solves for theinterestrateswhenthereisnodefaultinequilibrium. We aim to treat debt-deflation as a monetary phenomenon. We show that a lower circulation of money in the first period leads to debt-deflation in the second period. We also show how default leads to an inefficient allocation of capital. Our analysis proceeds in the following three steps. First, we want to examine the relationship between monetary policy in t = 0 and agent b’s decision to default (Proposition 4). Will a contractionary money supply in the initial period lead to default in the next period? Second, we want to see how the money supply and default lead to a change in interest rates (Proposition 4.6). Due to cash-in-advance constraints, interest rates are the "price" for liquidity, and they play an important role in the allocation of the capital good. Finally, we want to study the effects of interest rate variations on total production in t =1 due to monetary policy change and subsequentdefaultbyagents(Proposition4.7). Fiat money is the stipulated means of exchange, and it is exchanged for the acquisition of capital and commodities, while receipts from sales are used to pay back loans and possibly transfer wealth from one period to the other. However, we maintain all the structural characteristics of rational expectations modeling and since money does not enter into the utility function, agents will not hold money idle in the end. All available liquidity will be channeled in the capital and commodity markets at t = 1. This means that all the central bank money supply (i.e.M ,M ,M ,m¯) would exit the system via short-term and collat- 0 1 2 eralized loan repayments. This is captured by proposition 4.1. However, at t = 0 due to missing financial markets agents may opt to hold precautionary savings, to hedge against futureuncertainty. Proposition 4.1. Term Structure of Interest Rates. At t = 0, the aggregate money that exits the system is equal to the short-term loan repayment att =0 plus any precautionary saving, while the aggregate money that enters the system is equal to the collateralized 17

loan extension by the central bank plus the short-term loan credit extension. At t = 1, the aggregate money that exits the system is equal to the repayment on the short-term and collateralized loans, while the aggregate money that enters the system is equal to the precautionarysavingsplustheshort-termloanextension. Thus, (4.11) M r +d =m¯ 0 0 (4.12) M r +min[p C,µ¯]=d 1 1 12 (4.13) M r +min[p C,µ¯]=d 2 2 22 The above proposition shows that the liquidity provision by the central bank and the default decision by agent b may produce an intricate relationship among interest rates. Two important aims of our paper are to examine how liquidity and default affect interest rates,andhowaggregateoutputfluctuateswithinterestratelevels.14 Nevertheless, our thesis suggests that not only the interest rate, but also the quantity of moneyareimportantforthedeterminationofthepriceandoutputlevels. Thequantitytheory of money (Proposition 4.2) provides the intuition for the result. Reducing the quantity of money at t=0 does not only affect prices, but also quantities sold, since it has an effect on the ability of the poor in capital agent to leverage up and purchase capital (unlike the representative agent’s sell-all assumption). This, in turn, affects the price of capital in the second period, since the quantity sold will depend on the stock of the durable good that agentsholdfromthepreviousperiod. Proposition4.2. QuantityTheoryofMoneyProposition. InaMCE,theaggregateincome at t =0, namely the value of all capital good and commodity sales, is equal to the sum of totalshort-termcreditandcollateralizedloanextensionprovidedbythecentralbankminus theprecautionarysavings. Instatesatt =1,ifagentbdoesnotdefault,aggregateincome 14Weconsiderliquiditytobetheabilitytoborrowintheshort-termloanmarkets. Whentheinterestrateis higher,itismorecostlytoborrowmoneyandliquidityislower. 18

equalsthesumoftotalshort-termcentralbankmoneysupplyandofprecautionarysavings minus the collateralized loan repayment. If agent b defaults, aggregate income equals the sumoftotalshort-termcentralbankmoneysupplyandofprecautionarysavings. TheQTM holdsforeachpointintime. Inparticular, period0, p qb +p qa =M +m¯ −d 01 01 02 02 0 period1, ifagentbdoesnotdefaultinstates: p qb +p qa =M +d−µ¯ s1 s1 s2 s2 s ifagentbdefaultsinstates: p qb +p (qa +C)=M +d s1 s1 s2 s2 s 4.1 Interest Rates and Production We first show how individual production varies with the interest rate level (Lemma 4.1) andfinallyhowthelatteraffectstheallocationofcapitalandaggregateoutput(Proposition 4.3). We then distinguish between the default and no default cases. Proposition 4.4 solves for the interest rate in the case of no default, whereas proposition 4.6 corresponds to the case where default is present in equilibrium. When there is no default, production will be efficient/optimal in the last period (Proposition 4.5). As already discussed, production will not be optimal at t=0 and will depend on the available liquidity at that point in time, i.e. M . We show later that monetary policy in the initial period can increase aggregate 0 output, but at the same time affect prices as well. It is the credit friction of collateralized loans that allows this relationship to exists. When agent b chooses to default, capital gets reallocatedandproductionseizestobeoptimaleveninthelastperiod. Thisinefficiencyof 19

default is shown in proposition 4.7. The inefficiency stems from a change in interest rates, whichcreatesawedgebetweenbuyingandsellingcapital(Proposition4.6). Insection4.2 we show how contractionary monetary policy can create debt deflationary pressures in the value of collateral, which result in default in the last period and a reduction in aggregate output. Inthefollowinglemma,weformallyexaminetheimpactofmoneystockonproduction via interest rate changes. The agent who demands the capital good will purchase it from the agent who is rich in it, financing his purchase partly with short-term credit. A change in the price of short-term credit will have an impact on the trade of capital goods, thus it willaffecttheallocationofthecapitalgoodandoutput. Lemma 4.1. Relative prices, allocations and short-term interest rates. For agent b who borrows in the short-term money market, purchases capital goods and sells commodities, wehave: att =0 [1B σ(yb ) σ−1 +π 1B σ(yb ) σ−1 +π 1B σ(yb ) σ−1 ] (4.11∗) x 0 b 0 02 1 x 1 b 1 12 2 x 2 b 2 22 = p 02 (1+r 0 ) 1 p 01 xb 0 att =1,∀s∈S 1B σ(yb )σ−1 (4.12∗) x s b s s2 = p s2 (1+r s ) 1 p s1 xb s Foragentawhoborrowsintheshort-termmoneymarket,purchasescommoditiesandsells capitalgoods,wehave: att =0 (cid:104) (cid:105) 1A σ(ya )σ−1+π 1A σ(ya )σ−1+π 1A σ(ya )σ−1 (4.13∗) x 0 a 0 02 1x 1 a 1 12 2x 2 a 2 22 = p 02 1 p (1+r ) xa 01 0 0 att =1,∀s∈S 20

(4.14∗) x 1 s a A s σ(ya s2 )σ−1 = p s2 1 p (1+r ) xa s1 s s Equation(4.11∗)showsthetrade-offbetweenpurchasingcapitalgoodsandsellingcommodities. The numerator of the LHS is the marginal utility of agent b from the use of the durable capital to produce commodities. The denominator of the LHS is the marginal utility of his consumption. The RHS is the relative price of the capital good and commodity, including the interest rate wedge, since the purchase of the capital good is financed by short-term borrowing and thus is costly. The same discussion follows for the other three equations. The above lemma 4.1 shows that interest rates have intricate effects on the allocation of commodity and capital good, as well as production and final consumption. We are particularly interested in how interest rate variation affects the allocation of capital good andtotalproduction,whichisexaminedinthefollowingproposition. Proposition 4.3. Interest Rate’s Redistribution Effect on Capital Good. Att =1, there is aninterestratewedgebetweenthemarginalproductivityofagentaandagentb. B σ(yb ) σ−1 s s2 =(1+r )2 A σ(ya )σ−1 s s s2 The change of r ,s ∈ S is positively related to the change of ya and negatively related to s s2 thechangeofyb . s2 Both agents produce using the capital good. Agent a, who is rich in it, does not need topurchaseanycapitalandthusavoidsthefinancingcost. Agentb,whopurchasescapital, borrows short-term and has to pay the financing cost. The interest rate acts as a wedge between the marginal productivities of the two agents. In other words, there is a financing premium. Whentheinterestrateincreases,itismoreexpensiveforagentbtopurchasethe capital good. An increase in the marginal productivity of agent b is needed to compensate forthehigherfinancingcost,otherwiseitwouldnotbeprofitabletopurchaseanadditional 21

unit. Due to a concave production function, this results in a lower capital input for agent b. Since the total amount of the capital good is fixed in the economy, agent a will hold moreofitafteranincreaseintheinterestrate. Proposition4.3showsthatanincreaseinthe interest rate in state s will redistribute capital from the (initial) buyer to the (initial) seller duetotheinterestratewedgebetweenthemarginalproductivityofthesellerandthebuyer. Hence, the level of the interest rate determines the allocation of capital. When agent b does not default on his obligations, all interest rates are zero as shown in the following proposition. Proposition 4.4. Interest Rates under no default. When agent b does not default on the collateralized loan, the interest rates on short-term loans and the collateralized loan are allequaltozero,i.e.r s∗ =0,∀s∗ ∈S∗ andr¯=0. This proposition says that, if there is no default, all interest rates and the collateralized loan rate are zero, even if the central bank alters the money supply. This is contradictory withrealitywheremoneysupplyhasaninverserelationshipwithinterestrate. Theintuition isasfollows. Intheendoft =0bothagentswillrepayalltheirshort-termdebtsinfull. In theendoft =1bothagentswillrepayalltheirshort-termloansandthecollateralizedloan fully. Thetotalamountofrepaymentinthetwoperiods,includingprincipalandinterest,is M (1+r )+M (1+r )+m¯(1+r¯). Thetotalamountofmoneyavailableforthemtorepay 0 0 s s (i.e. allthemoneyavailableinthesystem)isequaltototalamountofmoneysupplyinjected bythecentralbank,i.e. M +M +m¯. Intheabsenceofdefault,onlywhenallinterestrates 0 s and the collateralized loan rate are zero will the money available be sufficient for agents to fulfill their obligations. Since they do not have monetary endowment themselves, the only way possible to repay each loan is to pay back an amount exactly equal to what they borrowed. Giventhatshort-terminterestratesarezero,wecanconcludefromproposition4.3that productionisefficientandtotaloutputismaximizedinthelastperiod. Proposition4.5. OptimalProductionintheAbsenceofDefault. AssumeA =B . Ifagent s s 22

bdoesnotdefaultonthecollateralizedloan,theproductionintheeconomyisoptimizedat t =1. Due to cash-in-advance constraints, agent b who is short in capital in period t = 1 needs to borrow short-term to finance additional purchases of capital. When there is no default and the interest rate is zero, there is no financing cost and the economy allocates capital efficiently. In other words, there is no interest rate wedge between the marginal productivitiesinthelastperiod. Themarginalproductivitiesofthetwoagentsaretherefore the same, which results in optimal production and maximum aggregate output. Otherwise, it is always welfare improving to transfer some of the capital from one agent to the other. Thisisnotthecaseintheinitialperiodregardlessofthezerointerestratewedge. Asitwill be more obvious in section 4.2, the money stock at t=0 affects the quantity of the capital goodsoldduetothefinancingfrictionintroducedbytheneedforcollateral. Wenowturntothedeterminationoftheinterestratesunderthepresenceofdefaultand showtheinefficiencyinproductionthatdefaultyields. Proposition4.6. InterestRatesUnderDefault. Consideranequilibriuminwhichagentb defaultsonthecollateralizedloaninstate2,butnotinstate1. Then,theshort-terminterest m¯ −p C 22 rateinstate2ispositive,i.e.r = >0,whiletheshort-terminterestratesatt =0 2 M 2 and in state 1, and the collateralized loan rate are all equal to zero, i.e. r =0, r =1 and 0 1 r¯=0. We can see that when agent b defaults in state 2 (and does not do so in state 1), the short-term interest rate is no longer zero. Agent b defaults and the collateral is foreclosed and sold. The proceeds go to the central bank as a form of repayment. However, this repayment is not in full, so there is some money left in the system. As we discussed above, in the end all money will exit the system, hence the extra money left in the system will exit as an additional interest payment for the short-term credit provided by the central bank. The intuition is that when agent b decides to default, the central bank cannot do 23

anything except foreclosing the collateral, which is less valuable than the full payment of collateralized loan. To compensate for the money lost in the collateralized loan extension, thecentralbankwillchargeapositiveinterestrateontheshort-termcreditasapenaltyfor default. Wenowshowtheinefficiencythatapositiveinterestratebringsinproductiondue todefault. Proposition 4.7. Suboptimal Production in the Presence of Default. When agent b defaultsonthecollateralizedloan,productionintheeconomyisnotefficient. Wecanseethatafterdefault,althoughallthecapitalgoodisstillfullyutilized,itisnot allocatedinanoptimalway. Duetoapositivefinancingcost,capitalisnolongerallocated efficiently. The positive interest rate acts as a wedge between the two agents’ marginal productivities, so that agent a has a lower productivity than agent b, or agent b ends up holding less capital good than agent a. It is welfare improving to transfer some capital fromagentatoagentb,sincebhasahighermarginalproductivity. Thetotalproductionin theeconomyisreducedduetotheinefficiencythatdefaultbringsalong. 4.2 Contractionary Monetary Policy and Default In this section we study the endogenous decision to default and examine when agent b decidestodefaultonthecollateralizedloan. Itisamarketconsensusthatagentwilldefault andsurrenderthecollateralwhenthevalueofcollateralislowerthanthevalueofloan. The followingpropositionofferseconomicintuitiononagent’sdecisionondefault. Proposition 4.8. Marginal Decision of Default Agent b will marginally default on the collateralized loan when the marginal gain from default equals to the marginal loss from default. Formally,wehave: (1+r s )(µ¯−bb s2 ) 1 =(C− bb s2)B σ(yb ) σ−1 1 p xb p s s2 xb s1 s1 s2 s1 24

TheLHSisthemarginalgainfromdefault. Iftheagentbdefaultsonthemortgageloan, thenitdoesnotpayµ¯ anditwillspendbb topurchasesomecapitalgoodafterforeclosure. s2 So agent b will end up having an increment of money amount equals to (µ¯−bb ). Since s2 cash-in-advance constraint is assumed, agent b needs to borrow (µ¯−bb ) less amount of s2 short-term credit to repay the inter-temporal loan. This means that it will sell (1+r )(µ¯− s bb )/p less amount of commodity to repay the loan. This means that it will have an s2 s1 incrementalutilityof(1/xb )(1+r )(µ¯−bb )/p . s1 s s2 s1 The RHS is the marginal loss from default. In default, agent b losses the collateral C and then buys back some bb /p amount of capital. However, he cannot buy back all s2 s2 of them, so he has (C−bb /p ) amount less of capital, evaluated at the marginal utility s2 s2 obtainedfromthecommodityproducedbycapitalgoodB σ(yb ) σ−1 (1/xb ). s s2 s1 One of the purpose of this paper is to study the relationship between the money supply and the default condition. The following provides a formal study on this topic. To simplify the proof, from now on we assume A s =B s , e=2, σ=0.3, and M s∗ >m¯,∀s∗ ∈S∗. We first derive the necessary conditions for agent b to default (lemma 4.2) and then show how contractionary monetary policy can lead to this condition. Given the production inefficiencythatdefaultbringsalong(Proposition4.7),weprovetheexistenceofasuboptimal equilibriumduetodebtdeflationarypressuresinproposition4.9. Lemma 4.2. Default Condition. Consider an equilibrium where agent b does not default 2m¯ onthecollateralizedloan. Wesaythatheisonthevergeofdefaultingifqa = and 02 M +m¯ 2 2m¯ willstartdefaultingifqa < . 02 M +m¯ 2 This lemma provides the equilibrium solution for the default condition that an agent will default on the collateralized loan if the collateral is less valuable than the amount of loan. It says that when the capital good sold by agent a in t = 0 (equivalent to the capital good purchased by agent b in t = 0) is lower than a certain threshold specified by the fundamentals of the economy, then agent b will default on the collateralized loan. Answering the question whether monetary policy has an impact on the default decision 25

is equivalent to seeing whether there is a money supply such that qa is smaller than this 02 threshold. Thisisexaminedinthefollowingproposition. Proposition 4.9. Debt deflation channel of Monetary Policy. Consider an equilibrium ∂qa where agent b does not default on the collateralized loan. Then, 02 > 0. Also, ∃M∗, ∂M 0 0 2m¯ 2m¯ such that qa = > and for M <M∗ agent b starts defaulting in state 2. 02 M +m¯ M +m¯ 0 0 2 1 Finally, default occurs due to debt deflationary pressures on the price of the collateral, ∂p since p =bb /(1−yb )=(M −m¯)/(2(1−qa ))and 22 >0. 22 22 02 2 02 ∂M 0 This proposition shows that in an initial equilibrium where agent b does not default in either state, there is a positive relation between the money supply at t =0 and qa . When 02 themoneysupplyatt =0isreduced,qa goesdownaswell. Also,thereisacertainmoney 02 supply M∗ at which qa reaches the default threshold in state 2, but not in state 1 where 0 02 there is a relatively higher money supply. In another words, agent b is on the verge of default in state 2. This proposition says that a contractionary monetary policy int =0 will leadagentbintodefaultinstate2. ∂qa Since 02 > 0, we can see that when the central bank reduces the money supply in ∂M 0 period t = 0, agent b purchases less capital. However, agent b will still borrow the same amount of collateralized loan, m¯, extended by the central bank. Thus, the same amount of collateralizedloanisbackedbylesscapital,orequivalentlyleverageishigherorthemargin islower. Moreover,wecanseethatwithalowerqa ,thepriceofthecapitalgoodinstates, 02 p =bb /(1−yb )=(M −m¯)/(2(1−qa )), is lower. To sum up, a lower money supply s2 s2 02 2 02 in t =0 leads to a lower qa and a lower p . Since the default decision in state s is given 02 s2 by p C < m¯, which is equivalent to p qa < m¯, we can see that a lower M will drive s2 s2 02 0 agentbclosertodefault. Infact,lemma4.2pointsoutthatwhenqa isreducedtoacertain 02 point,itwillleadagentbintodefaultinstates. Proposition4.9showsthatwhenthemoney supply int =0 is lower, agent b is closer to default. When the money supply is reduced to M∗, agent b is on the verge of defaulting in state 2, since qa =2m¯/(M +m¯), but agent b 0 02 2 willstillbeawayfromdefaultinstate1,sinceqa >2m¯/(M +m¯). Whenthecentralbank 02 1 26

reducesthemoneysupplyevenmore,thenagentbwillstartdefaultinginstate2. The above lemma 4.2 and proposition 4.9 show debt-deflation and default as monetary phenomena: a lower circulation of money in the first period leads to debt-deflation in the second period. We proxy the circulation of money with money supply. The debt-deflation heremeansrelativedeflation,i.e. alowerratioofcollateralvaluetothecorrespondingloan value. Itshowsthatadecreasingmoneysupplybythecentralbankinthefirstperiodleads to a lower ratio of collateral value to the corresponding loan value in the second period monotonically, i.e. there is a positive correlation between the money supply at t = 0 and theratioofcollateralvaluetoloanvalueatt=1. Thelowerthemoneysupply,thelowerthe ratio of collateral value to loan value. We coined this term "relative deflation." The lemma 4.2 points out the condition for default. When a money supply is reduced to a certain point, the ratio of collateral value to the loan value is equal to one. If the money supply is reduced further, the value of the collateral is less than the loan value, and the agent finds it profitable to default on the loan repayment. This is what we call a debt deflation channel of monetary policy, since in the presence of default capital gets reallocated and aggregate outputdecreases. 5 Conclusion We build a monetary general equilibrium model with collateral and production and have a formal treatment of the Fisher debt-deflation effects of monetary policy. We see that the usualpropositionsinamonetarygeneralequilibriummodelholdinthismodel,namelythe quantity theory of money and the term structure of interest rate. Since this is a model with production, we also show that money and interest rates have an effect on total production (realoutput). Oneimportantresultofthismodelisthatinterestrateasthecostoffinancing has a redistribution effect on investment. When the interest rate is higher, then the capital goodwillberedistributedfrommoreproductiveagentstolessproductiveones. 27

WearguethatFisheriandebt-deflationcanbeexplainedasamonetaryphenomenon. We examined how a negative shock in money supply in the initial period can lead to default in the second period through over-indebtedness and deflation. It is straightforward that a reduction in the money supply in the second period after the shock hits would result in debt-deflation dynamics and default. On the contrary, we focus on the pre-crisis money supply, presumably when the economy is on a stable path, and we advocate that future default and collateral prices are not independent of current monetary policy. Following Fisher, the two dominant diseases for debt-deflation is too-much debt (in our case high leverage)andsubsequentdeflation. Weshowthatwhenthecentralbankreducestheshortterm money supply in the first period, the leverage ratio in that period increases: the agent still borrows the same amount of collateralized loan while put less amount of capital good as collateral. Furthermore, when the initial money supply is reduced, we find that the price of the collateral (i.e. the capital good) is lower in the second period. The higher leverage and deflation are the lower ratio of collateral value to loan value becomes in the second period and this brings the agent closer to default. In fact, we find when the money supply in the initial period is lower than a threshold level, agents will default. If initially the agent does not default and the money supply is close to the threshold, then a small negativemoneysupplyshockcreatesrelativedeflationandgeneratesdefault. Thissuggests that considerations about the price of durables used as collateral should be included in the determinationofpolicyapartfrominflationandGDPgrowth. However,thisdoesnothave to be a continuous target, but rather a binary objective monitoring the incentive of agents todefaultoncollateralizedloans. Onewouldimaginethatifaneconomyisatitspotentialoutput,thenitwouldnotmatter significantly whether or not there is default. However, this turns out not to be the case. In ourmodel,theotherimportantresultisthatafterdefault,theinterestrateinstate2increases significantly,whichresultsinaredistributionofcapitalgoodfromthemoreproductivefirm to the less productive one. The production in state 2 is reduced and deviates from optimal 28

production. These variations in interest rate, investment and output do not have significant impact in an equilibrium without default. Note that agent’s default creates an externality to the economy by driving the short-term interest rate up that finally results into output contraction. Theupshotisthat,givenalltheproductionfactorsarefullyutilizedafterdebt-deflation, we still manage to show the reduction in production and the misallocation of resources. That is, we allow agent b to bid for the capital good in the market. Alternatively, had we put b into bankruptcy and forbid him from any further activities, then all the production factorwillbeinthehandofagentaandtheadverseeffectontotalproductionwillbeeven worse. ThisiswherewedifferentiateourselvesfromFisher’sdebtdeflationtheory. Recall, in his 1933 paper, Fisher considered the extreme case when defaulters get into bankruptcy after debt-deflation. This naturally leads to lower production since agents that default stop producingaltogether. However,inourmodel,wemanagetoshowinefficiencyofthedebtdeflation without forcing defaulters into bankruptcy. Here, all the production factors are stillinuse. Theexternalityisthattheyarenotusedasoptimallyaspreviously. Indeed,due tothehigherfinancingcost,thepoorincapitalagentproduceslessthantheinitiallyricher. Thus,deflationfavorsinasensethe"creditor"andharmsthe"debtor". Insum,Fisher’sdebtdeflationargumentcruciallydependsonbothliquidityanddefault as it is shown in proposition 4.9. It is precisely the interplay of liquidity and default that activatesthedefaultchannelthatdistortsoptimalcapitalinvestments. References Adrian, T. and H.S. Shin (2009), ‘Money, liquidity, and monetary policy’, American EconomicReview: Papers&Proceedings99(2),600–605. 29

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Geanakoplos, J. and W.R. Zame (2013), ‘Collateral equilibrium, I: a basic framework’, EconomicTheory,forthcoming. Gertler,M.andP.Karadi(2011),‘Amodelofunconventionalmonetarypolicy’,Journalof MonetaryEconomics58(1),17–34. Gilchrist, S., W.S. Jae and E. Zakrajs˘ek (2013), ‘Misallocation and financial market frictions: Some direct evidence from the dispersion in borrowing costs’, Review of EconomicDynamics16,159–176. Goodhart, C.A.E, D.P. Tsomocos and A.P. Vardoulakis (2010), ‘Modelling a housing and mortgagecrisis’,inFinancialStability,MonetaryPolicy,andCentralBanking,edited byRodrigoA.Alfaro.CentralBankofChile. Goodhart, C.A.E, P. Sunirand and D.P. Tsomocos (2006), ‘A model to analyse financial fragility’,EconomicTheory27,107–142. Kiyotaki, N. and J. Moore (1997), ‘Credit cycles’, Journal of Political Economy 105(2),211–248. Korinek,A.(2011),‘Systemicrisk-taking: Amplificationseffects,externalitiesandregulatoryresponses’,ECBWP1345. Lorenzoni, G. (2008), ‘Inefficient credit booms’, Review of Economic Studies 75(3), 809– 833. Mendoza,E.G.(2006),‘Lessonsfromthedebt-deflationtheoryofsuddenstops’,American EconomicReview96(2),411–416. Mendoza, E. G. (2010), ‘Sudden stops, financial crises and leverage’, American Economic Review100(5),1941–1966. Mendoza, E. G. and K.A. Smith (2006), ‘Quantitative implications of a debt deflation theoryofsuddenstopsandassetprices’,JournalofInternationalEconomics70,82–114. 31

Mishkin, F.S. (2009), ‘Is monetary policy effective during financial crises?’, American EconomicReview99(2),573–577. Shleifer, A. and R.W. Vishny (1992), ‘Liquidation values and debt capacity: A market equilibriumapproach’,JournalofFinance47,1343–1366. Stein, J.C. (2012), ‘Monetary policy as financial-stability regulation’, The Quarterly JournalofEconomics127(1),57–95. Tsomocos, D.P. (2003), ‘Equilibrium analysis, banking and financial instability’, Journal ofMathematicalEconomics39(5-6),619–655. Appendix Prooftoproposition4.1 Proof. Fromthebindingbudgetconstraints(01a),(02a),(01b),(02b)andmarketclearconditions p = ba /qb , p = bb /qa , 1+r¯ = µ¯/m¯ and 1+r = (µa+µb)/M , we have 01 01 01 02 02 02 0 0 0 0 M r +d = m¯; In state s ∈ S, if µ¯ ≤ p C, then agent b does not default; from the bind- 0 0 s2 ing budget constraints (s1a),(s2a),(s1b),(s2b) and market clear conditions p =ba /qb , s1 s1 s1 p =bb /qa and1+r =(µa+µb)/M ,wehaveM r +µ¯ =d;if p C<µ¯,thenagentbdes2 s2 s2 s s s s s s s2 faults;fromthebindingbudgetconstraints(s1a),(s2a),(s1b),(s2b)andmarketclearconditions p =ba /qb , p =bb /(qa +C)and1+r =(µa+µb)/M ,wehaveM r +p C= s1 s1 s1 s2 s2 s2 s s s s s s s2 d. Tosumup,wehaveM r +min[p C,µ¯]=d. s s s2 Prooftoproposition4.2 Proof. Theequationfort =0comesfromcombidingthebindingequation(01a)and(01b) andmarketclearconditions p =ba /qb , p =bb /qa ,1+r¯=µ¯/m¯ and1+r =(µa+ 01 01 01 02 02 02 0 0 µb)/M . Theprooffortheothertwoequationsareonthesameline. 0 0 Prooftolemma4.1 32

Proof. Equation (4.11∗) comes from combining the first order conditions of agent b’s optimizationproblemw.r.t. bb ,µb andqb ,wehave: 02 0 01 (cid:34) (cid:35) λb = 1 1 B σ(yb ) 1−σ +π 1 B σ(yb ) 1−σ +π 1 B σ(yb ) 1−σ , 1 =λb p , λb 01 = 01 p xb 0 02 1 xb 1 12 2 xb 2 22 xb 02 01 1+r 02 0 1 2 0 0 λb and. Likewise,wecangetequations(4.12∗),(4.13∗)and(4.14∗)fromotherfirstorder 02 equations. Prooftoproposition4.3 Proof. From the equations (4.12∗) and (4.14∗) , we have (B /A )(yb /ya )1−σ =(1+r )2. s s s2 s2 s Because B /A is fixed and positive, when r increases, we have yb /ya reduces. Because s s s s2 s2 yb +yc =e,wehaveyb increasesandya decreases. s2 s2 s2 s2 Prooftoproposition4.4 Proof. Given that the short-term interest rates r s∗ ≥0,∀s∗ ∈S∗ and the collateralized loan rater¯≥0. Whenagentbdoesnotdefaultonthecollateralizedloaninanystates∈S,then there must be p C≥µ¯, with market clearing condition 1+r¯=µ¯/m¯, equations (4.12) and s2 (4.13)become (4.41) M r +m¯ +m¯r¯=d 1 1 (4.42) M r +m¯ +m¯r¯=d 2 2 From (4.11), we can see d ≤m¯ since otherwise M r <0. If d <m¯, then from (4.41), we 0 0 havem¯r¯+M r =d−m¯ <0,whichcontradictswiththefactthtr¯≥0andr ≥0. Soonly 1 1 1 d¯= m¯ is possible. Hence we have M r = 0 and M r +m¯r¯ = 0. With the nonnegative 0 0 1 1 interest rates, we can see that r = 0, r = 0 and r¯= 0. The proof for r = 0 follows the 0 1 2 sameline. Prooftoproposition4.5 Proof. Whenagentbdoesnotdefault,fromproposition4.3,weknowthat(B /A )(yb /ya )1−σ= s s s2 s2 1. We can see that (yb /ya )1−σ = A /B . If the interest rate r is not zero, let y ˆb and s2 s2 s s s s2 33

yˆa be the capital good owned by agent b and a in state s respectively, then we have s2 (y ˆb /yˆa )1−σ =(A /B )(1+r )2. We have (y ˆb /yˆa )1−σ >(yb /ya )1−σ. Thus we can see s2 s2 s s s s2 s2 s2 s2 (y ˆb /yˆa )<(yb /ya ). SinceA =B ,ya +yb =eandyˆa +y ˆb =e,wehavey ˆb <yb =e/2 s2 s2 s2 s2 s s s2 s2 s2 s2 s2 s2 and yˆa >ya =e/2. So when interest rate is positive, the capital good owned by agent b s2 s2 is lower than the capital good owned by agent a, and the productivity of agent b is higher than the productivity of agent a. We can always distribute some capital good from agent a to agent b to achieve higher total production. When the interest rate is zero, we can see that monetary policy, any of M s∗ ,∀s∗ ∈ S∗ or m¯ or the combination of the above, has no impactoftheallocationofcapitalgoodint =1. Thecapitalgoodisevenlyallocatedtothe two agents. Since the production function is concave, we can see that evenly distributed capitalsgoodleadstothemaximumproductionintheeconomyatt =1. Prooftoproposition4.6 Proof. The proof for r = 0, r = 0 and r¯ = 0 follow the proposition 4.4. Since agent b 0 1 defaults on the collateralized loan, with the market clear conditions 1+r¯=µ¯/m¯, we have p C<µ¯b=m¯(1+r¯)=m¯. From(4.13),wehaveM r =d¯−p C>d¯−m¯ =0,sor >0. s2 2 2 s2 2 Q.E.D. Prooftoproposition4.7 Proof. Followstheproofforproposition4.5. Prooftoproposition4.8 Proof. Since we know agent b is on the verge of default when µ¯ =CP . Subtracting both s2 sides with bb , we have µ¯−bb =CP −bb . Combining the above equation and equation s2 s2 s2 s2 (4.12∗),wegettheproof. Prooftolemma4.2 34

Proof. When agent b does not default, from proposition 4.4, we have r s∗ = 0,∀s∗ ∈ S∗ and r¯ = 0. From proposition 4.5, we have ya = yb = 1. The capital good sold by s2 s2 bb agent a in state s ∈ S is qa = s2 = yb −yb = 1−yb . And with the binding buds2 p s2 02 02 s2 bb get constraints, (s1a), (s2a), (s1b) and (s2b) and market clearing conditions: s2 = qa , p s2 s2 ba µ¯b µa+µb M +m¯ s1 = qb , = m¯ and s s = M , we have bb = µa, ba = µb, µb = s , p s1 1+r¯ 1+r s s2 s s1 s s 2 s1 s M −m¯ µa = s . In state s, agent b will default on the collateralized loan in state s when s 2 bb bb p C <µ¯. From the market clearing condition 02 =qa , we haveC = 02 =qa . Hence s2 p 02 p 02 02 02 bb Ms−m¯ Ms−m¯ 2m¯ p C = s2qa = 2 qa = 2 qa < m¯, which is equivalent to qa ≤ . s2 qa s2 02 1−yb 02 02 1−qa 02 02 02 M s +m¯ Q.E.D. Wealsoderiveherethefollowingresultswhichwillbeimportantlaterinthepaper: p Since ya = yb = 1, σ = 0.3, and B = 1, from equation (4.12∗), we have s2 = s2 s2 s p s1 0.3(y ) −0.7 =0.3, ∀s∈S,. s2 M −m¯ Since p =(M −m¯)/[2(1−qa )],wehave p = p /(p /p )= s s2 s 02 s1 s2 s2 s1 0.6(1−qa ) 02 µb M +m¯ qb = s =0.3(1−qa ) s s1 p 02 M −m¯ s1 s M +m¯ xa =(ya )0.3+qb =1+0.3(1−qa ) s s s2 s1 02 M −m¯ s xb =(yb ) 0.3 +qb =1−0.3(1−qa ) M s +m¯ s s2 s1 02 M −m¯ s Prooftoproposition4.9 Proof. Step1: from the first order conditions of agent b’s optimization problem w.r.t. µ¯, bb ,µb andqb ,wehave: 02 0 01 λb 01 =λb +λb , 1+r¯ 03 05 (cid:34) (cid:35) λb = 1 1 B σ(yb ) σ−1 +π 1 B σ(yb ) σ−1 +π 1 B σ(yb ) σ−1 01 p xb 0 02 1 xb 1 12 2 xb 2 22 02 0 1 2 λb 01 =λb 1+r 02 0 1 λb p = 02 01 xb 0 35

From the first order conditions of agent b’s optimization problem w.r.t. µb, qb, µb and qb, 1 1 2 2 wehave λb 03 =λb (1+r ) 04 1 1 λb p =π 04 11 1 xb 1 λb 05 =λb (1+r ) 06 2 1 λb p =π 06 21 2 xb 2 1 Sincer s∗ =0,∀s∗ ∈S∗,π 1 =π 2 = ,σ=0.3,andB s∗ =1,∀s∗ ∈S∗ Wehave 2 1 1 1 1 1 1 1 1 = + p xb 2 p xb 2 p xb 01 0 11 1 21 2 1 1 1 (cid:104) 1 1 1 1 1 (cid:105) =0.3 (qa )−0.7+ + p xb p xb 02 2xb 2xb 01 0 02 0 1 2 Let 1 1 1 1 1 1 k= + 2 p xb 2 p xb 11 1 21 2 1 1 1 1 t =0.3( + ) 2xb 2xb 1 2 Theabovetwoequationsbecome: 1 1 (4.91) =k p xb 01 0 1 1 1 (cid:104) 1 (cid:105) (4.92) = 0.3 (qa ) −0.7+t p xb p xb 02 01 0 02 0 Step2: from the results that r s∗ =0,∀s∗ ∈S∗ and r¯=0, and the budget constraints and bb µb µ¯ µ¯ market clearing conditions: µa = p qa , p = 02, bb = 0 + , 1+r¯= and 0 02 02 02 qa 02 1+r 1+r¯ m¯ 02 0 µa+µb M +m¯ M −m¯ µa (M +m¯) 1+r = 0 0,wehaveµa= 0 andµb= 0 . Wehave pa = 0 = 0 , 0 M 0 2 0 2 02 qa 2qa 0 02 02 µb (M −m¯) p = 0 = 0 andxb =(qa )0.3−qb . 01 qb 2qb 0 02 01 01 01 Step3: substitute p , p andxb intothetwoequations(4.91)and(4.92)instep1. We 01 02 0 have k(M −m¯)(qa )0.3 qb = 0 02 01 2+k(M −m¯) 0 36

0.3(qa )0.3+(qa )1.3t qb = 02 02 (M −m¯) 01 (M +m¯)+qa t(M −m¯) 0 0 02 0 Combinetheabovetwo,wehave (4.93)k[0.7M +1.3m¯]=0.6+2tqa 0 02 LetJ =[0.7M +1.3m¯] 0 Step4: substitute k and t into the equation (4.93) and simplify, and let h=1−qa , we 02 haveaquadraticequationswithoneunknown, a h2−a h+a =0 1 2 3 a ,a anda areallexogenousvariables,where: 1 2 3 (cid:18) 1 1 a = (M +m¯)(M +m¯)0.32+ (M −m¯)(M +m¯)0.3+ (M −m¯)(M +m¯)0.3+ 1 1 2 1 2 2 1 2 2 (cid:19) 1 1 +J (M +m¯)0.3+J (M +m¯)0.3 2 1 2 2 (cid:18) (cid:18) (cid:19) (cid:18) (cid:19) 1 1 a = (M +m¯)(M −m¯)0.3 1+ +(M −m¯)(M +m¯)0.3 1+ +(M −m¯) 2 1 2 1 2 1 2 2 (cid:19) 1 1 (M −m¯)+J (M −m¯)+J (M −m¯) 2 2 1 2 2 a =2(M −m¯)(M −m¯) 3 1 2 Step5: WeresorttoMathematicatosolvetheaboveequationsandhavetworootsh and 1 2m¯ h . WewanttocheckthatthereexistsanM∗ where0<qa = <1,whichisequiv- 2 0 02 M +m¯ s (−m¯ +M ) ∂qa ∂h alentto0<h=1−qa = 2 <1. And 02 >0,whichisequivalentto <0. 02 (m¯ +M ) ∂M ∂M 2 0 0 Undertherestrictionm¯ >0,M −m¯ >0,M −m¯ >0,M −m¯ >0,andM >M . First, 0 1 2 1 2 wetakeh andverifythath ispositive. Andthereexists{m¯,M ,M ,M }suchthath <1. 1 1 1 0 2 1 (−m¯ +M ) 2 We then also verify that there exist {m¯,M ,M ,M } such that 0<h = <1. 1 0 2 1 (m¯ +M ) 2 ∂h 1 Thenwealsoverifythat <0. ∂M 0 Under the same restriction, we can also verify that h is positive and there exists 2 {m¯,M ,M ,M } such that h < 1. However, there does not exist {M ,m¯,M ,M } such 1 0 2 2 1 0 2 37

(−m¯ +M ) 2 that h = . What whatever h might be, agent b will never be on the verge of 2 2 (m¯ +M ) 2 default. 38