On Default and Uniqueness of Monetary Equilibria

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. On Default and Uniqueness of Monetary Equilibria Li Lin, Dimitrios P. Tsomocos, and Alexandros P. Vardoulakis 2015-034 Please cite this paper as: Li Lin, Dimitrios P. Tsomocos, and Alexandros P. Vardoulakis (2015). “On Default and Uniqueness of Monetary Equilibria,” Finance and Economics Discussion Series 2015-034. Washington: Board of Governors of the Federal Reserve System, http://dx.doi.org/10.17016/FEDS.2015.034. 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.

∗ On Default and Uniqueness of Monetary Equilibria † ‡ § Li Lin Dimitrios P. Tsomocos Alexandros P. Vardoulakis May 8, 2015 Abstract Weexaminetherolethatcreditriskinthecentralbank’smonetaryoperationsplaysin the determination of the equilibrium price level and allocations. Our model features tradeinfiatmoney,realassetsandamonetaryauthoritywhichinjectsmoneyintothe economythroughshort-termandlong-termloanstoagents. Short-termloansareriskless,butlong-termloansarecollateralizedbyaportfolioofrealassetsandaresubject tocreditrisk. Theprivatemonetarywealthofindividualsiszero,i.e.,thereisnooutsidemoney. Whenthereisnodefaultinequilibrium, thereisindeterminacy. Positive defaultineverystateoftheworldonsomelong-termloanendogenouslycreatespositiveliquidwealththatsupportspositiveinterestratesandresolvestheaforementioned indeterminacy. Hence, a non-Ricardian policy across loan markets can determine the equilibriumallocationswhileitallowsthecentralbanktoearnprofitsfromseigniorage inordertocompensateforanylosses. Keywords: Determinacy,Liquidwealth,Default,Collateral,Monetarypolicy JELClassification: D5,E4,E5 ∗We are grateful to the seminar participants of the XXIII European Workshop on General Equilibrium Theory in Paris and to John Geanakoplos, Charles Goodhart, Udara Peiris, Skander Van den Heuvel and especiallyHeraklesPolemarchakisforhelpfulcomments.Allremainingerrorsareours.Theviewsexpressed in this paper are those of the authors and do not necessarily represent those of Federal Reserve Board of Governors,anyoneintheFederalReserveSystem,oranyoftheinstitutionswithwhichweareaffiliated. †InternationalMonetaryFund;email: llin@imf.org ‡Saïd 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 Introduction Theanalysisoftheinteractionbetweentherealandmonetarysectorsoftheeconomycalls for theories of how the price level is determined in equilibrium. In general equilibrium models where all transactions take place in real terms, the general price level is indeterminate and only relative prices can be determined; a consequence of Walras law. The introduction of money or other monetary assets not only does not resolve the issue of indeterminacy, but can also lead to indeterminacy of real equilibrium allocations when asset marketsareincomplete(BalaskoandCass,1989;GeanakoplosandMas-Colell,1989). The presenceofamonetarysectorisessentialtoprovidetheadditionalrestrictionstodetermine the price level. Thus, the dichotomy between the nominal and real sectors of the economy vanishes when policy resolves the real as well as the nominal indeterminacy of monetary equilibria(seeGrandmont(1985)foranextensivediscussionoftheclassicaldichotomy). The focus of this paper is on how the monetary authority can determine the price level ratheronhowitchoosespolicyoptimallytoachievecertaineconomicoutcomes. Thus,we suggest and examine a sufficient condition such that monetary equilibria are determinate, which has not been studied in the existing literature. Many authors have suggested sufficient conditions to pin down the price level. Using a variant of the sell-all model of Lucas and Stokey (1987), Magill and Quinzii (1992) show that the monetary authority can determine the price level if it buys all the endowments of agents in exchange for a fixed supply of money, which subsequently the agents use to buy back goods. Determinacy is achieved becausethequantitytheoryofmoneyholdsbyconstructionandthemonetaryauthoritycan choose the price level by exchanging a fixed amount of money with the total endowments intheeconomy,i.e.,moneyandpricesmoveinlockstep. However, the non-trivial quantity theory of money can fail when we move towards a bid-offer model where agents endogenously decided the quantity of goods they buy and sell. Again,themonetaryauthoritycaninjectafixedamountofmoneyintheeconomy,for example in the form of a loan to agents, but agents may choose to spend only part of it to 2

purchasegoodsandkeeptherestas“idle"cash. Thehoardingbehaviorcanbediscouraged if there is an opportunity cost of holding cash, i.e., if the central bank charges a positive interest rate for the amount of money it injects in the economy. Dubey and Geanakoplos (2006)showthatasufficientconditiontoguaranteedeterminacyistheexistenceofprivate monetary endowments that can support positive interest rates in equilibrium (Dubey and Geanakoplos, 1992; Dubey and Geanakoplos, 2003a; Dubey and Geanakoplos, 2003b). These monetary endowments (outside money) are free and clear of any offsetting obligation,thustheyaredifferentfromthemoneystock(insidemoney)thatthemonetaryauthority injects into the economy and which needs to be repaid in the end.1 A key distinction between outside and inside money is that inside money is an asset of the central bank and a liability of agents, while outside money is an asset that belongs to the balance sheet of only one agent (see Gurley and Shaw (1960) for more on the distinction between inside andoutsidemoney). Our purpose is to establish an alternative sufficient condition to guarantee the determinacyofmonetaryequilibriaintheabsenceofoutsidemoney. Forthatpurpose,weconsider a two period monetary model introduced by Dubey and Geanakoplos (2006), which featuresbothshort-termandlong-termnominalloansissuedbythecentralbank,andextendit infouraspects. First,weintroducerealassets,akinto“Lucastrees"thatpromiseuncertain dividends in the second period. Second, we require that long-term loans are collateralized by a portfolio of these Lucas trees. The central bank has the authority to choose the collateral requirements. Third, we allow the central bank to offer a menu of long-term loans parametrized by the different set of collateral requirements and different promised interest rates or loan supplies. Finally, we allow agents to endogenously default on a long-term 1Numerouspapersusethisframework: Tsomocos(2008)appliestheargumenttoshowthedeterminacy of international monetary equilibria, while Giraud and Tsomocos (2010) prove uniqueness under the limitpriceexchangeprocess. Espinoza,GoodhartandTsomocos(2008)andEspinozaandTsomocos(2014)use a similar framework to connect the supply of liquidity by the central bank to the term structure of interest ratesandprovideanexplanationforthetermpremium.Goodhart,SunirandandTsomocos(2005),Goodhart, Sunirand and Tsomocos (2006), Goodhart, Peiris, Tsomocos and Vardoulakis (2010), Tsenova (2014) and Tsomocos(2003)usethesamemodelofmoneytoanalyzethefinancialstabilityinamonetaryeconomy. 3

loan,ifthevalueofthecollateralislowerthatthepromisedloanrepayment. Our result can be summarized as follows: When the central bank undertakes some credit risk in its intertemporal monetary operations, i.e., if agents default on some loan in everystateoftheworldinthesecondperiod,thenitcanendogenouslycreateprivateliquid wealth for the agents in the economy which can be used to support positive interest rates in all states and periods (see also Tsomocos (1996)). In the initial period agents borrow both in the short-term and long-term money markets. At the end the period, agents return tothecentralbanktheprincipalfromshort-termmoneyoperationsandcanpaytheinterest by using part of the money they borrow long-term, while they transfer the rest as deposits to the second period. Agents are able to act this way, because they do not need to repay thelong-term loansin full,but candefault onthem. Theextra privateliquidwealth, which constitutes a loss on long-term money market operations by the central bank, can be used torepaypositiveinterestratesonshort-termloansineverystateandperiod. We show that the central bank can choose interest rates and collateral requirements for a menu of long-term loans such that there is default on some loan in every state in the second period and additionally positive short-term interest rates can be supported within each state and period. In turn, this guarantees that agents will not hold “idle" cash, hence thequantitytheoryofmoneyobtains. Inotherwords,themonetaryauthoritycandetermine monetaryequilibriabyappropriatelychoosingthelevelofshort-termandlong-terminterest rates and additionally the level of collateral requirements on long-term loans (see section 6). The conditions to obtain determinacy when the monetary authority targets the money supplyinsteadoftheinterestrateslevelaremorerestrictiveandarediscussedinsection7. We should note that the fact that the central bank assumes credit risk and eventually losses in its long-term monetary operations does not imply that its budget constraint at the end of time is violated. On the contrary, the losses in the long-term loans are compensated by seigniorage revenues through the short-term monetary operations. This introduces the notion of non-Ricardian policy not only across time, but also across loan markets. Put 4

differently, the accounting inflows and outflows need not be balanced either across time or within each loan market. This is sufficient to create positive liquid wealth and guarantee determinacy. The presence of outside money in Dubey and Geanakoplos (2006) makes theirmodelnon-Ricardianaswell.2 Recapitulating, indeterminacy obtains whenever the government satisfies its budget constraint period by period, i.e., when it follows a Ricardian policy. Such a policy can be achievedeitherbyzerointerestratesorbytheredistributionofseignioragerevenuesasdividendstoagents(DrèzeandPolemarchakis,2000;Bloise,DrèzeandPolemarchakis,2005). On the contrary, a non-Ricardian policy can be achieved by violating the period by period governmentbudgetconstraint. NotethatinDubeyandGeanakoplos(2006)determinacyis assuredbynotdistributingtheinterestratepaymentsasdividendsinthesameperiod,while in our argument determinacy relies on the fact that the government/central bank does not taxinadvancetheagentssoastooffsettherealizedlossesaccruingfromfuturedefault. As an aside note, the monetary authority in our model can choose collateral requirements to create a full set of state-contingent bonds if the payoff matrix of the real assets has full rank. McMahon, Peiris and Polemarchakis (2014) and Peiris and Polemarchakis (2014) show that the equilibrium cannot be determined even in the presence of outside money, unless the monetary authority specifies the composition of the long-term loans, or usesaterm-structureorforwardguidanceruleasinAdão,CorreiaandTeles(2014),Magill andQuinzii(2014a)andMagillandQuinzii(2014b). Theneedfor“comprehensivemonetary policy" has been pointed out in Drèze and Polemarchakis (2000). Monetary policy in our model is comprehensive, since the monetary authority chooses long-term interest rates (ormoneysupplies)andcollateralrequirementsforeachlong-termcontractindependently, andinadditionsetstheshort-terminterestrates(ormoneysupplies). The rest of the paper proceeds as follows. In section 2 we describe the structure of the economy and the market interactions, while in sections 3 and 4 we present agents’ 2See Buiter (2002), Cochrane (2001), Sims (1994) and Woodford (1994) for the importance of non- Ricardianpolicyindeterminingtheequilibriumpricelevel. 5

optimization problems and market clearing conditions. In section 5 we prove a number of lemmas that we use in sections 6 and 7 to prove the determinacy of monetary equilibria under interest rate and money supply targeting by the central bank, respectively. Finally, section8concludes. 2 The Economy We consider a two-period economy,t ∈T ={0,1}, with the set of states of nature as S∗ = {0,1,...,S}. State0occursinperiod0,andnatureselectsoneofthestatess∈S={1,...,S} whichoccurinperiod1. Wealsodenotes∗ asonestateinS∗. ThereareL={1,...,L}perishablecommodities. Thecommodityspacemaybeviewed asRS∗L. The setofdurable,realassetsisK ={1,...,K},which,withoutlossof generality, + pays out in terms of good (cid:96)=1. Denote the real payoffs of an asset k in state s as Xk. The s assetpayoffs’spacemaybeviewedasXSK,whichisaS×K matrix. + Let the price for commodity (cid:96) in state s∗ ∈S∗ be p(cid:96) , and the price for asset k at t =0 s∗ be pk. Thus,thenominalpayoffofassetk instatesisXk·p1. s s ThesetofagentsisH ={1,...,H}. Theagentsareendowedwithboththecommodities and real assets. Let the initial endowment of commodity (cid:96) in state s∗ ∈ S∗ by agent h be e(cid:96) (h). We assume that no agent has the null endowment of commodities in any state, i.e., s∗ fors∗ ∈S∗ andh∈H: e s∗ (h)=(e1 s∗ (h),...,eL s∗ (h))(cid:54)=0. Moreover,eachnamedcommodityisactuallypresentintheaggregate,i.e., ∑ e(cid:96) (h)>>0, ∀(cid:96)∈L. s∗ h∈H Let ek(h) denote agent h’s endowment of asset k in period 0. Agent h has utility of consumption uh :RS∗L →R. We also assume that each uh is concave and smooth (i.e., second + 6

partialderivativesexistandarecontinuous),andstrictlymonotonic. Agents trade on both commodities (cid:96) ∈ L and real assets k ∈ K. In each state s∗ ∈ S∗, agenthspendsb(cid:96) (h)amountofmoneytopurchasecommodity(cid:96)orsellsq(cid:96) (h)amountof s∗ s∗ commodity (cid:96). Trades on the real asset k occurs only at t =0. Agent h∈H will purchase therealassetbyspendingbk(h)amountofmoneyorsellqk(h)amountofassets. We introduce the demand for money through cash-in-advance constraints. All commodities and assets are traded exclusively for money, which is fiat, and thus does not provideanydirectutilitytoagents. Moneyisnotonlythestipulatedmeansofexchange,butit is also a store of value, i.e., it is perfectably durable and can be carried forward for future use. Money can enter the economy either in the form of a loan, which creates a liability for agents to repay a certain amount in the future (inside money), or in the form of monetary endowments,whicharefreeandclearofanyoffsettingobligation. DubeyandGeanakoplos (2006), McMahon, Peiris and Polemarchakis (2014) and Tsomocos (2008) all consider both types of money, and show that positive outside money is necessary (but not always sufficient) to determine the price level. Our most important departure from the previous literatureisthatweonlyconsiderinsidemoney. Yet,wearestillabletoprovedeterminacy ofmonetaryequilibria. Inparticular,weconsideracentralbankwhichextendsloanstoprivateagents. Forsimplicity, we consider two types of bank loans. A short-term loan is traded at the beginning of each state s∗ ∈ S∗ and promises µ s∗ (h) dollars at the end of the period. Let r s∗ denote the interest rate for the short-term loan in state s∗ ∈S∗. Long-term loans are traded at the beginningofstate0andpromiseµ¯c(h)dollarsbeforecommoditytradeineveryfuturestate s∗ ∈S in period 1. The long-term loans are collateralized. Let c∈C≡{1,...,C} index the long-termbankloanthathascorrespondingcollateralrequirementsgivenbyγc,k∈K,and k r¯ denotetheinterestrateonbondc. c Each unit of long-term loan c needs to be collateralized by a bundle of assets where γc k 7

is the value of assets k ∈K that an agent should hold if he takes a long-term loan indexed by c ∈C. Note, that γc are the collateral requirements per one unit of loan, i.e., a longk term loan of size µ¯c(h) requires assets k worth µ¯c(h)·γc to be pledged as collateral. In this k sense,γc isakinto(theinverseof)aloan-to-valueratio,andthequantityofassetk pledged k as collateral is equal to γc/pk. The monetary authority chooses γc, ∀k ∈ K,c ∈C. The k k collateralconstraintshouldholdatt =0foreachassetindividually. Thus, γc bk(h) ∑µ¯c(h) k ≤ek(h)−qk(h)+ ∀k∈K.3 (1) pk pk c Agent h can choose to default at t = 1 if the value of her collateral is lower than the loanobligation(GeanakoplosandZame,2013),i.e.,shedelivers µ¯c(h)fc ∀c∈C, (2) s (cid:34) (cid:35) p1 where fc =min 1,∑γcXk s . s k s pk k Define Ic(h)an indicator variable for every c∈C, s∈S and h∈H, such that Ic(h)=1 s s 3Thecollateralconstraintlinksthemonetaryvalueofrequiredcollateraltothetotalvalueofassetholdings for each asset k. Hence, the amount of collateral is endogenously determined, i.e., the monetary authority neither specifies the quantity of assets to be pledged as collateral nor their nominal price, but only its total value. It should be noted that an alternative collateral requirement that links nominal loans to real asset holdingswouldresolvetheindeterminacyofmonetaryequilibriaifbinding,whichistypicallythecasewhen defaultobtainsinequilibrium.Insuchanarrangement,thepriceofcollateralwouldbedeterminedapriorias inShubikandTsomocos(1992)whoallowthemonetaryauthoritytosettheexchangepriceoffiatmoneyto gold(thedurableassetinourframework). Onthecontrary,wefocusonacollateralconstraintthatspecifies the loan-to-value ratio for long-term loans and emphasize the role of default on some loans in supporting positive interest rates on other non-defaultable loans due to capability of rolling over loans. The argument behindourdeterminacyresultreliescruciallyonthefactthatdefaultendogenouslycreates“outside"money suchthatpositiveinterestratescanbesupportedratherthanonthefactthatdefaultresultsinbindingcollateral constraints. Put differently, it is the positive interest rate and not the binding collateral constraint that pins downprices.However,wehastentoemphasizethatourargumentrequirestheoverlappingofbothdefaultable and non-defaultable loans, which are the long-term and short-term loans in our framework respectively, so that to allow for rolling over of debts. Most importantly, the non-defaultable loans need to carry positive interestratesandthedemandforthemshouldbepositiveinequilibrium. Naturally, agentswouldpreferto firsttakeloansthatdefault,buttheborrowingcapacityisrestrictedbythecollateralconstraint. Ifthebenefits ofborrowingatpositiveinterestratesarehighenough,i.e.,ifthe"gains-to-trade"hypothesisissatisfied(see DubeyandGeanakoplos(2003b)),thentherewillbedemandfornon-defaultableloansaswell.Nevertheless, ourargumentmaynotberobusttoaspecificationwherebytheseadditionalintra-periodloansdonotexist. In principle,ourresultshouldalsoobtainwithonlydefaultableandnon-defaultableoverlappingintetermporal loans. 8

whenthereisnodefaultandIc(h)=0whenthereisdefault. s Themonetaryauthoritycanusetwoalternativewaystoimplementmonetarypolicy. At one extreme we may suppose that the central bank sets quantity targets and pre-commits tothesizeofitsborrowingorlending,lettinginterestratesbedeterminedendogenouslyat equilibrium. At the other extreme we may suppose that the central bank sets interest rate targets, and pre-commits to supply whatever amount of money demanded at those rates. Weshowthatbothpoliciesleadtodeterminacy,buttheconditionsfortheformerpolicyare morerestrictive. 3 Agents Optimization Problem For convenience, we list prices as (r,p) where r is the (S∗+C) -dimentional vector of interestratesfortheshort-termloansandlong-termcollateralizedloans;and pisthe(S∗L+ K) -dimentional price vector, including the prices for commodities, p(cid:96) and for real assets, s∗ pk. The agents take (r,p) as given. Given (r,p), with r ≥ 0 and p >> 0, the budget set B(r,p)(h)availabletoagenth-specifyingthesequenceofmarketactionsandconsumption choicesσ(h)={b(cid:96) s∗ (h),q(cid:96) s∗ (h),bk s∗ (h),qk s∗ (h),µ¯c(h),µ s∗ (h),d¯(h),c(cid:96) s (h)}thatarefeasiblefor her-isdepictedinthefollowingtable: Steps Descriptions (i) borrowshort-termandlong-term,tradeincommoditiesandrealassets (ii) repaytheshort-termloanandconsume (iii) borrowshort-term,repayordefaultonlong-termloans,andtradeincommodities s (iv) repayshort-termloanandconsume s We require that the outflow of money at each point in time cannot exceed its stock on hand, letting ∆ s∗ (h) and ∆¯(h) denote the difference between the right-hand-side and the left-hand-side of the corresponding budget constraints below. ∆ s∗ (h) represent intratemporal deposits or “idle" money, while ∆¯(h) are intertemporal deposits or money “carried 9

forward". Every agent h ∈ H tries to maximize her expected utility from the consumption of L goods,4 (cid:16) (cid:17) (cid:16) (cid:17) maxUh =∑uh c(cid:96)(h) +∑∑uh c(cid:96)(h) , (3) 0 0 s s σ(h) (cid:96) s (cid:96) subjecttothefollowingconstraints: µ (h) µ¯c(h) ∑bk(h)+∑b(cid:96)(h)+∆ (h)= 0 +∑ , (4) 0 0 1+r 1+r¯ k (cid:96) 0 c c (i.e., Money spent on purchase of real assets and commodities in state 0 + money unspent =moneyborrowedonshortandlongloans), µ (h)+∆¯(h)=∆ (h)+∑pkqk(h)+∑p(cid:96)q(cid:96)(h), (5) 0 0 0 0 k (cid:96) (i.e., Money repaid on short loan + money unspent = money unspent in (4) + money obtainedfromsalesofrealassetsandcommodities), b(cid:96)(h) c(cid:96)(h)≡e(cid:96)(h)−q(cid:96)(h)+ 0 ∀(cid:96)∈L, (6) 0 0 0 p(cid:96) 0 (i.e.,Consumptionofcommodity(cid:96)≡endowmentof(cid:96)-salesof(cid:96)+purchaseof(cid:96)), γc bk(h) ∑µ¯c(h) k ≤ek(h)−qk(h)+ ∀k∈K, (7) pk pk c (i.e., Amount of real asset k required as collateral for long loans ≤ endowment k - sales of k +purchaseofk ), µ (h) ∑b(cid:96)(h)+∑Ic(h)µ¯c(h)+∆ (h)= s +∆¯(h), (8) s s s 1+r (cid:96) c s 4Tosimplifytheintricateequilibriumequationsthatarisewithincompletemarketsweconfineattention to“active”equilibriainwhicheachagentchoosestobuysomethingineverystate. SeeTsomocos(2008)for aformaltreatmentofindeterminacyinthepresenceofinactivemarkets. 10

(i.e.,Moneyspentonpurchaseofcommoditiesinstates+moneyrepaidfullytosomelong loans+moneyunspent=moneyborrowedshort-term+moneyunspentfrom(5)), µ (h)≤∆ (h)+∑p(cid:96)q(cid:96)(h), (9) s s s s (cid:96) (i.e., Money repaid on short loan ≤ money unspent at (8) + money obtained from sales of commodities), (cid:20) bk(h) (cid:21) γc ek(h)≡ ek(h)−qk(h)+ −∑(1−Ic(h))µ¯ c (h) k ∀k∈K, (10) s p s pk k c (i.e.,realassetk ownedatstates≡assetownedatstate0-assetforeclosed), b(cid:96)(h) c(cid:96)(h)≡e(cid:96)(h)−q(cid:96)(h)+ s ∀(cid:96)∈L/{1}, (11) s s s p(cid:96) s (i.e.,consumptionofcommodity(cid:96)atstates≡endowmentofcommodity(cid:96)atstates-sales ofcommodity(cid:96)atstates+purchaseofcommodity(cid:96)atstates), and b1(h) c1(h)≡∑ek(h)Xk+e1(h)−q1(h)+ s for(cid:96)=1, (12) s s s s s p1 k s (i.e., consumption of commodity 1 in state s ≡ commodity 1 produced by real assets + endowmentofcommodity1-salesofcommodity1+purchaseofcommodity1). Ifagenthchoosestodefaultoncontractcinstates,thenthemonetaryauthorityseizes the collateral and puts it up for sale in the market for good 1. Recall that assets deliver in terms of this good in the beginning of period 1. Instead of receiving µ¯c, the monetary authorityreceives∑γcXkp1/pk. k s s k 11

4 Market Clearing and Equilibrium 4.1 Goods markets Totalsalesshouldbeequaltototalpurchases,i.e., ∑ b(cid:96) (h) p(cid:96) = h s∗ ∀(cid:96)∈L\{1},s∗ ∈S∗, (13) s∗ ∑ q(cid:96) (h) h s∗ ∑ b(cid:96)(h) p(cid:96) = h 0 for(cid:96)=1, (14) 0 ∑ q(cid:96)(h) h 0 ∑ b(cid:96)(h) p(cid:96) = h s for(cid:96)=1. (15) s ∑ q(cid:96)(h)+qM h s s whereqM≡∑∑∑(1−Ic(h))µ¯c(h)γcXk/pk,istheamountofcollateralthatthemonetary s s k s h k c authorityliquidatesintheeventofdefault. 4.2 Assets markets Totalsalesbyagentsshouldbeequaltototalpurchases,i.e., ∑ bk(h) pk = h ∀k∈K. (16) ∑ qk(h) h 4.3 Short-term money markets Totalshort-termloansdemandshouldbeequaltomoneysupply,i.e., ∑ h µ s∗ (h) 1+r s∗ = . (17) ∑ h M s∗ 12

4.4 Long-term money markets Totallong-termloansdemandforeachc∈C shouldbeequaltomoneysupply,i.e., ∑ µ¯c(h) h 1+r¯ = ∀c∈C. (18) c M¯c Definition of Equilibrium: ((r,p),(M s∗ ,M¯ c c ∈C ),σ(h) h∈H ) is a Monetary Collateral Equi- (cid:18) (cid:19) (cid:16) (cid:17) librium (MCE) for the economy E= Uh,e s∗ (h),ek(h) ,X,(γc k ) k∈K,c∈C if and h∈H,s∗∈S∗ only if equations (13)-(18) hold, σ(h)∈argmax Uh,∀h ∈H. In sum, all mar- σ(h)∈B(r,p)(h) kets clear, expectations are rational, i.e. future prices and interest rates are correctly anticipated,andagentsoptimizegiventheirbudgetsets. 5 Equilibrium Analysis In this section we prove a number of lemmas that we will use to prove the determinacy of monetaryequilibriainsections6and7. We,first,establishsomerestrictionsonequilibrium variables. In particular, we show that interest rates cannot be negative (lemma 1), that all money will return to the central bank at the end of the final period (lemma 2), how the term-structureofinterestratesisdetermined(lemma3),howtheintretemporaldepositsare determined(lemma4and5),andthattherearenowashsalesofcommoditiesandassetsin equilibrium(lemmas6and7). Lemma 1: Atanymonetaryequilibrium,r s∗ ,r¯ c ≥0,∀s∗ ∈S∗ andc∈C Proof. Let r s∗ < 0, then agents could infinitely arbitrage the central bank. The collateral constraint puts a bound on the demand for long-term loans, hence r¯ can be less than r c 0 in equilibrium when positive, while agents continue to borrow in both markets. However, when there is no default in any state on loan c, the collateral constraint does not bind, and 13

r¯ (cid:29)r . Otherwise,agentswouldnotborrowshort-termatt =0andthemarketwouldnot c 0 clear. Lemma 2: Noworthlesscashatend. Proof. Supposethatagenthkeepsworthlesscashattheendofperiod1,thusconstraint(9) is non-binding. Then h can borrow a little more on r , use the money to buy more coms modities in state s (leaving all his other actions unchanged), without violating constraint (9), i.e., with enough money at hand to repay the extra loan. This improves his utility, a contradiction. Lemma3: Term-structureofinterestrates. Proof. Sumequations(4)and(5)overallagentsandapplymarketclearingconditions(13), (14)and(17)toget: r M +∆¯ =∑M¯c, (19) 0 0 c where ∆¯ =∑∆¯(h). Similarly, sum equations (8) and (9) over all agents (realize that Ic is s h independentoftheidentityoftheborrowingagenth)andapplymarketclearingconditions (13),(15),(17)and(18)toget: r M +∑ fc∑µ¯c(h)=∆¯ s s s c h ⇒r M +∑ fcM¯c(1+r¯ )=∆¯ ∀s∈S (20) s s s c c Combineequations(19)and(20)toget r M +r M +∑[fcM¯c(1+r¯ )−M¯c]=0 ∀s∈S. (21) 0 0 s s s c c 14

Lemma4: Supposer >0andr >0,∀s∈S. Then,∆ (h)=0,∀s∈Sandh∈H;∆ (h)=0 0 s s 0 ifagenthborrowsshort-terminstate0;however,∆ (h)maybepositiveifagenthdoesnot 0 borrowinperiod0. Proof. Suppose that agent h hoards money within state s, i.e., ∆ (h)>0. Then, she could s have reduced her money holdings by ε, borrowed ε(1+r ) less from the short-term loan s εr s marketwithoutviolatingconstraint(8)andreducethesaleofgood(cid:96)by withoutviolatp(cid:96) s εr ing constraint (9). This results in a utility gain s ∇h , hence a contradiction with optimalp(cid:96) s(cid:96) s ity. Similarly,∆ (h)=0ifagenthborrowsontheshort-termloaninperiod0. However,if 0 agenthdoesnotborrowshort-term,i.e.,µ (h)=0,then∆h canbepositive. 0 0 Lemma5: Salesinstatesimplyshort-termborrowingbutsalesinstate0maynot. Proof. Agenthwouldnotsellins∈Sifshedoesnotborrowonr becausethesalesrevenue s for s∈S are too late for anything except repayment of loans. Att =0, an agent might sell inordertodepositlong-term. Meanwhile,shemaynotneedtoborrowshort-termsinceshe may borrow long-term to purchase goods. Note that she may choose to borrow long-term andholdcashacrossperiodsatthesametime(seelemma10). Lemma 6: No wash sales of commodities. Suppose r >> 0. Then b(cid:96)(h)q(cid:96)(h) = 0, ∀ s s h∈H,s∈S∗ and(cid:96)∈L. Proof. Suppose b(cid:96)(h)q(cid:96)(h)>0 and µ (h)>0 for some s∈S∗. Let h borrow s less on the s s s short-term loan in state s (i.e. reduce µ (h) by (1+r )ε), spend ε less on the purchase of s s (cid:96), and sell (1+r )ε/p(cid:96) less of (cid:96). This would increase her consumption of (cid:96) by εr /p(cid:96) and s s s s improveherutilitywithoutviolatingthebudgetconstraints. Note that, since we consider an active economy where all agents buy some goods in every period, they also sell some other goods given that b(cid:96)(h)q(cid:96)(h) > 0. Hence, µ (h) > 0, bes s s causetheonlyreasontosellins∈S istopayofftheloan. Suppose µ =0, but µ¯c(h)>0 for some long-term loan c and b(cid:96)(h)q(cid:96)(h)>0. Again, let 0 s s 15

h borrow ε less on the loan term loan c, spend ε less on the good (cid:96) and reduce her sales revenue from (cid:96) by (1+r¯)ε. This would improve her consumption of (cid:96) by εr¯/pl, a contras diction. Finally, if µ (h) = µ¯(h) = 0, then h does not have money to purchase anything, thus 0 b(cid:96)(h)q(cid:96)(h)=0. s s Lemma 7: No wash sales of real assets. Suppose r >> 0. Then bk(h)qk(h) = 0, for all h∈H andk∈K. Proof. Similaraslemma6. We now turn to prove how the choice of collateral requirements determines default in equilibriumandwhattheeffectoninterestratesis. Ifthemonetaryauthoritysetscollateral requirements such that there is no default in equilibrium, then all interest rates are zero (lemma 8). This leads to indeterminacy of monetary equilibria in the absence of outside money which is extensively discussed in Dubey and Geanakoplos (2006). In lemma 9, we showthatthemonetaryauthoritycanchoosecollateralrequirementstoachieveanyprofile of default it wants and in lemma 11 we show that positive default is a necessary and sufficientconditiontosupportpositiveshort-termintereststatesineverystateandperiodofthe world. Theunderlyingreasonisthatdefaultcreatesendogenouslyliquidwealth(equivalent tooutsidemoney),whichcanbedistributedbetweenperiod0andperiod1short-termloan markets. This requires that agents can take both long-term loans and deposit intertemporallyinequilibrium,whichisshowninlemma10. Lemma8: If∑γcXkp1/pk >1∀c∈C,s∈S,thenr =r =r¯ =0. k s s 0 s c k Proof. Fromequation(2),thisimpliesthatthereisnodefaultonanycontractcandstates. Thus,Ic=1forallc∈C ands∈Sandequation(21)becomesr M +r M +∑r¯ M¯c=0. s 0 0 s s c c Hence,r =r =r¯ =0forallstatessandcontractsc. 0 s c 16

Lemma 9: ∃ γc, k∈K, c∈C such that there is default on some long-term loan c in every k states. Proof. The monetary authority needs to choose collateral requirements γc for every asset k k such that ∑γcXkp1/pk < 1 for some c in any state s. Also, the collateral requirement k s s k for at least one k needs to be strictly positive for every contract c, such that demand from long-term loans is bounded by the collateral constraint (1). Given that prices are bounded, themonetaryauthoritycanalwayschooseγc >0,suchthatdefaultoccursineverystatefor k some contract. To illustrate this, suppose that K =1. Then, the monetary authority could 1 choose 0 < γ < , which yields default in every state, while k (cid:2) (cid:3) max Xksup(p1)/inf(pk) s s s demand is bounded by (1). Obviously, with complete markets the monetary authority can choosecollateralrequirementssuchthateverycontractdefaultsinexactlyonestate. Let ∇h = ∂Uh/∂c(cid:96) (h), ∀s∗ ∈ S∗,(cid:96) ∈ L,h ∈ H. Given lemma 6, we can distinguish s∗(cid:96) s∗ between goods that agent h buys and goods that she sells. K(+) s∗ (h) and K(−) s∗ (h) are definedsimilarly. (cid:110) (cid:111) DenotebyL(+) s∗ (h)= (cid:96)∈L:b(cid:96) s∗ (h)>0andq(cid:96) s∗ (h)=0 andL(−) s∗ (h)= (cid:110) (cid:111) (cid:96)∈L:b(cid:96) s∗ (h)=0andq(cid:96) s∗ (h)>0 . Moreover,denoteby(cid:96)+ s∗ (h)oneelementofL(+) s∗ (h) andby(cid:96)− s∗ (h)oneelementofL(−) s∗ (h). Lemma10: Agenthmayholdcashacrossperiodsandborrowinthelongtermcollateralizedloanmarketsimultaneously,i.e.,∆¯(h)>0andµ¯c(h)>0forsomec∈C cancoexist. Proof. If agent h takes a long-term loan c and does not default on it later, she would not carrycashacrossperiod. Otherwise,if∆¯(h)>0,sincer¯c >>r ,∀c∈C,hwouldbebetter 0 ε off reducing the amount of inventory by ε, borrowing amount more on the short- 1+r 0 ε term loan and borrowing amount less on the long term loan. In the next period, she 1+r 0 ε(1+r¯c) will receive ε amount less of inventory, while repay less on the long-term loan. (1+r ) 0 17

ε(1+r¯c) Since >>ε,agenthwillbebetteroff. (1+r ) 0 However, if h does default on the long-term loan c, she may choose to carry cash across periodsatthesametime. Toseethis,considerthefirst-orderconditionfor∆¯(h): ∇h ∇h − 0(cid:96)+ 0 (h) +(1+r )∑ s(cid:96)+ s (h) =0 for∆¯(h)>0 p (cid:96)+ 0 (h) 0 s p (cid:96) s + s (h) 0 or,ifanagentdoesnotborrowshort-termatt=0, ∇h ∇h − 0(cid:96)+ 0 (h) +∑ s(cid:96)+ s (h) =0 for∆¯(h)>0. p (cid:96)+ 0 (h) s p (cid:96) s + s (h) 0 Thefirst-orderconditionsforlong-termloansrequire: ∇h 0(cid:96)+ 0 (h) −(1+r¯ )∑ ∇h s(cid:96)+ s (h) fc−∑λ ¯k γc k =0 ∀c∈C p (cid:96)+ 0 (h) c s p s (cid:96)+ s (h) s k pk 0 Tohavetheagentborrowinglongtermandinventorycashsimultaneouslyrequires: (1+r )∑ ∇h s(cid:96)+ s (h) =(1+r¯c)∑ ∇h s(cid:96)+ s (h) fc+∑λ ¯k γc k 0 s p (cid:96) s + s (h) s p (cid:96) s + s (h) s k pk or ∑ ∇h s(cid:96)+ s (h) =(1+r¯c)∑ ∇h s(cid:96)+ s (h) fc+∑λ ¯k γc k s p (cid:96) s + s (h) s p (cid:96) s + s (h) s k pk whereλ ¯k isthemultiplierrelatedtothebindingcollateralconstraintsk. Since fc issmaller s thanonefordefaultedloans,theaboveequationcanbesatisfied. Lemma 11: ∃ interest rates r¯ >0 and γc, k∈K, c∈C such that all r and r are strictly c k 0 s positivewhen0<∆¯ <∑M¯c. c Proof. From equation (19), r >0 when ∆¯ <∑M¯c. ∆¯ is the amount of money that can be 0 c collected as seigniorage att =1 on short-term loan and long-term loans. By appropriately 18

settingthecollateralrequirementγc foreachcontractc,themonetaryauthoritycanchoose k which long-term loans that will be repaid in full, i.e., Ic = 1, such that equation (20) is s satisfied for some r ,r¯c > 0 as long as ∆¯ > 0, which can happen in equilibrium (lemma s 10). 6 Determinacy with interest rate targets The monetary authority can set collateral requirements such that there is active default in every state of the world in the last period (lemma 9). In turn, this is sufficient to support positiveinterestratesineveryperiodandstate(lemma11),whichthecentralbankchooses as well. Thus, the quantity theory of money holds and agents do no hoard money within each period while borrowing short-term (lemma 4). This resolve the indeterminacy of monetaryequilbriaasweproveintherestofthesection. The strategy of our proof is to represent a monetary equilibrium with default as the solution to a system of simultaneous equations with the same number of unknowns, and then to apply the transversality theorem to prove that “generically" the solution to this systemisazero-dimensionalmanifold. (cid:0) (cid:1) The monetary authority targets interest rates r= (r¯ c ) c∈C ,(r s∗ ) s∗∈S∗ (cid:29)0 and chooses thecollateralrequirementsγ=(γc) . k k∈K,c∈C (cid:18) (cid:19) (cid:16) (cid:17) Ourexogenousvariablesare(u)= u(cid:96) ,aswehold s∗ s∗∈0∪S,(cid:96)∈L,h∈H (cid:18) (cid:19) (cid:16) (cid:17) r,X,γ, e(cid:96) (h),ek(h) fixed. s∗ s∗∈S∗,(cid:96)∈L,k∈K,h∈H (cid:18) (cid:16) (cid:17) (cid:16) (cid:17) Ourendogenousvariablesare (M s∗ ) s∗∈S∗ ,(M¯c) c∈C , p(cid:96) s∗ , pk , s∗∈S∗,(cid:96)∈L k∈K (cid:19) (cid:16) (cid:17) c(cid:96) s∗ (h),b(cid:96) s∗ (h),bk s∗ (h),q(cid:96) s∗ (h),qk(h),µ s∗ (h),µ¯c(h),λ ¯k(h) . s∗∈S∗,(cid:96)∈L,k∈K,c∈C,h∈H (cid:18) (cid:16) (cid:17) (cid:16) (cid:17) Givenlemma2,theendogenousvariables p(cid:96) , pk , s∗ s∗∈S∗,(cid:96)∈L k∈K 19

(cid:19) (cid:16) (cid:17) (M s∗ ) s∗∈S∗ ,(M¯c) c∈C , c(cid:96) s∗ (h) will be forced by equations (13)-(15), (16), s∗∈S∗,(cid:96)∈L,h∈H (17),(18),(6),(11),(12),andwillbefunctionsoftheremainingfreeendogenousvariables. DenotethesetofthefreeendogenousvariablesbyσwithdomainD(σ). Thefreevariableswillbedeterminedbythefollowingfirstorderconditionsandtheremainingbudgetconstraints(4),(5),(8)and(9). Thefirst-orderconditionsforcommodities require: ∇ p h s (cid:96) s ∗ ∗ (cid:96) − ∇ p h s (cid:96) ∗ + s (cid:96) ∗ + s ( ∗ h ( ) h) =0 for(cid:96)∈L(+) s∗ (h)\ (cid:8) (cid:96)+ s∗ (h) (cid:9) , (22) s∗ ∇ p h s (cid:96) ∗(cid:96) − ∇h s (cid:96) ∗ − (cid:96)− s ( ∗ h ( ) h) =0 for(cid:96)∈L(−) s∗ (h)\ (cid:8) (cid:96)− s∗ (h) (cid:9) , (23) s∗ p s∗ s∗ ∇h ∇h s∗(cid:96)+(h) s∗(cid:96)−(h) (cid:96)+ s ( ∗ h) −(1+r s∗ ) (cid:96)− s ( ∗ h) =0 p s∗ p s∗ s∗ s∗ or,ifanagentdoesnotborrowshort-term, ∇h ∇h s∗(cid:96)+(h) s∗(cid:96)−(h) s∗ − s∗ =0. (24) (cid:96)+(h) (cid:96)−(h) p s∗ p s∗ s∗ s∗ Thefirst-orderconditionfor∆¯(h)requires: ∇h ∇h − s∗(cid:96)+ 0 (h) +(1+r )∑ s∗(cid:96)+ s (h) =0 for∆¯(h)>0 p (cid:96)+ 0 (h) 0 s p s (cid:96)+ s (h) 0 or,ifanagentdoesnotborrowshort-termatt =0, ∇h ∇h − s∗(cid:96)+ 0 (h) +∑ s∗(cid:96)+ s (h) =0 for∆¯(h)>0. (25) p (cid:96)+ 0 (h) s p s (cid:96)+ s (h) 0 20

Thefirst-orderconditionsforlong-termloansrequire: ∇h s∗(cid:96)+ 0 (h) −(1+r¯c)∑ ∇h s∗(cid:96)+ s (h) fc−∑λ ¯k γc k =0 ∀c∈C. (26) p (cid:96)+ 0 (h) s p (cid:96) s + s (h) s k pk 0 Thefirstorderconditionsforassetpurchasesandsales,respectively,att =0require: − p (cid:96) p + 0 k (h) ∇h s∗(cid:96)+ 0 (h) +∑ s ∇ p h s s (cid:96) ∗ + s (cid:96)+ s (h ( ) h) X s kp1 s +λ ¯k =0 fork∈K(+) s∗ (h), (27) 0 p (cid:96) p + 0 k (h) ∇h s∗(cid:96)+ 0 (h) −(1+r 0 )∑ s ∇ p h s (cid:96) s ∗ + s (cid:96)+ s (h ( ) h) X s kp1 s −λ ¯k =0 fork∈K(−) s∗ (h). (28) 0 Finally,thecomplementarityslacknessconditionsrequire: (cid:20) bk(h) γc(cid:21) λ ¯k ek(h)−qk(h)+ −∑µ¯c(h) k =0 ∀k∈K. (29) pk pk c Thefollowingtablematchesfreevariableswithequations: Freevariable Equation b(cid:96) ,q(cid:96) (cid:96)∈/ (cid:96)+(h),(cid:96)−(h) Equations(22),(23) s∗ s∗ s∗ s∗ q(cid:96) (cid:96)∈(cid:96)−(h) Equations(24) s∗ s∗ b(cid:96) (cid:96)∈(cid:96)+(h) Equations(4),(8) s∗ s∗ qk,bk Equations(27),(28) 0 0 µ s∗ Equations(5),(9)orµ 0 (h)=0 µ¯c Equations(26) λ ¯k,k∈K Equation(29) ∆¯ Equations(25)or∆¯ =0 ∆ s∗ Lemma4orequation(5) The space Uh of utilities of agent h ∈ H consists of all linear perturbations of some fixedutilityu¯h :RTL →R,. i.e.,uh(c)=u¯h(c)+δ·c,whereδ∈RTL. LetU=× Uh. + + h∈H Consider a matching of the free variables in σ to equations, as in table 6. Consider the map ψ : U×D → Rd given by ψ(u,σ) = LHS of the d equations in the matching. If σ ∈ D is an active equilibrium of u ∈ U, then ψ(u,σ) = 0. i.e., σ ∈ ψ−1(0) where u 21

ψ ≡ψ(u,σ). Since dimension D=d, ψ−1(0) will be a zero-dimensional manifold prou u vided that ψ : D → Rd is transverse to 0. This follows from the transversality theorem u for almost all u ∈ U if each map ψ is transverse to 0. Theorem 1 proves that this is the case when the monetary authority sets strictly positive interest rates and chooses collateral requirementssuchthatthereisdefaultonsomelong-termloanineverystateinperiod1. Theorem1: Assumethatforeveryagenth,andforthecollateralizedloansandrealassets traded by the agent (denoted by C(h) and K(h)), the real payoff of a riskless bond 1/p, the real payoff of the long-term collateralized loans fc/p,c∈C(h) and the real payoff of the asset Xkp1/p,k ∈K(h) are linearlly independent. The set of equilibrium outcomes is s determinate for generic u in U when the monetary authority targets positive interest rates r = (cid:0) (r¯c) c∈C ,(r s∗ ) s∗∈S (cid:1) (cid:29)0 and chooses collateral requirements γc k c∈C such that there isdefaultinatleastonecontractcineverystates. Proof. To perturb (22) or (23), adjust ∇h (cid:96)∈/ (cid:96)+(h),(cid:96)−(h), and leave all other equations s∗(cid:96) s∗ s∗ undisturbed. To perturb (24), adjust ∇h (cid:96)∈(cid:96)−(h), which disturbs (23). To restore (23), adjust ∇h s∗(cid:96) s∗ s∗(cid:96) (cid:96)∈L(−) s∗ (h)\ (cid:8) (cid:96)− s∗ (h) (cid:9) ,whichleavesallotherequationsundisturbed. Next consider the set of first-order conditions for inventory, the long loan, and the assets. Note, following lemma (10), (25) and (26) can either invoked together or not. 1 Recall that the vectors , fc/p,c ∈ C(h) and Xkp1/p,k ∈ K(h) are linearlly indepenp s dent. Therefore, we can adjust ∇h /p (cid:96)+ s (h) in a direction perpendicular to all but one s∗(cid:96)+ s (h) s of these vectors, and thereby unilaterally perturb any one of the equations from this set. In the process (22) or (24) will be disturbed. We restore these by adjusting ∇h , for s∗(cid:96) (cid:96) ∈ L(+) s∗ (h)\ (cid:8) (cid:96)+ s∗ (h) (cid:9) and ∇h s∗(cid:96)− s (h) . The latter further disturb (23), which is restored via∇h s∗(cid:96) ,for(cid:96)∈L(−) s∗ (h)\ (cid:8) (cid:96)− s∗ (h) (cid:9) To perturb (7), adjust γc without affecting Ic. This disturbs (22) and (22). Adjust as k s describedabove. 22

ε To perturb (4), increase µ (h) by ε>0, thus h spends more on his good. Then, 0 1+r 0 (5)isdistortedforsomeotheragentsh∈H ,whoreceiveεh moremoney. Assumethatone ε agenttakeitall,thusεh = . Inthefollowing,weproceedbyconsideringtwocases: 1+r 0 Case 1: assume that the agent h increases his borrowing, which means that she can ε increase her purchases by . Continuing with this logic and assuming that every (1+r )2 0 ∞ ε 1+r subsequent agent borrows at r , the total increase in borrowing is ∑ = ε 0 . 0 (1+r )t r t 0 0 ε Thus,totalexpendituresongoodsgoupby ,whichisfinite. r 0 Case 2: alternatively, assume that the agent h does not borrow at r . Then, she would 0 carry the amount εh into the next period and will increase her purchases of goods by εh. Then, (9) would be distorted for some agents, who receive εh more money. Assume that ε oneagenth(cid:48) againtakeitall,thusεh(cid:48) = . Agenth(cid:48) willincreaseherborrowinginthe 1+r s same way as described in case 1. Note that agent h(cid:48) as well as all subsequent agents will borrow,followinglemma(5). Itfollowsthattheforcedvariables p(cid:96),c(cid:96)(h)alsochangeinfinitelysmallforall(cid:96)∈L,h∈H. s s ∇h Perturbingutilitiesasabove,wecanrestoretheoldratios s∗(cid:96). p(cid:96) s∗ Theproofsfor(5),(8)and(9)areonthesameline. Toperturb(29),adjustµ¯c forsomecwhichperturbs(4),and(8)or(22)or(23)(through (11)and(12)). Adjustthewaydescribedabove. Thusweseethatthemapψisindeedtransverseto0,provingthedeterminacyofactive monetaryequilibriumoutcomes. 7 Determinacy with money supply targets Instead of targeting interest rates, the monetary authority could target the money supplies (cid:0) (M s∗ ) s∗∈S∗ ,(M¯c) c∈C (cid:1) andlettheinterestratesbeendogenouslydeterminedfromthemarket clearing condition (17) and (18). The remaining variables and optimality condition are the same as in interest rate targeting. If short-term interest rates r s∗ are positive in equi- 23

librium in every state, then the analysis in Theorem 1 goes through and the equilibrium outcomes are determinate. However, we also need to account for the possibility that shortterminterestratesarezero,i.e.,intratemporalmoneyholding∆ s∗ (h)cannotbedetermined fromlemma4. We show in lemma 12 below that collateral requirements exist such that short-term interest rates in all states s ∈ S at t = 1 are positive. Hence, ∆ (h)·µ (h) = 0 and the only s s indeterminacyisduetotheinitialpricelevel,i.e.,∆ (h)cannotbedeterminedwhenr =0. 0 0 We,then,discussawaytodeterminetheinitialpricelevel. Lemma12: ∃interestratesr¯ ≥0andγc,k∈K,c∈C suchthatallr arestrictlypositive c k s whenr =0. 0 Proof. Fromequation(19),r =0requiresthat∆¯ =∑M¯c. Then,(21)impliesthat: 0 c r M +∑ fcM¯c(1+r¯ )=∑M¯c s s s c c c ∑λ r M +∑∑λ fcM¯c(1+r¯ )=∑∑λ M¯c, (30) s s s s s c s s c s c s ∇h where λ = s∗(cid:96)+ s (h) . Combining the first order condition (25) and (26) for ∆¯(h) and µ¯c(h) s p (cid:96)+ s (h) s yields: ∑λ fc(1+r¯ )+∑λ ¯kγc(1+r¯ )/pk =∑λ s s c k c s s k s ∑∑λ fcM¯c(1+r¯ )+∑∑λ ¯kγcM¯c(1+r¯ )/pk =∑∑λ M¯c. (31) s s c k c s c s c k c s Combining(30)and(31),wegetthat ∑λ r M =∑∑λ ¯kγcM¯c(1+r¯ )/pk >0. (32) s s s k c s c k 24

If r = 0 for some state s, then the monetary authority can choose a different collateral s requirement to reduce the interest rate r¯ that an agent is willing to pay for some constract c c that delivers fully in that state s. In the extreme, the monetary authority can choose collateral requirements such that all contracts c∈C default in that state, thus the shortfall intherepaymentoflong-termdebtaccruestotheshort-termmoneymarket. It remains to pin down the price level at t =0. In particular, we need a way to unilaterally perturb the period 0 budget constraints of agents that choose to hoard cash within theperiod. The presenceof privatemonetaryendowments (outsidemoney)plays thisrole. The period 0 budget constraints can be perturb by varying the level of outside money for each agents that hoards liquidity, while all other equations are not disturbed. We refer the reader to Dubey and Geanakoplos (2006) for a detailed analysis given that our focus has been to show the determinacy of monetary equilibria in the absence of outside money. Note, however, that we require outside money only in the initial period, while Dubey and Geanakoplos (2006) also need outside money in every future state sl when the monetary authoritytargetsthemoneysupply. 8 Conclusions We examine how the credit risk that the monetary authority undertakes in its operations relates to the determinacy of monetary equilibria. Our model features trade in fiat money, real assets and a monetary authority that either injects money into the economy through short-term and long-term loans to agents, or sets interest rates. Short-term loans are safe, but long-term loans are collateralized by a portfolio of real assets and are subject to credit risk,i.e.,agentscanchoosetodefaultifthevalueofthecollateralislessthanthepromised loanrepayment. If the monetary authority chooses collateral requirements such that there is no default inequilibriumallinterestratesarezero. Asaresult,theamountofmoneyusedbyagentsto 25

purchase goods and assets cannot be pinned down by monetary policy and the equilibrium outcomes manifest indeterminacy. Alternatively, the monetary authority can set collateral requirementsonlong-termloanssuchthatthereisdefaultonsomecontractineveryfuture state of the world. The shortfall for the central bank endogenously creates private liquid wealth for agents, which can be used for trade. Eventually, all money will end up with the central bank, thus the presence of private liquid wealth can support positive short-term interest rates. Consequently, agents do not hold “idle" cash in equilibrium, a non-trivial quantitytheoryofmoneyobtainsandtheequilibriumisdeterminate. Thecentralbankmakeslossesonitslong-termoperations,butmakesequalprofitsfrom seigniorage on its short-term ones. Nevertheless, its policy is non-Ricardian not only over time, but also across loan markets. The latter is a novel feature of our analysis, which is a consequenceofdefault. Absentdefault,non-Ricardianpolicyrequiresthatthecentralbank injectsoutside money inthe economyin theform ofprivate monetaryendowments, which arefreeandclearofanyoffsettingobligation. Our analysis is focused on how the monetary authority can determine the equilibrium pricelevel. Suchapolicywillalsoaffecttherealequilibriumallocationsunlesstheindeterminacywaspurelynominaltobeginwithandtheeconomywaspopulatedbyarepresentative agent. The classical dichotomy between the real and monetary sectors breaks with the resolution of real and nominal indeterminacy. However, we have not addressed whether theundertakingofcreditriskinthecentralbanksmonetaryoperationsisanoptimalpolicy. Goodhart, Tsomocos and Vardoulakis (2010), Lin, Tsomocos and Vardoulakis (2014) and Peiris and Vardoulakis (2014) present monetary models with collateralized debt where the monetarypolicyinteractswithdefaultandaffectsequilibriumallocationand,thus,welfare. 26

References Adão, Bernardino, Isabel Correia and Perdo Teles (2014), ‘Short and long interest rate targets’,JournalofMonetaryEconomics66,95–107. Balasko, Yves and David Cass (1989), ‘The structure of financial equilibrium with exogenousyields: thecaseofincompletemarkets’,Econometrica57(1),135–162. Bloise, Gaetano, Jacques H. Drèze and Herakles Polemarchakis (2005), ‘Monetary equilibriaoveraninfinitehorizon’,EconomicTheory25,51–74. Buiter, Willem H. (2002), ‘The fallacy of the fiscal theory of the price level: A critique’, EconomicJournal112(481),459–480. Cochrane, John H. (2001), ‘Long-term debt and optimal policy in the fiscal theory of the pricelevel’,Econometrica69(1),69–116. Drèze, Jacques H. and Herakles Polemarchakis (2000), Monetary equilibria, in G.Debreu, W.NeufeindandW.Trockel,eds,‘EconomicEssays: AFestschriftforWernerHildenbrand’,Springer,pp.83–108. Dubey, Pradeep and John Geanakoplos (1992), ‘The value of money in a finite-horizon economy: A role for banks’, in P. Dasgupta, D. Gale, D. Hart, and E. Maskin (eds.), Economic Analysis of Markets and Games, Essays in Honor of Frank Hahn. Cambridge,MA:MITPresspp.407–444. Dubey,PradeepandJohnGeanakoplos(2003a),‘Insideandoutsidemoney,gainstotrade, andis-lm’,EconomicTheory21(2),347–397. Dubey,PradeepandJohnGeanakoplos(2003b),‘Monetaryequilibriumwithmissingmarkets’,JournalofMathematicalEconomics39(5-6),585–618. Dubey,PradeepandJohnGeanakoplos(2006),‘Determinacywithnominalassetsandoutsidemoney’,EconomicTheory27(1),79–106. 27

Espinoza, Raphael A., Charles A.E. Goodhart and Dimitrios P. Tsomocos (2008), ‘State prices,liquidity,anddefault’,EconomicTheory39(2),177–194. Espinoza, Raphael A. and Dimitrios P. Tsomocos (2014), ‘Monetary transaction costs and thetermpremium’,EconomicTheory,forthcoming. Geanakoplos, John and Andreu Mas-Colell (1989), ‘Real indeterminacy with financial assets’,JournalofEconomicTheory47(1),22–38. Geanakoplos, John and William R. Zame (2013), ‘Collateral equilibrium, I: a basic framework’,EconomicTheory56(3),443–492. Giraud, Gaël and Dimitrios P. Tsomocos (2010), ‘Nominal uniqueness and money nonneutralityinthelimit-priceexchangeprocess’,EconomicTheory45(1-2),303–348. Goodhart, Charles A.E., Dimitrios P. Tsomocos and Alexandros P. Vardoulakis (2010), ‘Modelling a housing and mortgage crisis’, in Financial Stability, Monetary Policy, andCentralBanking,editedbyRodrigoA.Alfaro.CentralBankofChile. Goodhart, Charles A.E, Pojanart Sunirand and Dimitrios P. Tsomocos (2005), ‘A risk assessmentmodelforbanks’,AnnalsofFinance1(2),197–224. Goodhart, Charles A.E, Pojanart Sunirand and Dimitrios P. Tsomocos (2006), ‘A model to analysefinancialfragility’,EconomicTheory27(1),107–142. Goodhart, Charles A.E, Udara Peiris, Dimitrios P. Tsomocos and Alexandros P. Vardoulakis (2010), ‘On dividend restrictions and the collapse of the interbank market’, AnnalsofFinance6(4),455–473. Grandmont,Jean-Michel(1985),MoneyandValue,CambridgeUniversityPress. Gurley, John G. and Edward S. Shaw (1960), ‘Money in a Theory of Finance’, Brookings Institution,Washington,D.C.. 28

Lin, Li, Dimitrios P. Tsomocos and Alexandros P. Vardoulakis (2014), ‘Debt deflation effectsofmonetarypolicy’,FinanceandEconomicsDiscussionSeries2014-37.Board ofGovernorsoftheFederalReserveSystem. Lucas, Robert E. and Nancy L. Stokey (1987), ‘Money and rates of interest in a cash-inadvanceeconomy’,Econometrica55(3),491–513. Magill, Michael and Martine Quinzii (1992), ‘Real effects of money in general equilibrium’,JournalofMathematicalEconomics21(4),301–342. Magill, Michael and Martine Quinzii (2014a), ‘Anchoring expectations of inflation’, JournalofMathematicalEconomics50,86–105. Magill, Michael and Martine Quinzii (2014b), ‘Term structure and forward guidance as instrumentsofmonetarypolicy’,EconomicTheory56(1),1–32. McMahon, Michael, Udara Peiris and Herakles Polemarchakis (2014), ‘Perils of quantitativeeasing’,WarwickWP. Peiris,UdaraandAlexandrosP.Vardoulakis(2014),‘Collateralandtheefficiencyofmonetarypolicy’,workingpaper. Peiris,UdaraandHeraklesPolemarchakis(2014),‘Monetarypolicyandquantitativeeasing inanopeneconomy: Prices,exchangeratesandriskpremia’,WarwickWP. Shubik, Martin and Dimitrios P. Tsomocos (1992), ‘A strategic market game with a mutual bank with fractional reserves and redemption in gold’, Journal of Economics 55(2),123–150. Sims,ChristopherA.(1994),‘Asimplemodelforthestudyofthedeterminationoftheprice level and the interaction of monetary and fiscal policy’, Economic Theory 4(3), 381– 399. 29

Tsenova,Tsvetomira(2014),‘Internationalmonetarytransmissionwithbankheterogeneity anddefaultrisk’,AnnalsofFinance10(2),217–241. Tsomocos, Dimitrios P. (1996), ‘Essays on Monetary Theory, Banking and General EconomicEquilibrium’,Ph.DThesis,YaleUniversity. Tsomocos, Dimitrios P. (2003), ‘Equilibrium analysis, banking and financial instability’, JournalofMathematicalEconomics39(5-6),619–655. Tsomocos, Dimitrios P. (2008), ‘Generic determinacy and money non-neutrality of internationalmonetaryequilibria’,JournalofMathematicalEconomics44(7-8),866–887. Woodford, Michael (1994), ‘Monetary policy and price level determinacy in a cash-inadvanceeconomy’,EconomicTheory4(3),345–380. 30