100% money as �nancial stability

regulation

David Lindequist

University of Mannheim

March 17, 2014

This paper discusses and models Irving Fisher's (1936) notion of 100%

money as a means of �nancial stability regulation. Extending the model in

Stein (2012) to capture both a universal banking system and a system of

100% money, I analyze how these two banking systems compare in terms

of e�ciency and �nancial stability. Similar to Stein (2012), I �nd that the

universal banking system may issue too much money leaving the system prone

to crisis. Additionally, universal banks may fail to hold enough liquidity in

times of crises which leads to ine�ciency. This case is found to be especially

likely in the presence of governmental deposit insurance. A system of 100%

money implies larger liquidity creation than the universal banking system.

Furthermore, the amount of money issuance under 100% money is always

optimal. However, 100% money is neither able to preclude the lack of liquidity

in times of crises. My results call for the implementation of a minimum

liquidity reserve requirement for banks as a potential ex ante arrangement to

achieve e�ciency. Apart from that, ex post e�ciency may be restored if the

central bank acts as a lender of last resort in times of crises.

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Contents

1. Introduction: the �nancial system and economic instability 3

2. 100% money, dual banking and various supporters 5

3. Related literature 9

4. Main results 11

5. Model: Preliminaries 12

5.1. Modifying the role of patient investors . . . . . . . . . . . . . . . . . . . . . 125.2. Model setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.3. General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

6. Model I: Universal banking 14

6.1. How to pay out short-term creditors . . . . . . . . . . . . . . . . . . . . . . 146.2. Upper bound for private money creation . . . . . . . . . . . . . . . . . . . . 156.3. Bank behavior on asset market . . . . . . . . . . . . . . . . . . . . . . . . . 166.4. Banks' maximization problem in period 0 . . . . . . . . . . . . . . . . . . . . 206.5. Equilbrium on asset market . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.6. Social planner solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.7. Sources of ine�ciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.8. The danger of too little liquidity and potential remedies . . . . . . . . . . . . 29

7. Model II: Dual banking and 100% money 31

7.1. Universal banking vs. dual banking . . . . . . . . . . . . . . . . . . . . . . . 317.2. Household's maximization problem . . . . . . . . . . . . . . . . . . . . . . . 327.3. Dual banking and e�ciency . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

8. Concluding remarks 35

Appendix 37

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1. Introduction: the �nancial system and economicinstability

The recent �nancial crisis starting in 2008 has resuscitated interest in the interaction be-tween the �nancial system and the real economy. Particularly, the fact that a relatively smallshock on the subprime housing market led to a massive �nancial crises with severe real ef-fects challenges economists to come up with models reasonably linking the �nancial with thereal sphere of the economy. This task has been neglected for many years as the economicsresearch agenda was shaped by the work of Modigliani and Miller (1958) who argued that�nancial structure was basically irrelevant for real economic activity, and the work by Malkieland Fama (1970) famously promoting the idea of e�cient �nancial markets. Departing fromthis line of thought, Bernanke, Gertler and Gilchrist (1999) developed their famous ��nancialaccelerator" model which lies at the heart of many modern DSGE models including �nancialmarkets. They introduce credit-market frictions into a general equilbrium model and showhow a worsening in credit market conditions may a�ect the real economy. Their model relieson changes in the net worth of potential borrowers. This net worth is procyclical (as assetprices and pro�ts are procyclical) and propagates changes in economic activity due to changesin credit market conditions. The model of Kiyotaki and Moore (1997) further explores thisavenue of research. They develop a model in which assets are both factors of productionand serve as collateral for loans. Borrowers' credit limits are then endogeneously a�ected bychanges in the price of assets. An initial temporary negative shock a�ects the net worth ofconstrained and leveraged borrowers over several periods leading to persistence of the tem-porary shock. This persistence leads to an even further decline of asset prices causing anampli�cation of the initial shock.

Both these contributions are cornerstones in clarifying the role �nancial markets play in prop-agating and amplifying real shocks1. However, they do not comprise an explanation of howthe �nancial system itself may cause shocks a�ecting the real economy. Thus, the role of the�nancial system not just as a propagator and ampli�er of shocks, but as a source of shocksremains unaddressed. Schularick and Taylor (2012) make a strong point for the developmentof such a theory. They explore �nancial crises from 1870-2008 and �nd that �past growth ofcredit emerges as the single best predictor of future �nancial instability" (p. 1057). They �ndmost �nancial crises to be �credit booms gone bust�.2 Schularick and Taylor, based on theirempirical investigation into 130 years of �nancial crises, sympathize with the idea that �thecredit system may not be merely an ampli�er of economic shocks [...] but is quite capableof creating its very own shocks� (p. 1058).3 Further elaborating on this view, the challenge

1See Krishnamurthy (2010) for a deeper discussion of balance sheet and information multipliers.2In a famous study on the Great Depression starting in 1929, Eichengreen and Mitchener (2003) also�nd that the credit boom view might very well explain the formation of the 1920s boom and bust.

3Kindleberger (2011) points in the same direction when stating that �the pattern [of �nancial crises] wasthat investor optimism increased as economies expanded, the rate of growth of credit increased andeconomic growth accelerated [...] The increase in the supply of credit [...] led to economic booms asinvestment spending increased in response to the more optimistic outlook and the greater availabilityof credit� (p. 275).

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is to develop models which feature �nancial markets endogenously prone to instability andgenerating shocks a�ecting the real economy.

There have been some attempts at constructing such models.4 In times of �nancial crises,commentators and media frequently refer to Minsky (1992) and the �nancial instability hy-pothesis he put forward. Minsky distinguishes between di�erent forms of �nancing (hedge,speculative and Ponzi �nance) and argues that during the business cycle, the �nancial regimeendogeneously switches from stable �nancing to highly speculative �nancing regimes �nallyleading to instability and �nancial distress. In a remarkable e�ort to mimic the main featuresof the current �nancial crisis, Gorton and Ordonez (2014) develop a theory of �nancial criseswhich is based on the existence of informationally-insensitive debt, namely short-term debtclaims. In their model, an economy relying on such informationally-insensitive debt witnessesa credit boom as �rms with low quality collateral can borrow against their collateral. A crisisis triggered if agents start gathering information on debt claims. They are induced to do so incase of a possibly small exogenous shock rendering some good collateral into bad collateral.Gorton and Ordonez refer to that mechanism as a �sudden informational regime switch�. Ina regime of informationally-sensitive debt, fewer borrowers can borrow and a credit crunchmay arise. The sharpness of the economic downturn crucially depends on how long debt hasbeen informationally-insensitive. Thus, in the setup of Gorton and Ordonez, �nancial fragilitybuilds up endogenously.

In a recent series of theoretical papers, Jeremy Stein (2012) and his coauthors5 explorethe interrelatedness of monetary policy and �nancial stability. Stein (2012) develops a modelof excessive private money creation causing an externality which renders the unregulated out-come of private money creation ine�cient. He explicitly refers to his model as being ableto show that �unregulated banks may engage in excessive money creation and may leave the�nancial system overly vulnerable to costly crises� (p.3). In a second step, he shows how con-ventional monetary policy tools may be used to regulate this externality in a cap-and-tradesystem of money creation.

Just as the �nancial crisis starting in 2008 provoked a variety of research on its causes andtriggers, the Great Depression starting in 1929 served as a starting point for research intobusiness cycle dynamics and economic downturns. The most prominent work written duringthat time certainly is Keynes' General Theory (1936) which massively in�uenced the wayeconomists after WWII thought about business cycle �uctuations. However, there were othercontributions. In 1936, the famous economist Irving Fisher published a book called �100%Money� in which he presented his view on business cycle dynamics and the basic mechanismsof economic crises. Fisher argues that over-indebtedness and de�ation lie at the heart of

4As yet, theoretical research on how the �nancial system itself generates economic instability andbusiness cycles is rather heterodox. Only very few articles concerned with such issues are wellpublished and generally accepted as valuable contributions to economic research.

5See Greenwood, Hanson and Stein (2010) and Kashyap and Stein (2012).

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economic crises.6 His basic argument is that in a situation of over-indebtedness, individualstrying to lower their debt burden cause the price level to decrease which increases the realdebt burden. A debt-de�ation spiral kicks in and the economy undergoes a crisis. Fisher de-scribes endogenous �nancial instability being rooted in a malfunctioning banking system. Heattributes both origin and ampli�cation of debt-de�ation dynamics to the fractional reservebanking system. Under a fractional reserve system, banks are very �exible in their moneyissuance. As a result, in times of widespread optimism they might cause a situation of over-indebtedness by granting many loans. If over-indebted individuals then try to pay back theirloans, �check-book money� (money created under fractional reserve banking) is destroyedwhich puts pressure on the price level and may start a de�ation spiral. Fisher argues that asystem where private money creation was forbidden, i.e. where the fractional reserve ratiowould be 100%, is able to dampen business cycles7.

2. 100% money, dual banking and various supporters

Irving Fisher was not the �rst economist promoting the idea of implementing a system of fullreserve banking. Interestingly, the �rst elaborated treatment of 100% money can be foundin Soddy (1926). Frederick Soddy was a Nobel prize winner in chemistry and started todevelop various lines of heterodox research in economic issues. Out of the economists circle,Knight (1933) and Simons (1933) were famous early proponents of such a reform proposaland were both in�uenced by and in�uencing the work of Fisher8. Alongside Irving Fisher,famous Chicago economist Frank Knight was one of the most prominent proponents of fullreserve banking. Together with some colleagues from Chicago University, he wrote a shortmemorandum on banking reform which later became known as the �Chicago Plan� (see Knight(1933)). This plan basically coincides with the conception of Fisher's 100% money.

All these authors share the opinion that a fractional reserve system leads to �nancial in-stability and that full reserve banking may help to stabilize the �nancial system. Simons(1948) addresses the key problem of fractional reserve banking in the following way:

There is likely to be extreme economic instability under any �nancial systemwhere the same funds are made to serve at once as investment funds forindustry and trade and as the liquid cash reserves of individuals. Our �nancialstructure has been built largely on the illusion that funds can at the same timebe both available and invested. (cited in Phillips (1992))

Fisher (1936) distinguishes between �check-book money� and �pocket-book money�. The �rstis money that is created when banks grant a loan, while the second is physical money. Fisher

6This theory later became known as the Fisherian Debt-De�ation. After the �nancial crises from 2008,this concept became more visible in research again, see for example Brunnermeier and Sannikov(2011) and Eggertson and Krugman (2012).

7In the following, such a banking system will be called 100% money, full reserve banking or 100%banking. This variety of expressions re�ects the variety of terms used in the literature on this reformproposal.

8See Phillips (1992) for a good overview on di�erent proposal and the corresponding history of thought.

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argues that the supply of money is unstable given that �check-book money� is destroyedwhenever a bank calls a loan. In fact, his main explanation of the Great Depression startingin 1929 is shrinkage of the money supply as a consequence of banks neither prolonging oldnor awarding new loans. Fisher (1936) �nds it unacceptable that the �national circulatingmedium is now at the mercy of loan transactions of banks� (p. 13). According to his view,money supply is an issue of national interest and should thus be performed by governmentalauthorities only. Fisher proposes to establish a �currency comission� which, in a �rst step,transforms the current fractional reserve system into a full reserve system by buying assetsfrom banks with newly created money. Banks would have to use this money to meet the100% reserve requirement. After that, banks would be permanently required to hold a cashreserve of 100% against its demand deposits: �The checking deposit department of the bankwould become a mere storage warehouse for bearer money belonging to its depositors.� (p.15) In order to keep the supply of credit alive, Fisher has some sort of dual banking in mind.Alongside deposit banks, there would be loans banks which may lend credit either from theirown capital, from money repaid on maturing loans or from money received by customerswilling to lend money to the bank. Deposit banks would not be allowed to grant credit asthey are required to have the money from their customers always readily available. Fisherpoints out that such a system would prevent banks from creating money: �No money couldbe lent unless there was money to lend; that is, the banks could no langer overlending bymanufacturing money out of thin air so as to cause in�ation and a boom.� (p. 19) Hence,Fisher is convinced that 100% money will eliminate great in�ations and de�ations as thevolume of circulating medium of exchange would be una�ected by increases or decreases inbanks lending activity (which may be overly pessimistsic or optimistic). As a result, boomsand depressions would be dampened since Fisher regards in�ation and de�ation as majorsources of economic �uctuations and crises. Apart from that, there will not be any bankruns since depositors can be sure that they will always obtain their cash from deposit bankswhich are not allowed to lend out these funds. Furthermore, implementing a system of 100%money would greatly simplify the monetary system and banking. There would be checkingand savings deposits. Depositors would always know what a bank does with their deposits.Simons (1948) nicely summarizes the envisioned banking system with 100% money:

First, there would be deposit banks which, maintaining 100 per cent reserves,simply could not fail, so far as depositors were concerned, and could not createor destroy e�etcive money. These institutions would accept deposits just aswarehouses accept goods. Their income would be derived exclusively fromservice charges [..]A second type of institution, substantially in the form of investment trust,would perform the lending functions of existing banks. Such companies wouldobtain funds for lending by sale of their own stock; and their ability to makeloans would be limited by the amount of funds so obtained. [...] In a word,short-term lending would be managed in much the same way as long-termlending; and the creation and destruction of e�ective circulating medium byprivate institutions would be impossible. (cited in: Phillips 1992)

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Since its publication in 1936, Fisher's 100% money proposal was supported from both aca-demic and non-academic backgrounds. One of the most famous proponents is Milton Fried-man. He included a system of 100% money into his �Monetary and Fiscal Framework forEconomic Stability� (Friedman (1948)). Friedman outlines a proposal which touches uponfour main elements of governmental activity: the monetary system, government expenditureson goods and services, transfer payments, and tax structure. His main purpose is to precludediscretionary action by governmental authorities which might interfere with market outcomesleading to ine�cient allocations. To achieve this, Friedman proposes a strictly rule-basedsystem of government activity. With regard to the monetary system, he wants to �eliminateboth the private creation or destruction of money and discretionary control of the quantityof money by central bank authority� (p. 247). To do so, Friedman suggests separating thedepositary from the lending function of the banking system - just like Fisher (1936) proposes.In addition to this redesigning of the monetary system, Friedman recommends a �scal policywhich should not react to cyclical �uctuations in business activity. Neither should the tax northe transfer system do. Instead, government expenditures which are �xed at a certain levelare �nanced by either tax revenues or the creation of money if necessary9. Thus, Friedmanuses the concept of 100% money as a part of a broader framework for governmental activity.Technically, he proposes the same reform as Fisher. However, with regard to the motivationto implement such a system, Friedman greatly di�ers from Fisher. Friedman aims at pre-cluding discretionary governmental interference into market outcomes, while Fisher aims atstabilizing an ine�cient monetary system.

In the aftermath of the current �nancial crisis, interest in alternative monetary systems re-gained momentum. In Germany, for example, Joseph Huber (2013) propagates a system of100% money which he calls �Vollgeld� (full money). Like the proposals of Fisher and Fried-man, Huber stands in the tradition of the currency school which, in contrast to the bankingschool, argues in favor of central banks as the only emitters of money.10 However, Huber'sconcept implies the abolishment of both reserves and the distinction between check moneyand cash. Rather, all kinds of money would be �Vollgeld�. This new form of money wouldbe passed to the public by government expenditures which are �nanced by newly created�Vollgeld�. Money would stop being a liability for banks, but would be held as a liquid asset.That is, money would no longer be created as debt.

As a direct response to the current �nancial crisis, Laurence Kotliko� (2010) proposes theimplementation of limited purpose banking which is a variant of the 100% money proposalby Irving Fisher.11 Kotliko� de�nes his reform proposal as a �very low-cost change to our�nancial system, which limits banks to their legitimate purpose, namely connecting (interme-diating between) borrowers and lenders and savers and investors� (Kotliko� 2010, p. 123).His basic idea is to transform all banks (which he de�nes as �nancial and insurance compa-

9�De�cits or surpluses in the government budget would be re�ected dollar for dollar in changes in thequantity of money.�(p. 251)

10See Daugherty (1942) for a detailed discussion of di�erences between the banking school and thecurrency school.

11See also Chamley, Kotliko� and Polemarchakis (2012).

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nies with limited liability) into pass-through mutual fund companies. After the reform, bankswould sell various mutual funds, both safe as well as risky ones12. Banks themselves wouldnot invest into these mutual funds. Importantly, one type of mutual funds would be a cashmutual funds. These are funds that only hold cash and thus represent the demand depositsunder limited purpose banking. Note that a cash mutual fund basically is a �bank� holding100% reserves on checking deposits.Kotliko� (2010) most clearly describes the di�erence between universal banking as it is prac-ticed now and a dual banking system in which deposit banks (or cash mutual funds equiv-alently) operate under a system of full reserves. Under limited purpose banking13, saversthemselves would decide into which assets to invest depending on their risk appetite, whereasin a universal banking system generally banks decide about how to invest the funds theyobtain from depositors14. My understanding of both universal banking and dual banking with100% money quite closely follows that of Kotliko� (2010).

Dual banking has a prominent historical record. In the US, the Glass-Steagall Legislationwas in force from 1932 to 1999 and basically stipulated the separation of deposit businessand loan granting (�commercial banking�) on the one hand and securities business (�invest-ment banking�) on the other hand. The main argument for introducing such a regulationwas the notion that con�icts of interest arise when commercial banks are also engaged intosecurities business as a universal bank may fool its customers to invest into securities of lowquality15. Another concern after the impressions from the Great Depression was that bankstook too much risk in investment activities with the money from deposit banking activities.Both these concern led to the establishment of a dual banking system which was abolishedin 1999 by the Gramm-Leach-Bliley Act. Under the impression of the current �nancial crisis,there have been attempts in several industrialized countries to revive at least parts of theGlass-Steagall legislation. The �Dodd-Frank Act� in the US was a direct reaction to the re-cent �nancial crisis and was enacted in 2010. Besides changing the regulatory structure, itlimits the amount of risk banks can take when trading for own account. According to the socalled Volcker Rule (named after former Fed Chairman Paul Volcker), proprietary trading andinvestments into hedge-funds or private equity funds and other risky investments are limited.In Europe, the so called �Liikanen report� (elaborated by a group of experts arranged by theEuropean Commission) recommends that banks' proprietary trading should be assigned to aseparate entity (amongst other recommendations, see Liikanen 2012). This would establisha reduced version of a dual banking system16.Note that neither the original Glass-Steagall Act nor the Volcker Rule or the recommendations

12�These mutual funds include traded equity funds, private equity funds, real estate investment trusts,commercial paper funds, private mortgage funds, credit card debt funds, junk bond funds, fonds thatinvest in put options in U.S. Treasuries, in�ation-indexed bond funds, currency funds - you name it�(p. 126).

13Sometimes limited purpose banking is also referred to as �narrow banking�.14As Kotliko� puts it: �Under limited purpose banking, banks would let us gamble, but they would not

themselves gamble.� (p. 126)15See Kroszner and Rajan (1994) for a short description of the reasoning underlying the Glass-Steagall

legislation.16In Germany, for example, such a reduced dual banking system was enacted in 2013.

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of the Liikanen report meet the requirements of a dual banking system outlined by Kotliko�(2010). Kotliko�'s reform proposal is much more radical as he suggest to transform everybank into a pure intermediary which issues a variety of mutual funds, but does not itselfundertke any investments. Kotliko� (2010) emphasizes that simply reenacting Glass-Steagalllegislation would not be su�cient as the nonbank/shadow bank sector would have a competi-tive advantage over the then regulated banking sector which may lead to regulatory arbitrageand would not fully stabilize the �nancial system.

3. Related literature

The literature on 100% money, limited purpose banking or monetary reform in general is notvery voluminous.17

Wallace (1996) uses the famous banking model by Diamond and Dybvig (1983) to ana-lyze what this model implies about narrow banking proposals18. He models narrow bankingas requiring banks to accomodate any pattern of withdrawal. Wallace shows in this modi�edDiamond-Dybvig model that narrow banking basically eliminates the role of banking. How-ever, Wallace misses a key feature of narrow banking proposals as in Fisher (1936) or Kotliko�(2010) since he does not model narrow banking as a dual banking system where householdsinvest into both deposit banks and investment banks. Instead, Wallace only models one bankwhich he requires to be liquid in any case and without reliance on liabilites subordinate todeposits to show that narrow banking does not lead to an improved allocation. The modelI present later on better describes the structure of the banking system envisioned by Fisherand Kotliko�.

Bossone (2001) �nds that the implementation of narrow banking would eliminate the keyfunctions and bene�ts of conventional banking. He admits that narrow banking would de-crease banks' liquidity and credit risks and that the payment system in general would be morerobust to crises. Bank runs would not happen anymore. In addition, Bossone points outthat much less regulation and supervision would be needed for narrow banks and that theproblem of too-big-too-fail reduces since banking would become deconcentrated. However,according to Bossone the disadvantages of introducing narrow banks outweigh the advantagesfrom doing so. This is for three reasons. First, he underlines the social bene�ts banks createby improving economy-wide risk sharing. Banks are able to channel funds from risk-averseimpatient individuals to risk-inclined patient �rms willing to invest. According to Bossone,this risk and maturity transformation are the most important features which will be lost undernarrow banking19. Second, joint production of deposit-taking and lending may lead to lowerprices fot both activities which increases consumer welfare. These synergetic e�ects wouldnot be existent in a narrow banking system. Third, Bossone argues in favor of private money

17This might be explained by the fact that - unlike in the 1930s - research into monetary reform issomewhat heterodox and is rarely published in prestigious journals.

18The term narrow banking is used interchangeably with 100% money and limited purpose banking.19Kotliko� (2010) quite extensively argues that this is not necessarily the case under limited purpose

banking.

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creation by banks. He explains that banks stand at the beginning of the production cycle andcan conveniently create money to lend it to �rms. Bossone fears that credit supply mightshrink under narrow banking as it is totally brought back to the central bank. On the groundof these arguments, Bossone doubts that the increase in terms of �nancial stability is justi�edto accept the e�ciency losses associated with narrow banking.

In an IMF working paper which attracted considerable attention in the media, Benes andKumhof (2012) use a calibrated DSGE model of the U.S. economy to study the impacts ofimplementing the Chicago Plan reform proposal. They base their model on the assumption ofendogenous money20. They then model the implementation of 100% money as banks beingrequired to always hold 100% reserves mt against their deposits dt. Banks can make loanslt only by using their own equity nt or by demanding treasury credit ft. It then holds thatlt + mt = ft + dt + nt with mt = dt. Under this scheme the government is able to controlseparately both the aggregate volume of credit lt and the money supplymt as the governmentdirectly controls the money supply and the amount of treasury credit. The decision on howto allocate funds to di�erent investments projects, however, is still left to banks. Benes andKumhof are able to verify all the claims given in Fisher (1936). They show that business cycle�uctuations are dampened due to much better control of increases and contractions in bankcredit. Furthermore, bank runs are eliminated and public debt is dramatically reduced. Thisreduction in government debt is due to the fact that when implementing 100% money, bankshave to acquire reserves from the treasury to back their liabilities (deposits). Governmentdebt can be dramatically reduced if the government swaps government bonds against thesereserves. Private debt may also be substantially reduced if the goverments uses part of itwindfalls revenues when implementing 100% money to buy back private debt from banks. Inaddition, Benes and Kumhof �nd longer-term output gains approaching 10 percent. Thesegains are due to both reductions of the real interest rates as a consequence of lower net debtlevels and the potential of lowering distortionary taxes as seignorage income increases. Apartfrom that, Benes and Kumhof argue that under 100% money liquidity traps are now longerpossible. The central bank can e�ectively control the aggregate quantity of broad moneywithout banks being potentially reluctant to lend more. Additionally, negative nominal inter-est rates on treasury credit during �nancial crises are possible as only banks have access to itin order to fund investment loans.

The study by Benes and Kumhof is powerful as it integrates both endogenous money and itsregulation by 100% money in a modern DSGE model calibrated to the U.S. banking system.The result the authors obtain are very strong and are clearly in favor of 100% money. How-ever, Benes and Kumhof construct their model under a key premise, namely that the bankingsystem's credit is completely funded by non-monetary liabilities in the form of government

20�The critical feature of our theoretical model is that it exhibits the key function of banks in moderneconomies, which is not their largely incidental function as �nancial intermediaries between depositorsand borrowers, but rather their central function as creators and destroyers of money. A realistic modelneeds to re�ect the fact that under the present system banks do not have to wait for depositors toappear and make funds available before they can on-lend, or intermediate, those funds. Rather, theycreate their own funds, deposits, in the act of lending.� (p. 9)

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treasury credit. These liabilities are not subject to run (which precludes bank runs from themodel), but it must be assured that these liabilites do not become �near-monies�. Near-monies are liabilities other than money which do have the same features as money and areconsidered by private agents to be as good as money. Benes and Kumhof avoid this problemby assuming that households are all the same (especially with regard to their debt levels) sothat there is no lending between private agents. As a result, there is no incentive to developor trade �near-monies� in this economy and loans are completely �nanced by treasury credit.This is clearly unrealistic. If private agents were homogenous, there were net borrowers andnet savers. Consequently, banks could also �nance loans with funds from net savers. Thenthe emergence of �near-monies� may become a real concern in a 100% money system. Benesand Kumhof argue that investment trusts which are funded by savers' equity may be a solu-tion. Investment trusts which issue debt claims against borrowers' money may not solve thisproblem as the debt claims may become �near-monies� foling the idea of 100% money.

Besides these rather extensive studies on the consequences of introducing 100% money, therehave been several contributions touching upon a speci�c aspect of the reform or the mone-tary system in general. Krainer (2011) doubts that fractional reserve banking with risk-freegovernment sponsored deposit insurance constitutes an e�cient banking system. He derivesa CAPM type model in which he shows that fractional reserve banking generates a probabilitydistribution around future income that is more volatile than the probability distribution createdby a narrow banking system. Thus, in line with Fisher (1936), Krainer �nds that fractionalreserve banking ampli�es business cycle �uctuations due to private money creation �nancingexcessive investments in good times. Singh (2009) argues that narrow banking is ine�cient ascompetitive banks have a comparative advantage over the central bank in allocating credit.In a system of 100% money the central bank would gain total control over money supplywhich Singh fears to cause ine�ciencies. Azariadis et al. (2001) show that in a dynamicgeneral equilbrium model with privately-issued liabilities (i.e. money), an optimal monetaryarrangement is given by both public and private circulating liabilites. The authors concludethat neither a system of free banking without public money nor a system of 100% moneywithout private money lead to e�ciency. A regulatory authority might restore e�ciency exante by requiring su�ciently high minimum liquidity reserves. Additionally, the central bankcan achieve ex post e�ciency by acting as a lender of last resort.

4. Main results

The model presented in the following sections speaks to the debate on 100% money in avariety of ways. First, my model indicates that universal banking may indeed lead to sociallynon-optimal excessive private money creation. Besides, my model shows that times of eco-nomic distress are also times of liquidity crises in a system of universal banking. This is dueto the fact that universal banks may fail to hold enough liquidity in times of crises. I showthat this problem becomes even more urgent if a governmental deposit insurance is in placeinducing banks to take the risk of going bankrupt in times of economic distress. Minimumliquidity requirements may solve that problem, but may themselves lead to ine�ciency.

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In contrast to universal banking, a system of 100% money guarantees an e�cient amountof private money creation. Besides, my model demonstrates that liquidity supply is largerunder 100% money than under universal banking rendering objections against 100% moneyon the basis of potentially lower liquidity creation invalid. In addition, governmental depositinsurance does not lead to any ine�ciencies under 100% money. Actually, such an insur-ance is not needed anymore since under 100% money banks can not go bankrupt in times ofeconomic distress. Nevertheless, my model demonstrates that 100% money not necessarilyimplies e�ciency. It might still be the case that liquidity holdings in the economy are too lowin times of crises leading to an ine�cient allocation of resources.

5. Model: Preliminaries

In the following, the model of Stein (2012) is modi�ed to address the implementation of adual banking system based on 100% money in deposit banking. In a �rst step, I generalizethe model to cover universal banking and then I analyze how the solution with universal bankscompares with the social planner solution. In a second step, I change the institutional settingand universal banks are split into savings banks on the one hand and deposit banks on theother. Again, I compare the outcome of such a dual banking system with the social plannersolution. Finally, the outcomes of both systems are compared with respect to e�ciency and�nancial stability.

5.1. Modifying the role of patient investors

In Stein (2012), banks raise funds from households in period 0 to invest into an early pro-duction technology. They can issue long-term claims (bonds) or short-term claims (money)to raise these funds. In case of a bad signal in period 1, banks are forced to pay out theirshort-term creditors. They can do so by selling assets (which are backed by investments intothe production technology) to patient investors who hold exogenous resources W . In mymodel, these liquidity resources W will be endogenized. When modeling the universal banksolution, I consider patient investors as part of a bank which decides in period 0 how muchto invest into the production technology and how much liquidity to carry over to period 1.When discussing the dual banking solution, patient investors are assumed to be deposit banksinto which households may put their representative good as liquidity.

5.2. Model setup

The economy lasts for three periods: 0, 1 and 2. The economy is inhabited by H householdswho are endowed with one unit of a representative good in total. Household utility is givenby

U = c0 + βE(c2) + γM (1)

12

with γ ∈ (0, 1), β ∈ (0, 1) and γ+β < 1. That is, households only get utility from consump-tion in period 0 and period 2. In addition, there is a liquidity premium on money γ. 21 Therepresentative good is perishable, i.e. households can not store it themselves. Rather, theymay deposit their resources into banks which can invest into a production technology. Therationale for inserting a private banking sector into the model arises from the introduction ofa key friction: The representative household can not directly invest into either the productiontechnology nor money22. Only banks can issue money (or bonds) to raise funds and investresources into the production technology.

Period 1 serves as an intermediate phase of production which is the rationale for neglect-ing period 1 consumption on the household side. Two things are revealed in period 1: thestate of the economy and the quality of late investment projects.First, the economy is either in a �good state� or in a �bad state�. With probability p the goodstate realizes which means that an investment of I translates into f(I) > I. With probability(1− p), the economy will be in a "bad state". This means that with probability q the returnof an investment of I is only λIq , and with probability (1− q) the return will be 0. Period 1discloses whether the economy is in good or in bad state. Second, if the economy is in badstate, there will be �good� late investment projects g() for a fraction t of banks and �bad�late investment projects h() for a fraction 1− t. If the economy is in good state, every bankfaces good late investment projects.

5.3. General considerations

The return on money and bonds

Given the special form of household's utility

U = c0 + βE(c2) + γM

with γ ∈ (0, 1), β ∈ (0, 1) and γ + β < 1, the real returns on bonds and money arepinned down by a simple argument: Households can always consume their endowment ofthe representative good in period 0. The real return of doing so is 1. Holding bonds andmoney must compete with this use of endowments. Bonds are a means of transferring therepresentative good to period 2. Consumption in period 2 is discounted by β. As a result,the real return on bond holding must be RB = 1β > 1, because the household would preferconsumption in period 0 for a lower return. Holding money promises repayment in period 2and delivers a liquidity premium γ. The real return on holding money will be lower than RB

due to the liquidity premium which makes money more attractive relative to bonds. However,the return on holding money can not be lower than RM = 1β+γ > 1 as a lower return wouldinduce household to consume their endowment directly in period 0. Summing up, we have

21This money-in-utility approach has a long histoy starting with Sidrauski (1967).22One might motivate this assumption by arguing that households have neither the monitoring expertise

needed to invest into the production technology nor the possibility to create money outside thebanking sector.

13

that

RB =1

β(2)

RM =1

β + γ(3)

implying that RM < RB. For the banking sector, RB and RM represent �nancing costs.Thus, banks generally prefer issuing money over bonds due to the lower cost of issuing money.

6. Model I: Universal banking

There are N banks. Banks raise funds (i.e. real resources) from households in period 0 andissue either short-term debt (i.e. money) or long-term debt (i.e. bonds) in return. Unlikehouseholds, banks can store the real resources in form of liquidity holdings W . Banks decidewhether to invest the real resources into the production technology f() or whether to stockpilethem as liquidity W . In period 1, banks learn about the state of the economy. In case ofbeing in the bad state, a fraction t of banks faces good projects g, a fraction 1− t faces onlybad projects h.I assume the following functional forms for the production technologies23:

f(x) =3

2

√x (4)

g(x) = 2√x (5)

h(x) =√x (6)

All these speci�cations feature positive, but decreasing marginal returns to investment.

6.1. How to pay out short-term creditors

If the economy is in bad state, short-term creditors still have to be paid out. Hence, bankshave to make these resources available. They have two possibilities to meet their paymentpromises. First, they can draw down liquidity reserves built up in period 0. Second, they maysell assets backed by investments into the production technology on the asset market. Theasset market allows banks to trade assets with other banks. There is a discount k ∈ (0, 1)for assets being traded on the asset market.

LEMMA 1 Banks will always use their liquidity reserves in order to pay out short-term

creditors before selling assets.

To see this, note that the marginal contribution from selling assets to paying out creditorsis kλ which is smaller than 1 which is the marginal contribution from using liquidity. Thus,from the view point of the bank it is cheaper to use liquidity instead of proceeds from assetsales to pay out creditors.

23I need to specify functional forms to analyze the asset market behavior of banks which would not bepossible using general expressions only.

14

DEFINITION 1 The amount of liquidity a bank has left over after having paid out short-

term creditors is given by

WRes ≡W −m(I +W )RM (7)

where m(I +W )RM is the amount of resources a bank has to pay out in the bad state.

Note that all banks choose the same values for I, W and m which is due to the fact thateach bank faces the same maximization problem in period 0. As a consequence, WRes willbe the same for every bank. This has consequences for the sign of WRes.

LEMMA 2 For WRes < 0 the banking system goes bankrupt.

For WRes < 0 all banks would like to �re sell assets, but no bank is willing to buy assets.There will be no asset trade and banks can not pay out-short term creditors24. In order tocircumvent this problem, I assume in the following that

WRes ≥ 0 (8)

for all banks. This means that banks have enough liquidity to pay out short-term creditors inthe bad state. Note that WRes < 1 since there is only one unit of the representative goodand N banks.

6.2. Upper bound for private money creation

Consider a single bank. If in period 1 the bad state is revealed, short-term creditors requestthe disbursement of money claims which are worth m(I + W )RM . The assumption thatWRes ≥ 0 implies that W ≥ m(I +W )RM . This is only ful�lled if m ≤ W

(I+W )RM.

LEMMA 3 The maximum amount25 of money a bank can issue is given by

mmax =W

(I +W )RM(9)

Eq. (9) is a collateral constraint. Banks' issuance of money clams m is backed by theirliquidity holdings W . That is, holding more liquidity loosens the constraint in (9) on moneycreation. In contrast, more investments I into the production technology in period 0 reducethe potential to create money claims as investments I draw upon resources which are notreadily available to pay out short-term creditors in bad state. Note that if banks end upchoosing mmax, there will be no asset trade in the bad state since there will not be anyliquidity in the asset market after paying out creditors. Only for m < mmax, banks will beable to trade assets.

24This situation mirrors a bank run. There are more money claims than money (liquidity) is available.If all short-term creditors want to withdraw their money at the same time, the banking system isbankrupt.

25Literally, m is the fraction of money �nancing.

15

The assumption that banks will stockpile enough liquidity to pay out short-term creditorsin the bad state, i.e. WRes ≥ 0, has a main advantage. It circumvents a modeling problemin Stein (2012) which he is silent on. Basically, Stein's model setup implies that if banksissue the maximum amount of money claims in period 0, the banking system goes bankruptin case of a bad signal in period 1. Given the assumption that WRes ≥ 0, my model doesnot have this feature. See Appendix A for further details.

6.3. Bank behavior on asset market

6.3.1. The nature of the asset

Banks invest I into the production technology in period 0. In period 1, banks can tradeclaims on the period 2 output of this production technology. If the economy is in bad state,the (expected) output of the technology will be λI. An asset is de�ned as a claim on thisoutput. If a bank purchases such an asset, it obtains the right to acquire the output from thetechnology. Assets are traded on the asset market with a discount k which each bank treatsas exogenous.

6.3.2. The motivation to trade

In contrast to Stein (2012), bank behavior on the asset market in my model is not driven bybanks �re selling assets to pay out short term creditors. If the economy is in the good state,no bank is forced to pay out short-term creditors. So there is no need to �re sell assets. Ifthe economy is in bad state, the assumption that WRes ≥ 0 for all banks ensures that bankshave enough liquidity to pay out short-term creditors. So again, there is no need for anybank to �re sell assets. In case of a bad signal in period 1, banks are split into two groups:banks with good late projects (fraction t) and banks with late projects (fraction 1− t). Theasset market dynamics will be determined by the asset market behavior of these two groups.There is room for asset trade as banks with good projects may sell assets to banks with badprojects in order to raise additional funds to invest into their good late projects.26 Thus,there is trade on the asset market in the bad state of the economy which is not induced by�re sales, but rather by banks optimizing their investment behavior. Note that trade is onlypossible if WRes > 0, since for WRes = 0 there is no liqudity in the asset market which may�nance asset trades. The following Lemma greatly simpli�es the analysis of banks' behavioron the asset market.

LEMMA 4 If the bad state is revealed in period 1, a bank will either buy or sell assets, but

it will not do both.

This is due to the fact that both the return on selling assets and on buying assets is z ≡ 1−kk .Thus, selling assets to buy assets delivers a net return of 0, so there is no incentive for anybank to engage in some sort of arbitrage trading.

26It will not be the other way around since banks with bad late projects do not have the same incentiveto obtain additional funds to invest as their late projects are not as pro�table as those from bankswith good projects.

16

6.3.3. Banks with good late projects

If a bank faces good late investment opportunities in period 1, it might want to sell assets inorder to increase investments into these projects. The maximum amount of resources it canrelease is kλI by selling its whole investment into the production technology from period 0. Ifit sells a fraction AS ∈ [0, 1], it obtains ASkλI of resources to invest into late projects. Dueto the discount k, the bank occurs a net cost of selling assets which is z ≡ 1−kk and whichresults from the fact that the bank sells its investment into the early production technologybelow par. In addition, selling an asset implies losing the payo� λI in period 2.If the bad state is revealed in period 1, the bank faces the following maximization problem:

maxAS

g(WRes +ASkλI)−ASzλI + (1−AS)λI

⇐⇒ maxAS

g(WRes +ASkλI)− 1kASλI + λI

(10)

as 1 + z = 1k . The �rst-order condition for this problem reads as

g′(WRes +ASkλI) =1

k2(11)

Using g(x) = 2√x, this expression simpli�es to

1√WRes +ASkλI

=1

k2(12)

⇐⇒ AS = k3

λI− W

Res

kλI(13)

which is the fraction of assets a bank with good late projects will sell.The amount of resources a bank recovers when selling a fraction of assets AS is given by

ASkλI = k4 −WRes (14)

Thus, banks with good late projects will invest

glpi ≡WRes +ASkλI = k4 (15)

where glpi denotes good late project investments.Bank's pro�t from selling assets and investing into good projects in the bad state of theeconomy is then given by

g(WRes +ASkλI)− 1kASλI + λI = 2

√k4 −

(k2 − W

Res

k2

)+ λI

= k2 +WRes

k2+ λI

(16)

which will later be called payo� good project.

17

6.3.4. Banks with bad late projects

If a bank faces bad projects in period 1, it may use its liquidity to buy assets from banks withgood projects. The bank has resources ofWRes left after short-term creditors have been paidout and the price of the asset on the asset market is kλI. Banks with bad projects then solvethe following maximization problem

maxAP

h(WRes −APkλI) + 1kAPkλI + λI (17)

where AP is the amount of resources spent on buying assets. The �rst-order condition forthis problem reads as follows

h′(WRes −APkλI) = 1k

(18)

Given that h(W ) =√W , this simpli�es to

AP =WRes

kλI− k

4λI(19)

Buying asset from banks with good projects must compete with investments into the badproject. 1k is the marginal return from buying assets on the asset market and

1

2√WRes

is the

marginal return from investing the remaining liquidity into the bad project. Banks treat k asexogenous. The marginal return from buying assets must be su�ciently large (i.e. k mustbe su�ciently low) so that it is attractive for banks with bad projects to buy assets insteadof investing directly into the late project.The amount of resources a bank with bad projects put into buying assets is given by

APkλI = WRes −(k

2

)2(20)

Hence, banks invest

blpi ≡(k

2

)2(21)

into their late projects. AP is bounded from above since banks can not spend more resourceson buying assets than they have liquidity left. That is, it must hold that

WRes > APkλI

Note that this is true since(k2

)2> 0.

The payo� from purchasing assets and investing into bad late projects is then given by

h(WRes −APkλI) + 1kAPkλI + λI =

√(k

2

)2+WRes

k− k

4+ λI

=k

4+WRes

k+ λI

(22)

which will later be called payo� bad project.

18

6.3.5. A condition for asset trade

Note from (13) that AS is positive only if

k > (WRes)14 (23)

Further, AP in (19) is only positive if

k < 2√WRes (24)

Equation (23) and (24) determine conditions for k such that banks with good projects sellassets (asset supply) and banks with bad projects buy assets (asset demand). Intuitively,the discount on assets must not be too high (i.e. low k) as otherwise banks with good lateprojects would not be willing to sell assets to raise additional funds. On the other hand, thediscount must not be too low (i.e. high k) as otherwise banks with bad late projects wouldnot want to buy assets. If one of these two conditions is not ful�lled, the asset market breaksdown and there will be no trade in assets. As a consequence, it must hold that

k ∈(

(WRes)14 , 2√WRes

)This leads to the following Lemma.

LEMMA 5 The asset market can only be in equilibrium if

(WRes)14 < 2

√WRes

⇐⇒WRes > 116

= 0.0625(25)

is true.

Equation (25) is a lower bound for WRes. That is, if WRes is smaller than 0.0625, thenthere will be no trade on the asset market27. Rather, banks with good projects invest all theirliquidity in the good project, and banks with bad projects invest their liquidity into the badproject. Banks are not aware of this lower bound when choosing WRes as they are atomisticand do not take into account the impact of their choices on equilibrium outcomes.

27The exact number here is due to the assumption of speci�c functional forms for g and h. However,the result that there exists such a lower bound carries over to general functional forms.

19

6.4. Banks' maximization problem in period 0

6.4.1. Banks only issuing bonds

As a benchmark case, consider a bank that �nances its investments I and liquidity holdingsW solely by issuing bonds, i.e. m = 0. The bank's expected net pro�ts at time 2 then aregiven by

π = pf(I) + pg(W )− (I +W )RB +m(I +W )(RB −RM )+(1− p) [t(payoff good project) + (1− t)(payoff bad project)]

(26)

where pf(I) + pg(W ) is the return arising in the good state when a bank invests I intoprojects in period 0 and invests liquidity holdings W in late projects. Financing costs whenonly issuing bonds are (I + W )RB. If the bad state realizes, banks do not have to pay outany short-term creditors as m = 0. It then holds that WRes = W . Banks will use theirliquidity holdings to buy assets if they face bad projects or they will sell assets if they facegood projects, see above. Plugging in the expressions for payo� good project and payo� badproject from (16) and (22), the bank's maximization problem then is the following

maxW,I

pf(I) + pg(W )− (I +W )RB

+ (1− p)[t

(k2 +

W

k2+ λI

)+ (1− t)

(k

4+W

k+ λI

)] (27)The �re-sale discount k is an exogenous parameter for a single bank. That is, banks donot take into account the e�ect of their asset market behavior on k as there is perfectcompetition in the banking sector and each bank is very small. The �rst-order conditions forthis maximization problem are

∂π

∂I= pf ′(I)−RB + (1− p)λ != 0 (28)

∂π

∂W= pg′(W )−RB + (1− p)

[t

k2+

1− tk

]!

= 0 (29)

These two equations may be combined to get

pf ′(I) + (1− p)λ = pg′(W ) + (1− p)[t

k2+

1− tk

](30)

relating investment level I and liquidity holdings W in optimum. The left-hand side ofequation (30) is the marginal return from using resources to invest into investment projectsin period 0. The right-hand side represents the marginal return from using resources to holdliquidity. With probability p, the economy is in the good state and banks can invest all theirliquidity holdings into late projects. With probability (1 − p), the economy is in the badstate and banks invest into their late projects after having traded assets on the assets market.Liquidity W to invest into late projects is more valuable in case of having good projects(marginal product of 1

k2) than in case of having bad projects (marginal product of 1k ).

28

28Note that 1k2

> 1kfor k < 1.

20

6.4.2. Banks issuing both bonds and money

Consider now a bank that invests I in the production technology in period 0, holds W asliquidity and �nances (I + W ) with some fraction m ≤ mmax of money. The maximizationproblem for bank's expected net pro�ts at time 2 then is

maxI,W,m

pf(I) + pg(W )− (I +W )RB +m(I +W )(RB −RM )

+(1− p)[t

(k2 +

WRes

k2+ λI

)+ (1− t)

(k

4+WRes

k+ λI

)] (31)s.t.

m ≤ mmax = W(I +W )RM

Note that this formulation of the maximization problem crucially depends on the existenceof an asset market in the bad state of the economy. This in turn requires that banks setWRes > 0.0625 as argued above. Banks do not consider this requirement explicitly, butrather assume that they will be able to sell and buy assets in the bad state of the economyirrespective of their own behavior. Apart from that, this problem mirrors the problem in (27),with a few important di�erences. First, the term m(I + W )(RB − RM ) denotes �nancingcosts saved associated with using money rather than bonds to raise funds from the household.Second, banks will have to pay out short-term creditors in case of the bad state if m > 0.As a result, WRes = W −m(I +W )RM instead of WRes = W .

The Lagrangian for the maximization problem in (31) is

L = pf(I) + (1− p)λI + pg(W )− (I +W )RB +m(I +W )(RB −RM )

+(1− p)[t

(k2 +

WRes

k2

)+ (1− t)

(k

4+WRes

k

)]− µ

[m− W

(I +W )RM

](32)

The corresponding �rst-order conditions are

∂L

∂I= pf ′(I) + (1− p)λ−RB +m(RB −RM )

+ (1− p)[−tmR

M

k2− (1− t)mR

M

k

]− µ

[WRM

[(I +W )RM ]2

]!

= 0(33)

∂L

∂W= pg′(W )−RB +m(RB −RM )

+ (1− p)[t1−mRM

k2+ (1− t)1−mR

M

k

]− µ

[WRM − (I +W )RM

[(I +W )RM ]2

]!

= 0

(34)

21

∂L

∂m= (I +W )(RB −RM ) + (1− p)

[−t(I +W )R

M

k2− (1− t)(I +W )R

M

k

]− µ != 0

(35)

∂L

∂µ=

[m− W

(I +W )RM

]!≤ 0 (36)

µ∂L

∂µ= µ

[m− W

(I +W )RM

]!

= 0 (37)

6.4.3. The optimal relation between investment and liquidity

a) case of m < mmax

If a bank issues m < mmax, µ = 0 must hold to satisfy (37). Equation (33) and (34)may then be combined to obtain

pf ′(I) + (1− p)λ = pg′(W ) + (1− p)[t

k2+

1− tk

](38)

which is equal to eq. (30) and has a similar interpretation. That is, the last term on the right-hand side of (38) represents the marginal return from having one additional unit of liquidityleft after paying out short-term creditors. In contrast, eq. (30) is based on the assumptionthat there are no short-term creditors.

b) case of m = mmax

If a bank issues m = mmax, it follows that µ > 0. Using (35) to combine (33) and (34)yields29

pf ′(I) + (1− p)λ = pg′(W ) + RB −RM

RM(39)

which then determines the optimal relation between investments into the production tech-nology I in period 0 and liquidity holdings W . R

B−RMRM

is the net rate of costs savingsassociated with issuing money rather than bonds. If banks set m = mmax, there will be noliquidity left after short-term creditors have been paid out. So there will be no investmentsinto late projects. The marginal return from having one additional unit of liquidity left afterhaving paid out short-term creditors is now given by the rate of costs savings associatedwith issuing money. This means that banks would use an additional unit of excess liquid-ity to back further money claims, that is, to loosen their collateral constraint given by eq. (9).

29See Appendix B for derivation.

22

Note that m = mmax implies that banks do not invest into any late projects in the badstate since WRes = 0. Besides, there will be no asset trades since there is no liquidity in theasset market which allows for trade.

6.4.4. When do banks issue m = mmax?

The bank sets m = mmax (with µ > 0) if30

RB −RM

RM> (1− p)

[t

k2+

1− tk

](40)

This condition is readily interpreted. The left-hand side is the marginal return from increas-ing the fraction of money �nancing m. The right-hand side gives the marginal return fromhaving additional liquidity left in the bad state to potentially buy assets and invest into lateprojects. This re�ects the underlying money-liquidity trade-o� banks face in my model. Thatis, banks have an incentive to issue as much money as possible since they can save �nancingcosts by doing so. This cost savings are R

B−RMRM

. On the other hand, issuing money comesat the cost of having less liquidity available in the bad state of the economy as short-termcreditors have to be paid out with liquidity. This reduces investments into late projects andpotential asset purchases. The marginal return from having additional liquidity in the badstate is (1− p)

[tk2

+ 1−tk]. Only if the marginal return from issuing an additional fraction of

money is larger than the marginal return from having one additional unit of liquidity left inthe bad state, banks will issue the maximum amount of money mmax. This is exactly what(40) reveals.

If banks decide to set m = mmax, this implies that WRes = 0. This leads to a collapseof the asset market as no liquidity is left to trade assets. However, then the banks solveda maximization problem in (31) which contained an expected payo� in the bad state whichdoes not materialize for WRes = 0. For WRes = 0, there are no investments into the lateprojects and thus, the payo� from late projects to an individual bank is 0 in the bad state.

The spread RB−RMRM

is �xed as both interest rates only depend on preference parameters(see above). If condition (40) is ful�lled, banks will issue m = mmax, there will be no tradeon the asset market and the relation between investment I and liquidity holding W is asdescribed in (39). If condition (40) is not ful�lled, banks will choose m < mmax, there willbe trade on the asset market and the relation between investment I and liquidity holding Wis as described in (38).

The spread required for banks setting m = mmax, spread ≡ RB−RMRM

, has the followingproperties:1) ∂spread∂t > 0: The higher the probability for an individual bank to end up with goodprojects in the bad state, the higher must be the �nancing costs savings to induce banks toset m = mmax. This is due to the fact that issuing more money implies less liquidity available

30Check equation (35) to see that this is true.

23

to invest into the good project in bad state as more short-term creditors have to been paidout. If the probability of facing good projects in bad state increases, issuing money becomesrelatively more expensive then.2) ∂spread∂k < 0: The higher the discount on traded assets in bad state (i.e. the lower k),the higher must be the �nancing cost savings associated with issuing money to have bankssetting m = mmax. A high discount means that it becomes relatively expensive for banksto raise resources to invest into the good project as selling assets delivers a relatively smallreturn. A bank would then rather be in a situation in which more liquidity is left after pay-ing out creditors. Such a situation arises if banks do not issue so much money in the �rst place.

Note that k is an endogenous variable. It will be determined on the asset market and itwill hold that k(WRes, t) (see below).

In Stein (2012), there is an equation similar to (40). Stein obtains RB−RMRM

> (1 − p)zas the condition for banks setting m = mmax. It holds that31

z ≡ 1− kk

<t

k2+

1− tk

Thus, my model requires a larger spread than Stein's model. This is due to the fact thatin Stein's model banks do not face a liquidity-investment trade-o� as they do not stockpileliquidity. Instead, they invest all their funds into early projects. As a result, issuing money isrelatively cheaper than in my model as it comes not at the cost of reducing liquidity to investinto late projects.

6.5. Equilbrium on asset market

In the bad state of the economy, a fraction t of banks will sell assets and a fraction 1− t ofbanks will buy these assets. The asset market clears if the amount of resources spent onassetsales and asset purchases is equated:

tASkλI = (1− t)APkλI (41)

Note from (25) that AP and AS are only well-de�ned if WRes > 0.0625. Given eq. (14)and (20), we can rewrite (41) as

t(k4 −WRes

)= (1− t)

(WRes −

(k

2

)2)(42)

withWRes ≡W −m(I+W )RM . k will be such that the asset markets clears for the optimalvalues of W , I and m from bank's maximization problem.Eq. (42) rewrites as

WRes = tk4 + (1− t)(k

2

)2(43)

31See Appendix C for derivation.

24

with WRes = W −m(I + W )RM . This may be solved for k to obtain as the only positivevalued solution32:

k =

√t− 1

8t+

√(1− t)2 + 64tWRes

8t(44)

This expression is always positive for t > 0. Eq. (44) relates equilbrium k to bank's deci-sion onW , I andm sinceWRes = W−m(I+W )RM . It holds that k > 0 only ifWRes > 0.

Note that there are two potential scenarios under which the asset market breaks down.First, as already noted above, there will be no trade on the asset market if banks choosem = mmax. This becomes clear when evaluating eq. (44) for WRes = 0 which is the casewhen banks set m = mmax. It follows that k = 0. Second, for WRes < 0.0625 the assetmarket breaks down as either supply or demand for assets will be zero (see explanation above).Equilibrium k is not de�ned in this case as asset supply or asset demand is not de�ned. Only

if WRes > 0.0625, both k4 > WRes and(k2

)2< WRes hold at the same time such that

there is both demand and supply of assets in equilbrium. These considerations will becomeimportant when discussing the e�ciency of the universal banking solution.

6.6. Social planner solution

In order to analyze the e�ciency of the universal banking solution, I compare it to the outcomeof the social planner solution. The social planner maximizes the utility of a representativehousehold when choosing I, W and m. Banks' pro�ts �nally accrue to the households (whoown the bank). The social planner does not care about losses or gains accruing to individualbanks when interacting on the asset market since that market constitutes a zero sum game:the pro�t of one bank is the loss of another bank. Rather, the social planner is interested inthe output from the di�erent production technologies after assets have been traded. If thebad state realizes in period 1, investments into the good project deliver 2

√glpi = 2k2 and

investments into the bad projects deliver√blpi = k2 .

A key di�erence between the social planner problem and the problem an individual bankfaces is that the social planner realizes the dependence of k from the chosen values of W , Iand m. That is, the social planner maximizes

maxI,W,m

pf(I) + pg(W )− (I +W )RB

+m(I +W )(RB −RM ) + (1− p)[2tk2 + (1− t)k

2

] (45)s.t.

m ≤ mmax = W(I +W )RM

k =

√t− 1

8t+

√(1− t)2 + 64tWRes

8t

32See Appendix D for derivation.

25

Thus, the social planner takes into account that k will be determined by eq. (44).The Lagrangian for this problem is

L =pf(I) + (1− p)λI + pg(W )− (I +W )RB +m(I +W )(RB −RM )

+ (1− p)[2tk2 + (1− t)k

2

]− µ

[m− W

(I +W )RM

](46)

So we get the following �rst-order conditions

∂L

∂I= pf ′(I) + (1− p)λ−RB +m(RB −RM )

+ (1− p)[4tk

∂k

∂I+ (1− t)1

2

∂k

∂I

]− µ

[WRM

[(I +W )RM ]2

]!

= 0(47)

∂L

∂W= pg′(W )−RB +m(RB −RM )

+ (1− p)[4tk

∂k

∂W+ (1− t)1

2

∂k

∂W

]− µ

[−IRM

[(I +W )RM ]2

]!

= 0(48)

∂L

∂m= (I +W )(RB −RM ) + (1− p)

[4tk

∂k

∂m+ (1− t)1

2

∂k

∂m

]− µ != 0 (49)

∂L

∂µ=

[m− W

(I +W )RM

]!≤ 0 (50)

µ∂L

∂µ= µ

[m− W

(I +W )RM

]!

= 0 (51)

From (44) we get for the derivatives of k that

∂k

∂I=

1

2

(t− 1 +

√(1− t)2 + 64tWRes

8t

)− 12 1

8t

1

2

((1− t)2 + 64tWRes

)− 12 (−64tmRM ) < 0

(52)

∂k

∂W=

1

2

(t− 1 +

√(1− t)2 + 64tWRes

8t

)− 12 1

8t

1

2

((1− t)2 + 64tWRes

)− 12[64t(1−mRM )

]≶ 0

(53)

26

∂k

∂m=

1

2

(t− 1 +

√(1− t)2 + 64tWRes

8t

)− 12 1

8t

1

2

((1− t)2 + 64tWRes

)− 12[−64t(I +W )RM

]< 0

(54)

Hence, the equilibrium discount on asset sales k negatively depends on the marginal amountof resources invested in early projects and the marginal amount of money issued. More earlyinvestments lead to a higher supply of assets in the bad state as for a given fraction of money�nancing m more investments mean more payouts to short-term creditors in the bad state.This depresses the price of assets, i.e. decreases equilibrium k. Also, if the fraction of fundsraised with money is large, banks have relatively little liquidity left in the bad state of theeconomy as they have to pay out many short-term creditors. Banks with good projects willthen supply relatively many assets to raise additional funds to invest into late projects. Thisagain lowers the price of assets, i.e. decreases k. The marginal e�ect of holding more liquidityon equilibrium k is ambiguous. On the one hand, more liquidity holdings W lead to fewerasset supply in the bad state as banks with good late projects already have relatively manyfunds to invest into late projects lowering their asset supply. This tends to increase k. On theother hand, for a given fraction of money �nancing m more liquidity holdings implies morepayouts to short-term creditors in the bad state. This tends to decrease k. The overall e�ectis ex ante unclear.

The social planner sets m = mmax with µ > 0 if33

(I +W )(RB −RM ) > −(1 + p)[(

4tk + (1− t)12

)∂k

∂m

](55)

where ∂k∂m < 0. Note that this condition is not equal to the condition for the maximizationproblem of a single bank. This condition was given in (40) as

RB −RM

RM> (1− p)

[t

k2+

1− tk

]Comapring (55) and (40), note that (55) can be rewritten using (54) as

RB −RM

RM> (1− p)

[4tk + (1− t)1

2

]2√

k((1− t)2 + 64tWRes)(56)

It is obvious that (56) and (40) do generally not coincide. That is, the decision of individualbanks to set m = mmax likely deviates from the decision of the social planner.

Consider the case where (55) is ful�lled. Banks then set m = mmax with µ > 0. Using(52)-(54) to combine the three �rst-order conditions gives34

pf ′(I) + (1− p)λ = pg′(W ) + RB −RM

RM(57)

33Check equation (49) to see that this is true.34See Appendix E for complete derivation.

27

which coincides with the universal banking soluton in (39) and has the same interpretation.That is, if the social planner sets m = mmax, he chooses the same relation of I and Wuniversal banks do. However, the condition for setting m = mmax is not the same as I justhave shown.

Next, consider the case where condition (55) is not ful�lled and the social planner setsm < mmax with µ = 0. The �rst-order conditions may then be combined to obtain

pf ′(I) + (1− p)λ = pg′(W ) + (1− p)[(

4tk + (1− t)12

)(∂k

∂W− ∂k∂I

)](58)

The last term on the right-hand side of equation (58) represents the marginal value of holdingadditional liquidity in the bad state for the social planner. The social planner is only interestedin the amounts of investment into late projects if the bad state realizes. If the planner decidesto set m < mmax, there will be investments into late projects. These amounts depend onlyon k as (45) shows. The marginal value of additonal liquidity is determined by comparing itsmarginal e�ect on equilibrium k with the marginal e�ect of doing early investments instead,i.e. ∂k∂W −

∂k∂I > 0. The optimal relation between I andW described in (58) does not coincide

with the optimal relation an individual bank sets for m < mmax as given in (38).

Note that the social planner will chooseWRes > 0.0625 if he sets m < mmax since otherwisethe asset market would break down and k would not be de�ned. This is a key di�erence tothe universal banking solution. Individual banks do not take into account that a too lowWRes may cause the asset market to break down, whereas the social planner does.The following Proposition summarizes ine�ciencies in the universal banking system.

PROPOSITION 1 The universal banking system is potentially ine�cient. Banks are likely

to issue a non-optimal amount of money and run into liquidity problems in the bad state of

the economy.

First, the condition for an individual bank to set m = mmax (implying that there are noinvestments into late projects since WRes = 0) does not coincide with the condition from thesocial planner solution. As a result, it might be the case that the universal banking solutionissues too much money m and features no investments into late projects although this wouldbe e�cient. Second, even if both the social planner solution and the universal bankingsolution imply setting m < mmax, the social planner would always set WRes > 0.0625 as herealizes that otherwise the asset market breaks down. The universal banking problem doesnot ensure that WRes > 0.0625 and thus may lead to a break down of the asset marketwhich is ine�cient if m < mmax.

6.7. Sources of ine�ciency

In the baseline model of Stein (2012) where banks do not decide about liquidity holdings W ,an increase in money creation by a single bank reduces equilibrium k which then tightens

28

the collateral constraint of money creation when m = mmax is true. This is the source ofine�ciency in Stein's model. In contrast, in my model the potential to create money for asingle bank is not a�ected by equilbrium k when WRes > 0 is true. Given that WRes > 0,banks will never �re sell assets and so their potential to create money is una�ected by changesin the discount k.There is a di�erence source of ine�ciency in my model. In fact, there are two sources asaddressed above. Both rely on the fact that atomistic banks do not take into account thee�ects of their behavior on the market outcome. The market outcome, however, is indirectlyshaped by a single bank's decision due to homogeneity of banks in period 0. First, if bankschoose to set m < mmax there will be only trade in the asset market if WRes > 0.0625(as derived in (25)) is true. For smaller values of WRes the market will collapse. The socialplanner considers this fact while a single bank does not. Second, banks might set m = mmax

when the social planner does not. This is due to the fact that a single bank does not takeinto account that its own behavior changes equilibrium k - a fact the social planner does takeinto account.

6.8. The danger of too little liquidity and potential remedies

As previously argued, for WRes < 0 the universal banking system outcome certainly is inef-�cient as it implies that the banking system is bankrupt in the bad state. However, it mightbe ex ante advantageous for banks to end up in a situation of WRes < 0 in bad state. Oneexample particularly relevant in the real world is governmental deposit insurance. In the U.S.,for example, the Federal Deposit Insurance Corporation (FDIC) guarantees the safety of bankdeposits up to 250 000 $. Such a system may lead to the ine�cent result that WRes < 0 inthe bad state.

6.8.1. governmental deposit insurance

Consider a situation in which the deposits of the household in the banking sector are insuredby the government in case of the bad state of the economy. This means that governmentpromises to pay out money holders if the bad state realizes. Such a regulation implies inmy model that banks have a strong incentive to issue as much money as possible knowingthat they only have to repay debt if the good state realizes. Consequently, banks would setm = mmax and would solve the following maximization problem

maxW,I

pf(I)− p(I +W )RM + (1− p) 0 + pg(W ) (59)

That is, banks would only repay debt in case of the good state. In the bad state, banks sell alltheir assets to the government which in response guarantees to pay out short-term creditors.Consequently, banks have no resources left to invest into late projects and will make zeropro�ts in the bad state. However, banks were able to �nance very cheaply by issuing manymoney claims in period 0.

PROPOSITION 2 A system of governmental deposit insurance increases the probability

that the universal banking solution is ine�cient.

29

To see this, note that by issuing m = mmax and running into WRes < 0 banks save

(1−m)(I +W )RB + (1− p)(I +W )RM

of resources compared to setting WRes > 0. This comes at the cost of

(1− p)λI + (1− p)[t

(k2 +

WRes1k2

)+ (1− t)

(k

4+WRes1k

)]where WRes1 again denotes the remaining liquidity after paying out short-term creditors incase of WRes > 0. Thus, banks will choose to set WRes < 0 instead of WRes > 0 under aregime of governmental deposit insurance if

RB >1− p

(1−m)(I +W )

[λI + t

(k2 +

WRes1k2

)+ (1− t)

(k

4+WRes1k

)]− 1− p

1−mRM

(60)

This condition can be satis�ed if bond �nancing is very expensive, i.e. if RB is very high35.In this case, governmental deposit insurance would induce banks to run the risk of beingbankrupt in the bad state knowing that the deposit insurance scheme will help out. Thisre�ects the danger of moral hazard asscociated with deposit insurance.

6.8.2. Remedies: minimum liquidity reserve and central bank as lender of last

resort

There are two potential remedies for ine�ciencies arising in the universal banking system anddue to governmental deposit insurance.

PROPOSITION 3 A minimum liquidity reserve scheme can preclude the universal banking

system from being ine�cient. However, it can also cause ine�ciencies to an otherwise e�cient

solution. The central bank acting as a lender of last resort can restore ex post e�ciency in

the bad state.

First, a regulatory authority may require banks to always hold WRes > 0.0625, i.e. requirebanks to be able to pay out short-term creditors with stockpiled liquidity in the bad stateand still have some resources left. Such a minimum liquidity requirement would also precludethe problem of moral hazard arising from governmental deposit insurance. However, such aregulation comes at the cost of causing ine�ciences when it would be in fact e�cient if banksset m = mmax with WRes = 0. This is the case if the interest rate spread between moneyand bond �nancing is su�ciently large so that the social planner would also set WRes = 0.Second, the central bank could serve as a lender of last resort if the bad state realizes. Itcould then maintain the asset market even if banks end up in a situation of WRes < 0.0625or even WRes < 0. The central bank could use newly created liquidity to buy assets frombanks with good projects, thereby reviving the otherwise collapsing asset market.

35Note that RB is pinned down by the discount factor in househould utility, RB = 1β.

30

7. Model II: Dual banking and 100% money

Universal banking means that a single bank is allowed to raise funds to both invest intothe production technology and to hold liquidity. In a dual banking system, these tasks areconducted by seperated institutions which I call savings banks and deposit banks. Savingsbanks issue long-term debt claims (bonds) in period 0 and invest resources into the productiontechnology. A deposit banks issues short-term debt claims (money) and holds the acquiredresources as liquidity. They are not allowed to invest deposits. Thus, deposit banks operateunder a system of 100% reserve requirements. In contrast, in a system of universal bankingbanks can issue many short-term debt claims without holding a correspondent amount ofresources as liquidity.

7.1. Universal banking vs. dual banking

A dual banking system implies a regime shift. In order to better understand the consequencesof dual banking, consider balance sheets of representative banks in both systems.

Balance Sheet in Universal Banking

assets liabilities

liquidity W money m(I +W )investments I bonds (1−m)(I +W )

In the universal banking system, banks raise funds from households by either issuing money orbonds. Banks then stockpile these funds as liquidity or invest them into the early productiontechnology. They are free to choose how to allocate funds between liquidity and investments.There is also no restriction on their decision on whether to �nance by money or by bondsapart from m ≤ mmax as in (9). The dual banking system grants all decision power directlyto the households. They themselves decide over how much liqudity to have and how muchto invest in period 0. There are two types of banks.

Balance Sheets in Dual Banking

deposit banks

assets liabilities

liquidity W deposits (�money�)

savings banks

assets liabilities

investments I savings (�bonds�)

A savings bank is a bank which receives the endowment of the household and invests it intothe early production technology. Households investing into savings bank obtain a long-termclaim (�bond�) in return. Households can not withdraw their resources from savings banksbefore period 2. However, they might sell the claim on period 2 returns from early invest-ments on the asset market. That is, both the dual banking and the universal system share akey common feature: There is an asset market where assets can be traded. In the universalbanking system, banks trade assets backed by investments I in the early production technol-ogy. In the dual banking system, households may trade assets backed by deposits in savings

31

banks. These deposits are ultimately also backed by investments into the early productiontechnology. A deposit bank would accept the representative good from the household andwould store it for him. Thus, it creates liquidity W (�money�) which is readily available tothe households. Deposit banks would not be allowed to invest the resources they raise fromhouseholds. The money claims such a deposit bank issue may be called 100% money. Thehousehold can use this liquidity in period 1 to invest into late projects or to buy assets on theasset market.

Note that in a dual banking system, banks' choice set basically collapsed. They act aspure intermediaries. Banks can either invest into early production (savings bank) or stockpileliqudity (deposit bank). They can not do both. Even more importantly, banks can not choosethemselves how to �nance their fund raising. That is, they have no choice over the fraction ofmoney �nancing m. Rather, the issuance of money claims results from household demandingliquidity holdings from deposit banks.

7.2. Household's maximization problem

Consider H households which are all identical in period 0. The representative householdsolves the following maximization problem

maxI,W

U = c0 + βE(c2) + γW (61)

where

E(c2) = pf(I)+(1−p)λI+pg(W )+(1−p) [t (good project) + (1− t) (bad project)] (62)

andY = c0 + I +W (63)

Noting that the asset market in the dual banking system works in the same way as the assetmarket in universal banking withWRes = W , it holds that good project = payo� good projectand bad project = payo� bad project. That is, households with good late projects sell savingdeposits to households with demand deposits to obtain more liquidity to invest into theirgood projects. Households with bad projects buy assets and invest their remaining liquidityinto their bad projects.Thus, (62) may be rewritten using (16) and (22) to obtain

E(c2) = pf(I) + (1− p)λI + pg(W ) + (1− p)[t

(k2 +

W

k2

)+ (1− t)

(k

4+W

k

)](64)

Expressed in terms of period 2 consumption, the maximization problem rewrites as

maxI,W

U = (Y − I −W )RB + pf(I) + (1− p)λI + pg(W )

+ (1− p)[t

(k2 +

W

k2

)+ (1− t)

(k

4+W

k

)]+ γRBW

(65)

32

The �rst-order conditions are as follows

∂U

∂I=−RB + pf ′(I) + (1− p)λ != 0 (66)

∂U

∂W=−RB + pg′(W ) + (1− p)

[t

k2+

1− t4

]+ γRB

!= 0 (67)

Note that the �rst-order condition with respect to W implies that W > 0. Combining (66)

with (67) and noting that γRB = RB−RMRM

it follows that

pf ′(I) + (1− p)λ = pg′(W ) + (1− p)[t

k2+

1− t4

]+RB −RM

RM(68)

Comparing this equation relating I and W in optimum with the solution from universalbanking in (38) for m < mmax and in (39) for m = mmax, the following Propositionemerges.

PROPOSITION 4 The dual banking system solution implies a larger amount of liquidity

W relative to investments I than the universal banking system.

This result is due to the fact that the household receives utility directly from holding liquidityW as a consequence of the money-in-utility approach. In the universal banking solution,banks' optimal investment/liquidity decision only hinges on the spread R

B−RMRM

if m = mmax.Then banks like to loosen their collateral constraint on m by holding more liquidityW relativeto investments I. If m < mmax, banks do not wish to loosen their collateral constraint, butonly care about how additional liquidity might be used in the bad state afor asset purchasesand late project investments. The spread between money and bonds interest rates does nota�ect this decision and thus, it does not appear in the equation relating liquidity holdings Wand investments I in optimum. In the dual banking solution, households always care about theinterest rate spread since they gain utility from holding liquidity (�money�). In addition, theycare about how much liquidity is worth in the bad state of the economy where it may be usedfor asset purchases or investments into the late project. As a result, liquidity is more valuableto households than to banks. This is re�ected in the optimal relation between liquidity Wand investments I where dual banking implies more liquidity holdings than universal banking.

7.3. Dual banking and e�ciency

Just like individual banks in the universal banking solution, the individual household does nottake into account the e�ect of his behavior on equilibrium k. Equilibrium k will be determinedin a way analogue to the universal banking system since there will be a fraction t of householdswith good projects trying to sell as