MACROECONOMICS OF INCOMPLETE MARKETS†

Monetary Policy, Banking Crises, and the Friedman Rule

By BRUCE D. SMITH*

rgrscili

issl7

nr-eltos

ninenesrec.de

hirc

yy

lslns-t-n

ldllyer-

sel

gdalce.

-g

wn-

Banking crises are frequent events. GeraCaprio and Daniela Klingebiel (1997) cataloover 80 banking crises during the last 25 yeaInterestingly, some banking crises are assoated with no output losses whatsoever, whothers involve massive recessions. Finally, itknown that the probability of a banking crisrises as the rate of inflation rises (see ADemirguc-Kunt and Enrica Detragiache, 199John Boyd et al., 2001a, b).

While received wisdom exists about the coduct of monetary policy while a crisis is undeway, there is no formal treatment of how thconduct of monetary policy during “normatimes” affects the potential for banking crisesoccur. To fill that gap, I consider economiewhere spatial separation and limited commucation create a transactions role for money, arandom shocks to agents’ liquidity preferenccreate a role for banks. In addition, banks cofront randomness in withdrawal demand. Whwithdrawal demand is sufficiently high, bankexhaust their cash reserves. There are goodsons to associate this with a banking pani1

The output lost during a banking panic depenon how great withdrawal demand is. Sombanking crises generate no output losses, wothers generate large reductions in resouavailability.

In addition, the conduct of monetary policduring “normal times” affects the probabilit

-

t

ert

† Discussants: Stephen Williamson, University of Iowa;Jesus Fernandez-Villaverde, University of Minnesota; Beatrix Paal, Stanford University.

* Department of Economics, University of Texas, Aus-tin, TX 78712. I thank Costas Azariadis for encouraging meto write this paper, and Beatrix Paal and Stephen Williamson for helpful comments.

1 Alexander Noyes (1909) listed the distinguishing fea-tures of a banking panic as: suspension of cash paymentsdepositors, depletion of cash reserves, the emergence ocurrency premium and “emergency expedients” to providadditional media of exchange. This analysis gives rise to aof these.

128

d

.i-es

i;

-

-ds-n

a-

s

lee

of a banking crisis. The higher the nominainterest rate (the inflation rate), the higher ithe probability of a panic. Driving the nominainterest rate to zero (following the Friedmarule) eliminates bank panics. This constitutea new rationale for the Friedman rule. Nonetheless, conventional methods of implemening the Friedman rule never produce aoptimal resource allocation. In particular, lownominal interest rates induce banks to holarge cash reserves, thereby forgoing sociamore productive investments. In effect, thFriedman rule induces banks to become narow banks voluntarily. The banking system ivery safe, but it undertakes a suboptimal levof investment.

Less conventional methods for implementinthe Friedman rule, such as allowing unrestricteaccess to the discount window at a zero nomininterest rate, lead either to the nonexistenof equilibrium, or to massive indeterminaciesNone of the equilibria will be consistent withfull optimality.

I. Environment

Consider an economy populated by an infinite sequence of two-period-lived, overlappingenerations. Lett � 1, 2, ... index time.

There are two islands. At each date a neyoung generation appears on each island cosisting of a continuum ofex ante identicalagents with unit mass.

There is a single good. All agents are endowed withw units of this good when young,and nothing when old.2 In addition, there is astorage technology whereby one unit stored atgeneratesR � 1 units of consumption att � 1.Storage investments can be “scrapped” afttheir initiation. Scrapping a storage investmen

-

-

to

f aell 2 The initial old are endowed with the initial per capita

money supply,M0.

129VOL. 92 NO. 2 MACROECONOMICS OF INCOMPLETE MARKETS

generates r � 1 units of consumption in theperiod the investment was initiated. Finally,agents care only about old age consumption, c,and have lifetime utility u(c) � ln c.

To create a transactions role for money, I em-phasize limited communication between spatiallydistinct markets. In each period, trade occurs onlyamong agents who inhabit the same location. Atthe beginning of a period, all young agents makea bank deposit.3 The bank uses deposits to acquirethe economy’s primary assets: storage invest-ments and money. After bank portfolio allocationoccurs, all goods not stored are consumed (by oldagents).

After banks allocate their portfolios at t, andafter consumption occurs, some fraction �t ofyoung agents is selected at random to move to“ the other location.” The value �t is commonacross locations. However, �t is a random vari-able whose value is unknown when bankschoose their portfolios. After banks allocatetheir portfolios, �t is drawn from a distributionF with probability density function f. The sup-port of f is the interval [0, 1]. The distribution Fis common knowledge.

Agents who are not relocated remain in con-tact with their bank. Moreover, the agents theytransact with can also contact their bank. Hence,agents who are not relocated can make pur-chases with checks, credit cards, or other creditinstruments. However, relocated agents losecontact with their bank (which is in their orig-inal location). Limited communication preventsthe agents they transact with from contact-ing their bank. Hence, relocated agents cannotuse credit instruments. Rather, relocated agentswithdraw deposits prior to relocation. There aretwo things they might receive. One is currency,which is used to buy goods in an agent’s newlocation. In addition, the bank might scrap stor-age investments. The consumption goods ob-tained can be transported by an agent to his newlocation. However, unliquidated storage invest-ments cannot be transported (they must remainin their original location until “maturity” ).Thus, relocation forces agents to liquidate otherassets in favor of currency. Stochastic reloca-tion, coupled with limited communication,yields an explicit model in which transactions

3 As in Douglas Diamond and Philip Dybvig (1983), allsavings are intermediated.

can and cannot be accomplished without cash.Finally, a random relocation probability gener-ates randomness in the aggregate volume ofwithdrawal demand.

Let Mt denote the time-t per capita money sup-ply. The money supply grows at the exogenouslyselected gross rate �, chosen once and for all inthe initial period. Monetary injections/withdraw-als are accomplished via lump-sum transfers toyoung agents. Let �t denote the real value of thetransfer received by young agents at t, let pt denotethe time t price level, and let zt � Mt /pt denote realbalances at t. The government budget constraintrequires that �t � [(� � 1)/�]zt. The initial moneysupply, M0 � 0, is given.

II. Bank Behavior

At date t, young agents deposit their after-taxendowment w � �t. Banks acquire cash reservesand invest in storage. Let st denote the real valueof storage and mt denote the real value of cashreserves. The bank’s balance sheet constraint is

(1) st � mt � w � � t .

The bank also chooses a schedule of returns ondeposits. Let dm(�t) denote the gross real returnoffered to agents who are relocated between t andt � 1, and let d(�t) denote the corresponding ratefor agents who are not relocated. These returnsdepend on �t. Since relocated agents must eitherbe given cash or the proceeds of liquidated storageinvestments, the bank faces the following con-straints on its choice of deposit returns. Let �(�t)� [0, 1] denote the fraction of reserves the bankliquidates at t, and �(�t) � [0, 1] denote thefraction of storage investments liquidated. Then,

(2) � t dm��t ��w � �t �

� ���t �mt �pt /pt � 1 � � ���t �rst

(3) �1 � � t �d�� t ��w � � t �

� 1 � ���t �mt �pt /pt � 1 �

� 1 � ���t �Rst .

The bank is a coalition of ex ante identicalyoung agents at t. The bank then chooses func-tions dm(�t), d(�t), �(�t), and �(�t) and

130 AEA PAPERS AND PROCEEDINGS MAY 2002

values mt and st to maximize �01 [� ln dm(�) �

(1 � �)ln d(�)] f(�) d�, subject to equations(1)–(3). Proposition 1 partially describes thesolution to this problem.4

PROPOSITION 1: Define t � mt/(w � �t) tobe the bank’s reserve deposit ratio, It �R( pt � 1/pt) to be the gross nominal interestrate, and

(4) �*t � t / t � �1 � t �It

� H�t , It � � 0, 1

(5) �� t � t / t � �1 � t �It �r/R�

� Q�t , It � � �*t , 1

for It � [1, R/r].5 Then, (i) the bank’s optimalreserve liquidation strategy sets

�6� ��� t �

�� t�1 � It �1 � t �/ t � �t /�*t �t � �*t1 �t � �*t .

(ii) The bank’s optimal storage liquidationstrategy sets

�7� ��� t �

� �0 �t � �� t

�R/rIt �t /�1 � t ���t � �� t �/�� t �t � �� t .

(iii) The bank’s optimal deposit return sched-ules satisfy

�8� dm��t � d��t � t �pt /pt � 1 � � R�1 � t �

�t � �*t

dm��t � �t /�t ��pt /pt � 1 �

d��t � R�1 � t �/�1 � �t �

�t � ��*t , �� t �

dm��t � �t /�t ��pt /pt � 1 �

4 Proofs are in the appendix of Smith (2001).5 As I discuss in more detail in what follows, if It � R/r,

then money will be driven out of circulation (scrappingstorage investments will be superior to holding cash re-serves).

� ���t �/�t �1 � t �r

d��t � �1 � ���t �/�1 � �t �

� �1 � t �R �t � �� t .

Proposition 1 asserts that banks exhaust their cashreserves if and only if withdrawal demand exceeds�*t. I associate this event with a banking panic.Second, banks provide complete insurance againstrelocation whenever they do not exhaust their cashreserves. When cash reserves are exhausted, relo-cated agents get lower returns than non-relocatedagents. Third, if �t � (�*t, �� t), a banking crisisoccurs, but banks do not liquidate investments.There are no output losses due to the crisis. How-ever, if �t � �� t, banks liquidate investments.Since the return on scrapped storage is low, bank-ing crises associated with sufficiently high with-drawal demand are also associated with outputlosses. Finally, one can associate the incompleteinsurance offered when �t � �*t with a suspensionof convertibility of deposits (see Bruce Champ etal., 1996) and the liquidation of investments withthe provision of currency substitutes. Thus, themodel reproduces all features of a banking panicwhen withdrawal demand is sufficiently high.

Proposition 1 also implies the following result.

PROPOSITION 2: The bank’s objective func-tion, as a function of t and It alone, is

�9� �0

1

� ln dm��� � �1 � ��ln d��� f ��� d�

FH�t , It �ln�t� pt

pt � 1� � R�1 � t ��

� �1 � FQ�t , It � lnt �pt /pt � 1 �

� r�1 � t � � ln�R/r��1 � FQ�t , It �

� ln�R/r��1 � Q�t , It �FQ�t , It �

� ln�R/r� �Q�t ,It �

1

F��� d�

� �H�t ,It �

Q�t ,It � �� ln��t

��� pt

pt � 1��

� �1 � ��ln�R�1 � t

1 � ���� f��� d�

� M� t , It �.

6 Since bank portfolio choices occur prior to the realiza-tion of �t, the equilibrium price level, nominal interest rate,and bank portfolio allocation display no randomness.

7 There exists a steady state with positive real balancesonly if � � 1/r. For higher values of �, scrapping storagedominates the use of reserves as a means of providing forthe consumption of relocated agents.

131VOL. 92 NO. 2 MACROECONOMICS OF INCOMPLETE MARKETS

Smith (2001) then establishes the following.

PROPOSITION 3: (i) Let (It) denote the op-timal reserve-deposit ratio. Then (It) is im-plicitly defined by

�10� M1 � t , It �

�1 � �r/R�It

�It � � 1 � �It ��r/R�It

� � 1

�It �1 � �It �� �

H�It �,It

Q�It �,It

F��� d� � 0.

(ii) The reserve–deposit ratio satisfies (1) � 1and (R/r) � 0. For It � (1, (R/r)), (It) � (0, 1)holds. Also, �(It) � 0 holds for all It.

Thus, the optimal reserve-deposit ratio is de-creasing in It. Higher opportunity costs of hold-ing reserves induce banks to economize onthem. In addition, if It � 1 (the Friedman ruleis being followed), banks hold 100-percent re-serves. Effectively, banks voluntarily become“narrow banks” and hold only government lia-bilities. When cash reserves earn the same rateof return as other investments, banks hold onlycurrency, as this has the advantage of insuringagainst high withdrawal demand.

Proposition 3 has a corollary. It is easilyshown that H1(t, It) � 0 � H2(t, It).Therefore �*t � H[(It), It] is decreasing in It,so that increases in the nominal interest rate (theinflation rate) reduce �*t and increase the prob-ability of a panic [1 � F(�*t)]. That bankingcrises may cause either no output loss or someoutput loss and that higher inflation increasesthe probability of a crisis are consistent with theobservations cited previously. In addition, theprobability that some output loss will result atany date is 1 � F{Q[(It), It]}, which is alsoincreasing in It. Thus low nominal interest rates(low inflation) imply low probabilities of outputlosses due to a crisis. Nonetheless, it is neveroptimal to follow the Friedman rule.

III. General Equilibrium

An equilibrium occurs when the demandfor money and the supply of money are

equal.6 Since all beginning-of-period moneydemand derives from banks’ demands forcash reserves, the demand for money and thesupply of money are equal if

(11) zt �It ��w � � t � t � 1.

Moreover, �t � [(� � 1)/�] zt. Using thisrelation in equation (11) yields

(12) zt �It �w

1 � �� � 1�/��It �t � 1.

Then, the net-of-tax income of young agents att is

(13) w � � t w

1 � �� � 1�/��It �.

By definition, ( pt/pt � 1) � ( zt � 1/�zt) andIt � R( pt � 1/pt) � �R( zt/zt � 1). Then, fromequation (12), the money market clears att if

�14� zt�1 �Rzt ��1� �zt

�w � �� � 1�zt� t � 1.

Equation (14) describes the equilibrium evolu-tion of real balances. If It � 1 holds for all t,there is a continuum of perfect foresight equi-libria. One is a steady state in which real bal-ances are positive.7 Here It � �R, and

�15� z w��R���1 � ��� � 1�

� ���R�� .

Notice that I � 1 if and only if � � 1/R.Moreover, when � � 1/R, there is a continuumof perfect-foresight equilibria with zt 2 0 andIt 1 R/r as t 3 �. This indeterminacy isconventional in pure-exchange overlapping-generations models. I henceforth confine atten-tion to steady states.

The Friedman rule is implemented, in asteady state, by setting I � �R � 1. Since

132 AEA PAPERS AND PROCEEDINGS MAY 2002

(1) � 1, equation (15) implies that the Fried-man rule leads to steady-state real balancesz � w/R.

I now ask what choice of a steady-statenominal interest rate8 maximizes steady statewelfare.

IV. Welfare

The gross rate of money creation that sup-ports I as the steady-state gross nominal interestrate is � � I/R @I � [1, (R/r)]. Moreover, if� � I/R, then � � [(I � R)/I] z, and the net oftransfer income of a young agent is

�16� y�I� � w � � w��1 � ��R � I�

I ��I�� .

LEMMA 1: Some properties of y(I) are thaty(1) � w/R, y(R/r) � w, and

(17)Iy��I�

y�I�

R�I��1 � �R � I�/RI��I�/�I�

I � �R � I��I� � .

Proposition 2 implies that the steady-stateexpected utility of a young agent is given by

(18) W�I� � M�I�, I � lny�I�.

PROPOSITION 4: (i) W(1) � ln w; (ii)W�(1) � 0.

Proposition 4 states that it is never optimal tofollow the Friedman rule. Steady-state welfareis always increased by raising the nominal in-terest rate above zero.9

Intuitively, the Friedman rule is not optimalbecause, when I � 1, money becomes a verygood asset. Indeed, it becomes such a good asset

8 Choosing a steady-state nominal interest rate is equiv-alent to setting � � I/R.

9 The initial old always prefer the Friedman Rule, as thismaximizes the value of their real balances. This is not com-pelling, however. The initial old like the Friedman rule notbecause it has desirable allocative properties, but because itmaximizes the demand for money. This could be accomplishedby other means, such as imposing 100-percent reserve require-ments. In fact, 100 percent reserve requirements were advo-cated by Milton Friedman (1960).

that banks hold it to the exclusion of everythingelse. In effect, banks voluntarily become 100-percent reserve entities. They therefore becomevery safe: the probability that banks exhaustreserves is zero under the Friedman rule. None-theless, this is not optimal. When banks makeno socially productive investments, this com-pletely attenuates their intermediation function,which is socially undesirable.

Indeed, when the Friedman rule is followedand banks hold 100-percent reserves, they pro-vide complete insurance against the event ofrelocation in every state of the world. Whenagents are completely insured, they are locallyrisk-neutral. As a consequence, young agentswill prefer to take “small bets” that have posi-tive expected payoffs. Investing in storageyields a positive expected return, although itinvolves some risk that banks will exhaust theirreserves. Young agents will prefer that at leastsome storage occur, but to induce banks tomake storage investments, I � 1 must hold.

I next ask whether there is an interior op-timum for the nominal interest rate. The an-swer is yes, as shown by the followingproposition.

PROPOSITION 5:(i) W(R/r) � ln w � [�0

1 F(�) d�]ln R � [1 ��0

1 F(�) d�]ln r.(ii) W�(R/r) � 0.

Thus, it is never optimal to drive real balancesto zero. Therefore, an optimal choice of thesteady-state nominal interest rate exists in theinterval (1, R/r). In addition, Propositions 4and 5 imply that the Friedman rule generates alower level of steady-state utility than the policyof driving money out of circulation (maximiz-ing inflation) if and only if

� �0

1

F��� d�� ln R

� � 1 � �0

1

F��� d�� ln r � 0.

Thus, the Friedman rule can be an extremelybad policy.

133VOL. 92 NO. 2 MACROECONOMICS OF INCOMPLETE MARKETS

V. The Discount Window

Other methods exist for implementing theFriedman rule. One has the government lendfreely at the discount window at a zero nominalinterest rate. An absence of arbitrage opportu-nities then implies It � 1 @t.

Since in each period banks repay the amountthey borrowed from the discount window lastperiod, discount-window activity does not af-fect the time path of Mt. Of course the moneysupply can still be increased or reduced vialump-sum transfers. Thus the money stock canstill grow at the gross rate �. Since ability toborrow from the discount window at a zeronominal interest rate implies that It � 1 or that( pt/pt � 1) � R, it follows that per capita realbalances evolve according to

(19) zt � 1 �Rzt .

There are now three possibilities. One is that�R � 1. Then, equation (19) implies that realbalances grow without bound. This is inconsis-tent with the existence of an equilibrium. Thesecond possibility is that �R � 1. Then realbalances shrink over time. Moreover, the initialstock of real balances is indeterminate. Hence,the situation with �R � 1 closely resemblesthat obtaining when the government sets � �1/R.

If �R � 1, equation (19) implies that theeconomy is always in a steady state. Steady-state real balances are indeterminate, as is theprice level. Indeed any value z � (0, w/R]constitutes an equilibrium.10 There is a massiveindeterminacy of equilibrium.

VI. Conclusion

Historical banking panics were associatedwith high withdrawal demand and exhaustion ofbank reserves. Here, a banking system confrontsstochastic withdrawal demand. Members of thissystem hold cash reserves to insure depositorsagainst relocation, and banks choose reservesoptimally given their opportunity cost as re-flected in the nominal interest rate. The lower

10 If z � 0 holds, then the price level is infinite, anddiscount window lending (which is done with currency)cannot be of benefit to the economy.

this opportunity cost, the more reserves bankshold. Thus, low nominal interest rates make thebanking system relatively safe. As nominal in-terest rates go to zero, banks hold 100 percentreserves and become completely safe.

Nonetheless, the Friedman rule is not opti-mal. It makes money too good an asset, so thatbanks do not make other investments. This issocially wasteful. Moreover, this finding is con-sistent with observation: in periods with lownominal interest rates (as in the United Statesduring the Great Depression or in Japan over thelast decade) relatively little investment occurs.These episodes have been regarded as charac-terized by poor economic performance. Theanalysis here suggests why this happens.

Much literature shows the Friedman rule tobe optimal. Relative to that literature, I intro-duce a role for financial intermediaries. Theresults suggest that, when intermediation istaken seriously, pursuing the Friedman rule ei-ther is not optimal or leads to massive indeter-minacies, or both.

REFERENCES

Boyd, John H.; Gomis, Pedro; Kwak, Sungkyuand Smith, Bruce D. “A User’s Guide toBanking Crises.” Working paper, Universityof Texas, 2001a.

Boyd, John H.; Kwak, Sungkyu and Smith, BruceD. “The Real Output Losses Associated withModern Banking Crises.” Working paper,University of Texas, 2001b.

Caprio, Gerard, Jr. and Klingebiel, Daniela.“Bank Insolvency: Bad Luck, Bad Policy, orBad Banking?” in M. Bruno and B. Ples-kovic, eds., Annual World Bank Conferenceon Development Economics. WashingtonDC: World Bank, 1997, pp. 79–104.

Champ, Bruce; Smith, Bruce D. and Williamson,Stephen. “Currency Elasticity and BankingCrises: Theory and Evidence.” CanadianJournal of Economics, November 1996,29(4), pp. 828–64.

Demirguc-Kunt, Asli and Detriagiache, Enrica.“The Determinants of Banking Crises: Ev-idence from Industrial and DevelopingCountries.” Working paper, World Bank,Washington, DC, 1997.

Diamond, Douglas W. and Dybvig, Philip. “BankRuns, Liquidity, and Deposit Insurance.”

134 AEA PAPERS AND PROCEEDINGS MAY 2002

Journal of Political Economy, June 1983,91(3), pp. 401–19.

Friedman, Milton. A program for monetary stabil-ity. New York: Fordham University Press, 1960.

Noyes, Alexander D. “A Year After the Panic of

1907.” Quarterly Journal of Economics, Feb-ruary 1909, 23(2), pp. 185–212.

Smith, Bruce D. “Monetary Policy, Banking Cri-ses, and the Friedman Rule.” Working paper,University of Texas, 2001.