Liquidity, credit and bank money.*∗

Alistair Milne††

(Incomplete draft, not for quotation). November, 2010

Abstract

Abstract: Many researchers are now extending standard macromone-tary models to include financial intermediation and to allow for the impactof financial sector stresses on the wider economy. This is being done inmany different ways and the resulting models exhibit widely varying dy-namics and sensitivities. This paper takes a step backward, examiningthe somewhat simpler setting of a world with a single period of employ-ment and with uncertainty in both the level and the timing of subsequentoutputs and income. Financial frictions are imposed which create a rolefor both banks and for a central bank. Banks monitor and insure risks ofproduction, this is the credit side of the model, and provide bank money,used a means of payment to overcome transcations costs. Because banksare unable to borrow freely from each other, the central bank provision ofreserves and loans against collateral, plays a crucial role in determiningthe willingness of the banks to both create credit and money. This settingappears appropriate for investigation of central bank policies, includingreserve creation, reserve remuneration and ’lender of last resort’, and theireffects on the nominal price level (the price of money) and rates of returnon financial assets.

1 Introduction

This paper proposes a general equilibrium model of central bank provision ofliquidity to commercial banks. Banks provide ‘loan’(insurance) contracts thatprovide partial insurance against idionsyncratic risk. They also provide depositsused as means of payment. Uncertainty in the timing of payment flows createsshort term exposures between banks. The central bank can then play a crucialrole in intermediating these exposures, through the provision of reserves (whichcan be used as a means of payment between banks), either through purchase(in exchange for financial securities) or through temporary loans.

∗* The views expressed here are not necessarily those of the Bank of Finland. Any re-maining errors are my own responsibility. I am grateful for valuable discussion with WillRoberds.†† Faculty of Finance, Cass Business School, City University, London; and Monetary Policy

and Research Department, Bank of Finland, Helsinki email: amilne@city.ac.uk.

1

This set up is explored in order to address some basic questions about the op-eration and role of central banks. Standard monetary textbooks ([Woodford(2003)],[Walsh(2010)]) partition the activities of the central bank from the rest of theeconomy, by appeal to the standard central bank practice of imposing a ‘cor-ridor’system of demand for central bank reserves. with penalties imposed bythe central bank whenever an individual bank’s reserves with the central bankexceed or fall short of a relatively narrow range. This creates a demand for re-serves and the central bank can then control very short term (overnight) interestrates through its supply of reserves.This is a useful expositional device for analysing monetary policy, but it

pushes some obvious concerns into the background. First, while the ‘corridor’analysis does allow the central banks to temporarily lend reserves to the market(this is the mechanism thorugh which it reduces overnight interest rates), itdoes not consider the implications of differences in individual bank borrowingfrom the central bank. A bank may borrow by offering collateral, but doing thisreduces the claims of other bank liability holders, notably depositors. If costlybank failure is possible, and there is a government backed deposit insurancescheme, then this increases the exposure of taxpayers.Second, there is the possibility of bank depositors withdrawing deposits from

a bank, not because of underlying payments needs, but because of concernsabout the safey of their deposits. This is of course formerly modelled by [?] andothers; but those analyses have paid little attention to the role of the centralbank. Can the central bank eliminate such runs by providing the facility toborrow reserves (the more recent work of [Martin(2006)] suggests it can)? Arethere limits on the effectiveness of such policy or circumstances in which, despitereserve borrowing, bank runs of this kind can still occur?Third, does the actual level of central bank reserves matter? Recently, in

the wake of the global financial crisis, many central banks, notably the FederalReserve and the Bank of England, have suspended their corridor systems. Theyhave expanded their reserve base considerably, through purchase of governmentbonds and other securities; and also, in order to offset the impact on bank in-come, offered compensation (at relatively low rates of interest) on bank reserves.Overnight rates of interest on interbank borrowing are effectively maintained atclose to the rate of remuneration on bank reserves; and in any case there isrelatively little need for banks to borrow interbank. Does this imply that thetotal level of reserves are not, ultimately, of any economic relevance, and thatmonetary policy can be effectively operated, through the rate of remunderationon reserves? Or does large scale reserve creation, if it is not reversed,In order to address some of these questions, it is necessary to develop a fuller

characterisation of the role of the central bank in the macroeconomy. In orderto achieve this goal, this paper proposes a relatively simple, two period model(but with uncertainty of payment flows within the second period). This makesit possible to run simulations of payment uncertainty, based on frictional costsof both acquiring means of payment and, in an extension, of dealing with thefailure of financial institutions.The remainder of the paper is arranged as follows.

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Section 2 discusses related literature.Section 3 outlines the model.Section 4 (not yet complete) presents baseline simulations of payments flows,

without taking into acocunt the possibility of bank failure. Here all banks areessentially the same.Section 5 (ommitted entirely from this early draft) presents simulations in

which there are differences in the performance of banks and as a result a possi-bility of bank failure.Section 6 summarises and concludes.

2 Related literature

The work presented here can be related to both to the substantial historical andinstitutional literature on the creation and role of central banks; to the morerecent technical literature on interbank lending and the provision of central bankliquidity; and to recent attempts to include financial intermediation in dynamicstochastic general equilibrium models. This section discusses the connectionsto each of these literatures.

2.1 The historical and institutional literature on centralbanking

While there has until recently been relatively little formal modelling of theeconomic role of central banks, outside of their role in the operation of monetarypolicy, there is a substantial historical and institutional literature, dating backto the late 18th century on the relationship between central banks and privatecommercial banks.1

One central theme of this literature are the services provided by centralbanks to commercial banks.2 Banks need arrangements to make payments toeach other. Historically this could be achieved by settlement in preciuos metal(specie), but bank early realised that substantial cost savings could be achievedby depositing specie with a single bank, and settling payments on the accountsof this bank. Not only does this reduce costs of security and transport, it alsoreduces the amount of unremunerated reserve assets that banks need to hold.Another theme has been the role of the central bank in influencing short

term rates of interest, in particular through the practice of regularly discountingacceptable collateral, i.e. lending reserves for short periods, typically overnightagainst the pledge of good quality collateral and in expectation of immediaterepayment.

1 [Congdon(2009)] provides an insightful review of this literature.2Such services are central in the longstanding debate on whether central banking can be

seen as a response to the commercial needs of private banking institutions pursuing profit fromdeposit taking and lending activities; or as an undesirable imposition by central government.Many observers, notably [Smith(1936)], have argued that commercial banks should be free ofrestrictions, such as those that have limited the right to issue bank notes to a single centralbank.

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Finally much attention has been paid to the role of the central bank as‘lender of last resort’, in order to stem banking panics. The classic statementof this doctrine is provided by [Bagehot(1873)] as the provision of liquidity bythe lender of last resort freely, but at a penalty rate, high relative to marketrates of interest in normal times. This can be distinguished from the routinepractice of influencing money market interest rates, because such lender of lastresort facility is provided at a time when other sources of funding are beingwithdrawn. Thus, unlike the case of routine money market discounting, thereare real doubts about whether borrowing institutions will be able to repay asagreed.It is not possible to incorporate all of this historical and institutional lit-

erature into a single formal model. The focus here is on the role of centralbanks deposits as the means of payments settlement between banks. Thus, inline with this traditional literature, but in contrast to much standard monetarymodelling, central bank liquidity provision plays a central role. In this set uphowever is it also possible to consider arrangements not much addressed in thetraditional literature, for example the possibility of remuneration of reservesand the implications this has for monetary transmission.

2.2 Interbank lending and central bank liquidity

Formal modelling, over recent years, has begun to develop a number of insightsinto the role of the central bank as a provider of liquidity, but largely focussedon interbank markets, and with relatively sketchy treatment of the wider econ-omy. Contributions include [Freixas and Holthausen(2005)], who examine thesustainability of interbank markets in cross-border lending when there are rel-atively large information assymetries, and [Martin(2006)], who shows that the[Diamond and Dybvig(1983)] run equilibrium can be eliminated through theprovision of central bank liquidity, and that unlike when deposit insurance isused to combat bank runs, this need not create moral hazard.The global financial crisis, and the problems that this created for central

bank operations, have triggered a considerable amount of new work on interbankmarkets and central bank provision of liquidity. These include [Freixas et al.(2010)Freixas, Martin, and Skeie],[MARTIN(2009)] and [Allen et al.(2009)Allen, Carletti, and Gale]; as well asthe empirical study of [Ashcraft et al.(2008)Ashcraft, McAndrews, and Skeie]which uses detailed microdata on federal funds market lending to investigate amodel of interbank liquidity provision.There are however limitations with this work. They are all partial equi-

librium analyses, usually based on exogenous payment flows, and an inelasticdemand for credit by commercial banks in order to settle payments. There hasnot been much work relating payments institutions, interbank markets, or therole of the central bank as a provider of liquidity to macroeconomic outcomes.

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2.3 The incorporation of financial intermediation into mod-els of dynamic stochastic general equilibrium

The global banking crisis has also produced a large number of studies whichintroduce banks and financial intermediation into the standard New Keynsianmacroecomic modelling framework.3 Many of these papers conclude that bankbalance sheets have a major impact on monetary transmission. However thesensitivities that emerge from these models vary considerably (see for examplethe simlations reported by [Group(2010)] who report output impacts of changingbank capital standards that vary, according to model, by a factor of at least ten.There are several diffi culties with these amendments of the standard New

Keynsian models of monetary policy. First, most obviously, are the challengesof fitting these models to data. As the number of parameters are increased itbecome harder to pin down. A second problem is a lack of correspondence be-tween the microeconomics of these models and actually banking arrangements,for example some papers, such as [Meh and Moran(2010)], assume that capitalaccumulation depends on bank financing (more accurately the amount of capitaldepends on the net worth of the combined banking and entrpreunerial sectors).Also, while bank balance sheets play a role, the linearisation routinely appliedto these models . Finally, while models of this kind can generate substantialfeedbacks from financial intermediary balance sheets onto, the models them-selves are relatively complex and it can be diffi cult to understand the economicintuition underlying the reported simulation results. Another serious omissionis any discussion of payments flows and hence of potential illiquidiity.The approach taken here is to develop a single period general equilibrium

model of the macroeconomy, but one in which payment flows are explicitlymodelled, and hence in which the central bank liquidity provision matters.

3 Specification of the model

3.1 Overview

There are two periods t = 0, 1. In period 0, N households provide labour toM firms, with the labour of household n represented by l and the employmentof firm m represented by e (because households and firms are ex-ante identicallabour supply and employment are the same for all households.) The firms areowned locally by the labour force that works for them, so each firm employmentis given by e = (N/M) l = µl.

3There seem to be in excess of thirty examples of DSGE models (avaialable injournal articles, working paper series and unpublished mimeo) extended to incor-porate either banking sectors or interest rate spreads. Prominent contributions in-clude [Christiano et al.(2010)Christiano, Motto, and Rostagno], [Meh and Moran(2010)],[Andrea Gerali Stefano Neri and Signoretti(2010)], [Curdia and Woodford(2010)],[Dib(2010)], [?], [Gertler and Karadi(2010)], [Gertler and Kiyotaki(2009)]and[Gilchrist et al.(????)Gilchrist, Ortiz, and Zakrajsek]. Earlier models, predating thecrisis, that introduce similar frictions include [Olover Huelsewig and Wollmershaeuser(2006)]and [Goodfriend and McCallum(2007)]

5

In period 1 this labour yields non-storeable output, unique to each firm,represented by ym. This output is traded and households consume these goods.Household utility depends on the consumption of goods from all firms.Altogether there are four sectors:

• N households (or just households), providing labour and consuming goods.

• M co-operative firms, narrowly held by a group of households, each pro-ducing good labelled m = 1 . . .M

• Commercial Banks, possessing a monitoring technology that allows them,at a cost, to evaluate the performance of firms; and a payments technologythat allows them to offer liabilities (deposits) that are accepted as meansof payment

• Government, employing household labour in periods 0 and 1 to producea public good, issuing bonds to finance this activity, and levying taxes inperiods 1 on households to pay for its activities.

The model includes the following seven financial assets and liabilities

1. One period government bonds, issued at t = 0 and held by commercialbanks and households.

2. Firm equity, narrowly held

3. Commercial bank equity, broadly held by households

4. Commercial bank loans to households

5. Bank deposits, held by households (this is ‘money’, we discuss the demandfor money below)

6. Central bank reserves held as a means of payment between banks

7. Interbank loans

3.2 Household-entrepreneurs

3.2.1 Technology

There are M firms indexed by m = 1, . . .M . The log normally distributedoutput of firm m is represented by:

ym = e exp [θ + εm] = e exp θm

Where e is period 0 employment by the firm (by symmetry this is the samefor all firms), θ is a normally distributed aggregate productivity shock,and εm abinomial firm specific productivity shock. The aggregate shock has a standarddeviation of σθ. and σε respectively. The firm specific shock is assumed to takethe form:

6

3.2.2 Preferences

Household-entrepreneurs (household) preferences are represented by the follow-ing constant elasticity of substitution/ constant relative risk aversion utilityfunction:

u(ln, cn (θ)) =1

1− δE

(1− ln)1−δ

+

(N∑m=1

(cn,m (θ))1−γ−1

) 1−δ1−γ−1

where ln is labour supply by household n in period 0 and cn,m is consumptionby household n of the output of firm m in period 1. γ is the elasticity ofsubstitution between the different goods m. δ is the constant coeffi cient ofrelative risk aversion to uncertain consumption.Omitting the subscript n, the marginal rate of substitution between two

differentiated goods at time 1 for given aggregate output shock θ is given by:

MRSjk =∂U

∂cj/∂U

∂ck=

(cjck

)−γ−1(1)

and this must be the same for every pair of goods for every household (otherwisethere are unexploited gains from trade).

3.2.3 Relative prices and nominal output

The reason for choosing the CES utility function is so that the distribution ofincome amongst housholds does not affect relative prices. This standard resultfollow because in equilibrium (1) is the same for every household, implyingthat each individual household consumes different goods in the same commonproportions, differing only in their shares of aggregate consumption; equilibriumrelative prices then satisfy:

pnpm

=

(ynym

)−γ−1(2)

Here bank money serves as the unit of account and so pn represents the priceof good n at time t in terms of bank money.

The assumption of a log-normal distribution of output is a further conve-nience. The aggregate shock θ has no impact on relative prices (θ increases allindividual outputs by the same percentage, so the lhs of equation (2) remainsunchanged).Then, under the CES assumption, the consumption by household n of good

m can be represented as cn,m (θ) = χnym (θ) where χn is the expenditure shareof household n of all goods, and thus utility of household m can be representedas:

u(χn) = u(ln, cn (θ)) =1

1− δE[(1− ln)

1−δ+ χ1−δ

n Z (θ)1−δ]

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where Z (θ) represents aggregate real output (see further discussion below).The income of household n from sale of output in period t is ym(n)pm(n) ×

(N/M). We further assume that the elasticity of substitution γ > 1. Thisensures that each household’s share of aggregate income from output is alwaysincreasing in the level of household output. (Proof. this income share of firmsm can be written, using (2), as: sm = pmym/

∑pjyj = 1/

∑(pjyj/ (pmym)) =

1/∑

(yj/ym)1−γ−1 , so γ > 1 implies that ∂sm/∂ym = −(1−γ−1)

(−1/ (ym)

2)∑

(yj/ym)−γ−1

/[∑

(yj/ym)1−γ−1

]2>

0).We can say more about the impact of output shocks on relative prices and

income from production (without having to analyse either the nominal price levelor the distribution of wealth and consumption across households). Substitutingin for the shocks into (2), shows that:

pjpm

=

(exp [θ + εj ]

exp [θ + εm]

)−γ−1= exp

[−γ−1 (εj − εm)

]We can also state the price of good m (in terms of units of money) as:

pm = p (θ) e−γ−1

exp[−γ−1εm

](3)

where here p (θ) is a scaling factor for goods prices that also represents the gen-eral money price level (the level of p(θ) is expected to be a function of the statevariable (θ), it may also depend on individual firm shocks ε = {ε1, . . . εM}′).The income from production of good m is then given by:

pmym = p (θ) exp θ exp[(

1− γ−1)εm]e1−γ−1

so aggregate nominal output is given by:

Y = p (θ) exp θ∑m

exp[(

1− γ−1)εm]e1−γ−1

and aggregate real output (nominal output deflated by the price index p (θ)) by:

Z = exp θ∑m

exp[(

1− γ−1)εm]e1−γ−1 (4)

3.2.4 Risk sharing

Households are exposed to aggregate and idiosyncratic risks. Aggregate risksare undiversifiable. These can be priced, but there is no gain from trading ofthis risk. In addition there is incomplete insurance against idionsyncratic firmproductivity shocks εm. This insurance is provided only when banks monitorand confirm that a household idiosyncratic productivity is less than some given(possibly state dependent) threshold εm < ε∗ in which case the firm’s income isprotected.

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Specifically, in terms of the numeraire bank money, the bank provides inperiod 0 a one period ‘loan’to household m with an amount rAa repayable attime t = 1, when the market m meets.The repayment given by min [rAa, pmym] so firm income from production

(taking into account this insurance) is given by:

xm =

{pmym − rAa when pmym ≥ rAa

0 when pmym < rAa(5)

We assume a binomial (good/ bad) output shock exp[(

1− γ−1)εm]∈{

exp[(

1− γ−1)ε̄], 1− exp

[(1− γ−1

)ε̄]}, with probabilities {1− φ, φ} and ε̄ <

0 . We impose the further restiction, so the expected multiplicative shock isunity) This implies that:

xm =

{emp (θ) exp (θ) exp

((1− γ−1

)εm)e1−γ−1 − rAa probability 1− φ

0 probability φ

altogether there are φM success firms and (1− φ)M failure firms.Monitoring of a loan absorbs ζ/M units of nominal aggregate output Z, this

is offset by charging a relatively high interest rate on bank borrowing i.e. bor-rower share of wealth is reduced by ζ (and other borrower shares correspondinglyincrease).

3.3 Sequence of events

Time periods (0, 1) are represented by superscripts on financial assets and liabil-ities (denoted in upper case) and on gross financial returns, labour, output andconsumption flows (denoted in lower case). Decisions are made in the followingorder:Period 0

1. [Period 0 government spending, government financing and monetary pol-icy announcements] At the beginning of period 0 government decides howmuch labour to employ lG (with labour per household provided to govern-ment given by lG/N .) The central bank decides (in terms of an arbitraryunit of account that it itself declares) how many reserves to create in pe-riod 0, H, what interest they will credit on these reserves at the beginningof period 1, r, and how much they will purchase of government bonds(promising repayment of BC in period 1 units of the arbitrary unit ofaccount declared by the central bank.). Equity in the central bank ECis held by the government and so the central bank balance sheet satisfiesthe balance sheet identity EC = BC −H. Total government bond issue isB = BC +BB +BH . The market value of these bonds is given by B/rG.

2. [Period 0 insurance against idiosyncratic shocks] Commercial banks andfirms then negotiate period 0 lending there is per firm, in order to pro-vide partial insurance against idiosyncratic shocks. Each firm promises to

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repay an amount A (θ) in period 1 (as specified in (5) this is an insuredrepayment, it pays less if output is low). As discussed above each firmborrows an amount A0 = rA(θ)−1A(θ) in order to secure this insurance.

3. [Period 0 labour market ] Labour of e0n is employed by household n at a

wage rate of w0. Wages are paid in bank money, but part of this bankmoney is then used to pay wages for the firm (that part which is notborrowed from the bank), so together with government employment of l0G,net of household purchase of government bonds, this generates depositsfor household n of rD−1D = rA(θ)−1A(θ) +

(lG/N − rG−1Bh/N

).

4. [Period 0 financial markets] There are two other assets traded in period 0.Bank equity capital is initially distributed to housholds without payment.Its period 0 market value E is established by purchase and sale betweenhouseholds. For our baseline analysis we will assume that there is perfectcompetition within bank sectors so that profits are driven to zero and E =0 (later we will justify this). Households can also purchase governmentbonds B0

H paid through by reduction of deposits (later we will consideradditional sale of bank equity to households i.e. making bank capitalendogenous and creating a role for bank capital regulation.).

Period 1

5. [Period 1 productivity revealed ] At the beginning of period 1, aggregateand sectoral productibity are revealed and firms learn their individualidiosyncratic productivity. Deposit balances are incremented to D.

6. Firms are chosen in random order for the conduct of an m-market, wherethe output of the firm is assessed and sold. This ordering becomes publicknowledge at the beginning of period 1.

7. [m - market] At each m-market in turn:

• firm m announces its productivity and price of good m is established

• bank monitors output• households that need to finance their share of sales by selling assetsdo so, in bank equity and gov bond markets

• output is sold for cash, bank and household owners of the firm taketheir shares of the proceeds in cash

8. Government taxes are paid, again in cash, and government bonds arerepaid.

9. Bank equity holders are repaid (in equilbrium, with no aggregate bankshocks, I think bank equity turns out to have zero value)

10. Period 1 output is consumed. The enforcement of individual budget con-straints imply that all assets and liabilities are extinguished at the end ofperiod 1 and hence there is no settlement problem between banks.

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3.4 Solution

Endogenous variables, to be determinedpm for m = {1, . . .M} simplification: choice of CES and shocks so

relative prices determined, only p(θ) remains undetermined, assumed p(θ) =exp−θ

EF total private sector employment (e = EF /M is employment by firms,lF = EF /N is non-government employment per household).

cn,m for n = {1, . . . N} simplification: choice of CES and distribu-tion of shocks so relative consumption is same for all households, only χn, theconsumption shares of individual households remains to be determined.

w the wage rate. We anticipate that this will be determined by a homogene-ity in terms of level of initial reserves R

A the amount of borrowing by firms (so implicity rAA is the amount ofinsurance)

D stock of household wealth held as depositsB stock of government bond issued (promised repayments in money terms)Break down of this debt holding households, banks, central banks BH , BB ,

BCrA interest rate on depositsrD interest rate on depositsrB market yield on government debtExogenous variables/ decided directly by shocksym/e productivity of the M firmsgovernment employment lgPolicy variablesR reservesLG government employmentτ taxes levied in period 1 (subject to public sector terminal budget con-

straint)rR interest rate credited on central bank reservesParametersξ monitoring cost as a proportion of aggregate GDPN number of householdsM number of firms, owned by these householdsUseful summary variablesZ =

∑pmym is nominal output in period 1. z = Z/N is per capita nominal

output. zn = χnZ is the nominal consumption of household n. ym = pmym isthe revenue from production of firm m. xm is the revenue of firm m.

Y = Z/p (θ) is an index of real output in period 1. Note that this is deter-mined by CES/ relative price assumptions. y = Y/N is per capita real outputoutput.Counting:

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Variable Number Commente 1χn N − 1 Outcome of portfolio choices and random orderingw 1 Depends on homogeneity wrt RA 1D 1B 3rA 1rD 1rB 1

Accounting identitiesAccruals based balance sheets (period 0, assumes no "capital injection" for

banks)

Identity SectorwL = D + r−1

B BH + EF HouseholdA+ EF = wLF Firmsr−1B B = wLG GovernmentD = A+R+ r−1

B BI Intermediaries (banks)R = r−1

B BC Central BankCross-sectoral identiesL = LF + LGB = BH +BI +BCUsing these two identities the five balance sheets then sum, as they should,

to zero). The main issue is determining (jointly) the holding of bonds by thethree groups of institutions and the determination of the three interest rates r,and also the holding of deposits and extend of loans.In addition there is a terminal identity for government (here revenues are

per capita taxes τ plus profits from central bank E1C) τN+E1

C = B which, sincerRR + EC = BC , implies for the entire public sector, government plus centralbank:

τN = rRR+BH +BIBudget constraint for houshold sector (recall that all "frictions" are modelled

as distributional changes amongst households) can be thought of as rDD +BH + ∆F + ∆I − T =

∑pmym where ∆F , and ∆I are dividends paid by firms

and by intermediaries, respectively. Note however that this budget constraintnecessarily collapses to the identity

∑pmym =

∑pmym once all cash flows are

passed through the economy i.e. the budget constraint does not determine thenominal price level p (θ). There are two possible solutions.

1. First (but this is only possible in a multperiod setting) we can have slug-gish wage and/or price adjustment.

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2. Second (adopted here) we can have implicit indexing of all nominal fi-nancial contracts with period 1 payoffs (rD, rA, B). We assume our pricelevel can be written as p (θ) = p̄

w exp−θ and that all nominal contracts arewritten corrected for (undetermined) period 1 real price level p̄/w. Thishowever comes with a cost, we are unable to analyse inflation (but we cananalyse the determination of the wage rate w and hence of the price level,we expect homogeneity in R).

Resource constraint is that∑χn = 1. This is determined automatically be-

cause various period 1 frictions affecting households, i.e. costs of bond and bankequity sales, are specified in terms of shares of aggregate nominal income. Fric-tions affecting banks (bank borrowing costs) do not directly affect householdsbecause households all hold equal shares in bank equity.

FrictionsThese are the key ingredients of the model. We have (upto) three frictions.

First the monitoring costs required when financial intermediaries insure againstlow output. Second the frictional costs that arise when a household needs tomake a purchase at an m-market, but does not have suffi cient funds in orderto do so. Third the frictional costs that arise when a bank has to borrow ininterbank markets, especially if there are doubts about the quality of its assets,and when a bank has to be resolved (with transfer of assets and deposits toother banks).We fully model the first friction. The second and third frictions we model

in a a stylised fashion, imposing a fixed cost of sale of bonds (so bonds are onlysold once by household) and a fixed cost of borrowing interbank (so again thebank only borrows once), and finally a fixed cost of resolving a failed bank.It appears that only the second and third frictions are crucial to our mod-

elling.We simplify (cut a corner) by assuming that all uncertainty is resolved at

beginning of period 1, so households can see immediately, their own wealth andpayments needs (but crucially banks cannot make the same calculations.... sostill a problem of interbank borrowing).OptimisationsBegin with the optimal ‘insurance’contract against a low idiosyncratic pro-

ductivity shock. We can look at this in two ways. First as a pure insurancearrangement, with a payment made (from the firm to the bank) of λ+z in thecase of high income z

(1 + φ−1ψ

)a probability φ event, and a payment recieved

(from the bank to the firm) of λ−z in the case of low income z(

1− (1− φ)−1ψ)

a probability 1 − φ event, and with a monitoring cost in this latter case givenby ζz. Here z is percapita nominal output.We have four conditions to be met:

1. Incentive compatibility, avoiding lying about being in a good state: 1 −(1− φ)

−1ψ − λ+ ≤ 1 − (1− φ)

−1ψ + λ−, which is satisfied since the

payments are positive λ+ + λ− > 0

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2. Non-negative profits for the bank: φλ+ ≥ (1− φ)(λ− + ζ

). If we will as-

sume competition reduces these profits to zero, then this implies φ1−φλ

+ =

λ− + ζ. This sets a constraint on ζ.

3. Highest possible utility for the household-firm, E[u(λ+, λ−, ζ)

], subject

to the non-negative profit constraint for the bank

4. Participation constraint for the household-firm, E[u(λ+, λ−, ζ)

]≥ E [u(0, 0, 0)]

which sets an additional constraint on the value of ζ.

One possible outcome is full insurance: λ+ + λ− =(φ−1 + (1− φ)

−1)ψ, in

which case (from the zero profit condition) we have λ+ = (1− φ)((φ−1 + (1− φ)

−1)ψ + ζ

)and λ− = φ

(φ−1 + (1− φ)

−1)ψ − (1− φ) ζ. This will be the case if ζ is small

enough. Another case is where ζ is so high that the participation constraintdoes not apply and no insurance is possible.The intermediate case is partial insurance, with λ+ +λ− > 2ψ , so that now

payoff to the household in the two states is still unequal. We will impose limitson ζ so that we obtain this case.We can re-express this insurance contract as a loan contract, as follows.

With a loan contract the household-firm pays a = A/N in the event of a goodoutcome (high revenues), but pays only pmym in the event of a poor outcome(low revenues). Also the firm pays rA > rB on its borrowing, so this is aneffective transfer of resource to the bank. Thus in terms of the correpondingnotation, we have pmym − rAa =

(λ+ + λ−

)y.

The period 0 financial wealth of household n obtained from employment canbe divided into its share of insurance (lending), deposits and bond holdings asfollows w0

n = lnw = dn + r−1B bn − am(n). Its corresponding period 1 wealth is

w0n = rDdn + bn +

(p+m(n)y

+m(n) − rAam(n)

)µI+ − J (m) where p+

m(n)y+m(n) =

y(1 + φ−1γ

)Output is partially insured. This is represented by the indicator

function I+ which takes the value 1 if output of firm m(n) is high, and 0otherwise. J (m) y rerpresents teh costs of monetary frictions. The consumptionshare of household n is χn = w0

n/Y where Y is aggregate nominal output.µ = M/NExpected utility of household n can then be written (in terms of the con-

sumption fractions when the firms partially insured is high (χ+n ) or low (χ

−n ))

as:

u(χn) =1

1− δ φ[(1− l)1−δ

+ χ+n

1−δZ (θ)

1−δ]+

1

1− δ (1− φ)[(1− l)1−δ

+ χ−n1−δ

Z (θ)1−δ] 11−δ

so:∂un

∂χ+n

= φ χ+n−δZ (θ)

1−δ and∂un

∂χ+n

= φ χ−n−δZ (θ)

1−δ .

Consider a marginal increase in insurance (borrowing) am(n). The cost to thehousehold is a slightly lower level of consumption when the output of firm m(n)

14

is high, lowering χ+n . The benefit to the household is that its gross financial

wealth, available for investment in government bonds, is higher, increasing bothχ+n and χ

−n . Note also that χ

+n = χ−n +

(1 + θ−1ψ − rAam(n)

)Formally the first order condition for optimal insurance is that

∂un∂am(n)

= (rB − rA)∂un

∂χ+n

+ rB∂un

∂χ−n

= (rB − rA)φ χ+n−δZ (θ)

1−δ+ rB (1− φ) χ−n

−δZ (θ)

1−δ= 0

Which becomes:[rB − rArB

φ

1− φ

]− 1δ (χ−n +

(1 + φ−1γ

)− rAa

)+ χ−n = 0

yielding a demand function for borrowing:

a = r−1A

[χ−n

(1−

[rB − rArB

φ

1− φ

] 1δ

)+(1 + φ−1γ

)]Now we have a closed form solution which is independent of aggregate output

and of , but does depend upon the output share when income is low χ−n as wellas relative interest rates and the probabilities of high versus low output.A demand to hold bank liabilities (money) arises because only bank money

can be used for payments. However, in order to analyse demand for money, weneed to consider both the frictional costs of exchanging wealth into a monetaryform and also the implicit or explicit charging by banks for monetary trans-mission services. Since this is not central to the model, we will capture thisusing two simple parameters. First we assume that the bank offers a depositrate of interest, between period 0 and period 1, rD = rB − ξ, so that there isan opportunity cost of holding deposits. If rD = rB then households will holdonly money, not bonds.Second we will assume that the selling of bonds imposes a fixed ‘dealer

spread’, in which there is a reduction of the output share of the household,given by $bt where bt is the remaining value of the bond.A household learns, at the beginning of period 1, both its claim on revenues

from output and the timing of this output. At the commencement of period 1it holds two financial assets deposits of rDd and bonds of bh. Between then,and the timing of its output at t = 1 + m, deposits decline steadily, so dt =rDd− (t− 1) (bh + χn) and will fall to zero at time t̂ = 1 + rDd

bh+χn.

If m < t̂, then there is a jump upwards in its deposit holdings at t = m,from rDd− (t− 1) (bh + χn) to rDd− (t− 1) (bh + χn)

To obtain a simple expression for money demand we will assume that thereare a relatively large number of firms, so that households can ignore the riskthat a large number of firms . In the continuous limit, theBanks generate a return from both their loans to firms and from their mon-

etary deposits. We need to think about their price setting behaviour. But to

15

simplify initially we assume that profits are driven to zero because there aremany banks competing for the same customers (think of the firms as being intwo regions, with firms located in those regions, eventually we can have depositservices, so the market power is all with the firms/households and intermediaryi (expected) profit is zero).

4 Simulations with homogenous banks

To be added

5 Simulations with bank failure

To be added.

6 Conclusions

This is still a very preliminary draft. Conclusions to be drawn at this point intime are limited and subject to confirmation from subsequent simulation work.A first conclusion is about the relatioship between the provision of central

bank liabilities and the period 0 wage level. Banks must face frictional costs ofraising additional finance, whenever they have insuffi cient reserves to settle allintra-period payments. Such frictional costs inhibit the amount of bank lendingand the insurance that banks can provide against idiosyncratic uncertainty.The anticiapted level of reserves, available to banks when they have to settle

payments, determines their exposure to such frictional costs; but a higher levelof anticiapted reserves in turn allows banks to lend more to customers thanthey otherwise would. So in equilibrium the central bank can only play only alimited role in reducing frictional costs of settlement.At the same time it has to be acknowledged that the existence of such fric-

tional costs of interbank borrowing and settlement of payments, is inconsistentwith the assumed setting, in the baseline simulations of this paper, that banksnever fail. In such circumstances they should always be able to borrow enoughfrom other banks to meet their payments needs (with the further implicationthat the price level is no longer under the control of the central bank).Therefore it is necessary to consider how these conclusions are amended in

a setting where bank failure can arise. A basic assumption made here is thatthe price level can adjust in response to all aggregate productivity shocks. Butthis does not rule out the possibility of heterogenous shocks at the level of thebank, or the possibility that these coudl trigger a bank failure.This is still some way from being fully worked out, but a potential intuition

(as yet unconfirmed) is that this has two major impacts. The first is thatit constrains the ability of the central bank to provide liquidity (by acting asdiscounter of collateral or lender of last resort, these two functions are not clearlydistinguished in the present setting). Not only does the provision of liquidity to

16

banks expose the central bank to potential losses, but the willingness to provideliquidity encourages excessive period 1 lending by individual banks, because itprovides them with protection from risk of loss.The second potential related intuition concerns the ability of the central

bank to provide a monetary stimulus. While nominal neutrality will still hold,in the sense that an increase in the aggregate supply of reserves will lead to aone for one increase in the wage rate, policies of responding to liquidity shortageby providing additional period 1 liquidity will have a relatively small impact onperiod 0 wage rates.Finally it is to be hoped that the set up developed here is simple enough

to be extended to a multiperiod setting, with sluggish wage adjustment. Inthis case it will be possible to develop an analysis of how bank balance sheetsconstrain monetary transmission and how this relates to central bank liquiditypolicy.

17

References

[Allen et al.(2009)Allen, Carletti, and Gale] Allen, F., Carletti, E., Gale, D.,2009. Interbank market liquidity and central bank intervention. Journal ofMonetary Economics 56 (5), 639—652.URL http://www.sciencedirect.com/science/article/B6VBW-4W8VW69-1/2/454d2d962ac4a324ecad7607d1fac76b

[Andrea Gerali Stefano Neri and Signoretti(2010)] Andrea Gerali Stefano Neri,L. S., Signoretti, F. M., 2010. Credit and Banking in a {DSGE} Model ofthe Euro Area. Tech. Rep. 740, Banca D’Italia.

[Ashcraft et al.(2008)Ashcraft, McAndrews, and Skeie] Ashcraft, A. B., McAn-drews, J., Skeie, D. R., 2008. Precautionary Reserves and the InterbankMarket.URL http://ssrn.com/paper=1364704

[Bagehot(1873)] Bagehot, W., 1873. Lombard Street: a description of themoney market. H. S. King & Co, London 1873.

[Christiano et al.(2010)Christiano, Motto, and Rostagno] Christiano, L.,Motto, R., Rostagno, M., 2010. Financial Factors in Economic Fluctua-tions. Tech. Rep. 1192, European Central Bank.

[Congdon(2009)] Congdon, T., 2009. Central Banking in a Free Society. Insti-tute of Economic Affairs, London.

[Curdia and Woodford(2010)] Curdia, V., Woodford, M., 2010. Credit Spreadsand Monetary Policy. Journal of Money, Credit and Banking 42, 3—35.

[Diamond and Dybvig(1983)] Diamond, D., Dybvig, P., 1983. Bank runs, de-posit insurance and liquidity. Journal of Political Economy 91, 401—419.

[Dib(2010)] Dib, A., 2010. Banks, Credit Market Frictions, and Business Cycles.Tech. rep., Bank of Canada.

[Freixas and Holthausen(2005)] Freixas, X., Holthausen, C., 2005. InterbankMarket Integration under Asymmetric Information. The Review of Finan-cial Studies 18 (2), 459—490.URL http://ssrn.com/paper=900671

[Freixas et al.(2010)Freixas, Martin, and Skeie] Freixas, X., Martin, A., Skeie,D. R., 2010. Bank liquidity, interbank markets and monetary policy. Tech.rep., Federal Reserve Bank of New York, New York.

[Gertler and Karadi(2010)] Gertler, M., Karadi, P., 2010. A Model of Uncon-ventional Monetary Policy. Tech. rep.

[Gertler and Kiyotaki(2009)] Gertler, M., Kiyotaki, N., 2009. Financial Inter-mediation and Credit Policy in Business Cycle Analysis. Tech. rep.

18

[Gilchrist et al.(????)Gilchrist, Ortiz, and Zakrajsek] Gilchrist, S., Ortiz, A.,Zakrajsek, E., ???? Credit Risk and the Macroeconomy: Evidence froman Estimated {DSGE} Model. Tech. rep.

[Goodfriend and McCallum(2007)] Goodfriend, M., McCallum, B. T., 2007.Banking and Interest Rates in Monetary Policy Analysis: A QuantitativeExploration. Journal of Monetary Economics 54, 1480—1507.

[Group(2010)] Group, T. M. A., 2010. Interim Report. Tech. Rep.http://www.bis.org/publ/othp10.pdf, Bank for International Settlements.

[Martin(2006)] Martin, A., 2006. Liquidity provision vs. deposit insurance:preventing bank panics without moral hazard. Economic Theory 28 (1),197—211.URL http://dx.doi.org/10.1007/s00199-005-0613-x DO -10.1007/s00199-005-0613-x

[MARTIN(2009)] MARTIN, A., 2009. Reconciling Bagehot and the Fed’s Re-sponse to September 11. Journal of Money, Credit and Banking 41 (2-3),397—415.URL http://dx.doi.org/10.1111/j.1538-4616.2009.00210.x DO -10.1111/j.1538-4616.2009.00210.x

[Meh and Moran(2010)] Meh, C. A., Moran, K., 2010. The Role of Bank Capitalin the Propagation of Shocks. Journal of Economic Dynamics and Control34 (3), 555—576.

[Olover Huelsewig and Wollmershaeuser(2006)] Olover Huelsewig, E. M.,Wollmershaeuser, T., 2006. Bank Behaviour and the Cost Channel ofMonetary Transmission. working paper 1813, CESifo.

[Smith(1936)] Smith, V., 1936. The rationale of central banking and the freebanking alternative (1990 reprint), reprint Edition. Liberty Press, India-nopolis.URL http://www.getcited.org/pub/102882655

[Walsh(2010)] Walsh, C. E., 2010. Monetary Theory and Policy. MIT Press.

[Woodford(2003)] Woodford, M., 2003. Interest and Prices: Foundations of aTheory of Monetary Policy. Princeton University Press.

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