DRAFT — NOT FOR DISTRIBUTIONVERSION 10

Macroeconomic Effects of Credit Shocks:Loan Rigidities and System Memory

Benjamin Honig, Michael Honig1, John Morgan2, Ken Steiglitz3

1Department of EECS, Northwestern University,2Haas School of Business, U.C. Berkeley,3Department of CS, Princeton University

December 7, 2016

2016-12-07 01:45

Abstract

A foundational assumption in many macro-economic models is that continuouslyoptimizing agents can, in principle, immediately compensate for changes in modelparameters. Transient effects in such models stem from unexpected disturbances, pri-vate information, or frictions in price adjustments. We highlight an alternative sourceof transient effects: the inertia associated with an aggregate loan portfolio held bythe banking sector. Specifically, lags and rigidities in loan contracts that make upthe portfolio limit the speed of response to policy shifts or other shocks—even withforward-looking agents. We explicitly model credit in a model economy with differentsectors and monetary flows as state variables, and show that the dynamic effects ofcredit expansions and contractions are asymmetric: while expansions generally exhibitsmooth transitions, contractions with downward wage friction cause widespread un-employment and a loss in production. The memory in the loan portfolio also affectssteady-state behavior. In particular, increasing the inflation rate discounts the valueof past loans, causing a change in the real interest rate.

Contents

1 Introduction 31.1 Motivation of the model and its departures . . . . . . . . . . . . . . . . . . . 41.2 Features of the Baseline Model . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Additional Features with a Central Bank . . . . . . . . . . . . . . . . . . . . 8

2 Summary of Results 8

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3 Baseline Model 103.1 Sectors and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Dynamic Recursions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Equilibrium of the Baseline Model 22

5 Comparative Statics 225.1 Various variables vs. fractional reserve rate . . . . . . . . . . . . . . . . . . . 235.2 Loan rate and some other variables vs. fraction of transactions in cash . . . . 245.3 Equilibrium loan rate vs. maximum loan term, varying reserve rate and propen-

sity to save . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6 Model with Central Bank 25

7 Policy Experiments and Shocks 327.1 Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.2 Transient and Steady-State Effects of Central Bank Injections . . . . . . . . 337.3 Credit Expansion and Contraction . . . . . . . . . . . . . . . . . . . . . . . 367.4 Loan Defaults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407.5 CB Injections Through the Bank versus Household Sectors . . . . . . . . . . 42

8 Transient versus Steady-State Behavior 46

9 Model Limitations and Future Work 47

Appendices 48

Appendix A Calculation of production and other measurements 48

Appendix B Financing calculations 50

Appendix C Details of the baseline equilibrium calculation 52C.1 Equilibrium calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52C.2 Parameterizing the equilibrium manifold by r . . . . . . . . . . . . . . . . . 54C.3 Household debt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55C.4 The equilibrium manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Appendix D Uniqueness of fixed point for τm = 1 58

Appendix E Adaptive Update for s(t) 59

References 60

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1 Introduction

The problem of fluctuations in business activity has been studied intensively for at least thepast two centuries, and as recent events have shown, disruptive downturns and attendantbooms have become neither less severe nor less frequent. In this paper we study the macroe-conomic effects of the aggregate bank loan portfolio, which appears relatively neglectedin current macroeconomic modeling, but, because it is a source of long-term memory, maycontribute significantly to the boom-bust cycle. We begin by arguing that such system mem-ory is inherently difficult to incorporate into the DSGE formalism, currently the dominantmethodology.

The DSGE model has by now been widely adopted by mainstream macroeconomists, andprovides a valuable framework for understanding equilibrium as well as dynamic behavior(see, for example, [1, 2]). It is frequently used as a starting point and augmented to in-clude such features as banking and shadow banking, information asymmetries on the partof optimizing agents, incomplete information (for example, rational inattention), and riskassociated with excessive leveraging. While this work has provided some insight into thepossible causes and consequences of large-scale contractions, the ability to explain—not tomention predict—large-scale downturns remains quite limited.

In this paper we propose (1) that the system memory embodied in the bank loan portfoliocan play an important role in determining the dynamics of contractions and expansions, in-cluding especially what Stiglitz has termed “deep downturns” [3]; and (2) that incorporatingthe bank loan portfolio into the DSGE framework inevitably introduces a curse of dimen-sionality that further exacerbates the already formidable mathematical and computationalproblems in solving the DSGE model.

Consider the usual core DSGE formulation (see, for example [4]):

maxuE0 [∑∞

t=0 βtU(xt, ut)] , subject to (1)

xt+1 = F (xt, ut, zt) , (2)

where ut is the control vector (to be chosen by the representative agent), E0 the expectationoperator with scope starting at 0, β the discount factor, U the agent utility function, xt thestate vector, F the system update function, and zt is an exogenous random disturbance. Thedimension of the state space of this (normally nonlinear) dynamic system is typically on theorder of 10, and its solution, except for unrealistic examples, usually involves approximationtechniques, even to find the equilibrium.

Consider now what happens when we add a bank loan portfolio. We have in mind herean array of contractual loans that extend backwards in time over the maximum loan term,which in our examples will be on the order of 60 periods, or 5 years if we consider a periodto be one month. We can fix the distribution of loans over terms, but we still must carry thehistory of their implied contractual obligations as part of the system state. This increasesthe dimensionality of the dynamic system dramatically, from about 10 to about 70, whichrenders the resulting stochastic equations essentially intractable with present techniques.

We can gain further insight into the difficulty of including long system memory in theDSGE model by considering the continuous-time version of DSGE. Such a model is describedin [5], for example, where a simple DGSE model is shown to lead to an ordinary differentialequation of standard type and low dimension (actually, eight). However, the continuous-timelimit of a model with long (with respect to the period) delays leads to a delay-differential

3

equation (DDE), also called differential equations of the retarded type.1 This results from thefact that the delays in the bank loan portfolio are on a larger scale than the update increment(period), and remain macroscopic when the differential limit is taken. The same phenomenonsurfaces in [7], where modeling the effects of rare Poisson shocks make it necessary to addconsiderable memory to the state vector. As they put it, “... the simple awareness of largeand rare ‘Poisson jumps’ leads to an adjustment of households’ optimal consumption plans.One crucial difference to business cycle shocks is that an econometrician may not observerare events for a longer period, and thus households might appear to be irrational.” A similarjustification can be made for the long delays that naturally result from loan contracts.

1.1 Motivation of the model and its departures

A foundational assumption of most modern macro-economic models, including especiallythe family of Dynamic Stochastic General Equilibrium (DSGE) models, is that agents con-tinuously optimize their particular objective (usually profit or utility) subject to imposedconstraints. Without appropriate constraints, the transient behavior following a policy ortechnology shock, or random disturbance, is highly muted or non-existent, because the agentsimmediately compensate for those disturbances and quickly drive the economy to a newequilibrium. Transients arise with the addition of constraints on available information, mis-matched beliefs, and delays and frictions in adjusting state variables such as prices. Priceand wage stickiness, in particular, have played a central role in causing and shaping tran-sient behavior in the vast majority of models. As is well known, without such frictions theclassical monetary model generally predicts neutrality or near neutrality of monetary policywith respect to real variables.

Here we present a model in which persistent transients due to shocks can occur evenwhen prices are unconstrained and equilibrium prices are known by the agents. The causeof the transients is the inertia embedded in an aggregate loan portfolio held by the bankingsector. Specifically, the loan portfolio comprises loans spread over different durations, whichintroduces rigidities due to contractual interest rate agreements, along with inertia associ-ated with the speed with which total credit can expand and contract. State variables in thebaseline model consist of monetary flows between four sectors: aggregated households, con-sumer goods firms, capital firms, and banks.2 In this way, it is possible to model particulartypes of loan contracts and banking rules, such as a fractional reserve requirement. Thisenables a characterization of cash and credit flows as functions of the reserve rate and thestructure of the loan portfolio (total amount and distribution of loan windows). The totalmoney (cash) supply can be controlled by adding a central bank that can buy or sell loans.

A key property of the model we present is that the interest rate is adapted to balance thesupply and demand for loanable funds. The demand for loans arises from consumer goodsfirms, which borrow to finance capital purchases. The supply of loans is determined by thewillingness of banks to expand credit, as reflected by the fractional reserve requirement,accounting for current liabilities (existing demand-deposit accounts). Hence the interest

1In general DDEs have an infinite number of eigenvalues and lead to dynamic behavior that is inherentlymore complex than standard differential equations of low dimension. For example, a first-order DDE with asingle delay can easily lead to oscillatory behavior. Erneux [6] provides a good introduction and survey.

2Prices are then inferred by the associated flow variables. For example, the consumer price index is theaggregate amount spent on consumer goods (a state variable) divided by total consumer goods produced.

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rate in equilibrium is determined endogenously,3 and effectively determines the allocation oflabor resources between the capital and consumer goods sectors. This is in contrast with

Discuss,check

the usual DSGE models, in which the interest rate is set exogenously by a central bank, andthe “natural” interest rate (corresponding to the equilibrium interest rate in our baselinemodel) is determined by the growth rate and personal time-discount rate, both of which arespecified exogenously.

There are two ways to introduce credit expansions and contractions in this model: (1)lowering (respectively, raising) the fractional reserve requirement, and (2) increasing (re-spectively, decreasing) the rate at which a central bank injects cash. In both cases creditexpansions push the interest rate down, and contractions push the interest rate up. We showthat the dynamics (transients) associated with expansions and contraction are asymmetric:whereas expansions correspond to relatively smooth transitions in which the monetary flowsincrease, contractions combined with downward wage friction cause widespread unemploy-ment across both the consumer goods and capital sectors, with an associated transient loss inproduction. Moreover, the durations of the transients generally increase with loan durations,and can be quite long (that is, significantly longer than the maximum loan duration). Thisis an inherent property of the model due to the memory embedded in the aggregate loanportfolio, and cannot be eliminated via forecasting on the part of agents.4

Empirical support for such asymmetries has been presented in [8]. Others?

Results from simulated policy experiments are presented in which the central bank con-tinuously injects cash at a fixed percentage of the monetary base per iteration, increasingthe money supply at a given rate. Perhaps surprisingly, increasing the rate of injectionslowers the real interest rate and hence shifts some labor from the consumer goods sector tothe capital sector. (The nominal interest rate increases.) This occurs even though the agentsaccount for the associated inflation. This is again due to the memory in the loan portfolio:inflation discounts the liabilities associated with past loans, creating additional room for thebanks to increase the supply of loans relative to demand. Conversely, decreasing the rateof central bank injections increases the real interest rate in steady state. Additional exper-iments illustrate the effects of cash injections by the central bank versus credit expansionsby the banks, injecting cash through the banking sector versus giving the cash directly tohouseholds, and loan defaults.

Reference related work?

The paper is organized as follows. The next section highlights the main features ofthe model and Section 2 summarizes our main results. Section 3 describes the baselinemodel without the central bank, and specifies the associated set of recursions (time updates).Section 4 derives the equilibrium for the baseline model, with the details in an appendix.

3Alternatively, the interest rate can be set by a central bank. The corresponding amount of steady-statecash injections is then determined endogenously.

4In practice, transients will be influenced by traders and a financial sector, not modeled here. Whilethat will strongly influence dynamics, and may serve to shorten transients in some situations, it must alwaystake significant time to re-populate the loan portfolio after a shock.

5

Section 5 then presents comparative statics, illustrating the equilibrium effects of varyingfractional reserve rate, fraction of transactions in cash, and maximum loan term. Section 6then extends the baseline model to include a central bank along with loan defaults. Section 7presents results from policy experiments, followed by some additional discussion in Section 8.Possibilities for future work, building on the model presented here, are discussed in Section 9.

1.2 Features of the Baseline Model

We begin by constructing a baseline model with four sectors of the economy: Household (H),Consumer Goods (CG), Capital (K), and the Bank (B) sector. (See Fig. 1.) Throughout thepaper the sector (H, CG, K, B) will be capitalized, whereas constituent households, consumergoods, capital firms, and banks will be written in lower case. The CG sector is assumedto produce both private and public goods, so that we do not explicitly model a separategovernment sector funded by taxation. Each sector should not be viewed as an individualagent, but rather as consisting of a continuum of many heterogeneous entities. The monetaryflows between sectors then represent aggregate amounts summed over all transactions withina period among different types of firms, banks, and household agents. The model is dynamicin that state variables representing monetary flows, as well as the interest rate, are updatedeach period. In the baseline model the total cash (monetary base), is fixed, although the

Cashnotmoneysupplyis fixed,right?

Bank sector is able to generate additional credit. We subsequently introduce a Central Bank(CB) that can control the money supply. The baseline model has the following key features.

The interest rate is determined by a market for loans. The CG sector employs laborfrom the H sector and finances the purchase of capital from the K sector to produce output(consumer goods). All capital purchased by the CG sector is financed and the financing costis added to the cost of capital.

This feature has appeared in other models (Christiano?), but without the explicit marketfor credit presented here.

“Capital” then refers to any input to the CG sector that requires financing. The fraction ofrevenue the CG sector spends on labor versus capital, or labor-capital split, then determinesthe Demand for Loanable Funds (DLF). The Supply of Loanable Funds (SLF) is determinedby the amount of credit generated by the Bank sector (described next). The SLF can alsodepend on the savings-consumption split, which is the fraction of household wealth that is

Notehere Hsplitswealth,notincome?

saved versus consumed. In equilibrium the Bank sector sets the interest rate so that SLFequals DLF, and the labor-capital split, selected to maximize output, is balanced againstthe savings-consumption split.

The Bank sector generates credit for loans subject to a fractional reserve re-quirement. The Bank can generate credit in the form of deposit accounts (owned by theCG sector) up to a maximum limit determined by the reserve requirement. That in turndetermines SLF. We explicitly distinguish between cash flows from one sector to another,and credit flows. In particular, the Bank can generate loans by transferring cash to the CG

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sector (taken from its reserves), and/or by creating credit (demand-deposit accounts) with-out exchanging cash. The characteristics of a credit expansion and contraction then dependon the fraction of transactions that are cash versus credit.

Loans are issued and repaid over a distribution of loan terms. Loans to the CGsector are partitioned into different loan durations (windows) modeling a range of short- tolong-term bonds. Here each loan is repaid with interest when it expires, thus acting as azero-coupon bond. To meet its loan obligations the CG sector is able to make paymentseach period into an Escrow demand-deposit account, held by the Bank sector. The interest

Principalplus in-terest?

on all loans is passed along to both the H and CG sectors (for Escrow accounts) as a returnon savings. Furthermore, any cash deposited in Escrow adds to Bank reserves, allowing afurther expansion of credit and an increase in SLF. The continuing cycle of capital financingintroduces memory in the model that influences both the equilibrium and dynamics.

Unemployment can occur when total wage flow decreases. This is a direct con-sequence of introducing downward wage-price friction. In the absence of such friction, adecrease in total wages into a sector with a fixed number of laborers implies a corresponding

Wage-price?

decrease in wage rate averaged over the sector. Downward friction in the wage-price thencauses the total wages to be spread over a smaller subset of laborers. Unemployment cantherefore be sector-specific (that is, in CG or K) depending on the flow of wages into eachsector.

Dynamics and stability are controlled by interest rate adaptation and sectorforecasting. Shocking a system parameter, such as the fractional reserve requirement,for example, leads to a disequilibrium in which the interest rate and other state variablesare mismatched to the new set of system parameters. The system can adapt to the newequilibrium by adjusting the interest rate, using, in the present model, a tatonnement process.In addition, side-information and forecasts regarding state variables within each sector canbe incorporated in state updates, reflecting assumptions about expectations and any stateinformation that may be available to each sector.

We will illustrate the features of the model assuming all parameters are fixed, or changeaccording to a deterministic shock. That is, we do not introduce random fluctuations insystem parameters, as is typically done in DSGE models. This is because such randomness

Is“DSGEmodels”correcthere?

is not necessary for creating and prolonging disequilibrium states associated with transientsand convergence to a new equilibrium. More specifically, starting from a disequilibrium statewith fixed parameters, the system generally cannot immediately reach a new equilibrium inresponse to a deterministic shock—even in the absence of price frictions. This includes beingable to set the interest rate instantaneously to clear the loan market. The reason is that thememory in the system introduced by the loan portfolio held by the Bank sector preventsother state variables from instantaneously moving to their equilibrium values. Rather, thesystem evolves through disequilibrium states as the interest rate is adjusted and the sectorsinteract. Hence the system memory adds inertia, limiting the rate at which the systemis able to reach the new equilibrium. This inertia cannot be overcome by forward-lookingagents with perfect information about the model parameters; time is still needed to updatethe loan portfolio with the new equilibrium values. The associated transient increases withthe duration of loans, and can be long-lasting.

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Because the model is nonlinear, transient effects due to a shock can be complicated anddepend crucially on how the interest rate is adjusted. For example, rapid adaptation of theinterest rate often leads to instability without appropriate filtering or forecasting of otherstate variables. Here our focus is on identifying consistent dynamic trends caused by creditexpansions and contractions, as opposed to studying how dynamics change with differentassumptions concerning side-information and forecasts. Those trends should apply over arange of system parameters (such as the adaptation step-size) and reasonable assumptionsconcerning the information available to sectors.

1.3 Additional Features with a Central Bank

To gain insight into the effects of monetary policy, we introduce a CB that can inject liquidity(cash) into the economy represented by the baseline model. (See Fig. 5.) Key additionalfeatures of the expanded model follow.

Moneysupplyincludescredit aswell ascash?

The CB controls the money supply by purchasing CG debt. The CB can injectliquidity by purchasing CG debt, meaning that part of the loan portfolio owned by theBank is transferred to the CB in exchange for cash (computed as the total principal plus apremium that reflects accumulated interest).5 Liquidity may also be effectively injected by

DoesthiscoverBen’scom-ment?

decreasing the time for repayment. If the CB injects cash so that the money supply increasesat a fixed rate, then the system reaches a “steady-state” (assuming stability) in which allmonetary flows increase at the same rate as the money supply, reflecting a constant rate ofinflation.

Loan repayments to the CB can produce deflation. The net flow of cash into theeconomy is the injection rate minus the rate of cash flowing out due to loan repayments to theCB. A direct consequence is that steady-state injections with full repayments of principal plusinterest cause deflation in steady-state. To maintain a positive steady-state inflation rate,the CB must either continuously buy an increasing amount of debt (relative to the currentmoney supply) to compensate for the interest repayments, or collect only partial repayments,effectively gifting the CG sector with additional cash, corresponding to a “helicopter drop”.

Loan defaults are imposed exogenously. Loan defaults eliminate the correspondingBank assets (and liabilities), which reduces the return on savings to the Household sector.We will assume that a given fraction of loans default each period. In practice, the default ratevaries according to the level of risk incurred by the Bank sector. Such a relation, althoughnot included here, could easily be be incorporated into the model.

2 Summary of Results

Stillmention“prop-ertiesof themodel”?

The results described in this section are in addition to the transient properties alreadydiscussed, and are illustrated in the numerical examples shown in the sequence of figures in

5In practice, the CB typically buys government debt. Because both public and private sectors are mergedin CG, we do not need to distinguish explicitly between government and private CG debt, although we willassume that CB-owned debt is repaid directly from CG revenues, as opposed to being funded through regularescrow payments.

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Section 7.

Inflation due to CB injections has a real effect on interest rate and resourceallocation. This pertains to the scenario in which the CB injects cash at a fixed rateknown to all agents. In steady-state SLF therefore increases at the inflation rate (same

Moneysupplyor mon-etarybase(through-out)?

as the net injection rate), and the CG sector anticipates the nominal increase in targetcapital expenditures due to inflation when determining DLF. Even so, injections cause SLFnormalized by the money supply to increase relative to the normalized DLF, causing the realinterest rate to fall in steady-state in order to clear the market. (The steady-state nominalinterest rate increases, but by less than the inflation rate.)

This effect is due to the memory in the system introduced by the distribution over loanwindows. The injections have the effect of discounting past loans issued by the Bank sector,thereby reducing bank liabilities. Given a fixed reserve requirement, that leaves more roomfor banks to expand credit by increasing lending, causing an increase in SLF and a dropin the real interest rate. That produces a shift in resource (labor) allocation from CG tothe K sector. There is an associated increase in production due to the reduced friction forfunding capital expenditures. That increase effectively represents the continual transfer offuture consumption to the present, corresponding to a decrease in time-value of money, asreflected by the drop in real interest rate. This is illustrated in Fig. 6.

The preceding scenario occurs when the CB injections are inflationary, meaning the flowof cash from the CB into the economy exceeds the flow of cash out of the economy fromloan repayments to the CB. Conversely, deflationary injections (for example, with full loanrepayments) causes the real interest rate to rise, in turn causing a shift in resource allocationfrom the K to the CG sector, and a decline in production. Here the decline represents acontinual deferment of consumption to the future, as reflected by the increase in time-valueof money. This is illustrated in Fig. 7.

Credit contractions cause unemployment across the entire economy. A creditcontraction in the model can be caused by an increase in reserves held by the Bank sector,or a net decrease in the rate at which the CB injects cash. or a change in the reserve rate f .In any case that causes a decrease in monetary flow into the entire labor pool, belonging toboth the CG and K sectors. The wage rate (total wages divided by number of laborers) thendecreases in both sectors, and adding friction results in unemployment and a decrease in totalproductivity. The severity of the decrease increases with the size of the credit contraction.This is illustrated in Figs. 8 and 9. In contrast, a credit expansion does not generally causeunemployment since wages flowing to both sectors increase. Additional numerical examplesindicate that the transient effects due to credit contractions are often much more pronounced

Volatile?than those associated with credit expansions.

Widespread unemployment due to a credit contraction is different from sector-specificunemployment that may be caused by other types of shocks. For example, a sudden increasein return on capital (output elasticity) causes an increase in K wages and a decrease in CGwages, so that unemployment is confined to the CG sector. This is illustrated in Fig. 10.

Is thispara-graphnowclearandcorrect?

CB injections through the Bank sector cause a larger increase in loans and capi-tal expenditures than injections through the Household sector. Injecting throughthe Household sector refers to the scenario in which the CB transfers cash directly to the

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Household sector, adding to its nominal wealth, instead of to the Bank sector by purchasingloans. For this comparison, we assume the cash injections are not repaid, corresponding toa “helicopter drop”. Both types of injections increase capital expenditures in steady-state,although injections through the Bank sector cause a greater shift, which increases with theinjection rate. This effect can lead to over-investment in capital and a consequent decreasein production, because the CG sector acquires more loans to meet its target capital expen-ditures. In contrast, injections through households cannot cause such an over-allocation ofcapital, since the fraction of cash injections that are consumed contribute directly to CGrevenue. That revenue is then split between capital and labor to maximize production.

Loan defaults cause a credit contraction only by triggering an increase in theBank sector reserve ratio. Although loan defaults shift the equilibrium, they do notcause a credit contraction. In fact, to maintain the same reserve ratio, or equivalently, theamount of credit generated prior to the defaults, the Bank would need to increase loans toreplace those that default. A more likely scenario is that the Bank would react to the defaultsby increasing the reserve ratio, thereby causing a credit contraction. This is illustrated inFigs. 11 and 12.

3 Baseline Model

Figure 1 shows the flow diagram for the baseline model. We start by describing the monetaryflows between sectors and basic behavioral assumptions associated with each sector. Thebaseline model is then specified by a set of updates for the state variables.

Figure 1: Money transfers in the baseline model.

3.1 Sectors and Assumptions

Referring to the figure, the H sector supplies labor to the CG and K sectors, and the CGsector employs labor and capital to produce its output, consumer goods. The CG sectorfinances all capital purchases from the K sector via loans through the Bank sector. Because

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we do not introduce random disturbances, an equilibrium associated with a set of inputparameters corresponds to a set of state variables that is constant from period to period.Those include the interest rate, return on savings, flows between sectors, and savings andescrow accounts held at the Bank. The interpretation is that in equilibrium the sectors makethe same aggregate decisions each period.6 Hence, in equilibrium, the algebraic sum of flowsin and out of each sector must be zero; that is, markets clear in aggregate. In addition,we assume that no sector except for the Bank stores any cash or credit. Hence all revenuescollected by the CG and K sectors are passed to other sectors, and all household income iseither saved or consumed. (Implications are discussed in what follows.) All cash or creditwhich is not flowing between sectors stays in the Bank sector as reserves or deposit accounts.Next we describe each sector in more detail.

Household: At each period households in the H sector (split across those employed by CGand K firms) receive income consisting of wages WC and WK from the CG and K sectors,respectively, and also a return on savings I, which is added to total household wealth W(including savings). The H sector then decides what fraction of its aggregate wealth to keepin savings S in the B sector, and spends the remainder C on consumption. We emphasizethat this savings-consumption split is determined from total wealth, rather than from incomeper period. In principle, this decision can take any functional form of past, current, andforecast state variables. Here we make the simplifying assumption that the fraction saved isa nondecreasing function of the return on savings at the current period. While that limits theinformation used by the H sector to determine savings, which can affect transient dynamics,introducing forecasting would not change the equilibrium, since in an equilibrium the statevariables remain constant.

Because all income received by the H sector is either consumed or saved, there is no“cash-on-hand”, as often assumed in other models. Rather, here it is assumed that savingsaccounts are liquid in aggregate, so that households can withdraw whatever cash or creditis in the account at any time. With fractional-reserve banking, household cash-on-handcould be motivated by a concern that the Bank sector has insufficient reserves (possiblyprecipitating a bank run). Here we do not explicitly consider such scenarios, and assumethat households trust that their deposits are redeemable.

Consumer Goods: Firms in the aggregated CG sector receive revenue C (equal to con-sumption) from the sale of the consumer goods they produce, and must decide how much toallocate across capital and labor inputs. We assume that this decision maximizes production

DiscusswithMikeand Ben

for a fixed budget. In the particular implementation described here, the production functionis Cobb-Douglas with constant returns to scale, so that the CG sector allocates in equilib-rium a fixed fraction of its revenue to capital expenditures and wages. Maximizing outputimplies that the CG sector contains competitive firms, although the firms are interpreted asmanufacturing an array of heterogeneous consumer goods. Since all of the CG revenue isallocated across wages WC and capital expenditures KC , the CG sector can be interpretedas being zero-profit.7

6The uniqueness of the equilibrium will be discussed in Section 4.7It is possible to relax this assumption. For example, if firms in the CG sector have the ability to

manipulate prices, they may increase revenue by raising their prices, causing households to spend more onconsumption and to save less, causing a shift in the equilibrium. Profits in that scenario would be returned

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The average CG price is defined implicitly as the consumption C divided by CG outputWe omit friction in average CG price adjustments, and assume that in aggregate the CGmarket clears each period. Hence the average CG price is a function of C, a state variable,but is not a state variable itself. This again is due to the fact that transients in this model areprimarily due to the memory associated with the loan portfolio as opposed to sticky prices.The flexible CG price also removes the need to model explicitly any market dominance amongfirms (monopolistic competition) that is typically associated with sticky prices.

The CG sector finances all capital purchases, and in equilibrium takes out loans L fromthe Bank each period. Those loans are used to purchase capital from the K sector, andare allocated over a distribution of loan windows. We assume that the CG sector repayseach loan with a single balloon payment when the loan becomes due. In order to avoidmaking such balloon payments directly from the CG revenue stream, the CG sector makesan aggregate deposit each period into an aggregate Escrow-savings account held by theBank sector. That account, along with Household savings, then earns the current returnon savings. The demand for loans (DLF), which is L in equilibrium, is computed so thatthe capital expenditures (escrow deposits) per period, KC , meets the target that maximizesoutput.

Capital: We assume that capital is produced solely by labor as an input, and that thereis a fixed number of labor hours required to make one unit of capital.8 Hence the amountof capital manufactured per period increases linearly with the fraction of the labor poolallocated to the K sector. The Capital sector receives L per period for capital purchasesfrom the CG sector, and passes that to the Household sector to manufacture capital. Thefraction of households (laborers) that manufacture capital is then WK/(WK + WC), whereWK = L denotes wages for labor in the K sector. This therefore determines the labor-capitalsplit, which determines CG output. (Note that labor is the only resource that is allocatedin the economy.)

There are, of course, many ways in which the expiration of capital can be modeled.Here we make the simplifying assumption that the total amount of capital, which entersthe production function, is proportional to the amount of labor employed by the Capitalsector. Hence there is no inertia built into the production and consumption of capital. This

DiscusswithMikeand Ben

is in part to focus on the effect the loan portfolio has on dynamics. (In fact, one mightargue that inertia in the loan portfolio already captures the inertia in the capital sector,since the loan durations should roughly correspond to the time needed to put the capitalin service.) In addition, the amount of capital only affects CG production, which is not astate variable, hence the model for capital expiration affects neither transient behavior norequilibrium values.9

Bank: The Bank sector is the sole source of credit, which is issued to the Household sectoras Savings accounts S, and to the CG sector as Escrow accounts E. We do not explicitlymodel Bank profits or fees, but assume that any net revenue that is generated by the Bank

to households (shareholders) as part of the return on investment. This change in the model would not affectthe main results we present here.

8We could assume a Cobb-Douglas model for capital production. That would increase the overall demandfor capital, but would not affect our main results. CHECK THIS.

9An additional advantage of omitting inertia in capital production is that it eliminates the need for theCG sector to forecast capital inventories when determining DLF.

12

sector is distributed over all of its accounts. The credit issued is limited by a fractionalreserve requirement, and the Bank holds an amount Bc(t) as cash reserves. Specifically, thereserves Bc(t) must be at least a fraction f of total liabilities, which includes all accounts (Sand E) held by the Bank sector. SLF, equal to L in equilibrium, is then determined by thereserve requirement along with the amount of credit the Bank sector can generate.

A distinction is made throughout the model between flows of cash and credit. We assumethat loans are a fraction ck cash, which is deducted from Bank reserves and transferred to CG.The rest is credit, which is issued to CG without changing Bc(t) (e.g., in the form of checksdrawn against bank accounts). Consumption is also paid for with a fraction c cash, andthe rest credit, which flows between the corresponding demand deposit accounts held by theBank. (As we shall see, wages to labor in the Capital sector are paid with a fraction ck cash,and wages to labor in the CG sector with a fraction c cash.) We can therefore distinguishbetween an all-cash model (c = ck = 1) in which all flows in the economy represent cashtransactions, and an all-credit model (c = ck = 0) in which all cash resides at the Banksector as reserves, and all flows in the economy represent credit transactions.

In the all-credit model, the total credit issued by the Bank is limited to Bc(t)/f , and themoney multiplier is therefore 1/f . In the all-cash model, SLF is limited to (1−f) ·Bc(t) eachiteration and the usual argument then shows that the total credit is limited to Bc(t)·(1/f−1),so the multiplier is 1/f − 1.10 The model allows any 0 ≤ c, ck ≤ 1, and we will see that theparticular mix of cash and credit has a real effect on economic activity. Given a fixed money

Monetarybase?

(cash) supply and reserve requirement, the all-credit model generally creates more creditthan the all-cash model, but then may potentially experience deeper credit contractions dueto a shock.

The fractional reserve requirement f will be exogenously specified, and we will assumethat SLF always satisfies this requirement with equality. In practice, banks would attemptto optimize their loan portfolios (both amounts and durations) to maximize their revenue,taking into account the risk of default. We do not model risk here (although we will includedefaults), but rather assume that f is set by either the banks themselves or regulators as aproper threshold that balances returns against risk.

The banks competitively adjust the loan interest rate so that SLF matches DLF inequilibrium. Because of the memory in the loan portfolio, the interest rate is assumed tobe “sticky”. That is, it is adjusted incrementally. This adjustment plays a central role inbalancing flows and bringing the economy to an equilibrium. The interest associated withall loan repayments is allocated proportionally across Household Savings and CG Escrowaccounts. Interest on Household Savings I flows back to the households and adds to totalHousehold wealth. The principal part of the repayment replenishes the original accountsfrom which loans were made, which subsequently allows the Bank sector to make new loans.

3.2 Dynamic Recursions

Pathinsteadof tra-jectory?

Our goal is to simulate the trajectory of the state variables in the model rather than justsolve static equations for an equilibrium. We therefore seek update equations that move thestate of the system at discrete time t to the state at time t + 1. Those updates representan aggregation of actions taken by the collection of agents within each sector during each

10The all-cash model is often used to explain the expansion of credit in fractional-reserve banking (see[9], for example). However, in practice most bank accounts are created by issuing new credit [10].

13

period. Table 1 lists the main program state variables. References to model flow variableswill be capitalized (Savings (S), Escrow (E), Consumption (C), etc.).

From a formal point of view, if we represent the system variables at period t by the statevector ~x(t), we seek the state-transition rule

~x(t+ 1) = Φ(~x(t)) . (3)

The response to shocks can then be simulated by varying the input parameters to the modelshown in the lower section of Table 1. We defer the discussion of initialization–it is in factan interesting question exactly what initial values of the state variables across the economycorrespond to realistic and reachable states.

Loan rate: The loan rate r(t) is adjusted each period with the update

r(t) = r(t− 1) + kr · (Dlf (t− 1)− Slf (t− 1))/G0 . (4)

Dlf (t) and Slf (t) are DLF and SLF, respectively, kr is a constant step-size, and G0 is thetotal cash, fixed in the Baseline model. This is a simple tatonnement adjustment at thebank (equivalently, a gradient update) that moves the rate up when the demand for loansexceeds supply, and vice versa. Notice that the step size is normalized by the (for now fixed)size of the monetary base, G0, and is governed by kr. This step-size governs the speed withwhich the banking sector adjusts the loan rate. In general, when kr is too large the economyas a whole becomes unstable, and when it is too small the convergence to equilibrium iscorrespondingly sluggish. Ultimately, its determination is part of the calibration process.

Other methods for updating r(t) could be used. For example, it is possible to adjustr(t) myopically to balance Slf (t) and Dlf (t) at each iteration. That, however, can lead toinstability.11 This is effectively due to a mismatch in the adjustment rate for r(t) and thespeed with which the loan portfolio can adapt to shocks. While adding such features will

CheckwithBen

affect dynamics, they will not change the main results we present here.

Return on investment: This is the fractional interest earned per period, and is deter-mined de facto:

s(t) =LBPI(t− 1)

S(t− 1) + E(t− 1)(5)

where LBPI(t) is the total interest payments from all loans that are due at time t (to bedetermined), S(t) is the total amount in Savings at time t, and E(t) is the total amount inEscrow. This reflects the fact that the Bank sector is zero-profit, and the Household sectorreceives the distribution of all income from financing operations as their return on savings.(An alternative adaptive version for determining s(t) is discussed in Section 6 and AppendixE. That will not affect the equilibrium.)

Savings-Consumption split: Total Household wealthW(t) (to be determined) is dividedbetween Savings S(t) and consumption C(t) as:

S(t) = g(s(t)) · W(t) (6)

C(t) = [1− g(s(t))] · W(t) , (7)

11It may be possible to maintain stability by forecasting other state variables, although that would noteliminate the transient behavior.

14

r loan interest rates return on investment, sometimes savings rateW total household wealthDlf demand for loanable fundsSlf supply of loanable fundsG total cash in the economy, usually fixed at G0

Wc wages of labor in consumer goods sectorWk wages of labor in capital goods sectorS household savingsB demand-deposit Household accountEt target demand-deposit Escrow accountE demand-deposit Escrow accountC household consumptionL loans from Bank sector to finance capitalUcg unspent funds held by CGR bank cash reservesI interest earned by Household depositsKtar target capital expenditures, per periodKc actual capital expenditures, per periodξ continuing debt, total outstanding loansBc total current Bank cashΦ total debt of Household sector

f fractional reserve requirementτm maximum loan term, (minimum is 1)e propensity-to-save parameterβK output elasticity of capitalβL output elasticity of laborkr adaptation coefficient for loan rate adjustmentc fraction of CG wages in cashck fraction of loans in cash

Table 1: The main state variables in the baseline model. Structural parameters are in thelower section.

15

where g(·) is the propensity-to-save function, assumed to be a nonnegative and nondecreasingfunction of the current savings rate s(t). In the examples given later in this paper we willuse the simple form

g(s) =es

1 + es(8)

where the constant e is the propensity to save, which ensures g(s) ≤ 1, and g(s) ≈ es forsmall es. The savings deposit at period t is then

∆S(t) = g(s(t)) · W(t− 1)− S(t− 1). (9)

Note that ∆S(t) < 0 signifies a withdrawal of savings for consumption.The total Revenue received by the CG sector is equal to Household Consumption,

Rv(t) = C(t). (10)

As noted earlier, we will distinguish between cash and credit transactions. The fraction ofdeletedhouse-holdloans,addlater?

cash versus credit flowing between sectors will depend on the mix of cash and credit in loansissued by the Bank sector. The cash portions of the total amounts spent on consumptionand deposited by Households are given by

Cc(t) = c · C(t) (11)

∆C(t) = Wcash(t− 1)− Cc(t) (12)

where c and ck are constants (system parameters), and Wcash is the cash part of total (Kand CG) wages. That is, the fraction of cash spent on consumption can be different fromthe fraction of cash making up loans, although in the numerical examples, to be presented,we will assume c = ck.

Labor-Capital split: The firms that produce consumer goods must decide on the splitbetween expenditures for labor and investment in capital. This depends on the productionfunction Y (QL, QK), where QL is the quantity of labor and QK is the quantity of capital.We will take

Y (QL, QK) = QβLL Q

βKK (13)

where βL and βK are the output elasticities for labor and capital, respectively, and βL+βK =1.12 The amounts QL(t) and QK(t) are the only resources being allocated in the economy,and are determined by the relative amounts of wages for CG labor WC(t), and wages forCapital labor WK(t). That is, QL(t) = WC(t)/[WC(t)+WK(t)], and QK(t) = 1−QL(t). Theamount of capital is therefore memoryless in the sense that it varies instantaneously withthe amount of labor allocated to the Capital sector, as discussed in the previous subsection.

The CG firms split their revenue across labor wages and capital purchases to minimizetheir combined labor plus capital cost to produce a given output. Given the preceding

DiscusswithMikeand Ben

assumptions, this implies that the firms spend a fixed fraction of revenue on capital:

Ktar(t) = h · C(t) , (14)

12Growth due to technology and other factors is not explicitly included. This simplifies the model anddoes not affect our main results.

16

where Ktar(t) is the target capital expenditure and h = βK/(βL + βK). We assume thatcapital purchases are financed13 so that Ktar(t) represents payments on loans taken out inprevious periods (to be discussed next). In equilibrium, the loans issued to the CG sectoreach iteration must result in payments Ktar per period.14

Demand for loans and financing: Each period the CG sector computes its Demandfor Loans, Dlf (t), in order to meet its target capital expenditures Ktar(t) in the future.Specifically,

Dlf (t) = Ktar(t)/Θ(r, s) (15)

where Θ(r, s) is the discount factor associated with finance charges, i.e., the averaged per-unit, per-period cost of financing a loan. To compute the discount factor, we must specifythe structure of loans, which consists of their durations and scheduled repayments.

We assume that when an aggregate amount of loans L(t) is issued at time t, the loanterms are distributed as wτ for duration τ between 1 and τm, where

∑τ wτ = 1 and wτ ≥ 0.

That is, loan amounts of wiL(t) are taken with terms i, i = 1, ..., τmax. We can think of thisas the result of randomly selected loan terms in the limit as the number of CG firms becomeslarge. Were all loans to have one-period terms, the velocity of money would be infinite, all

Zeroperiod?Isn’t thevelocitystillfinite?Discuss

loans would be repaid with minimal delay, and there would be no continuing outstandingcredit. This is clearly unrealistic, and is the main motivation for introducing longer-termloans. With some abuse of notation we will denote the amount of loans of term τ at time tby L(τ, t).

The loan repayment agreement is modeled after a zero-coupon bond: loan repayments,including both principal and interest, are due when the loan term expires. No paymentsare required in the interim, although we assume that to avoid covering the loan balloonpayments from their revenue streams, CG firms make per-period payments into an Escrowsavings account. The payments are determined so that for a fixed return on savings, theamount in Escrow accumulates to the desired balloon payment when the loan becomes due.Other loan structures could, of course, be modeled. The zero-coupon format is assumedhere since this typically applies to corporate bonds, and also gives rise to extensive creditexpansions and contractions due to the creation of CG Escrow accounts.

Calculation of the Escrow payments is given in Appendix B. Specifically, the totalaccumulation in Escrow after i payments due to a unit loan with term τ is, from (82),

A(r, s, i, τ) = (1 + r)τ(1 + s)i − 1

(1 + s)τ − 1, (16)

where r and s are evaluated at the time the loan is initiated, assuming contractual rates atthat time. The target amount needed in Escrow at time t is then

Etar(t) =τmax∑τ=1

τ∑i=1

A(r(t− i), s(t− i), i, τ)L(τ, t− i) . (17)

13During a transient CG may have “unspent funds” that can be used to purchase capital directly, to bediscussed.

14Firms could subtract interest from revenue, and split the remainder, which would maximize production.That would sever the connection between time preference, deferred consumption, and investment. Hencethe increased production should not be sustainable in steady-state causing additional defaults. DISCUSS.

17

To meet this target the CG sector makes an escrow deposit given by

∆E(t) = Etar(t)− E(t− 1) . (18)

We will also need the cash portion, which is

∆Ec(t) = c ·∆E(t) . (19)

Returning to the determination of Dlf (t) in (15), the discount factor Θ(r, s) can becomputed by by adding up payments per period over all loans (see (84) in Appendix B). In

Checkequilibrium, the corresponding capital expenditures per period is equal to the escrow depositin (18).

Loan (balloon) repayments: The sum of balloon payments on all CG loans can becomputed from L(τ, t) using (87), (89), and (90):

LBP (t) =τmax∑τ=1

(1 + r(t− τ))τL(τ, t− τ) (20)

LBPP (t) =τmax∑τ=1

L(τ, t− τ) (21)

LBPI(t) = LBP (t)− LBPP (t) . (22)

where LBP is the total balloon payment, and LBPP and LBPI are the associated principal andinterest, respectively. These are included in total Bank assets, which are then distributedacross the Savings and Escrow accounts.

Bank assets, updated Savings, Escrow: The total bank assets after loan repaymentsare allocated to Savings and Escrow in proportion to their size, since this allocation isa return on investment. The following intermediate variables are introduced, which areupdated Savings and Escrow after deposits and loan repayments:

S ′(t) = S(t− 1) + ∆S(t) (23)

E ′(t) = E(t− 1) + ∆E(t)− LBP (t). (24)

The total balloon repayment is subtracted from Escrow, and is then allocated across all bankaccounts (Savings and Escrow). Note, however, that only the interest on repayments needsto be allocated since the principal amounts are already included in S(t) and E(t). Accordingto the proportional allocation rule, we have

B(t) = S ′(t) + E ′(t) (25)

L(t) = B(t) + LBPI(t) (26)

S(t) = L(t)S ′(t)

B(t), E(t) = L(t)

E ′(t)

B(t). (27)

where B and L are respectively total Bank assets (or equivalently, liabilities) before and afterthe loan repayments.

18

Bank assets, liabilities, and the supply of loans: The supply of loans, Slf (t), is deter-mined by applying the fractional reserve constraint to total Bank reserves (cash), accountingfor current liabilities. The Bank cash is given by

Bc(t) = [Bc(t− 1)− ckL(t− 1) + ∆C(t) + ∆Ec(t)]+, (28)

where ck is the fraction of loans issued in cash. Total Bank assets at period t are then Bc(t)Notright. Γdefinedonly inequil.Seebelow

plus the sum of all loans in the loan portfolio, given by Ltot(t) = Γ(t)L(t) where

Γ(t) =τm∑τ=1

τ−1∑i=1

wi. (29)

Total Bank liabilities at time t consist of its total credit obligations to the other sectors,and is given by S(t) +E(t) + (1− c) ·Wc(t) = L(t) + (1− c) ·Wc(t), where c is the amount ofCG wages paid in cash. That is, total Bank liabilities are the sum of all Savings and Escrowaccounts credited to households and CG firms, respectively, plus the additional credit thatis circulating as wages to the CG sector. The latter must be included since those credittransactions represent transfers between demand deposit accounts owned by the CG andHousehold sectors. Those accounts must be held by the Bank, hence add to the Bank’scredit obligations. (In contrast, the credit part of Capital wages ...??) We will assume thatBank assets are always equal to Bank liabilities. This need not be the case in practice, andin fact, it is possible to add an offset to Bank assets or liabilities, which then persists inequilibrium. This is discussed further in Section 4.

The net amount of Bank reserves, which serves as the base for issuing credit (loans), istherefore

ρ(t) = Bc(t)− f · [L(t) + (1− c) ·Wc(t)]. (30)

Consider first the all-cash model (ck = c = 1). Loans in that scenario are made by deductingthe loans from Bank reserves, and adding the that cash to CG accounts. Hence ρ(t) is theamount of “excess reserves”, all of which can be loaned out, so Slf (t) = ρ(t). With c = 1, andignoring loan repayments, Bank liabilities build in successive periods (i.e., credit expands) byloaning out increments of reserves according to the classical description of fractional reservebanking. (That is, ρ(t+i) = (1−f)·ρ(t+i−1), i ≥ 1, and summing over i gives the asymptoticamount of credit (total loans) as (1/f − 1) · ρ(t).) The actual amount of outstanding creditis reduced by loan repayments, and in equilibrium the credit revolves so that all expiringloans in period t are replaced by new loans with total amount Slf . Credit expansion in theall-cash model introduces some inertia, since credit builds over successive periods. However,the associated time-constant is relatively small compared with that associated with the loanportfolio (say, for τm > 50). This time-constant inherently limits the velocity of money.

Now consider the all-credit model. Here the Bank sector keeps Bc(t) in reserves, andcan create new accounts with total amount ρ(t)/f . Hence Slf (t) = ρ(t)/f , the reserverequirement is binding, and the Bank cannot issue more credit in successive periods untilloans are repaid. This eliminates the inertia associated with credit expansion in the all-cash model. Numerical examples comparing the all-cash and all-credit models show thatthe all-credit model generates substantially more credit in equilibrium. Although this hasa noticeable effect on equilibrium and dynamics for the particular cases considered, overallboth models seem to exhibit similar behavior.

19

In practice, a mix of cash and credit circulates in the economy, so that we allow c and ckto take any value between zero and one. It can be shown that with such an arbitrary mix ofcash and credit, the supply of loans becomes

Slf (t) =1

f + ck(1− f)ρ(t). (31)

An alternative form for Slf (t) can be derived by using the fact that Bank liabilities equalsUseequilvaluesfor this.Seeabove

assets, given by Bc(t) plus total outstanding loans, Γ(t)L(t), where Γ(t) is given by (29).Substituting in (31), replacing L(t) with Slf (t), which holds in equilibrium, and solving forSlf (t) gives

Slf (t) =(1− f) ·Bc(t)

f + ck(1− f) + f · Γ(t). (32)

This is equivalent to (31) in equilibrium, and assumes that the Bank knows the equilibriumloan window distribution {wi}. Numerical examples indicate that using (32) instead of (31)tends to reduce volatility during transients.

CG Wages, unspent capital budget: In analogy with (14), the target amount CGwishes to spend on labor is (1−h) ·Rv(t). In equilibrium, this will be equal to Rv(t)−∆E(t)(revenue minus current capital expenditures). However, because ∆E(t) depends on loansfrom previous periods, in general it will not be the same as the target capital expenditureduring a transient. We therefore let

Wc(t) = min{(1− h) ·Rv(t), Rv(t)−∆E(t)}. (33)

That is, if ∆E(t) exceeds the target capital expenditures h · Rv(t), then the CG sector firstmakes its loan payments, and then passes along the remaining revenue to the Householdsector as CG wages.15 Otherwise, if ∆E(t) ≤ h ·Rv(t), then Wc(t) is given by the first term(target wages). CG then retains an additional (unspent) amount for capital:

Ucg(t) = Rv(t)−∆E(t)−Wc(t). (34)

We assume that the CG sector uses this to purchase additional capital directly (withoutfinancing), so that this amount is added to capital wages (see (36)). Of course, in equilibriumUcg(t) = 0.16

Loans, Capital wages: The amount of loans issued at period t is

L(t) = min{Slf (t), Dlf (t)} . (35)

The loans are passed through the Capital sector as wages, so that capital wages are

Wk(t) = L(t) + Ucg(t) (36)

15It is possible that in a severe credit contraction ∆E(t) > Rv(t) in which case the CG sector goesbankrupt. The recursions can be re-initialized by defaulting on loans or by having a CB inject cash, as willbe discussed in Section 6.

16CG could instead spend Ucg(t) to purchase additional labor. This has the property, perhaps unrealistic,that CG under-invests in capital, relative to labor, during a credit expansion when loan obligations areincreasing.

20

where Ucg is the unspent CG revenue available to purchase additional capital. The cashportion of total wages is then

Wcash(t) = ckWk(t) + cWc(t) + ckUcg(t) , (37)

which appears in (12).

Household wealth: Household wealth consists of all assets owned by households, namely,

W(t) = Wc(t) +Wk(t) + S(t) . (38)

This is used to determine the Savings-Consumption split in (6).

Production and other measurements: Certain variables of interest can be consideredmeasurements, in the sense that they relate directly to commonly used metrics, but do notdirectly affect the economy’s evolutionary path, at least in the current development. Thecalculation of several important measurements, including GDP, labor apportionment, andprices, is described in Appendix A. These seem important enough to put in the main body.Also need to add unemployment.

Initialization: The state variables can be initialized in different ways depending on howthe cash is initially distributed across the sectors. For example, one possibility is to letBc(0) = B0 and Wcash(0) = W0. The total cash (money supply) is G0 = B0 +Wcash(0). Allother state variables can be initialized at zero. Provided that the loan rate step-size kr isnot too large, the state variables then typically converge to their equilibrium values wheniterating the recursions as t becomes large. To simulate the response to a parameter shock,it is also possible to initialize all state variables at their equilibrium values, as computed inSection 4. Add reference to discussion in Section 4.

Remarks: Should we summarize the Baseline recursions in a Table? We conclude thisEssentiallydone atstartof Ap-pendix C

section with the following observations:

• The duration of a period can be calibrated against the loan window distribution. Forexample, if we assume that the maximum loan duration is five years (after whichlonger-term loans are refinanced or traded), then taking τm = 60 implies that eachperiod corresponds to one month.

• Each state variable appears on the left-hand side of a dynamic equation in the (tem-porally ordered) loop exactly once, which means that a state variable x(t) is assignedits value exactly once. The state variable’s equilibrium value is then x = limt→∞ x(t),if that limit exists.

OK,Benagrees

• The total cash at the end of each period is Bc(t) + cWc(t) + cUcg(t), which can beshown to equal G0. check

21

4 Equilibrium of the Baseline Model

The equilibrium of the model just described can be calculated analytically in a straightfor-ward way, but the derivation is not without some subtleties. The details are laid out inAppendix C, and we summarize the results here.

The key fixed-point condition is, of course, the equality of supply and demand of loanablefunds, Slf and Dlf . This equality determines the equilibrium loan rate r, the single price inthe central money market of the model. However, when the consequences of this fixed-pointcondition are worked out, we learn that there is one more variable than equation. That is,up to this point, we have exactly one extra degree of freedom left in the model. Thus, if weconsider the baseline model with no further condition, there is a one-dimensional manifoldof equilibria, a fact that is easily verified in simulations.

Adding a constraint results in an equal number of equations and variables, and we chooseto specify a quantity that has a natural interpretation—the net debt (or, if negative, assets)of the household sector, which we denote by Φ.17 Because the balance of money and creditflow is enforced at every period, Φ is constant throughout a simulation. In the terminologyof computer science, Φ is a loop invariant, and it is established by the initial conditions ofthe model. For the purposes of this paper, we choose Φ = 0, although it will be of interestin future work to study how Φ affects the equilibrium.

When we do add the condition that household debt is zero, it turns out that the resultingnumerical problem reduces to that of solving two nonlinear equations in the two unknowns rand s, the loan rate and return on investment, respectively. The two equations are Eqs. 180and 181 in Appendix C.

5 Comparative Statics

Having the baseline equilibrium available without simulations makes it easy to study thecomparative statics of the model, and there are several reasons for doing this:

1. They can confirm our intuitive expectations of how equilibrium state variables depend onparameters, thus lending some confidence in the model’s economic realism.

2. They check that the simulations and the equilibrium analysis refer to the same system ofdynamic equations, an aid to debugging.

3. They can provide starting points for simulations, thus saving the time and possible diffi-culty of arriving at an equilibrium from a cold start.

4. They facilitate calibration.

In the next three subsections we present some representative studies of comparative statics inline with the first item, illustrating the dependence of equilibrium variables on the fractionalreserve rate, the fraction of transactions in cash, and the maximum loan term.

Addedcom-mentaboutproduc-tion tofigurecaption

17In our many equilibrium calculations and corresponding simulations we have always found the resultingequilibria to be unique. A proof of uniqueness for the special case τm = 1 is given in Appendix D. We leavea search for more general results for future work.

22

Reserve Rate0 0.5 1

Per

cent

0

1

2

3

4

5

6Loan Rate, Savings Rate (--)

Reserve Rate0 0.5 1

Per

cent

of M

oney

Sup

ply

0

1

2

3

4

5

6Savings, Consumption (--)

Reserve Rate0 0.5 1

Per

cent

of M

oney

Sup

ply

0

2

4

6

8

10

12Escrow

Reserve Rate0 0.5 1

Per

cent

of M

oney

Sup

ply

0

0.1

0.2

0.3

0.4

0.5

0.6Loans

Reserve Rate0 0.5 1

Per

cent

of L

abor

ers

0

20

40

60

80

100

Labor in Consumer Goods Sector,Labor in Capital Sector (--)

Reserve Rate0 0.5 1

Pro

duct

ion

Uni

ts

0.35

0.4

0.45

0.5Production

Equilibrium Values vs Reserve Rate

Model parameters: e=500 =max

=60 -K=-

L=0.5 c=c

k=0.1Model parameters: e=500 =

max=60 -

K=-

L=0.5 c=c

k=0.1

Figure 2: Various variables vs. fractional reserve rate f . The plot of production, lower right,uses a Cobb-Douglas production function with constant returns to scale.

5.1 Various variables vs. fractional reserve rate

The first set of plots, shown in Fig. 2, shows the effects of varying the fractional reserve ratef on several important system variables. The results are in complete conformity with ourintuition: as the reserve rate increases, available credit contracts, the loan rate and return

CheckwithBen

on investment therefore increase; savings, consumption, escrow, loans, and production asfractions of the money supply decrease; and labor shifts away from the capital labor sectoras investment shrinks.

Note that the savings rate (return on investment) in the upper-left plot in Fig. 2 exceedsthe loan rate for small (but reasonable) values of reserve rate. Actually, it is not uncommonfor this to occur in simulations. At first, it might seem problematic for the savings rate toexceed the loan rate, but the numerical result can be explained by the fact that the loanrate is a nominal, contractual rate set by the (zero-profit) Bank, whereas what we term thesavings rate is actually the ex post return on investment received by the Household and CGsectors, and therefore benefits from at least some compounding, especially for small reserverates, when more credit is available (see Eq. 5).

23

Fraction of Transactions in Cash0 0.2 0.4 0.6 0.8 1

Per

cent

of M

oney

Sup

ply

250

300

350

400

450Household Wealth, Loans

6.4

6.6

6.8

7

7.2

Per

cent

of W

ealth

Fraction of Transactions in Cash0 0.2 0.4 0.6 0.8 1

Per

cent

0.85

0.9

0.95

1

1.05

1.1

1.15Loan Rate, Savings Rate (--)

Equilibrium Values vs Fraction of Transactions in Cash

Model parameters: f=0.1 e=500 =max

=60 -K

=-L=0.5

Figure 3: Experiment 2 1.

5.2 Loan rate and some other variables vs. fraction of transactionsin cash

Figure 3 shows the effects of varying the fraction of transactions in cash, c = ck on theloan and return-on-investment rates, as well as household wealth and loans. Once againthis confirms our intuition. As more cash is used, available credit contracts, and householdwealth and loans contract. What is a bit surprising, perhaps, is that the effects are as smallas they are, especially on the rates. The equilibrium loan rate, for example, goes from about0.9% to about 1% as the fraction of cash goes from 0 (all-credit) to 1 (all-cash).

5.3 Equilibrium loan rate vs. maximum loan term, varying reserverate and propensity to save

Do you buy the following explanation?The last example, shown in Fig. 4, illustrates a less obvious effect. We plot the equilib-rium loan rate vs. the maximum loan term τm, where, as usual, loan terms are uniformlydistributed from 1 to τm. To explain the observed increase, peaking, and ultimate decreaseof the loan rate, consider first what happens if the maximum loan term increases suddenlyfrom a low value. The immediate effect, off equilibrium, is that actual capital expendituresKc(t) suddenly decreases, because payments to Escrow are now spread over a longer span.This means that the CG sector has more to spend on capital to meet its target, Ktar(t), thusincreasing demands for loans and putting upward pressure on the loan rate. With no initialcorresponding first-order increase in the supply for loanable funds, the loan rate rises, andeventually reaches a new, higher equilibrium.

At some point, however, as τm increases, the part of the balloon payments that is interestbecomes comparable to the part that is principal. This is easy to see from the fact thatLBPI/LBPP = X−1,18 and X = (1/τm)

∑τmi=1(1 + r)i grows with τm. This causes the supply

18See Appendix C.

24

Loan Window20 40 60 80 100 120 140 160

Per

cent

0

0.5

1

1.5Loan Rate (e=500)

f = 0.05f = 0.1f = 0.2

Loan Window20 40 60 80 100 120 140 160

Per

cent

0

0.5

1

1.5Loan Rate (f=0.1)

e = 250e = 500e = 1000

Equilibrium Loan Rate vs Maximum Loan Duration

Model parameters: -K

=-L=0.5 c=c

k=0.1Model parameters: -

K=-

L=0.5 c=c

k=0.1

Figure 4: Equilibrium r vs. τm for: (left) various values of fractional reserve rate f , e = 500;(right) various values of propensity-to-save e, f = 0.1.

of loanable funds to increase faster than the demand, whose rate of increase is slowing, andCheckwithBen

the equilibrium loan rate then turns around, and there is a value of τm at which the loanrate peaks.

The plots show curves for various values of reserve rate for fixed propensity-to-save, andvarious values of propensity-to-save for fixed reserve rate. The displacements of the curvesare all consistent with expectations: the rate goes up with increasing reserve rate, and downwith increasing propensity-to-save (when the supply of loanable funds increases).

6 Model with Central Bank

We now expand the Baseline model by introducing injections of new money into the economyby a CB. This is illustrated in Fig. 5, which shows the cash flows to and from the CB. The CBcan inject cash either through the Bank, by buying CG debt, or by giving it to the Householdsector. Loans owned by the CB are then repaid by the CG sector along with other loansowned by the Bank. The following recursions assume that the CB expands the moneysupply at a fixed rate by injecting an amount of cash equal to a fixed fraction κ of the moneysupply each period through purchases of CG debt. The amount that is repaid to the CB canbe varied, so that part of the injected cash is a gift to the CG sector (corresponding to a“helicopter drop”). We also introduce a constant loan default rate δ, which applies uniformly

25

Figure 5: Money transfers in the model with central bank.

to all loans. Policy experiments can then be simulated by changing these parameters eitheracross runs to examine steady-state behavior,19 or within a simulation to study dynamicbehavior.

Initialization As in the Baseline model, there is an initial distribution of cash amongsectors:

Bc(0) = B0, Wcash(0) = W0, G(0) = B0 +Wcash(0) (39)

where Bc(0) is the initial cash, which the Bank keeps as reserves, Wcash(0) is the additionalamount of cash in the economy flowing between sectors (or held in the CG sector), and G(0)is the initial (cash) money supply. For the policy experiments in the next section, all otherstate variables are initialized at zero.

The following iterations for the model state variables are then computed for t = 1, 2, · · · .

Interest rate The update for loan interest rate is

r(t) = max

{0, r(t− 1) + kr

[Dlf (t− 1)− Slf (t− 1)]

G(t− 1)

}(40)

where G0 in (4) is now replaced by G(t), the total cash in the economy (bank cash plus cashflowing among and held by other sectors), which now varies with period t.

Return on Investment It will be convenient to rewrite the expression for measured s(t)in the Baseline model as

s(t) = [L(t− 1)− B(t− 1)]/L(t− 1) (41)

19With CB injections we refer to “steady-state”, as opposed to “equilibrium”, to emphasize that if thesystem does reach a steady-state, then all state variables representing monetary flows increase at the injectionrate, as opposed to remaining constant. As we later emphasize, the corresponding normalized state variables,relative to the money supply, converge to constants in steady-state. Also, as in the Baseline model, we donot include Household debt, so that a unique steady-state is always observed.

26

where

L(t) = S(t) + E(t) (42)

B(t) = S ′(t) + E ′(t) (43)

and S ′ and E ′ are intermediate updates after withdrawals and deposits (see (58)-(59)). Thevariable L(t) represents total liabilities (equivalently assets) owned by the Bank at the endof the period, whereas B(t) represents total liabilities (or assets) before interest paymentsare made. Hence the numerator L(t− 1)−B(t− 1) is the total return on outstanding loansfrom the preceding period. (The total return includes assets left over from defaults, intereston Household debt, and premiums on loans purchased by the Fed.)

Because (41) is a myopic estimate, short-term fluctuations can lead to high volatility andinstability, especially when the denominator (corresponding to outstanding credit) is small.In those scenarios, the trajectory of s(t) can be smoothed, which can reduce volatility. Analternative approach is to incorporate information about the expected loan portfolio in orderto adapt the estimate of s(t) to match expected total returns with actual returns. Detailsare presented in Appendix E.

Savings-Consumption split: This is the same as for the Baseline model (see (6)) exceptthat the nominal return on savings s(t) is now replaced by the real return on savings s(t).Specifically, s(t) satisfies 1 + s(t) = [1 + s(t)]/Π(t), where Π(t) is the current inflationmultiplier (1 + inflation rate in loans). With this substitution the savings deposit is thengiven by (9). Total revenue received by the CG sector is again given by (10) and the cashportions of the total amounts spent on consumption and deposited by Households are givenby (12).

Inflation: The real return on savings depends on Π(t), which is the rate of increase of loans.Similarly, the update for Dlf (t) will depend on the rate at which CG revenue increases. Insteady state, Π(t) = Z(t + 1)/Z(t), where Z can be taken to be any model flow variable(Revenue, Loans, Wages, etc). This will be the same as the rate of increase in the price ofCG goods (total revenue divided by quantity purchased) and the wage-price (total wagesdivided by the number of households). Although these inflation rates will generally bedifferent outside of steady state, for simplicity we will assume a single inflation rate as therate at which Revenue (Consumption) increases. This is due to the central role this variable

CheckwithBen

plays in estimating Dlf (t), and the observation that the inflation rate for Loans tends totrack the inflation rate for Revenue. We also note that wage and price inflation is a directconsequence of the inflation in total Wages and Revenue, and in that sense plays a secondaryrole in understanding the model behavior.20

For policy experiments standard techniques can be used to obtain an estimate of Π(t),for example, log-smoothing of Rv(t+1)/Rv(t). The associated time constant then representsanother source of inertia. In some scenarios shocks combined with such an inflation estimatorcan cause short-term volatility and instability. For the policy experiments in Section 7, weinstead take Π(t) to be a smoothed version of the actual rate of cash injections by theCB. That implicitly assumes households and CG firms have access to that information andforecast the effects by applying a smoothing filter.

20This is due to the assumption of a closed system. Also, note that Dlf is a fixed fraction of total revenue,hence loans should approximately track revenue.

27

Loan defaults: If a loan defaults, then that loan is deducted from the corresponding valueof the loan array L(τ, t − i; t), where τ is the loan term, and t − i is the time at which theloan was made. We have added dependence on t, since with defaults existing elements of theloan array can change over time. Here we also have a loan array for loans owned by the CB,LCB(τ, t− i; t). In aggregate we will assume that a fraction δ of all loans default so that

L′(τ, t− i; t) = (1− δ)L(τ, t− i; t− 1) (44)

L′CB(τ, t− i; t) = (1− δ)LCB(τ, t− i; t− 1) (45)

for each i = 1, 2, · · · , τ and τ = 1, · · · , τm. We define the intermediate variables L′ andL′CB since the loan arrays will be subsequently modified due to the purchase of loans bythe CB. In general, δ could depend on i and τ ; however, for simplicity, we will assume thatdefaults occur uniformly over all elements of the loan array. Since a loan can default atan arbitrary time within the duration of the loan, it is assumed that the CG sector makesescrow payments up until the time of default. The bank then seizes the escrow account anddistributes what has accumulated in that account across Savings and Escrow. Note that theamount that is seized must be less than the full amount that the CG sector owes to the Banksector (for the loan that defaulted).

Here we assume that loans default at a constant rate δ. That is, at each iteration afraction δ of all loans default. With this assumption the aggregate amount left in escrowfrom loans that default each iteration is δ ·E(t). The return on household savings will theninclude the balloon repayments for all non-defaulting loans plus δ · E(t).

Escrow deposit (capital expenditures): This is computed as before for the Baselinemodel. Namely, the target amount in escrow and escrow deposit are given by (17) and (18),respectively, and the cash portion of the escrow deposit is given by (19).

Loan (balloon) repayments: With CB injections loans are split between the two loanarrays L(τ, t− i; t) and LCB(τ, t− i; t). Loan repayments to the Bank are the same as before,

LBP (t) =τm∑τ=1

[1 + r(t− τ)]τ · L′(τ, t− τ ; t) (46)

LBPI(t) =τm∑τ=1

{[1 + r(t− τ)]τ − 1} · L′(τ, t− τ ; t) (47)

where LBP (t) is the total amount of loan repayments, and LBPI(t) is the correspondinginterest.

Similarly, loan (balloon) repayments for CB-owned loans are given by

LCBP (t) =τm∑τ=1

[1 + r(t− τ)]τ · L′CB(τ, t− τ ; t) (48)

LCBI(t) =τm∑τ=1

{[1 + r(t− τ)]τ − 1} · L′CB(τ, t− τ ; t) (49)

where LCBP (t) is the total repayment and LCBI(t) is the corresponding interest. The loansare paid back to the CB in cash, reflecting the fact that the CB may attempt to control

28

the amount of cash reserves in the economy. Also, we assume that CG pays the CB out ofcurrent revenues, so does not keep an escrow account for loan repayments to the CB. Thiscorresponds to the scenario where the CB buys government debt (here included in total CGdebt), and loan repayments come from current tax revenues. Since the revenue contains amix of cash and credit (according to the parameters), the CB cashes in the credit part, whichis removed from Bank reserves. The fractions of the principal and interest to be paid backto the CB are, respectively, ζP and ζI so that the cash flow to the CB is given by

Bout(t) = ζP · LCBP (t) + ζI · LCBI(t). (50)

Target capital expenditures, demand for loans: As before, target capital expendi-tures are determined by maximizing output given labor and capital costs. For the Cobb-Douglas production function this gives Ktar(t) = h ·Rv(t) (see (14)). The demand for loansDlf (t) is then set so that the expected capital expenditures per iteration (escrow depositsplus repayments to the CB) in steady state is Ktar(t), assuming that Dlf (t) is in fact theamount of loans made in steady state.

In contrast to the scenario without a CB, here we have to distinguish three types ofloans: (1) CG loans with escrow accounts (as before), (2) loans owned by the CB, which arepaid back out of current revenues, and (3) loans owned by the CB, which are not paid back.Letting Λ(t) = LBP (t) +LCBP (t) +LCBI(t) be the total amount of loan repayments to boththe bank and the CB, the fractions of loans in each of these categories, are, respectively,

a1 =LBP (t)

Λ(t)a2 =

Bout(t)

Λ(t)a3 =

LCBP (t) + LCBI(t)−Bout(t)

Λ(t)(51)

For category (1), a unit loan taken out each iteration in steady state results in an escrowdeposit given by

du,1(t) =τm∑

τ=τmin

τ∑i=1

wτP (r, s, τ)Π−i(t− 1) (52)

where wτ is the fraction of loans with duration τ , and P (r, s, τ) is the escrow payment for aunit loan of duration τ made at interest rate r and return on investment s given by (81). Forcategory (2), a unit loan taken out each iteration in steady state results in loan repaymentsto the CB given by

du,2(t) =τm∑τ=1

wτ [1 + r(t− τ)]τ . (53)

Loans in category (3) are not included in the computation of Dlf (t).The demand for loans is then

Dlf (t) = Ktar(t)/[a1 · du,1(t) + a2 · du,2(t)] (54)

Strictly speaking, this should apply only in steady state; however, we will assume that theCG sector always has access to the corresponding amounts of loan repayments, and computesDlf in the same way even outside of steady state.21

21In practice, the loan amounts in (51) would likely be smoothed and forecast.

29

CG wages, unspent capital budget: These must be modified from the Baseline modelto account for payments to the CB. Specifically,

Wc(t) = min{(1− h) ·Rv(t), Rv(t)−∆E(t)−Bout(t)}. (55)

If the amount of capital expenditures ∆E(t) + Bout(t) is less than the target, then the firstterm is smaller, and CG retains the unspent amount

Ucg(t) = Rv(t)−∆E(t)−Wc(t)−Bout(t). (56)

We again assume that the CG sector uses this to purchase additional capital directly, so thatthis amount is added to capital wages (see (36)). In steady state Ucg(t) = 0.

Bank cash: As in the Baseline model, the updated Bank cash accounts for loans made atthe preceding iteration, savings withdrawals, and escrow deposits,

B′c(t) = [Bc(t− 1)− ckL(t− 1) + ∆C(t) + ∆Ec(t)]+ . (57)

This assumes that none of the cash reserves are needed to repay CB-owned loans. That is,there is enough revenue to cover those repayments, or if not, then the CB allows those loansto default. The new intermediate variable B′c is needed since the updated Bc will includecash from the CB.

Bank assets, updated Savings, Escrow The total bank assets after loan repaymentsare again allocated to Savings and Escrow in proportion to the size of those accounts Thefollowing intermediate variables are introduced, which are updated Savings and Escrow:

S ′(t) = S(t− 1) + ∆S(t) (58)

E ′(t) = (1− δ) · E(t− 1) + ∆E(t)− LBP (t). (59)

Total Bank liabilities are then given by (43). This is the same as the Baseline model, but nowdefaults are subtracted from Escrow, which is a reduction of Bank liabilities. The banks seizewhat it is in those defaulted accounts, and add those to Bank assets, which are distributedproportionally across the updated Savings and Escrow accounts.

Since all repayments to the CB are taken from the current revenue stream, the loanrepayments from Escrow, LBP , do not include loans owned by the CB. Note that in practicethe Bank will own additional government bonds for which no escrow is kept. That is, herewe are assuming that escrow is kept only for bank-owned loans, and ignore the fact thatthere are many loans to the combined CG-government sector for which no escrow accountswould be held.

Defaults on the bank assets side are subtracted from the total amount of loans owned bythe Bank at time t, Ltot(t) (see (70)), and affect both Savings and Escrow. Note that this doesnot change the Bank balance (assets less liabilities) since total assets are distributed acrossSavings and Escrow accounts as liabilities. In what follows, L(t) = S(t) +E(t) denotes totalbank liabilities for Savings and Escrow only, and does not include liabilities for the creditpart of Wages. Collecting those liabilities and allocating proportionally across Savings andEscrow gives

L(t) = B(t) + LBPI(t)− δ · Ltot(t− 1) + δ · E(t− 1) + pCB(t− 1) (60)

S(t) = L(t)S ′(t)

B(t), E(t) = L(t)

E ′(t)

B(t)(61)

30

where pCB(t) is the premium the CB pays to purchase loans, in addition to the loan principal,which covers the accrued interest up to that time (see (69)).

Note that these updates do not include loans owned by the CB. Any CB-owned loans thatdefault are simply not repaid, i.e., reduce the value of Bout in (50), resulting in an effectivegift to the CG sector. The defaults are therefore combined with government bonds, whichare not repaid from revenue streams; rather the treasury prints the money and transfers itdirectly to the CB.

CB injections, supply of loans: The CB is assumed to inject cash into the economyat a fixed rate by purchasing a constant fraction of CG debt each iteration. The amountinjected, money supply, and bank cash are given by

F (t) = κ ·G(t− 1), G(t) = G(t− 1) + F (t) (62)

Bc(t) = B′c(t) + F (t). (63)

As before (without CB injections), the supply of loans is then given by (31). That is a myopicestimate, which does not take into account side information the Bank may have. We canagain include information about the existing loan array to obtain the alternative estimateof Slf (32) that is expected in steady-state, given current state variables, hence is moreforward-looking. Here, however, the amount of total outstanding loans, which contribute tototal Bank assets, is the sum of all loans in the CG loan array (excluding CB-owned loans),and in steady-state is Γ(t)L(t), where

Γ(t) =τm∑τ=1

τ−1∑i=1

wi[1− δ(t)]iΠ−i(t)[1− η(t)]i (64)

and where η(t) is the fraction of loans bought by the CB (see (66)). That is, total loansare reduced by the default rate across the loan array, the inflation rate, and the fraction ofloans transferred to the CB. The same derivation then again gives (32), where Γ(t) is givenby (64).

Fraction of loans bought by CB: For simplicity we assume that the CB purchases afixed fraction of all CG loans. Let π(τ, t− i; t) denote the amount paid for loans L(τ, t− i; t).We determine π(τ, t− i; t) by setting the amount accumulated for the remainder of the loanduration τ − i at interest r(t) equal to the balloon payment of the original loan:

[1 + r(t)]τ−iπ(τ, t− i; t) = [1 + r(t− i)]τL(τ, t− i; t) (65)

for i < τ . (We assume that the CB only purchases loans that expire after time t.) Giventhat the CB wishes to spend a total of F (t), the fraction of loans it buys is then

η(t) = F (t)/

[τm∑τ=1

τ−1∑i=1

π(τ, t− i− 1; t)

](66)

where π(τ, t− i; t) is given by (65). The new loans purchased by the CB are then transferredfrom the CG loan array to the CB loan array:

L(τ, t− i; t) = [1− η(t)]L′(τ, t− i; t) (67)

LCB(τ, t− i; t) = L′CB(τ, t− i; t) + η(t) · L′(τ, t− i; t) (68)

31

for τ = 1, · · · , τm and i = 1, · · · , τ − 1.The difference π(τ, t − i; t) − L(τ, t − i; t) is the premium the CB pays for purchasing

the loans corresponding to L(τ, t − i; t). Hence the total premium amount the CB pays atiteration t is

pCB(t) =τm∑τ=1

τ−1∑i=1

η(t)[π(τ, t− i; t)− L′(τ, t− i; t)]. (69)

This is included as part of Bank liabilities in (60).22

Loans, Wages, and Wealth: These are the same as for the Baseline model, given by(35)-(38). The total amount of outstanding loans, excluding those owned by the CB, is

Ltot(t) =τm∑τ=1

τ∑i=1

L(τ, t− i; t). (70)

This is needed to compute the total amount of defaults at the next iteration. (See (60).)

Measurements: Measurement variables such as GDP, the proportion of labor allocatedto CG and K, and unemployment depend on the same state variables as in the Baselinemodel, hence can be computed in the same way.

7 Policy Experiments and Shocks

We now show a series of plots that illustrate some basic features of credit expansion andcontraction associated with the model. We begin by discussing the choice of system param-eters used to generate the plots, and subsequently show four sets of plots that are organizedaccording to the main results discussed in Section 2.

7.1 Model Parameters

Our intent in this section is to show a representative set of experiments that illustrate somebasic properties exhibited by the model. While there is a range of parameters that couldbe selected, we have chosen parameters that may correspond to a “reasonable” set of statevariables (e.g., a loan interest rate in the range of 0.5 to 1% per period). Unless specifiedotherwise, the plots assume the parameters shown in Table 1. The parameters above thedouble line are sufficient to determine the equilibrium and steady-state values, whereas theparameters below the double line affect dynamics (transients). Changing the latter set ofparameters can speed up or slow down transients, and mute or enhance transient swings instate variables due to shocks. As previously discussed, selection of those parameters has acritical effect on the duration of transients and stability. However, in practice the dominantinfluence on transient dynamics would likely be driven by traders in the financial sector, notmodeled here. As for the baseline model, the duration of a single time period is linked tothe interpretation of loan durations. If we assume a maximum loan duration of five years(after which the loans are refinanced or traded), then taking τm = 60 means that the eachperiod represents approximately one month.

22We assume that F (t) never exceeds the entire amount of CG loans.

32

f = 10% fractional reserve requiremente = 500 propensity to saveτm = 60 maximum loan durationwτ = 1/τm, τ = 1, · · · , τm loan window distributionc = ck = 0.1 cash fraction of loans, consumption (mostly-credit)βK = 0.5, βL = 0.5 return on capital, return on laborκ = 0.005 CB injection rate (fraction of money supply) (63)

kr = ... step-size for loan rateks = ... step-size for return on investmentki = ... step-size for inflation estimate

Table 2: Default parameters used to generate the results in Section 7.

7.2 Transient and Steady-State Effects of Central Bank Injections

Figure 6 shows the effects of injections by the CB. The CB injects money by buying out-standing CG sector debt from the Bank sector at at period 400 after the system reachesan initial equilibrium. Thereafter, each period the amount of loans purchased by the CB isequal to 0.5% of the current money supply. In this set of plots, the CB does not requirerepayments from the CG sector on the loans it purchases so that the cash it injects adds tothe existing money supply and never leaves the economy. (The next figure illustrates whathappens when the loans are paid back to the CB.) As a consequence, as shown in the first(upper left) plot, both the amount of cash injected per period (amount shown on the left)and the money supply (amount shown on the right) increase exponentially.

The top right plot shows both nominal and real loan interest rates. The real interest rateis the nominal rate minus the estimated inflation rate.23 The CB injections create a transientin which the nominal rate increases and the real interest rate decreases relative to the initialequilibrium rate. The system reaches a steady-state in which the nominal and real interestrates each converge to their final values, different from the preceding equilibrium value.Hence this figure indicates that the CB injections have a real effect on the economy even insteady-state. This is further illustrated in the lower left plot, which shows the allocation oftotal labor capacity to the K and CG sectors. The decrease in the real interest rate makescapital less expensive, so the CG demand for capital increases. That increases the flow oftotal wages into the K sector relative to the flow of wages going to the CG sector, whichcauses laborers to shift from the CG sector to the K sector. This shift in turn causes a slightincrease in production, as shown in the lower right plot. The increase in K production alsoshown in the figure is more pronounced than that for CG, since it is more directly influencedby the labor shift. The relative insensitivity of CG production shown in this example (andthe other examples in this section) is partially due to the choice of parameters,24 and also dueto the restrictive nature of the assumed Cobb-Douglas production function with constantreturns to scale.

The rise in nominal rate shown in the top right plot is due to inflation expectations.That is, demand for loanable funds Dlf (t) accounts for the anticipated rise in revenues

23In all plots with CB injections the inflation estimate Π(t) is taken to be a smoothed version of the actualrate of CB injections. In steady-state the estimated inflation rate converges to the actual injection rate.

24Decreasing the propensity to save e would accentuate the shift in labor-capital split, but would alsosignificantly increase the interest rate r.

33

due to the CB injections. (Recall that Dlf (t) is determined by the CG sector so thatcapital expenditures per iteration (escrow deposits) are a constant pre-determined fractionof revenue.) However, the increase in nominal rate from the preceding equilibrium value isless than the inflation rate, causing a drop in the real interest rate. In other words, if thereal interest rate were held constant, then when injections commence, the supply of loanablefunds Slf (t) would exceed Dlf (t). The reason for this is that the CB injections effectivelydiscount the loan obligations held as liabilities by the Bank, providing more room for theBank to expand credit.

To illustrate the real change in interest rate with a simple example, suppose that τm = 2,and that without CB injections the loan amount per period is L, split evenly between loandurations τ = 1 and τ = 2. At time t, loans initiated at time t − 1 with τ = 1 andloans initiated at time t − 2 with τ = 2 are repaid, so are not counted as liabilities whendetermining loans at time t. The total loan liabilities in equilibrium at time t are then L/2from period t−1, corresponding to τ = 2. To determine Slf (t) those liabilities are subtractedfrom the total amount of credit ξ the Bank can generate at time t, which is determined bythe reserve requirement (see (31)). Hence in equilibrium the ratio of total loan liabilities tototal credit generated is L/(2ξ). Now suppose that the CB injects cash at a rate of Π − 1per period. Then as t increases, in steady-state the loans per period and credit increase asL(t) = L(t0) · Πt−t0 and ξ(t) = ξ(t0) · Πt−t0 , respectively, where t > t0. At time t the totalloan liabilities are then L(t − 1)/2 = L(t)/(2Π), and the ratio of total loan liabilities tototal credit is L(t)/[2Πξ(t)] < L(t)/[2ξ(t)] when Π > 1. Hence the effect of the injections isto decrease the ratio of total loan liabilities to total outstanding credit, thereby increasingSlf (t) relative to Dlf (t), and decreasing the real interest rate. This effect is accentuated asthe duration of loans increase (i.e., as τm increases).

We point out that an important assumption in this example is that the Bank maximizesits loans subject to the reserve requirement. The increase in Slf (t) due to injections thereforehas a direct effect in increasing the loan amount. In practice, loan amounts will also dependon perceived risk of default, with the loans made at the margin incurring the largest risk.A steady-state influx of cash would therefore not necessarily induce the Bank to make moreloans (relative to the money supply) if it has already selected the loan target to optimize itsrisk-return tradeoff. Although that would seem to mute the real effects seen here, in thatscenario the excess cash reserves accumulated by the Bank (above the legal limit) wouldin steady-state be passed along to shareholders in the Household sector and subsequentlycirculate in the economy, again causing a real effect. (Injections through the Household sectorare illustrated in a subsequent example.) Finally, there may be reasons why an increase inSlf would induce the Bank to increase loans even taking risk into account. One is depositinsurance, which may cause the Bank to discount the probability of default, and another isthe time constant needed to quantify the risk-return tradeoff over the loan portfolio once theinjections commence.

Figure 7 is analogous to Figure 6, the difference being that all of the debt purchased bythe CB is repaid with interest to the CB. The first plot shows that the CB again begins tobuy debt at period 400 after the system reaches an initial equilibrium. The money supplythen increases initially, but once the purchased loans come to term the money supply startsto fall since the interest on each of the loans purchased by the CB, which leaves the economy,must exceed the premium paid by the CB. Hence there is a net flow of money out of theeconomy and into the CB. The cash injected per period also falls, since each period the CBinjects 0.5% of the decreasing money supply.

34

Period200 400 600 800

Mon

etar

y U

nits

0

2

4

6Central Bank Injections, Money Supply (--)

0

500

1000

1500

Mon

etar

y U

nits

Period200 400 600 800

Per

cent

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3Real Loan Rate, Nominal Loan Rate (--)

Period200 400 600 800

Per

cent

of L

abor

ers

44

46

48

50

52

54

Labor in Consumer Goods Sector,Labor in Capital Sector (--)

Period200 400 600 800

Pro

duct

ion

Uni

ts

48.5

49

49.5

50

Consumer Goods Production,Capital Production (--)

4.4

4.6

4.8

5

Pro

duct

ion

Uni

ts

Central Bank Injections to Bank Without Repayments

Model parameters: f=0.1 e=500 =max

=60 c=ck=0.1 -

K=-

L=0.5 5=0.005

Figure 6: Transient and steady-state effects of CB injections with no repayments. Quantitiesare shown as a function of the period t. The CB begins buying CG debt at time t = 400 afterthe system has reached an initial equilibrium. The top left plot shows the money supply(solid) and cash injections (dashed). The top right plot shows that the CB injections causethe real interest rate to decrease, which in turn affects the allocation of labor to the CG andK sectors (lower left) and production of consumer goods and capital (lower right). In each ofthe plots in this section, the y-axis for the solid (dashed) curve is shown on the left (right).

The top right plot shows nominal and real loan interest rates. Here the effects of theCB injections are the opposite of what is shown in Figure 6: the nominal rate decreasesand the real interest rate increases relative to the initial equilibrium rate. The decrease innominal rate is now due to deflation expectations (as opposed to inflation expectations inFigure 6) when computing Dlf (t). Here the outstanding nominal debt shrinks as t increases,so that interest payments corresponding to prior loans issued at t− t0 increase with t0. Bankliabilities relative to the total credit generated then increase with t, which decreases Slf (t),causing the real interest rate to rise.

The lower left plot shows the shift in labor with the injections. The increase in thereal interest rate makes capital more expensive, which decreases the flow into the Capitalsector relative to total wages, causing some labor to switch from the capital sector to theCG sector. This results in a slight decrease in output shown in the last plot. (Again, theshifts in labor and production are slight due to the parameter set and the restrictive nature

35

Period200 400 600 800

Mon

etar

y U

nits

0

0.5

1Central Bank Injections, Money Supply (--)

50

100

150

Mon

etar

y U

nits

Period200 400 600 800

Per

cent

0.75

0.8

0.85

0.9

0.95Real Loan Rate, Nominal Loan Rate (--)

Period200 400 600 800

Per

cent

of L

abor

ers

44

46

48

50

52

54

Labor in Consumer Goods Sector,Labor in Capital Sector (--)

Period200 400 600 800

Pro

duct

ion

Uni

ts

48

48.5

49

49.5

50

Consumer Goods Production,Capital Production (--)

4.5

4.6

4.7

4.8

4.9

Pro

duct

ion

Uni

ts

Central Bank Injections to Bank With Repayments

Model parameters: f=0.1 e=500 =max

=60 c=ck=0 -

K=-

L=0.5 5=0.005

Figure 7: Transient and steady-state effects of CB injections with full repayments. Quantitiesare shown as a function of the period t. The CB begins buying CG debt at time t = 400after the system has reached an initial equilibrium. The top left plot shows the moneysupply (solid) and cash injections (dashed). The money supply decreases in steady-statesince the cash outflow including interest payments to the CB exceeds the cash inflow neededto purchase the CG debt. The top right plot shows that the CB injections cause the realinterest rate to increase, which in turn affects the allocation of labor to the CG and K sectors(lower left) and production of consumer goods and capital (lower right).

of the production function.)

7.3 Credit Expansion and Contraction

Figures 8-9 show the effects of credit expansion and contraction. In Figure 8 the Bankinitially expands and subsequently contracts credit by changing the credit multiplier m =1/f . Specifically, in the first (upper left) plot m (solid curve) increases linearly from 10 to20 (f decreases from 10% to 5%) starting at period 400 after the system has reached aninitial equilibrium. After the system reaches a new equilibrium with expanded credit m issubsequently decreased linearly from 20 back to 10. Also shown is total Household money,or wealth, (dashed) given by (38). The changes in credit are reflected in the nominal wealth,which respectively increases and decreases as credit expands and contracts. (The effect on

36

real wealth, normalized by CG price, is related to the change in actual production.) Thetop right plot shows that the expansion of credit lowers the loan rate to a new equilibriumvalue, and the subsequent contraction increases the loan rate back to its initial equilibriumvalue. (The overshoots can be muted by decreasing the step-sizes in Table 2, but then thedurations of the transients increase.)

The lower left plot in Figure 8 shows total sector Wages Wc and Wk given by (33)and (36). The credit expansion and contraction respectively increases and decreases totalwages across CG and K sectors. The decrease in total wages associated with the creditcontraction causes unemployment, shown as the dotted curve. Specifically, the fall in totalwages creates a large spike in unemployment, which subsides once the system returns tothe original equilibrium. We emphasize that because wages fall across both the CG andK sectors, unemployment is spread across the entire economy (i.e., over both CG and Ksectors). This causes the transient shifts in labor allocations shown in the lower right plots.Note that the contraction causes the downward spike in labor allocations, and an associatedloss in productivity, between periods 700 and 800.

The severity of the transients shown in this figure again depends on the choice of step-sizesin Table 2 along with the parameter for sticky wages in (??) used to determine unemploy-ment. In particular, the overshoot in wages due to the contraction shown in the lower leftplot causes mild unemployment during the expansion phase. There is also a transient shiftin which the allocation of labor across CG and K reverses (again causing a mild transientdecrease in production). While changing the step-size and unemployment parameters canmute those transients, those changes do not affect the total fall in wages across both sectorsduring the contraction and the associated loss in productivity.

Figure 9 shows the analogous plots as in Figure 8, but where the credit expansion/contractionis due to increasing/decreasing CB injections. The top left plot shows the CB injections(solid) and the money supply (dashed). The injections are ramped up to 0.5% of the moneysupply per period at period 400, and subsequently ramped down to zero after period 800.The money supply therefore increases exponentially during the steady-state injections andthen levels out at a constant higher value when the injections stop. The top right plots showthe real and nominal interest rates. The real interest rate decreases to a new steady-statevalue caused by the injections, and then increases back to its original (pre-injection) value.The bottom left plots show total CG (solid) and K (dashed) wages, along with unemploy-ment (dotted). As in the previous figure, the credit expansion due to the injections increasenominal wages, which fall to their original level after the injections stop. The decrease intotal wages combined with sticky wage prices cause a spike in unemployment during thecontraction, which is again across both the CG and K sectors. The lower right plot showsthe labor-capital split. The injections cause labor to move from CG to K, corresponding toa drop in the real interest rate, which is reversed when the injections stop.

Comparing Figure 9 with Figure 8, the CB injections in Figure 9 cause a larger shift inthe real interest rate than that shown in Figure 8. Hence the CB injections have a somewhatmore pronounced effect on labor-capital split. In contrast, the unemployment spike duringthe contraction is less pronounced in Figure 9 than in Figure 8. That is because duringthe contraction, wages in Figure 8 fall approximately by 50% whereas in Figure 9, wagesfall about 10%. The increase and decrease in wages in Figure 8 correspond directly to theincrease and decrease in the credit multiplier m, whereas the increase and decrease in wagesin Figure 9 appear subdued since wages are normalized by the money supply, which increasesduring the expansion, and levels out during the contraction. Again we emphasize that the

37

Period200 400 600 800

Uni

ts

10

15

20Multiplier m, Household Money (--)

0

500

1000

Per

cent

of M

oney

Sup

ply

Period200 400 600 800

Per

cent

0.6

0.7

0.8

0.9

1

1.1

1.2Loan Rate

Period200 400 600 800

Per

cent

of M

oney

Sup

ply

20

40

60

80

Wages in Consumer Goods Sector,Wages in Capital Sector (--), Unemployment (" ")

-10

0

10

20P

erce

nt o

f Lab

orer

s

Period200 400 600 800

Per

cent

of L

abor

ers

40

42

44

46

48

50

52

54

Labor in Consumer Goods Sector,Labor in Capital Sector (--)

Credit Expansion and Contraction Due to Reserve Rate Shocks

Model parameters: f=0.1 e=500 =max

=60 c=ck=0.1 -

K=-

L=0.5 5=0.005

Figure 8: Dynamic effects of Bank credit expansion and contraction. The top left plots showthe credit multiplier m = 1/f (solid) and total Household money, or wealth (dashed). Themultiplier m is ramped up from 10 to 20 at period 400, and is subsequently ramped backdown at period 700. The top right plot shows the loan rate r, which decreases to a newequilibrium after m is increased, and increases back to its original value when m is rampedback down. The lower left plots show total wages WC (solid) and WK (dashed), and alsounemployment (dotted, y-axis on right). Unemployment is caused by a decrease in totalwages. The lower right plot shows the fraction of total labor allocated to the ConsumerGoods and Capital sectors (labor-capital split). Unemployment causes the downward spikein total labor utilization.

particular dynamics shown are affected by the particular choice of system parameters.25

We next contrast the preceding credit expansion and contraction examples with Fig. 10,which shows the effect of an output elasticity shock. Specifically, the top left plots showthat the return on capital βL (solid) is ramped from 0.5 down to 0.4 starting at period 400,which means βK (dashed) is ramped from 0.5 to 0.6. Hence the return on capital increases,the demand for loans increases relative to the supply, the loan rate increases, as shown inthe upper right plot, which in turn shifts the labor-capital split, as shown in the lower rightplot. That is, labor shifts from the CG sector to the K sector. (The labor-capital split

25In particular, increasing the step-sizes in Table 2 can produce instabilities with long-term oscillationsand crashes in which the CG sector debt exceeds its revenues.

38

Period400 600 800 1000 1200

Mon

etar

y U

nits

0

2

4

6Central Bank Injections, Money Supply (--)

0

500

1000

1500

Mon

etar

y U

nits

Period400 600 800 1000 1200

Per

cent

0.7

0.8

0.9

1

1.1

1.2

1.3Real Loan Rate, Nominal Loan Rate (--)

Period400 600 800 1000 1200

Per

cent

of M

oney

Sup

ply

26

28

30

32

34

Wages in Consumer Goods Sector,Wages in Capital Sector (--), Unemployment (" ")

-2

0

2

4

6P

erce

nt o

f Lab

orer

s

Period400 600 800 1000 1200

Per

cent

of L

abor

ers

47

48

49

50

51

52

53

Labor in Consumer Goods Sector,Labor in Capital Sector (--)

Credit Expansion and Contraction Due to Central Bank Injection Shocks

Model parameters: f=0.1 e=500 =max

=60 c=ck=0.1 -

K=-

L=0.5 5=0.005

Figure 9: Dynamic effects of credit expansion and contraction due to Central Bank injections.The top left plots show the Central Bank injections (solid) and money supply (dashed). Theinjections ramp up to 0.5% per period starting at period 400, and then decrease back tozero after period 800 when the system has reached a steady-state. The top right plot showsthe real (solid) and nominal (dashed) loan rate r. The real loan rate decreases to a newequilibrium after the CB injects, and increases back to its original value when the CB stopsinjecting. The lower left plots show wages WC (solid) and WK (dashed), normalized by themoney supply, and also unemployment (dotted, y-axis on right). Unemployment is causedby a decrease in wages. The lower right plot shows the fraction of total labor allocated tothe Consumer Goods and Capital sectors (labor-capital split). Unemployment causes thedownward spike in total labor utilization.

which maximizes output becomes 40% labor / 60% capital instead of 50-50.) The lower leftplot again shows wages and unemployment. In contrast to Figs. 8 and 9, here the fall inwages occurs entirely in the CG sector (ignoring the small overshoot in K wages after theshock). Hence unemployment is confined to the CG sector and does not occur in the Ksector in contrast with the preceding two figures. This example then serves to emphasizethe important distinction between the effects of a credit contraction, which spread from theK sector (by reducing loans) to the CG sector, and the effects of other kinds of parametershocks, which may have different qualitative effects on the different sectors in the economy.

39

Period300 400 500 600

Ela

stic

ity

0.4

0.45

0.5

0.55

0.6

0.65Labor and Capital Elasticity, -

l, -

k (--)

Period300 400 500 600

Per

cent

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04Loan Rate

Period300 400 500 600

Per

cent

of M

oney

Sup

ply

20

25

30

35

Wages in Consumer Goods Sector,Wages in Capital Sector (--), Unemployment (" ")

-2

0

2

4P

erce

nt o

f Lab

orer

s

Period300 400 500 600

Per

cent

of L

abor

ers

40

45

50

55

60

Labor in Consumer Goods Sector,Labor in Capital Sector (--)

Output Elasticity Shock

Model parameters: f=0.1 e=500 =max

=60 c=ck=0.1

Figure 10: Effects of an output elasticity shock. The top left plots show output elasticities βL(solid) and βK (dashed), which are ramped down and up, respectively, starting at period 400.The upper right plot shows that the loan rate increases to a new equilibrium value, and thelower right plots show the corresponding shift in the labor-capital split. The lower left plotsshow wages WC (solid), WK (dashed), and unemployment (dotted). Here the unemploymentis caused by the decrease in WC , and is therefore confined to the CG sector.

7.4 Loan Defaults

We now illustrate the effects of loan defaults in Figures 11-12. In both figures defaultsare ramped up exogenously to 1% of all loans from an initial equilibrium with no defaults.The defaults then persist for the remainder of the example. Once the system reaches anequilibrium with defaults, a credit contraction is introduced. Specifically, the top left plotsin Fig. 11 show that defaults (dashed) are ramped up starting at period 400, and that thecredit multiplier m (solid) is initialized at 20, and is ramped down to 10 starting at period700. This is motivated by a scenario in which a large credit multiplier causes the Bank sectorto make excessively risky loans, eventually increasing the fraction of loans that default inequilibrium. In response to the defaults, the Bank subsequently increases the reserve rate,causing a contraction in credit. In practice, the credit contraction would likely occur alongwith the defaults whereas here we separate these events. This is to illustrate the differenteffects caused by these events, which would be difficult to observe if the events overlapped.

40

The top right plots in Fig. 11 show the loan rate r (solid, y-axis on left), and total loansper period (dashed, y-axis on right). It is perhaps surprising that the introduction of defaultscause loans per period to increase and the interest rate to decrease. The reason for this isthat a loan default deletes the loan from the Bank’s liabilities, which are subtracted from thetotal credit the Bank can generate when determining Slf (t) in (??). In other words, the Bankincreases its loans to make up for the defaulting loans, so that it achieves its target creditexpansion m. Because the defaulted loans are not repaid in full, however, they reduce thereturn on investment s. (Recall that the Bank seizes whatever has accumulated in Escrowand passes that on to the Households, so the defaulted loans are partially repaid.) Of course,in practice, the Bank would not attempt to maintain the same target credit expansion m atthe onset of defaults. Hence m is subsequently reduced causing the interest rate to rise, andloans per period to fall. (Note that the final equilibrium interest rate is different from theinitial rate due to the presence of defaults.)

The bottom left plots shows the labor-capital split (solid and dashed), and unemployment(dotted, y-axis on the right). The defaults introduce 1% unemployment (corresponding tothe default rate), and shift the labor-capital split towards more capital due to the increase inloans. The bottom right plots show that the defaults cause a slight increase in the productionof capital (dashed), but a decrease in the production of CG (solid). The subsequent creditcontraction then causes a large spike in unemployment and downward spike in both CGand capital production. These results indicate that while the defaults themselves causea moderate increase in unemployment and reduction in return to savings, the subsequentcredit contraction triggered by the defaults cause a much more significant disruption toproductivity.

Fig. 12 shows the analogous results as in Fig. 11 with CB injections. The top left plotsshows that the CB injects 0.5% of the money supply per period (solid), and that defaultsare introduced at period 700 (dashed) after the system has reached an initial steady-state.This is motivated by a scenario in which the injections cause the Bank to increase the riskassociated with its loan portfolio leading to an increase in default rate (1% per period).The increase in default rate subsequently causes the Bank to reduce the credit multiplierm (dotted) from 20 to 10 starting at period 1100. As in Fig. 11, the system converges toa new equilibrium with defaults before the credit contraction is introduced to highlight theassociated effects.

The top right plots show the real loan rate r (solid, y-axis on left), and total loansper period (dashed, y-axis on right). Again the introduction of defaults cause loans perperiod to increase and the interest rate to decrease. The subsequent credit contraction thencauses loans to fall by about 50%, and the interest rate rises accordingly. The effects onlabor-capital split and unemployment (lower left) and production of consumer goods andcapital (lower right) are qualitatively the same as in Fig. 11. Namely, the defaults againintroduce 1% unemployment (corresponding to the default rate), and shift the labor-capitalsplit towards more capital due to the increase in loans. This causes a slight increase inthe production of capital, but a decrease in the production of CG. The subsequent creditcontraction then causes a large spike in unemployment and downward spike in both CG andcapital production.

41

Period200 400 600 800

Mul

tiplie

r

10

12

14

16

18

20Multiplier m, Defaults (" ")

0

0.2

0.4

0.6

0.8

1

Per

cent

Period200 400 600 800

Per

cent

0.2

0.4

0.6

0.8

1Loan Rate, Loans

30

40

50

60

70

Per

cent

of M

oney

Sup

ply

Period200 400 600 800

Per

cent

of L

abor

ers

35

40

45

50

55

Labor in Consumer Goods Sector,Labor in Capital Sector (--), Unemployment (" ")

0

5

10

15

20P

erce

nt o

f Lab

orer

s

Period200 400 600 800

Pro

duct

ion

Uni

ts

40

45

50

Consumer Goods Production,Capital Production (--)

2

4

6

Pro

duct

ion

Uni

ts

Credit Contraction Due to Reserve Rate Shock with Defaults

Model parameters: f=0.1 e=500 =max

=60 c=ck=0.1 -

K=-

L=0.5

Figure 11: Effect of defaults and subsequent credit contraction. The top left plots showthe default rate (dashed) and the credit multiplier m (solid). Defaults ramp up to 1% perperiod, and m subsequently ramps down from 20 to 10. The top right plots show the loanrate (solid) and loans per period (dashed). Due to the target credit expansion, defaults alonecause loans to increase and the loan rate to decrease. The bottom left plots show the labor-capital split (solid, dashed), and unemployment (dotted). Defaults cause 1% unemploymentand the contraction causes the corresponding spike in unemployment. The bottom rightplots show production of capital and consumer goods. Defaults cause capital production toincrease slightly, but production of consumer goods to decrease.

7.5 CB Injections Through the Bank versus Household Sectors

The final set of plots compare the effects of CB injections through the Bank sector, asassumed in the previous section and experiments, versus injections (cash) given directlyto the H sector. We first show in Fig. 13 how the equilibrium is affected by the creditmultiplier m. We will contrast this with credit expansion through the CB in subsequentplots. As m increases, the Bank sector is able to make more loans with the same cashreserves. As in previous experiments, transactions are made mostly by exchanging credit(since c = ck = 0.1), so almost all of the cash in the economy remains in bank reserves,making them roughly constant. As a result, SLF and therefore loans increase nearly linearlyin the upper left plot. The upper right plot shows that the loan rate decreases to allow DLF

42

Period600 800 1000 1200

Per

cent

0

0.2

0.4

0.6

0.8

1CB Injections, Defaults (--), Multiplier m ( " ")

10

12

14

16

18

20

Mul

tiplie

r

Period600 800 1000 1200

Per

cent

0.2

0.4

0.6

0.8Real Loan Rate, Loans (--)

20

40

60

80

Per

cent

of M

oney

Sup

ply

Period600 800 1000 1200

Per

cent

of L

abor

ers

30

40

50

60

Labor in Consumer Goods Sector,Labor in Capital Sector (--), Unemployment (" ")

-10

0

10

20P

erce

nt o

f Lab

orer

s

Period600 800 1000 1200

Pro

duct

ion

Uni

ts

40

45

50

Consumer Goods Production,Capital Production (--)

2

4

6

Pro

duct

ion

Uni

ts

Credit Contraction Due to Central Bank Injection Shock with Defaults

Model parameters: f=0.1 e=500 =max

=60 c=ck=0.1 -

K=-

L=0.5 5=0.005

Figure 12: Effect of defaults and subsequent credit contraction with Central Bank injections.The top left plots show the injection rate (solid), default rate (dashed), and the credit multi-plier m (solid). Defaults ramp up to 1% per period, and m subsequently ramps down from 20to 10. The top right plots show the real loan rate (solid) and loans per period (dashed), thebottom left plots show the labor-capital split (solid, dashed), and unemployment (dotted),and the bottom right plots show production of capital and consumer goods. The effectsshown here with injections are analogous to those shown in Fig. 11.

to adjust accordingly (*Real rate should equal the nominal here, need to investigate plot.Also more resolution would look nicer*). This increase in loans leads to a shift of labor fromCG to the K sector, pushing the output closer to the maximum as shown in the lower leftand right plots. As shown in the upper right plot, the interest right converges to a positive,lower bound as m becomes large even though loans continue to increase. The reason is thatthe part of the credit that the bank issues circulates in Escrow and CG Wages, which areBank liabilities, and therefore serve to limit SLF.(*Not complete, still dont really understand how r converges*)

Fig. 14 shows steady-state values versus the rate of CB injections introduced through theBank sector. In the upper left plot, loans increase with these injections due to two effects:(1) the increase in cash reserves at the Bank, and (2) the devaluation of bank liabilities dueto inflation, as discussed for Fig. 6. These effects are also present in Fig. 15, which showsa similar experiment, except the injections are introduced through the Household sector

43

Multiplier m10 20 30 40 50

Per

cent

of M

oney

0

20

40

60

80

100

120

140Loans

Multiplier m10 20 30 40 50

Per

cent

0.5

1

1.5

2

2.5

3Real Loan Rate, Nominal Loan Rate (--)

Multiplier m10 20 30 40 50

Per

cent

of L

abor

42

44

46

48

50

52

54

56

58

Labor in Consumer Goods Sector,Labor in Capital Sector (--)

Multiplier m10 20 30 40 50

Pro

duct

ion

Uni

ts

49.5

50

Consumer Goods Production,Capital Production (--)

4

5

Pro

duct

ion

Uni

ts

Equilibrium Values vs Multiplier

Model parameters: f=0.01 e=500 =max

=60 c=ck=0.1 -

K=-

L=0.5

Figure 13: Equilibrium values versus credit multiplier m. The top left plot shows that loansincrease with m (almost linearly for small m). The top right plot shows that the interestrate decreases with m and converges to a positive value (above 0.5%) as m → ∞. Hencethe labor-capital split (lower left) does not converge to half-half, which would maximizeproduction (lower right), as m becomes large.

rather than through the Bank sector. Both figures (upper right plots) show that the realinterest rate decreases due to the increase in SLF, whereas the nominal rate increases withthe inflation rate. Here, however, the CB is able to drive the real interest rate much lowerthan the Bank sector can with credit expansion in Fig. 13. Comparing the lower plots inFigs. 14 and 15, Fig. 14 shows a significantly larger shift in labor from CG to K as injectionsincrease compared to Fig. 15. In Fig. 15 the labor-capital split converges to half-half, whichmaximizes output. In Fig. 14 the shift of labor from CG to K becomes excessive as theinjection rate increases. This causes the production (lower right plot) to peak at the pointwhere the labor-capital allocation curves cross, in contrast to Fig. 15 where the productionincreases to the maximum.

The difference in results for these two experiments is due to the assumption that in Fig.14, the CG sector does not repay any of the debt bought by the CB, including principal(corresponding to a “helicopter drop”). For a given loan portfolio that considerably reducesthe total amount the CG sector pays for capital expenditures. Conversely, for a target capitalexpenditure that increases the amount of loans that can be carried. This effect is absent

44

Injections (Fraction of Money Supply)0 0.01 0.02 0.03

Per

cent

of M

oney

Sup

ply

25

30

35

40

45

50Loans

Injections (Fraction of Money Supply)0 0.01 0.02 0.03

Per

cent

0

0.5

1

1.5

2

2.5

3

3.5Real Loan Rate, Nominal Loan Rate (--)

Injections (Fraction of Money Supply)0 0.01 0.02 0.03

Per

cent

of L

abor

47

48

49

50

51

52

53

Labor in Consumer Goods Sector,Labor in Capital Sector (--)

Injections (Fraction of Money Supply)0 0.01 0.02 0.03

Pro

duct

ion

Uni

ts

49.9

49.92

49.94

49.96

49.98

50

Consumer Goods Production,Capital Production (--)

4.7

4.8

4.9

5

5.1

5.2

Pro

duct

ion

Uni

ts

Equilibrium Values vs Central Bank Injections to Banks without Repayments

Model parameters: f=0.1 e=500 =max

=60 -K

=-L=0.5

Figure 14: Steady-state values versus rate of CB injections through the Bank. The top leftplot shows that loans increase nearly linearly with the injection rate. The top right plotshows that the real interest rate (solid) can be driven to zero, in contrast to Fig. 13. Alsoshown is the nominal rate (dashed). The lower left plot shows the labor-capital split, andthe lower right shows production of consumer goods (solid) and capital (dashed). As theinjection rate increases, so does investment, causing an excessive allocation of labor to theK sector (where the curves cross), and a decrease in production.

when the CB injects through the households. The result is a small decrease in loans in Fig.15 relative to Fig. 14. (The absolute amount may appear insignificant, but because the ratioof total K wages to CG wages is nearly one in equilibrium, a relatively small change in eitherof those flows can cause a significant shift in labor.) In this case, loans become cheap enoughin Fig. 14, relative to Fig. 15, so that as injections increase, the CG sector overinvests incapital, lowering production.

A key assumption here is that the CG sector only borrows to invest in capital, not labor.Hence as borrowing costs decrease, the CG sector continues to skew the labor-capital splitmore towards capital. In practice, the CG sector would eventually observe this overinvest-ment, and would instead use loans to finance labor costs as well as capital. Accounting forsuch a change in investment strategy would likely push the labor-capital split back towardsthe split that maximizes production, although it is not straightforward to analyze how thatwould effect the interest rate and other state variables.

45

Injections (Fraction of Money Supply)0 0.01 0.02 0.03

Per

cent

of M

oney

Sup

ply

25

30

35

40

45Loans

Injections (Fraction of Money Supply)0 0.01 0.02 0.03

Per

cent

0

0.5

1

1.5

2

2.5

3

3.5Real Loan Rate, Nominal Loan Rate (--)

Injections (Fraction of Money Supply)0 0.01 0.02 0.03

Per

cent

of L

abor

47

48

49

50

51

52

53

Labor in Consumer Goods Sector,Labor in Capital Sector (--)

Injections (Fraction of Money Supply)0 0.01 0.02 0.03

Pro

duct

ion

Uni

ts

49.85

49.9

49.95

50

Consumer Goods Production,Capital Production (--)

4.7

4.8

4.9

5

Pro

duct

ion

Uni

ts

Equilibrium Values vs Central Bank Injections to Households

Model parameters: f=0.1 e=500 =max

=60 -K

=-L=0.5

Figure 15: Steady-state values versus rate of CB injections through the Household sector.The top left plot shows that loans increase nearly linearly with the injection rate, althoughthe maximum amount is somewhat less than that shown in Fig. 14. The top right plot againshows that the real interest rate (solid) can be driven to zero. Also shown is the nominalrate (dashed). In contrast to Fig. 14, the labor-capital split (lower left) approaches half-half,which maximizes production (lower right).

8 Transient versus Steady-State Behavior

Examining Figs. ??, we see that transients associated with credit expansions and contrac-tions can be quite long, i.e., at least τm = 60 iterations, corresponding to a few years.Contractions tend to be longer-lasting (¿ 100 iterations) and more severe than expansionsdue to the associated drop in total wages and unemployment. This is most evident in thetransient leading to the deflationary scenario depicted in Fig. 7. This is in contrast to theshock in elasticity shown in Fig. 10, where the economy tends to adapt relatively quicklyand smoothly. The dynamics are, of course, strongly influenced by the choice of step-sizefor adapting the interest rate along with the extent to which other variables are filtered(smoothed). The longer the loan durations, the more sensitive the dynamics are to theway variables are smoothed. Given insufficient smoothing, the transients can be oscillatory,chaotic, and unstable, leading to a crash in which the CG sector cannot service its loansfrom revenues. Of course, in practice, additional smoothing of state-flow variables, such as

46

Slf , Dlf (along with capital expenditures), and savings S is likely to occur. That introduces“viscosity” in flow-state variables (in turn leading to price frictions), which are additionalsources of inertia not explicitly included here.

The long transients shown here suggest a marked departure from the view that in practice,an economy is never far from an equilibrium, or steady-state, in which prices accurately reflectlong-term fundamentals. Rather, shifting CB policies combined with changing fundamental(model input) parameters can cause state variables to deviate substantially from steady-statevalues for extended periods of time. This may have important implications for assessing theeffects of CB policies on short-term traders. For example, traders following short-term trendsmay in fact amplify deviations from fundamentals during a long-lasting transient caused bya policy shift. Even traders with knowledge of fundamentals (i.e., model input parameters)may be motivated to follow inconsistent trends, when those trends are expected to persist.26

Incorporating a financial sector with such traders, and studying the effects on dynamics isleft for future work. We emphasize, however, that although traders will strongly influencedynamics, they cannot effectively suppress transients associated with re-populating the loanarray in response to a policy shift.

The view that in practice, an economy at any particular time is more likely to be in atransient state, as opposed to near steady-state, complicates calibration of the model. Thatis, the model input parameters for the simulations shown here were selected to produce whatmight be considered reasonable steady-state behavior. However, while such steady-state be-havior (specifically, stationary loan portfolio with fixed inflation rate) is a useful idealization,it is unlikely to occur in practice for a long enough time to enable reliable empirical mea-surements of model state variables. Hence the real steady-state effects of monetary policyindicated here may be difficult to identify in practice, since it is more likely that any realobserved effects are associated with a transient.

9 Model Limitations and Future Work

We conclude by discussing some of the main limitations of the model, and how those mightbe addressed in future work. First, as previously discussed, to understand dynamics in moredetail, it will be necessary to add a financial sector with traders, thus allowing arbitrage,trend-based speculation, and carry trades. That could potentially include a bond market,in which traders buy/sell CG debt, and perhaps stock and commodities markets in whichtraders buy/sell particular sub-sectors of the CG sector. Aggregate prices would then be setto balance flows. This could be combined with models for account expectations, forecasting,and emotional factors. An overall objective is to identify combinations of trading rules,parameters, and CB policies for which introducing traders either amplifies volatility, orenhances stability. In particular, pushing prices towards steady-state values before the loanarray has settled may not necessarily dampen transients.

One of the main limitations of the model presented here is that it assumes a closedeconomy. Hence it cannot be used to assess the effects of foreign trade imbalances, tariffs,and differences in CB policy with associated effects on currency exchange. Foreign tradecould be introduced by subdividing the CG sector into sub-sectors representing differentclasses of goods, and adding foreign sub-sectors with exogenous parameters (e.g., productionand prices). More interesting and realistic would be to embed a foreign CG sub-sector in

26This complements the explanation for bubbles and crashes given elsewhere, e.g., see [?, ?].**check

47

its own economy with a CB and an independent set of parameters. The flow of cash andcredit across the two economies would then be determined endogenously and could be usedto study the dynamic effects of varying model parameters and changing CB policies acrossthe two economies.

In addition to including foreign trade, sub-sectors of CG could also include government,manufacturing, services, and materials. Similarly, the H sector can be divided into corre-sponding sub-sectors of labor (along with a financial sub-sector). That might be done togain insight into the effects of taxation, fiscal policy, allocation of resources and wealth, andmovement of labor across economic sectors in response to shocks and policy decisions.

Another direction is to account explicitly for risk of loan defaults. That is, the Bank sectorpresented here maximizes loans subject to an exogenous reserve requirement. By modelingrisk, the Bank would be able to optimize its loan portfolio (both amount and loan windowdistribution) to maximize expected returns or a related objective. Hence the fractional re-serve requirement f would be determined endogenously, and could provide additional insightinto the effect of leveraging and elevated risk on dynamics.

Finally, there are, of course, many other ways in which the model can be refined and ex-tended, which can potentially provide additional insight into both steady-state and dynamicbehavior. For example, those include introducing viscosity in flow-state variables, refin-ing the model for production (both CG and capital goods), including inventories, includingleisure in the model for unemployment, and introducing random fluctuations in state vari-ables. Random fluctuations, in particular, can be combined with particular types of tradingalgorithms, capturing a notion of bounded expectations. Incorporating the features men-tioned here represents no small task, but may provide further insights into macro-economicdynamics that cannot be directly obtained from prior models.

**mention growth in 3.1

Appendix A Calculation of production and other mea-

surements

In the model developed to this point, the GDP is just the sum of the consumption of consumerDiscussGDPand gov-ernmentexpen-ditureswithMikeand Ben

goods and capital, C + KEc, since the economy is closed and there are no governmentexpenditures.27

To find the CG price, we need the CG output. To this end, let the total number oflaborers (for now fixed and equal to the number of households) be denoted by N = Nc +Nk,with Nc and Nk laborers in the CG and K sectors, respectively. We assume a perfectlycompetitive labor market, which implies Wc/Nc = Wk/Nk, the per-period wages per workerin dollars.

To use the Cobbs-Douglas production function, we assume capital is measured in numberof machines in service. To determine the amount of active capital in equilibrium, we needto take into account the fact that a machine with lifetime τ is going to provide service thatincreases with τ ; for simplicity we assume that a τ -machine provides τ times as much serviceas a 1-machine. In a perfectly competitive market for capital, the price of machines needs to

27In this and the remaining technical appendixes we adhere strictly to the following convention: Whenwe indicate the dependence on period t explicitly, as in “r(t)”, we mean the dynamic variable. The plainsymbol, as in “r”, will refer to the equilibrium value.

48

reflect this, or the more effective machines will drive the less effective ones out of the capitalmarket. We model this by assuming that a τ -machine, being superior to a 1-machine forτ > 1, requires q(τ)T periods of labor to produce, where q(τ) is nondecreasing in τ withq(1) = 1, and therefore costs q(τ)TWk/Nk dollars per machine. For the remainder of this

Discusspro-ductionwithMikeand Ben

paper we make the simplifying assumption that q(τ) = τ , which amounts to assuming thatthere are no economies of scale in the use of longer-lasting machines.

Next, we must think about what proportion of workers are engaged in producing machineswith τ = 1, 2, ..., τmax. For now, assume that the distribution of workers over τ -machinesis uniform—there is no reason to believe that short-term loans are used to produce short-life machines. Assume then that the distribution of labor over machine terms is uniform,so a fraction 1/τmax of the workers are engaged in making τ -machines.28 There are then(1/τmax)Nk/(τT ) τ -machines produced each period. But machines in general last morethan one period, and for every τ -machine produced there are at any given period τ suchmachines in service. The total number of machines in service at a given period is therefore,in equilibrium,

K =∑τ

1

τmax

Nk

τTτ =

Nk

T. (71)

The production of CG is then

Yc = A

(Nk

T

)βKNβLc (72)

= AW βKk W βL

c

(Wc +Wk)βK+βL,

using Nc = NWc/(Wc + Wk) and Nk = NWk/(Wc + Wk).29 In the case of constant returns

to scale, βL + βK = 1 and

Yc = AW βKc W 1−βK

k

(Wc +Wk), (73)

where A = AT−βKN . Finally, the CG price per unit is Cc/Yc, the consumption divided bythe number of units produced.

The same arguments go through to find the GDP and prices as functions of time. Con-sider the numbers of laborers in the CG and K sectors, Nc(t) and Nk(t). To mimic thediscussion above of equilibrium values, we assume that the wage rate in the CG and Kmarkets equilibrate instantaneously30 each period, which enables us to write

Wc(t)/Nc(t) = Wk(t)/Nk(t) (74)

for every t, and not just in equilibrium. It then follows in the same way that

Nc(t) = NWc(t)/(Wc(t) +Wk(t)) and (75)

Nk(t) = NWk(t)/(Wc(t) +Wk(t)) (76)

28Right now we assume that the ranges of machine lifetimes and loan terms are the same; perhaps weshould use two separate ranges.

29Notice that maximum production occurs when Nk = βKN and NL = βLN , assuming constant returnsto scale, βK + βL = 1.

30Or at least in a time much less than a period. We should revisit this assumption.

49

for every t.However, the amount of active capital as a function of time cannot be written simply as

Nk(t)/T , as in Eq. 71, because there is in general some time-varying distribution of machineswith different lifetimes; off equilibrium, history matters. To calculate the total active capitalK(t) at period t, note first that the number of τ -machines manufactured at period t is

L(t)/τmaxτTWk(t)/Nk(t)

, (77)

since L(t)/τmax is spent for τ -machines and τTWk(t)/Nk(t) is the cost of a τ -machine. Thetotal number of τ -machines active at time t is therefore

1

τ

t−1∑j=t−τ

Nk(j)L(j)/τmaxTWk(j)

, (78)

and the total active capital is

K(t) =τmax∑τ=1

1

τ

t−1∑j=t−τ

Nk(j)L(j)/τmaxTWk(j)

. (79)

We note that in equilibrium (see below), L = Wk, and Eq. 79 reduces to K = Nk/T , Eq. 71.The dependence of active capital on history will be important for the dynamics; it means,for example, that the full effects of a sudden change in investment will be delayed by a timeon the order of τmax periods.

The production of CG is then

Yc(t) = AK(t)βKNc(t)βL , (80)

and the price of CG per unit is simply C(t)/Yc(t).

Appendix B Financing calculations

Suppose a unit loan for term τ at rate r is initiated at period 0, uniform payments of Pper period are made to an Escrow account starting at period 1, and the Escrow accountearns at the rate s per period. We assume that when the loan is originated the rates r and

Discusstime-of-loans vs.current

s are contractually fixed at the then-current values for the entire term of the loan. After iperiods, the firm owes (1 + r)i to the Bank, and has an accumulation of P

∑i−1j=0(1 + s)j =

(P/s)[(1 + s)i − 1] in Escrow. Thus, for the firm to make the balloon payment of (1 + r)τ

that is due at period τ , we require that the payments be

P =s(1 + r)τ

(1 + s)τ − 1, P (r, s, τ) , (81)

which is the generalization of the usual mortgage amortization formula to the case wherethere is a spread between the loan and savings rates. To check the end cases, when s = 0this corresponds to setting aside (1 + r)τ/τ per period with no reinvestment (the limit ass → 0) for τ periods. When s = r this corresponds to the usual mortgage formula, where

50

the reinvestments earn at the Bank’s lending rate. For reference the total accumulation inEscrow after i payments due to a unit loan with term τ is

A(r, s, i, τ) = (1 + r)τ(1 + s)i − 1

(1 + s)τ − 1. (82)

The actual capital expenditures per period, which we call Kc(t), can now be written

Kc(t) =τmax∑τ=1

τ∑i=1

P (r(t− i), s(t− i), τ)Larray(τ, t− i) . (83)

In equilibrium this yields the appropriate version of the functional Θ, the per-unit, per-periodcost of loans:

Kc = L

τmax∑τ=1

τwτP (r, s, τ) , Θ(r, s)L . (84)

The balance in the Escrow account after t periods is

E(t) =τmax∑τ=1

τ∑i=1

A(r(t− i), s(t− i), i, τ)Larray(τ, t− i) , (85)

using Eq. 82 for the accumulation in Escrow. This expression includes payments of intereston non-mature and maturing loans but does not include new loans. The maturing loans willdrop off the end of the summation and comprise the balloon payments. The functional J forequilibrium Escrow is

E = Lτmax∑τ=1

wτ(1 + r)τ

(1 + s)τ − 1

τ∑i=1

[(1 + s)i − 1

](86)

= Lτmax∑τ=1

wτ(1 + r)τ

(1 + s)τ − 1

[(1 + s)τ+1 − 1

s− τ − 1

], J(r, s)L .

The balloon payments at time t are in total

LBP (t) =τmax∑τ=1

A(r(t− τ), s(t− τ), τ, τ)Larray(τ, t− τ) (87)

=τmax∑τ=1

(1 + r(t− τ))τLarray(τ, t− τ) ,

which is in equilibrium

LBP = Lτmax∑τ=1

wτ (1 + r)τ . (88)

The balloon payments can be broken down into principal and interest parts:

LBPP (t) =τmax∑τ=1

Larray(τ, t− τ) (89)

LBPI(t) = LBP (t)− LBPP (t) , (90)

51

and in equilibrium

LBPP = L (91)

LBPI = LBP − L . (92)

We also introduce the functional X that gives us the equilibrium balloon payments:

LBP = L

τmax∑τ=1

wτ (1 + r)τ , X(r)L . (93)

Thus, in equilibrium, the interest part of the balloon payments is

LBPI = LBP − L = (X − 1)L . (94)

Appendix C Details of the baseline equilibrium calcu-

lation

C.1 Equilibrium calculation

In equilibrium31, the market for loanable funds equilibrates, so that we know that Dlf =Slf = L = Wk. Begin by collecting the dynamic equations in subsection 3.2 with equilibriumvalues for the variables:

Dlf = Slf = L = Wk (95)

s = LBPI/(S + E) (96)

∆S = gW − S , g(s) a known function (97)

Rv = C = (1− g)W (98)

Cc = cC (99)

∆C = Wcash − Cc (100)

Ktar = hC (101)

Etar = known linear function of L (102)

∆E = Etar − E (103)

∆Ec = c∆E (104)

Dlf = Ktar/Θ(r, s) (105)

Wc = C −∆E (106)

LBP = known linear function of L (107)

LBPI = LBP − L (108)

B′c = Bc − ckL+ ∆C + ∆Ec (109)

S ′ = S + ∆S (110)

E ′ = Etar − LBP (111)

B = S ′ + E ′ (112)

L = B + LBPI (113)

31We will make no distinction between an equilibrium and a fixed-point, and use the terms interchangeably.

52

S = LS ′/B (114)

Bc = B′c (115)

E = LE ′/B (116)

Slf = 1f+ck(1−f)

· [Bc − f · (S + E + (1− c)Wc)] (117)

Wcash = ckWk + cWc (118)

W = Wc +Wk + S . (119)

We can now express all the state variables, except r and s, as functions proportional toL, and ordered in such a way that they can be computed sequentially. As we will see below,the value of L is determined by the monetary base, an arbitrary scale. First, though, it iseasy enough to eliminate the variables Cc, ∆C, Wcash, ∆Ec, and B′c, by direct substitution:

Dlf = Slf = L = Wk (120)

s = LBPI/(S + E) (121)

∆S = gW − S (122)

Rv = C = (1− g)W (123)

Ktar = hC (124)

Etar = known linear function of L (125)

∆E = Etar − E (126)

Dlf = Ktar/Θ(r, s) (127)

Wc = C −∆E (128)

LBP = known linear function of L (129)

LBPI = LBP − L (130)

S ′ = S + ∆S (131)

E ′ = Etar − LBP (132)

B = S ′ + E ′ (133)

L = B + LBPI (134)

S = LS ′/B (135)

E = LE ′/B (136)

Slf = 1f+ck(1−f)

· [Bc − f · (S + E + (1− c)Wc)] (137)

W = Wc +Wk + S , (138)

Observe that the bookkeeping for the bank cash Bc, Eq. 109, yields the equilibrium cash-balance condition

Bc = Bc − ckL+ ckWk + cWc − cC + c∆E , (139)

or

ck(L−Wk) = c(Wc − C + ∆E) = 0 , (140)

which tells us nothing new, except that the cash ratios 0 ≤ c, ck ≤ 1 can be chosen arbitrarily.

53

We will use the following known functions of r, s:

LBP = L∑τmax

i=1 wi(1 + r)i , XL (141)

LBPI = (X − 1)L (142)

Etar = L∑τmax

τ=1 wτ∑τ

i=1 A(r, s, i, τ) , Y L (143)

Θ =∑τmax

τ=1 τwτP (r, s, τ) (144)

g = given function of s (145)

Q , Θ/(h(1− g)) (146)

Z , (Y − 1 + gQ)/(Y −X + gQ) . (147)

We can now write all the state variables (except the rates) explicitly as multiples of L,using known factors, the factors, however, depending on the rates.

Dlf = Slf = L = Wk (148)

Etar = Y L (149)

Ktar = ΘL (150)

LBP = XL (151)

LBPI = (X − 1)L (152)

C = (Θ/h)L (153)

W = QL (154)

E ′ = (Y −X)L (155)

S ′ = gQL (156)

B = (Y −X + gQ)L (157)

L = (Y − 1 + gQ)L (158)

S = gQZL (159)

E = (Y −X)ZL (160)

∆E = [1− gQ(1− Z)]L (161)

∆S = [gQ(1− Z)]L (162)

Wc = [Q(1− gZ)− 1]L (163)

s = (X − 1)/(Y − 1 + gQ) . (164)

We are thus left with the problem of calculating two unknowns: r and s. The explicitequation for s, Eq. 164 immediately gives us one condition:

s =X − 1

Y − 1 + gQ...Condition 1 . (165)

Notice that this gives, implicitly, the relationship between equilibrium s vs. r, and thatthis relationship does not depend on f , c, or ck. We are thus left with one unknown—but,without further assumptions, we have run out of conditions.

C.2 Parameterizing the equilibrium manifold by r

It is particularly easy to resolve the extra degree of freedom in our model by simply choosingr, although that choice is not forced. This enables us to find the corresponding fixed point

54

by solving (using binary search, for example) the one-dimensional search problem Condition1, Eq. 165. Once the pair (r, s) is determined, the remainder of the state variables can befound by direct evaluation of the equilibrium conditions, Eqs. 141-163.

We can solving Condition 1 numerically using binary search, writing Eq. 165 in the form“increasing function of s = left-hand side = g = right-hand side”:

g =1

1 + Θ/hX−1

s−Y+1

, (166)

using Q = Θ/(h(1− g)). The binary search narrows the search down to a region where theright-hand side is decreasing, after which locating the crossing point is very easy. Figure C16shows the (increasing) left- and right-hand sides of Eq. 166 for the case τmax = 60, r = 2%,c = ck = 0, and e = 500. There is a pole in right-hand side at about 4%, which makes itnecessary to exercise caution in the binary search. One way to make the search reasonablyrobust is to start with the right end of the initial search bracket very small, and expand it tothe right (geometrically) until the value of the right-hand side is below that of the left-handside, and then stop.

Percent0 1 2 3 4 5 6

Uni

ts

0

0.5

1

1.5

2

LHS and RHS of a Version of Condition 1 vs Return on Investment

Model parameters: e=500 =max

=60 -K=-

L=0.5 c=c

k=0 r= 2%

Figure C16: An example showing the left- and right-hand sides of Eq. 166, a version ofCondition 1. There is a pole in the right-hand side, which we need to exclude from theinitial binary search bracket.

C.3 Household debt

Households are in debt to the bank a total Φ(t) (normalized value ϕ(t) , Φ(t)/G0). Asdescribed below, we will choose the household debt exogenously (rather than the loan rate

55

r) to render the equilibrium unique. We now calculate its value in terms of the other variablesin the economy.

The total bank assets32 are

Bc(t)− ckL(t) + ξ(t) + Φ(t) , (167)

since Bc(t)− ckL(t) is the cash at the bank, and we define ξ(t) to be the total outstandingloans at the end of period t, (which thus includes the loans made in that period):

ξ(t) ,τmax∑τ=1

τ∑i=1

Larray(τ, t− i+ 1) . (168)

The equilibrium value of ξ(t), used below, is

ξ = Lτmax∑τ=1

τwτ , ΓL . (169)

We note that the cash at the bank can also be written33

Bc(t)− ckL(t) = f(S(t) + E(t) + (1− c)Wc(t) + (1− ck)Wk(t)) . (170)

The corresponding total bank liabilities are34

L(t) + (1− ck)Wk(t) + (1− c)Wc(t) . (171)

We model the bank as a zero-profit sector, so its total assets and liabilities are equal at theend of every period. Therefore,

Φ(t) = L(t) + L(t) + (1− c)Wc(t)−Bc(t)− ξ(t) , (172)

using Wk(t) = L(t). Now use Bc from Eq. 137:

Slf =1

f + ck(1− f)[Bc − f(S + E + (1− c)Wc)] , (173)

or

Bc = [f + ck(1− f)]L+ f(S + E + (1− c)Wc) , (174)

to write Φ as

Φ = L − ξ − [f + ck(1− f)]L− f(S + E + (1− c)Wc) + L+ (1− c)Wc (175)

= L − ξ − [f + ck(1− f)]L+ (1− f)(1− c)Wc − f(S + E) + L .

Using the equilibrium conditions Eqs. 141-163, this can be written, as usual, as a multipleof L, yielding

Φ/L = (1− f)(1− c) [Q(1− gZ)− 1] + (Y − 1 + gQ) (176)

−Γ + 1− [f + ck(1− f)]− f [(Y −X) + gQ]Z

= (1− f)(1− c) [Q(1− gZ)− 1] + (1− f)(Y − 1 + gQ)

−Γ + 1− [f + ck(1− f)] ,

32Explain.33Check numerically. Explain.34Explain.

56

simplifying using [(Y − X) + gQ]Z = Y − 1 + gQ, which follows from the definition of Z.To check: Notice that when f = 1, the case when total reserve is required, Φ = −ΓL = −ξ,which is correct; household debt is the negative of bank assets, which are in this case justoutstanding loans ξ. When c = ck = 1, the all-cash case, Φ = (1− f) [Y − 1 + gQ]L−ΓL =(1− f)L − ξ.35

We may want to normalize Φ by G0, not L, so we write G0 in terms of L by combiningthe fact that G0 = Bc + cWc and Eq. 174. The result is

G0/L = (1− f)ck + f [Y + gQ+ (1− c)(Q− 1− gQZ)] + c [Q(1− gZ)− 1] (177)

, V , (178)

which, like X, Y , and Z, is a well defined function of r and s. We thus have ϕV =(Φ/G0)(G0/L) = Φ/L, and we have ϕ in terms of known functions of r and s:

ϕ =(1− f) [Y − 1 + gQ] + (1− f)(1− c) [Q(1− gZ)− 1]− Γ + 1− [f + ck(1− f)]

V.(179)

C.4 The equilibrium manifold

So far, we have left one extra degree of freedom in our model, and this means there is acurve, a one-dimensional manifold, in the state space, along which all possible equilibria areconstrained to lie.36

As we shall see later, if the loan rate is shocked, the model economy is knocked off theequilibrium manifold. When the shock is removed, it returns to exactly the same point onthe equilibrium manifold. Why? The answer can be found in the fact that net bank assetsϕ(t), and hence the household debt, is a conserved quantity, and fixing ϕ pins down theequilibrium to a unique point.

Remark 1 Household debt ϕ(t) is constant for every t, so long as there is no exogenousinjection or extraction of money. This follows from the fact that the bank sector balancesaccounts precisely at the end of every period. The economy will therefore return to the samepoint on the equilibrium manifold after state variables or parameters are perturbed, provided,of course, that system parameters are restored to their original values after the shock.

35Is this right?36How is this related, if it is related at all, to the “equilibrium manifold” in [11]?

57

Figure C17: Illustrating the fact that the system will return to the same equilibrium afterstate variables or parameters are perturbed, provided that the parameters are returned totheir original values. This is true regardless of how the equilibrium manifold is parameter-ized, and follows from the fact that, with no exogenous injections or extractions of money,household debt is invariant.

This is illustrated diagrammatically in Fig. C17.Household debt will not play an important role in the policy experiments in this paper,

and for simplicity we now stipulate that it is zero, thereby making the equilibrium unique.We then have two equilibrium conditions for the two unknowns r and s, the first fromEq. 165, and the second from the assumption that Φ = 0. We collect them here:

s =X − 1

Y − 1 + gQ...Condition 1 , (180)

Φ/L = 0 from Eq. 176 ...Condition 2 . (181)

To calculate equilibria numerically we now need to solve two equations in two unknowns,a two-dimensional search problem instead of the one-dimensional problem if we choose r.This is a consequence of our choosing Φ instead of r to disambiguate the equilibrium. Wefound the Nelder-Mead search algorithm [12] to be quite effective in this application.

Appendix D Uniqueness of fixed point for τm = 1

DRAFT, NEEDS TO BE CHECKEDWhile the functions of r and s that appear in Conditions 1 and 2 get complicated quickly

for τm > 1, they are simple enough for the case τm = 1 to allow us to prove that, dependingon a condition involving f, h, c , and ck (assuming 0 < f, h, c, ck < 1 and e > 0), there iseither exactly one solution with positive r and s, or none. We can thus conclude that, atleast when τm = 1, a fixed point with positive loan rate must be unique.

The proof is direct. When τm = 1, the functions simplify to X = Y = Θ = 1 + r , andΓ = 1. For reference, the fixed-point conditions become(

r +1 + r

hes

)s = r ...Condition 1′ , (182)

58

and

(1− c)(1 + r)(1/h− 1) +

(r +

1 + r

hes

)− ck −

f

1− f= 0 ...Condition 2′ , (183)

respectively. Solving Condition 2′ for r + (1 + r)(es)/h yields

r +1 + r

hes =

ck + f/(1− f)

(1− c)(1 + r)(1/h− 1),

1

λ(r), (184)

where λ(r) is linear in r with positive coefficients. Use this back in Condition 1′ to solve fors ,

s = rλ(r) , (185)

and use that to eliminate s from Condition 2′, which becomes

(1− c)(1 + r)(1/h− 1) + r +1 + r

herλ(r)− (ck +

f

1− f) = 0 , (186)

a cubic in r.It is easy to verify that the coefficients of r3, r2, and r are all positive. The constant is

(1− c)(1/h− 1)− (ck + f/(1− f)), and so, by Descartes’ Rule of Signs with sign sequence+ + +∓, the condition

(1− c)(1/h− 1) < ck + f/(1− f) (187)

ensures that there is exactly one fixed point, and it corresponds to a positive rate r. WhenCondition 187 is reversed, there is no positive root of Eq. 186, and hence a fixed point, ifone exists, must correspond to a negative loan rate.

Using (−x) in Eq. 186, the sequence of signs for Descartes’ Rule of Signs is − + −∓.Thus, if Condition 187 holds, there are two or zero negative roots, and both possibilities areallowed. If the reverse of Condition 187 holds, there are either three negative roots or one,and again, both possibilities are allowed.

To sum up, the important conclusion is that, for the case τm = 1, if there is a fixedpoint with positive loan rate, it must be unique, but there may be none, depending onCondition 187.

Appendix E Adaptive Update for s(t)

An adaptive update is given by

s(t) = s(t− 1) + ks [RBPI(t− 1)− s(t− 1) · L(t− 1)] /G(t− 1) (188)

where ks is the step-size, RBPI is an expectation of total returns to the bank in steady-stategiven the state variables at time t, including interest on loans from balloon payments, plusreclaimed credit from defaults. It is assumed that this aggregate information is known, orcan be estimated and distributed to the banks. In steady-state s(t) remains constant, andhence the second term on the right converges to zero. The total return to the bank is then

CheckwithBen59

given by37

RBPI(t) =LBPI(t− 1)

LBPI(t− 1) + LCBI(t− 1)L(t− 1)

τm∑i=1

wiΠ(t− 1)−i+1(1− δ)i{[1 + r(t)]i − 1}

+ δ · E(t− 1) + pCB(t− 1) (189)

where L(t) is total outstanding loans (owned by both the Bank and the CB), and Π(t) is theinflation multiplier (1 + inflation rate in loans). The last two terms on the right in (189)are then reclaimed credit from defaults and pCB, the total premium on loans, which theCB purchases, given by (69). The remaining (first) term is then the expected return on alloutstanding loans to be repaid to the bank. The terms inside the sum include discounts dueto defaults and inflation. The fraction preceding L is the fraction of interest repaid to theBank, relative to the total repaid to the Bank and the CB. In steady state, L(t) = Slf (t),the supply of loanable funds, which equals Dlf (t), the demand for loanable funds. However,in a transient this is no longer true, so that for purposes of estimating RBPI(t), we can takeL(t) = [Slf (t) +Dlf (t)]/2.

37This is used in the H sector savings calculation, but the de facto version in the baseline model is usedeverywhere else in the simulation because it is more stable.

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