collaterals and macroeconomic volatility

Research in Economics 64 (2010) 146–161

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Research in Economics

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Collaterals and macroeconomic volatilityI

Riham Barbar a,1, Stefano Bosi b,∗a EPEE, Université d’Evry, 4 bd F. Mitterrand, 91025 Evry CEDEX, Franceb THEMA, Université de Cergy-Pontoise, 33 bd du Port, 95011 Cergy-Pontoise CEDEX, France

a r t i c l e i n f o

Article history:Received 4 June 2009Accepted 8 March 2010

Keywords:Cash-in-advanceBalanced growthSuperneutralityEndogenous fluctuations

a b s t r a c t

In this paper, we study the effects of collaterals on business cycles and growth inmonetaryeconomies with credit market imperfections. We consider an endogenous growth modelwith a partial cash-in-advance constraint. It is assumed that the share of consumptionpurchases paid on credit depends positively on the collaterals available to the agent. Inthis case, money is no longer superneutral. We find that, under mild inflation rates, ahigher money growth rate is welfare-improving and, surprisingly, it makes the occurrenceof expectations-driven fluctuations less likely. The shape of credit share in consumptionpurchases, as outcome of regulatory policies, has an impact on both welfare and stability.In particular, the higher the sensitivity of the credit share to collaterals, the more stablethe economy under rational expectations. These analytical findings are complemented byeconomic interpretations.

© 2010 University of Venice. Published by Elsevier Ltd. All rights reserved.

1. Introduction

In the last few years, the US subprime crisis has spread out over the global economy, spilling from the financial sphereover real activities and becoming the focus of economists’ concerns.In the past three decades, housing financial systems experienced deep changes in some advanced economies.

Deregulation introduced new lending practices that allowed easier access to mortgage credits. Financial innovationscontributed to the rapid growth of mortgage credits in these countries. For example, in the US, the share of subprimemortgages in total consumer loans rose from slightly below 5% in 1994 to nearly 9% in 2001, andmore than doubled reaching20% by 2005 (Federal Reserve Bank, 2008). According to Braunstein (2007), from 1995 to 2006, the homeownership shareincreased from 65 to about 69% of all households.The subprime loans were proposed to consumers with a poor credit history or insufficient collaterals at higher and

adjustable interest rates. Following the rapid expansion of the subprime credit market, the default rate has increased,resulting in a sharp decline in house prices.Once house prices started to fall, the mortgage market began to experience higher levels of delinquency and foreclosure.

The share of mortgage loans that were seriously delinquent, increased from 0.7% in 1979 to 2.4% in 2002. In addition, theshare of delinquent loans rose from about 5.6 in mid-2005 to over 21% in mid-2008 (Federal Reserve Bank, 2008).With the deepening of the mortgage crisis, a large number of financial firms shut down and stock indices dropped

sharply. Moreover, the crisis in the housing sector spread over the rest of the economy. The fall in house prices, decreasing

I This work was supported by the French National Research Agency (ANR-05-BLAN-0347-01). We would like to thank Michael Brei, Eleni Iliopulos andOmar Licandro for helpful comments and the participants of the ASSET conference held in Florence on November 2008. All remaining errors or omissionsare our own.∗ Corresponding author. Tel.: +33 1 34 25 22 54; fax: +33 1 34 25 62 33.E-mail addresses: [email protected] (R. Barbar), [email protected] (S. Bosi).

1 Tel.: +33 1 69 47 70 78; fax: +33 1 69 47 70 50.

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R. Barbar, S. Bosi / Research in Economics 64 (2010) 146–161 147

households’ wealth and the value of collaterals, restricted loans to consumers and reduced their spending. Increasing riskperception and uncertainty about the size and scope of crisis translated into a sudden closure of financial wholesalemarkets,including interbank borrowing.Contagionwas reinforced by new banking practices such as the titrization of (bad) credits. In the last decade, outsourcing

the risk has allowed the banking system to elude accounting rules, while infecting also the stock market at the end.Both the credit crunch and bankruptcies, together with falling prices in the assetmarket, and (possibly) bursting bubbles,

have spread out from the financial sphere. They have affected the growth rate of the world economy through financialrationing and general wealth effects. This cascade has been magnified by overpessimistic animal spirits and coordinationfailures.In the US, the decline in the housing sector was reflected in the deceleration of the real GDP. According to estimates

released by the National Bureau of Economic Analysis, real GDP increased only 0.6% in the fourth quarter of 2007 followingan increase of 4.9% in the third quarter. Annual growth in 2007 was 2.2%, significantly below 2.9% in 2006 (Schnure, 2005).In light of the subprime crisis, we address the stability issue considering the role of collaterals in a monetary economy

with credit market imperfections. The role of collaterals is usually introduced through a credit constraint faced by agents;individuals can borrow as long as the repayment does not exceed the market value of their collateral (Kiyotaki and Moore,1997; Kiyotaki, 1998; Cordoba and Ripoll, 2003). In the spirit of Grandmont and Younès (1972), we depart from thesemodels by introducing collaterals in a partial cash-in-advance constraint. Namely, we assume that agents pay a part of theirconsumption purchases in cashwhile they finance the rest on credit. Furthermore, the amount of credit depends on the valueof the collaterals available to the agent. For a similar approach, the reader is also referred to Bosi and Seegmuller (2009).Notice however that they focus on the occurrence of expectation-driven rational bubbles in an overlapping generationsmodel.One of the goals of the paper is to find the conditions for equilibrium indeterminacy and self-fulfilling prophecies under

rational expectations.Along this line of research, some papers have shown that a partial cash-in-advance constraint promotes the occurrence

of multiple equilibria under a sufficiently low elasticity of intertemporal substitution in consumption. Expectation-drivenfluctuations also arise in themore general class ofmoney-in-the-utilitymodels with cash-in-advance timing (Carlstrom andFuerst, 2003).Bosi and Magris (2003), among others, have shown that, under sufficiently small liquidity constraints (a slight departure

from the traditional Ramsey model), the equilibrium is indeterminate for any value of the elasticity of intertemporalsubstitution. This conclusion is robust under the inclusion of investment in the liquidity constraint which modifies thecritical parameter values giving rise to indeterminacy (Bosi and Dufourt, 2008).Our model is also close to the cash-credit models pioneered by Lucas and Stokey (1987). Following their approach, we

refer to goods purchased with money, as ‘‘cash goods’’ and goods purchased with credit, as ‘‘credit goods’’.In order to extend the Bosi and Magris (2003) model, we assume that the fraction of consumption of credit goods (the

credit share), depends positively on the capital–consumption ratio (the degree of collateralization). That is, if the value of theagent’s capital is high relative to her current consumption, she can easily afford the repayments of the credit. The positivesensitivity of the credit share to collaterals is a feature of the credit market which depends either on lenders’ willingnessto grant credit to solvable borrowers or on legal constraints. On the one hand, we do not account for the lenders’ attitudeon the ground of asymmetric information existing between lenders and borrowers. On the other hand, we do not tackle thetricky question of incentive compatible regulations.In addition to characterizing the monetary equilibrium when collaterals matter, we highlight the role of public policies.

We see the impact of the money growth rate and the effects of credit market regulation on the equilibrium.Both the policies can be studied either from a welfare perspective or from a stability point of view.Regarding the monetary rule, money is no longer superneutral: not only along the transition path, but also at the steady

state. An increase in the monetary growth rate lowers the return on money, leading agents to substitute capital for cash.Hence, a higher rate of money growth is associated with a larger capital stock and a higher growth rate. Under a widercollateralization, agents are allowed to pay more on credit.Our results are in line with the recent literature. Lucas and Stokey (1987) have shown that an increase in the nominal

interest rate leads agents to substitute credit goods for cash goods. For an economywith capital formation and credit-goodsproduction, Aiyagari and Eckstein (1995) and Aiyagari et al. (1998) have clarified how the nominal interest rate determinesthe price of credit good relative to the price of cash good.We find awelfare-maximizing growth rate because of the trade-off between the initial consumption and the growth rate.

This rate is implemented by an optimal rate of money growth, which is different from the Friedman rule because inflationraises the opportunity cost of holding money and the substitution of money with capital lightens the CIA burden.From a stability point of view, there is also a positive impact of monetary growth under a weak inflation regime. More

precisely, a higher inflation results in a higher nominal interest rate and, henceforth, reduces the elasticity of future inflationwith respect to the current consumption. Low sensitivity of future inflation prevents the market from recovering the initialjump in expectations and rules out the occurrence of self-fulfilling prophecies.2

2 Under sufficiently low sensitivity to collaterals, indeterminacy arises when the income effect dominates the intertemporal substitution. This result isconsistent with the existing literature (among others, Bosi and Magris (2003)).

148 R. Barbar, S. Bosi / Research in Economics 64 (2010) 146–161

Regarding regulation, it is possible to design the credit share as an optimal function of collaterals. More precisely, firstand second-order elasticities of the credit share (sensitivity to collaterals and concavity) can capture the legal constraintsto the credit introduced by the policy maker to promote loans to solvable agents. The optimal growth rate determines alsothe optimal values of these elasticities.Eventually, regulation can immunize the economy against the risk of macroeconomic volatility. We prove that, under an

isoelastic credit share, a higher sensitivity of the credit market to collaterals reduces the indeterminacy range (in terms ofthe elasticity of intertemporal substitution).Themodel is developedwithin an endogenous growth framework. The qualitative results do not change in an exogenous

growth context, but there is a gain in terms of formal simplicity because the dynamic system is of a lower dimension. Ireland(1994) and Hromcova (2003) also address the issue of monetary equilibria in an endogenous growth framework. They showthat agents can use the costly services of financial intermediaries as an alternative to cash. As the economy grows, theseservices become cheaper and so money is relatively less used.This paper is organized as follows. In Section 2, we present themodel and derive the intertemporal equilibrium. Section 3

is devoted to the analysis of the steady state and the comparative statics. Section 4 studies the local stability and theoccurrence of local bifurcations. In Section 5, we provide a numerical example, while, in Section 6, the conclusion.

2. The model

We have developed an endogenous growth model with a partial cash-in-advance constraint and an inelastic laborsupply. The economy is populated by a large number of identical infinite-lived households acting under perfect foresight.There are also a large number of firms sharing the same CRS technology. All these agents are price-takers. The key featureof the model is that consumers are allowed to purchase a consumption good either in cash or on credit: only a fraction ofconsumption expenditures is subject to a cash-in-advance constraint. Our framework is an extension of Bosi and Magris(2003). While they suppose that a constant share of the consumption goods is paid cash, we assume that this share isendogenously determined by the amount of collaterals in the consumer’s hand.

2.1. Consumers

The lifetime utility function of a representative agent is given by∞∑t=0

β tu (ct) (1)

where β ∈ (0, 1) denotes the discount factor, ct is the consumption demand and the instantaneous utility function u verifiesthe following assumption.

Assumption 1. The single-period utility function u (c) is twice continuously differentiable for all strictly positive values ofc and satisfies, for any c > 0, u′ (c) > 0, u′′ (c) < 0, limc→0 u′ (c) = +∞ and limc→+∞ u′ (c) = 0.

In each period t , the household’s portfolio is constituted by the stock of capital kt and the amount of money balancesMt .She is subject to the budget constraint:

pt (ct + kt+1 −∆kt)+Mt+1 ≤ Mt + pt rtkt + ptwt lt + τt (2)for t = 0, 1, . . . ,where∆ ≡ 1− δ ∈ [0, 1] and δ is the depreciation rate of capital, τt ≡ Mt+1−Mt are nominal lump-sumtransfers ‘‘helicoptered’’ by themonetary authority, r is the real interest rate on capital andw is the real wage. For simplicity,labor supply is assumed to be inelastic, namely, lt = 1 for every t ≥ 0.In addition, the representative household faces a partial cash-in-advance constraint.In the spirit of Grandmont and Younès (1972), we assume that a share of purchases is paid cash, while the rest on credit.

Thus, our model is also close to Lucas and Stokey (1987), an economy with a cash and a credit good.We define credit share the fraction of consumption good paid on credit. We further assume that the amount of

disposable credit depends on the collaterals the household owns. More precisely, the individual’s capital stock relative toher consumption habits matters in order to open a credit line.Formally, the credit share is given by

γ

(kc

)≡

credit goodcredit good+ cash good

.

An augmented version of the partial cash-in-advance considered by Bosi and Magris (2003) is now provided in order toaccount for the role of collaterals:

[1− γ (kt/ct)] ptct ≤ Mt (3)for t = 0, 1, . . . .The shape of credit share γ can be viewed as a restriction due to lenders’ or sellers’ prudential attitude towards borrowers

in presence of asymmetric information, but also as a creditmarket regulatory policy, that is a legal constraint to credit grantsin order to ensure borrowers’ solvability.


We observe that the household is aware of the credit function, which is an institutional constraint, and maximizes theLagrangian also with respect to the arguments (kt , ct) appearing in the credit share.Introducing an endogenous credit share in a model of capital accumulation is the added value of our paper. We notice

that the velocity of money becomes endogenous as well and, therefore, variable: v (k/c) ≡ [1− γ (k/c)]−1. In this sense,our formulation takes into account one of the criticisms addressed to the cash-in-advance models: the implausibility of aconstant velocity of money. 3Let us introduce the first and second-order elasticities of credit share: ε1 (x) ≡ xγ ′ (x) /γ (x) ∈ [0, 1) and ε2 (x) ≡

xγ ′′ (x) /γ ′ (x) < 0 where xt ≡ kt/ct denotes the degree of collateralization. The credit function meets also reasonablerestrictions.

Assumption 2. The credit share γ (x) is twice continuously differentiable and satisfies γ (x) ∈ [0, 1], ε1 (x) ≥ 0, −1 ≤ε2 (x) ≤ 0 for every x ≥ 0.

In Assumption 2, γ ∈ [0, 1] simply means that credit and money velocity are non-negative; ε1 ≥ 0 captures the role ofcollaterals relative to consumption habits: the higher the ratio, the larger the credit; while−1 ≤ ε2 ≤ 0 ensures the second-order conditions in the consumer’s program to be satisfied and reconciles smoothness with the existence of an upper bound(γ ≤ 1).The representative agent maximizes (1) subject to (2) and (3), that is an infinite-horizon Lagrangian, with respect to the

consumption path ((ct)∞t=0) and the saving path (capital and balances: (kt ,Mt)∞

t=1).Deriving the infinite-horizon Lagrangian with respect the demand for balances and capital gives the portfolio arbitrage,

while deriving with respect to the demand for capital and consumption gives the intertemporal arbitrage, the consumptionsmoothing over time. Deriving the Lagrangian with respect to the multipliers, we recover the constraints, now binding.More precisely, at each period, a strictly positive marginal utility implies a binding budget constraint, while the

augmented CIA constraint is also binding under a strictly positive interest rate (It > 1).We observe that positive interest rates are necessary to the existence of a monetary equilibrium. Under no uncertainty,

when balances are dominated by a capital asset, the opportunity cost of holdingmoney induces agents to keep in the pocketsthe minimal amount of money for transaction purposes. As a consequence, the liquidity constraint becomes binding.After eliminating the multipliers, we get also a sequence of Euler equations:

u′ (ct)u′ (ct+1)

=β

πt+1

It+1 − γ ′ (xt+1) πt+11− γ ′ (xt) πt

1− γ ′ (xt) πt + (It − 1)[1− γ (xt)+ γ ′ (xt) xt

]1− γ ′ (xt+1) πt+1 + (It+1 − 1) [1− γ (xt+1)+ γ ′ (xt+1) xt+1]

(4)

(t = 0, 1, . . .), where πt+1 ≡ pt+1/pt and It ≡ (∆+ rt) πt denote for the inflation and the nominal interest gross rates,respectively.When collaterals plays no role (γ ′ = 0), the intertemporal consumption smoothing (4) writes

u′ (ct)u′ (ct+1)

= βIt+1πt+1

1+ (1− γ ) (It − 1)1+ (1− γ ) (It+1 − 1)

(5)

as in Bosi and Magris (2003). When the cash good disappears (γ = 1), the distortion due to the inflation rate also vanishesand the traditional Euler equation is recovered: u′ (ct) /u′ (ct+1) = β (∆+ rt+1). In this case, money cannot perturb theconsumption path. Conversely, the transition path generated by (5) is distorted by monetary savings, but balances remainssuperneutral at the steady state. The novelty of our paper is that money matters also at the steady state in the case of Eq. (4)because of the collaterals and fails to be superneutral in any regime.Eventually, a rational household takes into account the initial and the final condition, that is the initial endowments

M0, k0 ≥ 0 and the transversality condition:

limt→+∞

β tu′ (ct) (kt+1 +Mt+1/pt) = 0. (6)

2.2. Firms

On the production side, there is a large number of identical firms participating to competitivemarkets. The representativefirm rents capital and labor in order to produce the good under constant (private) returns to scale. External effects of capitalintensity spill over the other firms. Technology is rationalized by a Romer-type (1986) production function.

Assumption 3. F (K , L) ≡ Ak̄1−sK sL1−s, s ∈ (0, 1).

The TFP (Ak̄1−s) is affected by productive externalities of average capital intensity k̄, such as knowledge spill-overs. Eachproducer is price-taker and maximizes the profit.

3 Money-In-the-Utility (MIU) models are immunized against this criticism: the functional equivalence highlighted by Feenstra (1986) between CIA andMIU, no longer holds with a Cash-When-I’m-Done (CWID) timing or a CIA timing and strictly positive elasticity of substitution between consumption andreal balances (see Carlstrom and Fuerst (2003)).


The assumption of a representative household and a representative firm implies at equilibrium k̄ = k = K/L, while,under Assumption 3, profit maximization gives:

rt = sA (7)wt = (1− s) Akt . (8)

The equilibrium interest rate is constant over time as in many Romer-like models of endogenous growth.

2.3. Monetary authority

Money supply follows a simple rule. Lump-sum money is ‘‘helicoptered’’ to consumers:

τt = Mst+1 −Mst (9)

and the rate of money growth is kept constant over time by the monetary authority:Mst+1 = µMst for t = 0, 1, . . ..

2.4. General equilibrium

The economy under study is a system of three markets: money, labor and goods.The money market clears when the demand for real balancesmt ≡ Mt/pt , supported by the liquidity constraint, equals

the amount supplied by the monetary authority:

mt = [1− γ (xt)] ct = Mst /pt . (10)

The dynamic version of (10):

µ = πt+1ct+1ct

1− γ (xt+1)1− γ (xt)

(11)

highlights the decomposition of the nominal growth in inflation and real growth.In the labor market, the demand lt is determined by profit maximization (8), while, for simplicity, the supply is supposed

to be inelastic: lt = 1.By the Walras’ law the good market clears too. The equilibrium is obtained, by replacing (9), (7) and (8) in the

representative agent’s budget constraint (2).

ct + kt+1 −∆kt = rtkt + wt lt = Akt .

Dividing both the sides by kt , we obtain the growth rate:

gt+1 ≡kt+1kt= ∆+ A−

1xt. (12)

In order to compute the intertemporal equilibrium, let us introduce a popular utility function which is suitable to studyendogenous growth.

Assumption 4. The instantaneous utility function is given by:

u (c) ≡ ln c iff σ = 1

u (c) ≡ c1−1/σ / (1− 1/σ) iff σ 6= 1 (13)

where σ > 0 is the constant elasticity of intertemporal substitution. 4

Replacing (12) in (4) and (11), we obtain the dynamic system of a CIA economy with collaterals. Formally:

Definition 1. An intertemporal equilibrium with perfect foresight is a sequence (xt , πt)∞t=0 that satisfies (i) the initialconditions (M0, k0), (ii) the (transitional) dynamics:(

∆+ A−1xt

)xtxt+1

=

(β

πt+1

It+1 − γ ′ (xt+1) πt+11− γ ′ (xt) πt

×1− γ ′ (xt) πt + (It − 1)

[1− γ (xt)+ γ ′ (xt) xt

]1− γ ′ (xt+1) πt+1 + (It+1 − 1) [1− γ (xt+1)+ γ ′ (xt+1) xt+1]

)σ(∆+ A−

1xt

)xtxt+1

=µ

πt+1

1− γ (xt)1− γ (xt+1)

(14)

for t = 0, 1, . . . ,where It = (∆+ r) πt and r = sA, and (iii) the transversality condition (6).

4 The CES preferences rationalized by (13), satisfy Assumption 1.


3. Steady state

Growth is regular at the steady state5: g ≡ kt+1/kt = ct+1/ct = mt+1/mt .Dropping the time index in (14) and solving the system gives the stationary state (x, π). The balanced growth factor is

obtained from (12):

g = ∆+ A− 1/x. (15)

Replacing

(x, π) =(

1∆+ A− g

,µ

g

)(16)

in (14) gives the steady state Euler equation, where, as seen above, money is no longer superneutral because of the marketimperfection γ ′ > 0:

g =

β gµ

∆+ r − γ ′(

1∆+A−g

)gµ− γ ′

(1

∆+A−g

)σ . (17)

Using the elasticities of credit share is an appropriate way of capturing the credit market regulation and taking intoaccount the role of collaterals in the effectiveness of monetary policy. Eq. (17) becomes:

g =

[βgµ

∆+ r − γ ε1 (∆+ A− g)gµ− γ ε1 (∆+ A− g)

]σ. (18)

The equilibrium existence requires the term into the brackets to be strictly positive: γ ′ < min {g/µ,∆+ r} or γ ′ >max {g/µ,∆+ r}. The existence of a monetary equilibrium requires also I > 1 and, therefore, g/µ = 1/π < ∆+ r . Then,the strict positivity of g becomes equivalent to γ ′ < 1/π or γ ′ > ∆+ r . For simplicity, we assume, as sufficient conditions,the first inequality together with the strict positivity of the nominal interest rate in a neighborhood of the steady state.

Assumption 5. πγ ′ (x) < 1 < I .

Eventually, the balanced growth path (mt , kt , ct) = (m0, k0, c0) g t must satisfy the transversality condition: limt→+∞β tu′ (ct) (kt+1 + πmt+1) = 0. Substituting the path and taking the limit, we obtain an explicit parametric restriction6:

β < g1/σ−1. (19)

3.1. Comparative statics

As money is no longer superneutral, we try to explain how the monetary policy affects growth and welfare in the longrun. In the next section, we will analyze also the short-run effects of the rule on the business cycles and the sense of astabilization policy.Differentiating Eq. (17) with respect to µ and g gives the impact of the monetary rule on the growth rate:

εgµ ≡dgdµ

µ

g=

(1+

1σ

1− πγ ′

πγ ′− xgε2

I − 1I − πγ ′

)−1> 0 (20)

where εgµ is the monetary policy elasticity of the growth factor. This elasticity is strictly positive under Assumption 5.The higher the monetary growth (µ), the higher the inflation (π ) and the opportunity cost of holding money (I). Savers

will keep more capital and less money in the portfolio. The growth rate of capital will increase. In addition, when thecapital/consumption ratio goes up, consumption purchases are more collateralized, agents are willing to substitute cashby credit, the credit share increases, while the need for money lowers.Elasticity (20) writes as follows:

εgµ =

(1+

1σ

1− γ ε1π/xγ ε1π/x

− xgε2I − 1

I − γ ε1π/x

)−1> 0 (21)

5 In the long run, an unbalanced growth should violate either the positivity of variables or the transversality condition. However, we notice that growthrates may be unbalanced along the transition path.6 Inequality (19) is usual in the endogenous growth literature and it is verified for instance by a logarithmic utility.


and, ceteris paribus, the impact of collaterals on the effectiveness of monetary policy is captured through the followingderivatives:

∂εgµ

∂ε1> 0 iff σ < −

1xgε2

1I − 1

(I − γ ε1π/xγ ε1π/x

)2and ∂εgµ/∂ε2 > 0.Let us interpret the first derivative. Given the second-order moment (ε2), the higher the sensitivity to collaterals (ε1),

the larger the impact of money growth on the growth rate (εgµ), when households find hard smoothing consumption overtime (sufficiently low σ ). A weaker elasticity of intertemporal substitution makes more difficult to elude the CIA constraintthrough higher savings: in this case, a higher sensitivity to collaterals can free individuals from the burden of the constraintand promotes capital accumulation.The second derivative is interpreted as follows: households prefer a higher capital/consumption ratio under a faster

monetary growth. The flatter the credit share (the higher and closer to zero ε2), the sharper their response. To understandthe point, fix a value x of the ratio and the slope γ of the credit share in x. The flatter the credit share around x, the steeperthe credit share on the right-hand side of x. Rational households moves on the right and accumulate more capital to takeadvantage of the easier credit.

3.2. Optimal monetary policy and regulation

Under the assumption of representative agent, maximizing a welfare functionW is equivalent to maximizing her utilityfunction. Under the transversality condition (19), the series converges along the balanced growth path:

W ≡∞∑t=0

β tu (ct) =u (c0)

1− βg1−1/σ.

Proposition 2. The optimal (welfare-maximizing) monetary policy is given by

µ∗ =g∗

∆+ A

(1+

1− sγ ε1

A∆+ A− g∗

)(22)

where

g∗ ≡ [β (∆+ A)]σ (23)

is the optimal growth rate. Moreover, the higher the sensitivity to collaterals, the lower the optimal money growth rate:

∂µ∗

∂ε1= −

1− sγ ε21

Ag∗

(∆+ A) (∆+ A− g∗)< 0. (24)

Given the monetary rule, the optimal regulation (elasticity of credit share) is given by

ε∗1 =1γ

(1− s) Ag∗/µ(∆+ A− g∗) (∆+ A− g∗/µ)

. (25)

Proof. See the Appendix.

As shown in theAppendix, not onlyW depends onµ (because c0 depends on g and g , in turn, onµ), but alsoW ′ (µ) > 0 iffµ < µ∗. In other terms, increasing a monetary growth rate from a sufficiently low value can be beneficial to the society.The existence of an interior solution results from the trade-off between the initial consumption c0 and the growth rate g .

Renouncing to consume today, raises individual saving, capital accumulation and, finally, the growth rate. On the one side,too low growth rates supported by a slowmonetary growth, are inefficient. But, on the other side, a fastmonetary expansioncan entail capital overaccumulation, which requires an excessively low level of initial consumption.Contrary to the standard CIA model, the Friedman monetary policy is no longer optimal in our model because of the

imperfection due to the introduction of credit share as a function of collaterals. Recall that a Friedman rule with a zeronominal interest rate (that is It = 1) induces agents to hold money rather than capital, resulting in low saving levels.On the one hand, this reduces the production level and the growth rate. As shown above, the representative agent’s utility

can increase with the growth rate. Thus welfare-loss are generated from applying a zero nominal interest rate and a policyof strictly positive nominal interest rate is recommended.On the another hand, holding capital and collateralizing more their credit, agents can pay their consumption rather on

credit than in cash, and reduce the burden of the CIA constraint. This promotes the consumption smoothing and raises thewelfare level.


4. Local dynamics

In order to characterize the stability properties of the steady state and the occurrence of local bifurcations, we proceedby linearizing the dynamic system (14) around the steady state (x, π) defined by (16) and (17) and computing the Jacobianmatrix J , also evaluated at the steady state. Local dynamics are represented by a linear system (dxt+1/x, dπt+1/π)T =J (dxt/x, dπt/π)T . More explicitly, taking in account the elasticities of credit share, we obtain

[σε2 (A1 + B1)− 1]dxt+1x+ σB2

dπt+1π= [σε2 (A2 + B1)− C1]

dxtx+ σ (A2 + B2)

dπtπ

(1+ C2)dxt+1x−dπt+1π= (C1 + C2)

dxtx

(26)

where, under Assumption 57:

[A1 B1 C1A2 B2 C2

]=

γ ε1

πx

I − γ ε1 πx(>0)

γ ε1 (I − 1)− γ ε1 πx[1− γ (1− ε1)] (I − 1)+ 1− γ ε1 πx

∆+ Ag

(>1)

γ ε1πx

1− γ ε1 πx(>0)

[1− γ (1− ε1)] I − γ ε1 πx[1− γ (1− ε1)] (I − 1)+ 1− γ ε1 πx

γ ε1

1− γ(>0)

. (29)

Since the variables xt and πt are independently non-predetermined, the equilibrium is locally determinate if and only ifthe steady state is a source, while local indeterminacy requires a saddle point or a sink.The trace and the determinant of the Jacobian matrix:

J =[σε2 (A1 + B1)− 1 σB2

1+ C2 −1

]−1 [σε2 (A2 + B1)− C1 σ (A2 + B2)

C1 + C2 0

]are, respectively:

T (σ ) = 1+ D (σ )−

( 1σ+ A2

)(1− C1)− ε2 (A1 − A2)

1σ− B2 (1+ C2)− ε2 (A1 + B1)

(30)

D (σ ) = −(A2 + B2) (C1 + C2)

1σ− B2 (1+ C2)− ε2 (A1 + B1)

. (31)

In the following,we exploit the fact that the trace T and the determinantD are the sumand the product of the eigenvalues.The stability properties of the system, that is, the location of the eigenvalues with respect to the unit circle, is equivalentlyand conveniently characterized in the (T ,D)-plane (the reader is referred to Figs. 1 and 2).8In the spirit of Grandmont et al. (1998), we apply the geometrical method and we characterize the locus Σ ≡

{(T (σ ) ,D (σ )) : σ ≥ 0} obtained by varying the elasticity of intertemporal substitution in consumption σ in the (T ,D)-plane.The following lemma provides some technical results involved in the main proposition.

Lemma 3. (i)Σ is linear with origin (T (0) ,D (0)) = (C1, 0), endpoint

(T (∞) ,D (∞)) =(1+ D (∞)+

ε2 (A2 − A1)− A2 (C1 − 1)ε2 (A1 + B1)+ B2 (C2 + 1)

,(A2 + B2) (C1 + C2)

ε2 (A1 + B1)+ B2 (C2 + 1)

)(32)

7 Assumption 5 writes equivalently:

0 < 1− γ ε1π/x < I − γ ε1π/x (27)

and implies

[1− γ (1− ε1)] (I − 1)+ 1− γ ε1π/x > 0. (28)

8 More explicitly, we evaluate the characteristic polynomial P (z) ≡ z2 − Tz + D at −1 and 1. Along the line AC , one eigenvalue is equal to 1,i.e. P (1) = 1 − T + D = 0. Along the line AB, one eigenvalue is equal to−1, i.e. P (−1) = 1 + T + D = 0. On the segment [BC], the two eigenvalues arecomplex and conjugate with unit modulus, i.e. D = 1 and |T | < 2. Therefore, inside the triangle ABC , the steady state is a sink, i.e. locally indeterminate(D < 1 and |T | < 1 + D). It is a saddle point if (T ,D) lies on the right sides of both AB and AC or on the left sides of both of them (|1+ D| < |T |). Itis a source otherwise. A (local) bifurcation arises when an eigenvalue crosses the unit circle, that is, when the pair (T ,D) crosses one of the loci AB, ACor [BC]. (T ,D) depends on the structural parameters. We choose and vary a parameter of interest and we observe how (T ,D)moves in the (T ,D)-plane.More precisely, according to the changes in the bifurcation parameter, a saddle-node bifurcation (generically) occurs when (T ,D) goes through AC , a flipbifurcation (generically) arises when (T ,D) crosses AB, whereas a Hopf bifurcation (generically) happens when (T ,D) goes through the segment [BC].


-4 -2 2 4

-4

-2

2

4

A

B CT

D

(C ,0 )1

σ F

Fig. 1. Case (i).

-4 -2 2 4

-4

-2

2

4

A

B CT(C ,0 )1

σ Fσ H

D

Fig. 2. Case (iii).

and slope

S =[2− C1 +

A2 − A1 − (C1 − 1) (A1 + B1)(A2 + B2) (C1 + C2)

(ε2 − ε

C2

)]−1(33)

where

εB2 ≡ εC2 −

4 (A2 + B2) (C1 + C2)A2 − A1 − (A1 + B1) (C1 − 1)

εC2 ≡ − (C1 − 1)A2C2 + (A2 + B2) (C1 − 1)A2 − A1 − (A1 + B1) (C1 − 1)

are the critical points such that the lineΣ goes through the vertices B and C, respectively.(ii) Under Assumption 5, D′ (σ ) < 0, that is, the point (T (σ ) ,D (σ ))moves downwards in the (T ,D)-plane when σ goes up.

Proof. See the Appendix.As in the case of comparative statics, Assumption 5 plays a crucial role in order to characterize the local dynamics.

Proposition 4. Let Assumption 5 hold and

σF ≡1+ C1

(A2 + 2B2) (1+ C1 + 2C2)+ (A1 + A2 + 2B1) ε2(34)

σH ≡1

B2 (1− C1)− A2 (C1 + C2)+ (A1 + B1) ε2where σF and σH are solutions of D (σ ) = −T (σ )− 1 and D (σ ) = 1, respectively.


(i) If A2 − A1 − (C1 − 1) (A1 + B1) < 0 or (A2 − A1 − (C1 − 1) (A1 + B1) > 0 and εC2 < ε2 < 0), then the steady state isa saddle point for 0 < σ < σF and a source for σF < σ . The system generically undergoes a flip bifurcation at σ = σF(Fig. 1). 9

(ii) If A2− A1− (C1 − 1) (A1 + B1) > 0 and ε2 < εB2 , then the steady state is a saddle point for σ < σF and a sink for σF < σ .The system generically undergoes a flip bifurcation at σ = σF .

(iii) If A2 − A1 − (C1 − 1) (A1 + B1) > 0 and εB2 < ε2 < εC2 , then the steady state is a saddle point for σ < σF , a source forσF < σ < σH and a sink for σH < σ . The system generically undergoes a flip bifurcation and a Hopf bifurcation at σ = σFand σ = σH , respectively (Fig. 2).

Proof. See the Appendix.

Eq. (34) underlines the link between the indeterminacy range (0, σF ) and the shape of credit share (the sensitivity tocollaterals ε1 and the degree of concavity ε2). Focusing on the right-hand side of (34), one easily verifies that, in case (i)of Proposition 4, the indeterminacy range (0, σF ) widens when the credit share becomes more concave (that is ε2 (<0)decreases) if and only if A1 + A2 + 2B1 > 0.

4.1. Economic intuition for cycles

Case (i) of Proposition 4 is closer to Bosi and Magris (2003) (set ε2 = 0 with a constant credit share). This case showsthat, when collaterals matter, cycles of period two appear for a sufficiently low elasticity of intertemporal substitution inconsumption.Let us explain the intuition. Assume that kt increases from its steady-state value. Then, the income Akt increases aswell. If

the intertemporal substitution σ is weak, the income effect prevails and raises current consumption ct . If the intertemporalsubstitution is sufficiently weak, the response in terms of ct exceeds the increase of (∆+ A) kt and, according to the budgetconstraint, kt+1 = (∆+ A) kt−ct decreases. Thus, an increase of kt is followed by a decrease of kt+1; eventually, two-periodcycles arise.This mechanism can be reinforced by the sensitivity of credit share to collaterals: when kt goes up, xt and the credit

share also move up. The positive effect on the current consumption adds to the income effect entailing a deeper fall of kt+1.In case (i) of Proposition 4, cycles of period two are consistent with lower values of intertemporal substitution and thusmore plausible.The range (σF ,+∞) where supercritical cycles occur, widens, if A1 + A2 + 2B1 > 0. In this case, given ε1, increasing ε2

makes the credit share flatter and, thus, steeper on the right-hand side of x. Increasing kt , moves xt on the right of x. On theright, the credit share is now larger and households prefer to consume more. The mechanism described above takes place;thus, kt+1 decreases more. In other terms, given ε1, the closer ε2 to zero, the lower the critical value of flip bifurcation σF :because of the additional effect of ε2, cycles become consistent with lower values and, therefore, withmore plausible valuesof intertemporal substitution.In Bosi and Magris (2003), the additional effects due to the elasticities of credit share were not taken into account. These

effects amplify the traditional bifurcation mechanism which generates period-doubling cycles through a flip bifurcationwhen the intertemporal substitution is sufficiently weak.

4.2. Economic intuition for indeterminacy

In order to explain why prophecies can be self-fulfilling in cash-in-advance economies, let us consider three progressivemodels: (1) a full CIA economy as in Svensson (1985), (2) a partial CIA economy as in Bosi andMagris (2003) and, eventually,(3) a collateralized CIA economy.Assume that households anticipate an increase in the future price pt+1, that is, in the inflation factor πt+1. We want to

show that this expectation can be self-fulfilling.In case (1), γ = γ ′ = 0: the CIA constraint (10) and Eq. (4) simplify

ct+1 = mt+1 (35)

ct+1ct=

(βR

πt

πt+1

)σ(36)

where Rt = R ≡ ∆+ sA is the gross real interest rate. From (36), we obtain the elasticity of the inflation factor w.r.t. futureconsumption ct+1:

επc ≡dπt+1dct+1

ct+1πt+1

= −1σ. (37)

9 In this case, if the flip bifurcation is subcritical, the cycle is unstable and there is room for indeterminacy around the saddle point which is now stable.If the flip bifurcation is supercritical, stable cycles arise around the source. Multiple equilibria converging to the cycle as well as sunspot equilibria aroundcan occur.


An increase in the expected price pt+1 reduces the real balances mt+1 ≡ Mt+1/pt+1 and, according to Eq. (35), the futureconsumption. The decrease of ct+1 determines an increase ofπt+1 according to (35). However, such increase is required to belarge enough in order to make the initial prophecy of higher inflation self-fulfilling. This happens if and only if the elasticityof intertemporal substitution σ in the left-hand side of (37) is sufficiently low, as shown, among the others, by Cooley andHansen (1989).Focus now on case (2). An exogenous CIA constraint means γ ′ = 0. (10) and (4), respectively, reduce to:

(1− γ ) ct+1 = mt+1 (38)

ct+1ct=

[βR1+ (1− γ ) (Rπt − 1)1+ (1− γ ) (Rπt+1 − 1)

]σ. (39)

As above, the decrease of mt+1 following higher future inflation, entails a decrease of ct+1 according to (38). (39) givesthe elasticity of inflation with respect to future consumption:

επc ≡dπt+1dct+1

ct+1πt+1

= −1σ

(1+

γ

1− γ1It+1

). (40)

Still as above, prophecies of higher inflation become self-fulfilling if and only if this elasticity is sufficiently large.If γ = 0, we recover the case (1), while, if γ is sufficiently close to 1 (corresponding to an arbitrary small liquidity

constraint in Bosi and Magris (2003)), local indeterminacy can occur for whatever degree of intertemporal substitution σ .10Indeed, in order to obtain self-fulfilling beliefs, the rise of πt+1 following a decrease of ct+1, must be sufficiently large torecover the initial expectation. This happens iff επc is sufficiently negative: given σ (possibly very large), a sufficiently highγ < 1 always exists such that επc lowers enough below a critical value.Case (3) is also of interest, not only because it is more general, but also because collaterals play a role.11As above, ct+1 decreases following a higher expected inflation πt+1 because of the CIA constraint. However, now the

consumption elasticity of the inflation factor επc incorporates the elasticities of credit share. Differentiating (3) with respectto ct+1, πt+1 and xt+1 around the steady state, we get:

επc = −1σ

1

1− 1−ε1ε1

B1I−1−π/x + (A1 + B1) εxπε2

where εxπ ≡ (π/x) dxt+1/dπt+1 while, using (26) with dxt = 0, we obtain εxπ = 1/ (1+ C2) and, finally,

επc = −1σ

1

1− 1−ε1ε1

B1I−1−π/x +

A1+B11+C2

ε2.

Setting ε1 = ε2 = 0 and It+1 = I , we recover exactly (40).If (A1 + B1) / (1+ C2) > 0, then lowering ε2 (<0) (while keeping ε1 fixed) decreases επc below a critical threshold,

making self-fulfilling prophecies more likely.

4.3. Monetary policy and macroeconomic volatility

Wewant to study the effect of monetary policy on the occurrence of expectation-driven fluctuations. The indeterminacyrange is (0, σF ).Focus first on the benchmark of a constant credit share: γ ′ = 0 (Bosi and Magris, 2003). Then (34) reduces to

σF =12

(1+

γ

1− γ1I

)where I = (∆+ r) µ/g . Since µ has no effects on the growth factor g = [β (∆+ r)]σ , money is superneutral at the steadystate. Howeverµ affects the transition because the inflation factor modifies the Euler equation. In addition, σF depends alsoon µ and the occurrence of endogenous fluctuations depends on the monetary policy:

∂σF

∂µ= −

12

γ

1− γ1µI

< 0.

Surprisingly, monetary growth has a stabilizing power, making the occurrence of expectation-driven fluctuations lesslikely. In order to understand this result, focus on (40). Monetary growth raises the inflation factor π = µ/g and, in turn,the nominal interest factor I = (∆+ r) π . According to (40), the sensitivity of the expected inflation (in absolute value) tofuture consumption decreases. This rules out the occurrence of self-fulfilling prophecies.Introducing sensitivity to collaterals does not change this result from a qualitative point of view. At the end of the

following section we will see how σF behaves under the assumption of an isoelastic credit share in response to a change inthe monetary growth factor or in the elasticity of credit share.

10 In Bosi and Magris (2003), self-fulfilling inflation can arise even when the substitution effect dominates the income effect.11 Our model generalizes the case (2) which, in turn, generalizes the traditional CIA model (1).


5. A numerical example

Let us consider an explicit formulation of credit share: γ (x) ≡ η0 + η1xε , with ε ∈ [0, 1], η0, η1 ≥ 0. We obtain asrequired: ε1 = εη1xε/ (η0 + η1xε) ∈ [0, 1] and ε2 = ε − 1 ≤ 0.

5.1. Steady state

The steady state is defined by (18). We calibrate the TFP parameter A in order to get a plausible growth rate, that is weexpress A as an inverse function of g:

A =

[gµ− ε1γ (∆− g)

]g1/σ − [∆− ε1γ (∆− g)]β

gµ

γ ε1g1/σ + (s− ε1γ ) βgµ

. (41)

In addition, we require two kinds of restrictions to be satisfied.(1) First, the very essential: x = 1/ (∆+ A− g) > 0 (strict positivity of quantities), π = µ/g > 0 (strict positivity of

prices), 0 < γ < 1 (credit share), β < g1/σ−1 (transversality condition), I = (∆+ sA) µ/g > 1 (monetary equilibrium).(2) Second, optional restrictions: π ≥ 1 (positive inflation), g > 1 (growth), 0 < 1 − (∆+ A− g) γ ε1µ/g

(Assumption 5).In the following, for simplicity, we focus on the (locally) isoelastic case: η0 = 0 implies ε1 = ε.We set some preliminary parameters according to quarterly data: β = 0.99,∆ = 0.94574. In the spirit of Mankiw et al.

(1992), since capital encompasses the human capital, we fix s = 2/3. For simplicity, we set also γ = 1/2: cash and creditgood weight the same.

5.1.1. Comparative staticsFix now: ε = 0.5 (intermediate sensitivity to collaterals),µ = g = 1.01 (monetary growth accompanies growth), σ = 1

(logarithmic utility). Applying formula (41), we calibrate A = 0.11134, to implement a quarterly growth rate of 1%.12

To compute the effectiveness of monetary policy, we apply (21),

dgdµ

µ

g=

[1+

1σ

1− γ ε (∆+ A− g) µ/gγ ε (∆+ A− g) µ/g

+(1− ε) g∆+ A− g

I − 1I − γ ε (∆+ A− g) µ/g

]−1= 0.011741 > 0.

The impact of a monetary acceleration is strictly positive (money is no longer superneutral), but weak: increasing by 1%the monetary growth factor, raises by about 0.01% the growth factor. Moreover, a higher sensitivity to collaterals lowersthe effectiveness of monetary policy: if we increase ε from 0.5 to 0.9, living the other parameters unchanged, and wecalibrate the TFP parameter to ensure 1% of growth: A = 0.11105, we get (dg/dµ)µ/g = 0.0046039 > 0. Accordingto this parametrization, we can conclude that, even if the monetary policy matters, its effects are rather small.

5.1.2. Optimal monetary policyIn the two scenarios ε = 0.5 and ε = 0.9, the optimal growth factors g∗ ≡ [β (∆+ A)]σ are respectively given by

g∗ = 1.0465 and g∗ = 1.0462 and the corresponding optimal monetary rules by µ∗ = 14.878 and µ∗ = 8.6753 (see (22)and (24)). Because of the small impacts of µ on the growth factor, passing from g = 1.01 to g∗ requires a large injectionof balances. The policy maker is recommended to handle these solutions carefully. If, on the one hand, they confirm theconsistency of the model and point out a direction, on the other hand, their specific values can be politically unacceptable:on the empirical ground, more sophisticated models that incorporate our assumptions, can be considered to take also intoin account some undesirable effects of inflation.

5.2. Local dynamics

Let us show the impact of intertemporal substitution on local dynamics.We provide two parametrizations correspondingto the source and the saddle case. In the following, fix µ = g = 1.01 and ε = 0.5.Before computing the Jacobianmatrix and the eigenvalues,weneed to find blocks (29). Under the assumption of isoelastic

credit share, blocks and the Jacobian matrix simplify to:

12 The essential and optional restrictions (respectively, (1) and (2)) are fulfilled: x = 21.24 > 0, π = 1 ≥ 1, 0 < γ = 0.5 < 1, β = 0.99 < g1/σ−1 = 1,I = 1.0200 > 1, g = 1.01 > 1, 0 < 1− (∆+ A− g) γ εµ/g = 0.98823.

158 R. Barbar, S. Bosi / Research in Economics 64 (2010) 146–161[A1 B1 C1A2 B2 C2

]

=

γ ε (∆+ A− g) µ/gI − γ ε (∆+ A− g) µ/g

γ ε [I − 1− (∆+ A− g) µ/g]1+ (1− γ ) (I − 1)+ γ ε [I − 1− (∆+ A− g) µ/g]

∆+ Ag

γ ε (∆+ A− g) µ/g1− γ ε (∆+ A− g) µ/g

(1− γ ) I + γ ε [I − (∆+ A− g) µ/g]1+ (1− γ ) (I − 1)+ γ ε [I − 1− (∆+ A− g) µ/g]

γ ε

1− γ

(42)

(where I = (∆+ sA) µ/g) and

J =[σ (ε − 1) (A1 + B1)− 1 σB2

1+ C2 −1

]−1 [σ (ε − 1) (A2 + B1)− C1 σ (A2 + B2)

C1 + C2 0

](43)

respectively.

5.2.1. Equilibrium uniqueness vs multiplicity(1) In order to find equilibrium determinacy, we require sufficiently high intertemporal substitution effects. In this

respect, we set σ = 1. A calibrated TFP parameter implements the 1% growth rate (quarterly): A = 0.11134 accordingto formula (41).13 Using (42), we compute the eigenvalues of (43): λ1 = 1.0472, λ2 = 9.1038. Both the eigenvalues areunstable, the steady state is a source and the jump variables adjust: (xt , πt) = (x, π) for t = 0, 1, . . . Shocks on thefundamentals are neutralized by the rational expectations.(2) It is known that small elasticities of intertemporal substitution promote indeterminacy in CIA models with capital

accumulation (Cooley and Hansen, 1989). Fixing σ = 1/3, we calibrate the TFP parameter to maintain a quarterly growthrate of 1%: A = 0.14128.14 Still using (42), we compute the eigenvalues of (43): λ1 = −0.60269 and λ2 = 1.0767. Weobserve that convergence to the steady state is non-monotonic (−1 < λ1 < 0). When λ1 approaches−1, a flip bifurcationgenerically arises (σ = σF ).

5.2.2. Monetary policy, collaterals and macroeconomic volatilityLet us focus on the case (i) of Proposition 4 and assume the credit share to be isoelastic with η1 = 0: γ (x) ≡ η1xε . Fix

g = 1.01 and calibrate the TFP parameter as a function of (µ, ε, σF ):

A (µ, ε, σF ) ≡

[gµ− εγ (∆− g)

]g1/σF − [∆− εγ (∆− g)]β g

µ

γ εg1/σF + (s− εγ ) β gµ

(44)

that is, given themonetary policy, the degree of sensitivity to collaterals and the critical value of intertemporal substitution,set A to implement a quarterly growth factor g = 1.01. In addition, fix also the scaling factor η1 = γ /xε in order to keep γat an intermediate value, say γ = 1/2.Using (29) and (44), we represent the implicit equation (34), with ε2 = ε− 1, first in the (µ, σF )-plane, in order to study

the stabilizing role ofmonetary policy, then in the (ε, σF )-plane, in order to understand the role of collaterals. In other terms,we plot the behavior of the critical value σF (1) as a function of the monetary growth factor µ (see Fig. 3), (2) as a functionof the elasticity of credit share ε (see Fig. 4).(1) We plot the bifurcation value σF as a function of the monetary policy µ, setting ε = 1/2.For a given µ, the vertical segment between the axis of abscissas and the downward-sloped curve represent the

indeterminacy range (0, σF ) of Proposition 4.Roughly speaking, a higher money growth rate makes the occurrence of self-fulfilling expectations less likely. The higher

the inflation rate, the higher the elasticity επc (that is the lower its absolute value: see Eq. (40) in the particular case of aconstant credit share). This implies that the inflation factor does not adjust enough to recover the initial expectation.(2) Now, let us plot the bifurcation value σF as a function of the elasticity ε with µ = 1.01.According to our parametrization, the indeterminacy range shrinks with the credit market sensitivity to collaterals. In

other terms, the greater the role of collaterals, the less likely the macroeconomic volatility.Numerical analyses can contribute to understand some mechanisms at work during the financial crisis and its

propagation to the real economy. Our model can help to shed a light on the subprime crisis and give some insights tobridge two features of US economy arisen in the past decade: insufficient credit collateralization (low ε) and excess ofmacroeconomic volatility due to self-fulfilling prophecies (high σF ).

13 The essential and optional restrictions hold because the parametrization is the same than in the first case (comparative statics).14 Even in this case, the essential and optional restrictions hold: x = 12.984, π = 1 ≥ 1, 0 < γ = 0.5 < 1, β = 0.99 < g1/σ−1 = 1.0201,I = 1.0399 > 1, g = 1.01 > 1, 0 < 1− (∆+ A− g) γ εµ/g = 0.98075.


1 2 30.35

0.40

0.45

σ

μ

F

Fig. 3. Monetary policy and indeterminacy range.

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

σ

ε

F

Fig. 4. Sensitivity to collaterals and indeterminacy range.

6. Conclusion

In the last few years, the global economy experienced the real effects of the (subprime) financial crisis started in theUS. Considering the role of collaterals in the occurrence of endogenous fluctuations contributes to shed a light on themechanisms at work during the crisis.For this purpose, we have developed an endogenous growth monetary model where agents are subject to a partial

cash-in-advance constraint. In addition, we have assumed that the share of consumption purchases paid on credit dependspositively on the collateral available to the individual.Focusing on the role of monetary policy, we have shown that money is no longer superneutral: the increase in the

monetary growth rate decreases the return on money and promotes capital accumulation as a consequence of assetssubstitution in the portfolio. Moreover, a higher degree of collateralization allows the agents (less constrained by the cash-in-advance) to pay more on credit so as to better smooth consumption over time. From a welfare point of view, we provethat the optimal money growth is different from the Friedman rule. Indeed, inflation raises the opportunity cost of holdingmoney and the substitution of money with capital lightens the CIA burden.Monetary growth under a weak inflation regime has also a positive impact on the ground of stability: higher inflation

raises the nominal interest rate and reduces the elasticity of future inflation with respect to the current consumption. A lowdegree of sensitivity of future inflation prevents the market from recovering the initial jump in expectations. It also rulesout the occurrence of self-fulfilling prophecies.We also suggest an optimal design of credit market regulation, introducing legal constraints that shape the credit share

as a function of collaterals. More precisely, in order to discipline the granting of credits, the policy maker can fix the rulesassociated to the optimal (first, second or higher-order) elasticities of credit function. Finally, in the case of an isoelasticcredit share, we illustrate through a numerical example how a higher sensitivity of credit market to collaterals reduces theindeterminacy range (in terms of the elasticity of intertemporal substitution).

Appendix

Proof of Proposition 2. Wenotice that, at the steady state, c0 = (∆+ A− g) k0. ThenW = V (g) ≡ u((∆+A−g)k0)/(1−βg1−1/σ ). Deriving w.r.t. g gives

V ′ (g) =[(1−

1σ

)βg−1/σ

1− βg1−1/σ−

1∆+ A− g

c0u′ (c0)u (c0)

]V (g) .


Using (13), we obtain

V ′ (g) =(

βg−1/σ

1− βg1−1/σ−

1∆+ A− g

)(1−

1σ

)V (g)

and, therefore, V ′ (g) > 0 if and only if15 g < g∗.On the one hand, we can view the welfare as a function of the monetary policy:W (µ) ≡ V (g (µ)). Since g ′ (µ) > 0,

according to (20), we getW ′ (µ) > 0 iff V ′ (g) > 0, that is, iff g < g∗. Welfare is maximized by µ∗ = g−1 (g∗), where g−1is a well-defined function from the monotonicity of g (see again (20)). More explicitly:

µ∗ =g∗ − βg∗1−1/σ [∆+ sA− γ ε1 (∆+ A− g∗)]

γ ε1 (∆+ A− g∗)

and, replacing (23), we obtain (22). Deriving (22) w.r.t. ε1 gives (24).On the other hand, given amonetary ruleµ, we can also compute the optimal regulation (25) by solving (18)with respect

to ε1 and setting g = g∗. �

Proof of Lemma 3. The origin and the endpoint are obtained taking the limit of (30) and (31) as σ approaches 0 from aboveand+∞ from below, respectively. The slope is obtained computing T ′ (σ ), D′ (σ ) and the ratio S = D′ (σ ) /T ′ (σ ):

S =(1+

[A2 − A1 − (C1 − 1) (A1 + B1)] ε2 − (C1 − 1) (A2 + B2 + B2C2)(A2 + B2) (C1 + C2)

)−1and reduces to the compact form (33). Moreover

D′ (σ ) = −[D (σ ) /σ ]2

(A2 + B2) (C1 + C2)< 0

because C1 + C2 > 0 and, under Assumption 5,

A2 + B2 =I − γ ε1 πx1− γ ε1 πx

1− γ (1− ε1)[1− γ (1− ε1)] (I − 1)+ 1− γ ε1 πx

> 0 (45)

(see Eqs. (27) and (28) in the footnote). �

Proof of Proposition 4. (C1, 0), the origin ofΣ , lies on the T -axis, on the right of the line AC .Consider the endpoint (32). Since Assumption 5 holds, (27) holds as well and

A2 − A1 =(I − 1) γ ε1 πx(

I − γ ε1 πx) (1− γ ε1 πx

) > 0.Then, according to (45), D (∞) > 0 ⇔ ε2 (A1 + B1) + B2 (C2 + 1) > 0 ⇔ D (∞) > T (∞) − 1. This implies that Σnever crosses the line AC (the eigenvalues never cross +1 and there is no room for saddle node, transcritical or pitchforkbifurcations).Focus now on the impact of ε2 on the location of Σ . When ε2 (<0) increases, Σ rotates clockwise around the origin

(C1, 0) if and only if ∂S/∂ε2 < 0, where

∂S∂ε2= −

A2 − A1 − (C1 − 1) (A1 + B1)(A2 + B2) (C1 + C2)

S2. (46)

(1) If A2 − A1 − (C1 − 1) (A1 + B1) < 0, then ∂S/∂ε2 > 0 and Σ rotates counterclockwise around (C1, 0). We noticethat S (−∞) = 0+ and 0 < εC2 < εB2 . If 0 < S

(εC2), then S (ε2) ∈

(0, S

(εC2))for every ε2 < 0. If S

(εC2)< 0, then

S (ε2) 6∈(S(εC2), 0)for every ε2 < 0. In both the cases,Σ never crosses the triangle. Indeed:

(1.1) if S ∈ (0, 1), then D′ (σ ) < 0 and Σ ∩ AC = ∅ prevent Σ from entering ABC (Σ is a segment included in the cone{(T ,D) : D ≤ min {0, T − 1}});

(1.2) if S 6∈ (0, 1), since the line includingΣ intersects D = 1 on the right of C , it can not cross the triangle.In both the cases (1.1) and (1.2), there is room for a flip bifurcation when Σ crosses the line AB below A. More

precisely, the steady state is a saddle point for σ < σF and a source for σF < σ . The system generically undergoes aflip bifurcation at σ = σF .

15 Notice that the transversality condition ensures the convergence of the welfare series and it is equivalent to (βg)σ < g . Therefore, (23) is equivalentto (βg)σ < g < [β (∆+ A)]σ . The existence of a nonempty range for g requires g < ∆+ A, which is satisfied by (15).


(2) If A2 − A1 − (C1 − 1) (A1 + B1) > 0, then ∂S/∂ε2 < 0 and Σ rotates clockwise around (C1, 0). We notice thatS (−∞) = 0− and εB2 < εC2 < 0.

(2.1) When ε2 < εB2 , the steady state is a saddle point for σ < σF and a sink for σF < σ . The system generically undergoesa flip bifurcation at σ = σF .

(2.2) When εB2 < ε2 < εC2 , the steady state is a saddle point for σ < σF , a source for σF < σ < σH and a sink for σH < σ .The system generically undergoes a flip bifurcation and a Hopf bifurcation at σ = σF and σ = σH , respectively.

(2.3) When εC2 < ε2 < 0, then the steady state is a saddle point for σ < σF and a source for σF < σ . The system genericallyundergoes a flip bifurcation at σ = σF .

In all these cases, the existence of bifurcations requires strictly positive critical values: σF , σH > 0. Cases (1) and (2.3)correspond to case (i) in the proposition. Cases (2.1) and (2.2) correspond to cases (ii) and (iii), respectively. �

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