Endogenous information revelation in a competitive credit market and credit crunch

Journal of Mathematical Economics 68 (2017) 127–141

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Journal of Mathematical Economics journal homepage: www.elsevier.com/locate/jmateco

Endogenous information revelation in a competitive credit market and credit crunch Yuanyuan Li a,b , Bertrand Wigniolle c,∗ a

University of Bielefeld, Germany

b

University Paris 1 Pantheon-Sorbonne, France

c

Paris School of Economics, University Paris 1 Pantheon-Sorbonne, France

article

info

Article history: Available online 11 October 2016 Keywords: Credit crunch Endogenous information revelation

abstract In this paper, we propose a new mechanism able to explain the occurrence of credit crunches. Considering a credit market with an asymmetry of information between borrowers and lenders, we assume that borrowers have to pay a cost to reveal information on the quality of their project. They decide to be transparent if it is necessary for getting a loan or for paying a lower interest rate. Two types of competitive equilibria may exist: an opaque equilibrium in which all projects receive funding without revealing information; a transparent one in which only the best projects reveal information and receive funding. It is also possible to get multiple equilibria. Incorporating this microeconomic mechanism in an OLG model, the economy may experience fluctuations due to the change of regime, and indeterminacy may occur. © 2016 Elsevier B.V. All rights reserved.

1. Introduction With the global financial crisis of 2007–2008, originally driven by mortgage-backed securities, the world economy has experienced a strong financial instability. The pre-crisis period was a time of easy credit conditions, low interest rates, and decline in lending standards. The diffusion of structured investment vehicles, the extension of securitization, the development of shadow system banking have increased opacity into the financial markets. The crisis produced a credit crunch related to a crisis of confidence in all the borrowers. The financial crisis was transmitted to the real economy and the credit crunch led to a contraction of the economic activity: asset prices dropped, unemployment increased, and output growth bogged down. This paper proposes a theory that may explain a sudden credit crunch, associated with the transition from a high income equilibrium to a low income equilibrium. As many previous works, this theory is based on information asymmetry between borrowers and lenders, borrowers having an information advantage on their project. But it departs from these works in assuming that information revelation induces a cost that is borne by the borrower. If the relevance of this assumption has been recognized

∗ Correspondence to: C.E.S., 106-112, boulevard de l’hôpital, 75647 Paris Cedex 13, France. E-mail address: [email protected] (B. Wigniolle). http://dx.doi.org/10.1016/j.jmateco.2016.09.008 0304-4068/© 2016 Elsevier B.V. All rights reserved.

by different authors (see Tirole, 2006 for a general survey), its consequences have not been subject to a general analysis. More precisely, we assume that a borrower can choose to be ‘‘opaque’’ or ‘‘transparent’’, and that he must incur a cost to reveal the quality of his project. This cost can be explained by the existence of a direct cost of information revelation: auditing and advertising. It can also be justified by indirect costs, mainly the fact that being transparent implies to quit offshore activities, and to support higher taxes. In taking into account costly information revelation, we build a signaling model of credit markets. A borrower chooses to be transparent either if it is necessary to get a loan, or if he can get a lower interest payment and earn a higher profit. Moreover, as being transparent reveals the quality of the project to the lenders, only good projects have an incentive to be transparent as bad ones would pay a cost without being financed. We first consider a simple partial equilibrium model, with the assumption of an exogenous safe interest R0 at which banks can be refinanced. We show that two threshold levels Rˆ and Rˇ play a role in ˇ For a low safe interest (R0 < R), ˆ banks the equilibrium, with Rˆ < R. are willing to offer loans at low rates. All projects can be financed and no borrowers have incentive to reveal information. This leads to an opaque equilibrium of high activity where all projects obtain ˇ banks ask for a loan. When the safe interest is high (R0 > R), high repayments. At equilibrium, only the good projects reveal information and are financed. Projects of low quality cannot be financed. This leads to a transparent equilibrium with low activity.

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ˇ both Finally, when the safe interest is middle-valued (Rˆ < R0 < R), types of equilibria exist together. In this case, there exists some interdependence between good and bad projects. If good projects remain opaque, the average gain on all projects allows financing a return for the bank higher than R0 . Consequently, bad projects can be financed. If good projects are transparent, the remaining opaque projects offer an average gain that is too low to be financed. A simple extension of the static model is obtained in assuming the existence of a supply curve for savings that is increasing in R0 . With this assumption, the safe interest rate is endogenously determined. Two types of equilibria may exist, opaque or transparent.  Itis also possible to obtain multiple equilibria in the interval

Rˆ , Rˇ . In this interval, the economy may experience a

jump from an opaque to a transparent equilibrium that leads to a credit crunch. The credit crunch generates a sudden fall in loans accompanied by transparency requirements. It also leads to a fall in production. Finally, the static model is incorporated in an overlapping generations model that allows to endogenize the savings function and to study the dynamics of output. At each period, the economy can be in an opaque or transparent equilibrium. The equilibrium regime of period t determines the amount of output then the quantity of savings available to finance future projects. The demand for loans also depends on the type of equilibrium that will occur in t + 1, opaque or transparent. So the economy may experience transitions between the different regimes. The dynamics is studied with respect to the interest factor Rt −1 , which is a predetermined variable as it is determined by the credit market in t − 1. The intertemporal equilibrium can lead to different types of dynamics and some numerical examples are provided. Depending on the value of the parameters, the dynamics can be determinate or not. In the case of determinacy, for any value of Rt −1 there exists only one value for Rt . Even in the case of determinacy, the economy may experience endogenous fluctuations corresponding to a change of regime between opaque and transparent equilibria. We also present examples of indeterminacy for which the two regimes may be possible at a given period. The coordination of agents on one regime would need some selection mechanism such as some selffulfilling prophecy. Our model is based on previous works. First the static model can be viewed as an extension of Drees et al. (2013), who consider a model with investors that can choose between more or less opaque projects. They show that investors favor more transparent projects when the interest rate is higher. Our static model is based on a simplified version of their model, but it makes endogenous the choice for a firm to be opaque or transparent. Since the pioneer work of Stiglitz and Weiss (1981), a large literature has studied the role played by asymmetric information in determining the credit market equilibrium, considering adverse selection or moral hazard. Lenders’ lack of information on the relevant characteristics of the borrowers may result in underinvestment, and credit is said to be rationed. This seminal article has been extended in various directions, e.g. Diamond (1984), Williamson (1987), De Meza and Webb (1987) and Gale and Hellwig (1985). In all these contributions, acquiring information may induce a cost for the lender but not for the borrower. As for signaling problems in credit market, Jaffee and Russell (1976) and Leland and Pyle (1977) are the first contributions that consider the incentive for loans applicants to signal their quality either by choosing a particular contract or by investing in their own projects. The possibility of screening by the banks has been studied in various studies, see e.g. Milde and Riley (1988) and Besanko and Thakor (1987).

Bencivenga and Smith (1991, 1993) and Azariadis and Smith (1998) have investigated the macroeconomic consequences of imperfect capital markets. They have developed overlapping generations models in which imperfection of information may generate fluctuations and low activity equilibria. A recent contribution in this vein is Alberto and Filippo (2013). With respect to this literature, we propose an original mechanism that is able to generate fluctuations and indeterminacy. The paper is organized as follows. Section 2 presents the signaling problem framework in the static model and optimal decisions of agents. Section 3 characterizes the static equilibrium. Section 4 incorporates the static framework in an OLG model and presents various examples of dynamics that may lead to endogenous fluctuations and indeterminacy. Section 5 develops the macroeconomic implications of the model. Acknowledgments are provided, and the most demanding proofs are presented in the Appendix. 2. The model 2.1. Agents and gains Consider a credit market populated by two kinds of agents: entrepreneurs and investors. The entrepreneurs, also termed as ‘‘borrowers’’, are endowed with one project that needs to raise capital. Investors, also termed as ‘‘lenders’’ or ‘‘banks’’, are financial intermediaries that collect savings and invest in projects. All agents are risk-neutral. Each borrower needs to raise 1 unit of fund to proceed a project, which yields a random return of v . The return of the project varies across borrowers and is private information of the owner of the project. Lenders only know the cross-sectional distribution H (v) of v . The associated density function h(v) is positive and continuous for v ∈ [v, v¯ ] and zero elsewhere. Borrowers, when facing a certain loan contract proposed by a lender, have the option to choose either to publish information on their project or to remain silent. Publishing information is costly; the cost is c > 0 and borne solely by the borrower. It may correspond both to direct costs (auditing, advertising), and indirect ones (no offshore activities). We call the borrowers who reveal information transparent borrowers and those who do not reveal information opaque ones. The fact that transparency has a cost for the borrower is the main assumption of the model. The borrowers who choose to be transparent reveal full information about the return of their projects and lenders know the realization of the return as well as the project owner. Otherwise, lenders have no more information on the return other than the distribution of v , H (v). The market for loans is competitive and composed of two submarkets: one for transparent projects and one for opaque ones. Lenders offer loans that must be repaid with interest at the end of the period. The repayments required by the lenders are different for transparent and opaque borrowers: R1 for opaque borrowers and R2 for transparent ones. The contract between a borrower and a lender is a debt contract. Repayment Ri is fulfilled only when the realization of the return exceeds the corresponding amount, i.e. v ≥ R1 for an opaque project, and v − c ≥ R2 for transparent project. Otherwise lenders could only get what is realized, v or v − c. Williamson (1987) proved that the debt contract is an optimal arrangement between borrowers and lenders when informational problems exist and monitoring is costly. Such contracts serve to economize on monitoring costs. They are optimal if borrowers have no initial endowment to be pledged. Thus, the payoff of a transparent borrower for a project with a return v is

πBT = max{v − R2 − c , 0},

(1)

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Fig. 1. Decisions and related payoffs.

if he could successfully get the loan; otherwise, πBT = 0. The payoff of a borrower that obtains a loan without revealing information is

If opaque borrowers are financed at equilibrium, R1 must satisfy

πBO = max{v − R1 , 0}.

ZO (R1 ) :=

(2)

If v < R1 , the borrower obtains a null gain if he undertakes or not the project. It is assumed in this case that the borrower always undertakes the project if he can get a loan. This assumption can be explained by the fact that the entrepreneur also gets non market outcomes or private outcomes from leading a project. As a consequence of this assumption, when an equilibrium exists in which opaque projects get loans, all opaque projects are financed. Assuming that the funds are supplied by depositors at a safe interest rate r0 , and denoting R0 = 1 + r0 , the profit of a bank offering a loan to a transparent borrower with a project of return v , is

πLT = min{R2 , v − c } − R0 .

(3)

The expected profit of a bank, if it provides a loan to an opaque borrower, is

πLO = E [min{R1 , v}|v is opaque] − R0 .

(4)

To summarize, we have the expected payoff of both borrowers and lenders shown below in Fig. 1. Fig. 1 also provides the sequence of decisions: borrowers first choose to be opaque or transparent; secondly, lenders decide if they invest or not in the projects. 2.2. Optimal decisions Now we consider successively the optimal decisions of lenders and borrowers. Lenders For a lender, an investment in a transparent borrower is made if min{R2 , v − c } − R0 ≥ 0 ⇒ R2 ≥ R0 and v ≥ R0 + c . Let hO and hT be the density functions corresponding to opaque and transparent projects. hO and hT are two non-negative functions defined on [v, v¯ ], with ∀v ∈ [v, v¯ ], h(v) = hO (v) + hT (v). They are endogenous functions that will be characterized at equilibrium.  v The associated distribution  v functions are given by: HO (v) = v ) is the fraction of v hO (v)dv and HT (v) = v hT (v)dv . HO (¯ opaque borrowers. These notations allow expressing the expected profit of a lender investing in an opaque borrower

 R1 π = O L

v

v hO (v)dv + R1 (HO (¯v ) − HO (R1 )) HO (¯v )

 R1

v hO (v)dv + R1 (HO (¯v ) − HO (R1 )) HO (¯v )

≥ R0 .

If for all values of R1 , ZO (R1 ) < R0 , this implies that there does not exist any market for opaque project funding. These results can be summarized by the following lemma: Lemma 1. For given densities hO and hT characterizing the distribution of opaque and transparent projects, for given repayment R1 and R0 ,

• If R2 ≥ R0 , banks grant loans to the transparent projects with v ≥ R0 + c. • If R1 is such that ZO (R1 ) ≥ R0 , banks grant loans to all opaque projects. Borrowers Concerning borrowers, the decision to be transparent can be taken in two cases: either the borrower can get the loan only if he reveals information; or he can get the loan in any case, but the payoff of being transparent is higher. Case 1: the borrower can get the loan only if he reveals information:



min{R2 , v − c } ≥ R0 ZO (R1 ) < R0 .

The first inequality means that the borrower can get the loan when being transparent; the second one that he could not in remaining opaque. These inequalities can be expressed as: R0 ≤ R2 R0 ≤ v − c ZO (R1 ) < R0 .



Case 2: the borrower can get the loan in any case, but the payoff of being transparent is higher: min{R2 , v − c } ≥ R0 ZO (R1 ) ≥ R0 max{v − R2 − c , 0} ≥ max{v − R1 , 0}.



The first inequality means that the borrower can get the loan when being transparent; the second one that he can also get a loan in remaining opaque. The third one expresses that the profit is higher when transparent. These conditions can be simplified through: R0 ≤ R2 R0 ≤ v − c ZO (R1 ) ≥ R0

 − R0 .

v

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and if v ≥ R2 + c , R1 ≥ R2 + c if v < R2 + c , R1 ≥ R0 + c . The cases in which borrowers choose to be transparent are now known. There will also be two cases in which they will choose opacity: if revealing information would imply no funding as v < R0 + c; if remaining opaque implies a higher profit, which is the case if v ≥ R0 + c and ZO (R1 ) ≥ R0 , with either v ≥ R2 + c and R1 < R2 + c, or v < R2 + c and R1 < R0 + c. Note that there exists a limit case, if v = R2 + c or if R1 − c = R2 , such that the borrower is indifferent between revealing or not information. Remark 1. The analysis seems intricate as it is necessary at this stage to study all possible cases even if some of them will never happen at equilibrium. For a competitive equilibrium, the condition R2 = R0 will be fulfilled that will rule out the last subcase: it will be impossible to get both R0 ≤ v − c and v < R2 + c. Following this remark, it is only necessary to sum up the optimal borrowers’ behavior in the case R2 = R0 . We have the following lemma: Lemma 2. Assuming R2 = R0 , the behavior of a borrower depends on the value of his project v . 1. if v < R0 + c, he remains opaque. 2. if v ≥ R0 + c, two cases may arise: a if ZO (R1 ) < R0 , he is transparent; b if ZO (R1 ) ≥ R0 , he is opaque if R1 < R0 + c, transparent if R1 > R0 + c. When R1 = R0 + c, he is indifferent between revealing information or not.

The definition corresponds to the optimal decisions in the different possible cases for equilibrium repayments (R∗1 , R∗2 ). If Supp(hT ) ̸= ∅, there exists a market for transparent projects with a loan repayment R∗2 = R0 (zero profit condition). 2.a corresponds to Lemma 2(2.a), where there is no market for opaque projects. A borrower can obtain a loan only if he is transparent. 2.b is obtained in the case of Lemma 2(2.b), with R1 ≥ R0 + c, where a market for opaque projects exists with a loan repayment R∗1 . But as R∗1 ≥ R0 +c, revealing information increases the profit of the borrower. Finally, 3 is reached when there does not exist a market for transparent projects (Lemma 2 (2.b), with R1 ≤ R0 + c). No borrower reveals information as R∗1 ≤ R0 + c. 3.2. Characterization As we have discussed above, no borrower with a project v < R0 + c reveals information under any circumstance, since to disclose the insufficient quality of the project would disable them from being financed. We will refer to the borrowers with v ≥ R0 +c as good borrowers, and those with v < R0 + c as bad ones. To avoid the triviality, we assume that: Assumption 1. v¯ > R0 + c. This assumption means that at least some borrowers are good. Besides, it also implies that the cost of information disclosure is relatively small compared to the maximum value of the possible return on a risky project. By this assumption, we also have H (R0 + c ) < 1. The following propositions allow to completely characterize the different types of equilibria that may exist in this model.

3. Market equilibrium

Proposition 4 (Opaque Equilibrium). Assume that the following inequality holds:

3.1. Definition



R 0 +c

v

This part defines the equilibrium for the credit market with free entry in the banking sector: profits of financial intermediaries cancel out at equilibrium. The equilibrium is captured by a loan contract proposal (R∗1 , R∗2 ), with optimal decisions of borrowers endowed with a project v about being transparent or opaque, and optimal decisions of lenders to offer loans according to the type of borrowers (transparent or opaque). The equilibrium of the loan market is solved under the assumption that borrowers always prefer to implement their project rather than doing nothing when the profit in both cases is zero. As a consequence, when there is a market for opaque projects, all projects are financed. Under this assumption, the equilibrium can be characterized as follows: Definition 3. An equilibrium of the credit market is characterized by density functions for opaque and transparent projects hO and hT , loan repayments for opaque and transparent projects R∗1 and R∗2 such that: 1. h = hO + hT 2. if Supp(hT ) ̸= ∅, then R∗2 = R0 and Supp(hT ) ⊂ [R0 + c , v¯ ]. Moreover, a either ∀R1 ≥ R0, ZO (R1 ) < R0 ; b or ∃R∗1 ≥ R0 + c such that ZO (R∗1 ) = R0 . 3. if Supp(hT ) = ∅, then an equilibrium is characterized by a value

 R∗

R1 ≤ R0 + c such that v v h(v)dv + R1 (H (¯v ) − H (R1 )) = R0 . ∗

1

v h(v)dv + R0 (1 − H (R0 + c )) ≥ R0 .

Then, there exists a unique market equilibrium that is opaque: no borrower is revealing information and all projects are financed at a repayment loan R∗1 < R0 + c defined as the solution of R∗ 1

 v

v h(v)dv + R∗1 (1 − H (R∗1 )) = R0 .

Proposition 5 (Transparent Equilibrium). Assume that: R0 +c

 v

v h(v)dv + (R0 + c ) (1 − H (R0 + c )) < R0 .

Then, there exists a unique market equilibrium that is transparent: all borrowers with v ≥ R0 + c reveal information and are financed at a repayment loan R0 . The borrowers endowed with a project v < R0 + c choose to be opaque and are not financed. Proposition 6 (Multiple Equilibria). Assume that the two following inequalities hold: R 0 +c

 v R 0 +c



∗

v

v h(v)dv + R0 (1 − H (R0 + c )) < R0

v h(v)dv + (R0 + c ) (1 − H (R0 + c )) ≥ R0 .

Y. Li, B. Wigniolle / Journal of Mathematical Economics 68 (2017) 127–141

Then there exist 3 types of equilibria: 1. a transparent equilibrium in which borrowers with v ≥ R0 + c reveal information and are financed with a repayment loan R0 , whereas borrowers with v < R0 + c remain opaque and are not financed 2. an opaque equilibrium in which no borrower is revealing information and all projects are financed at a repayment loan R∗1 ≤ R0 + c such that R∗ 1

 v

v h(v)dv + R∗1 (1 − H (R∗1 )) = R0 ∗

3. a multiplicity of (unstable) equilibria such that R1 = R0 + c. All borrowers with v < R0 + c remain opaque. Borrowers with v ≥ R0 + c are split in opaque and transparent projects. Transparent projects are financed at the repayment loan R0 and opaque borrowers at the repayment loan R0 + c. hO is such that R 0 +c

 v

v h(v) (HO (¯v ) − HO (R0 + c )) dv + (R0 + c ) = R0 . HO (¯v ) HO (¯v )

Proof. see Appendix.



The three propositions are based on inequalities that rely on two functions:

φ(R) ≡



ψ(R) ≡



R +c

v h(v)dv + R(1 − H (R + c ))

v R +c

v

v h(v)dv + (R + c ) (1 − H (R + c )).

It is easy to check that both functions are increasing and that

131

there exists a multiplicity of unstable equilibria with R∗1 = R0 + c. All these equilibria have the same macroeconomic features. There exists an infinity of these equilibria as there is an infinity of ways to split the good borrowers between opaque and transparent ones leading to R∗1 = R0 + c. They are unstable in the sense of the static ‘‘Walrasian tâtonnement’’: a small variation in a price R∗1 or R∗2 leads to a jump in either the transparent or the opaque equilibrium. 3.3. Characterization with respect to the safe interest factor R0 This part shows that the different types of equilibria can be characterized with respect to the value of R0 , if we introduce additional assumptions on the distribution of the projects. The following functions are now introduced: F (R) ≡ G(R) ≡

R+c



v h(v)dv + R(1 − H (R + c )) − R

v R +c

 v

v h(v)dv + (R + c ) (1 − H (R + c )) − R.  v¯

¯, v v h(v)dv + c < v ˆ ˆ ∃!R > v − c such that F (R) = 0.

Assumption 2.

Lemma 7. Under Assumption 2, there exists a unique value Rˇ such that G(Rˇ ) = 0 and a unique value Rˆ such that F (Rˆ ) = 0. Moreover, ˇ Rˆ < R. Proof. G is a decreasing function of R with G′ (R) = −H (R + c ),  v¯ G(v − c ) = c > 0 and G(¯v − c ) = v v h(v)dv − v¯ + c. Under

φ(R) < ψ(R). Proposition 4 corresponds to the case φ(R0 ) ≥ R0 (and thus ψ(R0 ) > R0 ). Proposition 5 is obtained when ψ(R0 ) < R0 (and thus φ(R0 ) < R0 ). Finally, Proposition 6 corresponds to the intermediate case φ(R0 ) < R0 ≤ ψ(R0 ). The inequality ψ(R0 ) ≥ R0 as a simple interpretation. It means

Assumption 2, as G(¯v − c ) < 0, there exists a unique value Rˇ such that G(Rˇ ) = 0. F has the following properties: F ′ (R) = ch(R+c )−H (R+c ) with  v¯ F (v − c ) = 0 and F ′ (v − c ) = ch(v) > 0; F (¯v − c ) = v v h(v)dv −

that the expected gain for a lender when the repayment loan is R0 + c is higher than R0 , when no borrower reveals information, Therefore, there may exist a repayment loan R∗1 ≤ R0 + c that ensures the equilibrium of the opaque market when all projects are opaque. The inequality φ(R0 ) < R0 has also a simple interpretation when it is written under the form:

Proposition 8. Assume that the preceding assumptions hold. Then,

 v

R 0 +c

v h(v) dv < R0 . H (R0 + c )

It means that the expected gain for a lender when the repayment loan is R0 + c is smaller than R0 , for borrowers with v < R0 + c (these that have never interest to reveal information). Therefore, if all borrowers with v ≥ R0 + c choose to reveal information, there cannot exist a market for opaque projects. In Proposition 4, it is optimal for all investors to remain opaque, as they obtain a loan at a price R∗1 < R0 + c. The bad borrowers can be financed even if they are alone on the opaque market. The good borrowers have no incentive to be transparent as they would have to pay a cost c that would be higher than their gain R∗1 − R0 . In Proposition 5, it is not possible to have an equilibrium on the opaque market at a price R∗1 ≤ R0 + c. Therefore, all good borrowers reveal information and are financed at the cost R0 . The bad ones offer an average gain that is too low to be financed. Finally, in Proposition 6 there is some interdependence between good and bad borrowers. If all good borrowers remain opaque, all projects can be financed at a cost R∗1 ≤ R0 + c. If the good borrowers choose to reveal information, bad projects cannot be financed when they are alone on the opaque market. Between these two equilibria,

  v¯ + c < 0. Therefore, the existence of Rˆ ∈ v − c , v¯ − c such that F (Rˆ ) = 0 is obtained. But, as F is not monotone, the uniqueness of   Rˆ is assumed. Moreover, ∀R v − c , v¯ − c , F (R) < G(R), which ˇ  implies Rˆ < R.

ˆ there exists a unique opaque equilibrium in which all 1. if R0 < R, projects are financed, and the total amount of loans is H (¯v ) = 1; ˇ there exists a unique transparent equilibrium in which 2. if R0 > R, projects such that v ≥ R0 + c reveal information and are financed; the total amount of loans is 1 − H (R0 + c ); ˇ there exist two stable equilibria. One equilibrium 3. if Rˆ < R0 < R, is opaque, all projects are financed and the total amount of loans is H (¯v ) = 1. The second one is transparent. Projects such that v ≥ R0 + c reveal information and are financed, and the total amount of loans is 1 − H (R0 + c ). Proof. The proof results from the properties of F and G. We have the properties: R0 ≤ Rˆ ⇔ F (R0 ) ≥ 0 ⇔ φ(R0 ) ≥ R0 ; R0 ≤ Rˇ ⇔ G(R0 ) ≥ 0 ⇔ ψ(R0 ) ≥ R0 . Then, the 3 cases respectively correspond to Propositions 4–6.  When the funding cost is low, banks tend to lower their lending standards and invest in all projects. This leads to a high activity equilibrium. On the other hand, when the funding cost is high, only the best projects are financed and they reveal full information. Finally, for intermediate values of the interest rate, both types of equilibria may exist.

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is invested through financial intermediaries in the projects of the next generation, with an interest factor Rt . Financial intermediaries have no operating costs and are in perfect competition. In period t + 1, the agent consumes the proceed of savings. For a worker, the life cycle is very simple. The wage w earned in t is spent for consumption and savings. Savings is also invested through financial intermediaries with the interest factor Rt and allows consumption in t + 1. The utility function of a generation t agent is given by a CES function:



Fig. 2. Different types of equilibria.

(ct )

σ −1 σ

+ β (dt +1 )

σ −1 σ

 σ σ−1

(5)

with ct the consumption level when adult, and dt +1 the consumption level when old. Moreover, it is assumed that σ > 1. st is the amount that is saved between periods t and t + 1 and transferred to the financial intermediaries. For an entrepreneur, the budget constraints are: ct + st = It (v)

3.4. Credit supply and market transparency

dt +1 = Rt st In this part, the risk free interest factor R0 is endogenized in a static macroeconomic model. Let us assume that the supply for loans, corresponding to consumers savings, is given by an increasing function S (R0 ). The equilibrium of the market for loans can lead to two types of equilibria: opaque  ortransparent. If the equilibrium is obtained in the interval

Rˆ , Rˇ , there may exist

multiple equilibria. The opaque equilibrium is associated with a high activity level, when the transparent equilibrium corresponds to a low aggregate income.   When the equilibrium is in the interval Rˆ , Rˇ , the funding of

bad projects (those with v < R0 + c) depends on the presence of good ones on the opaque market for loans. If good projects choose to be transparent, bad ones cannot be funded. A jump from the opaque to the transparent equilibrium can be interpreted as a credit crunch that produces both a decrease in the supply of credits and a decrease in the interest rate. Fig. 2 gives a simple illustration of these properties. The next part will make endogenous the supply of loans in incorporating the model in a dynamic framework. 4. Macroeconomic dynamics 4.1. An OLG model In this section, an OLG model is developed that incorporates the financial markets described in the previous part. At each period, two types of agents are born, workers and entrepreneurs. Each type is represented by a continuum with a mass equal to 1. Agents are living during 3 periods: youth, adulthood and old age, and they consume in the two last ones. All workers are identical and receive an exogenous labor income in their second period of life equal to w . Entrepreneurs are heterogeneous. During youth, each entrepreneur is endowed with a project of value v . All agents, workers or entrepreneurs, are retired during their old age. A generation t agent is living in t − 1, t and t + 1. For an entrepreneur, the life cycle can be summarized as follows. In period t − 1, she/he decides to be transparent or opaque, and may borrow (if possible) an amount of 1 from financial intermediaries to finance his project, at the risk free factor of interest Rt −1 (determined in t − 1). In period t, she/he earns an income that is equal either to max(v − R1t −1 , 0) if the firm is opaque, or to max(v − c − Rt −1 , 0) if the firm is transparent, with R1t −1 the factor of interest for opaque projects. The income is spent for consumption and savings. Savings

with It (v) the income, that can take the values max(v − R1t −1 , 0) or max(v − c − Rt −1 , 0). The optimal behavior gives savings as: st =

It (v)

.

1 + β −σ Rt1−σ

For a worker, the budget constraints are: ct + st = w dt +1 = Rt st and the optimal behavior gives savings as: st =

w

.

1 + β −σ Rt1−σ

Aggregating the savings of both types of agents, total savings is determined by: St =

Yt 1 + β −σ R1t −σ

where Yt is the aggregate income earned by generation t during adulthood. We now consider the equilibrium on the credit market in period t that determines Rt . The total supply of loans is given by the amount of aggregate savings St . Therefore, it depends on the type of equilibrium that occurs in t, opaque or transparent. Moreover, the total amount of loans also depends on the equilibrium that will occur on the market in t + 1. Therefore, four regimes may arise for the equilibrium. Considering the supply of loans in t. Yt can take two values, depending on the type of equilibrium that occurs in t. For an opaque equilibrium, aggregate income is: YtO = w +

v¯

 v

v h(v)dv − Rt −1

as all projects have been financed. The average/aggregate cost of loans is Rt −1 : the good projects (v ≥ R1t −1 ) reimburse R1t −1 > Rt −1 when the bad ones (v < R1t −1 ) reimburse v , the average payment

ˇ being Rt −1 . This equilibrium exists if Rt −1 < R. For a transparent equilibrium, aggregate income is: YtT = w +



v¯ Rt −1 +c

v h(v)dv − (Rt −1 + c ) [1 − H (Rt −1 + c )]

Y. Li, B. Wigniolle / Journal of Mathematical Economics 68 (2017) 127–141

as only projects such that v ≥ Rt −1 + c have been financed. Each project bears two costs: Rt −1 to the lender and c to be transparent. ˆ It is possible to compare YtO and This equilibrium exists if Rt −1 > R. T Yt : YtO − YtT =

Rt −1 +c

 v

ˇ = Rˇ , YtO = YtT , and YtO > YtT ⇔ Rt −1 < R.

ˇ the aggregate income for generation t in period t When Rt −1 > R, is higher when only the good projects are financed. The total amount of loans on the credit market also depends on the equilibrium that will occur in t + 1. If the equilibrium in t + 1 is opaque, the total amount of loans is equal to 1. If the equilibrium in t + 1 is transparent, the total amount of loans is equal to 1 − H (Rt + c ). As a consequence, the dynamics of Rt −1 is determined by four conditions corresponding to the four regimes: Opaque equilibrium in t and t + 1: 1 + β −σ R1t −σ = w + Rt −1

v¯

 v

[1 − H (Rt + c )] 1 + β −σ Rt

 1−σ

=w+

v¯

 v

v h(v)dv − Rt −1 Rt > Rˆ .

Transparent equilibrium in t, opaque equilibrium in t + 1: 1 + β −σ Rt1−σ = w +



v¯



v h(v)dv − (Rt −1 + c )

Rt −1 +c

× [1 − H (Rt −1 + c )] > Rˆ and Rt < Rˇ .

Rt −1

Transparent equilibrium in t, transparent equilibrium in t + 1: [1 − H (Rt + c )] 1 + β −σ R1t −σ = w +







v¯

v h(v)dv

Rt −1 +c

R t −1

− (Rt −1 + c ) [1 − H (Rt −1 + c )] > Rˆ and Rt > Rˆ .

For the dynamics, Rt −1 is a predetermined variable in t, as it results from the equilibrium on the savings market in t − 1. It is possible to write the dynamics under a simple form, introducing the following functions:

F T (R) = w +

Remark 2. As all functions F Σt −1 (Rt −1 ) and GΣt (Rt ) are decreasing, the dynamics is monotonic when no change of regime occurs. A transition from one regime to the other one is necessary to have non monotonicity. 4.2. Existence of stationary states In this part some simple conditions are derived for the existence of a steady state in both opaque and transparent regimes.

σ (σ − 1) β

Rt −1 < Rˇ and

F (R) = w +

The state of the economy in period t is determined in t − 1, as for the cost of credit Rt −1 . Therefore, it is denoted by Σt −1 .

< Rˇ and Rt < Rˇ .



O

F Σt −1 (Rt −1 ) = GΣt (Rt )

Proposition 10. • In the opaque regime, a necessary condition for the existence of a steady state is:

v h(v)dv − Rt −1

Opaque equilibrium in t, transparent equilibrium in t + 1:



Proposition 9. The dynamics is characterized by a sequence (Rt −1 , Σt −1 ), where Σt −1 ∈ {O, T } is the state of the economy in period t such that:

and if Rt −1 < Rˆ , Σt −1 = O, if Rt −1 > Rˇ , Σt −1 = T , if Rˆ ≤ Rt −1 ≤ Rˇ , Σt −1 = O or T .

v h(v)dv + (Rt −1 + c )

× [1 − H (Rt −1 + c )] − Rt −1 . Then, for Rt −1

v¯

 v

v h(v)dv − R

GO (R) = 1 + β

v h(v)dv − (R + c ) [1 − H (R + c )]  −σ 1−σ Rt


Proof. See Appendix.

 v

v¯

v h(v)dv − 1.



This proposition shows that high values of w and β favor the existence of a steady state in the opaque regime as they increase savings. In contrast, a stable steady state always exists in the transparent regime for w = 0, as a result of the adverse selection mechanism. When only a few numbers of projects are financed, this implies a low aggregate income for the next period, that induces a low supply of credit and a tighter selection of projects. This process converges to a steady state in which only the best project is financed and aggregate income tends to zero. Incorporating an exogenous additional income w allows to obtain a transparent steady state associated with a positive aggregate income. For w low enough, this steady state always exists in a neighborhood of R = v¯ − c. 4.3. Study of the dynamics with a uniform distribution h is assumed to be a uniform distribution on [γ , γ + δ ], with a density 1/δ . In this case, F O (R) = w + γ +

δ 2

− R,



for R ∈ γ , Rˇ



  (γ + δ − R − c )2 F (R) = w + , for R ∈ Rˆ , γ + δ − c 2δ     O −σ 1−σ G (R) = 1 + β R , for R ∈ γ , Rˇ  (γ + δ − R − c )  1 + β −σ R1−σ , δ   for R ∈ Rˆ , γ + δ − c

GT (R) =

GT (R) = [1 − H (R + c )] 1 + β −σ R1t −σ .



σ

• In the transparent regime, if w = 0, R = v¯ − c is a stable steady state. For w low enough, a stable steady state exists in a neighborhood of R = v¯ − c.

R +c



1−σ

T

v¯



133



All these functions are decreasing. Moreover, F O (R) > F T (R) ⇔ R < Rˇ and G0 (R) > GT (R). Finally, it is possible to summarize the dynamics by the following proposition:

with: Rˆ = γ + c Rˇ = γ − c +

√

2δ c .

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Y. Li, B. Wigniolle / Journal of Mathematical Economics 68 (2017) 127–141

Fig. 3. Well defined dynamics.

Fig. 4. The dynamics does not exist for some values of R−1 .

Determinacy of the dynamics The following proposition shows that the dynamics may be determinate, indeterminate or may not exist for some initial conditions, depending on parameters values. Proposition 11. If

w ≤ 1 + β −σ γ 1−σ − δ/2    √ 1−σ δ 2c ≤ β −σ (γ + c )1−σ (δ − 2c ) − γ − c + 2δ c

(6) (7)

for any Rt −1 ∈ [γ , γ + δ − c] and any Σt −1 ∈ {O, T }, there exist Rt ∈ [γ , γ + δ − c] and Σt ∈ {O, T } such that F Σt −1 (Rt −1 ) = GΣt (Rt ). Moreover, if (7) holds with a strict inequality, indeterminacy occurs as there  may exist two possible values for {Rt , Σt }. If



2c > β −σ (γ + c )1−σ (δ − 2c ) − γ − c +

√

2δ c

1−σ

δ , the

dynamics may be defined only on a subset of [γ , γ + δ − c]. Proof. See Appendix.



Fig. 5. Indeterminacy of the dynamics for some values of R−1 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Condition (6) determines a limit value for w . If the labor income is too high, the supply of loans (equal to savings) may be higher than the demand, for low values of the current interest rate. From now, we assume that (6) holds.

Existence and uniqueness of a stable steady state in both regimes. From now we assume the following conditions:

Assumption 1. w ≤ 1 + β −σ γ 1−σ − δ/2.

Assumption 2.

Condition (7) limits the jump between the two regimes. When it is not fulfilled, it means that for some values of Rt −1 , the supply of loans is too low to get an equilibrium value Rt in the opaque regime and too high for the existence of a transparent equilibrium. Following this proposition, the dynamics  is well defined only

= β (γ + c ) (δ − 2c ) −  √ 1−σ  γ − c + 2δ c δ . In general, the dynamics is either indeter-

in a very special case: 2c

−σ

1−σ

minate, or Rt may not exist for some initial values of Rt −1 . Nevertheless, the dynamics is monotonic as long as there is no regime change, as it was emphasized in Remark 2. Therefore, it is now useful to characterize the existence of stable steady states in both regimes. To illustrate the 3 cases, we present a simple numerical simulation with γ = 0.9, δ = 4, β = 0.5, σ = 1.5. The limit value of c that allows a well defined dynamics is c = 0.357578 (Fig. 3). In Fig. 4 with c = 0.6 higher than the limit value, it is possible that the dynamics is not defined for some initial condition R−1 . In Fig. 5 with c = 0.2 smaller than the limit value, it is possible to have indeterminacy of the equilibrium. This case is illustrated with an example such that, for an initial condition R−1 , two paths are possible and represented in green and purple.

1 + β −σ

γ < (σ − 1)1/σ /β δ − 2c . (γ + c )1−σ >

(8) (9)

2

Let us define the following parameters:

σ (σ − 1)(1−σ )/σ δ −γ − +1 β 2 w2 = 1 + β −σ γ 1−σ − δ/2  √ 1−σ √ w3 = 1 + β −σ γ − c + 2δ c − c + 2δ c − δ/2 w1 =

 (δ − 2c ) δ − 2c  1 + β −σ (γ + c )1−σ − . δ 2δ

(10) (11) (12)

2

wm =

(13)

The proposition below characterizes the existence and uniqueness of a stable steady state in both regimes. Proposition 12. There exists a value βl such that, if β < βl , at most one stable steady state in each regime may exist. Moreover,

• assume that wm < w1 , then there exists at most one stable steady state.

Y. Li, B. Wigniolle / Journal of Mathematical Economics 68 (2017) 127–141

– If w ∈ [0, wm ], there exists a stable steady state RT∗ in the transparent regime. – If (σ − 1)1/σ /β < Rˇ and w ∈ [w1 , w2 ], there exists a stable steady state RO∗ in the opaque regime. For w ∈ (wm , w1 ), there does not exist any steady state. – If (σ − 1)1/σ /β > Rˇ and w ∈ [w3 , w2 ], there exists a stable steady state RO∗ in the opaque regime. For w ∈ (wm , w3 ), there does not exist any steady state. • Assume that wm > w1 , then there may exist two stable steady states, one in each regime.

ˇ for w ∈ [0, w1 ], there exists one – If (σ − 1)1/σ /β < R, stable steady state RT∗ in the transparent regime; for w ∈ (w1 , min(wm , w2 )), there exist 2 stable steady states RO∗ and RT∗ , one in the opaque regime and one in the transparent one; if wm < w2 , for w ∈ (wm , w2 ), there exists one stable steady state RO∗ in the opaque regime. ˇ two cases may arise: – If (σ − 1)1/σ /β > R, * if w3 < wm , for w ∈ [0, w3 ], there exists one stable steady state RT∗ in the transparent regime; for w ∈ (w3 , min(wm , w2 )), there exist 2 stable steady states RO∗ and RT∗ , one in the opaque regime and one in the transparent one; if wm < w2 , for w ∈ (wm , w2 ), there exists one stable steady state RO∗ in the opaque regime. * if w3 > wm , for w ∈ [0, wm ], there exists one stable steady state RT∗ in the transparent regime; for w ∈ (wm , w3 ), there exists no stable steady state; for w ∈ (w3 , w2 ), there exists one stable steady state RO∗ in the opaque regime. Proof. See Appendix.



The proof shows that condition (8) is useful to obtain the existence of a steady state in the opaque regime, whereas condition (9) allows to get a steady state in the transparent regime. Some general properties can be drawn from this proposition. A stable steady state may exist in the opaque regime if w is high enough (w > w1 ), when it may exist in the transparent regime if w is low enough (w < wm ). Indeed, a low value for w implies a low value for savings that favors the existence of an equilibrium with credit rationing. A high value of w favors an equilibrium in which all projects are financed. The condition β < βl allows to limit the number of steady states to one in the transparent regime. As it is proved in the Appendix, the value βl is the solution of an equation with no explicit solution. But a numerical value can be calculated in the examples. Long run cycles with alternate regimes. As a consequence of Proposition 12, it is possible that no stable steady state exists in the case wm < w1 when w ∈ (wm , w3 ). So it is possible to think on more complex long run equilibria relying on transitions between regimes. We focus in this part on the existence of cycles in which the economy experiences at each period a transition between the opaque and the transparent regime. We call such an equilibrium a cycle with alternate regimes.   Assume that RO , RT





γ , Rˇ the value

is a cycle with RO ∈

in the opaque regime and RT ∈





Rˆ , γ + δ − c the value in the

transparent one. By definition, we have F O (RO ) = GT (RT ) and F T (RT ) = GO (RO ):

   1−σ  γ + δ − RT − c  w+γ + −R = 1 + β −σ RT 2 δ δ

O

(14)

 w+

γ +δ−R −c 2δ T

2   = 1 + β −σ (R0 )1−σ .

(15)

We define H1 and H2 as: H1 (R) =



O −1

G



135



 T −1

G

◦ F O (R) and H2 (R) =

◦ F (R). A cycle with alternate regimes is obtained through a   T

couple of values RO , RT such that RT = H1 (RO ) and RO = H2 (RT ), or RT = H1 (H2 (RT )). We define then the function X (R, w) ≡ H1 ◦ H2 . As the study will be based on the value of w , we make explicit the dependency with respect to this parameter. H1 and H2 are both increasing functions of R and decreasing functions of w . Therefore, X is increasing in R and decreasing in w . The following proposition gives a method to build a cycle with alternate regimes in the case where the dynamics is well determined: (7) holds with an equality (cf. Proposition 11). We retain this case to have a simpler analysis. All results remain true in a neighborhood of this case. Proposition 13. Assume that w1 > wm and that





2c = β −σ (γ + c )1−σ (δ − 2c ) − γ − c +

√

2δ c

1−σ  δ .

• There exists a value w ¯ of w such that RT = Rˆ , RO = H2 (Rˆ ) is a cycle with alternate regimes.

• We define A ≡ X ′ (Rˆ , w) = H1′ (H2 (Rˆ ))H2′ (Rˆ ) (see Box I) – If A < 1, there exists a neighborhood of w, ¯ ]w l , w] ¯ , there exists l ˆ ˆ a neighborhood of R, [R, R [, such that, ∀w ∈]w l , w], ¯ ∃R ∈ [Rˆ , Rl [ such that R = X (R, w) and R decreases with w. {R, H2 (R)} is a cycle with alternate regimes. Moreover, the dynamics around this cycle is stable. – If A > 1, there exists a neighborhood of w, ¯ [w, ¯ wh [, there exists h ˆ ˆ a neighborhood of R, [R, R [, such that, ∀w ∈ [w, ¯ wh [, ∃R ∈ [Rˆ , Rh [ such that R = X (R, w) and R increases with w . {R, H2 (R)} is a cycle with alternate regimes. Moreover, the dynamics around this cycle is unstable. Proof. See Appendix.



4.4. Some numerical examples We provide some examples of the dynamics that can be observed. First we consider parameters such that wm < w1 : γ = 0.9, δ = 4, β = 0.5, c = 0.357578, σ = 1.5. The value of c allows to have a well defined dynamics. With these parameters, βl = 0.574701, wm = 1.54368, w1 = 1.87976. Starting from w = 0, Fig. 6 presents the dynamics that is monotonic and converges to R = v¯ − c. An increase of w lets unchanged the two functions GO (R) and GT (R), but induces an upward move for the two functions F O (R) and F T (R). Increasing w to 0.5 (Fig. 7) also gives a monotonic dynamics converging to a non trivial steady state. In these two cases, the dynamics is assumed to start from an initial opaque equilibrium. But there is a quick transition to a transparent one. When w takes the value 1.3 (cf. Fig. 3), the transparent  

steady state equilibrium now belongs to the interval Rˆ , Rˇ . The

transition between an opaque and a transparent state induces a non monotonic dynamics that converges to a transparent steady state. This illustrates a property of the model: the dynamics is always monotonic in a given regime, but a change of regime may induce non monotonicity. For w = 1.75 ∈ (wm , w1 ) (Fig. 8), there does not exist any steady state equilibrium. A possible long run equilibrium is a cycle in which the economy experiences at each period a transition between an opaque and a transparent equilibrium. Proposition 13 has provided a general method to build such cyclical dynamics with alternate regimes. In the case w = 1.95 (Fig. 9), there exists a unique steady state opaque equilibrium as w > w1 . Around this steady state,

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Y. Li, B. Wigniolle / Journal of Mathematical Economics 68 (2017) 127–141

δ − 2c A=   −σ .  1 −σ 1 + β −σ (γ + c ) + (δ − 2c ) β −σ (σ − 1) (γ + c )−σ β −σ (σ − 1) H2 (Rˆ ) Box I.

Fig. 6. Dynamics with w = 0.

Fig. 9. Dynamics with w1 < w .

Fig. 7. Dynamics with w < wm .

Fig. 10. An example with two stable steady states in the case wm > w1 .

The case wm > w1 may lead to a lot of cases. We choose to concentrate on one particular case, in which two stable steady states exist together, one in the opaque regime and one in the transparent one. Moreover, we assume that indeterminacy may occur as GT (Rˆ ) > GO (Rˇ ). Parameters   are such that the two steady states belong to the interval Rˆ , Rˇ . We take c = 0.2, γ = 0.9,

δ = 4, β = 0.5, σ = 2 and w = 2.4. With these figures, βl = 0.552782, w1 = 2.1, and wm = 2.55273 > w1 .  Fig. 10 is enlarged in order to focus on the part Rˆ , Rˇ . Starting

Fig. 8. Dynamics with wm < w < w1 .

the dynamics is monotonic and convergent. But, if the starting point is in the transparent regime, the economy may experience a complex oscillatory dynamics before reaching the opaque regime and converging to the steady state.

from the opaque steady state, it is possible to go to the transparent regime and converge to the transparent steady state. Conversely, starting from the transparent steady state, it is possible to go to the opaque regime and converge to the opaque steady state. The presence of indeterminacy may imply fluctuations among regimes, with problems for the agents to coordinate their expectations. In this case, sunspot equilibria may be a way to coordinate agents. Nevertheless, as this case of indeterminacy arises for low values of c, this implies that the two steady states are close and the  interval Rˆ , Rˇ is relatively small. So we may expect endogenous fluctuations in this interval with a small amplitude.

Y. Li, B. Wigniolle / Journal of Mathematical Economics 68 (2017) 127–141

5. Macroeconomic implications of the model In this part, we first discuss the optimality properties of the different equilibria. Then, we emphasize some features of the model that may play a role in financial crisis. 5.1. Optimality of the steady states We begin in determining the first best stationary state of the economy. As usual, we assume that a social planner maximizes the average utility of all agents of a generation at a stationary state under the resource constraint of the economy. If c and d denote the stationary level of consumption of each agent, the resource constraint of the economy for a stationary state is: 2c + 2d = w +

v¯



v h(v)dv −



v¯

h(v)dv.

x

x

As we have two continuums of agents with a mass equal to 1, total consumption is 2c + 2d. About resources, x is the minimum quality level of a project that will be financed. If v < 1, it is obvious that the optimal value of x is 1 (if v ≥ 1, x = v ). As the investment cost of each project is 1, it is optimal to invest only on projects for which the gain is higher than the cost. Finally, the resource constraint is:

   v¯ 1 c+d= w+ (v − 1)h(v)dv . 2

1

Now, with a general utility function, it is clear that the first order condition gives: Uc′ (c , d) = Ud′ (c , d) and, with the utility function (5), c β σ = d. We compare now this optimal stationary state with the two types of competitive stationary equilibria that we can get: an equilibrium in the opaque regime, or in the transparent one. The utility function (5) is weakly concave as it is linear homogeneous. So, the social planner has no aversion for inequalities. The indirect utility of each agent is proportional to her income. Therefore, aggregation is simple and the average utility level is equal to the indirect utility of an agent endowed with the average income of the economy. This average income is equal to: O

y =

1 2

  w+

v¯ v



v h(v)dv − R

in an opaque steady state, and to yT =

1 2

  w+

v¯

 v h(v)dv − (R + c ) [1 − H (R + c )]

R +c

in a transparent one. The intertemporal budget constraint is c+

d R

= yΣ with Σ = O or T .

137

• The transparent equilibrium leads to the optimal selection of projects only in the case R = 1 − c, which is incompatible with the optimal intertemporal arbitrage between c and d. If R = 1, the transparent equilibrium eliminates productive projects (those with v ∈ [1, 1 + c [) and leads to a sunk cost as a consequence of the cost of information revelation. In general, the selection of the projects is not optimal and the cost of information revelation is a deadweight loss. To go further in the comparison of the two competitive equilibria, we can calculate the average of the indirect utility of agents. Denoting by V Σ the indirect utility in the state Σ , we get: V Σ = y Σ 1 + β σ R σ −1



 σ −1 1

.





For a given value of R ∈ Rˆ , Rˇ where both regimes exist together, we know from Section 4.1 that yO > yT and thus V O > V T . Now, assume that there exist two competitive equilibria associated with two different values of R, RO∗ in the opaque regime and RT∗ in the transparent one. Using (18) and (19), we get:

 σ −1  σ −1 1



V O = 1 + β σ RO∗



V T = 1 − H (RT∗ + c )



 σ −1  σ −1 1

1 + β σ RT∗

.

These equations show that the transparent equilibrium leads to a loss in the income 1 − H (RT∗ + c ). However, this effect may be compensated when RT∗ > RO∗ . To make the analysis simple, we have retained a linear homogeneous utility function (5). If this function was strictly concave, the social planner would have some aversion for inequalities. This would be another source of sub-optimality of the competitive equilibria that always lead to unequal income among entrepreneurs. What could be the role of a macroeconomic policy in this framework? First, we see that it is not possible to reach the first best when v < 1: in an opaque equilibrium, some unproductive investments are undertaken; in a transparent equilibrium, the information revelation cost is a deadweight loss. So, a social planner is limited to second best policies. Two main variables could play a role: an intergenerational transfers and a tax/subsidy on savings. An intergenerational transfer (positive or negative) may be a way to control the level of savings and of the equilibrium interest rate. A tax/subsidy on savings may be a tool to eliminate indeterminacy. A tax τ (positive or negative) changes the interest factor for the agents in R(1 − τ ). Therefore, condition (7) becomes 2c ≤ β −σ



(γ + c ) (1 − τ )

1−σ

(δ − 2c ) 1−σ   √  δ − γ − c + 2δ c (1 − τ )

and an appropriate choice of τ allows to obtain an equality that implies the determinacy of the equilibrium.

The first order condition gives Uc′ (c , d) = RUd′ (c , d) and, with the utility function (5), c (β R)σ = d. From these results, we can derive the different sources of suboptimality of the competitive equilibria:

• When R ̸= 1, agents do not make the optimal intertemporal arbitrage between c and d.   • When v < 1, unproductive projects those with v ∈ [v, 1[ are financed in the opaque equilibrium.

5.2. Some macroeconomic features of the model The crisis in 2007–2008 shows different features that the model is able to generate: increase in the interest rate, credit rationing, increase in the volatility of interest rates and tightening of credit standards. For instance, the cost of borrowing for corporations in the Euro area increased from 4.91% in January 2007 to 6.03% (peak value) in September 2008. Meanwhile, the volume of loans to corporations decreased: the volume of loans to corporations up to 1 million Euros in the euro area dropped from over 9000

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Y. Li, B. Wigniolle / Journal of Mathematical Economics 68 (2017) 127–141

With a uniform distribution, we get: R1 = γ + δ −

Fig. 11. A negative shock on w induces a transition from an opaque to a transparent equilibrium.

million Euros in July 2007 to below 6000 in September 2008. (Data sources: FRED Economic Data; ECB Statistical Data Warehouse.) The indicator of the volatility of short-term interest rates rose since late 2007 up to nearly 300 in the summer of 2008, compared to an average level of below 100. The indicator of credit standards for loans to non-financial corporations in the Euro area increased up to 40 in the summer of 2008 from a level at around 0 at the beginning of 2007 (ESRB Risk Dashboard, March 2016). Our model allows understanding some mechanisms that may play a role in financial crisis. A first mechanism is related to an exogenous shock on income. Assume that the economy starts from a steady state in the opaque regime, as in Fig. 11. If the economy experiences an exogenous negative shock on w (in Fig. 11, w falls from 1.95 to 1.75), it is possible that the steady state opaque equilibrium does no more exist and that the economy converges with oscillations toward a transparent equilibrium. There is an amplification mechanism of the crisis by the financial markets due to the information problem. In the new equilibrium, all projects are no more financed that leads to an additional contraction of output. Moreover, the interest rate is higher and there is the information revelation cost that is lost. Indeed, for an opaque equilibrium, total GDP is given by: GDP O = w +

v¯

 v

v h(v)dv.

(16)

An amount R remunerates the savings of the old generation. The remaining income is used for the consumption of the young generation and for savings. For a transparent equilibrium, total GDP is: GDP = w + T



v¯

v h(v)dv − c [1 − H (R + c )] .

(17)

R +c

There is a loss of output coming from the project selection: only the projects with v ≥ R + C are financed. c [1 − H (R + c )] appears as a deadweight loss resulting from the cost of information revelation. From this GDP, an amount R [1 − H (R + c )] remunerates the savings of the old generation. The remaining income is used for the consumption of the young generation and for savings. In the opaque equilibrium, the interest rate R1 borne by entrepreneurs is higher than the return R earned by savers, following: R1

 v

v h(v)dv + R1 (1 − H (R1 )) = R.

 δ 2 + 2δ(γ − R).

ˇ where For R = γ , R1 = γ . Then, the difference is increasing till R, R1 − R = c. So this difference in general is small. In the example of Fig. 11, R1 and R have been drawn and the difference is negligible in the opaque steady state. We have considered that the economy after the negative shock converges to a new steady state in the transparent regime. It may also be the case that no steady state exists after the shock as in Fig. 8, and that the economy experiences endogenous fluctuations between regimes. Another interesting property of the model is that indeterminacy is possible, which may imply endogenous fluctuations driven by expectations. Two examples have been provided in Figs. 5 and 10. In Fig. 10, we have two steady states, RO∗ in the opaque regime and RT∗ in the transparent one. From Eqs. (16) and (17), GDP O > GDP T . But interest payments to the old generation are higher in the opaque regime. So, the total income earned by the young generation is slightly lower in the opaque regime. It is easy to see this property in Fig. 10, as this income is given by the functions F O and F T in each regime. In this example, starting from RO∗ , if agents coordinate on a transition to the transparent regime converging to RT∗ , this can be interpreted as an endogenous financial crisis that leads to some credit shortage. Some projects do no more receive investments and GDP falls. Acknowledgments The authors are grateful to two referees for their insightful remarks. They would like to thank participants at the conference ‘‘Financial and real interdependencies: volatility, inequalities and economic policies’’, May 28–30, 2015 in Lisbon, and particularly Frédéric Dufour, for helpful comments. They are also grateful to participants of the seminar CREM Université de Rennes 1 and of the Atelier ‘‘Methods of economics dynamics’’ in Paris School of Economics. Appendix Proof of Propositions 4–6. We first show that any equilibrium must be of one of the three types presented in Propositions 4–6. Starting from Definition 3, if an equilibrium satisfies conditions 2.a, there is no market for opaque projects. Then, only the good projects v ∈ [R0 + c , v¯ ] reveal information and are financed at the cost R0 . The density hO of opaque projects corresponds to the density h reduced to the interval [v, R0 + c [. There is no market for these projects if the maximal expected gain for a lender is smaller than R0 , or: R0 +c

 v

v h(v) dv < R0 . H (R0 + c )

If an equilibrium satisfies conditions 2.b, we can have either R∗1 > R0 + c or R∗1 = R0 + c. The case R∗1 > R0 + c is impossible as it implies that all the good projects v ∈ [R0 + c , v¯ ] reveal information. Therefore, all the opaque projects such that [v, R0 + c [ makes default and the expected gain for a lender does not rise in increasing R∗1 at a higher value than R0 + c. The only possible equilibrium is then for R∗1 = R0 + c. In this case, the good projects v ∈ [R0 + c , v¯ ] are indifferent between revealing information or

Y. Li, B. Wigniolle / Journal of Mathematical Economics 68 (2017) 127–141

not. They must be shared between opaque and transparent ones in such a way that the expected gain for opaque projects is exactly R0 when R∗1 = R0 + c. This property leads to: R 0 +c

 v

(HO (¯v ) − HO (R0 + c )) v h(v) dv + (R0 + c ) = R0 . HO (¯v ) HO (¯v )

There is an infinite way to satisfy this equality in splitting the good projects between opaque and transparent ones. This equality can be obtained if two conditions are fulfilled: the opaque projects v ∈ [v, R0 + c [ must give an average return smaller than R0 ; if all projects are opaque (hO = h), the average return is not lower than R0 . These two conditions give the inequalities: R 0 +c

 v R 0 +c

 v

v h(v) dv < R0 H (R0 + c )

v h(v)dv + (R0 + c )(1 − H (R0 + c )) ≥ R0 .

Finally, the equilibrium satisfies condition 3 if the expected return is not lower than R0 when all projects are opaque and R∗1 = R0 + c, or: R 0 +c

 v

v h(v)dv + (R0 + c )(1 − H (R0 + c )) ≥ R0 .

Indeed, under this condition there exists a value R∗1 ≤ R0 + c such that: R∗ 1

 v

v h(v)dv + R1 (1 − H (R1 )) = R0 . ∗

∗

For this value, no good project has an interest to reveal information. Considering the conditions for the existence of the different types of equilibria, it is possible to conclude. If the condition  R 0 +c v h(v)dv + R0 (1 − H (R0 + c )) ≥ R0 holds, as this condition v

R

R

+c

If v 0 v h(v)dv + (R0 + c )(1 − H (R0 + c )) < R0 holds, as this  R +c condition implies v 0 v h(v)dv + R0 (1 − H (R0 + c )) < R0 , the only possible equilibrium is the first one, in which only the good projects are financed at the cost R0 (Proposition 5). Finally, if the two following conditions are satisfied,

v



Proof of Proposition 10. In the opaque regime, a stationary state is a value of R < Rˇ such that 1 + β −σ R1−σ = w +

v¯

 v

v h(v)dv − R

(18)

or R + β −σ R1−σ = w +

v¯

 v

v h(v)dv − 1.

The left-hand side is a U-shape function that tends to be infinite in 0 and +∞, with a minimum in R = (σ − 1)1/σ /β . A necessary condition for the existence of a stationary state is that the minimum is below the right-hand side value:

σ (σ − 1) β

1−σ

σ


v¯

 v

v h(v)dv − 1,

and in this case, two steady states may exist. In the transparent regime, a stationary state is a value of R > Rˆ such that [1 − H (R + c )] 1 + β −σ R1−σ



=w+



v¯



v h(v)dv − (R + c ) [1 − H (R + c )] .

(19)

R+c

It is straightforward to see that, for w = 0, R = v¯ − c is a steady state that is stable. For w low enough, this steady state always exists in a neighborhood of R = v¯ − c. 

v h(v)dv + R0 (1 − H (R0 + c )) < R0

v h(v)dv + (R0 + c ) (1 − H (R0 + c )) ≥ R0

Condition (7) is obtained from: GO(Rˇ ) ≤ GT (Rˆ ). If GO (Rˇ ) < GT (Rˆ ), for Rt −1 such that F Σt (Rt −1 ) ∈ GO (Rˇ ), GT (Rˆ ) , there exist

R 0 +c

v R 0 +c

 R∗

cost of credit R∗1 such that v 1 v h(v)dv + R∗1 (1 − H (R∗1 )) = R0 .

Proof of Proposition 11. We first recall that all functions F Σ and GΣ are decreasing, with F O (R) > F T (R) with F O (Rˇ ) = F T (Rˆ ), GO (R) > GT (R) with GO (γ + δ − c ) = GT (γ + δ − c ). Condition from: FO (γ) ≤ GO (R). Under this  condition, (6)O is obtained T F (γ ) , F (γ + δ − c ) ⊂ GO (γ ) , GT (γ + δ − c ) .

 

c )) < R0 , it is impossible to have an equilibrium on this market. So, the resulting equilibrium is the transparent one. Assume now a small decrease in R∗1 , R∗1 = R0 + c − ε , with ε > 0. In this case, all good borrowers leave the transparent market as their total cost on this market is R0 + c > R∗1 . Therefore, all borrowers belong to the opaque market and the cost of credit on this market goes to good projects with v ≥ R0 +c leaves the opaque market. On the opaque market only remain the bad projects with  R +c v < R0 + c. But, as v 0 v h(v)dv + R0 (1 − H (R0 + c )) < R0 , it is impossible to have an equilibrium on this market. So, the resulting equilibrium is the transparent one associated with an equilibrium

+c

implies v 0 v h(v)dv + (R0 + c )(1 − H (R0 + c )) > R0 , the only possible equilibrium is the last one, in which all projects are opaque and are financed at a cost R∗1 < R0 + c (Proposition 4).

139

three types of equilibria may occur: the opaque equilibrium  R +c (similar to the one of Proposition 4 as v 0 v h(v)dv +(R0 + c )(1 − H (R0 + c )) ≥ R0 ), the transparent equilibrium (similar to the one  R +c of Proposition 5 as v 0 v h(v)dv + R0 (1 − H (R0 + c )) < R0 ), and

the multiple equilibria of Proposition 6 with R∗1 = R0 + c. Finally, we prove that the equilibria such that R∗1 = R0 + c are unstable in the sense of the static ‘‘Walrasian tâtonnement’’: a small variation in a price R∗1 or R∗2 leads to a jump in either the transparent or the opaque equilibrium. Assume for instance a small increase in R∗1 , R∗1 = R0 + c + ε , with ε > 0. In this case, all good borrowers earn more if they reveal information as their total cost is R0 + c < R∗1 . Therefore, all good projects with v ≥ R0 + c leave the opaque market. On the opaque market only remain the  R +c bad projects with v < R0 + c. But, as v 0 v h(v)dv+ R0 (1 − H (R0 +

two values of Rt , one in the O regime and one in the  T regime. If GO (Rˇ ) > GT (Rˆ ), for Rt −1 such that F Σt (Rt −1 ) ∈ GT (Rˆ ), GO (Rˇ ) , there does not exist values for Rt and Σt +1 such that F Σt (Rt −1 ) = GΣt +1 (Rt ).  Proof of Proposition 12. The following lemma characterizes the existence and uniqueness of a stable steady state in the opaque regime. Lemma 14. A necessary condition for the existence of a stable steady state in the opaque regime is:

γ < (σ − 1)1/σ /β. √

ˇ there exists Then, when (σ − 1)1/σ /β < γ − c + 2δ c = R, a unique stable steady state RO∗ ∈

γ , Rˇ if w ∈ (w1 , w2 ), with

140

Y. Li, B. Wigniolle / Journal of Mathematical Economics 68 (2017) 127–141

lim RO∗ = (σ − 1)1/σ /β and lim RO∗ = γ . When (σ − 1)1/σ /β > w→w1

γ −c +

w→w2

√





2δ c, there exists a unique stable steady state RO∗ ∈ γ , Rˇ

ˇ Moreover, when if w ∈ (w3 , w2 ), with lim RO∗ = γ and lim RO∗ = R. w→w2

w→w3

RO∗ exists, it decreases with w .

B > B¯ ≡

Proof of Lemma 14. A steady state is solution of R + β −σ R1−σ = w + γ +

δ 2

− 1.

(20)

The left-hand side is a U-shape function that tends to be infinite in 0 and +∞, with a minimum in R = (σ − 1)1/σ /β . This equation may have either zero or two solutions, or one solution in the very special case in which the solution is obtained for the value (σ − 1)1/σ /β . As w ≤ 1 + β −σ γ 1−σ − δ/2 which means that F O (γ ) ≤ GO (R), when two steady states exist, the left one is the stable one. Therefore, we need to express the conditions for the existence of 2 solutions   for Eq. (20), the smaller one RO∗ belonging to the interval γ , Rˇ . This is possible as we assume that γ < (σ − 1)1/σ /β . The  function H (R) ≡ R +β −σ R1−σ on the interval γ , (σ − 1)1/σ /β is

decreasing. It implies that H (γ ) > H (σ − 1)1/σ /β , or w2 > w1 .





Moreover, in the case Rˇ ∈ γ , (σ − 1)1/σ /β , H (γ ) > H (Rˇ ) >





H (σ − 1)1/σ /β , or w2 > w3 > w1 . For w > w1 , there exist 2 solutions for Eq. (20), as the minimum of the left-hand side is smaller than the right-hand side. Moreover, the smaller solution √ ˇ the RO∗ is decreasing with w . If (σ − 1)1/σ /β < γ − c + 2δ c = R, O 1/σ solution R∗ varies from (σ − 1) /β to γ when w increases from





√

1/σ

ˇ the solution RO∗ w1 to w2 . If (σ − 1) /β > γ − c + 2δ c = R, ˇ varies from R to γ when w increases from w3 to w2 .  The following lemma characterizes the existence and uniqueness of a stable steady state in the transparent regime. Lemma 15. There exists a value βl such that, if β < βl , there exists a unique steady state RT∗ in the transparent regime for w ∈ [0, wm ] with

wm =

 (δ − 2c )2 δ − 2c  1 + β −σ (γ + c )1−σ − . δ 2δ

The parameter βl satisfies βl > [2(σ − 1)]1/σ / (γ + δ − c ) .RT∗ is stable and increases with w . Proof of Lemma 15. R ∈



′′ ′ If J ′′ (γ + δ − c ) > 0, this implies that  J (R) > 0 ∀R. Then J (R) is increasing with J ′ (γ + δ − c ) = − 1 + B(γ + δ − c )1−σ < 0. Thus, J ′ (R) is negative and J is a decreasing function. This is true for J ′′ (γ + δ − c ) > 0 or



Rˆ , γ + δ − c is a steady state in the

transparent regime if:

 (γ + δ − R − c )2 (γ + δ − R − c )  1 + β −σ R1−σ = w + . δ 2δ Let us define the function:

 (γ + δ − R − c )2 (γ + δ − R − c )  J (R) ≡ 1 + β −σ R1−σ − . δ 2δ From (9), J (Rˆ ) = J (γ + c ) > 0 and J (γ + δ − c ) = 0. We want to prove that J is a decreasing function when β is small enough. To simplify the notations, we set B = β −σ . We calculate:   δ J ′ (R) ≡ − 1 + BR1−σ − (σ − 1)BR−σ (γ + δ − R − c ) + (γ + δ − R − c ) δ J ′′ (R) ≡ 2(σ − 1)BR−σ + (σ − 1)σ (γ + δ − R − c ) BR−σ −1 − 1. J ′′ (R) is a decreasing function of R and an increasing function of B with J ′′ (γ + δ − c ) = 2(σ − 1)B(γ + δ − c )−σ − 1.

(γ + δ − c )σ . 2(σ − 1)

B > B¯ gives a first upper bound on β equal to [2(σ − 1)]1/σ / (γ + δ − c ): if B > B¯ , J is a decreasing function. But, this condition is very restrictive and it is possible to enlarge the interval of admissible β . If J ′′ (γ + δ − c ) ≤ 0 and J ′′ (γ + c ) ≥ 0, this means that B is ¯ with such that B ≤ B ≤ B, B=

(γ + c )σ +1 . (σ − 1) [2(γ + c ) + σ (δ − 2c )]

In this case, there exists a value Ri ∈ [γ + c , γ + δ − c] such that J ′′ (Ri ) = 0 or 2(σ − 1)BR−σ + (σ − 1)σ (γ + δ − Ri − c ) BRi−σ −1 = 1. i

(21)

From (21), Ri is an increasing function of B, with Ri (B) = γ + c and Ri (B¯ ) = γ + δ − c. We are looking for a condition on B such that J ′ (Ri ) < 0. In this case, J ′ (R) will be negative and J (R) a decreasing function of R. We will now prove that the condition J ′ (Ri (B)) < 0 defines a unique value Bl < B¯ such that J ′ (Ri (B)) < 0 if B > Bl . (21) allows to write B as a function of Ri : B=

Rσi +1

(σ − 1) [2Ri + σ (γ + δ − Ri − c )]

.

Replacing B in the equation J ′ (Ri ) < 0, we get: 0 < 1 + Ri − γ − δ + c +

Ri [Ri + (σ − 1) (γ + δ − Ri − c )]

(σ − 1) [2Ri + σ (γ + δ − Ri − c )]

≡ H (Ri ). [R +(σ −1)(γ +δ−R −c )]

i i It is easy to check that [2R is an increasing function i +σ (γ +δ−Ri −c )] of Ri . Therefore H (Ri ) is increasing with H (γ + δ − c ) > 0. If H (γ + c ) < 0, the equation H (Ri (B)) = 0 defines a unique value Bl of B in (B, B¯ ) and H (Ri (B)) > 0 for B > Bl . If H (γ + c ) > 0, we take Bl = B. In both cases, we obtain that J (R) is a decreasing function of R for B > Bl .

−1

The condition B > Bl is equivalent to β < βl with βl = (Bl ) σ . Finally, for β < βl , ∀w ∈ [0, wm ], there exists a unique steady state RT∗ solution of J (RT∗ ) = w as J is decreasing. This steady state is stable. For w = 0, RT∗ = γ + δ − c and it is stable as dRt +1 /dRt |Rt −1 =γ +δ−c = 0. For w > 0, we

have F T (γ + δ − c ) = w > GT (γ + δ − c ) = 0. Moreover, dRt +1 /dRt |Rt −1 =RT = F T ′ (RT∗ )/GT ′ (RT∗ ) with −F T ′ (RT∗ ) < −GT ′ (RT∗ ). ∗ Therefore, dRt +1 /dRt |Rt −1 =RT < 1.  ∗

Finally, the proof of Proposition 12 results from Proposition 11 and Lemmas 14 and 15.  Proof of Proposition 13. For the first result, we start from the ˆ Indeed, when w = wm , we have property that X (Rˆ , wm ) > R. T ˆ T ˆ F (R) = G (R). Under the assumption that the dynamics is well determined, we have GT (Rˆ ) = GO (Rˇ ). Therefore, H2 (Rˆ ) = GO



−1

◦

ˇ F T (Rˆ ) = R. Now, denoting R = X (Rˆ , wm ) = H1 (Rˇ ), we want to show that ˆ This is equivalent to: GT (R) < GT (Rˆ ) and by definition of R > R. T R, G (R) = F O (Rˇ ). Finally we get: F O (Rˇ ) < GT (Rˆ ) = GO (Rˇ ). This is true as wm < w1 , which implies that ∀R, F O (R) < GO (R). So, we ˆ have proved that X (Rˆ , wm ) > R.

Y. Li, B. Wigniolle / Journal of Mathematical Economics 68 (2017) 127–141

As X (R, w) is a decreasing function of w , we just have to ˆ Such a value increase w till some value w ¯ such that X (Rˆ , w) ¯ = R. w ¯ exists and is the unique solution of the equation in w obtained ˆ in combining Eqs. (14) and (15) for RT = R:

w+

  δ (δ − 2c )2 (δ − 2c ) = 1 + β −σ w + γ + − 2δ 2 δ 1−σ    . × 1 + β −σ (γ + c )1−σ

ˆ the left-hand side of this equation For w = wm , as X (Rˆ , wm ) > R, is smaller than the right-hand side. Increasing w leads to a unique ˆ and RO = H2 (Rˆ ) defines a cycle solution w ¯ . For this value, RT = R, with alternate regimes. For the second result, we want to solve the equation R = X (R, w) in a neighborhood of (w, ¯ Rˆ ). We can use the implicit function theorem when A ̸= 1. When A < 1, the function X (R, w) − R is decreasing in R in a neighborhood of (w, ¯ Rˆ ) and decreasing in w. T ˆ As R must remain higher than R, we define the function on an interval ]w l , w] ¯ , taking its value in [Rˆ , Rl [. R decreases with w . When A > 1, the same method can be used. In both cases, R = X (R, w) defines a stationary cycle, with alternate regimes. It is possible to study the local dynamics around such a cycle. The stability of the cycle is obtained if R is a stable steady state of the dynamics: Rt = X (Rt −1 , w). Therefore, the cycle is stable when A < 1 and unstable when A > 1. 

141

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