Theory of Banking(2004-05)

Marcello Messori

Dottorato in Economia Internazionale April, 2005

Definition of financial intermediaries

• A financial intermediary is:

- Economic agent specialized in selling or purchasing financial contracts/financial assets

• Financial assets can be: - tradeable (shares, bonds, …) - non tradeable before the end of the contract (credits, deposits, …) (New tools: e.g. securitization)

• FIs/Banks: Financial contract (SDC)

Non-existence of financial intermediaries in the GEM

• GEMs: - complete markets; - no asymmetric information; as if…; - full divisibility.

• This makes it possible to design a: risk sharing contract between a lender and a borrower with perfect diversification

• It is the optimal contract (without intermediaries)

Existence of financial intermediaries (I)

Real world (Tobin 1958; Gurley-Shaw 1960):• Incompleteness of markets

transaction costs (Arrow)• Imperfect divisibility

economies of scale, economies of scope.Hence: Empirical explanation for the

existence of financial intermediaries (FIs).Theoretical point of view: FIs still based on

exogenous assumptions (constraints).

Existence of financial intermediaries (II)

• Asymmetries of information (AI) as a first principle (Arrow 1963; Akerlof 1970).

• AI FIs improve market efficiency and lead to the dominant solution (often: second best).

Several models. Here:

Diamond (1984);

Diamond-Dybvig (1983)

Basics on contract theory with asymmetric/imperfect information

• Ex ante AI: Adverse selection; Moral hazard with hidden action;

• Ex post AI: Moral hazard with hidden inform. (costly state verification models).

• Imperfect information: incomplete contracts

Diamond model (1984)

• Assumptions: ≥ mn agents with a monetary endowment = 1/m; n firms, each endowed with an indivisible project; each project ex ante identical with I=1, so that L=1; expected gross return on each project, ỹ, is stochastic (ỹi independent of ỹx V i ≠ x; and f(ỹi) distribution function of yi);

• Ex post firm i (i=1,2,…,n) can observe yi without costs; mn agents can observe yi only with a positive cost K (verification cost); each agent is not endowed with a private information technology.

DM: Form of the contracts

• Debt contract (L=1) = 2 types

SDC with ex post monitoring (1+r) = R if y ≥ R y if y < R;

DC with a non pecuniary cost C (exog.) R if y ≥ R C if y < R

where (by assump.): K < C < m K

DM: Contracts design

• Implementation of the 2 debt contracts:SDC: m agents do not monitor each firm (free riding problem, and C < mK); hence, each firm = incentive to declare y = 0.Hence: DC is the most convenient contract

XI = R if ya ≥ R and R < C XI = C if ya < R or R > C.

• Let assume that both contracts are dominated by a contract between each firm and a FI (delegated monitoring)

DM: FI and SDC contract

• Delegated monitoring: FI prefers SDC to the other contract since nK < nC.

• However, SDC between a given FI and each of the n firms is not sufficient; Also, nm contracts between the FI and nm investors: Each investor is promised RD/m in exchange for a deposit 1/m; if E(XFI) < nRD, the bank is liquidated. Given a “reserve return” of each investor equal to R: RD = E [min (∑ỹi - nK), nRD)] = nR

• Formally:

DM: Expected returns of FI

n

i

R

R

iiiFI dyyfRdyyfKyXE1 0

)()()()(

DM: Expected returns of investors

• E(XI) = min [E(XFI), nRD]

DM: Total cost of delegation

• In case of FI’s bankruptcy:

CT = E (max [nRD – (∑ỹi – nK); 0]).

• Hence: Delegated monitoring more efficient than direct lending if:

nK + CT < nmK.

DM: Why delegated monitoring increases efficiency

• The last condition: nK + CT < nmK (1) is fulfilled if:K < C (by assumption state verification is efficient);

m > 1, and the number n of investors is large enough (diversification by i.i.d.)E (y) > K + R (investments are socially efficient).

• (1) becomes K + CT/n < mK with CT/n 0 since n is large.

Diamond-Dybvig model (1983)

• A simplified version of DD.• Assumptions:

Economy characterized by 1 good and three periods: t=0, t=1, t=2; at t=0 n agents endowed by 1 unity of good and a long-term production technology, whose output is: X1 < 1 in t=1 X2 > 1 in t=2

DDM: information structure

• Two types of agents earlier consumers (1), with C in t=1 and later consumers (2), with C in t=2.

• Utility of type1 agents : U(C1) Utility of type2 agents : t U(C2) where t<1 is a discount factor

• Imperfect information: Agents learn their own type at t=1, but the probability distribution of types (p1 and p2, respectively) is common knowledge at t=0

DDM: agents’ choice set

• Hence, at t=0, the expected utility of agent i (i = 1, 2,…, n) is: Ui = p1 U(C1i) + p2 t U(C2i) with U’(C) > 0, U”(C) < 0.

• At t=0, agent i can choose: (a) to store the endowment so that C1 = C2 = 1 (b) to use the long-term technology so that C1<1 (= X1) but C2>1 (= X2);

• (a) is a dominant strategy for type 1 agents, (b) is a dominant strategy for type 2 agents. Mixed strategy is allowed (1 - I)

DDM: our aim

• We analyze this model in order to show that: the introduction of a FI as a depository institution improvement in the efficiency of the economy.

• Three different institutional structures: Autarky; Market economy; Financial Intermediation.

DDM: analytical setting

(1) Max Ui= max [p1 U(C1i) + p2 t U(C2i)]

s.t. (2) p1C1i = 1 – I

(3) p2 C2i = X2 I

The sum of (2) and (3) leads to

(4) p1C1i + (p2iC2i/X2) = 1

(4) thus becomes the constraint in the max. problem

DDM: FOC

• Given (1) and (4), determination of FOC by means of a Lagrangian

L = p1U(C1i) + p2tU(C2i) + λ [1-p1C1i-(p2iC2i/X2)]

(5) δL/δC1: p1U’(C1i) - λ p1 = 0

(6) δL/δC2: t p2U’(C2i) - λ (p2/X2) = 0.

From (5) and (6):

(7) (U’(C1i)/U’(C2i) = t X2; and then:

(8) U’(C*1) = t X2 U’(C*2) (FOC)

DDM: Autarky

At t=1 (9) C1= 1–I+X1 I = 1–I(1-X1) < 1 if I > 0

At t=2 (10) C2= 1–I+X2 I = 1+I (X2-1) > 1 if I > 0

< X2 if I < 1

Hence, suboptimal consumption: FOC is not fulfilled.

DDM: market economy

• It is sufficient to open a financial market at t=1 where agents can trade goods against a riskless bond (promise to obtain 1 unit of good at t=2)

• Type 1 agents, at t=1, sell the bond X2I at a price pT (≤ 1, to be determined). Hence:

(11) C1= 1 – I + pTX2I

• Type 2 agents, at t=2, purchase the bond (1-I) at a price 1/pT. Hence:

(12) C2= X2I + (1 – I)/ pT = 1/ pT (1 – I + pTX2I)

DDM: market economy

• Given C1= 1 – I + pTX2I (11) C2= 1/ pT (1 – I + pTX2I) (12)

it is possible to obtain: pT = C1/C2

• Moreover, (11) and (12) if pT > 1/X2, then all agents = sellers if pT < 1/X2, then all agents = purchasersHence, equilibrium in financial market requires pT = 1/X2

This C1= 1 (11*) and C2 = X2 (12*)

Autarky v/s market economy

• (11*) and (12*) dominate (9) and (10). But: are (11*) and (12*) compatible with(8) U’(C*1) = t X2 U’(C*2) (FOC) ?

According to DD assumptions (U functions are increasing and concave):

U’(1) > t X2 U’(X2)

Hence: (11*) and (12*) do not fulfill FOC: C1 = 1 < C*1

C2 = X2 > C*2

DDM: Financial intermediation

• (C*1, C*2) can be implemented by a FI which offers a deposit contract subject to a zero-profit condition.

• The contract is: At t=0 n agents deposit their unities of good, and they can get either C*1 at t=1 or C*2 at t=2.

• In order to fulfill this contract (n large enough): FI stores: p1 C*1

FI invests in the long-term technology: n - p1 C*1

DDM: Financial intermediation

• Problem: do later consumers always find it convenient to wait for consumption at t=2? Two conditions: (1) Sound expectations that FI can meet its obligations; (2) C*1 < C*2, that is t X2 ≥ 1 given the concavity of the utility functions and eq. 7: (U’(C1i)/U’(C2i) = t X2

DDM: Financial intermediation

• New assumption: later consumers adopt a strategic behavior. This possibility of bank run (it is a Nash equilibrium).

It is sufficient that a given later consumer has the expectations that other later consumers defect asking for liquidation at t=1.