Theory of Banking ( ) 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 (ỹ i independent of ỹ x V i ≠ x; and f(ỹ i ) distribution function of y i ); Ex post firm i (i=1,2,…,n) can observe y i without costs; mn agents can observe y i 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 X I = R if y a ≥ R and 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 R D /m in exchange for a deposit 1/m; if E(X FI ) < nR D, the bank is liquidated. Given a “reserve return” of each investor equal to R: R D = E [min (∑ỹ i - nK), nR D )] = nR Formally:

DM: Expected returns of FI

DM: Expected returns of investors E(X I ) = min [E(X FI ), nR D ]

DM: Total cost of delegation In case of FI’s bankruptcy: C T = E (max [nR D – (∑ỹ i – nK); 0]). Hence: Delegated monitoring more efficient than direct lending if: nK + C T < nmK.

DM: Why delegated monitoring increases efficiency The last condition: nK + C T < 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 + C T /n < mK with C T /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: X 1 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(C 1 ) Utility of type2 agents : t U(C 2 ) where t<1 is a discount factor Imperfect information: Agents learn their own type at t=1, but the probability distribution of types (p 1 and p 2, 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: U i = p 1 U(C 1i ) + p 2 t U(C 2i ) with U’(C) > 0, U”(C) < 0. At t=0, agent i can choose: (a) to store the endowment so that C 1 = C 2 = 1 (b) to use the long-term technology so that C 1 1 (= X 2 ); (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 U i = max [p 1 U(C 1i ) + p 2 t U(C 2i )] s.t. (2) p 1 C 1i = 1 – I (3) p 2 C 2i = X 2 I The sum of (2) and (3) leads to (4) p 1 C 1i + (p 2i C 2i /X 2 ) = 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 = p 1 U(C 1i ) + p 2 tU(C 2i ) + λ [1-p 1 C 1i -(p 2i C 2i /X 2 )] (5) δ L/ δC 1 : p 1 U’(C 1i ) - λ p 1 = 0 (6) δ L/ δC 2 : t p 2 U’(C 2i ) - λ (p 2 /X 2 ) = 0. From (5) and (6): (7) (U’(C 1i )/U’(C 2i ) = t X 2 ; and then: (8) U’(C* 1 ) = t X 2 U’(C* 2 ) (FOC)

DDM: Autarky At t=1 (9) C 1 = 1–I+X 1 I = 1–I(1-X 1 ) 0 At t=2 (10) C 2 = 1–I+X 2 I = 1+I (X 2 -1) > 1 if I > 0 < X 2 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 X 2 I at a price p T (≤ 1, to be determined). Hence: (11) C 1 = 1 – I + p T X 2 I Type 2 agents, at t=2, purchase the bond (1-I) at a price 1/p T. Hence: (12) C 2 = X 2 I + (1 – I)/ p T = 1/ p T (1 – I + p T X 2 I)

DDM: market economy Given C 1 = 1 – I + p T X 2 I (11) C 2 = 1/ p T (1 – I + p T X 2 I) (12) it is possible to obtain: p T = C 1 /C 2 Moreover, (11) and (12)  if p T > 1/X 2, then all agents = sellers if p T < 1/X 2, then all agents = purchasers Hence, equilibrium in financial market requires p T = 1/X 2 This  C 1 = 1 (11*) and C 2 = X 2 (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 X 2 U’(C* 2 ) (FOC) ? According to DD assumptions (U functions are increasing and concave): U’(1) > t X 2 U’(X 2 ) Hence: (11*) and (12*) do not fulfill FOC: C 1 = 1 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: p 1 C* 1 FI invests in the long-term technology: n - p 1 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 X 2 ≥ 1 given the concavity of the utility functions and eq. 7: (U’(C 1i )/U’(C 2i ) = t X 2

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.