Banking, Capital Markets, and the Macroeconomy by hua83081

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									  Banking, Capital Markets, and the Macroeconomy


                            Hans Gersbach                             Jan Wenzelburger

        CER-ETH—Center of Economic Research                      Centre for Economic Research
                    at ETH Zurich and CEPR                             Keele University
                                  ZUE D7                                Keele, ST5 5BG
                     8092 Zurich, Switzerland                          United Kingdom
                          hgersbach@ethz.ch                     j.wenzelburger@econ.keele.ac.uk




                                 Preliminary Version: September 2007


                                                     Abstract

             This paper investigates the macroeconomic consequences of the interplay be-
         tween capital markets and banks. We embed a competitive financial system
         consisting of commercial banks and capital markets into an OLG economy. En-
         trepreneurs whose production is subject to macroeconomic shocks may seek fi-
         nance either from commercial banks or on the capital market which is operated
         by investment banks. We investigate how the accessibility of capital financing
         impact on the allocation of risk, the aggregate variables, and on the volatility of
         output. We discuss two further applications of our setup: how sophistication in
         rating affects the default probabilities of banks and how bank-capital crises can
         be worked out.


         Keywords:          Financial intermediation, macroeconomic risks, banking crises,
                            risk premia, banking regulation.

         JEL Classification: D41, E4, G2.

First version: Jan. 2006, this version: Sep. 2007.
1    Introduction

The role of financial intermediaries and capital markets in aggregate economic activity
has been an important topic addressed in modern literature ever since the early contri-
butions of Gurley & Shaw (1960) and Patinkin (1965). However, embedding financial
intermediaries in a dynamic general equilibrium setting has proved to be difficult since
a well-founded microeconomic model of a banking industry has to be combined with
standard features of a macroeconomic model. The purpose of this paper is to develop a
model that allows to investigate the macroeconomic consequences of the interplay be-
tween commercial banks and capital markets which are operated by investment banks.


The paper considers an overlapping generation model in which each generation consists
of two types of private agents, consumers and entrepreneurs, who live for two periods.
Entrepreneurs can undertake production only if they obtain additional resources. In
each period, the output of entrepreneurs is subject to a macroeconomic productivity
shock. The quality of the investment project of a particular entrepreneur is private
information and investment decisions of entrepreneurs are not contractible. As a conse-
quence, private agents who offer resources to a particular entrepreneur face a combined
adverse selection and moral hazard problem.

Financial intermediaries with different specializations can alleviate contractual fric-
tions between lenders and borrowing entrepreneurs. Commercial banks alleviate moral
hazard of entrepreneurs. They offer deposit and loan contracts. If a loan contract be-
tween a commercial bank and an entrepreneur comes about, the bank acts as a costly
delegated monitor by increasing the willingness of entrepreneurs to invest after the
contract has been signed. Defaults of entrepreneurs spill over to the balance sheet of
commercial banks and thus may cause their default.

Investment banks operate capital markets and act as screeners by assessing the quality
of investment projects. The costs have to be borne by entrepreneurs who seek funds
from the capital market. Investment banks publish the result of the assessment if it is
sufficiently favorable and private agents are invited to lend to those entrepreneurs for
whom investment banks issue a debt contract. Defaults of such entrepreneurs do not
affect the balance sheet of investment banks as private lenders bear the credit risk.

This paper studies the macroeconomic consequences of such an economy with the focus
on the interplay between capital markets and commercial banks. We first characterize
the temporary equilibria in such an economy. In a temporary equilibrium, the pool of


                                          2
entrepreneurs in each generation is endogenously divided into three groups. The first
group of entrepreneurs, who has low-quality investment projects, invests its resources in
the capital market and renounces own production. The second group of entrepreneurs,
who has intermediate-quality projects, seeks loans from commercial banks. The third
group of entrepreneurs, who has high-quality projects, obtains financing in the capital
market. Consumers typically save by depositing their funds at commercial banks.
Equilibrium in the market for commercial banks determines their risk premia.

We compare an economy in which capital-market financing is developed and investment
banks are operating with an economy in which capital-market financing is not developed
such that no investment banks operate. We show that deposit interest rates as well
as the average return on equity of commercial banks is higher in the economy without
investment banks. In the presence of investment banks the loan volume decreases as
fewer entrepreneurs apply for loans. Moreover commercial banks will lose a portion of
their low-risk investments an will attract less equity when they have to compete with
investment banks. This means that their capital buffer against adverse productivity
shocks may shrink in the presence of investment banks. While entrepreneurs with high
quality projects get more favorable loan contracts from investment banks, entrepreneurs
with intermediate-quality projects face a higher default risk as loan interest rates at
commercial banks rise. We will also show that aggregate output in an economy without
investment banks is on average higher, while the volatility of output when measured by
its coefficients of variation remains unchanged. We conclude by outlining how a crisis
of the commercial banking system affects future generations, how such crises may be
prevented and how they can be worked out.

The paper is organized as follows. In the next section we discuss related literature. In
section three we set out the model. In section four we define a temporary equilibrium
whose existence is studied in section five. In section six, we discuss how the accessibility
of capital market financing impacts on the macroeconomy. Section seven concludes with
a discussion of two applications of our setup.


2     Relation to the Literature

The goal of the paper is to develop a macroeconomic model with commercial banks
and capital markets. Our investigation is motivated by four strands of the literature.
First, our paper departs from earlier business cycles models with financial intermedia-
tion with the contributions of Boyd & Prescott (1986) and Williamson (1987) and the



                                            3
more recent approaches by Schreft & Smith (1997). We extend earlier work Gersbach
& Wenzelburger (2007a) in which risk premia for a commercial banking system are
endogenized in a macroeconomic setting by introducing capital markets which coexist
with the banking system. Second, a wealth of research has addressed the co-existence
of bank-lending and bond-financing from a microeconomic perspective. Well-known
contributions are Besanko & Kanatas (1993), Hoshi, Kashyap, & Scharfstein (1993),
                                                               o
Chemmanur & Fulghieri (1994), Boot & Thakor (1997), Holmstr¨m & Tirole (1997),
von Thadden (1999), Repullo & Suarez (2000), Bolton & Freixas (2000), and Allen &
Gale (2004). A precursor of our own microeconomic model is Gersbach & Uhlig (1999).
An embedding of these models into a dynamic macroeconomic setting is missing.
Third, our approach to banking crises is different from a wealth of research that has
concentrated on bank runs, pioneered by Bryant (1980) and by Diamond & Dybvig
(1984). These static models of intertemporal liquidity have been extended to the realm
of dynamic macroeconomic models. In particular, Bhattacharya & Padilla (1996) show
that financial intermediaries can no longer improve on the stock market equilibrium.
The focus of our model is different. Bank runs cannot occur, since deposit contracts
last for one period only and informational externalities are absent. There is no uncer-
tainty about deposit withdrawals. In our set-up, a banking crisis is caused by negative
macroeconomic shocks which may leave commercial banks with insufficient capital to
carry out the intermediation services for the next generation. A collapse may occur,
unless intervention takes place.
Fourth, the issue of banking regulation in the presence of macroeconomic shocks has
received relatively little attention. One important contribution is Blum & Hellwig
(1995), who have shown that strict capital adequacy rules may reinforce macroeconomic
fluctuations. Some countries have responded to banking crises and associated economic
downturns by lowering short-term interest rates, as advocated by Hellman, Murdock
& Stiglitz (2000) and others. The most prominent empirical example is Japan. A
central policy issue, therefore, is to what extent capital requirements or interest-rate
intervention rules are appropriate measures to prevent or to resolve banking crises. We
will discuss this question in one of our applications.


3    The Model

Consider an overlapping generations (OLG) model with one physical good that can be
used for consumption and investment. Time is infinite in the forward direction and
divided into discrete periods indexed by t. Each generation consists of a continuum



                                           4
of agents with two-period lives, indexed by [0, 1]. Each agent of each generation re-
ceives a stochastic endowment w ∈ [w, w] of goods when young and none when old.
Generations are divided into two classes. One fraction of agents, indexed by [0, η] are
potential entrepreneurs, the other fraction, indexed by (η, 1], are consumers. Potential
entrepreneurs and consumers differ in that only the former have access to investment
technologies. The endowments are driven by an exogenous stochastic process which
will be specified in the sequel.1
Consumers are endowed with intertemporal preferences over consumption in the two
periods of their lives. Let u(c1, c2 ) be a standard intertemporal utility function of a
                               t t
                   1 2
consumer, with ct , ct denoting youthful and old-age consumption of a consumer born
in period t, respectively. Given the endowment w when young and a deposit interest
rate r d , each young household saves the amount s(w, r d). Aggregate savings of all
households are then denoted by S(w, r d) and given by S(w, r d) = (1 − η)s(w, r d).
Potential entrepreneurs are assumed to be risk-neutral and consume only when old.
Each entrepreneur decides whether to save her endowment or to invest into a production
project that converts period-t goods into period-t + 1 goods. The loan size required
for an investment project is fixed to I and each entrepreneur is required to invest all
of her initial endowment w in order to obtain a loan contract.
Entrepreneurs are heterogeneous in the sense that the quality of their investment
projects depends on their index i. The quality parameter of entrepreneur i is assumed
to be a private signal given by 1 + i. If an entrepreneur of type i obtains additional
resources I and decides to invest, her output y in the next period is determined by

                                   y = q (1 + i)f (w + I),                                  (1)

where f denotes a standard atemporal neoclassical production function and q ≥ 0
describes the macroeconomic shock affecting the productivity of the entrepreneurs and
thus causing fluctuations of aggregate output.


3.1     Agency Problems and financial institutions

Throughout this paper we make use of the fact that financial institutions tend to
specialize in activities such as alleviating moral hazard problems or reducing adverse-
selection problems. Commercial banks, for example, are typically specialized in mon-
itoring activities. This includes the capability of inspecting a firm’s cash flow when
  1
    The endowment may be thought of being obtained from short-term production with inelastically
supplied labor.




                                               5
customers pay or the ability of collateralizing assets which are created in a firm’s process
of investing or selling products. Other financial institutions, such as investment banks,
are typically specialized in acquiring knowledge about certain industries in which bor-
rowers are engaged. They are therefore capable of assessing the credit worthiness of
an investment project.
In our model a creditor who in period t provides an entrepreneur with funds I faces
a combined adverse selection and moral hazard problem. Firstly, he does not know
the quality i of the entrepreneur’s project and thus it may be unclear whether the
entrepreneur can pay back even if he invests. Secondly, the entrepreneur could decide
not to invest and simply try to consume his funds. We assume that such a shirker can
consume wt + I − zI in the next period. The term zI, 0 ≤ z ≤ 1, denotes the funds
that a shirker loses when attempting to divert resources. We allow z to be positive
which, for example, is justified when immobile resources such as land are financed with
the loan. Moreover, z may represent transportation and storage costs when resources
are diverted from investment projects for consumption purposes.
To alleviate these two agency problems in financial contracting, we assume that two
types of stylized banks are operating, commercial banks and investment banks.


  1. Commercial banks. These banks act as delegated monitors in the sense of Di-
     amond (1984) and are owned by entrepreneurs. Each commercial bank j (j =
                                                 d                d
     1, . . . , n) can sign deposit contracts D(rtj ), where 1 + rtj is the repayment of-
     fered for 1 unit of resources that bank j receives in period t. Loan contracts of
                                     b                  b
     bank j are denoted by C(rtj , I), where 1 + rtj is the repayment required from
     entrepreneurs for 1 unit of funds. All deposit and loan contracts last for one
     period. Commercial banks carry out two monitoring tasks:

       (a) Securing investments. Given a loan contract of size I, each commercial
           bank can secure a repayment of λb I from an entrepreneur who does not
           invest, where z < λb ≤ 1. This monitoring task costs β ≥ 0 units of the
           consumption good and has to be paid before the bank observes any shirking
           of entrepreneurs. The private benefits of a shirking entrepreneur are thus
           reduced to
                                       wt − β + I − λb I.                        (2)
           Here, monitoring may take several forms. For instance, banks could col-
           lateralize parts of the credit, or may release the funds sequentially upon
           observing the actual investment of an entrepreneur.



                                            6
       (b) Securing liquidation values. The second monitoring task of commercial
           banks is the verification of output. Commercial banks are capable of secur-
           ing the repayments if entrepreneurs invest. Monitoring in order to secure
           repayments takes many forms: inspection of firms’ cash flow when customers
           pay or collateralization of assets which have been created in the production
           process as, for example, the products which are to be sold to customers. For
           simplicity, we assume that the costs of verifying cash flows are zero if the
           entrepreneur has invested.

       The role of commercial banks in ensuring investment activities promised by bor-
       rowers and in preventing funds from being diverted is well-documented in the
       empirical literature (e.g., see James (1987), Lummer & McConnell (1989), Pe-
       tersen (2004) and Berger & Udell 2002).
       By setting λb = 1, we will assume that commercial banks can completely eliminate
       the shirking problem. In this case depositing funds is always better than shirking
       with borrowed funds.2 Moreover, we will assume throughout the remainder of
       the paper that β = 0 and that the costs of securing the liquidation value are
       zero.3 These assumptions are made for ease of exposition as we will focus on
       the relative costs between capital-market and commercial-bank financing. Notice
       that commercial banks still face the default risk of entrepreneurs who may fail
       to produce the promised repayment as their quality in relation to the realized
       macroeconomic shock may be too low.
       The justification for the contractual arrangement between entrepreneurs and com-
       mercial banks is provided in Gersbach & Uhlig (2006) who show that debt con-
       tracts arise endogenously in such models when banks compete for borrowers. In
       this paper we therefore work directly with debt contracts.

   2. Investment banks. There is a finite number of investment banks which operate
      the capital markets. In contrast to commercial banks, investment banks can only
      interact with entrepreneurs at the stage when debt contracts are issued and do
      not engage in continuous monitoring of investment behavior. Investment banks
      are owned by entrepreneurs and are specialized in alleviating adverse-selection
      problems in financial markets. For simplicity, we assume here that they act also
   2
      This assumption is tighter than necessary when deposit rates are positive. An interesting variant
is the case with low values of λb . Then banks face default risks from two sources: non-investment of
entrepreneurs and investing entrepreneurs with low quality.
    3
      These costs can easily be introduced into our model. Notice that β > 0 could always be incor-
porated by deducting the monitoring costs in the profit function of commercial banks. This would
widen the equilibrium interest rate margin.



                                                  7
        as rating agencies which can detect the credit worthiness of an entrepreneur by
        investing γ ≥ 0 per credit. Investment banks first rate the credit worthiness of
        entrepreneurs and then decide whether or not to issue debt contracts on behalf
        of entrepreneurs.
        We assume in particular that investment banks will issue debt contracts only
        for those entrepreneurs who will be able to pay back lenders with certainty.
        This involves two conditions: First, no further monitoring is required when an
        entrepreneur invests. Second, an investing entrepreneur is always better off by
        paying back than by defaulting. The formalization of these conditions will be
        given in the next section.
        Entrepreneurs are charged with the costs of the credit-worthiness test up front
        so that they need to pay the test out of their initial wealth wt . The costs γ
        include advertisement costs as well as placement and settlement costs. Different
        levels of γ may thus be associated with different levels of financial development.
        High levels of γ correspond to economies in which capital-market financing is less
        developed. Low levels of γ correspond to economies with fully developed financial
        markets.
        The use of debt contracts has to be justified for capital markets as well. Credit-
        worthiness tests will reveal the quality of entrepreneurs’ investment projects.
        Since entrepreneurs are risk neutral and repayments are expected to be certain,
        debt contracts are an optimal arrangement between lenders and borrowers in
        capital markets.4


3.2       The sequential structure

Let us first outline the sequential nature of the economy within a time period. The
timing in the intermediation part within a typical period t is as follows:


  1. Old entrepreneurs pay back with limited liability. Banks pay back old depositors.
     Bank capital is realized and distributed among old shareholders.
                 d        b
  2. Given wt , rt , and rt , young entrepreneurs decide whether to invest in equity of
     commercial banks, in the capital markets, or to apply for loans at commercial or
     investment banks to finance their production project. Consumers decide whether
     to take deposits or to invest in capital markets. Funded commercial banks decide
  4
      Other arrangements are also conceivable.




                                                 8
      on offering their intermediation services. Investment banks decide on offering
      their services.

    3. Resources are exchanged.

    4. Funded entrepreneurs produce subject to a macroeconomic shock.

We assume that deposit rates are guaranteed for all possible macroeconomic shocks.
This assumption is adopted for two reasons. First, often deposits are insured by gov-
ernments or may be implicitly insured by bail-out incentives of the future generation.
Second, without deposit insurance the willingness to lend may decline when the bank-
ing system faces a positive default probability and thus could accelerate an actual
default. Hence the assumption that deposits are insured by the next generation is
most favorable for a commercial banking system.


4     Temporary Equilibria

All entrepreneurs are price takers who operate under limited liability. Their technology
is described as follows. The expected profit of entrepreneur i conditional on wt at an
               b
interest rate rt is
                                     q
                 b                                                       b
           Π(i, rt , wt , I) =           max{q(1 + i)f (wt + I) − I(1 + rt ), 0} h(q)dq.   (3)
                                 q

Here h denotes the probability distribution of the macroeconomic productivity shocks.
                                                                                    b
Note that Π is non-decreasing in quality levels i and non-increasing in loan rates rt .
Entrepreneurs in a typical period t have the following options. First, they can deposit
                                                            d                        d
their funds at commercial banks with the interest rate rt which yields wt (1 + rt ) in
the subsequent period t + 1. Second, they can supply their funds in the capital market
                     c                  c
with interest rate rt yielding wt (1 + rt ) in the subsequent period t + 1. Third, they
can invest in equity of commercial banks. Fourth, they may apply for a loan contract
    b
C(rt , I) at a commercial bank that offers intermediation services. Having received a
                  b
loan contract C(rt , I), entrepreneurs will have to decide whether to produce or to shirk.
Their production is subject to a macroeconomic shock, the expected profit being given
by (3). Since we assume λb = 1, an entrepreneur can consume wt + (1 − λb )I = wt
if he decides to shirk. This implies that only those entrepreneurs apply for loans at
commercial banks for whom
                                               b                      d
                                         Π(i, rt , wt , I) ≥ wt (1 + rt ).                 (4)


                                                        9
Finally, entrepreneurs can apply for a credit-worthiness test at investment banks. In-
vestment banks will issue debt contracts on behalf of an entrepreneur by pledging his
capacity to pay back lenders only if full repayments are certain. We assume that in-
                           a
vestment banks compete ` la Bertrand for screening entrepreneurs. As entrepreneurs
are charged up front with the costs of the test, the operation is risk-less for investment
banks and the competitive equilibrium price of the test will be equal to its costs γ so
that their profits will be zero.5 Investment banks are willing to perform such tests for
any entrepreneur who applies, since they are payed in advance.
The situation for entrepreneurs is as follows. If investment banks issue a debt of size
I c on behalf of the entrepreneur, his expected profit is
                                       q
             c
       Π(i, rt , wt − γ, I c ) =                                                     c
                                           max{q(1 + i)f (wt − γ + I c ) − I c (1 + rt ), 0} h(q)dq,   (5)
                                   q

       c
where rt is the prevailing interest rate in the capital market. Hence, an entrepreneur has
an average consumption of (5) if he decides to invest or he can consume wt −γ +(1−z)I c
with 0 ≤ z ≤ 1 if he decides not to invest.
Given their options and knowing that investment banks will only issue debt contracts
if repayment is certain, entrepreneurs will seek financing in capital markets through
investment banks only if the following three conditions hold simultaneously:

   1. No-fraudulent default condition. An entrepreneur with a positive default risk
      could always claim that his project has been unsuccessful. To rule out such
      fraudulent defaults, investment banks will only admit entrepreneurs whose default
      risk is zero. In view of (5), these entrepreneurs must have a quality level i which
      is above
                             c
                                                                 c
                                                       I c (1 + rt )
                     i(wt , rt ) := max 0, min η,                     −1  .           (6)
                                                   qf (wt − γ + I c )
         Such entrepreneurs could only default by diverting resources.6 Investment banks
         will not issue debt on behalf of entrepreneurs with a positive default risk as
         nobody would offer funds.

   2. No-shirking condition. Investment banks ensure that entrepreneurs will invest
      before issuing debt. Otherwise they will renounce issuing debt as nobody will
   5
      If an investment bank serves a fraction x of entrepreneurs at price p, its profit is (p−γ)x. Bertrand
competition yields p = γ.
    6
      We implicitly assume that a legal system is in place which can penalize entrepreneurs who commit
a fraudulent default with large private costs. An alternative setting would be that investment banks
can verify the output of funded entrepreneurs as well. This would allow for the possibility that
entrepreneurs with non-zero default risk be funded.



                                                          10
      offer funds. The condition that an entrepreneur who obtains funds from the
      capital market will not shirk and invest is
                                     c
                               Π(i, rt , wt − γ, I c ) ≥ wt − γ + (1 − z)I c .               (7)

   3. Capital-market financing is more profitable. Entrepreneur will only apply for
      credit-worthiness tests if investment banks offers better credit terms than com-
      mercial banks. This is the case if the market conditions for an entrepreneur with
      quality level i is such that
                                        c                         b
                                  Π(i, rt , wt − γ, I c ) ≥ Π(i, rt , wt , I).               (8)

Summarizing, only entrepreneurs who meet all three conditions will seek funding from
capital markets. To facilitate the further analysis we assume from now on that the size
of the debt contract issued by investment banks is

                                             I c = I + γ.

This implies that entrepreneurs independently of the source of their funding invest the
total amount wt + I into their production projects.7 Using (6), condition (8) holds for
            c
i ≥ i(wt , rt ) if and only if the costs of borrowing at commercial banks are higher than
the borrowing costs in the capital market or, equivalently, if
                                              b               c
                                       I(1 + rt ) ≥ I c (1 + rt ).                           (9)

We define the critical quality level above which entrepreneurs seek financing from
capital markets by
                                                    c              b              c
                                            i(wt , rt ) if I(1 + rt ) ≥ I c (1 + rt ),
                              b c
               ic = ic (wt , rt , rt ) :=
                t
                                            η           otherwise.

If ic < η, then for any entrepreneur ic ≤ i ≤ η capital-market financing is more
    t                                     t
profitable than a loan from a commercial bank. Condition (9) shows that investment
banks are only in business if the costs of the credit-worthiness test in relation to the
interest rates is sufficiently low so that ic < η.
                                          t

Throughout the paper we assume that the no-shirking condition which is embodied in
(7) will hold. Notice to this end that, given (6), the expected profit of entrepreneur ic
                                                                                       t
is
                       Π(ic , rt , wt − γ, I c ) = E[q] − 1 I c (1 + rt ).
                          t
                               c
                                                    q
                                                                      c


  7
    Alternatively, one could presume that all production projects require the same fixed amount of
                                          c
funds F , so that F = wt + It = wt − γ + It for all periods t.



                                                   11
A sufficient condition that no entrepreneur who receives funding on the capital market
shirks in period t is
                                E[q]
                                 q
                                     − 1 I c ≥ wt + I.                          (10)
We assume (10) to hold throughout the remainder of the paper. Note that (10) is
independent of 0 ≤ z ≤ 1. The investment behavior of entrepreneurs may now be
characterized as follows.
Lemma 1
                 d b c               c    b
For given {wt , rt , rt , rt } with rt < rt there exists ib and ic such that the following holds:
                                                          t      t


  (i) All entrepreneurs with 0 ≤ i < ib become lenders to banks.
                                      t

 (ii) All entrepreneurs with ib ≤ i < ic apply for loans at commercial banks.
                              t        t

 (iii) All entrepreneurs with ic ≤ i ≤ η apply for credit worthiness at investment banks.
                               t


In order to derive a temporary equilibrium, we assume that a perfectly competitive
banking industry with a large number of commercial banks which take deposit and
loan rates as given. However, commercial banks freely decide whether or not to offer
their intermediation services. This polar assumption seems to be best suited to ex-
amine the role of risk premia for the vulnerability of a perfectly competitive banking
system to banking crises.8 Depositors randomly choose a bank that offers its interme-
diation services in order to save. Similarly, entrepreneurs applying for a loan contract
choose banks randomly. Throughout the paper we assume that aggregate uncertainty
is canceled out when depositors and entrepreneurs randomly choose banks. That is,
each active bank obtains the same amount of deposits and loans.9
We assume that savings are never sufficient to fund all entrepreneurs. Since

                  S(w, r d) = (1 − η) s(w, r d) < (1 − η) w for all r d ≥ 0,

this is the case if (1 − η)w < ηI. Otherwise, all entrepreneurs independently could re-
ceive funds independently of their quality. Thus loans are constrained by the amount of
deposits obtained. If entrepreneurs applying for loans were rejected, they will randomly
choose a bank and save. If dt denotes total equity which the commercial banking sys-
tem has acquired in period t, an individual commercial bank has an amount of equity
of dtj = dt as all commercial banks are assumed to be identical.
          n
   8
     The free-entry free-exit framework is a standard concept in industrial economics, e.g., see Vives
(2004) who uses free-entry models to describe innovation activities.
   9
     The exact construction of individual randomness so that this statement holds can be found in
  o
Al´s-Ferrer (1999). We could also rely on the weaker forms of the strong law of large numbers
developed in Al-Najjar (1995) and Uhlig (1996), where independence of individual random variables
can be assumed and aggregate stability is the limit of an economy with finite characteristics.



                                                 12
Commercial banks lend ic − ib I to entrepreneurs which they have to finance out of
                            t   t
                                          d
equity dt they raise and deposits S(wt , rt ) they receive. These funds have to be payed
back with interest in the subsequent period t+1. At the end of period t+1, commercial
banks will receive payments Pt+1 from entrepreneurs who received funds in period t.
These repayments are given by
                                                     ic
                                                      t
     Pt+1 =   P (qt+1 , wt , ib , ic , rt )
                              t t
                                        b
                                              :=                                           b
                                                          min qt (1 + i)f (wt + I), I 1 + rt      di,   (11)
                                                   ib
                                                    t


where qt+1 denotes the productivity shock in period t + 1 Given the endowment wt
                              d b
and a pair of interest rates rt , rt , future equity of the commercial banking system is
therefore determined by the function

                                d b                               b             d        d
                 G(qt+1 , wt , rt , rt ) = P qt , wt , ib , ic , rt ) − S(wt , rt )(1 + rt ),
                                                        t t                                             (12)

                                             d b                       d b
such that for any shock qt+1 and each wt , rt , rt ≥ 0, G(qt+1 , wt , rt , rt ) is the capital
base of the commercial banking system at the beginning of period t + 1 which is payed
out to their equity holders. Their return on equity in period t + 1 then is
                                                                  d b
                                                   G(qt+1 , wt , rt , rt ) − dt
                                      gt+1 =                                    .
                                                              dt

Investment banks lend [η − ic ]I c to those entrepreneurs who successfully applied for a
                            t
credit worthiness test.
A temporary equilibrium with financial intermediation in period t consists of a pair of
                d b
interest rates rt , rt , an equity level dt and a pair of critical quality levels (ib , ic ) such
                                                                                    t t
that

  (i) no commercial bank exits and no bank enters the market;

 (ii) entrepreneurs take optimal decisions;

(iii) investment banks compete in prices and take optimal debt issuance decisions;

 (iv) loan demand equals loan supply.

Denote by
                                                                  q
                                       d b                                       d b
                          E G(· , wt, rt , rt ) :=                    G(q, wt , rt , rt )h(q)dq
                                                              q

the expected capital of the commercial banking system in period t + 1. More formally,
a temporary equilibrium with financial intermediation is defined as follows.



                                                             13
Definition 1
Let dreg > 0 denote the minimum capital level prescribed by some regulatory frame-
work and wt ∈ [w, w] denote the endowment of an agent in period t in a competitive
banking system under full liability as described above, where I c = I + γ. A tempo-
rary equilibrium with financial intermediation (TEFI) in period t consists of an equity
level dt ≥ dreg , a pair of critical entrepreneurs (ib , ic ) with ic > ib , and interest rates
                                                       t t          t    t
  d b c
(rt , rt , rt ) such that the following conditions hold:

                                           d b                   c
                              E G(· , wt, rt , rt )   = dt (1 + rt )                           (13)
                                           b                      c
                                   Π(ib , rt , wt , I) = wt (1 + rt )
                                      t                                                        (14)
                                           d
                                   S(wt , rt ) + dt =     ic − ib I
                                                           t    t                              (15)
                                         ib wt − dt =
                                          t               η − ic I c
                                                               t                               (16)
                                                  d    c
                                                 rt = rt                                       (17)


In aggregate terms, condition (13) is the no-exit and no-entry condition for commercial
banks. The corresponding condition for an individual commercial bank is obtained by
dividing both sides in (13) by the number of banks. It states that the expected return
on commercial-bank equity has to be equal to the interest rate which prevails in the
capital market. Since it is more convenient we will use the aggregate condition (13)
throughout the paper.
Condition (14) states that all entrepreneurs i ≥ ib are borrowers, while all entrepreneurs
                                                  t
                                 b   d
i < ib are lenders. The spread rt − rt represents the premium commercial banks obtain
     t
for bearing macroeconomic risks. Equation (15) states that aggregate demand for loans
 ic − ib I at commercial banks is equal loan supply on the left hand side of equation
  t    t
(15). Equation (16) states aggregate demand for loans η − ic I c at investment banks
                                                                t
is equal to loan supply on the left hand side of equation (16).
Equation (17) reflects the fact that since there is no credit risk for lenders in the
                                                                       c
capital market, the equilibrium interest rate in the capital market rt must be equal
                     d                         c
to the deposit rate rt at commercial banks. rt cannot be lower because otherwise no
entrepreneur would become a lender in the capital market leaving investment banks
without funds. A higher equilibrium interest rate on the capital market would induce all
depositors at commercial banks to switch to the capital market. Moreover, commercial
banks would be better off by investing in the capital market than by granting loans.10
  10
    If the access to the capital market is costly for consumers (or bank deposits provide additional
advantages) and commercial banks cannot switch costlessly from their loan business to investing in
the capital market, deposit rates and capital market rates could differ.




                                                 14
The temporary equilibrium notion of Definition 1 presumes that banks operate under
full liability reflecting a situation in which bank managers have fully internalized the
possibility of an insolvency. The case with limited liability can be defined by replacing
the l.h.s. in (13) with
                                                d b
                                E max G(·, wt, rt , rt ), 0 ,
cf. also Gersbach & Wenzelburger (2007a). However, establishing existence and unique-
ness is technically more difficult and left for future research.


5      Existence of Temporary Equilibria

We first establish the existence of temporary equilibria with financial intermediation
before analyzing their properties. Given an exogneous endowment wt ∈ [w, w], a
temporary equilibrium with financial intermediation is completely described by an
                                               d b
equity level dt and a pair of interest rates (rt , rt ). Notice to this end that the two
resource constraints (15) and (16) in Definition 1 can be rearranged to yield

                                   ηI − S(wt , rt ) γ[η − ic (wt , rt , rt )]
                                                     d               d b
                       d b
        ib = ib (wt , rt , rt ) :=
         t                                             +                      ,                  (18)
                                        wt + I                wt + I
                                                         wt + I + γ
       dt             d b                   d b
            = d(wt , rt , rt ) := ib (wt , rt , rt )wt −                             d b
                                                                       η − ic (wt , rt , rt ) I. (19)
                                                           wt + I

It is straightforward to verify that ib ≤ ib < η and dt ≤ ib wt , where the lower bound
                                            t               t
     b
for it is given by
                                         ηI − (1 − η)w
                                   ib :=               .
                                              w+I
Assume for the remainder of the paper that the productivity of the entrepreneur with
quality level ib is on average greater than unity, that is,

                                      E[q](1 + ib )f (w + I)
                                                             > 1.
                                                I
The following existence theorem is the foundation of our further analysis.
Theorem 1
Consider a competitive banking system operating under full liability with minimum
capital requirement dreg > 0. Let the following conditions be satisfied:

    (i) The average productivity of the entrepreneur with quality level ib is greater than
        unity and there exists an interest rate rd with

                                                E[q](1 + ib )f (w + I)
                                     0 < rd <                          −1                       (20)
                                                          I

                                                   15
       such that

                   Π(0, 0, w, I) > w(1 + r d ) > Π(η, rd , w, I) for all w ∈ [w, w].

 (ii) The distribution of shocks satisfies

                                         E[q]     w+I
                                              −1>      .
                                          q       I +γ

Then for each w ≤ wt ≤ w, there exists a unique temporary equilibrium with financial
                                                                              b    d
intermediation (TEFI), given by an equity level dt ≥ dreg and interest rates rt > rt > 0.


The proof of Theorem 1 is given in the appendix. Condition (i) states that all en-
trepreneurs will prefer to invest into production at a zero loan interest rate given that
the deposit rate is below the upper limit rd . All entrepreneurs prefer to save or invest
in the capital market if deposit and loan interest rate exceed r d . Condition (ii) is the
no-shirking condition. An equity level dt above some prescribed regulatory level dreg
can always be achieved be allowing commercial banks to invest excess funds in the
capital market.
Having established existence and uniqueness of a temporary equilibrium, we discuss its
properties in the next section.


6      The Impact of Capital-Market Financing

In this section we examine how the easiness of capital market financing impacts on the
allocation of risk, the aggregate variables, and on the volatility of output.


6.1      Conditions for capital market financing

Our first result is a simple characterization.
Lemma 2
Under the hypotheses of Theorem 1 the following holds true.

                                                                          d            b
    (i) If the cost γ of credit worthiness test is such that (I + γ)(1 + rt ) < I(1 + rt ) in
        a temporary equilibrium of period t, then investment banks operate with a loan
        volume [η − ic ](I + γ) > 0.
                      t




                                               16
                                                                        d            b
 (ii) If the cost γ of credit worthiness test is such that (I + γ)(1 + rt ) ≥ I(1 + rt ) in
      a temporary equilibrium of period t, then investment banks do not operate, i.e.,
      ic = η.
       t


The proof of Lemma 2 follows directly from the proof of Theorem 1. Since the interest-
              b    d
rate margin rt − rt is always positive, it follows from Lemma 2 that investment banks
will operate if the costs of the credit worthiness tests γ are sufficiently close to zero.
However, in economies in which such tests are unavailable or too costly, that is, for a
high γ, commercial banks operate without the presence of investment banks.


6.2     Interest rates and investments

We are now in a position to compare the two situations, one in which investment
banks operate and another in which no investment banks operate. To simplify the
analysis, we consider two economies, one in which the costs of credit worthiness tests
are zero, γ = 0, and investment banks are allowed to operate and one in which the
costs of credit-worthiness test γ are prohibitively high so that investment banks do not
operate. This occurs for sufficiently high γ for which i(w, r c) as given in (6) is always
equal to η. It then follows from (18) that in both cases the critical entrpreneur ib who
                                                                                   t
is indifferent between lending and borrowing is given by
                                                         d
                                            ηI − S(wt , rt )
                                    ib =
                                     t                       .                             (21)
                                               wt + I
                                d b
For a given wt denote by (dt , rt , rt ) a temporary equilibrium of the first economy with
                            ˜ ˜d ˜b
investment banks and by (dt , rt , rt ) the corresponding temporary equilibrium of the
second economy without investment banks. As a first result we obtain the following.
Theorem 2
Under the hypotheses of Theorem 1, let w ≤ wt ≤ w be a given random endowment in
                                        d b           ˜ ˜d ˜b
an arbitrary period t. Denote by (dt , rt , rt ) and (dt , rt , rt ) the corresponding temporary
equilibria with and without investment banks, respectively. Then the following holds:

                                 d
                                     ˜d
                            (i) rt < rt ,                    b
                                                                 ˜b
                                                       (ii) rt > rt .

The proof of Theorem 2 is given in the appendix. It shows that in the presence of
investment banks deposit interest rates and hence the average return on equity of
commercial banks will decrease while loan interest rates will rise. The intuition of
this result is as follows. If high-quality entrepreneurs may apply of loan contracts
at investment banks, the loan volume of commercial banks will shrink. Hence their


                                                17
demand for funds will decrease so that in equilibrium deposit interest rates and the
return on equity must be lower. On the other hand, as will be shown in the proof of
the theorem, the expected return on loan volume of commerical banks will decrease
if they lose high-quality entrepreneurs. To compensate for this decline, commercial
banks require higher loan interest rates in equilibrium.
Under the assumption that the aggregate savings function S(w, r d) is non-decreasing
in deposit interest rates r d , Theorem 2 (i) implies that in any temporary equilibrium
                                 d        d
                                                       ˜d       ˜d
                         S(wt , rt )(1 + rt ) < S(wt , rt )(1 + rt ).                      (22)

Hence the liabilities of commercial banks in an economy with investment banks de-
crease. At the same time depositors and equity holders of commercial banks are worse
off in the presence of investment banks. The second results states that loan interest
rates at commercial banks increase in the presence of investment banks. Together with
the first result this implies that commerical banks receive higher interest-rate margins
when investment banks do not offer their services.
Using Theorem 2, the next result compares the investment behavior of entrepreneurs
in both economies.
Theorem 3
Under the hypotheses of Theorem 1, let w ≤ wt ≤ w in an arbitrary period t be given.
Then the following holds true:

                                                           ˜            ∂S
 (i) ib ≥ ˜b ,
      t   it      (ii) η − ˜b > ic − ib ,
                           it    t    t         (iii) dt < dt     if       (wt , r) = 0, r ≥ 0.
                                                                        ∂r

The proof of Theorem 3 follows directly from Theorem 2 and the definition of the
critical entrepreneur (21) and is given in the appendix. The first result of Theorem
3 states that fewer entrepreneurs will apply for loans in an economy with investment
banks. This entails three consequences. Firstly, the loan volume in an economy with
investment banks will decrease. Secondly, fewer low quality entrepreneurs will receive
loans from commercial banks. The second result in Theorem 3 implies that the loan
volume at commercial banks will be lower in the presence of investment banks and that
they will lose a portion of their low-risk investments to investment banks. By Theorem
2, commercial banks are thereby forced to increase the default risk of intermediate-
quality entrepreneurs who are confronted with higher loan interest rates. The third
result states that commercial banks will attract less equity when they have to compete
with investment banks because these require funds as well. The result is stated for
the case in which interest-rate elasticity of consumers is zero but in view of (19) will
prevail for low interest-rate elasticities.


                                             18
6.3     Default of commercial banks

As a consequence of Theorem 3 (iii), the capital buffer against adverse shocks of com-
mercial banks may shrink in the presence of investment banks. A smaller capital buffer
increases the default risk of the commercial banking system as less funds are available
to compensate for losses due to potential bankruptcies of entrepreneurs.
Whether or not the default probability of the commercial banking system will increase
or decrease when investment banks operate is yet ambiguous. The no-entry no-exit
condition (13) together with the balance condition (15) implies

                           E P (· , wt, ib , ic , rt ) = I[ic − ib ] (1 + rt )
                                         t t
                                                   b
                                                            t    t
                                                                           d



for the economy with investment banks and

                            E P (· , wt, ˜b , η, rt ) = I[η − ˜b ] (1 + rt )
                                         it      ˜b           it        ˜d

for the economy without investment banks. It follows from Theorem 2 (ii) and Theorem
3 (ii) that
                    E P (· , wt, ib , ic , rt ) < E P (· , wt, ˜b , η, rt ) ,
                                  t t
                                            b
                                                               it      ˜b
so that expected repayments to commercial banks are higher when no investment banks
operate.
Analogously to (12), the capital base of the commercial banking system in an economy
without investment banks is determined by the function

                 ˜
                 G(qt+1 , wt , rt , rt ) = P qt , wt , ˜b , η, rt ) − S(wt , rt )(1 + rt ),
                               ˜d ˜b                   it      ˜b            ˜d       ˜d        (23)

where qt+1 is some macroeconomic shock and wt is some realization of the endowment
process. It follows from Theorem 2 (i) and Theorem 3 (iii) that at least for sufficiently
inelastic savings functions
                        d b                 d     ˜       ˜d        ˜             ˜d ˜b
      Et G(qt+1 , wt , rt , rt ) = dt (1 + rt ) < dt (1 + rt ) = Et G(qt+1 , wt , rt , rt ) .   (24)

This means that a commercial banking system will on average accumulate more capital
when investment banks are absent. Equation (24) reflects that fact that commercial
banks will lose the profits from high-quality entrepreneurs when investment banks are
in business.
A default of the commercial banking system will occur if the future capital as deter-
mined by (12), respectively (23) is negative. It is straightforward to verify that

                               P (q, wt, ib , ic , rt ) < P (q, wt, ˜b , η, rt )
                                          t t
                                                    b
                                                                    it      ˜b                  (25)


                                                     19
for all low shocks q > 0 which cause a sufficiently high portion of entrepreneurs to de-
fault. In other words for low productivity shocks, equation (25) states that repayments
to commercial banks are lower in the economy with investment banks. However, since
by (22) the liabilities of commercial banks in such an economy are lower as well, the
effect on the default probability of the commercial banking system is ambiguous.


6.4    Aggregate output

In view of the aggregate consequences of the presence of investment banks, let us com-
pare aggreate output of both economies. After ecountering the shock qt+1 , aggregate
period-(t + 1) output of the economy with investment banks is given by
                                                              η
                            Yt+1 = qt+1 f (wt + I)                (1 + i)di
                                                         ib
                                                          t

while aggregate period-(t + 1) output of the economy without investment banks is
                                                          η
                            ˜
                            Yt+1 = qt+1 f (wt + I)            (1 + i)di.
                                                         ˜b
                                                         it

The last theorem of this paper now states that aggreate output in an economy without
investment banks is always higher, while the volatility of output when measured by its
(conditional) coefficients of variation will remains unchanged. Formally, we have the
following.
Theorem 4
Under the hypotheses of Theorem 1, let w ≤ wt ≤ w be given. Then the following
holds true:
                               ˜
                   (i) Yt+1 ≤ Yt+1 for all shocks qt+1 ∈ [q, q],
                                            ˜
                  (ii) Vart [Yt+1 ] ≤ Vart [Yt+1 ],
                                     ˜
                              Vart [Yt+1 ]          Vart [Yt+1 ]
                    (iii)                  =                     .
                                  ˜
                              Et [Yt+1 ]            Et [Yt+1 ]

The proof of Theorem 4 is given in the appendix. The results of the theorem exclusively
caused by a volume effect and the multiplicative structure of the production technol-
ogy (1). Since by Theorem 3 more entrepreneurs are financed in a system without
investment banks, aggregate output is higher.


6.5    Income distributional effects

When captial markets are effective, a variety of distributional effects occur. While
high-quality entrepreneurs benefit form investment banking as their costs of financing


                                               20
are lower, intermediate-quality borrowers are worse off as they are charged higher
loan interest rates. Moreover, consumers and low-quality entrepreneurs who become
equity holders of commercial banks are worse off as the return on savings and bank
equity declines. The model thus illustrates how capital markets may be responsible
for a income distribution which is more unequal than in an economy in which only
commerical banks finance production.
The finance literature points out a number of undoubted advantages of capital-market
financing over bank loans. The model in this paper, however, demonstrates that it is
by no means granted that every agent in the model is better off when capital-market
financing is present. Despite the fact that capital-market financing enlarges the set of
available monitoring techniques, aggregate production may decline.


6.6    Sophisticated banks

It is worthwhile to discuss a more sophisticated financial intermediary who has access
to both the monitoring technology of a commercial bank and the monitoring technology
of an investment bank. Such an universal bank may behave like a commercial bank, like
an investment bank, or it may apply both technologies on an individual entrepreneur.
If the costs of screening are sufficiently low, an universal bank behaves like an in-
vestment bank for high-quality borrowers as alleviating moral hazard is not needed.
They will test the credit worthiness on all other borrowers applying for a loan contract
and will monitor them like a commercial bank. In such a system, the separation of
markets will take place internally within the universal bank. However, the negative
effects of capital-market financing in such a scenario will be the same, as the profits on
high-quality borrowers of the univeral bank remain zero.


7     Conclusions and Outlook

We investigated the macroeconomic effects of a mixed system with investment banks
operating capital markets and commercial banks. Numerous issues deserve further
attention which we discuss in this section.
Until now there is no dynamics in the model as we have developed an overlapping gen-
eration model in which investment and trading takes place within the same generation.
Two sources of dynamic linkages can easily be introduced into the model. Firstly, as
deposits are implictly guaranteed by institutional arrangements, a future generation
will bear at least some of the costs of a commercial bank bail-out should commercial


                                          21
banks default. The costs of such a bail-out would reduce their intial endowments.
Secondly, there may be altruistic behavior of current owners of banks with regard to
their children in the sense of a warm glow, in which individuals derive utility from
the bequest to their heirs. As a consequence, commercial bank capital in a particular
period depends on past bequest decisions. In a future refinement we will introduce
a third dynamic linkage by by integrating a labor market into the model so that the
endowment process is endogenized.
The special case of the present model without investment banks is covered in Gersbach
& Wenzelburger (2007a) in which a dynamic linkage is modeled and government in-
tervention to work out commercial banking crises are examined. There we show that
without intervention, the commercial banking system runs the risk to collapse. This
risk is caused by the asymmetric impact of repeated macroeconomic shocks. Adverse
shocks may cause a decline in the repayment capacity of firms and cause credit losses
for commercial banks. Repeated bad spells may cause a downward spiral of bank
capital and a collapse of the system becomes inevitable, as banks do not recapitalize
sufficiently with positive macroeconomic shocks.
Gersbach & Wenzelburger (2007a) also established precise conditions under which a
banking crisis can be resolved by means of interest-rate interventions which help a
commercial banking system to recapitalize. However, even with the most drastic
interest-rate intervention an economy with a commerical banking crisis may experi-
ence long-lasting recessions. These finding may help to explain why banking systems
with characteristics similar to those of the Japanese system might exhibit persistent
weakness. In a future refinement these findings will have to be investigated for the case
in which investment banks are in business.
Until now commercial banks are assumed be unable to rate entrepreneurs. A recent
development is the application of sophisticated rating tools by commercial banks. From
the perspective of a single institution, it is clear that such sophistication of risk man-
agement techniques is always beneficial. However, whether or not this holds for the
economy is unclear. This issue has been addressed in Gersbach & Wenzelburger (2007b)
by looking at commercial banks only. There we compare a simple commercial banking
system in which an average rating is used with a sophisticated commercial banking
system in which banks are able to assess the default risk of entrepreneurs individually.
There we have shown that more sophistication in the assessment of individual en-
trepreneurs’ default risk may decrease banking stability if entrepreneurs’ productivity
is sufficiently high. This may be a serious concern for the impact of banking regula-
tion as sophistication in risk management is usually associated with stability enhanc-



                                           22
ing techniques. The reason is that sophistication in rating rewards high-quality en-
trepreneurs with low interest rates and penalizes low-quality entrepreneurs with high
interest rates. When a negative macroeconomic shock occurs, banks cannot benefit
anymore from cross-subzidation among borrowers. The repayments from high-quality,
non-defaulting borrowers are lower than for simple banks as the interest rates are lower
and the repayments from low-quality, defaulting borrowers are the same.
The goal of this research is to investigate the dynamic interplay between commercial
banks and investment banks and to discuss the financial system’s capability to protect
itself from such crises. Ultimately, this steps should help to design appropriate policies
to prevent or work out commercial banking crises from a macroeconomic perspective.




                                           23
8     Appendix

Proof of Theorem 1
We will show that for each w ∈ [w, w] the two equilibrium conditions (13) and (14)
define curves in the r d − r c plane whose unique intersection point is the TEFI. Notice
to this end that for each w ∈ [w, w], the two resource constraints (15) and (16) in
Definition 1 can be rearranged to yield

                                              ηI − S(w, r d) γ[η − ic (w, r b , r d)]
                 ib = ib (w, r d) :=                        +                         ,    (26)
                                                  w+I              w+I
where, using the interest rate parity r c = r d ,

                c      c      b   d           i(w, r d) if I(1 + r b ) ≥ I c (1 + r d ),
               i = i (w, r , r ) :=                                                        (27)
                                              η         otherwise,

Step 1. Consider first Condition (14) which for each w ∈ [w, w] takes the form
                                                                                 !
                    H(w, r d, r b ) := Π ib (w, r d), r b , w, I − w (1 + r d ) = 0.       (28)

By Assumption (i), for each w ∈ [w, w], r d ∈ [0, r d ],

              Π ib (w, r d), 0, w, I ≥ Π(0, 0, w, I) > w (1 + r d ) ≥ w (1 + r d ).

On the other hand, for each i ∈ [0, η], Π(i, r b , w, I) = 0 for r b ≥ r b with
                                              (1 + η)qf (w + I)
                                      rb :=                     −1                         (29)
                                                      I
Notice that by Assumption (i), r b > 0. and Π is strictly decreasing for r b < r b . Thus,
for each w ∈ [w, w], r d ∈ [0, r d ], there exists a uniquely determined continuous function
r b = h(w, r d) solving (28). For each w ∈ [w, w], r d → h(w, r d) is a curve in the r d − r c
plane. Since Π is strictly increasing in i and ib (w, r d is strictly decreasing in w and
r d , h(w, r d) must be strictly decreasing in r d . Moreover, for each w ∈ [w, w] we have
h(w, 0) > 0 > h(w, rd ).
Step 2. Consider now the no-entry no-exit condition (13). This condition takes the
form
                           E P ·, w, ib(w, r d), ic (w, r d), r b)              !
                 d b
          F (w, r , r ) :=        c (w, r d ) − ib (w, r d )
                                                                   − (1 + r d ) = 0. (30)
                               I i
Since repayments P are always less that I ic (w, r d) − ib (w, r d) (1 + r b ), we obtain

                       F (w, r d, r d) ≤ 0 for all w ∈ [w, w], r d ∈ [0, rd ].


                                                     24
On the other hand, for r b ≥ rb , we have
                                                                        1
     E P ·, w, ib(w, r d), ic (w, r d), r b )        E[q]f (w, I) 1 +   2
                                                                            ic (w, r d) + ib (w, r d )
                                                =
             I ic (w, r d) − ib (w, r d)                                     I

and Assumption (i) implies

                 F (w, r d, r b ) > 0 for all w ∈ [w, w], r d ∈ [0, rd ],         rb ≥ rb.

Since F is increasing in r b , the Intermediate Value Theorem then implies for each
w ∈ [w, w] and each r d ∈ [0, rd ] the existence of a unique interest rate r b = g(w, r d)
such that (30) holds. It is straightforward to verify that F is decreasing in r d , so
that g(w, r d) is increasing in r d . Moreover, g(w, r d) is a continuous function of both
arguments, where each g(w, ·), w ∈ [w, w] describes a curve in the r d − r c plane.
Step 3. Existence and uniquenesss of a TEFI. We show that for each w ∈ [w, w], the
two curves h(w, ·) and g(w, ·) defined in (28) and (30), respectively, have a unique
                    d       b
intersection point r∗ (w), r∗(w) which satisfies
                                      d             d        b
                                h w, r∗ (w) = g w, r∗ (w) = r∗ (w)
                                                          d      b
and hence defines a TEFI. If it exists, uniqueness of r∗ (w), r∗(w) follows from the
strict monotonicity of each h(w, ·). So we are left to prove existence.
                                                                             d
By construction of F in Step 2. for each w ∈ [w, w], there exists a unique rw ∈ [0, rd ]
                d      d                                    d
such that g(w, rw ) = rw . Using the definition of F , this rw has to satisfy
                                            d                      d
                              q 1 + ib (w, rw ) f (w + I) = I(1 + rw ).

By Assumption (i), rw < r d and it is straightforward to verify that
                    d


                          d    d               d     d                  d
                    H(w, rw , rw ) = Π ib (w, rw ), rw , w, I) − w(1 + rw ) > 0

Hence
                                   d           d      d
                             h(w, rw ) > g(w, rw ) = rw ,      w ∈ [w, w].
Since h is strictly monotonoically decreasing and g is increasing in r d , there exists a
                 d     b             b       d
unique TEFI r∗ (w), r∗(w) , where r∗ (w) < r∗ (w).
Given a random endowment wt ∈ [w, w], we have
                                   d    d            b    b
                                  rt = r∗ (wt ) and rt = r∗ (wt ),

as well as
                                               d
                  d               ηI − S(wt , rt )         wt + I + γ
     dt = d(wt , rt ) := wt                            −                                    d
                                                                             [η − ic (wt , rt )]I.       (31)
                                     wt + I                  wt + I


                                                     25
This proves existence and uniqueness of a TEFI.




Proof of Theorem 2.
For the two special cases in which either no investment banks operates or investment
banks operate at zero costs γ = 0, we have ic (w, r d, r b ) ≡ η and we we may redefine
(18) as
                                              ηI − S(w, r d)
                               ib (w, r d) :=                .                     (32)
                                                  w+I
This implies that for both economies the indifference condition (28) remains unchanged.
As in Step 2. of the proof of Theorem 1, the no-entry no-exit condition (30) for the
case without investment banks becomes

                                  E P ·, w, ib (w, r d), η, r b)                   !
               ˜       d   b
               F (w, r , r ) :=                                        − (1 + r d ) = 0,   (33)
                                        I η−    ib (w, r d)

defining a curve r b = g (w, r d). Since
                      ˜

               ˜
               F (w, r d , r b) > F (w, r d, r b ) for all w ∈ [w, w], r d ∈ [0, rd ],

we have
                   g (w, r d) < g(w, r d) for all w ∈ [w, w], r d ∈ [0, rd ].
                   ˜
                                               d
                                                   ˜d     ˜b    b
Given wt these curves, it is easy to see that rt < rt and rt < rt .




Proof of Theorem 3.
We only prove (iii). Using (31) we see that

                                                   ˜
                                              dt < dt

provided that S is sufficiently inelastic.




Proof of Theorem 4.
(i) Output of the economy with investment banks is
                                                              η
                               Yt+1 = qt+1 f (wt + I)             (1 + i)di                (34)
                                                          ib
                                                           t




                                                 26
while output the economy without investment banks is
                                                     η
                           ˜
                           Yt+1 = qt+1 f (wt + I)        (1 + i)di                 (35)
                                                    ˜b
                                                    it


By Theorem 3 (i) ib ≥ ˜b . Hence (34) must be smaller than (35).
                  t   it
(ii) The conditional variance of the output of the economy with investment banks is
                                                                             2
                                                             η
                  Vart [Yt+1 ] = Vart [qt+1 ] f (wt + I)         (1 + i)di         (36)
                                                            ib
                                                             t


while the conditional variance of the output of the economy without investment banks
is
                                                                             2
                                                             η
                        ˜
                  Vart [Yt+1 ] = Vart [qt+1 ] f (wt + I)         (1 + i)di         (37)
                                                            ˜b
                                                            it

The second statement then follows again from Theorem 3 (i).
(iii) Regarding the coefficients of variation, we see from taking expectations of (34) and
(35) that
                               ˜
                        Vart [Yt+1 ]     Vart [qt+1 ]   Vart [Yt+1 ]
                                     =                =              .
                            ˜t+1 ]
                        Et [Y           Et [qt+1 ]      Et [Yt+1 ]




                                            27
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