School of Economics, Mathematics and Statistics
Birkbeck College
MSc Finance, MSc Economics
Topics in Finance
Lecture 2: Financial Intermediation
A Model of Liquidity Insurance1
1
1. Economy with one good and a continuum of individuals that are exante identical. We analyze the economy over three periods, t = 0, 1, 2.
2. Each individual is endowed with one unit of the good in period 0.
They use this to finance their consumption (C1 , C2 ) in periods 1 and
2.
3. Individual receive ‘liquidity shocks’ in period 1. At t = 1, an individual
will found out
• with probability π1 that she is of type-1. Individuals of this type
are impatient and care only for consumption in period 1. Their
utility function is of the form u(C1 ),
• with probability π2 that she of type 2. Individuals of this type are
patient and care only for consumption in period 2. Their utility
function is of the form ρu(C2 ), where ρ < 1 is a discount factor.
In ex-ante terms, the consumer’s expected utility can be written as
U (.) = π1 u(C1 ) + π2 ρu(C2 ).
We assume u(.) is increasing and concave.
4. At time t = 0, each individual faces a choice of two investment technologies.
1
This section is based on Bryant (1980), as discussed in Frexias and Rochet, Microeconomics of Banking, pages 16-27.
1
• The first one is a one-period ‘storage’ technology: you get back
one unit for every unit you store away.
• The second technology is an investment technology that returns
R > 1 after two periods or L < 1 after one period.
Thus, relative to the storage technology, investment is more productive
over two period but there is a penalty for premature liquidation.
1.1
Autarky
Suppose first that there is no trade between agents. This is called autarky.
Each consumer must choose, independently of others, her level of investment
in the illiquid technology. Recall that the investment choice is made at time
t = 0, before the individual find out her type. Suppose the individual
invests I and stores the rest 1 − I. Then, if she discovers herself to be
impatient, she consumes whatever she had stored and the liquidated value
of her investment.
C1 (I) = 1 − I + LI.
If she discovers herself to be patient, she can consume her stored wealth (this
requires re-storing the good from period 1 to 2) and the maturity value of
her investment.
C2 (I) = 1 − I + RI.
Given these, each consumer chooses I to maximize expected utility.
Note that C1 (I) = 1 − I(1 − L) ≤ 1, with equality only when I = 0.
Similarly, C2 (I) = 1 + I(R − 1) ≤ R, with equality when I = 1. In general,
the consumption profile (C1 , C2 ) is bounded above by (1, R).
1.2
Market Economy
Suppose consumers can trade in period 1. Assume there is a riskless bond,
which involves the following contract: you pay p in period 1 (the price of
the bond), which entitle you to one unit of the good in period 2. It is easy
to see that p ≤ 1: if not, bonds would be dominated by storage.
The ability to trade alters the consumption possibilities relative to autarky. Consider an individual who has invested I and stored 1 − I. If she
discovers herself to be the impatient type she need not liquidate her investment: she could instead sell RI bonds at price p so that total consumption
is
C1 (I) = 1 − I + pRI.
2
If she discovers herself to be patient, her liquid, stored assets, 1 − I, will
allow her to buy 1−I
p bonds. Her consumption is
C2 (I) =
1−I
+ RI.
p
How do these consumption possibilities compare with autarky? The answer
clearly depends on p, the price of bonds. We argue that market clearing in
the bond market requires that p = R1 Why?2 Evaluating C1 and C2 at this
value, we get the market solution
C1M = 1
and
C2M = R,
and to enable this solution we must have I M = π2 . Note that the market
solution Pareto dominates the autarky allocation: no type is worse off and
at least one type is better off.
1.3
Pareto Optimal Allocation
While the market solution Pareto dominates the autarkic solution, it may
not be Pareto optimal (i.e., we could do even better). Consider the following
problem:
π1 u(C1 ) + ρπ2 u(C2 )
subject to
C2
= 1.
R
The last relation describes an aggregate resource constraint. The optimal
allocation (C1∗ , C2∗ ) satisfies the first-order condition,
π1 C1 + π2
u0 (C1∗ ) = ρRu0 (C2∗ ).
Does the market allocation (1, R) satisfy this? Only in the very special
condition
u0 (1) = ρRu0 (R).
How does optimal consumption (C1∗ , C2∗ ) compare with market-based consumption (1, R)? Suppose the expression Cu0 (C) is decreasing. (This is
2
1
Suppose not. If p > R
, consumption in both periods in increasing in I. This will
1
create excess demand in the market for bonds. If p < R
, consumption in both periods in
decreasing in I. This will create excess supply in the market for bonds.
3
equivalent to assuming the the coefficient of relative risk aversion is greater
than 1). In that case, since R > 1, we have
ρRu0 (R) < ρu0 (1) < u0 (1),
so that it would be welfare improving to increase C1M above 1 and decrease
C2M below R.
The problem here is that the market economy does not provide perfect
insurance against liquidity shocks.
1.4
Financial Intermediation
A financial intermediary could implement the efficient solution directly. Say,
everyone deposits all their endowment with the financial intermediary, in
exchange for the following contract. Either C1∗ in period 1 or C2∗ in period
2, where these are as given in the previous section.
In the presence of independent, individual liquidity shocks, the market
allocation can be improved upon by a financial intermediary.
This solution does not work if bond markets co-exist with financial intermediary. We shall develop this in a later lecture.
Exercises, based on the text
1. Solve for the autarkic allocation and the optimal allocation in Section
2.2.4, specializing the utility function to
u(ci ) = ln ci
2. Next solve for the optimal allocation in Section 2.2.4, this time using
the utility function
c1−λ − 1
u(ci ) = i
1−λ
3. Section 2.2.5 What is the solution with intermediation? Why does it
matter if ρR > 1
4