THE WELFARE COST OF BANK CAPITAL REQUIREMENTS
Skander Van Den Heuvel
1
The Wharton School, University of Pennsylvania
January, 2005
Abstract
This paper estimates the welfare cost of bank capital requirements and
finds that the macroeconomic stakes involved with the design of capital
adequacy regulation are potentially large. A general equilibrium model
with capital accumulation and a preference for liquidity is developed.
Banks provide liquidity services by accepting deposits. A capital
requirement plays a role, as it limits the moral hazard on the part of
banks that is induced by the presence of a deposit insurance scheme.
However, ceteris paribus, a higher capital requirement implies that banks
can accept fewer deposits and thus provide fewer liquidity services to
households. It is shown that equilibrium asset returns reveal the strength
of households’ preferences for liquidity and this allows the derivation of
a formula for the welfare cost of capital requirements that is a function
only of observable variables. Using U.S. banking data, the cost of
increasing the capital requirement by 10 percentage points is equivalent
to a permanent loss in consumption of at least 0.1 to 0.2 percent.
1
Finance Department, Wharton School, 3620 Locust Walk, Philadelphia, PA 19104, USA. Email:
vdheuvel@wharton.upenn.edu. The author has benefited from helpful comments from Andy Abel,
Joao Gomes, Gary Gorton and seminar participants at Wharton and the Federal Reserve Board. All
mistakes are mine.
This paper asks, and provides an answer to, the following question: How
large are the welfare costs of bank capital requirements? While there are a number
of papers on the theoretical benefits of capital adequacy regulation, based on
limiting the moral hazard involved with deposit insurance
2
or externalities
associated with bank failures, much less is known about whether there are also
costs involved with imposing restrictions on the capital structure of banks. Recent
work by Diamond and Rajan (2000) and Gorton and Winton (2000), among
others, suggests that capital requirements may have an important cost in that they
reduce the ability of banks to create liquidity by accepting deposits. After all, a
capital requirement limits the fraction of bank assets that can be financed by
issuing deposit-type liabilities. Unfortunately, the models in these papers do not
easily lend themselves to a quantification of this cost.
This paper’s contribution is to model the benefit of liquidity creation in a
flexible way, embed the role of liquidity creating banks in an otherwise standard
general equilibrium growth model, and use that framework to quantify the welfare
cost of increasing the capital requirement. This cost depends crucially on
households’ preferences for liquidity. A key insight from the model is that
equilibrium asset returns reveal the strength of these preferences for liquidity and
this allows us to quantify the welfare cost. The model also incorporates a rationale
for the existence of capital adequacy regulation, based on a moral hazard problem
associated with deposit insurance. The resulting benefits are characterized, but are
harder to quantify.
In many countries capital adequacy regulation is currently based on the
Basel Accords. In response to perceived shortcomings in the original Accord,
practitioners have added more and more detailed refinements, culminating in the
soon-to-be implemented Basel 2, while attempting to keep the required ratio of
capital to risk weighted assets for a typical bank approximately the same. But is
the 8% of the original Basle Accord a good number for the total risk-based capital
ratio? This fundamental question remains unaddressed in the literature. At the
same time the changes involved in Basel 2 seem likely to greatly increase the cost
of compliance and supervision of banks.
If we find that the welfare cost of capital requirements are trivial, this
could be an argument for creating a simple, robust system of capital adequacy
2
See, for example, Giammarino, Lewis, and Sappington (1993) and Dewatripont and Tirole
(1994). See Allen and Gale (2003) for a more skeptical view. Diamond and Dybvig (1983) is often
viewed as a theoretical justification for deposit insurance.
2
regulation, with low compliance and supervision costs, but with relatively high
capital ratios so as to make bank failure a sufficiently infrequent event. On the
other hand, if we find a high welfare cost of capital requirements, this could be an
argument for lowering them, by either accepting a higher chance of bank failure,
or by designing a more risk-sensitive system with the associated increased
supervision and compliance costs, which seems to be the trend in practice.
1. The model
The most important respect in which the model deviates from the standard
growth model is that households have a need for liquidity, and that certain agents,
called banks, are able to create financial assets, called deposits, which provide
liquidity services. Since a central goal of the model is to provide a framework not
just for illustrating, but for actually measuring the welfare cost of capital
requirements, it is important to model the preferences for liquidity in a way that is
not too restrictive. In as much as possible, we would like the data to provide the
answer, not the specific modeling choices. To that end, I follow Sidrauski (1967)
in adopting the modeling device of putting liquidity services in the utility
function. This has two disadvantages and one advantage.
One disadvantage is that it does not further our understanding of why
households like liquid assets, but this is not the topic of this paper, so this concern
can be dismissed.
3
A second disadvantage is that if one needs to specify a
particular functional form for the utility function, one is on loose grounds. For
example, is the marginal utility of consumption increasing or decreasing in
deposits?
Fortunately – and this is the advantage of this approach – there is no need
to make unpalatable assumptions of this kind. I will show that it is possible to
derive a first-order approximation the welfare cost of raising the capital
requirement without making any assumptions on the functional form of the utility
function, beyond the standard assumptions that it is increasing and concave. A
trade-off involved with modeling liquidity in this flexible way, and embedding it
in a general equilibrium analysis, is that the modeling of the banks’ assets is not
rich enough to incorporate the details of risk-based capital requirements.
3
Nonetheless, see Feenstra (1986) for how optimizing models of money demand based on a
Baumol Tobin transaction technology can be approximately rewritten as maximization problems
with money in the utility function.
3
The environment and the agents’ decision problems
Time is discrete and there are infinitely many periods. The economy
consists of households, banks, (nonfinancial) firms, and a government or
regulatory agent. Households own both the banks and the nonfinancial firms.
These firms combine capital and labor to produce the single good which
households consume. I now discuss the assumptions for each of these agents, and
analyze their decision problems in turn.
Households: There is a continuum of households with mass one. Households are
infinitely lived dynasties and have identical preferences. They value consumption
and liquidity services. Households can obtain these liquidity services by
allocating some of their wealth to bank deposits, an asset created by banks for this
purpose. As mentioned, the liquidity services of bank deposits are modeled by
assuming that the household’s utility function is increasing in the amount of
deposits.
Besides holding bank deposits, denoted d
t
, households can store their
wealth by buying and selling shares, or equity, e
t
. They supply a fixed quantity of
labor, normalized to one, for a wage, w
t
. Taxes are lump-sum and equal to T
t
.
There is no aggregate uncertainty, so the representative household’s problem is
one of perfect foresight:
0
{,}
0
max ( , )
ttt
t
tt
cd
t
uc d
β
∞
=
∞
=
∑
(1)
11
s.t. 1
DE
tttttttt
decwRdReT
++
++=+ + −
t
1
0
lim ( ) 0
T
E
sTT
T
s
Rde
−
→∞
=
⎛⎞
+
≥
⎜⎟
⎝⎠
∏
00 0
givendea+≡
where c
t
is consumption in period t,
D
t
R
is the return on bank deposits,
E
t
R
is the
return on (bank or firm) equity, and
β
is the subjective discount factor. The
returns
D
t
R
and
E
t
R
, and the wage are determined competitively, so the household
takes these as given. The same applies for the taxes. There is no distinction
between bank and nonbank equity, since, in the absence of risk, they are perfect
4
substitutes for the household and will thus also command the same return. The
second constraint is a no-Ponzi game condition, the third an initial condition.
The utility function is assumed to be concave, at least once continuously
differentiable on
and increasing on its domain in both arguments, and strictly
increasing in consumption:
2
++
(, ) (, ) 0
c
ucd ucd c≡∂ ∂ > and (, ) (, ) 0
d
ucd ucd d
≡
∂∂≥
The first-order conditions to the household’s problem are:
(c) ( , )
ct t t
ucd
λ
=
(d)
1
1
(, ) 0
D
dtt tt t
ucd R
λβλ
−
−
+− =
(e)
1
1
0
E
tt t
R
λβλ
−
−
−=
where
λ
t
is the Lagrange multiplier associated with the intertemporal budget
constraint (1). Rewriting this,
1
11
(( , )/(,))
E
tcttctt
R
uc d ucd
β
−
−−
=
(2)
(3)
(, ) (, )( )
ED
dtt ctt t t
ucd ucd R R=−
Equation (2), which determines the return on equity, is the standard Euler
equation for the intertemporal consumption-saving choice in a deterministic
setting, with one difference: the marginal utility of consumption may depend on
the level of deposits. Equation (3) relates the spread between the return on equity
and the return on bank deposits to the marginal value of the liquidity services
provided by deposits, expressed in units of the consumption good. If
,
then the return on equity will be higher than the return on deposits to compensate
for the fact that equity does not provide any liquidity services.
( , ) 0
d
ucd>
Banks: There is a continuum of banks with mass one, which make loans to
nonfinancial firms and finance these loans by accepting deposits from households
and issuing equity. The ability of banks to create liquidity through deposit
contracts is their defining feature. All contracts are resolved in one period. Every
period new banks are setup with free entry into banking. The balance sheet, and
the notation, for the representative bank during period t is:
5