ï»¿ WPS6059
Policy Research Working Paper 6059
Should Wall-Street be Occupied?
An Overlooked Price Externality
of Financial Intermediation
Maya Eden
The World Bank
Development Research Group
Macroeconomics and Growth Team
May 2012
Policy Research Working Paper 6059
Abstract
Does an unregulated financial system absorb too many inputs in excess of their internal funds, by borrowing
productive inputs? This paper studies this question in from unproductive agents. However, intermediation
the context of a dynamic model with heterogeneous requires the use of costly monitoring services. In
producers. In the absence of a financial system, the only equilibrium, intermediation increases the money in
way to purchase inputs is using internal funds. Producers circulation and raises nominal prices, thereby reducing
are subject to idiosyncratic productivity shocks, and the value of internal funds and making producers
will decide to produce only if their productivity is high increasingly reliant on costly monitoring services. For
enough. Otherwise, they will hold money. A financial this reason, society is better off when intermediation is
intermediation technology allows producers to purchase restricted.
This paper is a product of the Macroeconomics and Growth Team, Development Research Group. It is part of a larger
effort by the World Bank to provide open access to its research and make a contribution to development policy discussions
around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The author
may be contacted at meden@worldbank.org.
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issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the
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Produced by the Research Support Team
Should Wall-Street be Occupied?
An Overlooked Price Externality of Financial
Intermediation
Maya Edenâˆ—
World Bank
October 3, 2012
Abstract
This paper highlights a price externality that causes excessive re-
sources to be spent on ï¬?nancing. Intermediated funds increase the
demand for capital inputs, which raises their price (in both real and
nominal terms). This implies that producers can buy less inputs with
their internal funds, and must rely more heavily on ï¬?nancial inter-
mediation. As intermediation is costly, this feedback is ineï¬ƒcient. I
illustrate this mechanism in a simpliï¬?ed real model, and extend it to
a dynamic model with money.
JEL Classiï¬?cation: E44, G20
Keywords: Costs of the ï¬?nancial sector, ï¬?nancial intermediation
âˆ—
Please send comments to meden@worldbank.org. Earlier versions of this paper cir-
culated under the titles â€œThe Ineï¬ƒciency of Financial Intermediation in General Equilib-
riumâ€?, â€œThe Wasteful â€˜Money Creationâ€™ Aspect of Financial Intermediationâ€? and â€œExces-
sive Financial Intermediation in a Model with Endogenous Liquidityâ€?. This work is based
on a chapter of my thesis at MIT. I thank Fernando Broner, Ricardo Caballero, Ben Eden,
Anton Korinek, Guido Lorenzoni, Luis Serven and Ivan Werning for helpful comments. I
also beneï¬?ted from comments made by various seminar and conference participants. This
paper does not necessarily reï¬‚ect the views of the World Bank, its Executive Board or the
countries it represents. All errors are mine.
1
â€œHas the contribution of the modern world of ï¬?nance to economic
growth become so critical as to support remuneration to its par-
ticipants beyond any earlier experience and expectations? Does
the past proï¬?tability of and the value added by the ï¬?nancial in-
dustry really now justify proï¬?ts amounting to as much as 35 to
40 percent of all proï¬?ts by all US corporations? Can the truly
enormous rise in the use of derivatives, complicated options, and
highly structured ï¬?nancial instruments really have made a paral-
lel contribution to economic eï¬ƒciency?â€?
-Paul Volcker, â€˜The Time We Have Is Growing Shortâ€™, The New York
Review of Books, June 24th 2010.
1 Introduction
The recent ï¬?nancial crisis resurfaced the concern that too many production
inputs are being absorbed by the ï¬?nancial sector.1 At the same time, stan-
dard theoretical arguments suggest that ï¬?nancial intermediation improves
the allocation of inputs across producers, and that this eï¬ƒciency gain is
enough to oï¬€set the costs of intermediation.2 In this paper I argue that,
while ï¬?nancial services are sometimes socially valuable, the idea that they
are unambiguously so may be a partial equilibrium view: while ï¬?nancial in-
termediation may appear vital when input prices are ï¬?xed, in general equilib-
rium, when input prices are adjustable, even the most basic form of ï¬?nancial
intermediation may be a wasteful use of productive resources. Bluntly, it
is theoretically ambiguous whether or not the existence of a ï¬?nancial sector
1
See Benjamin Friedman, â€œOvermighty Finance Levies a Tithe on Growthâ€?, The Fi-
nancial Times, August 26th, 2009, and Paul Volcker, â€˜The Time We Have Is Growing
Shortâ€™, The New York Review of Books, June 24th 2010, for op-eds on this topic.
2
See Gorton and Winton [2003] for a survey of the literature on ï¬?nancial intermedia-
tion, as well as Levine [2005] and McKinnon [1973] on the role of the ï¬?nancial sector in
promoting growth and an eï¬ƒcient allocation of resources.
2
creates social surplus. Regardless, there is a price externality that implies
that an unrestricted ï¬?nancial system absorbs too many inputs.
To develop intuition, consider a benchmark economy with identical pro-
ducers. Land is the only input of production and is supplied inelastically.
Producers are liquidity constrained in the sense that they cannot fully pledge
post-production output for the purpose of buying land. The constraint is
binding: given the equilibrium price of land, producers would like to buy
more of it, but are unable to pledge the resources to do so. However, since
the aggregate supply of land is ï¬?xed, prices adjust so that all land is used
in equilibrium, and output is at its ï¬?rst best level. The price of land is
determined by the aggregate constraint: the amount of pledgeable output
determines the aggregate payment to land.
In this setup, consider the consequences of introducing an intermediation
technology that allows each producer to increase the amount of output that
he can pledge by employing â€œmonitoringâ€? services. Since liquidity constraints
are binding, it is privately optimal to make use of this technology, in hope to
buy more land at its depressed price. However, in general equilibrium, since
the supply of land is ï¬?xed, the increase in pledgeable output merely bids up
the price of land, in a way that does not allow producers to expand their
production. The costs of intermediation are purely wasted.
From the producersâ€™ perspective, there is a coordination failure: faced
with constrained aggregate demand and low input prices, each individual
producer tries to â€œbeat the systemâ€? by spending resources in order to cir-
cumvent the constraint. However, in general equilibrium, when everyone
tries to â€œcircumvent the constraintâ€?, the aggregate constraint becomes less
binding and, consequently, land prices appreciate. This un-internalized price
externality leaves all producers worse oï¬€ as they face higher land prices and
must pay intermediation costs.
In this stylized framework, the stark conclusion that intermediation is
purely wasteful relies on two stark assumptions: (a) that production inputs
3
are supplied inelastically; and (b) that producers are homogeneous. When
these assumptions are relaxed, there are some beneï¬?ts associated with the
appreciation of input prices. When there is some input supply elasticity,
higher input prices increase input supply. When producers are heterogeneous,
higher input prices deter less-productive producers from hiring inputs, which
may improve the eï¬ƒciency of input allocation. The robust conclusion is
therefore less extreme: ï¬?nancial intermediation is not always purely wasteful.
However, the price externality implies that in the unregulated equilibrium,
the social expenditure on ï¬?nancial intermediation is ineï¬ƒciently large.
The price externality is exacerbated by its monetary implications. I em-
bed this mechanism in a monetary model, in which money is used both as a
store of value and as a medium of exchange. The price externality operates
through an additional nominal channel: when producers borrow money from
idle money holders (savers), they are increasing the money in circulation.
This leads to an inï¬‚ation in nominal prices and a decline in real balances.
Interestingly, a steep increase in nominal input prices need not coincide with
a large real increase in input prices. In this case, the conclusion that ï¬?-
nancial intermediation is purely wasteful generalizes to settings with elastic
input supply and heterogeneous producers, as the â€œpositiveâ€? eï¬€ects of the
(real) price appreciation are no longer relevant. In this case, intermediation
is similar to a counterfeit money printing machine: it increases the money in
circulation at some real cost, but, in general equilibrium, this merely trans-
lates into higher nominal prices, leaving real prices unchanged (and wasting
resources on intermediation services, which, in the counterfeit money analogy,
are analogous to resources spent on counterfeit money printing machines).
The implications of the monetary model are broadly consistent with re-
cent trends in the US economy. The increase in the size of the ï¬?nancial
system in the US (as documented by Philippon [2008]) has been accompa-
nied by a decline in the amount of real balances: the average value of M1 as a
M1
fraction of nominal output ( P Y
in standard notation) dropped from around
4
0.18 between the years 1960-1989 to around 0.12 between the years 1990-
2007.3 This is consistent with the view that, at the aggregate level, ï¬?nancing
reduces the equilibrium amount of real balances.4
2 Related literature
In most models of ï¬?nancial intermediation, the price of inputs is essentially
ï¬?xed, and pledgeability directly determines the quantity of inputs that can be
employed. For example, in Holmstrom and Tirole [1997], external funds are
direct inputs in investment - implicitly, the price of investment goods in terms
of funds is ï¬?xed at 1. Under similar assumptions, Philippon [2008] considers
a model of costly ï¬?nancial intermediation, and shows that (despite its costs)
ï¬?nancial intermediation is eï¬ƒcient. The key diï¬€erence in this paper is the
endogenous determination of the relative price of inputs, that may reverse
this conclusion.5
This paper is related to other papers that study the welfare implications of
ï¬?nancial intermediation. Arcand et al. [2011] show empirically that, beyond
a certain threshold, there is a negative relationship between the size of the
ï¬?nancial sector and economic growth. They present a model that attributes
excessive ï¬?nancing to an implicit subsidy in the form of bailouts. This paper
emphasizes a diï¬€erent mechanism through which the ï¬?nancial system may
become overblown: in the spirit of Tobin [1984], the model highlights the dif-
3
The corresponding numbers for M2 are 0.59 and 0.51 respectively. These ï¬?gures are
calculated based on data from the Federal Reserve Bank of St. Luis (FRED) database.
4
In this paper, banks increase the amount of nominal bills in circulation - this will
increase M1, but lead to a decline in real balances. There are other (more positive) ways
to think about the role of banks in money creation. Stein [2012] presents a model in which
real riskless claims provide money-like services. The ability of banks to diversify risk allows
them to create riskless claims, which is equivalent to the creation of real balances.
5
Similar to Philippon [2008] and other standard models, in this model, the ï¬?nal good
can be used both for consumption and investment. However, physical capital is diï¬€erent
from the ï¬?nal good; the emphasis is on the relative price of physical capital in terms of
the ï¬?nal good. In this model, agents are not constrained in investment, but rather in the
amount of production inputs that they can employ.
5
ference between the private return to ï¬?nance and the social return, that leads
to too many productive inputs being employed on ï¬?nancial intermediation.
The main critique in Tobin [1984] is that only a fraction of the ï¬?nancial sys-
temâ€™s activities are consistent with its social objectives, such as transferring
funds from surplus ï¬?rms to ï¬?rms that need ï¬?nancing, or diversifying risk.
In related work, Simsek [2012] illustrates that ï¬?nancial innovation may be
driven by speculative trading motives, rather than by traditional risk sharing
motives. Here, I argue that even the traditional roles of ï¬?nance may lead to
general equilibrium ineï¬ƒciency.
The literature on the interaction between the price of capital and borrow-
ing has largely focused on the collateral channel emphasized in Kiyotaki and
Moore [1997]. When capital is used as collateral, a higher price of capital
enables constrained producers (who are also capital owners) to borrow more.
Through this channel, the appreciation of capital prices may improve the
allocation of inputs. In this paper, I abstract from this well-known channel
by assuming that capital has no collateral role.6 However, the spirit of the
results are incorporated into this model, as less-productive users of capital
ï¬?nd it less proï¬?table to employ capital at higher capital prices, implying a
favorable reallocation of inputs to more productive agents.
The mechanism in this paper is conceptually related to the Friedman
Rule (Friedman [1969]). In a monetary economy with constant money sup-
ply, binding liquidity constraints cause consumers to waste resources on trips
to the bank. The mechanism here is similar in spirit: binding liquidity con-
straints (on the producerâ€™s side) cause agents to spend resources ineï¬ƒciently
6
In this model, introducing a collateral role for capital may create â€œincreasing re-
turnsâ€? in the intermediation technology: the use of ï¬?nancial intermediation increases the
price of capital. Through the collateral channel, this price increase may make moni-
toring â€œcheaperâ€?; thus, as the amount of intermediation increases, the marginal cost of
intermediation declines. While this channel is potentially important, there are standard
considerations to do with operation scale that would suggest a decreasing returns inter-
mediation technology. In this model I assume that there is a ï¬?xed cost to intermediation
(section 3) or a constant return to scale intermediation technology (section 4), and reserve
this important discussion for future work.
6
on ï¬?nancial intermediation. The common theme is that when agents are
against a constraint, actions taken to relax the constraint may be socially
ineï¬ƒcient.
Other papers with related mechanisms include Bolton et al. [2012] and
Glode et al. [2012]. In these papers, intermediaries ineï¬ƒciently compete
to extract rents from agents faced with binding constraints. These papers
focus on the micro structure of the problem, and endogenize the costs of
intermediation. Here, I take the costs of ï¬?nancial intermediation basically as
exogenous, and focus on the general equilibrium price externality.
The nominal price externality highlighted in this model is similar in spirit
to Bewley [1987], who considers a model in which agents with idiosyncratic
income shocks hold money. In equilibrium, agents behave as if their marginal
utility of holding money is constant, and use money reserves to smooth con-
sumption. The mechanism here is similar: in the absence of ï¬?nancial in-
termediation, agents can respond to productivity shocks by using money
reserves. The nominal price level is low (and real balances are high) because
not all agents decide to use all of their money all of the time. In other words,
â€œidleâ€? money is important; the ineï¬ƒciency of ï¬?nancial intermediation in this
model stems, in part, from the reduction in â€œidleâ€? money, that compromises
its ability to buï¬€er shocks.7
3 A simpliï¬?ed single period model
Setup. Consider a single period economy with a unit measure of producers,
indexed i âˆˆ [0, 1], and a unit measure of capital suppliers. Capital is the only
input of production. Each producer is endowed with an AK production tech-
nology, and no capital. Note that all producers share a common technology,
and a common productivity parameter (A). Each capital supplier is endowed
7
This paper illustrates that money can buï¬€er idiosyncratic shocks. Allen et al. [2012]
illustrate that, with ï¬‚exible prices, money may help buï¬€er aggregate liquidity shocks and
even mitigate ï¬?nancial crisis.
7
with K units of capital, but no production technology.
In order to produce, producers must purchase capital from capital suppli-
ers. For this purpose, producers issue promises on post-production output.
The price of capital (in terms of promises) is denoted R. Capital suppliers
always sell their capital at the market price.8 The timing of the model is
summarized in ï¬?gure 1.
The amount of capital employed by producer i is denoted ki . Producers
maximize proï¬?ts, which are given by production revenues minus capital costs
(Aki âˆ’ Rki ).
The liquidity constraint. Producers are liquidity constrained, in the sense
that they are unable to fully commit post-production output.9 For simplicity,
I assume that the maximum amount of output that a producer can promise
to repay is some constant l (for â€œliquidityâ€?).10 I assume that l < AK , which
will guarantee that the liquidity constraint binds in equilibrium.
Equilibrium without ï¬?nancial intermediation. Consider the producerâ€™s
optimization problem:
max Aki âˆ’ Rki (1)
ki
s.t. Rki â‰¤ l
To see that the liquidity constraint is binding, note that l is the maximum
aggregate payment to capital. It follows that R < A:
RK â‰¤ l < AK â‡’ R < A (2)
8
The consumption value of capital is assumed 0.
9
This friction can be seen as a reduced form formulation of a moral hazard problem as
in Holmstrom and Tirole [1997].
10
Similar results can be obtained in more elaborate frameworks, in which the interme-
diation technology is arbitrary and the amount of output that producers can pledge is an
increasing function of post-production output.
8
Figure 1: The timeline of the model
â€¢ Producers born Market: Production,
with AK technology capital is sold repayment
â€¢ Capital suppliers for promises on
born with capital post-production output
As R < A, proï¬?ts are increasing in ki ; absent the liquidity constraint, the
producers would choose ki = âˆž. As the capital supply is ï¬?nite, the liquidity
constraint must bind.
The aggregate liquidity constraint pins down the equilibrium price of
capital, R. To see this, note that each producerâ€™s demand for capital is
pinned down by his liquidity constraint:
l
Rki = l â‡’ ki = (3)
R
1
Capital market clearing requires that 0 ki di = K . Thus, given the ex-
l
pression above we can conclude that K = R , and therefore the equilibrium
l
price of capital is given by R = K .
Note that the price of capital is constrained by the aggregate supply of
liquidity, and is increasing in l. Despite the fact that the liquidity constraints
are binding, output is at its â€œï¬?rst bestâ€? level, as all capital is employed in
equilibrium at its most productive use:
1
Y = Aki di = AK (4)
0
The reason that output is unaï¬€ected by the binding liquidity constraints
is that capital is supplied inelastically. The binding liquidity constraints
merely aï¬€ect the price of capital (R).
Financial intermediation. In the economy described above, assume a ï¬?-
nancial intermediation technology that allows producers to be â€œmonitoredâ€?,
9
thereby increasing the amount of post-production output that they can pledge
from l to l > l. However, monitoring is costly.11 For simplicity, I assume
that there is a ï¬?xed cost of monitoring: a monitored producer must forgo
Î¹ units of output, which are lost on intermediation activities (in section 4,
the costs of intermediation will depend the value of intermediated funds;
similar results can be obtained if the costs of intermediation are speciï¬?ed
in terms of capital inputs rather than output). To summarize, the ï¬?nancial
intermediation technology (when used) modiï¬?es the producerâ€™s problem as
follows:
max Aki âˆ’ Rki âˆ’ Î¹ (5)
ki
s.t. Rki â‰¤ l
I will continue to assume that l < AK , so that the aggregate liquidity
constraint continues to bind.
Equilibrium with ï¬?nancial intermediation. For Î¹ suï¬ƒciently small, I
conjecture an equilibrium in which all producers use ï¬?nancial intermediation.
Intuitively, liquidity-constrained producers are willing to pay a real cost in
order to relax their constraints.
To prove this conjecture, recall that the assumption l < AK implies that
in equilibrium, it must be the case that R < A. Thus, the liquidity constraint
is binding for every producer.
It is left to show that given R, producers ï¬?nd it optimal to use the ï¬?-
nancial intermediation technology. When a producer uses the intermediation
technology, he increases the amount of capital that he can employ (given R).
If intermediation were free (Î¹ = 0), this would be a strict gain, as A > R, so
proï¬?ts are strictly increasing in capital. Continuity implies that producers
11
Intermediation costs are modeled as the cost of monitoring (as in, for example, Dia-
mond [1984]). However, the setup is suï¬ƒciently general to allow for a richer interpretation
of intermediation costs, such as search costs as in Duï¬ƒe and Strulovici [2011], or informa-
tion acquisition costs such as in Farhi and Tirole [2012].
10
Figure 2: The economyâ€™s equilibria with and without intermediation
K K d=l/R K d=lâ€²/R
e eâ€²
KS
R=A R
Capital supply is constant at K s . The solid declining curve is the constrained demand
for capital without intermediation. The intersection (point e) is the equilibrium in the
no-intermediation economy. The use of a ï¬?nancial intermediation technology causes an
outward shift of the constrained demand curve, which is illustrated with the dotted line.
The equilibrium with ï¬?nancial intermediation is the point e .
opt for intermediation even when Î¹ > 0 but suï¬ƒciently small.
It follows that for Î¹ > 0 suï¬ƒciently small, all producers use ï¬?nancial
intermediation in equilibrium. Despite the fact that producers privately ï¬?nd
intermediation proï¬?table, aggregate output is lower compared to the no-
intermediation economy, as the costs of intermediation are socially wasted:
Y = AK âˆ’ Î¹ < AK (6)
The intuition is straightforward: privately, each producer would like to
increase his liquidity in order to buy more capital. However, as the entire
capital stock is already employed, intermediation works only towards increas-
ing the price of capital without expanding production. Figure 2 illustrates
graphically the equilibrium of this economy with and without ï¬?nancial inter-
mediation.
11
The partial equilibrium view. The result that ï¬?nancial intermediation
reduces equilibrium output may seem counter-intuitive. It is therefore useful
to illustrate how the standard intuition can be recovered from partial equi-
librium analysis of this model, that ignores the endogenous determination of
the price of capital (R).
Consider an economy that uses ï¬?nancial intermediation. The equilibrium
capital bill is given by RK = l . Holding R ï¬?xed, one would erroneously
conclude that ï¬?nancial intermediation increases the employment of capital:
absent ï¬?nancial intermediation, the capital bill would be bounded by l < l ,
leaving part of the capital stock unemployed. More broadly, ignoring the
endogenous determination of the costs of inputs leads to an overestimate of
the extent to which ï¬?nancial intermediation increases eï¬ƒciency, as, absent
ï¬?nancial intermediation, input prices would be lower and producers would
be able to employ more inputs with their internal funds.
Financial crises. While the partial equilibrium analysis leads to mislead-
ing conclusions regarding the social value of ï¬?nancial intermediation, it may
be useful for thinking about â€œpartial equilibriumâ€? situations, such as ï¬?nancial
crises. Consider a simple model of ï¬?nancial crises, in which the intermedia-
tion technology ceases to â€œworkâ€?, while the price of inputs remains ï¬?xed in
the short run. In other words, liquidity drops from l to l < l , but R remains
at its pervious level.
This type of ï¬?nancial shock would lead to a drop in employment and
output, as the available liquidity is insuï¬ƒcient to employ all inputs at the
price R. This suggests that while there is an argument for reducing the
â€œsteady-stateâ€? size of the ï¬?nancial system, there is still a case for bailing out
the ï¬?nancial system during ï¬?nancial crises.
The non-monotone implications of ï¬?nancial development. It is in-
teresting to note that this model predicts a non-monotone relationship be-
tween output and ï¬?nancial development. When ï¬?nancial intermediation is
12
prohibitively costly (Î¹ is suï¬ƒciently large), producers refrain from using it
and output is at its ï¬?rst best level. For smaller values of Î¹, ï¬?nancial inter-
mediation is used in equilibrium, and output is lower.
However, conditional on Î¹ low enough such that intermediation is used,
the resources spent on intermediation are increasing in Î¹ (equation 6).12 Thus,
a smaller value of Î¹, which can be interpreted as a more eï¬ƒcient ï¬?nancial
technology, implies a smaller output loss. The picture that emerges is there-
fore a non-monotone one: output is maximized either when Î¹ = 0, or when Î¹
is in the range that ï¬?nancial intermediation is not used at all. In the interim,
there are resources wasted on ï¬?nancial intermediation, but the resource cost
is smaller when the cost of intermediation (Î¹) is smaller.13
3.1 A generalized real model
Before presenting the dynamic monetary model, it may be useful to take an
interim step and generalize the modelâ€™s conclusion on two important dimen-
sions while still abstracting from monetary concerns.
Consider the following two extensions: ï¬?rst, assume that capital supply is
increasing in the price of capital, R. There is an increasing cost of supplying
capital; thus, as the demand for capital increases, there is an equilibrium
increase both in the supply of capital and in its price.
Second, assume that there are heterogeneous producers. The productivity
of producer i is denoted by Ai .
It is easy to cook up examples where, given these extensions, introducing a
ï¬?nancial intermediation technology increases output. For example, consider
an economy where R is ï¬?xed. In this case, the producerâ€™s decision to use
12
The expression in equation 6 is true only for Î¹ > 0 suï¬ƒciently small; in the interim,
there is a region in which intermediation is used but not by all agents.
13
More generally, in this model, the output implications of ï¬?nancial development (or
technological progress in the ï¬?nancial industry) depend on the elasticity of the use of
ï¬?nancial services. If ï¬?nancial development makes agents spend more resources on ï¬?nancial
services, it reduces output. Alternatively, if it makes agents reduce their expenditure on
ï¬?nancial intermediation, it increases output.
13
ï¬?nancial intermediation is eï¬ƒcient: he is right to take the price as given, and
tradeoï¬€ the cost of intermediation with the beneï¬?ts of increased pledgeability
(at the given price). There is no price externality.
Alternatively, maintaining the assumption that the capital supply is ï¬?xed,
ï¬?nancial intermediation may improve the allocation of capital in the presence
of heterogeneous producers. Notice that producers ï¬?nd it optimal to hire
capital only when Ai > R. Financial intermediation leads to an increase in
pledgeability and hence an increase in the demand for capital, that translates
into an increase in R. This increase in R discourages producers with low
productivity from hiring capital, freeing up capital for more productive users.
Depending on the the distribution of productivities and the cost of ï¬?nancial
intermediation, intermediation may increase or decrease output.
Given these extensions, the price externality discussed in the previous
section may not be strong enough to deem ï¬?nancial intermediation entirely
un-useful. However, regardless, it creates an ineï¬ƒciency, and there are gains
from restricting the amount of intermediation.
Proposition 1 In the above environment, consider a planner who can set
a tax on ï¬?nancial intermediation (the cost of intermediation that producers
face is (1 + Ï„ )Î¹). The plannerâ€™s objective is to maximize output net of inter-
mediation costs. Whenever the supply of capital is not entirely elastic, the
planner chooses some Ï„ > 0.
The proof of this proposition, together with other omitted proofs, is in
the appendix. The proposition implies that there is excessive ï¬?nancial in-
termediation in an â€œunregulatedâ€? equilibrium, in which producers face the
undistorted price of ï¬?nancial services. This result is due to the price exter-
nality discussed in detail in the previous section: producers fail to internalize
that, when they use ï¬?nancial services, they cause an increase in the price
of capital, that makes it more costly for everyone to produce. They are
therefore â€œtoo quickâ€? to spend resources on ï¬?nancial services.
14
4 A dynamic monetary model
In the model in the previous section, liquidity was determined only by pledge-
ability constraints. There was no notion of producers dynamically accumu-
lating liquid reserves, in anticipation of binding liquidity constraints. How
does the incentive to accumulate liquid reserves alter the conclusions, if at
all?
In this section, I allow producers to accumulate money that can be used
for the purchase of inputs. While the accumulation of money and the accu-
mulation of capital are privately substitutes, from a social perspective, they
are not: holding large amounts of liquid reserves is socially desirable.14
The analysis of the dynamic monetary model highlights that, in addition
to a â€œrealâ€? price externality, ï¬?nancial intermediation is associated with a
nominal price externality. The decision to borrow money from idle money
holders increases the money in circulation, and hikes up the nominal demand
for capital. This causes the nominal price of capital to increase, and leads
to a decline in equilibrium real balances. When intermediation is costly, the
increase in the price of capital may be entirely nominal; thus, the beneï¬?ts of
a real price appreciation in terms of allocative eï¬ƒciency and capital supply
disappear, and we are left again with a purely wasteful ï¬?nancial system.
Compared to the model in section 3, the model in this section allows for
richer welfare analysis, that incorporates distributional concerns. It is fairly
straightforward to show that in the simpliï¬?ed framework in the previous sec-
tion, intermediation may be wasteful in terms of output but is not necessarily
Pareto ineï¬ƒcient, as capital suppliers beneï¬?t from a higher price of capital.
In the dynamic version of the model, this is no longer the case: intermedi-
ation may reduce both equilibrium output and equilibrium consumption for
14
This result is in the spirit of Friedman [1969]. Recently, there has been some research
focused on the idea that â€œliquidity hoardingâ€? is socially harmful. Most of this literature
focuses on crisis situations, in which the price level is eï¬€ectively ï¬?xed in the short run.
A notable exception is Malherbe [2012], who illustrates that low real balances may be
socially desirable as they mitigate adverse selection.
15
every agent.
4.1 Setup
Consider an economy with a unit measure of inï¬?nitely lived agents, indexed
i âˆˆ [0, 1]. Unlike the model in section 3, the decision whether to become a
â€œproducerâ€? or a â€œcapital supplierâ€? will be determined endogenously in every
period. Time is discrete and indexed t âˆˆ {0, 1, ..}. In every period, each
agent receives an i.i.d productivity shock, Ai,t , which allows him to operate
an Ai,t K production technology. Denote by F (Â·) the cumulative density of
Ai,t , and denote by f (Â·) the probability density function of Ai,t , where f (Â·)
takes positive values on [0, AÂ¯].
Each agent (i) has some initial endowment of capital, ki,0 , and some
initial endowment of money, mi,0 . The within-period timing of the model
(without ï¬?nancial intermediation) is summarized in ï¬?gure 3. Denote by
ki,t and mi,t the amounts of capital and money with which agent i enters
period t. As agentsâ€™ productivity shocks are i.i.d, the distributions of ki,t
and mi,t are independent from Ai,t ; thus, there is a mismatch between the
owners of capital and its most productive users. To reduce this mismatch, a
market opens in which agents can trade capital for money, before production
takes place. In other words, productive agents can use their money to buy
additional capital for production; unproductive agents can sell their capital
in exchange for money. In this market, there is essentially a cash in advance
constraint on the purchase of existing capital.15 The nominal price of capital
in this market (at time t) is Rt . The net amount of capital purchased by
agent i at time t is denoted k Ëœi,t indicates that the agent is a
Ëœi,t . A negative k
net seller of capital. The nominal value of capital that an agent can buy is
bounded by his money, mi,t . The amount of capital that an agent can sell is
15
See Abel [1985] or Stockman [1981] for models of cash in advance constraints on the
purchase of inputs. However, money can be interpreted here more broadly as transferable
claims on output.
16
bounded by his capital (ki,t ):
mi,t Ëœi,t â‰¥ âˆ’ki,t
â‰¥k (7)
Rt
This setup departs slightly from the setup in section 3, in that absent ï¬?-
nancial intermediation, no post-production output can be pledged: the only
way that producers can purchase additional capital is by using their liq-
uid reserves. The results trivially extend to more similar settings in which
producers can pledge some post-production output even without ï¬?nancial
intermediation.
After production takes place, a second market opens in which agents can
buy and sell the ï¬?nal good at the nominal price pt . This means that producing
agents can sell the ï¬?nal good in exchange for money, and unproductive agents
can use their money holdings to buy goods.
At the end of the period, agents decide how much to consume, and how
much to save towards the next period. Agents can save in two ways: they
can carry over money to the next period (mi,t+1 ), or use current output to
install physical capital to be used or sold in the next period (ki,t+1 ).
Denote the consumption of agent i at time t by ci,t . The agentsâ€™ utility
âˆž
function is given by Ui ({ci,t }âˆž
t=0 ) = E (
t
t=0 Î² ln(ci,t )).
Social welfare is the average of expected utilities, where all agents are
1
weighted equally ( 0 E (Ui ({ci,t }âˆž t=0 ))di).
Let M denote the aggregate money supply, and let Kt denote the aggre-
gate supply of capital at time t. The money supply is assumed to be constant
across time. Capital fully depreciates after one period.16
16
The analysis can be generalized to allow for a depreciation rate 1 â‰¤ Î´ < 1, as long
Î²
as 1 âˆ’ Î´ < 1âˆ’ Î² E (A) (this condition is necessary for a solution in which there is a capital
market in equilibrium with R < âˆž). In this case, the analysis carries through with a
modiï¬?ed productivity distribution, A Ë† = A + 1 âˆ’ Î´ . An agent that employes k units of
capital can sell A + 1 âˆ’ Î´ units of output at the end of the period.
17
Figure 3: The within-period timeline of the no-intermediation economy
Agents enter Ai,t 1st market: Production 2nd market: Consumption
the period revealed capital sold output sold (ci,t) and
with mi,t, ki,t for money (R) for money (p) saving (mi,t+1, ki,t+1)
4.2 The no-intermediation economy
I restrict attention to recursive equilibria. A recursive equilibrium is deï¬?ned
as follows. There are three vectors of state variables, {Ai }iâˆˆ[0,1] , {ki }iâˆˆ[0,1] and
{mi }iâˆˆ[0,1] . To save on notation, I will denote the set of state variables by S =
{(Ai , ki , mi )}iâˆˆ[0,1] . A recursive equilibrium of the no-intermediation econ-
omy is deï¬?ned as a price function p(S ), a capital price function R(S ), a set
of consumption plans {ci (S )}iâˆˆ[0,1] , capital accumulation plans {ki (S )}iâˆˆ[0,1] ,
money accumulation plans {mi (S )}iâˆˆ[0,1] and capital purchases {k Ëœi (S )}iâˆˆ[0,1]
that jointly solve the following:
1. Agent iâ€™s optimization problem:
V (Ai , ki , mi , R, p) = max ln(ci ) + Î²EAi (V (Ai , ki , mi , R , p )) (8)
Ëœi ,ci
ki ,mi ,k
s.t. the inequalities in equation 7; mi â‰¥ 0; ki â‰¥ 0; and the budget
constraint:
Ëœi )
Ëœi + pAi (ki + k
pci + pki + mi = mi âˆ’ Rk (9)
1 Ëœi di = 0
2. Capital market clearing: 0
k
1
3. Money aggregation: 0
mi di = M
1 Ëœi )di = 1
4. Goods market clearing: 0
Ai (ki + k 0
(ci + ki )di
It is fairly straightforward to show that any equilibrium is characterized
by a cutoï¬€ A such that all agents with productivities Ai > A produce as
18
much as they can (given their capital and money supplies), and all agents
with productivities Ai < A do not produce. To see this, note that an agent
ï¬?nds it optimal to produce only if the following condition holds:
pAi > R (10)
The left hand side is the nominal return from employing one unit of
capital. The right hand side is the nominal price of capital. If this condition
holds, the agent can generate more revenue from employing his own capital
than from selling it to another producer. The same condition also implies
that the agent ï¬?nds it optimal to use his money to buy capital: with one
1
unit of money, the agent can buy R units of capital, that generate a nominal
pAi
revenue of R . The alternative strategy of holding money yields a within-
period return of 1. Dividing both sides of the above inequality by R yields
the equivalent condition, pA
R
i
> 1.
The following lemma characterizes the unique17 recursive equilibrium of
the no-intermediation economy:
Lemma 1 There is a unique recursive equilibrium in the no-intermediation
economy, in which:
R
1. The real price of capital, p
, and the production threshold, A, are time
AÂ¯
R Î² A Af (A)dA
invariant and jointly satisfy: A = p
= 1âˆ’Î²F (A)
.
(1âˆ’F (A))M
2. The nominal price of capital R is given by: R = F (A)K
.
3. The consumption of an agent with Ai â‰¥ A, ki and mi is: ci = (1 âˆ’
Î² )Ai (ki + m
R
i
). The consumption of an agent with Ai < A, ki and mi
is: ci = (1 âˆ’ Î² )A(ki + mR
i
).
17
Typically in models of ï¬?at money, there are at least two equilibria: one in which
money is valued and one in which the value of money is 0. Here, the equilibrium in which
money is not valued is ruled out mechanically by the assumption that the nominal prices of
capital and goods p(S ) and R(S ) are real-valued (if money were not valued, these nominal
prices would be âˆž).
19
AÂ¯
A Af (A)dA
4. Output is given by: Y = 1âˆ’F (A)
K.
R
5. Capital accumulation follows K = p
K.
In equilibrium, there is a time-invariant cutoï¬€, A. Log utility implies
that all agents consume a fraction 1 âˆ’ Î² of their wealth, and save the rest in
money and capital (in equilibrium, agents are indiï¬€erent between these two
saving facilities).
Agents with Ai â‰¥ A produce as much as they can: they employ their
own capital (ki ) and use their money holdings to purchase additional capital
Ëœi = mi ). Their wealth is their production revenue.
(k R
Agents with Ai < A do not produce. They sell their physical capital to
productive agents (k Ëœi = âˆ’ki ). Their real wealth is the sum of the revenue
from their capital sales ( Rk
p
i
) and the value of their money holdings ( m p
i
).
Capital accumulation is a function of the real price of capital. In other
words, a low nominal price of capital does not lead to less capital accumu-
lation, provided that the nominal price of the ï¬?nal good is proportionately
lower as well. This is why, from a social perspective, real balances (measured
as mR
or mp
) do not necessarily crowd out physical capital.
Unlike the simpliï¬?ed model in section 3, output is not at its â€œï¬?rst bestâ€?
level (deï¬?ned as the level of output that would be produced in an economy
with no liquidity constraints18 ). Capital is misallocated, since it is employed
at an entire range of productivities [A, A Â¯]19 ; output would increase if capi-
tal were reallocated from relatively unproductive producing agents to more
productive ones. Traditionally, ï¬?nancial intermediation is thought of as a
remedy to this type of misallocation. However, it turns out that misalloca-
tion may be just as bad in the presence of an unregulated ï¬?nancial sector;
18
In this framework, the â€œï¬?rst bestâ€? allocates the entire capital stock to the agent with
the highest productivity - here, output would be AK Â¯ .
19
The misallocation of capital as an equilibrium outcome of liquidity constraints is in
the spirit of Kiyotaki and Moore [1997].
20
Figure 4: The within-period timeline with ï¬?nancial intermediation
Agents enter Ai,t 1st market: Production Loans 2nd market: Consumption
the period revealed â€¢ capital sold are output sold (ci,t) and
with mi,t, ki,t for money (R) repaid for money (p) saving (mi,t+1, ki,t+1)
â€¢ borrowing
and lending
of money
in fact, if we take into account the fact that capital is employed ineï¬ƒciently
on ï¬?nancial intermediation, the allocation of capital may be even worse.
4.3 The intermediation economy
Consider an economy identical to the one described above, in which there is
a technology that allows for ï¬?nancial intermediation. Agents can use a mon-
itoring technology that allows them to pledge post-production sales. Moni-
tored producers can borrow money from non-producing agents to ï¬?nance the
purchase of additional capital.
Formally, the timing of the model is modiï¬?ed as follows (see ï¬?gure 4).
During the ï¬?rst market, in which agents exchange money for capital, agents
can also exchange money for promises on post production output (where en-
forcing such promises requires capital inputs). After production takes place,
borrowers repay lenders at the agreed upon rate.
Similar to the model in section 3, operating the intermediation technology
I
is costly in real terms. Denote by ki,t the amount of capital that agent i
purchases with borrowed (â€œintermediatedâ€?) funds. With a slight departure
from the model in section 3, I assume that the cost of intermediation is
I
proportional to ki,t . Formally, I assume that a fraction Î¸ of each unit of capital
employed through intermediation is absorbed on intermediation activities.
To illustrate, consider an agent with productivity Ai that uses intermediation
I
to ï¬?nance the purchase of ki units of capital. A fraction Î¸ of each unit of
I
capital is lost, so his net production is Ai (1 âˆ’ Î¸)ki .
21
Unproductive agents can lend both their money reserves (mi ) and their
nominal revenues from capital sales (âˆ’Rk Ëœi ). A way to think about this is
that there are several â€œroundsâ€? (within each period) in which agents can
trade capital for money, and lend the revenues from capital sales: at the
beginning of the ï¬?rst round, agents can lend mi and put up their capital for
sale. Some of the capital will be sold in the ï¬?rst round; the nominal revenues
from sold capital can be lent. A second round opens, in which productive
agents use intermediated funds to buy more capital. Revenues from second-
round capital sales are lent to productive agents, who use it to buy more
capital, and so on and so forth, until the market clears. The bottom line is
that an unproductive agentâ€™s loanable funds are mi + Rki (assuming that all
of his capital is sold, so ki = âˆ’k Ëœi ).
Compared to the no-intermediation economy, agents have an additional
choice variable which is how much money to borrow or lend. Denote the
demand for borrowing by m Ëœ i (a negative value implies that the agent is a
lender). It is assumed that agents (who are, in this model, doubling as
ï¬?nancial ï¬?rms) are subject to a reserve requirement: only a fraction 1 âˆ’ Î³
(where 0 < Î³ < 1) of their loanable funds can be lent, while the rest must be
held in liquid reserves. Thus, the maximum amount that agent i can lend is:
âˆ’m Ëœi,t ))
Ëœ i,t â‰¤ (1 âˆ’ Î³ )(mi,t + Rt (âˆ’k (11)
The return to intermediated funds is denoted Rm (in nominal terms; the
real return in terms of output is Rp
m
).
Borrowers bear the costs of intermediation. An agent with productivity
A will want to borrow as long as the returns to intermediated funds exceed
their costs:
p(1 âˆ’ Î¸)A
â‰¥ Rm (12)
R
The left hand side is the nominal revenue generated by one unit of bor-
1
rowed money: one unit of borrowed money can ï¬?nance the purchase of R
22
units of capital. A fraction Î¸ of each unit is lost on intermediation, so the
â€œnetâ€? production is (1âˆ’R
Î¸)A
units of output, that are sold at the nominal price
p. The right hand side is the cost of one unit of borrowed money, Rm .
Note that as long as the inequality above is strict, the agent will want to
borrow an inï¬?nite amount. Thus, competition among constrained producers
would necessitate an equality for A = A Â¯. Only agents with A = A Â¯ will
borrow in equilibrium, and they will be indiï¬€erent between borrowing and
not borrowing. The equilibrium return to intermediation is therefore given
by:
p(1 âˆ’ Î¸)AÂ¯
Rm = (13)
R
For unproductive agents to be willing to lend, it must be the case that
the nominal return to intermediated funds is greater than the within-period
nominal return on holding money, which is 1. Thus, for intermediation to
take place in equilibrium it must be the case that:
Â¯
p(1 âˆ’ Î¸)A
= Rm â‰¥ 1 (14)
R
This condition restricts the values of Î¸ under which the intermediation
technology will be used in equilibrium. For high values of Î¸, the equilibrium
without ï¬?nancial intermediation is a stable one: for example, if Î¸ = 1, no
unproductive agent will be willing to lend, as the nominal return to inter-
mediation is 0, which is lower than the return to holding money. The exact
condition under which intermediation will be used in equilibrium is as fol-
lows. Denote with superscript N I equilibrium values of the no-intermediation
economy. A suï¬ƒcient condition for intermediation to be used in equilibrium
is:
Â¯
pN I (1 âˆ’ Î¸)A
>1 (15)
RN I
Denote by Î¸0 the value of Î¸ for which the above condition holds with
23
NI 0
âˆ’Î¸ )A Â¯
equality ( p (1
RN I
= 1). If Î¸ = Î¸0 , agents are indiï¬€erent between using
intermediation or not; if Î¸ < Î¸0 , the equilibrium without ï¬?nancial intermedi-
ation is not stable, as at no-intermediation equilibrium prices, unproductive
agents strictly prefer lending. I will therefore focus on the parameter range
Î¸ < Î¸0 .
A recursive equilibrium of the intermediation economy is deï¬?ned as a price
function p(S ), a capital price function R(S ), a price function for intermedi-
ated funds Rm (S ), a set of consumption plans {ci (S )}iâˆˆ[0,1] , a set of capital
accumulation plans {ki (S )}iâˆˆ[0,1] , money accumulation plans {mi (S )}iâˆˆ[0,1] ,
capital purchase plans {kËœi (S )}iâˆˆ[0,1] , intermediated money plans {m Ëœ i (S )}iâˆˆ[0,1]
I
and plans for capital purchased with intermediation {ki (S )}iâˆˆ[0,1] that jointly
solve the following:
1. Agent iâ€™s optimization problem:
V (Ai , ki , mi , R, p, Rm ) = max ln(ci )+Î²EAi (V (Ai , ki , mi , R , p , Rm ))
Ëœi ,m
ki ,mi ,k I ,c
Ëœ i ,ki i
s.t. the inequalities in equation 7; the inequality in equation 11; mi â‰¥ 0;
I
ki â‰¥ 0; mËœ i âˆ’ Rki â‰¥ 0; and the budget constraint:
pci + pki + mi = mi + m Ëœi + k I ) + pAi (ki + k
Ëœ i (1 âˆ’ Rm ) âˆ’ R(k Ëœi + (1 âˆ’ Î¸)k I )
i i
1 Ëœ I
2. Capital market clearing: 0
(ki + ki )di = 0
1
3. Money aggregation: 0
mi di = M
1 Ëœi + (1 âˆ’ Î¸)k I )di = 1
4. Goods market clearing: 0
Ai (ki + k i 0
(ci + ki )di
The following proposition characterizes the properties of the recursive
equilibrium:
Proposition 2 For Î¸ < Î¸0 ,
24
1. There exists a unique recursive equilibrium. For Î¸ suï¬ƒciently large, the
equilibrium is welfare inferior to the no-intermediation economy.
2. The return to intermediated funds (Rm ) is decreasing in Î¸, and ap-
proaches 1 as Î¸ â†’ Î¸0 .
3. Assume that mi and ki are equally distributed across agents. Denote
by ci (Î¸) the equilibrium consumption of agent i given Î¸, and let cN i
I
denote the agentâ€™s consumption in the no-intermediation economy. At
the limit Î¸ â†’ Î¸0 , the values of R
p
and K converge to their levels in the
no-intermediation equilibrium, and:
cN
i
I
âˆ’ ci F (A)(1 âˆ’ Î³ )
lim 0 = (16)
Î¸â†’âˆ’ Î¸ ci 1 âˆ’ F (A)
The welfare comparison with the no-intermediation economy depends on
Î¸. For Î¸ close to 0, the â€œstandardâ€? intuition holds, and ï¬?nancial intermedia-
tion increases output. The transfer of money to the most productive agents
improves the allocation of capital within the productive sector. When this is
done at a negligible real cost, equilibrium output increases. Intermediation
may also imply a favorable redistribution of surplus, as unproductive agents
can realize high returns to their intermediated funds.
However, these acclaimed beneï¬?ts of ï¬?nancial intermediation are relevant
only when Î¸ is small; for Î¸ close to Î¸0 , welfare is lower than in the no-
intermediation economy. The intuition behind the welfare loss is as follows.
For intermediation to improve the allocation of capital, it must induce inef-
ï¬?cient producers to switch over from self-ï¬?nancing to lending. However, if
the costs of intermediation are high, self-ï¬?nancing producers will continue to
opt for production; the funds channeled through intermediation will originate
primarily from unproductive agents, who would otherwise choose to hold idle
money reserves. The capital purchased with these funds would be relatively
unproductive, as a large fraction of it would be lost on intermediation.
25
In fact, as Î¸ â†’ Î¸0 , the price externality of ï¬?nancial intermediation is
purely nominal. At the limit, the real price of capital is the same as in
the no-intermediation economy, and the decision whether or not to become
a producer remains the same as well. The only prices that change are the
nominal prices: instead of holding idle reserves, unproductive agents lend
their money (at a rate of return close to 1). This inï¬‚ates the nominal price
of capital. In turn, the higher nominal price of capital implies that the return
to carrying over money to the next period is lower: one unit of money will be
able to purchase less capital. The lower return to holding money translates
into a higher nominal price of the ï¬?nal good (p).
At the limit Î¸ â†’ Î¸0 , the ï¬?nancial intermediation technology is analogous
to a counterfeit money printing machine. Financial intermediation increases
the amount of money in circulation, which raises nominal prices. As the
real price of capital remains unchanged, production decisions and capital
accumulation decisions - that depend the real price of capital - are the same
as in the no-intermediation economy. From a social perspective, the resources
spent on ï¬?nancial intermediation are analogous to the operating costs of a
counterfeit-money printing machine.
How realistic is the case Î¸ â†’ Î¸0 ? One way to gauge at this is by looking
at the return to deposits. By Proposition 2, the return to deposits approaches
1 only as Î¸ â†’âˆ’ Î¸0 . As a deposit rate of 1 is close to observed levels in the
US,20 the choice Î¸ = Î¸0 does not seem unrealistic.
Under this assumption, we can use equation 16 to quantify the implied
consumption gain from eliminating the ï¬?nancial system. To calibrate the
expression in equation 16, note that, in this model, 1âˆ’ F (A)
F (A)
is the ratio of
producers to lenders. Let L denote the average amount that non-producers
lend. The total amount of lending is therefore LF (A). This amounts to
20
In the US, most checking accounts bear no interest. However, there are some services
associated with checking accounts (such as access to ATMs) that are provided to depositors
free of charge. This can be thought of as a small return to deposits.
26
LF (A)
1âˆ’F (A)
units of â€œloansâ€? per producer, which is also the average size of loans.
For the average loan, I use $550, 000, the average size of corporate and
industrial loans in the US in 2007.21 For L, I use $400, 000 times (1 âˆ’
Î³ ). $400, 000 is the average size of US household wealth in 2007, excluding
housing wealth.22 A fraction 1 âˆ’ Î³ of this wealth can be lent.
We therefore have:
400, 000(1 âˆ’ Î³ )F (A) (1 âˆ’ Î³ )F (A) 550, 000
550, 000 = â‡’ = = 1.375 (17)
1 âˆ’ F (A) 1 âˆ’ F (A) 400, 000
In other words, by equation 16, the estimated consumption gains from
eliminating the ï¬?nancial system are 137.5%. This is obviously an unrealisti-
cally high number.
Part of the problem is that, in this model, Î¸ captures both the marginal
cost of ï¬?nancial intermediation and its average cost. In the presence of a
decreasing returns intermediation technology, the average cost of intermedi-
ation is lower than its marginal cost, implying some productivity gains from
using ï¬?nancial intermediation and some proï¬?ts associated with intermedi-
ation activities. However, adding decreasing returns to the intermediation
technology is not a trivial extension.23
To summarize, while this model highlights a price externality that po-
tentially leads to quantitatively signiï¬?cant welfare losses, it is too soon to
conclude that the ï¬?nancial system should be eliminated entirely. A richer
quantitative investigation that takes into account decreasing returns (both
in intermediation and in production) would be necessary in order to assess
the optimal scope of ï¬?nancial intermediation.
21
Data source: Average Loan Size for All C&I Loans, All Commercial Banks (EAANQ),
Federal Reserve Bank of St. Luis (FRED database)
22
Data source: table 1.B, Wolï¬€ [2010].
23
When proï¬?ts from intermediation are included in income, the model is no longer linear
and becomes harder to solve analytically. I therefore reserve this important quantitative
exercise for future work.
27
Excessive ï¬?nancial intermediation in an unregulated economy. While
it is diï¬ƒcult to use this model to quantitatively assess the â€œnetâ€? welfare im-
plications of ï¬?nancial intermediation, the insight of Proposition 1 extends
to this richer framework: an unrestricted ï¬?nancial system absorbs too many
inputs. The proposition below summarizes this result:24
Proposition 3 (Generalization of Proposition 1) when Î³ â†’ 0, there
are excessive resources spent in equilibrium on ï¬?nancial intermediation. There
exists Î³ 0 such that for every Î³ < Î³ 0 , the equilibrium allocation given Î³ is wel-
fare inferior to the allocation given Î³ 0 .
5 Conclusion
The prevailing sense that the ï¬?nancial sector is â€œtoo largeâ€? may at ï¬?rst seem
at odds with traditional economic theory. Typically, in hyper-competitive
environments such as the ï¬?nancial sector, income reï¬‚ects the contribution to
output.
This paper suggests that the ï¬?nancial sector is inherently diï¬€erent. The
income generated by the ï¬?nancial sector does not reï¬‚ect its contribution to
output, because of the partial equilibrium nature of this income. Unlike
other sectors, if ï¬?nancial intermediation were restricted, the economyâ€™s need
for it would decline as well, as producers would face lower input prices and
would be better able to self-ï¬?nance. As agents do not internalize this price
externality, there is excessive ï¬?nancing in an unregulated equilibrium.
24
Unlike Proposition 1, the proposition is framed in terms of a reserve requirement
rather than a tax on intermediation services. The implications are the same: both are
instruments to restrict the usage of ï¬?nancial services.
28
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30
A Proofs
Proof of Proposition 1. The productivity threshold above which produc-
ers ï¬?nd it optimal to pay for ï¬?nancial services is given by:
Â¯ âˆ’ R) l Â¯ âˆ’ R) l âˆ’ Î¹
(A = (A (18)
R R
The LHS is the producerâ€™s proï¬?ts without paying for intermediation ser-
vices, and the RHS is the producers proï¬?ts with paying for intermediation
l
services (to see this, notice that R is the amount of capital that the producer
l
can hire absent ï¬?nancial intermediation, and R is the amount of capital that
the producer can hire with ï¬?nancial intermediation).
Assume that there is a small positive measure of agents with A = A Â¯, who
are indiï¬€erent between hiring capital or not. Consider ï¬?rst the case where
K is completely elastic: in other words, the price R is ï¬?xed. In this case,
the planner is indiï¬€erent whether or not these agents use the intermediation
technology. To see this, note that since R is ï¬?xed, the amount of capital that
l
other producers can hire remains unaltered at R (or, if using intermediation,
l
R
). Since the costs of ï¬?nancial services are exactly oï¬€set by the additional
amount of output that agents with A = A Â¯ can produce, output is the same
regardless of whether or not these agents choose to use ï¬?nancial services.
Consider next the case in which K s is not completely inelastic: in other
words, a decline in the demand for capital translates into a lower price of
capital (lower R). In this case, output is increasing if agents with threshold
productivities (A = A Â¯) decide not to use intermediation: if they refrain from
using intermediation, the demand for capital declines and R declines as well.
This decline in R will increase output in two ways: (a) all agents with A â‰¥ A
l l
will be able to hire more inputs, as the decline in R means that R and R are
higher, and (b) There will be new producing agents, who ï¬?nd it optimal to
hire capital given the lower R. Note that while these agents will lower the
average productivity of capital, total output will be higher (as the amount
31
l l
of capital that agents can hire is given by R or R , and R is lower).
Thus, while agents with A = A Â¯ are indiï¬€erent with respect to using
ï¬?nancial services, output is strictly higher when they decide not to.
Proof of Lemma 1. The equilibrium is characterized by a cutoï¬€ A, such
that agents with productivity A â‰¥ A produce as much as they can, and
agents with productivity A < A do not produce. Denote Ï? = 1 âˆ’ F (A).
First, I show that agents consume a fraction 1âˆ’Î² of their wealth. To prove
this, I am going to calculate âˆ‚EV
âˆ‚m
and âˆ‚EVâˆ‚k
. Let cu denote the consumption
given that the agent is unproductive (he has Ai < A), and let cA denote
consumption in case the agent is productive and his productivity is A.
Â¯
A
âˆ‚EV 1 A
= (1 âˆ’ Ï?) u (cu ) + f (A) u (cA )dA (19)
âˆ‚m p A R
Note that for all agents R âˆ‚EV
âˆ‚m
= âˆ‚EV
âˆ‚k
. The monetary cost of an ad-
ditional unit of k is the current price of output, p. Thus, for agents to
carry both money and capital into the next period, it must be the case that
âˆ‚EV 1 âˆ‚EV
âˆ‚m
=p âˆ‚k
.
Substituting the above,
âˆ‚EV R âˆ‚EV R
= â‡’ =1 (20)
âˆ‚m p âˆ‚m p
Consider the FOCs of the agentâ€™s problem, where Î» is the Lagrange mul-
tiplier on the budget constraint. The FOC with respect to c is u (c) = pÎ»,
and the FOC with respect to m is Î² âˆ‚EVâˆ‚m
= Î». Combining the two, we get
that:
Â¯
A
1 âˆ‚EV 1 A
u (c) = Î² = Î² ((1 âˆ’ Ï? ) u (cu ) + f (A) u (cA )dA) (21)
p âˆ‚m p A R
Denote the agentâ€™s end-of-the-period nominal wealth by W . Wu denotes
the agentâ€™s next period wealth if he is unproductive (Ai < A), and WA
32
denotes the agentâ€™s next period wealth if he is a producer with productivity
A. Under the conjecture that agents consume a fraction 1 âˆ’ Î² of their wealth,
we have that:
Â¯
A
1 1 1 1 A 1
W
=Î² ((1 âˆ’ Ï? ) + f (A)( )dA) (22)
p (1 âˆ’ Î² ) p p (1 âˆ’ Î² ) W
p
u
A R (1 âˆ’ Î² ) WA
p
Â¯
A
1 1 Ap 1
=Î² ((1 âˆ’ Ï? ) + f (A)( )dA) (23)
W Wu A R WA
If the agent is unproductive next period, his nominal wealth is Wu = m +
R k . If the agent is productive next period with productivity A, his nominal
wealth is WA = p A(k + m R
). The above condition can therefore be rewritten
as:
A Â¯
1 1 1 1
=Î² ((1 âˆ’ Ï? ) + f (A)( )dA) (24)
W m +Rk A R (k + m
R
)
1 1 Î²
=Î² ((1 âˆ’ Ï? ) +Ï? )= (25)
m +Rk Rk +m Rk +m
Or, R k + m = Î²W . As R = p, we have that:
pk + m = Î²W (26)
So the conjecture that the agent saves a fraction Î² of his income is veriï¬?ed.
Note the market clearing condition that determines R:
Ï?M
Ï?M = (1 âˆ’ Ï?)RK â‡’ R = (27)
(1 âˆ’ Ï?)K
Note that Ï? < 1, as there are agents with 0 productivity that will sell
their capital at any positive R. By equation 20, the ratio of next periodâ€™s
Ï?M
money and capital is given by the current price level p ( (1âˆ’ Ï? )K
= p).
To ï¬?nish the characterization of the equilibrium, we need to ï¬?nd the
equilibrium p. Note that the income of the producing agents is the entire
33
sales revenues from production, denoted Y . The savings of producing agents
is a fraction Î² of ï¬?nal sales.
The wealth of unproductive agents is the entire money supply (the money
they held at the beginning of the period, and the revenues from selling their
capital). Their savings are equal to a fraction Î² of the money stock.
Combining the two, and using M = M , we get that:
Î² (pY + M ) = M + pK (28)
Ï?M
Rewriting using the relationship (1âˆ’Ï? )K
= p:
Ï? M
Î² (pY + M ) = M + M= (29)
1âˆ’Ï? 1âˆ’Ï?
Substituting for Y , after some manipulations:
Ï?M 1 âˆ’ Î² (1 âˆ’ Ï? )
p= Â¯ ( ) (30)
K
A
Af ( A ) dA Î² (1 âˆ’ Ï? )
A
I conjecture an equilibrium with a time invariant A (and hence time
p
invariant Ï?). Recall that A is characterized by R
p
= A. Note that R is time
invariant:
AÂ¯
Ï?M
( 1âˆ’ Î² (1âˆ’Ï?)
Î² (1âˆ’Ï?)
)
p K A Af (A)dA 1 âˆ’ Î² (1 âˆ’ Ï?) 1 âˆ’ Î²F (A)
= Ï?M
= AÂ¯ = AÂ¯ (31)
R (1âˆ’Ï?)K Î² Af (A)dA Î² Af (A)dA
A A
It follows that A is time invariant and equal to the solution of the following
equation:
Â¯
A
(1 âˆ’ Î²F (A))A = Î² Af (A)dA (32)
A
Â¯).25
Note that this equation has a unique solution for some A âˆˆ (0, A
25
For A = 0, the LHS is 0 and the RHS is strictly positive. For A = A Â¯, the LHS is
strictly positive and the RHS is 0. By continuity, there is an interior solution. To show
34
To solve for K (and check that it is positive and less than Y ), equate
output supply and output demand:
M M
Y = (1 âˆ’ Î² )(Y + ) + K = (1 âˆ’ Î² )(Y + M 1âˆ’Î² (1âˆ’Ï?)
)+K (34)
p ( Î² (1âˆ’Ï?) )
Y
Î²Ï?
After some manipulations, this yields K = 1âˆ’Î² (1âˆ’Ï?)
Y . It follows that
Y > K > 0.
To ï¬?nish solving for K , substitute for Y :
AÂ¯ AÂ¯
Î²Ï? A
f (A)dA Î² A
f (A)AdA R
K = K= K=K = K (35)
1 âˆ’ Î² + Î²Ï? Ï? 1 âˆ’ Î²F (A) p
Proof of Proposition 2. Considering the partial derivatives of EV with
respect to m and k , it is possible to carryout the analysis as in the proof
of Lemma 1 (noting that the return to one unit money for an unproductive
agent is Î³ + (1 âˆ’ Î³ )Rm ), and obtain R = p. Similarly, analogous arguments
to the proof of Lemma 1 imply that agents save a fraction Î² of their wealth.
The value of A is given by the following indiï¬€erence condition:
pA Â¯
p(1 âˆ’ Î¸)A
= Î³ + (1 âˆ’ Î³ ) (36)
R R
Â¯
Note that p(1âˆ’R
Î¸ )A
â‰¥ 1. When the relation holds with equality, the reserve
requirement is not binding and unproductive agents are indiï¬€erent between
lending and holding cash. It turns out that this leads to a contradiction:
assuming that unproductive agents lend a constant fraction of their money
and solving for the equilibrium R (through the capital market clearing con-
that it is unique, note that the derivative of the LHS minus the RHS is positive throughout:
âˆ‚ (LHS âˆ’ RHS )
= âˆ’Î²Af (A) + 1 âˆ’ Î²F (A) + Î²Af (A) = 1 âˆ’ Î²F (A) > 0 (33)
âˆ‚A
Thus, there is a unique intersection LHS âˆ’ RHS = 0, and hence a unique A that solves
this equation.
35
dition) yields K = AK . Equating the supply of savings with the demand
for savings in a manner similar to the steps taken in the proof of Lemma 1
implies that A must be equal to its no-intermediation level, and that the real
price of capital ( R
p
) is also the same. This is a contradiction, as at these prices
agents are not indiï¬€erent between lending or not (by the assumption Î¸ < Î¸0 ).
We now continue to solve for the equilibrium, with the knowledge that the
reserve requirement is a binding constraint. Using this fact, we conclude that
non-producing agents lend a fraction 1 âˆ’ Î³ of their loanable funds, and the
nominal price of capital is given by the following market clearing condition:
M (Ï? + (1 âˆ’ Ï?)(1 âˆ’ Î³ ))
(1 âˆ’ Ï?)RK = Ï?M + (1 âˆ’ Ï?)(1 âˆ’ Î³ )(M + RK ) â‡’ R =
Î³ (1 âˆ’ Ï?)K
(37)
The demand for savings is given by:
Â¯
A
M Î³ Â¯
(1 âˆ’ Î¸)A
Î²( f (A)AdA(K + ) + (1 âˆ’ Ï?)( + (1 âˆ’ Î³ ) )(M + RK )) (38)
A R p R
The supply of savings is given by:
M M (Ï? + (1 âˆ’ Ï?)(1 âˆ’ Î³ )) M M 1
K + = + = (39)
p Î³ (1 âˆ’ Ï?)p p p Î³ (1 âˆ’ Ï?)
Equating supply and demand yields:
Â¯
A
pM pA M
Î²( f (A)AdA(pK + ) + (1 âˆ’ Ï?) (M + RK )) = (40)
A R R Î³ (1 âˆ’ Ï?)
Substituting for R, after some manipulations this yields:
M (Ï? + (1 âˆ’ Ï?)(1 âˆ’ Î³ ))
p= AÂ¯ (41)
Î³ (1 âˆ’ Ï?)Î²K ( A
f (A)AdA + (1 âˆ’ Ï?)A)
36
p
Note that R
is given by:
M (Ï?+(1âˆ’Ï?)(1âˆ’Î³ ))
AÂ¯
p Î³ (1âˆ’Ï?)Î²K ( A f (A)AdA+(1âˆ’Ï?)A) 1
= M (Ï?+(1âˆ’Ï?)(1âˆ’Î³ ))
= AÂ¯ (42)
R Î²( f (A)AdA + (1 âˆ’ Ï?)A)
Î³ (1âˆ’Ï?)K A
When is this value of Rp
consistent with a binding reserve requirement?
When the return on intermediated funds is greater than 1:
Â¯
Â¯
p(1 âˆ’ Î¸)A A
Â¯ â‰¥ Î²(
â‰¥ 1 â‡’ (1 âˆ’ Î¸)A f (A)AdA + (1 âˆ’ Ï?)A) (43)
R A
Where A solves:
R Â¯â‡’
A =Î³ + (1 âˆ’ Î³ )(1 âˆ’ Î¸)A (44)
p
Â¯
A
A =Î³Î² ( Â¯
f (A)AdA + F (A)A) + (1 âˆ’ Î³ )(1 âˆ’ Î¸)A (45)
A
The above equation has a solution: for A = A Â¯, the LHS is greater than
the RHS. For A = 0, the LHS is smaller than the RHS. From continuity, there
exists a solution. This solution is unique, and consistent with the condition
in equation 43. To see that it is unique, consider the derivative of the LHS
minus the RHS:
âˆ‚LHS âˆ’ RHS
= 1 âˆ’ Î³Î² (âˆ’f (A)A + f (A)A + F (A)) = 1 âˆ’ Î³Î²F (A) > 0 (46)
âˆ‚A
It follows that there is a unique solution. To see that it complies with the
condition in equation 43, note that the solution to A is increasing in Î³ : for
Î³ = 1, this is the â€œno-intermediationâ€? level of A. For Î³ = 0, the solution is
A = (1 âˆ’ Î¸)A Â¯. The solution is therefore somewhere in the middle. It follows
that A is a weighted average of (1 âˆ’ Î¸)A Â¯ and something that is smaller than
(1 âˆ’ Î¸)AÂ¯: thus, the condition in equation 43 is satisï¬?ed.
37
Note that A is decreasing in Î¸; to see that Rm is decreasing in Î¸, note
that, using the above, Rm can be written as:
Aâˆ’Î³ R
Â¯
(1 âˆ’ Î¸)A 1âˆ’Î³
p
1 A
Rm = R
= R
= ( Â¯ âˆ’ Î³) (47)
p p
1 âˆ’ Î³ Î²( A
f (A)AdA + F (A)A)
A
It is easy to see that this expression is increasing in A; since A is decreasing
in Î¸, it follows that Rm is decreasing in Î¸.
To ï¬?nish the characterization of the equilibrium, note that, as A is time
invariant, it follows that RK is time invariant. Thus, pK = R K = RK â‡’
K =R p
K.
Let A(Î³, Î¸) denote the equilibrium value of A. Note that for Î¸ â†’âˆ’ Î¸0 ,
A â†’ A(Î³ = 1) (A converges to its â€œno-intermediationâ€? equilibrium level).
To see this, note that for A = (1 âˆ’ Î¸0 )A Â¯, the equation characterizing A is
consistent with the no-intermediation equilibrium:
Â¯
A
A(1 âˆ’ Î³Î²F (A)) = Î³Î² Â¯
f (A)AdA + (1 âˆ’ Î³ )(1 âˆ’ Î¸0 )A (48)
A
Replacing (1 âˆ’ Î¸0 )A Â¯ = A, we get an equation that characterizes the
equilibrium value of A in the no intermediation economy. Thus, as Î¸ â†’âˆ’ Î¸0 ,
A converges to the no intermediation economy. Using this, it is easy to show
that Rp
converges to its no-intermediation level, as does K . It follows that
return to intermediated funds converges to Rm = 1.
However, the equilibrium prices of goods and capital are higher:
M (Ï? + (1 âˆ’ Ï?)(1 âˆ’ Î³ )) Ï?M Ï?M
R= > > = RN I (49)
Î³ (1 âˆ’ Ï?)K Î³ (1 âˆ’ Ï?)K (1 âˆ’ Ï?)K
Thus, as the limit Î¸ â†’ Î¸0 of R
p
is the same as in the no intermediation,
it can be concluded that the limiting p is higher as well. The increase in
38
nominal prices implies lower welfare. To see this, note that the consumption
of an agent with Ai = A Â¯, ki and mi is given by:
Â¯(ki + mi )
Â¯) = (1 âˆ’ Î² )A
c(ki , mi , A (50)
R
Â¯ â‰¥ Ai â‰¥ A is given by:
Note that the consumption of an agent with A
mi Ai Â¯)
c(ki , mi , Ai â‰¥ A) = (1 âˆ’ Î² )Ai (ki + ) = Â¯ c(ki , mi , A (51)
R A
The consumption of an agent with ki , mi and Ai < A is given by:
1 R mi
c(ki , mi , Ai < A) =(1 âˆ’ Î² ) (Rki + mi ) = (1 âˆ’ Î² ) (ki + ) (52)
p p R
R Â¯) = A c(ki , mi , A
Â¯)
= Â¯ c(ki , mi , A Â¯ (53)
pA A
Â¯) in the no-intermediation econ-
Consider the diï¬€erence between c(ki , mi , A
Â¯) in the economy with ï¬?nancial intermediation:
omy and c(ki , mi , A
Â¯) âˆ’ cI (ki , mi , A 1
Â¯) = (1 âˆ’ Î² )Am
Â¯ i(1
cN I (ki , mi , A âˆ’ ) (54)
RN I RI
Â¯ mi K (1 âˆ’ Ï?)( 1 âˆ’
= (1 âˆ’ Î² )A
Î³
) (55)
M Ï? Ï? + (1 âˆ’ Ï?)(1 âˆ’ Î³ )
Substituting mi = M and ki = K , with some manipulations equation 16
follows.
Proof of Proposition 3. By equation 44, when Î³ â†’ 0, A â†’ (1 âˆ’ Î¸)A Â¯.
In this range, capital employed through ï¬?nancial intermediation is the least
productive capital: all agents that are self ï¬?nancing realize returns that are
greater than A = (1 âˆ’ Î¸)A Â¯.
The rest of the proof is similar to the proof of Proposition 1: agents with
Ai = AÂ¯ are indiï¬€erent whether to borrow or not (the cost of capital is exactly
39
oï¬€set by its returns). However, the social planner is not indiï¬€erent: if the
agent with Ai = A Â¯ refrains from borrowing, the nominal price of capital
would be lower and self-ï¬?nancing producers (which are more productive on
average since A â‰ˆ (1 âˆ’ Î¸)AÂ¯) would be able to hire more capital. Thus, while
the borrowers are indiï¬€erent, the planner strictly prefers less borrowing.
40