USING  PIGOUVIAN  TAXES  TO  CORRECT  BANKING  EXTERNALITIES:      

A  CAUTIONARY  TALE*

Enzo Dia Ricercatore, Dipartimento di Economia, Metodi Quantitativi e Strategie di Impresa

Università degli Studi di Milano-Bicocca Piazza Ateneo Nuovo 1, Milano 20126, Italy

E-mail: enzo.dia@unimib.it

David VanHoose

Professor of Economics and Herman Lay Professor of Private Enterprise Hankamer School of Business, Baylor University

One Bear Place #98003, Waco, TX 76798 E-mail:David_VanHoose@baylor.edu

January 17, 2013

Abstract

This paper examines a framework in which banks fail to incorporate into their

individually optimal balance-sheet decisions combined effects on aggregate market lending that feed back via a strategic complementarity to influence the size of loan losses that they confront. It shows that in a setting with identical banks that fail to monitor their loans to mitigate such losses, a Pigouvian tax on lending can correct the resulting excessive-credit-expansion externality. In addition, however, the paper shows that in an expanded version of the model, in which banks can choose whether or not to incur a monitoring cost to eliminate the loan losses, the Pigouvian tax also has a perverse impact on the composition of lending. On the one hand, the impact of the tax on non-monitoring banks is smaller when their margins are thinner, while the tax falls more heavily on monitoring banks when their margins are greater. On the other hand, implementation of the tax compels monitoring banks to internalize the negative externality that falls on non-monitoring banks but not on them, which makes monitored loans less profitable. Consequently, although the tax reduces the supply of loans of both classes of banks, it tends to add to a growing market share for non-monitoring banks to a greater extent when their activity poses increased profit risks along their internal margins even as it tends to depress along the external margin the share of banks that voluntarily monitor loans to mitigate loan losses. Thus, the paper highlights unintended effects of applying Pigouvian taxation to banking markets that have not received careful attention in the literature to date. *Initial work on this paper by David VanHoose was completed during the term of a Baylor University research sabbatical, for which the author is grateful. Key Words: Pigouvian banking taxes, over-lending externality JEL Codes: G21, G28

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USING  PIGOUVIAN  TAXES  TO  CORRECT  BANKING  EXTERNALITIES:      

A  CAUTIONARY  TALE

1. Introduction

There has been a recent upsurge in interest in the idea of utilizing

traditional public-sector-style tax instruments to address financial market

externalities arising from divergences between social and private marginal

costs—that is, marginal social damages—in financial markets. A number of

economists have suggested trying to eliminate marginal damages via Pigouvian

taxes set exactly equal to the values of those marginal damages. In theory,

imposing such a tax on participants in financial markets might bring private

marginal cost into alignment with social marginal cost and thereby induce sellers

to reduce production, thereby alleviating negative externalities.

A number of authors have proposed the potential for several types of

financial-sector externalities. As Longworth (2011) notes, Brunnermeier et al.

(2009) provide a particularly extensive list. Recently, however, De Nicolòet al.

(2012) have developed three general categories: (1) externalities arising from

strategic complementarities, such as a tendency for banks to engage in

competitive interactions that generate decisions exposing the institutions to

greater loss risks (what Brunnermeier et al. call “excessive credit expansion”); (2)

externalities related to fire sales, or forced asset liquidations during times in

which potential buyers of those assets also are experiencing difficulties; and (3)

externalities related to interconnectedness of financial firms, most commonly

called “systemic risk.”

As noted by De Nicolò et al., most specific proposals for application of

Pigouvian taxes are aimed at addressing externalities perceived to arise from

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systemic interconnectedness. For instance, Acharya et al. (2011), Perotti and

Saurez (2011), and Shin (2010) focus attention on directing Pigouvian taxes in an

effort to mitigate negative externalities associated with systemic risks created by

financial firms’ interconnected activities. Nevertheless, others, such as the

International Monetary Fund Staff (2010), Keen (2011), Lockwood (2011), and

Shackelford et al. (2010) have discussed and evaluated the merits of applying tax

policies more broadly. Furthermore, Bianchi et al. (2011), Bianchi and Mendoza

(2011), and Jeanne and Korinek (2010) have studied the possibility of preventing

excessive credit expansion via imposition of Pigouvian taxes on debts issued by

financial firms to private borrowers.

The analysis conducted in this paper focuses on Pigouvian taxation of bank loans

aimed at addressing a form of strategic-complementarity externality identified by

Brunnermeier et al. (2009) and De Nicolò et al. (2012). The paper’s analysis is based on

the competitive-heterogeneous-banks model developed by Kopecky and VanHoose

(2006), which is extended to allow both for loan losses that banks confront to increase as

the aggregate volume of lending increases and for the possibility of imposing taxation of

bank lending. A negative externality arises if banks that fail to monitor loans in order to

eliminate such loan losses regard their individual contributions to this aggregate effect on

loan losses as negligible and consequently engage in excessive credit expansion. In

principle, imposing an appropriate tax on lending by banks that expose themselves to

such losses can bring their loan volumes back into line with the level consistent with

recognition of this externality.

Another feature of the model, however, is the capability of banks to incur

monitoring costs—which vary across institutions—in order to mitigate loan losses.

Nevertheless, this first-best, private-sector solution to the over-lending externality is

pursued only by a fraction of banks that opt to monitor their loans. The remainder of the

banking system does not monitor, however. Consequently, the non-monitoring portion of

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the banking system experiences loan losses that are boosted by the strategic-

complementarity externality.

In this expanded framework with two classes of banks, the imposition of a

Pigouvian tax continues to mitigate the negative over-lending externality that is created

by all banks but falls only on the non-monitoring banks. For these banks, imposing the

tax also has a smaller effect when their margins are narrower but a greater impact when

their margins net of loan losses are larger. Monitoring banks confronting the tax are

compelled to internalize the negative externality that adversely affects non-monitoring

banks but which they incur costs to eliminate, which reduces their profitability from

monitoring loans. Hence, although the tax addresses the negative externality by reducing

aggregate loan supply, it also tends to adversely influence the composition of aggregate

lending across both groups of banks. It does so by tending to add to a growing market

share for non-monitoring banks to a greater extent when their activity poses increased

profit risks on their internal margins even as it depresses along the external margin the

share of banks that voluntarily monitor loans to eliminate loan losses. Thus, the paper

highlights unintended effects of applying Pigouvian taxation to banking markets that

have not received careful attention in the literature to date.

Section 2 lays out a basic banking model in which identical banks engage in

individually optimal behavior that generates an externality that lead to socially excessive

credit expansion and shows how a Pigouvian tax can correct the externality. Section 3

expands the model to allow for a fraction of institutions in the banking system to opt to

employ heterogeneous monitoring technologies mitigate loan losses. A solution for

aggregate lending in this mixed banking system is developed and used to examine special

cases in which either all banks leave their loans unmonitored or choose to incur

differential costs to eliminate loan losses. Section 4 shows that implementation of the

type of Pigovian tax scheme examined in Section 2 would, in the banking system made

up of both non-monitoring and monitoring banks, have the intended effect of correcting

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the over-lending externality while having the unintended immediate effect altering the

composition of total lending in a way that tends to reduce aggregate loan quality. Section

5 concludes with a brief summary of the paper’s conclusions and a discussion of

implications for further research on the application of Pigouvian taxes to banking.

2. The Banking Framework and a Pigouvian Tax without Monitoring

This section discusses the basic banking framework examined in the paper,

introduces a simply strategic-complementarity externality, and shows how a Pigouvian

tax could mitigate this externality when banks fail to monitor their loans. Complications

arising from loan monitoring by a portion of institutions in the banking system are

considered in subsequent sections.

The banking system is made up of numerous institutions that take interest rates on

securities, loans, and deposits as given and that initially are assumed—for expository

purposes in this section—to face identical cost conditions. An individual bank i that does

not monitor loans to eliminate identifiable and avoidable sources of loan losses faces the

following (potentially after-tax) profit function:

[ ]{ } ( ) ( ) ( )2 2 2( ) ,2 2 2

NMi L i S i D i i i iL r L r S r D L S Dζ σ γπ α κ τ= − − + − − − − (1)

where L≡ loans that earn the gross market loan rate of return rL ; S≡ securities that earn

the gross and net market security rate of return rS ; D≡ deposit funds for which a bank

pays the gross and net market deposit rate of return rD ;α ,δ , ζ , σ , and γ are

nonnegative parameters discussed immediately following; andτ is a per-dollar tax rate

that may be assessed on the volume of a bank’s lending.

The parameters ζ , σ , and γ govern the magnitudes of quadratic resource costs

for individual balance-sheet items. Positive values of these parameters ensure that

marginal resource costs are upward-sloping in relation to the levels of loans, securities,

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and deposits, respectively. For the sake of simplicity, it is assumed that these resource

costs are separable.

A proportion 1-α of loan losses arise for unsystematic—primarily

macroeconomic—reasons outside of the control of the banking system. A fraction of

loans given by δ <α , however, are lost as a consequence of actions by borrowers that

erase part of the loan proceeds. Thus, a bank that fails to monitor in order to prevent such

actions by borrowers earn an effective net pre-tax rate of return on lending equal to

( ) Lrα κ− .

An additional assumption is that the loan-loss parameter depends positively on the

aggregate market quantity of loans, L. From the perspective of a social planner, we

assume that the effect of a credit expansion by an individual non-monitoring bank on the

per-bank loan loss is given by the linear relationship Lκ δ β= + where β ≡ ∂κ∂L

⎛⎝⎜

⎞⎠⎟

and

NM NM NMi i i

L LL L L Lκ κ β βλ φ

⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂⎛ ⎞= = = =⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠ ⎝ ⎠, which is the same for each of the banks

under the assumption that NMi

LL

λ⎛ ⎞∂= ⎜ ⎟∂⎝ ⎠

is identical across banks. Hence, there is a

strategic-complementary externality: When banks expand lending, on an individual basis

they fail to account for the aggregate effect that their combined credit expansions have on

the propensity for borrowers to engage in actions that generate greater loan losses. Given

that there are numerous institutions, the value ofλ is very small, and thus alsoφ is

similarly small. Indeed, the magnitude of φ is assumed to be sufficiently miniscule that

an individual bank regards its value as insignificant. The individual bank, therefore,

chooses its optimal balance-sheet configuration regarding φ as equal to zero. In contrast,

a social planner desires to take into account the fact that each bank’s contribution to total

credit expansion sums to a potentially substantial expansion of aggregate lending, L, and

thereby can induce an economically significant increase in the magnitude of κ .

From the point of view of an individual bank i, however, with φ perceived as miniscule,

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and with the tax rateτ initially equal to zero, the individually optimal profit-maximizing

quantity of loans is given by

LNM τ =0 =σ + γ( ) α −δ( )rL −γ rS −σ rD

ζσ + γ ζ +σ( ) ≡ LNM* , (2)

where LNM τ =0 ≡ L

NM* is the zero-tax lending level—that is, the optimal value withφ =τ =

0 for the individual non-monitoring bank. Thus, left to its own individually optimal,

untaxed choices, the bank engages in excessive credit expansion from the social planner’s

point of view. The optimal quantity for the social planner is in fact:

L^ NM

τ =0 =σ + γ( ) α −δ( )rL −γ rS −σ rDζ + 2φrL( )σ + γ ζ + 2φrL +σ( ) ≡ L

^ NM*

. (3)

In theory, this externality provides a rational for the social planner to utilize the

tax rate ( )Lτ in an effort to engage in a Pigouvian policy aimed at correcting the

externality. To see how this could be accomplished in the simple banking system

considered in this section, note that the general profit-maximizing balance-sheet

configuration, treating the values of φ and ( )Lτ as positive quantities and taking into

account the balance-sheet constraint Li + Si = Di (where each bank’s equity capital is

assumed fixed and equal to zero for the sake of simplicity) is given by

( ) ( )

( ) ( )( )L S DNM NM

iL L

r L r rL L

r rσ γ α δ τ γ σ

ζ φ σ γ ζ φ σ+ − − − −⎡ ⎤⎣ ⎦= =

+ + + +, (4a)

( ) ( ) ( )

( ) ( )( )L L S L DNM NM

iL L

r L r r r rS S

r rγ α δ τ ζ φ σ ζ φ

ζ φ σ γ ζ φ σ− − − + + + − +⎡ ⎤⎣ ⎦= =

+ + + +, (4b)

and

( ) ( ) ( )

( ) ( )( )L L S L DNM NM

iL L

r L r r r rD D

r rσ α δ τ ζ φ ζ φ σ

ζ φ σ γ ζ φ σ− − + + − + +⎡ ⎤⎣ ⎦= =

+ + + +. (4c)

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Because the solutions in (4) take into account a small positive magnitude of φ and allow

for the inclusion of the tax rate τ , they correspond to the profit-maximizing solutions as

perceived by the social planner. Of course, from the perspective of an individual bank

that ignores the externality and that confronts no Pigouvian taxes, φ is ignored and hence

treated as equal to zero, and τ also is equal to zero.

Our analysis below focuses on utilizing a positive tax rate to address the

externality by generating an appropriate adjustment in the supply of loans. Note,

however, that equations (4b) and (4c) indicate that imposition of a positive tax rate has a

negative impact on a bank’s desired deposits but induces a positive adjustment in its

securities holdings. Hence, the tax generates a further relevant distortion in banking

markets by favoring investment-banking activities at the expense of commercial banking

operations. A universal bank, for example, might adjust to imposition of the tax by

satisfying the demand for loans of a large client by underwriting a bond issue and

retaining a substantial part of the issuance in the balance sheet. Furthermore, its incentive

might be to finance this issuance by selling some other less risky securities in light of its

incentive to reduce its deposit scale. Thus, even though our emphasis in the rest of the

paper is placed on the partial-equilibrium question of how the magnitude and

composition of bank loan supply adjusts to Pigouvian taxation, in are additional general-

equilibrium issues that also might be explored using this model or a variant of it.

In this setting, there is a straightforward Pigouvian solution to the externality. If

the social planner imposes the positive tax rate

τ * = φrLL (5)

on each identical non-monitoring bank, L^ NM

τ * = LiNM* is achieved. That is, imposing the

appropriate tax rate τ * that takes into account each bank’s marginal contribution φ to the

aggregate effect on loan losses induces banks to reduce their lending to the externality-

9

corrected level. In this simple banking system, this appropriately specified Pigouvian tax

rate that removes the incentive for banks to engage in excessive credit expansion is equal

to the product of the externality parameter, φ , and revenues generated by lending, rLL .

Hence, the social planner achieves the social optimum by taxing away the externality’s

contribution to revenues for the marginal dollar of lending, thereby removing the

incentive for banks to lend beyond the socially optimal quantity.

If we suppose that the numerous banks are distributed uniformly over a

continuum of unit length, then integrating over the identical banks yields L^ NM

as the

market quantity of loans. For the sake of expositional simplicity, assume that the public’s

loan demand function is given by

L = l0 − l1rL , (6)

with the intercept l0 assumed to be substantially larger in magnitude than the loan-rate

semi-elasticity parameter l1 . Under this assumption, given exogenous security and

deposit rates, the market loan rate is equal to

r^L τ =0=

l0 ζσ + γ ζ +σ( )⎡⎣ ⎤⎦ + γ rS +σ rDζ + γ( ) α −δ( )+ l1 γσ + γ ζ +σ( )⎡⎣ ⎤⎦

. (7)

Equating L^ NM

τ * = LNM*with (4) yields the after-tax equilibrium loan rate

consistent with the Pigouvian-tax-corrected level of credit in the banking system. This is

a complicated expression involving a non-imaginary quadratic root but must exceed the

loan rate in (5) since LNM* <L^ NM

. Thus, correcting the externality via imposing on each

bank the tax rate τ *prevents an inefficient overexpansion of credit, thereby reducing the

market supply of loans by all banks and pushing up the market loan rate.

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3. Bank Heterogeneity and Monitoring

In the environment considered in the previous section, all banks behaved equally

“badly” from the perspective of the social planner. All banks did not just fail to take into

account their contribution to the overlending problem; they also failed to monitor loans in

order to mitigate loan losses. Now suppose that we alter the setting considered in Section

2 by following Kopecky and VanHoose by assuming that if bank i so chooses, it can

incur, in addition to the other expenses in (1), a monitoring cost equal to i1− i⎛⎝⎜

⎞⎠⎟c2Li( )2 to

eliminate entirely the δ loan-loss component. If bank i opts to monitor, therefore, its

(potentially after-tax) profit is given by

[ ] ( ) ( ) ( ) ( )2 2 2 2( ) .2 2 2 1 2

Mi L i S i D i i i i i

i cr L L r S r D L S D Li

ζ σ γπ α τ ⎛ ⎞= − + − − − − −⎜ ⎟−⎝ ⎠ (8)

Maximization of (8) with respect to the balance-sheet constraint yields the following

balance-sheet choices for the bank if it were to take into account its contribution to the

aggregate lending and hence the magnitude of the loan losses faced by non-monitoring

banks while opt to incur the monitoring cost to eliminate its own exposure to such losses

while potentially facing a Pigouvian tax:

( )[ ]( ) ( ) ( )

( )

1

L S DMi

L L

r L r rL

ir r ci

σ γ α τ γ σ

ζ φ σ γ ζ φ σ γ σ

+ − − −=

⎛ ⎞+ + + + + +⎜ ⎟−⎝ ⎠

, (9a)

[ ]

( ) ( ) ( )

( )1 1

1

L L S i L DMi

L L

i ir L r c r c r ri iS

ir r ci

γ α τ ζ φ σ ζ η

ζ φ σ γ ζ φ σ σ γ

⎛ ⎞ ⎛ ⎞− − + + + + − + +⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠=⎛ ⎞+ + + + + +⎜ ⎟−⎝ ⎠

, (9b)

and

[ ]

( ) ( ) ( )

( )1 1

1

L L S L DMi

L L

i ir L r c r r c ri iD

ir r ci

σ α τ ζ φ ζ φ σ

ζ φ σ γ ζ φ σ σ γ

⎛ ⎞ ⎛ ⎞− + + + − + + +⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠=⎛ ⎞+ + + + + +⎜ ⎟−⎝ ⎠

. (9c)

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Each monitoring bank will choose a different amount of lending given its own

unique marginal monitoring cost, c i1− i

. In this framework every bank has a choice if to

monitor or rather adopt the non-monitoring technology and it will decide on the basis of

its specific individual marginal monitoring cost. Note, however, that when the value of

the tax is different from zero the solutions for individual banks, either monitoring or non-

monitoring ones, depend on aggregate loans L .

The aggregate supply of loans, as discussed by Kopecky and VanHoose (2006), is

equal to the sum of loans originated by non monitoring banks and those by monitoring

banks. If Ω is the share of banks that opt to monitor their loans, the total supply of loans

in the market is given by

Ls = LiM di

i=0

Ω

∫ + 1−Ω( )LNM . (10)

As a first step to analyzing this expanded model, we obtain solutions for the private

market supply of loans in two polar cases: a first case in which all lending is conducted

by non-monitoring banks, so that Ω = 0 , and a second case in which all lending is

conducted by non-monitoring banks, so that Ω =1 .

For the case in which all loans are extended by non-monitoring banks, Ω =1 , and

total private market loan supply is

( )

( )( )^

2

NML

SL

A rL L

rσ γ δ

σ γ φ ζ γσ− +

= =+ + +

. (11)

where A ≡ A φ( )+ σ + γ( )φrLL = σ + γ( )αrL −γ rS −σ rD is the net interest margin. For the

second, all-monitoring case, after substituting the optimal value of the tax rate reported in

(5) into (9a), we define the following variables:

A φ( ) ≡ σ + γ( ) αrL −φrLL⎡⎣ ⎤⎦ −γ rS −σ rD ,

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B φ( ) ≡ ζ +φrL( )σ + γ ζ +φrL +σ( ) ,

C ≡ c + γ +σ( ) ,

so that

LiM =

σ + γ( ) αrL −τ (L)⎡⎣ ⎤⎦ −γ rS −σ rD

ζ +φrL( )σ + γ ζ +φrL +σ( )+ c i1− i

⎛⎝⎜

⎞⎠⎟γ +σ( )

=A φ( )

B φ( )+C i1− i

⎛⎝⎜

⎞⎠⎟

. (12)

Total lending by monitoring banks therefore is equal to

LiM di

i=0

Ω

∫ =A φ( ) B φ( )−C( )Ω +C log B φ( )− B φ( )−C( )Ω⎡

⎣⎤⎦{ }− log B φ( )( ){ }

B φ( )−C( )2. (13)

Hence, for the case in which all lending is conducted by monitoring banks, so that Ω =1 ,

total private market loan supply when all banks monitor their loans is

LS = L^ M

=

A B φ( )−C +C log CB φ( )

⎛

⎝⎜

⎞

⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

B φ( )−C( )2 + σ + γ( )φrL B φ( )−C +C log CB φ( )

⎛

⎝⎜

⎞

⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

. (14)

Comparing the all-monitoring solution in (14) with the all-non-monitoring

solution (11) indicates that for either a banking system composed of banks that do not

monitor their loans or consisting of all-monitoring banks, aggregate loan supply is a

nonlinear function of φ . The nonlinear complexity of the dependence of loan supply on

φ is clearly more pronounced in the case of the all-monitoring banking system. This is so

because heterogeneous decisions on the part of the banks utilizing monitoring

technologies of diverse efficiencies imply a distribution of loan responses across all

banks that, when combined, yields a complicated impact of φ on the total quantity of

loans supplied by monitoring banks at any given loan rate. It is important to note,

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however, that even though the existence of the externality effect embodied in φ

influences aggregate loan supply in an all-monitoring banking system, the banks’

differentially more or less efficient monitoring activities serve to eliminate loan losses

from influencing the banking system’s performance.

Thus, as in the case of a banking system made up entire of non-monitoring banks,

a banking system populated solely by monitoring banks creates an over-lending

externality. Nevertheless, in the all-monitoring banking system, the privately optimal

actions of monitoring banks provides a first-best solution to the externality, provided that

available resources exist to make the solution viable with at a nonnegative value for the

quantity of lending reported in (14).

Considerable computation for the case in which Ω lies between 0 and 1 yields

the general solution for the aggregate supply of loans as a function of Ω , the loan rate

and other interest rates, and all additional parameters, which is given by

LS =A B φ( ) B φ( ) −C( )Ω +CD φ( )⎡

⎣⎤⎦ + 1−Ω( ) B φ( ) −C( )2{ }− δ rL 1−Ω( ) B φ( ) −C( )2

B φ( ) −C( )2 B φ( ) + B φ( ) B φ( ) −C( )Ω +CD φ( )⎡⎣

⎤⎦ + 1−Ω( ) B φ( ) −C( )2{ } σ + γ( )φrL

, (15)

whereD φ( ) ≡ log B φ( )− B φ( )−C( )Ω⎡

⎣⎤⎦ − log B φ( )( ) . Naturally, this expression is a

particularly complex function of the externality effect embodied in φ .

4. The Effect of a Pigouvian Tax on the Overall Quality of Bank Lending

In principle, the general loan supply expression in (15) could be equated with (6)

to solve for the market-clearing loan rate and the equilibrium volume of lending. The

resulting loan-rate and loan-volume solutions would be highly nonlinear (for instance, the

obtaining solution for the market loan rate would involve solving a quartic equation),

however, and not readily amenable to a tractable policy analysis.

Consequently, we focus our policy analysis on the two extreme cases of either all-

14

non-monitoring or all-monitoring banking systems. Because the value of the optimal tax

rate in (5) that a social planner would employ to eliminate the over-lending externality

multiplies loan interest revenues, to analyze the impact of imposing such a tax on the two

classes of loans, we differentiate the solutions obtained for the marginal value of the

externalityφ . In the case in which no banks monitor, so that Ω = 0 and (11) applies, it

follows that the effect of a variation in the magnitude of the externality—and hence an

externality-combatting tax rate—is given by

( ) ( )( )

^

2

2

2

NM

L L

L

r A rd Ld r E

σ γ σ γ δφ σ γ φ

⎡ ⎤+ − +⎣ ⎦= −+ +⎡ ⎤⎣ ⎦

(16)

In the case in which all banks do not monitor, so that Ω = 0 and (14) applies, the effect

of a variation in a Pigouvian tax rate is given by

d L^ M

dφ= − A

σ + γ( )φ 2rL× (17)

2 σ + γ( )2φ 2rL

2 + 4 σ + γ( )φ +1⎡⎣ ⎤⎦C φ( )+ (E −C) 4 σ + γ( )φrL + 4C φ( )− 3(E −C)⎡⎣

⎤⎦{ }

2 σ + γ( )φrL + 3 E −C( )+C φ( ){ }2

,

where log CC CB

⎡ ⎤⎛ ⎞= ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦ and ( )E γ σ ζ γσ= + + . Whenever loan losses δ are below

the threshold value that permits non-monitoring lending to be viable, so that the

numerator of (11) is larger than zero, the derivative in (16) is unambiguously negative.

This result implies that the imposition of the tax unequivocally reduces the loan supply

toward the optimal level that eliminates the over-lending externality when all banks do

not monitor. This conclusion parallels the result obtained in Section 2 for the non-

monitoring banking system that consequently was exposed fully to externality-magnified

loan losses.

15

For the all-monitoring banking system, it is straightforward to demonstrate that

whenever monitoring costs c are below the threshold value that makes positive total

lending viable, so that the numerator of (14) exceeds zero, the derivative of (14) is

unambiguously negative. This result implies that the imposition of the tax unequivocally

reduces the optimal loan supply when all banks monitor.

Since the general solution reported in (15) is a weighted average of the two polar

cases, the result holds for the general case also.

Clearly, the imposition of the Pigouvian tax is an efficient mechanism for curbing

lending in order to avoid unforeseen loan losses arising from the strategic-

complementarity externality. Inspection of (15) and (16) reveals that the impact of the

tax depends on net interest margins of the two types of lending. In the case of non-

monitoring banks, it is evident that the impact of imposing the tax is lower for higher

values of δ . This result is not surprising. Because the supply of loans is a function of

net interest margins, the effect of imposing the tax rate is stronger when margins are

healthy and loan supply thereby is robust.

Note, however, that for the non-monitoring banking system, (16) indicates that the

impact of imposition of the tax is stronger for a smaller value ofδ , the component of

loan losses that is independent of the volume of lending, when higher interest margins

make the banking system more profitable and hence help to offset loan losses. In

contrast, the tax impact is hardly relevant when margins are thin and a resulting moribund

non-monitored banking industry simultaneously faces loan losses that further weaken its

performance

An even more serious problem with the tax, however, is that it causes a reduction

of lending on the part of monitoring banks, which because they invest in monitoring

technologies are not subject to the loan-loss function κ L( ) . The same basic argument

provided in the prior paragraph holds for monitoring banks that face the tax effect

reported in (17), although the aggregation makes the result less intelligible. As net

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interest margins for monitoring banks increase, the optimal supply of loans rises, and

hence the impact of imposing the tax rate is stronger. In the all-monitoring banking

system, the magnitude of the tax effect hinges in part on the size of the marginal

monitoring-cost coefficient c, but there is an added complication due to the fact that

monitoring costs are convex in the volume of loan supplied. Hence, in the all-monitoring

banking system, the Pigouvian tax also interacts with these cost factors. Additionally,

lower monitoring costs boost profit margins, resulting in greater loan supply and an

enlarged tax effect.

An unintended outcome of imposing the Pigouvian tax rate is that it generates a

strong negative policy externality on monitoring banks that choose to engage in costly

elimination of loan losses. Moreover, this negative effect on monitoring banks’ loan

supply caused by imposing the tax rate to mitigate the over-lending externality becomes

more pronounced the lower are monitoring costs. That is, the greater the degree of

efficiency with which monitoring banks act to eliminate loan losses on their own, the

stronger is the negative effect of the tax on their lending.

This analysis suggests that in a banking system comprised of both non-monitoring

and monitoring banks, imposition of a Pigouvian tax rate on banks could have perverse

impact on the composition of lending. On the one hand, the impact of the tax on non-

monitoring banks is smaller when their margins are thinner, while the tax falls more

heavily on non-monitoring banks when their margins net of loan losses are greater. On

the other hand, implementation of the tax compels monitoring banks to internalize a

strategic-complementary negative externality that falls on the other class of banks but not

on them, which makes monitored lending less profitable. Consequently, although the tax

reduces the supply of loans of both classes of banks, it tends to add to a growing market

share for non-monitoring banks to a greater extent when their activity poses increased

profit risks along their internal margins even as it tends to depress along the external

margin the share of banks that voluntarily monitor loans to mitigate loan losses.

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As noted in the introductory paragraph in this section, the intractability of

solutions forthcoming from a mixed banking system containing both non-monitoring and

monitoring banks prevents full policy analysis of such an environment. Nevertheless, the

above logic suggests that over at least some ranges of values of interest rates and

parameter values, the perverse effects created by imposing the Pigouvian tax rate likely

would generate higher aggregate loan losses than would have been observed without the

tax. At a minimum, over all parameter ranges, any net benefits arising from mitigation of

the over-lending externality would be reduced by these perverse spillover effects that the

tax generates.

5. Conclusion

This paper has provided a cautionary tale regarding the application of Pigouvian

taxation to banking markets. It has utilized a model in which the individually optimal

behavior of banks gives rise to an over-lending externality to demonstrate how a tax that

has the intended effect of correcting the externality can also have the unintended impact

of adversely altering the composition of aggregate lending. Even though the tax offsets

the strategic-complementarity effect on loan losses that fall on banks that fail to monitor

their loans, it has a smaller impact on these banks when their profit margins are narrower

and hence when they pose greater threats to the health of the banking system. At the

same time, the tax falls with relatively greater weight on banks that incur costs to monitor

loans in order to eliminate loan losses yet must also bear the tax. Hence, imposing the

tax tends to have the perverse effect of reducing aggregate loan quality via alteration of

the composition of lending even as it offsets the over-lending externality.

To date, most analyses of applying Pigouvian tax rates to banking markets have

not taken into account the fact that, as illustrated in this paper, differential burdens of

taxation can arise that in turn create diverging incentives for heterogeneous banks. Our

analysis in this paper has considered a competitive banking system containing a group of

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identical non-monitoring banks and a group of monitoring banks using heterogeneous

monitoring technologies to highlight the potential for unintended effects of imposing a

tax on bank lending to correct an over-lending externality. We anticipate that

qualitatively analogous results would be forthcoming in a model with imperfectly

competitive banking markets or in which a social planner opted to treat an over-lending

externality as an over-borrowing problem requiring a tax on banks’ loan customers.

We also suspect that unintended consequences likely would emerge in the same

type of setting if banks were to face other sources of externalities. Consider, for instance,

a systemic-risk-style externality arising from failure of one set of banks to consider along

their internal margins how their borrowing in a wholesale funds market to cover

commitments to a second group of institutions might expose the other banks to potential

failures to honor commitments to third group. Within a framework such as the one

considered in our paper, imposing a tax on banks’ wholesale borrowing intended to

contain the systemic-risk externality, such as the bank levy introduced in the United

Kingdom, would affect not only their borrowing choices on the liability sides of their

balance sheets but also their lending decisions. As in the analysis conducted in this

paper, the tax consequently would tend both to induce banks to restrain their lending and

to alter the relative profitabilities they would derive depending on whether or not they

monitoring their lending, thereby ultimately influencing, again, the composition of

aggregate lending.

Alternatively, suppose that we abstract from the sorts of asset-composition effects

considered in this paper. A systemic-risk tax aimed to influence banks’ wholesale

borrowing levels would, in the context of a model of realistically heterogenous banks,

exert effects on the composition of banks’ sources of borrowings. The likely result

would be an alteration in composition of aggregate liabilities. The adjusted overall

balance-sheet risk configuration might or might not reinforce the intended effect of the

systemic-risk tax.

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Thus, our conclusion is that the literature to date on subjecting banking markets to

Pigouvian taxation has not given sufficient attention to unanticipated incentive effects of

the imposition of taxes aimed at correcting alleged banking externalities. It has long been

recognized in the field of public economics that imposing taxes and subsidies can create

secondary behavioral distortions. Proposals for substantial alterations in banking policies

via the application of taxation instruments should not be contemplated without

consideration of the distortionary effects that the policies themselves may create. We

regard the analysis conducted in this paper as an initial step in this direction.

20

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