Bank reserves ; Banks and banking - Taxation

Scott Freeman

English

Research Papers in Economics

Federal Reserve Bank ofDallas
presents
RESEARCH PAPER
No. 9334
TheInefficiency of Seigniorage from Required Reserves
by
Scott Freeman.ResearchDepartment
Federal ReserveBank of Dallas
and
Department of Economics
University of Texas- Austin
August 1993
This publication was digitized and made available by the Federal Reserve Bank of Dallas' Historical Library (FedHistory@dal.frb.org) The Inefffciency of Seigniorage  fmm  Required Reserves
Scott Freeman, Research  Dq)artment
Federal Reserve  Bank of Dallas
and
Department of Economics
Universitv of Texas-Austin
The views expressed  in this article are solely those of the authors and should not be attributed
to the Federal Reserve  Bank of Dallas  or to the Federal Reserve  System.The Inelficiency of Seigniorage  from Required  Reserves
Scott Freeman
Departrnent  of Economics
University of Texas
Austin,  TX 78712
and
Research  Associate
Federal  Reserve  Bank  of Dallas
June  1993
Seigniorage  from required  reserves  is shown  to lead to lower capital
accumulation  than  a tax on investrnent  raising the same  rcvenue.  The result
is obtained  in an infinitely lived representative  agent  model  to avoid  the
intergenerational  wealth Eansfers  found in Freeman's  (1987) study of
seigniorage  from required  resewes  in an  overlapping  generations  economy.
I have  benefited  from conversations  with Joseph  Haslag  and  Gregory  Huffman. The views
expressed  are  not necessarily  those  of the Federal  Reserve  Bank of Dallas or the Federal
Reserve  System.Expansion  of the fiat money  stock  taxes  fiat money  balances  held  by the public.
Black  (1970),  Fama  (1980),  Calvo  and  Femandez  (1983),  and  Romer  (1985)  have  all
noted  that  r€quiring  banks  to hold reserves  offiat rnoney  then  functions  as  a tax on
banking.  Freeman  (1987),  using  an  overlapping  generations  model,  demonstrated  that the
tax on banking  implied by required  reserves  and  a finite rate  of money  expansion  leads  to
lower holdings  of capital  and  lower steady-state  welfare  than  would a direct  tax on banking
yielding the same  revenue.
Replacing  reserve-requirement  seigniorage  widr a dir€ct  tax on barking in an
overlapping  generations  model  hura the  initial generation  by rendering  wonhless  its initial
holding  of fiat money.  This intergenerational  Eansfer  of wealth  makes  welfare  comparisons
diffrcult to interpreLl  To avoid  intergenerational  wealth  transfers  this paper  takes  up the
optimality of resewe  requiremens  in an  infinitely lived representative  agent  framework.
Seigniorage  from required  reserves  is shown  unambiguously  to generate  lower steady-state
capital  and  welfare  than  a  direct tax on banking.
The  model: There  exists  a  continuum  of infinitely  lived agents  whose  population
is normalized  to 1.  Each  has  preferences  over  consumption  at date  r>1 described  by the
function  Z  P ,kr).  The  function  u(c) is increasing,  concave,  and  twice-continuously
,=o
differentiable  wift  lim6--e6u(c)  = ...
The  economy's  sole  good  can  be  consumed  or made  into capital  that  lasts  a single
period.  If t1 goods  are  made  into capital  at t, flft)  goods  will be produced  at t+1. The
lAn altemative  approach,  taken  by Bacchetta  and  Caminal  (1990),  retains  the ovedapping  genrations  mode!
and  requires  the govemmcnt  to fund intergeneradonal  transfers  o the initial old with distortionary  Exation.production  function/(&) is increasing,  concave,  and  twice-continuously  differentiable  with
lim*-+of(k) = o".  In the initial period  each  agent  starts  with/(ko) goods.
In the  initial period  the agent  also  owns  Mg unis of fiat money.  The nominal  stock
of fiat money  in any  period  ris Mr, which  is expanded  at  a  constanttate  z> I' implyng
lhzt  Mt= zMrt.Fiat  money  is held to satisfy  a legal  requirement  that  an agent  owning tl
units of capital  at t must also  hold fiat money  balances  worth {k1  goods'  The goods  value
of a unit of fiat money  at r will be  denoted  v1  and  an agent's  nominal  demand  for fiat money
balances  at r will be denoted  m1.  The seigniorage  from the expansion  of the  fiat money
stock  is used  o help finance  a fixed level of wasteful  govemnrent  expendiffes g. The
govemment's  only rwenue alternative  is a tax  collecting q goods  for each  good  made  into
capial at r. Like seigniorage,  this tax is a distorting  tax tlnt discourages  investrnenL
Because  it does  not  tax  existing  capital,  it introduces  no time-consistency  issues.
(fo  keep  the analysis  as straighdorward  as  possible,  I have  assumed  drat  the
rcserve  requfuement  and  invesunent  tax are  applied  to all capital.  The  real world case  of
reserve  requirements  imposed  only on some  foms of intemrediated  capital  is a trivial
extension  of this  analysis.)
Equitibrium: Equilibrium  is defined  to be  a sequence  lq,k)for  t2l  and  given
valdes  of g, z, t, and  @  such  that  each  agent  maximizes  his utility subject  to his budget
constraints  and  the  reserve  requirement,  taking  as  given  the  price sequence  [v1J;  each  agent
has  perfect  foresighq  the market  for frat money  clean; and  the government  budget
constraint  is met  with equality.
An individual's  budget  constraint  in each  period  r )l  is
f(k1-)+v2m2-1=ct*kt+v{4+rtkt,  (l)
which with the legal  requiremen  t v  lU  = @*1  can  be written
.(&r-t)+ (vlvriQktt=ct  -t'  kt +  Sfu +  tftj  .  A)
The  resulting  first order  condition  for the  maximization  of utility by an  individual isu'(c,)  _  (vlvri0  +f(kt)
Bu'(c*t)  I  +  O  +  11 
(3)
The  clearing  of the  market  for fiat money  requires  that
vtMt  =  Qh  (4)
which in a steady  state  (&,  = &  for t ) 0) implies  that
Yt+l  -  !  ttl vt  z
The govemment's  budget  constraint  in period  r is
't1k1  +  v[M1 -My1l
=  ttkr +  vMtll-Uzl
=  7&t  + $fulr-Llzl  (6)
Compartng steaily states: To compare  investment  taxation to seignioragc  from
reserves  let us examine  the steady-states  of two benchmart  policies.  First suppose  that
there  is no reserve  requirement  (@  = 0) so  ttrat  all revenue  comes  from investnent taxation'
The steady-state  first-order  condition  simplifies  to
!  _  "f(/c)  (7)
p  - 
l+t
When  the govemment  budget  constraint  (6) is also  included,  we find
I  -  f(k)
B 
-  r+stk 
(8)
Now suppose  that  all revenue  somes  from seigniorage  (t=  0).The steady-state
equilibrium  conditions  now simplify to
I  -  0/z+f'(k)
P 
-  r+o 
(e)
where the government budget constraint (6) implies that the reserve  requirement be such
that
^=  I  (ro) v  - 
1t-t1z1kNotice  from (9) and  (10) that  for any  finite rate  of fiat money  creation,  seigniorage
from required  reserves  is not equivalent  to investrnent  taxation.  Solving (10) for 1/z  and
substituting  the  expression  into  (9)  we  find
Q  -stk  +f'(k)
(11)
r+O
Comparing  (8)  with (ll)  [ and  noting  from (6) that  Q> glk], we  see  that  the  steady-state
marginal  product  of capital  is greater  with required  rpserves,  implyrng a lower steady-state
capial stock  for any  given  revenue.  Notice that  even  when  there  is no seigniorage[  z = I
and  g= 91,  a reserve  requirernent  lowers  steady-state  capital.  By requiring  investors  to hold
unprrductive  reserves  of fiat money  in order  to invest,  a reserve  requirement  discourages
investrnent  even  more  than  does  a direct  tax on investrnenL  To understand  this, combine
(9) and  (  l0) for an  expression  in z.
I
B
o
|  (z  - r)k + "f'(e)
B-  r*  8 -  (1  - llz)k
(12)
As z increases  (and  @  decreases)  the steady-state  marginal  product  of capital  decreases  and
steady-state  capital  increases.  As z goes  to infinity, the  equilibrium  condition  approaches
that  of a tax on investrnent.  The  larger  is z,  the more  that  a given  level of rcserves  is taxed
and  the less  that it is used  for saving,  This reveals  that  the social  cost  of a reserve
requircment  comes  from those  rcserves  that  are  held  but not taxed  away.
The  welfare  implications  of reserve  requirements  follow direcdy  from the  steady-
state  analysis.  In a steady-state  with positive  resewe  requircments,  the marginal  product  of
capital  exceeds  the  marginal  rate  of subsritution.  Therefore,  a policy that  increases  the  stock
of capital  in every  period  while raising  the same  revenue  increases  welfare.References
Black,  Fischer,  1970,  "Banking  and  Interest  Rates  in a  World withoutMoney,"  Jownal of
Bank  Research  l:8-2O.
Calvo,  Guillermo  A. and  Rogue  B. Fernandez,  1983,  "C-ompetitive  Banks  and  fte Inflation
Tax,"  Economic  Letters  12:313-317  .
Fama,  Eugene  F., 1980,  "Banking  in the Theory  of Firarrn,"  Journal of Monctary
Economics,  6,39-57  .
Freeman,  Scott  198?,  "Reserve  Requirements  and  Optimal  Seignorage,"  ,/ownal of
M  onetary  Economics,  19,  307-  14.
Romer,  David, 1985,  "Financial  Intermediation,  Resewe  Requirements,  and  Inside
Money,"  Journal of Monetary  Economics  16:.  175'194'RESEARCH PAPERS  OF TI{E RESEARCH  DEPARTMENT
FEDERAL RESERVE BANK OF DALI-AS
Available,  at no charge,  from the Research  Department
Federal  Reserve  Bank  of Dallas.  P.O.  Box  655906
Daflas,  Texas 7526i-5906
92O1, Are Deep Recessions  Followed  by Strong  Recoveries?  (Mark A  Wynne  and
Nathan  S. Balke)
92V2 The Case  of the "Missing  M2" (John  V. Duca)
9203  Immigrant  Unks to the Home Country: Implications  for Trade,  Welfare and
Factor  Rewards  (David  M. Gould)
9204  Does  Aggregate  Ou@ut  Have  a Unit Root? (Mark A" Wynne)
9205  Inflation and Its Variability: A Note (Kenneth  M. Emery)
9206  Budget  Constrained  Frontier Measures  of Fiscal  Equality and Efficiency  in
Schooling  (Shawna  Grosskopf,  Kathy Hayes,  I-ori Taylor, William Weber)
9207  The Effects  of Credit Availability, Nonbark Competitioq and Tax Reform on
Bank Consumer  I*nding  (John  V. Duca and  Bonnie  Garrett)
9208  On the Future Erosion  of the North American  Free Trade Agreement  (William
C. Gruben)
9209  Threshold  Cointegration  (Nathan  S. Balke and Thomas  B. Fomby)
9210  Cointegration  and Tests  of a Classical  Model of Inflation in Argentin4 Bolivi4
Brazil, Mexico,  and Peru (Rarll Anibal Feliz and  John  H. Welch)
92Il  Nominal Feedback  Rules  for Monetary  Policy:  Some  Comments  (Evan  F.
Koenig)
9212  T\e Analysis  of Fiscal  Policy  in Neoclassical  Models' (Mark Wynne)
92L3  Measuring  the Value of School  Quality (Lori Taylor)
921.4 Forecasting  Turning Points:  Is a Two-State  Characterization  of the Business
Clcle Appropriate?  (Kenneth  M. Emery  & Evan  F. Koenig)
9215  F;nerg Security:  A Comparison  of Protectionist  Policies  (Mine K  Yiicel  and
Carol Dahl)9216  An Analysis of the Impact of Two Fiscal Policies on the Behavior of a
Dynamic Asset Market (Gregory W. Huffrran)
9301  Human Capital Externalities, Trade, and Economic Growth
(David Gould and Roy J. Rutrin)
9302  The New Face of latin  America: Finaricial Flows, Markets, and Institutions in the
1990s  (John Welch)
9303  A General Two Sector Model of Endogenous  Growth with Human and
Physical Capital (Eric Bond, Ping Wang, and Chong IC Yip)
93M  The Political Economy of School Reform (S. Grosskopf, K  Hayes, L  Taylor,
and W. Weber)
9305  Money, Output, and Income Velocity (Theodore Palivos and Ping Wang)
9306  Construcring an Altemative  Measure of Changes  in Reserve Requirement
Ratios (Joseph H. Haslag and Scott E. Hein)
9307  Money Demand and Relative Prices During Episodes of Hlperinflation
(Ellis W. Tallman and Ping Wang)
9308  On Quantity Theory Restrictions and the Signalling Value of the Money
Multiplier  (Joseph  Haslag)
9309  The Algebra of Price Stability (Nathan S. Balke and Kenneth M. Emery)
9310  Does It Matter How Monetary  Policy  is Implemented? (Joseph  H. Haslag  and
Scott  E. Hein)
9311  Real Effects of Money and Welfare Costs of Inflation  in an Endogenously
Growing Economy with Trarsactions Costs (Ping Wang and Chong I{- Yip)
9312  Borrowing Constraints, Household Debt  and Racial Discrimination in Loan
Markets (John V. Duca and Stuart Rosenthal)
9313  Default Rislq Dolarization,  and Currency Substitution in Mexico
(William  Gruben and John Welch)
9314  Technological Unemployment (W  Michael Cox)
9315  OutpuL Inflation,  and Stabilization in a Small Open Economy: Evidence From
Mexico (John H. Rogers and Ping Wang)9316  Price Stabilizatioq  Output Stabilization and Coordinated Monetary Policy Actions
(Joseph  H. Haslag)
93L7  AL Altemative  Neo-Classical  Growth Model with Closed-Form Decision Rules
(Gregory W. Huffman)
9318  Why the Composite Index of Irading  Indicators Doesn't l-ead
(Evan F. Koenig and Kenneth M. Emery)
9319  Allocative  Inefficiency and I-ocal Government: Evidence Rejecting the Tiebout
Hypothesis (Lori  L  Taylor)
9320  The Output Effects of Government Consumption: A Note (Mark  d  Wynne)
9321  Should Bond Funds be Included in M2? (John V. Duca)
9322  Recessions  and Recoveries  in Real Business  Clcle Models: Do Real Business
C.lcle Models Generate Clclical  Behavior? (Mark A  Wynne)
9323r Retaliatioq  Liberalizatioq  and Trade Wars: The Political Economy of
Nonstrategic Trade Policy (David M. Gould and Graeme L  Woodbridge)
9324  A General Two-Sector Model of Endogenous  Growth with Human and Physical
Capital: Balanced Growth and Transitional Dynamics (Eric W. Bond, Ping Wang,
and Chong IC Yip)
9325  Growth and Equity with Endogenous  Human Capital: Taiwan's Economic Miracle
Revisited (Maw-Un  L.ee,  Ben-Chieh Uu,  and Ping Wang)
9326  Clearinghouse Banls  and Banknote Over-issue (Scott Freeman)
9327  Coal. Natural Gas and Oil Markets after World War II:  What's Old, What's
New? (Mine K  Yticel and Shengyi  Guo)
9328  On the Optimality  of Interest-Bearing Reserves  in Economies of Overlapping
Generations (Scott Freeman and Joseph Haslag)
9329* Retaliatior4 Liberalization, and Trade Wars: The Political Economy of
Nonstrategic Trade Policy (David M. Gould and Graeme L. Woodbridge)
Reprint  of 93?3
9330  On the Existence of Nonoptimal Equilibria  in Dynamic Stochastic  Economies
(Jeremy Greenwood and Gregory W. Huffman)
9331  The Credibility  and Performance of Unilateral  Target Zones: A Comparison of
the Mexican and Chilean Cases  (Raul A  Feliz and John H. Welch)9332
9333
9334
Endogenous Growth and International Trade (Roy J. Rutrin)
Wealth Effects, Heterogeneity and Dynarnic Fiscal Policy (Zsolt Becsi)
The Inefficiency of Seigniorage  from Required Reserves  (Scott Freeman)

4 Forecasting Turning Points: Is a Two-State Characterization of the Business Clcle Appropriate? (Kenneth

A General Two Sector Model of Endogenous Growth with Human and Physical Capital (Eric Bond, Ping Wang,

A General Two-Sector Model of Endogenous Growth with Human and Physical Capital: Balanced Growth and Transitional Dynamics (Eric

Allocative Inefficiency and I-ocal Government: Evidence Rejecting the Tiebout Hypothesis (Lori L Taylor)

Altemative Neo-Classical Growth Model with Closed-Form Decision Rules (Gregory W. Huffman)

and Income Velocity (Theodore Palivos and Ping Wang)

and Relative Prices During Episodes of Hlperinflation (Ellis W. Tallman and Ping Wang)

and Stabilization in a Small Open Economy: Evidence From Mexico (John H. Rogers and Ping Wang)9316 Price Stabilizatioq Output Stabilization and Coordinated Monetary Policy Actions (Joseph H. Haslag)

Are Deep Recessions Followed by Strong Recoveries? (Mark A Wynne

Banking and Interest Rates in a World withoutMoney,&quot;

Clearinghouse Banls and Banknote Over-issue

Cointegration and Tests of a Classical Model of Inflation

Construcring an Altemative Measure of Changes in Reserve Requirement Ratios

Does Aggregate Ou@ut Have a Unit Root? (Mark A&quot; Wynne)

Does It Matter How Monetary Policy is Implemented? (Joseph H. Haslag and

Dolarization, and Currency Substitution in Mexico (William Gruben and

Equity with Endogenous Human Capital: Taiwan's Economic Miracle Revisited (Maw-Un L.ee, Ben-Chieh Uu, and Ping Wang)

Externalities, Trade, and Economic Growth (David Gould and

F;nerg Security: A Comparison of Protectionist Policies (Mine K Yiicel and Carol Dahl)9216 An Analysis of the Impact of Two Fiscal Policies on the Behavior of a Dynamic Asset Market (Gregory W. Huffrran)

Frontier Measures of Fiscal Equality and Efficiency in Schooling (Shawna Grosskopf, Kathy Hayes, I-ori Taylor,

Funds be Included in M2? (John V. Duca)

Household Debt and Racial Discrimination in Loan Markets (John V. Duca and

Il Nominal Feedback Rules for Monetary Policy: Some Comments (Evan F.

Immigrant Unks to the Home Country: Implications for Trade, Welfare and Factor Rewards (David M. Gould)

Inflation and Its Variability: A Note (Kenneth M. Emery)

Liberalization, and Trade Wars: The Political Economy of Nonstrategic Trade Policy

Liberalizatioq and Trade Wars: The Political Economy of Nonstrategic Trade Policy

Natural Gas and Oil Markets after World War II: What's Old, What's New? (Mine K Yticel and Shengyi Guo)

On the Existence of Nonoptimal Equilibria

On the Future Erosion of the North American Free Trade Agreement (William C. Gruben)

On the Optimality of Interest-Bearing Reserves in Economies of Overlapping Generations (Scott Freeman and Joseph Haslag)

Real Effects of Money and Welfare Costs of Inflation in an Endogenously Growing Economy with Trarsactions Costs (Ping Wang and Chong I{- Yip)

Recessions and Recoveries in Real Business Clcle Models: Do Real Business C.lcle Models Generate Clclical Behavior? (Mark A Wynne)

T\e Analysis of Fiscal Policy in Neoclassical Models'

The Algebra of Price Stability

The Credibility and Performance of Unilateral Target Zones: A Comparison of the Mexican and Chilean Cases (Raul A Feliz and

The Effects of Credit Availability, Nonbark Competitioq and Tax Reform on Bank Consumer I*nding

The New Face of latin America: Finaricial Flows, Markets, and Institutions

The Output Effects of Government Consumption: A Note (Mark d Wynne)

The Political Economy of School Reform

Theory Restrictions and the Signalling Value of the Money Multiplier (Joseph Haslag)

V2 The Case of the &quot;Missing M2&quot; (John V. Duca)

Why the Composite Index of Irading Indicators Doesn't l-ead

The inefficiency of seigniorage from required reserves

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The inefficiency of seigniorage from required reserves

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