Ferreira, Pedro Cavalcanti Gomes

English

Research Papers in Economics

Ensaios Econ￿micos
Escola de
P￿s-Gradua￿ªo
em Economia
da Funda￿ªo
Getulio Vargas
N 431 ISSN 0104-8910
In￿ation, Welfare and Public Expenditures
Pedro Cavalcanti Gomes Ferreira
Agosto de 2001
URL: http://hdl.handle.net/10438/353Os artigos publicados sªo de inteira responsabilidade de seus autores. As
opiniıes neles emitidas nªo exprimem, necessariamente, o ponto de vista da
Funda￿ªo Getulio Vargas.
ESCOLA DE P￿S-GRADUA˙ˆO EM ECONOMIA
Diretor Geral: Renato Fragelli Cardoso
Diretor de Ensino: Luis Henrique Bertolino Braido
Diretor de Pesquisa: Joªo Victor Issler
Diretor de Publica￿ıes Cient￿￿cas: Ricardo de Oliveira Cavalcanti
Cavalcanti Gomes Ferreira, Pedro
Inflation, Welfare and Public Expenditures/
Pedro Cavalcanti Gomes Ferreira ￿ Rio de Janeiro : FGV,EPGE,
2010
(Ensaios Econ￿micos; 431)
Inclui bibliografia.
CDD-330Inﬂation, Welfare and Public Expenditures
Pedro Cavalcanti Ferreira∗
Graduate School of Economics,
Funda¸ c˜ ao Getulio Vargas
Abstract
This paper studies welfare eﬀects of monetary policy in an overlap-
ping generations model with capital and no form of taxation other than
in￿ation. Public expenditures have a positive eﬀect on labor productiv-
ity. The main result of the paper is that an expansive monetary policy
can be welfare improving, at least for ￿small enough￿ in￿ation rates, and
that there is an optimal in￿ation rate. Growth maximization, however,
is never optimal. Steady-state capital and output increase with in￿ation,
reproducing the so-called Tobin eﬀect. For large in￿ation rates, however,
the government authorities cannot aﬀect real variables and there are only
nominal eﬀects.
1 Introduction
This note presents a simple exercise in in￿ation, welfare and growth, using
a parameterized overlapping generations model with capital in which govern-
ment expenses positively aﬀect the growth rate of human capital ( following
Boldrin(1993) and Glomm and Ravikumar(1992), among others) and conse-
quently the growth rate of productivity. In the model it is assumed that money
creation is the only source available for the ￿nancing of public expenditure,
although results would not change qualitatively with the introduction of other
distortionary taxes. In this set-up it is shown that in￿ation does have posi-
tive welfare eﬀects by increasing the equilibrium level of private capital and
income (a result that reproduces the so-called Tobin eﬀect (e.g., Tobin (1965)
and Mundell (1965)) so that price stability is never optimal. Agents end up
being better oﬀ under in￿ationary policy because of the impact of public ex-
penditures, ￿nanced through seigniorage, on savings and capital accumulation.
In a certain sense in￿ation solves a problem of under-accumulation of capital.
∗Graduate School of Economics, Funda¸ c˜ ao Getulio Vargas. Praia de Botafogo, 190, Rio
de Janeiro, RJ, 22253-900, Brazil. Email address: ferreira@fgv.br. The author would like to
acknowledge his gratitude for the ﬁnancial support of CNPq and PRONEX.
1It is also shown that there is an optimal in￿ation rate (which is always below
the rate that maximizes productivity growth). When the rate of money creation
is small enough, the positive impact of public expenditures on the economy more
than compensates the distortionary eﬀects of in￿ation. As the rate of in￿ation
increases, seigniorage decreases and so does its impact on growth and capital
accumulation. At a certain point the two eﬀects cancel each other out. Hence,
there is a non-monotonic relationship between in￿ation and welfare.
This paper relates to Weiss(1980), Drazen(1981) and other articles if the lit-
erature on optimal monetary policy in life-cycle models with production. How-
ever, while this literature is mainly theoretical, we present a simulation exercise
and the transmission mechanism from in￿ation to capital and welfare is rather
diﬀerent from these papers, since public expenditure, ￿nanced though seignior-
age, is a factor of production (more precisely, it enhances labor productivity).
The model is presented in the next section, while in Section 3 the simulation
results are presented. Some brief concluding remarks are made in Section 4..
2 The Model
Consider an overlapping generations economy with no population growth. Each
generation is composed of a large number of individuals who live for two periods,
except the ￿rst generation, which only lives for one period. In the ￿rst period of
their lives -￿youth￿ - the individuals are endowed with one unit of labor which
they supply inelastically. When young, the individuals work, receive a wage,
consume the only good of this economy, and save. In the second and last period
of their lives, ￿old￿ people do not work, but consume the proceedings of their
savings.
There are two diﬀerent assets in the economy. One is ￿at money issued by
the government, and the other is a capital asset issued by ￿rms. Money may or
may not be valued in equilibrium; and in the latter case individuals will hold
only capital in their portfolios. The capital and money levels in the ￿rst period
(time zero) are given by history.
The budget constraint of a young person born at time t is given by:
C
y
t = wt − St
and his budget constraint when he is old is:
C0
t+1 = StRt+1 +( Πt − Rt+1)Mt/Pt
where Rt+1 is the gross return on capital, Πt the gross return on money (the
inverse of in￿ation factor), C
y
t and C0
t+1 are the consumption of a young person
and a old person born at period t, respectively, St stands for savings, Mt is the
nominal money demand, Pt the price level and wt wages. The problem of the
2consumer is to maximize his life-time utility by choosing the savings level as
well as its distribution between capital and money:
MaxC
y
t ,Mt,C0
t+1 (C
y
t )γ(Co
t+1)(1−γ) (1)
s.t. : C0
t+1/Rt+1 + C
y
t +( Πt/Rt+1 − 1)Mt/Pt = wt (2)
The ￿rst-order conditions for this problem are:
γ
C
y
t
=
(1 − γ)
C0
t+1
Rt+1 (3)
Rt+1 > Πt, = if Mt > 0( 4 )
Equation (3) is the usual Euler equation, while equation (4) is a non-arbitrage
condition. Competitive ￿rms maximize pro￿ts choosing optimally capital and
labor. A Cobb-Douglas production function is assumed, Yt = Kα
t L
1−α
t , where
Kt is capital stock at time t (which depreciates completely upon use) and Lt
the ￿ow of eﬃciency units of labor of a worker born at time t. Labor force was
normalize to one. The ￿rst-order conditions of this problem, in eﬃcient units,
are given by:
wt =( 1 − α)kα
t , (5)
qt = αk
α−1
t , (6)
where q is the rental price of capital. In equilibrium, and under the hypothesis
of full depreciation, qt is equal to Rt.
In this economy, the government budget constraint is PtGt = Mt − Mt−1 ,
where Gt is real government expenditures at time t. Assuming a constant and
pre-announced rate of money creation (￿), we obtain Mt =( 1+￿)Mt−1 ,w h i c h
implies
gt =
￿
1 + ￿
mt, (7)
where gt is real government expenditures per eﬃcient unit of labor and mt are
real money holdings per eﬃciency units of labor.
3In this economy, public expenditures enhance the productivity of labor, in-
creasing the ￿ow of labor services per unit of time. The idea is that by investing
in public education, infrastructure, health services, sanitation, and so on, the
government increases the quality of the labor force. In particular, assume that:
Lt+1 = λ(gt)Lt, (8)
where the function λ(gt) is the government expenditure function. It transforms
each unit of public investment in infrastructure, by a relative increase of Lt+1/Lt
in labor productivity. The government expenditure function λ will be of the
form λ(gt)=2−exp(−gt/φ), where φ is a real number greater than one.1 Note
that λ(0) = 1, so that labor productivity does not change when the government
investment is zero.
Given the above functional forms, the equilibrium saving function of this
economy does not depend on the interest or in￿ation rate, but only on income,
and is given by (1−α)(1−γ)kα. The equilibrium in the assets market in eﬃcient
units is given by:
(1 − γ)(1 − α)kα
t = mt + λ(gt)kt+1. (9)
The dynamic system given by equations (3)-(7) and (9) can be reduced to
two equations:
αkα−1
t+1 =
2 − exp
‡
−
￿
1+￿
mt
φ
·
(1 + ￿)
, (10)
(1 − γ)(1 − α)kα
t = mt + λ(gt)kt+1. (11)
At the steady state (actually, the balanced-growth path), equations (10) and
(11) become:
m =( 1 − α)(1 − γ)kα −
•
2 − exp
￿
−
￿
1 + ￿
m
φ
¶‚
k, (12)
k =


2 − exp
‡
−
￿
1+￿
m
φ
·
(1 + ￿)α


1
α−1
. (13)
1In Ferreira(1999) the existence of monetary equilibrium was proved for general functional
forms. Results do not depend on this particular form of λ.
4In this model, money creation has two opposite eﬀects on the well-being of
consumers. The ￿rst and usual one is that in￿ation tax distorts the optimal
allocation of the economy. The second eﬀect is that money-creation ￿nances
public investment, thus increases the growth rate of output and consumption,
and consequently improves consumer utility.
We will study the optimal monetary policy for a government that wants to
maximize the utility of its subjects and take their actions as given. The criterion
of optimality is the discounted steady-state welfare of the present and all future
generations. Hence, the government goal is to choose the rate of money-creation,
￿, that maximizes the ￿ow of utility from consumption in the economy. It solves
Max
￿
∞ X
t=0
βt(C
y
t )γ(Co
t )(1−γ).
Given that along the balanced growth path all variables grow at a common
rate λ, we can rewrite the consumption of the young and the old at time t
as λtCy and λtCo. Assuming that the term (λβ)
t is smaller than one (which
will always be the case for all simulations below), the government￿s problem
becomes:
Max
￿
1
βλ
(Cy)γ(Co)(1−γ). (14)
Note that Cy and Co in the above expression stand for the ￿rst-period and
second-period equilibrium consumption functions, respectively.
3R e s u l t s
Simulations were concentrated on steady-state equilibria. Speci￿cally, the be-
havior of the monetary steady state will be investigated, assuming diﬀerent
combinations of the parameters α, γ and φ, when the government changes the
monetary policy.
Although the corresponding values are diﬀerent, the behavior of money and
capital vis-a-vis the in￿ation rate2 for a wide range of parameters displays a
similar pattern. For ￿ equal or close to zero, the agents hold similar quantities of
capital and money on their portfolios. For successively higher rates of in￿ation,
the steady-state level of money decreases until it reaches zero while the capital
per eﬃciency unit of labor increases until the economy reaches the non-monetary
steady state. The positive correlation between in￿ation and capital reproduces
the so-called Tobin eﬀect: as in￿ation increases, agents substitute money for
capital in their portfolios until the return on both assets are the same. Overall,
savings are higher for higher rates of in￿ation.
2Note that in this model the inﬂation rate at the steady state is given by {(1+￿)/λ(g) - 1},
which is equal to ￿ only at g = 0. However, it follows ￿ very closely.
5In other worlds, through an in￿ationary ￿nancing scheme the government
can stimulate the economy and higher levels of steady-state in￿ation correspond
to higher levels of capital stock. At higher stationary in￿ation rates, money-
demand is very small or null. Thus, there is a bound on the ability of the
government to use in￿ation to boost capital accumulation.
Table I below presents the steady-state level of money holdings for diﬀerent
values of money growth rates, using six combinations of parameters: α equal
to one third and one quarter, and γ equal to 0.5, 0.45 and 0.35. For all com-
binations, φ was set to be equal to 10, as changes in this parameter did not
signi￿cantly aﬀect the results.
Table I
Steady State Money Stock Levels
￿ m
α=1/4 α=1/3
γ=0.35 γ=0.45 γ=0.50 γ=0.35 γ=0.45 γ=0.50
0.00 0.1496 0.1021 0.0788 0.0611 0.0223 0.0029
0.01 0.1485 0.10100 . 0 7 7 40 . 0 5 9 50 . 0 2 0 50 . 0 0 10
0.02 0.1475 0.1005 0.0761 0.0578 0.0187 1E-06
0.03 0.1463 0.0990 0.0761 0.0562 0.0168 0
0.11 0.1369 0.0887 0.0636 0.0422 0.00130
0.12 0.1357 0.0870 0.0622 0.0404 1E-06. 0
0.20 0.1294 0.0751 0.0502 0.0251 00
0.32 0.1087 0.0574 0.0311 1E-07 0 0
0.40 0.0968 0.0440 0.0177 0 0 0
0.50 0.0811 0.0276 1E-05 0 0 0
0.64 0.0575 1E - 0 5 0000
0.90 0.0097 0 0 0 0 0
0.95 1E - 0 7 00000
The steady-state value of money holdings falls monotonically with ￿ for
all the combinations of parameters. However, the level at which it will reach
zero depends crucially on the values of α and γ. The smaller their values,
the higher the in￿ation rates for which monetary steady state will exist. For
the lowest combination (capital share of one quarter and γ equal 0.35) there
will be a monetary steady state for rates of money-creation up to 0.95, which
corresponds roughly to an in￿ation rate of 94% per period in the model. On the
other hand, for α and γ equal to one third and one half, respectively, there is no
monetary steady state for ￿ larger than 0.02. This is not unexpected because
the higher the γ the higher the importance for the individual utility function
of consumption in the ￿rst period of his life, and consequently the lower the
propensity to save. Furthermore, steady-state capital increases with α so that
the participation of money in total savings, everything being the same, decreases
with this parameter.
6The behavior of capital is, in a certain sense, a mirror image of the behavior
of money: its level on the steady state is higher for higher values of α and
γ,f o rag i v e n￿, and it increases monotonically with ￿ and in￿ation for given
capital share and γ.A sα and γ fall, there will exist monetary steady states for
increasingly higher in￿ation rates. Consequently, the capital stock level at the
non-monetary steady state decreases with α and γ:i ti s0 . 3 8 6f o rα equal to
0.25 and γ e q u a lt o0 . 3 5a n d0 . 197 for α equal to one third and beta equal to
one half.
The welfare eﬀects of money-creation are presented in ￿gure 1 below, where
discounted welfare of all generations and growth rates were plotted against in-
￿ation rates. The welfare line was obtained from (14) for the diﬀerent ￿0s,a n d
α = 1/4a n dγ =0 .35 were used in this exercise.
0,0000
0,0005
0,0010
0,0015
0,0020
0,0025
0,0030
0%
3%
6%
9%
12%
15%
18%
21%
24%
27%
30%
33%
36%
39%
42%
45%
48%
51%
54%
57%
60%
63%
66%
69%
72%
75%
78%
81%
84%
87%
90%
93%
Inflation
G
r
o
w
t
h
 
R
a
t
e
0
5
10
15
20
25
30
35
W
e
l
f
a
r
e
Growth Rate Welfare
Figure 1: Welfare and growth as functions of in￿ation
The above result has interesting implications. In a model where government
expenditures are not ￿wasted￿ or lump-sum transferred, but directly aﬀect the
productivity growth of the economy, the best policy always implies some in-
￿ation. Positive in￿ation is always optimal because, up to a certain level, the
bene￿ts of money-creation over accumulation are greater than the distortion
costs. In these cases, through their impact over productivity, positive in￿ation
rates increase the rate of return of capital and consequently its equilibrium level.
This in turn boosts consumption in the ￿rst period of life, which dominates the
reduction in second-period consumption due to in￿ation. For higher rates of
money-creation, however, the distortionary eﬀect dominates and in￿ation de-
creases welfare. The optimal ￿ is such that these two eﬀects cancel each other
out.
7T h em e c h a n i s ma b o v ec a nr a t i o n a l i z es o m ei n ￿ationary episodes: if the gov-
ernment is unable to increase tax collection (say, for social or political reasons),
the importance of public expenses for the economy and their eﬀect over pro-
ductivity, forces the authorities to resort to money-creation. Even more impor-
tantly, it is optimal to do so.
The second implication is that the in￿ation rate which maximizes the util-
ity of the consumer is lower than that which maximizes growth (35% and 42%
respectively). As the rate of money-creation rises, the increase in seigniorage
becomes progressively small, and so do the welfare gains from economic growth.
At a certain point, the loss due to the distortionary eﬀect of in￿ation tax over-
comes the gains from the growth eﬀect. So if the government wants to maximize
the welfare of the present and all future generations, it should consequently op-
erate on the left side of this Laﬀer curve, below the maximum revenue it can
obtain from money creation, and should not maximize economic growth. This
result holds for a large number of combinations of γ and α.B a r r o ( 1990) ob-
tained a similar result in the context, however, of endogenous-growth models
with representative, in￿nite-lived agents. Moreover, this model has no money
but a ￿at-rate income tax and, in addition to public capital in the produc-
tion function, there are government consumption services that enter into the
household￿s utility function.
Results would not change qualitatively - but only quantitatively - if any other
form of distortionary taxation is introduced in the model. This follows Ramsey￿s
rule, so that given the present functional forms, the optimal tax combination
would always imply positive in￿ation. For instance, with a proportional tax on
labor income of 0.25, welfare would still be maximized with positive in￿ation,
but now the optimal rate of money-creation is only 13%, considerably lower
t h a ni nt h ec a s ew i t ho n l yi n ￿ation tax. However, it remains true that growth
maximization is not optimal.
4 Concluding remarks
In this note we have presented a simple model in which optimal monetary policy
always implies positive in￿ation. The basic idea is that if public expenditure
is an essential factor of production and seigniorage an important source of rev-
enue, for in￿ation rates ￿not large enough￿, money-creation positively aﬀects
steady-state levels of real variables such as capital stock and income, thereby
increasing welfare. The result is still valid in the presence of other forms of
distortionary taxation. For high in￿ation rates, however, in￿ation decreases
welfare as its distortionary impact on allocations outweighs the positive eﬀect
on capital accumulation.
8References
[1]B a r r o ,R .J . ,1990, ￿Government spending in a simple model of endogenous
growth,￿ Journal of Political Economy, 1990, v98(5), Part 2, S103-S125
[2] Boldrin, M., 1993, Public education and capital accumulation, Manuscript,
Northwestern University.
[3] Drazen, A., 1991,I n ￿ation and capital accumulation under a ￿nite horizon,
Journal of Monetary Economics 8, 247-260.
[4] Ferreira, P. C., 1999, In￿ationary ￿nancing of public investment and eco-
nomic growth, Journal of Economic Dynamics and Control, 23, 539-563.
[5] Glomm, G. and B. Ravikumar, 1992, Public versus private investment in hu-
man capital: endogenous growth and income inequality, Journal of Political
Economy 100, 818-834.
[6] Mundell, R., 1965, Growth, stability and in￿ationary ￿nance, Journal of
Political Economics 73, 97-109.
[7] Tobin, J., 1965, Money and economic growth, Econometrica 33, October,
671-684.
[8] Weiss, L., 1980, The eﬀects of money supply on economic welfare in the
steady state, Econometrica 48, 565-576.
9

Inflation, Welfare and Public Expenditures

http://bibliotecadigital.fgv.br/dspace/bitstream/10438/353/2/1279.pdf

Inflation, Welfare and Public Expenditures

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics