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Journal of Banking & Finance xxx (2008) xxx–xxx
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A note on ‘‘Inflation and Welfare” q
Rubens Penha Cysne
Graduate School of Economics of the Getulio Vargas Foundation (EPGE/FGV), Praia de Botafogo 190, 11 Andar, CEP 22250-900 Rio de Janeiro, Brazil
Received 17 July 2007; accepted 7 December 2007
Abstract
This note provides an analytical confirmation and a refinement of [Lucas Jr., R.E., 2000. Inflation and welfare. Econometrica 68 (62),
247–274 (March)] numerical findings regarding the characterization of optimality in the shopping-time model presented in that paper.
The original numerical analysis concludes that a coefficient of risk aversion (r) greater than 0.01 is sufficient for optimality. Here we use
Arrow’s sufficiency theorem to confirm this result and, more importantly, to show without more calculations how changes in parameters
can affect it.
Ó 2007 Elsevier B.V. All rights reserved.
JEL classification: E40; E50
Keywords: Arrow’s sufficiency theorem; Optimal control; Inflation; Welfare; Shopping-time
1. Introduction
Lucas (2000) has looked comprehensively at past
research on the welfare costs of inflation and provided
new estimates based on US time series for 1990–94. Posterior work in this area has dealt with some of the challenges
ahead pointed out by Lucas. Just to mention some of them,
Cysne (2003) and Jones et al. (2004), e.g., have investigated
the use of Divisia indices in the measurement of the costs of
inflation. Following another line of investigation, Ireland
(2007), Jones et al. (2005) and Dutkowsky et al. (2006)
are examples of papers concentrating on retail and
demand-deposit sweep programs to explain the difficulty
(also reported by Lucas) of obtaining a good fit of M1
money demand in the 1990s.
This note addresses a particular point of Lucas’ paper.
In Section 5, Lucas shows that the transactions-technology
model proposed by McCallum and Goodfriend (1987) can
provide a general-equilibrium rationale for measurements
q
This paper was reviewed and accepted while Prof. Giorgio Szego was
the Managing Editor of The Journal of Banking and Finance and by the
past Editorial Board.
E-mail address: [email protected]
of the welfare costs of inflation based on Bailey’s areaunder-the-inverse-demand-function formula. A crucial
problem of the optimization carried out in this section is
that the value function need not be concave. Lucas pursues
a solution for this problem relying on numerical analyses in
order to determine the conditions under which the consumer utility is maximal.
Cysne (2006) has shown, in general, how some categories of models, including the n-dimensional shopping-time
model, could benefit from an application of Arrow’s
(1968) sufficiency theorem regarding the characterization
of optimality. Here we apply such results directly to Lucas’
1-dimensional shopping-time model, providing a simpler
example of an application of Arrow’s theorem.
The main contribution of this note is to provide an
expression which allows us to see, without further numerical calculations, how changes in parameter values affect the
conclusions about optimality in Lucas’ analysis.1 In his
numerical simulations, Lucas (2000) found that ‘‘in this
non-convex problem the first-order conditions can fail to
hold under optimal behavior” for values of the coefficient
1
We refer here to inequality (16).
0378-4266/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.jbankfin.2007.12.012
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of risk aversion (r) below 0.01. Based on the minimal value
of real balances as a fraction of output in the US time series
used by Lucas, we confirm his results and show that,
indeed, r P 0.0085 suffices to characterize optimality.
The (present-value) Hamiltonian describing the consumer’s
problem is (we omit the argument t of the functions):
H ðm; s; kÞ ¼ eðgcð1rÞÞt U ðmf ðsÞÞ þ keðgcð1rÞÞt ð1 mf ðsÞ
2. Applying Arrow’s theorem to Lucas’ model
where U is given by (4). Lucas’ product-equilibrium equation (5.7) reads:
1 s ¼ mf ðsÞ:
ð7Þ
The steady-state interest rate is given by
r ¼ p þ g þ rc:
ð8Þ
This value can be easily obtained by including the market
for bonds in the initial problem.
The usual (present-value) Hamiltonian first-order conditions Hs = 0 and k_ ¼ H m in this case lead to:
(
0
ðsÞÞmf 0 ðsÞ
k ¼ eðgcð1rÞÞt U ðmf
1þmf 0 ðsÞ
ð9Þ
k_ ¼ eðgcð1rÞÞt U 0 ðmf ðsÞÞf ðsÞ þ kðf ðsÞ þ p þ cÞ
Therefore
k_
f ðsÞ
¼
þ p þ c:
ð10Þ
k mf 0 ðsÞ
Let
U 0 ðmf ðsÞÞmf 0 ðsÞ
a :¼ keðgcð1rÞÞt ¼
:
ð11Þ
1 þ mf 0 ðsÞ
Lucas’ shopping-time model has been originally presented in discrete time. The Euler equation is derived from
the Bellman equation using the first-order condition with
respect to the control variable and the (assumed) differentiability of the value function. The first-order condition
obtained by Lucas, for the purpose of developing the
empirical analysis of the balanced-growth path, is the same
that we derive below in a continuous-time framework,
using the Maximum Principle of Pontryagin et al. (1962).
The analysis presented here, therefore, loses no generality
by considering a continuous-time framework.
Let r be the nominal interest rate, p the rate of inflation,
s the fraction of the initial endowment spent as transacting
time (the total endowment of time being equal to the
unity), y real output, m real balances as a fraction of output, c consumption as a fraction of output, U(cy) a strictly
increasing and strictly concave function of real consumption, v the (exogenous) fraction of output transferred to
the government by the household (lump-sum tax), g > 0 a
discount factor and c > 0 the (exogenous) rate of real output growth (so that y(t) = y0ect).2
The consumer maximizes the discounted utility:
Z 1
egt U ðcyÞdt;
ð1Þ
0
subject to the household’s budget constraint (2) and to the
transacting technology constraint (3):
m_ ¼ 1 ðc þ sÞ m ðp þ cÞm;
ð2Þ
c ¼ mf ðsÞ:
ð3Þ
As usual, the dot over a variables denotes its derivative
with respect to time.3 Eq. (3) stands for the constraint given
by the transacting technology, with f0 (s) > 0 and f00 (s) 6 0.
Lucas (2000) assumes the utility function to be given by
the (1 r)-degree homogeneous function:
1r
ðcyÞ
; r 6¼ 1
ð4Þ
1r
in which case, by normalizing y0 to one, (1) can the written
as a function of c alone:
Z 1
eðgþcð1rÞÞt U ðcÞdt:
ð5Þ
U ðcyÞ ¼
0
2
Note that in this model the potential real output (obtained when s = 0)
is equal to y and the effective real output (which, in equilibrium, equals
consumption) is given by (1 s)y.
3
To obtain Eq. (2), make P stand for the price index, M for nominal
balances and write:
_ ¼ Py cPy sPy vPy
M
Next, divide both members by Py and use the formula of the derivative of
a fraction with respect to M/Py.
s m ðp þ cÞmÞ
ð6Þ
_
Since m and s are constant in the steady-state, 0 ¼ aa_ ¼ kk þ
g cð1 rÞ: Using (8) and (10):
f ðsÞ
þr ¼0
mf 0 ðsÞ
This leads to Lucas’ equation (5.6):
f ðsÞ ¼ rmf 0 ðsÞ:
ð12Þ
From Lucas’ original analysis, for r > 0 the equilibrium
values of s and m are bounded below by zero (and s
bounded above by one, as well), in which case one can as 2 B; B an open and convex
sume limt!1 ðsðtÞ; mðtÞÞ ¼ ðs; mÞ
set in R2 and 0 < s < 1.
Proceeding to the main point of this note, observe in (6)
that H is not jointly concave in the state and control variables, as required by the usual Mangasarian’s (1966) theorem. This happens due to the term mf(s). We now turn to
the application of Arrow’s theorem.
The reader is invited to look either at Seierstad and Sydsaeter (1977), Cysne (2006) or Kamien and Schwarz (1991,
p. 222) for a formal version of Arrow’s sufficiency theorem
written in terms of the present-value Hamiltonian H
(exactly as in (6)).4
4
For the reader interested in connecting our previous analysis to
Arrow’s sufficiency theorem, as presented in Cysne (2006) note that u and
x, the control and state variables in that theorem, correspond, respectively,
to s and m in the problem we are analyzing. Moreover, using the terms of
the theorem, n = 1, U ¼ ½0; 1; f 0 (x, u, t) = e (gc(1r))t U(mf(s)),
f1(x, u, t) = 1 (mf(s) + s) m (p + c)m, [t0, t1] = [0, 1]. Eq. (5) in the
_ ¼ H m ðs; mÞ where i = 1. The transversality condition
theorem reads kðtÞ
limt!1 p1 ðxðtÞ xðtÞÞ ¼ 0 in the theorem is satisfied by assumption of the
original paper.
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The correct application of Arrow’s theorem requires,
first, obtaining the value of s (call it s*) which maximizes
(6) for the given values of k and m; next, plugging this value
in (6). Call the new (maximized) Hamiltonian so obtained
H*. If this new Hamiltonian can be shown to be (strictly)
concave with respect to the state variable m, for the given
value of k, then (by Arrow’s theorem) the first-order conditions characterize a (unique) optimum.
From now on, as in Lucas (2000, p. 265), we particularize the transacting technology to:
f ðsÞ ¼ ks;
k > 0:
ð13Þ
In order to derive the maximized Hamiltonian used in the
theorem, use the first equation in (9) and (11) to obtain:
1=r
1 km þ 1
s¼
a
:
ð14Þ
km
km
Note that here a = U0 mk/(mk + 1) > 0. This value of s
maximizes the Hamiltonian, since Hss = e(gc(1r))t
U00 (c)k2m2 < 0. Plug (14) into (6) to obtain the maximized
Hamiltonian:
"
ð1rÞ=r
r
km
ðgcð1rÞÞt
H ðm; aÞ ¼ e
1 r að1 þ kmÞ
#
það1 m ðp þ cÞmÞ :
ð15Þ
The next step in the application of the theorem is showing
that the maximized Hamiltonian is concave with respect to
the state variable m. Note that the derivative of (15) with
respect to m is given by:
H m ðm; aÞ ¼ eðgcð1rÞÞt
"
#
ð1rÞ=r
km
1
ðp þ cÞa
að1 þ kmÞ
mð1 þ kmÞ
Taking the derivative of the above expression, one easily
concludes that H mm < 0 iff:
1
:
ð16Þ
r>
2 þ 2 km
Inequality (16) is the main point of this note. It is important because it shows without more calculations how
changes in parameters can affect the optimality condition
in Lucas’ original model.
In Lucas’ original paper, Figs. 9 and 10 are used to
report numerical calculations designed to check if consumer utility is in fact maximized along the balanced path
constructed from the first-order conditions of the dynamic
program. In all numerical simulations (see p. 267) the constant k is assumed to be equal to 400.
Make k = 400. Since the right hand side of condition
(16) is stated as a function of m, it can assume different values. One way to use (16) is to resort to the original empirical data. Since we are looking for a sufficient condition,
one possible strategy is to use the most biding condition
(the one which requires the highest coefficient of risk aversion r). This is carried out by plugging into (16) the lowest
3
value of m found in the time series. Considering the data
provided by the author, this value runs around 0.1451 (real
balances as 14.51% of output). Using this value of m leads
to the sufficiency condition r > 0.0085.
Regarding his numerical simulations, Lucas uses the
equilibrium value of m under different sets of parameter
values. Lucas reports that problematic behavior concerning
the characterization of the optimum by the first-order conditions occurs in the case of linear utility (r = 0) and
‘‘emerges at positive but very small (smaller than 0.01) values of r”. Lucas does not report how close to 0.01 he
assigned values to r and found a problematic behavior.
For this reason, all we can say based on the previous quote
is that, following his calculations, r > 0.01 would suffice for
optimality. In this way, the difference between our condition (r > 0.0085) and Lucas’ can be explained in terms of
one sufficient condition being a little finer that the other.
Given the differences in approaches, methods, and analyses, the results are surprisingly consistent.
For r > 0.0085, and provided that m P 0.145, H* turns
out in practice to be a strictly concave function of m and,
by Arrow’s sufficiency theorem (since Hss < 0 as well) the
interior balanced path is optimal and unique. In contrast
with the sufficiency condition based on Mangasarian’s theorem, which would demand the concavity of the original
Hamiltonian (6) for all pairs of s and m considered in their
respective domain, Arrow’s theorem implicitly takes into
consideration the fact that the choices of s (by (2) and
(3)) determine m, thereby allowing for a less demanding
sufficiency condition.
Since km > 0, a sufficient condition that does not depend
on the endogenous variable m is given by:
r > 1=2
ð17Þ
repeating the general result displayed in Cysne (2006).
If the curvature of utility is high enough the non-convexity of the domain in the optimization problem poses no
problem. On the other hand, when the coefficient of risk
aversion is too close to zero and utility is close to linear,
the consumer does not mind transferring consumption
from one period to another, and having s = 0 for a while
can be an optimum policy. As Lucas points out (see p.
269), ‘‘the optimum policy in this case is to set s = 0 for
a while, consuming nothing, earning maximum income
and accumulating cash, and then enjoy a consumption orgy
in which all cash is spent at once”.
Acknowledgements
I am thankful to Robert E. Lucas Jr. for the provision of
the original data used in Lucas (2000) as well as for conversations about this topic during my stay as a Visiting Scholar in the Department of Economics of the University of
Chicago. I am also thankful to the Referee for her/his suggestions to make the paper easier to follow. The usual disclaimer applies.
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References
Arrow, K.J., 1968. Applications of control theory to economic growth.
In: Dantzig, G.B., Veinott Jr., A.F. (Eds.), Mathematics of the
decisions sciences, American Mathematical Society, Providence,
RI.
Cysne, Rubens P., 2003. Divisa index, inflation and welfare. J. Money
Credit Bank. 35 (2), 221–238 (The Ohio State University).
Cysne, Rubens P., 2006. A note on the non-convexity problem in some
shopping-time and human-capital models. J. Bank. Fin. 30 (10), 2737–
2745.
Dutkowsky, D., Cynamon, B., Jones, B., 2006. US narrow money for the
twenty-first century. Econ. Inquiry 44, 142–152.
Ireland, P., 2007. On the welfare cost of inflation and the recent behavior
of money demand, Boston College Working Paper.
Jones, B., Asaftei, G., Wang, L., 2004. Welfare cost of inflation in a
general equilibrium model with currency and interest bearing deposits.
Macroecon. Dyn. 8 (4), 493–517.
Jones, B., Dutkowsky, D., Elger, T., 2005. Sweep programs and optimal
monetary aggregation. J. Bank. Fin. 29 (2), 483–508.
Kamien, M., Schwarz, N., 1991. Dynamic Optimization: The Calculus of
Variations and Optimal Control in Economics and Management.
Advanced Textbooks in Economics. Elsevier Ltd.
Lucas Jr., R.E., 2000. Inflation and welfare. Econometrica 68 (62), 247–
274 (March).
Mangasarian, O.L., 1966. Sufficient Conditions for the Optimal Control
of Nonlinear Systems. Siam J. Control 4 (February), 139–152.
McCallum, Bennet T., Goodfriend, Marvin S., 1987. In: Eatwell, John,
Milgate, Murray, Newman, Peter (Eds.), Demand for Money: Theoretical Studies in the The New Palgrave: A Dictionary of Econometrics. MacMillan/Sotckton Press, London/New York, pp. 775–781.
Pontryagin, L.S., Boltyanskii, V., Gamkrelidze, R., Mischenko, E., 1962.
The Mathematical Theory of Optimal Process. Interscience, New York
and London.
Seierstad, A., Sydsaeter, K., 1977. Sufficient conditions in optimal control
theory. Int. Econ. Rev. 18 (2), June.
Please cite this article in press as: Cysne, R.P., A note on ‘‘Inflation and Welfare”, J. Bank Finance (2008), doi:10.1016/
j.jbankfin.2007.12.012