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HB31
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working paper
department
of economics
On Inflation and Output with Costly Price Changes:
A Simple Unifying Result
Roland Benabou
Massachusetts Institute of Technology and
National Bureau of Economic Research
Jerzy D. Konieczny
Wilfrid Laurier University
No.
586
Sept.
1991
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
On Inflation and Output with Costly Price Changes:
A Simple Unifying Result
Roland Benabou
Massachusetts Institute of Technology and
National Bureau of Economic Research
Jerzy D. Konieczny
Wilfrid Laurier University
No.
586
Sept.
1991
DEWEY
MJ.T. LIBRARIES
NOV
1 4 1991
REUeivtu
On
Inflation
and Output with Costly Price Changes:
A
Simple Unifying Result.
by
Roland Benabou
Massachusetts Institute of Technology
and
National Bureau of Economic Research
and
Jerzy D. Konieczny"
Wilfrid Laurier University
September 1991
Department of Economics
Department of Economics
Massachusetts Institute of Technology
Cambridge,
02 139
Wilfrid Laurier University
USA
Canada
MA
Waterloo, Ont,
N2L 3C5
I.
Introduction.
We
study the effect of inflation on the average output of monopolistic firms facing
fixed costs of changing their nominal prices.
analyzed by several authors,
literature offers a
simple,
explicit
formula which unifies
all
special results.
is
As a
In this paper
result,
we
the
derive a
the effects identified in earlier literature.
finds, for constant elasticity
that the average of log-output
slope of this Phillips curve has been
usually with specific functional forms.
somewhat confusing array of
Rotemberg (1983)
The
demand functions and quadratic costs,
not affected by inflation.
Kuran (1986)
finds effects of
opposite signs for nonincreasing elasticity and concave increasing elasticity
demand
and constant
elasticity
functions, assuming constant marginal costs. Naish (1987) uses linear
demand and
cost functions to demonstrate the important role played by asymmetries in the
profit function
around
its
peak. Konieczny (1990) shows,
depends not only on the skewness of
He
function.
dominates
negligible.
profits,
more
generally,
that the effect
but also on the curvature of the
demand
does not, however, provide a way of determining which of these two factors
and incorrectly claims
Finally,
Danziger
that,
(1988)
at small inflation rates,
stresses the role played
the profit effect
by discounting
is
at small
inflation rates.
This paper provides a unifying but very simple treatment of
all
the effects described
above, for the case where the costs of price adjustment are small. This framework, which
seems to us the most
solution
relevant, allows the use of Taylor expansions to obtain a closed
(Proposition 2), which can easily be applied to any specification.
form
The Model.
II.
We
consider a monopolistic firm which produces a single, perishable good and
expects the inflation rate to remain constant over time.
any time but
disseminated,
it
c> 0;
incurs a cost,
c
etc.
new
the
price
1
The firm can change
its
price at
must be decided upon, the information
also proxies for the adverse reaction of customers
and competitors,
not captured by the model.
price.
It is
convenient to express profit and demand functions in terms of the log of the real
The
firm's problem, at the time of the first price change,
is
to
choose the sequence
CO
CO
of times of adjustment,
and
{t } T .
T
so as to maximize the present value of
V
(1)
-
t-O
where
F
is
(s,S)
and
assuming:
type
r
its profits:
-g(t-Q)e' ndt -ce'
r
is
F(z
n*
L
the profit function, assumed to be
inflation rate
price,
^F(P
£
{PT } T -o
each adjustment,
(logs of) real prices set at
We shall
the discount rate.
m
)
>
- F'(z
(Sheshinski and Weiss,
nominal price so that the (log
m
)
> F"(z m ).
1977):
of) real price
has eroded the real price to s < z™ < S.
strictly
is
at
quasiconcave and
C3
;
g >
is
7
denote by z " the (log) real monopoly
The optimal
pricing policy
is
of the
each price change the firm chooses the
5 >
z™; a
new change occurs when
inflation
2
1
Benabou (1991) analyzes firms which compete through customer search. In this
demand and profit functions are endogenous and the effect of inflation on output
depends on characteristics of the market such as search
2
The
case of deflation
is
the
case
also
costs.
obtained by permuting S and s throughout the paper.
The optimal
policy satisfies the
(s,S)
(2a)
F(S) -re - F(s)
(2b)
F(s) -
order conditions:
first
rV
Strict quasiconcavity implies that the
optimum
dS/dg >
> ds/dg
F'(s) >
(3)
> F'(S)
;
unique and
is
satisfies:
We now consider an economy consisting of many such firms which stagger their price
changes uniformly over time.
firm over
its
Y
.
the
is
"
(5)
then equal to the average output of a
function,
S-s
<
y'(-)
s
-L
y(z)dz
\
J*
0.
and rearranging, we obtain:
y(s) + v(5)
-
Y
F'(s)
-
F'(s)(S-s) dg
Since both
inflation
(4)
y(S-P )d, -
dS
dg
F'(s)
and
y(s) -
Y
are positive,
(5)
m±m [m
makes
_
?]
clear that the effect of
on output depends on:
the skewness of the profit function (represented by the asymmetry in marginal profits,
F'(s) + F'(S)
s
s>
demand
Differentiating
(i)
r"
J-
S-s Jo
v(-)
is
price cycle:
(4)
where
Aggregate output
and
5;
)
which, through
this will
(2),
be called the
determines the effect of inflation on the price bounds,
profit effect :
4
the
(ii)
y\s ) + y\
)
_
the
of
curvature
demand
(represented
function
y) which determines the output
the
output
difference
effects of those changes; this will
be called
2
the
demand
effect
.
Following Konieczny (1990)
for every z 1
< z" < z 2 such
inequality holds. If F(-)
1
slower than z"
-
s.
As a
is
that
we
call
F^)
= F(z2 ), and
strongly left
result,
(see
explicit
skewed
S
< -F'(z 2 )
if
the opposite
-
z™ increases
skewed towards lower values, so that
(5) as well as
We now
derive,
formula which shows the effect of
Small Adjustment Costs or Low
(11)
below).
The
Inflation:
A
its
implicit nature
for the case of small
all
makes
opposite
simplify the exposition
we
incorporated in Section
V
(2a')
F(s) - F(S)
(2b')
F(s) -
The
first
a
c,
Simple Formula.
consider the case where r = 0;
below.
rather
underlying parameters.
to derive closed-form solutions for
first
it
g and/or
concentrate on the most relevant case, where g or especially c
and use Taylor expansions
3
F'iz-^)
as inflation rises,
(5) reveals the qualitative effects at work,
opaque and unwieldy. Below we therefore
III.
if
strongly right skewed.
holds for F(-)
simple,
is
skewed
strongly right
skewed then,
each price cycle
the profit effect tends to increase output
While
F(-) strongly left
s,
S
and
is
small,
dY /dg 3
.
To
a positive discount rate
is
order conditions become:
F
Numerical simulations
in related
models (Benabou (1991) and Dixit (1991)) show
that such approximations are generally quite accurate over a wide range of parameters.
5
where
F
denotes average profits per unit of time, net of price adjustment
both price bounds, s and
We
costs.
By
(2'),
approach the monopoly price as c or g approach zero.
5,
therefore define:
- zm
S
=
S
«1
and
z
where, throughout the paper, ^T « J^"
order
n
in
x,
relative rate at
X
stands for
i.e.
which
t
= zm
-
-
m - s
r
an *
a^x',
'Y^\<*
« ax
x* <:
xn
0,
a262
+
<S
represents a Taylor expansion of
a x + a2
Thus
.
i
goes to zero in comparison to
s
6
= S
represents the
<S
-
z
m
,
when
c
or
g become very small.
Now,
for all z
F(z)
(6)
in
[5,5]:
a F(z m )
+
F"
—
(z
2
Expanding
(2a')
a = -F"'(z m )/2F"(z m )
z
(7)
Next,
m -
s
m (z~z m
)
f +
to the third order in
Till
i— (2 m
6
)
(z-2
m
3
)
6
leads to
c^
=
1,
a 2 = -2a/3,
measures the skewness of the profit function around zF. Thus:
a 5(l-2fl5/3)
inserting
(6)
and
(7) into (2b')
gives the following expression for
1V3
5 - zm = S «
(8)
as in
3cg
-2F"(z m )
Mussa (1981) and Rotemberg
Finally,
we
where
return to output.
(1983);
Using
its
interpretation
(7)-(8)
to
is
expand
straightforward.
(4)
we
get:
6:
Y«
and
y(z
m
)
+ y'(z
m
)a8 2 /3
+ y"(z
m S2
/6
)
so:
Y*y(z m ) -y\z m )
(9)
—
6
Proposition
Let
F'"(z
[
m
_ y"(z
)
m
m
y'{z
m
F"(z )
)
)
1.
=
r
0;
then,
for small adjustment costs
m
F'"(z
(10)
y"(z
)
dg
(gc
<-
F'Xz™)):
m
m
y'(z
m
F"(z )
or/and inflation rates
)
)
1/3
where
m
-y'{z
A
Interpretation
IV.
The use
)
\%F"{z m )
>
0.
_
and Examples.
of Taylor approximations yields explicit measures of the effect of skewness
of profits and the curvature of
demand on
the average output.
not change signs in the neighbourhood of z™, F(-)
if
F"'{z
from
m
)
<
(
>)
0.
Thus two measures of the
is
Assuming
that
strongly left (right)
effect of
F'"{z)
does
skewed near
1
z"
skewness on price bounds are,
(6)-(8):
s +
S
(11)
The
-z™*^
or
effect of changes in the price
the curvature of the
demand
function.
f'(s)
+ f'(S)
*
^
bounds on average output depends,
A measure
of this effect
is,
from
in turn,
(7)-(9):
on
^LLI^l -Y
(12)
=y"(z m )
—
2
By Jensen's
function
is
inequality,
and
convex,
Proposition
is
demand
it if
shows how the net
1
effects.
negligible as
effect tends to increase output
demand
effect of inflation
depends on the
Konieczny (1990) claims
compared
demand
to the
this is
is
not that of
profits,
demand
the
relative strength
low inflation
at
that,
rates,
the
because the profit function
effect,
not correct: both effects are of order S
the relevant asymmetry
if
concave.
is
hence symmetric up to a third order approximation.
(log) quadratic,
and (12) that
the
to decrease
demand
of the profit and
profit effect
3
2
This
.
but of marginal
It is
is
is
from (11)
clear
due to the
fact that
can be seen from
profits, as
the term F'(S) + F'(s) in (5).
We
conclude
commonly used
When
the
if
is
effects for
some of
the most
functional forms.
demand
function
zero and so sign (dY/dg) =
profit effect
by identifying the two
this discussion
-
is
linear (in the log of real price) the
m
sign (F'"(z )).
zero and sign (dY/dg) =
both conditions are met;
the profit function
m
3A and
y(z) =
costs the profit effect
the latter
is
-A;
is
effect
is
(log) quadratic the
not affected by inflation
is
considered by Rotemberg (1983).
in the real price:
and either constant or linear marginal
Output
sign (y"(z )).
this is the special case
For demand functions linear
as the former equals
If
demand
thus
—
A
t
-
A 2e
z
,
A
x
>
0,
A2
dominates the demand
=
2A >
>
0,
effect,
0.
dg
For
is
is
isoelastic
demand and
the cost function and a >
Ap
0,
p >
cost functions: y(z) =
1,
the profit effect
and so dY/dg = A(l-ap). The
is
A
t
e'Pz
A(l
-
*
,
p
-
(y)
a/3);
= ya
the
where
demand
*(•)
effect
profit effect tends to decrease output while the
8
demand
the
>
1;
this is true in particular for constant
the
demand
V.
A
The
effect operates in the opposite direction.
effect to dominate,
and
profit effect
is
stronger for
a&
(a >
For
for increasing marginal costs
marginal cost must decrease fast enough (a <
1).
1/fi).
General Formula with Discounting.
When
the discount rate
is
more about
firms care
positive,
profits closer to the
beginning of each cycle than those in the later phase. Therefore, in comparison to the case
considered in section
level,
z™.
The
they set the
III,
price
is
initial real price,
then allowed to deteriorate more before the next adjustment.
+
Discounting therefore creates a discontinuity at g =
Y;
average output
to
F'(s)
•
(1988),
makes S
Therefore,
the terminal one
falls.
This
is
resulting in
,
dY/dg
to
by
because,
closer to /", for small c or
as the inflation rate increases,
the
(at
(2),
g
it
initial
an upward jump
any given g >
0),
it
4
in
is
F'{S)dS/dg - F'(s)ds/dg.
reduces
|
F'(S)
price increases
|
relative
more than
This raises the average price and lowers average output (Danziger
Konieczny (1990)).
To show
precisely
generalize Proposition
positive value,
1
how
discounting combines with the profit and
to the case of a positive discount rate.
and consider the case where c
m
(13)
As
see Danziger (1988).
negatively affected by discounting.
Since discounting
closer to the profit maximizing
5,
f*F(z)e*-'
,Sr/g
Mdz
is
small.
_ p -rr/g
-F(s)
We
let
demand
effects
we
g take any given
(2b) can be rewritten as:
- cge Sr/s
r/g
4
=
In particular, as g
F(s) + re.
-* 0,
5 tends
to z™
while s remains bounded away since F(S)
where,
first
as before,
S - zm = S <
1;
m
z
=
- s
« axS
r
a2S2
+
order conditions (2a) and (13) to the third order in
(2/3)(r/g
-
a).
5
l{ -r -a\s 2
S +
<S
or
3U
given by
is still
at =
obtain:
1,
a2 =
z
m
£J-S
_
—
-
a
18
(8).
Expanding average output as before leads
Y~y(z m ) -y\z m )
(15)
we
6
Hence:
(14)
where
Expanding again the
.
—
2r
+
F'"(z
[g
m
_ y"(z
)
F"(z m
m
m
y'(z
)
to:
)
) J
Equations (14)-(15) clearly show that positive discounting lowers the average price
and
raises average output,
inflation tradeoff,
Proposition
If
*Ag-&
dg
is
the
same
F'"{z
is
m
[F"(z m
all
small
(gc
< -F
its
effect
on the slope of the output-
(z
m
)
and
re
< -F
y"{z
)
m
U-,m\
y'(z )
is
m
)),
then
)
g)
1.
three factors which affect the slope of the inflation-output
solely in terms of the underlying parameters of the model.
A detailed proof
(z
m
)
as in Proposition
This result embodies
tradeoff,
to
2.
dY
A
As
given by:
it is
the cost of price adjustment
(16)
where
as argued above.
available
from the authors upon request.
It
incorporates
10
Proposition
1
as a special case
and also shows
that, for small
falls
with inflation regardless of the strength of the
VI.
Conclusions.
enough
demand and
inflation rates, output
profit effects.
We analyzed the effect of inflation on the average output of monopolistic firms facing
a small fixed cost of changing nominal prices.
Using Taylor expansions,
closed form solution which can be applied to any specification.
we
derived a
This extremely simple
formula allows us to evaluate the relative impact of the three factors which affect the
inflation-output tradeoff: the asymmetry of the profit function, the convexity of the
function,
and discounting.
clarifies the
It
previous results of
demand
thus provides a unifying framework which substantially
this literature.
11
References:
Benabou, R. (1991),
Studies,
Inflation
and Efficiency
in
Search Markets,
Review of Economic
forthcoming.
Danziger, L. (1988), Costs of Price Adjustment and the Welfare Economics of Inflation and
Disinflation, American Economic Review, 633-646.
Dixit,
A. (1991), Analytical Approximations in Models of Hysteresis, Review of Economic
Studies,
Konieczny,
J.
141-151.
(1990), Inflation, Output and
Infrequently, Economica,
Labour Productivity when Prices are Changed
201-218.
Kuran, T. (1986), Price Adjustment Costs, Anticipated Inflation and Output, Quarterly
Journal of Economics, 407-18.
Mussa, M. (1981), Sticky Prices and Disequilibrium Adjustment in a Rational Model of
the Inflationary Process, American Economic Review, 1020-27.
Naish, H.F. (1986), Price Adjustment Costs and the Output-Inflation Tradeoff, Economica,
219-30.
Rotemberg, J.J. (1983), Aggregate Consequences of Fixed Costs of Price Adjustment,
American Economic Review, 433-436.
Sheshinski, E. and Y. Weiss (1977), Inflation and the Costs of Price Adjustment, Review
of Economic Studies, 287-303.
i
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