Faro, Clovis de

English

Research Papers in Economics

Ensaios Econ￿micos
Escola de
P￿s-Gradua￿ªo
em Economia
da Funda￿ªo
Getulio Vargas
N 323 ISSN 0104-8910
The E￿ect of In￿ation on Growth Invest-
ments: A Note
Clovis de Faro
Abril de 1998
URL: http://hdl.handle.net/10438/540Os 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
de Faro, Clovis
The Effect of Inflation on Growth Investments: A Note/
Clovis de Faro ￿ Rio de Janeiro : FGV,EPGE, 2010
(Ensaios Econ￿micos; 323)
Inclui bibliografia.
CDD-330THE EFFECT OF INFLATION ON GROWTH
INVESTMENTS : A NOTE
CLOVIS DE FARO
1
Revised
May, 1998
                                                       
1 Graduate School of Economics, Fundação Getulio VargasTHE EFFECT OF INFLATION ON GROWTH
INVESTMENTS:  A NOTE
I. Introduction
One of the most traditional issues in the theory of capital, is the choice of the optimal duration
of growth investments.  This problem, also known as capital deepening (cf.  Baumol, 1972 and
Hirshleifer, 1970) and which can be traced back to the 19
th century work of the English mathematician
W.S. Jevons, is typical of the so called point-input, point-output process.  Illustrated, in   an  idealized
way, by the activities of wine aging and tree growing, it consists of determining the length of time that
an investor should keep an asset whose value increases with time.
Although a standard topic in text-books presentation of capital theory (besides the above
mentioned works of Baumol and Hirshleifer, see, for instance: Allen, 1938; Bierman and Smidt, 1993;
Henderson and Quandt, 1971; Simon and Blume, 1994 and Varian, 1990), the effects of inflation and
taxation are usually not included in the analysis.  Brenner and Venezia (1983) appear to be the first
authors to take into account the joint effect of taxes and inflation on the optimal duration.  However,
despite the importance of their analysis, the very fact of considering the joint effect of taxes and
inflation rendered some of the results somewhat confusing.
A previous note focused attention on the isolated effect of taxation on the optimal duration in
a non-inflationary environment (de Faro, 1996).  Complementing the analysis, the present note is
aimed at the investigation of the effects of  inflation by itself. Specifically, considering the no-
reinvestment case only, our objective is to include in the analysis both a numerical illustration of the
peculiar results of BV as well as some indexation procedures that have been used in some high
inflation economies.
II. Basic Model
Adapting the notation in BV, let
F(T) - net present value of the investment cash-flow, as measured in monetary units referred
to the date of the investment;2
f(T)  - pre tax net receipt, as measured in monetary units referred to the date of the
investment, derived from termination at time T;
h(T) - pre tax net receipt, as measured in nominal terms, derived from termination at time T;
R    - real rate of interest (as measured in instantaneous terms and assumed to be positive);
C     - cost of investment, as measured in monetary units referred to the date of the investment;
T     - time to termination of investment;
q     - constant rate of inflation (as measured in instantaneous terms and assumed to be
positive);
j      - tax rate.
Assuming that the asset can be depreciated, for tax purposes, in a lump sum at the time of
termination, with no indexation of the investment cost, and that the pre tax net receipt is fully indexed
(i.e. h(T) = e
qT f (T)), the cash flow associated with the investment, as expressed in current prices (i.e.
nominal terms), is as depicted in Figure I
                                               e ( ) ( ) ( )
q q j
T T f T e f T C - -   
0
        T time
C
    Figure I
Cash-Flow for the Basic Model
The above cash-flow corresponds to what will be called the basic model, and is the one that
was addressed by BV.
Working with constant prices, as referred to the date of the investment, it follows that the
function whose maximization determines the optimal duration is:
1
F(T)= - C + e ( ) ( ) ( ) { } C   T
T T j j
q q + -
+ - f   e 1
       R                    (1)
                                                       
1 Observe that, if we denote by i=R+q the corresponding nominal rate of interest, if j=0 and if we work directly with
h(T), equation (1) corresponds to the classical text-book presentation of the problem (c.f. Hirshleifer, 1970, Simon and
Blume, 1994 and Varian, 1990).3
Repeating here, for completeness, the presentation of BV, we have:
( ) ( ) ( ) ( ) ( ) { }
( ) dF T
dT
f T Rf T e R C e
T R T ( )
= - ¢ - - +
- + 1 j q j
q q                           (2)
Thus
( ) ( ) ( ) ( ) ( ) ( )
dF T
dT
f T Rf T e R
T
= ￿ - ¢ - - + 0 1 0       =
 
j q j
q
C                           (3)
On the other hand
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) { }
( ) d F T
dT
f T Rf T e f T Rf T e e
T T R T
2
2 1 1 = - ¢¢ - ¢ + - ¢ -
- + j q j
q q q
( ) ( )
- + R
dF T
dT
q                                                                              (4)
Therefore, denoting by T* the solution of (3), we have:
( ) ( ) ( ) ( ) ( ) ( ) ( ) { }
d F T
dT
f T Rf T f T Rf T e
T
RT
2
2 1
ø
ß
œ = - ¢¢ - ¢ + ¢ -
-
*
* * * *
*
j q           (5)
Thus, as 1-j>0 and e
-RT  >0,  it follows that T* will be the optimal duration if
( ) ( ) ( ) ( ) ( ) ¢¢ - ¢ + ¢ - < f T Rf T f T Rf T * * * * q 0                                              (6)
To investigate the effect of changes in the inflation rate  q on the optimal duration  T*,
differentiate (3) holding j and R as constants.  It follows that:4
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) { }
dT
d
C T f T Rf T e
f T Rf T f T Rf T e
T
T q
j j
j q
q
q =
- - ¢ -
- ¢¢ - ¢ + ¢ -
1
1
                                (7)
Taking into account (6), we know that, at  T*, the denominator of (7) is negative.
Accordingly, the sign of dT* / dq is the opposite of the sign of the numerator of (7), which depends on
T*.
Rather than pursuing the examination of the sign of the numerator of (7), a cumbersome   task
indeed, BV    turned   their    attention   to the  investigation of the sign of ¶
2 F(T) / (¶ T ¶ q), at T*,
as it is the same as the sign of dT/dq, also at T*.
1
From (2), it follows that
( ) ( ) ( ) ( ) ( ) { }
( ) ¶
¶ ¶q
j j
q q
2
1
F T
T
T f T Rf T e C e
T R T = - ¢ - -
- +
                              -T ( ) ( ) ( ) ( ) ( ) { }
( ) 1- ¢ - - +
- + j q j
q q f T Rf T e R C e
T R T
Thus, taking into account (3), we have:
           ( ) ( ) ( ) ( ) ( ) { }
( ) . * * *
*
* * ¶
¶ ¶q
j j
q q
2
1
F T
T
T f T Rf T e C e
T
T R T ø
ß
œ = - ¢ - -
- +
or, given that  ( ) ( ) ( ) ( ) ( ) 1- ¢ - = +
- j q j
q f T Rf T e R C
T * *
*
( ) ( ) { }
( ) ¶
¶ ¶q
q j j
q
2F T
T
T R C C e
T
R T ø
ß
œ = + -
- +
*
* *
                                ( ) { }
( ) = + -
- + j q
q C T R e
R T *
* 1                                       (8)
                                                       
1 This result follows from the so-called envelope theorem (cf. Varian, 1992).5
Therefore, the sign of dT/dq at T* is the same as the sign of T* (R+q) - 1. Thus, as R+q was
assumed to be positive, it follows that:
dT
d
*
q
>
<
0  if   T*  
>
< +
1
R q
                                                                              (9)
or
dT
d
*
q
>
<
0   if  q
>
<
-
1
T
R
*
                                                                                (10)
Quoting BV, we can say that “for investments with long duration increased inflation increases
duration and the opposite holds for short duration.  Alternatively, as expressed by (10), at high levels
of inflation, increased inflation calls for longer duration and the opposite holds at low levels of
inflation.  In particular, a change from no inflation to some low level of inflation should decrease
optimal duration”.
With the sole purpose to give at least some evidence of the magnitude of the numerical results
that may be obtained, let us consider the case where, following an adaptation of an example that was
presented in Hirshleifer (1970, pg, 87), we have :
f (T) = 90 log (1 + T) + 120
If C = 100, j = 20%, R = 10% and if we have no inflation (q = 0), it is easily verified that the
optimal duration is T
* @ 2,493902. On the other hand, if we move to the situation where we start to
have inflation, at the very moderate rate  q = 1%, the optimal duration will be slightly reduced to T
* @
2,475614.
Supposing now that C = 56, and mantaining the values of j and R, consider the situation
where we have rampant inflation, with q = 50%. In this state of affairs we will have T
* @ 2,503765.6
On the other hand, if inflation increases even further, say q = 60%, we will also have an increase
(though small)  in the optimal duration, as T
* @ 2,529576.
III. Variations of the Basic Model
The persistent presence of high inflation in some countries, has caused the implementation of
indexation procedures.  In particular, in the Brazilian case, the so called monetary correction scheme,
1
which was made official in 1964 and is in use even nowadays, makes indexation a crucial factor for
tax purposes.  Accordingly, based primarily in the Brazilian experience, we are going to consider the
following variations of the basic model.
a) absence of indexation
As the first variation of the basic model, one which is more appropriate for low inflation
economies, let us consider the case where neither the pre tax net receipt nor the investment, the latter
for tax purposes, are indexed to the inflation rate.  In this situation, the cash flow associated with the
investment, as expressed in current prices, is as depicted in Figure II.
                  h( ) ( ) ( ) T T C - - j h
                                                          0
                                                            C      time
                        Figure II
Cash-Flow in the Absence of  Indexation
Thus, at constant prices, as referred to the date of the investment, we have:
F( )
( ) ( ) ( ) { } T C e h T C
R T = - + - +
- +q j j 1                                                         (11)
T7
or, working with the nominal interest rate i = R + q
F( ) ( ) ( ) { } T C e h T C
iT = - + - +
- 1 j j                                                                  (11¢ )
That is, if the classical Fisher equation relating i, R and q holds, and if we do not have
indexation, the optimal duration is a function of the nominal interest rate i.  In other words, with h(T)
playing the role of f(T), we have, formally, exactly the case treated in equation (2) in BV.  The
difference is that, rather than analyzing the effects of changes in j on the optimal duration T*, we are
interested now in investigating the effects of changes in the inflation rate q.
Proceeding with the maximization of F(T), we have:
            ( ) ( ) ( ) ( ) ( ) { }
dF T
dT
h T ih T i C e = - ¢ - - 1 j j
-iT*
                               (12)
Thus
( ) ( ) ( )
( ) ( )
dF T
dT
i
h T
h T C
= ￿ =
- ¢
- +
0
1
1
j
j j
                                                            (13)
That is, the optimal duration T* has to satisfy the extended version of the classical Jevon’s
formula, as given by (13).
With regard to the second order condition, we have:
( ) ( ) ( ) ( ) ( ) { }
d F T
dT
h T ih T e
T
iT
2
2 1
ø
ß
œ
œ
= - ¢¢ - ¢
-
*
* * * j                                                 (14)
Thus, if the nominal pre tax net receipt h(T) is a well behaved function of T, i.e. increasing
and concave, the considered solution T* is indeed optimal.
                                                                                                                                                                         
1 For assessments of the monetary correction scheme see, besides the early work of Fishlow (1974), Barbosa (1993) and
the very comprehensive surveys of Simonsen (1983 and 1995).8
On the other hand, to investigate the effects of changes in the inflation rate q on the optimal
duration T*, let us start considering the effects of changes in the nominal interest rate i.  To do this,
differentiating (13), holding the tax rate j constant, we have:
    ( ) ( )
( ) ( ) ( ) ( )
dT
di
h T C
h T ih T
=
- +
- ¢¢ - ¢
1
1
j j
j
                               (15)
Therefore, given the assumptions on the behavior of  h(T), it follows that increases in the
nominal rate of interest  i shortens the optimal duration  T*.  Making use of the chain rule of
differentiation, and taking into account that di/dq > 0, it follows then that:
dT
d
dT
di
di
d q q
= ￿
Ł ￿
￿
ł ￿
￿
Ł ￿
￿
ł ￿ < 0                                                                                          (16)
Thus, if there is no indexation at all, increases in the rate of inflation shortens the optimal
duration.
b) full indexation
In the Brazilian case, the usual procedure with regard to taxation, at least up to the
implementation of the Real Plan in 1994, was to have the investment value indexed to the inflation.
Therefore, if the nominal value of the pre tax net receipt h(T) is also indexed, i.e. h(T) = e
qT f(T), we
have a situation that will be called of full indexation, and the cash flow associated with the investment,
as expressed in current prices, is as depicted in Figure III
                                    e ( ) ( ) ( )
q q q j
T T T f T e f T e C - -
T          time
Figure III
Cash-Flow in the Case of Full Indexation
C
  09
Thus, as measured in constant prices, as referred to the date of the investment, the present
value function to be maximized is:
F(T) = - C + e
-RT { } ( ) ( ) 1- + j j f T C                                                                 (17)
It is then obvious that the optimal duration does not depend on the inflation rate q.
c) partial indexation
Concluding the analysis, let us now consider the case where while the investment value is
indexed for tax purposes, the nominal value of the pre tax net receipt h(T) is not.  In this situation,
assuming that h(T) > Ce
qT , for T>0, it follows that the cash flow associated with the investment, as
expressed in current prices, is as depicted in Figure IV.
1
                                                                                                 T       time
                                                      C
          Figure IV
Cash-Flow in the Case of Partial Indexation
In terms of constant prices, as expressed at the date of the investment, the correspondent
present value is:
F(T) = - C + e
- + ( ) R T q ( ) { } ( ) 1- + j j
q h T Ce
T                                                        (18)
                                                       
1 Note that, more generally, the value of the tax is given by max { } ) Ce   -   (h(T)     ; 0
T q j
  0
h( ) ( ) ( ) T T Ce
T - - j
q h10
with
( ) ( ) ( ) ( ) ( ) ( ) { }
( ) dF T
dT
h T R h T R Ce e
T R T = - ¢ - + -
- + 1 j q j
q q                                  (19)
Therefore, the optimal duration T* has to satisfy the following equation:
( ) ( ) ( ) ( ) ( ) 1 0 - ¢ - + - = j q j
q h T R h T R Ce
T                                                          (20)
On the other hand, as
( ) ( ) ( ) ( ) ( ) ( ) { }
( ) d F T
dT
h T R h T R Ce e
T
T R T
2
2 1
ø
ß
œ = - ¢¢ - + ¢ -
- +
*
* * * * j q q j
q q                 (21)
T* will be indeed the optimal duration if
( ) ( ) ( ) ( ) ( ) 1 0 - ¢¢ - + ¢ - < j q q j
q h T R h T R Ce
T * *
*                                                          (22)
Also, differentiating (20), holding constant R and j , we have that:
( ) ( )
( ) ( ) ( ) ( ) ( )
dT
d
h T T R Ce
h T R h T R Ce
T
T
* * *
* *
*
* q
j j
j q q j
q
q =
- +
- ¢¢ - + ¢ -
1
1
                                         (23)
Thus, as the numerator of (23)  is obviously positive, it follows from (22) that an increase in
the rate of inflation decreases the optimal duration.11
IV.  Conclusion
Taking into account some indexation procedures that have been used in some high inflation
economies, particularly in the Brazilian case, we have extended the basic model presented by BV in
order to cover three additional situations. The fundamental conclusion is that while the effect of an
increase in the inflation rate is somewhat ambiguous in  the basic situation, as it depends on the
relative  size of the optimal duration itself, the effect is very much predictable on the other three
situations. Thus, while the optimal duration is independent of the inflation rate in the case of full
indexation, an increase in the rate of inflation always shortens the optimal duration both in the case of
partial indexation and in the case of absence of indexation.12
References
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Englewood Cliffs, N.J.).
Barbosa, Fernando de Holanda, 1993. La Indización de Los  Activos Financieros: La Experiência
Brasileña, in Luis  Felipe  Jiménez, Editor.   Indización  de  Activos  Financieros :  Experiências
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New York).13
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30 Anos de Indexação (Editora da Fundação Getulio

Mathematical Analysis for Economists (St.

Mathematics for Economists (W.W.

The Effect of Taxes on Growth Investments: A Note,

The Effects of Inflation and Taxes on Growth Investments and Replacement Policies,

The Effect of Inflation on Growth Investments: A Note

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The Effect of Inflation on Growth Investments: A Note

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