6891302.dvi


Research Article
A Note on Discrete Multitime Recurrences of
Samuelson-Hicks Type

Bruno Antonio Pansera1,2 and Francesco Strati1,2,3

1Department of Law and Economics, University Mediterranea of Reggio Calabria, Reggio Calabria, Italy
2Decisions Lab, University Mediterranea of Reggio Calabria, Reggio Calabria, Italy
3Department of Economics and Statistics, University of Siena, Siena, Italy

Correspondence should be addressed to Francesco Strati; francesco.strati@unisi.it

Received 11 August 2016; Accepted 5 October 2016

Academic Editor: Gafurjan Ibragimov

Copyright © 2016 B. A. Pansera and F. Strati. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.

By this work, we aim at fostering further research on the applications of multitime recurrences. In particular, we shall apply this
method by generalizing the Samuelson-Hicks model so as to make the new concept of time that this method proposes clear. In
particular, the multitime approach decomposes a point of time into a vector, taking into account how different coordinates of time
referring to the same date can affect the dynamics of a model.

1. Introduction

This work belongs to a series of papers (see [1–4]) which
explore the notion of multitime in different branches of
mathematical analysis, such as economics and physics. Since
1932, the adjective multitime, introduced by Paul Dirac,
appears so as to cope with an environment in which there
might be more than one dimension of time.

We make some considerations concerning the physical
idea of the multitime notion: we recall that a coordinate
space is an index numbering degrees of freedom, and the
coordinate of time is the usual physical time in which a
system evolves. This model is satisfactory, unless we turn our
attention to relativistic problems. Moreover, some physical
phenomena (and social sciences too) are observed in a two-
time environment 𝑡 = (𝑡1, 𝑡2), in which 𝑡1 is the intrinsic time
and 𝑡2 is the observer time. In some real phenomena, there
is no reason to prefer one coordinate to another. Following
these assumptions, we refer to multitime as vector parameters
of evolution in multitime geometric evolution and multitime
optimal control problems.

The second relevant aspect to analyze is the multitime
wave functions considered by Paul Dirac in 1932 via 𝑚-time
evolution equations 𝑖ℎ(𝜕𝜓/𝜕𝑡𝛼) = 𝐻𝛼𝜓. The Dirac PDEs

system is consistent (completely integrable) if and only if[𝐻𝛼,𝐻𝛽] = 0 for 𝛼 ̸= 𝛽. The consistency condition is easy
to achieve for noninteracting particles, while it turns out to
be harder in the presence of interactions. All of these laws
cannot be applied to relevant problems, if one aims at a plain
covariant formulation of relativistic quantum mechanics.

Eventually, following the literature, we can divide the laws
of evolution of physical theories into two branches: the single-
time evolution laws (ODEs) and the multitime evolution laws
(PDEs). It is clear that, in order to turn a single-time evolution
into a multitime evolution, changing the ODEs into PDEs is
enough by accepting that time 𝑡 is 𝐶∞ function of certain
parameters; let us say that 𝑡 = 𝑡(𝑠1, . . . , 𝑠𝑚).

Many authors studied the different applications of these
notions: we recall some significative papers as [4–10] in which
the PDEs constraints usually show significant challenges
for optimization principles. The multivariable maximum
principle was studied in the presence of PDE constraints (see
[1]).

The multivariate recurrences are based on multiple
sequences; they come from areas like analysis of algorithms,
computational biology, information theory, queueing theory,
filters theory, statistical physics, and so forth. We consider a
lattice of points with positive integer coordinates in R𝑛. A

Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2016, Article ID 6891302, 4 pages
http://dx.doi.org/10.1155/2016/6891302



2 Discrete Dynamics in Nature and Society

multivariate recurrence is a set of rules which transfer a point
to another, together with initial conditions, which is capable
of covering the hole lattice.

A multivariate recurrence relation is an equation that
recursively defines a multivariate sequence, once one or more
initial terms are given: each further term of the sequence
is defined as a function of the preceding terms. Some
defined recurrence relations can have very complex (chaotic)
behaviors, and they are part of a field of mathematics known
as nonlinear analysis.

We shall extend the Floquet theory which had been firstly
utilized for periodic linear ODEs [11] and then extended
to difference equations [12–15]. In [3, 16], the authors have
extended this theory to the multitemporal first-order PDEs.
In Floquet theory, it is necessary to find the associated
monodromy matrix and its eigenvalues (called Floquet mul-
tipliers) in order to pass from a constant coefficient state to
a periodic one. It can be crucial in a recurrence formulation
(as noted in [14]) in which time-independent coefficients are
unrealistic. An application of the Floquet theory can be found
in [14]; here, we shall discuss a further possible extension.
In this case, there is a generalization of the periodicity to a
sheet of time rather than a point of time approach. In few
words, it can be stated that, by a multitime approach, a point
of time opens up to a vector of multiple times at that date (see
Figure 1).

2. The Model

2.1. A Hint on the Samuelson-Hicks Model. The standard
Samuelson-Hicks model has been made of two crucial com-
ponents: 𝛾, the propensity to consume, for which amount𝐶𝑡 = 𝛾𝑌𝑡−1 is consumed, in which 𝑌𝑡−1 is the income at𝑡 − 1, and the rest is saved. The second component is that
productive capital needs to aggregate production in fixed
proportion 𝛼: 𝐾𝑡 = 𝛼𝑌𝑡−1. Defining the investment as a
change in capital stock 𝐼𝑡 = 𝐾𝑡−𝐾𝑡−1, then 𝐼𝑡 = 𝛼(𝑌𝑡−1−𝑌𝑡−2),
that is, the standard principle of acceleration. Furthermore,
since𝑌𝑡 = 𝐶𝑡+𝐼𝑡, then a single recurrence equation in income
is

𝑌𝑡 = (𝛼 + 𝛾)𝑌𝑡−1 − 𝛼𝑌𝑡−2. (1)
In [17], Samuelson applied the accelerator to consumption
only, while, in [18], Hicks described its application to all
expenditures including floors to depreciation levels and
ceilings to incomes growth.

The equations of the model encompass autonomous
factors 𝐶 (consumption independent from income) and 𝐼
(investment not dependent on business cycles):𝐶𝑡 = 𝛾𝑌𝑡−1+𝐶
and 𝐼𝑡 = 𝛼(𝑌𝑡−1 − 𝑌𝑡−2) + 𝐼. Notice that these autonomous
components are crucial for the determination of the cycle.
The induced investment depends on the change in output
(on the cycle), while the autonomous part depends on
technological progress, innovation, or population growth. If
the autonomous investment increases at a regular rate, then
the economy is in a progressive equilibrium.

A problem may arise: autonomous components grow at
a constant rate, and 𝛼 and 𝛾 do not depend upon the cycle

t

t
1

i

t
2

i

Figure 1: An example of multitime sheet.

and are assumed to be constant too. In [14], these constant
factors become periodic and then they are floquetized in
the sense of the following section (even if they, and we,
detrended the model since growing autonomous investments
have been removed). Moreover, we shall discuss a further
generalization of the Floquet periodicity in which every
period is decomposed into a vector of times rather than a
simple point.

2.2. From Floquetizations to Multitime. Define element 𝑡 =(𝑡1, . . . , 𝑡𝑚) ∈ N𝑚 as a discrete multitime and function 𝑥 :
N
𝑚 → R𝑛as a multivariate sequence. Moreover, denote𝜇(𝑡) =

min{𝑡1, . . . , 𝑡𝑚} and 1 = (1, . . . ,1) ∈ N𝑚. It seems intuitive to
see the discrete multitime as a series of vectors in which the
time is not seen as a simple straight line in R and every time as
a point on that line; with the multitime approach, every point
of time is turned into a vector. In general, in economics, a
model is defined in either discrete or continuous time: in the
former case, it is usual to say that a generic event happens
at 𝑡 = 1 and spreads its effects at 𝑡 = 2 and so forth. For
a multitime approach, an event does not come into being at𝑡 = 𝑖 ∈ N but at 𝑡 = 𝑡𝑖 ≡ (𝑡𝑖1, 𝑡𝑖2, 𝑡𝑖3, . . . , 𝑡𝑖𝑚); it can be thought
of as a sheet of time instead of a point (see Figure 1). An event
that occurred on a sheet of time 𝑡1 spreads its effects on the
subsequent sheet of time 𝑡2 and so on.

Periodic coefficients are a proper generalization of con-
stant ones, since the former take into account the periodical
characteristic. This generalization can be applied to the
Samuelson-Hicks model with constant coefficient and try
to understand if generalizing it to a periodic coefficient
model may be of some interest. This approach is called
floquetization, in particular for coefficient ℎ𝑡

ℎ𝑡+𝜃 = 𝑧ℎ𝑡; (2)
then it can be defined as a floquetian of type (𝜃,𝑧), in which
when 𝑧 = 1, it is nothing more than 𝜃-periodicity. If𝑧 ̸= 1, then floquetian coefficiencies are a straightforward
generalization of the 𝜃-periodic ones. The discrete multitime



Discrete Dynamics in Nature and Society 3

periodic coefficient model is a further generalization to a
multitime approach, where

ℎ(𝑡 + 𝜃) = 𝐴(𝑡)ℎ(𝑡) , (3)
ℎ(𝑡 + 1) = 𝐴(𝑡)ℎ(𝑡) with 𝜃 ≡ 1, (4)

where, in case of 𝑚 ⩾ 2, 𝐴 : N𝑚 → M𝑛(R). In general, for
any 𝑇 ∈ N, the form

𝐴(𝑡 + 𝑇 ⋅ 1) = 𝐴(𝑡) (5)
is called 𝑇-diagonal periodic.
2.3. Samuelson-Hicks Meets Multitime. The discrete multi-
time periodic coefficient Samuelson-Hicks model can be
written as a first order diagonal recurrence system of the form
of (4)

(𝑌(𝑡 + 1)𝐶(𝑡 + 1)) = (
𝛾(𝑡) + 𝛼(𝑡) − 𝛼(𝑡)𝛾(𝑡 − 1)𝛾(𝑡) 0 )(

𝑌(𝑡)
𝐶(𝑡)),
𝑡 ∈ N𝑚

(6)

that satisfies the following conditions:

𝑌(𝑡)|𝑡𝛼=0 = 𝑌0𝛼(𝑡1, . . . , 𝑡𝛼, . . . , 𝑡𝑚),
𝐶(𝑡)|𝑡𝛼=0 = 𝐶0𝛼(𝑡1, . . . , 𝑡𝛼, . . . , 𝑡𝑚),
𝑌0𝛼 (𝑡1, . . . , 𝑡𝛼, . . . , 𝑡𝑚)󵄨󵄨󵄨󵄨󵄨󵄨𝑡𝛽=0
= 𝑌0𝛽 (𝑡1, . . . , 𝑡𝛽, . . . , 𝑡𝑚)󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑡𝛼=0 ,

𝐶0𝛼 (𝑡1, . . . , 𝑡𝛼, . . . , 𝑡𝑚)󵄨󵄨󵄨󵄨󵄨󵄨𝑡𝛽=0
= 𝐶0𝛽 (𝑡1, . . . , 𝑡𝛽, . . . , 𝑡𝑚)󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑡𝛼=0 .

(7)

The marginal propensity to consume 0 < 𝛾 < 1;
moreover, 𝛼 can be defined as decelerator if 0 < 𝛼 < 1, keeper
if 𝛼 = 1, and accelerator if 𝛼 > 1. National income 𝑌 and its
endogenous variable𝐶 (consumption) are assumed as strictly
positive multitime sequences 𝑌(𝑡) and 𝐶(𝑡). Notice that, in
the original model, 𝛾 and 𝛼 are constant factors. In [14], they
have been used as (𝜃,𝑧)-periodic parameters 𝛾𝑡 and 𝛼𝑡 as the
autonomous consumption 𝐶 turned into 𝐶𝑡. This periodicity
has been extended to (𝜃,𝐴)-periodic parameters 𝛾(𝑡), 𝛼(𝑡),𝑌(𝑡), 𝐶(𝑡).

The model may be written as the second order homoge-
neous diagonal recurrence equation:

𝑌(𝑡 + 2) − (𝛾(𝑡 + 1) + 𝛼(𝑡 + 1))𝑌(𝑡 + 1)
+ 𝛼(𝑡 + 1)𝑌(𝑡) = 0, 𝑡 ∈ N𝑚, (8)

satisfying the following conditions:

𝑌(𝑡 + 2) − (𝛾 + 𝛼)𝑌(𝑡 + 1) + 𝛼𝑌(𝑡) = 0, 𝑡 ∈ N𝑚,
𝑌(𝑡)|𝑡𝛼=0 = 𝑌0𝛼(𝑡1, . . . , 𝑡𝛼, . . . , 𝑡𝑚),
𝑌(𝑡)|𝑡𝛼=1 = 𝑌1𝛼(𝑡1, . . . , 𝑡𝛼, . . . , 𝑡𝑚),
𝑌0𝛼 (𝑡1, . . . , 𝑡𝛼, . . . , 𝑡𝑚)󵄨󵄨󵄨󵄨󵄨󵄨𝑡𝛽=0
= 𝑌0𝛽 (𝑡1, . . . , 𝑡𝛽, . . . , 𝑡𝑚)󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑡𝛼=0 ,

𝑌1𝛼 (𝑡1, . . . , 𝑡𝛼, . . . , 𝑡𝑚)󵄨󵄨󵄨󵄨󵄨󵄨𝑡𝛽=1
= 𝑌1𝛽 (𝑡1, . . . , 𝑡𝛽, . . . , 𝑡𝑚)󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑡𝛼=1 ,

𝑌0𝛼 (𝑡1, . . . , 𝑡𝛼, . . . , 𝑡𝑚)󵄨󵄨󵄨󵄨󵄨󵄨𝑡𝛽=1
= 𝑌1𝛽 (𝑡1, . . . , 𝑡𝛽, . . . , 𝑡𝑚)󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑡𝛼=0 .

(9)

A prominent role is played by the so-called Floquet mul-
tiplier of (8), that is, first reconverting (8) into an equivalent
diagonal recurrence system:

(𝑌(𝑡 + 1)𝑍(𝑡 + 1))
= ( 0 1−𝛼(𝑡 + 1) 𝛾(𝑡 + 1) + 𝛼(𝑡 + 1))⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=𝐴(𝑡+1)

(𝑌(𝑡)𝑍(𝑡)),

𝑡 ∈ N𝑚,

(10)

where 𝐴(𝑡 + 1) is 𝑇-periodic matrix. Then, find the Floquet
multiplier, which is an eigenvalue of the matrix:

𝐶(𝑡) = 𝑇−1∏
𝑘=0

𝐴(𝑡 + (𝑇 − 𝑘 − 𝜇(𝑡) ⋅ 1)) , (11)
that is called the monodromy matrix. It is a 2 × 2 matrix with
determinant

det𝐶(𝑡) = 𝛼(𝑡 + (𝑇 − 𝜇(𝑡)) ⋅ 1) × ⋅ ⋅ ⋅
× 𝛼(𝑡 + (1 − 𝜇(𝑡)) ⋅ 1) . (12)

Thus, the multipliers are the roots of the quadratic equation:

𝑧2 − 𝑧(Tr𝐶) + det𝐶 = 0 (13)
and these multipliers will depend on 𝑡1 − 𝜇(𝑡), . . . , 𝑡𝑚 − 𝜇(𝑡).
3. Discussion and Conclusions

The multiplier-accelerator model shows how cycles can
occur in economics. The Floquet theory can be used to



4 Discrete Dynamics in Nature and Society

generalize the constant coefficient framework of the standard
Samuelson-Hicks model. By floquetizing this model, the
behavior of the system around their constant value can
be studied and thus it is possible to study the periodic
cycle. This can happen by the use of the monodromy matrix
which is the link between the coefficients and their periodic
generalization. These kind of matrices study the behavior of a
dynamic system around a singularity, in this case a constant
coefficient.

It can be stated that each motion of national income 𝑌
is bounded in (8) if and only if this motion is bounded and𝛼 ⩽ 1 in (12). For 𝛼 > 1, it is unbounded and, moreover, each
motion of 𝑌 vanishes if 𝛼 < 1.

The floquetized Samuelson-Hicks model has been further
generalized by taking into account a discrete multitime model
generalizing the point of time to a sheet of time, as mentioned
above. By a sheet of time, the dynamic system may be
decomposed into different summable time vectors in which
it is possible to study the effect of each coefficient separately.
It can be possible to explode the model into different current
times, like those paintings from the analytical cubism. It
is important to apply the model so as to study recursively
the effect of lagged coefficients to the subsequent ones, but
the new effect which may become crucial is that in each
sheet one can observe a decomposed model: on every time
encompassed by a sheet, one can see how modifications of
a coefficient affect the coefficients in the same current time
sheet and so how this vector affects the model in the other
sheet by understanding that model thoroughly.

Competing Interests

The authors declare that there are no competing interests
regarding the publication of this paper.

Acknowledgments

The authors thank Massimiliano Ferrara for his invaluable
advice and guidance. This research was partially supported
by a grant of the Department of Law and Economics of the
University Mediterranea of Reggio Calabria.

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