1. INTRODUCTION

This paper is about the mathematization of Economics^{1}. It aims to analyze the consequences of the gradual
incorporation of the dynamic systems techniques (DST) in Economics during the last
three decades^{2}. In particular, the
paper shows how mainstream economists adopt specific assumptions and mathematical
tools to translate economic concepts into dynamic formats.

The choice of topic and time is relevant for at least two reasons. First, because the
link between DST and Economics shapes the new mathematical orientation of the latter
over the recent past (^{Weintraub, 2002}).
Through DST’s extensive use, some branches of Economics have been able to expand
their scope by studying properties of market mechanisms that are inherently unstable
in their dynamics and not only deterministic and stable^{3}. The second reason is that the future destiny of
Mathematical Economics (ME) appears to be also linked to DST since, according to
^{Holt, Rosser, and Colander (2011)}, the
“era of complexity” is intended to replace the “neoclassical era” of ME. This new
era, which is barely in its infancy, comprises the work of economists with
approaches that assume interactions between heterogeneous agents, perpetual novelty,
and dynamics without an optimal equilibrium (^{Arthur,
Durlauf, and Lane, 1997}). In other words, they are works that demand new
ways of applying DST in Economics because their authors are reluctant to accept the
neoclassical view of dynamics.

Both reasons make the study of the relation between DST and Economics a relevant issue to understand the price paid by the latter for having to conform to specific dynamic formats. The strong opposition of some economists to the particular way this pairing takes place is a warning that should be taken into account, mainly because any attempt at formalization involves the risk of leaving aside some concepts (mostly non-quantifiable) in the dynamic analysis. Which criteria do mainstream economists consider to pick up or discard determined elements? Why do they prefer to keep the mathematical structure unchanged rather than relaxing restrictive assumptions? We shall address these sorts of questions below.

The document has four additional sections. The second presents data from a classification of research papers to illustrate the consequences of the penetration of DST into specific fields of Economics. The following two sections explain these consequences through an analysis of the degree of realism of the assumptions and the conceptual validity of the dynamic models. In particular, the third section concentrates on how assumptions, like bounded rationality, are considered by mainstream and non-mainstream economists. The fourth develops two models to show how variations in dynamic techniques may lead to establishing different perspectives on the same economic problem. In the fifth section, conclusions summarize the main findings.

2. ADVANCES OF DST IN ECONOMICS

The start of the third and most recent phase of the mathematization in Economics
occurred around the 1960s and its main feature is the turn of the me literature
towards dynamic analysis (^{Varian, 1991}; ^{Weintraub, 1991})^{4}. ^{Ramírez and Juárez
(2009)} find, in effect, that 24.48% of the 2,835 core-articles published
between 1990 and 2004 in the most influential journals of the discipline
(*American Economic Review*, *Econometrica*,
*Journal of Political Economy* and *Journal of Economic
Theory*), incorporated dynamic analysis. This result means that, in less
than fifteen years, the percentage of articles with some dynamic content more than
doubled that recorded between 1980 and 1990 in *the American Economic
Review* and *Econometrica* journals (11%)^{5}.

Among the reasons given to explain the rapid expansion of DST, there are two
particularly illustrative. The first is that, unlike the early stages of economic
models, economists have to cope now with non-linear mathematical specifications that
require professional training in areas relatively foreign to Economics (such as
Topology, Stochastic Optimization, and Differential Games; see ^{Boldrin and Woodford, 1992}). The flourishing of Econophysics
during this period, together with its range of techniques derived from
Thermodynamics, is an example of the new influence of these fields on different
areas of Economics, particularly Finance (^{Bali,
2011}).

The second reason is that DST have spread rapidly throughout most of Economics by
venturing into areas typically dominated by static analysis. In this regard, ^{Ramírez and Juárez (2009)} indicate that between
1990 and 2004, articles in the 14 sub-disciplines classified as static, which had
incorporated dynamic techniques, represented a little more than 50% of the sample; a
surprising percentage considering that ten years earlier, the other two remaining
naturally dynamic sub-disciplines, Macroeconomics and Economic Growth, accounted for
77% of the total. After updating the sample originally elaborated by ^{Ramírez and Juárez (2009)}, we find that this
proportion remained roughly the same for the period 2005-2010 (see Table 1)^{6}.

Sample | Ramírez and Juárez | Updated sample |

Sub-disciplines | 1990-2004 | 2005-2010 |

Traditionally dynamic | 335 (11.82%) | 139 (11.95%) |

Non-traditionally dynamic | 359 (12.66%) | 142 (12.20%) |

Sample size | 2,835 | 1,163 |

Note: Traditionally dynamic sub-disciplines are Macroeconomics and Economic Growth. Non-Traditionally dynamic sub-disciplines consist of all the others, mentioned in Table 2. These samples include only core-articles that represent 65% (1990-2004) and 67% (2005-2010) of the original sample size in both periods.

Source: Own elaboration.

The rapid spread of DST has fostered connectivity between sub-disciplines. Authors
now seek to combine various elements from different sub-disciplines not only to make
models more realistic but also to explore the behavior of critical points under new
constraints. Let us think, for instance, in the development of International Trade
and Economic Growth. Both sub-disciplines recently merged again after having
remained split for more than a century, as a result of a radical change in the
long-standing view on international trade (^{Afonso,
2001}). With the advent of the endogenous growth theory in the 1980s, some
economists started to realize that innovation was a fundamental part of
international trade. In particular, economists began to consider new theoretical
elements affecting the dynamics of innovation, like trade openness, geographic
structure of international trade or capacity for internal technological adaptation,
as drivers of human capital and, consequently, of economic growth (^{Keller, 2002}). The introduction of these
elements into economic growth models allowed economists to build up more realistic
scenarios, but at the expense of complicating the calculation of equilibria that
fitted the new concept of “open economy.”

As a by-product of this connectivity, optimization problems have become progressively
more complex. In addition to the typical saddle points, centers, spirals, or nodes,
there are articles containing equilibria that differ from the two traditional types
of motion: Steady state (when the system ceases its motion) and periodic (when the
system enters limit cycles). Now it is common to find researchers dealing with
quasi-periodic and complex dynamic motions in some parts of the long-term paths
before reaching the definitive equilibrium (^{Day,
1983}). The treatment of these new critical points demands more
sophisticated motion equations (logistic maps or higher-order equations differential
systems) as well as more complicated ways of optimizing sequential decision making
over time and under uncertainty.

However, it is worth noting that this technical sophistication is not the same
throughout the literature, mainly because most articles treat the concept of
dynamics differently. There are articles in which the aim is to test differential or
difference equations econometrically but without considering any analytical or
qualitative treatment of the trajectories. Examples of these articles are found in
the first group of Table 2, which includes
studies of Economic History, Welfare Economics, and another six sub-disciplines.
There are also studies in the fields of Game Theory and Labor Economics, such as
those in the second group of Table 2, that go
beyond a simple econometric estimation and seek to calculate deterministic,
stochastic equilibria based on evolutionary strategies for multiple stages games
(^{Binmore, Piccione, and Samuelson, 1998}).
Lastly, there is a third group, the most numerous and heterogeneous of all,
characterized by maintaining diverse positions. This group is composed, on the one
hand, by authors in the purest neoclassical tradition who develop dynamic models
favoring unique, stable equilibria (^{Howitt,
1999}) and, on the other hand, by economists interested in predicting the
existence of multiple, unstable equilibria under the same assumptions of traditional
growth models (^{Backus, Kehoe, and Kydland,
1992}). Between these poles, there is a sub-group, not as large but
extremely representative of authors who recognize the inadequacy of traditional
analysis due to the restrictive nature of the assumptions (see ^{Mitra and Nishimura, 2001}).

Sample | Ramírez and Juárez | Updated sample |

Groups | 1990-2004 | 2005-2010 |

I | 95 | 36 |

II | 264 | 106 |

III | 335 | 139 |

Note: Group I: Economic History, Environmental Economics, Welfare Economics, Industrial Economics, Political Economy, Regional Economics, Development Economics, and Institutional Economics. Group II: Game Theory, Labour Economics, Experimental Economics, Microeconomics, Finance, and International Trade. Group III. Macroeconomics and Economic Growth.

Source: Own elaboration.

Despite these differences, it is clear that the incorporation of DST has
significantly benefited the development of Economics. The availability of new DST
has allowed some fields of Economics to flourish, such as the two-sex population
theory, Econophysics, endogenous growth theory, or the new generation of matching
models (^{Noldeke and Van Damme, 1990}).
Likewise, some practical problems linked to dynamic financial derivatives- pricing
would be unthinkable without the support of dynamic stochastic optimization
techniques. By using optimal control or stochastic dynamic programming techniques,
authors are now capable of solving more theoretically-oriented problems of risk
measures that would have been impossible to model utilizing the traditional tools of
corporate finance. For these kinds of reasons, modern me would be inconceivable
without the support of DST, either for generating new ideas or for rejecting other
long-accepted ones.

2.1. Alternative points of view

As it is common in different fields of knowledge, not all share the same optimism for new developments, especially in Economics, where the application of Mathematics is viewed with suspicion by many. One can realize this immediately as soon as one begins to review the works of economists who publish in less orthodox journals. In particular, two conflicting points stand out when comparing the journals of the sample with others.

The first point has to do with the concept of motion of an economy used by most
macroeconomic models of the third group in Table
2. In such models, the word motion is deemed by non-mainstream
economists as extraordinarily narrow and instead linked to the way physicists
formalize inanimate physical entities. With that concept at work, ^{Lorenz (2009)} says that dynamic equilibria
do not seem derived from a model, in the sense that they appear more as
displacements from a fixed point than as a result of internal adjustments to the
system. Dynamic models need to include feedback systems in which larger equation
systems record the new information acquired by agents. Modeling the dynamics of
an economy with fixed, deterministic laws of motion and under idealized
conditions, is then seen as a contradiction in itself. According to ^{Velupillai (2011)}, modern economies must be
analyzed with the help of complex dynamic equilibria because they are the most
complex of all dynamic systems.

The second point deals with the way neoclassical authors define the set of state
variables *X*. As explained in ^{Hirsch, Smale, and Devaney (2004)}, there must be a close
correspondence between *X* and the rule of change in order to set
the correct dynamics of any phenomenon under study; otherwise, its dynamical
nature would be ill-specified. For this reason, it is wrong to combine, for
instance, non-linear differential equations with state variables in which
bifurcations are absent since the range of *X* would not be
possible to identify.

In most articles of the four mainstream journals, the range of *X*
considers only the values along the stable branch of a saddle point. The reason
lies in neoclassical economists’ stressed tendency to favor equilibria whose
nature is asymptotically stable according to Lyapunov’s criteria. They think of
the remaining states or points on the *X* path over time as
temporary disturbances near the steady-state. Hence, the introduction of systems
of high-order differential equations, that supposedly intend to capture more
complex dynamic behavior, should not always be seen as a genuine attempt to make
the dynamic analysis more realistic but as a merely formal way to show
sub-optimum unstable equilibria. Authors are more concerned with stable orbital
equilibria rather than structural stability and so tend to use smooth
differential dynamic systems that yield equilibria around an isolated critical
point. The rules of change involving logistic maps or Duffing-like equations are
regularly discarded because they produce tent maps, manifolds, or different
kinds of chaos that prevent reaching stable critical points.

This peculiar correspondence between the space of state variables and the rules
of change is not uncommon in Economics. It is a practice that is rooted in the
axiomatic method, mainly fostered by ^{Debreu
(1984}, ^{1991)} and the
particular methodological formalization adopted by mainstream economists. In
both cases, the idea of unique and stable equilibria is omnipresent. In fact,
without that concept of equilibrium, the use of mathematics in Economics would
lose meaning, since otherwise the chain of reasoning of any model could no
longer be broken down into is elementary steps. Mathematics helps the models of
equilibrium make the chain of reasoning credible (^{Backhouse, 1998}).

How does this breaking down process work? The most straightforward answer comes
from the axiomatic method. According to ^{Duppe
(2010)}, this method presupposes the separation of economic content
from mathematical reasoning because the formal structure does not require any
interpretation. During the five critical steps of the process of axiomatization
(selecting primitive concepts, representing these concepts as mathematical
objects, specifying assumptions, deriving consequences, and interpreting), the
only thing that matters is the mathematical structure. The fifth step,
interpretation, is foreign to the first four ones because it is not considered
an activity that belongs to the stages of logical rigor. It is instead a thing
to be discovered (^{Boyland and O’Gorman,
2007}; ^{Duppe, 2010}).

Each of the first four steps shaping the formal structure is subject to a
rigorous deductive process free of logical contradictions. Neither stage is
independent of the other. However, insofar as these steps are empty of economic
content, the economist’s task reduces thus to fill the formal structures by
making their interpretations pass the “acid test.” Interpretations contradicting
the rules of logic cannot be called rigorous and epistemically equivalent (^{Duppe, 2010}). Examples of erroneous
interpretations are the so called theories of disequilibrium whose postulates
violate axioms or assumptions that are equilibrium determinations in themselves:
Disequilibrium points are only equilibrium points under new constraints.

In terms of our discussion, this means that just as no economic model is
logically consistent if it is not in equilibrium, no dynamic model makes sense
if it uses non-smooth rules of change or if its set *X* includes
disequilibrium values. The economist’s function must be, therefore, particular
and limited to find an appropriate pair of mathematical objects
(*X* and rules of change) that fit the formal structure. This
corollary seems to be inherited nowadays by the Slutsky-Frisch-Tinbergen
approach, a dominant methodological formalization in economic growth and
macroeconomic models. Under this approach, shocks affecting any economy tend to
be propagated in a muted fashion along the planning horizon because of the
existence of filtration mechanisms that prevent the economy from continually
wavering. Models that take the basis of this approach, such as business cycle
models with linear specifications, set rules of change that restore
disequilibrium in much the same way as mechanical oscillations models with
damped motions do. That is to say; they are models that, after specifying the
differential equation with perturbations, try to find the equilibrium trajectory
with determined amplitude in some phases of the business cycle; as it happens
with a differential equation that models the motion of a body with mass m
suspended from a spring and subject to friction, resonance or external forces.
This position is sharply criticized by other scholars who assert that even under
the same assumptions of stable equilibrium and in the absence of exogenous
shocks, the economy may oscillate indefinitely. In essence, trajectories of
business cycles do not necessarily converge to stable attractors (^{Boldrin and Woodford, 1992}).

As it is to be supposed, the adherence to a particular approach does condition
the use of specific methods of dynamic optimization. Those who adhere to the
Slutsky-Frisch-Tinbergen methodology assume that exogenous shocks do not destroy
the toroidal resonant region and, therefore, that Hamiltonian equations remain
integrable^{7}. For them,
shocks only affect the scale of the control variables, not their symplectic or
conservative structure. Consequently, these authors have no difficulty in
adopting some variants of the Turnpike Theorem to continue using the
optimization and stability schemes of Hamiltonian mechanics, without altering
the integrable nature of the equations. On the contrary, those who adopt the
alternative approach think that shocks create zones in the phase spaces that are
occupied by perturbed resonant toroids or fat fractals. They reject the tenets
of the Turnpike Theorem by using Hamiltonians of disturbed systems or physical
models of complex dynamics (^{Ramírez and Juárez,
2009})^{8}.

While the first approach has dominated the discipline for a long time, the second
is still emerging. The relative importance of both approaches has been, however,
changing as the dominant trends in me have shifted from the traditional
Hilbert’s formalist program to a paradigm based on the extensive use of
simulated models. In the formalist program, economists have regarded it as
essential to prove the existence of unique, stable equilibria in general
equilibrium models deduced from unquestioned axioms (^{Weintraub, 2002}). In contrast, simulation model-oriented
economists have questioned the existence of these equilibria by adopting DST
that make intensive use of digital computer programs, such as chaos or fractal
theory (^{Weintraub, 2002}).

3. ON THE REALISM OF ASSUMPTIONS

In addition to the previous reasons, there are others of particular nature explaining the differential expansion of DST in Economics. Two of them refer to the realism of assumptions and the conceptual validity of the translation of Economics into dynamic formats.

Lack of realism in assumptions is an essential point of the criticism of neoclassical
dynamic models, and of the theory in general. Critics insist that it is untenable to
draw valid conclusions from unrealistic or simplifying assumptions since, under
these conditions, far from being successful, the deductive approach becomes
misleading. The tendency to model dynamic systems with a high level of abstraction
in their assumptions, sets Economics away from the usual practice of other sciences,
consisting only of formalizing long-established results that are grounded on
empirically-tested assumptions (^{Sarukkai,
2012}). Contrary to this practice, Economics follows a similar route to
Mathematics, in which it only models what can be reduced to a formal project,
virtually ignoring the factual reliability of assumptions. In neoclassical dynamic
models, assumptions do not necessarily need to have real descriptive content for the
simple reason that they are only predictive tools. Hence, as a result of this
instrumentalist view, it is not possible to expect the same unreasonable
effectiveness of Mathematics in Economics as in other sciences (^{Velupillai, 2005}). In this context, ^{Sarukkai (2012)} says that the relationship
between Mathematics and Economy is one of subordination, not cooperation.

In order to make this point clearer, let us draw our attention to one assumption that
is now very important in dynamic modeling: Bounded rationality. It aroused in the
realm of the theory of organization as H. Simon’s reaction to the traditional view
of modeling decision-making employing rational optimization (^{Barros, 2010}). Unlike the original concept of ideal rationality
in which economic agents are fully-informed maximizers of utility or profits, Simon
placed the assumption of bounded rationality at the core of a different
decision-making process. He says that an agent learns about his decisions in a
search process guided by aspiration levels or values of variable goals (^{Selten, 1999}). This process, named satisficing,
is not fixed, as aspiration levels change with different situations: They can be
raised or lowered depending on the ease of finding satisfactory alternative
decisions.

These ideas of aspiration-adaptation gradually spread to many areas of Economics
where decision making is a significant concern, in particular to Game Theory, where
the assumption is currently of great importance either in mathematical theorizations
(Evolutionary Game Theory) or in non-mathematical ones (Evolutionary Economics). As
in many other parts of Economics, the meaning of bounded rationality depends on the
dynamic view of authors. In Evolutionary Economics, the assumption helps model
agents’ behavior in a world with constant technological, organizational, and
structural changes. In this world, there is a persistent emergence of innovations
redefining economic structures and a complex dynamic involving nonlinear
interactions (^{Witt, 2008}). Therefore, the use
of bounded rationality is more in the Simon tradition because agents are allowed to
acquire information to obtain superior goals in an indeterminate process of
satisficing. It hints that bounded rational decision making economic behavior has a
non-optimizing character, but rather a flavor of continually adaptive learning. For
this reason, evolutionary economists use the assumption as a means of finding
evolutionarily stable strategies, in which new information enables the agents to
search for more realistic options in calculating rational choices. This calculation
has not to do with agents’ perfect knowledge of a set of lotteries (^{Hodgson and Huang, 2010}).

In neoclassical models, where bounded rationality is required, the situation is quite
different. Unlike evolutionary economists, neoclassical use the assumption devoid of
social considerations because the concept of dynamics has a strict quantitative
meaning. Dynamic variables that are not measurable in terms of probability
distributions or do not meet suitable convexity properties for optimization are
expendable. As part of the mainstream, evolutionary game theorists give the
assumption only a quantitative role in modeling boundedly rational economic behavior
as optimizing. They are not interested in accounting for all the implications
mentioned above but instead in finding an optimization method that allows
calculating the optimal payoff rates in principal-agent models or the Bayesian-Nash
pooling and separating equilibria in games with asymmetric information (^{Klaes, 2004}).

These differences in perception have kept the two approaches apart, especially after
evolutionary economists denied the existence of automatic adjustment mechanisms by
which consumers or producers can relentlessly be a Bayesian or statistically
adjusted rational maximizers of utility or profits. They say that people do not
always obey Bayes rule as their probabilities judgments fail to meet the
monotonicity requirements for the set of inclusion. In other words, they do not know
how to choose the rational option when the situation is not familiar, and time is
scarce. As a consequence of this, evolutionary economists have distrusted
traditional methods of optimization based on Game Theory and opted for a more
empirical approach using agent-based modeling (^{Selten, 1999}).

The neoclassical economists’ responses to these criticisms have been minimal as they
consider that there are no common grounds for discussion. They insist that no
argument about bounded rationality or any other assumption is valid if it does not
fit a formal program. The reluctance to accept any kind of criticism is what critics
consider a narrow idea of formalization or an inadequate translation of the language
of Economics into Mathematics. Modeling what can only be expressed in equations
means admitting that mathematics imposes its narrowness on economic analysis and,
therefore, that ‘the big picture -society’s long- term transformation- is excluded
of the analysis on the grounds that its dynamics cannot be sufficiently
mathematized’ (^{Hudson, 2000, p. 293}).

4. THE TRANSLATION OF ECONOMIC CONCEPTS INTO DYNAMIC FORMATS

Regarding the conceptual validity of the translation of Economics into dynamic
formats, we present two versions of the Malthusian population principle to
illustrate how old ideas can be modified or updated using alternative
techniques^{9}. The idea behind
this exercise is to show that each technique is subject to the concept involved (in
this case, the principle of population) and, therefore, there is no established
method to associate it with definitive mathematical tools or a neutral way of doing
me.

The first version is the widely known standard Malthusian model in which the stationary state is the inevitable destination of all possible trajectories of the population and means of subsistence. In its treatment, we use conventional DST. The second version is a new approach to the way oscillations can delay the convergence of those trajectories on the stable attractor. The study of oscillations is a hidden aspect of the principle, very little studied, that is at the core of Malthusian thinking, especially because oscillations, or retrograde and progressive movements experienced by the population’s welfare around the “subsistence floor,” are linked to Malthus’s idea that a highly stratified society produces different demographic regimes. In its modeling, we use delay differential equations, which to our best knowledge, have not been used before to this purpose.

4.1. The traditional view of the principle

The typical Malthusian path of population growth assumes an economy that works
under two assumptions. First, means of subsistence are determined by a
production function *K(t)* that depends on the population
*P(t)*, the exogenous technological parameter A>0, and the
coefficient of decreasing marginal returns 0<α<1^{10}. Second, the population grows logistically
as an inverse function of the reciprocal of the *per capita*
product *k(t)=K(t)/P(t)*, expanded by a constant
*s*. This constant is the lower limit of the per capita
product’s growth rate. In formal terms:

where: K(*t*)=AP(*t*)^{α}

To express equation [1] in terms of the *per capita* product, we
assume that *k(t)=K(t)/P(t)=AP(t)*
^{α-1} grows according to the differential equation:

After separating the variables and replacing
*k*(*t*) and [1] in [2], we have:

This result eventually produces the trajectory of the *per capita*
product:

or expressed in terms of birth α and mortality *b* rates, with
α=r_{0} and *b*=r_{0}s

Thus, if we introduce [5] or [6] into our definition of
*k(t)=AP(t)*
^{α-1} and solve *P(t)* then we will obtain the equation
for the target population:

Equation [7] shows that the population’s trajectory will converge to a defined value by birth and mortality rates and the technological constant, in other words:

since *b*/α.

Likewise, if we replace [7] in the production function and apply limits, we find
that the *K(t)* and *P*
_{
e
}
*(t)* attractors are regulated by the same constants:

The convergence of attractors in [8] and [9] will be faster as the value of( α
decreases when *t*→∞. In the limit case, the instantaneous rates
of all variables will be nil^{11}:

4.2. The principle with delays and the existence of oscillations

Most neoclassical economists believe that the stationary state of equation [10]
is the only possible result of the Malthusian population principle. We claim
that a complete analysis of the principle requires introducing parameters into
equation [1] that capture the presence of oscillations. One way to do so is
assuming that population growth is not instantly affected by the birth rate but
rather that there is a period of delay during which *P*(t-τ)
influences *P*(*t*) through mean birth and death
rates. Since subsistence levels also affect population growth after a delay has
elapsed, the non-linear effect created by oscillations in the means of
subsistence is transmitted to the *P*(*t*)
variable through a delay in the inhibiting term of the logistic equation (^{Kuang, 1993}):

where:
*K*(*t*)=*AP*(*t*-τ)^{α}

The resulting Hutchinson-like equation has a known stable limit value
(P_{s}(t)) provided that

In particular, ^{Kuang (1993)} shows that if
r_{0}t<π/2 then [11] will converge on a stable limit value, which
in our case is the attractor Ps(t)=(s/A)^{1/(α-1)}.

By comparing the attractors of systems [1] and [11], it is possible to conclude
that both coincide but only in its limit value, since if the
r_{0}t<π/2 condition does not hold, then oscillations produced by
[11] will alter the overall asymptotic stability of trajectories. Changes in the
values of r_{0} and τ will produce quasi-periodic behaviors or
fluctuations in the P(t-τ) term (^{Gopalsamy,
1992})^{12}. To stress
the impact of r_{0} on oscillations, we will assume that this behaves
according to [12]:

where k_{1} and k_{2} are, respectively, the lower and upper
asymptotes of the birth rate; α_{1} is the birth level and α_{2}
the speed of change of *r*(*t*). The new equation
becomes:

The limit value of [13] is similar to that of [11] given that

The fundamental difference between [11] and [13] lies in the population’s paths
since the disturbance produced by r_{0}(*t*) further
accentuates the oscillations produced by the τ delay in the relation established
by and k’(*t*)/k(*t*) and
P’(*t*)/P(*t*)^{13}.

4.3. What does the introduction of the new DST reveal?

The exposition of the two previous models shows that there is no unique way to formalize the Malthusian population principle and, therefore, to find the equilibrium path. Different equilibrium paths need different DST to formalize them. In any case, the selection of a specific technique has advantages and disadvantages. A significant advantage of the smooth dynamic systems, like the first model, is that they provide unique stable equilibria in closed-formulas that make the numerical calculation of equations easier. This advantage, however, comes at a cost: They cannot discover, for example, the existence of oscillations. Similarly, an advantage of non-smooth dynamic systems, such as the second model, is to show that the steady-state equilibrium is only one result among many. The disadvantage is that the model cannot predict the equilibrium solution since oscillations are differential responses of populations to changes in their economic environment.

The rationale for choosing one or another DST is a matter that continues to be
debated since there are always excesses and arbitrariness, not exempt from the
ideological burden. As far as mainstream economists are concerned, they consider
it pointless use non-smooth dynamic systems because they assume that the
stationary state is the only equilibrium possible to extract from Malthus’s
Essay. However, this a limited reading of that book, as his author repeatedly
insisted on the need to consider the delays in the population’s growth responses
to means of subsistence growth. Far from looking for a stable equilibrium point,
Malthus instead sought to emphasize the progressive or retrograde movements that
differentiate the well-being of the population dedicated to different productive
activities. Since not all the inhabitants experience these movements in the same
way, there are no foundations to associate them with the same demographic
behavior (^{Waterman, 1998}). This
demographic diversity cannot be shown with smooth dynamic systems that
standardize the population’s response to means of subsistence.

5. CONCLUSIONS

The paper argues that the use of DST has guided the growing mathematization of economics in recent decades. Not only has it made the dynamic analysis more complex on topics traditionally considered static, but it has also encouraged the development of new areas of knowledge and allowed substantiating little-studied results in Economics. However, there are flaws in how DST are applied. Specifically, the reductionism of the economic analysis to the formal program stands out. The adoption of ad hoc techniques by the neoclassical economists has led other authors to criticize the inclusion of assumptions and types of equilibria within the formats of mainstream Economics.

The overall conclusion of the paper is that any pairing between DST and Economics should be cautious because it is not realistic to assume that there is a general mathematical approach to Economics. Not all economic phenomena can be formalized or explained with equations. Nor is it true that there is always a unique way to model a phenomenon. For these reasons, it is important to decide in which sense the mathematization is useful to enrich the explanation of the economic problem at stake. Otherwise, economists will continue to perpetuate preconceived and abstract schemes in which form generally takes precedence over economic content.