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Calculus of variations on time scales: applications to economic models
 Małgorzata Guzowska^{1},
 Agnieszka B Malinowska^{2}Email author and
 Moulay Rchid Sidi Ammi^{3}
https://doi.org/10.1186/s1366201505370
© Guzowska et al. 2015
Received: 22 February 2015
Accepted: 10 June 2015
Published: 4 July 2015
Abstract
The time scale calculus theory can be applicable to any field in which dynamic processes are described by discrete or continuoustime models. On the other hand, many economic models are dynamic models. Therefore it is natural to relate those two subjects. This work is intended to motivate the use of the calculus of variations and optimal control problems on time scales in the study of economic models. We show that a phenomenon known from the theory of behavioral economics may be described and analyzed by dynamical systems on time scales.
Keywords
MSC
1 Introduction
The origins of the idea of time scales calculus date back to the late 1980s when S Hilger introduced this notion in his PhD thesis and showed how to unify discrete and continuoustime dynamical systems [1, 2]. With time this unification aspect has been supplemented by the extension and generalization features. Currently, several researchers are getting interested in the time scale calculus, contributing to its development and showing applications of the theory and methods in biology (see, e.g., [3, 4]), engineering (see, e.g., [5–10]), physics (see, e.g., [11]), and economy (see, e.g., [12–15]).
The calculus of variations on time scales was introduced in 2004 by M Bohner who used the delta derivative and delta integral [16], and it has since then been further developed by several different authors in several different directions, e.g., [17–22]. Many classical results of calculus of variations as necessary or sufficient conditions of optimality have been generalized to arbitrary time scale. The aim of the present work is to apply some of those results to economic models and to show advantages of using time scales tools. We present two economical models: a ‘cakeeating’ problem and a simple model of household consumption. Both were already considered in the time scale literature, but here the use of them is different from the previous works. We show that besides unification and generalizations aspects the study of dynamical systems on time scales allows one to observe the model behaviors which are different from those known in the classical economic theory but rather coincide with facts of the behavioral economics. Models of the life cycle or, more generally, almost all economic models assume the intertemporal choice model proposed by Samuelson [23] as it is not possible to analyze human decisions concerning consumption and saving without making certain assumptions as regards their time preferences. The models presented in this paper are also based on this concept. However, we have to remember that in many economic models a rationality of people is assumed, their computing skills, iron will and farsightedness, since such assumptions help to solve optimization problems. Yet, the search for increasingly reliable modeling of reality and the desire to use more and more advanced econometrical and statistical techniques force researchers to adopt numerous premises about human behavior, particularly in terms of their time preferences. Therefore, a crucial question arises: do the rules of behavior attributed to individuals reliably reflect this behavior? Studies and tests to verify the accuracy of predictions based on currently dominant theories, i.e., on M Modigliani’s and M Friedman’s hypotheses as well as on works that support them, indicate that the predictive power of these models is often weak. In fact, the observed human behavior patterns seem to differ from the conclusions drawn from the approach based on the model of a rational consumer who optimizes their decisions across time. For instance, the results of the studies by Shapiro and Slemrod [24] or Parker [25] show that the expected changes in income affect the consumption rate in a shortterm period. It means that people’s spending on nondurable goods increases in line with the expected rise in income. Such phenomena are rejected in all the lifecycle models, but they remain an integral part of the behavioral models. Relations between temporal choice and behavior patterns can be observed during experiments, but are not taken into account in classical models and approaches. The results obtained in this paper, i.e., the classical economic models developed on nonstandard time scales prove that the time scale analysis can explain the phenomena in this part of the behavioral economics which deals with the intertemporal choices.
The paper is organized as follows. In the next section, we recall basic terminology related to the time scale calculus. Section 3 provides a detailed exposition of the ‘cakeeating’ problem on time scales. In Section 4 we apply the results of time scales optimal control to a simple model of household consumption. We end the paper with some conclusions in Section 5.
2 Time scales
In this section we give a brief exposition of the time scale calculus. For a more complete presentation we refer the reader to the books [26, 27].
A nonempty closed subset of \(\mathbb{R}\) is called a time scale and it is denoted by \(\mathbb{T}\). Thus, \(\mathbb{R}\), \(\mathbb{Z}\), and \(\mathbb{N}\), are trivial examples of time scales. Other examples of time scales are: \([4,3] \cup\mathbb{N}\), \(h\mathbb{Z}:=\{h z  z \in\mathbb{Z}\}\) for some \(h>0\), \(q^{\mathbb{N}_{0}}:=\{q^{k}  k \in\mathbb{N}_{0}\}\) for some \(q>1\), and the Cantor set. We assume that a time scale \(\mathbb{T}\) has the topology that it inherits from the real numbers with the standard topology.
The mapping \(\sigma:\mathbb{T}\rightarrow\mathbb{T}\), defined by \(\sigma(t)=\inf{\{s\in\mathbb{T}:s>t\}}\) with \(\inf\emptyset=\sup\mathbb{T}\) (i.e., \(\sigma(M)=M\) if \(\mathbb{T}\) has a maximum M) is called the forward jump operator. Accordingly, we define the backward jump operator \(\rho:\mathbb{T}\rightarrow\mathbb{T}\) by \(\rho(t)=\sup{\{s\in\mathbb{T}:s< t\}}\) with \(\sup\emptyset=\inf\mathbb{T}\) (i.e., \(\rho(m)=m\) if \(\mathbb{T}\) has a minimum m). The following classification of points is used within the theory: a point \(t\in\mathbb{T}\) is called rightdense, rightscattered, leftdense and leftscattered if \(\sigma(t)=t\), \(\sigma(t)>t\), \(\rho(t)=t\), and \(\rho(t)< t\), respectively. The functions \(\mu,\nu:\mathbb{T}\rightarrow[0,\infty)\) are defined by \(\mu(t)=\sigma(t)t\) and \(\nu(t)=t\rho(t)\).
Example 1
If \(\mathbb{T} = \mathbb{R}\), then \(\sigma(t) = \rho(t) = t\) and \(\nu(t)=\mu(t)= 0\). If \(\mathbb{T} = h\mathbb{Z}\), then \(\sigma(t) = t + h\), \(\rho(t) = t  h\), and \(\nu(t)=\mu(t)= h\).
For two points \(a,b\in\mathbb{T}\), the time scales interval is defined by \([a,b]_{\mathbb{T}}=\{t\in\mathbb{T}:a\leq t\leq b\}\).
We shall state the definition of differentiability on time scales. Throughout we will frequently write \(f^{\sigma}(t)=f(\sigma(t))\).
Definition 1
Example 2
If \(\mathbb{T} = \mathbb{R}\), then \(f^{\Delta}(t)=f'(t)\). If \(\mathbb{T} = h\mathbb{Z}\), then \(f^{\Delta}(t) = \frac{f(t+h)  f(t)}{h} := \Delta_{h} f(t)\), where \(\Delta_{h}\) is the usual forward difference operator.
Now we shall define Δintegration on time scales.
Definition 2
A function \(f:\mathbb{T}\rightarrow\mathbb{R}\) is called rdcontinuous if it is continuous at rightdense points and if the leftsided limit exists at leftdense points.
We denote the set of all rdcontinuous functions by \(C_{\mathrm{rd}}\) or \(C_{\mathrm{rd}}(\mathbb{T})\), and the set of all Δdifferentiable functions with rdcontinuous derivative by \(C_{\mathrm{rd}}^{1}\) or \(C_{\mathrm{rd}}^{1}(\mathbb{T})\).
Definition 3
A function \(F:\mathbb{T}\rightarrow\mathbb{R}\) is called a delta antiderivative of \(f:\mathbb{T}^{\kappa}\rightarrow\mathbb{R}\) provided \(F^{\Delta}(t)=f(t)\) for all \(t\in\mathbb{T}^{\kappa}\).
Example 3
Let \(a, b \in\mathbb{T}\) with \(a < b\). If \(\mathbb{T} = \mathbb{R}\), then \(\int_{a}^{b}f(t)\Delta t =\int_{a}^{b}f(t) \, dt\), where the integral on the righthand side is the classical Riemann integral. If \(\mathbb{T} = h\mathbb{Z}\), then \(\int_{a}^{b}f(t)\Delta t = \sum_{k=\frac{a}{h}}^{\frac{b}{h}1}hf(kh)\).
In order to define the delta exponential function, first we introduce the concept of regressivity.
Definition 5
A function \(p:\mathbb{T}^{\kappa}\rightarrow\mathbb{R}\) is regressive provided \(1+\mu(t)p(t)\neq0\) holds for all \(t\in\mathbb{T}^{\kappa}\). We denote by \(\mathcal{R}\) the set of all regressive and rdcontinuous functions.
Theorem 6
(Theorem 1.37 in [27])
Definition 7
Let \(p\in\mathcal{R}\) and \(t_{0}\in\mathbb{T}\). The exponential function on time scales is defined by the solution of the IVP (1) and is denoted by \(e_{p}(\cdot,t_{0})\).
The reader can find several properties of the delta exponential function in [26]. We also recommend this book as a source of material on dynamic equations on time scales via the delta derivative.
3 The ‘cakeeating’ problem
Let us now formulate the time scale ‘cakeeating’ problem.
The ‘cakeeating’ problem with the delta derivative
Theorem 8
Proof
Example 4

In the case of the time scales \(\mathbb{T}=\mathbb{Z}\cap[0,60]\) and \(\mathbb{T}=10\mathbb{Z}\cap[0,60]\) the consumer behavior meets the expectations. Figure 1 shows that the time paths of consumption have the same tendency. We can also observe that consumers eat small amounts when they eat more often and they have large meals when they eat seldom. This observation is also confirmed by the analysis of the percentage share of consumption in the time t in relation to the remaining resources (Table 1), which shows a small variation in time.Table 1
The percentage share of consumption in the time t in relation to the remaining resources on \(\pmb{10\mathbb{Z}\cap[0,60]}\)
t
\(\boldsymbol{\frac{\tilde{c}(t)}{60\sum_{\tau\in[0,\rho(t)]}\tilde{c}(\tau )}100}\) (%)
0
50.794
10
51.613
20
53.333
30
57.143
40
66.667
50
100

In the case of the time scale \((\{2^{\mathbb{N}_{0}}\}\cup\{ 0\} )\cap[0,64]\), where the moment of consumption is step by step increasingly far we can see an increase in the consumption level. We observe a tendency to ‘accumulate’ food when it is known that a meal will be more and more delayed. It is clearly demonstrated in Figure 2 (right) and in Table 2 (left).Table 2
The percentage share of consumption in the time t in relation to remaining resources on \(\pmb{(\{2^{\mathbb{N}_{0}}\} \cup\{0\} )\cap[0,64]}\) (left) and on \(\pmb{T_{1}}\) (right)
t
μ ( t )
\(\boldsymbol{\frac{\tilde{c}(t)}{60\sum_{\tau\in[0,\rho (t)]}\tilde{c}(\tau)}100}\) (%)
1
1
9.348
2
2
17.187
4
4
29.649
8
8
46.827
16
16
67.742
32
0
100
0
2
9.9
2
8
36.718
10
5
17.767
15
15
84.034
40
25
89.245
55
0
100

For the time scale \(T_{1}\), in which consumption frequency varies over time the ‘accumulation’ described above can be seen even more clearly. Figure 3 (right) and Table 2 (right) show a considerable growth in consumption in relation to the remaining resources when the moment of the next meal is remote.

The analysis of the model dynamics on the proposed time scales has allowed for the observation of some theoretical findings known from the behavioral economics [34–38], and which are impossible to observe when analyzing the model dynamics on the traditional homogeneous time scales (that is, with \(\mu\equiv \mathrm{const.}\)).
4 The household problem
The household problem with the delta derivative
Theorem 9
[21]
 (i)if \(\mathbb{T}_{y}=\emptyset\), then the budget constraint is$$a^{\Delta}(t)=\frac{r(t)}{1+r(t)\mu(t)}a^{\sigma}(t)\frac{1}{1+r(t)\mu (t)}c^{\sigma}\bigl(m(t)\bigr); $$
 (ii)
if \(\mathbb{T}_{y}=\emptyset\) and \(\mathbb{T}_{a}=\emptyset\), then model (27)(28) coincides with the delta ‘cakeeating’ problem (2)(3).
Example 5
Consumption \(\pmb{c(t)}\) , in Example 5 , with \(\pmb{r=0.03}\) , \(\pmb{r=0.05}\)
t  c ( t ) with r = 0.03  c ( t ) with r = 0.05 

0  0  0 
2  7.905  7.905 
4  6.785  6.917 
6  13.077  13.834 
8  23.975  25.362 
10  20.578  22.192 
12  39.66  44.384 
The change in the interest rate from 0.03 to 0.05 does not affect the above mentioned behavior, but it indicates the higher propensity to save (see Table 3). After increasing r, the dynamics of consumer spending is the same as for \(r=0.03\). However, a greater rate of interest makes the consumer try to save more money for the last period.
5 Conclusions
Dynamic optimization in economics appeared in the 1920s with the work of Hotelling and Ramsey (see, e.g., [39, 40]). There are three major approaches to dynamic optimization problems: dynamic programming, calculus of variations, and optimal control theory. In this paper we have examined the last two approaches but in the more general framework, using the time scale theory. Economists model time as continuous or discrete. The time scale theory allows us to handle discrete and continuous models as being two pieces of the same framework. Moreover, as was shown in this paper the time scale approach enhances economic modeling by the possibility of working with more complex time domains. This possibility allows one to illustrate and confirm the theories dealing with preferences concerning the time and intermediate choices, which were discussed before in the neoclassical economic theory [23] and the behavioral economics [34–38].
Declarations
Acknowledgements
The authors sincerely thank the referees for their valuable suggestions and comments. AB Malinowska was supported by the Bialystok University of Technology grant S/WI/02/2011.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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