Fractional Calculus in Economic Growth Modeling. The Portuguese case* In´ es Tejado 1,2 , Duarte Val´ erio 1 , Nuno Val´ erio 3 Abstract— A variety of fractional order models have been proposed in the literature to account for the behaviour of financial processes from different points of view. The objective of this work is to model the growth of national economies, namely, their gross domestic products (GDPs), by means of a fractional order approach. The particular case of Portugal is addressed, and results show that fractional models have a better performance than the other alternatives considered and proposed in the literature. I. I NTRODUCTION Developing mathematical models to simulate stochastic processes has an important role in financial analysis. Be- cause fractional operators are non-local, they are suitable for constructing models for long series, possessing a memory effect — more so than models using integer derivatives and integrals alone [25]. This is the reason why fractional differential equations possess large advantage in describing economic phenomena over large time periods. Indeed, a variety of fractional order models have been proposed in the literature to account for the behaviour of fi- nancial processes from different points of view. For example, as diffusion or stochastic processes by means of L´ evy models [3], [6], [7], [8], [17] or continuous time random walks for the movement of log-prices (e.g. [10], [16], [19], [20], [22], [23]), respectively. A modified fractality concept was applied in [12] to describe the stochastic dynamics of the stock and currency markets. Likewise, a macroeconomic state space model was proposed in [24] for national economies consisting of a group of fractional differential equations. A similar form was used in [27] but with variable orders. The objective of this work is to model the growth of national economies, namely, their gross domestic products (GDPs), by means of a fractional order approach. In the lite- rature many models have been published, among which the classical papers [5], [21]. Yet, to the best of our knowledge, no fractional model of GDP as a function of a vector of inputs had yet been found. Here, the GDP of a national economy was modelled as function of a vector with nine *In´ es Tejado would like to thank the Portuguese Fundac ¸˜ ao para a Ciˆ encia e a Tecnologia (FCT) for the grant with reference SFRH/BPD/81106/2011. This work was partially supported by Fundac ¸˜ ao para a Ciˆ encia e a Tec- nologia, through IDMEC under LAETA, and under the joint Portuguese– Slovakian project SK-PT-0025-12. 1 I. Tejado and D. Val´ erio are with IDMEC, Instituto Superior T´ ecnico, Universidade de Lisboa, Portugal. email: {ines.tejado,duarte.valerio}@tecnico.ulisboa.pt. 2 I. Tejado is also with Industrial Engineering School, University of Extremadura, Spain. email: itejbal@unex.es. 3 N. Val´ erio is with Instituto Superior de Economia e Gest˜ ao, Universi- dade de Lisboa, Portugal. email: nuno.valerio@iseg.ulisboa.pt. variables. The particular case of the economy of Portugal along the last five decades is studied. The sources of data were the World Bank database [4], the Portuguese Historical Statistics [26], and the statistical appendix of [18]; for details, see the paper’s appendix. The remainder of this paper is organised as follows. Section II describes the proposed method to account for the behaviour of national economies. In Section III, the obtained results after fitting are given. Finally, Section IV draws the concluding remarks and perspectives future works. II. ECONOMIC GROWTH MODEL Consider a simple model of a national economy in the following form: y(t)= f (x 1 ,x 2 ...) (1) where the output model y is the GDP (in 2012 euros) and the x k are the variables on which the output depends. The inputs considered are the following: • x 1 : land area (km 2 ); • x 2 : arable land (km 2 ); • x 3 : population; • x 4 : school attendance (years); • x 5 : gross capital formation (GCF) (in 2012 euros); • x 6 : exports of goods and services (in 2012 euros); • x 7 : general government final consumption expenditure (GGFCE) (in 2012 euros); • x 8 : money and quasi money (M2) (in 2012 euros). The rationale behind this choice of variables is the following: • natural resources are represented by x 1 , and their quality by x 2 ; • human resources are represented by x 3 , and their quality by x 4 ; • manufactured resources are represented by x 5 ; • external impacts in the economy are represented by x 6 ; • internal impacts in the economy are represented by x 7 (budgetary impacts), x 8 (monetary impacts) and also by x 5 (investment). Rather than having x 5 play two roles, we will rather use another variable x 9 ≡ x 5 to represent the impact of investment in the economy, bringing the number of inputs up to 9. This choice of variables joins those traditionally considered in growth accounting [9], [13], [14] to those acknowledged by Keynesian models having short-term inputs related to im- pacts in the economy. The quality of manufactured resources is sometimes translated in a variable such as the number of patents filed each year. We did not use this variable, not