A Caputo fractional derivative of a function with respect to another function

In this paper we consider a Caputo type fractional derivative with respect to another function. Some properties, like the semigroup law, a relationship between the fractional derivative and the fractional integral, Taylor’s Theorem, Fermat’s Theorem, etc., are studied. Also, a numerical method to deal with such operators, consisting in approximating the fractional derivative by a sum that depends on the first-order derivative, is presented. Relying on examples, we show the efficiency and applicability of the method. Finally, an application of the fractional derivative, by considering a Population Growth Model, and showing that we can model more accurately the process using different kernels for the fractional operator is provided.

The economic consequences of Portuguese State’s movement towards business sector

Over the last 36 years, the relationship with the Portuguese state-owned enterprises registered several dynamics: nationalizations, privatizations and corporatization of public services. However, until now the State Business Sector from a national accounts perspective was never analyzed. Based on data collected and compiled for the first time at Statistics Portugal, this PhD thesis aims to test, analyzing in eight dimensions, whether the weight of the State Business Sector increased and if it contributed positively to the Portuguese economy, from 2006 to 2010. In addition to this analysis, an overview of the economic theory of state intervention in the economy, the paradigm changes of public policy in the international context, the evolution of the Portuguese State Business Sector since 1974, accompanied with a business and national accounting perspective between 2006 and 2010, are also presented. The results allow us to conclude that, in general, the weight of the State Business Sector in the Portuguese economy increased and had a tendency of a positive contribution to its economic growth. The State Business Sector also contributed positively to the nominal labour productivity (although with a decreasing trend of contribution to growth over the period under review) and the profitability of the non-financial corporations sector (although impairing the overall ratio of this sector). Nonetheless, the State Business Sector contributed negatively to the fairness in compensation of employees (although with an improvement trend) and to the competitiveness of labour cost, investment and sectorial sustainability of the Portuguese economy (reinforced by a falling trend). The results also suggest that the State Business Sector had an economic behaviour closer to a welfare maximizing model than to a profit maximizing model. This distinct performance with respect to the institutional sector in which is included, highlights the need to study and reassess the relationship of the state with public corporations, in light of agency theory using micro-data. Lastly, contributions to improve the economic performance of the State Business Sector and future prospects of evolution are presented.