Global Indeterminacy and Invariant Manifolds Near Homoclinic Orbit to a Real Saddle

In this paper we investigate the dynamic properties of the Romer model. We determine the whole set of conditions which lead to global indeterminacy and the existence of a homoclinic orbit that converges in both forward and backward time to a real saddle equilibrium point. The dynamics near this homoclinic orbit have been investigated. The economic implications are discussed in the conclusions

Chaotic solutions in non linear economic-financial models

Following Mulligan and Sala-i-Martin (1993) we study a general class of endogenous growth models formalized as a non linear autonomous three-dimensional differential system. We consider the abstract model. By using the Shilnikov Theorem statements, we determine the parameters space in which the condition for the existence of a homoclinic Shilnikov orbit and Smale horseshoe chaos are true. The Lucas model (1998) can be considered as an application of the general result. The series expression of the homoclinic orbit is derived by the undetermined coecient method. We show the optimality for the solutions path based on the Shilnikov Theorem. Some economic implications of this analysis are discussed. Keywords: homoclinic Shilnikov bifurcation, Smale horseshoe chaos.

Can bank interaction during rating measurement of micro and very small enterprises ipso facto Determine the collapse of PD status?

This paper begins with an analysis of trends - over the period 2012-2018 - for total bank loans, non-performing loans, and the number of active, working enterprises. A review survey was done on national data from Italy with a comparison developed on a local subset from the Sardinia Region. Empirical evidence appears to support the hypothesis of the paper: can the rating class assigned by banks - using current IRB and A-IRB systems - to micro and very small enterprises, whose ability to replace financial resources using endogenous means is structurally impaired, ipso facto orient the results of performance in the same terms of PD assigned by the algorithm, thereby upending the principle of cause and effect? The thesis is developed through mathematical modeling that demonstrates the interaction of the measurement tool (the rating algorithm applied by banks) on the collapse of the loan status (default, performing, or some intermediate point) of the assessed micro-entity. Emphasis is given, in conclusion, to the phenomenon using evidence of the intrinsically mutualistic link of the two populations of banks and (micro) enterprises provided by a system of differential equation

Fractal analysis of Dow Jones industrial index returns

The Dow Jones Industrial Average 30 (DJIA30) Index was analyzed to show that models based on the Fractal Market Hypothesis (FMH) are preferable to those based on the Efficient Market Hypothesis (EMH). In a first step, Rescaled Range Analysis was applied to search for long term dependence between index returns. The Hurst coefficient was computed as a measure of persistence in the trend of the observed time series. A Monte Carlo simulation based on both Geometric Brownian Motion (GBM) and Fractional Brownian Motion (FBM) models was used in the second step to investigate the forecasting ability of each model in a situation where information about future prices is lacking. In the third step, the volatility of the index returns obtained from the simulated GBM and FBM was considered together with that produced by a GARCH(1,1) model in order to determine the approach that minimizes the Value at Risk (VaR) and the Conditional Value at Risk (CVaR) of one asset portfolio where the DJIA30 index underlies an Exchange Traded Commodity (ETC). In the case observed returns could either follow a gaussian distribution or a Pareto distribution with a scale parameter equal to the inverse of the Hurst coefficient determined in the first step

Shilnikov Chaos, Low Interest Rates, and New Keynesian Macroeconomics

The paper shows that in a New Keynesian (NK) model, an active interest rate feedback monetary policy, when combined with a Ricardian passive fiscal policy, à la Leeper-Woodford, may induce the onset of a Shilnikov chaotic attractor in the region of the parameter space where uniqueness of the equilibrium prevails locally. Implications, ranging from long-term unpredictability to global indeterminacy, are discussed in the paper. We find that throughout the attractor, the economy lingers in particular regions, within which the emerging aperiodic dynamics tend to evolve for a long time around lower-than-targeted inflation and nominal interest rates. This can be interpreted as a liquidity trap phenomenon, produced by the existence of a chaotic attractor, and not by the influence of an unintended steady state or the Central Bank's intentional choice of a steady state nominal interest rate at its lower bound. In addition, our finding of Shilnikov chaos can provide an alternative explanation for the controversial “loanable funds” over-saving theory, which seeks to explain why interest rates and, to a lesser extent inflation rates, have declined to current low levels, such that the real rate of interest is below the marginal product of capital. Paradoxically, an active interest rate feedback policy can cause nominal interest rates, inflation rates, and real interest rates unintentionally to drift downwards within a Shilnikov attractor set. Policy options to eliminate or control the chaotic dynamics are developed

Shilnikov Chaos, Low Interest Rates, and New Keynesian Macroeconomics

The paper shows that in a New Keynesian (NK) model, an active interest rate feedback monetary policy, when combined with a Ricardian passive fiscal policy, à la Leeper-Woodford, may induce the onset of a Shilnikov chaotic attractor in the region of the parameter space where uniqueness of the equilibrium prevails locally. Implications, ranging from long-term unpredictability to global indeterminacy, are discussed in the paper. We find that throughout the attractor, the economy lingers in particular regions, within which the emerging aperiodic dynamics tend to evolve for a long time around lower-than-targeted inflation and nominal interest rates. This can be interpreted as a liquidity trap phenomenon, produced by the existence of a chaotic attractor, and not by the influence of an unintended steady state or the Central Bank's intentional choice of a steady state nominal interest rate at its lower bound. In addition, our finding of Shilnikov chaos can provide an alternative explanation for the controversial “loanable funds” over-saving theory, which seeks to explain why interest rates and, to a lesser extent inflation rates, have declined to current low levels, such that the real rate of interest is below the marginal product of capital. Paradoxically, an active interest rate feedback policy can cause nominal interest rates, inflation rates, and real interest rates unintentionally to drift downwards within a Shilnikov attractor set. Policy options to eliminate or control the chaotic dynamics are developed

Forest Area Changes in Cinque Terre National Park in the Last 80 Years. Consequences on Landslides and Forest Fire Risks

Cinque Terre, one of the most important Italian cultural landscapes, has not been spared from depopulation and agricultural abandonment processes, that involved many rural areas in Europe, as a consequence of socio-economic transformations that occurred after WWII. Depopulation of rural areas, especially in mountains or in terraced areas, caused significant environmental consequences, such as the decrease of biodiversity, the landscape homogenization, the increase of hydrogeological and forest fires risks. Cinque Terre National Park (5TNP) was established in 1999, and, differently from other Italian National Parks, not just for protecting natural habitats, but mainly to preserve, restore and valorize the historical terraced landscape. Moreover, the area is a UNESCO cultural landscape site and it is partly protected by three Sites of Community Importance. The research intended to investigate the transformations that have affected forested areas inside the 5TNP in the period 1936–2018, also highlighting the connections with hydrogeological and forest fires risks, as a support for the Park planning strategies and the conservation of the UNESCO site. Results highlighted that 37% of the current forests are the consequence of dry stones terraces abandonment that occurred in the twentieth century, with negative effects on the stability of steep slopes, hydrogeological risk, forest fires and on the conservation of a unique cultural landscape. This confirms the current national trend showing no deforestation occurring, but rather a continuous increase of forests on abandoned land. While 5TNP policies and actions are effectively aimed at pursuing an equilibrium between cultivated areas and forests, the Sites of Community Importance located inside the Park mainly focuses on the conservation of “natural habitats”, even if the current vegetation is also the result of secondary successions on former cultivated land. The research highlighted the need to valorize “cultural values” in forest planning as well as the importance of forest history for an accurate planning of forest resources in protected areas

Existence and implications of a pitchfork-Hopf bifurcation in a continuous-time two-sector growth model

This paper shows that the dynamics of the Lucas (J Monet Econ, 22:3–42, 1988) endogenous growth model with flow externalities may give rise to a 2-torus, a compact three-dimensional manifold enclosed by a two-dimensional surface. The implications of this result are relevant for many fields of economic theory. It is first of all clear that if we choose to initialize the dynamics in the basin of attraction of this trapping region, a continuum of perfect foresight solutions may be observed. A simple econometric exercise, linking the physical-to-human capital ratio (state variable) to the 5-years forward variance of the growth rate of an unbalanced sample of 183 countries, seems to provide empirical backing for the phenomenon. Other important consequences, relevant from the point of view of endogenous cycles theory, are also scrutinized in the paper