On projective systems of rational difference equations

We discuss first order systems of rational difference equations which have the property that lines through the origin are mapped into lines through the origin. We call such systems projective systems of rational difference equations and we show a useful change of variables which helps us to understand the behavior in these cases. We include several examples to demonstrate the utility of this change of variables

On competitive discrete systems in the plane. I. Invariant Manifolds

Let $T$ be a $C^{1}$ competitive map on a rectangular region $R\subset \mathbb{R}^{2}$. The main results of this paper give conditions which guarantee the existence of an invariant curve $C$, which is the graph of a continuous increasing function, emanating from a fixed point $\bar{z}$. We show that $C$ is a subset of the basin of attraction of $\bar{z}$ and that the set consisting of the endpoints of the curve $C$ in the interior of $R$ is forward invariant. The main results can be used to give an accurate picture of the basins of attraction for many competitive maps. We then apply the main results of this paper along with other techniques to determine a near complete picture of the qualitative behavior for the following two rational systems in the plane. $$x_{n+1}=\frac{\alpha_{1}}{A_{1}+y_{n}},\quad y_{n+1}=\frac{\gamma_{2}y_{n}}{x_{n}},\quad n=0,1,...,$$ with $\alpha_1,A_{1},\gamma_{2}>0$ and arbitrary nonnegative initial conditions so that the denominator is never zero. $$x_{n+1}=\frac{\alpha_{1}}{A_{1}+y_{n}},\quad y_{n+1}=\frac{y_{n}}{A_{2}+x_{n}},\quad n=0,1,...,$$ with $\alpha_1,A_{1},A_{2}>0$ and arbitrary nonnegative initial conditions.Comment: arXiv admin note: text overlap with arXiv:0905.1772 by other author

A theoretical framework for trading experiments

A general framework is suggested to describe human decision making in a certain class of experiments performed in a trading laboratory. We are in particular interested in discerning between two different moods, or states of the investors, corresponding to investors using fundamental investment strategies, technical analysis investment strategies respectively. Our framework accounts for two opposite situations already encountered in experimental setups: i) the rational expectations case, and ii) the case of pure speculation. We consider new experimental conditions which allow both elements to be present in the decision making process of the traders, thereby creating a dilemma in terms of investment strategy. Our theoretical framework allows us to predict the outcome of this type of trading experiments, depending on such variables as the number of people trading, the liquidity of the market, the amount of information used in technical analysis strategies, as well as the dividends attributed to an asset. We find that it is possible to give a qualitative prediction of trading behavior depending on a ratio that quantifies the fluctuations in the model

Testing near-rationality using detailed survey data

This paper considers the evidence of “near-rationality,†as described by Akerlof, Dickens, and Perry (2000). Using detailed surveys of household inflation expectations for the United States and Sweden, we find that the data are generally unsupportive of the near-rationality hypothesis. However, we document that household inflation expectations tend to settle around discrete and largely fixed “focal points,†suggesting that both U.S. and Swedish households gauge inflation prospects in rather broad, qualitative terms. Moreover, the combination of a low-inflation environment and an inflation target in Sweden has been accompanied by a disproportionately high proportion of Swedish households expecting no inflation. However, a similar low-inflation trend in the United States, which does not have an explicit inflation target, reveals no such rise in the proportion of households expecting no inflation. This observation suggests that the way the central bank communicates its inflation objective may influence inflation expectations independently of the inflation trend it actually pursues.inflation expectations, rationality, inflation targeting, Phillips curve, Bryan, Palmqvist