Abstract
In this work, we consider a classic international trade model with two countries and one firm in each country. The game has two stages: in the first stage, the governments of each country use their welfare functions to choose their tariffs either: (a) competitively (Nash equilibrium) or (b) cooperatively (social optimum); in the second stage, firms competitively choose (Nash) their home and export quantities under Cournot-type competition conditions. In a previous publication we compared the competitive tariffs with the cooperative tariffs and we showed that the game is one of the two following types: (i) prisoner's dilemma (when the competitive welfare outcome is dominated by the cooperative welfare outcome); or (ii) a lose-win dilemma (an asymmetric situation where only one of the countries is damaged in the cooperative welfare outcome, whereas the other is benefited). In both scenarios, their aggregate cooperative welfare is larger than the aggregate competitive welfare. The lack of coincidence of competitive and cooperative tariffs is one of the main difficulties in international trade calling for the establishment of trade agreements. In this work, we propose a welfare-balanced trade agreement where: (i) the countries implement their cooperative tariffs and so increase their aggregate welfare from the competitive to the cooperative outcome; (ii) they redistribute the aggregate cooperative welfare according to their relative competitive welfare shares. We analyse the impact of such trade agreement in the relative shares of relevant economic quantities such as the firm's profits, consumer surplus, and custom revenue. This analysis allows the countries to add other conditions to the agreement to mitigate the effects of high changes in these relative shares. Finally, we introduce the trade agreement index measuring the gains in the aggregate welfare of the two countries. In general, we observe that when the gains are higher, the relative shares also exhibit higher changes. Hence, higher gains demand additional caution in the construction of the trade agreement to safeguard the interests of the countries.

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For uniformly asymptotically affine (uaa) Markov maps on train tracks, we prove the following type of rigidity result: if a topological conjugacy between them is (uaa) at a point in the train track then the conjugacy is (uaa) everywhere. In particular, our methods apply to the case in which the domains of the Markov maps are Canter sets. We also present similar statements for (uaa:) and C-r Markov families. These results generalize the similar ones of Sullivan and de Faria for C-r expanding circle maps with r > 1 and have useful applications to hyperbolic dynamics on surfaces and laminations.

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We exploit ideas of nonlinear dynamics in a non-deterministic dynamical setting. Our object of study is the observed riverflow time series of the Portuguese Paiva river whose water is used for public supply. The Takens delay embedding of the daily riverflow time series revealed an intermittent dynamical behaviour due to precipitation occurrence. The laminar phase occurs in the absence of rainfall. The nearest neighbour method of prediction revealed good predictability in the laminar regime but we warn that this method is misleading in the presence of rain. The correlation integral curve analysis, Singular Value Decomposition and the Nearest Neighbour Method indicate that the laminar regime of flow is in a small neighbourhood of a one-dimensional affine subspace in the phase space. The Nearest Neighbour method attested also that in the laminar phase and for a data set of 53 years the information of the current runoff is by far the most relevant information to predict future runoff. However the information of the past two runoffs is important to correct non-linear effects of the riverflow as the MSE and MRE criteria results show. The results point out that the Nearest Neighbours method fails when used in the irregular phase because it does not predict precipitation occurrence. © 2007, Birkhäuser Verlag Basel/Switzerland.