A $N$-body system is simple if all associated physically relevant space-time correlations are short-ranged, meaning that their momenta of all orders are finite. Such systems typically have a total number $W$ of admissible microscopic configurations which grows with $N$ as $W(N) \sim \mu^N\;(\mu >1, \, N\to\infty)$. Their thermostatistical properties are known to be satisfactorily handled within Boltzmann-Gibbs (BG) statistical mechanics. In contrast, a system is complex if one or more associated correlations are long-ranged. The simplest examples of such systems are those for which $W(N) \sim N^\rho\;\;(\rho>0)$. A generalized statistical mechanics grounded on the nonadditive entropic functional $S_q(\{p_{i}\})=k\sum_{i=1}^W p_i \ln_q \frac{1}{p_i} \;\;(q\in \mathbb{R}, \;S_1=S_{BG})$ with $\ln_q z =\frac{z^{1-q}-1}{1-q}$ ($\ln_1 z=\ln z$) satisfactorily handles such systems. Indeed, for equiprobabilities, $S_{q=1-1/\rho}(\{1/W(N)\})=k\ln_{1-1/\rho} W(N) \propto N$. Furthermore, for complex systems, the size of the corresponding phase spaces can be related to the property $\ln_q (x\otimes_q y) =\ln_q x + \ln_q y$, $x,y\geq1$, $q\leq 1$, where $x\otimes_q y=[x^{1-q}+y^{1-q}-1]^{\frac{1}{1-q}}_{+}\;\;(x\otimes_1 y=xy)$, and the $q$-product $\otimes_q$ grounds the definition of a $q$-generalized algebra. Another class of complex systems corresponds to $W(N) \sim \nu^{N^\gamma}\;\;(\nu >1, \,\gamma > 0)$, which appears to be the case of black holes and other cosmological phenomena. This class can be handled with $S_\delta(\{p_{i}\}) = k\sum_{i=1}^W p_i \Bigl(\ln \frac{1}{p_i} \Bigr)^\delta \;\;(\delta>0)$. Finally, we can define $S_{q,\delta}(\{p_{i}\})=k\sum_{i=1}^W p_i \Bigl(\ln_q \frac{1}{p_i} \Bigr)^\delta$ $(q\in\mathbb{R},\delta>0)$, which unifies all the above ones. In this paper, we generalize the $q$-algebra to a new, promising one, namely the $(q,\delta)$-algebra.

I characterize stochastic non-tâtonnement processes (SNTP) and argue that they are a natural outcome of General Equilibrium Theory. To do so, I revisit the classical demand theory to define a normalized Walrasian demand and a diffeomorphism that flattens indifference curves. These diffeomorphisms are applied on the three canonical manifolds in the consumption domain (i.e., the indifference and the offer hypersurfaces and the trade hyperplane) to analyze their images in the normalized and the flat domains. In addition, relations to the set of Pareto optimal allocations on Arrow-Debreu and overlapping generations economies are discussed. Then, I derive, for arbitrary non-tâtonnement processes, an Attraction Principle based on the dynamics of marginal substitution rates seen in the "floor" of the flat domain. This motivates the definition of SNTP and, specifically, of Bayesian ones (BSNTP). When all utility functions are attractive and sharp, these BSNTP are particularly well behaved and lead directly to the calculation of stochastic trade outcomes over the contract curve, which are used to model price stickiness and markets' responses to sustained economic disequilibrium, and to prove a stochastic version of the First Welfare Theorem.

A well-known feature of overlapping generations economies is that the First Welfare Theorem fails and equilibrium may be inefficient. The Cass (1972) criterion furnishes a necessary and sufficient condition for efficiency, but does not address the matter of existence of efficient equilibria, and Cass, Okuno, and Zilcha (1979) provide nonexistence examples. I develop an algorithm based on successive approximations of a nonstationary, consumption-loan, prone to savings, overlapping generations economy with finite-lived heterogeneous agents to find elements of its set of equilibria as the limit of nested compact sets. These compact sets are the result of a backward calculation through equilibrium equations that departs from the set of Pareto optimal equilibria of well-behaved tail economies. The equilibria calculated through this algorithm satisfy the Cass (1972) criterion and are used to derive the existence results on efficient equilibria.