This comprehensive study investigates the intricate relationship between water scarcity and agricultural production, emphasizing its critical global significance. The research, through multidimensional analysis, investigates the various effects of water scarcity on crop productivity, especially the economic water scarcity (AEWS) which is the main factor of influence. The study stresses the possibility of vertical farming as a viable solution to the different kinds of water scarcity problems, hence, it emphasizes its function in the sustainable agricultural development. Although the study recognizes that some problems still remain, it also points out the necessity of more research to solve the issues of scalability and socio-economic implications. Moving forward, interdisciplinary collaboration and technological innovation are essential to achieving water-secure agriculture and societal resilience.

The concept of Nash equilibrium in behavioral strategies (NashEBS) was formulated By Nash~\cite{Nash (1951)} for an extensive-form game through global rationality of nonconvex payoff functions. Kuhn's payoff equivalence theorem resolves the nonconvexity issue, but it overlooks that one Nash equilibrium of the associated normal-form game can correspond to infinitely many NashEBSs of an extensive-form game. To remedy this multiplicity, the traditional approach as documented in Myerson~\cite{Myerson (1991)} involves a two-step process: identifying a Nash equilibrium of the agent normal-form representation, followed by verifying whether the corresponding mixed strategy profile is a Nash equilibrium of the associated normal-form game, which often scales exponentially with the size of the extensive-form game tree. In response to these challenges, this paper develops a characterization of NashEBS through the incorporation of an extra behavioral strategy profile and beliefs, which meet local sequential rationality of linear payoff functions and self-independent consistency. This characterization allows one to analytically determine all NashEBSs for small extensive-form games. Building upon this characterization, we acquire a polynomial system serving as a necessary and sufficient condition for determining whether a behavioral strategy profile is a NashEBS. An application of the characterization yields differentiable path-following methods for computing such an equilibrium.

This paper resolves Aaron's social insurance paradox, which suggests that introducing a pay-as-you-go (PAYG) pension system increases welfare when population growth plus average wage growth exceeds interest rates. Using a simplified overlapping generations model, we demonstrate this apparent advantage stems from asset reduction rather than inherent superiority. We analyze three pension systems - traditional PAYG, capital-funded, and capital-funded with bonus payments - and establish an equivalence between PAYG and the bonus-payment system. This equivalence reveals that systems with identical contributions and benefits differ only in accounting frameworks and asset positions, challenging the notion of PAYG superiority. Our analysis exposes a fundamental conceptual inconsistency in how sustainability is assessed across equivalent pension systems. As an alternative, we propose $\alpha$-stability, a framework using index shares to evaluate pension systems relative to economic indicators. These findings suggest that perceived advantages between pension systems often result from their formulation rather than substantive economic differences.

We introduce a model of infinite horizon linear dynamic optimization with linear constraints and obtain results concerning feasibility of trajectories and optimal solutions necessarily satisfying conditions that resemble the Euler condition and transversality condition. We prove results about optimal trajectories of strictly alternating problems, eventually conclusive problems, strongly eventually conclusive problems and two-phase problems.

We introduce a novel estimator for quantile causal effects with high-dimensional panel data (large $N$ and $T$), where only one or a few units are affected by the intervention or policy. Our method extends the generalized synthetic control method (Xu 2017) from average treatment effect on the treated to quantile treatment effect on the treated, allowing the underlying factor structure to change across the quantile of the interested outcome distribution. Our method involves estimating the quantile-dependent factors using the control group, followed by a quantile regression to estimate the quantile treatment effect using the treated units. We establish the asymptotic properties of our estimator and propose a bootstrap procedure for statistical inference, supported by simulation studies. An empirical application of the 2008 China Stimulus Program is provided.