This book is a comprehensive effort on the analytical foundation of environmental modeling. One of the central theoretical cornerstones of environmental economics is externality theory. How to implement our accurate understanding of externality in solving the environmental problem is an unsettled issue. We connect some crucial loose ends in the relationship between the environment and externality. In a unified modeling framework, we establish the triangular equivalence relationship among the Lindahl equilibrium without transfers, the Nash bargaining solution, and the social optimum under the Lindahl weights. Furthermore, we identify a wide range of applications of the above relationship in the areas of climate change and international environmental agreement (IEA). There should be a two-way bridge linking economic theory and its applications. The algorithms derived from constructive proofs of the theory and programmable algorithms in the specific modeling are construction materials of such a bridge. In this study, we produced the needed materials for the bridge. The algorithmic approach in this study is reflected in the analytical expositions and illustrated through numerical examples and empirical models. The comprehensiveness of this study is in bridging the theory and application, not omnipresent coverage of the broad topics of environmental externality"--