We consider a network model of production with externalities which describes a situation typical for many economic, social, and political systems. In the first period of time each of the agents in the network receives endowment and distributes it between consumption and investment. In the second period the agent’s consumption depends on its own investment as well as on investments of its neighbors. The agent’s benefit is determined by its consumption in the two periods. We introduce adjustment dynamics into this model and study the problem of stability of the game equilibrium. An important fact which we have discovered in our research is the special role of the conditions of the presence and the absence of productivity both in a static and in a dynamic framework. The specifics of the dynamics and the nature of the resulting equilibrium depend on the parameters of the model and on the character of the initial disturbance. We have found the instability of the inner equilibrium and have studied the convergence to a new corner equilibrium and the stability of the latter. The instability of the inner equilibria, which we found and the sources of which we study, is the property typical for social and economic systems. The presence of many social institutions can be explained by the wish of the members of the society to preserve the existing equilibria under the dynamic instability which would take place without such stabilizing institutions.