This research uses the optimal control theory to construct two kinds of dynamic economic systems, which are an economic system without externalities and with externalities. Within each economic system both the centralized economy model and the decentralized economy model are included. The centralized economy (a social planner) model representes a kind of ideal economy, and the goal what the social planner pursues is that the resource allocation of the society satisfies the Pareto Efficiency criteria. On the other hand, the decentralized economy model (consists of a representative producer and a representative consumer) demonstrates the real economy, where economic agents persue their own best interests. While constructing the models, goods market equilibrium, labors market equilibrium, the dynamic accumulation process of capital, and the dynamic accumulation course of pollution are under consideration. Then, I apply optimal control method to get the first order conditions, and compare these f.o.c¡¦.s to verify whether they are unanimous. This paper proves that when externalities of pollution does not exist in the dynamic economic system, the decentralized economy model can achieve the Pareto Efficiency. On the contrary, when externalities of pollution emerges in the dynamic economic system, the decentralized economy model cannot reach Pareto Efficiency. If the externalities of pollution is internalized by the dynamic decentralized economic system economy, it will coincide with Pareto Efficiency. Besides, Pigouvian tax is still an effective policy instrument. Finally, I discuss all dynamic models in this paper to find out whether there exists a long-term and stable steady state. I find stable steady state, saddle-point equilibria, do exist under certain restrictions.

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