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Introduction to Modern Economic Growth subject to (5.22) k˙ (t) = f (k (t)) − c (t) − δk (t) k (t) ≥ and given k (0) The objective function (5.21) is the direct continuous-time analogue of (5.18), and (5.22) gives the resource constraint of the economy, similar to (5.19) in discrete time Once again, this problem lacks one boundary condition which will come from the transversality condition The most convenient way of characterizing the solution to this problem is via optimal control theory Dynamic programming and optimal control theory will be discussed briefly in the next two chapters 5.10 Taking Stock This chapter introduced the preliminaries necessary for an in-depth study of equilibrium and optimal growth theory At some level it can be thought of as an “odds and ends” chapter, introducing the reader to the notions of representative household, dynamic optimization, welfare theorems and optimal growth However, what we have seen is more than odds and ends, since a good understanding of the general equilibrium foundations on economic growth and the welfare theorems should enable the reader to better understand and appreciate the material that will be introduced in Part below The most important take-away messages from this chapter are as follows First, the set of models we study in this book are examples of more general dynamic general equilibrium models It is therefore important to understand which features of the growth models are general (in the sense that they not depend on the specific simplifying assumptions we make) and which results depend on the further simplifying assumptions we adopt In this respect, the First and the Second Welfare Theorems are essential They show that provided that all product and factor markets are competitive and there are no externalities in production or consumption (and under some relatively mild technical assumptions), dynamic competitive equilibrium will be Pareto optimal and that any Pareto optimal allocation can be decentralized as a dynamic competitive equilibrium These results will be relevant 247