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The Basic Optimal Control Problem

The Basic Optimal Control Problem As was already mentioned, the rate of return in any period along a consumption-e¢ cient trajectory is the extra consumption that could be obtained next period from curtailing consumption in this period by one unit, but otherwise leaving the rest of the e¢ cient trajectory as is. We now make yet-another heroic assumption that the economy is moving along an e¢ cient trajectory where at all times the own rate of interest (or rate of return) on the single consumption good is the constant r. This might be considered a tolerable approximation for a great many practical purposes because it accords with a well-known ìstylized factîthat the real rate of return has been essentially trendless over time, at least throughout the measurable past.6 One could try to argue that the economic entity corresponding most closely to this concept is the annual after-tax real return to capital (because it approximately deÖnes the relevant intertemporal consumption trade-o§ faced by the average citizen in deciding how much to save). As a very rough approximation, a trendless round Ögure of r 4-5% per year might then be used for this real interest rate in the postwar period or even, perhaps, over longer historical periods.7 Some reáection reveals that such an economy moving along an e¢ cient trajectory with 5This is not quite true. Technically speaking, I need more convexity in the sense that (C(t); I(t); K(t))”B, where B is a time-independent (2n+ 1)-dimensional convex set of feasible alternatives. See Weitzman (2003) for further details. I implicitly assume that this stronger convexity condition is being met. 6Nordhaus (1994) in his section entitled ìEmpirical Evidence on the Return on Capitalî summarizes a large number of studies that are consistent with a trendless interpretation. Indeed, this is one of Kaldorís famous ìstylized factsî about the growth of advanced industrial economies. (For a discussion, see Solow (1970), p. 3.) 7Nordhaus (1995), Jorgenson (1994), or Feldstein (1997) could each be cited to justify trendless áuctuations in the rate of return averaging around 4-5% per year. 4 own rate of interest on consumption equal to the constant r must be a solution of the optimal control problem that maximizes the present discounted value of consumption at constant discount rate r. More formally, such a trajectory will maximize Z 1 0 e rt C(t) dt (2) subject to the feasibility constraints (1) and the capital accumulation equation K_ (t) = I(t); (3) and with the given initial condition K(0) = ; (4) where stands for the initially given vector of capital stoc