Email: support@essaywriterpros.com
Call Us: US - +1 845 478 5244 | UK - +44 20 7193 7850 | AUS - +61 2 8005 4826

# client consumption trajectory

. Because the maximand (2) is a weighted linear average of fC(t)g with time weights fe rtg, the rate of transformation along the optimal trajectory between consumption at time  and consumption at time  0 is guaranteed to be exp(r( 0  )). Thus, the solution of maximizing (2) (subject to constraints (1), (3), (4)) has the property of being an e¢ cient consumption trajectory all along which the rate of return is the constant r. Conversely, an e¢ cient consumption trajectory all along which the rate of return is r must maximize (2) subject to constraints (1), (3), (4). I note in passing that maximizing (2) (subject to constraints (1), (3), (4)) can be interpreted as an as-if optimal control problem. The underlying problem might be maximizing some complicated form of dynamic utility, but the solution is as if there is an e¢ cient consumption trajectory all along which the rate of return is the constant r. This way of looking at things seems to me to be a more intuitive characterization than in terms of dynamic utility maximization. There might be some form of complicated dynamic utility maximization in the background, but the reduced form of a solution is as if it results in an e¢ cient consumption trajectory along which the rate of return is the constant r. This as if form accords with the well-known ìstylized factîthat the real rate of interest has been essentially trendless over time, at least throughout the measurable past.8 I am e§ectively assuming that we are in a special case where instantaneous ìutilityîis as-if identiÖed with the single measurable consumption good whose rate of return is r. An application of the famous maximum principle of optimal control theory to the ìas ifî problem here takes the following form. Introduce the n-dimensional investment-price row 8See the last two footnotes for a further exposition of this poi vector P = fPjg. Henceforth we adhere to the convention that all prices are row vectors and all quantities are column vectors. The current-value Hamiltonian expression C + PI is in the form of net domestic product (NDP) with consumption as numeraire. A function of K and P that will play a critical role in the maximum principle is the current-value maximized Hamiltonian expression He(K; P)  max (C;I)”A(K) fC + PIg: (5) A feasible trajectory satisÖes (1), (3), (4). With underlying convexity, the necessary and su¢ cient condition for a feasible trajectory fC (t); I (t); K (t)g to be optimal is the existence of an investment-price trajectory fP (t)g satisfying for all t the condition9 He(K (t); P (t)) = C (t) + P (t) I (t)