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Economic Interpretation of the Maximum Principle

Economic Interpretation of the Maximum Principle What is the economic interpretation of conditions and ? Brieáy, are describing an economy-wide dynamic competitive equilibrium with a perfect capital market whose rate of return is r. There is a deep connection between maximization of (2) over time and competitive market equilibrium over time in an economy whose rate of return is r. Essentially, the basic invisible-hand theorem here says that an economic situation maximizing present discounted consumption (at discount rate r) is being optimally controlled over time if and only if the situation can simultaneously be interpreted as describing a dynamic perfectforesight competitive equilibrium with a perfect capital market paying rate of return r. , as applied to the formulation of this paper. For convenience, here I am omitting the transversality condition at inÖnity. Again, seefor a full treatment of this aspect. is being maximized subject to constraints (1), (3), (4), then optimal investment levels must be accompanied by investment or capital prices (relative to a consumption numeraire price of 1) in such a way that any observers would swear they are witnessing the evolution of an economy with a perfect capital market in a dynamic perfect-foresight competitive equilibrium having rate of return r. Conversely, if price and quantity trajectories are behaving as if the model were describing a dynamic perfect-foresight competitive equilibrium with a perfect capital market having rate of return r, then it must constitute a solution to the problem of maximizing (2) subject to constraints (1), (3), (4). A convenient thought experiment for conceptualizing a nation having a perfect capital market paying a real interest rate of r is to postulate an imaginary national bank where any amount of consumption may be borrowed or deposited at real interest rate r, under the condition that all loans must eventually be paid o§. The unit of account in this nation is the consumption good C, whose numeraire price is 1. In e§ect, the national bank is a place where extra consumption can be deposited or borrowed at real interest rate r. The perfectly competitive price of investment (or capital) good j at time t is P j (t). This means that at time t any representative agent can buy or sell any amount of capital Kj (t) (or net investment Ij (t)) at the given competitive price P j (t). The Örst condition of the maximum principle (6) signiÖes that at all times representative agents are acting to maximize NDP, taking prices as given. Envisioning the optimal control problem here as a movie unfolding over time, condition (6) represents a ìstill frameî photographed at instant t. It is a static description, indicating that at time t the economy is on its attainable possibilities frontier at a point where the marginal rates of transformation among current consumption and current investments, or among various forms of current investment, is exactly the current competitive prices of investments P (t).