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suu - Saving Under Uncertainty Peter Ireland EC720.01 Math...

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Saving Under Uncertainty Peter Ireland EC720.01 - Math for Economists Boston College, Department of Economics Fall 2010 This last example presents a dynamic, stochastic optimization problem that is simple enough to allow a relatively straightforward application of the Kuhn-Tucker theorem. The optimality conditions derived with the help of the Lagrangian and the Kuhn-Tucker theorem can then be compared with those that can be derived with the help of the Bellman equation and dynamic programming. 1 The Problem Consider the simplest possible dynamic, stochastic optimization problem with: Two periods, t = 0 and t = 1 No uncertainty at t = 0 Two possible states at t = 1: Good, or high, state H occurs with probability π Bad, or low, state L occurs with probability 1 π Notation for a consumer’s problem: y 0 = income at t = 0 c 0 = consumption at t = 0 s = savings at t = 0, carried into t = 1 ( s can be negative, that is, the consumer is allowed to borrow) r = interest rate on savings y H 1 = income at t = 1 in the high state y L 1 = income at t = 1 in the low state y H 1 > y L 1 makes H the good state and L the bad state c H 1 = consumption at t = 1 in the high state Copyright c 2010 by Peter Ireland. Redistribution is permitted for educational and research purposes, so long as no changes are made. All copies must be provided free of charge and must include this copyright notice.

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