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# notes2 - The Maximum Principle Peter Ireland EC720.01 Math...

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The Maximum Principle Peter Ireland * EC720.01 - Math for Economists Boston College, Department of Economics Fall 2009 Here, we will explore the connections between two ways of solving dynamic optimization problems, that is, problems that involve optimization over time. The first solution method is just a straightforward application of the Kuhn-Tucker theorem; the second solution method relies on the maximum principle. Although these two approaches might at first glance seem quite different, in fact and as we will see, they are closely related. We’ll being by briefly noting the basic features that set dynamic optimization problems apart from purely static ones. Then we’ll go on consider the connections between the Kuhn- Tucker theorem and the maximum principle in both discrete and continuous time. References: Dixit, Chapter 10. Acemoglu, Chapter 7. 1 Basic Elements of Dynamic Optimization Problems Moving from the static optimization problems that we’ve considered so far to the dynamic optimization problems that are of primary interest here involves only a few minor changes. a) We need to index the variables that enter into the problem by t , in order to keep track of changes in those variables that occur over time. b) We need to distinguish between two types of variables: stock variables - e.g., stock of capital, assets, or wealth flow variables - e.g., output, consumption, saving, or labor supply per unit of time * Copyright c 2009 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. 1

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c) We need to introduce constraints that describe the evolution of stock variables over time: e.g., larger flows of savings or investment today will lead to larger stocks of wealth or capital tomorrow. 2 The Maximum Principle: Discrete Time 2.1 A Dynamic Optimization Problem in Discrete Time Consider a dynamic optimization in discrete time, that is, in which time can be indexed by t = 0 , 1 , ..., T . y t = stock variable z t = flow variable Objective function: T X t =0 β t F ( y t , z t ; t ) Following Dixit, we can allow for a wider range of possibilities by letting the functions as well as the variables depend on the time index t . 1 β > 0 = discount factor Constraint describing the evolution of the stock variable: Q ( y t , z t ; t ) y t +1 - y t or y t + Q ( y t , z t ; t ) y t +1 for all t = 0 , 1 , ..., T Constraint applying to variables within each period: c G ( y t , z t ; t ) for all t = 0 , 1 , ..., T Constraints on initial and terminal values of stock: y 0 given y T +1 y * The dynamic optimization problem can now be stated as: choose sequences { z t } T t =0 and { y t } T +1 t =1 to maximize the objective function subject to all of the constraints. Notes: 2
a) It is important for the application of the maximum principle that the problem be additively time separable: that is, the values of F , Q , and G at time t must depend on the values of y t and z t only at time t .

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notes2 - The Maximum Principle Peter Ireland EC720.01 Math...

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