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Unformatted text preview: Chapter 3 Dynamic optimization and utility functions This chapter offers a brief account of one of the leading approaches to solving dynamic optimization problems and how the so called Euler equation can be derived. We will apply the Lagrange multiplier method (applying the KuhnTucker Theorem) and show how the Euler equation is derived within both OLG and Ramsey models. In addition, we briefly discuss some specific utility functions that we will use in the next chapter when studying consumption theory. For a more detailed discussion about dynamic optimization and other methods to solving optimization problems the reader should consult for example Dixit (1990), Sydsaeter och Hammond (1995) or Obstfeld and Rogoff (1996). 3.1 The Euler equation within a 2period model Let us consider a 2 period model with one consumer good. Assume also that the utility function is intertemporally additive, i.e., the rate of substitution between consumption on any two dates is independent of consumption on any third date. Thus, we rule out any intertemporal consumption dependencies such as habit formation. The individual maximizes the utility of consumption over the two dates given labor income: U = U ( C 1 ) + U ( C 2 ) < < 1 (3.1) where c is consumption and = 1 / (1 + ) is a measure of the individuals impatience to consume, where is the the subjective rate of time preference. The utility function U ( C ) is increasing in consumption and strictly concave: U ( C ) > 0, U 00 ( C ) < 0. In addition we assume that lim C U ( C ) = which implies that the individual always desire at least a little consumption in every period, i.e., consumption is always positive so we dont have to add the constraint that C i 0 to the maximization problem. The individual maximizes the utility in (3.1) with respect to the budget constraints C 1 = Y 1- B 1 (3.2) 28 Dynamic optimization and utility functions and C 2 = Y 2 + (1 + r ) B 1 (3.3) where Y is labor income and B is the value of net assets at the end of period t (savings). Use equation (3.2) to solve for B 1 and insert into (3.3) such that the budget restriction can be written as C 1 + C 2 1 + r = Y 1 + Y 2 1 + r which states that the present value of consumption is equal to the present value of labor income. Note that we rule out inheritance. To find a solution to the optimization problem, i.e., determine C 1 and C 2 , we form the Lagrangian L = U ( C 1 ) + U ( C 2 )- C 1 + C 2 1 + r- Y 1- Y 2 1 + r ....
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