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Unformatted text preview: LECTURE 3: DYNAMIC ROBINSON CRUSOE 1. Introduction In our previous Crusoe economy, Robinson has a technology for converting his time and effort into coconuts. He liked coconuts and he liked leisure (time not spent harvesting coconuts) and he was obliged to trade the two goods off against each other. We used economic theory to analyze his choice and distilled implications for the macroeconomics of labor supply. Now, we will give Robinson a different choice. Suppose that he lives for multiple periods t = 1 , . . ., T , where T is his last period of life. He now has a technology for converting coconuts today into coconuts tomorrow. In other words he can plant a coconut and harvest the produce from the tree in the next period. Robinson likes consuming in each period so his trade off is an intertemporal one, i.e. he must tradeoff coconuts today for coconuts in the future. We will formulate his problem carefully and (ultimately) hope to learn something about the macroeconomics of saving and investment from the endeavor. 2. An economic environment Let t denote the tth time period. Let c t denote Robinson’s consumption in period t and { c t } T t =1 Robinson’s profile of consumption over his life. { c t } T t =1 is just shorthand for { c 1 , c 2 , . . . , c T } . Each c t is nonnegative so { c t } T t =1 is a list or vector of T nonnegative variables. If we are being fancy, we write { c t } T t =1 ∈ R T + , where ∈ means “belongs to” and R T + , is the set of all lists of T nonnegative real numbers. 1 2 LECTURE 3: DYNAMIC ROBINSON CRUSOE 2.1. Preferences. Robinson’s preferences over consumption are given by: U ( { c t } T t =1 ) = u ( c 1 ) + βu ( c 2 ) + . . . + β T − 1 u ( c T ) . We usually write this more compactly as: U ( { c t } T t =1 ) = T summationdisplay t =1 β t − 1 u ( c t ) . where the notation ∑ T t =1 is shorthand for “sum the terms from t = 1 until t = T ”. Here β is called the discount factor. We assume that 0 < β < 1 (which I sometimes write as β ∈ (0 , 1)). The fact that β is less than one captures the fact that Robinson is impatient and other things equal prefers to consume today rather than in the future. The function u : R + → R is Robinson’s one period utility function so that u ( c t ) is the utility Robinson gets from consumption c t in time t . Preferences of the form ∑ T t =1 β t − 1 u ( c t ) are called time additively separable meaning that the utility Robinson gets from consumption in each period is a separate term which we add up to get his overall utility. We make various assumptions on u . To begin with: U1) u is continuously differentiable with derivative u ′ . U2) u is increasing and so u ′ > 0....
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 Fall '09
 Kyrkv
 Economics, Utility, Marginalism, Robinson C., DYNAMIC ROBINSON CRUSOE

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