01-savings-problem - Viktor Tsyrennikov Graduate...

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Unformatted text preview: Viktor Tsyrennikov Graduate Macroeconomics Savings Problem In this part we study the building block of macroeconomics the savings problem. Our subject is an economic agent a household or an individual. It receives variable income and decides how much to spend and how much to save for the future. Our aim is to show how to model individual decisions and how individual and aggregate (that is economy-wide) decisions are related. 1 Preferences Preference specification assigns a quantitative measure of agents satisfaction to a combination of goods that he or she consumes. 1.1 Time and discounting We assume that economies exist indefinitely. We also assume that time is discrete. That is agents in our economy are able to make decisions only at predetermined dates indexed by t N { , 1 , 2 , ... } . For example, a household decides how much to work, save and spend on the first day of every month (and cannot change its decision within a month). Crucially, we also assume that agents care about future consumption less than about present consumption. One convenient way to represent this fact formally is with time-separable life-time utility function: 12 U ( c ) = summationdisplay t =0 t u ( c t ) = u ( c ) + u ( c 1 ) + 2 u ( c 2 ) + ..., (1) 1 It is time separable because utility from consumption in period t does not depend on consumption in other periods. 2 A convention adopted here is that an agent represents a dynasty rather than a single finitely lived household. That is household decision maker is altruistic towards his off- springs. It can also be shown that preferences of a finitely-lived household with bequest motives can be represented by (1). 1 Viktor Tsyrennikov Graduate Macroeconomics where [0 , 1) is a discount factor , 3 u is a strictly increasing and a strictly concave ( i.e. u > , u < 0) one-period utility function and c = ( c 1 , c 2 , ... ) is a consumption stream. Assumption 1. a) u ( c ) > , u ( c ) < , c ; b) life-time utility function is time-separable with [0 , 1) ; c) utility function satisfies the Inada condition at zero: lim c du ( c ) /dc = . Assumption 1.a insures that the utility function conforms with standard axioms of individual behavior. It requires that each additional unit of con- sumption brings a strictly positive but a decreasing amount of utility. As- sumption 1.b is needed for tractability. Assumption 1.c insures that con- sumption in every period is strictly positive. So, we can ignore the require- ment that c t greaterorequalslant 0 in our analysis. A common choice for u in macroeconomics is the constant relative risk aversion (CRRA) utility function: u ( c ) = c 1 1 1 , [0 , ) . (2) This specification includes logarithmic preferences as a special case (can you prove this?): lim 1 c 1 1 1 = ln( c ) , c (0 , ) ....
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This note was uploaded on 11/10/2011 for the course ECONOMICS 601 taught by Professor Viktortsyrennikov during the Spring '11 term at Cornell University (Engineering School).

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01-savings-problem - Viktor Tsyrennikov Graduate...

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