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Unformatted text preview: Notes on Overlapping Generations Models Russell Cooper September 20, 2006 1 Introductory Comments and Overview These notes are on a series of overlapping generations models. These are dynamic equilibrium models and thus incorporate both dynamic optimization at the individual level and market clearing conditions. The individual optimization aspect of these models was covered in the notes on two period optimization. The lecture on competitive equilibrium in static economies sets the stage for the equilibrium aspect of the models summarized in these notes. 2 Simple OG model: Shell( JPE , 1971) We start with the overlapping generations model from Shell. This is an easy paper to read though be careful in thinking about the structure of markets for the individuals in the economy. 2.1 Assumptions infinite horizon; t = 1 , 2 ,.... Agents live for two periods youth and old age and then they die! One person, normalized, born each period. So that total population is 2. No population growth. Assume price taking behavior Individual born in period t is a member of generation t . Endowments: One unit of chocolate each period. Preferences of generation t : Indifferent between goods over time so that u t ( c t t ,c t t +1 ) = c t t + c t t +1 . (1) 1 With these preferences, consumption today and tomorrow are perfect substitutes. Note the notation in (1): superscripts will refer to the generation and subscripts will refer to the time period. Time starts with period one so we also need to assume that there is an initial old individual endowed with one unit of chocolate as well. This initial old generation meets generation 1 in period 1. We are interested in a competitive equilibrium in which all buyers and sellers take prices as given. An equilibrium has two components: individual optimization and a consistency requirement (market clearing in this case) across the choices of the individuals. We look at these in turn. 2.2 Individual optimization Agents maximize utility subject to a budget constraint taking prices as given . That is, agents do not consider whether their demands are consistent with supply and market clearing: they simply maximize given their budget constraints. We let p t be the price of period t chocolate in terms of period 1 chocolate. So these are present value prices in that the period one commodity is taken as the numeraire. The budget constraint of a (representative) generation t agent is p t c t t + p t +1 c t t +1 = p t + p t +1 . (2) The right side of this constraint is the total value of the endowment of a generation t agent. The left side is the level of spending over the two periods in which the generation t agent obtains utility from consuming, periods t and t + 1....
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