olg - 1 1.1 Notes on the OLG Model Introduction The...

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1N o t e s o n t h e O L G M o d e l 1.1 Introduction The overlapping generations (OLG) model, introduced by Sameulson (1958), is a dynamic economic model with many interesting properties. It contains agents who are born at di®erent dates and have ¯nite lifetimes, even though the economy goes on forever. This induces a natural heterogeneity across individuals at a point in time, as well as nontrivial life-cycle considerations for a given individual across time. These features of the model can also generate di®erences from models where there is a ¯nite set of time periods and agents, or from models where there is an in¯nite number of time periods but agents live forever. In particular, competitive equilibria in the OLG model may not to be Pareto optimal. A closely related feature of the model is that it has a role for ¯at money. This means we can use OLG models to address a variety of substantive issues in monetary economics. 1.2 The Basic Model Suppose that t =1 ; 2 ;::: , and that at every date t there is born a new generation G t of individuals who live for two periods. More realistic (longer) lifetimes can be studied, but two periods is the simplest case where the generations overlap. There is also a generation G 0 around at t =1who only live for one period, called the \initial old." For now, every generation consists of a [0 ; 1] continuum of homogeneous agents. Let c t 1 and c t 2 denote consumption of an individual from G t , t ¸ 1, in the 1st and 2nd periods of life, and let e 1 and e 2 denote his (time-invariant) endowments in the 1st and 2nd periods of life. His utility function u ( c t 1 ;c t 2 ) is strictly increasing and quasi-concave. Members of generation G 0 consume only c 02 and are endowed 1
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with only e 2 . One can de¯ne a Walrasian Competitive Equilibrium (WCE) for this econ- omy as follows. Let p t be the price of a unit of the consumption good at date t . Now clearly every member of generation G 0 simply consumes his endow- ment, c 02 = e 2 . For all t ¸ 1, every member of G t maximizes u ( c t 1 ;c t 2 ) subject to p t c t 1 + p t +1 c t 2 = p t e 1 + p t +1 e 2 (1) and c tj ¸ 0. 1 We always write budget constraints with strict equality because u is strictly increasing. Then a WCE is a sequence of prices and allocations f p t t 1 t 2 g such that: c 02 = e 2 ;g iven f p t g ,( c t 1 t 2 )so lvesthemax im izat ion problem of G t for all t ¸ 1; and markets clear in the sense that for all t c t 1 + c t ¡ 1 ; 2 = e 1 + e 2 : (2) One can also de¯ne a Recursive Competitive Equilibrium (RCE) as follows. Let s t denote savings or loans by a member of G t at t ,and R t the gross (principal plus interest) return on savings between t and t + 1. Then for all t ¸ 1, every member of G t maximizes u ( c t 1 t 2 )subjectto c t 1 = e 1 ¡ s t (3) c t 2 = e 2 + R t s t (4) and c tj ¸ 0. 2 ARCEisasequence f R t t 1 t 2 g such that: c 02 = e 2 1 We could write (1) more generally as P j p j c tj = P j p j e tj ,where c tj is the consump- tion and e tj is the endowment in period j of an agent born at t , but it is obvious from the speci¯cation of preferences and endowments that c tj and e
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olg - 1 1.1 Notes on the OLG Model Introduction The...

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