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cooperogree - Rational Expectations Equilibria in the OG...

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Rational Expectations Equilibria in the OG Model with Production Russell Cooper October 25, 2006 1 Stochastic Money Transfers Consider the overlapping generations model presented in class. The key aspects developed in other notes are: agents work in youth and consume in old age agents are endowed with a unit of time in youth and a technology which converts labor input into output generation t preferences are: u ( c t +1 ) - g ( n t ) where u ( · ) is strictly increasing and strictly concave and g ( · ) is strictly increasing and strictly convex population is constant and normalized at 1 the per capita money supply is a constant ¯ M time is infinite: t = 1 , 2 , ..... there is a unique interior monetary steady state at n * which satisfies u ( n * ) = g ( n * ). non-stationary equilibria are solutions to the difference equation: n t +1 u ( n t +1 ) = n t g ( n t ) (1) these are perfect foresight equilibria in that the prices which agents expect to prevail in the future are the actual equilibrium prices. money is neutral in this economy in that the solutions to (1) are indepen- dent of the fixed stock of money Our goal now is to introduce uncertainty through stochastic proportional mone- tary transfers. In this case, the per capita money supply follows M t +1 = M t x t +1 where x t is an iid random variable drawn from a known distribution f ( · ). These money transfers are proportional to money holdings so that generation t agents who work n units of time have a budget constraint of c t +1 = p t nx t +1 p t +1 . 1
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Substituting this into their objective function, the representative generation t agent chooses n to maximize: E ( x t +1 ,p t +1 | s t ) u ( p t nx t +1 p t +1 ) - g ( n ) (2) where s t is a vector of variables that are known in period t and used to forecast the random variables ( x t +1 , p t +1 ). 1 The first order condition is: E ( x t +1 ,p t +1 | s t ) p t x t +1 p t +1 u ( p t nx t +1 p t +1 ) = g ( n ) . (3) The left side is the marginal gain to working more, where p t x t +1 p t +1 is the random return to work and the right side is the marginal cost of additional work. A key issue here is formulating the expectations of future prices. The expec- tation of x t +1 is easy since the distribution f ( · ) is assumed to be known. But the distribution of p t +1 is determined as part of the equilibrium. As we shall see, this is key to the idea of a rational expectations equilibrium. In addition, to individual optimization, we need to formalize market clearing. This condition is given by: M t p t = n t . (4) The left side is the real demand by the old in period t and the right side is the output of generation t workers. This condition for market clearing holds in all periods for all realizations of the money supply.
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