huggett-aiyagari - General Equilibrium with Incomplete...

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General Equilibrium with Incomplete Markets Jonathan Heathcote Updated April 2006 1. Heterogeneity Consider an economy with a continuum of agents of total mass equal to 1 . Each agent’s productivity process is independent of all other agents current and past endowments and de f ned by a f rst order Markov process. How should we describe the equilibrium distribution of households across asset holdings and endowments? We should use a probability measure. Let a max denote the endogenous maxmum asset holdings in equilibrium. (In the iid shock case described above a max = a 0 ( z max ) ). Let ψ be a probability measure de f ned on ( S, β S ) where S =[ φ, a max ] × E and β S is the Borel σ algebra (an appropriate set of subsets of S ). Thus for any set B β S ( B ) is the mass of agents whose individual state vectors lie in B. Let P (( a, e ) ,B ) be the probability than an agent with state ( a, e ) today has in individual state in the set B β S tomorrow. Formally the transition function P : S × β S [0 , 1] is given by P (( a, e ) ,B )= X e 0 I ( a 0 ( a,e ) ,e 0 ) B × π ( e 0 | e ) Since we are in an economy with constant prices we would hope that ψ will be unchanged over time. A probability measure ψ is stationary provided that ψ 0 ( B )= Z P ((( a, e ) ,B ) = ψ ( B ) for all B β S Let x =( a, e ) be the individual state vector. Let q be the price of a non-contingent bond that pays one unit of consumption in the next period. Given a constant q, the individual’s problem, in recursive form is characterized by the following functional equation: v ( x ; q )= m a x { c,a 0 Γ ( x ; q ) } ½ u ( c )+ β P e 0 π ( e 0 | e ) v ( a 0 ,e 0 ; q ) ¾ Γ ( x ; q )= { ( c, a 0 ): c + qa 0 a + e ; c 0; a 0 ≥− φ } Thanks to the Principle of Optimality, a solution to this functional equation (as-
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General Equilibrium with Incomplete Markets 2 1.1. Equilibrium De f nition. A stationary equilibrium is a c ( x ) ,a 0 ( x ) ,q and ψ satisfying 1. c ( x ) and a 0 ( x ) are optimal given q 2. Markets clear. In Huggett’s (1993) economy this means R S c ( x ) = R S e ( x ) R S a 0 ( x ) =0 3. ψ is a stationary probability measure In Aiyagari’s 1994 and 1995 papers and Huggett’s 1997 paper the aggregate supply of assets is endogenous and derived from a production technology. This is an impor- tant di f erence with respect to Huggett 1993 paper in which assets are assumed to be in zero net supply. Aiyagari and McGrattan 1998 allow two sources of positive asset supply: capital used in production and government debt. These alternative models change the market clearing conditions, but do not a f ect the form of the individual consumption-savings problem. 1.2.
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huggett-aiyagari - General Equilibrium with Incomplete...

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