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Unformatted text preview: Consumption-Based Asset Pricing Craig Burnside Economics 702 University of Virginia 1 A Simple Asset Pricing Model & Based on Lucas (1978), Econometrica & There are M identical households with preferences ∞ X t =0 β t u ( c t ) ( 1 ) & There are N assets, indexed by i , each of which at time t pays a stochastic endowment (per unit owned) given by d it . A household&s beginning of period holdings of the asset are given by s it . The time t price of the asset is given by p it . & Normalize the M = 1, and assume that the net supply of each asset equals 1 (this just makes for clean notation and is not of any real consequence). & The single consumption good is the numeraire and is perishable. This implies that storage is not an option for the household. & Assume that households can borrow and lend from each other, and that they trade riskless claims to units of consumption in the future. As an example we will imagine that the household has b t claims to consumption at time t , at the beginning of time t . The price of a claim to a unit of consumption at time t + 1 is q t . Of course, since all households are identical and there are no other agents in the model, the equilibrium value of b t must be zero in all periods. & Given this description the household&s budget constraint is N X i =1 ( p it + d it ) s it + b t ≥ c t + N X i =1 p it s it +1 + q t b t +1 . (2) & Since the d it are stochastic processes, the household maximizes the time-0 expected value of its utility, (1), subject to the sequence of budget constraints, (2). 1 & The & rst-order conditions, where λ t is the Lagrange multiplier on the time t constraint are c t : β t u ( c t ) = λ t s it +1 : p it λ t = E t ( p it +1 + d it +1 ) λ t +1 b t +1 : q t λ t = E t λ t +1 . & In equilibrium, s it = 1 for all i and t , since we have assumed there is one unit of each asset in net supply and have normalized the number of households to 1. Also b t = 0 for all t , so that c t = P N i =1 d it . & Substituting out λ t from the & rst-order conditions we can rewrite them as p it u ( c t ) = E t β u ( c t +1 )( p it +1 + d it +1 ) q t u ( c t ) = β E t u ( c t +1 ) or p it = E t m t,t +1 ( p it +1 + d it +1 ) ( 3 ) q t = E t m t,t +1 (4) where m t,t +1 ≡ β u ( c t +1 ) /u ( c t ) is the marginal rate of subsitution between consumption at date t and date t + 1. & Notice that both (3) and (4) illustrate a basic principle of consumption-based asset pricing theory: that the price of any asset is the conditional expectation of the product of the marginal rate of substitution and the payo f to holding that asset between t and t +1. If we generically denoted the price of an asset as p t , and the payo f as x t +1 , then p t = E t ( m t,t +1 x t,t +1 ). In the case of the assets paying stochastic dividends the payo f to holding the asset is the dividend plus the right to future dividends. In the case of the riskless asset, each unit is a claim to one unit of consumption, so the payo f is 1....
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This note was uploaded on 08/27/2011 for the course ECON 101 taught by Professor Hal during the Spring '11 term at Ewha Womans University.
- Spring '11