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Unformatted text preview: Stochastic Growth Model Prof. Lutz Hendricks August 10, 2009 L. Hendricks () Stochastic Growth Model August 10, 2009 1 / 35 Stochastic Growth Model We add shocks to the growth model. Recursive methods are needed. The resulting model is used to study business cycles asset pricing L. Hendricks () Stochastic Growth Model August 10, 2009 2 / 35 Planning solution The history is of shocks is θ t . Preferences: ∞ ∑ t = β t ∑ θ t Pr & θ t j θ ¡ u & c ¢ θ t £¡ (1) Technology: for all θ K ¢ θ t , θ £ = F & K ¢ θ t £ , L ¡ + ( 1 & δ ) K ¢ θ t £ & c ¢ θ t £ (2) L. Hendricks () Stochastic Growth Model August 10, 2009 3 / 35 Bellman equation De&ne k = K / L . V ( k , θ ) = max k 2 [ 0, f ( k , θ )+( 1 & δ ) k ] u & f ( k , θ ) + ( 1 & δ ) k & k ¡ + β E ¢ V & k , θ ¡ j θ £ (3) L. Hendricks () Stochastic Growth Model August 10, 2009 4 / 35 Firstorder conditions Verify that A1A5 hold ... Theorems 16 apply. FOC u ( c ) = β EV k & k , θ ¡ Envelope V k ( k , θ ) = u ( c ) [ f k ( k , θ ) + 1 & δ ] Euler u ( c ) = β E ¢ u & c ¡ £ f k & k , θ ¡ + 1 & δ ¤ j θ ¥ (4) Solution: V ( k , θ ) and π ( k , θ ) that "solve" the Bellman equation L. Hendricks () Stochastic Growth Model August 10, 2009 5 / 35 Characterization Now for the bad news ... there really isn&t much one can say about the solution analytically. But see Campbell (1994) for a discussion of a loglinear approximation. L. Hendricks () Stochastic Growth Model August 10, 2009 6 / 35 Competitive equilibrium The model comes in 2 &avors. 1 Complete markets for every history, there exists an asset that pays in that state of the world the implication is complete risk sharing: all idiosyncratic risks are insured aggregate risks remain 2 Incomplete markets some securities are missing there is no representative agent L. Hendricks () Stochastic Growth Model August 10, 2009 7 / 35 Trading arrangements With complete markets, date 1 ArrowDebreu trading is convenient Uncertainty essentially disappears from the model. With incomplete markets, it is easiest to specify the set of securities available at each date. Sequential trading. L. Hendricks () Stochastic Growth Model August 10, 2009 8 / 35 Complete markets  Arrow Debreu trading The environment is standard. The history is of shocks is θ t . Arrow securities pay out 1 unit of the good exactly when state θ t occurs. The point: This looks like a static model without uncertainty. L. Hendricks () Stochastic Growth Model August 10, 2009 9 / 35 Household: Preferences ∞ ∑ t = β t ∑ θ t Pr & θ t j θ ¡ u & c ¢ θ t £¡ (5) L. Hendricks () Stochastic Growth Model August 10, 2009 10 / 35 Household: budget constraint Expenditures in state θ t : x & θ t ¡ = p...
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