Solutions_to_PS6_Fall2011-1

# Solutions_to_PS6_Fall2011-1 - Solutions to PS6 Econ 202A...

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Unformatted text preview: Solutions to PS6 Econ 202A - Second Half Fall 2011 Prof. David Romer GSI: Victoria Vanasco 1 Romer 8.13. Habit formation and serial correlation in con- sumption growth Utility of the representative consumer is given by: U = T summationdisplay t =1 parenleftBigg 1 1 + ρ parenrightBigg t 1 1 − θ parenleftbigg c it z it parenrightbigg 1 − θ 1.1 External Habits Suppose Z it = C φ t − 1 , where φ ∈ [0 , 1] , so the reference point is aggregate consumption in the previous period, which an individual takes as given. i) Euler Equation. Using the perturbation approach, we know that: U ′ ( c it ) dc = U ′ ( c it +1 ) (1 + r ) dc parenleftbigg c it z it parenrightbigg − θ 1 z it = 1 1 + ρ (1 + r ) parenleftBigg c it +1 z it +1 parenrightBigg − θ 1 z it +1 parenleftbigg c it z it parenrightbigg − θ z it +1 z it = 1 1 + ρ (1 + r ) parenleftBigg c it +1 z it +1 parenrightBigg − θ c it +1 c it = parenleftBigg 1 + r 1 + ρ parenrightBigg 1 θ parenleftbigg z it +1 z it parenrightbigg θ- 1 θ c it +1 c it = parenleftBigg 1 + r 1 + ρ parenrightBigg 1 θ parenleftBigg C t C t − 1 parenrightBigg θ- 1 θ 1 ii) Consumption growth in equilibrium. So we know that in equilibrium C t = c it , ∀ t. Plugging this into our Euler equation: C t +1 C t = parenleftBigg 1 + r 1 + ρ parenrightBigg 1 θ parenleftBigg C t C t − 1 parenrightBigg φ θ- 1 θ Taking logs and denoting by Δ c t +1 = ln ( C t +1 /C t ) Δ c t +1 = 1 θ [ln (1 + r ) − ln (1 + ρ )] + θ − 1 θ φ Δ c t Using the following approximation: ln (1 + r ) ≃ r : Δ c t +1 = 1 θ ( r − ρ ) + θ − 1 θ φ Δ c t When θ = 1 (and in this case the utility function becomes u ( c ) = ln ( c )) we get: Δ c t +1 = r − ρ So habit formation does not a ect the behavior of consumption. In this model θ re ects the elasticity of intertemporal substitution. With log utility the marginal utility of consumption is not a ected by the habit, and we're back in the model with no habit formation. To see this, note that from Euler we would get: parenleftbigg c it z it parenrightbigg − 1 1 z it = 1 1 + ρ (1 + r ) parenleftBigg c it +1 z it +1 parenrightBigg − 1 1 z it +1 1 c it = 1 1 + ρ (1 + r ) 1 c it +1 Which is the same Euler equation we would get with log utility and no habit formation, habit does not a ect my marginal utility and thus does not a ect my allocation of consumption between today and tomorrow. When θ > 1 , the elasticity of intertemporal substitution is such that habit does impact my marginal utility. In particular, by decreasing my consumption today I decreases my marginal utility 2 of consumption tomorrow through the habit term, and thus habit would impact my decisions. Note that in this case consumers do not internalize their impact on the habit....
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## This note was uploaded on 02/28/2012 for the course ECON 202A taught by Professor Akerlof during the Fall '07 term at Berkeley.

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Solutions_to_PS6_Fall2011-1 - Solutions to PS6 Econ 202A...

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