This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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....
View
Full
Document
This note was uploaded on 02/28/2012 for the course ECON 202A taught by Professor Akerlof during the Fall '07 term at Berkeley.
 Fall '07
 AKERLOF
 Utility

Click to edit the document details