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 + parenrightBigg t 1 1 parenleftbigg c it z it parenrightbigg 1 1.1 External Habits Suppose Z it = C t 1 , where [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 1 z it = 1 1 + (1 + r ) parenleftBigg c it +1 z it +1 parenrightBigg 1 z it +1 parenleftbigg c it z it parenrightbigg z it +1 z it = 1 1 + (1 + r ) parenleftBigg c it +1 z it +1 parenrightBigg c it +1 c it = parenleftBigg 1 + r 1 + parenrightBigg 1 parenleftbigg z it +1 z it parenrightbigg - 1 c it +1 c it = parenleftBigg 1 + r 1 + parenrightBigg 1 parenleftBigg C t C t 1 parenrightBigg - 1 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 + parenrightBigg 1 parenleftBigg C t C t 1 parenrightBigg - 1 Taking logs and denoting by c t +1 = ln ( C t +1 /C t ) c t +1 = 1 [ln (1 + r ) ln (1 + )] + 1 c t Using the following approximation: ln (1 + r ) r : c t +1 = 1 ( r ) + 1 c t When = 1 (and in this case the utility function becomes u ( c ) = ln ( c )) we get: c t +1 = r So habit formation does not a ect the behavior of consumption. In this model 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 + (1 + r ) parenleftBigg c it +1 z it +1 parenrightBigg 1 1 z it +1 1 c it = 1 1 + (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 > 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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Solutions_to_PS6_Fall2011-1 - Solutions to PS6 Econ 202A -...

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