Lecture 6: Recursive Preferences Simon Gilchrist Boston Univerity - PowerPoint PPT Presentation

Lecture 6: Recursive Preferences Simon Gilchrist Boston Univerity and NBER EC 745 Fall, 2013

Basics Epstein and Zin (1989 JPE, 1991 Ecta) following work by Kreps and Porteus introduced a class of preferences which allow to break the link between risk aversion and intertemporal substitution. These preferences have proved very useful in applied work in asset pricing, portfolio choice, and are becoming more prevalent in macroeconomics.

Value function: To understand the formulation, recall the standard expected utility time-separable preferences are defined as � ∞ β s − t u ( c t + s ) , V t = E t s =0 We can also define them recursively as V t = u ( c t ) + βE t V t +1 , or equivalently: V t = (1 − β ) u ( c t ) + βE t ( V t +1 ) .

EZ Preferences EZ preferences generalize this: they are defined recursively over current (known) consumption and a certainty equivalent R t ( V t +1 ) of tomorrow’s utility V t +1 : V t = F ( c t , R t ( V t +1 )) , where R t ( V t +1 ) = G − 1 ( E t G ( V t +1 )) , with F and G increasing and concave, and F is homogeneous of degree one. Note that R t ( V t +1 ) = V t +1 if there is no uncertainty on V t +1 . The more concave G is, and the more uncertain V t +1 is, the lower is R t ( V t +1 ) .

Functional forms Most of the literature considers simple functional forms for F and G : � (1 − β ) c 1 − ρ + βz 1 − ρ � 1 1 − ρ , ρ > 0 : F ( c, z ) = 0 : G ( x ) = x 1 − α α > 1 − α. For example: � � 1 � � 1 − ρ 1 − ρ (1 − β ) c 1 − ρ E t V 1 − α V t = + β 1 − α . t t +1

Limits Limits: 1 : F ( c, z ) = c 1 − β z β . ρ = α = 1 : G ( x ) = log x. Hence � � 1 1 − α , V 1 − α α > 0 : R t ( V t +1 ) = E t t +1 α = 1 : R t ( V t +1 ) = exp ( E t log ( V t +1 )) .

Proof Define f ( x ) F ( c, z ) = cf ( x ) where x = z/c and � 1 − β + βx 1 − ρ � 1 f ( x ) = 1 − ρ So f ′ ( x ) βx − ρ f ( x ) = 1 − β + βx 1 − ρ and f ′ ( x ) f ( x ) = β lim X . ρ → 1 Since f continuous this implies ρ → 1 f ( x ) = X β lim (note this is simply the proof that a CES function converges to a Cobb-Douglas as ρ → 1 ).

Risk Aversion vs IES In general α is the relative risk aversion coefficient for static gambles and ρ is the inverse of the intertemporal elasticity of substitution for deterministic variations. Suppose consumption today is c today and consumption tomorrow is uncertain: { c L , c, c, .... } or { c H , c, c, .... } , each has prob 1 2 . Utility today: � � 1 �� 2 G ( V L ) + 1 c, G − 1 V = F 2 G ( V H ) where V L = F ( c L , c ) and V H = F ( c H , c ) . Curvature of G determines how adverse you are to the uncertainty. If G is linear you only care about the expected value. If not, this is the same as the definition of a certainty equivalent: G ( � V ) = 1 2 G ( V L ) + 1 2 G ( V H ) .

Special Case: Deterministic consumption If consumption is deterministic: we have the usual standard time-separable expected discounted utility with discount factor β and IES = 1 ρ , risk aversion α = ρ. Proof: If no uncertainty, then R t ( V t +1 ) = V t +1 and V t = F ( c t , V t +1 ) . With a CES functional form for F, we recover CRRA preferences: � � 1 (1 − β ) c 1 − ρ + βV 1 − ρ 1 − ρ V t = t t +1 � ∞ W t = (1 − β ) c 1 − ρ β j c 1 − ρ + βW t +1 = (1 − β ) t + j , t j =0 where W t = V 1 − ρ . t

Special Case: α = ρ Similarly, if α = ρ, then the formula � � 1 � � 1 − ρ 1 − ρ (1 − β ) c 1 − ρ E t V 1 − α V t = + β 1 − α t t +1 simplifies to � � V 1 − ρ = (1 − β ) c 1 − ρ E t V 1 − α + β t t t +1 Define W t = V 1 − ρ , we have t W t = (1 − β ) c 1 − ρ + βE t ( W t +1 ) , t i.e. expected utility.

Simple example with two lotteries: Lotteries: lottery A pays in each period t = 1 , 2 , ... c h or c l , the probability is 1 2 and the outcome is iid across period; lottery B pays starting at t = 1 either c h at all future dates for sure, or c l at all future date for sure; there is a single draw at time t = 1 . With expected utility, you are indifferent between these lotteries, but with EZ lottery B is prefered iff α > ρ. In general, early resolution of uncertainty is preferred if and only 1 if α > ρ i.e. risk aversion > IES . This is another way to motivate these preferences, since early resolution seems intuitively preferable.

Resolution of uncertainty For lottery A, the utility once you know your consumption is either c h , or c l , since � � 1 1 − ρ . (1 − β ) c 1 − ρ + βV 1 − ρ V h = F ( c h , V h ) = h h The certainty equivalent before playing the lottery is � 1 � � 1 � 1 2 G ( c h ) + 1 + 1 1 − α G − 1 2 c 1 − α 2 c 1 − α 2 G ( c l ) = . h l For lottery B, the values satisfy � 1 � 1 − ρ + 1 1 − α W 1 − ρ = (1 − β ) c 1 − ρ 2 W 1 − α 2 W 1 − α + β , h h h l � 1 � 1 − ρ + 1 1 − α W 1 − ρ = (1 − β ) c 1 − ρ 2 W 1 − α 2 W 1 − α + β , l l h l

Resolution of uncertainty We want to compare G − 1 � 1 � 2 G ( W h ) + 1 2 G ( W l ) to G − 1 � 1 � 2 G ( c h ) + 1 2 G ( c l ) . 1 − ρ 1 − α is concave if 1 − ρ < 1 − α , Note that the function x → x i.e. ρ > α, and convex otherwise. As a result, if ρ > α, � 1 � 1 − ρ � � 1 − ρ � � 1 − ρ + 1 1 1 − α + 1 1 − α 2 W 1 − α 2 W 1 − α W 1 − α W 1 − α ≥ 1 − α h l h l 2 2 1 + 1 2 W 1 − ρ 2 W 1 − ρ = h l Also � 1 � + 1 W 1 − ρ (1 − β ) c 1 − ρ 2 W 1 − ρ 2 W 1 − ρ ≥ + β h h h l � 1 � + 1 W 1 − ρ (1 − β ) c 1 − ρ 2 W 1 − ρ 2 W 1 − ρ ≥ + β l l h l

Continued These results imply that if ρ > α then W 1 − ρ + W 1 − ρ ≥ c 1 − ρ + c 1 − ρ h l h l . 2 2 in which case the certainty equivalent of lottery A is higher than the certainty equivalent of lottery B and agents prefer late to early resolution of uncertainty. Technically, EZ is an extension of EU which relaxes the independence axiom. Recall the independence axion is: if x � y, then for any z , α : αx + (1 − α ) z � αy + (1 − α ) z. With EZ, “Intertemporal composition of risk matters”: we cannot reduce compound lotteries.

Euler’s Thereom: We have � + βR t ( V t +1 ) 1 − ρ � 1 (1 − β ) C 1 − ρ 1 − ρ V t = t where � � �� 1 V 1 − α R t ( V t +1 ) = E t 1 − α t +1 Since V t is homogenous of degree one, Euler’s thereom implies V t = MC t C t + E t MV t +1 V t +1

Euler equation: Taking derivatives: MC t = ∂V t = (1 − β ) V ρ t C − ρ t ∂C t and ∂V t ∂R t ( V t +1 ) MV t +1 = ∂R t ( V t +1 ) ∂V t +1 where ∂V t ∂R t ( V t +1 ) = V ρ t βR t ( V t +1 ) − ρ and ∂R t ( V t +1 ) = R t ( V t +1 ) α V − α t +1 ∂V t +1 This implies t R t ( V t +1 ) α − ρ V − α MV t +1 = βV ρ t +1

IES Define the intertemporal marginal rate of substitution as MV t +1 MC t +1 S t,t +1 = MC t � C t +1 � − ρ � � ρ − α V t +1 = β C t R t ( V t +1 ) The first term is familiar. The second term is next period’s value relative to its certainty equivalent. If ρ = α or there is no uncertainty so that V t +1 = R t ( V t +1 ) this term equals unity.

Household wealth: Start with the value function: V t = MC t C t + E t MV t +1 V t +1 Divide by MC t : � MV t +1 MC t +1 � V t V t +1 = C t + E t MC t MC t MC t +1 Define V t W t = MC t then W t = C t + E t S t,t +1 W t +1 is the present-discounted value of wealth.

The return on wealth Define the cum-dividend return on wealth: W t +1 R m,t +1 = W t − C t Note that = V 1 − ρ t +1 C ρ V t +1 t +1 W t +1 = MC t +1 1 − β Hence � ρ � � � C t +1 V 1 − ρ t +1 C ρ V 1 − ρ t +1 t +1 R m,t +1 = = V 1 − ρ C ρ V 1 − ρ − (1 − β ) C 1 − ρ C t t − C t t t t Now use fact that V 1 − ρ = (1 − β ) C 1 − ρ + βR t ( V t +1 ) 1 − ρ t t to obtain � � 1 − ρ � − 1 � C t +1 � − ρ � R t ( V t +1 ) R m,t +1 = β C t V t +1

Certainty Equivalent Use this equation to solve for the value function relative to the certainty equivalent: � � 1 − ρ � � C t +1 � − ρ � R t ( V t +1 ) R − 1 = β m,t +1 C t V t +1 � � − ρ � 1 / (1 − ρ ) � C t +1 V t +1 R t ( V t +1 ) = βR m,t +1 C t Comment: we can use this to directly evaluate the cost of uncertain returns and consumption.

SDF: From above: � C t +1 � − ρ � � ρ − α V t +1 S t,t +1 = β C t R t ( V t +1 ) − θ C t +1 ψ β θ R θ − 1 = m,t +1 C t where 1 − α θ = 1 − ρ and ψ = 1 /ρ Note if ρ = α we have � C t +1 � − ρ S t,t +1 = β C t Now take logs log S t,t +1 = θ log β − θ ψ ∆ c t +1 − (1 − θ ) r m,t +1

Expected returns The return on the ith asset satisfies: E t S t,t +1 R i t,t +1 = 1 Taking logs: � � E t R i t,t +1 − cov (log S t,t +1 , log R i log = t,t +1 ) R f t +1 θ = ψ ( cov (∆ c t +1 , r i,t +1 )) + (1 − θ ) cov ( r m,t +1 , r i,t +1 ) Epstein-Zin is a linear combination of the CAPM and the CCAPM model.

Market return: For the market return we have � ER m � = θ ψcov (∆ c, r m ) + (1 − θ ) σ 2 log m R f We can write as: r m + σ m 2 = r f + θ ψcov (∆ c, r m ) + (1 − θ ) σ 2 m