1 Riskaversion and intertemporal substitution under external habits.

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1 Riskaversion and intertemporal substitution under external habits. Reall that e de ne the oe ient of relative riskaversion as R r u00 ( ) u 0 ( ) so for CRRA utility e have u () R r t ; hih determines the prie of risk. Reall that e an use the Euler equation to derive the onsumption CAPM, external habit) ov dut+ d+ ; ~r t+ E t ~r t+ r t+ du E t+ t d+ ov t+ r ; ~r t+ t+ E t t+ r t+ + ov t+ ; ~r t+ E t t+ : A high, leads to a high risk premium. At the same time, the high, leads to a lo intertemporal elastiity of substitution. Consider the ase of no unertainty and a onstant interest rate. The Euler equation is t ( + r) t+ t ( + r) ( + g) t ( + r) ( + g) + g (( + r) ) here g is the groth rate of onsumption. Clearly, a high value of R r ; implies a lo sensitivity of groth rates to interest rates or onversely, that interest rates hange a lot as the groth of onsumption hanges. With standard preferenes, the same parameters determines both riskaversion and (the inverse) of the intertemporal elastiity of

2 onsumption. Let s de ne the intertemporal elastiity of substitution (IES) as I dg ( + r) dr ( + g) d (( + r) ) A + r dr + g d (( + r) ) A + r dr (( + r) ) : Clearly, I R r : Let us see ho habits hange this. No, suppose there is a habit ith the standard property that it a ets the marginal utility positively and utility negatively. Consider the additive habit, u ( ; t ) ( ( t) if 6 and > 0 ln ( k t ) if : () Then, e have u 00 ( t ) u 0 ( t ) d 2 ( t ) d 2 t t (t d t ) t d This means that hen onsumption is lose to the habit, riskaversion is very high. The inverse of the seond fator, namely s t is often alled the surplus ratio. No, let s look at the intertemporal elastiity of onsumption under the additive external habit, so that Then, t t k : du t d ( t ) ( k ) 2

3 No, e "guess" the solution is a onstant groth rate g: Then, the Euler equation is ( k ) ( + r) (+ k ) k + g ( + r) (( + g) k ) k t ( + r) ( + g) + g ( + r) ( + g) + g (( + r) ) k t + g Note that this is the same as under no habit formation at all, i.e., the intertemporal elastiity of substitution is regardless of k: On the other hand, risk aversion an be arbitrarily high in this ase. We an alulate it expliitly, ( + g) t ( + g) k + g + g k ; implying that I R r + g + g k :. Externalities When the habit is external, there an externality. Consider no the referene utility model, t kc t ; (2) here C t is aggregate onsumption. Then, ( kc t ) ( + r) (+ kc t+ ) : Suppose an individual hooses loer and higher + : Then, she ontributes to reduing the marginal utility of onsumption for everyone else in t; and inreasing it at t + : Everyone else should then respond by also reduing period t onsumption and inreasing it in period t + : Could this generate multiple equilibria? In priniple, yes, but not ith these preferenes. Again, guess the solution is a onstant groth rate, ( kc t ) ( + r) (( + g) k (( + g)) C t ) ( + r) ( + g) + g (( + r) ) This is a unique solution, oiniding ith the no habit ase. 3

4 To analyze this a bit further in a simple ay. Suppose there is to periods only, that feliity is given by () and the referene level by (2). Individuals then solve max fu ( ; ) + u (( + r) ( ) ; 2 )g here is their initial endoment. The rst order ondition is ( v ) ( + r) (( + r) ( ) v 2 ) 0: To onsider a simple ase, set :We then get the folloing system of equations or implying ( v ) ( + r) (( + r) ( ) v 2 ) 0 k 2 k ( + r) ( ) ( k ) ( + r) (( + r) ( ) k ( + r) ( )) 0 + ; regardless of k: Clearly, the deentralized alloation under habits oinides ith the one under no habits. Is this an e ient alloation? Consider the planning problem max fu ( ; k ) + u (( + r) ( ) ; k ( + r) ( ))g ( ) ( ( k)) max + (( + r) ( ) ( k))! ( k) max + (( + r) ( )) : From this e onlude that the planner also ants the alloation implied by the deentralized equilibrium. i.e., ( + ). Why is that? To shed some more light. Consider the ase of "onspiuous onsumption". Suppose that referene utility arise on goods onsumption but not in leisure l: No, e just need one period to illustrate the point. Individuals solve max u (; ; l) s:t: ( l) Suppose ( v) u (; ; l) + l 4

5 Then, the private rst order ondition is ( v) Again, onsidering the log ase, this yields 0: + v + + k +! + ( k) hih inreases in k; and more so, the higher is : The planning solution is max ith rst order ondition ( k) ( k) + ; hih in the log ase, yields + : Comparing the deentralized outome ith the optimal, e see that onsumption in the deentralized ase is too high by a term ( + ( k)) ( + ) k: Introduing a tax on labor ould orret the externality. Clearly, our examples do extend to dynami ases. What ould it apply for asset aumulation? Could there be more reasonable asymmetries than leisure/onsumption? Returning nally to the issue of multipliity. Let s the solution to the private rst order ondition ith referene utility as (C) : Clearly, a private equilibrium is a xed point of : Therefore a neessary ondition for multipliity is that 0 (C) in some range. Let s alulate it in the additive ase. Here, (C) is de ned by the solution to ( kc) 0 5! (C) + kce ln + e ln

6 ith 0 (C) k + < : It is, hoever, in priniple possible to have multipliity. Consider the folloing extreme preferenes xa ( u (; C; l) x b ( C) + l if C C) + l if < C The marginal utility of leisure is onstant at unity hile the marginal of onsumption is x b if individual onsumption is belo the average C and x a if it is above. No, suppose x b > > x a : Then, the private optimum is learly C; and there is an in nite number of private equilibria, hile the only e ient one is 0: 6

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