Theory of Inverse Demand: Financial Assets

Transcription

1 Theory o Inverse Demand: Financial Assets Felix Kubler University o Zurich Swiss Finance Institute Larry Selden Columbia University University o Pennsylvania Aril 2, 2011 Xiao Wei Columbia University Abstract While the comarative statics o asset demand have been studied extensively, surrisingly little work has been done on the behavior o equilibrium asset rices and returns in resonse to changes in the sulies o securities. This is desite considerable interest in the equity remium and interest rate uzzles. In this aer, we seek to ill this void or the classic case o a reresentative agent economy with a single risky asset and risk ree asset in both one and two eriod settings. It would seem natural to suose that in resonse to an increase in the suly o the risky asset, its rice would all and the gross equity risk remium would increase. We show that in standard settings where reerences are reresented by requently assumed orms o exected utility, one can obtain the oosite result. The necessary and sui cient condition or rices (gross equity remium) to increase (decrease) with suly is determined by the sign o the sloe o the asset ngel curve. This observation allows us to derive (i) sui cient conditions directly in terms o the reresentative agent s risk aversion roerties or general utility unctions and (ii) necessary and sui cient conditions or the widely used HARA (hyerbolic absolute risk aversion) class. 1

2 1 Introduction In this aer, we examine the behavior o equilibrium asset rices and returns in resonse to changes in the sulies o securities or the classic case o a single risky asset and risk ree asset in both one and two eriod settings. We show that in standard single eriod and two eriod reresentative agent exchange economies where reerences are reresented by requently assumed orms o utility, it might very well be the case that an increase in the suly o the risky asset leads to a decrease o the gross equity remium. Assuming the reresentative agent s reerences satisy the aroriate exected utility axioms, we show that in a single eriod exchange economy there is a close linkage between the demand and equilibrium rice comarative statics. The only related work o which we are aware, addresses the relationshi between demand and equilibrium rice comarative statics in a certainty ramework. Nachbar 19] shows that a necessary condition or the equilibrium rice o a good to increase with suly is the normal-inerior good behavior o demand (see also Nachbar 19] and Quah 24]). 1 It is natural to wonder i similar results hold in the uncertain inancial asset setting, given the recent indings o Kubler, Selden and Wei 14], that the risk ree asset can be an inerior good i either short selling o the risk ree asset is allowed or relative risk aversion is decreasing. We identiy the necessary and sui cient conditions or when asset rices (gross equity remium) increase (decreases) with asset sulies. The single eriod case turns out to be quite secial in that there is a ull equivalence between the comarative statics o demand and the equilibrium rice ratio. The necessary and sui cient condition or rices (gross equity remium) to increase (decrease) with suly is determined by the sign o the sloe o the asset ngel curve. This observation allows us to derive (i) sui cient conditions directly in terms o the reresentative agent s risk aversion roerties or general utility unctions and (ii) necessary and sui cient conditions or the widely used HARA (hyerbolic absolute risk aversion) class. 2 We extend our single eriod analysis to two eriods. Once eriod one consumtion is introduced as a third good, the equivalence between the demand and equilibrium rice comarative statics breaks down. While the demand comarative statics become signiicantly more comlex, most o the single eriod equilibrium comarative statics results based on the reresentative agent s general and HARA reerences extend rom the single eriod setting. Moreover by selecting eriod one consumtion as the numeraire, one can derive simle sui cient conditions or the comarative statics o the equilibrium exected return and the risk ree rate. In Section 2, we derive single eriod comarative statics corresonding to changes in asset sulies. Section 3 extends our analysis to two eriods and considers an alternative seciication or reerences. The inal Section contains concluding comments. 2 Single Period Comarative Statics In this Section, we consider a single eriod setting in which the reresentative agent with a given endowment o assets selects asset demands so as to maximize exected utility or end o eriod consumtion. In the next Section, we consider the natural extension to a two eriod setting where the reresentative agent at the beginning o eriod one chooses both the level o certain consumtion 1 In a distribution economy setting, see Malinvaud s classic (16], chater 5) and the later contribution o Kohli 13]. 2 See 8], , or a discussion o the HARA class and the roerties o its members. 2

3 c 1 as well as asset holdings the returns on which und eriod two consumtion, c 2. The notational conventions and reerence structure o the current Section acilitate a simle transition to the two eriod roblem in Section conomy Assume the classic risky asset and risk ree asset setting. Let the risky asset ) have random ayo ξ > 0 and a corresonding arbitrary cumulative distribution unction F ( ξ. There also exists a risk-ree asset with ayo ξ > 0. Let n and n denote the number o units o the risky asset and risk ree asset, resectively. The rices o the risky and risk ree assets are given by and, resectively. In the current single eriod setting, the reresentative agent s reerences are deined over random c 2 ξn + ξ n > k, where k is 0 or some ositive constant, and satisy the standard exected utility axioms where the NM (von Neumann-Morgenstern) index W (c 2 ) satisies W C 3, W > 0 and W < 0. 3 The exected utility unction W ( c 2 ) given by ) ) ( ξ) W (n, n ) W ( ξn + ξ n W ( ξn + ξ n df. (1) Then as is well-known, W will be strictly increasing and concave in both n and n. The reresentative agent can be viewed as solving the otimization roblem ) max W (n, n ) max W ( ξn + ξ n n,n n,n (2) subject to n + n n + n, (3) where n and n denote, resectively, endowments o the risky and risk ree assets. irst order condition is given by 4 W n W n The resulting )] ξw ( ξn + ξ n )] ξ W ( ξn. (4) + ξ n In this reresentative agent setting, the no bankrutcy condition is given by Throughout this aer, we also assume that n in ξ + ξ n > k. (5) ξ > ξ, (6) rom which it ollows that n > 0. 5 n > 0. As a result we assume a ositive endowment o the risky asset 3 These single eriod NM reerences are extended in Section 3 to the two eriod exected utility W (c 1, c 2 ) W 1 (c 1 )+W 2 ( c 2 ), where the consumer is choosing over both c 1 and (n, n ). The single eriod NM utility considered here can be viewed as corresonding to W 2 ( c 2 ) in the two eriod W (c 1, c 2 ). 4 Throughout this aer we deine W n W ( ξn+ξ n ), W n,n 2 W ( ξn+ξ n ) 2. Other terms such as W n, W n,n and W n,n are deined similarly. 5 To see that n > 0, note irst that rom eqn. (4) ξ ( ξn ) ] ξ W + ξ n 0. 3

4 On the other hand, we allow n 0, which runs contrary to the conventional assumtion that the suly o bonds is zero (e.g., 3]). In recent years, a number o aers have aeared which consider the case o a ositive net suly o bonds. Heaton and Lucas 10] consider the existence o an outside suly o bonds, treating bonds as a Lucas 15] "tree" technology with a dividend equal to the equilibrium interest rate. Cochrane, Longsta and Santa-Clara 5] generalize the Lucas tree structure emloying a two tree model and briely consider one tree being a bond. Parlour, Stanton and Walden 22] require a ositive suly o bonds in order to make rogress on resolving the equity remium, risk ree rate and excess volatility uzzles. They state The assumtion that bonds are in zero net suly is consistent with an ininitely lived reresentative agent in an economy absent any rictions...by contrast, in a world with initely lived investors, or with rictions, it may be ossible or the current generation to borrow against the consumtion o uture generations, leading to a ositive suly o bonds and risk-ree consumtion or the current generation over a signiicant time eriod. Indeed, in any economy in which Ricardian equivalence ails, government bonds can be in ositive net suly. (22],. 3) While the above reerences assume a ositive suly o bonds, Favilukis, Ludvigson and Van Nieuwerburgh 7] assume a negative suly o bonds which they motivate by oreigners holding US debt. Thus, in the case where n > 0, the issuer o the bonds (or examle, the Government) is outside the model. Analogously i n < 0, it is the lender that is outside the model. Finally, Cass and Pavlova 4] observe that while nonnegativity assumtions or commodity endowments are very deensible, there is nothing contradictory in droing this assumtion when considering inancial assets, esecially when there are no restrictions on asset trade. In this aer, we dro the zero net suly assumtion or the risk ree asset. By not restricting n 0, we will be able to derive a number o interesting dierences in the comarative statics o equilibrium returns and gross equity remium corresonding to changes in asset sulies. Given the reresentative agent setting seciied above, it is clear that there will be a unique equilibrium deined by (,, n, n ). This equilibrium corresonds to the ixed arameter set (n, n, ξ, ξ ) where equilibrium rices are endogenous. 2.2 Inverse Demand and quity Premium Behavior Following Katzner 11] when solving the reresentative agent s demand roblem eqns. (2)-(3), one can think o ixing the budget constraint based on a given endowment and rices and inding the maximum utility value. On the other hand, when solving or equilibrium rices, one ixes the seciic indierence curve assing through the endowment oint and then solves or the equilibrium rice ratio equal to the sloe o the tangent to the indierence curve at that oint. In both instances, the otimal oint corresonds to the tangent oint (n, n ) in Figure 1. Clearly, we have ] ( ξ ξ W ξ ) ξ W n 0. Thereore, our assumtion that ξ > ξ imlies that n > 0. This argument continues to hold in the two eriod case considered in Section 3 since the irst order condition or n and n remains the same. 4

5 Figure 1: Next we argue that in the single eriod case, the necessary and sui cient conditions or signing the equilibrium rice ratio and demand comarative statics resulting rom changes in n and n are equivalent in the ollowing sense. Corresonding to changes in these sulies, the sign eect on the equilibrium rice ratio deends on and only on the sign eect on the reresentative agent s demand. In the ollowing result it will be useul to deine income or wealth I as I n + n. (7) Then changes in the endowments n or n can equivalently be viewed as changes in I or, in terms o standard demand theory, income eects. Proosition 1 Assume a single eriod, reresentative agent asset exchange economy, where the otimization roblem is given by eqns. (2)-(3). Then (8) n n and 0 0. (9) nn Proo. Dierentiating the irst order condition eqn. (4) with resect to n, we can obtain W n,nw n W n,n W n Wn 2. (10) Dierentiating the irst order condition with resect to the income I, we have W n,n + W n,n W ( ) n W n,n W n + W n,n 0. (11) 6 In an exchange economy since only relative rices matter, i one uses the normalization 1 as in 9] and 21] then the comarative statics or will be the same as or. 5

6 Dierentiating the budget constraint with resect to the income I, we have Thereore, we have + 1 (12) 1 W n,n W n W n,n W n n n W n 2W n,n W n,n W n,n. (13) Since W (n, n ) is concave, we always have W n,n W n,n W 2 n,n 0 Then we have 2W n,n W n,n ) W n,n 2 ( W n,n W n,n + W n,n Thereore, we have Similarly, we can rove that 0 W n,n W n,n W n,n 0. (14) 2 ( W n,n W n,n ) W n,n 0. (15) 0. (16) n n 0 0. (17) nn Remark 1 It is imortant to note that the condition n n > 0 does not corresond to the risk ree asset being a normal good since n need not be ositive. The derivatives o n resect to I should be viewed as the sloe o the risk ree asset ngel curve. When goods can be negative, the aroriate deinition o a normal good should be n > 0. Indeed there are our dierent sign combinations or the sloe and whether the risk ree asset is being held long or short. In a certainty setting, Nachbar 19] shows that i there are L commodities and good L is selected as the numeraire, then the rice o good 1 will increase with its suly only i the comosite commodity ormed by the other L 1 commodities is an inerior good. Because or Nachbar commodities must be ositive, the condition that the sloe o a good s ngel curve is ositive is equivalent to it being a normal good. The geometric intuition or Proosition 1 can be exressed very simly in terms o Figure 2. Let n (0), n be the initial equilibrium oint and denote the budget constraint assing through the oint L 0. Suose the corresonding income level is I (0) n (0) + n. Now move to a new equilibrium oint n (1), n characterized by a larger endowment o the risky asset. Label the new constraint tangent to the indierence curve assing through n (1), n as L1. Using the aroach in 11], the sloe o the reresentative agent s indierence curve in the n n lane is given by dn dn. (18) W const I > 0 then L 1 will be steeer than L 0. Now consider another budget constraint L 2, which asses the oint n (1), n and arallel to L0. Clearly L 2 corresonds to the same rice vector as L 0, but the corresonding income level I (1) is larger than I (0). Since L 1 is also steeer than L 2, 6

7 Figure 2: the indierence curve tangent to L 1 will intersect with L 2 below n n. Due to the convexity o the indierence curve, the otimal oint on L 2 is below the n n line, imlying < 0. n n Thereore, we have the Proosition 1 conclusion. The case o increasing n can be discussed similarly. Given Proosition 1, we can determine the comarative statics or the equilibrium rice ratio once we seciy the seciic conditions determining the sign o the demand comarative statics. First deining the classic Arrow-Pratt 1]-23] absolute and relative risk aversion measures corresonding to the reresentative agent s reerences as τ A de W (c 2 ) W (c 2 ) and τ R de c 2 W (c 2 ) W (c 2 ), (19) it ollows rom Arrow 1] that in a single risky asset and risk ree asset setting, the risky asset will be a normal good, i.e. > 0, i (i) both assets are held long and (ii) τ A < 0. I one additionally assumes (iii) τ R 0, then Aura, Diamond, and Geanakolos 2] oint out that the risk ree asset will also be a normal good, i.e. > 0. Thereore under (i)-(iii), we always have (n, n ) (n, n ) < 0 and > 0. (20) (n, n ) (n, n ) (See 14] or an in deth discussion o the signs o and when conditions (i) - (iii) do not hold in a comlete market setting.) The above discussion including Proosition 1 can be directly alied to the comarative statics results or the gross equity remium Z which is deined as Z de R R (n, n ) ξ (n, n ) ξ. (21) Corollary 1 Assume a single eriod reresentative agent asset exchange economy, where the otimization roblem is given by eqns. (2)-(3). I τ A 0, then Z 0 0 (22) n n 7

8 and Z 0. (23) Proo. This ollows directly rom the deinition o Z and Proosition 1. The seeming asymmetry o the Z result is a direct consequence o τ A 0 imlying that the risky asset is a normal good. It ollows rom Kubler, Selden and Wei 14] that i one allows either n < 0 or τ R < 0, then the ngel curve or the risk ree asset can be downward sloing and it ollows rom Corollary 1 that increasing the suly o the risky asset n can result in a decrease in the gross equity remium. It will be noted that the comarative statics with resect to asset sulies or Z and always change in an oosite direction. Throughout this aer, we will ocus rimarily on the comarative statics o. 2.3 Simle Preerence Restrictions Given that the necessary and sui cient condition or the sign eects o changing the suly o the risky asset deends on whether risk ree asset ngel curve is increasing or decreasing in income, it is natural to ask when this occurs given exlicit restrictions on the reresentative agent s reerences. In rincile one can use Kubler, Selden and Wei 14] to settle this question. However, we choose a dierent aroach to rove the results since the roos here will then extend directly to the two-eriod setting in the next section. The ollowing roosition generalizes the above corollary. Proosition 2 Assume a single eriod reresentative agent asset exchange economy, where the otimization roblem is given by eqns. (2)-(3). Then we have 0 τ A 0. (24) I τ A 0, the we have I τ A < 0 < 0. (25) (i) τ R 0 and n 0, then we have (ii) τ R 0 and n 0, then we have 0, (26) 0, (27) where or both (i) and (ii) the equal sign can be reached i and only i n 0 and τ R 0. Proo. Dierentiating the irst order condition eqn. (4) with resect to n and noticing that ] ξ ξ W 0, we can obtain W ( ξ ( ) ] ξ W )]) 2. (28) W ( ξn + ξ n 8

9 It ollows rom the generalized Chebyshev s Algebraic Inequality (see 8], Proosition 15 (2)) that i τ A 0, or equivalently, W W 0, then Thereore, we have ] W ξ ξ W W ] ξ ξ W ] W 0. (29) W 0 τ A 0. (30) Moreover, we have ) ] ) ] W ξ ( ξ ξ W W ( ξ ξ W ξw W )]) 2 )]) 2. (31) ξ ( W ( ξn + ξ n ξ ( W ( ξn + ξ n I τ A 0, then ξw W is a strictly increasing unction o ξ and hence we have ) )] ( ξw ( ξ ξ W ] ] W > ξ ξ W ξw W 0, (32) which imlies that When τ A < 0, deining c 2 ξn + ξ n, we have ] ( ξ ξ ξ W ξ ξ < 0. (33) ) W c 2W W ] ] ξ n ξ ξ W. (34) Thereore, i τ R 0 and n 0, then we have 0 (35) and i τ R 0 and n 0, then we have where the equal sign can be reached i and only i n 0 and τ R 0. 0, (36) Remark 2 We have argued that under the assumtion ξ > ξ, we have n > 0. Since the sign o τ A will ully determine whether the risky asset is a normal good or not, we have τ A (37) Moreover, i τ A 0 and τ R 0, then the risk ree asset is a normal good when n > 0, imlying > 0 and < 0. Similarly, i τ A 0 and τ R 0, then the risk ree asset is a normal good when n < 0, imlying < 0 and > 0. 9

10 While Proosition 2 rovides sui cient conditions or the cases where τ R n 0, it is silent on the cases where τ R n 0. However or the widely used HARA class, we are able to rovide necessary and sui cient condition or each member. Proosition 3 Assume the reresentative agent s reerences satisy the standard exected utility axioms and the NM index W (n, n ) corresonds to the HARA class and that (5) and (6) are satisied. Then (i) I then (ii) i then (iii) I then (iv) I then (v) I W (c 2 ) c δ 2, δ > 1, (38) δ 0 n 0, (39) W (c 2 ) (c 2 a) δ, δ > 1, a > 0 (40) δ 0 n a ξ, (41) W (c 2 ) (c 2 + a) δ, δ > 1, a > 0, (42) δ 0 n a ξ, (43) W (c 2 ) ex ( λc 2), λ > 0, (44) λ < 0, (45) W (c 2 ) qc 2 c 2 2, q > 0, (46) then < 0. (47) Proo. We aly a similar method as in the roo o Proosition 2, which does rely on the corresonding demand roerties. For case (i), we have ) ] 1 δ ξ ( ξn + ξ n ) 1 δ ]. (48) ξ ( ξn + ξ n 10

11 Thereore, (1 + δ) ξ A ( ) ]) 1 δ 2, (49) ξ ( ξn + ξ n where A ) ] 2 δ ) ] 1 δ ξ ( ξn + ξ n ξ ( ξn + ξ n ξ2 ( ξn ) ] 2 δ ) ] 1 δ + ξ n ( ξn + ξ n. Ater some algebra, we can rewrite A as ) ] 2 δ ) ] 1 δ A n ξ ( ξn + ξ n ξ ( ξn + ξ n Noticing that we can rewrite A as A n n ξ ( ξn + ξ n ) 2 δ ] (50) (51) ) ] 1 δ ξ ( ξn + ξ n. (52) ) ] 1 δ ξ ( ξn + ξ n ) ] 1 δ ξ ( ξn + ξ n, (53) ) ] 1 δ ) ] 2 δ ξ ( ξn + ξ n ξ ξ ( ξn + ξ n. (54) It ollows rom a non-monotonic version o Chebyshev s Algebraic Inequality that ) ] 2 δ ξ ξ ( ξn + ξ n < 0. (55) Thereore, we can conclude that For case (ii), deining n 0 A 0 n new it ollows rom the the same stes as above that For case (iii), deining n new 0 n a ξ n new it ollows rom the the same stes as above that n new 0 n a ξ 0. (56) n a ξ, (57) 0. (58) n + a ξ, (59) 0. (60) Case (iv) and (v) have been roved in Proosition 2. Although the roo o Proosition 3, does not rely at all on the demand roerties o the HARA utility unctions, the ollowing xamle rovides intuition into the connection to the ngel curve roerties given in Proosition 1. 11

12 Demand Demand Income I (a) Income I (b) Figure 3: xamle 1 Assume the reresentative agent s reerences are reresented by the HARA utility W (c 2 ) (c 2 a) δ, (61) δ where a > 0 and δ > 1. For simlicity, let the random variable ξ take the values ξ 21 with robability π 21 and ξ 22 with the robability π 22 1 π 21. The corresonding ngel curves or two dierent sets o arameters are lotted in Figure 3. It is clear that 0 n a ξ. (62) Thereore in equilibrium, the comarative statics o corresonding to changes in the suly o the risky asset, n, are comletely determined by a comarison between n and a ξ. The ollowing illustrates the imortant case where asset ngel curves are non-linear and hence not covered by Proosition 3. xamle 2 Assume the reresentative agent s reerences satisy the standard exected utility axioms and the NM index is given by W (c 2 ) c δ1 2 δ 1 c δ2 2, δ 1 > δ 2 1. (63) δ 2 It can be veriied that τ A < 0 and τ R < 0. Thereore, i n < 0, we have > 0. (64) Next we consider the case when n > 0. Assume that { ξ ξ 21 (π π 21 ). (65) ξ 22 (π π 22 ) 12

13 Demand Income I (a) (b) Figure 4: The irst order condition gives that ( 2 (ξ2i i1 π ) 1 δ1 ) 1 δ2 2iξ 2i n + ξ n + ξ2i n + ξ n ( 2 (ξ2i i1 π ) 1 δ1 ) 1 δ2. (66) 2iξ n + ξ n + ξ2i n + ξ n We lot the ngel curves in Figure 4(a) and versus n in Figure 4(b). From Figure 4(b), clearly is not monotone in n. The intuition can be understood as ollows. In Figure 4(a), we consider the line n 0.4 and there are two income levels that corresond to this level o risk ree asset demand. At the lower income level, we have > 0 and at the higher income level, we have < 0. Thereore, i we choose n 0.4 in equilibrium, we need also seciy the value o n to determine which equilibrium we are considering. This is very di erent rom the HARA class discussed in Proosition 3 since the ngel curves are linear there. In Figure 4(b), we draw a line corresonding to 1. The n values corresonding to the intersection oints between the line and the curves are exactly the two n values we need seciy or the equilibrium. For the smaller n value, we are at the low income level equilibrium in Figure 4(a) and hence > 0, which is consistent with the behavior that < 0 in Figure 4(b). For the larger n is value, we are at the high income level equilibrium in Figure 4(a) and hence < 0, which is consistent with the behavior that ( ) > 0 in Figure 4(b). 3 Two Period Comarative Statics In the two eriod setting, the equivalence between the single eriod demand and equilibrium rice ratio comarative statics breaks down. While the ormer becomes more comlex, ortunately the latter which is our rimary interest tends not to. 13

14 3.1 conomy The reresentative agent s two eriod reerences are deined over certain eriod one and random eriod two consumtion airs (c 1, c 2 ) and satisy the aroriate exected utility axioms. The NM index W (c 1, c 2 ) is additively searable and W (c 1, c 2 ) is given by W (c 1, n, n ) W 1 (c 1 ) + W 2 ( ξn + ξ n ). (67) The eriod two index W 2 exhibits the same roerties as the single eriod NM utility, W 2 C 3, W 2 > 0, W 2 0. The unction W 1 satisies W 1 C 2, W 1 > 0, W 1 < 0. Finally, we assume the asset return roerties introduced in Section 2 continue to hold. The otimization roblem o the reresentative agent becomes ( )) max W (c 1, n, n ) max W 1 (c 1 ) + W 2 ( ξn + ξ n, (68) c 1,n,n c 1,n,n subject to the exchange economy constraint 1 c 1 + n + n 1 c 1 + n + n, (69) where 1 and c 1 > 0 denote resectively the rice and endowment o eriod one consumtion. The resulting irst order conditions are given by )] ξw 2 ( ξn + ξ n W n W c1 W n W c1 W n W n ξ W 2 W 1 (c, (70) 1) 1 )] ( ξn + ξ n W 1 (c 1) )] ξw 2 ( ξn + ξ n ξ W 2 1, (71) ( ξn + ξ n )]. (72) Since eqn. (72) is identical to the single eriod irst order condition eqn. (4) assuming W 2 is ai nely equivalent to the one eriod NM index, it ollows immediately that the equilibrium rice ratio and gross equity remium Z remain the same as in the single eriod setting. Thus Proositions 2 and 3 extend without change. It should be noted that the act that Proosition 1 may change even when W is additively searable is o no ractical signiicance given that Proositions 2 and 3 remain the same quilibrium Return and quity Premium Behavior By moving to two eriods, we are able to derive comarative statics results or the equilibrium risk ree return and the risky asset exected return. 8 7 It can be shown that i the comosite commodity n+ n is a normal good, then Proosition 1 can be extended to the two eriod case. (See Nachbar 19] or a dierent alication o the comosite commodity in considering when the rice o a good increases with its suly.) However, testing whether the comosite commodity is a normal good may not be easy in ractice. But, i one assumes comlete markets then n + n + 21 c c 22 and this is always a normal good. 8 It is straightorward to show that i the reresentative agent s otimization is given by eqns. (68) - (69), then ollowing the roo o Proosition 4, we have R c 1 < 0 and R c 1 < 0 14

15 Proosition 4 Assume a two eriod reresentative agent asset exchange economy, where the otimization roblem is given by eqns. (68) - (69) and W satisies the conditions seciied. Taking irst eriod consumtion as the numeraire, we have R > 0 and R > 0 (73) and R > 0 and R > 0. (74) Proo. Using the normalization 1 1, it ollows directly rom the irst order conditions eqns. (70)-(71) that R ξ W 1 (c 1 ) ) (75) ( ξn + ξ n W 2 and R ξ W 1 (c 1 ) ξ )]. (76) ξw 2 ( ξn + ξ n Since we assume that ξ, ξ > 0, we have W 2 ] ξw 2 < 0 and W 2 ξ W ] 2 < 0. (77) and Thereore, we have and ] ξw 2 ] ξ2 W 2 < 0 and ] ξw 2 ξ ξw 2 ] < 0. (78) R > 0 and R > 0 (79) R > 0 and R > 0. (80) One may wonder why the equilibrium return comarative statics do not require restrictions aralleling those or Z. It ollows rom the the concavity o W 2 that an increase in any asset s suly will increase the relative abundance o eriod two consumtion versus eriod one consumtion. The marginal utility o eriod two consumtion will decrease while that or eriod one will remain unchanged causing the rice o either asset to decline. Since the ayo on the asset does not change, it ollows rom eqns. (70) and (71) that or each asset its return will increase. 3.3 Non Additively Searable xected Utility We consider the habit ersistence generalization o additively searable NM utility (67) which was originally introduced to reconcile the equity remium uzzle (see Constantinides 6]). 9 Following and Z c See 25] or a summary o the alications o this orm o utility to asset ricing, macro, monetary olicy and business-cycle theory. 15

16 the classic certainty literature, the NM utility (67) is tyically modiied as ollows resulting in W no longer being additively searable W (c 1, n, n ) W 1 (c 1 ) + W 2 ( ξn + ξ n αc 1 ), (81) where α > 0 and W c 1 > 0. The standard interretation or the habit ersistence term αc 1 ollows rom the simle certainty setting. Assuming W is a unction o current and uture consumtion, the corresonding intuition or habit ersistence is that "the more I eat in eriod one the hungrier I get in eriod two" 25]. Continuing to maintain the assumtions made or the two eriod additively searable W, the otimization roblem o the reresentative agent becomes ( )) max W (c 1, n, n ) max W 1 (c 1 ) + W 2 ( ξn + ξ n αc 1 (82) c 1,n,n c 1,n,n subject to the exchange economy constraint The resulting irst order conditions are given by 1 c 1 + n + n 1 c 1 + n + n. (83) W n W c1 ξ W 2 W 1 (c 1) αw 2 1, (84) and W n W n W n W c1 ξw 2 W 1 (c 1) αw 2 (85) 1 )] ξw 2 ( ξn + ξ n αc 1 ξ W 2 ( ξn + ξ n αc 1 )]. (86) Since at equilibrium, αc 1 is a constant in eqn. (86), i we deine n new n αc 1 ξ, (87) Proositions 2 and 3 extend to the current setting. the modiication o conditions (i)-(iii) given below. This can be seen in the case o the latter rom Corollary 2 Assume a two eriod reresentative agent asset exchange economy, where the otimization roblem is given by eqns. (82) - (83). Further assume the reresentative agent s reerences satisy the standard exected utility axioms and the NM index W 2 (c 2 αc 1 ) corresonds to the HARA class and that (5) and (6) are satisied. I then W 2 (c 2 αc 1 ) (c 2 αc 1 a) δ, δ > 1, a 0, (88) δ 0 n a + αc 1 ξ. (89) Proo. The roo directly ollows rom the roo o Proosition 3. It is interesting to note that increasing c 1 increases the range o risk-ree asset suly over which the equity remium decreases with suly. Also note that i a < 0, the comarative statics result will deend on the relative size o a and α c 1. 16

17 4 Concluding Comments In this aer, we investigate the comarative statics o equilibrium rices, returns and the gross equity remium corresonding to changes in asset sulies in both single eriod and two eriod reresentative agent exchange economies. In the single eriod case, we demonstrate an equivalence between the comarative statics o asset demand and equilibrium rices. Several sui cient conditions are given or the resonse o equilibrium rices to changes in asset sulies. Also necessary and sui cient conditions are given or the oular HARA class o utilities. We show that surrisingly the rice o the risky asset can increase with increasing suly or erectly standard orms o exected utility. When considering the two eriod case, unortunately the equivalence between the demand and equilibrium rice comarative statics breaks down. However most o the equilibrium rice and gross equity remium comarative statics carryover rom the one eriod case. And i we make the erectly natural choice o eriod one consumtion as the numeraire, we can derive simle sui cient conditions or the comarative statics o equilibrium asset returns. These results suggest two areas or uture research. First, having based our analysis on the reresentative agent model, it would seem natural to consider similar comarative statics analyses or economies with heterogeneous agents. Second, given the results obtained in this aer and the recent aearance o a number o equilibrium models with a ositive suly o risk ree bonds, such as those discussed earlier, it would seem otentially ruitul to extend our analysis to Macro settings in which one endogenizes increases in the suly o bonds by a Government and equities by cororations. Reerences 1] Arrow, K., ssays in the Theory o Risk Bearing, Markham, Chicago (1971). 2] Aura, S., Diamond, P. and Geanakolos, G., "Savings and Portolio Choice in a Two-eriod Two-asset Model", American conomic Review 92 (2002), ] Barsky, R., "Why Don t the Prices o Stocks and Bonds Move Together?", American conomic Review 79 (1989), ] Cass, D. and Pavlova, A., "On Trees and Logs", Journal o conomic Theory 116 (2004), ] Cochrane, J., Longsta, F. and Santa-Clara, P., "Two Trees", Review o Financial Studies 21 (2008), ] Constantinides, G., "Habit Formation: A Resolution o the quity Premium Puzzle", Journal o Political conomy 98 (1990), ] Favilukis, J. S. Ludvigson and S. V. Nieuwerburgh, "The Macroeconomic ects o Housing Wealth, Housing Finance, and Limited Risk-Sharing in General quilibrium", SSRN (2011). 8] Gollier, C., The conomics o Risk and Time, MIT Press (2001). 9] Gollier, C. and Schlesinger, H., "Changes in Risk and Asset Prices", Journal o Monetary conomics 49 (2002)

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