When is a Risky Asset "Urgently Needed"?

Transcription

1 When is a Risky Asset "Urgently Needed"? Felix Kubler University o Zurich Swiss Finance Institute Larry Selden Columbia University University o Pennsylvania Xiao Wei University o Pennsylvania March 2, 23 Abstract The demand or the risk ree asset in the classic ortolio roblem is shown to decrease with income i and only i the consumer s uncertainty reerences over assets satisy the condition that the risk ree asset is more readily substituted or the risky asset as the quantity o the risky asset increases. In this case, the risky asset is said to be "urgently needed" ollowing the terminology o Johnson in his classic 93 certainty analysis [9]. However we show that the asset and certainty settings dier in critical ways which result in a much greater likelihood or the urgently needed reerence roerty to be satisied in the ortolio roblem. We rovide several sui cient conditions or when the risky asset will be urgently needed and a surrisingly simle, comlete characterization or widely oular members o the HARA hyerbolic absolute risk aversion class. For more general reerences, two examles are given where it is ossible to ully describe the region o asset sace in which the risky asset is urgently needed. Finally, using a standard reresentative agent model we show that the risky asset being urgently needed is equivalent to the equilibrium relative rice o the risky asset increasing with its own suly. JEL Codes: D, D, D53. Kubler: University o Zurich, Plattenstrasse 32 CH-832 Zurich kubler@gmail.com; Selden: University o Pennsylvania, Columbia University, Uris Hall, 322 Broadway, New York, NY 27 larry@larryselden.com; Wei: Sol Snider Entrereneurial Research Center, Wharton School, University o Pennsylvania, 3733 Sruce Street Philadelhia, PA xiaoxiaowx@gmail.com. Kubler acknowledges inancial suort rom NCCR-FINRISK and the ERC. Selden and Wei thank the Sol Snider Research Center or its suort. Each author declares that he has no relevant or material inancial interests that relate to the research described in this aer.

2 Introduction While the ossibility o a good being inerior is discussed in every introductory economics class, it turns out that or most utility unctions used in ractice the demand or each commodity actually increases with income. However in the classic single eriod ortolio model with one risky asset and one risk ree asset, we have recently shown in [22] that the demand or the risk ree asset can decrease with income and the risk ree asset can even be a Gien good. Moreover, this can occur or erectly standard orms o uncertainty reerences such as members o the widely oular HARA hyerbolic absolute risk aversion amily o utility unctions. Unortunately the necessary and sui cient condition derived in [22] or risk ree asset demand to decrease with income is not articularly intuitive and is not helul in connecting the certainty and ortolio settings. comlete asset markets. Moreover it is based on the restrictive assumtion o In this aer we ind a universal condition which overcomes each o these shortcomings and also allows us to address the ollowing three questions. Q: Why does demand decrease with income more readily or the risk ree asset than or commodities? Q2: What imlications does the necessary and sui cient condition or the risk ree asset demand decreasing with income have or equilibrium rices in a standard reresentative agent economy? Q3: Also in a reresentative agent setting, is it ossible to characterize the set o asset endowments such that the demand or the risk ree asset decreases with income? In considering each o these questions, it will rove invaluable to ocus on consumer reerences deined over assets rather than over contingent claims as in [22]. By so doing, we can emloy a result rom the certainty demand analysis in a remarkable 93 aer by Johnson [9]. He roved that the necessary and sui cient condition or a commodity to be an inerior good is that as the quantity o the good increases, holding the amount o a second good ixed, the marginal rate o substitution between the goods increases. 2 Because the consumer is willing to give u more o the second good to get more o the irst good even This condition assumes two states o nature and involves a comarison o the ratio o the Arrow-Pratt absolute risk aversion measure in each state versus the ratio o asset ayos in the two states. 2 As a historical note, Johnson s rimary motivation or deriving this result was to use it in introducing an alternative characterization o comlement and substitute goods to that o Pareto. The Pareto deinition was recognized not to be invariant to increasing monotonic transorms o the utility. However Johnson s marginal rate o substitution based condition is ordinal in this sense. See Allen [2] and the comments o Samuelson in his seminal aer on comlementarity [32]. 2

3 as the quantity o the irst good increases, he reerred to the irst good as being "urgently needed" or more urgently needed than the second good. We show that in the standard ortolio roblem, alying Johnson s condition to asset reerences can result in risk ree asset demand decreasing with income. We ollow Johnson s terminology in reerring to the risky asset as being "urgently needed" given that the consumer desires the risky asset so much that her willingness to give u the risk ree asset to get more o the risky asset increases even as the quantity o the risky asset increases. It should be stressed that this is a ure reerence roerty not deendent on asset rices or income levels. In resonse to Q, we show that there are three critical dierences between the commodity and asset settings when considering whether a commodity or an asset is urgently needed. First assuming ositive asset ayos and concave utility, the cross artial derivative o the Exected Utility unction with resect to the risky and risk ree asset holdings is shown to always be negative, satisying a necessary condition or the MRS marginal rate o substitution to increase with the quantity o the risky asset. This is in contrast to the certainty case where the cross derivative o utility with resect to two goods can be assumed to take any sign. Second whereas tyically assumed commodity reerences or two goods are essentially symmetric, the induced utility or assets is ar rom symmetric. Indeed the standard assumtion o decreasing absolute risk aversion ensures that the demand or the risky asset always increases with income, but does not imly that the risk ree asset behaves in the same way. The third dierence is that commodity demands are required to be ositive whereas it is standard to allow negative holdings or short-selling o the risk ree asset. In characterizing when the risky asset is urgently needed, we rovide two general sui cient conditions in which the quantity o the risk ree asset lays a surrisingly imortant role. The irst condition involves the Arrow-Pratt measures o absolute and relative risk aversion and the sign o the risk ree asset holdings. The second requires reerences to satisy the Inada conditions ensuring that the risk ree asset Engel curve begins at its origin. the HARA class o Exected Utility reerences as well as or all homothetic reerences, we derive necessary and sui cient conditions, which very surrisingly deend only on the quantity o the risk ree asset. For To bridge the ga between the latter conditions and the more general sui cient conditions we analyze several examles in which it is ossible to ully characterize regions o asset sace where the risky asset is and is not urgently needed. In resonse to Q2, we show that in a standard reresentative agent exchange economy the risky asset s relative equilibrium rice increases with its suly i and only i the asset is urgently needed. 3 For the secial case o HARA reerences, it is ossible to rovide a 3 In certainty equilibrium analyses, the suly o each good is assumed to be ositive. However, in uncertainty setting although the risky asset is tyically assumed to be in ositive suly, the risk ree asset is most oten assumed to have zero net suly e.g., [4]. Recently a number o aers have relaxed this 3

4 articularly shar resonse to Q3. Whether or not the equilibrium relative rice increases with the suly o the risky asset deends solely on the quantity o the assumed suly o the risk ree asset. In [22], we investigate when the risk ree asset will be an inerior good. However because the asset can be held short, inerior good behavior is not equivalent to demand decreasing with income see Section 3.2 below. This aer shows that the latter roerty is equivalent to the ure reerence restriction associated with the risky asset being urgently needed. This new ocus rovides interesting insights into the comarative statics o equilibrium rices esecially the crucial role layed by the suly o the risk ree asset. The rest o the aer is organized as ollows. In Section 2, we review the certainty analysis o Johnson and rovide a concrete examle illustrating the relationshi between one good becoming urgently needed and the demand o a second good decreasing with income. Section 3 investigates when the risky asset becomes urgently needed and comares the conditions with those in the certainty case. In Section 4, we discuss the equilibrium imlication o our results. The inal Section contains concluding comments. 2 Urgently Needed Good: Certainty Case In certainty settings, the ossibility that demand decreases with increasing income is tyically recluded by standard reerence assumtions such as additive searability or the weaker roerty o suermodularity see [8] and concavity. Indeed inding otherwise well behaved utility unctions that generate such behavior is oten viewed as being dii cult. In this Section, we irst review the reerence characterization o inerior good behavior derived by Johnson [9] and then rovide an examle which satisies his condition. In the next Section we show how alication o the Johnson result in an uncertainty setting diers in several critical ways rom the certainty case. Consider a two good setting in which x and y denote the units o the goods. Assume a consumer whose reerences over x, y airs deined on a convex subset Ω o the ositive orthant are reresentable by a strictly quasiconcave utility U x, y which is increasing in each good, and satisies U C 3. roblem subject to The consumer can be viewed as solving the otimization max x,y U x, y I = x x + y y, 2 assumtion, allowing aggregate suly to be ositive e.g., [5], [] and [28] or negative e.g., [2]. While the motivation or these dierent suly assumtions is unrelated to our analysis, the imlications o the assumtions or equilibrium rice comarative statics are ar rom innocuous. 4

5 y x Figure : where x and y denote the rices o the goods and I is initial income or wealth. As is standard, y is said to be an inerior good i and only i y/i <. Deine the MRS by Ux U y. 4 Then we have the ollowing result. Proosition Johnson [9] Assume the otimization roblem given by eqns. -2. Then y I U x U y x. 3 The intuition or Proosition can be exressed very simly in terms o the two indierence curves lotted in the x y choice sace in Figure. 5 tangent to each indierence curve corresonds to the MRS = Ux tangency oint P to Q in Figure, x is increased while y is held constant. Minus the sloe o the dashed U y. When moving rom Corresonding to this move, the sloe o the new indierence curve is seen to become steeer and the MRS increases, U x U y x = U x Uxx U yx >, 4 U y U x U y and ollowing Johnson s terminology good x would be reerred to as being more "urgently needed" than y. It ollows rom Proosition that y < and good y is an inerior good.6 I 4 Throughout this aer artial derivatives will be denoted by subscrits. For examle, we deine U x = U x.5 See [25] or a similar discussion. 6 I good y is an inerior good, it ollows rom the budget constraint that good x must be a normal good. 5

6 Johnson viewed the oosite case rom Figure, where the MRS decreases between the two oints and y is a normal good, to be the "standard" case. As good x becomes relatively more abundant relative to the ixed good y, the consumer should be more willing to give u less o good y to obtain one more unit o good x along the shited indierence curve as relected by the decreased MRS. This is consistent with the standard assumtion o suermodularity Ux Uy and concavity since i U yx and U xx <, then rom eqn. 4 <. x Next we consider a simle non-traditional, although well behaved orm o utility, which results in good y being an inerior good and x being urgently needed. Examle Consider the ollowing strictly quasiconcave utility unction U x, y = β x + y a δ where a >, β > β 2 > and δ >. and y and U yx <. 7 We have δ β 2x + y a δ, 5 δ It can be veriied that U is strictly increasing in x U x = β β x + y a δ + β 2 β 2 x + y a δ. 6 U y β x + y a δ + β 2 x + y a δ Since U x U y = x + δ β β 2 β x + y a 2 δ β 2 x + y a δ x+ y a x+ y a β β 2 β x + y a δ + β 2 x + y a δ 2, 7 it ollows that U x U y y a, 8 x imlying that x is urgently needed and y is an inerior good in a region o the commodity sace deined by < y < a. Whereas the orm o utility in Examle is non-traditional in certainty demand analysis, as we will see this orm is quite standard in the uncertainty asset demand setting. Using the terminology o Hirshleier [6], good x can be reerred to as being an ultrasuerior good since corresonding to an increase in income, the incremental demand or good x not only increases corresonding to its being a normal good but increases by more than the ull increase in income, which results in good y becoming an inerior good, i.e., y I <. 7 It should be noted that or the utility 5 to be well-deined or all ossible δ, we require that β 2 x+y a >. This imlies that indierence curves are only deined or oints in the ositive region o the x y sace northeast o the line y = a β 2 x. 6

7 3 When the Risky Asset is Urgently Needed In this Section we derive a number o restrictions on Exected Utility reerences corresonding to the risky asset being urgently needed. 3. Preliminaries Consider a risky asset with random ayo ξ ξ > in every state and a corresonding arbitrary cumulative distribution unction F, which is indeendent o the amount invested. Suose there also exists a risk ree asset with ayo ξ >. Let n and n denote the units o the risky asset and risk ree asset, resectively. We consider ortolios consisting o a risk ree asset and a risky asset where ositive holdings o the ormer are not required. In a single eriod setting, the consumer s reerences are deined over random end o eriod wealth z = ξn + ξ n and satisy the standard Exected Utility axioms where the NM von Neumann-Morgenstern index W z satisies W C 3, W > and W <. The Exected Utility unction is given by EW z = EW ξn + ξ n = The consumer can be viewed as solving the otimization roblem, n = max EW ξn + ξ n n,n subject to max n,n W ξ ξn + ξ n df. 9 n + n = I, where and are the rices o the risky and risk ree assets and I denotes initial income or wealth. The no arbitrage condition su ξ > ξ > in ξ is assumed. Under our assumtions, it can be easily veriied that the unction W is strictly increasing and concave in both n and n and the otimization roblem has a unique solution in n and n. In order to guarantee non-negative income, a no bankrutcy condition is assumed. We deine minimum income by I min = n +n, where n and n are the otimal asset holdings satisying in ξn +ξ n = and assume I > I min. For the comlete market case, an analytical orm or I min is given in [22]. It will rove convenient to also assume that E ξ > ξ, which imlies that the otimal risky asset holding is ositive. 8 The demand or assets is a continuous unction in asset rices, and income I. Instead o writing n,, I and n,, I, we will suress the deendence on rices and income whenever ossible and simly use n, n to denote these unctions. 8 To see that n >, note that the irst order condition or the otimization roblem - is [ E ξ ξn ] ξ W + ξ n =. 7

8 3.2 General Case In the classic multicommodity certainty setting when demand decreases with income, a good is said to be an inerior good. However in the uncertainty ortolio case, because n can be negative, it is necessary to modiy this standard deinition. Given that the conventional income eect in the Slutsky equation corresonding to is deined by n, it is natural I to generalize the inerior good deinition to be n <. This deinition is consistent with I that o Hicks [4] where, in a multicommodity setting, an inerior good is characterized by having a negative income elasticity. Also see Kubler, Selden and Xiao [22]. When n >, one obtains the traditional deinition I <. Alternatively when n < and I <, borrowing can be viewed as being a normal good as it increases with income. We next show that, unlike the certainty case, in determining whether the MRS increases with the holdings o the risky asset, one needs to ocus on whether the risk ree asset Engel curve is downward sloing and not whether it is an inerior good. In order to characterize when I ortolio setting. <, we next extend Proosition to the uncertainty Proosition 2 Assume the otimization roblem given by eqns. -. Then W n I. 2 Proo. The irst order condition or the otimization roblem - is given by ] W E [ ξw ξn + ξ n n = ] = E [ξ W ξn. 3 + ξ n Dierentiating both sides o the above equation with resect to n and I yields, resectively, =,n,n 4 2 and,n I +,n I,n I + W n,n =. 5 I Clearly, we have [ E ξ ξn ] ξ W + ξ n E ξ ξn ξ EW + ξ n n. Thereore, the assumtion that imlies n >. E ξ > ξ 8

9 Dierentiating eqn. with resect to I, one obtains It ollows that I = Since, n is strictly quasiconcave, I + I =. 6,n,n 2,n,n W n,n. 7 2,n W 2 n,n W 2 n,n >, 8 or equivalently imlying that 2,n,n,n >, 9 I. 2 Remark Similarly, one can show that W n I. 2 Given the assumtions o n > and decreasing absolute risk aversion, 9 it ollows rom Arrow [3] that >. Thus rom 2, we always have I >. Comaring this result with Proosition 2, there is a natural asymmetry in the behavior o the two assets in the ortolio setting. This is dierent rom the certainty case, where there is no a riori reason to suose the two goods are asymmetric. Consider Figure 2, where two Exected Utility indierence curves are lotted in the n n choice sace and it is assumed that n, n >. When moving rom the tangency oint P to Q the risky asset becomes relatively more abundant and yet along the indierence curve through oint Q, the consumer is willing to give u more o the risk ree asset to obtain one more unit o the risky asset. risk ree asset. > and I Thus the risky asset is "more urgently needed" than the Unlike the analogous conditions in the certainty case, we will argue that < should not be dismissed as being "non-standard". 9 See the deinition in eqn. 24 below. In this examle since n >, the risk ree asset is an inerior good. However i in Figure 2 n < and one had increasing MRS with n imlying I borrowing would be a normal good. <, the risky asset would be urgently needed even though 9

10 n n Figure 2: Although when n >, the downward sloing Engel curve indicates inerior good behavior, the intuition or the risk ree asset to be an inerior good is very dierent rom that o certainty commodities. For the latter, as suggested by the exression "inerior" good, there is a long tradition o interreting such goods as ossessing inerior attributes or quality. As a result when income increases, the consumer switches to goods with suerior attributes. Classic examles include substituting rom otatoes to meat, rom unctional to stylish clothing and rom basic, low cost automobiles to models with greater unctionality. Gould [3] challenged the assumtion that one good must be o inerior quality. He illustrates this henomena with the case o excellent quality wine and cigars. At low levels o income, these goods may be consumed inrequently and at dierent times. But as income increases and the consumer seeks to enjoy both together, he may discover that increased smoking dulls the alate and intereres with the enjoyment o wine. Eventually with increasing income, the marginal utility or wine decreases with the consumtion o cigars and, as a result, the demand or cigars decreases and the demand or wine increases. In the uncertainty ortolio case it ollows rom =,n,n 22 2 that since, > and,n <,,n < is a necessary condition or > and <. But given our assumtion that ξ, ξ I > and the concavity o W or risk aversion o the consumer, we always have ] = E [ ξξ W ξn + ξ n <. 23,n

11 Since,n < is guaranteed by the assumtion o risk aversion, in contrast to the certainty case it is not obvious that one good should be interreted as ossessing inerior quality or that the goods conlict as in the wine-cigar examle. As a result, the MRS condition > would seem to occur more readily or the ortolio roblem versus the certainty case. But since,n < is not sui cient, what else must be assumed to ensure that the risky asset is urgently needed? As noted in Remark, the assumtion o decreasing absolute risk aversion makes the risk ree asset and risky asset asymmetric. To investigate this issue more careully, it will rove convenient to ormally introduce the classic Arrow-Pratt absolute and relative risk aversion measures τ A z = de W z W z and τ R z = de z W z W z. 24 We denote the derivatives o these unctions by τ A and τ R, and unless stated otherwise, suress the deendence on z. Returning to Figure 2, it is obvious that when moving rom oint P to Q, z = ξn + ξ n increases in each state. I as a result, the MRS increases and the risky asset becomes urgently needed imlying that I that reerences satisy τ A >, then it ollows rom [3] that it cannot be the case. However, decreasing absolute risk aversion is not enough or the risky asset to become urgently needed. We next rovide general sui cient conditions or when this is and is not the case and then subsequently necessary and sui cient conditions or secial orms o utility. 2 Proosition 3 Assume the otimization roblem given by eqns. -. I τ A <, i τ R and n, then ii τ R and n, then, 25, 26 It is natural to wonder what the intuition is or the cross artial derivative,n to always be negative. First, the Exected Utility orm W can be viewed as a concave transorm o the linear asset ayo n ξ +n ξ. This linear ayo structure is ully consistent with the intuition that the two assets are substitutes. Second,, n exhibits diminishing marginal utility in each o its arguments. Thus since n and n can be thought o as substitutes, it is natural that i one increases the quantity o one asset then the marginal utility o the other asset should decrease because it is almost like increasing the quantity o that asset. This argument is closely related to the classic notion o Edgeworth-Pareto comlementarity where or the certainty utility Ux, y, U xy < indicates that the goods are substitutes see Samuelson [32]. 2 Combined with Proosition 2, Proosition 3 can be viewed as a general roo o Theorem 2 in [22] where the assumtion o comlete markets in the latter can be droed.

12 where or eqns. 25 and 26 the equal sign can be reached i and only i n = and τ R =. Proo. Since W, W and W are always deined on z = ξn + ξ n, we will suress the argument or simlicity. Dierentiating the irst order condition 3 with resect to n and noticing that yields E [ ] ξ ξ W =, 27 = EW E [ ξ ξ ξ W ] ξ EW Given that z = ξn + ξ n, we have [ ] [ ] [ ] ne ξ ξ ξ W = E ξ zw ξ W ξ W n E ξ ξ W. 29 It ollows rom [?], Proosition 5 that i τ A <, or equivalently, W W >, then [ ] [ ] [ ] W E ξ ξ W W > E ξ ξ W W E =. 3 W It also ollows that i τ R, or equivalently, zw W, then E [ ξ ξ W Thereore, i τ R and n, then ] zw E W [ ξ ξ W ] E [ ] zw =. 3 W. 32 Similarly one can show that i τ R and n, then W n. 33 In both cases the equal sign can be reached i and only i n = and τ R =. In Proosition 3 since the equal sign can be reached only i n = and τ R =, it ollows rom i that i τ R < and n = we have W n >. 34 Thereore, by continuity, there must exist a region with n urgently needed or this case. 2 > where the risky asset is

13 One may argue that the assumtion o τ R in Proosition 3 is strong.3 Next we show that there always exists some region in n n sace such that the risky asset becomes urgently needed i the NM index W z satisies the well-known Inada conditions see [8],.2. lim z That is in addition to W > and W <, we assume that lim W z z =. Proosition 4 Assume the otimization roblem given by eqns. -. I W z satisies the Inada conditions, then there always exists some region in n n >. z W z z = and sace such that Proo. I we can show that or W z satisying the Inada conditions, there always exists some, I such that /I <, then Proosition 4 ollows immediately rom Proosition 2. The irst order condition is given by ] W E [ ξw ξn + ξ n n = ] = E [ξ W ξn ξ n When n =, we have = and i W ξ n is a ositive inite number, then E [ ξw ] ξ n E [ ξ W ξ n ] 36 = E ξ ξ >, 37 imlying that the irst order condition cannot be satisied. Thereore, we must have W ξ n = or when n =. Since W =, we can conclude ξ n =, or equivalently n =. The act that n = n = imlies that I = and the Engel curves or the risky and risk ree assets both start rom their resective origin. Assume is large enough such that We want to argue ξ in ξ. 38 in ξn + ξ n It should be noted that the roerty o decreasing relative risk aversion has received attention in emirical and exerimental aers e.g., Levy [23], Ogaki and Zhang [27], Meyer and Meyer [24], Calvet et al. [5] and [6]. Moreover, the multieriod NM index used in standard additive habit ormation models in asset ricing literatures also exhibits decreasing relative risk aversion. 3

14 The reason is as ollows. I W in ξn + ξ n is inite, then = ] E [ ξw ξn + ξ n [ ] > E ξ W ξn + ξ n in ξ ξ, 4 imlying that the irst order condition cannot be satisied. Thereore, we have W in ξn + ξ n, or equivalently in ξn + ξ n. 4 Since in ξ >, ξ > and n >, we must have n <. We have shown above that n = when I =. We have also argued that i is large enough such that eqn. 38 holds, then n <. Due to continuity, we must have /I < or some income levels. Thereore, it ollows rom Proosition 2 that there always exists some region in n n sace such that >. Remark 2 The intuition or Proosition 4 is very clear. The Inada conditions are both necessary and sui cient or the risk ree asset Engel curve to start rom the origin, I, n =,. Then i the risk ree asset rice is large enough, the consumer will short the risk ree asset, i.e., n <. Due to continuity, one must have /I < at low income levels, imlying that there exists some region in n n sace such that >. Whereas the conditions in Proositions 3 and 4 are only sui cient, we next rovide necessary and sui cient conditions or our oular members o the widely assumed HARA class o utilities. 3.3 HARA Class In general, one would exect the sign o to deend on both n and n. 4 However, it ollows rom Proosition 3 that, given decreasing absolute risk aversion and monotone relative risk aversion, the sign o may deend just on the value o n. Next we show that this is indeed the case or our widely assumed members o the HARA class and moreover or these utilities much stronger conditions can be derived. 4 I the risk ree asset always has an uward lat, downward sloing Engel curve, i.e., /I > =, <, no matter what rices and income are, then it ollows rom Proosition 2 that we always have < =, >. Otherwise, since n. is a unction o n, n, its sign will deend on the values o n and 4

15 Proosition 5 Assume the otimization roblem given by eqns. index W z is a member o the HARA class. Then - and the NM i i then ii i then iii i then W z = W z = z δ, δ >, 42 δ W n n, 43 W z = z a δ, δ >, a >, 44 δ n a ξ, 45 z + a δ, δ >, a >, 46 δ n a ξ, 47 iv i then ex λz W z =, λ >, 48 λ W n <. 49 Proo. We aly a similar method as in the roo o Proosition 3 which does not rely on the demand roerties imlied by the seciic orms o HARA utility. For case i, [ ] δ E ξ ξn + ξ n = W δ ]. 5 n E [ξ ξn + ξ n Thereore, = + δ ξ A ] δ 2, 5 E [ξ ξn + ξ n 5

16 where A = E [ ] [ 2 δ ] δ ξ ξn + ξ n E ξ ξn + ξ n E [ ξ2 ξn ] [ 2 δ ] δ + ξ n E ξn + ξ n. Ater some algebra, A can be rewritten as A = n n E n n E [ ] 2 δ ξ ξn + ξ n [ξ ξn + ξ n 2 δ ] E ] δ E [ξ ξn + ξ n 52 [ ] δ ξ ξn + ξ n. 53 Noticing that E [ ] δ ξ ξn + ξ n = ] δ E [ξ ξn + ξ n, 54 A can be rewritten as Since A = n n E [ξ ξn + ξ n δ ] E it ollows rom [?], Proosition 5 that [ ] 2 δ E ξ ξ ξn + ξ n < E Thereore, one can conclude that For case ii, deining [ ] 2 δ ξ ξ ξn + ξ n. 55 ξn + ξ n <, 56 ξ n A [ ] [ δ ] ξ ξ ξn + ξ n E ξn + ξ n = n new = n a ξ, 59 and ollowing the same stes as above, n new n a ξ. 6 For case iii, deining n new = n + a ξ, 6 6

17 and ollowing the same stes as above, n new n a ξ. 62 For case iv, we have τ A =. Following an argument similar to that in Proosition 3, it can be easily veriied that i τ A, the risky asset can never become urgently needed, or equivalently, W n <. 63 Remark 3 For the Proosition 5i and ii utilities, it can be easily veriied that τ A < and τ R. Thereore, we have 3i. For a discussion o why the MRS = rays, see Remark 5. > when n <, which is consistent with Proosition is constant along the n = or n = a ξ The geometric meaning o Proosition 5 can be illustrated by considering the Tye ii reerences reresented by eqn. 44. The n n lane in Figure 3 is divided by the n = a ξ horizontal line into two searate regions, which are characterized by dierent indierence curve roerties. Above below this line, when moving horizontally to the right, the sloe o the indierence curves becomes latter steeer. 5 Along the n = a in Figure 3, each o the indierence curves has the same sloe ξ [ ξ δ ] E ξ E horizontal line [ ξ δ ], imlying that =. It should be noted that the same argument alies or Tyes i and iii excet that the horizontal boundary lines corresond to n = and n = a ξ, resectively. Remark 4 It will be noted that or the HARA utility 44, the necessary and sui cient condition or the risky asset to be urgently needed is strikingly similar to the certainty Examle. Indeed the corresonding induced Exected Utility unction deined over assets arallels quite closely the non-standard certainty utility General Homothetic Preerences We next show that the MRS result 43 or the HARA Tye i utility readily extends to general homothetic reerences whether or not they are reresentable by an Exected Utility 5 The utility deined by 44 has also been used to create Figure 2, where the movement rom P to Q is in the region below the n = a ξ boundary line. 7

18 n n Figure 3: unction, where the term homothetic is deined as is customary see Deaton [], Because in the ollowing, reerences need not satisy the standard axioms or the existence o an Exected Utility reresentation, we denote the utility deined over assets by Un, n rather than, n. Proosition 6 Assuming reerences are homothetic and can be reresented by U n, n, then U n U n U Proo. Since reerences are homothetic, n U n U Thereore, along any ray n = kn, where k is a constant, n Un Un n. 64 = i n =. Noticing that along each indierence curve Un U n is a homogeneous unction o degree zero. n = kn is uward downward sloing or n > <, one can conclude that i n > <. Hence the result 64 holds. Remark 5 The intuition or Proosition 6 is as ollows. U n is a constant, imlying that decreases with n and Un Un < > I reerences are homothetic, then along each ray going through the origin, the MRS Un U U n is constant. In Figure 4, n U n is constant along the rays OA and OB, resectively. Let A and B corresond to two oints on the same indierence curve. Since indierence curves are convex, the Un U n value at oint A is greater than that at oint B. Point C is on the same ray as oint B and thus has 8

19 4 3 2 n n Figure 4: the same Un U n value. Thereore, the Un U n value at oint A is greater than that at oint C, imlying that U n U n <. 65 Using a similar argument, one can show that when n < U n U n >. 66 I reerences are quasihomothetic instead o homothetic, then the oint O shits to the corresonding translated origin. Along each ray going through the shited origin, the MRS Un U n is constant and the above argument can be alied. Thus, or instance, i the NM index or an Exected Utility W n, n corresonds to case ii in Proosition 5, then the shited origin is n, n =, a and ξ n a ξ. 67 For the negative exonential case iv in Proosition 5, the utility can be viewed as a translated 9

20 origin CRRA where the origin n, n =,. 6 And thus or this case, we always have < Decreasing Relative Risk Aversion: Two Examles For Proosition 5 Tye i-iii HARA utilities and or general homothetic reerences, the value o n clearly subdivides the n n asset sace into two discrete regions corresonding to being negative and ositive. More generally it ollows rom Proosition 3i that i reerences exhibit decreasing relative risk aversion, the risky asset is always urgently needed in the ortion o asset sace where n and by continuity in at least some ortion where n >. To characterize this latter region o asset sace, we next consider two examles, where in each case the orm o utility can be viewed as a natural extension o the Proosition 5i CRRA case. the roerties o For the utility assumed in the irst Examle, it is instructive to comare to those o CRRA utility. Examle 2 Assume Exected Utility reerences characterized by the ollowing NM index z δ W z = + z δ2, δ, δ 2 >. 69 δ δ 2 Comuting τ A and τ R yields and τ A z = W z W z = + δ z δ 2 z δ + z δ δ 2z δ2 2 z δ + z δ 2 7 τ R z = zw z W z = + δ z δ z δ + z δ δ 2z δ2 z δ + z δ 2. 7 It ollows immediately that the utility 69 satisies τ A < and τ R, where the equal sign can be reached i and only i δ = δ 2. There are two senses in which the utility 69 can be viewed as an extension o CRRA utility. First, it takes the CRRA orm as δ and δ 2 converge. Second, the relative risk aversion or 69 is a weighted average o the relative 6 It ollows rom Pollak [29] that negative exonential utility is characterized by an indierence ma homothetic to the oint, in contingent claim sace. Since we have the ollowing relation between the contingent claims c 2, c 22 and inancial assets n, n c 2 = ξ 2 n + ξ n and c 22 = ξ 22 n + ξ n, it can be easily seen that, in the contingent claim sace is equivalent to, in the inancial asset sace. 2

21 n n a δ = δ 2 = 2 n n b δ = 3.5, δ 2 =.5 Figure 5: risk aversion measures, + δ and + δ 2, or two CRRA utilities corresonding to δ and δ 2. And or this latter reason, 69 is reerred to as weighted average constant relative risk aversion WACRRA utility. It ollows rom Proosition 3 that the risky asset is always urgently needed when n <. Thereore we ocus on the region where n in the ollowing analysis. For simlicity, consider a risky asset with ayo ξ that takes the values ξ 2 with robability π 2 and ξ 22 with robability π 22 = π 2. Without loss o generality, let ξ 2 > ξ 22 >. Suose there exists a risk ree asset with ayo ξ >. Assume the ollowing arameter values ξ 2 =.2, ξ 22 =.8, ξ = and π 2 = In Figure 5, we lot contours corresonding to constant values o the MRS = or the ositive orthant o asset sace. The numbers on each contour corresond to dierent MRS values. For the δ = δ 2 CRRA case in Figure 5a, the constant MRS contours are rays starting rom the origin which is consistent with reerences being homothetic. In Figure 5b or the WACRRA utility where δ > δ 2, the n = ositive vertical axis is a constant MRS contour, as in the CRRA Figure 5a case, where = E ξ ξ = Also each MRS contour begins at the origin n, n =, as in the CRRA case. But corresonding to lower MRS values, the contours become more curved. Eventually as one moves along contours increasing n, the n value decreases. I one considers a horizontal ray between n = and n =.4, it is clear that as n increases along the ray the MRS 2

22 n.8.2 n n a δ = δ 2 = n b δ = 3.5, δ 2 =.5 Figure 6: irst declines and then increases imlying that there is a oint corresonding to =. Given the MRS contours in Figure 5, we next consider the attern o changes in the MRS associated with increases in n. Contours corresonding to constant values are dislayed in Figure 6. 7 For the CRRA δ = δ 2 utility in Figure 6a, one always has, where the equal sign can be reached only along the n = horizontal. This is consistent with Figure 5a and the conclusion o Proosition 5i. The = contour in Figure 6b corresonding to the WACRRA δ > δ 2 utility orms the boundary between the region o ositive and negative values o one always has. "Inside" the = contour, > with the risky asset being urgently needed and "outside" the = contour, one always has <. This is consistent with the observation above that along any ray in Figure 5b between n = and.4, as n increases the MRS value declines and then increases. To most clearly comare the CRRA and WACRRA boundaries o the region where the risky asset is urgently needed, see Figure 7. = Examle 3 Assume Exected Utility reerences characterized by the ollowing Exo-Power 7 Note that the horizontal n-axis in Figure 6 does not start rom since when n, very negative. To illustrate the ine structure close to argument alies to Figure 8b below. 22 becomes =, we let n begin at.. A similar

23 n n Figure 7: NM index 8 W z = ex βz α, α, β and αβ >. 74 It can be easily veriied that τ A z = W z W z = α βzα + z 75 and τ R z = zw z W z = α βzα The unctional orm 74 deines a amily o utility unctions corresonding to dierent values o the arameters α and β. It can easily be veriied that absolute risk aversion is decreasing, constant or increasing i and only i α <, =, >. increasing i and only i β < or >. 9 Relative risk aversion is decreasing or To satisy the conditions in Proosition 3i, assume that α = and β =.8. This imlies that the risky asset will always be urgently needed when n as well as or some region in the ositive orthant o asset sace. arameter values 72 are also assumed to hold or this examle. The Paralleling the WACRRA case, contours corresonding to dierent and values are lotted in Figure 8a and b, resectively. It can be veriied that the attern and shae o the contours in Figures 8a and b are similar to those in Figure 5b and 6b. The risky asset is urgently needed in the region inside the = contour in Figure 8b. To acilitate comarison with 8 The utility 74 was irst introduced by Saha [3] and subsequently used in dierent alications by Abdellaoui, Barrios and Wakker [] and Holt and Laury [7]. 9 Although it ollows rom the τ R unction 76 that relative risk aversion will be constant i β =, given that Saha [3] rules out the case where β = in the deinition o the amily, it is necessary to modiy the deinition as done in Abdellaoui et al. []. 23

24 n n n a n b Figure 8: n n Figure 9: 24

25 Figure 7, the = contour has isolated in Figure Equilibrium Price and Risky Asset Suly In this Section, we investigate the relationshi between the equilibrium rice ratio and the suly o the risky asset in a single "reresentative" agent economy. We show that when and only when the risky asset is urgently needed, the equilibrium relative rice increases with its suly. This seemingly counter intuitive equilibrium rice behavior can be exected to occur more readily in uncertainty than in certainty settings because, as discussed in Section 3.2, under uncertainty a good is more likely to be urgently needed. Extending the results to economies with heterogeneous agents is straightorward or the case o aggregation see the classic aers o Chiman [9] and Rubinstein [3] and need not be discussed here. Consider a standard single agent exchange economy setting, where the agent s reerences satisy the assumtions in Subsection 3.. subject to max n,n, n = max EW n,n The reresentative agent solves ξn + ξ n 77 n + n = n + n, 78 where n and n denote, resectively, endowments o the risky and risk ree assets. Equilibrium rices, ensure that markets clear. 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. The otimal oint corresonds to the tangency oint n, n in Figure. Given the single agent setting, 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. Without loss o generality, we will use the risk ree asset as the numeraire. Since, as noted in Section 3., our assumtion that E ξ > ξ imlies n >, a ositive endowment o the risky asset n > will be assumed, as is standard, throughout the remainder o this aer. On the other hand, we allow n, which runs contrary to the conventional assumtion that the net suly o bonds is zero e.g., [4]. In recent years, a number o aers have aeared which consider the cases o ositive and negative net sulies o bonds see ootnote 3. ollows In [28], the authors summarize the argument or not requiring n = as 2 It should be noted that unlike the WACRRA case in Figure 7, the never declines with increasing n. 25 = contour in Figure 9

26 Figure : 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. [28],. 3 In much o this literature the assumtion that n is ositive, zero or negative is made or analytic convenience or to acilitate a articular discussion such as deicits. 2 However, as we will see, the sign assumtion on n can signiicantly imact whether the equilibrium rice ratio / increases or decreases with the suly o the risky asset. Beore giving our general result relating the risky asset s equilibrium relative rice and its suly, we irst consider the rice-suly curve or the two Examles in Subsection 3.5. For the WACRRA and CRRA cases, assuming n =.4 and the same arameter values 72, we lot the equilibrium rice ratio / versus n in Figure a. It can be seen that or the δ = δ 2 case, the equilibrium rice ratio will always decrease with n and or the δ > δ 2 case the rice ratio will irst decrease and then increase with n. This is consistent with Figure 2 Cass and Pavlova [7] observe that while nonnegativity assumtions or commodity endowments are very deensible, there is nothing contradictory in droing these assumtions when considering inancial assets, esecially when there are no restrictions on asset trade. 26

27 a b Figure : 7 since i one draws a horizontal line at n =.4, it ollows that i or δ = δ 2 always negative and ii or δ > δ 2 as n increases the quantity and then becomes ositive. 22 is is at irst negative For the latter case, the zero sloe oint in Figure a corresonds to the oint on the = contour in Figure 7 where n, n = n, n. For the Exo-Power case, assuming n =., α =, β =.8 and the same arameter values 72, we lot the equilibrium rice ratio / versus n in Figure b. As in the δ > δ 2 WACRRA case, the rice ratio irst decreases and then increases with n. This attern is also consistent with the act that in Figure 9 along the n =. horizontal, as n increases the value o is irst negative and then becomes ositive. 23 Given the above discussion, the seemingly uzzling rice-suly behavior in Figures a and b can be easily exlained by the risky asset being urgently needed as summarized by the ollowing 22 It should be noted that in Figure 7 as n increases, the = contour eventually curves down and would intersect an n =.4 horizontal ray twice. Thereore, the equilibrium rice ratio / will decrease again when n is sui ciently large. 23 It should be noted that when τ A < and τ R >, it is not ossible to conclude rom Proosition 3ii whether or not > in the n < ortion o asset sace. However Proosition 5iii rovides one examle where this is the case. It is ossible to create an another examle utilizing the Exo-Power utility. Assuming α = β =.5, it is straightorward to show that there exists a region o the n < ortion o asset sace in which the risky asset is urgently needed and or a single agent economy the equilibrium rice ratio / increases with n. 27

28 straightorward Proosition. 24 Proosition 7 Assume a single agent exchange economy, where the otimization roblem is given by eqns Then. 79 Proo. Given that n, n = n, n in equilibrium, it ollows rom the irst order condition that the resulting equilibrium rice ratio is given by ] = W n E [ ξw ξn + ξ n = ]. 8 n,n =n,n E [ξ W ξn + ξ n Hence eqn. 79 holds. 25 Remark 6 Combining Proositions and 7, it ollows immediately that the rice o a commodity or asset increases with its suly i and only i its Engel curve is downward sloing. In the case o commodities where quantities are always assumed to be ositive, this is equivalent to a good being inerior. This is consistent with the argument o Nachbar [26] that in a multigood setting, rice can increase with suly only i the comosite commodity ormed by the other commodities is inerior. In the case o assets, as discussed above, an asset is normal i the quantity is ositive and increases with income or negative and decreases with income. Thus it ollows rom Proosition 7 that whether the risky asset s relative rice increases with its suly does not deend on whether the risk ree asset is an inerior good but rather whether its Engel curve is decreasing with income or the risky asset is urgently needed. 26 Combining Proositions 2-6 with 7, one can obtain alternative conditions or when the equilibrium rice ratio / increases with the risky asset suly n. In the cases o Proositions 5 and 6, the conditions deend solely on the suly o the risk ree asset n. examle, combining Proositions 5 and 7, one obtains the ollowing very simle necessary and sui cient condition. 24 An earlier version o this Proosition was roved in [2]. Proosition 7 subsumes the rior result and exresses the equilibrium comarative static result in terms o the urgently needed reerence roerty. 25 I an equilibrium exists, the no arbitrage condition is automatically satisied and the no bankrutcy condition in the equilibrium setting is given by in ξn + ξ n >. Note that or some secial orms o utility such as 44, the no bankrutcy condition needs to be modiied. For examle in this case, the condition is given by in ξn + ξ n > a. 26 Also see the discussion o Kohli [2] in a two commodity, certainty distribution economy setting where one commodity is assumed to be a Gien good. 28 For

29 Corollary Assume a single agent exchange economy, where the otimization roblem is given by eqns Let the agent s NM index W z be given by W z = z a δ, 8 δ where δ > and a is allowed to be negative, zero or ositive. Then n a. 82 ξ This result covers the constant, decreasing and increasing relative risk aversion members o the HARA class corresonding resectively to the Tye i, ii and iii utilities in Proosition 5. For Tye ii where a >, i one assumes, as is standard, a zero suly o the risk ree asset, then it is always the case that the equilibrium rice ratio / increases with the suly o the risky asset. Remark 7 Given the orm o reerences in Corollary, it ollows rom Kubler, Selden and Wei [22] that or the economy as a whole, risk ree asset demand can only decrease with income and the equilibrium rice ratio / increase with n i the agent is not too risk averse as characterized by where δ < δ critical, 83 δ critical = de ln π22ξ ξ22 π 2ξ 2 ξ ln ξ 22 ξ However eqn. 82 in Corollary seems to be indeendent o δ. I we suose that the reresentative agent is very risk averse with δ being very large, it seems quite counterintuitive that the risky asset can become urgently needed. To exlain this aradox, note that when δ is very large, there should be no reason or the reresentative agent to hold a small quantity o the risk ree asset or to even short it. However i the market suly n < a, the agent ξ has no choice but to hold a small amount o the risk ree asset in equilibrium. Note that the threshold risk aversion arameter δ critical in 84 is determined by the equilibrium rice ratio. When n = a, the reresentative agent s risk aversion arameter δ will match the threshold ξ δ critical exactly and hence =. I n < a, then the threshold δ ξ critical is greater than the reresentative agent s δ, imlying that the agent is not risk averse enough and hence >, or the risky asset becomes urgently needed. To show this more ormally, we 29

30 next establish the relation between δ critical and equilibrium asset sulies. condition gives The irst order = π 2ξ 2 ξ2 n + ξ n a δ + π22 ξ 22 ξ22 n + ξ n a δ π 2 ξ ξ2 n + ξ n a δ + π22 ξ ξ22 n + ξ n a, 85 δ imlying that π 22 ξ ξ 22 π 2 ξ2 ξ = = π 2 ξ π 2 ξ 2 n+ξ n a δ +π 22 ξ 22 ξ 22 n+ξ n a δ 22 ξ π 2ξ 2 n+ξ n a δ +π 22ξ 22 n+ξ n a δ π 2 ξ 2 π 2ξ 2 ξ 2 n+ξ n a δ +π22 ξ 22 ξ 22 n+ξ n a δ π 2ξ 2 n+ξ n a δ +π 22ξ 22 n+ξ n a δ ξ22 n + ξ n a +δ. 87 ξ 2 n + ξ n a Thereore, we have δ critical = In the secial case where n = a ξ, + δ ln ξ 22 n+ξ n a ξ 2 n+ξ n a. 88 ln ξ 22 ξ 2 δ critical = + δ lnξ 22/ξ 2 = δ, 89 lnξ 22 /ξ 2 which veriies the conclusion that i n = a, then δ = δ ξ critical. Moreover, i we consider the contingent claim setting assumed in Kubler, Selden and Wei [22] and deine it can be easily veriied that imlying that c 2 = ξ 2 n + ξ n and c 22 = ξ 22 n + ξ n, 9 ξ 22 n + ξ n a ξ 2 n + ξ n a = τ A c 2 τ A c 22, 9 τ A c 2 τ A c 22 ξ 22 n a δ critical δ ξ 2 ξ I which is consistent with Theorem ii in Kubler, Selden and Wei [22]., 92 n,n=n,n clear interretation or the unusual absolute risk aversion ratio τ A c 2 /τ A c 22. This rovides a 3