Income and Substitution Effects of Increases in Risk when Payoffs are Linear in the Random Variable. Carmen F. Menezes and X.

Transcription

1 Inome an Substitution Effets of Inreases in Risk hen Payoffs are Linear in the Ranom Variable Carmen F. Menezes an X. Henry Wang Department of Eonomis niersity of Missouri-Columbia Columbia, MO 65211

2 Abstrat Payoff funtions that are linear in the ranom ariable arise in a ie ariety of eision moels uner unertainty. The eomposition of the effet of inrease risk on eisions into inome an substitution terms for suh moels has reeie muh attention in the literature. This paper proies a Hiks-Slutsky eomposition of the effet of Rothshil-Stiglitz inreases in risk on the optimal eision. To measures of aersion to aitional risk are introue. Their behaior is shon to ontrol the signs of the inome an substitution effets. JEL Classifiation Numbers: D21, D81. Key Wors: Inome effet, substitution effet, unit-risk premium. 1

3 1. Introution Many eision problems uner unertainty gie rise to payoff funtions that are linear in the ranom ariable. 1 Consequently, there has been onsierable interest in the effet of inreases in the riskiness of the ranom ariable on the optimal eision in suh moels an in the eomposition of this effet into inome an substitution terms. While the literature has obtaine neessary an suffiient onitions on preferenes to sign the effet of inrease risk on the optimal eision, a Hiks-Slutsky eomposition of this effet has yet to be erie. Sanmo (1971) separate this effet into Slutsky-like inome an substitution terms. Dais (1989) proie a theoretial founation for Sanmo s eomposition by shoing that it an be erie using a ranom-ompensation metho, an Haar an Seo (1992) extene Dais analysis to general shifts in the riskiness of the ranom ariable. As Dais (1989, p. 133) note, there is no meaningful relationship beteen the Slusky-like inome effet an the inome eriatie (hih shos ho the optimal eision hanges hen inome hanges but risk is unhange). In a separate eelopment, Pope an Chaas (1985) erie the Hiks-Slutsky equation for an inrease in the expete alue of the ranom ariable in a proution moel ith a payoff that is linear in the ranom ariable. Their ompensation metho is the unertainty ounterpart of the Hiks-Slutsky inome ompensation uner ertainty. In this paper, e use uality to erie the Hiks-Slutsky equation for a Rosthshil-Stiglitz (RS) inrease in the riskiness of the ranom ariable. Our ompensation metho is similar to that use by Pope an Chaas (1985) an Chaas an Larson (1991). For expositional oneniene, an folloing Haar an Seo (1992) an Pope an Chaas (1985), our analysis is presente in terms of the stanar problem of a ompetitie firm faing prie unertainty. In setion 2 e present the firm s eision problem, its ual, an erie the Hiks-Slutsky equation for an RS inrease in prie unertainty. In setion 3 e present to measures of aersion to aitional risk, alle unit-ross an unit-enogenous risk premiums. In setion 4 e relate the behaior of these risk premiums to the signs of the inome an substitution effets an proie a geometri illustration of these 1 For examples of the ariety of ontexts in hih suh payoffs arise, see Bigelo an Menezes (1995). 2

4 effets. Conluing remarks are in setion The Moel Consier a ompetitie firm hih hooses its output q before market prie ~ p is knon so as to maximize the expete utility of final ealth ~, i.e., b Max q Eu( ~ ) u ( + pq (q))f(p, β. (1) a ) In (1), E is the expetation operator an u( ) is the firm s on Neumann-Morgenstern utility funtion, hih is inreasing an onae. The firm s initial (non-ranom) ealth is, an its total ariable ost funtion is (q). 2 The istribution funtion of ~ p is F(p, β) ith support [a, b] (a > ), here β is an inex of Rothshil-Stiglitz (RS) risk. The RS riskiness of ~ p inreases as β inreases. To proie insights about the effets of inrease prie risk on the output eision, e transform (1) into an equialent onstraine optimization problem. Let p enote the expete prie an p ~ p p. 3 The firm s expete ealth is + pq (q), an its total ealth is ~ +, here p q is the atuarially neutral risky omponent of total ealth. Sine epens on the hoie ariable q, e refer to as enogenous risk. The firm s expete utility funtion Eu( ) inues a preferene orering oer pairs (q, ), here q is the sale of enogenous risk. 4 This inue orering is represente by the erie utility funtion q (,, β ), efine by q (,, β ) u ( + pqf(p ), β ). From the properties of u, the erie utility funtion is ereasing in q an β, inreasing in an 2 The firm s initial ealth an be positie, negatie or zero. 3 The istribution of p is the same as that of ~ p but its support is [, ] ith a p an b p. 4 Changes in q inue enogenous hanges in the riskiness of reenues an profits, hile hanges in β represent exogenous hanges in risk. 3

5 onae in q an. Let I {(q, ) ( qβ,, ) } enote the set of (q, ) bunles hih hae expete utility. Consier to bunles ( q1, 1 ) an ( q, ) 2 2 from I. By efinition, the firm is inifferent beteen enogenous risk-expete ealth pairs ( p q 1, ) an ( p q 1 2, ). The graph of I 2 is therefore the lous of enogenous risk-expete ealth pairs among hih the firm is inifferent. The slope of an inifferene ure, the marginal rate of substitution beteen enogenous risk an expete ealth, is m( qβ,, ) q (, q, β) (, q, β ) Epu [ '( + pq)], (2) Eu'( + p q) here subsripts of funtions enote partial eriaties. 5 The marginal rate of substitution gies the minimum amount of expete ealth require to ompensate the firm for a unit inrease in the sale of enogenous risk. From the properties of, m is positie an is inreasing in q an along any inifferene ure of. The eision problem (1) an be reritten as a onstraine maximization in hih both q an are hoie ariables, i.e., Max ( q,, β ) {q, } (3) s.t. + pq (q). The solution to (3), ( (p,, β ), (p,, β) ) q, satisfies the neessary onitions m(q,, β) p ' (q), + pq (q). (4) Figure 1 illustrates the solution to (3). The optimal point ( q, ) is here the inifferene ure, 5 In setion 3, e interpret the marginal rate of substitution m as the risk premium per unit inrease in enogenous risk. 4

6 , an the opportunity onstraint, + pq (q), are tangent. 6 The firm s (unompensate) output supply is (p,, β ) an its expete ealth is (p,, β ). q <Insert Figure 1 here> The ual to the firm s expete utility maximization problem (3) is to hoose q an so as to minimize the amount of initial ealth require to attain a gien leel of expete utility, i.e., Min [pq (q)] {q, } (5) s.t. ( q,, β ). C C The solution to (5), ( q (p,, β ), (p,, β) ), satisfies the neessary onitions m(q,, β) p ' (q), (q,, β). (6) q C (p,, β ) is the firm s ompensate output supply. As in the original problem (3), the solution to the ual problem (5) ours here the erie inifferene ure is tangent to the opportunity onstraint. C To relate ompensate supply q (p,, β ) an (unompensate) supply (p,, β ), e efine the ompensation funtion q C(p,, β ) C C C (p,, β ) [pq (p,, β) (q (p,, β))]. (7) C is the minimum ealth in exess of require for the firm to attain expete utility. Supply an ompensate supply oinie hen C(p,, β ), i.e., q (p, C + C(p,, β), β) q (p,, β ). (8) This ientity proies the basis for eomposing the firm s output supply into inome an substitution effets. 6 In this an subsequent graphial illustrations, the ost funtion (q) is assume to be linear in q so that the opportunity onstraint is a straight line. 5

7 If ( q (p,, β ), (p,, β), β) gies the Hiks-Slutsky equation for an inrease in RS risk, ifferentiating (8) an using the fat that C(p,, β ) q C ( ) q ( ) q ( ) C( ). (9) β β β In (9), [ q / ] [ C/ β] is the inome effet; it is the negatie of the prout of the inome eriatie q / an the ompensation C / β require to keep expete utility onstant. The ompensate term q C / β is the substitution effet nit-risk Premiums In this setion e present to measures of aersion to aitional risk hose behaior respetiely ontrols the signs of the substitution an inome effets of inreases in RS risk. We first introue the unit-ross risk premium, hih is the Ross risk premium per unit inrease in RS risk. It is a measure of aersion to exogenous aitions in risk. We sho that the unit-ross risk premium is the marginal rate of substitution beteen the RS risk parameter an ealth orresponing to the erie utility funtion. We then onsier a measure of aersion to aitions to enogenous risk, alle unit-enogenous risk premium. It is the marginal rate of substitution beteen the sale of enogenous risk an ealth orresponing to the erie utility funtion. Ross (1981) efine the ounterpart of the Arro-Pratt risk premium for situations inoling unaoiable bakgroun risk ~ +q p. Let ~ ε be a ranom ariable ith E( ~ ε p ). The Ross premium π is efine impliitly by the equation E u(+q( p+ε ~ )) E ε u( π + qp ). (1) E p ~ p p π is the maximum amount that the iniiual is illing to pay to aoi the risk ~ ε hen +q p is 7 A Hiks-Slutsky equation an be erie for a hange in any parameter in the moel. For example, if β is replae by the expete prie (9) gies the Hiks-Slutsky equation erie by Pope an Chaas (1985, eq. (13)). 6

8 unaoiable. Rothshil an Stiglitz (1972) hae shon the equialene beteen seeral haraterizations of inreasing risk. Their ork implies the existene of β > suh that the istribution of p+ε ~ is F( p, β+ β). 8 We an therefore rerite (1) as u ( + qp )F(p, β + β) u ( π + qp )F(p, β). (11) From (11), in the limit as β, π/ β beomes Π π lim β β u( + qp )Fβ (p, β). (12) Eu' ( + qp ) We all Π the unit-ross risk premium. From (16) belo, the unit-ross risk premium is the eriatie of the ompensation funtion ith respet to the RS risk parameter. Hene, the unit-ross risk premium is the ompensation in initial ealth require to keep expete utility onstant hen there is an infinitesimal inrease in exogenous risk. In Figure 2, Π is the ertial istane beteen the inifferene ures for RS risk parameters β an β (> β) both haing the same expete utility. 9 <Insert Figure 2 here> The enogenous risk premium ( ) is the Arro-Pratt risk premium for the inrease in the sale ( q) of enogenous risk, efine by u ( + (q + q)p )F(p, β) u ( + qp )F(p, β). (13) From (13), in the limit as q, / q beomes m lim q q E[p u' ( + qp )]. (14) Eu' ( + qp ) We all m the unit-enogenous risk premium. From (2), the unit-enogenous risk premium is the marginal rate of substitution beteen enogenous risk an expete ealth, i.e., the minimum amount of expete ealth require to ompensate for a unit inrease in the sale of enogenous risk. 8 See Theorem 2 of Rothshil an Stiglitz (197) an Theorem 3 of Mahina an Pratt (1997). 9 Note that any to suh inifferene ures hae the same ertial interept an the one ith a higher RS risk parameter must lie aboe the other eeryhere else. 7

9 It shoul be note that the unit-ross risk premium is proportional to the unit-enogenous risk premium hen RS inreases in risk take the often use multipliatie form, ~ p p + βp. In this ase, Π β E[u' ( E[u' ( + βp q)p q] qm / β. (15) + βp q)] 4. Inome an Substitution Effets We no sho that the signs of the inome an substitution effets are respetiely ontrolle by the behaior of the unit-enogenous risk premium an the unit-ross risk premium. We first present expressions for the inome an substitution effets. Applying the enelope theorem to (7) gies C β β u( + qp )Fβ (p, β) Eu' ( + qp ). (16) Totally ifferentiating (4) gies q m J, (17) here J ( mq + m m ) ''( q) is the Jaobian eterminant for (4) an is negatie by the seon orer onition for (3). By (16) an (17), Inome effet q C β m β [ ] [ ]. (18) J From (18), the inome effet is the negatie of the prout of the inome eriatie q / an the unit-ross risk premium Π ( C/ β). Hene, the sign of the inome effet is opposite to the sign of the inome eriatie. This is beause the inome effet is obtaine by remoing the inome ompensation require to keep expete utility unhange hen the RS risk parameter inreases. Totally ifferentiating (6) gies Substitution effet q C m β m β J β. (19) 8

10 The theorem that follos relates the signs of the inome an substitution effets to the behaior of the to unit risk premiums. Theorem 1. (i) The inome effet of an RS inrease in prie risk is negatie (positie) if an only if the unitenogenous risk premium m is ereasing (inreasing) in ealth, an this happens if an only if absolute risk aersion u"() / u' () is ereasing (inreasing) in. (ii) The substitution effet of an RS inrease in prie risk is negatie (positie) if an only if the unit-ross risk premium Π is inreasing (ereasing) in q along a erie inifferene ure. Proof: (i) Sine J <, β <, (14) ith respet to yiels, >, the first part of (i) is immeiate from (18). Differentiating m in [ Eu' ][ E( p u")] [ Eu"][ E( p u' )] E[( m ~ p '( q)) u"]. ( Eu') 2 Eu' Sine, as is ell-knon, E[( ~ p ' (q))u"] is positie (negatie) if an only if absolute risk aersion is ereasing (inreasing), 1 the seon part of (i) follos. (ii) From (12), (16) an (19), Π q 1 ( ) 2 β q β q 1 ( [ + m ] [ + m ]) ( ) 2 qβ mβ m C β q J. β Sine J <, (ii) follos from the aboe equality. β β q 1 See Sanmo (1971, p. 69). 9

11 Theorem 1(i) shos that, uner ereasing absolute risk aersion, the unit-enogenous risk premium is ereasing in ealth an the inome effet is negatie. Theorem 1(ii) shos that if the unit- Ross risk premium is inreasing along a erie inifferene ure the substitution effet is also negatie. 11 That is, the to effets ork in the same iretion. Figure 3 illustrates the inome an substitution effets. The soli β-inifferene ure represents the firm s preferenes prior to an inrease in the RS riskiness of p. The ashe β -inifferene ures represent preferenes after the inrease in the riskiness of p. The firm is initially in equilibrium at point A on the β-inifferene ure. The moement from A to B is the substitution effet of the inrease in the riskiness of p. 12 It is obtaine by shifting upar the opportunity onstraint so that it is tangent to the β -inifferene ure (hih has the same expete utility as that of the β-inifferene ure). The moement from B to C is the inome effet of the inrease in the riskiness of p. 13 It is obtaine by moing from the ompensate optimal bunle (B) to the bunle (C) here a β -inifferene ure is tangent to the initial opportunity onstraint. The moement from A to C is the total effet of the inrease in the riskiness of p. <Insert Figure 3 here> 5. Conluing Remarks The approah an results obtaine for the ompetitie firm uner prie unertainty are ali for moels in hih the ranom payoff y ~ is linear in the ranom ariable z ~. Speifially, the payoff 11 Sine u <, Π must be inreasing immeiately to the right of q. Hene, Π is either non-monotone or it is XQLIRPO\ LQFHDVLQJ LQ T )RP ZH VHH WKDW DW OHDVW IR WKH PXOWLSOLFDWLYH IRP RI 56 LQFHDVHV LQ LVN LV inreasing in q along a erie inifferene ure sine m is inreasing in q along a erie inifferene ure. 12 The iretion of hange in expete ealth (namely, hether A is aboe or belo B in Figure 3) annot be etermine ithout further assumptions about preferenes. 13 Both effets ork to erease the firm s output. Speifially, the inome effet is negatie gien ereasing absolute risk aersion an the substitution effet is negatie gien that the unit-ross risk premium is inreasing in output along a erie inifferene ure. 1

12 funtion is of the form ~ y y + h(x) + g(x) ~ z (2) here y is non-ranom initial inome or ealth; g(x) an h(x) are onae in the hoie ariable x. Many eision problems hae this struture. An example is the to-asset (one safe, one risky) portfolio moel. In this moel, z ~ is the rate of return on the risky asset, h(x), an g(x) x enotes the amount ineste in the risky asset. Theorem 1(i) implies the ell-knon result that an inrease in y inreases risky inestment uner DARA. Theorem 1(ii) gies a ne result. It implies that a ompensate inrease in the riskiness of the return to the risky asset reues risky inestment an therefore inreases inestment in the safe asset if the unit-ross risk premium inreases along a erie inifferene ure. The ual approah use in this paper an be use to analyze the inome an substitution effets of hanges in other parameters in (2). For example, a hange in the expete alue of the ranom ariable ~ z simply rotates the opportunity onstraint in Figure 1. The effet of this rotation an be separate into inome an substitution effets. As another example, a tax on payoffs (e.g., profits) inues a rotation in the opportunity onstraint an a shift in the erie inifferene ures similar to that inue by an inrease in RS risk. 11

13 Referenes Bigelo, J.P. an Menezes, C.F. Outsie risk aersion an the omparatie statis of inreasing risk in quasi-linear eision moels. International Eonomi Reie, ol. 36, (1995), pp Chaas, J.P. an Larson, B.A. Eonomi behaior uner temporal unertainty. Southern Eonomi Journal, ol. 57 (1991), pp Dais, G.K. Inome an substitution effets for mean-presering spreas. International Eonomi Reie, ol. 3, (1989), pp Harar, J. an Seo, T.K. The effets of shifts in a return istribution on optimal portfolios. International Eonomi Reie, ol. 31, (199), pp Harar, J. an Seo, T.K. General hanges in unertainty. Southern Eonomi Journal, ol. 58, (1992), pp Mahina, M.J. an Pratt, J.W. Inreasing risk: Some iret onstrutions. Journal of Risk an nertainty, ol. 14, (1997), pp Pope, R.D. an Chaas, J.P. Prouer surplus an risk. Quarterly Journal of Eonomis, ol. 1, (1985), pp Ross, S.A. Some stronger measures of risk aersion in the small an in the large ith appliations. Eonometria, ol. 49, (1981), pp Rothshil, M. an Stiglitz, J.E. Inreasing risk I: A efinition. Journal of Eonomi Theory, ol. 2, (197), pp Sanmo, A. On the theory of the ompetitie firm uner prie unertainty. Amerian Eonomi Reie, ol. 61, (1971), pp