1646 B.Keren,J.S.Pliskin European Journal of Operational Research 174 (2006)1643-1650 be verified that for a risk-neutral newsvendor with a linear utility (ux)=a+bx),Eq.(6)converts back to the critical ratio of(4). Now we use (6)and(7)to show that O*for a risk-averse newsvendor is less than O*for a risk-neutral one.For simplicity of the proof,we set all the utility function values at the right hand side of(6)and(7)to be positive.This can be done since a Von Neumann and Morgenstern(1944)utility function is defined up to a positive linear transformation.We use the theorem that states that the ranking of a group of alternatives will remain constant,using the utility functions u(x)or u*(x)=a+bu(x),independent of the values of x,as long as b>0.Proof for this is presented by Levy and Sarnat(1984,pp.121-122). The ratio on the left hand side of (7)is larger than 1,1.That can be verified since this con- (πj(c+) dition can be written as(S-c(S+h)+Sπ>π·c.Under our pre-assumptionsπ>O,S>cand (c+h)>0,therefore Sn>c and (S-c)(S+h)>0.An interesting conclusion here is that the maxi- mum expected profit determines o*,so that (TP(D=Blo=O))>(TP(D=Al=O)),i.e.given that the newsvendor orders o*,the total profit under this condition will be larger if D=B occurs,than if D=A occurs. Now,since u(TP(D =Blo=O))>u(TP(D=AlO=O)),the rank of the utilities from the basic state- of-natural events can be ordered u(TP(D==))>u(TP(D=BlO=))>u(TP(D=Al0=0))>0. (8) Our solution of Eq.(6)is a special case of Eeckhoudt et al.'s(1995)result.It presents a simple closed form solution and hence can serve as a benchmark. Claim.If the demand for newspapers is uniformly distributed over A,B],a risk-averse newsvendor with a uniformly more concave utility function sets his O*to a lower value than a less risk-averse newsvendor.We note here that a more concave utility function is more risk-averse according to Pratt's (1964)measure of risk, r(x)=-'(x)/u'(x) Proof.Let us assume that for a risk-averse newsvendor with utility function ui(x),the optimal order quan- tity Oi was found and all the values of the right hand side of(6)were set to be positive.Under this assump- tion,the first order condition (6)can be written as follows: (h+S)(S-c+)u1[(S-c)Oi]-u[(S+h)A-Oi(c+h)] (9) (π)(c+h) [(S-c)Q1-[(S-c)Q+π(-B)] We use the concave utility function fx)to generate a more concave utility function ua(x)=fu(x)).Now we replace u()by u()on the right hand side of(6).The result of this operation,as shown in Lemma 1(see Appendix A),is that the right hand side of(6)increases and therefore we get (h+S)(S-c+π)2[S-c)2]-2I(S+h)A-21(c+h)] (10) (π(c+h) 2[(S-c)21]-[(S-c)21+π(21-B)] According to Lemma 2(see Appendix A),the right hand side of(6)increases monotonically with O1.There- fore,in order to attain equality for (10),O must be decreased.The final conclusion is that >3. Corollary.A risk-averse newsvendor with a concave utility function sets O*less than a newsvendor who is risk- neutral. For the right hand side ratio of ()the ratiohas three important properties: (1)This ratio is 0 for o=4 and oo for O=B. (2)Given any value of O that satisfies the condition of Ratio >1,from this point and up to O=B,the ratio increases monotonically with O.The proof is given in Lemma 2.be verified that for a risk-neutral newsvendor with a linear utility (u(x) = a + bx), Eq. (6) converts back to the critical ratio of (4). Now we use (6) and (7) to show that Q* for a risk-averse newsvendor is less than Q* for a risk-neutral one. For simplicity of the proof, we set all the utility function values at the right hand side of (6) and (7) to be positive. This can be done since a Von Neumann and Morgenstern (1944) utility function is defined up to a positive linear transformation. We use the theorem that states that the ranking of a group of alternatives will remain constant, using the utility functions u(x) or u*(x) = a + bu(x), independent of the values of x, as long as b > 0. Proof for this is presented by Levy and Sarnat (1984, pp. 121–122). The ratio on the left hand side of (7) is larger than 1, ðhþSÞðScþpÞ ðpÞðcþhÞ > 1. That can be verified since this con￾dition can be written as (S c)(S + h) + Sp > p Æ c. Under our pre-assumptions p > 0, S > c and (c + h) > 0, therefore Sp > p Æ c and (S c)(S + h) > 0. An interesting conclusion here is that the maxi￾mum expected profit determines Q*, so that ðTPðDe ¼ BjQ ¼ Q ÞÞ > ðTPðDe ¼ AjQ ¼ Q ÞÞ, i.e. given that the newsvendor orders Q*, the total profit under this condition will be larger if De ¼ B occurs, than if De ¼ A occurs. Now, since uðTPðDe ¼ BjQ ¼ Q ÞÞ > uðTPðDe ¼ AjQ ¼ Q ÞÞ, the rank of the utilities from the basic state￾of-natural events can be ordered uðTPðDe ¼ Q jQ ¼ Q ÞÞ > uðTPðDe ¼ BjQ ¼ Q ÞÞ > uðTPðDe ¼ AjQ ¼ Q ÞÞ > 0. ð8Þ Our solution of Eq. (6) is a special case of Eeckhoudt et al.s (1995) result. It presents a simple closed form solution and hence can serve as a benchmark. Claim. If the demand for newspapers is uniformly distributed over [A, B], a risk-averse newsvendor with a uniformly more concave utility function sets his Q* to a lower value than a less risk-averse newsvendor. We note here that a more concave utility function is more risk-averse according to Pratts (1964) measure of risk, r(x) = u00(x)/u0 (x)). Proof. Let us assume that for a risk-averse newsvendor with utility function u1(x), the optimal order quan￾tity Q 1 was found and all the values of the right hand side of (6) were set to be positive. Under this assump￾tion, the first order condition (6) can be written as follows: ðh þ SÞðS c þ pÞ ðpÞðc þ hÞ ¼ u1½ðS cÞQ 1 u1½ðS þ hÞA Q 1ðc þ hÞ u1½ðS cÞQ 1 u1½ðS cÞQ 1 þ pðQ 1 BÞ . ð9Þ We use the concave utility function f(x) to generate a more concave utility function u2(x)  f(u1(x)). Now we replace u1(Æ) by u2(Æ) on the right hand side of (6). The result of this operation, as shown in Lemma 1 (see Appendix A), is that the right hand side of (6) increases and therefore we get ðh þ SÞðS c þ pÞ ðpÞðc þ hÞ < u2½ðS cÞQ1 u2½ðS þ hÞA Q1ðc þ hÞ u2½ðS cÞQ1 u2½ðS cÞQ1 þ pðQ1 BÞ . ð10Þ According to Lemma 2 (see Appendix A), the right hand side of (6) increases monotonically with Q1. There￾fore, in order to attain equality for (10), Q1 must be decreased. The final conclusion is that Q 1 > Q 2. h Corollary. A risk-averse newsvendor with a concave utility function sets Q* less than a newsvendor who is risk￾neutral. For the right hand side ratio of (6), the ratio u½ðScÞQu½ðSþhÞAQðcþhÞ u½ðScÞQu½ðScÞQþpðQBÞ has three important properties: (1) This ratio is 0 for Q = A and 1 for Q = B. (2) Given any value of Q that satisfies the condition of Ratio > 1, from this point and up to Q = B, the ratio increases monotonically with Q. The proof is given in Lemma 2. 1646 B. Keren, J.S. Pliskin / European Journal of Operational Research 174 (2006) 1643–1650