Technical Note – Price-Setting Newsvendor Problems with Uncertain Supply and Risk Aversion Burak Kazaz [email protected] Whitman School of Management Syracuse University Syracuse, NY 13244 Scott Webster [email protected] W.P. Carey School of Business Arizona State University Tempe, AZ 85287 November 3, 2014 Please do not distribute this manuscript without the permission of the authors. Technical Note – Price-Setting Newsvendor Problems with Uncertain Supply and Risk Aversion The price-setting newsvendor problem, which models the economic tradeoffs associated with uncertain demand of a perishable product, is fundamental to supply chain analysis. However, in settings such as agriculture, there is significant economic risk associated with supply uncertainty. We analyze how risk aversion and the source of uncertainty—demand and/or supply—affect tractability and optimal decisions. We find that concavity of the objective function is preserved under the introduction of risk aversion if the source of uncertainty is demand, but is not necessarily preserved if the source of uncertainty is supply. We identify a structural difference that explains this result, and show that this difference can lead to opposing directional effects of risk aversion on optimal decisions. Subject classifications: inventory/production; marketing: pricing 1. Introduction The classic price-setting newsvendor problem (PSNP) is to determine the price and quantity of a perishable product assuming uncertain demand, deterministic supply, and a risk-neutral decision-maker. We study a generalization of the classic PSNP that includes uncertain supply and a risk-averse decisionmaker. Our motivation comes from an agricultural setting, and in particular, a winemaker who leases vineyard space and sells direct-to-consumer through wine-club memberships. The leasing of farm space is common among many agri-businesses, including winemaking. In comparison with selling wine through distributors, the direct-to-consumer channel loses the benefit of pricing flexibility in response to the realized crop yield but offers less volatile demand in the marketplace. Smith-Madrone Winery, for example, sells exclusively through its wine club and tasting room, and does not change its wine price according to crop yield (Heron 2010). This is equivalent to fixing the selling price of the wine before the growing season begins, at the time the lease quantity is determined, yielding a PSNP with uncertain supply. Compared to large diversified firms such as Constellation Brands that owns and distributes wines under many different brands, smaller wineries such as Smith-Madrone have a relatively narrow product range and limited financial resources and are therefore more sensitive to risk in their decision making. This note is most closely related to two papers on the PSNP with demand uncertainty. Kocabıyıkoğlu and Popescu (2011) consider a risk-neutral newsvendor. They introduce an elasticity measure and use this measure to identify conditions that assure the optimal price and quantity can be obtained from a solution to the first-order condition. The nature of the price-demand function and the probability distribution 1 determine whether or not the conditions are satisfied. Eeckhoudt et al. (1995) study the impact of risk aversion on the newsvendor’s quantity decision. They assume that price is exogenous and show that a risk-averse newsvendor will order less than a risk-neutral newsvendor. We analyze the impact of supply uncertainty and risk aversion on the classic PSNP. We generalize the elasticity measure of Kocabıyıkoğlu and Popescu (2011) to accommodate uncertain supply and we show that the tractability conditions they identify continue to hold under a risk-neutral objective when supply, as well as demand, is stochastic. We investigate conditions for tractability when a newsvendor is risk averse. Toward this end, we consider the question of whether a concave expected profit function is assured to remain concave under a risk-averse utility function. We prove that the answer is “yes” when the source of uncertainty is demand and “no” when the source of uncertainty is supply. We show that the underlying cause of this result also drives opposing directional effects of risk aversion on univariate optimal price and quantity decisions. In particular, for a given price, a risk-averse newsvendor will order less than a risk-neutral newsvendor when demand is uncertain whereas the opposite is generally the case when supply is uncertain. Alternatively, for a given quantity, a risk-averse newsvendor will price higher than a risk-neutral decision-maker when supply is uncertain, whereas the opposite is generally the case when demand is uncertain. 2. Model We consider the price-setting newsvendor problem with a risk-averse decision-maker and either supply or demand uncertainty. We model the newsvendor’s risk aversion through concave utility function U(x), U (x) > 0, and U (x) 0 for all x. If U (x) = k for all x, then the newsvendor is risk neutral and expected utility is maximized by maximizing expected profit. Otherwise, the newsvendor is risk averse. The firm’s decision variables are quantity q and price p, and we capture randomness in supply and demand through random variables Y and Z, respectively. We use lower case y and z to denote realizations of these random variables. Random yield and random demand functions are q Y and d(p, Z). The finite support of Y is denoted [yL, yH] with yL 0, and the mean of Y is normalized to 1. We assume that the demand function d(p, z) is strictly decreasing in price p and is strictly increasing in realization z of Z, which has finite support [zL, zH]. We write the inverse of d(p, z) with respect to z as z(p, d), i.e., d(p, z(p, d)) = d and z(p, d(p, z)) = z. As in Kocabıyıkoğlu and Popescu (2011), we assume that the revenue function pd(p, z) is strictly concave in price for any realization z. The pdf and cdf of Y and Z are i() and i() for i {Y, Z}, respectively. If supply is deterministic, then the yield function is simply q; if demand is deterministic, then we write the demand function as d(p). 2 The end-of-season salvage value per unit is s. The purchase cost per unit is c. We define price and cost as net-of-salvage-value. Accordingly, the random profit under uncertain supply and deterministic demand is S p, q, Y = p min d p , qY cq sq Y 1 .1 The expected utility is S(p, q) = E U S p, q, Y and the newsvendor’s problem is PS: max S p, q p , q X where X denotes a convex set of viable (p, q) values. The corresponding functions and problem for the model with deterministic supply and uncertain demand are D p, q, Z = p min d p, Z , q cq , D(p, q) = E U D p, q, Z , and PD: max D p, q . p , q X 3. Sufficient Conditions for Concavity A general expression for expected lost sales is E[(d(p, Z) – qY)+], which reduces to E[(d(p) – qY)+] for PS and to E[(d(p, Z) – q)+] for PD. We identify sufficient conditions for joint concavity of the expected utility function that use the price-elasticity of marginal lost sales: p (p, q) = 2 E d p, Z qY pq . E d p, Z qY q (1) This is a general function that applies to both PS and PD by substituting in the appropriate deterministic terms. This function was originally introduced by Kocabıyıkoğlu and Popescu (2011) for a risk-neutral analysis of PD, though with a slightly different interpretation. In particular, they define the price-elasticity of the lost-sales rate. For PD, the lost-sales rate is P d p, Z q = 1 Z z p, q . Note that the lostsales rate is, excepting the sign, the marginal lost sales, i.e., E d p, Z q = 1 Z z p, q . q Thus, for PD, the price-elasticity of marginal lost sales given in (1) reduces to the price-elasticity of the lost-sales rate. However, for PS, the price-elasticity of marginal lost sales and the price-elasticity of the lost-sales rate are different functions. For example, given price p and cost c, the net price and cost are p = p – s and c = c – s > 0, and S = pmin{d(p), qY} – cq + s(qY – min{d(p), qY}) = pmin{d(p), qY} – cq + s(Y – 1)q. 1 3 Before introducing the results in this section, we require additional notation. For PS, we define two measures of marginal utility, u(p, q) = U ' S p, q, d p / q and u+(p, q) = d p/ q yL d p/ q U ' S p, q, y yY y / tY t dt dy . The first function, u(p, q), is the marginal utility when yL demand is equal to supply. Because yY y / d p/ q tY t dt is a valid density function defined over yL [yL, d(p)/q] (i.e., nonnegative and integrates to 1), u+(p, q) is a measure of marginal utility when demand is more than supply. Thus, from the concavity of U, it follows that u p, q / u p, q 1, with equality if the decision-maker is risk-neutral. The random yield function has a more restrictive structure than the random demand function (i.e., supply has a multiplicative form). Consequently, one might anticipate that the tighter structural limitations on the yield function translate into simpler or less restrictive conditions for assurance of concave S(p, q) than for concave D(p, q). We find the opposite. Proposition 1. (a) If (p, q)u(p, q)/u+(p, q) ½ for all (p, q) X, (2) then S(p, q) is concave. (b) If E D p, q, Z is concave, then D(p, q) is concave. (c) If (p, q) ½ for all (p, q) X, (3) then D(p, q) is concave, E S p, q, Y is concave, but S(p, q) is not necessarily concave. Kocabıyıkoğlu and Popescu (2011) show that (3) assures that E D p, q, Z is concave. Proposition 1c shows that (3) is also sufficient for concave D(p, q). Indeed, (3) is a fundamental indicator of tractability because it assures that the expected profit function is concave even when both supply and demand uncertainty are present (see Corollary 1A in the appendix). However, (3) is not sufficient to assure concavity when supply is uncertain and the newsvendor is risk-averse. The more restrictive condition (2) assures concave S(p, q); this condition also assures concave expected utility when both supply and demand uncertainty are present (see Proposition 1A in the appendix). Why do we see this difference in sufficient conditions for concavity? The reason stems from the polar-opposite character of the realized profit functions for these two problems. Specifically, in the neighborhood of (p, q), S p, q, Y is not concave at low-profit realizations (i.e., Y < d(p)/q) and concave at high-profit realizations (i.e., Y > d(p)/q), whereas the opposite is true for D p, q, Z , i.e., D p, q, Z is concave at low-profit realizations (i.e., Z < z(p, q)) and not concave at high-profit realizations (i.e., Z > 4 z(p, q)).2 With the introduction of risk aversion, this structural difference is problematic for PS because the character of the function at low-profit realizations is amplified relative to the character of the function at high-profit realizations, due to the diminishing marginal utility of profit. This intuition is generalized in Lemma 2A of the appendix where we show that the expected utility of a multivariate function is concave if (i) the expected value of the function is concave and, informally speaking, (ii) any non-concave realizations of the function correspond to high-value realizations of the function. In the next section, we will see that this structural difference between PS and PD explains why the introduction of risk aversion can lead to opposite directional effects on optimal decisions. 4. Effect of Risk Aversion on Optimal Decisions We compare optimal decisions of risk-neutral and risk-averse newsvendors. We assume a strictly concave utility function; otherwise the inequalities in our results below would not be strict. We assume the riskneutral and risk-averse newsvendor objective functions have a unique stationary point that is a global maximum (e.g., conditions of Proposition 1 hold). For both PS and PD, we let po(q) denote the optimal price for a given quantity, and we let qo(p) denote the optimal quantity for a given price. We use superscript * to denote optimal values for risk-neutral models. Eeckhoudt et al. (1995) consider PD and show that a risk-averse newsvendor will order less than a risk-neutral newsvendor for a given price (stated as Proposition 2c below). Proposition 2 expands on this result and highlights how the source of uncertainty affects the impact of risk aversion on optimal univariate decisions. Proposition 2. For PS: c s c s o * , (a) If yL , yH , then q (p) > q (p). p s s (b) po(q) > p*(q). For PD: (c) qo(p) < q*(p). (d) If d(p, z) = d(p)z, then po(q) < p*(q). (e) If d(p, z) = d(p) + z, then po(q) < p*(q). When the conditions of Proposition 2 hold, we see that the directional impact of risk aversion on optimal univariate decisions is affected by the source of uncertainty. If the source of uncertainty is supply, then optimal univariate decisions increase. If the source of uncertainty is demand, then optimal univariate 2 At a more detailed level, PS and PD exhibit opposing single-switch properties, which are defined and discussed in Section A.2 of the appendix. 5 decisions decrease. As discussed above, the introduction of risk aversion amplifies the concern for low profits, which are associated with low realizations of Y in the case of random supply and low realizations of Z in the case of random demand. Low values of Y translate into low sales and low profit due to insufficient supply, and the risk-averse decision-maker increases the quantity (for fixed price) or increases the price (for fixed quantity) to protect against this risk. In contrast, low values of Z translate into low sales and low profit due to excess supply, and the risk-averse decision-maker lowers the quantity (for fixed price) or lowers the price (for fixed quantity) to protect against this risk. Proposition 2a specifies a sufficient condition for the quantity inequality for PS: the support of Y lies within the ratio of cost-to-price and the ratio of cost-to-salvage value (recall that p and c are defined as net of salvage value s). If the support extends beyond this range, then the realized profit per unit and the realized cost per unit (i.e., purchase cost less salvage revenue) are negative at realizations of Y outside of this range (i.e., margin is negative at low realizations of supply and cost is negative at high realizations of supply). Relative to the risk-neutral newsvendor, the risk-averse newsvendor is more concerned about the possibility of negative margins and is less concerned about the possibility of negative costs. Both of these differences put downward pressure on the order quantity, potentially to the point where the risk-averse quantity is lower than the risk-neutral quantity (see Example 1 in the appendix). Propositions 2d and 2e state that multiplicative random demand or additive random demand are sufficient for the price inequality to hold for PD. Interestingly, this price relationship does not hold under a combined multiplicative-additive form of demand, e.g., d(p, z) = (p)z + (p) (see Example 2 in the appendix). The reason is that the conclusions in propositions 2d and 2e derive from distinct underlying sufficient conditions on the revenue function pd(p, z), and both of these conditions break down under a multiplicative-additive form of random demand (see Lemma 4A and the proof of Proposition 2 in the appendix). We next consider the impact of the introduction of risk aversion on the joint optimal price and quantity, denoted (p*, q*) and (po, qo) for the risk-neutral and risk-averse models, respectively. When the conditions of Proposition 2 hold, there is a force that puts a pressure on price and quantity to move in the same direction when the other decision is held fixed, either up or down depending on the source of uncertainty. However, there may be a countervailing force that puts pressure on decisions to move in opposite directions. This force stems from the inverse relationship in the price-demand function (i.e., dp(p, z) < 0), e.g., an increase in price exerts downward pressure on quantity, and vice-versa. Thus, in general, the impact of the introduction of risk aversion on joint optimal decisions can go either way (see Example 3 in the appendix). However, the impact of risk aversion on the joint optimal decisions is determinate if the optimal risk-neutral price and quantity are increasing in their respective arguments. As an example, consider PS at the risk-neutral optimal price and quantity given that the conditions in 6 Proposition 2 hold. The optimal response to the introduction of risk aversion is an increase in price (for fixed quantity) and an increase in quantity (for fixed price). In a risk-neutral model, the optimal response to an increase in price (quantity) is an increase in quantity (price). This alignment of directional forces assures that the risk-averse price and quantity are larger than the risk-neutral price and quantity. Proposition 3. For PS: c s c s , (a) If E S p, q, Y and S(p, q) each have a unique global maximum, yL , yH * , p s s q*(p) 0 for all (p, q*(p)) X, and p*(q) 0 for all (p*(q), q) X, then po > p* and qo > q*. For PD: (b) If E D p, q, Z and D(p, q) each have a unique global maximum, either d(p, z) = d(p)z or d(p, z) = d(p) + z, q*(p) 0 for all (p, q*(p)) X, and p*(q) 0 for all (p*(q), q) X, then po < p* and qo < q*. 5. Summary In this note we extend technical results on the tractability of the classic PSNP in Kocabıyıkoğlu and Popescu (2011) to problems with supply uncertainty and risk aversion. In addition, we analyze the impact of risk aversion on optimal decisions. We show how demand uncertainty and supply uncertainty are structurally different, and that this difference affects some results but not others. In particular, conditions on the price-elasticity-of-marginallost-sales function that assure concavity of the risk-neutral PSNP objective function are robust to the introduction of supply uncertainty. However, there are two areas where results are affected by the source of uncertainty. First, if only demand is uncertain, then previously identified tractability conditions for the risk-neutral PSNP extend to the risk-averse newsvendor (see (3)), whereas this is the not case if supply uncertainty is introduced. In particular, the problem with supply uncertainty exhibits a polar-opposite single-switch property than the problem with demand uncertainty, and this difference inhibits tractability under a risk-averse objective. As a consequence, the problem requires a more restrictive condition for assurance of tractability (see (2)). We note that our conditions for tractability are general in the sense that they extend to the problem with both supply and demand uncertainty, i.e., (3) is sufficient for concave expected profit and (2) is sufficient for concave expected utility. Second, the two sources of uncertainty tend to drive opposing directional effects of risk aversion on optimal decisions. Our results lend insight into how risk mitigation tactics differ between traditional settings in the literature where main source of uncertainty is demand and settings faced by firms such as Smith-Madrone Winery where the main source of uncertainty is supply. 7 6. References Eeckhoudt, L., C. Gollier, H. Schlesinger. 1995. The risk-averse (and prudent) newsboy. Management Science 41(5) 786-794. Kocabıyıkoğlu, A., I. Popescu. 2011. The newsvendor with pricing: A stochastic elasticity perspective. Operations Research 59(2) 301-312. Milgrom, P., C. Shannon. 1994. Monotone comparative statics. Econometrica 62(1) 157-180. Heron, K. 2010. Try the red: Napa learns to sell. The New York Times. February 16. Appendix In this appendix, we use the following to denote expected profit for PS and PD: S p, q E S p, q, Y D p, q E D p, q, Z . We use subscripts on functions to denote partial derivatives, e.g., S p = S / p . A.1 Examples This section presents four examples. The first three examples are referred to in the manuscript. Example 4 is used in the proof of Proposition 1. Example 1: qo(p) < q*(p) with [yL, yH] [(c + s)/(p + s), (c + s)/s] for PS c = 40, p = 60, s = 0, d(p) = 100 – p, [yL, yH] = [0, 2] [40/60, ), Y Uniform[0, 2], U(x) = 1 – e-0.000001x: qo(p) = 24.4927 < q*(p) = 24.4949 Example 2: po(q) > p*(q) with combined multiplicative-additive form of demand for PD c = 1, q = 20, d(p, z) = (10200 – 100p)z + 101 – p1/2, Z Uniform[0, 2], U(x) = 1 – e-0.0000000001x: po(q) = 94.0430 < p*(q) = 93.9916. Example 3: Instances of (1) po < p*, qo > q* and (2) po > p*, qo < q* for PD and PS PD: (1) c = 70, d(p, z) = (100 – p)z, Z Uniform[0, 2], U(x) = 1 – e-0.000000005x: po = 89.187 < p* = 89.226, qo = 4.6590 > q* = 4.6430. (2) c = 10, d(p, z) = (100 – p)z, Z Uniform[0.5, 1.5], U(x) = 1 – e-0.0001x: po = 56.7841 > p* = 56.7230, qo = 56.5599 < q* = 57.2860. PS: (1) c = 70, d(p) = 100 – p, Y Uniform[0, 2], U(x) = 1 – e-0.0000001x: po = 85.3122 < p* = 85.3132, qo = 8.1080 > q* = 8.1069. (2) c = 40, d(p) = 100 – p, Y Uniform[0,2], U(x) = 1 – e-0.000001x: po = 71.3548 > p* = 71.3475, qo = 19.1286 < q* = 19.1334. 8 Example 4: Concave S p, q and non-concave S(p, q) for PS We define a problem instance where S p, q is concave and (p, q) 1/2 over convex set X and where S(p, q) is not concave over X, i.e., there are points in X where the determinant of the Hessian of S(p, q) is negative. Let d(p) = 1 – p Y(y) = 1/, y [1 – 0.5, 1 + 0.5], (1, 2] (i.e., Y is a uniform random variable) x, x k U(x) = k , x k s = 0. A decision-maker would never set q > d(p)/(1 – 0.5) (i.e., all realizations of supply are greater than demand) nor q < d(p)/(1 + 0.5) (i.e., all realizations of supply are less than demand). Similarly, a decision-maker would never set p c or p 1 (i.e., d(1) = 0). For any (p, q) {(p, q) : p (c, 1), q (d(p)/(1 + 0.5), d(p)/(1 – 0.5))}, pq 1 p 1 0.5 S p, q = p 1 p cq 2 q q S p =1 2 p 2 2 2 p 1 p 1 p 1 0.5 1 0.5 q q 1 p q 1 0.5 1 3 p q 1 0.5 =1 2 p 2 q S pp = 3 p 2q 1 0.5 2 q p Sq = 2 S qq = S pq = p 2 q 1 0.5 d p q 2 2 1 p 1 0.5 c q p 1 p q3 2 <0 1 1 p 1 3 p 2 1 0.5 2 2 q q 1 p 1 0.5 E d p qY = 2 q 2 2 2 1 1 p E d p qY = 1 0.5 ≤ 0 2 q q 9 <0 1 p 2 E d p qY = ≥0 q2 pq (p, q) = 2 p 1 p 2 2 2 1 p q 1 0.5 q . If k pq(1 – 0.5) – cq, then S(p, q) = k for all realizations of Y. If k pd(p) – cq, then S(p, q) = S p, q . Therefore, we examine the case of k (pq(1 – 0.5) – cq, pd(p) – cq); for any (p, q) {(p, q) : p (c, 1), q [d(p)/(1 + 0.5), d(p)/(1 – 0.5)} and k (pq(1 – 0.5) – cq, pd(p) – cq), k cq k cq k cq cq Y S(p, q) = pqE Y | Y k 1 Y pq pq pq pq k cq 1 k cq k k cq 1 0.5 cq 1 0.5 1 0.5 =k pq 2 pq pq pq k cq 1 0.5 =k 2 pq q Sp = 2 2 2 k cq 2 pq k cq k cq 1 0.5 1 0.5 p 2 pq pq pq 1 = 2 k cq k cq 2 1 0.5 q 1 0.5 k cq p pq pq 1 = 2 2 2 1 2 k cq q 1 0.5 p q 1 = 2 2 2 1 k 2 2 2ck c q q 1 0.5 p q 2 2 2ck c 2 q 1 k Spp = 3 3 3 < 0 p p q p p Sq = 2 2 k k cq pq k cq 1 0.5 2 1 0.5 2 pq 2 pq pq 1 = 2 2 k cq k cq 2 2k k cq 1 0.5 p 2 1 0.5 p 1 0.5 q pq pq q 1 = 2 2k k cq q pq k cq 2 p 2c 1 0.5 p 1 0.5 pq 2 10 2 c2 2 1 k = 2 2c 1 0.5 p 1 0.5 p 2 pq 2 1 k Sqq = 3 < 0 pq 1 Spq = 2 2 2 2 k c 1 0.5 pq p For our example instance, we set = 1.8, c = 0.1, k = 0.01, and X = {(p, q) : p (0.2, 0.9), q ((1 – p)/(1 + /2), (1 – p)/(1 – /2))}. Note that X is a convex set. By applying the above expressions, it can be numerically verified that S pp < 0, S qq < 0, S pp Sqq S pq 2 > 0 for all (p, q) X, and (p, q) 1/2 for all (p, q) X. Similarly, it can be numerically verified that there are points in X where S pp Sqq S pq 2 < 0. As one example, at p = 0.25 and q = 1.75, we have k = 0.01 (pq(1 – 0.5) – cq, pd(p) – cq) = (-0.131, 0.013) (i.e., S(p, q) S p, q and S(p, q) k), and thus we apply the above expressions to obtain S pp Sqq S pq 2 = 0.025 S pp Sqq S pq 2 = -0.002. A.2 Single-Switch Properties In this section we define two opposing properties, discuss the relationship between these properties and the single-crossing property, and present a result on the preservation of concavity of a multivariate function under risk aversion. We use this result in the proof of Proposition 1 that appears in the next section. Lemma 3A in the next section shows that the risk-neutral objective function of PS satisfies one of these properties and the risk-neutral objective function of PD satisfies the other property. As discussed above, this structural difference between these two objective functions helps explain the differing results for PS and PD that appear in propositions 1 – 3. Let x = (x1, …, xn) and let F(x, z) be a function defined on convex set X [zL, zH] that is nondecreasing in its last element for any x X. We say that the function satisfies a single-switch property (with respect to its last dimension) if the difference between a convex combination of the function evaluated at two points in X (for given z) and the function evaluated at the convex combination of the two points switches sign at most once as z increases over its interval [zL, zH]. This notion is formalized in the following definitions. 11 Increasing-Single-Switch Property (SS+). For any x1 X, x2 X, and (0, 1) with x = x1 + (1 – )x2, if F(x1, z1) + (1 – )F(x2, z1) – F(x, z1) 0 implies F(x1, z1) + (1 – )F(x2, z1) – F(x, z1) 0 for all z z1, and if F(x1, z1) + (1 – )F(x2, z1) – F(x, z1) > 0 implies F(x1, z1) + (1 – )F(x2, z1) – F(x, z1) > 0 for all z z1, then F(x, z) satisfies the increasing-single-switch property. Decreasing-Single-Switch Property (SS-). For any x1 X, x2 X, and (0, 1) with x = x1 + (1 – )x2, if F(x1, z1) + (1 – )F(x2, z1) – F(x, z1) 0 implies F(x1, z1) + (1 – )F(x2, z1) – F(x, z1) 0 for all z z1, and if F(x1, z1) + (1 – )F(x2, z1) – F(x, z1) < 0 implies F(x1, z1) + (1 – )F(x2, z1) – F(x, z1) < 0 for all z z1, then F(x, z) satisfies the decreasing-single-switch property. The single-switch properties are similar in spirit to the single-crossing property (SX) that is used in comparative statics (Milgrom and Shannon 1994). The function F(x, z) satisfies SX if, for any x2 > x1, F(x2, z1) – F(x1, z1) 0 implies F(x2, z) – F(x1, z) 0 for all z z1, and F(x2, z1) – F(x1, z1) > 0 implies F(x2, z) – F(x1, z) > 0 for all z z1. A function F(x, z) that satisfies SS+ exhibits similar character to a function with partial derivative Fx(x, z) that satisfies SX, as shown below. Lemma 1A. F(x, z) is twice-differentiable with respect to x. If F satisfies SS+ or if Fx satisfies SX, then Fxx(x, z1) 0 implies Fxx(x, z) 0 for all z z1, and Fxx(x, z1) > 0 implies Fxx(x, z) > 0 for all z z1. Proof. Let h be a positive number. Because F satisfies SS+, if 0.5F(x, z1) + 0.5F(x + 2h, z1) – F(x + h, z1) 0 then 0.5F(x, z) + 0.5F(x + 2h, z) – F(x + h, z) 0 for all z z1. Thus, if lim F x 2h, z1 2 F x h, z1 F x, z1 h2 h0 Fxx x, z1 0 then lim F x 2 h , z 2 F x h, z F x , z h2 h 0 Fxx x, z for any z z1. Following the same steps with strict inequalities, Fxx(x, z1) > 0 implies Fxx(x, z) > 0 for any z z1. Because Fx satisfies SX, if Fx(x + h, z1) – Fx(x, z1) 0, then Fx(x + h, z) – Fx(x, z) 0 for any z z1, Thus, if lim h0 Fx x h, z1 Fx x, z1 h Fxx x, z1 0 12 then lim Fx x h, z Fx x, z h0 h Fxx x, z 0 for any z z1. Following the same steps with strict inequalities, Fxx(x, z1) > 0 implies Fxx(x, z) > 0 for any z z1. Lemma 2A. If E F x, Z is concave on X and F x, Z satisfies SS+, then E U F x, Z is concave on X. Proof. Let (z) denote the pdf of Z with support [zL, zH]. For a given (0, 1), x1 X, and x2 X, let z* = max{z | F(x1, z) + (1 – )F(x2, z) – F(x, z) 0, z [zL, zH]}, b(z) = F(x1, z) + (1 – )F(x2, z), a(z) = F(x, z), H = E[F(x1, Z)] + (1 – )E[F(x2, Z)] – E[F(x, Z)], and K = E[U(F(x1, Z))] + (1 – )E[U(F(x2, Z))] – E[U(F(x, Z))]. Since E[F(x, Z)] is concave, we have H 0. (4) From the fact that U is a concave increasing function, it follows that for y1 > y2, U(y1) – U(y2) U(y)(y1 – y2) y y1 and U(y1) – U(y2) U(y)(y1 – y2) y y2. (5) We are now ready to show that H 0 implies K 0: K= U F x1 , z 1 U F x 2 , z z dz U F x, z z dz U F x1 , z 1 F x 2 , z z dz U F x, z z dz (due to concave U) = U b z U a z z dz = z z* U b z U a z z dz z z* U a z U b z z dz (where each integrand is nonnegative due to SS+) U ' a z* b z a z z dz U ' a z a z b z z dz * zz * (due to Fz(x, z) 0) z z* = U ' a z* H 0, (due to U > 0) i.e., K 0 for any given (0, 1), x1 X, and x2 X, and thus E[U(F(x, Z))] is concave. A.3 Proofs Lemma 3A shows that the expected profit functions of PS and PD satisfy opposing single-switch properties; we use this result in the proof of Proposition 1. Lemma 4A is used in the proof of Proposition 2. We then present the proofs of propositions 1 – 3. At the end of this section, we present Proposition 1A 13 and Corollary 1A. These results apply to a problem that is a generalization of PS and PD, so we place this content at the end of this section. A special case of Proposition 1A yields one of the results given in Proposition 1. Corollary 1A is referred to in the discussion that follows Proposition 1 in the body of the manuscript. We use r to denote the revenue function in this section, i.e., r(p) = pd(p) for PS and r(p, z) = pd(p, z) for PD. Lemma 3A. (a) S p, q, Y satisfies SS-. (b) D p, q, Z satisfies SS+. Proof. Part (a): The realized profit function of PS is S p, q, y = pmin{d(p), qy} – cq + sq(y – 1). For a given (0, 1), (p1, q1) X, (p2, q2) X, with (p, q) = (p1 + (1 – )p2, q1 + (1 – )q2), let g(y) = p1min{d(p1), q1y} + (1 – )p2min{d(p2), q2y} – pmin{d(p), qy}. We must show that g(y) switches from positive to negative no more than once as y increases over interval [yL, yH]. In order to do this, we consider all possible cases regarding inequalities among parameters. Without loss of generality we index prices such that p1 < p2, and we let p = p2 – p1 and q = |q2 – q1|. We refer to the case of q = q1 – q2 (i.e., q1 > q2) as Scenario 1 (S1) and we refer to the case of q = q2 – q1 (i.e., q2 > q1) as Scenario 2 (S2). We can write pi and qi in terms of p, q, , p, and q for each scenario. S1: (p1, q1) = (p – (1 – )p, q + (1 – )q), (p2, q2) = (p +p, q – q) S2: (p1, q1) = (p – (1 – )p, q – (1 – )q), (p2, q2) = (p +p, q + q) For a given p, q, , p, and q , which determine p1, q1, p2, q2, we define y0 = d(p)/q, y1 = d(p1)/q1, y2 = d(p2)/q2. Note that (p1, q1) and (p2, q2) (and thus (p, q)) are elements of X (otherwise the points are not relevant for assessing the single-switch property on X), and thus min{y0, y1, y2} 0. We consider how the sign of g(y) changes for values of 0 y < min{y0, y1, y2} to y > max{y0, y1, y2}, which is assured to span the support [yL, yH]. Note that g(y) = p1d(p1) + (1 – )p2d(p2) – pd(p) 0 (due to the strict concavity of pd(p)). We begin with S1, for which p1 < p < p2, d(p1) > d(p) > d(p2) q1 > q > q2 . For 0 y min{y0, y1, y2}, g(y) = [p1q1 + (1 – )p2q2 – pq]y = {[p – (1 – )p][q + (1 – )q] + (1 – )[p +p][q – q]}y – pqy 14 (6) = -(1 – )pqy 0. (7) For y max{y0, y1, y2}, g(y) = p1d(p1) + (1 – )p2d(p2) – pd(p) 0 (see (6)). We consider three possibilities for the ordering of y0, y1, y2, and for each possibility we examine g(y) for y (min{y0, y1, y2}, max{y0, y1, y2}]. We will see that g(y) 0 for all y in this range. Suppose that y0 < y1 y2 (the analysis is same for the case of y0 < y2 y1). For y (y0, y1], g(y) = [p1q1 + (1 – )p2q2]y – pd(p) = p1d(p1) + (1 – )p2d(p2) – p1[d(p1) – q1y] – (1 – )p2[d(p2) – q2y] – pd(p) < 0 (due to (6), d(p1) q1y, and d(p2) > q2y), and for y (y1, y2], g(y) = p1d(p1) + (1 – )p2q2y – pd(p) = p1d(p1) + (1 – )p2d(p2) – (1 – )p2[d(p2) – q2y] – pd(p) 0 (due to (6) and d(p2) q2y). Thus, g(y) 0 for all y [yL, yH] (i.e., SS- holds). Suppose that y1 < y0 y2 (the analysis is same for the case of y2 < y0 y1). For y (y1, y0], g(y) = p1d(p1) + (1 – )p2q2y – pqy = -{[pq – (p1q1 + (1 – )p2q2)]y + p1[q1y – d(p1)]} < 0 (due to (7) and q1y > d(p1)), and for y (y0, y2], g(y) = p1d(p1) + (1 – )p2q2y – pd(p) = p1d(p1) + (1 – )p2d(p2) – (1 – )p2[d(p2) – q2y] – pd(p) 0 (due to (6) and d(p2) q2y). Thus, g(y) 0 for all y [yL, yH] (i.e., SS- holds). Finally, suppose that y1 < y2 y0 (the analysis is same for the case of y2 < y1 y0). For y (y1, y2], g(y) = p1d(p1) + (1 – )p2q2y – pqy = [(p1q1 + (1 – )p2q2) – pq]y – p1[q1y – d(p1)] < 0 (due to (7) and q1y > d(p1)), and for y (y0, y2], and for y (y2, y0], g(y) = p1d(p1) + (1 – )p2d(p2) – pqy = [ (p1q1 + (1 – )p2q2) – pq]y – p1[q1y – d(p1)] – (1 – )p2[q2y – d(p2)] < 0 (due to (7), q1y > d(p1), and q2y > d(p2)). Thus, g(y) 0 for all y [yL, yH] (i.e., SS- holds). In summary, SS- holds under S1. We now consider S2, for which p1 < p < p2, d(p1) > d(p) > d(p2) q1 < q < q2 which implies d(p2)/q2 < d(p)/q < d(p1)/q1, and y2 < y0 < y1. For y y1, g(y) 0 (see (6)). For y [y0, y1), 15 g(y) = p1q1y + (1 – )p2d(p2) – pd(p) = p1d(p1) + (1 – )p2d(p2) – p1[d(p1) – q1y] – pd(p) < 0 (due to (6) and d(p1) > q1y). For 0 y y2, g(y) = [p1q1 + (1 – )p2q2 – pq]y = {[p – (1 – )p][q – (1 – )q] + (1 – )[p + p][q + q]}y – pqy = (1 – )pqy 0. Thus, we need to show that there exists y* (y2, y0] such that g(y) 0 for 0 y y*, g(y*) = 0, and g(y) 0 for y > y*. (8) For y (y2, y0], g(y) = d[p1q1y + (1 – )p2d(p2) – pqy]/dy = p1q1 – pq < 0 (due to p1 < p and q1 < q) g(y0) = p1q1y0 + (1 – )p2d(p2) – pd(p) = p1d(p1) + (1 – )p2d(p2) – p1[d(p1) – q1y0] – pd(p) < 0 (due to (6) d(p1) > q1y0), which ensures that there exists y* [y2, y0] such that (8) holds. Thus, SS- holds in both scenarios. Part (b): The proof follows the proof of part (a). We omit the details. Lemma 4A. For PD: If rp(p*(q), z) 0 for all z z(p*(q), q) or if rpz(p*(q), z) 0 for all z z(p*(q), q), then po(q) < p*(q). Proof. Note that z p* q , q * Dp(p (q), q) = U ' r p* q , t cq rp p* q , t Z t dt zL zH z p* q , q U ' p* q c q qZ t dt zH z p* q ,q = U ' p* q c q h t rp p* q , t Z t dt qZ t dt z z p* q , q L where h(t) = U ' r p* q , t cq U ' p q c q * . Note that h(t) > 1 and h(t) < 0 for t < z(p (q), q), and h(z(p (q), q)) = * 1. Therefore, from either rp(p*(q), t) 0 or rpz(p*(q), t) 0 for t z(p*(q), q) and from Dp p q , q = * z p* q , q zH rp p* q , t Z t dt z p q , q zL * qZ t dt = 0, it follows that Dp(p*(q), q) < Dp p* q , q = 0, which implies po(q) < p*(q). 16 * Proof of Proposition 1. Part (a) follows from Proposition 1A below. Part (b) follows from Lemma 2A and Lemma 3Ab. For part (c), Kocabıyıkoğlu and Popescu (2011) show that (3) is a sufficient condition for concave D p, q (see Proposition 2), and thus it follows from Proposition 1b that (3) is also a sufficient condition for concave D(p, q). If the newsvendor is risk-neutral, then u(p, q)/u+(p, q) = 1, and the concavity of E S p, q, Y follows from Proposition 1a. Finally, Example 4 in Section A.1 of the this appendix shows that (3) is not sufficient for concave S(p, q). Proof of Proposition 2. The proof of Proposition 1A shows that expected profit and expected utility functions are strictly concave in price and in quantity (keeping the other variable fixed). Thus, unique univariate optimal solutions are obtained from the first-order condition, and this fact allows us to analyze the signs of the partial derivatives in the following proofs. Part (a): Note that d p / q* p * Sq(p, q (p)) = U ' S p, q* p , t pt c s t 1 Y t dt yL yH d p / q* p U ' S p, q* p , t c s t 1 Y t dt . Let u d p/ q p pt c s t 1 Y t * * p, q p = U ' S p, q p , t d p / q* p dt yL pt c s z 1 Y z dz yL u yH c s t 1 Y t * * p, q p = U ' S p, q p , t yH dt . d p / q* p c s z 1 Y z dz * d p/ q p * Note that py – c + s(y – 1) > 0 for all y (yL, yH] (due to yL cs ) ps cs ) s (9) – c + s(y – 1) < 0 for all y [yL, yH) (due to yH d(p)/q*(p) (yL, yH) (due to S q p, d p / yH > 0, S q p, d p / yL < 0) (11) 17 (10) d p / q* p pt c s t 1 t dt 0 (due to (9) and (11)) Y (12) yL u p, q * p u p, q* p > 0 (due to U < 0, (9) – (11)) Therefore d p / q* p * Sq(p, q (p)) = u p, q p pt c s t 1 t dt * Y yL u p, q* p yH c s t 1 t dt Y d p / q* p yH d p/ q p > u p, q p pt c s t 1 Y t dt c s t 1 Y t dt yL d p / q* p * * = u p, q* p Sq p, q* p = 0, which implies qo(p) > q*(p). Part (b): d p* q / q * Sp(p (q), q) = U ' p* q qt cq qtY t dt yL yH U ' p* q d p* q cq rp p* q Y t dt d p* q / q Note that (due to S p p* q , q = 0) rp(p*(q)) < 0 U ' p* q qy cq U ' p* q d p* q cq for all y < d(p*(q))/q (due to U < 0). Therefore yH d p* q / q Sp(p , q) > U ' p q d p q cq qtY t dt rp p* q Y t dt y d p* q / q L * * * = U ' p* q d p* q cq S p p* q , q = 0, which implies po(q) > p*(q). Part (c): This result appears in Eeckhoudt et al. (1995). Part (d): If d(p, z) = d(p)z, then the support of Z must be nonnegative (i.e., to avoid the impractical possibility of negative demand). Let p1 denote the price that maximizes expected revenue E[pd(p)Z], i.e., 18 zH d p p d ' p t t dt 0 . 1 1 1 Z zL From zL 0, it follows that d(p1) + p1d (p1) = 0. Therefore, Dp p , q = 1 z p1 , q 0 d p1 p1d ' p1 tZ t dt z p1 , q qZ t dt 0 . Thus, p*(q) > p1, which implies d p* q p*d ' p* d p1 p1d ' p1 0 (due to rpp < 0). Therefore rp p* q , z d p* q p* q d ' p* q z 0 and the result holds by Lemma 4A. Part (e): If d(p, z) = d(p) + z, then rpz(p, z) = 1, and the result holds by Lemma 4A. Proof of Proposition 3. Before considering parts (a) and (b), we develop a result regarding the nature of the q*(p) and p*(q) curves that applies to both PS and PD. For this purpose, we generically represent the expected profit function for either problem as F(p, q). The proof of Proposition 1A shows that expected profit functions for PS and PD are strictly concave in price and in quantity (keeping the other variable fixed), i.e., Fpp < 0 and Fqq < 0. By the implicit function theorem, p (q) = * q (p) = * Fpq p* q , q Fpp p* q , q Fpq p, q* p Fqq p, q* p . From the fact that F has a unique stationary point (p*, q*) that is a global maximum, it follows that p (q )q (p ) = * * * * Fpq p* , q* 2 Fpp p* , q* Fqq p* , q* <1 (13) (i.e., the determinant of the Hessian of F is positive). We relate p*(q) to p*-1(p) via implicit differentiation of p*(p*-1(p)) = p, i.e., 0 d d 0 p* p*1 p p p* ' p*1 p p*1 ' p 1 , dp dp which can be rewritten as p*(q) = 1/ p*-1(p). (14) 19 By supposition, q*(p) 0, p*(q) 0, and the curves p*-1(p) and q*(p) intersect at a single point that is the optimum. Therefore, from (13) and (14) it follows that p*-1(p*) > q*(p*) 0, i.e., p*-1(p) intersects q*(p) once from below, as illustrated in Figure 1A. Figure 1A. Illustration of p*-1(p) and q*(p). Both curves have nonnegative slope with p*-1(p) intersecting q*(p) once and from below. Part (a): From the fact that S(p, q) has a unique stationary point that is a global maximum, it follows that curves qo(p) and po(q) intersect at a single point that is a global maximum. From propositions 2a and 2b, qo(p) > q*(p) for all p p*, and po(q) > p*(q). Thus, curve qo(p) is an upward shift of curve q*(p) for p p* and curve po(q) is an upward shift of p*(q) (equivalently, a right-shift of curve p*-1(p)), and it follows that the intersection point (po, qo) must be up and to the right of (p*, q*), i.e., po > p* and qo > q*. Part (b): From the fact that D(p, q) has a unique stationary point that is a global maximum, it follows that curves qo(p) and po(q) intersect at a single point that is a global maximum. From propositions 2c – 2e, qo(p) < q*(p) and po(q) < p*(q). Thus, curve qo(p) is a downward shift of curve q*(p) and curve po(q) is a downward shift of p*(q) (equivalently, a left-shift of curve p*-1(p)), and it follows that the intersection point (po, qo) must be down and to the left of (p*, q*), i.e., po < p* and qo < q*. Proposition 1A pertains to a problem that includes both supply and demand uncertainty. We define the joint pdf and cdf of random variables Y and Z are (y, z) and (y, z). As noted above, the marginal density and distribution functions are i() and i() for i {Y, Z}. The random profit under uncertain supply and demand is R p, q, Y , Z = p min d p, Z , qY cq sq Y 1 and the expected utility is R(p, q) = E U R p, q, Y , Z (the expectation operator is applied to all random variables within its brackets, in this case, both Y and Z). Functions S(p, q) and D(p, q) are special cases of R(p, q). Proposition 1A. If (p, q)u(p, q)/u+(p, q) ½ for all (p, q) X, then R(p, q) is concave. 20 If the newsvendor is risk neutral, then u(p, q)/u+(p, q) = 1 for all (p, q) X, which leads to the following corollary. Corollary 1A. If the newsvendor is risk neutral and (p, q) ½ for all (p, q) X, then S(p, q), D(p, q), and R(p, q) are concave. Proof of Proposition 1A. We develop the proof with random variable Y conditioned on the realization of Z, which by setting Z to be deterministic, leads to results for PS. The key to the proof is exposing a special structure in the determinant of the Hessian of R(p, q) and the application of different forms of the Schwarz inequality. We obtain this structure via substitution after deriving several identities. The structure contains four major elements. The sign of the first two elements have a sign that follows from inequalities derived earlier in the proof. The sign of the third element becomes apparent through application of a continuous form of the Schwarz inequality. The fourth element contains a term that can be both negative and positive (depending on parameter values). However, by application of the discrete form of the Schwarz inequality in combination with known properties of the third element along with inequalities obtained earlier in the proof, we are able to guarantee that the fourth element is nonnegative regardless of the sign of this particular term. A number of expressions required for the determinant of the Hessian are long, so we introduce some new notation, thereby obtaining expressions in a more compact form than would otherwise be possible. Let R p, q, y , z = r(p, z) – cq + sq(y – 1) (i.e., profit realizations where demand is less than supply) and R p, q, y , z = pqy – cq + sq(y – 1) (i.e., profit realizations where demand is more than supply). R p, q, d p, z / q, z = R p, q, d p, z / q, z = r(p, z) – cq + s(d(p, z) – q) R pp p, q, y, z = rpp(p, z) R pp p, q , y , z = 0 R q p, q, d p, z / q, z R q p, q, d p, z / q, z = pd(p, z)/q R qq p, q, y, z = 0 R pq p, q , y , z = 0 R pq p, q , y , z = y Rp(p, q | z) E U R p, q, Y , z / p = E U ' R p, q, Y , z R p p, q, Y , z Rq(p, q | z) E U R p, q, Y , z / q = E U ' R p, q, Y , z Rq p, q, Y , z Rp(p, q) = E U ' R p, q, Y , Z R p p, q, Y , Z 21 Rq(p, q) = E U ' R p, q, Y , Z Rq p, q, Y , Z zp(p, d) > 0 (15) zx(p, d) > 0 (16) (the last two inequalities can be obtained from dp(p, z) < 0, dz(p, z) > 0, and implicit differentiation of d(p, z(p, d)) = d). The Arrow-Pratt measure of absolute risk aversion at wealth x is (x) = -U (x)/U (x). Beginning with Rp(p, q | z), we obtain the cross-derivative Rpq(p, q | z) = E U ' R p, q, Y , z R pq p, q, Y , z R p, q, Y , z R p p, q, Y , z R q p, q, Y , z d p, z , z R p p, q, q d p, z d p, z d p, z U ' R p, q, , z ,z 2 q d p, z q q R p , q, ,z p q and beginning with Rq(p, q | z), we obtain the cross-derivative Rpq(p, q | z) = E U ' R p, q, Y , z R pq p, q, Y , z R p, q, Y , z R p p, q, Y , z R q p, q, Y , z d p, z , z R q p, q, q , d p z d p, z d p p, z , z ,z U ' R p, q, q q d p, z q ,z Rq p, q, q which implies d p, z , z R q p, q, q , , d p z d p z qd p p, z , z R p p, q, , z R p p, q, d p, z . q q , d p z R p, q, ,z q q Rpp(p, q | z) = E U ' R p, q, Y , z R pp p, q, Y , z R p, q, Y , z R p p, q, Y , z 2 22 (17) d p, z , z R p p, q, q d p , z d p, z d p p, z U ' R p, q, , z q , z q q d p , z R p , q, ,z p q = E U ' R p, q, Y , z R pp p, q, Y , z R p, q, Y , z R p p, q, Y , z 2 d p, z , z R q p, q, 2 q , d p z d p, z d p p, z , z , U ' R p, q, z q d p, z q d p, z ,z R p, q, q q (see (17)) 2 Rqq(p, q | z) = E U ' R p, q, Y , z R p, q, Y , z R q p, q, Y , z d p, z , z R q p, q, q d p , z d p, z d p, z U ' R p, q, , z ,z 2 q d p, z q q Rq p, q, ,z q We make use of the following notation and inequalities. d p, z d p, z d p, z d p, z B(p, q, z) = R q p, q, , z Rq p, q, , z ,z 2 q q q q d p, z pd p, z ,z = 0 q3 q 2 z = Y d p, z / q = event that demand is not more than supply z = Y d p, z / q = event that demand is more than supply p (p, q) = u(p, q) = 2 E d p, Z qY E B p, q, Z qd p, Z / d p, Z p pq = 0 E Y ; Z E d p, Z qY q E U ' r p, Z cq s d p, Z q B p, q, Z qd p p, Z / d p, Z E B p, q, Z qd p p, Z / d p, Z 23 B p, q, z qd p p, z / d p, z Z z = U ' r p, z cq s d p, z q zH dz zL B p, q, t qd p p, t / d p, t Z t dt z L zH u-(p, q) = E U ' R p, q, Y , Z R pp p, q, Y , Z E R pp p, q, Y , Z zH yH rpp p, z y, z = U ' r p, z cq sq y 1 zH yH dydz 0 zL d p , z / q r p, s s, t dsdt z d p,t / q pp L + u (p, q) = E U ' R p, q, Y , Z R pq p, q, Y , Z E R pq p, q, Y , Z E U ' R p, q, Y , Z Y ; Z E Y ; Z zH d p , z / q y y , z = U ' R p, q, y, z z d p ,t / q dydz 0 H zL yL s s, t dsdt zL yL 2 K1(p, q) = E U ' R p, q, Y , Z R p, q, Y , Z R p p, q, Y , Z 0 2 K2(p, q) = E U ' R p, q, Y , Z R p, q, Y , Z R q p, q, Y , Z 0 K3(p, q) = E U ' R p, q, Y , Z R p, q, Y , Z R p p, q, Y , Z Rq p, q, Y , Z d p, Z ,Z R q p, q, 2 q r p, Z cq d p , Z d p p, Z L1(p, q) = E U ' ,Z s d p, Z q q d p , Z d p, Z R p, q, ,Z q q r p, Z cq pd p, Z 2 d p, Z qd p, Z 2 p , Z = E U ' 3 q q d p, Z s d p, Z q 2 r p, Z cq qd p p, Z B p, q, Z = E U ' d p, Z 0 s d p, Z q 24 d p, Z ,Z R q p, q, q r p, Z cq d p , Z d p, Z L2(p, q) = E U ' ,Z 2 s d p, Z q q d p Z , q R p, q, ,Z q q r p, Z cq pd p, Z 2 = E U ' q3 s d p, Z q d p, Z , Z q r p, Z cq = E U ' B p, q, Z 0 s d p, Z q d p, Z ,Z R q p, q, r p, Z cq q d p , Z d p p, Z L3(p, q) = E U ' ,Z s d p, Z q q q R p, q, d p, Z , Z q q r p, Z cq pd p, Z 2 = E U ' q3 s d p, Z q d p, Z qd p p, Z , Z q d p, Z r p, Z cq qd p p, Z = E U ' B p, q, Z d p, Z s d p, Z q = u p, q E Y ; Z p, q 0. (18) Using the preceding notation, we get Rpp(p, q) = E Rpp p, q | Z = E U ' R p, q, Y , Z R pp p, q, Y , Z R p, q, Y , Z R p p, q, Y , Z 2 d p, Z ,Z R q p, q, 2 q r p, Z cq d p , Z d p p, Z E U ' ,Z s d p, Z q q d p , Z d p, Z R p, q, ,Z q q = u p, q E rpp p, Z ; Z K1 p, q L1 p, q < 0 Rqq(p, q) = E Rqq p, q | Z 25 (19) 2 = E U ' R p, q, Y , Z R p, q, Y , Z R q p, q, Y , Z d p, Z ,Z R q p, q, q r p, Z cq d p , Z d p, Z E U ' ,Z 2 s d p, Z q q d p Z , q R p, q, ,Z q q = K 2 p, q L 2 p, q < 0 (20) Rpq(p, q) = E Rpq p, q | Z = E U ' R p, q, Y , Z R pq p, q, Y , Z R p, q, Y , Z R p p, q, Y , Z R q p, q, Y , Z d p, Z ,Z R q p, q, q r p, Z cq d p, Z d p p , Z E U ' ,Z q q s d p, Z q R p, q, d p, Z , Z q q = u p, q E Y ; Z K 3 p, q L3 p, q = u p, q E Y ; Z K 3 p, q u p, q E Y ; Z p, q and the determinant of the Hessian is = Rpp(p, q)Rqq(p, q) – Rpq(p, q)2 = u p, q E rpp p, Z ; Z K 2 p, q L2 p, q u p, q E Y ; Z 2 u p, q 1 2 p, q u p, q K1 p, q K 2 p, q K 3 p, q L1 p, q L2 p, q L3 p, q 2 2 K1 p, q L 2 p, q K 2 p, q L1 p, q 2 K 3 p, q u p, q E Y ; Z L3 p, q Suppose that p, q u p, q u p, q 1 . 2 (21) The first line is nonnegative because it is the product of two nonnegative terms. The second line is nonnegative due to (21). Each of the parenthetical terms in the third line is nonnegative due to the Schwarz inequality, which for double and single integration is 26 2 2 f y, z dydz g y , z dydz f y, z f y, z dydz A A A b b b 2 2 f z dz g z dz f z g z dz a a a 2 2 In the case of the first parenthetical term, K1(p, q)K2(p, q) – K3(p, q)2, 1/ 2 R p p, q, y , z 1/ 2 R q p, q, y, z f y , z U ' R p, q, y , z R p, q, y , z y , z g y , z U ' R p, q, y, z R p, q, y, z y , z and in the case of the second parenthetical term, L1(p, q)L2(p, q) – L3(p, q)2, qd f z U ' r p, z cq s d p, z q B p, q, z Z z g z U ' r p, z cq s d p, z q B p, q, z Z z 1/2 1/ 2 p p, z / d p, z . Let us now consider the fourth line. The only term with indeterminate sign in the fourth line is K3(p, q) (all of the other terms are nonnegative). If K3(p, q) = 0, then the fourth line is clearly nonnegative. Suppose that K3(p, q) 0. To simplify the notation, we suppress p, q, and Z, and the fourth line is K = K1L 2 K 2 L1 2 K 3 u E Y ; L3 . Case 1. K3 > 0: A discrete version of the Schwarz inequality is a 2 1 a2 2 b12 b2 2 a1b1 a2b2 . 2 Therefore K1K 2 L1L2 K1L2 K 2L1 = 2 K1 L1 2 K 1K 2 L1L 2 K2 2 L2 2 2 K1K 2 L1L 2 2 K1K 2 L1L 2 Recall from analysis of the third line that K1K2 K32 and L1L2 L32. Substituting these inequalities into the above and simplifying, we get K1L 2 K 2 L1 2 K 3L3 0, which implies K = K1L 2 K 2 L1 2 K 3L3 2 K 3u E Y ; 0. Case 2. K3 < 0: We rewrite the fourth line as 27 u K = K1L 2 K 2 L1 2 K 3 u E Y ; 1 u . From (21) it follows that K K1L2 K 2 L1 K 3 u E Y ; = K1L 2 K 2 L1 K 3 L3 u / u (due to L3 uE Y ; ; see (18)) K 1L 2 K 2 L1 2 K 3 L3 (due to (21)) 0 (see Case 1). Note that Rpp 0 and Rqq 0 (see above). Thus, (21) assures that R is concave. 28
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