Designing supply contracts: buy-now, reserve, and wait-and-see

ABSTRACT We consider three types of purchase contracts a manufacturer could offer in order to maximize its profit when supplying a retailer that uses responsive pricing to sell in an uncertain market: buy-now before the selling season starts, reserve stock for possible future purchase, and wait-and-see the market before making purchases. The existing literature has shown that adding a recourse purchase—i.e., the wait-and-see alternative—is always beneficial for the retailer who faces an uncertain demand. We find that this is not necessarily the case for the manufacturer who supplies the retailer, as its optimal contract mix depends on the market uncertainty as well as its production characteristics. The manufacturer should offer only the buy-now alternative if its recourse production is much more costly than advance production. As the recourse production cost decreases, the manufacturer should add a second contract to the portfolio: initially the reserve contract and then the wait-and-see contract. However, when the recourse production is cheaper than advance production, the manufacturer should drop the buy-now contract from the mix. As such, it is only in a small region, which shrinks with decreasing uncertainty in demand, that the manufacturer finds it optimal to offer all three purchasing alternatives.


Introduction
Traditionally, for short-lived products, a manufacturer or supplier agrees to produce and deliver goods to the buyer before demand uncertainty is resolved. If demand turns out to be low, committing to a fixed quantity in advance may result in significant inventory write-offs for the buyer but helps the manufacturer lock-in profit upfront; alternatively, if demand turns out to be high, the pre-commitment may lead to a significant opportunity cost, due to the lost sales for both the manufacturer and the buyer. Fortunately, these adverse effects of mismatches between demand and supply can be mitigated by using a combination of traditional and more flexible contracts.
As an illustration, consider the benefit to Hewlett-Packard (HP) of using such a portfolio-based approach to make purchases. HP saved an estimated $50 million U.S. dollars using a combination of traditional long-term and more-flexible shortterm contracts according to Nagali et al. (2008) in an Edelman Award-winning paper. Specifically, after quantifying demand uncertainty, HP uses three types of contracts to procure supplies. For "nearly certain" segments of demand, HP commits to buying-now a fixed quantity at a fixed price through the traditional contract. For "intermediate certain" segments of demand, HP employs a flexible-quantity commitment contract, to reserve a pre-determined amount of stock for possible purchase after market uncertainty has been resolved. Finally, for "high uncertain" segments of demand, a wait-and-see strategy is adopted, deferring procurement until the uncertainty is resolved. Like CONTACT Jen-Yi Chen j.chen@csuohio.edu Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/uiie.
Supplemental data for this article can be accessed at www.tandfonline.com/uiie HP, many other high-tech companies, including IBM and Sun Microsystems (Chen and Shen, 2012), are adopting such a portfolio approach to purchasing, in order to better match supply with demand.
There is a complementary trend on the demand side to postpone pricing decisions until uncertainty has been resolved. Sometimes called responsive pricing (Van Mieghem and Dada, 1999), this ability to manage demand by postponing or adjusting price has proven quite effective. Recently, Granot and Yin (2008) analyzed supply chain issues that arose in designing a buy-now contract to supply a price-responsive retailer. Kleindorfer and Wu (2003) and Haksoz and Seshadri (2007) provided excellent reviews of the growing literature that examines research at the nexus of flexible procurement contracts and responsive pricing.
Although the value to the buyer of utilizing a portfolio of contracts has been documented (Nagali et al., 2008), it remains unclear whether the manufacturer benefits from offering more flexible contracts that include options (reserve) and recourse (wait-and-see) purchasing. If there exists a cost advantage in supplying an early commitment over an option contract, then the advance fixed-quantity contract (buy-now) could play a significant role for the manufacturer discussed in Wu and Kleindorfer (2005). However, should it offer all three contracting alternatives or just one or two of them? Are these alternative contracts substitutes or complements? How should the manufacturer price them? How do the manufacturer's production characteristics, such as production cost/flexibility and demand uncertainty, affect the answers to these questions? In this Copyright ©  "IIE" article, we take a step further toward addressing these issues by considering variants of the price-responsive retailer model whose decision problem captures the salient features identified above.
To this end, we formulate and analyze a bilateral game between the manufacturer and the retailer. We assume complete information and that each party is risk-neutral and seeks maximal expected profit. As depicted in Fig. 1, an uncapacitated manufacturer, as the Stackelberg leader, sells a product to a monopolist retailer that can influence demand by setting the retail price. In the commitment stage, before market uncertainty is resolved, the manufacturer offers two procurement contracts. In the buy-now or advance purchase (A) contract, the retailer purchases K a at unit price p a , and the manufacturer produces at unit cost c 0 for immediate delivery. In the reserve or option (O) contract, the retailer purchases K o options at unit price p o , each of which can be exercised, after uncertainty is resolved, at unit price p e .
In the recourse stage, after market uncertainty is resolved, the manufacturer offers the retailer the wait-and-see or recourse purchase (R) contract at recourse price p r . As such, analogously to Mendelson and Tunca (2007), the recourse price acts effectively as the endogenized proxy for the spot market, which is assumed exogenous by Serel et al. (2001), Wu et al. (2002), Spinler and Huchzermeier (2006), Dong and Liu (2007), Secomandi (2010), and Pei et al. (2011). Subsequently, the retailer decides the retail price p and the quantity to sell q. Equivalently, it decides whether to sell all units acquired in the commitment stage or hold back some inventory, how many of the K o options to exercise at unit price p e , and how much more to procure from the manufacturer at the recourse price p r . The manufacturer produces and delivers these additional units if q > K a at unit cost c, which may differ from the unit manufacturing cost c 0 in the commitment stage. We assume that the options are exercised after demand uncertainty is resolved as in Wu and Kleindorfer (2005); otherwise, should there be no demand updating before the maturation date, the option contract is essentially another advance purchase contract, either dominating advance purchase (p o + p e < p a ) or being dominated by it (p o + p e > p a ).
We begin our analysis in Section 2 by first formulating the model of the full-information noncooperative game between the manufacturer and retailer; we then precisely delineate the technical properties of the uncertain demand function that are invoked in order to facilitate the structural analysis of the model. The recourse stage is fully analyzed in Section 3; these results are used in Section 4 to solve the game in the commitment stage. Although this game has three procurement opportunities that yield the AOR model (A for Advance purchase, O for Options, and R for Recourse purchase), in certain circumstances the model simplifies to one of three natural variants with recourse purchasing. These three variants are the AR, OR, and R models, which are examined in the Online Supplement. The complementary case of the AO model where recourse purchasing is not available is examined in Section 5 with techniques similar to those in Sections 3 and 4. Its two natural variants, the A and O models, are also examined in the Online Supplement. Together the results of Sections 3, 4, and 5 allow us to provide insights into the structure of the game that characterizes the equilibrium for the underlying problems. Given the complexity of the analyses, the findings are necessarily technical in nature. Thus, to glean sharper insights, in Section 6 we specialize the results of Sections 3, 4, and 5 to the case where demand uncertainty is represented by a uniformly distributed random variable. Additional insight is found by conducting an extensive computational study that is summarized in Section 7.
We find that it is never optimal to offer either the Option or Recourse purchase contract by itself. We also find that when recourse production cost c is lower than c 0 , it is too costly for the manufacturer to offer advance purchasing, so the portfolio consists of O and R contracts. Offering both O and R contracts simultaneously allows the manufacturer to mimic some features of the A contract (payments in the commitment stage) but without significantly compromising the possibility of making some sales in the recourse stage at a price lower than the exercise price in low-demand states. This strategic complementarity of O and R contracts cannot occur if only one of them is offered.
However, when c is significantly greater than c 0 , which makes postponed production exorbitantly costly, it is optimal for the manufacturer to offer only the Advance purchase contract. As the cost of advance production c drops, the manufacturer complements the A contract by adding the O alternative contract. Offering options allows the manufacturer to raise the unit advance purchase price, shifting some sales from A to O. This increase in profit margin in the commitment stage and the incremental revenue from options is balanced against the contingent cost of the options exercised. For even smaller values of c, the A contract is complemented by the R contract instead of the O contract, allowing the manufacturer the flexibility to set higher prices in the recourse stage during high-demand states since there are fewer initial sales.
Lastly, only in the narrow parameter set where c is marginally higher than c 0 does the manufacturer offer all three procurement alternatives. Interestingly, as will be seen from the numerical results of Section 7, this set shrinks as demand variability decreases. Moreover, the relative benefit of offering a third alternative is rather low for the manufacturer, suggesting that for most practical settings, offering two procurement choices is sufficient.

The AOR model
We consider a manufacturer (M) that supplies a product to a retailer (R). The retailer is a monopolist that faces a market with demand function q = y(p) where y(p) is a decreasing deterministic function of price p and the multiplier is a non-negative random variable with probability density function f (·) and cumulative distribution function F (·). In the commitment stage, before market uncertainty is resolved, M offers R the opportunity to purchase in advance at unit price p a . In addition, it offers R an option contract under which R pays a unit price p o in the commitment stage for every option and pays the unit price p e for every option it exercises in the recourse stage. In response, R decides the advance purchase quantity, K a , and the option quantity, K o . We assume that in the commitment stage of the AOR model, M will make R aware of the recourse purchase opportunity for quantity beyond the units in the option contract. Upon receiving the order, M produces and delivers K a units to R at unit manufacturing cost c 0 .
In the recourse stage, after market uncertainty is resolved, M announces the recourse price p r at which R may acquire additional units. If p r = p e , we assume that R would exercise its options first before recourse purchasing. Once R decides how many options to exercise and the recourse purchase quantity, M produces and delivers these quantities at unit manufacturing cost c. Let q be the total quantity R sells in the consumer market, M and R be M's and R's respective expected profits in the commitment stage, and π M and π R be their respective profits in the recourse stage.
This full information noncooperative game between M and R can be formulated as follows. In the recourse stage, after market uncertainty is resolved, anticipating R's responses, the Stackelberg leader M sets recourse price p r to maximize its profit: where x + = max{x, 0}, K = K a + K o , and q = y(p). Subsequently, given p r and p e , R determines the retail price p, or equivalently, the total quantity q, to maximize its profit: Equations (1) and (2) imply that if p r < p e , R will never exercise its options; otherwise, it will exhaust its options first before making recourse purchases. In the commitment stage, anticipating R's responses K a and K o , M chooses p a , p o , and p e to maximize its expected profit: where E is the expectation operator. Subsequently, R determines K a and K o to maximize its expected profit: Since the model formulation is quite general, the technical conditions presented next are invoked to facilitate the analysis. However, the conditions that are invoked depend on the model variant that is under investigation.

The demand function y(p)
To ensure that the recourse problems for both M and R are wellbehaved, we must have both retail price elasticity of demand and recourse price elasticity of residual demand as nondecreasing in price (Properties 1 and 2, respectively, in the Appendix). Many demand functions possess these two properties, which are subsumed by the following differentiable form where α > 0. Since we have y(p) > 0 and y (p) < 0, the ratio in Equation (5) is negative. Thus, it is applicable if and only if 0 ≤ p < α/β. Since p is bounded below by c 0 and c, to avoid trivial cases, we assume that c 0 < α/β and c < α/β. As noted in Petruzzi (1995), this functional form accommodates many common demand functions, including isoelastic demand, y(p) = ap −b , price-proportional demand, y(p) = ae −bp , and linear demand, y(p) = a − bp with α = 0, 1/b, a/b, and β = 1/b, 0, 1 in Equation (5) The condition in Lemma 1(i) excludes the trivial case where the optimal retail price p(p r ) = ∞ since R's marginal revenue is positive for all p > 0 if β ≤ −1. Lemma 1(ii) characterizes the shape of the demand function. If β ≤ 1, then y (p) = (1 − β)/(−α + β p) 2 ≥ 0, so y(p) is a convex decreasing function. Lemma 1 facilitates the commitment stage analysis by further specializing the demand function. Henceforth, we assume that −1 < β ≤ 1 applies unless specified otherwise (e.g., no restriction on β when considering models without recourse purchasing since only Property 1 is invoked). For the recourse price p r satisfying Equation (A2), although there exists no closed-form expression for a general demand function, we can characterize how it behaves with respect to the parameters and the limits it can take.
Lemma 2(i) states that the quantity acquired in the commitment stage B puts downward pressure on M's recourse price, p r . For any fixed B, the equilibrium recourse price p r increases as manufacturing cost (c) or market size ( ) increases. These monotonic comparative statics properties can be used to establish an upper bound on the equilibrium recourse price. In Lemma 2(ii), the upper bound on p r is obtained at = ∞; i.e., the most optimistic market size, whereas the lower bound on p r ensures M a non-negative marginal profit from its recourse sales. Note that both the upper and lower bounds, like p r itself, are increasing in c.

The random multiplier
For the analysis of the commitment stage, depending on the underlying modeling specification, the cumulative distribution function of the random multiplier may be arbitrary or exhibit one of the following properties: (i) IGFR (Increasing General Failure Rate); (ii) IFR (Increasing Failure Rate); or (iii) PF n (Polya) distributions. Many common distributions satisfy the IGFR property, as shown in Lariviere (2006), and the prevalent IFR property is a special case of IGFR (Porteus, 2002). A further subsumed case, the Polya distributions (PF n ) include the group of exponential and the convolutions of exponential distributions-e.g., Erlang (Porteus, 2002)-has the variation diminishing property (Schoenberg, 1951). The PF 2 distributions, in particular, subsume many common distributions, including uniform, normal, truncated normal, exponential, and all their translations and convolutions.

The recourse stage
Since the recourse stage is reached after market uncertainty has been resolved, is known. Given , K a , K o , c, and p e , M sets p * r to maximize its profit π M , defined in Equation (1) where the superscript * refers to optimal. In response, R chooses the total quantity q * or, equivalently, the retail price p * , to maximize its profit π R , defined in Equation (2). In making its decision, R equates marginal revenue to marginal cost, which is either zero, p e , or p r , depending on q, K a , and K. Although p e is independent of the recourse stage, we know from Lemma 2(ii) that the value of p r increases from c to an upper bound with increasing so that p e may be greater than or less than p r .
When p e ≤ p r , depending on the realized value of , exactly one of five cases occurs. In Fig. 2 these tradeoffs are depicted for the case of the linear demand curve y(p) = (50 − p) with K a = 10 and K = K a + K o = 25 (K o = 15). When is low, we see from Fig. 2(a) that R holds back inventory so the marginal cost is zero, as only a portion of K a is used. For somewhat higher values of , we see from Fig. 2(b) that exactly K a units are sold, as the marginal revenue is between zero and p e . In Fig. 2(c), analogous to the case in Fig. 2(a), R uses some of the K o options at marginal cost p e . Figure 2(d) depicts the case where the marginal revenue is between p e and p r so that all K = K a + K o units are sold. Only when is sufficiently high, so that marginal revenue can be equated to p r , does q exceed K. The complementary case where p e > p r is omitted, as it is similar; however, we note that it is simpler since R never exercises its options as p e > p r . This allows us to conclude that since p r is bounded from above, p e is also bounded by the same value.
Given the best response of R that can be derived formally, we can determine the derived residual demand curve for M, from which M's optimal price, p * r , can be determined. Since both π R and π M are continuous and concave in a compact action space, this Stackelberg game has an equilibrium point. To proceed, we define thresholds for , which explicitly partition the action space into the cases 0 ≤ p e < c and c ≤ p e < (c + α)/(1 + β).
These thresholds identify breakpoints where the structure of equilibrium switches.
Theorem 1. The recourse stage subgame has a unique subgame perfect Nash equilibrium, as summarized in Table 1.

The commitment stage
Building upon the analysis of the recourse stage, in this section we examine the commitment stage of the game. We first establish that just as R will not purchase options if the exercise price p e is greater than a threshold, M will not set p e < c.
For any p e < c, R would be better off if M reduces the upfront price p o while raising the exercise price p e by an equal amount. This holds true because R would only pay the full price p o + p e if and only if the option is exercised. Otherwise, it could save by paying the reduced p o in advance. However, to avoid cannibalizing its A and R contracts, M may then increase p o just enough to make the options less attractive and sustain its profitability from all three alternatives. Due to of Theorem 2, henceforth, in this section we only consider the recourse stage equilibrium summarized in the upper part of Table 1, the case where p e ≥ c.

The retailer's problem
In the commitment stage, given the prices (p a , p o , p e ), R determines the advance purchase quantity, K a , and options quantity, K o , to purchase. It facilitates the analysis to work with decision variables K = K o + K a and K a , instead of K o and K a . Using Equation (4) and Table 1, we establish the following properties for R's expected profit function.
Lemma 3(i) establishes the additive separability of R into its arguments, allowing us to solve for K a and K separately. Lemma 3(ii) implies that the best response, K, is unique, and Lemma 3(iii) identifies a sufficient condition for the unimodality of R in K a . Consequently, when unimodality is ensured, the following two First-order Optimality Conditions (FOCs) of R with respect to K a and K can separately and uniquely determine the interior solution; i.e., K > K a > 0: It follows from Equation (6) that the advance purchase quantity, K a , depends only on the difference between p a and p o rather than on the price p a itself, as R weighs the upfront cost difference between the two alternative contracts against the potential gain/loss later. The last term in Equation (6) represents the savings that would ensue from not exercising options should advance purchases be made instead of purchasing options. Lemma 3 and the FOCs above yield R's best response K a and K.
then R has a unique best response K a and K that satisfies Equation (6) and Equation (7), respectively.
If the solution is not in the interior-i.e., K = K a > 0combining Equations (6) and (7) yields LetK a = K satisfy Equation (7), a function of p o and p e . Sub-stitutingK a into Equation (8)    options. Similarly, we can derive conditions on p a , p o , and p e for which K a = 0. These results are summarized below.
To graphically illustrate Theorem 3 and Corollary 1, we use y(p) = 50 − p, f ( ) = e − , c = 14, and p e = 23. Figure 3 shows the regions where R makes advance purchases and/or buys options in the commitment stage. Given p e , if both p a and p o are too high as in Region I, no purchases are made in the commitment stage. In Region II, p a is relatively too high compared with p o , so only options are purchased; the converse is true in Region III. It is only in Region IV where the advance purchase price and option prices are sufficiently close to each other so that both types of purchases are made in the commitment stage.
Since Theorem 3 presupposes the uniqueness of the best response, the full characterization of R's response yields the following comparative statics.

then: (i) as p a increases, K a decreases and K o increases, but K is invariant; (ii) as p o or p e increases, K o and K decrease, but K a increases.
Corollary 2 implies that both K a and K o are affected by each of the three prices. In particular, an increase of p a does not change the total purchase quantity K but merely shifts some purchases from the advance purchase to the option contract. However, with p a fixed, an increase of either p o or p e makes options more expensive; thus, R raises K a , but lowers K o . The net result is a lower K since the increase of K a is less than the fall of K o .

The manufacturer's problem
Although the irregularities inherited from the recourse stage can be smoothed away for the retailer's problem, M's decision problem is significantly more difficult since it takes R's response function into consideration. As with R's problem, we attack M's problem by a change of variables, replacing M's three pricing decisions (p a , p o , p e ) with (K a , K, p e ). Once p e , K a , and K are obtained, the imputed p a and p o can be calculated from Equation (6) and Equation (7), respectively. With this change of variables, a price schedule can be determined to induce R to choose exactly the K a and K that are optimal for M.
After changing the decision variables from p a and p o to K a and K, we get the rather complex expression for M given as Equation (A14) in the Appendix. Not surprisingly, we get results that are analogous to Lemma 3 but weaker (Lemma A2 in the Appendix).
The equilibrium point, if in the interior with K > K a > 0, satisfies the FOCs: Although Equation (9) and Equation (10) are similar to Equation (6) and Equation (7) with p o and p a replaced by c and c 0 , respectively, their complexity reveals why M's problem is much more difficult. However, like R's problem, K a and K can still be optimized separately using Equation (9) and Equation (10), respectively. We note that M's (indirect) quantity decisions, K a and K, depend on the respective unit manufacturing costs, c 0 and c in the commitment and recourse stages. Furthermore, M produces in the recourse stage with probabilitȳ ; that is, it produces only if R exercises an option or purchases more at the unit price p r . Thus, we can fully characterize M's decisions for the most general advance purchase and option contract with recourse as follows: Theorem 4. If K > K a > 0, then the equilibrium K * a and K * satisfy Equation (9) and Equation (10), respectively; the equilibrium exercise price p * e can be found by a bounded one-variable line search: Since Theorem 4 is analogous to Theorem 3, we omit its interpretation except to note that M's decision requires a bounded search for p e . Therefore, this two-stage game with up to 10 decision variables is reduced to a bounded line search in only one decision variable, p e . After obtaining p * e , the optimal prices p * o and p * a can be recovered from Equation (6) and Equation (7), respectively, yielding a complete solution for this complex game.
Moreover, based on the FOCs, we can derive boundary conditions for K * a ≥ 0 and K * ≥ 0. Analogous to Equation (8), if K a = K, then M's FOC with respect to K a or K becomes LetK a = K satisfy Equation (10), soK a is a function of p e and c whenever the equilibrium is unique. Then substitutingK a into Equation (12) Unlike the retailer's problem, which has sufficient structure that unimodality of the profit function in K a could be established as in Lemma 3, the manufacturer's profit function is not wellbehaved. However, for c 0 ≥ c, we can show in part (a) of Corollary 3 that an OR contact dominates an AR contract, so that it is not optimal to use a contract without recourse purchases; conversely for c o ≤ c, the derivative of M is positive at K a = 0, so that it is always optimal to make some advance purchases and K * a > 0. Part (b), although analogous, is weaker than Corollary 1, with (c, c 0 ) replacing (p o , p a ). Uniqueness can be established, for example, as shown in Section 6 when is uniform and y(p) is linear. Other specifications are also possible but not discussed here. With uniqueness, for high advance cost c 0 ≥ c, it can be further shown that A is too costly and hence, OR is the optimal contract mix. For low c 0 , AR performs equally well as AOR, due to the relatively high exercise price of O making the retailer consider only A and R. Only when c 0 falls between c 0 and c would AOR be utilized.
Having completed the analysis of the AOR model, we examine its three variants the AR, OR, and R models in the Online Supplement. Since there are two or fewer contract types, we can characterize the uniqueness of the equilibrium under rather general conditions for these simplified models. We next consider an AO model that does not have recourse purchases. Since there are fewer decisions in the second stage, the results are much sharper as in the AR, OR, and R models.

The non-recourse AO model
We now examine the AO model, in which after uncertainty is resolved, R does not have the recourse purchase opportunity. The recourse stage equilibrium for the AO model, easily specialized from the analysis of the AOR model, analogous to Theorem 1, is given below.
Theorem 5. In the AO model, the recourse stage has a unique subgame perfect Nash equilibrium, summarized in Table 2.
As recourse purchasing is no longer a factor, not surprisingly the equilibrium characterization is simpler than that in Theorem 1 for the AOR model. Consequently, the retailer's response in the commitment stage is unique. In fact, given (p a , p o , p e ), Equation (4) and Theorem 5 yield R's expected profit in the commitment stage (see Appendix), as a negative-definite function of K a and K. Thus, analogous to Lemma 3, we have the following stronger results.
With Lemma 4, the following two FOCs separately and uniquely determine the interior solution: wherep ≡ y −1 (K a / ) andp ≡ y −1 (K/ ). These FOCs, Equations (13) and (14), are simpler than their counterparts with recourse, Equations (6) and (7), because they have fewer breakpoints in the absence of the recourse opportunity. Moreover, analogous to Theorem 3 and Corollary 1, we obtain the following theorem.
Theorem 6. Given (p a , p o , p e ), R's unique best response, K a and K, satisfy Equation (13) and Equation (14), respectively.
If K = K a , combining Equations (13) and (14) yields Substituting theK a that satisfies Equation (14) into Equation (15) yieldsp a (p o , p e ), a threshold function on p a , below which K a = K. These results are summarized below.
For y(p) with β > 0, the price threshold for non-trivial solutions is greater than its counterpart in the AOR model. That is, y −1 (0) = α β > (c + αβ + 2α)/(1 + β) 2 , implying that without the recourse purchase alternative, R is less likely to forgo early procurement opportunities. In particular, since for the isoelastic and price-proportional demand curves, y −1 (0) = ∞, R always makes advance purchases or buys options. As the comparative statistics for K and K a are analogous to those in Corollary 2, these results are not presented.
Using Theorems 5 and 6, changing M's decision variables to (K a ,K,p e ) and using Equations (13) and (14) Having established unimodality, we now characterize the equilibrium for the AO model using the following FOC.
Theorem 7. The AO model has a unique equilibrium at which p * e = (c + α)/(1 + β), p * o = 0, and K * = ∞. Moreover, if M (K a , K, p e ) is unimodal in K a , K * a > 0 and it uniquely satisfies Equation (16) Without recourse sales, M forgoes the opportunity to make profits if the demand state turns out to be high. To compensate for this, M sets p * o = 0 and thus K * o = ∞, so any demand beyond the advance purchase quantity can still be satisfied by R exercising the options at p * e . Consequently, M's problem is reduced to a problem with only one decision variable, K a .
It is insightful to compare Theorem 7 with its counterpart, Theorem 4 for the AOR model. Unlike in the models with recourse purchasing that necessitate a search for the optimal p * e and thus the invoking of Property 2, here p * e is explicitly determined. Hence, we can conclude that it is the recourse purchasing alternative or, equivalently, the invoking of Property 2 that complicates the analysis. Note that in this AO model and its subsumed cases A and O, we presume that M would credibly commit to having no recourse, as it may play this game repeatedly so reputations have to be maintained (Fudenberg and Levine, 1989;Cachon and Feldman, 2015). Furthermore, in the AO equilibrium, such credibility may be self-sustained, due to Ms using a degenerate option contract (p * o = 0 and K * o = ∞) to recoup the profits that are forgone because of the absence of the recourse purchase alternative.
Although we show in the Online Supplement that specializing the AO model to its natural variants, the A and O models, does lead to additional insights, sharper results can be obtained when is uniformly distributed.

Equilibrium with a uniform multiplier
For our most general setting of the AOR model, we have formalized in Theorem 4 that equilibria may be found by performing a bounded search for p * e . As we show in the Appendix, although this bounded search can be used to compute equilibria, additional modeling restrictions have to be invoked to obtain analytical insights. In this section, to illustrate the analytical results, we obtain sharper results by modeling the multiplier as a uniformly distributed random variable between zero and U .

Theorem 8. If is uniform, for the AOR model, p * e = (3c + α)/(3 + β).
We first note that the determination of p * e in the uniform AOR model (denoted as p AOR e ) is independent of K a or K, which implies that for the uniform case, Theorem 4 and thus Corollary 3 characterize the equilibrium K * a (p * e ) and K * (p * e ) satisfying Equation (9) and Equation (10) at p e = p * e , respectively. Second, p * e is linearly increasing in c, but invariant with c 0 , due to the options, if exercised, products are produced at the unit recourse production cost c. Third, since in the AO model p * e (denoted as p AO e ) can also be solved explicitly as in Theorem 7 for general , the characterization of the option contract can be ranked.

Corollary 5. If is uniform, p
Compared with the AO model, offering a recourse purchase opportunity in the AOR model results in M creating competition for itself; in the sense that if p e is sufficiently high, M may set p r low enough for some low-demand states. No such contingency arises in the AO model, so M can price more aggressively and therefore p AO e > p AOR e . On the other hand, in the AOR model, to take advantage of the recourse selling opportunity, M sets p AOR o > 0 and thus K AOR < ∞, so that R would still make recourse purchases should the demand turn out to be sufficiently high.
Although the results above are based on the general demand specification, even sharper results can be found with a linear demand function; for example, the equilibrium quantities K * a and K * can be written explicitly leading to additional insights.

The case of linear demand
When the demand is linear-i.e., y(p) = a − bp-Theorem 9 yields explicit expressions for the decisions in the AOR and AO models and provides implicit unique expressions for other variants.  /(a − bc)) + 4 ln 2 − 2 Theorem 9. If y(p) = a − bp and f ( ) = 1/U for ∈ [0, U ], the equilibria under the AOR and AO models are as summarized in Table 3.
As seen from the derivation of the expression for K * in the AOR model, the constant 0.0554 is found by choosing the appropriate solution to a cubic equation that represents the first derivative of the profit function. Even in this highly specialized setting there exist two stationary points, one maximizer and one minimizer, so it is unlikely that unimodality can be established except under rather restrictive conditions. However, as can be seen, the qualitative results of Corollary 3 hold, so that unimodality is not necessary. Hence, in the Online Supplement we identify regularity conditions on the random multiplier (such as IGFR, IFR, and PF n ) and on the demand curve (iso-elastic, price-proportional, and linear) for which the profit functions may behave well in some special cases or have finite stationary points for the general case. Since our AOR/AO models subsume many special cases studied in the literature, our analytical results contribute to the literature by providing a more-general treatment. For instance, our A model is essentially the wholesale price contract with price postponement in Granot and Yin (2008). We generalize their assumption on iso-elastic demand and a specific multiplier into several rather general combinations of conditions. Next, we show how to exploit Theorem 9 to obtain even more structural insights into the model. To this end, as a complement to Corollary 5, which compares K AO and K AOR , we now compare the advance purchase quantities K * a 's in the two models.

Corollary 6. If y(p) is linear and is uniform, then K AOR
Surprisingly, even with the recourse opportunity, the retailer makes more advance purchases in the AOR model, due to the earlier result that the options are more costly to the retailer (p AOR o > p AO o = 0). We also note that K * a and K * are linear functions of the respective cost parameters, c 0 and c, which yield the following comparative statics.

Corollary 7. If y(p) is linear and is uniform, then:
(i) as c 0 increases, K * a decreases and K * o increases but K * is invariant; and (ii) as c increases, K * o and K * decreases but K * a increases.
Analogous to R's responses (Corollary 2), c 0 and c influence M's desired advance purchase quantity. A higher advance production cost does not change the total advance quantity K; it merely shifts some of the buy-now quantity to options. A higher recourse production cost dampens K, as the increase of K a is not enough to compensate for the fall of K o . Analogous to the AOR model, the AO model has similar qualitative results, and thus Theorem 10 shows that M tends to forgo the options part of the contract with the availability of recourse sales; thus, AOR is reduced to AR at a higher c 0 value than it is in the case of AO-to-A degeneration. This is intuitively pleasing, in the sense that at a higher value of c the transition point occurs at a higher value of c 0 . Interestingly, the convexity in c, which is equivalent to concavity in c 0 , shows diminishing returns of the option contract as seen in Fig. 4. As we are not able to directly compare profitability of the two "adjacent" AOR variants (AR and AO models) relative to each other, the results of Theorems 9 and 10 have to be complemented by computational work, which is presented next. The numerical study will also help identify the impact of uncertainty and incremental profit gains by providing these three contract alternatives, complementing the insightful structural results derived from the uniform multiplier case in this section.

Numerical studies
Having completed the technical analysis of the game between M and R, we perform a numerical study to gain insights that complement our analytical results. In particular, we compare the seven variants and the related integrated model to explore the following three issues: 1. How does the cost difference between c 0 and c affect M's contract preferences? 2. How does the recourse opportunity affect the decisions made by the two parties; and, how close is the channel's performance (M and R's joint profits) compared to the integrated channel profit? 3. How does M's contract preference change with demand variability?

... The impact of costs on contract portfolio
From Corollary 3, we know how M's preferences shift given c 0 and c for the four models with recourse, and from Theorem 7 how M's preferences shift among the three models without recourse. Here we rank these models against each other. To utilize Table 3, let the market uncertainty be uniformly distributed on [0, 2] and y = 50 − p. Since we have already established that it is the cost difference rather than the absolute cost values that determines the equilibrium type-e.g., K > K * a = 0 if c 0 > c-we set the advance production cost c 0 to 10 but vary the recourse production cost c. A complete analysis yields Fig. 4.
In Region I where both c 0 and c exceed 50, there is no demand since y = 50 − p and K * = K * a = 0. Thus, extremely high c 0 and c yield nominally the (degenerate) contract O or, equivalently, R, as M's preferred contract type. In Regions II and IIa, since c < c 0 , contracts with advance purchasing are not viable, and the choice between the OR model and the O model is resolved in favor of the OR model. Hence, M offers both reserve and wait-and-see alternatives to R.
We now consider the complementary case with c > c 0 . When c > 50 as in Region III, offering recourse purchasing is not viable, so M offers only the advance purchasing alternative, yielding the A model, the traditional or buy-now contract where M's profit is determined with certainty. For lower values of c, M prefers offering only the A contract as long as c is significantly greater than c 0 , as depicted in Region IIIa. As c gradually decreases, in addition to advance purchasing, it is optimal for M to add a second contract, first the option or reserve contract and then recourse purchasing or the waitand-see alternative, resulting in Regions IIIb and IIIc, respectively. Since p e is determined before uncertainty is resolved, the option contract allows M to raise the unit advance purchase price shifting some sales from A to O. Moreover, the revenue from offering options is balanced against the cost of them exercised, therefore benefiting M. In contrast, the R contract allows M to set prices optimally in response to market realization in the recourse stage, which is viable as c continues to drop because it allows M to charge a higher price for higher demand realizations. Only in the narrow Region IV, where c is marginally higher than c 0 , does M offer all three procurement alternatives Interestingly, it is never optimal to offer either the O or R contract by itself. Offering both O and R contracts simultaneously allows M to mimic some features of the A contract (payments in the commitment stage), however, without significantly compromising the alternative to make some sales in the recourse stage, at lower than the exercise price in low demand states. This strategic complementarity of O and R contracts cannot occur if only one of them is offered.

... The impact of c 0 and c on expected profits
Having fully characterized how c and c 0 influence the contract offered by M, we now explore how the manufacturer's profit changes as an additional procurement alternative is offered. We consider the representative case in which c 0 = 10 as depicted graphically in Fig. 5. The horizontal axis identifies M's preferred contract as c progressively increases from zero to 50.
It is notable that when the AOR contract is preferred, it is only marginally better than the AR, OR, or AR contracts as shown by the small dip in the AOR segment in the solid line in Fig. 5(a), which depicts the effectiveness of the two best contracts by normalizing against the optimal AOR contract (denoted by AR/OR/AO M / AOR M ). In fact, offering the two best alternatives achieves 100% of the value in all but a small range of c between 10 and 13 (99.7% at the lowest). In contrast, the dashed line, which depicts the effectiveness of only one best alternative (denoted by A/O/R M / AOR M ) may go as low as 82.9% at c = 17. This big dip around c = 17 is moderated by the type of uncertainty that is used to characterize the demand.
Additional numerical examples (in Section 7.2) suggest that the effectiveness of the best contract design falls for distributions with long right-tails and rises with decreasing variability. This observation is consistent with Fig. 4, in which the region where c is not too much higher than c 0 shows the most transitions between different contract mixes. In summary, we infer that although a second procurement opportunity has a big impact, a third alternative benefits M and R only marginally. However, consistent with the literature (Erhun et al., 2008;Nagali et al., 2008), the retailer weakly prefers having all three alternatives, although the incremental benefit of a third alternative is quite limited.
Due to the profits varying with c, it is hard to discern its impact on M, R, or the channel. To this end, we normalize all profits by dividing each by the integrated (INT) channel profit to develop the contract efficiency as seen in Fig. 5(b). As c increases, the relative value of the channel profit to that of the integrated channel falls but not necessarily monotonically, as can be seen when c is around 30. This implies that the benefit of providing multiple order opportunities is offset by higher recourse costs leading to a decreasing channel efficiency. It is also notable that the channel efficiency hovers around 80%. If this is considered too far from the first-best solution, as we discuss in the next section, there is a mechanism to set contracts that achieve the first-best solution.

The impact of demand variability on contract portfolio and profits
Although the previous results are for the uniform case, to examine the robustness of our findings, we also modeled as a realization from a Weibull distribution with shape parameter s = 1 (the exponential case), 2, and 3. We normalized the result by keeping the mean constant at one as we changed the shape parameter from one to two and then to three. This results in the variance decreasing from 1 to 0.2731 to 0.1321. Since the Weibull distribution with s ≥ 1 belongs to the PF 2 distribution class and the linear demand function satisfies Equation (5), we can use our analytical results to expeditiously compute equilibria. Although the relative preference switches from OR to AOR to AR to AO to A, as for the uniform distribution, there are qualitative differences as seen from Fig. 6(a). First, notice that the AOR region is nonempty only when c 0 is positive. Moreover, this region shrinks rapidly as variability decreases, thus confirming the earlier inference that a third purchase opportunity is of marginal benefit. It also confirms that neither the O nor the R alternative by itself is an effective mechanism for channel efficiency. We also noted that as variability decreases, the profits of M and R increase as shown in Fig. 6(b). It is worth noting that the effect is opposite if market uncertainty is in the additive form as discussed in next section.

Summary and extensions
This article examines how a manufacturer should design supply contracts when selling to a retailer that sets responsive prices via the combinations of three contract alternatives: buy-now in advance, reserve options to exercise in favorable demand states, and wait-and-see the market. At a technical level we show that finding the equilibrium in the non-cooperative game with up to 10 decisions is reduced to determining one variable, the option exercise price set by the manufacturer. For less general variants, we derive explicit expressions for this price and demonstrate the uniqueness of the equilibrium. Comparative statistics complemented by a numerical study reveal that for most of the parameter sets, the optimal choice is to offer at most the waitand-see alternative complemented by either the buy-now or reserve alternative. Only for a relatively small set of parameters where production costs in the commitment and recourse stages are relatively close to each other does the manufacturer prefer offering all three purchasing alternatives. Moreover, this parameter set shrinks as demand variability decreases. Also, typically when the equilibrium has all three purchasing alternatives, the relative benefit of offering all three elements is modest.
The numerical work also indicates that the contract offered by the manufacturer, achieves roughly 70−80% of channel efficiency relative to the integrated or first-best solution. Fortunately, it is possible to devise a mechanism that achieves coordination. In the Online Supplement we show that coordination can be achieved if M offers the following contract, which is a variant of those considered above: where A i ≥ 0 for i = 1, 2, 3, and I{·} is the indicator function, taking the value one if the argument is true and zero otherwise. Such a mechanism provides the retailer with the same marginal cost function as the integrated channel faces in each stage, thereby yielding coordination. Thus, a coordinating contract would need at least two contract types, A and R; our coordinating scheme has all three types to be comprehensive. In fact, only two contract types are needed; one aligns the commitment stage decisions and the other aligns the recourse stage decisions. Pei et al. (2011) also studied the efficiency of the optimal option contract with recourse purchase; however, they focused on the impact of asymmetric information regarding the retailer's type (willingness-to-pay) on supply chain efficiency, while offering an opportunity to purchase options coupled with recourse purchasing that have exogenously determined price.
Under the assumption of full information and that the manufacturer sells to one retailer only, we note that the manufacturer would not benefit from over-producing in the commitment stage to lower delivered cost in the recourse stage. This is because the rational retailer would anticipate such speculative investment and respond by ordering less in advance since it can anticipate a lower recourse price in the recourse stage. It is possible that under incomplete information, which might arise if the manufacturer serves more than one retailer, speculative overproduction may increase the manufacturer's profitability. Such a generalization may add additional insight into how to design supply contracts.
We close our discussion by noting that we have assumed that market uncertainty has a multiplicative effect on demand. When the uncertainty is additive, we find that qualitatively the preference structure is similar. However, under additive uncertainty, there is no upper bound on the recourse price, which helps explain why the firms' profits may increase with increasing variability in such cases.

Notes on contributors
Jen-Yi (Jay) Chen is an Assistant Professor at Cleveland State University. He received his Ph.D. in Operations Management from Purdue University. His recent research lies at the interface of marketing and operations management with a focus on sustainability issues. He teaches spreadsheet modeling and business analytics and advises regional medical centers and manufacturing companies.
Maqbool Dada, Ph.D., is Professor at Johns Hopkins Carey Business School. He received his expertise in the areas of operations management, supply chain management, and pricing models. He is also interested in health care operations. He is presently a Department Editor at IIE Transactions.
Qiaohai (Joice) Hu received her Ph.D. from Case Western Reserve University. Her research interests are in the areas of operations management, in particular of the interfaces between operations and finance, supply chain management, and pricing models. Before returning to academia, she worked as supply chain analyst at FedEx and Eastcom. Wu, D.J., Kleindorfer, P.R. and Zhang, J.E. (2002)  Also, y (p) = dy(p)/dp < 0 since y(p) is downward sloping. The elasticity η(p) is the percentage decrease in the demand for each percentage increase in the retail price. Property 1 implies that demand is more sensitive at higher prices and therefore R's marginal revenue is decreasing in price. Consequently, it ensures that the following implicit price function, p(A), uniquely satisfies Equation (A1) for any A ≥ 0: Since R sets the retail price (and thus quantity, q) in response to M's recourse price, p r , in order to ensure the unimodality of M's recourse stage problem, when applicable, we further assume the following property.
The elasticity ζ (p r ) is the percentage decrease in M's net recourse sales quantity for a percentage increase in the recourse price. Note that for ζ (p r ) to be meaningful, we must have q = y(p(p r )) > K. That is, R makes recourse purchases. The responsive retail price p(p r ) (derived in Section 3) characterizes R's response to M's recourse price p r . Since Property 2 implies that M's marginal revenue is decreasing in the recourse price, it ensures that the following implicit recourse price function, p r (B, , c), uniquely satisfies Equation (A2) for any B ≥ 0: Together, Properties 1 and 2 ensure that the recourse problems for both M and R are well behaved.
Proof of Lemma 1.

Proof of Lemma 2.
(i) It suffices to show the cross partial derivatives of π M w.r.t. K or K a and the corresponding parameter is positive (negative) for K or K a being increasing (decreasing) in the parameter. The lemma follows The lower bound of c ensures that M earns non-negative recourse profits.
Proof of Lemma A1. (i) Using Equation (5) to substitute for y(p)/y (p) in (A1) yields p(A) = (A + α)/(1 + β), which is increasing in A for β > −1, where A is the marginal cost that the retailer pays for a realized ; i.e., zero if only using the advance purchases and p e if exercising the options. Thus, The order of i s and i s then follows because y(p) is decreasing in p and 0 Proof of Theorem 1. We will analyze the players' decisions for different cases and then derive the corresponding boundary conditions on . Note that for M to make non-negative recourse profit, we must have p r ≥ c. We first prove the results in the top part of Table 1, which apply to the case of c ≤ p e .
(i) q < K a : π R = yp. Thus, dπ R /dp = y [p + y/y ] = y p[1 + y/(py )] where for brevity and henceforth in all subsequent proofs, the arguments of functions are suppressed so y ≡ y(p) and y ≡ dy(p)/dp unless confusion may arise. The partial derivative dπ R /dp changes sign from positive to negative if η(p) is increasing (Property 1). Thus, there exists a unique p * ≡ p(0) satisfying p + y/y = 0. The constraint q * = y(p(0)) < K a yields the boundary condition < K a /y(p(0)) ≡ 1 . Since R neither exercises options nor makes recourse purchase, π * M = 0. (ii) q = K a : q * = y(p) = K a yields p * = y −1 (K a / ). Substituting p * into π R yields π * R = K a y −1 (K a / ). Since M does not sell anything, π * M = 0. The threshold 2 will be derived in (iii). (iii) q > K a and p r < p e : π R (p) = yp − p r ( y − K a ) and which uniquely determines p * = p(p r ) by Property 1. Note that the Left-Hand Side (LHS) of Equation (A3) is increasing in p, so p * (p r ) is increasing in p r , which implies a positive comparative static: p ≡ dp(p r ) dp r = [y (p(p r ))] 2 2[y (p(p r ))] 2 − y(p(p r ))y (p(p r )) > 0. (A4) Substituting p(p r ) into π M and setting its partial derivative w.r.t. p r to zero yields which uniquely determines p * r = p r (K a ; , c) if ζ (p r ) is increasing (Property 2). Note that the LHS of Equation (A5) is increasing in p r , increasing in K a , and decreasing in since y (p(p r ))p (p r ) < 0. As a result, ∂ p r /∂K a < 0, ∂ p r /∂ > 0, and ∂ p r /∂c > 0. The constraint p * r ≥ c yields ≥ 2 ≡ K a /y(p(c)) while the other constraint p * r < p e yields (iv) q > K a and p r = p e : R's problem is as in (iii) with M setting p * r = p e . Thus, p * ≡ p * (p e ) and q * = y(p * (p e )). The threshold 5 will be derived in (v).
The analysis is the same as in (iii) with K replacing K a . The constraint p * r ≥ p e yields ≥ 5 = K y(p(p e )) 1 − ((p e − c)/(α − β p e )) .
Next, we will prove the bottom panel results in which p e < c. (i) q < K a : same as in p e ≥ c. (ii) q = K a : almost identical to its p e ≥ c counterpart except for threshold 2 , which will be derived next. (iii) K a ≤ q < K: the equilibrium prices, quantity, and profits are the same as in (iv) of the case p e ≥ c. The constraint K a ≥ q * yields ≥ 2 ≡ K a /y(p(p e )), whereas q * < K yields < 3 ≡ K/y(p(p e )). (iv) q = K: q * = y = K yields p * = y −1 (K/ ) and 4 will be derived next. (v) q > K: the equilibrium is the same as in p e ≥ c. For M to make recourse profit, p * r ≥ c, which yields ≥ 4 = K/y(p(c)). Proof of Theorem 2. For p e < c: where p(p r ) is defined in Equation (A1) and p r (K ) is defined in Equation (A2) with K replacing K a . Its first-order partial derivatives are wherep ≡ y −1 (K a / ) andp ≡ y −1 (K/ ). For ease of presentation, unless confusion may arise, we again omit the argument of functions; e.g., p r ≡ p r (K ), and y ≡ y(p(p r (K ))), the convoluted demand function of K. Subsequently, the respective second-order partial derivatives are Note that Equation (A9) is negative if the partial of the term in square brackets; that is, where s(p r ) = −β(1 + β)p 2 r + (4αβ − 2β 2 c + 2βc)p r + (−2α 2 + β 2 c 2 − βc 2 ). Recall that c ≤ p r < (c + α)/(1 + β) and s (p r ) = 0 leads to the only stationary point, p r = (2α + (1 − β)c)/(1 + β), of the quadratic function s(p r ). Since s(c) = −2(α − βc) 2 < 0: for β ≤ 1; thus, s(p r ) must be non-positive for p r ∈ [c, (c + α)/(1 + β)). Therefore, R is concave in K and K a . Setting Equations (A6) and (A7) to zero yields Differentiating both sides of Equations (A10) and (A11) w.r.t. p e yields dp a dp e = dp o dp e +F ( 2 ) = 3 2 dF ( ) and dp o dp e = −F ( 3 ), y(p(p e )) + (p e − c) y (p(p e )) dp(p e ) dp e dF ( ) > 0.
(ii) In this case: The negativity holds because (i) the integrand is negative as shown in the proof of Theorem 2, the second term in square brackets is positive since both the nominator and the denominator are negative, and (ii) d 4 /dK a = 4 /K a > 0. Therefore, R is concave and unimodal in K. (iii) In this case: whose sign remains undetermined because the first two integrals are negative while the third term is positive. Thus, R is not necessarily concave in K a . Let ξ = K a − , then Note that given K a , h(ξ ) is weakly decreasing in each ξ domain. Since h changes sign at most once from positive to negative (+, −), if f (·) is PF 2 , then ∂ R /∂K a also changes sign at most once, leading to the unimodality of R in K a by the Variation Diminishing Property of PF n distribution (Schoenberg, 1951).

Variation diminishing property of PF n (Schoenberg, )
If h : R → R changes sign j < n times, a is a PF n random variable with the density function f (·), and (k) := E[h(k − a)], then also changes sign at most j times. If changes sign exactly j times, then the changes must occur in exactly the same order as h.
Proof of Corollary 2. It suffices to show that the corresponding cross partial derivatives have the same signs. From Equations (A12) and (A13): (1 + β)p e − c − α 2 K a y (p e + α)/(1 + β) also changes sign at most three times and thus M has at most three local maxima w.r.t. K a .
Proof of Theorem 4. It follows directly from Lemma A2. LetK a = K at which the inequality is binding, soK a is a function of p e and c. Consequently, for condition (b) to hold, we must have
From the proof of (ii), K * > K * a iff _ c 0 < c 0 ; from (i), K * a > 0 iff c 0 < c.  Proof of Theorem 6. It follows immediately from Lemma 4.
Proof of Corollary 4. The proof is similar to that of Corollary 1 but with different conditions: