Coordination by Option Contracts in a Retailer-Led Supply Chain with

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WANG Xiaolong (王小龙) et al:Coordination by Option Contracts in a Retailer-Led … ∞ ∫ eq ( 1+ q2 −d)d F( x| ε ). q1+ q2 Here, ΠS( ε , q1, q2) represents his second-stage problem. After making the supplier’s second-stage problem independent of q 1 and after some algebra, the supplier’s problem is max Π ( q ) = ( c −c − o + o ) q + ( w − o ) d + q1d1 S 1 2 1 2 1 1 1 1 1 ∫ 0 ∞ Π ( ε, q )d G( ε), S 1 where ΠS( ε, q1) = max πS( ε, q) and qmax{ d, q1} π S( ε, q) = ( e− c2+ o2) q+ w2d2−ed−o2max{ d−q1,0} − q ( e−v) ∫ F( x| ε )d x . d N N Let ( q , q 1 ) denote the solution to this problem, and then we have Proposition 1. Proposition 1 Given an option contract specified by the retailer, the supplier’s optimal total production quantity is N ⎧ N −1⎛e− c2 + o2 ⎞⎫ q = max ⎨q1, F ⎜ ε ⎟⎬ ⎩ ⎝ e−v ⎠⎭ . The proof is relatively straight forward since the supplier’s problem follows a newsvendor structure. N Proposition 1 shows clearly that to coordinate q = c e− c2 + o2 p− c2 + s q , it is necessary to set = , i.e., e−v p− v+ s p− v+ s e= p+ s− o2 (5) c2−v Based on Eq. (5), e v e+ o > c to make > and 2 2 −1⎛ e− c2+ o2 ⎞ F ⎜ | ε ⎟ meaningful. Economically these two ⎝ e−v ⎠ conditions prevent the supplier from suffering losses. e+ o2is all the revenue that the supplier can get from one unit of product if it can be sold after the market demand is realized. If e+ o2 < c2, the supplier has no incentive to overproduce in the second period, and thus channel coordination is never reached. Since e= p+ s− p− v+ s o2 , the two conditions are equivalent with c2−v o2 < ( c2 − v) . Therefore, o2 < ( c2 − v) from now on. Next, consider the retailer’s problem. In all the fol- p− v+ s lowing parts, assume that e= p+ s− o2holds c2−v except where specified. The retailer’s problem is 575 ∞ max ΠR( d1) =−o1( q1−d1) − wd 1 1+∫ ΠR( ε, d1) d G( ε), d10 0 where ΠR( ε, d1) = max πR( ε, d) and dd1 π R( ε , d) = −w2d2 −o2( q2 −max{ d − q1,0}) + pE min{ D, q} + v( d −Emin{ D, d}) − eE min{max{ D −d,0}, q −d} − sED ( − Emin{ Dq , }). Rearranging, max Π ( d ) = ( o − o ) q + ( w − w + o ) d + where d10 R 1 2 1 1 2 1 1 1 ∫ ∞ Π 0 R( ε, d1)d G( ε), ΠR( ε, d1) = max πR( ε, d) and dd R 2 1 π ( ε, d) = ( e−w ) d −( e− v) F( x| ε)dx+ o max{ d − q ,0} + ( p+ s−e−o ) q− 2 1 2 ∫ ( p+ s−e) F( x| ε )dx−sED. 0 q N N Let ( 1 , d d ) denote the optimal solution to the retailer’s problem. Proposition 2 describes the retailer’s ordering behavior. e−w2 e− w2 + o2 Proposition 2 Denote α = , β = , e−v e−v N −1 N and γ ( ε) = max{ d1, F (max{ α,0}| ε)}. Define ˆ( ε d1 ) = N N sup{ ε: Fd ( 1 | ε) max{ α,0}} and allow ˆ( ε d1 ) = +∞ if ε max is infinite. For a given option contract, the optimal ordering behavior for the retailer has the following structure: (1) After the market signal ε has appeared, the optimal total order quantity is determined as follows: (i) If Fq ( 1 | ε ) β , N d = γ ( ε ) ; (ii) Otherwise, −1 ⎧ γε ( ), if πR( ε, F (max{ α,0}| ε)) > ⎪ N −1 d = ⎨ πR( ε, F ( β | ε)); ⎪⎩ −1 F ( β | ε), otherwise. (2) The optimal first-period order quantity is given by N ˆ( ε d1 ) N 2 1 1 0 2 1 ∫ w − w + o + [ e−w −( e− v) F( d | ε)]d G( ε) = 0 when e+ o1 > w1. Otherwise, N d1 ≡ 0 . Corollary For some second-period option price(s) −1 o 2 such that β > 0 and set A= { ε : πR( ε, F (max{ α, −1 0}| ε )) π ( ε, F ( β | ε))} is not empty, there exists a R ∫ 0 d
576 signal T ( 2) o ε such that N −1 only if T ( 2) o ε ε greater than 1 q and T ( 2) max o ε ε d = F ( β | ε) > q1if and N . Otherwise, d will never be = for such cases. Proof First, prove part (1). Omitting all the irrelevant parts, the retailer’s second-period problem can be rewritten as d ⎧( e−w2) d −( e−v) F( x| ε )d x, ⎪ ∫ 0 ⎪ if dq1; max πR( ε, d) = ⎨ dd d 1 ⎪( e− w2 + o2) d −( e−v) F( x| ε )d x, 0 ⎪ ∫ ⎪⎩ otherwise. Then, the unconditional optimal N d may not be unique but can be either −1 F (max{ α,0}| ε ) or −1 N −1 F (max{ β,0}| ε ) or both. However, if d = F ( β | ε ) , −1 there must be F ( β | ε) > q1, i.e., 0 < Fq ( 1 | ε ) < β , so that max{ d − q1,0} = d − q1 makes sense. This completes the proof of part (i) when Fq ( 1 | ε ) β and N d = γ ( ε ) . Now consider 0 < Fq ( 1 | ε ) < β . The criterion is sim- N ple but clear. Whether d = γ ( ε ) or N 1 − d = F ( β | ε ) depends on which can bring more expected profit, i.e., −1 ⎧ γε ( ), if πR( ε, F (max{ α,0}| ε)) > ⎪ N −1 d = ⎨ πR( ε, F ( β | ε)); ⎪⎩ −1 F ( β | ε), otherwise. This completes the proof of part (1) in Proposition 2. Next, show the link between the retailer’s second-period expected profit and the market signal to estimate the optimal total order from the beginning of period 1 based on the market signal conditions, which is the core idea of the corollary. −1 When β > 0, let π ( ε, F (max{ α,0}| ε)) = π ( ε, −1 F ( β | ε )) . Note that R R d ≡ d when β 0 because N N 1 β α . Suppose that there exists a solution T ( 2) o ε , and then for T ( 2) o ε the retailer is indifferent towards the choice of 1 − −1 F (max{ α,0}| ε ) and F ( β | ε ) . Obviously, for stronger signals T ( 2), o ε ε > N −1 d = F ( β | ε ) > q 1 . Otherwise, if such a T ( 2) o ε does not exist, or −1 −1 equivalently, πR( ε, F (max{ α,0}| ε)) > πR( ε, F ( β | ε )) for any signals, define T ( 2) max o N ε = ε . Thus, d = −1 F ( βε | ) > q1is impossible because the market signal cannot be greater than its maximum. Tsinghua Science and Technology, August 2008, 13(4): 570-580 Next consider part (2) which depicts the first-period ordering behavior. If o2=0, then the retailer’s second-period problem is given by the following: For any α , similar to the reasoning in proof of Lemma 1, d is implied by 2 1 1 N 1 N ˆ( ε d1 ) ∫ 0 2 N 1 α , i.e., e w2 N 1 w − w + o + [ e−w −( e− v) F( d | ε)]d G( ε) = 0. Particularly, when 0 , inequality Fd ( N | ε ) 0 holds for any signal ε , then ˆ( ε d1 ) must be equal to ε max , which is infinite in our model. Under such cases, we can rewrite the above equation as +∞ N 2 1 1 0 2 1 ∫ w − w + o + [ e−w −( e− v) F( d | ε)]d G( ε) = 0. Rearranging, +∞ N 1 1 0 1 ∫ e− w + o −( e− v) F( d | ε)d G( ε) = 0. Note that 0 ∞ N ∫ Fd ( 0 1 | ε)d G( ε) 1 special cases when e+ o1 w1 , then . There are some N N d = d1= 0. Note that e− w1+ o1 e−v cannot succeed because it requires o1 w1−v, which would incentivize the supplier to produce infinitely in period 1 and, thus, is ridiculous. In summary, the optimal first-period order quantity d is given by N 1 N ˆ( ε d1 ) N 2 1 1 0 2 1 ∫ w − w + o + [ e−w −( e− v) F( d | ε)]d G( ε) = 0 N when e+ o1 > w1. Otherwise, d1 ≡ 0 . □ Proposition 2 shows that the retailer’s ordering behavior is closely linked to the option price o 2 and the market signal ε . Different option prices result in different ordering behavior. Under some circumstances, the optimal total order may not even be unique. Signals such as ˆε and ε T can be used to categorize the retailer’s orders. These marking signals are important in models concerned with demand information update. The conclusion in the corollary is meaningful for disclosing the relationship between the retailer’s second-period profit curve and the market signal that optimal total order more than q 1 is possible if and only if T ( 2) max o ε ε < exists. T ( 2) o ε must be always greater than ε ( q1) for a given q 1 because the conservative retailer has incentive to choose an optimal total order greater than q 1 with T ( 2) o ε that he will never do
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