Coordination contract design for the newsvendor model

Coordination Contract Design for the Newsvendor Model

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Coordination Contract Design for the Newsvendor Model Linqiu Li, Ke Liu PII: DOI: Reference:

S0377-2217(19)30897-5 https://doi.org/10.1016/j.ejor.2019.10.045 EOR 16131

To appear in:

European Journal of Operational Research

Received date: Accepted date:

18 April 2019 31 October 2019

Please cite this article as: Linqiu Li, Ke Liu, Coordination Contract Design for the Newsvendor Model, European Journal of Operational Research (2019), doi: https://doi.org/10.1016/j.ejor.2019.10.045

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Highlights • We design contracts to coordinate the newsvendor setting with a supplier and a retailer under asymmetric demand information. • To characterize the structure of coordinating payment, we design mechanisms in a more general way by directly constructing payment schemes to satisfy incentivecompatibility and individual-rationality. • In the continuous case, we find that the set of coordinating contracts is restricted to special two-part tariffs where wholesale price equals unit production cost. • In the discrete case, contrary to expectation, linear wholesale price contracts achieve coordination.

1

Coordination Contract Design for the Newsvendor Model Linqiu Lia,b,c,∗, Ke Liua,b,c a

b

Academy of Mathematics and Systems Science, CAS, Beijing, 100190, China Key Laboratory Management, Decision and Information Systems, CAS, Beijing,100190, China c University of Chinese Academy of Sciences, Beijing, 100190, China

Abstract We design contracts to coordinate the newsvendor setting with a supplier and a retailer. Traditional approaches begin with a restricted set of contracts. However, we design mechanisms in a more general way by directly constructing payment schemes to satisfy incentivecompatibility and individual-rationality. Under symmetric information, coordinating contracts in the literature can be interpreted by this method. Under asymmetric information, we model the retailer’s private demand information as a space of either continuous or discrete states. In the continuous case, wholesale price contracts cause system inefficiency and there exists a unique optimal wholesale price for the supplier if the distribution of forecast error has IFR (Increasing Failure Rate) property. We further characterize the structure of coordinating payment, and find that the set of coordinating contracts is restricted to special two-part tariffs where wholesale price equals unit production cost. In the discrete case, contrary to expectation, linear wholesale price contracts achieve coordination. With demand forecast distributed more and more densely on its support, the interval of coordinating wholesale prices gradually shrinks to the unit production cost. Keywords: Supply chain management, Game theory, Asymmetric information, Mechanism design

∗

Corresponding author. Email addresses: [email protected] (Linqiu Li), [email protected] (Ke Liu)

Preprint submitted to Elsevier

November 7, 2019

1. Introduction Traditional research on supply chain management focuses on a firm’s internal operations to improve profitability. However, this individual decision is rarely in the best interest of the total system due to the divergence of objectives. Supply chain coordination has brought a new philosophy, which aims to align the motivations of independent channel members and coordinate their decisions and activities. This paper considers the classical newsvendor setting with one supplier and one retailer. Under symmetric information, market demand is stochastic with a distribution known by the firms. Due to a long production lead time, the retailer places an order in advance of the selling season and the supplier begins production after receiving the order. All the risks from stochastic demand and inventory are pushed onto the retailer. In fact, supply chain inefficiency often stems from the imbalance of risk allocation. If one corporation bears all the hazards but only obtains a portion of the channel’s profit, it will cause a discrepancy on marginal profit between the part and the whole, which is widely known as double marginalization. Costs and benefits should be allocated reasonably by a contract to achieve supply chain coordination. There are several coordinating contracts in the context of newsvendor model: buy-back contracts, revenue-sharing contracts, quantity discounts and quantity flexible contracts. The analysis recipe for all of those contracts is the same: begin with a given contract form, figure out the equilibrium of the game, and then determine the set of contract parameters that coordinate the supply chain. We analyze the problem in a somewhat reverse way. Now that the firms know all the information of the system (such as cost, retail price and demand distribution), both of them can calculate the expected profit under any designated contract. We only need to construct a transfer function under which both firms will choose the best decision for the collective and earn more than their reservation profits, that is, to satisfy the so-called incentive-compatibility and individual-rationality. We analyze the method in detail, and point out that some coordinating contracts in the literature can be obtained by this method. In fact, transfer functions give essential depictions of contracts. Specifically, contracts generated by the same transfer function are mathematically equiv3

alent, such as revenue-sharing contracts and buy-back contracts. Quantity discounts are equivalent to them in the sense of expectation. By choosing appropriate transfer functions, more contracts can be constructed to coordinate the supply chain. We further consider coordination mechanisms under asymmetric information. Due to the proximity to terminal market, the retailer tends to have better information about demand. When participants hold private information, they may make a false report about their private information in order to get more profits. Ineffective information sharing does harm to the interests of the collective. Egri and Vncza (2013) point out that designing applicable coordinating contracts under asymmetric information is one of the most compelling challenges of supply chain management. Wholesale price contracts prevail in practice. We make a detailed analysis of the contracts under asymmetric information, showing that the wholesale price contracts can neither achieve reliable information sharing nor coordinate the supply chain. Here we assume that wholesale price is endogenous. The supplier determines the wholesale price according to demand forecast information provided by the retailer. Lariviere and Porteus (2001) prove that there exists a unique equilibrium under symmetric information if the distribution of demand has IGFR(Increasing General Failure Rate) property. We prove the counterpart under asymmetric information: the supplier always ignores the forecast announced by the retailer and there exists a unique optimal wholesale price for the supplier, provided that the distribution of forecast error satisfies IFR (Increasing Failure Rate) condition. Although some kind of cost and profit sharing would be favorable for coordination, unfortunately it is not guaranteed in general networks with private information. We try to find coordinating contracts by the apparatus of mechanism design. The method comes from game theory with incomplete information. Its background is how to design mechanisms to specify public decision and monetary transfer when agents have private information. This is quite similar to supply chain coordination under asymmetric information. In contrast to most of the previous work that transfer functions depend on reported private information as well as realized demand, such as Egri and Vncza (2012), we assume there are no interactions between the supplier and the retailer after true demand information is revealed. Our 4

mechanism is actually a quantity discount contract. The supplier provides a menu, whereby the retailer is asked to choose a quantity or payment based on her private demand information. We characterize the structure of coordination mechanism, and we find that it is related to the distribution type of demand forecast. When private demand information is modeled as a continuous random variable, the set of coordinating contracts is restricted to a certain two-part tariff: wholesale price must be the unit production cost and upfront payment is arbitrary. In fact, the conclusion is partly intuitive. With the retailer’s marginal cost equal to the unit cost of the channel, double marginalization disappears. There’s only a constant difference between the retailer’s expected profit and the expected profit of the channel. Private information is no longer valuable and incentive-compatibility condition is satisfied naturally. This contract guarantees that the supplier always gets a fixed profit. We prove this is the unique possible coordinating contract under asymmetric information in the continuous case. However, it is much easier to coordinate the supply chain in the discrete case. We find that linear wholesale price contracts achieve coordination at this time. This result is somewhat surprising because researchers have long known that wholesale price contracts cause system inefficiency due to double marginalization. The set of coordinating wholesale price is bounded by incentive-compatibility constraints. With demand forecast distributed more and more densely on its support, the interval of feasible wholesale price will gradually shrink to unit production cost. This is consistent with the conclusion that wholesale price can only be the unit production cost in the continuous case. The rest of the paper is organized as follows. Section 2 reviews the related research that focuses on supply chain coordination and mechanism design. In Section 3, we interpret coordinating contracts under symmetric information by directly constructing transfer functions to satisfy incentive-compatibility and individual-rationality. In Section 4, we discuss supply chain coordination under asymmetric information by mechanism design. Section 5 concludes and provides extensions.

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2. Literature Review The main content of this paper is the theory of single-stage supply chain coordination. Researchers begin with the newsvendor setting under symmetric information. Firms have full information about each other’s business parameters, such as production cost and forecast information. It has long been known that wholesale price contracts cause system inefficiency due to double marginalization. Lariviere and Porteus (2001) analyzes the equilibrium solution of the wholesale price contract in detail, and further discusses the influence of distribution parameters on equilibrium and supply chain coordination. They point out that there exists a unique equilibrium solution for the Stackelberg game under a mild restriction that demand distribution satisfies IGFR condition. In fact, supply chain coordination is realized by several contracts. Pasternack (1985) proposes buy-back contracts: the supplier sells products to the retailer at a wholesale price and pays the retailer at a fixed unit price for every remaining good at the end of the season. Pasternack proves that if the wholesale price and the repurchase price are appropriately chosen, the buy-back contract achieves coordination. Cachon and Lariviere (2005) put forward revenue sharing contracts based on the inspiration from video rental market: the retailer pays the supplier a wholesale price for each unit purchased, plus a percentage of the retailer’s revenue. When the wholesale price and the revenue sharing proportion satisfy certain conditions, the revenue sharing contract also coordinates the supply chain. Both the buy-back contract and the revenue sharing contract arbitrarily distributes profits between the supplier and the retailer. In addition, channel coordination can be achieved by quantity flexibility contracts (Tsay (1999)), target rebate contracts (Taylor (2002)) and price-discount contracts (Bernstein and Federgruen (2005)). Cachon (2003) and Li and Wang (2007) provide excellent reviews for this part. The essential aspect of these contracts is to let the supplier and the retailer share the risk of uncertain demand and eliminate double marginalization. We use the method of mechanism design to construct payment functions, which can explain all the above coordinating contracts to a certain extent. Symmetric information is a strong assumption. In practice, information is often privately held by one member of the channel. The informed party may make a false report 6

about his private information to induce the other party to act in his favor. Hence, the uninformed party has a reason to doubt this forecast and make decisions based on his own inaccurate judgment. This incredible information sharing makes the supply chain more difficult to coordinate. The literature that explicitly models asymmetric information can be classified into two groups. A group of researchers focus on information asymmetry in market demand, such as Cachon and Lariviere (2001) and Ozer and Wei (2006). The other group focuses on information asymmetry in production cost, such as Ha (2001). The existing contracts under asymmetric information can also be divided into two categories from another point of view. The party with less information hopes to distinguish the private information of the other party through contracts, so as to make a corresponding optimal decision. Such contracts offered by the uninformed party are called screen contracts. On the other side, the informed party wants to differentiate from other types of participants in order to induce the other firm to act in his favor. Such contracts provided by the informed party are called signal contracts. For a survey of these two kinds of contracts, we refer the reader to Chen (2003). Cachon and Lariviere (2001) propose a signal contract with linear cost structure that enables reliable information sharing and coordinates the supply chain. Under their contract, the supplier’s expected profit is always zero. They assume that private information is a binary Bernoulli random variable. If this is extended to a multi-point distribution, it will be extremely complex. Cachon and Lariviere point out that any advance purchase contract that includes firm commitments is not a coordinating contract. Once final demand is less than the amount promised, it will cause a waste of capacity and production. Ozer and Wei (2006) put forward two contracts which achieve credible information sharing. The first one is capacity reservation contract. The supplier provides a menu of contracts composed by capacity and corresponding reservation payment. The supplier detects demand information through the manufacturer’s capacity reservation. This is a kind of screen contract. The second one is an advance purchase contract in which the manufacturer promises the supplier an order before the supplier invests in the capacity of component. The manufacturer sends his demand forecast to the supplier through his commitment capability. This is a sort of signal contract. 7

While these two contracts achieve effective information sharing, they don’t coordinate the supply chain. Ozer and Wei (2006) point out that the supply chain can be coordinated by combining payback contracts with advance purchase agreements. Egri and Vncza (2012) is the most related to ours. They suggest a coordination protocol that provides both partners right incentive to minimize the total cost. The customer is interested in sharing her demand expectation and forecast uncertainty. The supplier’s rational decision concurs with the overall optimum. Nevertheless, there are several differences. First, the payment depends on forecast report as well as final demand but our payment depends on forecast report only. Second, all the demand must be satisfied in their model while lost sales are allowed in our setting. Third, the demand follows logistic distribution in their paper but our model makes no assumption with specific distribution. Egri and Vncza (2013) consider a distributed network and design a coordination mechanism under Vendor Managed Inventory (VMI). The payment scheme still depends on forecast report as well as true demand. Another related research field is mechanism design. Supply chain coordination under asymmetric information and mechanism design problems have similar characteristics. Each agent has a private type. The utility of each agent is affected by public decision and his private type. A mechanism includes two parts: public decision and transfer function, which are both functions of reports of private types. According to Myerson’s Revelation Principle (Fudenberg and Tirole (1991)), the equilibrium of any mechanism can be generated by an incentive-compatible mechanism, in which all agents will truthfully report their private types. Thus, we only need to consider incentive-compatible mechanisms. A mechanism is called efficient if the sum of all agents’ utility is maximized by the public decision specified by the mechanism. If the total transfer of all agents specified by the mechanism equals zero, the mechanism is called budget-balanced. If the expected utility obtained by every agent under the mechanism is not less than his reservation utility, the mechanism is called individual-rational. The VCG (Vickrey-Clarke-Groves) mechanism establishes the existence of effective and incentive-compatible mechanism, but it does not balance the budget. Under the additional assumption that private information is independent across 8

agents, d’Aspremont and Grard-Varet (1979) construct an efficient, incentive-compatible and budget-balanced mechanism using the solution concept of Bayesian Nash equilibrium. Myerson and Satterthwaite (1983) consider a bargaining problem between one buyer and one seller for a single project. The seller’s valuation and the buyer’s valuation for the object are assumed to be independent random variables and each individual’s valuation is unknown to the other. Myerson and Sarthwaite characterize the set of allocation mechanisms that are Bayesian incentive-compatible and individual-rational, and show the general impossibility of ex post efficient mechanism without outside subsides. Following this methodology, we find a payment function that satisfies four conditions (incentive compatibility, individual-rationality, effectiveness and budget-balance) for the newsvendor model with private information, so as to achieve supply chain coordination. Lobel and Xiao (2017) use a relaxed approach proposed by Kakade et al. (2013) to consider the newsvendor model with asymmetric demand information over an infinite time horizon. When the excess demand is backlogged, the form of the optimal contract is surprisingly simple: the retailer only needs to select an appropriate wholesale price and prepay the corresponding deposit based on her private demand forecast in the first stage, and a wholesale price contract is executed at the selected price in each subsequent stage. However, the contract is designed to optimize the supplier’s profit and does not coordinate the supply chain. 3. Symmetric Information In the models under symmetric information, each participant knows all the information, including contract details and system parameters. The past relevant literature often fixes a contract form first, then lists the event flow under the contract rules. Next, the set of contract parameters is determined to coordinate the supply chain. Finally, the performance of the supply chain will be analyzed, such as efficiency and profit distribution. In this section, we devise coordinating contracts by directly constructing transfer functions to satisfy individual-rationality and incentive-compatibility. To a certain extent, the method explains all the coordinating contracts. In this paper, we will use ”she” to refer to the downstream retailer and ”he” to refer to the upstream supplier for convenience. 9

3.1. Centralized Model In this section, we consider a newsvendor setting composed by a supplier and a retailer. The retailer needs to place an order before the arrival of the selling season. We assume that the demand D is stochastic and its probability distribution function is Φ(·) (Φ = 1 − Φ) with density function φ(·). Let [D, D] be the support of D. The sequence of events is as follows: (1) The retailer places an order based on the possible demand state of the next selling season. (2) The supplier produces at unit cost c and delivers the order before the selling season. (3) The demand occurs in the selling season, and the retailer sells the products to the final customers at a fixed unit price p. (4) The supplier and the retailer allocate the cost and revenue according to the specific terms of the innitial contract. Since it is a model under symmetric information, all of the above information is transparent. We also assume that the supplier is able to acquire the retailer’s specific sales without incurring additional cost. This assumption will be removed in some cases. In order to clarify the essence of our innovative method, we demonstrate the analysis for the simplest case. The retailer has only one opportunity to place an order and the supplier can always provide the order quantity required by the retailer. Unmet demand is lost without additional stockout penalty and unsold inventory has zero salvage value. In order to avoid trivial solutions, we impose p ≥ c ≥ 0. Our goal is to find a set of contracts, under which no firm has a profitable unilateral deviation from the set of optimal actions for the entire supply chain. When the supplier and the retailer are vertically integrated, no payment between the firms is exchanged. The supply chain’s only decision is the production quantity q. The channel’s expected profit is given by: Πc (q) = pE D min(q, D) − cq, where the superscript c refers to coordination for notatinal simplicity. This is a classic

10

newsvendor model and the order quantity that maximizes the total profit is: c

−1

q =Φ

! c . p

(1)

Note that all the demand is satisfied by the order quantity qc with probability 1 − cp . The optimal quantity qc decreases with the unit production cost c and increases with the retail price p. We will use qc as a benchmark for supply chain coordination: a contract is said to coordinate the supply chain if the equilibrium order quantity of the game induced by the contract equals qc . 3.2. Wholesale Price Contract With a wholesale price contract the supplier charges the retailer w per unit purchased. Researchers have long known that the wholesale price contract does not achieve coordination because of double marginalization. The wholesale price contract induces a Stackelberg game led by the supplier. The event flow is as follows: (1) The supplier sets the wholesale price w. (2) The retailer chooses an order q ≥ 0, which could be equal to zero, indicating that the retailer refuses the contract. (3) The supplier fills the order at unit production cost c. (4) The retailer receives the order and sells at a fixed unit price p when the selling season comes. Suppose w ∈ [c, p], q ∈ [D, D]. We use the backward recursive method to determine the subgame perfect equilibrium of this game. For a given wholesale price w, the retailer’s expected profit as a function of q is given by: ΠwR (q) = pE D min(q, D) − wq. −1

The optimal order quantity for the retailer is q(w) = Φ ( wp ). The wholesale price w is usually expressed as a function of q: w(q) = pΦ(q) and the expected profit of the supplier is ΠwS (q) = (w − c)q = (pΦ(q) − c)q. Lariviere and Porteus (2001) prove that there exists a unique qw maximizing ΠwS (q) under the assumption that Φ has the IGFR property ( φ(q)q is Φ(q)

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weakly increasing for all q such that Φ(q) < 1). If the equation    c φ(q)q   = Φ(q) 1 − p Φ(q)

(2)

has a solution q∗ in [D, D], the optimal order quantity is q∗ . Otherwise, the optimal order quantity equals D. In order to make a positive profit, the supplier sets a wholesale price w strictly higher than c. It’s easy to find that qw < qc by comparing (1) and (2), so the wholesale price contract generally does not coordinate the supply chain. Taking the above analysis as an example, we see that the traditional way proceeds under a specific contract mode. We point out that we are able to design coordinating contracts by directly constructing proper transfer functions. 3.3. Construct Transfer Under symmetric information, both the supplier and the retailer know all the parameters and information of the system. Given any contract, the firms can calculate their specific expected profits and the loss of the channel. We only need to construct a transfer function under which both firms will choose the best decision for the collective and earn more than their reservation profits, that is, to satisfy the so-called incentive-compatibility and individual-rationality. Given a transfer function τ, the expected profits of the supplier and the retailer are: ΠτS

= τ − cq,

ΠτR = pmin(q, D) − τ. To guarantee that the final profits of the supplier and the retailer are nonnegative and the retailer will spontaneously choose the order quantity qc , the payment function τ should

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satisfy the following conditions: ΠτS

≥ 0,

ΠτR ≥ 0, qc = argmaxq≥0 ΠτR = argmaxq≥0 [pmin(q, D) − τ]. One of the simplest ideas is to let the retailer capture a fraction of the total profit. Then ΠτR = α(pmin(q, D) − cq), and τc = (1 − α)pmin(q, D) + αcq.

(3)

Under the payment function τc , the coordinating order quantity qc is optimal to the two firms. Thus, the transfer function τc coordinates the supply chain even under voluntary compliance. Here we show that the classical buyback contracts, revenue sharing contracts and quantity discount contracts are all special cases of this idea. Under a buyback contract (w, b), the retailer pays w to the supplier for each unit of the order, and the supplier pays b to the retailer per unit remaining at the end of the selling season. Then the transfer function of the buyback contract (w, b) is τ(w, b) = wq − b(q − D)+ , where a+ = max(a, 0). Denote S (q) = min(q, D). We have (q − D)+ = q − S (q), τ(w, b) =

wq − b(q − S (q)) = (w − b)q + bS (q) and τc = αcq + (1 − α)pS (q). Choose (w, b) satisfying w − b = αc, b = (1 − α)p. Then τ(w, b) = τc . In conclusion, the buyback contract (w, b) with 1 − the supply chain and the retailer captures α of the total profit.

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b c

=

w−b p

coordinates

Under a revenue sharing contract (w, ϕ), the retailer pays w to the supplier for each unit of the order, and the supplier shares 1 − ϕ of his revenue to the retailer at the end of the selling season. The transfer function of the revenue sharing contract (w, ϕ) is τ(w, ϕ) = wq + (1 − ϕ)pmin(q, D) = wq + (1 − ϕ)pS (q). Choose (w, ϕ) satisfying w = αc, ϕ = α. Then τ(w, ϕ) = τc . In conclusion, the revenue sharing contract (w, ϕ) with w = ϕc coordinates the supply chain and the retailer captures α of the total profit. Under the quantity discount contract, the supplier charges the retailer w(q) per unit purchased, where w(q) is usually a decreasing function. The transfer function of the quantity discount contract w(q) is τd = w(q)q. When the wholesale function is given by w(q) =

(1 − α)pE D min(q, D) + αcq , q

(4)

we can obtain τd = E D τc . The retailer will choose the order quantity qc in order to maximize her expected profit. In conclusion, the quantity discount contract with (4) coordinates the supply chain and the retailer captures α of the total expected profit. The above three contracts are very similar, but there are still subtle differences. The buyback contract and the revenue sharing contract are completely equivalent, and there is a one-to-one correspondence between their parameters. Under these two contracts, transfer functions have not been defined until the end of the selling season. The payment depends

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on specific demand and the retailer gets α of the total realized profit. However, the capital transfer of the quantity discount contract is completed before the selling season begins. In other words, no matter what the final demand is, the supplier’s profit will not be affected under the quantity discount contract, and the retailer obtains α of the total expected profit. The key question is when the payment actually happens. If it happens after the demand is realized, the supplier will also bear some risk of the uncertain demand. If it happens before the demand is realized, only the retailer will suffer the risk from the uncertain demand. The fact that the buyback contract and the revenue sharing contract coordinate the supply chain is based on the following assumption: the supplier has access to the retailer’s specific sales volume or the amount of the unsold inventory at the end of the selling season without incurring additional costs. However, the quantity discount contract does not need this assumption. In a word, a transfer function gives an essential depiction of a contract. Specifically, contracts generated by the same transfer function are mathematically equivalent, such as the revenue sharing contract and the buyback contract. And the quantity discount contract is equivalent to them in the sense of expectation. More generally, with a transfer function τ(q), the expected profit of the retailer is given by: ΠτR (q) = pE D min(q, D) − τ(q). In order

to make qc maximize the expected profit of the retailer, we need the following equation: τ0 (qc ) = c,

(5)

which is obtained by the first-order condition and Φ(qc ) = cp . If we further require −pφ(q) −

τ00 (q) < 0, the retailer’s expected profit ΠτR (q) is a strictly concave function and qc is the

unique optimal order quantity for the retailer. The above three contracts are all special cases when transfer functions take some form. We can explain more coordinating contracts by constructing new transfer functions, such as the quantity-flexibility contract proposed by Tsay (1999). Under a quantity-flexibility contract (w, δ), the supplier charges w per unit purchased but gives the retailer a full refund on the first δq units returned. The transfer function of the quantity-flexibility contract (w, δ)

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is τ(w, δ) = w[q − (δqF((1 − δ)q) + Z q = w[q − Φ(y)dy].

Z

q (1−δ)q

(q − y)dΦ(y))]

(1−δ)q

In order to satisfy τ0 (qc ) = c, we need w(δ) =

c Φ(qc ) + (1 − δ)Φ((1 − δ)qc )

.

Note that the wholesale price w(δ) increases with δ, δ ∈ [0, 1] and w(δ) ∈ [c, p]. Next, we check the second-order condition: d2 ΠτR (q) dq2

= −(p − w(δ))φ(q) − (1 − δ)2 w(δ) f ((1 − δ)q) < 0.

Other coordinating contracts can also be explained similarly by constructing proper transfer functions. 4. Asymmetric Information The previous discussions are all under symmetric information, where participants have the same information while making decisions. System inefficiency comes from the conflict of the firms’ interests. But there are different levels of asymmetric information about market demand in practice. The informed party may report his private information untruthfully in order to get more profits, so it is more difficult to coordinate the supply chain under asymmetric information. In the following model, we assume that the downstream company has more accurate information about market demand. Only the retailer knows some parameter u about demand distribution and the distribution increases stochastically with u, which means that for all u0 ≤ u, D(x|u0 ) > D(x|u) for all x > 0 and D(0|u0 ) ≥ D(0|u). Specifically, we work with additive model, which is also adopted by Lobel and Xiao (2017) to solve their optimal

16

contract design problem in multi-stage environment as well as by Ozer and Wei (2006) to investigate capacity procurement model. 4.1. Model Decripition We consider a supplier selling to a retailer, who subsequently sells to consumers at a given retail price p. Before a selling season, the retailer has a demand forecast u, which deviates from the true demand by a zero-mean forecast error . That is, the retailer’s demand D is u + , where u and  are independent random variables. Their respective cumulative probability distributions are F(·) and G(·), with densities f (·) and g(·). Let [u, u] and [, ] be the corresponding supports of u and , and u∗ be the expected value of u. To avoid negative demand, we assume u +  ≥ 0. Demand forecast u is the only asymmetric information in our model. All the other messages, such as production cost c, retail price p, the distribution of demand forecast u and forecast error , are all uncovered to both parties. The retailer perceives demand as the sum of a certain u and a random variable  with distribution G(·) and the supplier perceives demand as the sum of two random variable u and  with distributions F(·) and G(·) respectively. The other assumptions are the same as the ones in symmetric scenarios. We first calculate the optimal solution for the integrated system. When the supplier and the retailer are vertically integrated, no forecast information asymmetry exists and no payment between the firms is exchanged. Decision maker bases its order quantity q on demand forecast u and remaining market uncertainty. The expected profit of the channel is Πc (u) = pE min(q, u + ) − cq, where E means taking expectation with respect to forecast error  only. This is a classic newsvendor model and the order quantity that maximizes the total profit is: c

q = u+G

−1

! c . p

(6)

Note that (6) is equivalent to P (u +  ≤ qc ) = 1 − cp , which means that all the demand is 17

satisfied with probability 1 − cp . We will use the above results as a benchmark for supply chain coordination: a contract coordinates the supply chain if the equilibrium order quantity under the contract equals qc . 4.2. Wholesale Price Contract Wholesale price contracts are frequently used in practice. In this section, we make a detailed analysis of the equilibrium solution under wholesale price contracts, showing that the wholesale price contract neither achieves reliable demand information sharing nor coordinates the supply chain. We assume that the wholesale price is endogenous. The supplier determines the wholesale price according to demand forecast information provided by the retailer. Under a wholesale price contract, the sequence of events is as follows: (1) After obtaining the demand forecast information u, the retailer announces her forecast information to be uˆ , which could differ from her true forecast information u. (2) The supplier sets the wholesale price w according to the forecast information provided by the retailer, where w is assumed to be in [c, p]. (3) The retailer places an order q based on the wholesale price w and her true demand forecast u. (4) The supplier fills the order at unit cost c. (5) Demand D happens at the selling season and the retailer sells at unit price p. We determine the subgame perfect Nash equilibrium of this game by backward induction. After observing w, the retailer decides the order quantity before demand is realized. For a given w, the retailer’s expected profit is: ΠwR (q) = pE min(q, u + ) − wq, where the superscript w refers to quantities under the wholesale price contract. The optimal −1

order quantity is q(w) = u + G ( wp ). It is easy to see that q(w) decreases with the increase of w. To proceed recursively, we need the following lemma: Lemma 1 For the newsvendor model under asymmetric information, the retailer’s expected profit decreases as wholesale price w increases under a wholesale price contract. Proof: Let ΠwR (w, q) denote the expected profit of the retailer which depends on the 18

wholesale price w and the order quantity q. When w1 < w2 , it’s easy to check that ΠwR (w1 , q(w1 )) ≥ ΠwR (w1 , q(w2 )) > ΠwR (w2 , q(w2 )). So the retailer’s expected profit decreases as wholesale price w increases.  −1

The expected profit of the supplier is ΠwS (w; u) = (w − c)q(w) = (w − c)(u + G ( wp )). Nevertheless, the true forecast information u is not transparent to the supplier, then his expected profit is actually −1 w −1 w ΠwS (w) = Eu (w − c)(u + G ( )) = (w − c)(u∗ + G ( )). p p

There’s an inverse of distribution function and it’s not convenient to calculate the best −1

wholesale price directly, so we let y = u∗ + G ( wp ). The wholesale price w can be expressed as a function of y: w = pG(y − u∗ ) and the expected profit of the supplier is ΠwS (y) = (pG(y − u∗ ) − c)y. The optimal decision problem for supplier is y0 = argmaxy≥0 ΠwS (y).

g() weakly increases G() y0 maximizing ΠwS (y) and it is deter-

Lemma 2 If the distribution function G() has IFR property(i.e.

with  for all  such that G() < 1), there’s a unique mined by the following equation: ∗

G(y − u ) 1 −

yg(y − u∗ ) G(y −

u∗ )

!

=

c . p

(7)

If the equation has no solution in [u∗ + , u∗ + ], y0 = u∗ + . Proof: ΠwS (y) = (p − c)y for y ∈ [0, u∗ + ]. So ΠwS (y) is linear and strictly increasing in

[0, u∗ + ].

! dΠwS (y) g(y − u∗ )y ∗ − c. = pG(y − u ) 1 − dy G(y − u∗ )

Since G() satisfies IFR condition, Obviously, y ≥ u∗ + .

dΠw S (y) dy

g(y−u∗ )y G(y−u∗ )

∗

increases with y. Let y = sup{y| g(y−u ∗)y ≤ 1}. G(y−u )

strictly decreases with y in [u∗ + , y], so ΠwS (y) is strictly

concave for y ∈ [u∗ + , y]. ΠwS (y) is strictly decreasing for y ∈ [y, +∞], because

19

dΠw S (y) dy

< 0.

ΠwS (y) is strictly unimodal and y0 satisfies the following first-order condition: G(y − u∗ )(1 −

yg(y − u∗ ) G(y − u∗ )

)=

c . p

If the equation has no solution in [u∗ + , y](that is pG(y − u∗ )(1 − y ∈ [u∗ + , y]) , y0 = u∗ + . 

g(y−u∗ )y ) G(y−u∗ )

− c < 0 for all

Let w0 = pG(y0 − u∗ ), the next theorem characterizes the equilibrium under wholesale

price contract. Theorem 3 Under wholesale price contracts, the supplier always ignores the provided forecast information and sets wholesale price w0 . Proof: If the supplier believes the report of the retailer, the expected profit of the supplier −1

is ΠwS (w) = (w−c)q(w) = (w−c)(u+G ( wp )), which can also be expressed as (pG(q−u)−c)q in terms of q. According to the first order condition, G(q − u) 1 −

qg(q − u) G(q − u)

!

=

c . p

It’s easy to prove that q increases with u by contradiction from the above equation. By Lemma 1, the expected profit of the retailer decreases with w so increases with q. Hence, the retailer has an incentive to report a higher level. The supplier can’t distinguish the true demand state, and is better-off ignoring the report and setting wholesale price w0 based on his own information.  What drives the above results is the retailer’s underlying incentive to overforecast. It can be seen from the proof that a low-demand retailer would like to disguise herself as a high-demand customer to get a lower wholesale price. Expecting such an incentive to overforecast, the supplier can’t distinguish the real demand type of the retailer. Thus, the supplier will ignore the retailer’s demand forecast. Effective information sharing is not realized under the wholesale price contract. Similar results also appear in Ozer and Wei (2006), Ren et al. (2010). By comparing (6) and (7), we find that the order quantity under the wholesale price contract does not change with specific demand forecast, so it is not optimal for the whole 20

supply chain. Besides, even for a known demand forecast u, the order quantity under the wholesale price contract is less than the system optimal quantity. Inefficiency comes from information asymmetry and double marginalization. Now that wholesale price contracts cannot achieve reliable information sharing and coordinate supply chain, a natural question is whether such contracts exist. The next section gives an answer to the issue from the perspective of mechanism design. 4.3. Mechanism Design The method of mechanism design comes from game theory under incomplete information. Its background is how to design mechanism to arrange public decision and monetary transfer among agents when participants hold private information. This is quite similar to supply chain coordination under asymmetric information. We use mechanism design to find coordinating contracts under asymmetric information. Before the selling season begins, the retailer can obtain forecast information about demand. Then, the retailer reports her demand forecast. After that, a mechanism will specify order quantity and money that the retailer should pay to the supplier. That is to say, a mechanism is characterized by two functions q(·) and τ(·), denoted by (q, τ). When the retailer reports her demand forecast as uˆ , q(ˆu) denotes the order quantity specified by the mechanism and τ(ˆu) denotes the corresponding payment. The mechanism (q, τ) is actually a quantity discount contract. It can be interpreted as a menu, whereby the retailer is asked to choose a pair of order quantity and payment. In contrast to much of the work that has been done under asymmetric information, we assume that there are no interactions between the supplier and the retailer after demand realization. Transfer functions depend on the report of demand forecast only and have no relationship with the true demand. A mechanism is called incentive-compatible if it’s optimal for the retailer to report her demand forecast truthfully under the mechanism. Based on Myerson’s Revelation Principle(Fudenberg and Tirole (1991)) that the outcome of any mechanism can be realized by the equilibrium of an incentive-compatible mechanism, we restrict our attention to incentivecompatible mechanisms without any loss of generality.

21

Specifically, we say that a mechanism (q, τ) is incentive-compatible if and only if for every u and uˆ in [u, u], E pmin(q(u), u + ) − τ(u) ≥ E pmin(q(ˆu), u + ) − τ(ˆu).

(8)

The inequality asserts that the retailer can never expect to gain more by reporting uˆ when u is true. Furthermore, we assume that both firms are risk neutral and reservation profit is zero. To guarantee they are willing to participate in a mechanism, appropriate individual-rationality constraint is that the mechanism gives each individual nonnegative expected gain from the trade. So a mechanism (q, τ) is individual-rational if and only if ΠR (u) = E pmin(q(u), u + ) − τ(u) ≥ 0, ∀u ∈ [u, u],

(9)

ΠS = Eu τ(u) − cEu q(u) ≥ 0.

(10)

Note that we assume that the contract is signed just after the realization of demand forecast. Therefore, the expected profit of the retailer is calculated for any demand forecast while the supplier’s expected profit is computed in average. −1

Finally, a mechanism is called efficient if q(u) = u + G ( cp ). That is, order quantity attains optimality for the integrated system. We try to find efficient contracts that satisfy incentive-compatibility and individual-rationality. First, we give the following lemma: Lemma 4 If a mechanism (q, τ) is incentive-compatible, the retailer’s expected profit ΠR (u) increases with u. Proof: By the envelope theorem in Milgrom and Segal (2002) and IC condition (8), Π0R (u) = pG(q(u) − u) ≥ 0. So ΠR (u) increases with u.  The lemma is quite intuitive. When the incentive-compatibility condition is satisfied, the retailer will truthfully report the demand forecast. Demand forecast can be regarded as the unbiased estimation of final demand because they differ by a zero-mean error. With the increase in demand forecast, market condition becomes more promising and the expected profit of the retailer also increases. 22

It’s interesting to see that Π0R (u) is expressed only by order quantity q if the mechanism (q, τ) is incentive-compatible. Transfer function τ disappears in Π0R (u). If the mechanism (q, τ) is incentive-compatible and efficient,Π0R (u) = pG(q(u) − u) = −1

−1

pG(G ( cp )) = p[1 − G(G ( cp ))] = p − c. The expected profit of the retailer is linear in demand forecast u at this time. We can now state and prove our main result in the following. −1

Theorem 5 Suppose q(u) = u + G ( cp ). A mechanism (q, τ) is incentive-compatible if and only if τ(u) = cu + l (l is an arbitrary constant). Proof: Suppose τ(u) = cu + l. When u < uˆ , pE (min(q(u), u + ) − min(q(ˆu), u + )) Z q(ˆu)−u Z  = p( (q(u) − u − )g()d + (u − uˆ )g()d) q(ˆu)−u

q(u)−u

≥ p(u − uˆ )G(q(u) − u) = c(u − uˆ ) = τ(u) − τ(ˆu)

When u > uˆ , pE [min(q(u), u + ) − min(q(ˆu), u + )] Z q(u)−u Z  = p[ (u +  − q(ˆu))g()d + (u − uˆ )g()d] q(ˆu)−u

q(u)−u

≥ (u − uˆ )G(q(u) − u) = c(u − uˆ ) = τ(u) − τ(ˆu) So for any u, uˆ ∈ [u, u],

E pmin(q(u), u + ) − τ(u) ≥ E pmin(q(ˆu), u + ) − τ(ˆu). The sufficiency has been proved. −1

Suppose q = u + G ( cp ), 23

ΠR (u) = pE min(q(u), u + ) − τ(u) −1 c = pE min(u + G ( ), u + ) − τ(u) p −1 c = pu − τ(u) + pE min(G ( ), ) p Then Π0R (u) = p − τ0 (u). We have Π0R (u) = p − c based on the discussion after Lemma 4. So τ0 (u) = c, τ(u) = cu + l. The necessity is also proved. 

As can be seen from the above theorem, with order quantity maximizing the expected profit of the channel, the mechanism is incentive-compatible if and only if payment function is linear in demand forecast and the slope is unit production cost c. The constant l determines the profit allocation between the firms. This is similar to the classical result that double marginalization disappears when wholesale price equals the unit production cost. The −1

sufficiency of the theorem is intuitive. Under the mechanism (q, τ) with q(u) = u + G ( cp ) and τ(u) = cu + l, the expected profit of the retailer and the system differ by a constant. The retailer will make the optimal decision for the whole channel spontaneously. Private information is no longer meaningful and IC condition can be satisfied naturally. Only when market condition is promising, can the entire supply chain be profitable. Next, we analysis when the optimal expected profit of the whole system is nonnegative for all demand forecast. Theorem 6 If the following inequality holds, the expected profit of the whole supply chain will be nonnegative for all demand forecast.

(p − c)u + p

Z

−1

G ( cp ) 

g()d ≥ 0

(11)

Proof: The expected profit of the whole system is Π(q, u) = pE min(q, u + ) − cq. Denote q(u) = arg maxq Π(q, u). For u1 ≤ u2 , Π(u1 , q(u1 )) ≤ Π(u2 , q(u1 )) ≤ Π(u2 , q(u2 )). 24

To gurantee Π(u, q(u)) ≥ 0 for all u ∈ [u, u], we only need −1 c −1 c Π(u, q(u)) = pE min(u + G ( ), u + ) − c(u + G ( )) ≥ 0, p p

that is (p − c)u + p

R G−1 ( c ) p



g()d ≥ 0. 

Under the condition that the system can be profitable, the following theorem characterizes the set of coordinating contracts. Theorem 7 Suppose the channel can be profitable for all demand forecast. The mech−1

anism (q∗ , τ∗ ) satisfies IR conditions, where q∗ (u) = u + G ( cp ), τ∗ (u) = u(c + ul ) and the R G−1 ( c ) −1 −1 p constant l ∈ [cG ( cp ), (p − c)u + p  g()d + cG ( cp )]. Proof: According to Theorem 5, the mechanism (q∗ , τ∗ ) is incentive-compatible. By

Lemma 4, to guarantee the individual-rationality of the retailer, we only need −1 c ΠR (u) = pE min(q∗ (u), u + ) − τ∗ (u) = (p − c)u + pE min(G ( ), ) − l ≥ 0. p

To guarantee the individual-rationality of the supplier, we need −1 c ΠS = Eu τ∗ (u) − cEu q∗ (u) = cEu + l − c(Eu + G ( )) ≥ 0. p

The range of l is obtained by solving the above two inequalities, and the length of the interval is nonnegative by Theorem 6.  By the theory of mechanism design, we give the set of coordinating contracts under asymmetric information. The condition for the existence of coordinating contracts is mild, only requiring the profitability of the system. From the proof of the theorem, we can see −1

that the supplier’s expected profit equals l − cG ( cp ), which is a constant unrelated to u. So the mechanism (q∗ , τ∗ ) also satisfies the supplier’s ex post individual-rationality. The profit of the supplier is nonnegative for any demand forecast u ∈ [u, u] under the contract. We make a further explanation about how the contract is executed. Although it seems −1

that we force q∗ (u) = u + G ( cp ) in our analysis, the contract is actually a quantity discount contract: w(q) = c +

−1

l−cG ( cp ) . q

It just specifies the relationship between order quantity 25

and wholesale price. If the retailer chooses a higher quantity, the wholesale price will decrease. The efficient order quantity is chosen voluntarily by the type u retailer because IC condition is satisfied by our mechanism. This contract can also be seen as a two-part tariff: −1

τ(q) = cq + l − cG ( cp ). The payment scheme includes a wholesale price equal to the unit production cost and an appropriate upfront payment. It guarantees that the supplier always gets a fixed profit. And we prove this is the unique possible coordinating contract under asymmetric information if all the interactions are completed before demand realization. At the end of this section, we analyze the discrete case similarly. We first allow two values for demand forecast u. P(u = uL ) = p, P(u = uH ) = 1 − p, where uL < uH . Before signing the contract, the retailer knows whether the demand of the selling season is good (u = uH ) or bad (u = uL ), but the supplier only knows the distribution of u. Other assumptions and definitions are the same as in the case of continuity. A mechanism (q, τ) is incentive-compatible iff the retailer will truthfully report her demand forecast under the mechanism (q, τ). So a (q, τ) is incentive-compatible iff the following two inequalities are satisfied: ΠR (uL ) = pE min(q(uL ), uL + ) − τ(uL ) ≥ pE min(q(uH ), uL + ) − τ(uH ),

(12)

ΠR (uH ) = pE min(q(uH ), uH + ) − τ(uH ) ≥ pE min(q(uL ), uH + ) − τ(uL ).

(13)

A mechanism should further satisfy individual-rational conditions: ΠR (u) = E pmin(q(u), u + ) − τ(u) ≥ 0, u ∈ {uL , uH }, ΠS = Eu τ(u) − cEu q(u) ≥ 0. −1

Obviously, we should require q(u) = u + G ( cp ), u ∈ {uL , uH } in order to coordinate the supply chain. 26

First, the following result is easy to get: Theorem 8 Suppose that demand forecast occurs in two states. (1)ΠR (uH ) ≥ ΠR (uL ) if the mechanism (q, τ) is incentive-compatible. −1

(2)Once q(u) = u + G ( cp ), a mechanism (q, τ) is incentive-compatible iff

p c− uH − uL

Z

−1

G ( cp )+uH −uL −1

G ( cp )

Z

p ≤c+ uH − uL

! c τ(uH ) − τ(uL ) G ( ) + uH − uL −  g()d ≤ p uH − uL −1

−1

G ( cp ) −1

G ( cp )−(uH −uL )

! c  − (G ( ) − (uH − uL )) g()d p −1

Proof: (1) According to IC conditions (12) and (13), we get ΠR (uH ) − ΠR (uL ) ≥ pE [min(q(uL ), uH + ) − min(q(uL ), uL + )] ≥ 0 (2) It follows immediately from (12) and (13).  To guarantee that the whole supply chain is profitable even in the worst case, we need R G−1 ( c ) p g()d ≥ 0 by (11). The next theorem presents a coordinating contract (p−c)uL + p  that satisfies IC and IR conditions when the whole supply chain is always profitable.

Theorem 9 If demand forecast occurs in two states and the integrated channel is always profitable, the mechanism (q∗ , τ∗ ) coordinates the supply chain and satisfies IC and −1

−1

IR conditions. q∗ (u) = u + G ( cp ), τ∗ (u) = wq∗ (u) = w(u + G ( cp )), w ∈ [c, c + ∆], and ∆ = min{∆1 , ∆2 } > 0, where p ∆1 = uH − uL

Z

−1

G ( cp ) −1

G ( cp )−(uH −uL )

! c  − (G ( ) − (uH − uL )) g()d, p −1

27

and ∆2 =

(p − c)uL + p

R G−1 ( c ) p

 −1 uL + G ( cp )

g()d .

Proof: It’s clear that the mechanism (q∗ , τ∗ ) coordinates the supply chain. By theorem 8(2), it’s easy to check that the mechanism (q∗ , τ∗ ) satisfies IC conditions. Now we check IR conditions. The supplier is individual-rational because −1 c ΠS = Eu τ∗ (u) − cEu q∗ (u) = (w − c)Eu [q∗ (u)] = (w − c)(u∗ + G ( )) ≥ 0. p

To show the individual-rationality of the retailer, it suffices to prove ΠR (uL ) ≥ 0 by theorem 8(1). When w ∈ [c, c + ∆], ΠR (uL ) = pE min(q∗ (uL ), uL + ) − τ∗ (uL ) −1 c −1 c = pE [uL + min(G ( ), )] − w(uL + G ( )) p p Z G−1 ( c ) p −1 c −1 c = puL + cG ( ) + p g()d − w(uL + G ( )) p p  ≥ 0. It’s obvious that p ∆1 = uH − uL

Z

−1

G ( cp ) −1

G ( cp )−(uH −uL )

! c  − (G ( ) − (uH − uL )) g()d > 0, p −1

28

and

∆2 =

(p − c)uL + p

R G−1 ( c ) p

g()d

 −1 uL + G ( cp ) −1 c)uL + pG(G ( cp )) −1 uL + G ( cp )

>

(p −

=

(p − c)(u + ) −1

uL + G ( cp )

> 0. So ∆ > 0.  We next extend the model to include multiple demand states. Demand forecast u takes P values in {u1 , u2 , · · · , un }, u = u1 < u2 < · · · < un = u, P(u = ui ) = fi , ni=1 fi = 1.

Theorem 10 Suppose demand forecast occurs in mutiple states and the integrated chan-

nel is always profitable. (1)The mechanism (q∗ , τ∗ ) coordinates the supply chain and satisfies IC and IR con−1

−1

ditions, where q∗ (u) = u + G ( cp ), τ∗ (u) = wq∗ (u) = w(u + G ( cp )), w ∈ [c, c + ∆n ], R G−1 ( c )   p g()d R G−1 ( cp ) (p−c)u+p  −1 c p ∆n = min{ u j −ui −1 c  − (G ( p ) − (u j − ui ) g()d, }. −1 c i< j

u+G ( p )

G ( p )−(u j −ui )

(2)If u1 , u2 , · · · , un are uniformly distributed in [u, u], that is u2 − u1 = · · · = un − un−1 =

u−u n−1 ,

then limn→∞ ∆n = 0.

Proof: (1) The proof is similar to the proof of theorem 9.   R G−1 ( cp ) −1 c p ( (2)By the L’Hospital rule, lim u j −u  − (G ) − (u − u ) j i g()d = 0. −1 c p i

So limn→∞ ∆n = 0. 

u j →ui

G ( p )−(u j −ui )

This is nothing but a wholesale price contact. The retailer is asked to choose a quantity from the given options based on her private demand information. It is somewhat surprising that simple wholesale price contracts with w ∈ [c, c + ∆n ] coordinate the supply chain in the discrete case. With more and more possible demand states in the support [u, u], the length of the feasible interval [c, c + ∆n ] gradually shrinks to 0. The unit production cost becomes the unique coordinating wholesale price in the limit. This phenomenon is consistent with

29

the conclusion in the continuous case, because IC inequalities (8) should be satisfied for any two points u,ˆu that are sufficiently adjacent in [u, u]. Example Consider a supply chain with c = 3, p = 15. u and  are both uniformly distributed: u ∼ U[5, 15],  ∼ U[−5, 5]. By Theorem 10, the interval of feasible prices is [3, 3 + ∆n ]. ∆1 = 6, ∆n =

7.5 n−1 ,

and limn→∞ ∆n = 0. The order quantity q∗ = u + 3. With

only two values for u (uL = 5, uH = 15), coordinating wholesale prices are chosen from [3, 9] and the optimal expected profit for the integrated channel is 12u − 12. For a low type retailer, the supplier gets gets

3w−9 28

w−3 6

× 100% of the total profit. For a high type retailer, the supplier

× 100% of the total profit. Note that the wholesale price contract does not achieve

arbitrary profit allocation between the supplier and a high type retailer in this example. 5. Conclusions and Discussions This article investigates coordination mechanisms for the newsvendor model. We design mechanisms in a more general way by directly constructing payment schemes to satisfy incentive-compatibility and individual-rationality. When the private demand forecast of retailer is continuous, we prove that the supply chain can be coordinated if and only if it is a special two-part tariff where the wholesale price is the unit production cost. When the distribution of demand forecast is discrete, we find that linear wholesale price contracts achieve coordination. With demand forecast distributed more and more densely on its support, the interval of feasible wholesale prices will gradually shrink to the unit production cost. Any contract specifies certain rules for participants to comply with and firms play a game under the mechanism. The mechanism under asymmetric information consists of order quantity and transfer function, which depend only on the report of private information. Although it seems that the efficient order decision is enforced by the mechanism, the contract is actually implemented as a quantity discount contract. The supplier just provides a payment scheme or a menu, and the retailer chooses a quantity based on her private demand information. The efficient order quantity is chosen voluntarily by the retailer because our mechanism is incentive-compatible. The literature always assumes that the participants have commitment power. We also presume that all individuals will abide by the contract 30

once they accept it. The supply chain achieves the best performance under a coordination contract. In the increasingly competent market environment, it’s quite important for a company to stay in a sustainable supply chain. A provident firm has an incentive to care about the development of the whole system. To implement the coordination contract in our paper, the firms need to negotiate the parameter l in Theorem 7. Cachon and Lariviere (2005) point out that the particular profit split chosen probably depends on the firms’ relative bargaining power. In our paper, we prove the existence of the coordination contracts under asymmetric information and characterize the set of coordinating payments. We have to admit that our work focuses more on theoretical analysis. How to make the firms reach an agreement to execute the coordination contract in practice is still a common problem in the field of supply chain coordination. More future research is needed to solve this problem. We are extending the current conclusions to multi-stage contract design. For a multistage newsvendor model, order quantity and transfer function may depend on the whole history consisting of demand forecast in every stage and other information, such as inventory level and past demand realizations. This is a dynamic mechanism design problem. Lobel and Xiao (2017) use a relaxation method of Kakade et al. (2013) to design an infinite-stage optimal contract from the perspective of the supplier. Under a backlogging model, the optimal dynamic contract takes a simple form. The retailer only needs to select an appropriate wholesale price in the first stage and prepay an associated upfront fee, and then executes wholesale price contracts with the selected price in each subsequent stage. However, supply chain inefficiency appears under this contract. We are trying to find a coordinating contract in dynamic environment by using the existing dynamic mechanism design method. Acknowledgements The research is partially supported by the National Natural Science Foundation of China under grants 71390334.

31

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