Optimal Quantity Discount Policy: Asymmetric Info

A Supplier’s Optimal Quantity Discount Policy Under Asymmetric Information Charles J. Corbett Xavier de Groote Presented by Jing Zhou

Agenda • Introduction • The Basic Model: Full Information • Optimal Contract Under Asymmetric Information • Comparing the Contracts • Discussion

Motivation Supply Chain • Inefficiency occurs under self-interest Supplier Buyer Lot size Q Demand d ks + kb Setup costs: hb Holding cost:

Literature • Joint economic lot-sizing literature • The supplier wishes to induce the buyer to choose a higher lot size • Quantity discount scheme can achieve the jointly optimal lot size • The supplier has full information about the buyer’s cost structure. • Lal and Staelin (1984) • N groups of buyers of different sizes • Holding costs, order costs and demand rates vary between groups but not within groups • Optimal unified pricing policy (no closed-form)

In this paper • The buyer holds private information about her holding cost; The supplier knows the prior distribution of the holding cost • Arbitrary continuum of buyer types • Demand per unit time is known and constant, and is not affected by the lot sizing or contracting decision • The supplier can choose not to trade with buyers • Optimal contracts in both full information case and asymmetric information case (quantity discount)

Notation • cases considered (full information and asymmetric information) • range of buyer holding cost hb • supplier’s prior distribution and density over hb • payment from supplier to buyer, as a function of hb, under contract i • buyer, supplier and joint cost function (excluding quantity discount) • maximum net cost level acceptable to buyer and supplier (reservation net cost level) • buyer, supplier and joint net cost function in case i, after quantity discount • Partial derivative of P(hb) with respect to hb

The Basic Model • Two decisions: lot size Q and quantity discount P(Q) per unit time from supplier to buyer

A Necessary Condition • If the joint costs under jointly optimal lot size Qj exceed , there will be no trade. Therefore, we need (1)

Cut-Off Point h*b • Because of the reservation net cost level, , the supplier choose a cut-off point h*bsuch that he will choose not to trade with buyers with holding cost hb > h*b

The Principal-Agent Framework • The supplier is the principal and the buyer is the agent (adverse selection). • The supplier proposes a “menu of contracts” or discount scheme, specifying the discount P(q) offered for any lot size q. • The buyer decides whether or not to accept the contract. She chooses some order lot size Q if accepts. • The supplier gives the buyer a discount of P(Q).

The Supplier’s Problem

Optimal Contract Under Full Information (First-Best) • Payment from supplier to buyer: • The buyer will choose the joint economic lot size Qj(hb) • The supplier keeps all efficiency gains • Condition (1) guarantees that for all

Assumptions • Assumption 1 Decreasing reverse hazard rate: Many common distributions satisfy this assumption • Assumption 2 Rule out the problems caused by the distribution with thin tails • Neither of these assumptions on F(hb) are necessary conditions

Recall: The Supplier’s Problem

Contracting Procedure • The buyer knows hb, unobserved by the supplier • The supplier offers a menu for any the buyer announces To ensure that IRb and IRs are satisfied, the supplier can seth*b such that IRb and IRs are met for all hb <=h*b • The buyer chooses to accept or reject the contract. • Accept (hb <=h*b): buyer choose buyer and supplier incur net costs per period of • Reject (hb >h*b): both parties revert to their outside alternatives and incur net costs per period of , respectively

Revelation Principle • The revelation principle (Laffont and Tirole 1993) states that if there is an optimal contract for the supplier, then there exists an optimal contract under which the buyer will truthfully reveal her holding cost. • Formulate an IC constraint on PAI(hb): Let FOC be satisfied at and yields IC constraint:

Revelation Principle (Con’t) • IRb Assume that IRb is always met for any , and for IRb becomes • The supplier’s problem

Proposition 1

Proposition 1 (Con’t)

Some Findings • There is a probability 1 - of no trade taking place. • For , QAI(hb) is decreasing in hb. • In general, cannot be found explicitly as it is impossible to write PAI(hb) explicitly. • Both QAI(hb) and PAI(hb) are strictly decreasing in hb, so there is a one-to-one mapping PAI(Q) which isincreasing. • Supplier’s and buyer’s costs are increasing in hb. • The buyer’s net costs are equal to at . • At , the resulting lot size is equal to the jointly optimal lot size; as hb increases, the discrepancy between QAI(hb) and QFI(hb) increases.

Proposition 2

Discussion • Contribution • Consider asymmetric information in EOQ environment: buyer’s holding cost is not observable to the supplier • The supplier can choose not to trade with buyer with certain large holding cost • Introduce the cut-off point policy

Discussion (Con’t) • Limitation • The assumptions on prior distribution are too specific • Lack of necessary proof or explanation of major results or findings • Demand is not affected by the lot sizing or reduction of unit price • Only the uncertainty about buyer’s holding cost is considered. More dimensions can be analyzed.

Thank you!

Optimal Quantity Discount Policy: Asymmetric Info