Now, consider the asymmetric-information setting: the manufacturer does not observe the realization of the market state (i.e., η) when designing the contracts. However, it knows that the market could be in the high state, i.e., η = ηh, with probability φ, and in the low state, i.e., η = ηl with probability 1 − φ. In contrast, the retailer, who is better informed than the manufacturer, observes the realization of the market state when making the decision to choose and accept the contracts, as well as when setting the retail prices.
In this analysis, we assume that κL ≤ κ2 and κH ≥ κ2L + 2βα and let ηh ∈ [η∗, η2] and ηl ∈ [η, η∗)39. Under asymmetric information, the manufacturer will offer a menu of
39These conditions ensure that the {G, wG, FG} arises as a candidate of the optimal contract in equilibrium;
see Proposition 4.1. Further, under these conditions, consumers in the high segment always purchase via the
contracts, {G, wG, FG} and {P, wP, FP}, to the retailer for self-selection. Based on the result of Proposition 4.1, if the market is in the high state, then the G contract is more profitable;
in contrast, if the market is in the low state, then the P contract is better. Therefore, the {G, wG, FG} contract in the menu under asymmetric information is designed for the high state (i.e., ηh), and the {P, wP, FP} contract is designed for the low state (i.e., ηl).
In the asymmetric-information setting, the sequence of events unfolds in five stages as follows. In the first stage, the manufacturer designs a menu of contracts, knowing that the market could be in the high state with probability φ and in the low state with probability 1−φ. In the second stage, nature moves and the state of the market is realized (ηi ∈ {ηh, ηl}).
In the third stage, the retailer observes the realization of the market state and decides on which of the two contracts to accept, if any. In the forth stage, the retailer sets its retail prices according to the contract it accepts. Finally, in the last stage, demand is realized and profits made.
In the following, we describe the optimization problem that the manufacturer is facing when designing this menu of contracts. Note that this menu must ensure proper incentives for the retailer to participate (see the IR-h and IR-l constraints), as well as to choose the
‘right’ contract (i.e., {G, wG, FG} when the market is in the high state and {P, wP, FP} in the low state; see the IC-h and IC-l constraints). The characteristics of the optimal menu is summarized in Proposition 4.2. All the derivations are given in Appendix C.
The menu of contracts under asymmetric information regarding η solves the following constrained optimization problem:
traditional posted-pricing mechanism while those in the low segment will choose the group-buying option whenever this option is available (see Appendix C).
max
wG, FG, pPh|h, pGBh|h wP, FP, pl|l
π = φ πGh|h + (1 − φ) πl|lP
= φ πG(wG, FG, pPh|h, pGBh|h; ηh, ·) + (1 − φ) πP(wP, FP, pl|l; ηl, ·)
= φn
α − βpPh|h− γ(pGBh|h + κL)wG+ FGo + (1 − φ)n
α − (β + γ)pl|lwP + FPo
, (4.23)
subject to:
(FOC-G) : {pPh|h, pGBh|h} = argmax
pP, pGB
ΠGh|h = ΠG(pP, pGB, wG, FG; ηh, ·), (4.24) (FOC-P) : pl|l = argmax
p
ΠPl|l = ΠP(p, wP, FP; ηl, ·), (4.25) (IR-h) : ΠGh|h∗ = max
pP, pGBΠGh|h ≥ 0, (4.26)
(IR-l) : ΠPl|l∗ = max
p ΠPl|l ≥ 0, (4.27)
(IC-h) : ΠGh|h∗ ≥ ΠPl|h∗ = max
p ΠPl|h = ΠP(p, wP, FP; ηh, ·), (4.28) (IC-l) : ΠPl|l∗ ≥ ΠGh|l∗ = max
pP, pGB
ΠGh|l = ΠG(pP, pGB, wG, FG; ηl, ·). (4.29)
Proposition 4.2 Under asymmetric information, given that the cost of joining group-buying is significantly high for consumers in the high segment (i.e., κH ≥ κ2L + 2βα) and sufficiently low for those in the low segment (i.e., κL ≤ κ2), the manufacturer offers to the retailer a menu of two contracts, {G, wG∗, FG∗} and {P, wP∗, FP∗}, where:
The former contract is chosen by the retailer when the market is in the high state, and the latter in the low state.
This result implies that under asymmetric information regarding η, the manufacturer, by offering the menu of contracts as described above, can regain the first-best profitability. To understand the intuition underlying this interesting finding, consider the nature of market uncertainty and its impact on the profitability of the two contracts (i.e., the P and G contracts), as well as on the retailer’s incentive of choosing the contracts. Whether the market is in the high or low state regarding η, the size of the entire market remains fixed.
In other words, consumers just reallocate themselves from one segment to the other between the two states; the total demand of the market remains identical. Therefore, the P contract, which serves the entire market with a single price, provides the same profitability in both market states. Most importantly, having the information on the total market demand, the manufacturer can determine the fixed fee in the P contract (i.e., FP) to extract all the surplus from the retailer whenever this contract is chosen. Regarding the G contract, as discussed above, it is more profitable when the market is in the high state (due to the higher benefit and the lower cost of price discrimination). Designed to be chosen in the high state, the fixed fee in the G contract (i.e., FG) is set to extract all the gross profit that the retailer may earn upon the employment of the G contract in the high state.
It is the profitability of the two contracts in each market state that determines the retailer’s incentive to choose either one of them. The above-mentioned pattern of profitability
perfectly aligns the retailer’s incentive to that of the manufacturer, and therefore, allows M to recover the first-best results even under asymmetric information. In particular, when the market is in the low state, the retailer has no incentive to choose the undesired G contract—
choosing this contract in the low state results in negative profit to the retailer since the fixed fee, FG, is higher than the gross profit (i.e., profit prior to paying the fixed fee) it would earn from this contract in the low state40. Most importantly, when the market is in the high state, the undesired contract, which is now the P contract, is not attractive to the retailer due to the fact that its profitability is identical to that when the market is in the low state and equals to the fixed fee it would pay to the manufacturer.
In summary, the term on the pricing mechanism in the contracts serves as a perfect sensor for the manufacturer to detect the state of the market (regarding the relative sizes of the two consumer segments) via the behaviors of the retailer. It helps eliminate the adverse incentive that the retailer would have under asymmetric information. In this setting, the private infor-mation regarding the relative sizes of the two consumer segment does not reward the retailer for its (possible) adverse selection of the pricing mechanism. This arrangement effectively improves channel coordination and increases the efficiency of the entire distribution system.