General levels of pessimism

So far we have restricted our analysis to the case that the level of pessimism γ≡ Pr(θ = θL) = ¹₂. We now remove this assumption and allow γ to be any number within the interval (0, 1). The procedure of deriving the equilibrium behaviors when γ ̸= ¹₂ is almost identical to that when γ = ¹₂. As the only modification is to generalize the quantities N_jk, P_jk, Z_k, and the probability for the reseller to see the good signal to be functions of γ, we omit the details here. Unfortunately, even though the closed-form expressions of optimal contracts and efforts can still be derived, the generalization of γ greatly complicates our three-layer supply chain and prevents us from obtaining clear-cut analytical results in terms of the manufacturer’s profitability and preferences. Therefore, we resort to numerical experiments to generate insights. Our first observation is in line with Proposition 1 and characterizes how the reseller’s accuracy affects the manufacturer’s expected profit.

Observation 1. For any value of γ, the manufacturer’s expected profitM is either first decreasing and then increasing or monotonically decreasing in the reseller’s accuracy λ_R∈ [¹₂, 1]. In particular, M tends to be nonmonotone when γ is low but monotonically decreasing when γ is high.

Though generalizing the level of pessimism destroys the convexity in general, it qualitatively preserves the relationship between the manufacturer’s expected profit and the reseller’s accuracy.

This is because those conflicting effects still exist under any value of γ and thus the shape of the manufacturer’s profitability remains similar. More interestingly, we find that the value of γ plays a role in determining the shape of the manufacturer’s expected profit. For each curve in Figure 5, the manufacturer’s expected profit is monotonically decreasing in λ_R at the left-hand side but nonmonotone at the right-hand side. When γ decreases, the right-hand side enlarges and the manufacturer’s expected profit is more likely to be increasing in λ_R when λ_R is large. This is because if the reseller becomes more accurate under a low level of pessimism (small γ), it is more likely for her to observe the good signal more often, to offer a more generous contract, and to induce a higher sales effort. The manufacturer thus benefits from such an improvement. On the contrary, increasing γ enlarges the left-hand side makes the manufacturer’s expected profit more likely to be decreasing in λ_R when λ_R is large. This is due to the fact that under a high level of pessimism (large γ), being more accurate makes the reseller more pessimistic and drives the expected effort level down. Consequently, the manufacturer earns less in expectation.

Our next step is to investigate how different values of γ affect the manufacturer-optimal re-seller’s accuracy λ^∗_R. Recall that in Proposition 2 and Figure 3 we characterize and visualize the

1 1.2 1.4 1.6 1.8 0.5

0.6 0.7 0.8 0.9 1

The market condition ratio η Thesalesagent’saccuracyλA

γ = 0.4 γ = 0.2

γ = 0.5 γ = 0.6

γ = 0.8

Figure 5: Monotonicity of the manufacturer’s expected profit with various levels of pessimism.

two-dimensional cutoff structure when γ = ¹₂. As we summarize in the next observation, the same structure still applies to other values of γ. This observation is visualized in Figure 6, where we depict several cutoff curves under various values of γ. For each curve, the manufacturer-optimal reseller’s accuracy is λ^∗_R = ¹₂ at the left-hand side and λ^∗_R = 1 at the right-hand side. It is clear that all these curves have similar shapes and the insight we obtained for the special case γ = ¹₂ is still valid.

Observation 2. For any γ ∈ (0, 1), the manufacturer’s expected profit M is maximized at λ^∗_R= ¹₂ (respectively, λ^∗_R= 1) if η and λA are both small (respectively, large) enough. Moreover, it is more likely that λ^∗_R= ¹₂ (respectively, λ^∗_R= 1) when γ increases (respectively, decreases).

The second part of Observation 2 delivers more messages to us regarding the impact of the level of pessimism γ on the manufacturer-optimal reseller’s accuracy λ^∗_R. As γ increases, improving the reseller’s accuracy makes the reseller more pessimistic. The manufacturer thus prefers the uninformed reseller more. Decreasing γ introduces the opposite effect and makes the manufacturer prefer the precise reseller more. Interestingly, even if γ is extremely close to 0, it is still possible that delegating to an uninformed reseller is optimal for the manufacturer. The following proposition provides an analytical support.

Proposition 7. For any γ ∈ (0, 1) and η < 2, there exists a threshold ´λA(γ, η) ∈ (¹₂, 1] such

The market condition ratio η Thesalesagent’saccuracyλA

1 1.4 1.8 2.2 2.6 3

0.5 0.6 0.7 0.8 0.9 1

γ = 0.6

γ = 0.8 γ = 0.5

γ = 0.4 γ = 0.2

Figure 6: Manufacturer-optimal reseller’s accuracy with various levels of pessimism.

that delegating to the uninformed reseller uniquely maximizes the manufacturer’s expected profit if λ_A< ´λ_A(γ, η).

6 Conclusion

In this paper, we consider a three-layer supply chain with a manufacturer, a reseller, and a sales agent. While the manufacturer is uninformed about the realization of the random market condition, both the reseller and the sales agent can conduct demand forecasting to estimate the realized market condition. We show that the manufacturer’s profitability is hurt when the reseller or the sales agent improves her/his low accuracy. When the accuracy is high, however, an improvement may allow the manufacturer to earn more in expectation. From the manufacturer’s perspective, when the market demand is only slightly volatile and the sales agent is not accurate, the uninformed reseller is preferred; when the demand is highly volatile and the sales agent is pretty accurate, delegating to the precise reseller is optimal. We also find that the manufacturer’s interest may be aligned with the reseller’s when only the reseller can choose her accuracy. However, this alignment is never possible when both downstream players have the discretion to choose their accuracy.

Our study certainly has its limitations. In this study, we exclude the possibility for the manufacturer to communicate directly with the sales agent. While our three-layer supply chain

is pervasive in practice, there are also situations where the sales agent is hired or can be directly compensated by the manufacturer (for more details about this situation, see, e.g., [14]). New insights may be found under this alternative setting. We also assume that the forecasting accuracy is public to every supply chain member. Removing this assumption will introduce the necessity of a two-stage screening process (one for the accuracy and the other for the signal) and may change some of our results. Finally, a promising direction is regarding the interest alignment issue we point out in this study. When the downstream players are allowed to choose their own accuracy, it is highly possible that the equilibrium accuracy mix will be manufacturer-suboptimal. Whether there exists any mechanism to align the interests remains open and calls for further investigation.

Appendix

Proof of Lemma 1. First, we can apply AF k ≥ AF k(U ), N_{F k}² ≥ N_{U k}² , and AU k ≥ 0 to show that AF k ≥ 0 is redundant. If we ignore the constraint AU k ≥ AU k(F ), the remaining two constraints will be binding at the optimal solution. Therefore, we can replace αF and αU by α_F = ¹₂β_U²(N_{F k}² − N_{U k}² )−¹₂β_F²N_{F k}² and α_U = −¹₂β_UN_{U k}² in the objective function. The problem

Verifying the constriantAU k ≥ AU k(F ) is trivial and omitted. The reseller’s expected profitRk(t) can be found by plugging β_F^∗ and β_U^∗ into (8).

Proof of Lemma 2. First, we can applyRG ≥ RG(B), Z_G≥ ZB, andRB≥ 0 to show that RG≥ 0 is redundant. If we ignore the constraint RB ≥ RB(G), the remaining two constraints will be binding at the optimal solution. Therefore, we can replace u_Gand u_Bby u_G= ^v₂^B2(Z_G−ZB)−^v₂^2GZ_G

G through the first-order condition. Verifying the constraint

RB ≥ RB(G) is trivial and omitted. The manufacturer’s expected profit M can be found by

G)] by combining the results obtained in Lemmas 1 and 2. It then remains to solve for the first-best outcome. When the supply chain is integrated by the manufacturer, if it observes s_A= j and s_R= k, it will select the effort level by solving

maxa≥0 E[

The first-best effort level is a^{F B}_jk = N_jk, which implies that the expected first-best effort level is a^{F B} =∑

j∈{F,U}

∑

k∈{G,B}a^{F B}_jk P¯jk = ¹₂[PF GNF G+ PF BNF B+ PU GNU G+ PU BNU B]. As Yk < 1 for k∈ {G, B} if λA> ¹₂ and Z_B< Z_G if λ_R> ¹₂, we have a^{F B} > a^∗ unless λ_R= λ_A= ¹₂.

Proof of Proposition 1. First, we express the manufacturer’s contract design problem as deter-mining the optimal sales bonus for the type-B reseller, i.e., M = maxv∈[0,1]

{ZG

8 (1− v²) + ^Z₄^Bv }

. Therefore, M will be convex if ZG and Z_B are both convex. To prove the convexity of Z_k, note that the type-k reseller’s expected profit Rk = ut+ Zkv²_k can also be expressed as the problem of determining the optimal amount of downward distortion of the sales bonus for the type-(B, k) agent, i.e.,

This implies that Z_k is also the maximum of several functions. As all the coefficients are non-negative with y ∈ [0, 1], it suffices to show that PF kN_{F k}² , P_{U k}N_{U k}² , and P_{F k}N_{U k}² are all con-vex. Consider the case with k = G first. It is straightforward to verify that _∂λ^∂22

R

λ_A. For h(λ_A, 1), which is a third-degree polynomial function of λ_A, because its smaller stationary is continuous, it follows that when λR is sufficiently close to 1, _∂λ^∂

RYk will be positive. We may then define ¯λ_k(λ_A, θ_H, θ_L) as max

{

λ_R∈ (¹₂, 1)_∂λ^∂_RY_k = 0 }

if it exists or ¹₂ otherwise.

Proof of Proposition 2. Let M(λA, λ_R) be the manufacturer’s expected profit under λ_A and λR. With this notation, we have M(λA, 1) = ¹₄(θ²_H+ θ_L⁴/θ²_H) and root according to Descartes’ rule of signs. Therefore, p1(η) has a unique greater-than-one root η₁ ≈ 1.3954 and thus _∂λ^∂_Ax(λ¯ _A,¹₂)|λA=1 < 0 for all η < η₁. For η ≥ η1, we know M(λA,¹₂) <

M(λA, 1) for all λA if and only if M(¹₂,¹₂) > M(¹₂, 1), i.e., p2(η) ≡ η⁴− 2η³ − η²+ 2 > 0. By applying Descarte’s rule of signs again, we know there is only one root of p2(η) that is greater than 1, which is denoted by η₂ ≈ 2.2695. It then follows that M(¹₂,¹₂) > M(¹₂, 1) for all η > η₂. For key quantity to investigate is _∂λ^∂

RR|λR=1. As long as this is negative, we know λ^′_R < 1 and thus

is quadratic in λ_A. It can be verified that the coefficient of λ²_A is positive and g(λ_A) is convex for η > η₁. Because the roots of g(λ_A) = 0 are ^2η3−3η±

√−η(2η²−3)(η²−2)

2η³+η²−3η−η , they are complex if and only if η <

√3

2 < η1 or η > √

2. The convexity of g(λA) then implies that g(λA) > 0 and thus

∂

∂λRR|λR=1< 0 when η >√

2. Now we can focus on the intermediate region η∈ [η1,√

2], in which the two roots are real. In this case, the smaller root ˜λA(η) is the only root that is within [¹₂, 1].

The convexity of g(λ_A) then implies that _∂λ^∂

RR|λR=1 ≥ 0 if and only if λA≥ ˜λA(η). λ^′_R may then be 1. If λ_Ais also large enough so that λ^∗_R= 1 (cf. Proposition 2), then it is possible for us to have λ^′_R= λ^∗_R.

Proof of Proposition 4. The proof is very similar to that of Proposition 1 and is omitted.

Proof of Proposition 5. We follow Proposition 2 to prove this proposition. LetM(λA, λR) be the manufacturer’s expected profit under λ_Aand λ_R. With this notation, when η≤ η1,M(λA,¹₂)≥ M(λA, 1) for all λA and the inequality is strict unless λA= 1. Therefore, M(λA, λR) is uniquely maximized at (¹₂,¹₂). When η∈ (η1, η₂),M(λA,¹₂) > ¯x(λ_A, 1) for λ_A close enough to ¹₂. The fact that M(λA, 1) is a constant then implies that M(λA, λR) is also uniquely maximized at (¹₂,¹₂).

Finally, when η≥ η2,M(λA,¹₂)≤ M(λA, 1) for all λ_A and ¯x(λ_A, λ_R) is maximized when λ_R = ¹₂, regardless of the value of λA.

Proof of Proposition 6. While the sales agent’s limited liability affects how he selects a contract, it does not affect how he chooses his effort level for a given contract. Therefore, the sales agent’s effort decision will be the same and the reseller’s contract design problem is formulated by adding the limited liability constraints into her original problem defined in (1)–(3). For the new problem, α_j ≥ 0 implies Ajk ≥ 0 and the IR constraints are redundant. Adding the two IC constrains up results in βF ≥ βB. If we ignore AU k ≥ AU k(F ), it is clear that αB = 0 at optimality, βF ≥ βB

and α_G≥ 0 together show the redundancy of AF k ≥ AF k(U ), and thus α_G= 0 at optimality. The problem becomes unconstrained and is optimized at β^L_F = β_U^L = ¹₂vt, where vt is the sales bonus chosen by the reseller. Such a single contract satisfies the omitted IC constraint. The reseller’s expected profit isR^L_k(t) = ut+¹₂Wkv_t², where Wk ≡ ¹₂(PF kN_{F k}² + ¯PU kN_{U k}² ). For the manufacturer’s problem, note that it can be formulated by replacing Z_k by W_k in (5)–(7). Following Lemma 2, the sales bonuses offered to the reseller will be v_G^L= 1 and v_B^L = ^W_W^B

G.

To show thatM is convex in λR, we may follow the same arguments in the proof of Proposition 1 and reduce the convexity of M to the convexity of WG and W_B, which again reduces to the convexity of P_jkN_jk² , j ∈ {F, U}, k ∈ {G, B}. As this is verified in the proof of Proposition 1,

the convexity ofM is established. For the last part, let M^L(λ_R) be the manufacturer’s expected

The term outside the square bracket is always positive. When η < 2, the first term inside the square bracket is also positive. If η² ≥ 2, then the second term inside the bracket is nonnegative and M(¹₂,¹₂, γ) > M(¹₂, 1, γ). If η² < 2, the two terms inside the square bracket are jointly minimized at γ = 1 as 2(η− 1) > 0. Therefore, we still have ¯x(¹₂,¹₂, γ) > ¯x(¹₂, 1, γ). The desired result then follows from the continuity ofM(λA, λ_R, γ).

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