Complementarity between demand and services

Recall that when the sales outcome is in the additive form x = θ +a+ϵ, we have M^K = µ+_2(1+ρσ¹ 2), M^D = µ + ¹₂, and thus M^D− M^K does not depend on θ. In other words, M^K− M^D does not change when the manufacturer modifies its belief about the market condition. However, when the sales outcome is in the multiplicative form x = θa + ϵ, it follows from Proposition 2 that M^D − M^K now depends on the distribution of θ. This is because when demand and services are complementary, different realizations of the market condition drive the salesperson to choose different service levels. Among all possibilities for θ to change, we are particularly interested in the situation that µ alters. Studying this situation allows us to investigate the relative effectiveness of the two resellers in big markets (when µ is large) and small markets (when µ is small). The following analytical finding regarding uniform distribution provides a partial answer. Numerical experiments for other distributions also demonstrate similar qualitative results.

Proposition 9. When x = θa + ϵ and θ follows a uniform distribution with mean µ and a fixed variance, M^D− M^K increases in µ.

When the manufacturer delegates to the diligent reseller in a big market, the complementarity allows her to induce a high service level. However, if the delegation occurs in a small market, the diligent reseller’s monitoring will become less effective. On the other hand, the demand forecasting ability possessed by the knowledgeable reseller is not affected if the variance of θ does not change.

Therefore, when the market size goes down, the diligent reseller will become relatively less effective.

6 Conclusions

In this paper, we investigate the effect of demand forecasting and performance measurement on motivating salespeople and creating new demands. Within our three-layer supply chain, we jointly study the manufacturer’s partner selection problem and the resellers’ salesforce compensation prob-lem. Since decision making within this supply chain is decentralized, including a reseller with a certain monitoring ability is different from having the ability by the manufacturer itself. All intu-itions obtained from traditional two-layer supply chains therefore cannot be applied directly. We show that the manufacturer unambiguously prefers the reseller who is able to monitor the sales-person to the one that can monitor the market. This dominance result is not prone to our model characteristics regarding the degree of complementarity between service level and market condition, the relative importance between moral hazard and adverse selection, the reseller’s risk attitude, the observability of the resellers’ monitoring expertise, and the contract form. We further show that the performance gap between delegating to the two resellers decreases when adverse selection becomes relatively more important, the reseller becomes more risk-averse, or the market size goes down.

Moreover, because the efficiency of direct sales does not depend on the reseller, direct sales may outperform both indirect selling schemes when the reseller is too risk-averse.

Our study certainly has its limitations. In our study, we assume that the selling price is exoge-nously given. If the price is endogenized, the manufacturer and the reseller may distort the price or include it in the menu in order to induce better truth-telling. Price and commission rate can therefore serve as complementary screening tools. Another possible extension is to investigate a variety of contract forms (e.g., buy-back contracts, advance purchase contracts, and quantity flex-ibility contracts) and see how they affect the strategic decisions of the manufacturer and resellers.

Moreover, we exclude the effect of competition from other manufacturers, which may be inappro-priate in some contexts. Introducing the competition between manufacturers creates a common

agency problem, since these manufacturers may compete in contract offers in order to earn the collaboration opportunity with a specific reseller. This issue calls for future investigations.

Appendix

Proof of Lemma 1. It follows from the first-order necessary condition of the IC constraint (2) that _dθ^dCES(θ) = β(θ) ≥ 0 for all θ ∈ (−∞, ∞), which implies that CES(θ) is nondecreasing in θ. The IR constraint (3) implies that CES(−∞) = 0 at the optimal solution. Consequently, CE(θ) =∫_θ

−∞β(y)dy and the binding IR constraint lead to α(θ) = −β(θ)θ − 1

Replace the α(θ) in the objective function and ignore the IC constraint for a moment, we reduce the problem to

where the second equality comes from integration by parts. Maximizing the integrand pointwise yields β^∗(θ) = ^[1−H(θ)]_1+ρσ2⁺, and the maximum objective value M can be calculated by plugging β^∗(θ) back. The IC constraint can be easily verified and is omitted.

Proof of Lemma 2. We first observe that the constraint must be binding at the optimal solution.

If this is not the case, the reseller can reduce the fixed payment α by a sufficiently small amount such that the objective value increases while the constraint is still satisfied. Thus, the problem reduces to maximizing a quadratic function of β: R^K(θ) = max_β≥0

Proof of Lemma 3. Observing that at optimality the constraint must be binding, we can replace u by−vµ −_2(1+ρσ¹ 2)v² in the objective and reduce the problem into M^K = maxv≥0 the first-order condition. The corresponding induced service level is _1+ρσ¹ 2, regardless of θ.

Proof of Lemma 4. It follows from the first-order necessary condition of the IC constraint that _dθ^dCES(θ) = β(θ) ≥ 0 for all θ ∈ (−∞, ∞), which implies that CES(θ) is nondecreasing

in θ. The (IR) constraint implies that CE_S(−∞) = 0 at the optimal solution. Consequently, CE_S(θ) =∫_θ

−∞β(y)dy. Moreover, from (4), we have α(θ) =−β(θ)θ − β(θ)a(θ) +1

2[a(θ)]²+ 1

2(1− ρσ²) [β(θ)]²+

∫ _θ

−∞β(y)dy.

Substituting α(θ) in the objective with the right hand side and ignoring the (IC) constraint for a while, the reseller’s problem can be rewritten as

R^D = max

where the equality follows from integration by parts. Since the integrand is strictly decreasing in β(θ), the optimal commission rate is β^D(θ) = 0. The corresponding optimal service level is a^D(θ) = v, and the reseller’s maximum expected payoff is R^D = u+vµ+¹₂v². Given the commission rate β^D(θ) = 0, the corresponding fixed payment is α^D(θ) = ¹₂[a^D(θ)]² = ¹₂v², which is independent of θ. Since the reseller only offers a single contract, the IC constraint is satisfied.

Proof of Lemma 5. At optimality, the constraint should be binding. Thus, the manufacturer’s problem reduces to M^D = max_v≥0{

µ + v−¹₂v²}

, which gives rise to the optimal commission rate is v^D = 1 and M^D = µ + ¹₂. The corresponding service level is 1.

Proof of Proposition 2. Recall that the salesperson under a knowledgeable reseller gets payoff α + βx−¹₂a² by setting his service level to a. The corresponding certainty equivalent CES(θ|a) = α + βθa− ¹₂a²− ¹₂ρσ²β² is then maximized by a^K = βθ as CES(θ) = α + ¹₂(θ²− ρσ²)β². With service level βθ, the expected sales outcome is βθ². The knowledgeable reseller is then solving

R^K(θ) = max

At optimality, the constraint must be binding. Therefore, the problem reduces to R^K(θ) = max_β≥0{

u +¹₂(θ²− ρσ²)β²+ vθ²β− θ²β²}

. By the first-order condition, this is maximized by β^K(θ) = _θ2+ρσ^{θ2 2}v. We then have a^K= _θ2+ρσ^{θ3 2}v and R^K(θ) = u +_2(θ2^θ+ρσ^{4 2})v². DenotingEθ

As the constraint must be binding at optimality, we can replace u in the objective function by

−¹₂γv² and obtain that M^K = maxβ≥0{

Now consider the diligent reseller. Observing θ but choosing a contract by reporting ˜θ, the salesperson’s certainty equivalent is now CES(˜θ, θ) = α(˜θ) + β(˜θ)(θa(˜θ))− ¹₂a(˜θ)² − ¹₂ρσ²β(˜θ)².

With the definition CE_S(θ) = CE_S(˜θ, θ), the reseller’s problem is

−∞β(y)a(y)dy at optimality. The fact that the second constraint is binding then leads to R^D = max{β(θ)≥0,a(θ)≥0}Eθ

[u + vθa(θ)−¹₂a(θ)²−¹₂ρσ²β(θ)²− β(θ)a(θ)H(θ)] according to integration by part. Since the integrand is non-increasing in β(θ), we have β^D(θ) = 0.

It then follows that R^D = max_{a(θ)≥0}Eθ

By the same way as in the knowledgeable reseller case, this problem can be solved with maximizers v^D = 1 and u^D = −¹₂Eθ[θ²]. We then have M^D = ¹₂Eθ which concludes that delegating to a diligent reseller is more profitable for the manufacturer.

Proof of Proposition 3. Consider the knowledgeable reseller. Given contract (α, β), the sales-person’s certainty equivalent is CE(θ|a) = α + β(θ + a) −_2k¹a²−¹₂ρσ²β², which is maximized by a(θ) = βk. In the reseller’s optimal contract, the salesperson receives a zero certainty equivalent and therefore

Suppose the salesperson observes a market condition θ but chooses the contract (α(˜θ), β(˜θ), a(˜θ)) from the diligent reseller, his certainty equivalent is CE(˜θ, θ) = α(˜θ) + β(˜θ)(θ + a(˜θ))−_2k¹[a(˜θ)]²−

1

2ρσ²[β(˜θ)]². Define CE(θ)≡ CE(θ, θ). Again, the first order condition of the IC constraint and CE(−∞) = 0 imply that CE(θ) =∫_θ

−∞β(y)dy. Ignore the IC constraint for a moment, we rewrite the problem as

β^D(θ) = 0 and a^D(θ) = vk optimizes this problem and result in R^D = Eθ

[u + vθ + ^k₂v²]

as the reseller’s maximum expected payoff. With R^D = 0 at optimality, we have

M^D = max

Proof of Proposition 4. We start with the first case in which the manufacturer can observe the market condition θ. In this case, the manufacturer’s problem is equivalent to the knowledgeable reseller’s problem in Section 3.2 with u = 0 and v = 1. By substituting u by 0 and v by 1 in Lemma 2, we can conclude that the manufacturer will receive µ +_2(1+ρσ¹ 2) in expectation. Similarly, if the manufacturer can observe the service level a, its problem is equivalent to the diligent reseller’s problem in Section 3.3 with u = 0 and v = 1. It then follows that the manufacturer will receive µ +¹₂ in expectation if we replace u by 0 and v by 1 in Lemma 4.

Proof of Lemma 6. We first solve the reseller’s problem. At optimality, the constraint is binding, so the problem becomes

= 0 at optimality and reduce her problem to M_A^K= max

Proof of Lemma 7. First we must solve the reseller’s problem and derive the optimal menu {(α^D_A(θ), β_A^D(θ), a^D_A(θ))}. Applying the first order condition on the IC constraint, we obtain _dθ^dCE_S(θ) = β(θ)≥ 0. Ignoring the IC constraint for a moment, it is clear that the IR constraint must be binding at θ =−∞ and thus CES(θ) =∫_θ

−∞β(y)dy at optimality. The problem then reduces to CE_R^D = max verification of the IC constraint is straightforward. Accordingly, α^D_A(θ) can be computed by plugging a^D(θ) and β^D(θ) back into the binding IR constraint.

The diligent reseller using the optimal contract is said to be “strong”. However, we consider an alternative “weak” diligent reseller using a suboptimal contract a^D_A(θ) = v and ¯β^D(θ) = 0 for all θ. It is optimal for the weak diligent reseller to offer ¯α^D(θ) = ¹₂v² as the fixed payment. This single contract, though suboptimal, guarantees the participation of all-types of salesperson. She then obtains CE^D_R = u + vµ + ¹₂v²− ¹₂rv²σ² as her certainty equivalent by plugging β(θ) = 0 and a(θ) = v back into (13). To contract with the weak diligent reseller, the manufacturer solves

M^D_A = max

u urs,v≥0

{

(1− v)(µ + v) − u CE^DR ≥ 0}

. (14)

At optimality, the binding constraint reduces her problem to M^D_A = maxv≥0{

µ + v−¹₂(1 + rσ²)v²} . The manufacturer then receives M^D_A = µ + _2(1+rσ¹ 2) with the maximizer ¯v^D_A = _1+rσ¹ 2.

The last step is to show that M^D_A is a lower bound of the maximum expected profit M_A^D. To see this, note that the strong diligent reseller obtains CE_R^D. The manufacturer will then solve

M_A^D = max

u urs,v≥0

{

(1− v)(µ + v) − u CER^D ≥ 0}

. (15)

The fact CE_R^D ≥ CE^D_R implies the feasible region of (15) is no smaller than that of (14). Since these two problems also have an identical objective function, we have M^D ≥ M^D.

Proof of Proposition 5. First we observe from (8) and (9) that u^D_s + v_s^Dµ + 1

2(v_s^D)²≥ u^Ks + v_s^Kµ + 1

2(v^K_s )² ≥ u^Ks + v^K_s µ + 1

2(1 + ρσ²)(v^K_s )² ≥ 0,

which implies that (10) is redundant. Furthermore, if we ignore (7) for a moment, the relaxed manufacturer’s problem becomes:

We first observe that at optimality (18) must be binding; otherwise, decreasing u^K_s a bit yields a higher expected payoff for the manufacturer while relaxing (17). Likewise, we can show that (17) must be binding as well, for otherwise decreasing u^D_s would be profitable for the manufacturer.

Thus, we can replace u^K_s and u^D_s by

in the manufacturer’s objective and get the maximizers v_s^D = 1, and v^K_s = 1

1 + (1− p)ρσ²/p.

The corresponding fixed payments are u^K_s = −vs^Kµ− _2(1+ρσ¹ 2)(v_s^K)² and u^D_s = −µ − ¹₂ + (¹₂ −

1

2(1+ρσ²))(v^K_s )². The knowledgeable reseller receives zero expected payoff since (18) is binding, whereas the diligent reseller obtains an information rent u^D_s + v_s^Dµ +¹₂(v_s^D)²= (¹₂−_2(1+ρσ¹ 2))(v^K_s )². It is then straight forward to verify that (7) is satisfied under this menu of contracts. Finally, the induced service levels a^K_s (θ) and a^D_s (θ) follow from Propositions 3 and 5.

Proof of Proposition 7. Given B, it is straightforward to show that the optimal solutions for (11) is (u^K(B), v^K(B)) = (−µ−^η₂, 1) if B≤ −µ−^η₂ and is decreasing in B. In the last case, we have

M_L^D(B)− ML^K(B) = (1− µ)v^D(B)− (v^D(B))²− (η − µ)v^K(B) + η(v^K(B))². (19) Note that depending on the values of µ and B, there are four combinations of v^K(B) and v^D(B).

We will first show that (19) is nondecreasing in three combinations and then show that the last combination is not possible.

then follows from the monotonicity of M_L^D(B) and the fact that M_L^D(−∞) > M^∗. We may thus set ˆµ = 1 to complete the proof. For different combinations of parameters, better lower bounds may be found.

Proof of Proposition 9. When θ is uniformly distributed with mean µ and variance ξ², we have M^D = ¹₂E_θ[

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