In this section, we examine the situation in which the reseller is risk-averse and have a negative exponential utility function. Because the selling-the-business strategy exposes the reseller to the undesired risk, it will not be accepted by the reseller. Full profit extraction is thus no longer possible for the manufacturer. This is especially problematic when the manufacturer delegates to the diligent reseller. When the diligent reseller is risk-neutral, she is willing to accept a sell-out contract (vD = 1), offer a no-commission contract (βD = 0), and bear the risk for the whole supply chain. Without this risk neutrality, these two contracts will not be implemented and the moral hazard issue will not be eliminated completely. In this case, adverse selection does amplify efficiency loss in the supply chain. Whether the diligent reseller is still dominating now requires reinvestigation.
To address this question, we assume that a reseller is endowed with a negative exponential utility −e−ry, where y is her payoff and r > 0 is the coefficient of absolute risk aversion. We assume that both types of resellers, i.e., knowledgeable and diligent, have an identical risk aversion magnitude so that the comparison is fair. Moreover, within our three-layer structure we assume that the resellers are no more risk-averse than salespeople, i.e., r≤ ρ. Typically, a reseller is much
larger than a salesperson in terms of size/economic scale. This assumption is thus reasonable since the degree of risk aversion generally decreases as the size of a party increases. The risk-neutral case can be treated as a limiting case when r approaches 0.
We start our analysis with the risk-averse knowledgeable reseller. As in the risk-neutral case, the salesperson’s certainty equivalent is CES(θ) = α + βθ +12(1− ρσ2)β2 and the optimal service level is aKA(θ) = β (subscript A is used for “risk aversion” in this section). Getting payoff u− α + (v− β)(θ + β + ϵ), the risk-averse reseller’s utility is −e−r[u−α+(v−β)(θ+β+ϵ)] =−e−rCERK(θ), where CERK(θ) = u− α + (v − β)(θ + β) − 12r(v− β)2σ2 is the reseller’s certainty equivalent and the subscript R stands for “reseller”. To maximize her certainty equivalent, the risk-averse reseller’s problem now becomes
CERK(θ) = max
α urs,β≥0
{
u− α + (v − β)(θ + β) −1
2r(v− β)2σ2 s.t. CES(θ)≥ 0} . Let{
αKA(θ), βAK(θ)}
be the optimal contract. It then follows that the induced service level is βAK(θ) and the risk-neutral manufacturer’s problem is
MAK(r) = max
u urs,v≥0
{Eθ
[(1− v)(
θ + βAK(θ))
− u] s.t. Eθ
[CERK(θ)]
≥ 0} ,
where MAK(r) is the manufacturer’s maximum expected payoff and the subscript A stands for
“averse”. The solutions to the above two problems are summarized below.
Lemma 6. It is optimal for the manufacturer to offer vAK = 1+rσ1+rσ2+rρσ2 2 as the commission rate to the risk-averse knowledgeable reseller. Then the reseller should optimally offer βKA(θ) =
1+rσ2
1+ρσ2+rσ2vKA as the salesperson’s commission rate and induce service level aKA(θ) = 1+ρσ1+rσ2+rσ2 2vKA for all θ. The manufacturer receives MAK(r) = µ +2(1+ρσ2+rσ(1+rσ2)(1+rσ2)2 2+rρσ4).
Compared with Lemmas 2 and 4, the absolute risk aversion coefficient r now affects the optimal contracts, the induced service level, and the expected profit of the manufacturer. Consider the commission rate vKA first. The fact vAK < 1 shows that the manufacturer should not sell the business to the reseller. As the reseller becomes more risk-averse, she prefers a contract that is less risky, which consists of a lower commission rate. Therefore, vAK decreases in r. On the other hand, that the reseller is more risk-averse means the salesperson is relatively less risk-averse compared to the reseller. This drives the reseller to offer a higher commission rate (note that the coefficient1+ρσ1+rσ2+rσ2 2 increases in r). Because the induced service level aKA(θ) and the manufacturer’s expected surplus are jointly affected by these two opposite forces, under certain scenarios they are non-monotonic in r (first decreasing and then increasing). Though such non-monotonicity may
be an interesting issue to be further studied, at this moment we will continue on the comparison between the two types of resellers under the same risk magnitude r.
Now consider the risk-averse diligent reseller. With the definition of CES(˜θ, θ) and CES(θ) in (4), her problem is
CERD = max
{α(θ) urs, β(θ)≥0, a(θ)≥0} Eθ
[
u− α(θ) + (v − β(θ))(θ + a(θ)) −1
2r(v− β(θ))2σ2 ]
s.t. CES(θ)≥ CES(θ, ˜θ) ∀ θ ∈ (−∞, ∞) CES(θ)≥ 0 ∀ θ ∈ (−∞, ∞),
where the objective is to maximize her expected certainty equivalent. Though the complicated optimal menu of contracts {αAD(θ), βAD(θ), aDA(θ)} can be derived, it is difficult to solve the man-ufacturer’s problem optimally due to the complex structure of βD(θ). Nevertheless, the following lemma provides a lower bound of the manufacturer’s maximum expected profit.
Lemma 7. The manufacturer can obtain an expected profit MAD(r), which is at least MDA(r) = µ +2(1+rσ1 2), by contracting with the risk-averse diligent reseller.
The intuition of this lemma is as follows. Suppose there exists a “naive” diligent reseller who offers{¯αDA(θ), ¯βAD(θ) = 0, aDA(θ)} to the salesperson. Note that while the optimal commission rate βAD(θ) depends on the contract offered by the manufacturer and θ, the naive reseller simply ignores θ and offers no commission rate (with a necessary modification on the fixed payment to maintain the optimal service level aDA(θ) and induce the salesperson to participate). Therefore, we may interpret the suboptimal behavior as a result of lack of intelligence. The manufacturer then looks for a contract (¯uDA, ¯vAD) that is optimal when facing the naive reseller. It is clear that the “true”
reseller will also accept (¯uDA, ¯vAD) since she can always do better than the naive one. Because the two resellers assign the same service level aDA(θ), the manufacturer obtains the same expected profit MDA = (1− ¯vAD)(µ + aDA(θ))− ¯uDA by offering (¯uDA, ¯vDA) to the two resellers. Since (¯uDA, ¯vDA) is only one of the manufacturer’s available options facing the true reseller, it follows that MDA is a lower bound of the manufacturer’s maximum expected profit. Finally, MDA = µ + 2(1+rσ1 2) can be found by analyzing the manufacturer’s contracting problem with the naive reseller. All the details are provided in the appendix.
With our assumption r≤ ρ, we now have MAK = µ + (1 + rσ2)2
2(1 + ρσ2+ rσ2)(1 + rσ2+ rρσ4)
≤ µ + (1 + rσ2)2
2(1 + 2rσ2)(1 + rσ2+ r2σ4) < µ + 1
2(1 + rσ2) = MDA ≤ MAD,
where the first inequality comes from the assumption r < ρ and the second inequality comes from a direct comparison. Therefore, the diligent reseller is more profitable to the manufacturer. This conclusion, which is a generalization to Proposition 1, is summarized in the following proposition.
Proposition 6. Suppose that both resellers are risk-averse with the same risk aversion magnitude and less risk-averse than the salesperson, then the manufacturer prefers the diligent reseller to the knowledgeable reseller.
Proposition 6 shows that our main insight is not prone to the specific choice of the reseller’s risk attitude. In fact, the results in Section 3 are the limiting cases of those derived in this section as r approaches 0. This implies that Proposition 6 is a generalization of Proposition 1.
According to Lemmas 3 and 7, we have MAD > MK, which means delegating to the risk-averse diligent reseller is better than the risk-neutral knowledgeable reseller. In other words, the effectiveness of resolving moral hazard dominates the efficiency loss from risk aversion. As we already know that including the risk-neutral knowledgeable reseller is more profitable than direct sales, we obtain the following corollary.
Corollary 1. Suppose that both resellers are risk-averse with the same risk aversion magnitude and