Managing Consumer Referrals on a Chain Network

Appendix A

Proof of Proposition 1. From (4), the firm sets α2∗ to maximize π(α)=(P(α)–c)α for any α₁>0. The first-order condition is π′(α2∗)=0. The firm sets the standard monopoly output and price: α2∗=αm and p^*=p^m. At the optimal α2∗=αm, the profits are

(13)Π(α1,  αm)=(P(α1)−c+πm−ρ)α1. (13)

The first-order condition for α₁ is then:

(14)∂Π(α1,  αm)∂α1=π′(α1)+πm−ρ=0. (14)

Assuming π^m>ρ, we conclude that α1∗>α2∗ and α2∗r∗−ρ=P(α2∗)−P(α1∗)>0. Consumer 1 receives an expected payoff v−p+α2∗r∗−ρ, which is higher than the expected payoff v–p of consumer 2. Both payoffs are higher than those under no referrals, which are v–p for consumer 1 and 0 for consumer 2. The firm is also better off. Without referrals, the firm sells only to consumer 1. Its price and profits are p^m and π^m. Suppose that the firm supports referrals by setting a referral fee such that α2∗r∗−ρ=0 and charging p^m. Then, α1∗=α2∗=αm and the profits are π^m+(π^m–ρ)π^m. Assuming π^m>ρ, the firm would choose to support referrals.

Fully differentiating first-order conditions π′(α2∗)=0 and (14) with respect to c and ρ, we find that ∂α2∗∂c=1π″(α2∗)<0,∂α2∗∂ρ=0,∂α1∗∂c=1+α2∗π″(α1∗)<0, and ∂α1∗∂ρ=1π″(α1∗)<0. To determine how the expected referral payments and referral fee depend on production and referral costs, we use (3) to obtain

(15)d(α2∗r∗−ρ)dρ=d(P(α2∗)−P(α1∗))dρ=−P′(α1∗)∂α1∗∂ρ=−P′(α1∗)π″(α1∗)<0 (15)

and

d(α2∗r∗)dρ=α2∗dr∗dρ=1−P′(α1∗)π″(α1∗).

From π″(α)=αP″(α)+2P′(α), it follows that dr∗dρ>0 whenever α1∗P″(α1∗)+P′(α1∗)<0. Assuming this condition on demand holds, as in the case of the uniform distribution of values, we find that dr∗dρ>0.

Similarly, we can derive

(16)d(α2∗r∗−ρ)dc=d(α2∗r∗)dc (16)

(17)=P′(α2∗)∂α2∗∂c−P′(α1∗)∂α1∗∂c (17)

(18)=P′(α2∗)π″(α2∗)−P′(α1∗)π″(α1∗)(1+α2∗) (18)

and

(19)α2∗dr∗dc=P′(α2∗)π″(α2∗)−P′(α1∗)π″(α1∗)(1+α2∗)−r∗π″(α2∗). (19)

For the uniform U[0, 1] distribution of values, π(α₂)=(1–α₂–c)α₂. Then, α2∗=1−c2,p∗=1+c2,α1∗=12(1−c−ρ+πm),r∗=ρ+πm1−c=ρ1−c+(1−c4), and α2∗r∗−ρ=α1∗−α2∗=12(πm−ρ)>0, where πm=14(1−c)2. The firm would support consumer referrals because Π(α1∗,  α2∗)=(α1∗)2>πm=(α2∗)2 whenever π^m>ρ. Since π″(α)=–2, P′(α)=–1, we find that d(α2∗r∗)dρ=α2∗dr∗dρ=12>0,d(α2∗r∗)dc=−12α2∗<0, and dr∗dc=14(ρπm−1)<0.□

Proof of Lemma 1. Strict concavity of π(α)=α(P(α)–c–ρ) implies that the marginal revenue MR(α)=(αP(α))′=P(α)+αP′(α) is decreasing and guarantees that the second-order conditions are satisfied for any k≥2 in the optimization problem (12) and when we maximize π(α). We take four steps to prove Lemma 1.

1. Consider a chain of length n=1. The profit-maximizing solution γ^*(1)=α^m satisfies MR(γ^*(1))=c.

2. Next consider n=2. In the optimization problem

   • V(2)=maxα[α(P(α)−c−ρ)+αV(1)],

   • the optimal solution γ^*(2) is characterized by the following first-order condition γ^*(2):

   • MR(γ∗(2))−c−ρ+V(1)=0.

   • Since V(1)=π^m>ρ, MR(α) is decreasing, and MR(γ∗(1))−c=0, we conclude that γ^*(2)>γ^*(1).

3. Suppose that γ^*(1)< … <γ^*(k–1) for k≥3. We will show that γ^*(k–1)<γ^*(k) holds. By definition, we have

   • V(k)=maxα[α(P(α)−c−ρ)+αV(k−1)],

   • and the first-order condition for γ^*(k) is

   • MR(γ∗(k))−c+V(k−1)−ρ=0.

   • The first-order condition for γ^*(k–1) is

   • MR(γ∗(k−1))−c+V(k−2)−ρ=0.

   • Since MR(α) is decreasing and V(k–2)<V(k–1), we conclude that γ^*(k–1)<γ^*(k) holds.

4. By an induction argument, we complete the proof.□

Proof of Corollary 1. Since αk∗=γ∗(n−k+1) for all k=1,…,n, the proof is clear from Lemma 1.□

Proof of Lemma 2. Note that V(k) is described in the following manner:

(20)V(k)=∑h=1k−1[(P(αh∗)−c−ρ)(∏ℓ=1hαℓ∗)]+P(αk∗)∏ℓ=1kαℓ∗, (20)

where αh∗=γ∗(k−h+1) for all h=1,…,k. Let’s look at the monopoly problem with k+1 consumers. Consider the following policy for the firm: set the same purchase probabilities for the first k consumers as in the k-consumer problem (i.e. αh=αh∗ for all h=1,…,k) and make the kth consumer just willing to make a referral to the last (k+1)th consumer (by setting the expected referral benefit equal to the referral cost: αk+1rk+1∗=ρ). Since αk+1=αk∗=αm, the monopoly profit under such a policy is

Π(α1∗, …, αk∗,  αk+1=αm)=V(k)+(πm−ρ)(∏ℓ=1kαℓ∗),

where (∏ℓ=1kαℓ∗) is the unconditional probability that the kth consumer purchases the product. Assuming π^m>ρ, V(k+1)≥Π(α1∗, …,  αk∗,αk+1=αm)>V(k). It follows that V(k+1)>V(k).□

Proof of Proposition 2. From Lemma 2 and Corollary 1, we know that α1∗>α2∗> … >αn∗=αm>0 and p=p^m. From (8), αk∗rk∗=p−P(αk−1∗)+ρ for k=2,…,n, and since α1∗>α2∗> … >αn∗, we conclude that α2∗r2∗>α3∗r3∗> … >αn∗rn∗>ρ.□

Proof of Proposition 3. We prove the first statement by induction. Since γ^*(1)=α^m and V(1)=π^m, dV(1)dc=−γ∗(1)=−αn∗=−αm<0. From (11), we have dV(k)dc=−γ∗(k)(1−dV(k−1)dc) by the envelope theorem for k=2,…,n. Since γ∗(k)=αn−k+1∗>0, it follows that for any k=2,…,n, if dV(k−1)dc<0, then dV(k)dc<0. Hence, dV(k)dc<0 for k=1,…,n. The first-order condition for (12) is

P(γ∗(k))−c−ρ+V(k−1)+γ∗(k)P′(γ∗(k))=0.

Totally differentiating the above, we obtain

(2P′(γ∗(k))+γ∗(k)P″(γ∗(k)))dγ∗(k)dc=1−dV(k−1)dc>0.

From the second-order condition for profit maximization, 2P′(γ^*(k))+γ^*(k)P″(γ^*(k))<0 holds, and we conclude that dαn−k+1∗dc=dγ∗(k)dc<0 for k=1,…,n.

We use similar arguments to derive the comparative statics result for referral cost ρ. First, dV(1)dρ=0 and dV(2)dρ=−γ∗(2)<0. From (11), dV(k)dρ=−γ∗(k)(1−dV(k−1)dρ). Since γ∗(k)=αn−k+1∗>0, it follows that if dV(k−1)dρ<0, then dV(k)dρ<0 for k=2,…,n. Hence, dV(1)dρ=0 and dV(k)dρ<0 for k=2,…,n. From (12), we obtain (2P′(γ∗(k))+γ∗(k)P″(γ∗(k)))dγ∗(k)dρ=1−dV(k−1)dρ>0, and we can conclude that dαn∗dρ=dγ∗(1)dρ=0 and dαn−k+1∗dρ=dγ∗(k)dρ<0 for k=2,…,n.

From Proposition 2, the optimal price is p^*=p^m, and therefore it decreases in production cost c but is independent of referral cost ρ. Finally, from (8), αk∗rk∗−ρ=pm−P(αk−1∗). Hence, d(αk∗rk∗−ρ)dρ=−P′(αk−1∗)dαk−1∗dρ<0.□

Proof of Proposition 4. To see that rk∗<rk−1∗ holds for k=2,…,n in the case of the uniform distribution of values, note that the profit-maximizing solution in this case is γ(k)=arg max_α{α[1–α–c–ρ+V(k–1)]} for all k=2,…,n and γ(1)=αn∗=1−c2. Solving this, we have γ(k)=12(1−c−ρ+V(k−1)). Thus, V(k)=γ(k)[1−γ(k)−c−ρ+V(k−1)]={12(1−c−ρ+V(k−1))}2=(γ(k))2 for all k=2,…,n. It follows that γ(k)=12(1−c−ρ+(γ(k−1))2) for all k=2,…,n and γ(1)=αn∗=1−c2. From (8), we have

rk∗=P(αn∗)−P(αk−1∗)+ραk∗=P(γ(1))−P(γ(n−k+2))+ργ(n−k+1)=1−1−c2−1+12(1−c−ρ+(γ(n−k+1))2)+ργ(n−k+1)=12(ργ(n−k+1)+γ(n−k+1))=12(ραk∗+αk∗).

As is easily verified, rk∗ is increasing in αk∗ as long as (αk∗)2>ρ. This inequality holds for any k≤n because πm=(αn∗)2>ρ and V(1)=πm<V(n−k+1)=(αk∗)2 for any k≤n. Since by Proposition 2, αk∗ is a decreasing sequence, so is rk∗. Thus, we have r2∗>r3∗> … >rn∗>0. By Proposition 3, dαk∗dc<0 for k=1,…,n and since drk∗dc=αk∗2−ρ2αk∗2dαk∗dc, it follows that referral fees rk∗ are decreasing in production cost as well, drk∗dc<0 for k=2,…,n. Together, these results imply that the expected referral payments rk∗αk∗ are decreasing in production cost for any k=2,…,n.□

Appendix B: The Case of Uniform Referral Fees

Following Jun and Kim (2008), let us consider the case where the firm has to set a common referral fee and price for all consumers, regardless of whether they are early or late adopters. Jun and Kim show that when the second-to-last consumer has a strictly positive referral benefit, r(1–F(p))>ρ (their referral condition RC), the earlier a consumer is located in the chain, the higher is her probability of purchasing the product α₁>α₂>…>α_n (their Proposition 1). This result further implies effective price discrimination among consumers according to their position in the referral chain: although the firm charges a common price p and pays a referral fee r to all consumers, the firm effectively discriminates in favor of consumers located earlier in the chain because these consumers obtain higher expected benefits from making referrals.

We take a closer look at the optimal strategy of the firm. In particular, we examine the possibility of a stationary outcome being optimal. We allow for referral equilibrium to be consistent with a binding referral condition r(1–F(p))=ρ by assuming that when indifferent, consumers make referrals. We show that the firm’s profit can be improved by increasing both p and r in a right proportion starting from the optimal stationary outcome, implying that the stationary outcome is not even a local maximum for any finite n (Proposition B1). This result strongly justifies Jun and Kim’s analysis and also implies that at least for large finite n, the optimal solution is perhaps very close to the stationary outcome.

Denote by α_k the probability that consumer i buys the product conditional on being introduced to it, k=1,…,n. The firm chooses a strategy (p, r) to maximize its profits

(21)Π^(p,  r)=(p−rα2−c)α1+(p−rα3−c)α1α2+…+(p−rαn−c)α1…αn−1+(p−c)α1…αn (21)

where α₁,…,α_n are determined by (p, r) as follows: α_n=D(p)=1–F(p)≥0 and α_k= D(p–α_k+1r+ρ)≥0 for k=1,…,n–1. Denote by P(α)=D^–1(α) the standard inverse demand function. We assume that the profit function without referrals, π(α)≡α(P(α)–c), is strictly concave.

Assuming rα_k≥ρ for k=2,…,n, probabilities α₁,…,α_n, are determined by the following system of equations:

(22)P(αn)=pP(αn−1)=p−rαn+ρ⋯P(α1)=p−rα2+ρ, (22)

and α₁≥…≥α_n. Suppose that the referral condition is binding for the kth consumer: rα_k+1=ρ for some k=1,…,n–1. Then, P(α_k)=p, and we have P(α₁)=P(α₂)= …=P(α_n)=p and α₁=α₂=…=α_n. This is a stationary outcome, for which consumer referral conditions are all binding: rα_k+1=ρ for all k=1,…,n–1. We will show that this outcome is not locally optimal.

The firm’s profit can be written in terms of α_k s only:

(23)Π(α1, …,  αn−1,  αn; n)=α1(P(α1)−ρ−c)+α1α2(P(α2)−ρ−c)+…+α1⋯αn−1(P(αn−1)−c−ρ)+α1⋯αn(P(αn)−c) (23)

where (α₁,…,α_n) is a solution to system (22). Under the stationary outcome α₁= …=α_n–1=α_n=α, the profit when there are n≥1 consumers can be written as

(24)Π(α, α, …, α; n)=An(α)(π(α)−ρ)+ρ, (24)

where

(25)An(α)≡1+α+α2+…+αn−1=1−αn1−α (25)

and π(α)=α(P(α)–c).

Denote the optimal stationary policy for an n-consumer chain by β(n)≡arg maxαΠ(α, α, …, α; n). Proposition B1 states that β(n) cannot be a local maximum for small ρ.

Proposition B1.The optimal stationary policy β(n) is not the optimal policy if π(β(n))>ρ.

To prove Proposition B1, we will show that the firm’s profit is locally improvable [starting from β(n)] by choosing an appropriate policy change (dp, dr)>>0. We first provide a sketch of the proof of Proposition B1. First, in Lemma B1, we investigate the properties of the optimal stationary policy β(n). Then, we look at the profit function evaluated at the optimal stationary policy α₁=…=α_n–1=α_n=β(n). We show that there is some M (1≤M<n) such that profits increase with purchase probability for consumers located before M and decrease with purchase probability for consumers located after M: ∂Π∂αk|α=β(n)>0 for all k<M and ∂Π∂αk|α=β(n)<0 for all k>M (Lemma B2). We then show that there exists a policy change dΔ=(dp, dr)>>0 such that for any M (1≤M<n) the probability of buying increases for consumers located before M and decreases for consumers located after M. We prove this by showing that, starting at α₁=…=α_n=α, if α_n decreases while α_M is kept constant, dα_k>0 for all k<M and dα_k<0 for all k>M (Lemma B3). Using Lemmas B2 and B3, we conclude that the optimal stationary policy is not a local maximum.

We prove Proposition B1 by using a sequence of lemmas.

Lemma B1.

1. For all n and all α such that π(α)–ρ>0, Π(α, α,…,α; n+1)>Π(α, α,…,α; n).

2. The optimal stationary solution β(n)≡arg max_αΠ(α,α, …,α; n) satisfies the following condition:π′(β(n))=−A′n(β(n))An(β(n))(π(β(n))−ρ).(26)

3. Suppose π(β(n))–ρ>0. Then,β(n)>β(n−1)> … >β(1)=arg maxα˜π(α˜).

Proof. From (24), the difference in profits from (n+1)- and n-consumer chains is

(27)Δn(α)≡Π(α, α, …, α; n+1)−Π(α, α, …, α; n)=αn(π(α)−ρ). (27)

Hence, Δⁿ(α)>0 if π(α)–ρ>0. This proves (i).

The optimal policy α=β(n) is implicitly defined by the first-order condition

(28)dΠ(α, α, …, α; n)dα=A′n(α)(π(α)−ρ)+An(α)π′(α)=0. (28)

This proves (ii).

Finally, using (26), we find that at α=β(n)

(29)dΔndα=nαn−1(π(α)−ρ)+αnπ′(α)=αn−1(π(α)−ρ)(nAn(α)−αA′n(α)An(α))>0. (29)

The last inequality holds because

(30)nAn(α)−αA′n(α)=n(1+α+α2+…+αn−1)−α(1+2α+…+(n−1)αn−2)=n+(n−1)α+(n−2)α2+…+αn−1>0. (30)

Hence, if α=β(n)>0 and π(β(n))–ρ>0, then β(n+1)>β(n). Since β(1)=α^m>0, it follows that π′(α)<0 for all α>β(1). Thus, π(β(1))>π(β(2))>…> π(β(n)) holds, and we conclude that β(1)>β(2)>…> β(n) if π(β(n))–ρ>0.□

Notice that the profit Π(α₁, α₂,…,α_n; n) in equation (23) can be defined recursively:

(31)Π(αn; 1)=π(αn)=αn(P(αn)−c)Π(αn−1, αn; 2)=αn−1(P(αn−1)−c−ρ)+αn−1Π(αn; 1)⋯Π(α1, α2, …, αn; n)=α1(P(α1)−c−ρ)+α1Π(α2, …, αn; n−1) (31)

Using these formulas, we prove the following result.

Lemma B2.Suppose that π(β(n))–ρ>0 holds.At α₁=…=α_n=β(n), (i) ∂Π∂αn<0and∂Π∂α1>0; (ii) there exists M such that∂Π∂αk>0for any k<M and∂Π∂αk<0for any k>M.

Proof. The marginal profits with respect to buying probabilities α₁,…,α_n are

(32)1α1α2α3⋯αn−1∂Π∂αn=π′(αn)1α1α2α3⋯αn−2∂Π∂αn−1=π′(αn−1)+Π(αn; 1)−ρ1α1α2α3⋯αn−3∂Π∂αn−2=π′(αn−2)+Π(αn−1, αn; 2)−ρ⋯∂Π∂α1=π′(α1)+Π(α2, …, αn; n−1)−ρ. (32)

Thus, at α₁=α₂=…=α_n=α,

(33)∂Π∂αn=αn−1π′(α)∂Π∂αn−1=αn−2(π′(α)+Π(α; 1)−ρ)⋯∂Π∂α1=π′(α)+Π(α, …, α; n−1)−ρ. (33)

Note that ∂Π∂αn=αn−1π′(α)<0 at β(n)>β(1)=α^m by Lemma B1. By assumption, Π(α; 1)=π(β(n))>ρ, and by Lemma B1, Π(α,…,α; k) is increasing in k. Hence, if for some ℓ, ∂Π∂αℓ>0, then ∂Π∂αk>0 for any k<ℓ, and if ∂Π∂αℓ<0, then ∂Π∂αk<0 for any k>ℓ.

In the following, we will show that for α=β(n)>0, there exists M such that ∂Π∂αk>0 for any k<M and ∂Π∂αk<0 for any k>M. Recall that

(34)Ak(α)=1+α+α2+…+αk−1=1−αk1−α. (34)

For k≤n–1, we have

(35)∂Π∂αk=αk−2(π′(α)+Π(α, …, α; n−k+1)−ρ) (35)

where

(36)Π(α, α, …, α; n−k+1)=An−k+1(α)(π(α)−ρ)+ρ. (36)

Hence, ∂Π∂αk>0 as long as

(37)π′(α)+An−k+1(α)(π(α)−ρ)>0. (37)

Plugging in the expression for the optimal π′(α) from Lemma B1 and assuming (πβ(n))–ρ>0, the inequality is equivalent to

(38)An−k+1(α)An(α)>A′n(α), (38)

which is equivalent to

1−αn−k+11−α1−αn1−α>∂(1−αn1−α)∂α=1(1−α)2(1−αn−nαn−1+nαn).

Thus, we conclude that ∂Π∂αk>0 holds if and only if

(39)[nαn−11−α1−αn−αn−k+1]>0. (39)

The expression in the brackets is strictly decreasing in k. Note that for k=1, [nαn−11−α1−αn−αn]=(n1−α1−αn−α)αn−1>0 because n(1−α)−α(1−αn)>n(1−α)−α(1−α)=(n−α)(1−α)>0. Hence, ∂Π∂α1>0. Since ∂Π∂αn=αn−1π′(α)<0, there exists M (1≤M<n) such that ∂Π∂αk>0 for any k<M and ∂Π∂αk<0 for any k>M at α=β(n).□

In Lemma B3, we describe the effects of a policy change (dp, dr)>;0 at a stationary outcome α₁=…=α_n=α.

Lemma B3.Consider a policy of increasing p and r, starting at α₁=…=α_n=α. For any M(1≤M<n), there is a policy change (dp, dr)>;0 such that dα_k>0 for all k<M and dα_k<0 for all k>M.

Proof. Totally differentiating equations (22) and evaluating at α₁=…=α_n=α, we have:

(40)P′(α)dαn=dpP′(α)dαn−1=(dp−αdr)−rdαn⋯P′(α)dα1=(dp−αdr)−rdα2. (40)

When p is increasing (dp>0), we necessarily have dαn=1P′(α)dp<0.

Let x≡r−P′(α)>0. We choose (dp, dr)>;0 such that dα_M=0. From

(41)P′(α)dαM=(dp−αdr)−rdαM+1=(dp−αdr)(1+x)−rxdαM+2=(dp−αdr)(1+x+…+xn−M−1)−rxn−M−1dαn=(dp−αdr)(1+x+…+xn−M−1)+xn−Mdp=dp(1+x+…+xn−M)−αdr(1+x+…+xn−M−1)=0, (41)

it follows that dα_M=0 implies

(42)dp=αdr1+x+…+xn−M−11+x+…+xn−M (42)

Similarly,

(43)P′(α)dαk=dp(1+x+…+xn−k)−αdr(1+x+…+xn−k−1). (43)

Using (42),

(44)P′(α)dαk=αdr[1+x+…+xn−M−11+x+…+xn−M(1+x+…+xn−k)−(1+x+…+xn−k−1)]. (44)

Then, since P′(α)<0 and dr>0, dα_k>0 if and only if

(45)1+x+…+xn−M−11+x+…+xn−M<1+x+…+xn−k−11+x+…+xn−k. (45)

This inequality holds whenever k<M because 1+x+…+xa−11+x+…+xa=1−xa1−xa+1 and ∂(1−xa1−xa+1)∂a=xalnx(1−xa+1)2(x−1)>0. Similarly, dα_k<0 whenever k>M.□

From Lemmas B2 and B3, we conclude that, assuming π(β(n))>ρ, the optimal stationary outcome β(n) is not a local optimum for any finite n. This proves Proposition B1.□

It follows from Proposition B1 that the optimal strategy (p, r) is such that the referral condition is not binding for any consumer. This justifies the analysis of Jun and Kim (2008). As is known from Jun and Kim’s Proposition 1, this implies that the firm effectively price-discriminates by subsidizing consumer referrals and supporting a decreasing sequence of purchase probabilities, α₁>…>α_n.