# Managing Consumer Referrals on a Chain Network

Maria Arbatskaya and Hideo Konishi

## Abstract

We consider a monopoly that sets a price and differentiated referral fees to spread product information along a simple consumer communication network (a chain). The profit-maximizing solution involves standard monopoly pricing and referral fees that provide consumers with strictly positive referral incentives. Effective price discrimination among consumers based on their positions in the chain occurs both in the case of differentiated referral fees and in the case of uniform referral fees.

JEL codes: D4; D8; L1

Corresponding author: Maria Arbatskaya, Department of Economics, Emory University, Atlanta, GA 30322-2240, Phone: +(404) 727 2770, Fax: +(404) 727 4639, e-mail:

## Acknowledgments

We thank Jeong-Yoo Kim and an anonymous referee for their encouragement and helpful comments.

## 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*=pm. At the optimal α2=αm, the profits are

(13)Π(α1,αm)=(P(α1)c+πmρ)α1. (13)

The first-order condition for α1 is then:

(14)Π(α1,αm)α1=π(α1)+πmρ=0. (14)

Assuming πm>ρ, we conclude that α1>α2 and α2rρ=P(α2)P(α1)>0. Consumer 1 receives an expected payoff vp+α2rρ, which is higher than the expected payoff vp of consumer 2. Both payoffs are higher than those under no referrals, which are vp 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 pm and πm. Suppose that the firm supports referrals by setting a referral fee such that α2rρ=0 and charging pm. Then, α1=α2=αm and the profits are πm+(πmρ)πm. Assuming πm>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 α2c=1π(α2)<0,α2ρ=0,α1c=1+α2π(α1)<0, and α1ρ=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(α2rρ)dρ=d(P(α2)P(α1))dρ=P(α1)α1ρ=P(α1)π(α1)<0 (15)

and

d(α2r)dρ=α2drdρ=1P(α1)π(α1).

From π″(α)=αP″(α)+2P′(α), it follows that drdρ>0 whenever α1P(α1)+P(α1)<0. Assuming this condition on demand holds, as in the case of the uniform distribution of values, we find that drdρ>0.

Similarly, we can derive

(16)d(α2rρ)dc=d(α2r)dc (16)
(17)=P(α2)α2cP(α1)α1c (17)
(18)=P(α2)π(α2)P(α1)π(α1)(1+α2) (18)

and

(19)α2drdc=P(α2)π(α2)P(α1)π(α1)(1+α2)rπ(α2). (19)

For the uniform U[0, 1] distribution of values, π(α2)=(1–α2c)α2. Then, α2=1c2,p=1+c2,α1=12(1cρ+πm),r=ρ+πm1c=ρ1c+(1c4), and α2rρ=α1α2=12(πmρ)>0, where πm=14(1c)2. The firm would support consumer referrals because Π(α1,α2)=(α1)2>πm=(α2)2 whenever πm>ρ. Since π″(α)=–2, P′(α)=–1, we find that d(α2r)dρ=α2drdρ=12>0,d(α2r)dc=12α2<0, and drdc=14(ρπm1)<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)=αm satisfies MR(γ*(1))=c.

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

• V(2)=maxα[α(P(α)cρ)+αV(1)],

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

• MR(γ(2))cρ+V(1)=0.

• Since V(1)=πm>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)cρ)+αV(k1)],

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

• MR(γ(k))c+V(k1)ρ=0.

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

• MR(γ(k1))c+V(k2)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(nk+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=1k1[(P(αh)cρ)(=1hα)]+P(αk)=1kα, (20)

where αh=γ(kh+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)+(πmρ)(=1kα),

where (=1kα) is the unconditional probability that the kth consumer purchases the product. Assuming πm>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=pm. From (8), αkrk=pP(αk1)+ρ for k=2,…,n, and since α1>α2>>αn, we conclude that α2r2>α3r3>>αnrn>ρ.

Proof of Proposition 3. We prove the first statement by induction. Since γ*(1)=αm and V(1)=πm, dV(1)dc=γ(1)=αn=αm<0. From (11), we have dV(k)dc=γ(k)(1dV(k1)dc) by the envelope theorem for k=2,…,n. Since γ(k)=αnk+1>0, it follows that for any k=2,…,n, if dV(k1)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))cρ+V(k1)+γ(k)P(γ(k))=0.

Totally differentiating the above, we obtain

(2P(γ(k))+γ(k)P(γ(k)))dγ(k)dc=1dV(k1)dc>0.

From the second-order condition for profit maximization, 2P′(γ*(k))+γ*(k)P″(γ*(k))<0 holds, and we conclude that dαnk+1dc=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)(1dV(k1)dρ). Since γ(k)=αnk+1>0, it follows that if dV(k1)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=1dV(k1)dρ>0, and we can conclude that dαndρ=dγ(1)dρ=0 and dαnk+1dρ=dγ(k)dρ<0 for k=2,…,n.

From Proposition 2, the optimal price is p*=pm, and therefore it decreases in production cost c but is independent of referral cost ρ. Finally, from (8), αkrkρ=pmP(αk1). Hence, d(αkrkρ)dρ=P(αk1)dαk1dρ<0.

Proof of Proposition 4. To see that rk<rk1 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 maxα{α[1–αcρ+V(k–1)]} for all k=2,…,n and γ(1)=αn=1c2. Solving this, we have γ(k)=12(1cρ+V(k1)). Thus, V(k)=γ(k)[1γ(k)cρ+V(k1)]={12(1cρ+V(k1))}2=(γ(k))2 for all k=2,…,n. It follows that γ(k)=12(1cρ+(γ(k1))2) for all k=2,…,n and γ(1)=αn=1c2. From (8), we have

rk=P(αn)P(αk1)+ραk=P(γ(1))P(γ(nk+2))+ργ(nk+1)=11c21+12(1cρ+(γ(nk+1))2)+ργ(nk+1)=12(ργ(nk+1)+γ(nk+1))=12(ραk+αk).

As is easily verified, rk is increasing in αk as long as (αk)2>ρ. This inequality holds for any kn because πm=(αn)2>ρ and V(1)=πm<V(nk+1)=(αk)2 for any kn. Since by Proposition 2, αk is a decreasing sequence, so is rk. Thus, we have r2>r3>>rn>0. By Proposition 3, dαkdc<0 for k=1,…,n and since drkdc=αk2ρ2αk2dαkdc, it follows that referral fees rk are decreasing in production cost as well, drkdc<0 for k=2,…,n. Together, these results imply that the expected referral payments rkαk 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>…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 αk 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)=(prα2c)α1+(prα3c)α1α2++(prαnc)α1αn1+(pc)α1αn (21)

where α1,…,αn are determined by (p, r) as follows: αn=D(p)=1–F(p)≥0 and αk= D(pαk+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 kρ for k=2,…,n, probabilities α1,…,αn, are determined by the following system of equations:

(22)P(αn)=pP(αn1)=prαn+ρP(α1)=prα2+ρ, (22)

and α1≥…≥αn. Suppose that the referral condition is binding for the kth consumer: k+1=ρ for some k=1,…,n–1. Then, P(αk)=p, and we have P(α1)=P(α2)= …=P(αn)=p and α1=α2=…=αn. This is a stationary outcome, for which consumer referral conditions are all binding: 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 αk s only:

(23)Π(α1,,αn1,αn;n)=α1(P(α1)ρc)+α1α2(P(α2)ρc)++α1αn1(P(αn1)cρ)+α1αn(P(αn)c) (23)

where (α1,…,αn) is a solution to system (22). Under the stationary outcome α1= …=αn–1=αn, 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++αn1=1αn1α (25)

and π(α)=α(P(α)–c).

Denote the optimal stationary policy for an n-consumer chain by β(n)argmaxαΠ(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=…=αn–1=αn(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:Παk|α=β(n)>0 for all k<M and Παk|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=…=αn=α, if αn decreases while αM is kept constant, k>0 for all k<M and 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 maxαΠ(α,α, …,α; n) satisfies the following condition:π(β(n))=An(β(n))An(β(n))(π(β(n))ρ).(26)

3. Suppose π(β(n))–ρ>0. Then,β(n)>β(n1)>>β(1)=argmaxα˜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=An(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=nαn1(π(α)ρ)+αnπ(α)=αn1(π(α)ρ)(nAn(α)αAn(α)An(α))>0. (29)

The last inequality holds because

(30)nAn(α)αAn(α)=n(1+α+α2++αn1)α(1+2α++(n1)αn2)=n+(n1)α+(n2)α2++αn1>0. (30)

Hence, if α=β(n)>0 and π(β(n))–ρ>0, then β(n+1)>β(n). Since β(1)=αm>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,…,αn; n) in equation (23) can be defined recursively:

(31)Π(αn;1)=π(αn)=αn(P(αn)c)Π(αn1,αn;2)=αn1(P(αn1)cρ)+αn1Π(αn;1)Π(α1,α2,,αn;n)=α1(P(α1)cρ)+α1Π(α2,,αn;n1) (31)

Using these formulas, we prove the following result.

Lemma B2.Suppose that π(β(n))–ρ>0 holds.At α1=…=αn=β(n), (i) Παn<0andΠα1>0; (ii) there exists M such thatΠαk>0for any k<M andΠαk<0for any k>M.

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

(32)1α1α2α3αn1Παn=π(αn)1α1α2α3αn2Παn1=π(αn1)+Π(αn;1)ρ1α1α2α3αn3Παn2=π(αn2)+Π(αn1,αn;2)ρΠα1=π(α1)+Π(α2,,αn;n1)ρ. (32)

Thus, at α1=α2=…=αn=α,

(33)Παn=αn1π(α)Παn1=αn2(π(α)+Π(α;1)ρ)Πα1=π(α)+Π(α,,α;n1)ρ. (33)

Note that Παn=αn1π(α)<0 at β(n)>β(1)=αm 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, Πα>0, then Παk>0 for any k<ℓ, and if Πα<0, then Παk<0 for any k>ℓ.

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

(34)Ak(α)=1+α+α2++αk1=1αk1α. (34)

For kn–1, we have

(35)Παk=αk2(π(α)+Π(α,,α;nk+1)ρ) (35)

where

(36)Π(α,α,,α;nk+1)=Ank+1(α)(π(α)ρ)+ρ. (36)

Hence, Παk>0 as long as

(37)π(α)+Ank+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)Ank+1(α)An(α)>An(α), (38)

which is equivalent to

1αnk+11α1αn1α>(1αn1α)α=1(1α)2(1αnnαn1+nαn).

Thus, we conclude that Παk>0 holds if and only if

(39)[nαn11α1αnαnk+1]>0. (39)

The expression in the brackets is strictly decreasing in k. Note that for k=1, [nαn11α1αnαn]=(n1α1αnα)αn1>0 because n(1α)α(1αn)>n(1α)α(1α)=(nα)(1α)>0. Hence, Πα1>0. Since Παn=αn1π(α)<0, there exists M (1≤M<n) such that Παk>0 for any k<M and Παk<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=…=αn=α.

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

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

(40)P(α)dαn=dpP(α)dαn1=(dpαdr)rdαnP(α)dα1=(dpαdr)rdα2. (40)

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

Let xrP(α)>0. We choose (dp, dr)>;0 such that M=0. From

(41)P(α)dαM=(dpαdr)rdαM+1=(dpαdr)(1+x)rxdαM+2=(dpαdr)(1+x++xnM1)rxnM1dαn=(dpαdr)(1+x++xnM1)+xnMdp=dp(1+x++xnM)αdr(1+x++xnM1)=0, (41)

it follows that M=0 implies

(42)dp=αdr1+x++xnM11+x++xnM (42)

Similarly,

(43)P(α)dαk=dp(1+x++xnk)αdr(1+x++xnk1). (43)

Using (42),

(44)P(α)dαk=αdr[1+x++xnM11+x++xnM(1+x++xnk)(1+x++xnk1)]. (44)

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

(45)1+x++xnM11+x++xnM<1+x++xnk11+x++xnk. (45)

This inequality holds whenever k<M because 1+x++xa11+x++xa=1xa1xa+1 and (1xa1xa+1)a=xalnx(1xa+1)2(x1)>0. Similarly, 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>…>αn.

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Published Online: 2014-5-13
Published in Print: 2014-3-1