A Theory of Consumer Referral: Revisited

A Theory of Consumer Referral: Revisited Maria Arbatskaya Hideo Konishi October 23, 2013 Abstract Jun and Kim (2008) consider the optimal pricing and referral strategy of a monopoly that uses a consumer communication network to spread product information. They show that for any …nite referral chain, the optimal policy involves a referral fee that provides strictly positive referral incentives and e¤ective price discrimination among consumers based on their positions in the chain. We revisit this problem to strengthen Jun and Kim’s results by weakening their referral condition. Moreover, we characterize the …rst-best policy when individual-speci…c referral fees are available and show that it is qualitatively similar to the second-best solution of Jun and Kim (2008). Keywords: consumer referral policy, referral fee, price discrimination. JEL numbers: D4, D8, L1. Maria Arbatskaya, Department of Economics, Emory University, Atlanta, GA 303222240. Phone: (404) 727 2770. Fax: (404) 727 4639. Email: [email protected]. Hideo Konishi, Department of Economics, Boston College, 140 Commonwealth Avenue, Chestnut Hill, MA 02467. Tel: (617)-552-1209. Fax: (617)-552-2308. Email: [email protected]. Acknowledgments We thank Jeong-Yoo Kim for his comments. 1 A Theory of Consumer Referral: Revisited Highlights • We strengthen the analysis of consumer referrals by Jun and Kim (IJIO 2008). • In the model, a monopoly uses a referral chain to spread product information. • We examine the optimal pricing and referral strategy of the firm. • The firm indeed supports nonbinding referral conditions and price discrimination. • The outcome in the second-best problem resembles that in the first-best problem. 1 Introduction Jun and Kim (2008) consider the optimal pricing and referral strategy of a monopoly that uses a consumer referral chain to spread product information. In their model, one …rm sells a product to a …nite chain of n consumers. A consumer who purchases the product can refer it to the immediate successor. Each consumer’s willingness-to-pay v is an i.i.d. random variable drawn from a twice continuously di¤erentiable distribution function F (v) over [v; v] with density function f (v). The monopoly …rm can choose a price p and referral fee r that they pay for a successful referral: i.e., a consumer can receive a referral fee if the consumer refers the next-in-line consumer to the product, and that consumer purchases the product. Consumers need to pay a cost > 0 to make a referral. Consumers make their purchase and referral decisions to maximize their expected utility. In this model, the last consumer (consumer n) cannot make a referral, her purchase probability is n =1 F (p), and her purchase and referral of the product do not generate extra sales. The second-to-last consumer’s purchase has an externality since her purchase may lead to consumer n’s purchase, but the externality is limited only to sales made to her successor. For consumers positioned earlier in the chain, the externality is larger. That is, early buyers are more valuable to the …rm than later buyers since their purchase of the product is necessary for the referral chain to continue, and the potential gains from a longer chain are larger. With this model, Jun and Kim …rst show that when the second-to-last consumer has a strictly positive referral bene…t, r(1 F (p)) > (their referral condition RC), the earlier a consumer is located in the chain, the higher is her probability of purchasing the product 2 1 > 2 > ::: > n (their Proposition 1). This result further implies e¤ective price discrimination among consumers according to their positions in the referral chain: although the …rm charges a common price p and pays a referral fee r to all consumers, the …rm e¤ectively discriminates in favor of consumers located earlier in the chain because these consumers obtain a higher expected bene…t from making a referral.1 Then, Jun and Kim numerically calculate the optimal price and referral fee combinations. Somewhat surprisingly, the optimal product price and referral fee are non-monotonic functions of the chain’s length, and referral and production costs and c. We take a closer look at the optimal strategy of the …rm. When the referral chain is endless (n goes to in…nity), it is easy to show that the …rm’s optimal policy generates a stationary outcome (with equal purchase probabilities case, since n variables 1 ; :::; n 1 = 2 = ::: = 2 n ). In the …nite need to be controlled by two policy tools p and r, …nding an optimal choice of (p; r) is a second-best problem, and it can be a highly nonconvex problem with multiple local maxima. This means that although intuitively it may be bene…cial for the …rm to (e¤ectively) price discriminate between consumers based on their positions in a chain, the stationary outcome is another plausible candidate for the optimal solution, especially when n is a large …nite number. A stationary outcome is characterized by a binding referral condition and no price discrimination. However, Jun and Kim (2008) say nothing about the case where the referral condition is binding: r(1 F (p)) = (or r n = ).3 It is easy to show that if the refer- 1 Consumer k’s purchase probability k depends on consumer (k + 1)’s purchase probability k+1 since consumer k takes the expected net bene…t from referral r k+1 into account when she makes her purchase decision: k = 1 F (p + r k+1 )). 2 When there are in…nite consumers in a chain, an indeterminacy problem arises, and there is a continuum of equilibrium strategies (p; r) that support the unique optimal stationary . See Appendix C for details. 3 Jun and Kim (2008) assume that if a consumer is indi¤erent between making and not making a referral, 3 ral condition is binding even for one consumer, then purchase probabilities must be equal 1 = 2 = ::: = n, which also implies that there is no price discrimination among consumers. In this note, we examine the possibility of stationary outcome being optimal. We allow for referral equilibrium to be consistent with a binding referral condition r(1 F (p)) = by assuming that consumers make referrals when they are indi¤erent between making and not making referrals. We obtain two results that strengthen Jun and Kim’s …ndings. First, we show that the …rm’s pro…t 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 …nite n (Theorem 1). This result strongly justi…es Jun and Kim’s analysis, and also implies that at least for large n, the optimal solution is perhaps very close to the stationary outcome. Second, we …nd that when the …rm can charge di¤erent referral fees based on consumers’ positions (the …rst-best problem), both the probability of purchase k and the expected referral fee rk k are decreasing as k increases (Theorem 2). This implies that Jun and Kim’s second-best solution is qualitatively close to the …rst-best solution, since both k and r k are decreasing in Jun and Kim’s solution (their Proposition 1). Despite the simplicity of the Jun and Kim model, the proofs are rather involved. Techniques developed in this note may be useful for other purposes. 2 Jun and Kim’s Problem Here we show that the pro…t-maximizing stationary outcome is not a local maximum. Denote by i the probability that consumer i buys the product conditional on being introduced to then she will not make a referral. This tie-breaking rule is convenient since it directly implies that consumers make referrals if and only if there are positive incentives for referral (their Proposition 2). Thus, a stationary outcome is ruled out as it is not compatible with active referrals. Here, we are assuming a tie-breaking rule that allows for referrals to be given in the stationary outcome. We assume that if a consumer is indi¤erent between making and not making a referral, she refers. 4 it, i = 1; :::; n. The …rm chooses a strategy (p; r) to maximize its pro…ts ^ (p; r) = (p r + (p where k 1 ; :::; = D (p n c) 2 r 1 + (p c) n r 1 3 c) n 1 + (p are determined by (p; r) as follows: k+1 r + ) 0 for k = 1; :::; n n 1 2 c) + ::: 1 (1) n = D (p) = 1 F (p) 0 and 1. Denote by P ( ) = D 1 ( ) the standard inverse demand function. We assume that the pro…t function without referrals, k = 2; :::; n, 1 ; :::; n, (~ ) P( n) = p n 1) = p r n + = p r 2 + ; P( r k+1 1 ::: = ::: = P ( n. = p and for k 1) (2) Suppose that the referral condition is binding for the kth consumer: for some k = 1; :::; n n) c), is concave. Assuming r are determined by the following system of equations: P( and ~ (P (~ ) 1 = 2 1. Then, P ( = ::: = referral conditions are all binding: r n. k+1 k) = p, and we have P ( 1) = P( 2) = This is a stationary outcome, for which consumer = for all k = 1; :::; n 1. We will show that this outcome is not locally optimal. The …rm’s pro…t can be written in terms of ( 1 ; :::; n 1; n ; n) = 1 + (P ( 1 1) ks only: c) + n 1 (P ( 5 n 1) 1 2 c (P ( 2) )+ c) + ::: 1 n (P ( (3) n) c) where ( n 1 = 1 ; :::; n n) is a solution to system (2). Under the stationary outcome = , the monopoly pro…t when there are n 1 = ::: = 1 consumers can be written as ( ; ; :::; ; n) = An ( ) ( ( ) )+ ; (4) where An ( ) and ( )= (P ( ) 1+ + 2 + ::: + n 1 = 1 1 n (5) c). Denote the optimal stationary policy for an n-consumer chain by (n) arg max ( ; ; :::; ; n). Theorem 1 states that (n) cannot be a local maximum for small . Theorem 1. The optimal stationary policy (n) is not the optimal policy if ( (n)) > . For the formal proof, see Appendix A, where we show that the …rm’s pro…t is locally improvable (starting from (n)) by choosing an appropriate policy change (dp; dr) 0. It follows from Theorem 1 that the optimal strategy (p; r) is such that the referral condition is not binding for any consumer. This justi…es the tie-breaking rule adopted by Jun and Kim (2008). As is known from Jun and Kim’s (2008) Proposition 1, this implies that the …rm price-discriminates by subsidizing consumer referrals and generating 1 > ::: > n. We will provide a sketch of the proof of Theorem 1 here. First, in Lemma 1, we investigate the properties of the optimal stationary policy (n). Then, we look at the pro…t function evaluated at the optimal stationary policy is some M (1 1 = ::: = n 1 = n = (n). We show that there M < n) such that pro…ts increase with purchase probability for consumers located before M and decrease with purchase probability for consumers located after M : @ @ k = (n) > 0 for all k < M and @ @ k = (n) < 0 for all k > M (Lemma 2). We then show 6 that there exists a policy change d = (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 decreases while M is kept constant, d k 1 = ::: = > 0 for all k < M and d k n = , if n < 0 for all k > M (Lemma 3). Using Lemmas 2 and 3, we conclude that the optimal stationary policy is not a local maximum. 3 The First-Best Problem In this section, we consider the same n-consumer model as Jun and Kim (2008) but allow for referral fees to vary along the referral chain.4 That is, the policy tools are p and r2 ; :::; rn , where rk is the referral fee that consumer k 1 can get if consumer k purchases the product following her referral. m Let m (P ( m ) m be the standard monopoly output, = arg max [ (P ( ) c)], and m = c) be the associated monopoly pro…t. The monopoly pro…t with active con- sumer referrals is = n X k=2 where n k " (p c rk k Y1 k) `=1 ` !# + (p c) n Y ` (6) `=1 is consumer k’s probability of purchase when she is informed about the product: = D(p) and k = D(p k+1 rk+1 + ) for all k = 1; :::; n We can describe the problem in terms of ks p = P( 4 1. only by using n) (7) After we completed the current paper, Jeong-Yoo Kim let us know about his unpublished note (Kim 2006) that analyzes the …rst-best problem for the special case of n = 3. 7 and P( rk = n) P( k 1) + (8) k for k = 2; :::; n, where P ( ) = D 1 ( ). The …rm’s pro…t can be written as " !# n k n X Y1 Y = (P ( k 1 ) c ) + (P ( n ) c) ` k=2 = n 1 X k=1 " `=1 (P ( k) ) c k Y ` `=1 !# (9) ` `=1 + (P ( n) c) n Y (10) ` `=1 It is easy to see what the optimal probability of buying is for consumer n. Taking the …rst-order condition with respect to @ @ This implies that as long as k n m = n, = (P ( ) we obtain P 0( ) c) n Y1 `=1 and P ( n) ` ! = 0: (11) is the monopoly price for any n and k = 1; :::; n 1 > 0. This observation is quite sensible: no matter how many consumers are there, the last consumer does not make a referral, and the …rm should charge the monopoly price for her. The main result we have is the following: m Theorem 2. Suppose ::: > satisfy n = 2 r2 m , p = P( > 3 r3 m > . For all n, the …rm’s optimal policy satis…es 1 > ), and the expected referral bene…ts for consumers 1 through n > ::: > n rn 2 > 1 > 0. The formal proof of Theorem 2 is in Appendix B, but we provide a sketch of the proof here. To prove Theorem 2, we analyze the situation where k consumers are left in the chain, and we solve recursively by backward induction. Let V (k) be the optimal pro…t from the last k consumers, and let last consumer. Since (k) be the pro…t-maximizing purchase probability of the kth to (1) = m , the expected pro…t from the last consumer reached by referral is the simple monopoly pro…t, V (1) = 8 m . The optimal solutions V (k) and (k) when k consumers are left in the chain can be de…ned recursively: V (k) = max [ (P ( ) ) + V (k c 1)] (12) and (k) = arg max [ (P ( ) for all k ) + V (k c 1)] (13) 2. Proposition 1 in Appendix B shows that the optimal purchase probability for the kth to last consumer (k) is an increasing sequence of k: (k + 1) > (k) for all k, assuming the optimal pro…t is increasing with the number of consumers left, i.e. V (k + 1) > V (k) for all k. Under the assumption of increasing pro…t sequence, we can use Proposition 1 to characterize the optimal purchase probability sequence: k = (n k + 1) for any …xed n and all k = 1; :::; n because the kth consumer from the top is the (n from the bottom: i.e., 1 > 2 > ::: > n 2 k + 1)th consumer (Corollary 1 in Appendix B). Proposition 2 in Appendix B shows that the optimal pro…t sequence V (k) is indeed increasing as long as m > . This proposition is proved by induction. Suppose that V (k) > ::: > V (1). Then, looking at the monopoly problem with k + 1 consumers, we show that the …rm can achieve higher pro…ts V (k + 1) > V (k) if it applies the optimal policy for k to the …rst k consumers and provides consumer k with just enough incentives to make a referral to consumer k + 1 (which is pro…table because m and consumer k + 1 will > face the monopoly price). This proves that V (k) is an increasing sequence. By putting Corollary 1 and Proposition 2 together, we conclude that also implies that 2 r2 > 3 r3 > ::: > n rn > 0. 9 1 > 2 > ::: > n = m , which 4 Concluding Remarks In the framework of Jun and Kim (2008), two qualitatively di¤erent referral equilibria could possibly arise. The one described by Jun and Kim (the non-stationary outcome) is characterized by a nonbinding referral condition, unequal probabilities of purchase, and price discrimination among consumers. The other one (i.e., the stationary outcome) involves a binding referral condition, equal probabilities of purchase, and no price discrimination among consumers. We strengthen Jun and Kim’s …ndings by showing that even if we allow for the stationary outcome to arise by adopting a natural tie-breaking rule for referrals, we can show that it cannot be optimal. We also show that the equilibrium in the second-best problem (with a common referral fee) resembles the equilibrium in the …rst-best problem (when the …rm can set referral fees conditional on consumer location in the chain). References [1] Tackseung Jun and Jeong-Yoo Kim. 2008. A theory of consumer referral. International Journal of Industrial Organization, Volume 26, Issue 3, 662-678. [2] Jeong-Yoo Kim. 2006. Consumer referral with di¤erentiated referral fees. Draft, Kyun Hee University. 10 Appendix A We prove Theorem 1 by using a sequence of lemmas. Lemma 1. (i) For all n and all such that ( ) (ii) The optimal stationary solution > 0, (n) ( ; ; :::; ; n + 1) > arg max ( ; ; :::; ; n). ( ; ; :::; ; n) satis…es the following condition: 0 (iii) Suppose ( (n)) ( (n)) = A0n ( (n)) ( ( (n)) An ( (n)) > 0. Then, (n) > (n ): (14) 1) > ::: > (1) = arg max ~ (~ ). Proof. From (4), the di¤erence in pro…ts from (n + 1)- and n-consumer chains is n Hence, n ( ) ( ; ; :::; ; n + 1) ( ) > 0 if ( ) The optimal policy = ( ; ; :::; ; n) = n ( ( ) ): (15) > 0. This proves (i). (n) is implicitly de…ned by the …rst-order condition d ( ; ; :::; ; n) = A0n ( ) ( ( ) d ) + An ( ) 0 ( ) = 0: (16) This proves (ii). Finally, using (14),we …nd that at d d n =n = n 1 n 1 = (n) ( ( ) ( ( ) )+ ) n 0 ( ) nAn ( ) A0n ( ) An ( ) 11 (17) > 0: The last inequality holds because A0n ( ) = n 1 + nAn ( ) = n + (n Hence, if m = 1) (n) > 0 and 0 > 0, it follows that 2 + n 1 + ::: + + (n ( (n)) ( ) < 0 for all 2 2) + ::: + ( ( ( 1; (n). Since (1) = > 0: (n + 1) > n) n ; 1) = ( ( n 1; n ; 2) = n 1 (P ( n 2; n 1; n ; 3) = n 2 (P ( 2 ; :::; n ; n) = 1 (P ( ( (n)) > 0. in equation (3) can be de…ned recursively: ( 1; (18) 1) > (1). Thus, ( (1)) > ( (2)) > ::: > ( (n)) n ; n) 2 ; :::; n 1 > 0, then holds, and we conclude that (1) > (2) > ::: > (n) if Notice that the pro…t n 2 1 + 2 + ::: + (n = n) c) n 1) c )+ n 1 ( n ; 1) n 2) c )+ n 2 ( n 1; n 1) (P ( c )+ 1 (19) ( 2 ; :::; n; n n ; 2) 1) Using these formulas, we will prove the following result. Lemma 2. Suppose that @ @ 1 ( (n)) > 0; (ii) there exists M such that > 0 holds. At @ @ k 1 = ::: = n > 0 for any k < M and Proof. The marginal pro…ts with respect to buying probabilities 12 (n), (i) @@ = @ @ k n < 0 and < 0 for any k > M . 1 ; :::; n are 1 1 2 3 1 1 2 3 n 1 1 2 3 n @ n 1@ @ 2@ n @ 3@ n @ @ Thus, at 1 = 2 = ::: = n @ @ @ @ ( ; 1) = some `, @ @ @ @ n = 0 ( n) = 0 ( n 1) + ( n ; 1) = 0 ( n 2) + ( n 1; = 0 ( 1) 1 > 0, then n 1 0 @ @ k @ @ k n ( 2 ; :::; n; n 1) : = , = n 1 0 = n 2 = 0 ( ) (21) n ( 0( )+ ( ; 1) ) ( )+ ( ; :::; ; n 1) 1 ( ) < 0 at (n) > (1) = m by Lemma 1. By assumption, ( ; :::; ; k) is increasing in k. Hence, if for > 0 for any k < `, and if @ @ ` < 0, then @ @ k < 0 for any k > `. = (n) > 0, there exists M such that @ @ k >0 < 0 for any k > M . Recall Ak ( ) = 1 + For k + 1 In the following, we will show that for for any k < M and n ; 2) 2 ( (n)) > , and by Lemma 1, ` (20) n n 1 @ @ Note that = + 2 + ::: + k 1 = 1 1 k . (22) 1, we have @ @ = k 2 ( 0( )+ ( ; :::; ; n k 13 k + 1) ) (23) where ( ; ; :::; ; n Hence, @ @ k k + 1) = An k+1 ( )( ( ) )+ : (24) > 0 as long as 0 ( ) + An k+1 ( )( ( ) Plugging in the expression for the optimal 0 )>0 (25) ( ) from Lemma 1 and assuming ( (n)) > 0, the inequality is equivalent to An k+1 ( ) An ( ) > A0n ( ) ; (26) which is equivalent to n k+1 1 1 n 1 1 Thus, we conclude that > @ @ k @ n 1 1 = @ 1 )2 (1 n 1 n n 1 +n n > 0 holds if and only if n n 1 1 1 n k+1 n > 0: (27) n 1 1 1 The contents of the brackets is strictly decreasing in k. Note that for k = 1, n n 11 n 1 n 0. Hence, @ @ k @ @ 1 > 0 because n (1 > 0. Since @ @ > 0 for any k < M and n @ @ k = ) n 1 0 (1 n ) > n (1 ) (1 ) = (n ( ) < 0, there exists M (1 < 0 for any k > M at = 1 = ::: = n and d 0 at a stationary = . M < n), there is a policy change (dp; dr) k ) (1 (n). Lemma 3. Consider a policy of increasing p and r, starting at M (1 n n M < n) such that In Lemma 3, we describe the e¤ects of a policy change (dp; dr) outcome : < 0 for all k > M . 14 1 = ::: = 0 such that d k n = . For any > 0 for all k < M = )> Proof. Totally di¤erentiating equations (2) and evaluating at P 0 ( )d n = (dp dr) rd n P 0 ( )d n 2 = (dp dr) rd n 1 = (dp dr) rd 2: When p is increasing (dp > 0), we necessarily have d r P 0( ) > 0. We choose (dp; dr) P 0 ( )d n = 1 dp P 0( ) 0 such that d M < 0. = 0. From (29) = (dp dr) = (dp dr) (1 + x) = (dp dr) 1 + x + ::: + xn M 1 = (dp dr) 1 + x + ::: + xn M 1 M = , we have: M rd M +1 = dp 1 + x + ::: + xn it follows that d n (28) n 1 1 = ::: = = dp P 0 ( )d P 0 ( )d Let x 1 rxd M +2 M 1 rxn + xn M d dp dr 1 + x + ::: + xn M n M 1 = 0; = 0 implies dp = dr 1 + x + ::: + xn M 1 1 + x + ::: + xn M (30) Similarly, P 0 ( )d k = dp 1 + x + ::: + xn k 15 dr 1 + x + ::: + xn k 1 : (31) Using (30), P 0 ( )d k = dr 1 + x + ::: + xn M 1 1 + x + ::: + xn 1 + x + ::: + xn M k 1 + x + ::: + xn k 1 : (32) Then, since P 0 ( ) < 0 and dr > 0, d k > 0 if and only if 1 + x + ::: + xn M 1 1 + x + ::: + xn k 1 < : 1 + x + ::: + xn M 1 + x + ::: + xn k This inequality holds whenever k < M because xa (1 ln x xa+1 )2 (x 1) > 0. Similarly, d k 1+x+:::+xa 1 1+x+:::+xa = 1 xa 1 xa+1 (33) and @ 1 xa 1 xa+1 @a = < 0 whenever k > M . From Lemmas 2 and 3, we conclude that, assuming ( (n)) > , the optimal stationary outcome (n) is not a local optimum for any …nite n. This proves Theorem 1. 16 Appendix B Here we provide the proofs to Propositions 1 and 2, Corollary 1, and Theorem 2. Proposition 1. Suppose that (k + 1) > m (k) holds for all k > and V (k + 1) > V (k) for any k 1. Then, 1. Proof of Proposition 1. We have four steps to show this: 1. Consider a chain of length n = 1. The pro…t-maximizing probability satis…es M R( (1)) = c, where M R( ) = ( P ( ))0 = P ( ) (1) = m P 0 ( ) is the marginal revenue from extending sales. 2. Next consider n = 2. In this case, the optimization problem is V (2) = max [ (P ( ) c ) + V (1)] ; and V (2) > V (1) implies that there is an optimal solution (2) in this problem. Since M R( ) is decreasing, the following …rst-order condition characterizes M R( (2)) Since V (1) = (2) > m (2): + V (1) = 0. c > , M R( ) is decreasing, and M R( (1)) c = 0, we conclude that (1). 3. Suppose that (1) < ::: < (k 1) for k 3. We will show that holds. By de…nition, we have V (k) = max [ (P ( ) 17 c ) + V (k 1)] ; (k 1) < (k) and the …rst-order condition for (k) is M R( (k)) The …rst-order condition for (k M R( (k c + V (k 1) = 0: 1) is 1)) c + V (k 2) = 0: Since M R( ) is decreasing and V (k 2) < V (k 1), we conclude that (k 1) < (k) holds. 4. By an induction argument, we complete the proof. Corollary 1. Suppose m > and V (k + 1) > V (k) for any k 1. Then, for all n, under the optimal strategy, the probability of purchase declines along the referral chain: i.e., 1 > 2 > ::: > n. Proof of Corollary 1. If n consumers are left in the chain, then the probability of purchase for the …rst consumer is then n 1 = (n). If the referral for the second consumer is successful, 1 consumers are left in the chain, and the probability of purchase for the second consumer is 2 = (n 1). Similarly, k = (n k + 1) for all k = 1; :::; n. The probability of sales declines along the referral chain (i.e. (n) > (n 1) > ::: > Proposition 2. Suppose 1 > 2 > ::: > n) because (1) by Proposition 1. m > . Monopoly pro…t increases in the length of the consumer chain: V (k) < V (k + 1) for all k 1. Proof of Proposition 2. Note that V (k) is described in the following manner: " !# k 1 h k X Y Y V (k) = (P ( h ) c ) + P ( k) ` `; h=1 `=1 18 `=1 (34) where h = (k h+1) for all h. Let’s look at the monopoly problem with k +1 consumers. Consider the following policy for the …rm: set the same purchase probabilities for the …rst k consumers as in the k-consumer problem (i.e., h = h 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 bene…t equal to the referral cost: k+1 rk+1 = ). The monopoly pro…t under such a policy is ( k+1 ; k) 1 ; :::; = V (k) + ( k+1 P ( k+1 ) ) c k Y ` `=1 Note that Qk product, and the …rm pays with m = k+1 to let her make a referral. Since the maximum V is achieved , we have e = max ( k+1 ; m k) 1 ; :::; k+1 if and only if = V (k) + ( m = (P ( m Theorem 2. Suppose n m = satisfy 2 r2 , p = P( > m P( m ) ) c k Y ` `=1 m 3 r3 m ! > V (k); m = m > . For all n, the …rm’s optimal policy satis…es and since 1 > 2 n) > ::: > 1 > ), and the expected referral bene…ts for consumers 1 through n > ::: > and p = P ( e , it > . n rn = P( n, 2 > 1 > 0. Proof of Theorem 2. From Proposition 2 and Corollary 1, we know that n (35) c) > . Thus, we have e > V (k). Since V (k + 1) ) follows that V (k + 1) > V (k) whenever ::: > : is the unconditional probability that the kth consumer purchases the ` `=1 ! m ). From (8), we conclude that 19 k rk 2 r2 =p > P( 3 r3 k 1) > ::: > + 1 > 2 > ::: > > 0 for k = 2; :::; n, n rn > 0. Appendix C: Jun and Kim’s Model for an In…nite Referral Chain The monopoly pro…ts with no referrals are at m = 0 such that ( m ( )= (P ( ) c). They are maximized ) = 0. From (4) and (5), monopoly pro…ts with active consumer referrals in the case of n ! 1 are = where ( )= (P ( ) 1 1 ( ( ) c) and )+ = = D (p (P ( ) 1 ); c (36) r + ). Solving the last equality for r, we …nd that 1 r= (p P ( )+ ) (37) The pro…t-maximizing probability of purchase is de…ned by the …rst-order condition d d = = Assuming > m m ( m 1 ( ( ) 1 1 (1 ) 2 ) ( ( ) 0 (38) )+ 1 1 0 ( ) = 0: ) > , the …rst-order condition implies that d d j = (39) m > 0, and therefore . For the uniform U [0; 1] distribution of valuations, we can obtain the explicit solution: p =1 c+ (40) and r = 1 1 p c+ p p c+ + : Note that r is increasing in p: a higher price is associated with a higher referral fee. 20 (41) The optimal monopoly pro…ts with active consumer referrals are while the standard monopoly pro…ts (with no referrals) are m of pro…ts reveals that p p c+ 1 4 if and only if p 0 is satis…ed whenever p (1 m = 1 4 (1 p c)2 . The referral condition c+ and r = 1 4 1 2 c+ ) , c)2 . A comparison c + . Hence, for any a continuum of optimal strategies (p ; r ): p p = (1 p1 c+ r = (1 c)2 there is (p p c+ + ) that support referrals. The referral condition is binding at the lowest optimal price and referral fee, p = p lowest price p = p p c+ 1 2 c+ c+ and r = 1 p c+ , and not binding at higher p and r . The is below the standard monopoly price pm = (1 + c) whenever 1 4 (1 1 2 (1 + c) because c)2 . For the lowest optimal price and referral fee, we have monotonic comparative statics results – the optimal price and referral fee increase in referral and production costs and c. 21