Technical Appendix: When To Fire Customers? Customer Cost Based Pricing

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1 Technical Appendix: When To Fire Customers? Customer Cost Based Pricing Jiwoong Shin, K Sudhir, and Dae-Hee Yoon 1 May Associate Professor of Marketing, Yale School of Management, Yale University, 135 Prospect St New Haven, CT 0650 ( jiwoongshin@yaleedu, Tel: (03) ); James L Frank Professor of Private Enterprise and Management and Professor of Marketing, Yale School of Management, Yale University, 135 Prospect St New Haven, CT 0650 ( ksudhir@yaleedu, Tel: (03) ); Assistant Professor of Accounting, Yonsei School of Business, Yonsei University, 134 Shinchon-dong, Seodaemun-gu, Seoul, Korea ( dae-heeyoon@yonseiackr, Tel: (8) )

2 Table of Content 1 Numerical Example Analysis for endogenous case when τ is large (τ τ IC = ( s (v sh )δ)δ 4( δ) ) 3 Comparison between Price dierence and Cost dierence

3 Numerical Example In the main text, we set δ = 09, v = 1, s L =0 For the low and high service cost heterogeneity cases, we set s H =03 and s H =1, respectively Several diagnostic metrics for the rst and second periods are provided in Table A below Customers who buy in the rst period are referred to as Old customers and those who buy only in the second period are referred to as New customers First, consistent with propositions 1 and in the main paper, CCP leads to lower aggregate prots (Π CCP = 0946) than the benchmark prots without CCP (Π NoCCP = 1047) when the service cost heterogeneity is low ( s = 03) ; but CCP has higher prot (Π CCP = 0470 > Π NoCCP = 0466) when the service cost heterogeneity is high ( s = 1) One can also see the intuition about the ratchet eect being stronger when service cost hetergoenteity is low When service cost heterogeneity is low ( s = 03), second period prices rise by 57% for old customers (p 1 = 049 p = 077) No distinction is made between the high and low cost types; only the purchase information is used in setting prices (as in Villas-Boas 004 ) In contrast, when service cost heterogeneity is high ( s = 1), both cost and purchase information are used Second period prices rise by 65% for the high cost old customers (p H 1 = 067 ph = 11), but only by 1% for the low cost customers (p L 1 = 067 pl = 081) The weaker ratcheting eect for the old low cost customers coupled with the price discrimination eect between high and low cost customers makes CCP more protable Second, the optimal retention strategy depends on the service cost heterogeneity When the service cost heterogeneity is low, the average retention rate is 100% because there is no reason to re customers The average retention rate declines to 65% when the service cost heterogeneity is high, but this lower retention is dierentiated by high and low cost customers 100% of the low cost customers are retained, but only 5% of the high cost customers are retained In contrast to general exhortations to raise retention rates across customers, our results demonstrate that optimal retention rates should be managed based on customer characteristics Firms should lower retention rates among its higher cost customers in order to obtain a more favorable mix of low to high cost customers 1

4 Table A-1: Low Service Cost Heterogeneity ( s = 03) Period 1 Old New Benchmark 1 Benchmark Low Cost High Cost Low Cost High Cost Old New Price Margin Quantity Customer Share 50% 50% Period Old New Benchmark 1 Benchmark Low Cost High Cost Low Cost High Cost Old New Price Margin Quantity Customer Share 9% 9% 1% 1% Retention 100% 100% Π CCP = 0946 < Π NoCCP = 1047 Benchmark 1: traditional behavior-based price discrmination based only on the past purchase history Benchmark : the case without price discrimination in which the rms uses neither the past purchase information nor the customer cost type information

5 Table A-: High Service Cost Heterogeneity ( s = 1) Period 1 Old New Benchmark 1 Benchmark Low Cost High Cost Low Cost High Cost Old New Price Margin Quantity Customer Share 50% 50% Period Old New Benchamrk 1 Benchmark Low Cost High Cost Low Cost High Cost Old New Price Margin Quantity Customer Share 49% 13% 19% 19% Retention 100% 5% Π CCP = 0470 > Π NoCCP = 0466 Benchmark 1: traditional behavior-based price discrmination based only on the past purchase history Benchmark : the case without price discrimination in which the rms uses neither the past purchase information nor the customer cost type information 3

6 Analysis for endogenous case when τ is suciently large (τ τ IC = ( s (v s H )δ)δ 4( δ) ) Recall we only consider the case of s > δvmax H, where cost information was used by the monopolist in the exogenous case This condition ensures that ŵ L 1 satised in equilibrium We conrm that v + sl < ŵ L 1 > v + sl and ŵ H 1 and v + τ + sh < v + τ + sh τ > ŵ H 1 satised when s > δvmax H Therefore, if (IC) constraint is not binding, the rm charges τ are are indeed p L = ŵ1 L and p H = argmax p (p s H )(v + τ p) = v + τ + sh, (01) and D L(p) = min{v pl, v ŵl 1 } = v ŵl 1, DH (p) = min{v+τ ph, v + τ + v ŵh sh 1 } = v+τ v + τ s H Hence, the monopolist does not serve all of the previous customers = Like the exogenous model in the main paper, the rm sets price for customers who have not purchased in the rst period using the rst-order condition from Equation (15) in the main paper as follows: p O = argmax p (p s H )(ŵ1 H p + τ) + (p s L )(ŵ1 L p) = ŵh 1 + ŵl 1 + sh + s L + τ, (0) 4 We summarize the equilibrium outcomes in the following lemma Lemma 1 When service demanded by consumers (and therefore cost-to-serve) is endogenous and the utility from service τ is suciently large that (IC-H) constraint will be always satised, the marginal customers in the rst period dier by customer type: ŵ L 1 = v + s δsh + ( δ)τ (), v + s ŵh δsh 1 = (6 δ)τ () The equilibrium outcomes are as follows: p 1 = (4v ( δ)τ) ( δ) + 8s s(δ δ ), 4() p H = v + sh + τ, p L = ŵ L v + s δsh 1 = ( δ)τ + (), and po = ŵh 1 + ŵl 1 + sh + s L + τ 4 Proof The marginal consumer in the rst period, ŵ j 1 = ŵj 1 (p 1), can be calculated by using the fact that p O < p 1 and the marginal consumer does not get any surplus in the second period if she already 4

7 purchased it in the rst period:ŵ1 H + τ p 1 = δ ( ŵ1 H + τ po p 1 = δ ( ŵ1 L ) po ŵ L 1 = p 1 δp O 1 δ ) ŵ H 1 = p 1 τ δ(p O τ) 1 δ, and ŵ1 L Therefore, we can obtain the rst period cuto line for purchasing a product by using the fact that p O = ŵh 1 +ŵl 1 +sh +s L +τ 4 : ŵ1 H = p 1 δs δ τ, ŵ1 L = p 1 δs δ In the rst period, the monopolist maximizes the following total expected prot with a common discount factor δ < 1: Π(p 1 ) =(p 1 s H ) ( v ŵ H 1 ) + (p1 s L ) ( v ŵ1 L ) + δ{(p H s H )(v + τ p H ) + (p L s L )(v p L ) + (p O s H )(ŵ H 1 + τ p O ) + (p O s L )(ŵ L 1 p O )}, where ŵ1 H = p 1 δs δ τ,ŵ1 L = p 1 δs δ, p L = ŵl 1, ph = v + τ + sh Taking the rst order condition gives us the rst period price p 1 : And the rst period marginal consumer is now, p 1 = (4v ( δ)τ) ( δ) + 8s s(δ δ ) 4() ŵ L 1 = v + s δsh + ( δ)τ (), v + s ŵh δsh 1 = (6 δ)τ () Hence, p H = v + sh + τ, p L = ŵ L v + s δsh 1 = + ( δ)τ () Further, we can check that v + sl when δ ( v s H) < s : < ŵ L 1 and v + τ + sh τ > ŵ H 1 are satised in equilibrium ŵ1 L v + sl = δ(v sl ) + (1 δ) (s H s L ) + ( δ)τ > 0, () ( ) v + τ + s ŵ1 H H τ = δ(v sh ) (s H s L ) τ < 0 () Unlike the exogenous case, where customer type is xed, now the marginal customer in the rst period from the high and low type dier in their willingness to pay This is because the high type customer gets an extra utility τ from consuming the rm's augmented services 5

8 From Lemma 1, we identify the τ-condition for (IC-H) constraint to be satised: p L p H τ δ v + s δsh + τ τ IC = ( s (v sh )δ)δ 4( δ) ( δ)τ () v + sh + τ τ δ (03) That is, when τ is suciently large (τ τ IC ), the high type consumer will reveal his type in the rst period by choosing a high level of service Results Suppose that the service cost heterogeneity is suciently large that s > δvmax H, and the utility from service is large enough that τ τ IC = ( s (v sh )δ)δ 4( δ) 1 The monopolist uses CCP and charges dierent prices to the H and L-type customers: p H = v+s H +τ > p L = ŵl 1 Further, the total prot with CCP is greater than the total prot without price discrimination (Π CCP > Π No P D ) when s becomes large Proof First, we calculate the prot under no discrimination Without price discrimination, the monopolist simply maximizes the following per-period prot function Π S t = (p S s H )(v p S + τ) + (p S s L )(v p S ) The optimal price is p S = v+sh +s L +τ 4, and the total prot is Π S = (1+δ) [ Π S ] t = (1 + δ) [(v sh s L ) +(v 3s H +s L )τ+τ ] 8 Using the results of p 1,p H, and pl in Lemma 3 of the main paper, we get Π CCP = v (4+Ω)+(s H ) (+δ) +(s L ) Ω+s L Ω(v τ) s H (+δ)(4v s L ( δ))+ω(4vτ 6s H +τ ) 8(4 δ), where Ω = 4 + ( δ)δ It immediately follows that Π = Π CCP Π S = δ(v ( δ)+4vτ (s H ) (1+δ) (s L ) s L (4v τ) τ +s H (6s L 4v(1 τ)+6τ) 8(4 δ) ( (4 ) (v s L + τ) δ) (1 δ) (1 + δ)τ Therefore, Π 0 if and only if s 1 + δ Hence, as s becomes larger, Π CCP > Π S When τ τ IC, the (IC) constraint for H-type is not binding Therefore, the customers will reveal their types in the rst period even under optimal prices that the monopolist would have charged when the customer type is xed Hence, the equilibrium outcome is consistent with the our main model where the customer types are exogeneous and CCP can increase a rm's prot 6

9 Comparison between Price dierence and Cost dierence Exogenous case: 1 When the heterogeneity is small: there is no price discrimination so that price dierence (zero) is smaller than cost dierence When the heterogeneity is large, price dierence is always smaller than cost dierence as follows: p H = v + s H, p = v + s δs H, p H p L = s H( + δ) s L vδ ; () p H p L (s H s L ) = s L(3 δ) 3s H ( δ) vδ () Then, p H p L (s H s L ) > 0 if s < s L(3 δ) vδ 3( δ) = s L δ(v s L) 3( δ) < s L,which is a contradiction Therefore, p H p L < s H s L Endogenous case: 1 When τ < τ IC = ((s H s L ) (v s H )δ)δ 4( δ) ; p H p L = τ δ, p H p L (s H s L ) > 0 if τ > (s H s L )δ Therefore, for the condition to hold, it must be the case that τ IC > (s H s L )δ However, we can easily see that τ IC (s H s L )δ because τ IC (s H s L )δ = δ(s H(6 5δ) + vδ s L (6 4δ)) 4( δ) = δ((s H s L )(6 5δ) + δ(v s L )) 0 4( δ) Therefore, p H p L < s H s L When τ > τ IC = ((s H s L ) (v s H )δ)δ 4( δ) ; p H p L = s H( + δ) s L vδ + τ 8 δ 7

10 We rst note that p H p L (s H s L ) > 0 if And we can show that τ > τ IC since τ > τ = s H(6 3δ) + vδ s L (6 δ) = s H(6 3δ) + vδ s L δ s L (6 3δ) = (s H s L )(6 3δ) + δ(v s L ) τ τ IC = ()((s H s L )(6 5δ) + δ(v s L )) 4( δ) > 0 Therefore, p H p L > s H s L if τ > τ = (s H s L )(6 3δ) + δ(v s L ) 8