MANAGEMENT SCIENCE doi /mnsc ec

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1 MANAGEMENT SCIENCE doi 0287/mnsc ec e-comanion ONLY AVAILABLE IN ELECTRONIC FORM informs 2009 INFORMS Electronic Comanion Otimal Entry Timing in Markets with Social Influence by Yogesh V Joshi, David J Reibstein, and Z John Zhang, Management Science, doi 0287/mnsc

2 ec2 e-comanion to Joshi, Reibstein, and Zhang: Otimal Entry Timing in Markets With Social Influence E-Comanion for Otimal Entry Timing In Markets With Social Influence This e-comanion contains roofs and other suorting analysis for the above aer Aendix : Estimating the Parameters of Social Influence for Porsche In this Aendix, we estimate the cross market arameters of social influence for Porsche using the monthly sales data for the Boxster and Cayenne, assuming Porsche launched the Cayenne at the otimal time These monthly sales exhibit strong seasonality atterns, and also reflect external shocks in the market Since seasonality and external shocks that affect Porsche are also likely to affect the US automobile industry, we introduce two industry-wide variables in our model to serve as control variables: total cars sold, and total light trucks sold, er month within the US With these control variables, our model becomes: y = ( + q n m q n 2 m 2 )(m n ) + c x + ɛ, (EC) n 2 y 2 = ( 2 + q 2 + q + n )(m 2 n 2 ) + c 2 x 2 + ɛ 2, (EC2) m 2 m where ɛ, ɛ 2 are indeendently and normally distributed, y = Porsche Boxsters sold in the US er month, y 2 = Porsche Cayennes sold in the US er month, n = cumulative Porsche Boxsters sold in the US, till the revious month, n 2 = cumulative Porsche Cayennes sold in the US, till the revious month, x = total cars sold in the US er month, and x 2 = total light trucks sold in the US er month We control for auto-correlation by introducing a lag arameter into these equations (Franses 2002): n (t) ŷ (t) = ( + q q n 2(t) )(m n (t)) + c x (t), m m 2 y (t) = ŷ (t) + AR (y (t ) ŷ (t )) + ɛ, n 2 (t) ŷ 2 (t) = ( 2 + q 2 + q + n (t) )(m 2 n 2 (t)) + c 2 x 2 (t), m 2 m y 2 (t) = ŷ 2 (t) + AR 2 (y 2 (t ) ŷ 2 (t )) + ɛ 2 (EC3) (EC4) We constrain, 2, q, q 2 to be non-negative A summary of the nonlinear OLS model is reorted in Table A As a check on whether there is correlation across the two error terms, we first estimated these two equations using seemingly unrelated regression Since this correlation was very low (-00943), reorted results are with nonlinear OLS estimation Table A: Model summary for Porsche data Durbin-Watson statistic for Boxster 86 Durbin-Watson statistic for Cayenne 99 AdjustedR 2 for Boxster 068 AdjustedR 2 for Cayenne (n=34) 050 The values of Durbin-Watson statistic indicate that auto-correlation is no longer an issue The estimates for the model arameters are reorted in Table A2 Table A2: Cross market social influence arameter estimates for Porsche Boxster Cayenne Estimate t-statistic -value q m q c AR Estimate t-statistic -value q m q c AR

3 e-comanion to Joshi, Reibstein, and Zhang: Otimal Entry Timing in Markets With Social Influence ec3 Note, that these estimates are based on monthly sales data, and are contingent on assuming that Porsche entered the SUV market in March 2003 The low values of intra-market diffusion arameter estimates (esecially, 2 ) could be attributed to the resence of control variables and the existence of significant inter-market effects (coefficients of leverage and backlash) Observe that the cross-market social influences (q +, q ) are significant, as anticiated by Porsche executives Porsche s car market rovides a significant leverage for sales in the SUV market, and Porsche s SUV market exerts a significant backlash on sales in its sorts cars market Interestingly, a comarison of the relative magnitudes of the two cross-market influences indicates that for Porsche, the backlash effect is almost six times stronger than the leverage effect That may be the reason why intuitively some industry analysts and outside observers argued that Porsche may be better off not entering the SUV market Clearly, this analysis is with its caveats, including the need to control for numerous factors imacting sales in each market, such as advertising, distribution, romotions, etc Given data limitations, the arameter estimates we derive are urely suggestive, but nevertheless, serve as a good illustration of the cross market social influences at lay Aendix 2: Proof for Lemma Figure A: Dominant strategies for entering the new market Consider the ayoffs for the following two cases: π a = ayoff from serving only market 2 in stage and both markets in stage 2, π a = n 2 a + r( n 2 a + n 2 2 a ) π b = ayoff from serving both markets in both stages, π b = n b + n 2 b + r( n 2 b + n 2 2 b ) Note that n 2 b + r n 2 2 b n 2 a + r n 2 2 a, since market only contributes ositively to the adotion in market 2; and n b + r n 2 b > r n 2 a, since market 2 only contributes negatively A similar analysis with all Porsche cars sales for market (instead of sales of the Boxster only) gives q = 007 and q + = 006, both statistically significant at 0002

4 ec4 e-comanion to Joshi, Reibstein, and Zhang: Otimal Entry Timing in Markets With Social Influence to the adotion in market With time, adotions in market 2 are higher, resulting in a stronger negative influence on adotions in market, further decreasing the rate of adotion in market In addition, in case b, adotions take lace at an earlier time leading to a higher net resent value: = π b > π a Thus, entering only market 2 is always dominated by entering both markets Similar arguments hold for the case of serving only market 2 in stage 2 versus both markets in stage 2 Aendix 3: Two Stage Model: Profit Calculations Prior to stage, the cumulative adotion in both markets is 0 (ie, n 0 = n 0 2 = 0) With the late entry strategy, in stage the firm serves only market, and in stage 2 the firm serves both markets n = m, n 2 = 0, n 2 = ( + q)m( ), n 2 2 = min{, + q + ( + ( + q)( ))}m = π l = m + r(( + q)m( ) + min{, + q + ( + ( + q)( ))}m) With the early entry strategy, in both stages, the firm sells to both markets 2 n = m, n 2 = ( + q + )m, n 2 = max{0, ( + q) q ( + q + )}m( ), n 2 2 = min{, + q( + q + ) + q + ( + ( ) max{0, ( + q) q ( + q + )})}m( ( + q + )) These exressions rovide us with bounds on the values for leverage and backlash coefficients Based on ast observed values for, q, we assume 0 and 0 ( + q) (Lilien, Rangaswamy and Van den Bulte, 2000) First, consider the exression for n 2 If q + >, all consumers in segment 2 adot in the first stage itself Higher values of q + generate no new dynamics in the analysis Hence, conservative bounds on q + are 0 q + Next, consider the exression for n 2 If ( + q) q ( + q + ) < 0, consumers in market do not adot during stage For higher values of q such that ( + q) q ( + q + ) < 0, no new dynamics are generated Thus, the uer bound on q is determined as q +q Given q + [0, ], the maximum value that +q + the right hand side takes is + q Hence, conservative bounds on q are 0 q ( + q) Continuing with the rofit calculations, π e = (m + ( + q + )m) + r(max{0, ( + q) q ( + q + )}m( ) + min{, + q( + q + ) + q + ( + ( ) max{0, ( + q) q ( + q + )})}m( ( + q + ))) Let the difference in ayoffs be denoted by π: π = π l π e = m( + q + ) + rm(( + q)( ) + min{, + q + ( + ( + q)( ))} max{0, ( + q) q ( + q + )}( ) (EC5) min{, + q( + q + ) + q + ( + ( ) max{0, ( + q) q ( + q + )})}( ( + q + ))) 2 From n 2 : Based on historical observations we assume 0 (Lilien, Rangaswamy and Van den Bulte, 2000) Further, if (+q + ) >, all consumers in segment 2 adot in the first stage For values of q + such that (+q + ) >, no additional dynamics are generated Hence, this constraint serves as a conservative uer bound on q + : 0 q + From n 2 2 : Based on ast observed values for and q, we assume that ( + q) (Lilien, Rangaswamy and Van den Bulte, 2000) Further, if ( + q) q ( + q + ) < 0, consumers in segment do not adot during stage 2 For higher values of q such that ( + q) q ( + q + ) < 0, no additional dynamics are generated, since adotion in segment stos Thus, q values of interest are determined by ( + q) q ( + q + ) 0, or q +q Given +q + q + [0, ], the maximum value that the right hand side takes is + q This constraint serves as a conservative uer bound on q : 0 q ( + q)

5 e-comanion to Joshi, Reibstein, and Zhang: Otimal Entry Timing in Markets With Social Influence ec5 The exression for π can be further simlified using the three conditions listed below: Condition Boundary C = (( + q) q ( + q + )) C = 0 C2 = ( + q( + q + )) + q + ( + ( ) max{0, C}) C2 = C3 = ( + q + ( + ( + q)( ))) C3 = Deending on the values taken by C(C 0), C2(C2 ) and C3(C3 ) in the {q, q + } arameter sace, we have atmost eight distinct regions, within which the exression for π can be simlified For C < 0, the intersection of C2 > and C3 < is a null set This narrows down the regions to at most seven distinct regions {R R7} These conditions are lotted in Figure A2 (for = 05, and q = 04) The simlified exressions for π within each of these regions are as follows: R : C 0, C2, C3 : π R = m( + q + ) + rm(( + q)( ) + + q + ( + ( + q)( )) (( + q) q ( + q + ))( ) ( + q( + q + ) + q + ( + ( )(( + q) q ( + q + ))))( ( + q + ))) R2 : C < 0, C2, C3 : π R2 = m( + q + )(r( + ( + q)q + ) ) R3 : C < 0, C2, C3 > : π R3 = m( + q + ) + mr( + ( )( + q)) + mr( + q)( + q + )(( + q + ) ) R4 : C < 0, C2 >, C3 > : π R4 = m( + q + ( r) r( + ( )( + q))) R5 : C 0, C2 >, C3 > : π R5 = m( + q + )( ( + ( )q )r) R6 : C 0, C2, C3 > : π R6 = m( + q + ) + rm(( + q)( ) + (( + q) q ( + q + ))( ) ( + q( + q + ) + q + ( + ( )(( + q) q ( + q + ))))( ( + q + ))) R7 : C 0, C2 >, C3 : π R7 = m( + q + ) + rm(( )( + q)( + q + ) + q + + ( + q + )( + q ( )) + ( )( + q)) We solve for the oints of intersection of the three conditions: A {( + q) 2, (+q) }, (+q) B { q(+(+q)()), }, () 2 (+(+q)()), +(+q)() C { (+q)(+(+q)()) } (+(+q)()) Figure A2: Distinct regions for comaring the ayoffs from entering early vs late

6 ec6 e-comanion to Joshi, Reibstein, and Zhang: Otimal Entry Timing in Markets With Social Influence Aendix 4: Proof for Proosition It is otimal for a firm to enter early if π 0 From equation (EC5), this imoses the following condition on r : r ( + q + )/(( + q)( ) + min{, + q + ( + ( + q)( ))} max{0, ( + q) q ( + q + )}( ) min{, + q( + q + ) + q + ( + ( ) max{0, ( + q) q ( + q + )})}( ( + q + ))) Of the seven regions, the above right hand side exression is lowest in region R3, when C < (+q + ) +(+q)q + ((2+q + ) ) 0, C2, C3 > = r This uer bound on r takes its lowest value when q + is lowest within the oerating constraints (at ) Consequently, we have 0 r (+(+q)()) + = π < q 4(+q) Aendix 5: Proof for Proosition 2 Consider + r We first rove art (i) of this roosition For 2 6+4q 4(+q) q q(+(+q)()) () 2 (oint B in Figure A2), this area of low backlash consists of arts of three regions: R, R5 and R7 We now derive the conditions on model arameters within these sub-regions such that the difference in ayoffs is negative (ie, early entry is otimal) (a) In R, note that for q + (condition C3), π (+(+q)()) q + R 0 Also, 2 π q +2 R 0 π R is convex in q + Thus, if π R 0 at the uer bound of q and both uer and lower bounds of q + in this region, then π R 0 in the entire region 2 ))) 0 imlies r π R,q = q(+(+q)()) () 2 π R,q = q(+(+q)()) () 2,q + = (+(+q)()) = m(+r(q (+q(4+q) (+q),q + =0 () 0 imlies r +(2 )q+()q 2 (+q(4+q) (+q) 2 ) q We can show that Hence, π +(2 )q+()q 2 (+q(4+q) (+q) 2 ) q R 0 = r, the uer bound on r as described in the roosition +(2 )q+()q 2 (b)in R5, note that π q R5 0 and π q + R5 0 Thus, if π R5 0 at the uer bound of q and lower bound of q + in this region, then π R5 0 in the entire region π R5,q = q(+(+q)()) 0 imlies r () 2,q + = (+(+q)()) +(2 )q+()q 2 (c) In R7, note that π q R7 0 Also, + r 2 6+4q 4(+q) uer bounds of q and q + in this region, then π 0 in the entire region π R7,q = q(+(+q)()) 0 imlies r () 2,q + = (+(+q)()) π q + R7 0 Thus, if π 0 at the +(2 )q+()q 2

7 e-comanion to Joshi, Reibstein, and Zhang: Otimal Entry Timing in Markets With Social Influence ec7 From (a)-(c), when + r, and 0 q q(+(+q)()), early 2 6+4q 4(+q) +(2 )q+()q 2 () 2 entry is otimal We now turn our attention to art (ii) of this roosition For (+q)(+(+q)()) +(+q)() in Figure A2), this area of high backlash consists of arts of four regions: R,R2, R3 and R4 q (oint C (d) In R, as mentioned in (a) above, for q +, (+(+q)()) π R is convex in q + Thus, π R has utmost one zero Since π R is continuous, if π R changes sign within this region, it will also change sign at the extreme values of q + We can show that π R,q + =0 0 Further, along C = 0, we have q = +q This gives π +q + R 0 π q + R 0 Also, 2 π q +2 R 0 if q + r Thus, if leverage is low, early entry is always otimal in R r(+q) +q (e) In R2, since C < 0, we have < q = π +q + R2 0 if q + r, and π r(+q) R2 0 if r < r(+q) q+ < Thus, in R2, if leverage is low, early entry is otimal, but if leverage (+(+q)()) is moderate, late entry is otimal (f) In R3, solving for π R3 = 0 gives two roots for q +, of which only the smaller one lies within this region We have: π R3 0 = q + +( 2)(+q)r +(+q)r(2 (3 q+4()(+q))r) 2r(+q), π R3 0 = +( 2)(+q)r +(+q)r(2 (3 q+4()(+q))r) 2r(+q) Further, for r +()(+q) 2 q + < +( 2)(+q)r q + (+q) it is always the case that in this region: +(+q)r(2 (3 q+4()(+q))r) 2r(+q) This imlies that for q 4(+q) < r < +()(+q) 2 : (+q) q + < +( 2)(+q)r +(+q)r(2 (3 q+4()(+q))r) imlies π > 0, 2r(+q) +( 2)(+q)r +(+q)r(2 (3 q+4()(+q))r) < q + < (+q) imlies π < 0, 2r(+q) (+q) and, for < r, for all values of q + in this region, π > 0 +()(+q) 2 Thus, in R3, late entry continues to be otimal when leverage is moderate, and early entry is otimal when leverage is high (g) In R4, π R4 is indeendent of q and is decreasing in q + In R4, (+q) q + (+q) At q + = (+q), if r < then π < 0, ie, π < 0 for this entire region and early (+q) +()(+q) 2 entry is otimal (For < r, π > 0 if q + < r(+()(+q)), and π < 0 if r(+()(+q)) < q + ) +()(+q) 2 r r Thus, continuing the outcomes observed in R3, in R4 as well, late entry is otimal if leverage is moderate, and early entry is otimal when leverage is high Aendix 6: Proof for Proosition 3 In Aendix 4, we show that for + r 2 6+4q 4(+q) q(+(+q)()), and +(2 )q+()q 2 0 q, early entry is always otimal Consider ayoffs in these regions, R5, R7 and R, () 2 when atience is high ( r ) +(2 )q+()q 2 Note that in R5, π R5 0 when q r In R7, π r() q R7 0 Thus, if π R7 0 at the uer bound of q, then π 0 in the entire region Note that if q +, π (+(+q)()) R7 0, and since C3 0, it is always the case that q + Finally, in R, as discussed in Aendix 3, (+(+q)()) π R is convex in q + Thus, if π R 0 at the uer bound of q and both uer and lower bounds of q + in this region, then π R 0 in R = mr( )( + q) 0, which is always true π R,q = r r(),q+ =0 π R,q = r r(),q+ = (+(+q)()) 0 r +(2 )q+()q 2 (Note that this uer bound of q + that we evaluate in fact exceeds the value at the boundary of R, hence this would be a conservative check)

8 ec8 e-comanion to Joshi, Reibstein, and Zhang: Otimal Entry Timing in Markets With Social Influence Next, we turn to the case of high backlash, where (+q)(+(+q)()) q + q Here, high +(+q)() atience is characterized by r For +()(+q) q+, (Regions R3 and R4), (+(+q)()) late entry is always otimal For q +,the boundary searating the otimal strategies (+(+q)()) for early and late entry lies in regions R and R2 For R: q +r(q (+q)(+(2 )q+ )) π 0, r()(+q + )(q + ) +r(q (+q)(+(2 )q + )) q +q π 0 r()(+q + )(q + ) +q + For R2: q + < r π < 0, r(+q) q + > r π > 0 r(+q) Thus, at high backlash and atience, it is otimal to enter early if leverage is low and late otherwise References [] Franses, P H 2002 Testing for residual autocorrelation in growth curve models Technological Forecasting and Social Change