Pricing Multiple Products with the Multinomial Logit and Nested Logit Models: Concavity and Implications

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1 Pricing Mutipe Products with the Mutinomia Logit and Nested Logit Modes: Concavity and Impications Hongmin Li Woonghee Tim Huh WP Carey Schoo of Business Arizona State University Tempe Arizona USA Sauder Schoo of Business University of British Coumbia Vancouver BC Canada V6T Z2 A Onine Suppement A Proof of Theorem 2 Proof Taing first order partia derivative of the revenue function 5 with respect to q j Rq q j p j q M m m q m p mq q j Taing a partia derivative of p j q given in equation 4 p m q q j b m i Q τ m i b m τ m i Q i b m q {m and j} τ m m b m τ m Q {m} 4 m where represents a binary indicator function Thus Rq p j q q j K Mm q m i Q i b m m M τ q b b p j q p j q τ Q K i Q i m K i Q i m Q m b m i Q i i Q i K m K m τ m τ m τ m τ m Mm q m b m Q m b m b τ Q m 5 b m τ m b τ where the second inequaity foows from M m q m Q m Aso reca Cq q j c j Since Γq Rq Cq is concave Theorem we equate the partia derivative of Γ at q to zero and obtain p j q i Q i K m i Q m c j 6 b m τ m b τ

2 Now substituting 6 into Rq K M j p jq qj given in 5 it foows Γq M j Q i Q i q j i Q i K Since ρ Γq we obtain m K m Q m b m τ m ρ Q m b m τ m M j Q b τ i Q i K q j b τ i Q i M K c j qj M c j qj j K m Q m b m τ m j Q b τ 7 Thus substituting this expression into 6 we obtain p j p jq ρ b τ c j which is the required expression for p j Let I og M j expa j b p j represent the optima aggregate attractiveness of branch Using the above expression for p j it foows that I og M j expa j /τ b ρ b c j Then the optima aggregate maret share for branch is from 2 Q Therefore we obtain e τ I K eτ I Q M and it foows from 7 and 8 M ρ j expa j /τ b ρ b c j b τ j expa j /τ b ρ b c j K M j expa τ 8 j /τ b ρ b c j τ K M j expa τ j /τ b ρ b c j τ e M j expa j b ρ τ b c j b τ where the ast equaity foows from exp /τ τ e Thus from 7 we obtain 8 The vaue of ρ satisfying 8 is unique since the eft side is increasing and the right side is decreasing in ρ Finay et qj ea j b p j/ M ea b p as in 2 From 8 as we as p j ρ b τ c j q j Q q j e M e K M i ea i /τ b ρ τ b c i K i ea i b ρ b c i τ M i ea i /τ b ρ b c i τ M i ea i b ρ b c i τ which yieds the required expression for q j e a j b c j M i ea i b c i e a j /τ b ρ b c j M i ea i /τ b ρ b c i ii

3 A2 Proof of Theorem 3 Proof Taing a partia derivative of R q with respect to q j R q q j Recaing the expression of p q/ q j given in 4 we obtain q p q q j b b τ q i Q i q i Q i b τ q τ i Q i b {j} τ q b τ Q p j M q p q q j b {j} τ q b τ Q Since M q Q it foows R q p j Q q j b τ i Q τ i b b τ p j { } Q b τ i Q i By setting the partia derivative of Γ q R q C q to zero at ˆp and ˆq we obtain ˆp j b τ c j 9 i i The above identity 9 is expressed in terms of both the equiibrium price and the equiibrium quantity vector We use the reationship between the price vector and the maret share vector given in 4 to express it in terms of the equiibrium quantity vector ˆq ony By equating two expressions of ˆp j p j ˆq given in 4 and 9 b τ i i c j a j b b og og ˆq j b τ τ Mutipying both sides by b and rearranging terms a j b c j τ τ og og ˆq j i i og τ i i og ˆq j i i Exponentiate the above equation and sum over j to obtain M e a j /τ b c j M j exp og j i i τ i i og og τ ˆq j exp og i i og og τ i i where the ast equaity foows from M j ˆq j Thus M e a j /τ b c j τ exp og i i i i j i i exp i i Since the eftmost side of the above equaity is A see equation 7 it foows from the definition of the Lambert W function that W A i i 20 iii

4 By substituting this expression into 9 we obtain the required expression for ˆp j Now from 20 W A i i Appying this to 20 we obtain W A K i i K i i i K i K i i W A K i W A i which is the required expression for To get the expression for ˆq j 2 and 3 impy ˆq j ˆq j e a j b ˆp j M e a j b c j ea b ˆp M ea b c where the ast equaity foows from the fact that ˆp ˆp M which foows from the first part of this theorem A3 Proof of Lemma Proof We first show that V x < W x For any x 0 et w W x and v V x Then w expw x v expv/ v Since v 0 we have v/ v > v Thus w expw v expv/ v > v expv Since ye y is stricty increasing in y we concude that w > v Next we show that W x λv x/ λ for λ W x For any x 0 now et w W x and v V x/ λ Then w expw x λ v expv/ v We wish to show that w λv Let λ w Then by rearranging terms we obtain w/ λ w w which impies w expw/ λ w w expw λ v expv/ v Assume by way of contradiction that w > λv Then the above inequaity impies expw/ λ w expv/ v ie w/ λ w v/ v However the assumption w > λv impies both v < w/ λ and v > w/ λ λ w/ λ By dividing the first inequaity with the second inequaity we obtain v/ v < w/λ w which is a contradiction Thus we concude w λv as required Finay we show that λv x/λ x Again et v V x/λ Since expv/ v it foows that x λ v expv/ v λ v A4 Proof of Theorem 4 Proof Since q j q j Q it foows from 2 and the definition of I og M j ea j b p j q j p j q j p j Q q j Q q p j Q j I { b q j q j } q j τ Q Q b q j q j p j Q b q j q j b τ q j q j Q I p j iv

5 Simiary for i j obtain q i / p j b q i q j b τ q i q j Q Since Γ M j p j c j q j Γ M q j p i c j q M i q j b p i c j q j p i c j b q p j p i q j b τ q i q j Q j i Setting this expression to 0 and simpifying it it foows b p j c j b τ Q M i p i c j q i We use tide to indicate that it is the price-competition equiibrium soution Since the right side of this equation is independent of j it foows that p p M which we denote by p Since q q M we obtain b p c j b τ Q p j c j Thus p c j / b τ Q We caim p M og expa i og b b i τ To prove this caim substitute 2 and 3 into 4 to obtain p j a j Q e og Q a j og b b i M i ea i i b τ τ Then since a j ogexpa j the above equaity impies the caim Therefore b τ Q p c j M og expa i og b b τ From agebraic transformation and the definition of A in 7 og Q Q Q Q τ og i Q og Q og Q og Q Q og Q c j expa i b c j og A 2 Exponentiating the above equation and using Q 0 Q we obtain Q / Q 0 exp Q Q A Then from the definition of the V function we have Q V A Q 0 From Q 0 Q we verify that Q 0 satisfies 0 Finay for the required expression for q j use 2 and 3 aong with Q V A Q0 and the fact p p M For the required expression for p j use the above expression for Q as we as p c j /b τ Q A5 Proof of Coroary 2 Proof We first compare the quantity-competition equiibrium soution to the optima monopoy soution It foows from the expressions of p j and ˆp j given in the statements of Theorems 2 and 3 that it suffices to show b τ ρ > W A From equation 8 in Theorem 2 ρ e τ M τ e a j b c j b ρ > e M e a j b c j b ρ e M e a j b c j τ e b τ ρ b τ b τ b τ j j j v

6 Thus M b τ ρ exp b τ ρ > e j e a j b c j τ A W A exp W A where the first equaity foows from the definition of A given in 6 and the second equaity foows from the definition of the W function Now since ye y is increasing in y we have b τ ρ > W A 22 where the second inequaity above foows since W is increasing Thus we concude p j > ˆp j Since p j > ˆp j hods for any j and it foows that Î > I for each where Î og M j ea j b ˆp j and I og M j ea j b p j Thus from the definition of Q e τ I / K in equation 2 it impies that 0 K eτ Î < K eτ I Q Q 0 Now we compare the price-competition equiibrium to the quantity-competition equiibrium Let ψq 0 Q 0 V A Q 0 Since V is an increasing function ψ is aso increasing It foows from the definition of Q 0 that ψ Q 0 0 Since / K W A by Theorem 3 Thus ψ 0 ψ Q 0 W A A V i W A i A V W A i W A i i W A 0 i where the inequaity foows from Lemma by etting x A and λ i W A i Thus ψ 0 ψ Q 0 Since ψ is increasing we obtain 0 Q 0 Finay we show p j ˆp j for any j and From the above argument W A 0 W A W A V A W A V A 0 It impies aong with the definitions of V A 0 and W A V A 0 exp A 0 A V A 0 V A 0 W A expw A Since exp is an increasing function 0 W A V A 0 V A 0 b τ b τ {ˆp j p j } { } W A b τ b τ V A 0 where the ast equaity foows from the expressions of ˆp j and p j given in Theorem 3 and Theorem 4 Thus ˆp j p j eτ I vi

7 A6 Proof of Coroary 3 Proof Part a From Theorem 2 we now that the monopoy optima price of products within branch are the same ie p p M Simiary from Theorem 3 we aso now ˆp ˆp M Thus the conditiona probabiity q j e a j/ M i ea i remains the same under the oigopoy and under the monopoy Since q j q j Q it is sufficient to show the existence of ˆ Q { 2 K} such that { Q if { ˆ Q } Q if {ˆ Q K} We wi estabish this by proving i Q and that ii Q impies Q From the expression of q j in Theorem 2 and the fact that Q M j q j Q M i ea i b c i b ρ τ e K M j ea j b c j b ρ τ e M i ea i b c i τ e b τ ρ K e M j ea j b c j τ e b τ b τ ρ A e b τ ρ K A e b τ b τ ρ 23 where the ast equaity foows from the definition of A e M j ea j b c j τ given in 7 Aso from Theorem 3 W A K W A A exp W A K A i exp W A A expw A K A expw A W A 24 where the second equaity hods due to the definition of the Lambert W function that W z ze W z Suppose Reca from 22 that b τ ρ > W A Since both b τ b τ and A A hod by assumption we obtain e b τ b τ ρ e 0 e W A W A Thus by comparing 23 and 24 we have Q which is statement i Now for statement ii suppose { K } Then A Q { { expb τ ρ { expw A expb τ ρ { } A expb τ b τ ρ } A expw A W A } A expb τ b τ ρ expw A A Q vii } A i expw A W A

8 where the both equaities foow from 23 and 24 and the inequaity foows from the assumptions that A A and b τ b τ Thus if Q then this inequaity impies Q yieding statement ii This competes the proof of Part a Part b Foowing the same argument as a it suffices to show the existence of ˆ Q {0 2 K} such that { Q if { ˆ Q } Q if {ˆ Q K} Since Q Coroary 2 there exists at east one firm j such that j Q j Thus it suffices to show that Q impies Q for any { K } From 2 and 20 aong with the definition of the W function og Q Q Q Q Q Q og A og A og Assume by a contradiction we have Q but > Q Then since Q / Q < / we must have Q Q Q Q > Since Q Q < where the midde inequaity comes from Theorem 4 and the condition A A each of the expressions in any bracet is nonnegative Thus the ζ vaue defined beow shoud be positive: ζ Q Q Q Q The ζ vaue as a function of Q is inear and the vaue of Q shoud beong to the interva Q At the eft endpoint of this interva ζ Q Q Q This expression is non-positive since Q < impies both Q and Q Q Now at the right endpoint of the interva ζ Q Q Q Q Q Q which simpifies to Q 2 which is a non-positive quantity Then by the inearity of ζ in Q we concude that the ζ vaue is aways non-positive and this is a desired contradiction A7 Proof of Equation 2 Proof Now taing a partia derivative of we obtain J t x q Rq λ V t x λ p q q q b i q i b i q V t x viii

9 where the ast equaity foows from the proof of Theorem 2 Let ˇq t x denote the choice of q that maximizes J t x q for given x For notationa simpicity we omit the superscript t when there is no ambiguity By setting the above partia derivative to zero we obtain p ˇq V t x ˇq i b i i ˇq b Mutipying the above identify by ˇq and summing over a products we obtain p ˇqˇq ˇq ˇq b ˇq i ˇq ˇq V t x b i i ˇq ˇq b ˇq ˇq V t x ˇq b ˇq ˇq V t x Since we set řx ˇq { p ˇq V t x } ˇq it foows that řx ˇq ˇq b ˇq Thus we obtain p ˇq V t x řx ˇq b which is 2 ix