Multiproduct Price Optimization Under the Multilevel

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WORKING PAPER Vol. 00, No. 0, xxxxx 0000, pp. 000–000 issn xxxx-xxxx | eissn xxxx-xxxx | 00 | 0000 | 0001

doi 10.1287/xxxx.0000.0000 www.iOptimize.org

Multiproduct Price Optimization Under the Multilevel Nested Logit Model Hai Jiang, Rui Chen, He Sun Department of Industrial Engineering, Tsinghua University, Beijing 100084, China [email protected], [email protected], [email protected]

We study the multiproduct price optimization problem under the multilevel nested logit model, which includes the multinomial logit and the two-level nested logit models as special cases. When the price sensitivities are identical within each primary nest, that is, within each nest at level 1, we prove that the profit function is concave with respect to the market share variables. We proceed to show that the markup, defined as price minus cost, is constant across products within each primary nest, and that the adjusted markup, defined as price minus cost minus the reciprocal of the product between the scale parameter of the root nest and the price-sensitivity parameter of the primary nest, is constant across primary nests at optimality. This allows us to reduce the multidimensional pricing problem to an equivalent single-variable maximization problem involving a unimodal function. Based on these findings, we investigate the oligopolistic game and characterize the Nash equilibrium. We also develop a dimension reduction technique which can simplify price optimization problems with flexible price-sensitivity structures. Key words : differentiated products; multiproduct pricing; multilevel nested logit; oligopolistic competition

1. Introduction It is customary for a firm to offer a set of differentiated products, as differentiation makes each individual product more attractive to a particular customer segment and creates a competitive advantage for the firm. The firm, therefore, faces the multiproduct price optimization problem, whose goal is to determine the prices of these products so as to maximize the expected profit. Most of the literature in this field is based on the multinomial logit and the nested logit models, both of which are widely used in marketing, economics, transportation, and operations management (BenAkiva and Lerman 1985, Train 2009). For the price optimization problem under the multinomial logit model, Hanson and Martin (1996) show that the profit function is not concave with respect to the price variables and develop a path following procedure to find the global optimum. In the context of product line selection, Chen and Hausman (2000) discretize prices and formulate the problem as an integer program, whose linear programming relaxation is quasi-concave. Aydin and Porteus (2008) demonstrate that the optimal price vector satisfying the first-order condition is unique. When the price-sensitivity is identical across products, Dong et al. (2009) and Song and 1

Jiang, Chen, and Sun: Multiproduct Pricing Under the Multilevel Nested Logit Model c 0000 xxxxxxx Working Paper 00(0), pp. 000–000,

2

Xue (2007) show that the profit function is concave with respect to the market share variables. Anderson and de Palma (1992) and Hopp and Xu (2005) show that the markup, defined as price minus cost, is identical across all products at optimality. When products are substitutable of, or similar to, each other, which is often the case for differentiated products, the Independence from Irrelevant Alternatives (IIA) assumption of the multinomial logit model no longer holds. Researchers, therefore, resort to the nested logit model to capture the similarity among the products. Under the two-level nested logit model with identical price sensitivities, Kok and Xu (2011) show that at optimality the markup is constant for all products under centralized management and characterize the Nash equilibrium. Later, Li and Huh (2011) establish that the profit function is concave with respect to the market share vector even when the price sensitivity is nest-specific. They derive the unique equilibrium in a closed-form expression involving the Lambert W function (Corless et al. 1996). When the price sensitivities are product-specific, Gallego and Wang (2014) show that the market share transformation no longer results in a concave profit function. However, they find that at optimality the adjusted markup is constant across all products within a nest and the adjusted nest-level markup is constant across all nests. The authors also show that the profit function can be reduced to a single-variable unimodal function and study the oligopolistic competition. In this paper, we investigate the multiproduct price optimization problem under the multilevel nested logit model, which includes the multinomial logit and the two-level nested logit models as special cases. It is capable of handling situations where products are differentiated along multiple dimensions. For example, products in a clothing store may be differentiated first by brands (e.g., Calvin Klein, Dockers, Polo Ralph Lauren), then by categories (e.g., blazers, shirts, pants), then by materials (e.g., cotton, polyester, leather), and finally by color (e.g., white, yellow, blue). Although the importance of using the multilevel nested logit model has long been demonstrated by many empirical studies (Morey et al. 1993, Gabriel and Painter 2002, Liaw and Frey 2003, Coldren and Koppelman 2005, Hsiao and Hansen 2011), its application in price optimization is very scarce. Li et al. (2013) are the first to study the pricing problem under the multilevel nested logit model, where price sensitivities are product-specific. Since the profit function is not concave with respect to the price variables, the authors develop an iterative algorithm that finds the stationary points, or local maxima, of the profit function instead. In the multilevel nested logit model we investigate, we assume that price sensitivities are identical within each primary nest, that is, within each nest at level 1. In the clothing store example, it means that price-sensitivity parameters are identical within each brand. This modeling assumption is not unreasonable because the set of products that are similar along the primary dimension are likely to have comparable price sensitivities (Erdem et al. 2002, Li and Huh 2011). For example, Polo


3

Ralph Lauren branded products are often considered to have a low price sensitivity, because they are regarded as well-made quality clothing. We first build the one-to-one mapping from the price variables to the market share variables and prove that the profit function is concave with respect to the market share variables, which immediately generalizes the results in (Li and Huh 2011). We proceed to show that the markup, defined as price minus cost, is constant across products within each primary nest, and that the adjusted markup, defined as price minus cost minus the reciprocal of the product between the scale parameter of the root nest and the price-sensitivity parameter of the primary nest, is constant across primary nests at optimality. This allow us to reduce the multidimensional price optimization problem to an equivalent single-variable maximization problem involving a unimodal function. Based on these findings, we investigate the oligopolistic game. We show that it is equivalent to a log-supermodular game and there exists a unique Nash equilibrium. We also develop a dimension reduction technique that can simplify price optimization problems with flexible price-sensitivity structures under the multilevel nested logit model. The remainder of this paper is organized as follows. In Section 2, we develop the formulation for the multiproduct pricing problem under the multilevel nested logit model. In Section 3, we prove the concavity of the profit function with respect to the market share variables. In Section 4, we study the properties of the markups and the adjusted markups at optimality. In Section 5, we solve the oligopolistic game and present the dimension reduction technique. Finally, we conclude the discussion in Section 6.

2. Modeling Framework Consider a firm that offers a set of differentiated products that are substitutable of each other to some extent along multiple dimensions. Customers select these products following the multilevel nested logit model: They first choose the desired value along the first dimension, such as brand in the clothing store example, which gives rise to a subset of products within a brand; they then choose the desired value along the second dimension, such as category, which continues to narrow down the subset of products within the chosen brand. In general, the choice along each subsequent dimension further restricts the subset of eligible products. The above choice hierarchy is often visualized by an upside down tree shown in Figure 1. Let n be the number of levels in this tree. Each level of this tree (except levels 0 and n) represents a dimension, along which products are partitioned into nests, or groups. At the root nest, that is, nest h0i, all products are partitioned into D0 nests, denoted as h1i, h2i, · · · , hD0 i. For nest hi1 i

at level 1, it is further partitioned along the second dimension into Di1 nests, denoted as hi1 , 1i,

hi1 , 2i, · · · , hi1 , Di1 i. Generally speaking, a nest at level m (1 ≤ m ≤ n) is denoted by an m-tuple hi1 , i2 , · · · , im i, which refers to the set of products that are first classified into the i1 -th nest along

4


the first dimension, then into the i2 -th nest along the second dimension, · · · , and finally into the im -th nest along the m-th dimension. Products belonging to nest hi1 , i2 , i3 , · · · , im i are partitioned

along the (m + 1)th dimension into Di1 ,i2 ,··· ,im nests, denoted as hi1 , i2 , · · · , im , 1i, hi1 , i2 , · · · , im , 2i,

· · · , hi1 , i2 , · · · , im , Di1 ,i2 ,··· ,im i. Following this notation, a product can be represented by an n-tuple hi1 , i2 , · · · , in i, where i1 ∈ {1, 2, · · · , D0 }, i2 ∈ {1, 2, · · · , Di1 }, · · · , in ∈ {1, 2, · · · , Di1 ,i2 ,··· ,in−1 }, which

indicates that it is sequentially classified into nest hi1 i at level 1, nest hi1 , i2 i at level 2, · · · , nest

hi1 , i2 , · · · , in−1 i at level n − 1. And finally, it becomes a leaf at level n, which can be viewed as a

degenerated nest with no children. In addition to the actual products offered by the firm, customers have a no-purchase option denoted by hN i at level 1. A customer that chooses this option does not

purchase anything from the firm, which keeps the pricing problem bounded.

Let Nm denote the set of nests at level m, where m ∈ {0, 1, · · · , n}. That is, Nm =

{hi1 , i2 , · · · , im i|i1 ∈ {1, 2, · · · , D0 }, i2 ∈ {1, 2, · · · , Di1 }, · · · , im ∈ {1, 2, · · · , Di1 ,i2 ,··· ,im−1 }}. Following

this notation, Nn refers to all the products offered by the firm. There are two types of nests that

deserve special attention: primary nests and elementary nests. A nest at level 1 is alternatively referred to as a primary nest. That is, nest hi1 i, where i1 ∈ {1, 2, · · · , D0 }, is a primary nest. A

nest at level n − 1 is alternatively referred to as an elementary nest, because it cannot be further partitioned in the tree structure. Let µi1 ,i2 ,··· ,im be the scale parameter associated with nest hi1 , i2 , · · · , im i ∈ Nm . Let us agree that when m = 0, hi1 , i2 , i3 , · · · , im i = h0i and µi1 ,i2 ,··· ,im = µ0 ,

which corresponds to the scale parameter associated with the root nest h0i. As is shown in Jiang et al. (2014), the scale parameters must satisfy the condition that µi1 ,i2 ,··· ,im−1 ≤ µi1 ,i2 ,··· ,im for all

hi1 , i2 , · · · , im i ∈ Nm and m ∈ {1, 2, · · · , n}, so as to be consistent with random utility maximization.

It essentially implies that products belonging to the same nest at level m are more similar to each other than those belonging to different nests at this level.

A customer’s utility associate with purchasing product hi1 , i2 , · · · , in i ∈ Nn is given by Ui1 ,i2 ,··· ,in =

−bi1 · pi1 ,i2 ,··· ,in + ai1 ,i2 ,··· ,in + i1 ,i2 ,··· ,in , where pi1 ,i2 ,··· ,in is the selling price of the product; bi1 is

the price-sensitivity parameter, which is positive and identical for products in each primary nest;

ai1 ,i2 ,··· ,in is a constant representing the quality of the product; and i1 ,i2 ,··· ,in is a random variable representing the unobserved component of the utility. The term −bi1 · pi1 ,i2 ,··· ,in + ai1 ,i2 ,··· ,in is also referred to as the systematic component of the total utility and is denoted as Vi1 ,i2 ,··· ,in , that is, Vi1 ,i2 ,··· ,in = −bi1 · pi1 ,i2 ,··· ,in + ai1 ,i2 ,··· ,in .

(1)

Without loss of generality, the utility associated with the no-purchase option is conveniently set to zero, that is, VN = 0. According to the formula derived by Jiang et al. (2014), the probability to choose product hi1 , i2 , · · · , in i ∈ Nn can be expressed as follows qi1 ,i2 ,··· ,in =

n Y

m=1

P (hi1 , i2 , · · · , im−1 , im i|hi1 , i2 , · · · , im−1 i),

(2)


5 Scale parameter

Level h0i

0

1

2

h1i

h1, 1i

h2i

h1, 2i

hi1 i

hN i

µi1 ,i2 ,i3

µi1 ,i2 ,···,im

hi1 , i2 , · · · , im i

hi1 , i2 , · · · , im , 1i

h1, i2 , · · · , im , Di1 ,i2 ,···,im i

hi1 , i2 , · · · , im , 2i

n−2

µi1

µi1 ,i2

hi1 , i2 , i3 i

m

n−1

hD0 i

hi1 , i2 i

3

m+1

µ0

hi1 , i2 , · · · , in−2 i

µi1 ,i2 ,···,in−2

hi1 , i2 , · · · , in−2 , in−1 i

µi1 ,i2 ,···,in−1

n

hi1 , i2 , · · · , in−2 , in−1 , in i

Figure 1

The tree structure for a multilevel nested logit model. Note that the scale parameters are specific to the nests.

where P (hi1 i|h0i) =

exp(µ0 Vî1 ) D0 P i01 =1

(3)

exp(µ0 Vî01 ) + exp(µ0 VˆN )

and P (hi1 , i2 , · · · , im i|hi1 , i2 , · · · , im−1 i) =

exp(µi1 ,i2 ,··· ,im−1 Vî1 ,i2 ,··· ,im−1 ,im ) Di1 ,i2 ,··· ,im−1

P i0m =1

,

exp(µi1 ,i2 ,··· ,im−1 Vî1 ,i2 ,··· ,im−1 ,i0m ) ∀m ∈ {2, 3, · · · , n} ;

(4)


6

and the generalized utilities are given by  ! Di1 ,i2 ,··· ,im  P  1  , when 1 ≤ m ≤ n − 1; exp µi1 ,i2 ,··· ,im Vî1 ,i2 ,··· ,im+1  µi ,i ,··· ,i ln m 1 2 i =1 ˆ m+1 Vi1 ,i2 ,··· ,im =     Vi1 ,i2 ,··· ,in , when m = n.

(5)

and VˆN = VN . Let ci1 ,i2 ,··· ,in be the unit cost associated with product hi1 , i2 , · · · , in i and p be the set of prices

for all products, that is, p = {pi1 ,i2 ,··· ,in |hi1 , i2 , · · · , in i ∈ Nn }. The multiproduct price optimization problem under the multilevel nested logit model can be formulated as Equations (6) through (10), whose goal is to determine the prices for all products so as to maximize the expect total profit R(p): Di1 ,i2 ,··· ,in−1

D

(Formulation 1)

max R(p) =

i1 D0 X X

i1 =1 i2 =1

···

X in =1

(pi1 ,i2 ,··· ,in − ci1 ,i2 ,··· ,in ) · qi1 ,i2 ,··· ,in

(6)

subject to qi1 ,i2 ,··· ,in =

exp(µi1 ,i2 ,··· ,im−1 Vî1 ,i2 ,··· ,im ) , PDi1 ,i2 ,··· ,im D0 P î ,i ,··· ,i0 exp µ V m=2 0 ˆ i ,i ,··· ,i 1 2 m−1 1 2 im =1 m exp(µ0 Vi01 ) + 1 exp(µ0 Vî1 )

n Y

·

i01 =1

Vî1 ,i2 ,··· ,im =

1 µi1 ,i2 ,··· ,im



∀hi1 , i2 , · · · , in i ∈ Nn ;  exp µi1 ,i2 ,··· ,im Vî1 ,i2 ,··· ,im+1  ,

(7)

∀hi1 , i2 , · · · , im i ∈ Nm , m ∈ {1, 2, · · · , n − 1};

(8)

Di1 ,i2 ,··· ,im

X

ln 

im+1 =1

Vî1 ,i2 ,··· ,in = −bi1 pi1 ,i2 ,··· ,in + ai1 ,i2 ,··· ,in , pi1 ,i2 ,··· ,in ∈ R,

∀hi1 , i2 , · · · , in i ∈ Nn ;

∀hi1 , i2 , · · · , in i ∈ Nn .

(9) (10)

In the above formulation, Equation (7) is obtained by combining Equations (2) through (4) and the fact that VˆN = VN = 0. Equations (8) and (9) are obtained from Equation (5).

3. Concavity of the Profit Function It is a challenging task to maximize the profit function in Formulation 1 directly. Hanson and Martin (1996) show that R(p) is not quasi-concave in p even under the multinomial logit model. The authors develop a path-following procedure that does not require global concavity. The procedure works by creating a path of solutions between an easy to solve model and the difficult to solve true model. Later, Song and Xue (2007) and Dong et al. (2009) express the profit under the multinomial logit model as a function of the market share variables and show that it is jointly


7

concave. Recently, Li and Huh (2011) extend this result to the two-level nested logit model with identical price-sensitivity parameters within each nest. Their results, however, does not hold when the price sensitivities are product-specific (Gallego and Wang 2014). In this section, we show that the profit function is concave with respect to the market share variables under the multilevel nested logit model with identical price sensitivities in each primary nest, which immediately generalizes the result in Li and Huh (2011). Given a set of price variables p, the multilevel nested logit model can produce the corresponding set of market share variables, which is denoted as q, that is, q = {qi1 ,i2 ,··· ,in |hi1 , i2 , · · · , in i ∈ Nn }. Conversely, we can construct a mapping from the set of market share variables q to the set of price variables p as follows: pi1 ,i2 ,··· ,in (q) =

ai1 ,i2 ,··· ,in 1 + (ln qN − ln qi1 ,i2 ,··· ,in ) bi1 bi1 µi1 ,i2 ,··· ,in−1 n−1 X µi1 ,i2 ,··· ,im − µi1 ,i2 ,··· ,im−1 + (ln qN − ln qi1 ,i2 ,··· ,im ) , bi µi ,i ,··· ,im µi1 ,i2 ,··· ,im−1 m=1 1 1 2

(11)

where Di1 ,··· ,im ,im+1 ,im+2 ,··· ,in−1

Di1 ,i2 ,··· ,im Di1 ,i2 ,··· ,im ,im+1

X

X

im+1 =1

im+2 =1

qi1 ,i2 ,··· ,im =

···

X

qi1 ,i2 ,··· ,im ,im+1 ,im+2 ,··· ,in

(12)

in =1

is the market share for all products belonging to nest hi1 , i2 , · · · , im i ∈ Nm and 1 ≤ m ≤ n − 1; and PD qN = 1 − i10=1 qi1 is the market share of the no-purchase option. (For the proof of Equation (11), see Appendix C.1.) We can now re-write the profit function defined in Equation (6) in terms of the market share variables by plugging in Equation (11) and get Di1 ,··· ,in−1

D

R(q) =

i1 D0 X X

i1 =1 i2 =1

+

i1 =1 i2 =1

+

D0 Di1 X X i1 =1 i2 =1

··· ···

in =1

Di1 ,··· ,in−1

D

i1 D0 X X

···

X

X in =1

ln qN − ln qi1 ,i2 ,··· ,in · qi1 ,i2 ,··· ,in bi1 µi1 ,i2 ,··· ,in−1

Di1 ,··· ,in−1 n−1 X X in =1

ai1 ,i2 ,··· ,in − ci1 ,i2 ,··· ,in · qi1 ,i2 ,··· ,in bi1

ln qN − ln qi1 ,i2 ,··· ,im bi1 m=1

1 µi1 ,i2 ,··· ,im−1

−

1 µi1 ,i2 ,··· ,im

· qi1 ,i2 ,··· ,in .

(13)

In the literature of price optimization, there are a number of ways to prove the concavity of the profit function after it is expressed as a function of the market share variables. For example, under the multinomial logit model, Song and Xue (2007) and Dong et al. (2009) compute the Hessian matrix and show that it is negative semidefinite. Akcay et al. (2010) use the Lambert W function to solve the first order condition price in the multinomial logit model and show that the solution is unique and that the profit function is unimodal. Li and Huh (2011) decompose the


8

profit function into several components whose concavity can be independently proved. The authors establish the concavity of the profit function because concavity is preserved on summation (Boyd and Vandenberghe 2009). We follow the approach developed by Li and Huh (2011) and prove that the profit function is concave under the multilevel nested logit model in the following proposition. Proposition 1. The profit function R(q) defined in Equation (13) is joint concave in q. A direct benefit of the above proposition is that the profit maximization problem can now be solved fairly reliably and efficiently using convex optimization techniques, such as the interior point method, and we can be assured that the local maximum coincides with the global maximum. Once the optimal market share variables are determined, we can use Equation (11) to calculate the optimal price variables.

4. Unimodality of the Profit Function In this section, we investigate the properties of the profit function at optimality and develop a novel approach which simplifies the multidimensional profit function to an equivalent single-variable unimodal function. In Section 4.1, we show that the markups are constant across products within each primary nest. In Section 4.2, we show that the adjusted markups are constant across primary nests and that the profit function can be reduced to a single-variable unimodal function. In Section 4.3, we present results from a numerical example. 4.1. Constant Markup Within Each Primary Nest Consider product hi1 , i2 , · · · , in i ∈ Nn , its markup, denoted as θi1 ,i2 ,··· ,in , is defined as the difference between its price and cost. That is, θi1 ,i2 ,··· ,in = pi1 ,i2 ,··· ,in − ci1 ,i2 ,··· ,in . Let us also define si1 ,i2 ,··· ,in as si1 ,i2 ,··· ,in = ai1 ,i2 ,··· ,in − bi1 ci1 ,i2 ,··· ,in .

(14)

Let θ (n) = {θi1 ,i2 ,··· ,in |hi1 , i2 , · · · , in i ∈ Nn } be the set of markups for all products. We can re-write Equations (6) and (9) in Formulation 1 in terms of the markups and obtain the following equivalent formulation: Di1 ,i2 ,··· ,in−1

D

(Formulation 2)

max R(θ

(n)

)=

i1 D0 X X

i1 =1 i2 =1

···

X in =1

θi1 ,i2 ,··· ,in · qi1 ,i2 ,··· ,in

(15)

subject to qi1 ,i2 ,··· ,in =


·

n Y

i01 =1

∀hi1 , i2 , · · · , in i ∈ Nn ;

(16)


9





Di1 ,i2 ,··· ,im

Vî1 ,i2 ,··· ,im =

1 ln  µi1 ,i2 ,··· ,im

exp µi1 ,i2 ,··· ,im Vî1 ,i2 ,··· ,im+1  ,

X im+1 =1

∀hi1 , i2 , · · · , im i ∈ Nm , m ∈ {1, 2, · · · , n − 1}; (17)

Vî1 ,i2 ,··· ,in = −bi1 θi1 ,i2 ,··· ,in + si1 ,i2 ,··· ,in , θi1 ,i2 ,··· ,in ∈ R,

∀hi1 , i2 , · · · , in i ∈ Nn ;

∀hi1 , i2 , · · · , in i ∈ Nn .

(18) (19)

The following proposition says that for a given elementary nest hi1 , i2 , · · · , in−1 i ∈ Nn−1 , all products that belong to this elementary nest shall have a constant markup at optimality. Proposition 2. The markup, defined as price minus cost, is constant at optimality for all products in each elementary nest. Let θi1 ,i2 ,··· ,in−1 be the constant markup for products in elementary nest hi1 , i2 , · · · , in−1 i ∈

Nn−1 and θ (n−1) be the set of markups for all elementary nests, that is, θ (n−1) = {θi1 ,i2 ,··· ,in−1 |hi1 , i2 , · · · , in−1 i ∈ Nn−1 }. The profit function in Equation (15) can be re-written as Di1 ,i2 ,··· ,in−1

D

R(θ

(n−1)

)=

i1 D0 X X

i1 =1 i2 =1

=

i1 =1 i2 =1

θi1 ,i2 ,··· ,in−1 qi1 ,i2 ,··· ,in

in =1 Di1 ,i2 ,··· ,in−2

D

i1 D0 X X

X

···

X

···

in−1 =1

θi1 ,i2 ,··· ,in−1 · qi1 ,i2 ,··· ,in−1 .

(20)

In addition, the generalized utility for nest hi1 , i2 , · · · , in−1 i at level n − 1 can be expressed as a function of θ (n−1) as follows Vî1 ,i2 ,··· ,in−1 =

1 µi1 ,i2 ,··· ,in−1

Di1 ,i2 ,··· ,in−1

ln

X

exp µi1 ,i2 ,··· ,in−1 −bi1 θi1 ,i2 ,··· ,in−1 ,in + si1 ,i2 ,··· ,in−1 ,in

in

= −bi1 θi1 ,i2 ,··· ,in−1 +

Di1 ,i2 ,··· ,in−1

1 µi1 ,i2 ,··· ,in−1

ln

X

exp µi1 ,i2 ,··· ,in−1 si1 ,i2 ,··· ,in ,

(21)

in =1

where the first equality is due to the definition of Vî1 ,i2 ,··· ,in−1 and the second equality is obtained by plugging in the fact that θi1 ,i2 ,··· ,in = θi1 ,i2 ,··· ,in−1 for all in ∈ {1, 2, · · · , Di1 ,i2 ,··· ,in−1 }. Let us define the second term in Equation (21) as si1 ,i2 ,··· ,in−1 , that is, si1 ,i2 ,··· ,in−1 =

1 µi1 ,i2 ,··· ,in−1

Di1 ,i2 ,··· ,in−1

ln

X

exp µi1 ,i2 ,··· ,in−1 si1 ,i2 ,··· ,in .

(22)

in =1

In Formulation 2, if we replace Equation (15) with Equation (20) and replace Equation (18) with Equation (21), it can be re-written as Formulation 3 shown below, where θ (n−1) become the decision


10

variables. Note that Equations (24) and (25) are due to the definitions of the choice probability and the generalized utility, respectively. Di1 ,i2 ,··· ,in−2

D

(Formulation 3)

max R(θ

(n−1)

)=

i1 D0 X X

i1 =1 i2 =1

X

···

in−1 =1

θi1 ,i2 ,··· ,in−1 · qi1 ,i2 ,··· ,in−1

(23)

subject to qi1 ,i2 ,··· ,in−1 =

exp(µi1 ,i2 ,··· ,im−1 Vî1 ,i2 ,··· ,im ) , PDi1 ,i2 ,··· ,im D0 P î ,i ,··· ,i0 exp µ V m=2 0 ˆ i ,i ,··· ,i 1 2 m−1 1 2 i =1 m exp(µ0 Vi01 ) + 1 m exp(µ0 Vî1 )

·

n−1 Y

i01 =1

Vî1 ,i2 ,··· ,im =

1 µi1 ,i2 ,··· ,im

∀hi1 , i2 , · · · , in i ∈ Nn−1 ;  Di1 ,i2 ,··· ,im X ln  exp µi1 ,i2 ,··· ,im Vî1 ,i2 ,··· ,im+1  ,

(24)



im+1 =1

∀hi1 , i2 , · · · , im i ∈ Nm , m ∈ {1, 2, · · · , n − 2}; (25)

Vî1 ,i2 ,··· ,in−1 = −bi1 θi1 ,i2 ,··· ,in−1 + si1 ,i2 ,··· ,in−1 , θi1 ,i2 ,··· ,in−1 ∈ R,

∀hi1 , i2 , · · · , in−1 i ∈ Nn−1 ;

∀hi1 , i2 , · · · , in−1 i ∈ Nn−1 .

(26) (27)

It is worth-noting that Formulations 3 follows the same structure as Formulation 2, except that the number of decision variables is reduced from |Nn |, the number of products offered by the firm,

to |Nn−1 |, the number of nests at level n − 1. The following proposition further extends this idea and shows that the number of decision variables can be reduced to |N1 | = D0 . Let us generalize

Equation (22) and define si1 ,i2 ,··· ,im as follows si1 ,i2 ,··· ,im =

Di1 ,i2 ,··· ,im

1 µi1 ,i2 ,··· ,im

X

ln

exp µi1 ,i2 ,··· ,im si1 ,i2 ,··· ,im+1 ,

im+1 =1

∀hi1 , i2 , · · · , im i ∈ Nm , 1 ≤ m ≤ n − 1.

(28)

Proposition 3. The markup, defined as price minus cost, is constant at optimality for products in each primary nest. Let θi1 be the markup of all products in primary nest hi1 i and θ (1) be the set of markups for all primary nests, that is, θ (1) = {θi1 |hi1 i ∈ N1 }. Formulation 2 can be equivalently

written as max R(θ

(Formulation 4)

(1)

)=

D0 X i1 =1

θi1 · qi1

(29)

subject to qi1 =

exp(µ0 Vî1 ) D0 P i01 =1

exp(µ0 Vî01 ) + 1

Vî1 = −bi1 θi1 + si1 , θi1 ∈ R,

,

∀hi1 i ∈ N1 ;

∀hi1 i ∈ N1 ;

∀hi1 i ∈ Nn .

(30)

(31) (32)


11

The above proposition suggests the original price optimization problem that aims at determining the optimal prices for all products is equivalent to a markup optimization problem that aims at determining the optimal markup for each primary nest, which reduces the dimension of the search space to D0 . We also want to point out that it may appear that only the scale parameter of the root nest, that is, µ0 , enters the formulation and the other scale parameters do not. This is not true because the other scale parameters implicitly enter the formulation through si1 in Equation (31). 4.2. Constant Adjusted Markup Across Primary Nests Proposition 4. Consider a primary nest hi1 i ∈ N1 . Its adjusted markup, defined as φi1 = θi1 − 1 bi1 µ0

, is constant across primary nests at optimality, that is, φi1 = φ for all hi1 i ∈ N1 .

Consequently, we can re-write the profit function defined in Formulation 4 as D0 X 1 · qi1 , R(φ) = φ+ bi1 µ0 i =1

(33)

1

where qi1 = PD0

exp(µ0 Vî1 ) , exp(µ0 Vî0 ) + 1

i01 =1

and

1

∀hi1 i ∈ N1 ;

1 , ∀hi1 i ∈ N1 . Vî1 = −bi1 θi1 + si1 = −bi1 φ + si1 − µ0 The original multiproduct pricing problem is finally reduced to maximizing R(φ) with respect to

a single variable φ. The next proposition reveals an important property of R(φ): Proposition 5. R(φ) is strictly unimodal in φ. At its unique optimal solution φ∗ , we have R(φ∗ ) = φ∗ . The above proposition can speed up our search for the optimal φ considerably because the binary search procedure can be applied directly. Once the optimal value of φ is determined, we can obtain the optimal markup for product hi1 , i2 , · · · , in i using the following formula: θi1 ,i2 ,··· ,in = θi1 = φ +

1 . bi1 µ0

(34)

This formula suggests that at optimality, products in primary nests with larger price-sensitivity parameters shall have a lower markup than those in primary nests with smaller price-sensitivity parameters. Note that it may be tempting to conclude that the optimal markup for products in a primary nest is a decreasing function of µ0 , which, unfortunately, is not true because when µ0 changes, the optimal adjusted markup φ also changes. In Section 4.3, we give an example where the optimal markup for products in a primary nest first decreases and then increases when µ0 increases from zero.


12

Proposition 6. If the price sensitivities are identical across products, that is, if bi1 = b, for all hi1 i ∈ N1 , the optimal markup is constant across products and can be expressed as

θi1 ,i2 ,··· ,in =

1 . bµ0 qN

(35)

This generalizes the result in Kok and Xu (2011) under the two-level nested logit model with identical price sensitivities to the multilevel nested logit model. The optimal markup is a decreasing function of the price-sensitivity parameter, because when b increases, qN also increases. The optimal markup is also a decreasing function of qN . These results suggest that if customers are price sensitive and/or the total market share of the firm is low, this firm needs to lower its margin to maximize profitability. As is pointed out earlier, we shall avoid inferring the influence of µ0 over θi1 ,i2 ,··· ,in . 4.3. A Numerical Example To make Propositions 5 concrete, we solve the price optimization problem for the example shown in Figure (2). This figure shows the tree structure of a three-level nested logit model, where the root nest h0i has three children: two primary nests denoted as h1i and h2i; and the no-purchase option denoted as hN i. We have N1 = {h1i, h2i}, N2 = {h1, 1i, h1, 2i, h2, 1i, h2, 2i}, and N3 =

{h1, 1, 1i, h1, 1, 2i, h1, 1, 3i, h1, 2, 1i, h1, 2, 2i, h1, 2, 3i, h2, 1, 1i, h2, 1, 2i, h2, 1, 3i, h2, 2, 1i, h2, 2, 2i, h2, 2, 3i}.

The scale parameter associated with each nest is assumed to be identical for nests at the same

level. That is, µ0 = µ ˜0 = 1, µ1 = µ2 = µ ˜2 = 2 and µ1,1 = µ1,2 = µ2,1 = µ2,2 = µ ˜3 = 4. Note that µ ˜0 ≤ µ ˜1 ≤ µ ˜2 . Altogether, the firm offers 12 products. For product hi1 , i2 , i3 i ∈ N3 , the values

of ai1 ,i2 ,i3 and ci1 ,i2 ,i3 are reported at the bottom of the tree. For example, a1,1,1 = 14.5 and

c1,1,1 = 9.3. The price-sensitivity parameters for products in primary nests h1i and h2i are b1 = 0.7 and b2 = 1.1, respectively.

To determine the profit maximizing prices for these products, we need to first compute the value of si1 ,i2 ,··· ,im associated with nest hi1 , i2 , · · · , im i ∈ Nm ,where 1 ≤ m ≤ 3,

using Equations (14) and (28). Their values are reported in Table 1. For example, s1,1,1 = P3 1 −b1 · c1,1,1 + a1,1,1 = −0.7 × 9.3 + 14.5 = 7.99 and s1,1 = µ1,1 ln i3 =1 exp(µ1,1 s1,1,i3 ) = 14 × ln [exp(4 × 7.99) + exp(4 × 5.62) + exp(4 × 3.15)] = 7.99. According to Equation (33), we can reduce

the original price optimization problem with 12 price variables to a single variable maximization problem, whose formulation is given by Equation (36): max R(φ) = (φ + 1.43) q1 + (φ + 0.91) q2 , where q1 =

exp(Vˆ1 ) exp(Vˆ2 ) , q2 = , 1 + exp(Vˆ1 ) + exp(Vˆ2 ) 1 + exp(Vˆ1 ) + exp(Vˆ2 ) Vˆ1 = −0.7φ + 7.55, and Vˆ2 = −1.1φ + 3.40.

(36)


13 Scale parameter

Level h0i

0

µ ˜0 = 1

b2 = 1.1

b1 = 0.7 h1i

1

h2i

hN i

µ ˜1 = 2

No purchase

3 ai1 ,i2 ,i3 ci1 .i2 ,i3

Figure 2

h1, 2i

h1, 1i

2

h2, 1i

h1, 1, 1i h1, 1, 2ih1, 1, 3i h1, 2, 1i h1, 2, 2i h1, 2, 3i 14.5

10.8

7.0

15.0

11.0

8.5

9.3

7.4

5.5

9.5

7.5

6.3

µ ˜2 = 4

h2, 2i

h2, 1, 1i h2, 1, 2i h2, 1, 3i h2, 2, 1i h2, 2, 2i h2, 2, 3i 13.9

9.5

9.0

14.0

10.2

9.8

9.0

6.8

6.5

9.0

7.1

6.9

The tree structure for the illustrative example.

Level 3

Leve 2

Level 1

s1,1,1 = 7.99 s1,1,2 = 5.62

s1,1 = 7.99

s1,1,3 = 3.15

s1 = 8.55

s1,2,1 = 8.35 s1,2,2 = 5.75

s1,2 = 8.35

s1,2,3 = 4.09 s2,1,1 = 4.00 s2,1,2 = 2.02

s2,1 = 4.00

s2,1,3 = 1.85

s2 = 4.40

s2,2,1 = 4.10 s2,2,2 = 2.39

s2,2 = 4.10

s2,2,3 = 2.21 Table 1

The values of si1 ,i2 ,··· ,im in the numerical example (b1 = 0.7 and b2 = 1.1).

The optimal solution to Equation (36) can be easily solved using binary search. In Figure 3, we plot the profit function R(φ) and f (φ) = φ. We can clearly see that R(φ) is a strictly unimodal function as is stated in Proposition 5. when R(φ) and f (φ) intersect, that is, when φ = 8.27, R(φ) reaches its maximum value of 8.27. In addition, we observe that when φ < 8.27, R(φ) is above f (φ) and is strictly increasing; and when φ > 8.27, R(φ) is below f (φ) and is strictly decreasing. Figures 4 through 6 examine the sensitivity of the optimal markups and the optimal profit with respect to µ0 , VN , and b1 , respectively. In Figure 4, the scale parameter of the root nest, µ0 , is changed. Note that µ0 in this example needs to stay between 0 and 2 so as to be consistent with the assumptions of the multilevel nested logit model. The dashed line refers to φ, the optimal adjusted markup. Recall that R(φ) = φ at optimality, therefore, the dashed line also corresponds to the


14

20

25 φ, R θ1 θ2

R(φ) 15 20 10 ←− (8.27,8.27)

15

5 10 0

−5 −5

0

5

10

15

20

5

0

0.5

1 µ0

φ

Figure 3

Unimodality of R(φ).

Figure 4

20

1.5

2

Sensitivity with respect to µ0 .

20 φ, R θ1 θ2

φ, R θ1 θ2

15 15 10 10 5 5 0

−5 −5

0

5

10

15

20

0

0.5

VN

Figure 5

Sensitivity with respect to VN .

1

1.5

2

b1

Figure 6

Sensitivity with respect to b1 .

optimal profit. The thin solid line is the optimal markup for products in primary nest h1i and the thick solid line is that for products in primary nest h2i. We observe that for a given µ0 , θ1 is greater

than θ2 , which is consistent with what Equation (34) implies because b1 < b2 . We also notice that when µ0 increases from 0 to 2, that is, when the similarity among products within each primary nests weakens, the optimal markups, that is, θ1 and θ2 , first decrease and then increase, which is consistent with our comment earlier that the markups are not necessarily decreasing functions of µ0 . Figure 5 shows the sensitivity with respect to VN , the utility of the no-purchase option. When VN increases, consumers get more attracted by the no-purchase option. Therefore, the firm cannot


15

afford a high margin, which is evidenced by the decline in markups in this figure. The optimal profit decreases almost linearly when −5 < VN < 5 and gradually diminishes to zero when VN exceeds 10. Note that the three curves are sort of “parallel” to each other because the difference between φ and θ1 (or θ2 ) corresponds to the reciprocal of the product between µ0 and b1 (or b2 ), which remains as a constant when VN changes. Figure 6 shows the sensitivity with respect to b1 , the price-sensitivity parameter in primary nest h1i. When b1 is smaller than b2 , we have θ1 > θ2 , which implies that products in nest h1i can afford

a higher margin. When b1 = b2 , the markups in both nests are the same; and when b1 exceeds b2 , the markup in nest h1i is smaller than that in nest h2i. Note that when b1 increases, R, the total profit, declines sharply in the beginning and then stays nearly flat. This is understandable because when we examine the values of ai1 ,i2 ,i3 at the bottom of Figure 2, we notice that their values, or the quality of the products, in both nests are similar. When b1 is smaller than b2 , customers are less sensitive to the prices for the products in nest h1i. Therefore, the firm can set higher markups for products in this nest, and this nest contributes substantially toward the total profit. When b1 increases from 0.4 to 1.1, the profit coming from nest h1i gets negatively impacted, which translates to sharp loss in total profit. When b1 exceeds b2 and continues to increase, products in nest h1i would have very limited market share and their profit becomes negligible. At the same time, nest h2i gradually grows as the profit generator and this explains why the profit function stays flat when

b1 far exceeds b2 .

5. Applications In this section, we show how results obtained earlier can be applied. In Section 5.1, we study the oligopolistic price competition and prove the uniqueness of the Nash equilibrium. In Section 5.2, we develop a dimension reduction technique which can simplify pricing problems with flexible price-sensitivity structures, that is, when the price-sensitivity parameters are not identical within each primary nest. 5.1. Oligopolistic Game In the oligopolistic game under the multilevel nested logit model, each firm own a primary nest and the associated products in that nest. In our clothing store example, it means that there is an independent manager who makes pricing decisions for each brand. The brands are in direct competition with each other because the expected profit of one brand depends on the other brands’ pricing decisions. Existing literature on oligopolistic games primarily focuses on the multinomial logit model (Gallego et al. 2006, Aksoy-Pierson et al. 2013). Under the two-level nested logit model, Liu (2006) and Li and Huh (2011) study price competition with nest-specific price sensitivities,


16

while Gallego and Wang (2014) investigate the situation where product-specific price sensitivities are allowed. In our oligopolistic game, each primary nest corresponds to a firm and they are referred to as firms 1, 2, · · · , and D0 . For firm i1 ∈ {1, 2, · · · , D0 }, it decides the prices of all products under its control, that is, products belonging to nest hi1 i, to maximize its expected profit. The profit, or payoff, function for firm i1 can be written as D

Ri1 (pi1 , p−i1 ) =

Di1 ,··· ,in−1

D

i1 i1 ,i2 X X

i2 =1 i3 =1

···

X in =1

(pi1 ,i2 ,··· ,in − ci1 ,i2 ,··· ,in ) · qi1 ,i2 ,··· ,in (pi1 , p−i1 ),

where pi1 is the set of prices for products offered by firm i1 , that is, pi1 = {pi1 ,i2 ,i3 ,··· ,in |i2 ∈ {1, 2, · · · , Di1 }, i3 ∈ {1, 2, · · · , Di1 ,i2 }, · · · , in ∈ {1, 2, · · · , Di1 ,i2 ,i3 ,··· ,in−1 }}, and p−i1 is the set of prices

for products offered by all other firms, that is, p−i1 = {pi01 ,i02 ,i03 ,··· ,i0n |i01 ∈ {1, 2, · · · , D0 } \ {i1 }, i02 ∈ {1, 2, · · · , Di01 }, i03 ∈ {1, 2, · · · , Di01 ,i02 }, · · · , i0n ∈ {1, 2, · · · , Di01 ,i02 ,i03 ,··· ,i0n−1 }}.

According to Proposition 4, the price optimization problem for each firm can be reduced to a single decision variable, the constant markup for all products offered by that firm. Let θi1 be the constant markup for products offered by firm i1 , and θ −i1 be the set of constant markups for products offered by all other firms, that is, θ −i1 = {θi01 |i01 ∈ {1, 2, · · · , D0 } \ {i1 }}. We can simplify the profit function for firm i1 as follows Ri1 (θi1 , θ −i1 ) = θi1 · qi1 (θi1 , θ −i1 ) = θi1 ·

exp(µ0 Vî1 ) , D0 P ˆ exp(µ0 Vi01 ) 1+

(37)

i01 =1

where Vî01 = −bi01 θi01 + si01 ,

∀i01 ∈ {1, 2, · · · , D0 }.

For the above oligopolistic game, we have the following results: Proposition 7. The oligopolistic game is strictly log-supermodular and has a unique Nash equilibrium.

5.2. The Case of Flexible Price Sensitivities So far, we have assumed that the price sensitivities are identical for all products within each primary nest. In situations when flexible price sensitivities are desired, that is, when the price sensitivities vary within one or more of the primary nests, we can use the following proposition to reduce the complexity of the pricing problem.


17

Proposition 8. Consider products within nest hi1 , i2 , · · · , im i ∈ Nm , where m ∈ {0, 1, · · · , n − 1}, if their price sensitivities are identical and is denoted as bi1 ,i2 ,··· ,im , their markups are constant at optimality. If we denote this constant markup as θi1 ,i2 ,··· ,im , the profit contributed by these products can be equivalently replaced by a pseudo product whose price is θi1 ,i2 ,··· ,im , cost is zero, and systematic utility is Vi1 ,i2 ,··· ,im = −bi1 ,i2 ,··· ,im θi1 ,i2 ,··· ,im + si1 ,i2 ,··· ,im . The proof of the above proposition largely follows that of Proposition 3 and is available from the authors upon request. Let us return to the example shown in Figure 2 and assume that the price-sensitivity parameters for nests h1, 1i and h1, 2i are 0.7 and 0.5, respectively. This means that we no longer have identical price-sensitivity parameters for products in primary nest h1i. In this situation, we can leverage the above proposition to simplify the price optimization problem. Since the price-sensitivity parameters are identical within nests h1, 1i, h1, 2i, and h2i, respectively, we can replace products under each nest with a pseudo product. Table 2 shows the detailed steps that calculate the values of si1 ,i2 ,··· ,im . Note that si1 is not defined because products in nest h1i have different price-sensitivity parameters. The original tree structure can then be simplified to the one shown in Figure 7, which corresponds to a two-level nested logit model. Level 3

Leve 2

Level 1

s1,1,1 = 7.99 s1,1,2 = 5.62

s1,1 = 7.99

s1,1,3 = 3.15

—

s1,2,1 = 10.25 s1,2,2 = 7.25

s1,2 = 10.25

s1,2,3 = 5.35 s2,1,1 = 4.00 s2,1,2 = 2.02

s2,1 = 4.00

s2,1,3 = 1.85

s2 = 4.40

s2,2,1 = 4.10 s2,2,2 = 2.39

s2,2 = 4.10

s2,2,3 = 2.21 Table 2

The values of si1 ,i2 ,··· ,im in the numerical example (b1,1 = 0.7, b1,2 = 0.5, and b2 = 1.1).

The dimension reduction technique are useful in two ways: a) The simplified problem contains less number of decision variables, which shall reduce the size of the optimization problem and facilitate the solution process; and b) When the tree structure of the simplified problem becomes a multinomial logit or a two-level nested logit model, we can benefit from results in existing literature. For example, the simplified price optimization problem shown in Figure 7 corresponds to a two-level nested logit model with product-specific price sensitivities, which means that findings in


18

Scale parameter

Level h0i

0

b2 = 1.1

h1i

1

b1,1 = 0.7

µ ˜0 = 1

h2i b1,2 = 0.5

V2 = −1.1θ2 + 4.40

hN i No purchase

µ ˜1 = 2

2 h1, 1i

V1,1 = −0.7θ1,1 + 7.99

Figure 7

h1, 2i V1,2 = −0.5θ1,2 + 10.25

The tree structure for the simplified price optimization problem.

Gallego and Wang (2014) can be adopted directly. Since the majority of existing empirical studies, particularly those involving the three-level nested logit model, produce discrete choice models with identical price-sensitivity parameters either across products (Coldren and Koppelman 2005, Jiang 2009, Hsiao and Hansen 2011) or across product categories (Erdem et al. 2002), the proposed dimension reduction technique can be applied to many practical problems.

6. Conclusions We study the multiproduct price optimization problem under the multilevel nested logit model, which includes the multinomial logit and the two-level nested logit models as special cases. When the price sensitivities are identical within each primary nest, that is, within each nest at level 1, we show that there is a one-to-one mapping from the price variables to the market share variables and the profit function is concave with respect to the market share variables. We proceed to investigate the properties of the markups at optimality and show that: a) the markup, defined as price minus cost, is constant for products within each primary nest at optimality; and b) the adjusted markup, defined as price minus cost minus the reciprocal of the product between the scale parameter of the root nest and the price-sensitivity parameter of the primary nest, is constant across primary nests, which allows us to reduce the multidimensional pricing problem to an equivalent singlevariable maximization problem involving a unimodal function. We use these results to investigate the oligopolistic game and show that it is equivalent to a log-supermodular game and the Nash equilibrium is unique. In situations where flexible price-sensitivity structures are desired, that is, when the price sensitivities are no longer identical within one or more primary nests, we develop a dimension reduction technique that can simplify the corresponding pricing problems. Appendix A: Notation In this section, we summarize key notation used throughout the paper for ease of reading:


19

n

The number of levels in the tree structure for the multi-level nested logit model.

h0i

The root nest of the tree.

hi1 , i2 , · · · , im i

This is the identifier for a nest in the tree. When m = 0, it is defined as h0i.

hN i

The no-purchase option in the tree. When m = n, it becomes a leaf, or a degenerated nest with no children, and refers to a specific product.

Nm

The set of nests at level m, that is, Nm = {hi1 , i2 , · · · , im i|i1 ∈ {1, 2, · · · , D0 }, i2 ∈ {1, 2, · · · , Di1 }, · · · , im ∈ {1, 2, · · · , Di1 ,i2 ,··· ,im−1 } . When m = n, Nn refers to all the products.

Di1 ,i2 ,··· ,im µi1 ,i2 ,··· ,im ai1 ,i2 ,··· ,in pi1 ,i2 ,··· ,in bi1

The number of children nests belonging to nest hi1 , i2 , · · · , im i. The scale parameter associated with nest hi1 , i2 , · · · , im i.

The quality, or the price-independent utility, associated with product hi1 , i2 ,

· · · , in i.

The selling price of product hi1 , i2 , · · · , in i.

The price-sensitivity parameter for products in primary nest hi1 i ∈ N1 . Note that bi1 is positive.

ci1 ,i2 ,··· ,in θi1 ,i2 ,··· ,in Vi1 ,i2 ,··· ,in VN Vî ,i

1 2 ,··· ,im

VˆN qi1 ,i2 ,··· ,in qi1 ,i2 ,··· ,im qN θi1 ,i2 ,··· ,im φ Bi1 ,i2 ,··· ,im

The cost of product hi1 , i2 , · · · , in i.

= pi1 ,i2 ,··· ,in − ci1 ,i2 ,··· ,in , which is the markup for product hi1 , i2 , · · · , in i.

= −bi1 · pi1 ,i2 ,··· ,in + ai1 ,i2 ,··· ,in , which is the systematic utility for product hi1 , i2 , · · · , in i.

The systematic utility of the no-purchase option. The generalized utility for nest hi1 , i2 , · · · , im i.

The generalized utility for the no-purchase option. The market share of product hi1 , i2 , · · · , in i.

Defined in Equation (12), which is the market share for the products belonging to nest hi1 , i2 , · · · , im i. PD = 1 − i10=1 qi1 , the market share of the no-purchase option.

The constant markup for products belonging to nest hi1 , i2 , · · · , im i.

The constant adjusted markup across primary nests. We have φ = θi1 − bi 1µ0 for 1

all hi1 i ∈ N1 .

= P (hi1 , i2 , · · · , im−1 , im i|hi1 , i2 , · · · , im−1 i), which is the conditional market share

of nest hi1 , i2 , · · · , im i given nest hi1 , i2 , · · · , im−1 i. Note that we have Bi1 ,i2 ,··· ,im = i0 ,i0 ,··· ,i0 Λi11 ,i22 ,··· ,inn ,··· ,im ,··· ,in Ωii11,··· ,im

Ai1 ,i2 ,··· ,im

qi1 ,i2 ,··· ,im /qi1 ,i2 ,··· ,im−1 .

Defined in Equation (38). Defined in Equation (39). Defined in Equation (42).


20

Appendix B: Direct and Cross Price Elasticities of the Multilevel Nested Logit Model We derive the direct and cross price elasticities under the multilevel nested logit model in this section. We assume that the systematic utility is specified as that in Equation (1). Consider product hi1 , i2 , · · · , in i, the direct price elasticity measures the responsiveness of its market share to a change

in its own price, which is defined as qi ,i ,··· ,i

Epi11,i22,··· ,inn =

∂qi1 ,i2 ,··· ,in pi1 ,i2 ,··· ,in · . ∂pi1 ,i2 ,··· ,in qi1 ,i2 ,··· ,in

The cross price elasticity involves another product, denoted as hi01 , i02 , · · · , i0n i, and measures the responsiveness of its market share to a change in the price of product hi1 , i2 , · · · , in ip, which is defined as qi0 ,i0 ,··· ,i0

Epi11,i22,··· ,inn =

∂qi01 ,i02 ,··· ,i0n pi1 ,i2 ,··· ,in · . ∂pi1 ,i2 ,··· ,in qi01 ,i02 ,··· ,i0n

In addition, for these two products hi1 , i2 , · · · , in i and hi01 , i02 , · · · , i0n i, define i0 ,i0 ,··· ,i0

Λi11 ,i22 ,··· ,inn = l ∈ {0, 1, 2, · · · , n − 1} ,

(38)

where hi1 , i2 , · · · , il i = hi01 , i02 , · · · , i0l i and il+1 6= i0l+1 . It means that they both belong to

nest hi1 , i2 , · · · , il i, but then diverge into two different children nests hi1 , i2 , · · · , il , il+1 i and

hi1 , i2 , · · · , il , i0l+1 i, respectively. Let us further define  m −1 2  Q P hi1 , i2 , · · · , im0 , im0 +1 i|hi1 , i2 , · · · , im0 i, when 0 ≤ m1 ≤ m2 ≤ n − 1; i1 ,··· ,im1 ,··· ,im2 Ωi1 ,··· ,im1 = m0 =m1 (39)  1, when m1 = m2 .

This definition suggests that Ω0i1 ,...,in = P (hi1 , · · · , in i). The direct and cross price elasticities of the multilevel nested logit model are expressed in Equations (40) and (41), whose proofs can be found in Sections C.9 and C.10, respectively. qi ,i ,··· ,i

Epi11,i22,··· ,inn = − bi1 pi1 ,i2 ,··· ,in = − bi1 pi1 ,i2 ,··· ,in

n X

i1 ,i2 ,··· ,in ,i2 ,··· ,in µi1 ,··· ,im−1 Ωii11,··· − Ω i1 ,··· ,im−1 ,im−1 ,im

m=1 " n−1 X

m=1

,··· ,in µi1 ,i2 ,··· ,im−1 − µi1 ,i2 ,··· ,im Ωii11,i,i22,··· ,im # − µ0 qi1 ,i2 ,··· ,in + µi1 ,i2 ,··· ,in−1

(40)

and qi0 ,i0 ,··· ,i0

Epi11,i22,··· ,inn = −bi1 pi1 ,i2 ,··· ,in i0 ,i0 ,··· ,i0

where l = Λi11 ,i22 ,··· ,inn .

"

l X m=1

# ,··· ,in µi1 ,i2 ,··· ,im−1 − µi1 ,i2 ,··· ,im Ωii11,i,i22,··· ,im − µ0 qi1 ,i2 ,··· ,in ,

(41)


21

Appendix C: Proofs of Equations and Propositions C.1. Proof of Equation (11) Proof:

For ease of presentation, let us define Ai1 ,i2 ,··· ,im , where m ∈ {0, 1, · · · , n − 1}, as D i1 ,i2 ,··· ,im P    exp(µi1 ,i2 ,··· ,im Vî1 ,i2 ,··· ,im ,im+1 ), when 1 ≤ m ≤ n − 1;    im+1 =1 Ai1 ,i2 ,··· ,im =  D0  P    when m = 0. exp(µ0 Vî01 ) + 1, 

(42)

i01 =1

and Bi1 ,i2 ,··· ,im , where m ∈ {1, 2, · · · , n} as Bi1 ,i2 ,··· ,im =P (hi1 , i2 , · · · , im−1 , im i|hi1 , i2 , · · · , im−1 i). We then have Bi1 ,i2 ,··· ,im =

exp(µi1 ,i2 ,··· ,im−1 Vî1 ,i2 ,··· ,im−1 ,im ) Ai1 ,i2 ,··· ,im−1 " !# Di1 ,i2 ,··· ,im−1 ,im P µi1 ,i2 ,··· ,im−1 exp µi ,i ,··· ,i ln exp µi1 ,i2 ,··· ,im Vî1 ,i2 ,··· ,im ,im+1 ,i 1 2

m−1 m

im+1 =1

=

Ai1 ,i2 ,··· ,im−1 µi ,i ,··· ,i 1 2 m−1 µi ,i ,··· ,im 1 2

=

Ai1 ,i2 ,··· ,im , Ai1 ,i2 ,··· ,im−1

(43)

where the first equality is due to the definitions of Bi1 ,i2 ,··· ,im and Ai1 ,i2 ,··· ,im−1 ; the second equality is due to the definition of Vî1 ,i2 ,··· ,im−1 ,im in Equation (5); and the third equality is due to the definition of Ai1 ,i2 ,··· ,im . Re-arrange the terms in this equation, we can get µi ,i ,··· ,im 1 2 µi ,i ,··· ,i 1 2 m−1 i1 ,i2 ,··· ,im−1

µi ,i ,··· ,im 1 2 µi ,i ,··· ,i 1 2 m−1

·A

Ai1 ,i2 ,··· ,im = Bi1 ,i2 ,··· ,im

,

which, when applied recursively, can help us obtain the following result: µi ,i ,··· ,i 1 2 n−1 µi ,i ,··· ,i 1 2 n−2

µi ,i ,··· ,i 1 2 n−1 µi ,i ,··· ,i 1 2 n−2 i1 ,i2 ,··· ,in−2

Ai1 ,i2 ,··· ,in−1 =Bi1 ,i2 ,··· ,in−1 · A

n−1 Y

=

µi ,i ,··· ,i 1 2 n−1 µi ,i ,··· ,i 1 2 m−1 i1 ,i2 ,··· ,im

B

m=2

·A

µi ,i ,··· ,i 1 2 n−1 µi 1 i1

.

Therefore, we have exp(µi1 ,i2 ,··· ,in−1 Vî1 ,i2 ,··· ,in−1 ,in ) =Bi1 ,i2 ,··· ,in Ai1 ,i2 ,··· ,in−1 =Bi1 ,i2 ,··· ,in

n−1 Y

µi ,i ,··· ,i 1 2 n−1 µi ,i ,··· ,i 1 2 m−1 i1 ,i2 ,··· ,im

B

m=2

=

n Y m=2

µi ,i ,··· ,i 1 2 n−1 µi ,i ,··· ,i 1 2 m−1

Bi1 ,i2 ,··· ,im

µi ,i ,··· ,i 1 2 n−1 µi 1 i1

·A

µi ,i ,··· ,i 1 2 n−1 µi 1 i1

·A

,in−1 µi1 ,i2 ,··· ,in−1 µi1 ,i2 ,··· n µ0 Y qi1 ,i2 ,··· ,im−1 ,im µi1 ,i2 ,··· ,im−1 qi1 = , qi1 ,i2 ,··· ,im−1 qN m=2

(44)


22

where the first equality is due to the definitions of Bi1 ,i2 ,··· ,in and Ai1 ,i2 ,··· ,in−1 ; the second equality is obtained by plugging in the result in Equation (44); and the fourth equality is due to the fact and qi /qN = exp(µ0 Vî )/exp(µ0 VˆN ) = exp(µ0 Vî ) = that Bi ,i ,··· ,im = qi ,i ,··· ,i ,im /qi ,i ,··· ,i 1 2

1 2

exp( µµi0 1

ln Ai1 ) =

µ0 /µi Ai1 1 .

m−1

1 2

m−1

1

1

1

Finally, let us plug the fact that Vî1 ,i2 ,··· ,in−1 ,in = −bi1 pi1 ,i2 ,··· ,in + ai1 ,i2 ,··· ,in

into the L.H.S. of the above equation, take the logarithm on both sides, re-arrange terms, and get n

X ai ,i ,··· ,i 1 1 pi1 ,i2 ,··· ,in = 1 2 n + ln qi1 ,i2 ,··· ,im−1 − ln qi1 ,i2 ,··· ,im + (ln qN − ln qi1 ) bi1 bi µi ,i ,··· ,im−1 bi1 µ0 m=2 1 1 2 =

1 ai1 ,i2 ,··· ,in + (ln qN − ln qi1 ,i2 ,··· ,in ) bi1 bi1 µi1 ,i2 ,··· ,in−1 n−1 X 1 1 1 + − (ln qN − ln qi1 ,i2 ,··· ,im ) . bi µi1 ,i2 ,··· ,im−1 µi1 ,i2 ,··· ,im m=1 1

This completes the proof.

C.2. Proof of Proposition 1 Proof:

Before we prove the concavity of Equation (13), let us first prove that f (x, y) = x(ln y −

ln x) is concave over the region {(x, y)|0 < x, y < 1 and x + y ≤ 1}. Since the Hessian of f (x, y) is 1 1 −x y H(f ) = 1 − yx2 y and for any two real numbers α and β, we have √ 2 α β x α α β H(f ) =− √ − ≤ 0, β y x therefore H(f ) is negative semidefinite and f (x, y) is concave.

Now, let us prove that each of the three components in Equation (13) is concave. This ensures that the profit function is concave because concavity is preserved on summation (Boyd and Vandenberghe 2009). The first component is concave because it is linear. The second component is also concave because (ln qN − ln qi1 ,i2 ,··· ,in ) · qi1 ,i2 ,··· ,in is concave. For the third component, let us re-write it as follows: Di1 ,i2 ,··· ,im−1 Di ,i ,··· ,im 1 2

D

i1 D0 n−1 X X X

m=1 i1 =1 i2 =1

···

X

X

im =1

im+1 =1

Di1 ,i2 ,··· ,in−1

···

X in =1

ln qN − ln qi1 ,i2 ,··· ,im bi1

· Di1 ,i2 ,··· ,im−1

D

=

i1 D0 n−1 X X X

m=1 i1 =1 i2 =1

···

X im =1

ln qN − ln qi1 ,i2 ,··· ,im · bi1

1 µi1 ,i2 ,··· ,im−1 1

−

1

µi1 ,i2 ,··· ,im 1

· qi1 ,i2 ,··· ,im ,··· ,in

− µi1 ,i2 ,··· ,im−1 µi1 ,i2 ,··· ,im   Di1 ,i2 ,··· ,im Di1 ,i2 ,··· ,in X X · ··· qi1 ,i2 ,··· ,im ,im+1 ,··· ,in  im+1 =1

in

Jiang, Chen, and Sun: Multiproduct Pricing Under the Multilevel Nested Logit Model c 0000 xxxxxxx Working Paper 00(0), pp. 000–000, Di1 ,i2 ,··· ,im−1

D

=

i1 D0 n−1 X X X

m=1 i1 =1 i2 =1

X

···

im =1

1 bi1

1 µi1 ,i2 ,··· ,im−1

−

23

1

(ln qN − ln qi1 ,i2 ,··· ,im ) qi1 ,i2 ,··· ,im .

µi1 ,i2 ,··· ,im

In the above equation, (ln qN − ln qi1 ,i2 ,··· ,im )qi1 ,i2 ,··· ,im is concave, therefore the third component is concave.


Consider product hi1 , i2 , · · · , in i ∈ Nn , we can re-write the profit function defined in

Equation (15) as follows Di1 ,i2 ,··· ,in−1

R(θ

(n)

X

) =θi1 ,i2 ,··· ,in qi1 ,i2 ,··· ,in +

θi1 ,i2 ,··· ,in−1 ,i0n qi1 ,i2 ,··· ,in−1 ,i0n

i0n =1,i0n 6=in Di0 ,i0 ,··· ,i0

D0

+

i1 D0 X X

1 2

i01 =1 i02 =1

···

Di0 ,i0 ,··· ,i0

n−2

1 2

n−1

X

X

i0n−1 =1

i0n =1

θi01 ,i02 ,··· ,i0n qi01 ,i02 ,··· ,i0n ,

hi01 ,i02 ,··· ,i0n−1 i6=hi1 ,i2 ,··· ,in−1 i

where the first term is the profit from product hi1 , i2 , · · · , in i, the second term is the profit from

the other products in elementary nest hi1 , i2 , · · · , in−1 i, and the third term is the profit from the remaining products. Therefore, the partial derivative of R(θ (n) ) with respect to θi1 ,i2 ,··· ,in can be expressed as Di1 ,i2X ,··· ,in−1 ∂qi ,i ,··· ,i ,i0 ∂qi1 ,i2 ,··· ,in ∂R(θ (n) ) = qi1 ,i2 ,··· ,in + θi1 ,i2 ,··· ,in + θi1 ,i2 ,··· ,in−1 ,i0n 1 2 n−1 n ∂θi1 ,i2 ,··· ,in ∂θi1 ,i2 ,··· ,in ∂θi1 ,i2 ,··· ,in 0 0 in =1,in 6=in

D0

+

Di0 ,i0 ,··· ,i0

Di0 ,i0 ,··· ,i0

i1 D0 X X

1 2

i01 =1 i02 =1

···

n−1

1 2

n−2

X

X

i0n−1 =1

i0n =1

θi01 ,i02 ,··· ,i0n

∂qi01 ,i02 ,··· ,i0n ∂θi1 ,i2 ,··· ,in

.

(45)

hi01 ,i02 ,··· ,i0n−1 i6=hi1 ,i2 ,··· ,in−1 i i0 ,i0 ,··· ,i0

Consider the last term in the above equation and let l = Λi11 ,i22 ,··· ,inn , we have ∂qi01 ,i02 ,··· ,i0n ∂θi1 ,i2 ,··· ,in

=

qi01 ,i02 ,··· ,i0n θi1 ,i2 ,··· ,in

=−

qi0 ,i0 ,··· ,i0

Eθi 1,i 2,··· ,inn 1 2

"

qi01 ,i02 ,··· ,i0n

#

l X

,··· ,in Ωii11,i,i22,··· ,im

bi θi ,i ,··· ,i µi ,i ,··· ,i − µi1 ,i2 ,··· ,im − µ0 qi1 ,i2 ,··· ,in θi1 ,i2 ,··· ,in 1 1 2 n m=1 1 2 m−1 " l # X i1 ,i2 ,··· ,in = −bi1 µi1 ,i2 ,··· ,im−1 − µi1 ,i2 ,··· ,im Ωi1 ,i2 ,··· ,im qi01 ,i02 ,··· ,i0n − µ0 qi1 ,i2 ,··· ,in qi01 ,i02 ,··· ,i0n = −bi1

"m=1 l X m=1

µi1 ,i2 ,··· ,im−1 − µi1 ,i2 ,··· ,im "

= −bi1 qi1 ,i2 ,··· ,in

l X m=1

i0 ,i0 ,··· ,i0n Ωi11,i22,··· ,im qi1 ,i2 ,··· ,in

µi1 ,i2 ,··· ,im−1 − µi1 ,i2 ,··· ,im

i0 ,i0 ,··· ,i0n Ωi11,i22,··· ,im

# − µ0 qi1 ,i2 ,··· ,in qi01 ,i02 ,··· ,i0n

# − µ0 qi01 ,i02 ,··· ,i0n ,


24

where the first equality is due to the definition of cross elasticity; the second equality is due to i0 ,i0 ,··· ,i0

,··· ,in n 1 2 Equation (41); the fourth equality is due to qi1 ,i2 ,··· ,in /Ωii11,i,i22,··· ,im = qi01 ,i02 ,··· ,i0n /Ωi1 ,i2 ,··· ,im for all

m ≤ l. As a result, we can re-write the last term in Equation (45) as   Di0 ,i0 ,··· ,i0 Di0 ,i0 ,··· ,i0  Di0  D0 1 2 1 2 1 X X X n−2 X n−1 − bi1 qi1 ,i2 ,··· ,in ··· θi01 ,i02 ,··· ,i0n   0 0 0 0 i =1 i1 =1 i2 =1 in−1 =1 n  hi01 ,i02 ,··· ,i0n−1 i6=hi1 ,i2 ,··· ,in−1 i

" ·

l X

m=1

i0 ,i0 ,··· ,i0n µi1 ,i2 ,··· ,im−1 − µi1 ,i2 ,··· ,im Ωi11,i22,··· ,im − µ0 qi01 ,i02 ,··· ,i0n

 #  

.

  

In the above equation, the expressions inside the curly braces is a function of hi1 , i2 , · · · , in−1 i,

which we denote as F (i1 , i2 , · · · , in−1 ). Therefore, Equation (45) can be re-written as

Di1 ,i2X ,··· ,in−1 ∂qi ,i ,··· ,i ,i0 ∂qi1 ,i2 ,··· ,in ∂R(θ (n) ) = qi1 ,i2 ,··· ,in + θi1 ,i2 ,··· ,in + θi1 ,i2 ,··· ,in−1 ,i0n 1 2 n−1 n ∂θi1 ,i2 ,··· ,in ∂θi1 ,i2 ,··· ,in ∂θi1 ,i2 ,··· ,in 0 0 in =1,in 6=in

− bi1 qi1 ,i2 ,··· ,in · F (i1 , i2 , · · · , in−1 ).

Since

∂qi1 ,i2 ,··· ,in ∂θi1 ,i2 ,··· ,in

=

qi1 ,i2 ,··· ,in θi1 ,i2 ,··· ,in

qi ,i ,··· ,i

· Eθi 1,i 2,··· ,inn and 1 2

(46)

∂qi ,i ,··· ,i 0 1 2 n−1 ,in ∂θi1 ,i2 ,··· ,in

=

qi ,i ,··· ,i 0 1 2 n−1 ,in θi1 ,i2 ,··· ,in

qi ,i ,··· ,i

· Eθi 1,i 2,··· ,inn−1

,i0n

1 2

by the

definitions of direct and cross price elasticities, we plug the results in Equations (40) and (41) into Equation (46) and get ∂R(θ (n) ) ∂θi1 ,i2 ,··· ,in = qi1 ,i2 ,··· ,in − bi1 θi1 ,i2 ,··· ,in qi1 ,i2 ,··· ,in

n−1 X m=1

,··· ,in µi1 ,i2 ,··· ,im−1 − µi1 ,i2 ,··· ,im Ωii11,i,i22,··· ,im

(47)

! + bi1 µ0 θi1 ,i2 ,··· ,in qi21 ,i2 ,··· ,in − bi1 µi1 ,i2 ,··· ,in−1 θi1 ,i2 ,··· ,in qi1 ,i2 ,··· ,in 

Di1 ,i2 ,··· ,in−1

+ −

X

bi1 θi1 ,i2 ,··· ,i0n qi1 ,i2 ,··· ,in

i0n =1,i0n 6=in

n−1 X

i ,i ,··· ,i0n µi1 ,i2 ,··· ,im−1 − µi1 ,i2 ,··· ,im Ωi11,i22,··· ,im

m=1

+

(49)



Di1 ,i2 ,··· ,in−1

X

(48)

bi1 µ0 θi1 ,i2 ,··· ,in−1 ,i0n qi1 ,i2 ,··· ,in−1 ,in qi1 ,i2 ,··· ,in−1 ,i0n 

(50)

i0n =1,i0n 6=in

− bi1 qi1 ,i2 ,··· ,in · F (i1 , i2 , · · · , in−1 )

=qi1 ,i2 ,··· ,in − bi1 µi1 ,i2 ,··· ,in−1 θi1 ,i2 ,··· ,in qi1 ,i2 ,··· ,in Di1 ,i2 ,··· ,in−1

−

X i0n =1

bi1 θi1 ,i2 ,··· ,in−1 ,i0n qi1 ,i2 ,··· ,in

n−1 X

m=1

Di1 ,i2 ,··· ,in−1

+

X i0n =1

i ,i ,··· ,i

n−1 (µi1 ,i2 ,··· ,im−1 − µi1 ,i2 ,··· ,im−1 ,im )Ωi11,i22,··· ,im

bi1 µ0 θi1 ,i2 ,··· ,in−1 ,i0n qi1 ,i2 ,··· ,in−1 ,in qi1 ,i2 ,··· ,in−1 ,i0n

,i0n


25

− bi1 qi1 ,i2 ,··· ,in · F (i1 , i2 , · · · , in−1 ).

(51)

Of the five terms in Equation (51), the third term is obtained by combining the second term in Equation (47) with the term in Equation (49), the fourth term is obtained by combining the first term in Equation (48) with the term in Equation (50). Since all five terms in Equation (51) contain the expression qi1 ,i2 ,··· ,in , we can re-write it as h ∂R(θ (n) ) =qi1 ,i2 ,··· ,in · 1 − bi1 µi1 ,i2 ,··· ,in−1 θi1 ,i2 ,··· ,in ∂θi1 ,i2 ,··· ,in Di1 ,i2 ,··· ,in−1

−

X

bi1 θi1 ,i2 ,··· ,in−1 ,i0n

i0n =1

n−1 X

i ,i ,··· ,i

n−1 (µi1 ,i2 ,··· ,im−1 − µi1 ,i2 ,··· ,im−1 ,im )Ωi11,i22,··· ,im

,i0n

m=1

Di1 ,i2 ,··· ,in−1

+

X

bi1 µ0 θi1 ,i2 ,··· ,in−1 ,i0n qi1 ,i2 ,··· ,in−1 ,i0n

i0n =1

i − bi1 · F (i1 , i2 , · · · , in−1 ) . It turns out that the sum of the third and the fourth terms in the square brackets of the above equation is a function of hi1 , i2 , · · · , in−1 i, which we denote as G(i1 , i2 , · · · , in−1 ). Hence, at the stationary point, we shall have ∂R(θ (n) ) = 1 − bi1 µi1 ,i2 ,··· ,in−1 θi1 ,i2 ,··· ,in + G(i1 , i2 , · · · , in−1 ) − bi1 F (i1 , i2 , · · · , in−1 ) = 0, ∂θi1 ,i2 ,··· ,in because qi1 ,i2 ,··· ,in is positive. Let us re-arrange terms and get θi1 ,i2 ,··· ,in =

1 + G(i1 , i2 , · · · , in−1 ) − bi1 F (i1 , i2 , · · · , in−1 ) . bi1 µi1 ,i2 ,··· ,in−1

The R.H.S. of the above equation is identical for all products in elementary nest hi1 , i2 , · · · , in−1 i, which means that the markups for products belonging to this elementary nest are constant. Following the same logic, we can show that the markups are constant across products in each elementary nest.


Let us define M P OP (k) as the formulation shown in Equations (52) through (56),

where θ (k) = {θi1 ,i2 ,··· ,ik |hi1 , i2 , · · · , ik i ∈ Nk }, for any k ∈ {n, n − 1, · · · , 1}. We claim that in the optimal solution to M P OP (k), where k ∈ {n, n − 1, · · · , 2}, we have θi1 ,i2 ,··· ,ik−1 ,ik is constant across

ik ∈ {1, 2, · · · , Di1 ,i2 ,··· ,ik−1 } for all nest hi1 , i2 , · · · , ik−1 i ∈ Nk−1 . If we denote this identical value for nest hi1 , i2 , · · · , ik−1 i as θi1 ,i2 ,··· ,ik−1 , we can re-write M P OP (k) as M P OP (k − 1). Di1 ,i2 ,··· ,i

D

M P OP (k) :

max R(θ

(k)

)=

i1 D0 X X

i1 =1 i2 =1

···

X ik =1

k−1

θi1 ,i2 ,··· ,ik · qi1 ,i2 ,··· ,ik

(52)


26

subject to exp(µi1 ,i2 ,··· ,im−1 Vî1 ,i2 ,··· ,im ) , PDi1 ,i2 ,··· ,im D0 P î ,i ,··· ,i0 exp µ V m=2 0 ˆ i ,i ,··· ,i 1 2 m−1 1 2 im =1 m exp(µ0 Vi01 ) + 1 exp(µ0 Vî1 )

qi1 ,i2 ,··· ,ik =

k Y

·

i01 =1

Vî1 ,i2 ,··· ,im =



Di1 ,i2 ,··· ,im

1

X

ln 

µi1 ,i2 ,··· ,im

∀hi1 , i2 , · · · , ik i ∈ Nk ;  exp µi1 ,i2 ,··· ,im Vî1 ,i2 ,··· ,im+1  ,

(53)

im+1 =1

∀hi1 , i2 , · · · , im i ∈ Nm , m ∈ {1, 2, · · · , k − 1}; (54)

Vî1 ,i2 ,··· ,ik = −bi1 θi1 ,i2 ,··· ,ik + si1 ,i2 ,··· ,ik , θi1 ,i2 ,··· ,ik ∈ R,

∀hi1 , i2 , · · · , ik i ∈ Nk ;

∀hi1 , i2 , · · · , ik i ∈ Nk .

(55) (56)

Let us first look at the special case when k = n. It is not difficult to verify that M P OP (n) is the same as Formulation 2 defined by Equations (15) through (19) and M P OP (n − 1) is the same as

Formulation 3 defined by Equation (23) through (27). Therefore, Proposition 2 proves the special case when k = n.

Now, let us look at the general case when k ∈ {n, n − 1, · · · , 2}. We need to prove the in the

optimal solution to M P OP (k), θi1 ,i2 ,··· ,ik is constant across ik ∈ {1, 2, · · · , Di1 ,i2 ,··· ,ik−2 } for all

hi1 , i2 , · · · , ik−1 i ∈ Nk−1 . Consider the following auxiliary multiproduct pricing problem:

• The tree structure and the associated scale parameters are the same as the n-level nested logit

model in the original problem (see Figure 1) except that all the nests beyond level k are removed. In this problem, a product is now identified by a k-tuple: hi1 , i2 , · · · , ik i; and

• For product hi1 , i2 , · · · , ik i ∈ Nk , its price and cost are θi1 ,i2 ,··· ,ik and 0, respectively. In addition,

its systematic utility is specified as Wi1 ,i2 ,··· ,ik = −bi1 θi1 ,i2 ,··· ,ik + si1 ,i2 ,··· ,ik , where si1 ,i2 ,··· ,ik is defined in Equation (28).

It is not difficult to verify that the formulation for the auxiliary multiproduct price optimization problem is exactly M P OP (k). Following the logic of the proof for Proposition 2, we can prove that the optimal markups are constant across products within each elementary nest in the auxiliary problem, which implies that θi1 ,i2 ,··· ,ik−1 is identical across ik ∈ {1, 2, · · · , Di1 ,i2 ,··· ,ik−1 } for all

hi1 , i2 , · · · , ik1 i ∈ Nk−1 . Let θi1 ,i2 ,··· ,ik−1 be the constant markup for products in nest hi1 , i2 , · · · , ik−1 i,

we can re-write M P OP (k) as M P OP (k − 1) as follows:

Di1 ,i2 ,··· ,i

D

M P OP (k − 1) :

max R(θ

(k−1)

)=

i1 D0 X X

i1 =1 i2 =1

···

k−2

X ik−1 =1

θi1 ,i2 ,··· ,ik−1 · qi1 ,i2 ,··· ,ik−1

subject to qi1 ,i2 ,··· ,ik−1 =


i01 =1

·

k−1 Y

(57)


Vî1 ,i2 ,··· ,im =

27

∀hi1 , i2 , · · · , ik−1 i ∈ Nk−1 ;  Di1 ,i2 ,··· ,im X ln  exp µi1 ,i2 ,··· ,im Vî1 ,i2 ,··· ,im+1  ,

(58)



1 µi1 ,i2 ,··· ,im

im+1 =1

∀hi1 , i2 , · · · , im i ∈ Nm , m ∈ {1, 2, · · · , k − 2}; (59)

Vî1 ,i2 ,··· ,ik−1 = −bi1 θi1 ,i2 ,··· ,ik−1 + si1 ,i2 ,··· ,ik−1 , θi1 ,i2 ,··· ,ik−1 ∈ R,

∀hi1 , i2 , · · · , ik−1 i ∈ Nk−1 ;

(60)

∀hi1 , i2 , · · · , ik−1 i ∈ Nk−1 .

(61)

In the above formulation, Equation (57) is obtained from Equation (52) because Di1 ,i2 ··· ,i

D

R=

i1 D0 X X

i1 =1 i2 =1

=

i1 =1 i2 =1

=

D0 Di1 X X i1 =1 i2 =1

ik =1

θi1 ,i2 ,··· ,ik · qi1 ,i2 ,··· ,ik

Di1 ,i2 ··· ,i

D

i1 D0 X X

X k−1

···

X k−1

···

ik =1

θi1 ,i2 ,··· ,ik−1 · qi1 ,i2 ,··· ,ik

Di1 ,i2 ··· ,i

···

X k−2

ik−1 =1

θi1 ,i2 ,··· ,ik−1 · qi1 ,i2 ,··· ,ik−1 ;

Equations (58) and (59) are due to the definitions of qi1 ,i2 ,··· ,ik−1 and Vî1 ,i2 ,··· ,im ; and Equation (60) is obtained from Equation (55) because Vî1 ,i2 ,··· ,ik−1 =

=

1 µi1 ,i2 ,··· ,ik−1 1 µi1 ,i2 ,··· ,ik−1

Di1 ,i2 ,··· ,i

ln

k−1

X

exp µi1 ,i2 ,··· ,ik−1 Vî1 ,i2 ,··· ,ik

ik =1 Di1 ,i2 ,··· ,i

ln

k−1

X

exp µi1 ,i2 ,··· ,ik−1 (−bi1 θi1 ,i2 ,··· ,ik−1 ,ik + si1 ,i2 ,··· ,ik )

ik =1

= − bi1 θi1 ,i2 ,··· ,ik−1 +

1 µi1 ,i2 ,··· ,ik−1

Di1 ,i2 ,··· ,i

ln

X

k−1

exp µi1 ,i2 ,··· ,ik−1 si1 ,i2 ,··· ,ik−1

ik =1

= − bi1 θi1 ,i2 ,··· ,ik−1 + si1 ,i2 ,··· ,ik−1 , where the first equality is due to the definition of Vî1 ,i2 ,··· ,ik−1 ; the second equality is due to Equation (55); the third equality is due to the fact that θi1 ,i2 ,··· ,ik−1 ,ik = θi1 ,i2 ,··· ,ik−1 ; and the fourth equality is due to the definition of si1 ,i2 ,··· ,ik−1 . This proves our claim stated in the beginning. To prove this proposition, we just need to repeatedly apply our claim for k = n, n − 1, · · · , 2. In the last step we have k = 2 and in the optimal solution to M P OP (2), we know that θi1 ,i2 is identical across i2 ∈ {1, 2, · · · , Di1 } for all hi1 i ∈ N1 . If we denote this identical value as θi1 , we can re-write M P OP (2) as M P OP (1), whose formulation is the same as Formulation 4 defined by Equations (29) through (32). This completes the proof.


28


Let us look at Formulation 4 defined by Equations (29) through (32). Consider hi1 i ∈ N1 ,

the first order condition with respect to θi1 implies that ∂R(θ (1) ) = qi1 + θi1 ∂θi1

1 1+

  · − 1 + D0 X

+

PD0

i01 =1

D0 X

2 

exp(µ0 Vî01 ) exp(µ0 Vî1 )bi1 µ0 + exp(µ0 Vî1 ) exp(µ0 Vî1 )bi1 µ0 

i01 =1

θi01

i01 =1,i01 6=i1

= qi1 + θi1

exp(µ0 Vî01 ) 

1

1+

exp(µ0 Vî1 ) exp(µ0 Vî0 )bi1 µ0 1 ˆ 00 i00 =1 exp(µ0 Vi1 )

PD0 1

−bi1 µ0 qi1 + bi1 µ0 qi21

= qi1 − bi1 µ0 qi1 θi1 + bi1 µ0

D0 X

D0 X

+ bi1 µ0

θi01 qi1 qi01

i01 =1,i01 6=i1

θi01 qi1 qi01

i01 =1

= 0. Since qi is finite, from the last equality we can get D

θi1 −

0 X 1 = θi01 qi01 . bi1 µ0 0

i1 =1

Note that the R.H.S. of the above equation is independent of i1 , therefore θi1 −

1 bi1 µ0

is constant

for all hi1 i ∈ N1 .

C.6. Proof of Proposition 5 From Equation (33), we can write the derivative of R(φ) as follows D0 D0 X X 1 1 0 R (φ) = qi1 + φ+ 2 P b µ D i1 0 i1 =1 i1 =1 1 + i0 0=1 exp(µ0 Vî01 ) 1     D0 D0 X X exp(µ0 Vî01 ) exp(µ0 Vî1 )µ0 bi1 + exp(µ0 Vî1 ) · exp(µ0 Vî01 )µ0 bi01  · − 1 +

Proof:

i01 =1

=

D0 X

qi1 +

i1 =1

=

D0 X i1 =1

D0 X i1 =1

i01 =1

1 φ+ bi1 µ0

−bi1 µ0 qi1 + qi1



qi1 − φbi1 µ0 qi1 − qi1 + φ + D0

D0

=−φ



X i1 =1

bi1 µ0 qi1 +

X i1 =1

φ+

1 bi1 µ0

1 bi1 µ0

D0 X

 bi01 µ0 qi01 

i01 =1

qi1

D0 X

 bi01 µ0 qi01 

i01 =1 D0

qi1 ·

X i01 =1

bi01 µ0 qi01


= (R(φ) − φ) ·

D0 X

29

µ0 bi1 qi1 .

i1 =1

For ease of discussion, let g(φ) =

PD0

i1 =1 µ0 bi1 qi1

> 0 and we can re-write the above equation as

0

R (φ) = (R(φ) − φ) · g(φ). Now, we prove that R0 (φ) = (R(φ) − φ) · g(φ) is unimodal. First of

all, there does not exist an interval [φ1 , φ2 ], where φ1 < φ2 , such that R(φ) − φ = 0. Otherwise, R0 (φ) = (R(φ) − φ) · g(φ) = 0 for all φ in this interval and we get φ1 = φ2 , which contradicts the fact that φ1 < φ2 .

Secondly, R(φ) must contain at least one stationary point. From Equation (33), we know that R(φ) > 0 when φ = 0 and R(φ) → 0 when φ → +∞. Let us define f (φ) = φ. We know that R(φ) >

f (φ) when φ = 0 and that R(φ) < f (φ) when φ → +∞. Since both R(φ) and f (φ) are continuous,

these two functions must intersect at least once in (0, +∞). That is, ∃φ0 ∈ (0, +∞), such that R(φ0 ) = φ0 . Hence, R0 (φ0 ) = (R(φ0 ) − φ0 ) g(φ0 ) = 0, which is a stationary point of R(φ).

Finally, we show that R(φ) has exactly one stationary point. Suppose that there are more

than one stationary point and we arbitrarily pick two consecutive ones, which are denoted as φ1 and φ2 . Without loss of generality, assume that φ1 < φ2 . On one hand, consider φ− 2 in the − − − − − 0 left neighborhood of φ2 . If R(φ− 2 ) < φ2 , it implies that R (φ2 ) = (R(φ2 ) − φ2 ) · g(φ2 ) < 0. As a

− − result, φ− 2 > R(φ2 ) > R(φ2 ) = φ2 , which contradicts the fact that φ2 < φ2 . Hence, we must have + − − − − − 0 R(φ− 2 ) > φ2 and R (φ2 ) = (R(φ2 ) − φ2 ) · g(φ2 ) > 0. On the other hand, consider φ1 in the right

+ neighborhood of φ1 . Since R0 (φ1 ) = 0 and R0 (φ) is continuous, we know that R(φ+ 1 ) < φ1 and

− + + + + 0 R0 (φ+ 1 ) = (R(φ1 ) − φ1 ) · g(φ1 ) < 0. Since R (φ) is continuous between φ1 and φ2 , there must exist − 0 φ4 ∈ (φ+ 1 , φ2 ) such that R (φ4 ) = 0. This means that φ4 is a stationary point between φ1 and φ2 ,

which contradicts that fact that φ1 and φ2 are two consecutive stationary points. Therefore, R(φ) has exactly one stationary point. If we denote the stationary point as φ∗ , R(φ) is monotonically increasing when φ < φ∗ and monotonically decreasing when φ > φ∗ .


When bi1 = b for all hi1 i ∈ N1 , the profit function in Equation (33) can be written as R(φ) =

D0 X i1 =1

1 φ+ bµi0

1 qi1 = φ + bµi0

X D0 i1

1 qi1 = φ + bµi0 =1

According to Proposition (5), R(φ) = φ, so we have 1 φ= φ+ (1 − qN ). bµi0 Re-arrange terms in the above equation and get φ+

1 1 = . bµ0 bµ0 qN

(1 − qN ).


30

As a result, we have θi1 ,i2 ,··· ,in = φ +

1 1 = , bµ0 bµ0 qN

which is constant across all products. This completes the proof.


Consider firm i2 6= i1 , we can compute the cross derivatives of ln Ri1 (θi1 , θ −i1 ) as follows 1 ∂ln Ri1 (θi1 , θ −i1 ) = − µ0 bi1 (1 − qi1 ) , ∀i1 ∈ {1, 2, · · · , D0 }; ∂θi1 θ i1 ∂ 2 ln Ri1 (θi1 , θ −i1 ) = µ20 bi1 bi2 qi1 qi2 ≥ 0, ∀i1 , i2 ∈ {1, 2, · · · , D0 } and i1 6= i2 . ∂θi1 ∂θi2

(62)

Since the cross-derivatives are nonnegative and the strategy space for each firm is in a single dimension, the oligopolistic game is log-supermodular. According to the results in Vives (2001), the equilibrium set is a nonempty complete lattice. Now, let us prove that the Nash equilibrium is unique. Let A =

ˆ0

PD0

i01 =1 exp(µ0 Vi1 ).

Set Equation

(62) to zero and plug in the fact that qi1 = exp(µ0 Vî1 )/(1 + A), we can express A as a function of θi1 as follows A = ψi1 (θi1 ) =

µ0 bi1 θi1 exp(µ0 Vî1 ) − 1. µ0 bi1 θi1 − 1

(63)

Note that A = ψi1 (θi1 ) is strictly decreasing in θi1 because ψi01 (θi1 ) = −µ0 bi1 exp(µ0 Vî1 ) −

µ20 b2i1 θi1 exp(µ0 Vî1 ) (µ0 bi1 θi1 − 1)

2

< 0,

(A) where the last inequality holds because θi1 is positive. As a result, the inverse function θi1 = ψi−1 1 exists and is a decreasing function of A. Moreover, by setting Equation (62) to zero, we can express qi1 as a function of θi1 as follows qi1 = 1 −

1 1 =1− , µ0 bi1 θi1 µ0 bi1 ψi−1 (A) 1

As a result, we have D0 X

m

X 1 1 qi1 = qN + + 1− 1 + A l=1 µ0 bi1 ψi−1 (A) 1 i =1

! = 1.

1

Since the L.H.S. of the last equality in the above equation is strictly decreasing in A, it has a ∗ unique solution, which we denote as A∗ . Let θi∗1 = ψi−1 (A∗ ) and θ ∗ = θ1∗ , θ2∗ , · · · , θD is the unique 1 0 Nash equilibrium.


31

C.9. Proof of Equation (40) Proof:

Consider hi1 , i2 , · · · , im i ∈ Nm , where 1 ≤ m ≤ n − 1, we have ∂ Vî1 ,i2 ,··· ,im ∂pi1 ,i2 ,··· ,im ,im+1 ,··· ,in

=

∂ Vî1 ,i2 ,··· ,im ,im+1 i ,i ,··· ,im ,im+1 ,··· ,in · Bi1 ,i2 ,··· ,im ,im+1 = −bi1 Ωi11,i22,··· ,im , ∂pi1 ,i2 ,··· ,in

(64)

where Bi1 ,i2 ,··· ,im ,im+1 is defined in the beginning of Section C.1 and the second equality is obtained by the repeated application of the first equality. We also have that ∂Bi1 ,i2 ,··· ,im ∂pi1 ,i2 ,··· ,im ,im+1 ,··· ,in  =

∂ ∂pi1 ,i2 ,··· ,im ,im+1 ··· ,in

µi ,i ,··· ,i 1 2 m−1 µi ,i ,··· ,i 1 2 m−1 ,im i1 ,i2 ,··· ,im

A   Ai

1 ,i2 ,··· ,im−1

   

! ∂ Vî1 ,i2 ,··· ,im ,im+1 ∂ Vî1 ,i2 ,··· ,im − Bi1 ,i2 ,··· ,im · =µi1 ,i2 ,··· ,im−1 Bi1 ,i2 ,··· ,im Bi1 ,i2 ,··· ,im ,im+1 · ∂pi1 ,i2 ,··· ,im ,im+1 ,··· ,in ∂pi1 ,i2 ,··· ,im ,im+1 ,··· ,in i ,i ,··· ,im ,im+1 ,··· ,in i ,i ,··· ,im ,im+1 ,··· ,in − Ωi11,i22,··· ,im−1 , (65) = − bi1 µi1 ,i2 ,··· ,im−1 Bi1 ,i2 ,··· ,im Ωi11,i22,··· ,im where the first equality is due to Equation (43) and the last equality is due to Equation (64). As a result, we get ∂qi1 ,i2 ,··· ,in ∂pi1 ,i2 ,··· ,in =

∂

n Y

∂qi1 ,i2 ,··· ,in

m=1

! Bi1 ,i2 ,··· ,im

Qn n X Bi ,i ,··· ,i ∂Bi1 ,i2 ,··· ,im = · m=1 1 2 m ∂pi1 ,i2 ,··· ,in Bi1 ,i2 ,··· ,im m=1 n q X i ,i ,··· ,im ,im+1 ,··· ,in i ,i ,··· ,im ,im+1 ,··· ,in i ,i ,··· ,i · 1 2 n − Ωi11,i22,··· ,im−1 =− bi1 µi1 ,i2 ,··· ,im−1 Bi1 ,i2 ,··· ,im Ωi11,i22,··· ,im B i1 ,i2 ,··· ,im m=1 " n−1 # X i1 ,i2 ,··· ,in = − bi1 qi1 ,i2 ,··· ,in µi1 ,i2 ,··· ,im−1 − µi1 ,i2 ,··· ,im Ωi1 ,i2 ,··· ,im − µ0 qi1 ,i2 ,··· ,in + µi1 ,i2 ,··· ,in−1 . m=1

By the definition of direct elasticity, we can immediately obtain Equation (40). C.10. Proof of Equation (41) Proof:

i0 ,i0 ,··· ,i0

Let us first consider the case when Λi11 ,i22 ,··· ,inn = l ≥ 1, we then have ∂Bi1 ,i2 ,··· ,il ,i0l+1 ∂pi1 ,i2 ,··· ,il ,il+1 ,··· ,in  =

∂ ∂pi1 ,i2 ,··· ,il ,il+1 ··· ,in

µi ,i ,··· ,i 1 2 l µ i1 ,i2 ,··· ,il ,i0 l+1 i1 ,i2 ,··· ,il ,i0l+1

A    Ai1 ,i2 ,··· ,il

    


32 µi ,i ,··· ,i 1 2 l µ i1 ,i2 ,··· ,il ,i0 l+1 i1 ,i2 ,··· ,il ,i0l+1

exp(µi1 ,i2 ,··· ,il Vî1 ,i2 ,··· ,il+1 )µi1 ,i2 ,··· ,il ∂ Vî1 ,i2 ,··· ,il+1 − A2i1 ,i2 ,··· ,il ∂pi1 ,i2 ,··· ,il ,il+1 ,··· ,in

=A

= − µi1 ,i2 ,··· ,il Bi1 ,i2 ,··· ,il ,i0l+1 Bi1 ,i2 ,··· ,il ,il+1

!

∂ Vî1 ,i2 ,··· ,il+1 ∂pi1 ,i2 ,··· ,il ,il+1 ,··· ,in

,··· ,in =bi1 µi1 ,i2 ,··· ,il Bi1 ,i2 ,··· ,il ,i0l+1 Ωii11,i,i22,··· ,il ,

(66)

where the last equality is due to Equation (64). Therefore, we have ∂qi1 ,i2 ,··· ,il ,i0l+1 ,··· ,i0n ∂pi1 ,i2 ,··· ,il ,il+1 ,··· ,pin =

∂

l Y

∂pi1 ,i2 ,··· ,in

m=1

Bi1 ,i2 ,··· ,im · Bi1 ,i2 ,··· ,il ,i0l+1 ·

n Y

! Bi1 ,i2 ,··· ,il ,i0l+1 ,··· ,i0m

m=l+2

l X ∂Bi1 ,i2 ,··· ,im qi1 ,i2 ,··· ,il ,i0l+1 ,··· ,i0n ∂Bi1 ,i2 ,··· ,il ,i0l+1 qi1 ,i2 ,··· ,il ,i0l+1 ,··· ,i0n = + +0 ∂pi1 ,i2 ,··· ,in Bi1 ,i2 ,··· ,im ∂pi1 ,i2 ,··· ,in Bi1 ,i2 ,··· ,il ,i0l+1 m=1

= − bi1 qi1 ,i2 ,··· ,il ,i0l+1 ,··· ,i0n ·

l X m=1

i ,i ,··· ,il ,il+1 ,··· ,in i ,i ,··· ,il ,il+1 ,··· ,in µi1 ,i2 ,··· ,im−1 Ωi11,i22,··· ,im − Ωi11,i22,··· ,im−1 i ,i ,··· ,i ,il+1 ,··· ,in

= − bi1 qi1 ,i2 ,··· ,il ,i0l+1 ,··· ,i0n · = − bi1 qi01 ,i02 ,··· ,i0l ,i0l+1 ,··· ,i0n ·

l X m=1 l X m=1

+ bi1 qi1 ,i2 ,··· ,il ,i0l+1 ,··· ,i0n · µi1 ,i2 ,··· ,il Ωi11,i22,··· ,ill

!

,··· ,in µi1 ,i2 ,··· ,im−1 − µi1 ,i2 ,··· ,im Ωii11,i,i22,··· ,im − µ0 qi1 ,i2 ,··· ,in

! ,··· ,in µi1 ,i2 ,··· ,im−1 − µi1 ,i2 ,··· ,im Ωii11,i,i22,··· ,im − µ0 qi1 ,i2 ,··· ,in

,

(67)

where the third equality is due to Equations (65) and (66) and the last equality is due to hi1 , i2 , · · · , il i = hi01 , i02 , · · · , i0l i.

i0 ,i0 ,··· ,i0

Now, let us consider the case when Λi11 ,i22 ,··· ,inn = 0, we have ∂qi01 ,i02 ,··· ,i0n ∂pi1 ,i2 ,··· ,in

=

∂Bi01

qi01 ,i02 ,··· ,i0n

∂pi1 ,i2 ,··· ,in

= −µ0

Bi01 exp(µ0 Vî ) exp(µ0 Vî0 ) 1

(A0

1

+ 1)2

·

∂ Vî1

!

qi01 ,i02 ,··· ,i0n

∂pi1 ,i2 ,··· ,in 0 0 0 q i1 ,i2 ,··· ,in i1 ,i2 ,··· ,in

=µ0 Bi1 Bi01 · bi1 Ωi1

Bi01

=µ0 bi1 qi1 ,i2 ,··· ,in qi01 ,i02 ,··· ,i0n . Note that in Equation (67), when l = 0, we have

Bi01

(68)

Pl

m=1

,··· ,in µi1 ,i2 ,··· ,im−1 − µi1 ,i2 ,··· ,im Ωii11,i,i22,··· ,im = 0

and Equation (67) immediately reduces to Equation (68). Hence, we can combine results in these two equations and use Equation (67) as the expression for ∂qi01 ,i02 ,··· ,i0n /∂pi1 ,i2 ,··· ,in . By the definition of cross elasticity, we can easily obtain Equation (41) from Equation (67).


33

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