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Management Science
Product Differentiation, Store Differentiation, and Assortment Depth Stephen F. Hamilton∗ Cal Poly San Luis Obispo
Timothy J. Richards Arizona State University
April 6, 2009
Abstract This paper considers the relationship between product differentiation, store differentiation and the equilibrium depth of the product assortment. We find an inverted u-shaped relationship between product differentiation and assortment depth, with the depth of the assortment rising at first and then falling with the degree of product differentiation. For product categories that consist of relatively non-differentiatied variants, a positive relationship arises between assortment depth and category sales, whereas a negative relationship emerges between assortment depth and sales in categories with more differentiated variants. Both the extent and manner in which store differentiation changes has important implications for assortment depth. If retailer market power is augmented following the closure of rival retailers, product assortments become deeper; however, if retailers gain market power by investing in store attributes that facilitate customer loyalty, product assortments become shallower.
JEL Classification: L11; L13; D43 Keywords: Product differentiation; multi-product firms
∗
Correspondence to: S. Hamilton, Department of Economics, Orfalea College of Business, California Polytechnic State University, San Luis Obispo, CA 93407. Voice: (805) 756-2555, Fax: (805) 756-1473, email:
[email protected]. We would like to thank Ramarao Desiraju, Glenn Harrison, Robert Innes, Vincent Requillart, and the members of the 6th INRA-IDEI Conference for helpful comments on an earlier draft. Financial support from the FSRG at the University of Wisconsin is gratefully acknowledged.
Management Science
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Introduction
Retail product assortments have expanded substantially over the past quarter century. Among supermarkets, for example, the median number of stock-keeping units (SKUs) increased from 16,500 to 25,153 over the 1990-2004 period (Progressive Grocer).
An essential element of this trend is an increase in the depth of product
assortments, as measured by the number of variants within existing product categories.1
What are the market forces that drive retailers to increase the depth of
their product assortments? To what extent do product-specific attributes that differentiate variants in a category determine which assortments become “deep” and which assortments remain “shallow”? And how does the degree of store differentiation in the market influence assortment depth? The aim of this paper is to characterize how the equilibrium depth of the product assortment is determined across categories that vary in the degree of product differentiation and across retail environments that vary in the degree of store differentiation. There is little research to date on the nature of the incentives that underlie product assortment decisions (Draganska and Jain, 2005, 2006).
Messinger and
Narasimhan (1997) examine a number of seemingly plausible alternatives for why retailers stock more products in their stores (economies of scale, technological improvement, monopoly power, higher margins, and lower inventory cost) and rule out each of these factors in favor of a theory based on consumers’ desire to minimize shopping costs. We frame our analysis around a similar effect by considering transaction costs between consumers and retailers that generate economies of “one-stop shopping”. This feature, alone, is capable of explaining why retailers expand their product assortments; however, it falls short of providing insight to explain why the equilibrium assortment is deeper in some categories and shallower in others. A large and growing literature on efficient assortment seeks to identify the re1
We define the depth of a product assortment as the number of variants sold in a category, rather than as the size of the attribute space spanned by the variants within the category. This definition, though not uncontroversial, is standard in the literature (see, e.g., Draganska and Jain, 2005).
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lationship between assortment depth and sales in a product category.
This liter-
ature finds a positive relationship between assortment depth and category sales in some cases (Van Ryzin and Mahajan, 1999; Borle, Boatwright, Kadane, Nunes and Shmueli, 2005) and a negative relationship in others (Dreze, Hoch and Purk,1994; Broniarczyk, Hoyer and McAlister,1998; and Boatwright and Nunes, 2001).
Our
analysis contributes to this literature by distinguishing the circumstances under which a deeper product assortment is positively related to category sales. Indeed, we find the relationship between assortment depth and category sales depends critically on the degree of differentiation among variants in the category. Product categories with highly differentiated variants have higher prices and lower category sales levels than product categories with less differentiated variants; however, we find that equilibrium assortment depth follows an inverted u-shaped pattern over product differentiation, with deeper assortments in categories with an intermediary level of differentiation and shallower assortments “at the tails”. Hence, in cross-sectional data, our model predicts a positive relationship between assortment depth and sales in highly differentiated product categories, but a negative relationship between assortment depth and sales in less differentiated product categories. The inverted u-shaped pattern between assortment depth and product differentiation emerges in monopoly retail markets as well as in oligopoly retail markets due to offsetting incentives facing category managers. Highly differentiated product categories have wider retail margins than less-differentiated categories, and this favors a deeper assortment since category sales rise with assortment depth. But new variants introduced in a product category cannibalize a portion of their sales from existing variants, and cannibalization is more costly when retail margins are wide than when retail margins are narrow. The non-monotonic pattern emerges because the relative magnitude of these effects evolves with the degree of product differentiation. In categories with relatively non-differentiated variants, deepening the assortment has little impact on category demand.
Retailers respond by selecting a shallow assortment
since stocking new variants is costly. In categories with more differentiated variants,
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assortment depth has a larger impact on category demand. Retail margins are wider in these categories relative to categories with less differentiated variants, and retailers respond by deepening their assortments.
But profit per variant falls with assort-
ment depth since new variants cannibalize a portion of their demand from existing variants in the category. as retail margins rise.
Cannibalizing category sales becomes increasingly costly This effect eventually dominates, and, and, for categories
with sufficiently differentiated variants, retailers respond to an increase in product differentiation accordingly by reducing assortment depth. In retail markets that vary in the degree of store differentiation, the manner in which store differentiation alters the ability of consumers to switch between retailers has important implications for assortment depth.
We consider a locational model
of store differentiation in which the ability of retailers to exercise market power over consumers depends on how costly it is for consumers to switch from shopping with one retailer to shopping with another retailer. For given product assortments and prices, the cost of switching between retailers increases when there is a smaller number of retailers in the market (greater market concentration) or when the transaction cost of traveling to a rival retailer becomes larger, for instance in response to retailer investments in store attributes that generate customer loyalty.
When consumer
switching costs increase in response to a decline in the number of retailers serving the market, we show that retailers respond by raising prices and deepening their assortments; however, when consumer switching costs increase in response to greater customer loyalty, we find that prices rise and product assortments become shallower. The reason that the nature of the change in store differentiation matters for the equilibrium depth of the product assortment is that retailers must select prices and assortment depth jointly to serve existing customers and to attract new customers into the store. In terms of pricing, changes in store differentiation align these goals. A smaller number of retailers in the market allocates more customers to each retailer, and this makes raising prices attractive as a means to extract greater profit from existing customers.
An increase in customer loyalty makes it more difficult for
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retailers to acquire market share from rivals.
Greater customer loyalty also leads
to higher prices by dampening retailers’ ability to use selective price discounts as a means of attracting new customers into their stores. product assortment decisions.
But the opposite is true for
Providing a deeper product assortment raises sales
(and profit) per customer and also serves as a strategic tool for enhancing market share. Consequently, an increase in the customer base served by each retailer leads to deeper product assortments while a rise in customer loyalty results in shallower product assortments. Our observations on the relationship between the depth of product assortments, product differentiation, and store differentiation have several notable implications for retail management practices. Consistent with the private-label literature (Dhar and Hoch, 1997), our results suggest that it may be beneficial for strategic reasons to facilitate store differentiation through private label development, while at the same time limiting differentiation within the category by positioning private labels within close proximity to national brands in attribute space.
Moreover, our finding of a
non-monotonic relationship between product differentiation and assortment depth points toward a need for category managers to consider how category performance changes with assortment depth on a category-by-category basis, a result that lends cautionary support to emerging fact-based management methods. The remainder of the paper is organized as follows. In §2, we characterize the retail monopoly equilibrium and both the short-run and long-run retail oligopoly equilibria in a setting where consumers make discrete choices over retailers and purchase multiple goods on each shopping occasion. In §3, we position the main results of the paper by deriving the relationship between store differentiation, product differentiation, and equilibrium assortment depth for the case of constant elasticity of substitution (CES) preferences over variants in the category. The CES specialization, which has been used in previous models to explain product assortment decisions (see, e.g., Shugan 1987), is useful for expositional purposes because it encompasses a reasonably wide class of preferences with a scalar measure of product differentiation.
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This allows analytical solutions to be derived for equilibrium prices and equilibrium assortment depth in the monopoly case and lends itself readily to numerical techniques in the oligopoly case.
We summarize our main findings and elaborate on
some extensions of the model in §4.
2
The Model
We frame our analysis by synthesizing the consumption preferences of Spence (1976a, 1976b) and Dixit and Stiglitz (1977) over variants in the product category with the locational preference structure of Salop (1979) over retailers in the market. Consumers purchase multiple products at a time from a retailer and incur switching costs (either real or perceived) when changing their loyalties from one retailer to another retailer. Retailers jointly select assortment depths and prices to serve existing customers and to attract new customers into their stores. To focus the model on how assortment depth influences retail market outcomes, we suppress strategic considerations among manufacturers at the brand level in wholesale markets. Specifically, we suppose an arbitrarily large number of potential variants are available to retailers in the product category at a cost of $c per variant and that the demand facing retailers for any one variant is the same as for any other variant in the category.
While this formulation suppresses some potentially interesting nuances
in the role of brands, attributes, and flavors in providing better matches between consumers and retailers, it has the advantage of allowing assortment depth to be measured as a scalar index of the number of variants in the category, a feature that greatly clarifies our observations. We consider n retailers who differ in terms of their spatial proximity to consumers. The location of each retailer is represented as a point on a circle of unit length and consumers are distributed uniformly about the circle so that no one retail location is inherently superior to any other retail location. Retailers are situated such that their territories are of equal size in the symmetric allocation where retailers select identical product assortments and prices. As in Hamilton (2009), competition among 5
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retailers for market share is localized in the sense that consumers compare only the assortment depths and prices of neighboring retailers in deciding where to shop. Consumers incur increasing transaction costs over distance to visit retailers. The decision to shop with a given retailer commensurately depends on the transaction cost required to visit the retailer relative to the consumption opportunity afforded by that retailer’s product assortment and prices. Consumer preferences over retail products are represented by the class of utility functions described by U (z, y) = G (z) + y, where z is a composite good representing all variants in the product category of interest and y is the consumption level of all other retail goods. G(z) is an increasing function with constant elasticity (1 − ε) ∈ (0, 1). The consumption level of the composite good z is determined by the subutility Rv function z = i=0 f (xi )di, where xi is the amount consumed of variant i and f (x) is a smooth, increasing, and strictly concave function for all x > 0.
A natural
interpretation of v is the “number” of variants in the product category, but since we choose to measure variety continuously we refer to v as “assortment depth”. Inverse demand for product i for the representative consumer is p(z, xi ) = G0 (z) f 0 (xi ).
(1)
Equation (1) implicitly defines the demand functions for the representative consumer, xi (p, v), where p is the functional of prices. The demands can be used to recover indirect utility, u(p, v), which is increasing in v and decreasing in prices.2 Aggregate demand facing each retailer depends on the decisions made by consumers at all points on the circle regarding where to shop. Letting t denote consumer transaction cost per unit distance, a consumer at a distance of δ ∈ (0, 1) from the representative retailer could achieve surplus of u(p, v) − δt by purchasing from that retailer. If there are n retailers located about the circle, for consumers located on the interval 0 ≤ δ ≤ 1/n between a retailer and his nearest neighbor the surplus available by purchasing from the best alternative to the retailer is u(p, v) − t (1/n − δ), where 2
Bhatnagar and Ratchford (2004) provide empirical evidence that the likelihood of patronizing a supermarket is increasing in assortment depth.
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v is the assortment depth available at the nearest rival and p is the associated functional of prices. Letting δ ∗ denote the location of the consumer who is indifferent between these alternatives, δ ∗ solves u(p, v) − δt = u(p, v) − t (1/n − δ): δ ∗ (p,v; p, v) =
1 1 + [u(p, v) − u(p, v)] .3 2n 2t
(2)
All consumers located at a distance of δ ≤ δ ∗ prefer to shop with the representative retailer and more distant consumers prefer to shop with his rival. The demand for retail product i, accordingly, is Xi (p, v; p, v) = δ ∗ (p, v; p, v)xi (p,v) and total category Rv demand is X(p, v; p, v) = δ ∗ (p, v; p, v) i=0 xi (p, v)di.
Now consider the retailers’ problem. Each retailer pays a fixed set-up cost, F , and
a constant unit cost of c to stock an individual product. It follows that per-customer profit for the representative retailer is π(p, v) =
Z
v
i=0
(pi − c)xi (p, v)di,
(3)
and total retailer profit, counting consumers on both sides, is given by Π(p, v; p, v) = 2δ ∗ (p, v; p, v)π(p, v) − vF.
(4)
Differentiating (4) with respect to pi gives the first-order necessary condition π(p, v) ∂u(p, v) ∂π(p, v) + 2δ ∗ (p, v; p, v) = 0, t ∂pi ∂pi
(5)
where ∂u(p, v)/∂pi = −xi (p, v) < 0 holds by Roy’s identity. Notice that condition (5) decomposes the effect of a price increase into an inter -retailer margin, which reflects the role of prices in determining market share, and an intra-retailer margin, which represents the role of prices in augmenting profit per customer. The first term on the left-hand side defines the effect of a price change on the inter-retailer margin. A small increase in price of dpi units shifts −(xi /t)dpi customers away from the retailer and towards his nearest rivals. This term represents the oligopoly incentive to acquire market share. Because each customer accounts for π(p, v) in category profit, 3
This formula and the ones that follow hold in the range u(p, v) − t/n < u(p, v) < u(p, v) + t/n. To avoid outcomes where an equilibrium fails to exist, we assume these inequalities are always met.
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the retailer’s loss from an increase in price of dpi units is −(xi /t)π(p, v)dpi . The remaining term is the effect of a price change on the intra-retailer margin. For a given market share (2δ ∗ = 1/n in the symmetric case), the retailer selects prices to maximize per-customer profit in the category. A monopoly retailer would choose category prices so that ∂π(p, v)/∂pi = 0 to maximize profits on the intra-retailer margin. An oligopoly retailer selects prices below the monopoly level, that is ∂π(p, v)/∂pi > 0 in equation (5), because the first term on the left-hand side —the market share effect of a price increase— is negative. Notice that the terms in (5) take opposing signs. This implies that a strengthening of effects on the intra-retailer margin has qualitatively similar implications for retail prices as a weakening of effects on the inter-retailer margin. If retailer market power increases in response to an increase in the transaction cost of switching between retailers (t), prices rise since higher transaction costs weaken the ability of retailers to acquire market share from rivals. If retailer market power increases in response to a decrease in the number of retailers serving the market (n), prices rise because a greater number of customers is now allocated to every retailer, thereby strengthening monopoly pricing incentives. Thus, an increase in consumer switching costs raises retail prices irrespective of whether the resulting increase in retailer market power is driven by the development of greater consumer loyalty (higher t) or by the closure of existing retailers (lower n). The first-order necessary condition for profit maximization with respect to assortment depth (v) is π(p, v) ∂u(p, v) ∂π(p, v) + 2δ ∗ (p, v; p, v) − F = 0. t ∂v ∂v
(6)
This condition has a similar interpretation. The first term on the left-hand side of (6) is the effect of a deeper assortment on the inter-retailer margin of profit. Unlike the case of a price increase, an increase in assortment depth has a positive effect on market ³ ´ v) share; that is, ∂u(p, v)/∂v = 1−θ(x pv xv (p, v) > 0, where θ(xv ) = f 0 (xv )xv /f (xv ) θ(xv ) denotes the elasticity of f (xv ).4 4
Providing a deeper product assortment attracts
In a symmetric allocation, 1 − θ(x) measures of the degree of consumer preference for variety.
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new customers to the store. The second term on the left-hand side is the effect of assortment depth on the intra-retailer margin.
This term is also positive since a
deeper assortment raises retail profit per customer in the category, ∂π(p, v)/∂v ≥ 0. Notice that a deeper assortment has reinforcing effects on both the inter-retailer and intra-retailer margins of (6). This implies that the manner in which retailers acquire market power has important implications for assortment depth.
Retailer
investments in customer loyalty that raise the transaction cost of switching between retailers (t) weaken the ability of retailers to acquire market share on the interretailer margin. shallower.
Equilibrium prices rise and the equilibrium assortment becomes
A decrease in the number of retailers in the market (n) strengthens
monopoly incentives for category management on intra-retailer margin by allocating more customers to each retailer. Equilibrium prices rise, but the equilibrium product assortment becomes deeper. PROPOSITION 1.
An increase in t raises prices and leads to deeper product as-
sortments; an decrease in n raises prices but leads to shallower product assortments. The equilibrium price per product, pe , and the equilibrium number of products, v e , in the short-run allocation are determined by the simultaneous solution of equations (5) and (6).
The long-run equilibrium (p∗ , v ∗ , n∗ ) is determined by these
two equations and the entry condition, which states that profits are zero.
In the
symmetric allocation (pi = p), this implies (p − c)
3
x(p, v) − F = 0. n
(7)
Store and Product Differentiation
In this section we provide a profile of the equilibrium assortment depth at various levels of store and product differentiation. To do so, it is helpful to refine the equilibrium described above to a form that generates analytical solutions. We consider the Rv class of constant elasticity of substitution (CES) subutility functions, z = i=0 xθi di, It is the proportion of social benefits not captured by revenues when a new product is introduced, 1 − G0 f 0 x/G0 f = 1 − θ(x). For an excellent discussion, see Vives (1999, pp.167-176).
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where 0 < θ < 1 measures the degree of substitutability between products.5 This refinement allows the degree of product differentiation in the category to be represented by the constant θ. Products in the retail category are more highly differentiated for smaller values of θ and products are perfect substitutes when θ → 1. The CES form results in analytical solutions for assortment depth and prices in the case of a monopoly retailer. For the oligopoly case, analytical solutions do not arise under CES utility and we consequently resort to numerical techniques.6 Rv With z = i=0 xθi di, demand for product i is derived by substituting terms in equation (1) and inverting the resulting expression. Doing so yields 1 Ã µZ v −θ ¶−ε ! 1−θ(1−ε) −1 xi (p, v) = θ(1 − ε) pi1−θ di pi1−θ .
(8)
i=0
In the symmetric allocation (pi = p), demand per product reduces to ∙ ¸ 1 θ(1 − ε) −ε 1−θ(1−ε) x(p, v) = v , p
(9)
which is decreasing in both p and v. The retailer’s incentive to raise prices and deepen the product assortment depends on the elasticity of per-product demand with respect p to prices, ep = − ∂x ∂p x =
1 1−θ(1−ε) ,
v and assortment depth, ev = − ∂x ∂v x =
ε 1−θ(1−ε) .
Notice that the assortment elasticity of demand is proportional to the price elasticity of demand, ev = εep , and that both terms become less elastic in response to an increase in product differentiation (i.e., a decrease in θ). The interpretation of ev is the degree to which new variants cannibalize their demand from the demand for existing variants as the assortment deepens.
To see
this, note that category demand in the symmetric case is χ(p, v) = vx(p, v). When ev = 0, demand per variant is independent of assortment depth. A small increase in assortment depth of dv units increases category demand by x(p, v)dv units. When ev = 1, total category demand remains constant as the assortment deepens. Demand for a new variant is cannibalized entirely from the demand for existing variants. 5
Formally, the elasticity of substitution between any two variants is σ = 1/(1 − θ). In the on-line appendix, we consider an example with logarithmic subutility, G(z) = ln z, that results in analytical solutions for oligopoly retailers. These outcomes corroborate the essential qualitative features of the numerical model in the oligopoly case. 6
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As θ decreases, ev decreases smoothly to zero. A greater level of product differentiation reduces the rate of product cannibalization in the category since a larger share of category demand is able to survive the introduction of a new variant the more differentiated the variants in the product category. It is tempting to conclude that a high degree of product differentiation favors a high profit level in the category since prices are higher and the cannibalization rate is lower in categories with highly differentiated variants.
But this is not the case.
Category profitability depends on the opportunity cost of cannibalization, not on the level, and cannibalizing sales from existing variants in a category is more costly when retail margins are wide.
To see this, it is helpful to suppress for a moment the
inter-retailer margin of the model and consider the monopoly case.
3.1
The Monopoly Outcome
A monopoly retailer selects assortment depth and prices to maximize profit, Π(p, v) = π(p, v) − vF.
(10)
The model is identified by substituting xi (p, v) from (8) into (3) and making use of this term in (10). Differentiating the resulting expression with respect to pi and v, respectively, and then evaluating terms in the symmetric equilibrium gives ¶ ¶ µ µ ∂π(p, v) p−c ep = 0, = x(p, v) 1 − ∂pi p ∂π(p, v) = (p − c)x(p, v) (1 − ev ) − F = 0. ∂v
(11)
(12)
Conditions (11) and (12), which define the intra-retailer margin of the model, completely characterize the monopoly outcome.
Condition (11) implies the familiar
monopoly result that the price-cost margin (in percentage terms) be set equal to the reciprocal of the price elasticity of demand. Because ep increases with θ, the equilibrium price level rises with the degree of product differentiation in the category. Equilibrium prices are set higher for all variants in a more differentiated product category than in a less differentiated product category. 11
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In (12), an increase in assortment depth generates profit of (p − c)x(p, v) from sales of the new variant, but cannibalizes ev (p − c)x(p, v) in profit from existing variants. Category demand thus expands by x(p, v) (1 − ev ) with assortment depth, providing the monopoly retailer with a marginal return to a deeper assortment of (p−c)x(p, v) (1 − ev ). Condition (12) equates this marginal benefit with the marginal cost of deepening the assortment, which is the set-up cost per variant, F . Perhaps surprisingly, profit per variant, (p − c)x(p, v), must decrease with a rise in product differentiation (i.e., smaller θ) by condition (12). Despite the higher price level, profit per variant is lower in categories with more differentiated variants than in categories with less differentiated variants. The reason is that demand per variant, x(p, v), is forced downward to maintain (p − c)x(p, v)(1 − ev ) = F as prices rise and ev declines with the degree of product differentiation in the category. The monopoly equilibrium is given by the simultaneous solution to (11) and (12): pm
c , = θ(1 − ε)
"
v m = θ1+θ(1−ε) (1 − ε)2 c−θ(1−ε)
µ
(1 − θ) F
¶1−θ(1−ε) # 1ε
.
(13)
PROPOSITION 2. Equilibrium assortment depth follows an inverted u-shaped pattern with product differentiation. The inverted u-shaped relationship between product differentiation and assortment depth arises through the interplay of two effects, the relative magnitude of which determines whether a deeper or shallower assortment arises from an increase in product differentiation. The first effect is the “survival rate” of category demand, 1 − ev , as new variants are introduced in the category.
As the degree of product
differentiation rises, a greater share of demand for existing variants survives the introduction of a new variant. The survival effect favors a deeper assortment in product categories with more differentiated variants and a shallower assortment in categories with less differentiated variants.
The second effect is the reduction in profit per
variant, π(p, v)/v = (p − c)x(p, v), as the degree of product differentiation rises in the category. Profit per variant declines with the degree of product differentiation in 12
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the category, and this favors shallower assortments in product categories with more differentiated variants and deeper assortments in categories with less differentiated variants.
As profit per variant declines, introducing new variants in the product
category eventually ceases to be worthwhile. Column (a) of Figure 1 shows the profile of monopoly assortment depth at various levels of product differentiation (d = 1 − θ) for the parameters ε = 0.5, c = 0.01, and F = 0.01.
Notice that an inverted u-shaped relationship emerges in the upper
panel of column (a) between product differentiation and assortment depth. lower panel of the figure shows the forces driving this outcome.
The
The dashed line
plots the survival rate and the solid line depicts profit per variant.
The survival
rate increases while profit per variant decreases in categories with more differentiated variants. Assortment depth rises with product differentiation when profit per variant is high relative to the survival rate, then peaks and begins to decline shortly after the point where the these two curves cross. To better understand this outcome, consider the case in which θ is arbitrarily close to one (non-differentiated variants). With non-differentiated variants, adding depth to the product assortment has no effect on category demand, ev → 1, and the equilibrium depth of the product assortment tends to zero in (13). Because category demand must remain constant with changes in assortment depth, moreover, demand per variant in (9) becomes arbitrarily large as v → 0, which makes profit per variant accordingly large.7 Now suppose we decrease θ slightly from unity, injecting an element of product differentiation in the category. As θ decreases, ev falls, so that adding depth to the product assortment begins to increase category demand. Profit per variant is “high”, and the monopoly retailer responds by deepening his assortment to facilitate category sales. But profit per variant decreases with assortment depth due to cannibalization (recall that the product of these two effects is constant), and this dampens the return to expanding category demand by introducing new variants. As retail margins widen and the level of sales per variant falls, cannibalization 7
Formally, notice that pm converges to the constant pm = c/(1 − ε) > c as θ → 1.
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becomes increasingly costly, and the equilibrium depth of the product assortment ultimately becomes shallower.
3.2
The Oligopoly Outcome
In an oligopoly retail environment, consumers choose among retailers according to location and the relative attractiveness of the nearest retailers’ product assortments and prices.
Substituting the demands in (8) into the indirect utility function and
integrating the resulting expression yields v(p, v, y) = (1 − θ(1 − ε)) (θ(1 − ε))
θ(1−ε) 1−θ(1−ε)
µZ
v
i=0
−θ 1−θ
pi
¶ (1−θ)(1−ε) 1−θ(1−ε) di + y.
Next, make this substitution into (2) and (4) and differentiate retailer profit with respect to pi and m. Evaluating terms in the symmetric equilibrium gives ∙ µ ¶ ¸ p−c (p − c)vx(p, v) 1 = 1− ep , t n p ¶ ¸ ∙µ 1 − θ vpx(p, v) 1 − ev + = F. (p − c)x(p, v) θ t n
(14) (15)
Notice that conditions (14) and (15) introduce incentives on the inter-retailer margin that are absent from (11) and (12).
The term on the left-hand side of (14) is the
market share effect of a price change. This effect, which forces the right-hand side of the equation away from zero, leads the retailer to discount category prices from the monopoly level in an attempt to acquire market share from his rivals. The market share effect of a price discount is stronger for smaller values of t while the incentive to raise prices to augment profit per customer is stronger for smaller values of n. The terms in the square brackets of (15) represent the effects of a deeper product assortment on the inter-retailer and intra-retailer margins, respectively. Both these terms are positive since providing a deeper assortment attracts store traffic and also facilitates greater category demand per customer. The short-run oligopoly equilibrium is given by the simultaneous solution to (14) and (15).
Column (b) of Figure 1 shows the equilibrium assortment depth in a
duopoly market (n = 2) for the parameters ε = 0.5, c = 0.01, F = 0.01, and t = 10. 14
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Changes in the degree of product differentiation (d = 1 − θ) produce a qualitatively similar profile of assortment depth as in the monopoly case. The reason is that both the inter-retailer and intra-retailer margins of (15) are weighted by profit per variant, (p − c)x(p, v).
Profit per variant increases in θ, so that greater product differen-
tiation dampens the return to assortment depth simultaneously on both margins of retailer profit. Relative to the monopoly case, the additional market share effect of assortment depth in the oligopoly case accounts for the sharper rise in assortment depth at low levels of product differentiation and its more pronounced fall at higher levels of product differentiation. Figure 2 depicts the relationship between assortment depth and store differentiation for alternative measures of store differentiation (higher t or lower n). In the left panel of the figure, the product assortment becomes shallower as t increases. An increase in t weakens the market share effect of a deeper assortment, reducing retailers’ strategic incentives to deepen their assortments. In the right panel of the figure, the product assortment becomes deeper as market concentration ( n1 ) increases. An increase in market concentration allocates a larger share of customers to each retailer, strengthening retailers’ incentives to deepen assortments. Both category profits and category sales increase monotonically in θ. Given the non-monotonic relationship between product differentiation and assortment depth, this implies that the relationship between assortment depth and category sales depends on the degree of product differentiation in the category. For relatively nondifferentiated products, a decrease in θ leads to lower category sales and a deeper product assortment; however, for highly differentiated products, a decrease in θ leads to lower category sales and a shallower product assortment. The long-run oligopoly equilibrium is given by the simultaneous solution to (7), (14) and (15). These outcomes are qualitatively similar to the short run outcomes, with the exception that the equilibrium prices are lower and the equilibrium product assortment is shallower for all parameterizations of the model.
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4
Conclusion
Our analysis points to a rich relationship between store differentiation, product differentiation, and the equilibrium depth of the product assortment.
An inverted
u-shaped relationship emerges across product categories that vary in the degree of product differentiation, with the deepest assortment occurring for categories with an intermediate level of product differentiation and shallower assortments arising “at the tails”. The level of category sales falls with greater product differentiation, moreover, which suggests a negative relationship between category sales and assortment depth in categories with relatively non-differentiated variants, but a positive relationship between sales and assortment depth in categories with more differentiated variants. The manner in which store differentiation changes in a market also has critical implications for assortment depth. An increase in store differentiation facilitated by increased customer loyalty raises both prices and profits in the product category, but reduces assortment depth.
An increase in store differentiation precipitated by the
exit of retailers from the market also raises prices and profits in the product category; however, retailers respond by increasing assortment depth. Much of the empirical work to date on the strategic effects of product assortment decisions has considered only within-category effects. Our analysis suggests a role for empirical research that examines how product assortment decisions are used as a competitive tool among multi-product retailers to acquire market share. Recent developments in spatial econometric methods allow retailer behavior to be examined across a variety of formats (e.g., Super Target, Aldi, and Whole Foods) to directly test the strategic effects of store and category differentiation on prices and product assortments. Our results also suggest several directions for future theoretical research. It may be possible to gain a greater understanding of the forces driving assortment depth by considering alternative definitions of product differentiation.
In the oligopoly
equilibrium described in §2, the degree of product differentiation in the category, θ(x),
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depends on the volume of transactions per variant, x.
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Deepening the assortment,
in this case, implicitly changes the degree of differentiation among variants in the category. More generally, the degree of product differentiation in the category might depend both on the volume of sales per variant and on the number of variants, θ(v, x), as would be the case if the attribute space becomes crowded as new variants are introduced. Broadening the measure of product differentiation along these lines may lead to important new insights. Another potential research direction is to consider manufacturer decisions to add variants to their product lines. The explicit consideration of retailer-manufacturer interactions in determining assortment depth can expand the reach of the present model by encompassing a role for slotting allowances in retail transactions.
To
the extent that contracts between manufacturers and retailers that involve slotting allowances are used to facilitate higher retail prices in a product category (see, e.g., Shaffer 1991), such a practice would also increase retailer returns to deepening the assortment. A variation of the present model that considers the returns to slotting allowances in categories that differ in the degree of product differentiation seems promising in explaining why slotting allowances arise in some retail categories but not in others. A particularly interesting extension of the analysis would be to consider the returns to jointly expanding assortment breadth and assortment depth. To the extent that adding new categories to the product assortment increases retailer market share and stimulates sales in existing categories, retailer incentives to increase assortment breadth would entail a similar decomposition of effects on the inter-retailer and intraretailer margins of profit. An explicit consideration of assortment breadth in models of this kind seems fruitful in developing insights regarding the conditions under which retailers would find it profitable to increase the total number of variants in their stores by providing broader, as opposed to deeper, product assortments.
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5
References
References [1] Anderson, S. P. and A. de Palma. 1992. Multiproduct firms: A nested logit approach. Journal of Industrial Economics 40(3): 261-76. [2] Bhatnagar, A. and B. T. Ratchford. 2004. A model of retail format competition for nondurable goods. International Journal of Research in Marketing 21(1): 39-59. [3] Boatwright, P., and J. C. Nunes. 2001. Reducing assortment: An attributebased approach. Journal of Marketing 65(1): 50-63. [4] Borle, S., P. Boatwright, J. B. Kadane, J. C. Nunes, and G. Shmueli. 2005. The effect of product assortment changes on customer retention. Marketing Science 24(4): 616-622. [5] Broniarczyk, S. M., W. D. Hoyer, and L. McAlister. 1998. Consumers’ perceptions of the assortment offered in a grocery category: The impact of item reduction. Journal of Marketing Research 35(2): 166-176. [6] Dhar, S. K. and S. J. Hoch. 1997. Why store brand penetration varies by retailer. Marketing Science 16(3): 208-227. [7] Dixit, A. K. and J. E. Stiglitz. 1977. Monopolistic competition and optimum product diversity. American Economic Review 67(3): 297-308. [8] Draganska, M. and D. C. Jain. 2005. Product line length as a competitive tool. Journal of Economics and Management Strategy 14(1): 1- 28. [9] Draganska, M. and D. C. Jain. 2006. Consumer preferences and product-line pricing strategies: An empirical analysis. Marketing Science 25(2): 164-174. [10] Dreze, X., S. J. Hoch, and M. E. Purk. 1994. Shelf management and space elasticity. Journal of Retailing 70(3): 301-326. 18
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[11] Hamilton, S.F. 2009. Excise taxes with multi-product transactions. American Economic Review 99(1): 458-71. [12] Messinger, P. R. and C. Narasimhan. 1997. A model of retail formats based on consumers’ economizing on shopping time. Marketing Science 16(1): 1-23. [13] Progressive Grocer, 72nd Annual Report of the Grocery Industry, April 15, 2005. [14] Salop, S. 1979. Monopolistic competition with outside goods. Bell Journal of Economics 10(1): 141-56. [15] Shaffer, G. 1991. Slotting allowances and resale price maintenance: A comparison of facilitating practices.” RAND Journal of Economics 22(1): 120-135. [16] Shugan, S. 1989. Product assortment in a triopoly. Management Science 35(3): 304-320. [17] Spence, A. M. 1976a. Product selection, fixed costs, and monopolistic competition. Review of Economic Studies 43(2): 217-36. [18] Spence, A. M. 1976 b. Product differentiation and welfare. American Economic Review 66(3): 407-14. [19] Van Ryzin, G. and S. Mahajan. 1999. On the relationship between inventory costs and variety benefits in retail assortments. Management Science 45(11): 1496-1509. [20] Vives, X. 1999. Oligopoly Pricing: Old Ideas and New Tools. (Cambridge: MIT Press).
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FIGURE 1: The relationship between assortment depth (v) and product differentiation (d = 1-θ) under monopoly and oligopoly (a) Monopoly
(b) Oligopoly
v
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FIGURE 2: The relationship between assortment depth (v) and store differentiation in response to increased transaction costs (t) and increased market concentration (1/n) v
v
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Product differentiation, store differentiation, and assortment depth By Stephen F. Hamilton and Timothy J. Richards April 6, 2009 Across many important product categories, retailers have dramatically increased the depth of their product assortments by adding brands, sizes, flavors, and variants. We consider how the profit maximizing depth of the product assortment is altered by changes in product differentiation and store differentiation (retailer market power). We show that the equilibrium depth of the product assortment follows an inverted u-shaped pattern across categories, with the deepest assortment occurring in categories with an intermediate range of product differentiation and shallower assortments arising “at the tails”. The reason is that retail margins are higher in product categories with more differentiated variants, which raises the return to increasing category demand by deepening the assortment, but also raises the cost of adding new variants that cannibalize a portion of their demand from existing variants in the category. The manner in which store differentiation evolves also has critical implications for assortment depth. In response to a decrease in the number of retailers serving a market, equilibrium assortments become deeper; however, in response to retailer investments in store attributes that generate customer loyalty, for instance private label development, equilibrium assortments become shallower.
