A Methodology for Retailing Analysis
Harvard Business School Working Paper 98-030
David E. Bell
Graduate School of Business Administration
Harvard University
Boston, Massachusetts 02163
October 1997
Copyright © 1998 David E. Bell: March 7, 2016
A Methodology for Retailing Analysis
Abstract
We present a simple model for analyzing retail decisions. While undoubtedly
too simplistic to be used for operational purposes, it may provide insight into a wide
range of retail issues such as pricing, inventory, store location and customer
satisfaction. We explore in some detail the pricing and profitability implications of colocated competitors, determining conditions under which both may benefit from being
together. Central to our analysis is the idea of matching: a customer’s visit to a store
may not lead to a purchase, either because the store is out of stock or because it does not
have just the right kind of item, for example where fashion is important.
Keywords:
Retailing, Location Theory, Gravitation Model, Inventory, Store Siting
1
1.
INTRODUCTION
Recently, HMV Records decided to renovate its flagship store in Manhattan.
Seeking a temporary location, it found one—next door to the flagship store of its arch
rival, Tower Records.
Interviewed by The New York Times for a comment on this
disaster, the manager of the Tower store allowed that, if anything, sales were now
higher than before. Perhaps this is a victory for the principle of attraction: if you live in
Manhattan, where now do you go for a compact disk? But what do you suppose
happened to Tower’s margins and profitability after the arrival of HMV? When is it
preferable to be isolated from one’s competitors and when is it better to be co-located
with them such as in a suburban “auto-mile” or in streets of similar specialty stores
such as London’s Saville Row?
Attempts to distill retailing to its essence usually focus on its logistical dimension
(“retailing is distribution” or “location, location, location”) yet an equally important
aspect is matching, trying to arrange that the customer can select the most appropriate
item for his or her needs. The retailer achieves this, in part, by selecting from the
universe of available items those few to carry in the store, and in part by having
knowledgeable salespeople to assist the customer in making the final selection. To give
further assurance about the sustainability of the purchase, the retailer may also offer an
explicit or implicit guarantee of satisfaction, first by the quality of products she stocks
and second by offering a generous return policy. Finally, the store seeks to offer
convenience, not only in its choice of location but also by having a logical (and
2
consistent) layout and by being well staffed.
Retailers compete in their ability to
provide these services, and to do so at low cost, or at least, at low price to the customer.
But in order to be profitable, some tradeoffs must be made; very low prices are usually
associated with a lack of services, a distant location, or an incomplete assortment.
Most quantitative approaches to retailing focus on specific issues; using existing
store data to forecast performance at new sites, using past sales of a product to forecast
future demand, optimizing the layout of shelves, optimizing the performance of a
warehouse. (For recent examples, see Lal and Rao (1997), Padmonabhan and Png (1997)
and Tellis and Zufryden (1995).) But these models are very specific to the problem at
hand.
Bacon (1984) provides a comprehensive approach to retail problems that is
similar in spirit to our approach, but is quite complex.
The most widely known
quantitative models were developed by Reilly (1929) and Huff (1960) to predict demand
at retail locations. The very simplicity of these models is attractive to those seeking to
use quantitative explanations for insight into retailing dilemmas.
The model we propose here is similarly simple. Yet, as we will demonstrate, it is
robust enough to be applicable to a wide range of important retail problems. The
validity of the model may be judged in two ways; the first from the compelling nature
of its assumptions (which are transparent) and the second by its ability to generate
sensible recommendations and useful insights.
2.
THE BASIC MODEL
3
We begin with a single store offering a single product at price p. Potential
customers are distributed with uniform density around the store, as far in every
direction as we shall need. Customers each have the same regard for the product, it is
worth at most a to them; if they pay p for the product, they will be better off by an
amount a-p. We assume customers require at most one of the product “per period.”
A customer who finds himself at the store will buy the product if and only if pa,
but the customer who lives a distance d away must consider whether the time and effort
of traveling to the store is offset sufficiently by the prospect of the transaction to make
the trip worthwhile. Specifically, we assume each customer has a travel cost (per unit
distance) of t, so that a customer living at distance d from the store will undertake the
trip, and buy the product, if and only if a  p  2td the quantity 2td representing the total
amount of inconvenience of the round trip.
Trading
Area
.
d
.
Customer
Store
The retailer buys the product at price c and so makes a margin p-c on each sale.
She will select p so as to maximize the total margin collected from all those who buy.
For the sake of algebraic simplicity, we will assume that the number of potential
customers in a trading area of radius r is r2, an effective density of 1/.
4
Finally we assume the situation is “in equilibrium.” Potential customers know
the quality of the product (their willingness-to-pay a) and the price charged by the
retailer (perhaps through previous familiarity with the store, or through advertising).
Later, when we consider the possibility that the retailer may be out-of-stock, we will
assume the customer knows the historic rate at which the retailer is out-of-stock (say
10% of the time) but not whether she will be out of stock at the time of any particular
visit. Naturally there are a myriad of refinements that could be made to the basic
model, and some of these, like the possibility of out-of-stocks, we will analyze in this
paper. The basic model, described above, is intended only as a foundation.
Our primary focus will be on buying items at destination stores. These are items
and stores that determine the reason for the shopper’s trip. We might also have called
these planned purchases and planned trips. Items that are bought, or stores that are
visited incidental to some other activity (including, perhaps, a planned trip to a nearby
store), we will refer to as impulse purchases or impulse stores. As at a destination store,
customers will buy the store’s product so long as pa. We will reserve the term
convenience store to refer to stores whose primary attractiveness is their proximity.
Clearly, a real store will serve in all three of these capacities to various people
and at various times.
The Filene’s Basement discount apparel store in downtown
Boston is a destination store for many residents of the region, it is an impulse store for
the thousands of subway travelers who pass through it on the way to work (or to other
5
stores), and it is a convenience store for office workers in the area. Our desire here,
however, is to strive for simplicity. Unless noted, we assume all customers make a
planned trip to buy a store’s product.
2.1
Analysis of the Basic Model
A potential customer, living at distance d from the store, will undertake a trip to
the store, and buy the product if a - p  2td. The breakeven distance that defines the
trading area is thus (a-p)/2t. Total store demand (per period) is thus [(a-p)/2t]2. The
following is an elementary implication of the model.
Proposition 1
Demand (at a destination store) increases as the square of added value offered
to the customer.
The total profitability of the retailer is given by ( p  c )[( a p ) / 2t ]2 which is
maximized when p = (a + 2c)/3. The retailer’s profits at this price are (a-c)3/27t2. We
will describe the quantity a-c as the “added value created by the retailer.” When priced
appropriately, one-third of this added value is retained by the retailer (p-c = (a-c)/3) and
two-thirds is given to the customer as an inducement to travel (a-p = 2(a-c)/3). Demand
may also be calculated as (a-c)2/9t2.
Proposition 1A Demand (at a destination store) increases as the square of the added value
created by the retailer.
Proposition 2
Two thirds of the added value created by the retailer should be offered to the
customer as an inducement to visit a destination store.
Proposition 3 The profitability of a destination store increases as the cube of the added value
6
created by the retailer.
Contrast Proposition 3 with the case of an impulse store. At an impulse store,
passersby purchase the product so long as pa, since they incur no cost to visit the store.
The retailer should thus set the price at a. If the number of passersby per period is T the
profitability of the store will be T(a-c), which is linearly proportional to the value added.
As an illustration of these propositions, consider the question of providing
entertainment at a store. Suppose that a piano player would increase the enjoyment of
each customer by an amount e, at a total cost of E per period to the retailer. In effect the
value of the product has been increased to a+e, so the retailer should raise prices to (a+e
+2c)/3 leading to profits of (a+e-c)3/27t2. In accordance with Proposition 2, the retailer
passes on two-thirds of the added value e by charging only e/3 extra for the product.
The retailer should hire the piano player if [(a+e-c)3 - (a-c)3]/27t2 is greater than E.
As another illustration, suppose the retailer stocks a second product, say gum, as
an impulse item to customers. We will take it as a given that the retailer will charge the
maximum possible price for the gum (equal to the customers’ willingness-to-pay for the
product), and that all customers of the store’s principal product will also buy one unit
of gum at that price. Thus the retailer now makes a total margin of p-c+m per customer
(m being the margin on gum). Demand is still proportional to (a-p)2 and so the retailer’s
profits are proportional to (p-c+m)(a-p)2. This is maximized when p = [a+2(c-m)]/3. The
retailer’s best strategy, consistent with Proposition 2, is to return two-thirds of the added
7
value created by the gum in the form of a price reduction on the principal product
(which is what drives customer visits).
2.2 The Consequences of Being Out of Stock
An especially frustrating experience for the shopper is to travel to a store only to
find it out of stock. Let q be the probability that that store is able to satisfy any given
customer. In our simple model, 1-q is the probability that the store happens to be out of
stock at the time of the customer’s visit; more generally q may be thought of as the
conversion ratio, the fraction of customers who actually buy. In reality q is an interaction
between the store’s stocking decision, the unpredictability of demand, and in the case of
multiple products, the ability of salespeople to find a match between available products
and the customer’s needs. For now we will take q to be a constant.
The customer, who is aware of q, but not whether the retailer is in-stock on this
particular occasion, will the visit the store if (a-p)q  2td. The retailer will make a
margin of (p-c)q per visitor, and so should choose p to maximize (p-c)q[(a-p)q/(2t)]2 . The
optimal price is therefore unchanged by q, but the impact on profitability is
considerable, being proportional to q3. In fact this cubic relation between profitability
and q is a consequence of Proposition 3; average value created is now only (a-c)q and so
profitability is proportional to [(a-c)q]3. If q drops to 0.99q, profits drop by a factor of
0.993 = 0.97.
Proposition 4
A 1% increase in the conversion ratio leads to a 3% increase in (gross)
8
profits.
Wal-Mart pioneered the use of a greeter to welcome customers to the store; the
importance of the conversion ratio suggests that there is a role to be played in asking
people, as they exit the store, if they were able to find what they came for.
2.3 Return Policies
Suppose the retailer occasionally sells a product to a customer who later regrets
the choice; for ease of modelling we assume that such a customer finds the product
worthless (in retrospect). We assume customers know the average rate of returns, r, but
they cannot anticipate whether they will be more or less likely than the average to wish
to make a return.
We will consider three cases depending on whether the store does or does not
permit returns to be made, and if it does, on whether the customer needs to make a
special trip to do so. The value proposition to the retailer and the customer in each case,
the optimal price, and the retailer’s relative profits (relative because they are multiplied
by 27t2), are as follows:
Expected
Retailer
Margin
No Returns Allowed p-c
Returns Allowed
(p-c)(1-r)
No Special Trip
Returns Allowed
(p-c)(1-r)
Extra Trip Needed
Expected
Customer
Surplus
a(1-r)-p-2td
(a-p)(1-r)-2td
Optimal
Price
(a(1-r)+2c)/3
(a+2c)/3
(a-p)(1-r)-2(1+r)td
(a+2c)/3
9
Optimal
Profits
(a(1-r)-c)3
(a-c)3(1-r)3
(a-c)3(1-r)3/(1+r)2
We may use the results in this table to investigate whether a retailer should allow
returns and if so, when? If no special trip would be required then a retailer always does better
by permitting returns since (a-c)(1-r)>a(1-r)-c. When returns are permitted and require a
special trip (a small appliance might be a typical example) the profits, for small r, are
approximately (a-c)3(1-r)5. Let 1-r be the “rate of customer satisfaction.”
Proposition 5
If returning an item requires the customer to undertake a special trip then a
1% increase in the rate of customer satisfaction leads to a 5% increase in profitability.
This proposition is a reminder, if one were needed, of the deleterious effect on
profitability of mismatching customers with products. If a special trip is required,
returns should be allowed if
( a  c )3 (1  r )3
 ( a(1  r )  c )3
2
(1  r )
which for small r is equivalent to requiring (a  c )3 (1  5r )  (a  c )3  3ar(a  c)2 or
 5(a  c )   3a or c / a  0.4 . To get a sense of this ratio, note that the gross margin of
the product (in our base model, and also here if returns are allowed) is
pc
ac
1 ca


.
p
a  2c 1  2 c a
The constant c/a > .4 is equivalent to a gross margin under 33%.
Proposition 6
If returns require a special trip, the retailer should permit returns only if gross
margins are low.
10
To understand the logic of this result, consider the marginal customer, the one at
the edge of the trading area, for whom the expected added value of the product just
equals the round trip travel cost.
A trip to return the product is only worth
undertaking (rationally speaking) if the price (the amount to be received for returning
the product) exceeds the travel cost. In our base model this means p > 2td = a-p (at the
boundary of the trade area), or (a+2c)/3 > 2(a-c)/3 or a gross margin less than 50%. In
addition, from a systems point of view, it is inefficient for the retailer to “acquire” a
product by having a customer incur a travel cost of 2td if this exceeds c, the cost of
acquiring the product directly. The breakeven condition c=2td for the most distant
customer occurs in the base model when c=2(a-c)/3 or c/a > .4.
3.
INVENTORY POLICIES FOR UNCERTAIN DEMAND
So far we have assumed inventory policy to be out of our control; at first we
assumed the product was always available, and then that it was out-of-stock with a
given frequency 1-q. Now we will assume that demand can go up or down by as much
as a fraction k either way; a demand that before we assumed was D can now be
anywhere from (1-k)D to (1+k)D (with uniform distribution between). The retailer now
selects an inventory I, at a cost of cI, risking the prospect of losing sales if I is too low, or
discarding excess inventory if I is too high.
Expected (unit) sales may be calculated as follows:
11
E ( Sales )  E ( Sales | I  Demand ) P( I  Demand )  E ( Sales | I  Demand ) P( I  Demand )
  (1  k ) D  I

I  (1  k ) D
 I  (1  k ) D  

 I




2

  (1  k ) D  (1  k ) D   (1  k ) D  (1  k ) D 

2 I ( I  k ) D  I 2  (1  k ) 2 D 2
4kD
.
Let q(I) = E(Sales)/E(Demand) be the proportion of customers who are able to
buy the product under this stocking policy. Then
2 I (1  k ) D  I 2  (1  k )2 D 2
q( I ) 
4kD 2
so
I2 - 2(1+k)ID + (1-k)2D2 + 4kD2q = 0
or
I  D[1  k  2 k (1  q)] .
The retailer would like to maximize pE( Sales)  cI  D[ pq  c(1  k )  2c k (1  q)]
with
respect to p and q. Recall from section 2.2 that the expected demand D is proportional
to (a-p)2q2. Our “equilibrium” assumption punishes stores that are conservative in their
choice of I, leading to low values of q. The optimal solution in terms of p and q is
1 q  k c
2
a2
and
p
a( a 2  2ac(1  k )  5kc 2 )
3( a 2  kc 2 )
with profits of ( a  c )3 ( a  kc )3 / 27a 3 .
12
If the retailer has a choice among products to offer, she should evidently select
the one that maximizes (a-c)(1-kc/a). For small k this is equivalent to maximizing (a-c).
For k=1 (the largest value our uniform distribution assumption permits) the choice of
product should maximize (a-c)(1-c/a). So, while a-c is the main driver of profitability,
there is a necessary bias to products with a high ratio of added value, a/c, as the fashion
(obsolescence) risk increases.
Proposition 7 A retailer should select among items with predictable demand according to
their direct product profitability (related to a-c), but among items with variable demand
according to their gross margin (related to a/c).
The probability that some customer finds the store to be stocked-out is [(1+k)DI]/2kD = c/a. This very simple result is reminiscent of the newsagent problem (also
known as the critical fractile problem). It is well known that the inventory stocking
problem that focuses only on immediate gain and losses has the solution that the
optimal inventory is such that the chance of being out of stock for any customer is c/p,
(see, for example, Bell and Schleifer (1995)).
Proposition 8
A destination store should hold enough inventory so as to be stocked out only a
fraction c/a of the time.
Two observations are important. First, in terms of the “newsagent” view of this
problem, the retailer should act as if stock-outs represent lost profits of a-c, not p-c. That
is the retailer should act as if her fortunes and those of her customers are intertwined (cf.
Brandenburger and Stuart (1996)). The second observation is that this result is true for
13
all demand distributions (not just the uniform distribution). (This is shown in Appendix
1.)
So far we have assumed the inventory to be perishable. At the other extreme we
might assume any left over inventory has simply been purchased one period too soon.
If the holding cost is hc per period per product then the retailer should maximize
( p  (1  h )c ) E( Sales )  chI . To solve this problem, we may adapt the earlier solution by
making substitutions as follows:
Perishable Case
Non-Perishable Case
p
p - (1-h)c
c
hc
a
a - (1-h)c
For example, the retailer should now hold enough inventory so as to be stocked out
only with probability hc/(a-(1-h)c) (rather than c/a).
4.
MULTIPLE PRODUCTS
Now let us examine the basic model in the case where there are two
independently demanded products.
With products indexed in the normal way, a
consumer will travel to the store if ( a1  p1 )q1  ( a2  p2 )q2  2td .
We assume the
consumer will buy either or both products if they are available. The retailer should
14
maximize
( p  c )q ( p
Differentiating
1
1
1
with
 c2 )q2 (a1  p1 )q1  (a2  p2 )q2 
2
2
respect
to
p1
(a1  p1 )q1  (a2  p2 )q2  2 ( p1  c1 )q1  ( p2  c2 )q2  .
and
with respect to p1 and p2.
equating
to
zero
gives
The differential with respect to p2
gives the same equation.
The prices, therefore, are optimal so long as a1 - p1 0, a2 - p2  0 and
 a1  2c1

 a  2c2

 p1  q1   2
 p2  q2  0 .

 3



3
So while a range of possibilities exists, the price solution of the basic model remains
optimal in the multiple product case. The same conclusion is true for a general number
of products. That the basic solution remains optimal is useful not only because the
prices of one product do not depend on the portfolio of products it is sold with, and is
even independent of its own q, but also this price solution means the consumer’s
relative valuation of products 2(ai-ci)/3 is equivalent to those of the retailer (ai-ci)/3.
This alignment of incentives seems beneficial.
4.1
Two Independent Products with Uncertain Demand
If we analyze the problem of two independent products with uncertain demand,
we can show (but don’t here) that the optimal stocking strategies are also unchanged.
Proposition 9
For multiple independent products, with demand for each varying by a
positive uncertain factor, the optimal stocking decision for product i is such that the period outof-stock probability is
ci
ai .
15
As before, there is some latitude with pricing, but the decentralized, feasible
solution, is to set pi  ( ai  2ci x i ) / 3 where xi is the factor associated with the optimal
stocking level. In particular, if ci/ai = ½ then xi = 1.
4.2
Two Substitute Products with Uncertain Demand
Suppose now that the two products are perfect substitutes. A visitor to the store
will buy product 1 in preference to product 2 if a1-p1 > a2-p2, and either if only one is
available.
As might be expected, we can show that the retailer should price the
products so that a1-p1 > a2-p2 if a1-c1 > a2-c2. Let q1 be the probability that a typical
customer finds product 1 in stock. Let q2 be the probability that a customer, finding
product 1 to be out of stock, is able to buy product 2. In particular, the probability the
visitor finds the store completely out of stock is (1-q1)(1-q2). The expected demand is D
= [(a1-p1) q1 + (a2-p2)(1-q1)q2]2
x is a factor
with the actual demand = Dx~ where ~
distributed uniformly from 1-k to 1+k.
Using similar logic as before, we may calculate, for example, that expected sales
of product 1 are
2 I1 (1  k )D  I12  (1  k )2 D2
4kD
and of product 2 are
2 I2 ((1  k )D  I1 )  I22
.
4kD


The optimal solution may be expressed as I1  D (1  k )  2 k (1  q1 ) and
16


I 2  D 1  1  q2 2 k (1  q1 )

2
.Also
2
 c c 
c 
1  q1  k  1 2  and (1  q1 )(1  q2 )  k  2  .
 a1  a2 
 a2 
The probability that the retailer is completely out of stock for a particular visitor is c2/a2;
the probability of being out of stock in product 1 is (c1-c2)/(a1-a2).
The retailer profits are
3

1 
( c1  c2 )2 c22  
  .
( a  c )  k  c1 
27  1 1
a1  a2
a2  

The optimal prices are any that satisfy

 ( c1  c2 )2 c22  
( a1  p1 )q1  ( a2  p2 )q2 (1  q1 )  2 c1 (1  k )  2k 
  .
a2  
 a1  a2

From Section 3, if product 1 alone is stocked then profits are
3
1  ( a1  c1 )( a1  kc1 ) 
 ,
27 
a1

if only product 2 then
3
1  ( a2  c2 )( a2  kc2 ) 
 .
27 
a2

Therefore, product 1 alone is preferred to product 2 alone if (and only if)
a
1
c
c
 c1  1  k 1 a   ( a2  c2 )1  k 2 a  .
1
2
This confirms Proposition 7: in a certain (staple) world (k=0), products are ranked by a-c.
In an uncertain (fashion) world, selection is biased more in favor of products that are
high with respect to a/c, the “risk-reward ratio” being better for these products.
17
If product 1 is being stocked then it is beneficial to also offer product 2 (with a1p1>a2-p2) if 0  q2 1 , that is, if
 c2 
 
 a2 
2
a
c
1
 a2 
1  c2 
2
2
1
that is, if a2/c2 > a1/c1.
4.3
A Numerical Illustration
A perennial dilemma of the fashion retailer is how to stir up a lot of interest in a
new product, while mitigating the downside of dissatisfied customers if inventory is too
low. As the following numerical example shows, a second, substitute product, with
lower cost but higher relative value, may do the job. A private label dress may not be as
desirable as the designer label dress but it may assuage the anger of a customer who
would otherwise find the store to be out of stock.
Product 1 has a1=200 and c1=100. Product 2 has a1=150 and c2=60. Thus a1-c1>a2c2 but a2/c2>a1/c1.
No Uncertainty
The optimal retail strategy if each product is offered alone is as follows:
Price
Gross Margin %
Demand
Contribution
Product 1 Alone
133
25
4444
148,148
Uncertainty (k = ½)
18
Product 2 Alone
90
33
3600
108,000
The results of offering each product, again alone, are as follows:
Product 1 Alone
Product 2 Alone
143
2500
2500
2188
62,500
98
2534
2304
2120
55,296
50%
13%
40%
8%
Price
Inventory
E(Demand)
E(Sales)
Contribution
Stock-Out Frequency
Per Period
Per Customer
Because a2/c2 > a1/c1, the inventory is more aggressive for product 2 and profits are
almost as high as for product 1. (Indeed if c2=55, profits are higher for product 2 than
product 1.)
If both products are offered, the solution is as follows. All data are unique,
except for prices.
Price
Inventory
E(Demand)
E(Sales)
Contribution
Stock-Out Frequency
Per Customer
Product 1
140
1893
2704
1839
68160
32%
Product 2
103
1082
865
649
2144
25%
Total
-2974
2704
2488
70304
8%
As this example shows, not only are the combined profits higher with the
products jointly offered, even product 1’s profits are higher. This is largely due to the
retailer’s ability to have less safety stock of product 1, knowing that the customer can
19
always fall back on product 2, which is a less costly product in which to hold reserve
stock. This analysis suggests a strategic role for private label products when used in
conjunction with fashion sensitive brand names, that is, as a safety stock.
We have discussed earlier that a choice between either product 1 or product 2
depends on the quantity (a-c)(1-kc/a). Two products will be complementary when
offered together only if they differ in their rankings by a-c and a/c.
Proposition 10
Two products that are substitutes from the customer’s point of view, are
complementary from the retailer’s point of view if one offers higher absolute added value (a-c),
but the other offers higher relative added value (a/c). The retailer should offer better value on the
former product.
5.
CO-LOCATED STORES
Usually stores cluster together. Though zoning laws encourage this practice,
there may be advantages to both retailers and customers from co-location. We begin by
considering the case of two co-located stores that sell independently needed products,
and then move to co-located stores selling substitute products. In a later section we will
consider direct competitors who are located at a distance, but near enough to have an
impact on each other.
5.1
Co-Located, Non-Competing Stores
Since the products have independent demand the consumer will travel to the
stores and buy from both, if (a1  p1 )  (a2  p2 )  2td . The stores’ problem is to price
20
their own products to maximize profits:
Store 1: Max (p1-c1)[(a1-p1) + (a2-p2)]2
Store 2: Max (p2-c2)[(a1-p1) + (a2-p2)]2 .
These problems have an equilibrium solution with p1<a1 and p2<a2 so long as (a1-c1) <
3(a2-c2) and (a2-c2) < 3(a1-c1) :
p1 
a1  3c1 a2  c2
a  3c2 a1  c1

, p2  2

4
4
4
4
with profits to each store of [( a1  c1 )  ( a2  c2 )]3 / 64t 2 .
In this event, prices are higher at each store than when isolated, as is demand,
and as are the profits. The co-location serves as a magnet for traffic, and each store
shares the price burden of attracting customers.
If one store offers rather more added value than the other, say (a1-c1)>3(a2-c2),
then store 1 should revert to its isolation price of (a1+2c1)/3, and allow store 2 to become
an impulse store with p2=a2; effectively store 2 is a free rider on store 1’s traffic. Clearly
store 1 is no better off for store 2’s presence in this case.
Store 2, however, would prefer playing a parasitic role than being isolated so
long as ( a1  c1 )2 ( a2  c2 ) / 9  ( a2  c2 )3 / 27 or (a2  c2 )  3(a1  c1 ) . Moreover, store 2
prefers
the
parasitic
role
to
the
equilibrium
solution
when
( a1  c1 ) 2 ( a2  c2 ) / 9  ( a1  c1 )  ( a2  c2 ) / 64 or (a2  c2 ) .737(a1  c1 ) . Though there is
3
room for tactical maneuvering on the part of store 1, the most stable outcome would
appear to be as follows:
21
(a2-c2)  .737(a1-c1)
Store 1 acts in isolation
Store 2 is a free rider
.737(a1-c1)  (a2-c2)  1.357(a1-c1)
Both stores share in
attracting traffic
(a2-c2)  1.357(a1-c1)
Store 2 acts in isolation
Store 1 is a free rider
This conclusion generalizes easily if we introduce in-stock probabilities q1 and q2, simply
substitute (ai-ci)qi for (ai-ci) in the above inequalities.
The conclusion is not unexpected; smaller stores (those with less potential to add
value) will flock to the side of larger stores, raise prices and live off the larger store’s
traffic. One can understand the attraction of shopping malls to the larger (anchor) store,
since it offers a formal device (rent reduction) for compensating the large store for its
role, in effect a side payment from the specialty stores to the anchor. The demise of
downtown shopping centers may be forestalled if a similar practice developed there
too.
The analysis also suggests that two large stores side-by-side may have an uneasy
relationship as each eyes the other’s contribution toward the attraction of customers.
Proposition 11
The only stable arrangement of a cluster of stores offering non-competing
products is with one large store pricing as if isolated and one or more smaller stores charging
impulse prices.
5.2
Co-Located Competing Stores
Now let us consider two co-located stores that are competitors. A customer will
buy either, but not both of the two products, and will prefer product 1 if a1-p1 > a2-p2. If
22
both stores are always in stock, then the retailer with the higher added value, say a1-c1 >
a2-c2, can win the resulting price war by setting a1-p1 = a2-c2, thus ensuring herself a
profit but not for her competitor. At any higher price, store 2 has the temptation to steal
share by lowering its price. If the stores are not always in stock then the situation need
not be so cut throat. Assuming for the moment that a1-p1 > a2-p2, we assume a fraction q1
of the traffic will buy at store 1, and a fraction (1-q1)q2 at store 2.
Traffic at the stores will be proportional to
( a
1
 p1 )q1  ( a2  p2 )(1  q1 )q2  and optimal prices, assuming p1<a1, p2<a2 are
2
p1 
a1  3c1 (a2  c2 )(1  q1 )q2

4
4q1
and
a2  3c2 (a1  c1 )q1

.
4
4(1  q1 )q2
p2 
Again, both stores have equal profits, namely
(a  c )q  (a
1
1
1
 c2 )q2 (1  q1 ) / 64t 2 .
3
2
If we interpret qi as the ability of a store to match customers to its products, then we will
assume that if, by contrast, the prices are such that a1-p1<a2-p2 then a fraction q2 will buy
at store 2 and a fraction (1-q2)q1 at store 1.
In this case both stores find it in their interest to let store 1 price more
aggressively, i.e., take the lead, if
(a1  c1 )q1  (a2  c2 )(1  q1 )q2  (a1  c1 )(1  q2 )q1  (a2  c2 )q2
23
or (a1  c1 )  (a2  c2 ) . Our results now closely parallel those of the previous section,
for example, store 2 prefers to be a free rider if ( a2  c2 )(1  q1 )q2  .737( a1  c1 )q1 . Both
stores charge higher prices than when isolated if the equilibrium solution holds, but
only the free rider does otherwise.
When can it be good to have a competitor move next door? If you offer better
value then this will happen only if the new store is not a free rider, an unlikely event. If
instead the new store offers better value, then you, store 2 say, gain if
( a1  c1 )2 q12 ( a2  c2 )(1  q1 )q2 / 9  ( a2  c2 )3 q23 / 27
that is, if your free rider profits exceed your profits if isolated. This inequality may be
reduced to
( a1  c1 )2 q12 (1  q1 )  ( a2  c2 )2 q22 .
Interestingly, both stores share an interest in both q1 and q2 being reasonably high, in
part because this helps drive demand. In particular, as a free rider, store 2 would prefer
q1 to be 2/3, the maximum of q12 (1  q1 ) . And as the traffic generator, store 1’s only hope
that store 2 will cease free riding is if (a2-c2)q2 gets larger, in particular if q2 goes up.
The delicate balance of two similar stores is illustrated if we consider what
happens if a1=a2=a , c1=c2=c and q1=q2=q. For them to charge equal prices is unstable; the
stable solution is for one store to take the lead, with both stores making
( a  c )3 q3 ( 2  q )3 / 64t 2 . This is higher than their profits when isolated if
24
1
1
( 2  q )3 
64
27
that is, if q  2 / 3 .
Proposition 12
Competing stores profit from co-location when their conversion ratio is
relatively low.
This intuitive result matches our observation that auto dealers, rug salesmen,
and fashion apparel stores each like to cluster together, whereas bakeries, pizza shops
and hardware stores like to be separated. Category killers are “high q” stores that starve
competitors of “left-overs.” An exception to this observation is gasoline stations which
offer q=1, but often can be found in groups.
But for these retailers, convenience,
advertising and share are the battleground, not low price, which is a given.
6.
DISTANT COMPETING STORES
Consider two competing stores that are located a distance s apart. A customer
living at distance xi from store i will shop at store 1 if a1  p1  2tx1  a2  p2  2tx2
assuming of course that also a1-p1-2tx1  0.
For reasons of algebraic tractability we will take the view that a customer living
at distance x from store 1, also lives a distance s-x from store 2 and vice versa. This will
be approximately true if stores of type 1 and 2 are spread in grid-like fashion with
alternating store types at the nodes.
Such a customer will shop at store 1 if
a1  p1  2tx  a2  p2  2t (s  x) and a1  p1  2tx  0 . If the stores charge their isolation
25
prices of (ai+2ci)/3 then their trading areas will overlap if (a1  c1 )  (a2  c2 )  3ts. Store
1 will completely dominate store 2 if (a1  c1 )  2ts  (a2  c2 ) for it may choose to drive
out store 2 by charging a low enough price to bring store 2’s share to zero.
If the stores charge prices that are in equilibrium then the marginal customer will
be at
distance
x
a1  p1  2tx  a2  p2  2t (s  x)
from store 1 where
and
a1  p1  2tx  0 . Hence x = ((a1-p1)-(a2-p2)+2ts)/4t and (a1-p1) + (a2-p2)  ts. The stores’
problems are as follows:
Store1 : Max( p1  c1 )(a1  p1 )  (a 2  p 2 )  2ts 
2
Store2 : Max( p 2  c 2 )(a 2  p 2 )  (a1  p1 )  2ts 
2
with solutions
p1 
a1  3c1 a2  c2

 ts
4
4
p2 
a2  3c2 a1  c1

 ts
4
4
with profits to store 1 of (a1  c1 )  (a 2  c 2 )  4ts / 64t 2 . This equilibrium requires
3
(a1-c1) + (a2-c2)  4ts. In this case we see that prices are not necessarily lower than if the
stores are isolated, although profits will be lower.
But what if 3ts < (a1-c1)+(a2-c2) < 4ts? In this interval the stores’ pricing strategies
will be unstable. The most desirable situation will be to allow the other store to be the
price leader while you follow. Your competitor will be anxious to reverse the situation
26
and instability will result.
In the special case where a1-c1 = a2-c2 the optimal prices are pi = ci + ts with profits
of ts3/4 (assuming a-c  2ts). Note that this profitability is independent of a-c; that is, the
store makes a profit based on its relative convenience rather than because of its added
product value.
7.
CUSTOMER BENEFIT
Our calculations to date have focused on the retailers’ profits, with the customer
simply choosing to travel or not, and to buy or not. But how do the various store
locations affect the amount by which the customer benefits? We will focus on two
measures of performance. One is the average customer benefit, the second is the total
customer benefit.
The first may indicate how likely individual customers are to be
satisfied with a given configuration (but leaving out unserved customers who are not
near a store). The second gives an idea of how much society is served as a whole.
For a store with a trading area of radius d, the average benefit is experienced by a
customer living a distance 2d/3 from the store (not d/2 because the number of
customers increases with the distance from the store). The average benefit is thus (a-p)q
- 4td/3 and the total benefit is [(a-p)q-4td/3]d2. In all of the following results we will, for
simplicity of exposition, assume that all stores have the same parameters, a, c and q. We
will denote (a-c)q by v. The letter d will represent the trading radius of the isolated store
(d=v/3t) and r will be the distance between stores. We will use the isolated store as our
27
benchmark.
7.1
The Isolated Store
The average benefit is 2v/3 - 4td/3 = 2v/9. That is, of the added value created by
the store, v, one third is taken up by retailer profits, 2/9 as customer residual benefit,
and
the
remainder,
4/9
by
travel,
that
is,
for
every
dollar
of
Customer
Benefit
Customer
Travel
Retailer Profits
Allocation of Benefit at an Isolated Store (Per Customer)
added value gained by the customer, another $2 is spent in travel. The total customer
benefit is [2v/9]d2 = 2v3/81t2.
7.2
Distant Competing Stores
As trading areas begin to overlap, one might imagine that prices will drop and
convenience will go up, a double victory for the consumer. The following table tracks
key indicators for a store as the distance to its nearest competitor drops.
Isolation
(s>2d)
Price
Distant Competitors
(d<s<2d)
(a+2c)/3
a-ts
28
Convenience
(s<d)
c+ts
Average Benefit
Retailer Profit/Customer
Total Customer
Benefit (per store)
Total Profit (per store)
2v/9
v/3
2v3/81t2
ts/3
v-ts
ts3/12
v-5ts/3
ts
(v-5ts/3)s2/4
v3/27t2
(v-ts)s2/4
ts3/4
As an illustration we compare these values for four representative values of s,
namely
9
4
d , 7 4 d , 5 4 d and
Price
Average Benefit/Customer
Retail Profit/Customer
Total Profit/Store
3
4
d.
Isolated
Very
Distant
s  9d
s  7d
4
100
100
100
100
106
88
125
96
Somewhat
Distant
4
s  5d
119
63
175
68
4
Close
s  3d
4
94
263
75
11
All rows but price are independent of a, c, v and t. The table highlights the result that
stores do not compete for customers until they get quite close together.
Indeed,
somewhat counterintuitively, prices increase as stores get close enough to have
overlapping trade areas. This is because stores no longer have to give steep price
discounts to compensate distant customers for travel, since distant customers are lost to
the competitor anyway.
7.3
Co-Located Stores
In the following table, similar results are reported, this time comparing the
isolated store with co-located stores. Benefits accrue to the customer from both stores
when they are together.
Isolated
Non Competing
29
Competing
Average Customer Benefit
Total Customer Benefit
(Both Stores)
2v/3
v/3
v(2-q)/6
4v3/81t2
v3/12t2
v 3 ( 2  q )3 / 96t 2
Retailer Profits
(Both Stores)
v3/27t2
v3/2t2
v 3 ( 2  q )3 / 16t 2
The average customer prefers competing stores to be co-located if (2-q)/6 > 2/9 or
q<2/3 which accords with the retailer’s desire. Total customer benefit however, is
increased only if (2-q)3 > 384/81 or q<0.32. Co-location, while reducing the chance of a
wasted trip, increases travel expense unless q is quite low.
8. CONCLUSION
We have shown that our model may be applied to a wide range of retail
situations. For the most part, the model comes to conclusions that match intuition or
observed practice. This is good news if the model is to be an attractive vehicle for
prescriptions in non-intuitive situations, or for correcting established practices.
Appendix 1
Proposition 8A
x where ~
x is a positive relative error
Suppose demand equals (a  p) n q n ~
distribution and n is any positive number, perhaps fractional. If the cost of inventory is c, then
optimal pricing and inventory decisions imply that the chance of an understock is c/a.
Proof
The basic model of the paper corresponds to the case n=2.
30
The statement
generalizes the model in the paper by allowing demand to be proportional to any
power of the “attraction” (a-p)q. The generalization is included mostly because the
proof is no harder for general n, but also because the case n=1 may be of interest.
The retailer wishes to maximize pE(Sales) - cI. Define i by I = (a-p)nqni. Then the
retailer wishes to maximize (a-p)nqn[pq-ci]. Differentiating with respect to p and setting
to zero we have (n+1)pq - nci = aq. Differentiating with respect to i and setting to zero
we have (( n  1) pq  nci )dq / di  cq or
where
f()
is
the

density
dq / diif (i) f (i) f ( x)dx so
i
dq / di  c a . Now
function


i
of
~
x,
the
i

0
i
q(i) xf ( x)dxi  f ( x)dx
error
distribution.
So
f ( x )dx  c a .
(End of Appendix 1)
31
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32