Volume Discounts, Loss Leaders,

and Competition for More Profitable Customers

Patrick DeGraba

Federal Trade Commission

600 Pennsylvania Ave.

Washington, DC 20580

pdegraba@ftc.gov

Draft date

April, 2003

When some customers are more profitable to serve than others, one might expect 

sellers to compete more vigorously for the more profitable customers. One way sellers might 

do this is to sell goods that are purchased primarily by the most profitable customers using a 

lower mark-up than on other goods. This allows the firms to give discounts to more profitable 

customers without offering them to less profitable customers. 

This paper presents a model in which competing multi-product firms facing customers 

that purchase different quantities of goods, set prices in this manner. This model suggests a 

theory of multi-product pricing in which the markup on any particular product is inversely 

related to the average profitability of the customer that purchases the good. 

One interesting implication of this paper is that loss leader pricing might be viewed as a 

way for firms to compete more vigorously for more profitable customers. Such an explanation 

offers another characteristic of products that should be used as loss leaders. This explanation 

provides potentially testable implications about the types of goods that can (or ought to) be used 

as loss leaders and can explain why grocery stores sell turkeys as loss leaders at Thanksgiving 

and lower the price of eggs at Easter, but not flowers on Mother's Day, or candy on Valentines 

Day.

I thank Dave Reiffen for first introducing me to this topic as well as discussions about 

possible explanations. I also thank Cindy Alexander, Kyle Bagwell, Robert Frank, Robert 

Gibbons, Joe Harrington, Robert Mason and Michael Waldman for helpful comments. The 

views expressed in this paper do not necessarily reflect the views of the Commission
2

1. Introduction

Anyone who has purchased a turkey at Thanksgiving probably has noticed that at the 

time of year when demand is the greatest, the prices of turkeys are at their lowest. Discussions 

with a number of economists over the years about why this is true have primarily yielded the 

explanation that grocery store must be pricing the turkey as a "loss leader."

"Loss leader pricing is the practice of setting prices on selected products at low levels 

that generates less than the usual profit margins... For retailers the objective is to increase store 

traffic so they can sell other products at traditional profit margins... Products that are used as 

loss leaders are usually well known brands and frequently purchased."1 Thus the traditional loss 

leader story suggest that customer choose the store at which they shop based on the price of 

one good, and then purchase that good along with others, (on which store owners earn their 

profit).

This paper offers a somewhat different explanation for loss leader pricing. Rather than 

being priced to attract all shoppers, loss leaders can (or should) be used as a way of competing 

for higher profit customers (that shop at a particular store)2 by offering them discounts that are 

not available to less profitable customers. Thus, loss leader pricing can be a way of price

discriminating among groups of customers in a competitive setting. This explanation, while 

not inconsistent with the traditional explanations, implies that it is neither necessary nor sufficient 

for a product to be sold under a well known brand, nor purchased by many customers (as 

suggested above). Rather a product could be priced as a loss leader if, in a market in which 

some customers purchase bundles of products that are more profitable than bundles purchased 

by others, the product is purchased primarily by customers that purchase more profitable 

bundles.3 The reason is that, for more profitable customers, sellers have an incentive to 

compete more vigorously to keep them from purchasing another seller. I will provide conditions 

under which loss leader pricing is an equilibrium way to offer such a discount in competition.

Applying the explanation to turkey pricing, I will suggest that turkeys are effective loss 

leaders, not only because almost every customer purchases a turkey at Thanksgiving, but 

because the customers that do purchase turkeys, on average purchase more units of other 

 

1

See Busch and Houston (1985) pg. 498

2

It is not the absolute quantity of purchase that creates the incentive for a discount, but rather the relative 

difference across customers at a store. Thus, both grocery stores and convenience stores can offer 

quantity discounts even though the absolute quantity of low volume users at grocery stores is likely to 

exceed the quantity of high volume purchasers at convenience stores.
3

goods than do customers shopping at the same time that do not buy turkeys, and are thus more 

profitable. This explanation differs in a very important and potentially testable way from the 

traditional loss leader story. The logic of the traditional loss leader story suggests that (if turkeys 

are used as loss leaders at Thanksgiving then) we should expect to see candy or flowers to be 

priced as loss leaders before Valentine's Day or Mother's Day. However, we typically do not 

see such discounts. My explanation is consistent with this observation because while a 

purchaser of a turkey just before Thanksgiving is very likely to purchase more items than the 

average grocery shopper, a purchaser of candy before Valentine’s Day is not.4 Therefore, one 

should expect to see stores offering a discount in the form of discounted turkey prices at 

Thanksgiving, but no discount in the form of discounted candy prices around Valentines Day.

There are two other papers, Simester (1996) and Nagle and Novak (1988) that 

suggest a price discrimination explanation for loss leader pricing. Each of these papers looks at 

competition between two sellers in a market with three types of customers, those that are loyal 

to (will always buy from) the first seller regardless of the price he charges, those that are loyal to 

(will always buy from) the second seller regardless of the price he charges, and those that will 

price shop and purchase from the store whose prices offer them the higher customer surplus. 

They then hypothesize that there are some products that are purchased more often by price 

shoppers than by loyal customers. In such a case it can be shown that these products will be 

priced as loss leaders. 

As I explain in the next section, this explanation is fundamentally different from the 

explanation proposed in this paper. For now I simply note that for these explanations to explain 

why turkeys are priced as loss leaders at Thanksgiving, but candy is not priced as a loss leader

before Valentines Day, one must believe that the customers that buy turkeys at Thanksgiving are 

primarily customers that choose to shop at the store offering the most surplus, while those that 

buy candy before Valentine’s are primarily customers that are loyal to a particular store.

I present several models that look at the pricing of a "target" good (the turkey in our 

opening example) when the demand of other goods (called side goods) changes. I do this for 

three reasons. First, it helps explain the use of "seasonal" loss leaders such as turkeys at 

 

3

If the margin on each product is roughly the same, more profitable customers will be large volume customers.

4 Hosken and Reiffen (2002) report that eggs typically have a lower price at Easter than at other times. This 

is consistent with the hypothesis that the average person purchasing large number of eggs at Easter is likely 

coloring them for their children, and would likely be buying groceries for a family and is therefore a larger 

than average customer. Chevalier et al (2003) find empirical evidence that lower prices during peak demand 

period is consistent with loss leader pricing.
4

Thanksgiving. Second, it is the prediction that loss leader pricing will be used when the demand 

for other goods is high and that it won't when such demand is low that distinguishes this theory 

from the other theories in the literature. That is, my theory predicts that when overall demand 

for other goods increases for a group of customers, the price of goods bought primarily by these 

customers will fall. None of the other theories make this prediction. Finally, and most 

importantly it highlights the more general principle that applies to the more traditional or 

everyday loss leaders such as diapers and milk, which is that some goods are priced as loss 

leaders (i.e. close to or even below cost) on a regular basis because customers that purchase 

them regularly are more profitable than the average customer. 

Therefore this theory is not just a theory of loss leader pricing during holidays, rather it is 

a general addition to the theory of pricing based on a correlation between the total profitability of 

the bundle of goods purchased by consumers and the propensity of them to purchase a particular 

good. I use the "holiday setting" as an explanatory device because the change in price of the 

"holiday good" resulting from a change in the volume of demand highlights the theory, and 

provides a potential test.

This paper offers another contribution. Recently there has interest in explanations for 

why seller would grant customers quantity discounts.5 This paper offers one simple explanation. 

When the distribution of taste across small volume purchaser is the same as the distribution of 

taste across large volume users, each firm's demand from large volume users is more elastic, with 

respect to quantity6 than the demand from small volume customers. This results in lower per unit 

prices for larger volume customers.

2. Literature Review

2.1. Loss Leaders

There is a growing literature studying the phenomena of loss leader pricing. Hess and 

Gerstner (1987) argue that when customers purchase impulse goods (good purchased by a 

customer that does not compare the price at other stores), a store may price a shopping good 

(a good for which a shopper compares price) below marginal cost in order to attract customers 

who then purchase impulse goods. 

Bagwell and Ramey (1994) provide a coordination explanation for loss leader pricing. 

 

5

See for example Snyder 1996, Chae and Heidhues (1999) and Chipty and Snyder (1999).

6

 The elasticity with respect to customers is identical in this model
5

They conjecture that some stores can exploit economies of scale in retailing better than other 

stores. Then, if such a store obtains many customers, it can offer lower prices because it 

realizes economies of scale. For this to happen, customers must know which store can take 

advantage of the economies of scale (thus the need to coordinate). Loss leader pricing (or loss 

leader pricing along with dissipative advertising) can credibly signal which store has the 

economies of scale and act as a coordination device. 

 Lal and Matutes (1991) present a model in which two stores choose how many 

products to sell, which products to advertise (if any) and what prices to set for each good. 

Customers can choose the number of stores at which to shop and which good(s) to purchase at 

each store and can only know the price of an item if it is advertised. They show that there exists 

an equilibrium in which each store chooses to advertise only one good, which is sold below cost 

(the loss leader) and that customers make their "shopping decisions" based on this information 

and rationally formed expectations regarding the unadvertised price(s). In their equilibrium 

customers purchase the loss leader at a price below cost, but all other goods at a price equal to 

their reservation value.

Simester considers a market in which there are two sellers. Each seller has a set of 

customers that is loyal to him. Loyal customers are those that always buy from the same seller 

regardless of the price, although the quantity they purchase varies inversely with price. There is 

also a group of customers called price shoppers that will purchase from which ever seller's 

prices offer the customer the greater surplus. In this market setting Simester shows that in 

general goods that are priced as loss leaders are those that are purchased primarily by price 

shoppers and those for whom demand is most elastic. 

Nagle and Novak (1988) also hypothesize the existence of price shoppers over whom 

stores compete as well as a set of "convenience shoppers" associated with each store. They 

argue that there are certain demographic characteristics such as the number of children in a 

family, income, number of adults that work, etc., that will make a shopper more likely to be 

either a price shopper or a convenience shopper. Using grocery store data they then provide 

empirical evidence suggesting that items most often purchased by customers with demographic 

characteristics that make them more likely to be price shoppers have lower mark ups than items 

that are not. 

Perhaps a useful way to distinguish the model in this paper from those in Simester 

(1996) and Nagle and Novak (1988) is that in their works, some customers are more 
6

responsive to a price decrease (in terms of switching sellers) than other customers. That is, a 

one dollar price reduction offered to price shoppers generates more additional customers than a 

one dollar price reduction offered to loyal customers (which generates no extra customers). In 

our model a one dollar price reduction offered to low volume customers will cause just as many 

low volume customers to switch sellers as will a one dollar price reduction offered to high 

volume customers. The difference is that high volume customers (all else equal) are potentially 

more profitable to a seller than a low volume customer so sellers are willing to offer them deeper 

discounts to steal them from the competing seller.7

There are two other works related to loss leader pricing. Denekere, Marvel, and Peck 

(1993) show that a manufacturer might not want its good to be priced as a loss leader because 

in a market with uncertain demand, all retailers might not choose to stock the good. There 

analysis however does not explain why a retailer would choose to sell a particular good as a 

loss leader.8 Walters and McKenzie (1988) empirically study the effects of loss leader 

marketing on firm profit. They show that firms make the same profit when they use loss leader 

pricing as when they don't and that the use of a loss leader does not significantly increase the 

amount of customer traffic. As we will discuss later, our theory is consistent with both of these 

findings.

 

7As we will see competition will dissipate all of the additional potential profits.

8

In their model the retailer only sells one type of good. Thus, the model cannot capture the decision 

regarding which good to sell as a loss leader.
7

2.2. Quantity Discounts

Recent work explaining why large buyers can command lower prices fall into three main 

categories. First work by Snyder (1996, 1998) suggests that when sellers tacitly collude, it is 

more difficult to prevent a firm from shading price with respect to a large customer than with 

respect to a small customer. Thus colluding firms would set lower prices to large customers to 

reduce the incentive any one firm would have to lower price to a large firm. My results differ 

from those of Snyder in that there is no collusion between competing firms. Rather all prices are 

Nash equilibrium prices of a one shot game.

Second work by Horn and Wolinsky (1988) Stole and Zwiebel (1996) Chipty and 

Snyder (1999) and Chae and Heidhues (1999a, 1999b) look at bargaining between a 

monopolist and different size sellers. In these models joint surplus between buyers and the seller 

is increasing but strictly concave in total output. Each buyer views himself as the marginal buyer 

and so bargains over the marginal surplus given that all other buyers have completed their 

bargains. It is assumed that the sell and buyer split the (perceived) surplus, (a la the Nash 

bargaining solution). The surplus retained by the seller is Interpreted as a payment from the 

buyer to the seller. Because the surplus function is concave, the average surplus per unit of 

output is smaller for large buyers than for small buyers. Thus when the surplus is split, the seller 

receives a lower surplus per unit than he receives from small buyers. This lower surplus is 

interpreted as a quantity discount.

My results differ from these results in that because all customers inelastically demand a 

fixed number of units, and marginal cost is constant, the joint surplus between a firm and 

customers is linear. In addition these results rely in the assumption that buyer and seller divide 

evenly the surplus from a transaction. In my work the division of surplus is determined 

endogenously.

A third explanation DeGraba (2003) is that sellers who are risk averse and cannot view 

each customer’s valuation for a good but can observe the number of units they demand find 

larger customers riskier than smaller customers. To mitigate some of the risk associated with 

larger customers, sellers offer them lower prices. In the paper that follows I rely neither on risk 

averse sellers nor incomplete information.

3. The Basic Model

I now present a model of competition between two competing multiproduct firms and 
8

compute equilibrium prices for their products. I will compare prices of products in a “normal” 

period to those in a test period. I will define loss leader pricing as a price reduction on a given 

good during the test period9, and look for characteristic of goods that will be priced in this 

manner.

Consider two firms indexed by i ∈ {0, 1} located at opposite ends of a Hotelling line of 

length 1. Customers are located uniformly along the line and have linear travel cost normalized 

to 1 per unit traveled. Customers demand two goods, a “target” good (denoted t) and a side 

good, (denoted s). These goods are produced at zero marginal cost by the two firms.

On a "normal week" each customer demands one unit of the target good and sn units of 

the side good. On a test or "holiday week" each customer purchases one unit of t and sh units 

of s. Assume sh > sn to capture the notion that customers purchase more in preparation for the 

"holiday" than they do during a normal week. 

Given this structure I construct the following game. Firms initially set the prices of the 

two goods, Pti and Psi simultaneously. Customers observe Pti and Psi and purchase from the 

firm at which the total cost of purchasing, Pti + swPsi is lower, where w ∈ {n, h}. The firms’ 

payoffs are their profits.

In proposition 1 I compare the equilibrium prices charged by each firm during a normal 

week to the prices charged on a holiday. 

Proposition 1. Firms charge a lower per unit price during a holiday week than during a normal 

week.

Proof: Let z* be the location of the customer purchasing sw units of s (and 1 unit of t) who is 

indifferent between purchasing from firm 0 and firm 1. z* satisfies

Pt0 + swPs0 + z* = 1 - z* + Pt1 + swPs1. (1)

Thus firm i's objective function is

Õi = (Pti + swPsi)(1 + (Pt-i + swPs-i - Pti - swPsi))/2 (2)

 

9 More generally, one could calculate the Lerner indices for all equilibrium prices and interpret the products 

with the lowest index as the loss leader. This would allow for product with small but positive margins to be 

considered loss leaders.
9

where -i indicates firm i's competitor. The first order conditions indicate that 

Pti + swPsi = 1. (3)

Since sh > sn, firms charge a lower per unit price during holidays weeks than during normal 

weeks. QED.

This result establishes the idea that when demand per customer increases, competing 

firms have an incentive to lower their per unit prices in an effort to prevent customers at the 

margin from purchasing from their competitors. To understand this note that for any given level 

of customer demand, the profit maximizing price is set such that a marginal decrease in the 

prices of a seller's goods results in an increase in profits from sales to additional customers that 

just offsets the decrease in profit due to the reduction in the price/cost margin. When demand 

per customer increases, the marginal customer becomes more profitable (as long as he buys 

units whose price is greater than marginal cost) but the cost of the price decrease (the lower 

price/cost margins) needed to attract him does not change. Thus, after demand per customer 

increases, firms would strictly prefer to lower their per unit prices, which effectively amounts to 

a volume discount.

10

An alternative explanation for a monopolist's use of volume discounts is to discriminate 

among different customers who have differing valuation for the marginal unit. We use the 

Hotelling model to rule out this explanation since all customers demand all units inelastically. 

Thus, the use of volume discounts in this model is driven entirely by the fact that as customers’ 

demand increases they become more profitable at the margin so sellers have a greater incentive 

to lower price. 

In proposition 1 the additional price competition caused by higher per customer demand 

dissipates the entire potential profit from the increase in demand. Thus, customers have the 

same total payment during normal weeks and holiday weeks, even though they purchase more 

during the holiday week. The equality of payments across weeks is an artifact of the 

assumptions that i) quantity is demanded inelastically and ii) the increase in demand per 

customer occurs uniformly along the line. 

 

10 Lal and Matutes 1991 make a similar observation.
10

The effect of i) can be seen in a model in which the quantity demanded is strictly 

decreasing in price. In this case it can be shown that a customer's total payment in a holiday 

week is greater than his payment in normal weeks. I chose the Hotelling framework 

specifically to avoid this effect. The model highlights the fact that an increase in demand per 

customer results in lower prices precisely because firms compete more vigorously over the 

marginal customer, and is independent of the fact that customers buy more at a lower price.

To understand the effects of ii) note that if demand per customer increased only for 

those customers near the "ends" of the Hotelling line, then equilibrium prices would be higher, 

and a customer's total payment in a holiday week would be greater than his payment in a 

normal week. Similarly, if demand per customer increased only for customers in the "middle" 

of the line then equilibrium prices would be lower, and the total payment in a holiday week 

would be less than the payment in normal weeks.

This result is particularly noteworthy since in many models an increase in demand will 

cause an increase in price. This difference is due to the fact that in most models (where 

equilibrium price is greater than marginal cost) an increase in demand causes the demand curve 

to be more inelastic leading to an increase in price.11 In this model an increase in per customer 

demand causes each firm's demand curve with respect to each item's price to be more elastic 

(for reasons discussed above) leading to lower equilibrium per unit prices.12

While these results indicate that overall prices will fall when per customer demand 

increases, the previous model with only one type of customer, does not indicate which prices 

will fall. In the next section I introduce more than one type of customer and show that the 

distribution of customer types will uniquely determine which prices decrease as a result of a per 

customer increase in demand.

4. Multiple Customer Types

In the previous section I showed that an increase in demand per customer will lead to 

discounted prices. I now show why at Thanksgiving stores would offer this discount by 

reducing the price of turkeys as opposed to lowering the price of all goods. Essentially I will 

 

11 In models where P = MC (i.e. perfect comp etition) the increase in price is not driven by changes in 

demand elasticity but because the market equilibrium simply move up the supply curve.

12 When a customer buys good A and B in the same shopping trip, an increase in the per customer quantity 

demanded of good A, makes the demand for B faced by each firm effectively more price elastic because 

lowering the price of B now generates more revenue from the sale of A.
11

argue that on a given holiday week customers that buy a turkey will purchase more side goods 

than those that do not buy a turkey. Thus, by offering a discount on turkeys, sellers can give 

discounts to those who buy large quantities while charging small buyers higher prices. 

I expand the model in the previous section as follows. There are two types of 

customers T and S. Type S customers demand sS units of s on both normal weeks and holiday 

weeks and never buy t. They are distributed uniformly along the line and have mass mSn during 

a normal week and mSh during a holiday week. Type T customers always buy one unit of t, sTn

units of s during a normal week and sTh units during a holiday week. Type T's are uniformly 

distributed along the line and have mass mTn during a normal week and mTh during a holiday 

week. Assume sTn = sS -1 so that during a normal week type T's and S's purchase the same 

number of units. Let ct be the marginal cost of producing the target good and cs be the marginal 

cost of producing the side good.

Proposition 2. If sTh = sS then sellers offer a volume discount by lowering the price of t. 

Further P*t is a decreasing function of sTh/sS and is independent of the mass of type S's and type 

T's. 

Proof:

The objective function of firm i is 

Õi = (Pti - ct + sTw(Psi - cs))(1/2)(1 + Pt-i + sTwPs-i - Pti - sTwPsi)mTw + 

(sSPsi - cs)(1/2)(1 + sSPs-i - sSPsi)mSw. (4)

The first order conditions are

∂Õi/∂Pti = {(1 + Pt-i + sTwPs-i - Pti - sTwPsi) - (Pti - ct+ sTw(Psi - cs))}mTw = 0 (5.a)

∂Õi/∂Psi = {(1 + Pt-i + sTwPs-i - Pti - sTwPsi)sTw- (Pti - ct+ sTw(Psi - cs))sTw}mTw 

+ {(1 + sSPs-i - sSPsi)sS - (sSPsi - cs)sS}mSw = 0, (5.b)

which reduce to

P*ti - cti + sTw(P*si - cs) = 1 (6.a)
12

sS(P*si - cs) = 1. (6.b)

Since sTn = sS - 1, the price/cost margin during a normal week is 1/sS. During a holiday 

week (6.b) implies the price cost margin on s is still 1/sS. sTh = sS in (6.a) implies P*ti - cti = 0, 

yielding the result that the discount is granted solely by reducing the price of t. The fact that P*ti -

cti is decreasing in sTh/sS obtains immediately from substituting 

(P*si - cs) = 1/sS into (6.a). The lack of dependence of P*ti on mTw and mSw is immediate from 

(6.a). QED.

To understand why the discount is given solely by discounting t, simply note that (as 

proposition 1 suggests) sellers wish to give discounts relative to the normal week’s prices only 

to high volume customers (type T's during a holiday week). Since all high volume customers 

purchase t, the discount can be given to them without being given to the low volume customers. 

The fact that P*ti is decreasing in sTh/sS simply reflects the fact that the greater the 

number of units of s demanded by a type T customer (at a mark up of 1/sS) the more profitable 

he is at the margin. Thus competition causes sellers to offer a deeper discount in order to 

prevent him from switching sellers.

The mass of customers purchasing, mT, has no affect on the price of the loss leader (or 

even if the good should be used as a loss leader) in part because in this model an increase in the 

number of customers uniformly across the line has no effect on the elasticity of demand. This is 

because such an increase in the number of customers increases that quantity sold by the same 

proportion that it decreases the slope of the demand curve facing the seller.

The lack of dependence of P*ti on mT and mS is also due to the fact that all high 

volume customers and none of the low volume customers purchase the target good.

Note that in this case the monopolist's profit during the holiday week is the same as during the 

normal week, as found in Walters and McKenzie (1988). This is because the competition for 

the higher volume customers in the form of the loss leader price discount completely dissipates 

the potential additional profits from the higher per customer demand. Of course, in this model 

the traffic cannot increase since the number of customers is fixed. However, the fact that in such 

a model loss leader pricing is used in equilibrium does highlight the fact that loss leader pricing 

can be used by each firm to steal customers from competitors, rather than generate new 
13

customers in the market.

 I now consider pricing when some high volume and some low volume customers 

purchase the target good. As before the total number of those purchasing the good has no 

effect on the price, however the proportion of high volume to low volume customers will have 

an effect on P*ti. 

I expand the model of this section in the following manner. Assume that during a holiday 

week there are two types of type T customers, high volume customers and low volume 

customers. High volume customers purchase sThh units of the side good during a holiday week 

and low volume customers purchase sTlh, where sThh > sTlh. High and low types have mass mTl

and mTh respectively and each are distributed uniformly along the line.

Proposition 3. The price of the loss leader is a decreasing function of the proportion of type T

customers that purchase high volume.

Proof: The objective function for firm i during the holiday week is

Pi = (Pti - ct + sTlh(Psi - cs))(1/2)(1 + Pt-i + sTlhPs-i - Pti - sTwPsi)mTlh + 

 (Pti - ct + sThh(Psi - cs))(1/2)(1 + Pt-i + sThhPs-i - Pti - sTwPsi)mThh +

(sSPsi - cs)(1/2)(1 + sSPs-i - sSPsi)mSh. (7)

and arithmetic similar to that of proposition 2 yields

P*ti - ct = 1- (sTlh

2mTl + sThh2mTh)/sS . (8)

 _ ___ _ _ __________________________________________________________________________________________________________________________________________________________________________________________________________________________________ .

 (sTlhmTl + sThhmTh)

Since both the numerator and the denominator of (8) are weighted sums of mTl and mTh P*ti is 

a function of the ratio mTl/mTh and since sThh > sTlh, it is decreasing. QED

The intuition behind this result is an extension of the intuition developed in proposition 1. 

 In equilibrium competition causes sellers to give discounts to large (and therefore more 

profitable) buyers, but not to the small buyers. Thus the greater the proportion of type T's that 

are low volume customers, the more costly it is for the seller to offer volume discounts to the 

high volume customers, thus the smaller the discount. 

One might ask why a store would offer a volume discount implicitly through a reduction 
14

in the price of one good rather than explicitly by announcing a discount schedule that is a 

function of the quantity purchased. There are three potential reasons. First customers that were 

willing to make many small purchases over time and pay a high price, might find it in their 

interest to consolidate purchases to take advantage of the discount, thereby reducing the store's 

profits. The second is that low volume customers could potentially combine shopping to qualify 

for the volume discount. Offering the discount on a good prevents this behavior.

Third, in a market in which products have different marginal costs and different 

markups, discounts based on explicit quantity or revenue levels may be as imperfect a method 

of discrimination as loss leader pricing. Sellers would like to discount based on the profitability 

of each customer’s purchases. It would be difficult to establish a pricing policy that is clear to 

customers that is based on profit margins since customers cannot observe marginal cost. A plan

that gives a discount based on total value of the grocery bill will be an imperfect approximation 

of profit if some customers buy large baskets of goods with small profit margins, while others 

purchase smaller baskets of goods with higher mark-ups. Thus, it is not clear that an explicit

quantity discount based on revenue would necessarily better target profitable customers than a 

well chosen loss leader.

5. Most frequently purchased product.

As stated earlier a popular belief is that a loss leader should be a product that most 

people purchase. I now present a model that shows that being the most frequently purchased 

product by itself is not enough to cause a product to be priced as a loss leader.

Again consider two firms located at opposite ends of a Hotelling line. There are 3 

goods that customers can buy, T, s1, and s2. The marginal costs of producing these goods are 

ct, cs1 and cs2. There are three types of customers each of which is distributed uniformly along 

the line. The first type buys one unit of T and one unit of s1. The second type buys one unit of 

T and one unit of s2. The third type buys one unit of T, one unit of s1 and one unit of s2. 

As before firms set the prices of each good non-cooperatively. Proposition 4 shows 

that even though all customers purchase a unit of T, it is not priced as a loss leader.

Proposition 4. In this market T is the only good with a positive mark-up. 

Proof: Each firm faces a profit function of :

Pi = (Pti - ct + (Ps1i - cs1))(1/2)(1 + Pt-i + Ps1-i - Pti - Ps1i) +
15

 (Pti - ct + (Ps2i - cs2))(1/2)(1 + Pt-i + Ps2-i - Pti - Ps2i) +

(Pti - ct + (Ps1i - cs1) + (Ps2i - cs2))(1/2)(1 + Pt-i + Ps1-i + Ps2-i - Pti - Ps1i - Ps2i) . (9)

The first order conditions are

∂Pi/∂Pti = - (Pti - ct + (Ps1i - cs1)) + (1 + Pt-i + Ps1-i - Pti - Ps1i) -

(Pti - ct + (Ps2i - cs2)) + (1 + Pt-i + Ps2-i - Pti - Ps2i) -

(Pti - ct + (Ps1i - cs1) + (Ps2i - cs2)) + (1 + Pt-i + Ps1-i + Ps2-i - Pti - Ps1i - Ps2i) =0

∂Pi/∂Ps1i = - (Pti - ct + (Ps1i - cs1)) + (1 + Pt-i + Ps1-i - Pti - Ps1i) -

(Pti - ct + (Ps1i - cs1) + (Ps2i - cs2)) + (1 + Pt-i + Ps1-i + Ps2-i - Pti - Ps1i - Ps2i) = 0

∂Pi/∂Ps2i = - (Pti - ct + (Ps2i - cs2)) + (1 + Pt-i + Ps2-i - Pti - Ps2i) -

 (Pti - ct + (Ps1i - cs1) + (Ps2i - cs2)) + (1 + Pt-i + Ps1-i + Ps2-i - Pti - Ps1i - Ps2i) = 0

which reduce to

P*ti - cti = 1 (11.a)

P*s1i - cs1 = 0 (11.b)

P*s2i - cs2 = 0. (11.b)

QED

The easiest way to understand this result is to recall that in equilibrium when a customer 

type is uniformly distributed along the line, and travel cost equals 1, each customer of that type 

will pay a total mark up above marginal cost of 1 to a firm regardless of how many units that 

customer type will purchases. The only way this can be satisfied for all three groups is for T to 

be marked up 1 over marginal cost and for s1 and s2 to be priced at marginal cost. 

The important point of this example is that the good purchased by the most customers 

(and the good that sells the most units) is T, and in competition it has the highest mark up of all 

goods sold. Thus this result calls into question the popular notion that a good is priced as a loss 

leader if it is purchased by the most people. 

6. Conclusion

It is widely believed that a good loss leader is a product that is a well known brand or a 
16

product that is widely purchased. The argument is that lowering the price on such items will 

attract customers into the store who will then purchase other items at the regular price. I have 

shown that this reasoning by itself may not be complete. This paper presents a model 

suggesting that an important consideration for choosing a product as a "loss leader" is that it be 

purchased primarily by high profit producing customers.13 To the extent that more profitable 

customers purchase a larger basket of goods (in dollar terms) I argue that correlation with the 

size of customer purchases, should be an important characteristic that determines if a good 

should be priced as a loss leader.

A more general implication of this work is that the intuition developed should not be 

limited to products used as loss leaders. Rather every product should be priced taking into 

account not only the fact that this price affects the quantity of the product sold, but also the fact 

that at the margin it will affect the make up of customers that enter the store and therefore the 

quantity of other units that will be purchased as well. Thus, in general goods that are bought 

primarily by high profit (or large volume) customers should have lower mark ups (all else equal) 

than goods primarily bought by low profit customers. . All else equal the larger the basket of 

goods in which a good is purchased, the lower should be the mark up on it.

This paper suggest (at least) one interesting line of empirical research. The intuition 

behind the results could be generalized to suggest that if large volume purchasers 

systematically buy a set of goods that small volume purchasers do not, then we should expect 

the markup on these items to be less than the average markup through out the store. So for 

example if milk, bread and eggs, are purchased mostly by those shopping for families with 

large grocery orders, and less so by single people with small orders, this theory predicts these 

items would have smaller than average margins. Thus a testable implication of this theory (to 

distinguish it from the notion that a product simply need be purchased by most customers or 

be a popular brand) is that on average, the baskets of goods that contain the loss leader is 

larger than baskets of goods that do not contain the loss leader. 

 

13 Note that in the model of the previous section, since all side goods had the same price cost margin high 

revenue producing customers is synonymous with high volume customers.
17

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