Nonlinear Price Incentives and Dynamic Brand Choice: B2B
Purchasing Decisions with Quantity Discounts
James C. Reeder, III
Simon Graduate School of Business
University of Rochester
Abstract
I present an empirical framework that captures the dynamic decision-making process of a retailer
choosing both which manufacturer to order from and how much should be ordered when the manufacturers compete by oering quantity discounts. I apply the model to a rich data set provided
by a multinational medical device manufacturer, in which, I observe the daily ordering habits of
doctors to the focal manufacturer. Unique to this market, the doctor functions as a retailer and
faces a cost-minimization problem based on his choice of prescriptions made throughout the day.
Reduced-form evidence suggests that doctors align the number of prescriptions written in a day to
the focal manufacturer with the nonlinear price incentives oered. I replicate the full information
set of a marketing manager and select informative macro and micro moments for simulated method
of moments estimation. After estimation of the structural model, I nd the magnitude of the distortion. Conditional on one patient already receiving a product from the focal manufacturer, the
doctor is 12 times more likely to prescribe a second product from that manufacturer. Next, I analyze
dierent price schedules by adjusting the threshold levels and per-unit discounts. I nd one system
that increases rm protability by 7% without changing the initial and nal tier per-unit price of the
rm's products. The best single-price plan only increases protability by 3.5% in comparison. Also,
I examine the protability implications of volume discounts in two dierent counterfactual scenarios;
one in which 60% of the patients require a rell of their previous prescription and the other with 0%
rell patients. I nd volume discounts provide the greatest protability gains over a single-tier pricing
system in the setting with 60% rell patients. In the other scenario, some of the optimal nonlinear
price schedules actually result in less protability than a simple single-tier pricing system. Finally, I
conclude with comments on the eld implementation of a new price schedule based on the structural
estimation, which resulted in a 9% increase in year over year revenue.
1
1 Introduction
Marketing managers constantly struggle with how best to price their products to increase protability.
The choice of pricing has a direct, sometimes immediate, impact on the behavior of buyers in the market.
However, understanding the potential benets or consequences of using a more complex pricing system is
not always straight-forward. For example, consider the volume-based discounting strategy used frequently
in business-to-business (B2B) transactions.
The general purpose of this strategy is to induce retailers
to purchase greater quantities, by conditioning the nal per unit price on the total amount ordered. In
practice, volume-based discounting comes in the form of a price schedule, where a manufacturer presents
the list of products oered and the per-unit price for each product conditional on the overall order size,
with a discontinuity in prices at quantity threshold levels. If a purchasing agent has expectations of his
future product needs, then just the presence of these discontinuities in the marginal price results in a
dynamic decision-making process for the retailer.
Theoretical B2B literature has shown that quantity
discounts do indeed cause rms to purchase larger lot sizes from a manufacturer (Goyal 1976, Monahan
1984, Lee and Rosenblatt 1986).
However, to the practitioner and the general empirical literature, a
full understanding and measurement of the potential distortions in behavior caused by the nonlinear
price schedule is critical. A model of the complex decisions a retailer makes in choosing a manufacturer
to fulll an order must take into consideration not only the nonlinear price incentives oered by each
manufacturer, but also the retailer's belief about future customer demand for each product. The result
is a complex dynamic discrete-choice model for the retailer. However, the estimation of such a model is,
generally, complicated by the lack of transparency in B2B transaction data. A single manufacturer can
provide a wealth of information on its own sales, prices, and marketing practices, but rarely has any more
information on the competitor than its prices and aggregate market shares. This paucity in the observed
data for a single manufacturer creates an additional hurdle for empirical analysis.
The goals of this paper are two-fold.
First, I present a framework and estimation technique to
overcome these two major hurdles. Using only the readily available data from a multinational medical
device manufacturer, I create an estimation strategy to overcome the limitations of the observed data.
From this framework, I am able to recover structural primitives of interest, namely, the inuence of
preferred client status, sales force eectiveness, and a measure of price elasticity for the retailer.
The
specic empirical application is observing prescription choices made by a doctor in an international
market, where the doctor chooses between manufacturers that each oer nonlinear price incentives. The
unique characteristics of this market environment force doctors to act as retailers, and manufacturers
2
employ nonlinear pricing as a competitive tool.
The second goal of this paper is to gain a deeper
understanding of how nonlinear incentives inuence protability and purchasing behavior, or rather, to
nd the determinants of an optimal price schedule given a set of rm constraints. Through counterfactual
study, I am able to comment further on the nuances and insights gained from observing changes in
behavior as I adjust the key factors of the nonlinear price schedule. The results from structural estimation
were presented to the focal manufacturer and implemented during a restructuring program in the specic
international division. Preliminary evidence from the rst year after implementation shows a 9% increase
1
in revenue under the new price schedule, as opposed to the year-over-year 15% decline in previous periods.
As a result, this paper is another in a growing literature that showcases the value of structural modeling
in enhancing rms decisions (Mantrala et al. 2006, Cho and Rust 2008, Misra and Nair 2010).
The empirical context of his paper is concerned with understanding how a doctor chooses between
manufacturers in prescribing the medical device to a patient. In this market, the doctor functions as a
retailer, where by he fully internalizes the costs for ordering a specic prescription, and the patient pays
the doctor for the product ordered. Therefore, this business relationship mirrors the classic B2B set-up
between a buyer and a supplier. At the start of a business day, the doctor observes some information
on the potential arrival of patients into the oce, noting if certain patients are coming in for a rell of
a previous prescription or if the patient is new to the practice.
At this point, before the rst patient
even arrives, the doctor has some expectation of the total cost for all prescription choices made during
the day. Given the price schedules that each manufacturer oers, the doctor has incentives to allocate
more prescriptions to one brand rather than spreading out smaller orders across manufacturers, with one
notable exception presented in the motivating example section of this paper. However, the needs of the
patient and their preferences result in a non-deterministic process. Although the doctor does have sway
over what he prescribes each patient, it is the discussion between the patient and the doctor results in
the nal prescription. The doctor then must incorporate the prescriptions already written and his beliefs
about the preferences of future customers to try to nd an optimal prescription strategy for minimizing
the cost to his practice on a daily basis.
Although this paper's focus is on B2B transactions, the insights can be applied to the B2C setting as
well. Nonlinear incentives take a variety of forms, such as, a "Buy One, Get One Free" oer or a more
complex loyalty program that oers both points from purchases and thresholds for larger gains, such as
the application in Kapalle et al. (2012). The common thread in these two examples is a threshold level
1 Although other aspects of the business changed during this period as well, market analysts with the focal rm noted
the new prices have been well received by the market and are a large contributor to the rm's performance.
3
where agents have an opportunity to achieve a benet based on their choices.
The presence of these
nonlinear gains make agent dynamics an important part of modeling behavior.
In B2B transactions, manufacturers oer rms a menu of prices for their products.
Conditioning
on specic volume targets, such as total revenue in a quarter or overall order size, the cost per unit
is lower for the purchasing rm.
2
In Table 1, I present two examples from the empirical application.
In this example, Brand A has the rst nonlinear discount occurring after a doctor orders four boxes of
product and a subsequent discount after eight or more boxes. In addition, deeper discounts exist at both
levels compared to Brand B, which has a single discontinuity at eight or more boxes with a more modest
percentage discount. The inuence of the components of a price schedule is a critical consideration for
both practitioners and the pricing literature.
This paper's approach provides valuable insights into two broad areas of marketing: channel coordination through the use of nonlinear pricing and the impact of nonlinear price incentives on choice. A
broad literature exists on the usage of nonlinear pricing to both achieve greater rm protability and
better coordinate the ordering process between manufacturers and retailers. Nonlinear pricing is shown
to induce self-selection of customers into dierent quantity-allocation levels (Gerstner and Hess 1987,
Spence 1980).
The addition of multiple products oered by a monopolist using nonlinear pricing has
been addressed as well in Adams and Yellen (1976), McAfee, McMillan, and Whinston (1989), Armstrong (1996), Rochet and Choné (1998).
However, my application is based on a set of rms oering
competitive nonlinear price schedules to a single retailer (doctor), which is closer to the work of Spulber
(1979), Stole (1995), Armstrong and Vickers (2001), Rochet and Stole (2002), Yin (2004), who examine
competitive nonlinear pricing. Most recently, Armstrong and Vickers (2010) found that nonlinear pricing,
in the form of a two-part tari, can yield greater protability if a set of conditions hold: (i) demand is
elastic, (ii) heterogeneity exists in the preferences for the products of dierent manufacturers, (iii) the
mix of products contain correlated preferences, and (iv) shopping costs are suciently high. In the empirical application, all these conditions hold and I nd support for the theoretical claims of Armstrong
and Vickers, by showing how dierent price schedules increase protability.
The aforementioned works indicate that nonlinear pricing can inuence consumer demand for a product; however, the marketing literature also discusses this pricing strategy as a method of channel coordination. Goyal and Gupta (1989), Weng (1995) and Cachon (2003) provide a summary on the usage of
quantity discounts to manage the channel between a buyer and supplier. Jeuland and Shugan's (1983)
2 Due to condentiality reasons, I am unable present the actual prices. Instead, I show the percentage of the discounts
from the initial price and the volume targets to achieve said discounts.
4
seminal work in the area discusses how several dierent methods, including quantity discounts, are used in
channel coordination. Lal and Staelin (1984) look at optimal pricing levels, where groups of purchasers
are heterogeneous, and use the buyer's and supplier's costs to show the need for quantity discounts.
Moorthy (1987) and Ingene and Perry (1995) both present theoretical models founded on the demand
for products being price dependent and discuss the implications of choosing a quantity-discount strategy.
These two papers are important to my empirical study, because I nd doctors respond to changes in the
prices the manufacturer oers.
Though a large theoretical literature exists in this area, the empirical B2B literature is sparse. What
little is present, generally focuses on modeling the consumer's demand for a product from a retailer
and then using this information to make inferences about the supply side of the product. Villas-Boas
and Zhao (2005), Villas-Boas (2007), and Bonnet and Dubois (2010) take this approach. Instead, I use
proprietary information from a single manufacturer, which also includes the pricing information of all
competitors in the market, to model the demand of the retailer. Zhang (2011), similarly, looks at the
dynamics involved in B2B purchasing for the aluminum market, where customizable prices are given on
a per-order basis through the asking and accepting of price quotes. The use of price quotes is common
in situations with highly customized products. Although this pricing mechanism constitutes a portion of
the B2B transactions, I focus on another dominant pricing strategy, the use of nonlinear price schedules.
Finally, this paper falls within a broader class of literature, dynamic decision-making in the presence
of nonlinear incentives.
Three recent structural works highlight the impact of nonlinear incentives on
altering the behavior of an economic agent. Hartmann and Viard (2008) examine how a loyalty program
creates switching costs, based on the attainability of rewards given the customer's choices. Misra and Nair
(2011) examine a rm that uses a nonlinear incentive scheme for sales force compensation.The author
nds sales agents change their behavior based on their proximity to the discontinuity in the compensation
contract, and shift their eort accordingly. Finally, Kopalle et al. (2012) examine how both frequency
rewards and customer tier benets create a dynamic between the behavior of customers and their choices.
My contribution to this growing literature is through incorporating competing nonlinear price incentives
in the decision-making process and estimating the resulting distortion eect.
Descriptive analysis of the focal rm's data show that doctors shift their prescription choices based on
the nonlinear price incentives oered by the focal manufacturer. I nd that doctors distort their ordering
behavior at the quantity thresholds. This information not only aides in identication of the doctor's price
sensitivity, but also is an important feature to account for when determining an optimal price schedule.
Knowing that doctors adjust their ordering volume based on the marginal gains present in the price
5
schedules is useful in formulating the frequency of discounts and their respective discount levels in order
to increase manufacturer protability.
The estimation of a model with forward-looking agents that make quantity-allocation decisions across
multiple brands and multiple product oerings can become computationally intractable as the state space
expands with each choice opportunity. Further, the context of the empirical application has a terminal
period, where the cost savings of the decisions made are realized and all quantity allocations are reset.
The result is a nite-horizon problem that becomes conceptually easy to understand, but dicult to
estimate as the value function itself changes based on the total number of patients arriving in a day and
the doctor's knowledge of the future patient's states. In the empirical application, I track the choices
doctors make on a daily basis across 16 dierent brand/product combinations, which makes calculating all
the choice-specic value functions at each time period for all brand/product manufacturers and patient
states computationally burdensome.
Keane and Wolpin (1994) propose a method of simulation and
interpolation to approximate the value function to reduce the computational burden in estimating the
choices an agent facing a nite horizon makes. This estimation step is then nested within a simulated
method of moments estimator, to reconcile the empirically validated macro and micro moments with the
per patient choices made by a doctor on a daily basis.
This paper provides contributions in three key areas. First, I develop and describe a structural model
and an estimation technique that captures the behavior of a forward-looking agent being inuenced by
brands competing in nonlinear incentives. Second, through careful study of reduced-form evidence, I nd
a novel set of moments that identify the disutility of cost to an agent, even in a situation in which prices
are contractual and do not vary over long time horizons, which would otherwise cause the parameter of
interest, price sensitivity, to be unidentied separately from the intercept term.
I use the estimated structural primitives to explore the relative performance of dierent nonlinear price
schedules in both a new and a mature market. My ndings highlight four important insights. First, in
comparing nonlinear pricing with a standard, single-price option, the latter increases the number of orders
placed, but this gain is purely on more frequent, smaller orders. As a result, proper quantity discounts
provide the necessary incentives to obtain fewer, but larger, orders by comparison and to increase overall
rm protability by 7%, compared to 3% with the single-price option. Second, not all price schedules
provide adequate incentives to obtain greater protability; the choice of threshold location and the depth
of discounting is critical. I nd that a price schedule with a higher initial discount coupled with a smaller
discount on a larger quantity target is optimal.
The large initial discount is sucient to provide an
incentive for the doctor to increase his order volume to this initial-quantity threshold. However, once this
6
initial threshold is met, the manufacturer is better o oering a smaller discount on a larger order size,
and the multiplicative eect placed on the smaller discount yields the proper incentives to the doctor.
Third, in a new market, nonlinear pricing performs marginal at best, or in the case of some discount
schedules, worse than a single-price point schedule. The returning customers create an opportunity cost
for the doctor to consider. The innate quantity realized from the inux of patients returning to renew
their previous prescriptions causes doctors to allocate further units to that manufacturer; doing otherwise
would cause doctors to miss out on the potential cost savings. Finally, the ratio of rell patients impacts
the structure of the optimal nonlinear price schedule. When a greater proportion of new patients exists,
a low initial price with smaller discounts is advantageous. The lack of any persistence in the choices a
doctor makes causes an increase in competition for the rst patient, which places a downward pressure
on pricing. Conversely, when the ratio of rell patients is high, a high initial price coupled with deep
discounts is optimal.
The manufacturer has the ability to collect rents on the smaller orders, while
providing the necessary incentives to the doctor to achieve larger order quantities and increase overall
manufacturer protability.
I organize the rest of the paper as follows. I begin with describing the industry and the institutional
details that are important to capture in the structural model. Next, I present the structural model of
the doctor's prescription choices made on a daily basis, where the doctor is a forward-looking agent
and considers both patient needs and the price schedules oered by the manufacturers in his choices.
Subsequent to this model overview, I present an example where nonlinear pricing causes a distortion in
the prescriptions written that is not addressed in the theoretical literature. I then outline the dierent data
sources I synthesized together to recover the structural primitives, and report reduced-form evidence to
support the structural model. I detail the estimation steps and outline my identication strategy. Finally,
I present results and counterfactuals and conclude with a brief discussion.
2 Industry Overview
I focus on the interaction between a retailer and a set of manufacturers to measure the impact of competitive nonlinear pricing on the manufacturer and quantity choices made by the retailer. Specically, I
examine the daily prescription habits of a doctor prescribing a commonly used medical device in an international market. In this section, I detail the cost minimization problem faced by the doctor, the factors
that inuence the prescription decision, and industry specic characteristics that inform the structure of
the dynamic discrete choice model.
7
At the start of a business day, the doctor receives information on the number of patients arriving
into the oce and their individual characteristics.
Specically, if the patient has been seen by the
oce previously, then the doctor has information on both the product and the manufacturer that was
prescribed to the patient. Additionally, the doctor knows if the returning patient either wants a rell
of their prescription or intends to switch to a dierent manufacturer due to a mismatch in the product
itself.
The class of products under observation are horizontally dierentiated, so the best product for
an individual patient is made through the match between the manufacturer and the patient, which is
unknown a priori. As a result, the patient must try out the product to see if a match occurs. According to
industry experts, approximately 70% of the time a patient continues to use the manufacturer's product
they were rst prescribed, otherwise the patient requests to be prescribed a dierent manufacturer's
product. The patient incurs no cost in switching between manufacturers and the side eects are common
across all manufacturers; namely, mild discomfort and irritation. Each patient has a set of symptoms
that requires a specic product. Therefore, the doctor is unable to switch a patient between products,
unless the patient's symptoms change. However, given the constraints of the patient preferences and the
cost incentives presented by the manufacturer, the doctor's problem is to minimize the total costs of his
decisions on a daily basis, thereby maximizing his protability.
In this market, the doctor operates as a retailer. The doctor is responsible for not only prescribing
products, but also ordering these products from the set of manufacturers in the market. The doctor bears
the cost of the product ordered and charges a new price to the patient for the prescription. The prices
charged by the doctor for products in a given product class are virtually equal across manufacturers, with
only slight increases in the retail price for the higher cost manufacturers. As a result, the doctor subsidizes
some of the manufacturer's cost and does not pass the full amount through to the patient. In addition, the
doctor does not change the prescription price based obtaining volume-based discounts due to his ordering
process. Three of the four manufacturers in this market oer a nonlinear, volume-based discount price
schedule to doctors, where discounts are applied on a per-order basis. Historically, the doctors placed
orders to the manufacturers after every patient received a prescription. Manufacturers provided incentives
to the doctors, in the form of a price schedule that contained volume-based discounting, to collect the
prescriptions at the end of the day, rather than ordering on a per-patient basis.
At the end of each
business day, the doctor places an order to each manufacturer, detailing the product requirements based
on the prescriptions written throughout the day. The oered price schedules are xed for long periods of
time, sometimes remaining the same for years. Although manufacturers may oer infrequent, temporary
price promotions on a select number of products to select doctors, the prices remain otherwise xed. The
8
only dierence in the list price oered by manufacturers versus the nal price paid by the doctor is the
result of a doctor being a member of a buying groups in the market, where each buying group oers a
dierent at discount on prices charged by the manufacturer.
The prescribed product must be customized to the specications of the patient, which are determined
at the time the doctor meets with the patient. For the focal manufacturer under observation there are
over 15,000 SKUs. As a result, a doctor does not retain an inventory and operates under a just-in-time
delivery process. This feature of the market is advantageous for estimation; each prescription the doctor
writes is directly allocated to a single patient, without any changes in the order size attributed to holding
inventory. Rather than examining the entire scope of products oered by each manufacturer, I limit the
estimation to four separate product categories, where each product is oered by all manufacturers. The
combination of these 16 products accounts for roughly 70% of total patient share in the industry.
3 A Model of Dynamic Quantity Choice
I present a model where a doctor
i
chooses the prescription choice for patient
number of patients a doctor sees in a day is
sents the product
j
from manufacturer
k
T.
t
during day
For each patient, the doctor decides
that is prescribed to patient
t.
qjkt ,
τ.
The total
which repre-
The doctor's contemporaneous
prescription choice is impacted by three forces: the prescription choices already made in the day, the expected demand of products for patients that are arriving later in day
τ , and the nonlinear price schedules
oered by the manufacturers in the market. The doctor functions as a retailer in the market, the doctor
pays the manufacturer for the prescriptions written and sets a retail price to the patients, and is modeled
as such in this paper. For ease of reading, I remove the
i
and
τ
subscripts in what follows.
It is the impact of the nonlinear pricing on the manufacturer and quantity choices made by the doctor
that is the focus of this structural model. Manufacturers oer incentives for the doctor to accumulate
the prescriptions and order at the end of each business day. The mechanism that doctors internalize the
prices oered by the manufacturers is through the use of a price schedule. The price schedule oered by
a manufacturer takes the form of
Ck (Qk ),
total quantity ordered at the end of day
all products
j
the per unit prices that the doctor realizes is a function of the
τ,
which is expressed as
prescribed by the doctor to manufacturer
unit price is based on the total order volume,
Qk ,
k , qjk .
Qk
and
Qk =
PJ
j=1 qjk or the sum of
The quantity that determines the per
and not the brand/product specic volume,
function is a step function, where the realized per-unit price for manufacturer
k 's
product
j
qjk .
is
0
cjk ,
This
and
conditional on both the total amount ordered during the day and the quantity thresholds established by
9
the manufacturer,
0
Qk .
The prime superscript denotes the current price/quantity level the doctor has
3
achieved through the prescriptions written. More formally, the step function is :
Ck (Qk ) =




c1jk




if
c2jk






 c3jk
if
Qk < Q1k
(1)
Q1k ≤ Qk < Q2k
if
Q2k ≤ Qk
c3jk ≤
A manufacturer sets the levels of the quantity thresholds and the per-unit prices such that
c2jk ≤ c1jk .
Each of the superscripts, 1 through 3, corresponds to a dierent price level or price tier.
Conditional on meeting and exceeding a quantity threshold, the doctor then realizes a new per-unit price
corresponding to the new pricing tier. The choice of the specic quantity-threshold levels and prices a
manufacturer oers for its products is not in the scope of this model, because these change infrequently.
However, I analyze dierent structures of price schedules in a series of counterfactual exercises at the end
of this paper.
The problem the doctor faces is a nite-horizon problem. The per unit prices obtained through the
order size allocated to manufacturer
k
resets after each order is placed.
Doctors are known to places
their orders on a daily basis, which results in a nite horizon problem that starts anew each business day.
Qkt
is then the sum of all products
t, Qkt =
j
oered by manufacturer
k
that doctor
PJ
j=1 qjkt . I collect the price schedules for all manufacturers,
Ck
i
has ordered up to patient
into a vector
φ = {Ck }. Pt
are the patient characteristics that act as a constraint on the consideration set of product/manufacturer
combination the doctor can prescribe to patient
t.
The total amount ordered up to time
number of patients that have not yet arrived in the day,
current and future patient preferences,
Pt ..PT
Nt = T − t ,
t, Qkt ,
the
and the information about the
, are all state variables, which I collect into a vector
st = {Qkτ t , Nt , Pt , Pt+1 , .., PT }.
3.1 Actions and Per-Period Utility
When a doctor meets with patient
from a single manufacturer
k , Qkt = Qkt + qjkt .
k.
t,
the doctor observes his current state,
st ,
and chooses to order
qjkt
The doctor then updates the total quantity purchased from manufacturer
Based on this decision, the doctor then realizes any cost savings obtained from the
nonlinear price schedule of manufacturer
k
based on the current and expected quantity allotment.
The use of other marketing variables may sway the doctor to purchase from manufacturer
k.
Manu-
3 If manufacturers update their price schedules periodically, the price-schedule function and both the per-unit price and
quantity thresholds should have a subscript for time as well.
10
facturers routinely use sales force personnel or oer other services to an agent to gain some of its business,
Xkt .
Further, the agent considers not only its own cost sensitivity,
the ecacy of the product
αjk . jkt
β,
in his choices, but also his belief on
is the error term that represents the interaction between the patient
and the doctor, where additional information is revealed to the doctor from the patient that may alter
the doctor's choice specic utility. The total cost of all prescription choices are not realized by the doctor
until the terminal patient
T
is prescribed a product and the orders are sent to the manufacturers. As a
result, the choice specic utility for patient
t<T
is
ujkt = αjk + δXkt + jkt
(2)
3.2 State Variables and State Transitions
There is one source of dynamics involved in this structural model, the changes in the expected total cost
of prescription choices that is derived from both the prescriptions already written and the expectation of
future patient needs. To derive the optimal choice a doctor makes, there are three state variables that
must be considered: the total quantity allocated to each manufacturer, the expected preferences of the
patients, and the total number of patients left during the business day.
The rst state variable, the total quantity a doctor prescribes updates deterministically based on each
sequential choice the doctor makes throughout day
τ.
resets at the end of each day after the nal patient
Qkt
The total quantity allocated to each manufacturer
T
is prescribed. Therefore, the state transitions of
is:
Qkt+1 =



 Qkt + qjkt
if



if
0
t<T
(3)
t=T
The second state variable for the doctor to consider is the preferences of the patient,
Pt .
The pa-
tient preferences enter into the utility formulation as a constraint on the choice set of the doctor,
Pt ∈ {jt , rt , kt }.
The set of patient states,
manufacturer switcher
rt ,
Pt
contains his product needs
jt ,
if he is a new/rell/ or
and, if applicable, the manufacturer he was prescribed previously
patient is a new patient, then the doctor has the ability to prescribe any manufacturer
patient is a rell patient, then the patient receives only
kt .
k 's product.
If the
If the
kt , the manufacturer he was previously prescribed.
Finally, if the patient can be a manufacturer switcher, then the patient is prescribed any brand but the
one previously prescribed. At the start of the business day, the doctor knows the states of all patients.
11
The last state variable,
the terminal patient
T
Nt ,
the number of prescriptions the doctor has to write during the day, until
arrives, updates deterministically.
Nτ =








Nt
Nt − 1







0
if
t=0
if
t<T
(4)
if t=T
The number of patients arriving into the oce acts as a constraint on the ability to achieve certain
quantity thresholds, which has direct implications on the value function formulation.
One nal variable is the mix of patients arriving into the oce in a future quarter.
ξ 'jk , is the doctor's
internal patient share for a given manufacturer/product combination. The observed set of products is
on a six month rell cycle. Since product preferences in the market are generally xed, if a patient is
prescribed a particular manufacturer's product today, they are likely to ask for it when they come in for a
rell. The expected patient preferences evolve deterministically.
k 's
product
j
σjk
is the total amount of manufacturer
that has been ordered in the quarter. As a result, the prescriptions written today inuence
the doctor's choices in the future:
ξ 'jk =















P σjk +1
( k σjk )+qjkt
if
k=K
and
j=J
P σjk
( k σjk )+qikt
if
k 6= K
and
j=J
(
Pσjk
k σjk )
if
(5)
j 6= J
Model free evidence, explained in a subsequent section, suggests that doctors do not optimize over
this variable. As a result, it is not part of the value function formulation. Instead, it is used to show
persistence in the model across time to mimic how the market evolves based on changes in prescription
behavior during the current business quarter.
3.3 Optimal Actions
The per-period utility and the state transitions aide in characterizing the problem the doctor faces in
selecting the optimal manufacturer from which to order a specic set of products to maximize both
the per-period utility through the contemporaneous utility gains and the expected terminal-period cost
savings.
From the parameters that describe the per-period utility formulation,
space observed by the doctor
st ,
θ = {αk , β, δ},
the value function of optimal choices is formulated.
12
and the state
I start with the
value function at the terminal period,
VT (sT , φ, θ) = qmax
jkT


T
:
αjk + δXikT + jkT




XX
−β
(QjkT −1 + qjkT ) Cjk (QkT −1 + qjkT )

(6)
j∈J k∈K
s.t. qjkT ∈ PT
Because the quantity levels reset after time
T,
there is no continuation value in the value function,
which characterizes the agent's problem as a nite-horizon problem. Unlike the per-period utility presented in equation 2,
P
j∈J
P
k∈K
prescriptions written up to time
T.
(Qjkt + qjkT ) Cjk (Qkt + qjkT )
represents the total costs of all the
The total cost of the doctor's choices captures the doctor's dynamic
decision-making process. Recall from equation 1, the description of the nonlinear price schedule the manufacturer oers, that a threshold value
allocation of
qjkT
causes
QkT > Q1k ,
Q1k
c2jk . If the
c1jk − c2jk QjkT −1
exists where by the per-unit cost shifts from
that choice carries the specic benet of
P
j∈J
in cost savings to the doctor on all the other choices already made through time
c1jk
T − 1.
to
The result is
an incentive for the doctor to order larger quantities from the manufacturer, conditional on both the
expected quantities to the other manufacturers and their specic price schedules.
For any time
t < T,
the doctor takes expectations of both the patient preferences for the dierent
products oered by the manufacturers and the error term in the next period. The resulting value function
is:
Vt (st , φ, θ) = max
qjkt



ujkt + 

Xˆ
st+1


Vt+1 (st+1, |st , φ, θ)f ()f (st+1 |st )ddst+1 

(7)
s.t. qjkt ∈ Pt
Given the demand shocks realized when the doctor meets with a specic patient
future period is stochastic.
t,
the value in any
Also, for patients that are new to the practice, the doctor does not know
their product requirements and it free to assign any manufacturer's product to them.
Similarly, for
patients that want to switch to a dierent manufacturer, the doctor knows the product class but has
to take expectations over the prescription. Note that because the context of the doctor's problem is a
nite-horizon problem, the value function is dierent at each time period
a manufacturer at each time point
quantity allocated at time
t
t,
t.
As the doctor orders from
the total costs for all the choices are updated based on the total
and the expected future allocation based on the expected preferences of
13
patients from periods
t+1
to
T.
To illustrate the impact of incentive discontinuities on the behavior of
rms, the next section presents an analytical example that results in a rm exhibiting binning behavior.
3.4 Motivating Example - The Inuence of Price Schedules and Expected
Patient Preferences on Choice
In this section I present a series of examples to illustrate how the intersection of expected patient preferences and the discontinuities in price schedules impact the contemporaneous prescription choice of the
doctor. For this example, I assume there are two manufacturers in the market,
A
A
and
B.
Manufacturer
is the high cost product that has a deep discount at the quantity threshold. Manufacturer
low cost product, with a smaller discount at the threshold value.
B
is the
I use prices and discounts that are
representative, but not exact, of the costs of a product in the empirical setting for this simulation.
I
summarize the competing price schedules below:
Simulation Price Schedule
Qk < 2
Qk ≥ 2
Manufacturer
A
$44
$35
Manufacturer
B
$29
$28
Both manufacturers have a discontinuity at two units. For this simulation, I begin by assuming that
the doctor sees three patients during the day. In addition, I assume that the structure of the error term
is Type I Extreme Value, which is the distribution I use in the empirical application.
I calculate the probability of dierent order sizes based on the following scenarios. The rst scenario
assumes that all three patients are new patients, so they have no prior preferences on either manufacturer. The second scenario assumes that the last patient of the day requires a rell from manufacturer
(P3 = A).
A,
The third scenario assumes that patients two and three both require rells from manufacturer
A, (P2 = A, P3 = A).
The fourth scenario assumes that patient 3 requires manufacturer
2 requires manufacturer
requires manufacturer
For simplicity,
B , (P2 = B, P3 = A).
A
and patient
Finally, the fth scenario assumes that patient three
B , (P3 = B).
aA = aB = 0
and
β = .5.
The probability of a given order size based on each scenario
presented above is presented in the following table:
Simulation of
14
β = .5
QA = 0 QA = 1 QA = 2 QA = 3
0.9973
0.0010
0.0017
0
0
0.2283
0.7529
0.0187
0
0
0.9526
0.0474
(P2 = B, P3 = A)
0
0.3775
0.6225
0
(P3 = B)
0.9988
0.0007
0.0006
0
Scenario 1 (No Rell)
Scenario 2
(P3 =A)
Scenario 3 (P2 =A,P3 =A)
Scenario 4
Scenario 5
When there are no rell patients for manufacturer
A the probability of prescribing any amount of A's
product is virtually 0. However, once a patient requires manufacturer
A
of ordering two units from manufacturer
A's
product, then the probability
is the highest in all scenarios.
It is this distortion in the
prescription choice of a doctor that showcases the importance of the nonlinear incentive present in the
price schedule.
manufacturer
A
Due to the discontinuity in the per unit price at two units in the price schedule for
and the deep discount, it is advantageous for the doctor to prescribe
A
over
B
to reach
that threshold amount and purchase no further. The combination of a patient requesting a rell and the
ability to cross the quantity threshold creates an opportunity cost for the doctor. Next, I conduct the
same scenarios, but adjust the doctor's cost sensitivity to
Simulation of
β = .3
and
β = .7.
β = .3
QA = 0 QA = 1 QA = 2 QA = 3
0.9435
0.0233
0.0314
0.0017
0
0.2549
0.6882
0.0569
0
0
0.8581
0.1419
(P2 = B, P3 = A)
0
0.4256
0.5744
0
(P3 = B)
0.9732
0.016
0.0108
0
Scenario 1 (No Rell)
Scenario 2
(P3 =A)
Scenario 3 (P2 =A,P3 =A)
Scenario 4
Scenario 5
Simulation of
β = .7
QA = 0 QA = 1 QA = 2 QA = 3
0.9999
0
0.0001
0
0
0.1977
0.7963
0.006
0
0
0.9852
0.0148
(P2 = B, P3 = A)
0
0.3318
0.6682
0
(P3 =B )
1.0000
0
0
0
Scenario 1 (No Rell)
Scenario 2
(P3 =A)
Scenario 3 (P2 =A,P3 =A)
Scenario 4
Scenario 5
15
As the cost sensitivity parameter increases, the probability of prescribing two units of manufacturer
A's
product increases. The connection between the xed prices, the number of rell patients, and the
cost sensitivity of the doctor relates to the size of the distortion in order size at the threshold value for
the high cost manufacturer. This is a key result that must be present in the observed data for the focal
manufacturer to provide validity for the structural model. In the next section, I describe the data sources
and then perform model-free and reduced form analysis to investigate if this behavior is present.
4 Data and Reduced Form Evidence
In this section, I start by describing the various data sets that are synthesized together to replicate the
information set of the marketing manager for the focal manufacturer.
Next, I present model-free and
reduced-form evidence to demonstrate that doctors align their actions with nonlinear incentives. Finally,
I summarize with a brief discussion on the implications of this analysis for the structural estimation.
4.1 Data Source Details
To understand if doctors are aligning their prescription choices with nonlinear incentives, I compile a
variety of data sources that are readily available to the marketing manager. I use both limited to the
focal rm and syndicated data provided by a marketing research rm. Overall, my observation window
for all data sets, except where noted, is from June 2009 to May 2010.
Syndicated Data
Syndicated data are in the form of an 87-doctor study (known as Study A) of
patient arrivals and prescription habits and the aggregate patient shares for each manufacturer/product
combination in the market.
The observation of order sizes to the focal rm alone is not sucient to
identify the daily demand for products. Study A allows for the estimation of the arrival rate of patients
into a given doctor's practice. The observation level of Study A is the total number of patients arriving
into a doctor's oce during the three week period and the total number of days the doctor recorded
patient arrivals.
From this study, each patient receives, on average, 1.79 boxes of product, and 2.07
patients arrive at the practice per day. Each patient is assumed to receive two boxes of product, so I am
able to align the prescription process of the doctor to the aggregate patient share. I use the arrival-rate
information to create a mixture of Poissons that approximates the stochastic arrival of patients, which
acts as the constraint on order-size volume on a daily basis for the doctor.
The second set of syndicated data is the aggregate, quarterly patient shares of each manufacturer/product
16
combination in the market for the full year. I observe the aggregate percentage of patients that are new
versus the total patient visits.
Focal Manufacturer Data
Aggregate share data alone are insucient to estimate the structural
parameters of the dynamic discrete-choice model, since the variation in prices only occurs with the
prescription choices made on the daily level. I examine daily transaction data from 1,254 doctors provided
by the focal manufacturer. This group represents a fairly homogenous group of doctors. These practices
do not receive preferential pricing, they prescribe only the four products I measure in the empirical
application, and they are small, individual practices.To achieve this sample, I eliminate any doctors that
are chain retailers, hospitals, doctors with preferred pricing, doctors that participate in the bank program
oered by the focal manufacturer, and they must have ordered from the focal manufacturer at least twice
during each quarter. For these doctors, I observe the complete purchase history across eight-quarters. I
not only observe individual, daily order quantities, but also additional focal manufacturer information. I
merge daily sales force actions and the preferred client status of a doctor to the focal manufacturer with
aggregate sales data.
Sales-force agents are known to be an eective way to generate sales. For the focal manufacturer, I
observe the daily sales call log that details which doctors are visited by a sales-force agent on a daily
basis. On average, the doctor sees one sales-force visit in a quarter. Further, doctors are often assigned as
preferred clients to a manufacturer, where they are oered other non-price related services and incentives.
The focal manufacturer assigns each doctor a rank on a scale of A to E, which relates to how important
the doctor's business is to the manufacturer. The manufacturer sees those in Tier A as critical doctors
they must keep, and give them preferential, non-price related treatment. As such, including preferred
client tier information in the utility formulation as well is important. Table 11 presents information about
the purchasing habits and focal rm variables. Here, the number of Tier A clients is approximately 20%
of the sample and Tier B is 40% of the sample.
Pricing Data
The nal sets of data are the pricing lists and buying-group discount levels. Three of the
four rms employ a nonlinear pricing schedule, and I observe all manufacturer's pricing choice for the full
sample window. Figure 2 contains two examples of pricing schedules observed in the empirical exercise.
The volume-based discounting is based on a per-order basis, which is found to occur on a daily basis. The
price schedules are xed across the sample window. I also know the buying-group status of each doctor.
If a doctor is part of a buying-group, then the doctor receives an additional at discount on the nal cost
17
of their orders, which varies by manufacturer. The manufacturer provided me with information on their
own discount schedule to each buying group and estimates of the discount schedule for the competitors.
I use this information in construction of the doctor specic cost function.
4.2 Model Free and Reduced Form Evidence
The purpose of the structural model is to describe the doctor's dynamic ordering process.
As such,
doctors must exhibit three key behaviors for the structural model to encompass the daily prescription
ordering behavior. First, doctors must make their ordering decision on a daily basis and not accumulate
their orders across multiple days. Second, doctors must be cost conscious. If doctors do not evaluate
the cost of their choices, the choice process presented in equations 6 and 7 is invalid. Third, which is an
extension of the previous point, doctors must be not only cost conscious, but also align their purchasing
behavior with the nonlinear incentives in the price schedule.
4.2.1 Do Doctors Order on a Daily Basis or Accumulate their Prescriptions Across Days?
The incentives oered by manufacturers are on a per-order basis. The timing of when orders are placed
is critical in estimation. The original purpose of providing doctors with quantity discounts was to give
a reason to accumulate prescriptions and place one order at the end of the business day. As a result, I
look to see if doctors are collecting orders across two or more days, rather than just a single day. To test
for this possibility, I construct the transition probabilities of order sizes from one day to the next. More
formally, I construct the empirical distribution of
P (Qt |Qt−1 )for
all the given quantity sizes. Table 2
4
summarizes the results based on doctors accumulating prescriptions across two days.
If doctors are accumulating orders across business days, two patterns in transition probabilities should
be present. First, the probability of
probability of
Qt = 0
Qt > 0
should be higher when
should be highest when
Qt−1 > 0.
Qt−1 = 0.
On the other hand, the
I do not observe either eect. The pattern shown
in the empirically estimated transition probabilities is the opposite. The probability of
when
Qt−1 = 0.
Further, the probability that
Qt > 0
increases as
Qt−1
Qt = 0
is highest
increases. Therefore, I nd no
indication from the transition probabilities that doctors are accumulating orders across days.
4.2.2 Are Doctors Inuenced by Price?
Knowing that doctors order on a daily basis is only part of the story. I next explore whether they are
inuenced by the costs they incur from their prescription choice.
4 Checking
Met with \#\#\#.
She has t a
this behavior across multiple days shows a similar pattern and are omitted from this document.
18
few of her patients(sic) with our \#\#\#\#. Would t more if the price was right is a direct quote
from the sales-call log of a sales agent to a doctor's oce. This summary indicates that the doctor is
concerned with the current cost position of the focal manufacturer's product and, if the price oered by
the manufacturer was lower, the doctor would prescribe more of its product. In fact, 37.2% of the more
than 12,000 entries in the call log for this rm contained some information related to pricing.
Table
3 presents a few more interactions, where the cost of the product is a clear concern of the prescribing
doctor.
Although doctors may claim prices are too high, do they alter their prescriptions if prices change?
The focal rm conducted a pricing test for a subset of 85 doctors (Study B). For this group of doctors,
the price of the agship product was lowered for a quarter.
The change in price resulted in a 15.8%
increase in order size for the discounted product on the experimental group, as shown in Table 4.
To understand whether discounting the agship product resulted in an increased tendency to order
other products, I recover the conditional probability of purchasing other products, given that a unit
of the discounted product was purchased as well.
discounted product,
P (A),
I create the marginal probability of purchasing the
by summing all the orders of each group that contain at least one unit of
the discounted good and dividing by all days in the sample. Similarly, I construct the joint probability,
P (A, B),
by summing all orders containing the discounted product and any other product oered by the
brand, and dividing by all days. Using Bayes' rule, I create the conditional probability
P (B|A).
Shown
in Table 5, the initial probability of ordering the discounted product increases for the test group. What is
more interesting is that the test group shows a modest increase in the tendency to order other products,
conditional on ordering the discounted product, and the control group shows a large decrease in this
behavior. After the testing, both groups show a decrease in both marginal purchase probabilities and the
conditional probabilities. This model-free evidence suggests that, not only was the price test successful
in increasing the amount purchased of the discounted good, but it also increased the amount purchased
of other products.
Table 6 presents the results from a dierence-in-dierence (dif-in-difs) regression model. The estimated
parameters of Model 1, whose dependent variable is the total amount of the discounted product ordered on
each received order, indicate the pricing study achieved its goal. The interaction of importance, whether
or not the doctor is in the discount group and if the observed quarter was during the experimental period,
is statistically signicant and positive. Similarly, in Model 2, the interaction of importance is positive
again. In this case, doctors are ordering 0.16 more units of any other product during the promotional
period if they are in the group receiving the discount, or roughly 10% more.
19
Doctors did alter their
ordering behavior by prescribing more of both the discounted product and all other products of that
rm. However, a change in behavior outside of pricing may drive this result: either sales force agents
leveraged the discount in order to have more ecient visits or there may be a promotional eect that
5
caused the increase in the allocation of prescriptions to the focal rm.
4.2.3 Elimination of Alternative Hypotheses
Both model free-evidence and dif-in-difs regression analysis suggest that a doctor changes his behavior
in response to a change in the manufacturer's prices. Two alternative theories that might explain the
increase in quantity sold and bundling behavior during the price promotion are a sales-force eect or a
promotional eect. If sales-force agents use the change in price to promote the product more eciently,
then detailing eorts may drive the observed results. On the other hand, the doctors in the test group may
be purchasing more because the brand is now at the top of their minds. If they know the manufacturer's
products are on promotion, the brand the brand then has a greater presence in their mind that may yield
an increase in sales..
To test the sales-force eect, in Model 3, I run a regression of the order size by a doctor
τ
i
at time
as the dependent variable. Covariates that inuence the choice are whether the doctor is in the test
group(X1 ), whether a salesperson from the focal manufacturer visited the doctor (X2 ), whether the
doctor placed the order during the promotional window (X3 ), and interactions between all these dummy
variables.
To test the promotional eect, in Model 4, I test for the presence of a halo eect. If the promotion
causes the brand to be top of mind, then there should be a relationship between an order placed at
and
τ
during the promotional window.
τ −1
The model contains the same set of terms as before, save for
excluding sales-force visits and the addition of a lagged term for ordering (X4 ).
Table 7 summarizes the results from this analysis.
In both cases, the important interaction (test
variable, with the test group, in the promotional window) is not signicant. No statistical relationship
exists between the sales-force eect and a promotional eect during this time. The choice to increase the
amount ordered and the eect of bundling is then strongly associated with the change in price by the
focal manufacturer.
6
5 Additional analysis suggests the number of orders did not change. Instead, only the amount per-order changed as a
result of this pricing study
6 I perform additional testing of a logit model of purchase, where the dependent variable whether doctor i places an order
at time τ to determine whether the change in quantity is due to an increase in the order size or an increase in the frequency
of ordering. Sales force eort seems to drive the order process, but not necessarily the amount of product per order.
20
4.2.4 Are Doctor's Actions Aligned with Nonlinear Incentives?
The fact that manufacturers still provide doctors with price schedules hints at the possibility nonlinear
pricing is useful in driving the prescription choices of the doctor.
at best.
However, this evidence is anecdotal
To uncover this behavior, I rst examine the order-size frequencies of doctors in the sample.
Figure 2 is a collection of histograms that show the distribution of order sizes to the focal rm, and Table
8 summarizes the same information. All of these examples show a disproportionate frequency of orders
at the threshold levels of either two patients, four patients, or both.
This is a preliminary indication
that doctors are choosing to prescribe the focal rm's products in a manner explained in the motivating
example that illustrates bunching or binning behavior at the threshold values. Table 9 summarizes the
overall histogram of order sizes for the rm for all doctors in the sample.
The overall daily order-size distribution is a summary of all the individual choices made on a daily
basis.
The focal rm's position in the market is the lowest in overall patient share, so the number of
days with no orders, shown as an order with 0 prescriptions, is in line with expectations. The frequency
of larger order sizes does not clearly indicate whether doctors are, on average, taking the nonlinearities
of the pricing schedule into consideration when making decisions. If this behavior is occurring, I should
see spikes in the sequential, conditional probabilities of prescribing the focal rm's product.
I start
by calculating the continuation probability by using the empirical distribution of order sizes. Figure 3
summarizes these results.
The pattern in the continuation probabilities has spikes corresponding to prescribing one more unit
to cross each of the quantity thresholds, two and four respectively. This nding is an initial indication of
doctors adjusting their prescription choice to take advantage of the marginal cost savings. However, this
result may be driven by the distribution of patients that arrive into the doctor's practice on a daily basis.
To test for this possibility, I perform an additional analysis by merging a semi-parametric distribution of
patient arrivals to recover the sequential, conditional probabilities of quantity allocation.
To understand how I construct the estimate of the conditional probabilities, I present the following
example. Start with assuming that the maximum number of patients a doctor can see in a day is two.
Then, for a manufacturer to see an order containing two prescriptions, two patients must have arrived into
the oce that day and both received prescriptions for the focal manufacturer's products. More formally,
the probability of a brand receiving an order of size two in this example must be the probability of two
patients arriving into the oce,
product,
Pk (t1 = k, t2 = k). tn
π2 ,
times the joint probability of both patients receiving the brand's
is the patient number, and
21
k
is the brand choice.
The probability of a rm receiving an order with only one prescription arises from a combination of
events. An order of size one can occur in three ways: one patient arrives and receives the brand's product,
P (τ1 = k); 7
two patients arrive and the rst receives the brand's product but the second does not,
Pk (τ1 =
k, τ2 6= k); or two patients arrive and the rst does not receive the brand's product but the second one does,
Pk (τ1 6= k, τ2 = k).
Summing up all these events and conditioning on the probability of either one or two
patients arriving in the oce,
π1
order of size one of the brand,π1
where
Q
or
π2 , respectively, gives an estimate of the total probability of seeing an
[Pk (τ1 = k)] + π2 [Pk (τ1 6= k, τ2 = k) + Pk (τ1 = k, τ2 6= k)] = Pk (Q = 1),
is the total quantity ordered.
By minimizing the distance between the constructed order-size frequency,
order-size frequency,
Pk (Q = n),
P̂k (Q = n),
and the actual
I recover estimates of the conditional probabilities. I use the syndicated
doctor journal study (Study A) to provide an estimate of the distribution of patients arriving into the
oce on a daily basis, outlined in section 5.
I also provide a second distribution of order sizes based
on doctors prescribing the focal manufacturer's product through simple random assignment, where the
probability of prescribing the product is equal to the rm's market share. Figure 3 summarizes the t
between observed and estimated frequencies, as well as the estimated conditional probabilities.
8 The
measure of t is based on the percentage dierence between the observed frequency and the estimated
frequency of each order size.
The framework of simple assignment (blue) based on market shares does not match the observed
frequencies as well as the exible model (red). Moreover, the conditional probabilities for Patients 2 and
4 show dramatic increases when compared to the probabilities before and after these values. Patients 2
and 4 are the volume levels that correspond to the threshold values in the nonlinear schedule. From this
and the previous exercise, I assert that doctors are not only inuenced by price, but they also make their
9 Because the histogram of order sizes
quantity-allocation decisions based on nonlinear price incentives.
captures the desired behavior of doctors aligning the volume of prescriptions written on a daily basis with
the nonlinear incentives oered by the focal manufacturer, I use the distribution of order sizes as a set of
moments to identify cost sensitivity.
7 With formal expectations, the doctor internalizes the probability of the unit being allocated to a given brand based
on the patient's state space. Although I cannot incorporate that belief in this reduced form exercise, I can assume that
without a discount factor, the belief of prescribing a brand in the future is equivalent to prescribing one in the past.
8 The outside options, other manufacturers, are assumed to only change based on the conditional probability of ordering
for the focal rm. This assumption is based on three facts: I do not observe the actual order volumes of other products;
their price discontinuities occur at eight or more boxes; and the discounts are not nearly as large as those for the focal
manufacturer. I use this analysis only to show the potential changes in conditional probabilities and can be seen as a lower
bound on the actual probabilities.
9 I perform similar analysis on subsets of the panel of doctors, by both random selection and segmentation based on
preferred client status to determine whether results are robust to potential heterogeneity concerns. In both cases, the
results hold.
22
4.3 Discussion
The reduced-form analysis in this section highlights four key aspects that must be addressed when constructing a utility framework to model sequential brand choice. Doctors are not only inuenced by price,
but they also make choices that are aligned with nonlinear price incentives. A utility function must be
able to capture the potential gains from the sequential choice between the brands and induce a tendency
to bundle products together for a particular brand to benet from nonlinearities in the price schedule.
Sales force is an important factor in the doctor's choice to order from a brand at a given time. Finally,
the doctor's dynamic decision-making process occurs on the daily level.
5 Estimation and Identication
5.1 Econometric Assumptions
The intended use of this model is to quantify the eect of nonlinear pricing for both the doctor and
the manufacturer and then perform policy experiments to determine an optimal pricing structure. To
estimate the structural model, I have to make a few assumptions.
I present these assumptions in the
following section.
I assume doctors are homogenous in the distribution of patient arrivals, product requirements, and
rell types. Further, I assume doctors are heterogeneous in the manufacturer choice in the past, which
inuences the product preferences of returning patients in the observation window. Next, I assume that
up to six patients arrive in a day for this particular set of products and that each is prescribed two units
of product. I base these assumptions on conversations with industry experts and information provided
by Study A. I assume patient characteristics and preferences, outside of the state space that acts as a
constraint on the choice set of doctors, is completely contained within the error term of the doctor's
utility function. I do not observe any patient-specic characteristics, as a result I cannot identify any
patient-specic eects outside the patient's simulated state space.
IID error term, because each patient visit within a day
visiting on day
τ.
τ
This assumption allows for a clean
is unique and unrelated to any other patient
I assume the IID error term is Type I Extreme Value (Gumbel), so the doctor's choice
takes the form of a multinomial logit, conditional on the current patient's state, and creates a closed-form
solution for the continuation value. Finally, the choices a doctor makes at time
preferences in the future. If a doctor prescribes brand
time point will request brand
k
k
τ
inuence the patient
to patients, the returning patients at some future
due to the nature of rells in this market. However, the doctor does not
23
optimize the current prescription choice over this variable, as indicated from evidence in section 4.Table
11 presents the summary statistics of the variables in this model.
5.2 Patient Arrivals and Patient's States
I observe the brand/product choices for a specic Brand A made by 1,254 doctors across one year. In
addition, for Brand A, I observe the preferred client tiers of the doctors to Brand A and the dates of
sales force visits.
A particular brand may favor certain clients and oer them preferred client status.
Doctors are designated as Tier A or Tier B. Tier A contains the most favored clients and Tier B is the
second level. A doctor can be in a specic tier or none at all. I, also, know the sales history of Brand
A to those doctors for six months prior to the study window. To understand the dynamics of the other
brands, I know the aggregate market shares, in terms of patient visits, of all brands in the market across
all product categories for the entire year. I also know the aggregate probability of patients to either be
new patients or loyal patients in the market by product type. Finally, I know the full list of prices for all
products in the market across the brands. The items I need to then approximate, before estimating the
model, are: the arrival rate of patients and the patients' states.
5.2.1 Patient Arrivals
Industry experts suggest doctors cannot carry inventory of the products under observation due to the
specicity of the products and space constraints. Therefore, the daily ordering process is limited to the
number of patients that arrive to the oce on day
τ.
I observe the complete order records for the focal
manufacturer. However, this information provides no guidance on exactly how many patients arrived on
day
τ.
Study A contains information on the total number of patients arriving to the doctor's oce across
some time period. Table 9 and Figure 5 present a summary of the information from Study A.
I construct a semi-parametric estimator of the distribution of patients for simulation by using the
information provided by Study A. I assume the arrival rates of all doctors are homogenous and equal to
the distribution I estimate from the representative sample in Study A. In this section, I outline the steps
taken to construct the distribution of patient arrivals.
I assume patients arrive according to a Truncated Poisson process, with the truncation point at
six patients. Although this assumption is strong, it does allow me to use the information provided to
construct the needed distribution. For each doctor in Study A, I recover the average number of patients
arriving to the practice per day. This term is
λ̂d
to estimate the distribution of patients arrivals.
10 Another specication allows each doctor to have a dierent arrival rate,
24
10
which results in the simulation of each individual
I weigh the distribution of each individual doctor by the accuracy of the information in the study to
create a mixture of Poissons. The more days a doctor recorded patients arriving to the practice, the more
accurate the estimate of the average number of patients arriving to the practice. As a result, I weigh
each doctor-specic distribution by total number of days the doctor recorded patient arrivals divided by
the total number of study days. The nal distribution of patients is summarized as:
P̂ (N = n) =
D
1 X Daysd
× T P (n|λd )
D
Days
(8)
d
Where
D
represents the total number of doctors in the study,
doctor entered data for the journal study, and
Daysd
is the total number of days the
T P (n|λd ) is the truncated Poisson distribution probability
forn, conditional on the estimated arrival rate.
5.2.2 Patients' States
The state space of the patient,
Pt ,
comprises three parts: the product for the patient; if the patient is a
new, rell or switching patient; and the previous brand prescribed to the patient. I assume the distribution
of product preferences is homogenous across all doctors . With this assumption, aggregate quarterly level
data provide enough information to construct an empirical distribution of product preferences for a given
doctor's oce.
Similarly, I assume the distribution of new, rell, and switching patients is homogenous across doctors.
Aggregate share data contain information on the aggregate number of patient visits by product type and
whether the patient is a new patient or a loyal patient (the dierence between total patient visits and
new patient visits). Industry experts say 30% of all people prescribed a particular brand switch in the
next period. I approximate the number of switching customers from these two pieces of data.
Since the manufacturer/product requirements are determined in a previous period for rell and switching patients, there is an initial conditions problem that must be overcome. I compute the average number
of patients arriving in the oce in a given quarter by multiplying the average number of patients arriving
to the oce from the auxiliary study and the number of days in the quarter. Next, I use the proportional
allocation of products seen in the aggregate data to compute the number of patients of each product type
a doctor sees in a quarter. Looking at the data for Brand A, I observe the total number of each product
type ordered from Brand A in the six-month lagged quarter, because the rell cycle is six months. Taking this amount and dividing by the total a doctor should have seen for that product type produces as
doctor having a dierent λ. I use this specication, rather than assuming homogeneity among doctors, as a forthcoming
robustness check.
25
estimate of the market share, by doctor, for Brand A of each product type. For the other three brands, I
proportionally allocate them the left over patient share based on their average brand share amount from
the aggregate data for each product type.
5.3 Value Function Approximation
The dynamic discrete choice problem faces is built upon three sets of information: prescriptions already
written in the day, the anticipated needs of the current and future patients, and the potential cost savings
from the nonlinear incentives. The doctor faces a nite horizon problem as a terminal state occurs each
day at the
NT
patient. However, even though the decision is made across six periods, the state space
under consideration is large.
For a given patient, there are three rell states.
If the patient is a new
patient, the potential to allocate one of 16 dierent brand/product combinations exists. A rell patient
only accepts the previous brand chosen for the product category they are currently using. A switcher
patient requires any one of the three remaining brands they had not been prescribed in the previous
quarter. I assume doctors have perfect knowledge of the number of patients arriving in a day and the
state space of each. If six patients arrive in a given day, calculating the expected value of each state space
combination becomes computationally burdensome.
Keane and Wolpin (1994) provide an estimation technique to alleviate the curse of dimensionality
in nite horizon problems, through the use of an iterative process of estimating the value function for
a nite number of points in the state space and interpolating the rest.
In this section, I summarize
the estimation steps provided by Keane and Wolpin (1994) by borrowing heavily in the notation and
explanation of Crawford and Shum (2005) regarding the value function approximation algorithm.
I begin by xing a set number of points in the state space
S̃ ,
where the states are the patient states,
focal rm marketing variables, and the prescription choices made by the doctor. Given the states, and
some value of the structural primitives
the
NT ,
θ̂,
I compute the tted, value function, in the terminal state for
given the choices the doctor made during the day. The terminal value function is represented by
WN0 T (ST ),
which is calculated for each simulated set of states from the set of all possible states,
Next, I assume a exible linear relationship exists between the states
W 0NT (SNT ),
which is
ν
and the terminal value function,
g(SNT ).
W 0NT (SNT ) = g(SNT )ν +ιNT
Where
S
ST ∈S̃ .
(9)
is a vector of linear coecients and ιNT is a mean zero error term over patients. For the each
26
drawn state space, I obtain
ν0
(which corresponds to the terminal value function). I then approximate
the value function for all states in
S̃
by
W (S) = S 0 ν 0 .
The process then iterates to the patient before
the terminal value function, so the rst iteration of the value function becomes:
W 1NT −1 (SNT −1 ) = maxk uk (S) + ES 0 |S,k WN0 T −1 (S 0 )
Where
k
is the manufacturer choice and
manufacturer choice for patient
NT − 1.
presented in equation 11 and obtain
S0
(10)
is the new state conditional on the previous state and
Using this new approximation, I estimate a new linear regression
ν1.
I continue this process until I arrive at the value function for
the rst person arriving that day, which creates the continuation values at each stage using backwards
induction and the simulation/interpolation steps as necessary. Because each quantity of patients carries
a dierent potential for nonlinear savings, I need to compute this series of value function for
{2, 3, 4, 5, 6}.
NT =
I take uniform draws from the product requirements, rell state, and previous products
oered for the patient states used in value function formulation. I also take uniform draws of the doctorspecic states: buying group status, preferred client tier, and sales force visit.
5.4 Simulated Method of Moments Estimation
A full likelihood specication for this model is intractable due to the necessary simulation steps. As a
result, I use another estimation procedure that applications in models of path dependency have used
successfully, simulated method of moments (SMM). The parameters that minimize the SMM objective
function are those that provide simulated results that closely align with observed outcomes. I provide a
brief description of the estimation process and calculation of the weighting matrix based on the inuence
function approach. Borrowing heavily from the notation from Hennessy and Whited (2007), I summarize
the following step of the routine to recover parameters and standard errors.
Begin with obtaining data moments
m̂sn
M̂N ,
which are the sample counterparts to population moments.
are moments obtained from the simulated data set based on the current set of structural parameters.
Given these two vectors of moments, the minimization routine can be written as:
#0
"
#
S
S
1X
1X
b̂ = arg min M̂n −
m̂ ŴN M̂n −
m̂
b
S s=1
S s=1
"
ŴN
(11)
is a positive denite matrix of weights for each of the selected moments. To approximate the
optimal weighing matrix, I employ the inuence function method from Erickson and Whited (2000). In
27
brief,
Ŵ
is the inverse of the variance matrix of the actual moments. To achieve an estimate of variance
matrix, I estimate the dierence between the data moment value
M̂N
and the moment value for each unit
of observation. The inner product of the inuence function vectors approximates the variance matrix,
which is then inverted to obtain, the optimal weighing matrix
ŴN .
I use a modied version of this
technique. I have two sets of moments, both macro and micro, to estimate the parameters of interest
that do not share an explicit link, I use a block diagonal approximation to the true optimal weighing
matrix. One block contains the variance of the market share moments and the other block contains the
variance of all the moments from the focal rm data. Although this is not the true optimal weighing
matrix, the cross correlations between aggregate shares and individual doctor moments should be weak.
5.5 Estimation Algorithm Summary
I simulate potential days for each doctor to recover the model primitives, using simulated method of
moments (SMM). The rst three steps are simulated once and used as a basis for the estimation. For
each doctor, I simulate a potential 20 days for each observed doctor/day pairing.
1) Simulate the number of patients arriving in a day for each doctor, using the semi-nonparametric
truncated Poisson distribution obtained from the 87-doctor sample study.
2) Simulate the patient states for each doctor (their product requirements, their rell state, and any
previous product they used), using both aggregate shares and the computed, internal doctor market
shares.
3) Draw 1,000 hypothetical day/doctor type/sales force visit combinations to use in the nite horizon
approximation of the expected value function (as per Keane and Wolpin).
These next steps are employed to minimize the objective function value in SMM estimation. For a
given estimate of structural parameters, I then:
4) Estimate the value function approximation using linear regression for each set of simulated states.
5) For the 1,254 doctors, simulate the choice made by each doctor at the given parameters values for
the rst patient, and continue untilN̂T choices are made for each simulated day.
6) Update the internal market share for the doctor across the quarter for each brand/product types
prescribed in the day to use in future patient states.
7) Once all the choices have been simulated, calculate the moments and minimize the objective
function.
8) Repeat steps 4-7 until the minimization occurs by updating the parameter values of the model.
28
5.6 Identication
This section summarizes the method used to identify the structural parameters by using a novel, informative set of moments selected on the basis of both model-free and reduced form analysis. In essence, there
are three sets of parameters to identify: the manufacturer intercept terms, the cost sensitivity parameter,
and the sensitivity of the doctor to the focal rm's marketing mix variables. To identify these parameters,
I select moments that respond to changes in the structural parameters. If no linkage exists between the
structural parameters and the moment selection, the structural parameters are not properly identied
using the available data sources. I explain my moment selection for the intercept terms rst, then the
focal rm specic marketing variables, and last the cost sensitivity parameter.
To match the intercept terms for three of the four manufacturers, I use the aggregate patient share
of non-rell patients. I choose to use the patient share of the non-rell patients as the the doctor has
inuence over the manufacturer choice. Holding the other parameters xed in the structural model, the
intercept term then reconciles the aggregate patient share for each manufacturer.
To identify the focal rm specic eect of marketing mix variables, my moment selection are a set of
regression coecients. Specically, the dependent variable is the logged quarterly quantity of prescriptions
to the focal manufacturer. The explanatory variables are indicator variables for the preferential client tiers
and a sum of all the sales force visits in the quarter. The results of this regression are presented in Table
12. Since I lack information on the marketing activities of the other manufacturers, I focus on the data at
hand to tease out the eect of the focal rm's activities. If belonging to a preferred client tier increases the
contemporaneous choice utility for that manufacturer, then that doctor should, on average, order more
from that manufacturer. Therefore, the degree to which I match the dummy variables in the regression
between the observed data and the data recovered from SMM estimation identies the eect of being
in a preferred client tier for the focal manufacturer. A similar argument can be made in identifying the
structural parameter associated with a sales force visit. By matching the regression parameter associated
with the number of sales force visits in a quarter, I am able to identify the contribution of a sales force
visit to the doctor's value function.
The last variable to identify is the doctor's cost sensitivity.
The distribution of order sizes to the
focal manufacturer provides information on the degree of cost sensitivity. The intra-day choice based on
the assumed exogenous arrival of customers to the practice that provides information on how doctors
respond to the marginal cost savings contained in the price schedules. Recall the motivating example,
where a variety of scenarios are used to show how the anticipated patient needs and discontinuities
29
in the price schedules cause distortions in the number of prescriptions written by the doctor at the
threshold value.
The degree to which a doctor is cost sensitive inuences the importance of meeting
or exceeding the threshold values. The greater the cost sensitivity, the more the doctor has incentives
to meet the threshold value for the high cost product and go no further, creating a distortion in the
proportion of order sizes that matches the threshold values in the price schedule. I show that doctors
not only respond to prices, but also distort their behavior at the quantity thresholds. Further, I show
this empirical result cannot replicated by doctors randomly assigning patients to the focal manufacturer
based on the aggregate market share. The conditional probability of ordering the units that trigger the
nonlinear cost savings shows spikes at the corresponding discontinuities in the price schedule, which is the
underlying behavior found in the distribution of order sizes. Although prices are xed for a given doctor,
changes in the distribution of order sizes can only come from two sources: the average utility gained
by the doctor/patient in the consumption of the prescription, and the cost sensitivity.
Purely raising
the average utility results in a greater frequency of larger orders but would not create disproportionate
allocation at the threshold values.
Therefore, the degree to which a doctor is cost conscious provides
information on their ordering strategy conditional on the assumed exogenous arrival of customers to the
practice.
6 Results
In this section, I briey discuss the estimated coecients from the structural model and provide two
economically interesting measures. First, I estimate the elasticity of price from the reduced form data,
a myopic agent model, the full dynamic specication, and comment on the dierence. Second, I provide
a measure of the distortion eect on the prescription choice of the focal manufacturer's products. Next,
I present an exercise based on the needs of the focal rm in order to see if their current pricing policy
is optimal or if adjusting the quantity thresholds and discount levels can increase protability. Finally, I
showcase a counterfactual exercise the rm requested in order to understand how well nonlinear pricing
performs in both new and mature markets.
Table 13 presents the structural parameters and simulated moments. I nd that all parameters except
the intercept for Brand A are statistically signicant.
The parameter for price sensitivity is negative,
as expected. The estimates of the parameters associated with being a Tier A client or Tier B client, or
being visited by a sales force agent are all positive and signicant. In addition, Tier A provides more
utility than Tier B, which is in line with expectations. However, this brief description of the structural
30
primitives does not really explain the nuances of how nonlinear incentives inuence the decision making
process.
In terms of model t, the structural model provides a close approximation of the selected moments.
The structural results in a larger patient share for the new patients of Brand A, but this dierence can
be attributed to the sample of doctors in the empirical analysis. The regression coecient moments are
approximated well by the underlying parameters in the structural model. Finally, the structural model
over-estimates the proportion of order size 1 and under estimates the proportion of order size 2, but the
dierence is minimal. Further, the model also does a fair job matching the frequency of order sizes 3 and
4, which are not explicitly match moments, but rather used as an out of sample test of model validity.
I start with calculating the price elasticity of the doctor. Using a standard two point simulation of
the prices, I perturb all the price levels for each product simultaneously. I nd the average price elasticity
is 1.53.
If I solely use the information on aggregate changes in behavior from Study B, scaled by the
aggregate proportion of patients the doctors have the ability to prescribe a dierent product to, I nd
the estimated elasticity is 1.383. If I assume the agent is myopic and apply the sequential choice model,
the elasticity is 1.13.
The elasticity estimates obtained in the dynamic model are considerably higher
than those in the static model and the reduced form estimate. This nding is important as doctors are
shown to be cognizant of costs, but constrained number and characteristics of the patients arriving into
the oce during the day.
Next, I examine the eect of the nonlinear price schedule on distorting the prescription choices of
doctors. I examine the average sequential probability of prescribing the focal manufacturer's product,
conditional on the current order size, where
presentation, I present the following ratio,
Q
represents the current quantity of the order. For ease of
P (Q+1|Q)
P (Q=1) , which is the conditional probability of ordering one
more unit given the current order size compared to the probability of prescribing the initial unit to the
focal manufacturer:
P (2|1)
P (1)
P (3|2)
P (1)
P (4|3)
P (1)
P (5|4)
P (1)
12.013
3.545
109.866
11.438
The probability of prescribing two units, given that one unit is already prescribed,
P (2|1),
is 12 times
more likely than the initial probability of prescribing Brand A's product. This conditional probability
then decreases at
P (3|2),
but is still 3.5 times higher than the original probability of prescribing a single
unit. The spikes in probability are obvious. At the threshold values the doctor is able to capture the
opportunity cost of prescribing the necessary amount to achieve not only a lower per-unit price on the
31
current patient's prescription, but also on all other prescriptions written to that manufacturer.
Also,
even though the per-unit price at two units is higher than that of ve units, due to the gains in cost
savings achieved by crossing the quantity threshold at two units. Recognizing this distortion eect is an
important factor in developing a new nonlinear price schedule, presented in the next section.
6.1 Protability Improvement under Focal Firm Constraints
At its core, the framework and methodology presented is useful in aiding marketing managers in manufacturing rms to derive better pricing policies. The main motivation behind this empirical analysis is to
provide the focal rm with a better pricing policy for its products.
Cij (Qi )
is the representation of the
nonlinear price schedule, where the per-unit price to the purchasing rm is determined by the total order
size
qij
Qi
k . χj
allocated to manufacturer
is the total amount of product
j
is the variable cost of production to make a unit of product
doctor
i
orders from the manufacturer. Finally,
FC
j.
is the cost of
creating and shipping the order. Given that the order sizes are bounded at six units, the shipping and
processing costs are considered xed on an order level. Therefore, the protability for manufacturer
an order from doctor
i
on day
τ
k
on
is
πiτ =
X
Cij (Qiτ ) − χj qijτ − F C
(12)
j∈J
The original problem the manufacturer faced is the cost incurred from the doctor ordering after every
patient rather than accumulating orders at the end of the business day. The nonlinear incentives were put
in place to encourage doctors to adopt the desired behavior of accumulating prescriptions until the end
of the business day. The cost of fullling an order is generally invariant to the order size for this product
class.
The manufacturing rm prefers to send out fewer, larger orders.
The focal rm is concerned
about dramatically changing its prices, since that might cause the other competitors to adjust their
prices accordingly, which bounds the potential price schedules I analyze in this study. In this analysis, I
present a series of price schedules based on the guidance of the focal manufacturer. Using the focal rm's
constraints of preserving the nal tier prices as a guide, I construct dierent quantity thresholds and
discount levels. Short of a pure grid search over possible prices and discontinuities, there is no straight
forward method to assess the optimal pricing plan.
Instead, I select a set of price schedules that are
either commonly used or suciently interesting to study the change in protability, order frequency, and
order sizes. These three items characterize how the characteristics of the price schedule inuence both
channel coordination and protability.
32
To generate the price schedules, I take the following steps. I use the initial,
of each product as benchmarks. I then subtract
allowable in this exercise.
initial price
c1j
c1j − c3j = ωj ,
where
I discretize this value into fths, so
ωj
dj =
c1j ,
and nal tier price,
c3j ,
is the total dollar value discount
ωj
5 , and use these values and the
to construct the dierent pricing scenarios presented below.
Price Schedule
Q1
Q2
Q3
Q4
Q5
Q6
Schedule 1 -Initial Price
c1j
c1j
c1j
c1j
c1j
c1j
Schedule 2 - Linear
c1
c1 − dj
c1 − 2dj
c1 − 3dj
c1 − 4dj
c1 − 5dj
Schedule 3
− Hi
Lo
c1
c1 − 4dj
c1 − 4dj
c1 − 5dj
c1 − 5dj
c1 − 5dj
Schedule 4
− Lo
Hi
c1
c1 − dj
c1 − dj
c1 − 5dj
c1 − 5dj
c1 − 5dj
Schedule 5
− Even
c1
c1 − 2dj
c1 − 2dj
c1 − 4dj
c1 − 4dj
c1 − 4dj
Figure 5 is a graphical representation of the dierent price.
I examine the percentage change of
protability, order frequency, and order sizes from each dierent plans against those measures from the
current price schedule. Table 14 is a summary of these results.
I present the performance of four single-tier and ve nonlinear price schedules. The discounts in the
single-tier plans correspond to the amount of
dj
applied to the initial price. For the nonlinear plans, the
numbers correspond to the quantity level at which the discontinuity occurs, and the naming indicates the
discount structure. For example, 2 Low and 4 High is a price schedule with discontinuities at quantity
levels 2 and 4, where the initial discount at 2 is a small and the discount at 4 is large.
Initial evidence suggests that given the current economic conditions and constraints on the price
schedule, the rm's current plan is performing well.
Only two plans show an increase in protability
over the current plan, Single Tier - 60% Discount and NL - 2 Hi and 4 Lo. The nonlinear price schedule
provides the largest increase in protability, 6.96% as opposed to 3.45% for the single-tier price schedule.
It is worth investigating how the number of prescriptions per order change in each situation to understand
why the nonlinear price schedule has such a marked improvement over the single-tier pricing system.
Focusing just on these two pricing policies, I nd the single-tier price schedule increases the overall
order volume by 29.8%, whereas the nonlinear schedule increases order volume by 5.56%. Initially, one
would expect the single-tier system to perform better, given the higher order volume. However, I nd the
gains in order volume are predominately on increasing the frequency of single unit orders, with decreases
in all other order sizes.
In fact, the number of orders for one unit increases by over 50% under the
single-tier plan. The dramatic increase in the frequency of smaller orders is not in the best interest of
the manufacturer to coordinate the channel between itself and the doctors.
33
Although the nonlinear plan shows a modest increase in overall order frequency, the number of prescriptions on each order placed that matters more to the manufacturer. For all the order sizes, there is
an increase in the frequency when compared to the current plan. The order sizes of two and three show
the greatest gains in frequency: 18.08% and 17.9% respectively. A large, initial discount gives the doctor
immediate incentives to purchase more from that brand. Once the initial discount level is achieved, the
smaller amount of cost savings at the higher quantity level is attractive to the doctor as the per-unit cost
savings are multiplicative. At four units, the doctor realizes a total cost savings on not just the incremental incentive for crossing the threshold on the fourth prescription, but also across the three previous
prescriptions expected or written in the day already.
Next, I compare the price schedules where the quantity thresholds are shifted by one unit or the
quantity thresholds remain intact but the incentives are reversed, such as NL-2 Lo 4 Hi (2-4 Plan) and
NL-3 Hi 5 Lo (3-5 Plan). Even though the discontinunities match the optimal nonlinear plan, which the
3-5 contract does not have, the 2-4 plan performs worse, with an overall decrease in protability compared
to the current plan of -21.21% versus -13.76% for the 3-5 price schedule. The 2-4 Plan loses orders across
all order sizes, even where the discontinuity is at its greatest value. Although a large incentive is present,
the likelihood of attaining that goal is very small.
Conversely, the 3-5 Plan shows an increase at its
threshold size of three units and another at ve units, though it sacrices volume on some of the smaller
orders. Just that small change of where the discount levels are placed relative to the arrival of customers
has an impact on protability and the number of prescriptions per order.
There are two key insights obtained from this analysis. First, given the nature of quantity discounts
and current market conditions, a high initial price followed by a large discount at a low quantity threshold,
and a more modest discount at a high quantity threshold is optimal. This pattern of discounts provides
the necessary incentives to increase the average order size without increasing the overall order volume.
Second, the location of the the quantity thresholds matters in its relation to the number of patients
arriving into the practice. Given this exercise, the focal rm also wanted to understand the impact of the
percentage of rell and switcher customers on the structure of the nonlinear price schedule to understand
the characteristics of an optimal pricing plan in more or less mature markets.
6.2 Counterfactual Exercise - Market Maturity and Quantity Discounts
This section examines how the maturity of the market impacts the creation of a nonlinear pricing policy.
With markets in their infancy, no strong product preferences would exist and all customers would be
considered new; while more mature markets would have reached a state where preferences are fairly
34
stable and only a small percentage of new customers come into the market at a given point in time.
The rm wanted to understand how the price schedule should change in each market.
I apply the
same structure of nonlinear price schedules used before; however, I adjust the proportion of rell and
switching patients. In this exercise, I do not have the any constraints on the price schedule, so I have
more exibility in determining the optimal price schedule. I do not assume any competitor reaction in
this exercise; instead, I assume the competitor's behavior remains xed, and focus on the components of
the optimal price schedule. I detail the construction of this set of price schedules and market conditions
below, and then discuss the results of the counterfactual exercise.
6.2.1 Counterfactual Assumptions
Below is a table that summarizes the specics of each price schedule tested in the counterfactual analysis.
Each price schedule contains three components: the per-unit price, the number of discontinuities, and
the location of each quantity threshold. Each schedule starts with an initial price for one unit,
a nonlinear discount is applied if a certain quantity threshold is met,
have dierent quantities of
D
D.
I x
D
c1 .
Then
in each evaluation, but
at given threshold values depending on the nature of the price schedule
under observation. For example, a price schedule that has the same discount applied at each quantity
level would start o with
c1
at the initial quantity allocation.
equals two prescriptions, the per-unit price would be
c1 − D.
At
Then, at
Q3 ,
Q2 ,
when the order quantity
the per-unit price would be
c1 − 2D,
and so on.
Price Schedule
Q1
Q2
Q3
Q4
Q5
Q6
Schedule 1 -Single
c1
c1
c1
c1
c1
c1
Schedule 2 - Linear
c1
c1 − D
c1 − 2D
c1 − 3D
c1 − 4D
c1 − 5D
Schedule 3
− Hi
Lo
c1
c1 − 4D
c1 − 4D
c1 − 5D
c1 − 5D
c1 − 5D
Schedule 4
− Lo
Hi
c1
c1 − D
c1 − D
c1 − 5D
c1 − 5D
c1 − 5D
Schedule 5
− Even
c1
c1 − 2D
c1 − 2D
c1 − 4D
c1 − 4D
c1 − 4D
The rst, and most basic price schedule, is a single-tier (Schedule 1). In a single-tier price schedule,
the rm oers only one per-unit price per product. =Schedule 2 is a linear discount schedule. For each
additional unit purchased, the per-unit price decreases by
(Q − 1) × D.
This price schedule provides
frequent, but smaller incentives. Schedule 3 is a Hi-Lo discount schedule. At the threshold of two units,
a very large discount is oered; at four units a more modest discount is oered. Schedule 4 is a Lo-Hi
discount schedule. The quantity thresholds are the same as those in Schedule 3, but with the discount
35
amounts reversed:
a modest discount at two units and a large discount at four units.
another Hi Lo price schedule.
11
.
Schedule 5 is
I adjust the discount on the initial price, across all products in
$2 increments and have the nonlinear discount amounts increase by $0.50. I use the same protability
formulation in equation 14 and present the results of this analysis as a ratio of the simulated contract
protability against the focal manufacturer current plan's protability.
I keep the same arrival rate of customers and the same initial conditions of the doctors in the market
that I use in estimation.
I adjust the percentage of rell and switcher customers between the two
counterfactual exercises. On one extreme, I assume both the percentage of rell and switcher patients is
set to 0. This market represents one with no xed preferences, so the doctor experiences no constraints
on his prescription behavior, except for the product requirements of the patient. The other example has
a rell rate of 60% and a switcher rate of 30%. This market represents one that is mature in the product
life cycle. I select these two proportions based on the current market environment having approximately
30% rell patients.
6.2.2 Counterfactual Results
I summarize the results from the two counterfactual scenarios in Table 15, as well as present heat maps
that show the contours of relative protability levels in Figures 7 and 8. Comparing the relative prot
levels, the current plan would do poorly in a new market scenario. None of the optimal plans do worse than
2.18 times the protability achieved by using the current plan. Given that this economic environment
is completely dierent from the one the rm is currently in, this result is not surprising.
Three of
the ve nonlinear price schedules perform worse than a single-price system, with the 2 Lo - 4 Hi price
schedule performing the worst.
In a brand new market, the model suggests a pricing model based on
market penetration is the optimal strategy, a low initial price with small discounts. As a result, all price
schedules have a low initial value and minimal discounting. The 2 Hi - 4 Lo price schedule performs the
best, with a relative prot level of 2.313 versus the single tier relative protability of 2.233. Even in a
system where there is no potential for rell patients, the combination of the large initial discount with a
smaller discount for higher volume levels is optimal in providing enough incentives to have the highest
protability. To show the regions of protability, Figure 7 presents a heat map of the relative protability
levels for each price schedule. Areas of deeper blue indicate higher relative protability and red are areas
of lower relative protability. In all cases, the regions of the highest protability are clustered in the same
general area.
11 I have also tested using dierent threshold levels, but due to the limitation of only seeing six patients a day, the volume
thresholds provided above always provide greater protability.
36
On the other hand, in the mature market scenario, all nonlinear price system perform better than a
single-price system. In this market, some heterogeneity is present in the initial price and the discount
levels that are optimal for each plan, but the overall trend is the same. In more mature markets, a higher
initial price and deeper discounts enhance protability. This result is consistent with the original pricing
exercise performed for the focal manufacturer.
Again, the 2 Hi - 4 Lo price schedule performs better
than all the others. However, of the three that under performed the single-tier pricing system in the new
market, the 2 Lo - 4 Hi went from the worst price schedule to the third best price schedule, which shows
the impact of attainability on the performance of a price schedule. With a greater ow rate of people
preferring the focal rm's product and sticking with it, the probability of attaining the higher quantity
threshold and achieving the large discount is higher. However, practices with a higher volume dedicated
to a dierent manufacturer have no incentive to switch and prescribe the focal brand's products. The 2
Hi - 4 Lo provides just the right mix of incentives for the practices that currently prescribe the product.
Though it performs the best at optimal levels, the 2 Hi - 4 Lo price schedule is also the riskiest. Out of
all the price schedules, it has the greatest region of protability at or below the current plan.
6.2.3 Discussion of Counterfactual Analysis
In the scenario with all new patients, no persistence exists in the choices the doctor makes, since no
patients return as rell patients in this market. As a result, each manufacturer must compete for the
prescription of each patient.
With no expectation of a future patient needing a particular product,
the initial patient is even more critical than before in the quantity allocation process. The increase in
competition for that single patient creates a downward pressure on the manufacturer to set the initial
price as low as possible.
However, once the rst prescription is locked into a manufacturer, providing
the rest of the prescriptions that day to the same manufacturer does not require much in the way of
further incentives to do so. The process is more deterministic for the doctor, because each patient has
no previous product preferences that constrain his choice set.
For these reasons, a simple linear price
performs better than some of the nonlinear plans, and only marginally worse than the other two.
The mature market scenario provides a completely dierent view. The presence of a large number of
rells allows the manufacturer to extract economic rents from the doctor on the days when only a few
units are ordered.
This high initial price is then oset by the promise of achieving large cost savings
for just allocating a few more prescriptions to that manufacturer.
However, because the proportion
of prescriptions that can be allocated to any brand freely is relatively small, the discounts need to be
suciently large to change the doctor's behavior. The discounts act as a opportunity cost for the doctor.
37
If a doctor knows one patient is arriving that day for a given product with a high price, the choice of not
allocating an additional unit to the given manufacturer is analogous to an opportunity cost.
6.3 Epilogue
Initial evidence from the rst year of implementing the new price schedule based on the results from the
structural model shows success in the eld. Prior to implementation, the division was consistently showing
year over year share and revenue losses of 12% to 15% respectively. However, after the rm restructured
its international division, including the pricing structure, the past year has shown approximate gains of
9% in both year over year share and revenue. Without further research, I cannot estimate impact of the
new pricing system on the growth of the division, but after speaking with management, I can say that
the new pricing policy is a driving force behind the change.
7 Conclusion
This paper presented a framework and estimation technique for marketing managers to improve their
nonlinear pricing policies in B2B transactions and measure the distortions caused by the nonlinear incentives. Using SMM and the technique presented in Keane and Wolpin, I am able to estimate a structural
model of a doctor writing prescriptions, conditional on price schedules and anticipated daily demand,
using only the standard information set a marketing manager would have in this industry. While I focus
on B2B transactions in the building of the structural model and empirical analysis, the general framework
is easily extended to the nonlinear benets oered in B2C transactions. I show that nonlinearities in the
price schedules cause a disproportionate frequency of orders at the threshold values.
This behavior is
consistent given the prices chosen by the focal rm and aid in identication of the parameter associated
with the cost sensitivity of a doctor. I further show additional moments of interest to estimate the structural primitives of commonly used marketing mix variables employed in the B2B setting.
Examining
the model t in terms of the selected macro and micro moments, I nd the presented structural model
approximates the doctor's daily decision making process.
Through both pricing exercises, I uncover some interesting insights into the inuence of nonlinear
pricing on protability. First, an optimal nonlinear price schedule includes a large discount at a smaller
quantity level and a smaller discount at a larger quantity level. This alignment of incentives increases
the average order size sent to the rm without drastically increasing the order frequency, which are
the optimal conditions in channel coordination. Second, the presence of a rell patient in the doctor's
38
ordering process causes a re-assessment of the quantity gains. The more patients that come in for rells of
a particular brand, the greater the incentives are to allocate any new or, if possible, switching patients to
that brand to achieve the greater cost savings. Not allocating those additional units leaves considerable
cost savings behind, which is an opportunity cost to the doctor. The direct implication of this nding
comes from the second counterfactual exercise, which shows nonlinear pricing is more protable than a
single-tier price system in the presence of a higher percentage of rell patients.
Conversely, in newer
markets without the ow of rell patients to the doctor, a single-tier pricing system works better than
some of the nonlinear price contracts.
I conclude with a few comments on the limitations of the model and possible extensions. First, the
characteristics of the doctor's ordering process mimics a just-in-time inventory system.
In other B2B
ordering, the purchasing rm carries over an inventory and has greater exibility in how much it can
order at a given time, although with the potential costs of holding inventory and unsold units. Second,
this model is founded upon a quantity discount application, but it can be extended to a two part tari
design. Third, I assume all products have the same xed utility value in the market. However, dierent
rms are known to specialize in dierent products, which causes heterogeneity in the market preferences
for the dierent product types. Finally, as mentioned previously, this model can be extended to other
nonlinear benet contracts in B2C transactions.
In this empirical analysis, I focus only on the use of
price schedules, which are unique to B2B transactions. However, nonlinear incentives are utilized in a
variety of contexts that can be modeled in the B2C environment, where the nonlinear gains are more
intangible and lack a true dollar value.
39
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International Journal of Industrial Orga-
Table 1: Examples of Nonlinear Price Schedules
This table summarizes the nonlinearities in the pricing schedule for the products in the market under observation. I have
selected two schedules to present, the one for the focal firm, Brand A, and a second firm, Brand B. Due to confidentiality
reasons, I cannot present the actual dollar values. So, I have divided all the values by the initial price for each brand/product
combination. Brand A has discounts in two levels while Brand B has only one. In addition, Brand A offers larger discounts
than Brand B does.
Firm A
Product
Product
Product
Product
1
2
3
4
1-3 Boxes
(1 Patient)
1
1
1
1
4-7 Boxes
(2-3 Patients)
0.93
0.979
0.917
0.942
8 or more Boxes
(4+ Patients)
0.884
0.947
0.85
0.865
1
2
3
4
1-3 Boxes
(1 Patient)
1
1
1
1
4-7 Boxes
(2-3 Patients)
1
1
1
1
8 or more Boxes
(4+ Patients)
0.964
0.913
0.913
1
Firm B
Product
Product
Product
Product
Table 2: Order Size Transitions
This table displays the transition probabilities from day τ − 1 to day τ to examine whether or not doctors are accumulating
orders across days, rather than ordering on a daily basis. The left hand side is the order quantity on day τ − 1 and the
columns are the order sizes of day τ . The percentages in the middle are the probability of transitioning from one quantity
amount to the next, conditional on the order quantity size ordered at τ − 1
Lagged Order Size
0
1
2
3
4
5
0
0.905
0.835
0.805
0.729
0.709
0.625
1
0.064
0.108
0.105
0.148
0.134
0.195
Current Order Size
2
0.023
0.041
0.062
0.078
0.096
0.119
43
3
0.005
0.01
0.016
0.024
0.026
0.018
4
0.002
0.004
0.007
0.014
0.02
0.018
5
0.001
0.001
0.003
0.003
0.009
0.012
Table 3: Summary of Sales Force Responses from Customers
This table is a collection of a few comments observed in a collection of sales call visits. In each case, I omit any personal,
firm level, or product details to maintain confidentiality.
ID
Comment
Comment
Comment
Comment
Comment
Comment
Comment
Comment
Comment
1
2
3
4
5
6
7
8
9
Comment
Use Brand 3 right now not much Brand 2 because of price.
Need more of an incentive to move over.
She has a few of her patients with our product. Would fit more if the price was right.
BLANK said that due to price and design, he is going to make this his number one Product 3!!
Not interested in going to Product 2. Mostly Brand 3 now based on price.
Believes in Product 4 modality but too high priced.
Pricing is quite high.
Not doing much in Product 4 due to pricing but is willing to promote if can provide great price.
BLANK keen to try out Product 4. Price is a barrier
Table 4: Changes in Quantity Ordered Before, During, and After Study B
This table displays the descriptive statistics of a pricing study employed by the focal firm. 85 doctors were given an altered
pricing schedule, where the flagship product, Product A, was discounted to a flat price; all other prices remained the same.
The average quantity ordered by doctor of the discounted product is shown in three time frames for both the test and control
groups: before, during and after the price change.
Average Quantity Per Order
Standard Deviation
Minimum
Median
Maximum
Before
0.421
0.882
0
0.300
3.068
Control Group
During
0.432
0.989
0
0.310
3.833
After
0.399
0.933
0
0.250
2.50
Before
0.725
1.357
0
0.512
2.255
Test Group
During
0.840
1.484
0
0.680
2.667
After
0.775
1.353
0
0.585
2.820
Table 5: Probability of Ordering the Discounted Product Before, During, and After the Study B
This table compares the probability of ordering the flagship product P (A) and the probability of binning P (B|A). The
probability of binning is defined as the probability of ordering any other product offered by the firm, conditional on ordering
the discounted product.
Before
Test Group
Control Group
During
After
P (A)
P (B|A)
P (A)
P (B|A)
P (A)
P (B|A)
9.31%
2.56%
39.03%
21.07%
9.94%
2.55%
39.88%
19.97%
8.53%
2.26%
36.56%
15.41%
44
Figure 1: Flow Chart of a Doctor’s Ordering Process
This figure shows how the doctor’s ordering process evolves throughout the day. Starting at time t=0, the doctor observes
the future patients’ states. As choices are made throughout the day, the terminal period’s value is updated and alters the
contemporaneous choice the doctor has to make.
The Day Begins
Observes Information Arriving Patients
Observes Patient States and
Creates Expectation of Final
Costs based on them
Creates
Expected
Utilities
•New or Returning Patient
•If Returning – Refill State, Product
Need and Previous Brand
•If New – No information is known
Initial Quantities
Created based on
Patient States
First Patient Arrives
Doctor Observes Patient’s State
and Expected Value based on
Current Choice
Doctor Updates Order
for Brand k
Doctor Prescribes a Product for Patient 1
Brand k’s Quantity
Increases by 1
Updates
Expected
Utilities
Nth Patient Arrives
Doctor Observes Patient’s State
and Final Costs based on Current
Choice
Doctor Updates Order
for Brand k
Brand k’s Quantity
Increases by 1
Doctor Prescribes a Product for Patient N
Orders are Placed to All
Brands K and Final
Cost is Realized
The Day Ends
45
Table 6: Diff-In-Diffs Estimation of Units Ordered
This table presents the result of two Diff-In-Diffs regressions. Both models examine the order size of a given doctor. For a
subset of 85 doctors, the price was reduced for a single product, all other prices were kept constant. I examine the impact
of this decision to determine if the quantity ordered by this group changed during the discount window on two dimensions.
The first dimension is if the order size increased for the discounted product (Model 1). The second dimension is if the test
group purchased more of other products (Model 2). X1 is a dummy variable for the doctor being in the test group. X2 is a
dummy variable for if the observation is during the test period when the price is discounted. Finally, X1 *X2 is the interaction
between these two dummy variables. If the promotion is a success, then the interaction should be statistically significant.
The table summarizes the estimated parameter, the standard error (SE), and statistical significance (Sig.). If Sig. is ∗ ∗ ∗,
then it is significant to the 99% level. If Sig. is ∗, then it is significant to the 90% level
Model 1 - Discounted Product
Estimate
SE
Sig.
Intercept
X1
X2
X1 *X2
R Squared
.448
.303
-.009
.054
.005
.020
.007
.028
.005
Model 2 - All Other Products
Estimate
SE
∗∗∗
∗∗∗
1.083
-.014
.08
.081
∗
.007
.030
.010
.042
.001
Sig.
∗∗∗
∗∗∗
∗
Table 7: Excluding Alternative Hypotheses - Sales Force and Promotional Effects
This table presents the results from reduced form analysis of two competing hypotheses that may drive the increase in sales
and bundling. Both models are regression models where, Ydt is the quantity ordered by doctor d at time t. Model 3 is testing
the influence of sales force visits (X3 ), measured as a dummy variable if the doctor had been visited by a sales force agent
in the past 5 business days. Two other variables of interest are if the doctor is included in the test group(X1 ) and if the
observed day t is during the pricing study (X2 ). I then interact all the variables to determine if sales force activity during
the promotion for doctors receiving the promotion have a significant effect on increasing the probability of purchase. Model
2 uses the same variables Ydt , X1 , and X2 . I then include variable to measure the effectiveness of a promotional effect. To
capture this behavior, I use a lagged term of an order (X4 ). All the interactions between the variables are included as well to
determine if the promotional effect is strongest for doctors receiving the discount during the promotional period. The table
summarizes the estimated parameter, the standard error (SE), and statistical significance (Sig.). If Sig. is ∗ ∗ ∗, then it is
significant to the 99% level.
Model 3 - Sales Force Effect
Estimate
SE
Sig.
Intercept
X1
X2
X1 *X2
X3
X1 *X3
X2 *X3
X1 *X2 *X3
X4
X1 *X4
X2 *X4
X1 *X2 *X4
R Squared
.218
.149
-.015
-.007
.063
-.042
-.004
-.027
.002
.007
.002
.010
.015
.044
.020
.058
Model 4 - Promotional Effect
Estimate
SE
∗∗∗
∗∗∗
∗∗∗
Sig.
.198
.150
-.010
-.008
.002
.008
.002
.011
∗∗∗
∗∗∗
∗∗∗
.097
-.043
-.017
.007
.003
.009
.004
.012
.010
∗∗∗
∗∗∗
∗∗∗
∗∗∗
.003
46
Table 8: Frequency of Order Sizes to the Focal Firm
This table summarizes the order size frequencies to the focal firm. For each doctor, represented by an ID number, I summarize
the total amount of units ordered from the focal firm on a daily basis. The distribution of orders have peaks at the quantity
levels that correspond with nonlinear incentives in the price schedule. The frequency value, for ease of reading, is presented
as the total number of orders of a given size divided by the total number of business days.
ID
980
1530
2397
557
141
1204
12
1458
1428
0 Patients
53.42%
72.22%
73.93%
70.51%
73.50%
75.21%
74.36%
73.93%
74.36%
1 Patient
17.95%
6.84%
5.98%
7.26%
8.55%
9.83%
8.12%
5.98%
12.82%
2 Patients
21.37%
15.81%
15.38%
14.96%
14.10%
13.68%
13.25%
13.25%
8.97%
3 Patients
5.56%
2.99%
0.85%
2.14%
2.14%
0.43%
2.14%
1.71%
0.85%
4 Patients
1.28%
1.71%
2.99%
3.42%
1.71%
0.85%
1.28%
4.27%
2.56%
5 Patients
0.43%
0.43%
0.85%
0.85%
0.00%
0.00%
0.85%
0.43%
0.43%
Table 9: Descriptive Statistics of Order Size Frequency
This table displays the descriptive statistics of the frequency of order sizes. This summarizes the daily ordering pattern of
1,254 doctors for 252 business days.
0 Boxes (0 Patients)
1-2 Boxes (1 Patient)
3-4 Boxes (2 Patients)
5-6 Boxes (3 Patients)
7-8 Boxes (4 Patients)
9-10 Boxes (5 Patients)
Mean
85.45%
9.41%
3.8%
.84%
.34%
.11%
S. D.
.1
.0524
.0359
.0145
.008
.003
Minimum
23.9%
1.28%
0%
0%
0%
0%
Median
88.67%
8.12%
2.56%
.427%
0%
0%
Maximum
94.87%
30.34%
23.5%
14.96%
8.97%
3.41%
Table 10: Descriptive Statistics of 87 Doctor Arrival Rate Study (Study A)
This table displays the descriptive statistics of Study A. Each doctor recorded the number of patients arriving in a day,
the amount of product prescribed to the patient, and the prescribed brand. Due to confidentiality reasons, I only observe
the total number of days recorded, the total number of patients that arrived into the office, and the total number of boxes
ordered.
Days Recorded
Total Number of Patients
Average Number of Patients
Mean
8.14
16.1
2.06
S. D.
4.08
10.62
1.04
47
Minimum
3
5
1
Median
7
12
1.6
Maximum
21
72
5.25
Figure 2: Examples of Order Size Distributions from Doctors in the Sample
This is figure is a representation of the data summarized in Table 7. Note, I have omitted the density for the 0 patient option
to better show the distribution of order sizes for Brand A’s products. In each case, note the disproportionate number of
orders clustered around the threshold values of 2 and 4.
0.15
0.20
Freq
0.15
0.10
0.10
0.05
0.05
0.00
0.00
Freq
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
0.14
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
1
Freq
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
2
3
4
0.12
0.12
0.10
0.10
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0.00
0.00
5
1
2
3
4
5
0.12
0.12
0.12
0.10
0.10
0.10
0.08
0.08
0.08
0.06
0.06
0.06
0.04
0.04
0.04
0.02
0.02
0.02
0.00
0.00
1
2
3
Order Size
4
5
0.00
1
2
3
Order Size
48
4
5
Order Size
Figure 3: Recovery of Conditional Probabilities of Binning From the Aggregate Observed Distribution
of Order Sizes
In this figure, I present the percentage deviation from observed values for two competing hypotheses on the ordering process.
The first mode (Simple), in blue, is a simple model where the probability of ordering a unit from the focal firm is fixed at
the overall market share. The second model (Flexible), in red, is where the conditional probabilities of ordering additional
quantities are semi-parametrically estimated from the observed data. I, also, present a table that summarizes these estimated
conditional probabilities as well as the continuation probabilities computed from the empirical distribution of order sizes.
Comparison of Estimated vs. Actual Frequencies of Order Sizes
% Difference Between Estimated and Acutal Frequencies
0.5
0.0
Model
Flexible
Simple
−0.5
−1.0
0
1
2
3
4
Order Size
Continuation Probabilities
PO>0|0
14.5%
PO>1|1
35.2%
PO>2|2
25.8%
PO>3|3
36.4%
PO>4|4
31.7%
PO>5|5
31.4%
Recovered conditional probabilities
P1
8.5%
P2|1
55.35%
P3|2
39.38%
49
P4|3
87.69%
P5|4
60.78%
Figure 4: Constructing the Estimated Arrival Rate
This series of graphs show how the data in Table 9 is transformed into the distribution of the number of patients arriving into
a doctor’s office during the day using Study A’s data. I start with the distribution of average number of patients arriving into
the office, λ, shown in the first graph. The second graph presents the estimated Truncated Poisson distribution associated
with each estimated λ for each doctor. Each small bar in the graph represents an individual doctor, and I have ordered them,
from left to right, based on the smallest to largest λ. Finally, the last graph shows the estimated distribution based on the
method proposed in equation 10.
0.8
Density
0.6
0.4
0.2
0.0
1
2
3
4
5
6
Density
Average Number of Patients Arriving in a Day per Doctor
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
0
1
2
3
4
5
6
Number of Patients per Order
0.25
Density
0.20
0.15
0.10
0.05
0.00
0
1
2
3
4
Number of Patients Arriving in a Day
50
5
6
Table 11: Descriptive Statistics of Focal Manufacturer Data
This table presents the descriptive statistics of the focal manufacturer data. I observe the daily transactions of 1,254 doctors
for 252 business days. I view the daily orders, order sizes, and products prescribed by each doctor. In addition, I have details
on the total sales force activity for each doctor, in frequency of visits and the exact date a salesperson visited the doctor.
Finally, I know the preferred client status for each doctor to the focal firm, which is presented as a fraction of all 1,254 doctors
in the table below.
Orders per Doctor
Average Order Size per Doctor
Total Amount Ordered per Doctor
Total Sales Force Visits per Doctor
Tier A Doctors
Tier B Doctors
Mean
34.15
1.40
51.46
3.48
0.20
0.43
St.Dev.
23.48
0.27
46.80
3.85
Median
26.00
1.33
35.00
2.00
Table 12: Reduced Form Regression to Use as Moments in SMM Estimation
This table presents the results from reduced form analysis of the log of quarterly sales. This regression model contains dummy
variables for the preferred client tiers and another variable for the total number of sales force visits in the quarter. The table
summarizes the estimated parameter, the standard error (SE), and statistical significance (Sig.). If Sig. is ∗ ∗ ∗, then it is
significant to the 99% level.
Variable
(Intercept)
Tier A Dummy
Tier B Dummy
Number of Sales Force Visits (Quarterly)
R Squared
Estimate
1.816
1.201
0.49
0.056
51
SE
0.016
0.027
0.021
0.007
.314
Sig.
***
***
***
***
Table 13: Simulated Method of Moments Estimation
Estimates of the structural parameters of the model based on a sample of 1,254 doctors observed for 252 business days from
focal manufacturer data merged with quarterly patient shares of brand and product shares for four quarters (June 2009 May 2010). Estimation is done by SMM, which minimizes the distance between data moments and simulated moments. In
line with the literature, I present both the actual and the moments recovered from the structural model. . The second table
contains the estimates and standard errors of the estimated structural parameters: aA the brand intercept for brand A, aB
the brand intercept for brand B, aC the brand intercept for brand C, β the measure of price sensitivity, the utility gained
for brand 1’s products if the doctor is in tier A or B, and the effect of salesperson visits from brand A.
Measures of Model Fit
Market Shares
(in Patient
Visits)
log
Sales
Regression
Histogram of
Daily Order
Size (in
Patients)
Estimated Parameters
Group
Intercepts
Cost
Focal
Firm
Effects
Measure
Brand A
Brand B
Brand C
Tier A
Tier B
SF Visits
0 Patients
1 Patient
2 Patients
3 Patients
4 Patients
Actual
0.1339
0.2979
0.3562
1.2
0.49
0.056
0.854
0.094
0.038
0.0084
0.0034
Parameters
aA
aB
aC
β
Client Tier A Dummy
Client Tier B Dummy
Sales Force 1-Day Dummy
Estimates
-0.012
0.903
-2.556
-0.428
2.494
0.937
10.739
52
Structural
0.1413
0.297
0.3554
1.2
0.49
0.053
0.858
0.098
0.0325
0.0063
0.0033
SEs
0.008
0.045
0.154
0.101
0.015
0.068
0.222
Sig.
***
***
***
***
***
***
Figure 5: Examples of Different Price Schedules
Below are four, general examples of different price schedules I use in counterfactual analysis. The first is a single-tier price
schedule, where there is no incentive for purchasing additional units. The second is a linear discount schedule. For each
additional unit purchased, a smaller per unit discount is achieved. The third is a Hi - Lo price schedule, where the smaller
quantity is given a larger discount at the threshold and the larger quantity is given a smaller discount. The final is a Lo - Hi
price schedule, which is the opposite of the previous price schedule. C1 is the initial price point and d is the discount.
Price
Price
Single - Tier
C1
NL – Linear
C1
d
Quantity
Price
Quantity
Price
NL – Hi Lo
C1
NL – Lo Hi
C1
d
d
Quantity
Quantity
53
Table 14: Comparison of Single-Tier and Nonlinear Price Schedules to the Current Plan
Based on the constraints provided by the focal firm, I analyzed a variety of price schedules and compared them to their
current plan. Single-Tier is a plan with just one price per product; the discount is the percentage of the dollar value discount
allowable by the focal firm. NL is a plan with nonlinearities present in the price schedule. NL-Linear is a linear discounting
based on order size. The of the NL plans have a number, which corresponds to the quantity size where the break point
occurs, and a description of the discount given, either high, even or low. The column, Profit, showcases the change in profit
from using the new plan versus the current plan. No. of Orders corresponds to the percentage change in how many orders
are placed based on the new plan versus the current plan. The final five columns show the percent change of the order sizes,
conditional on the total number of orders placed, between the new plan and the current plan.
Price Schedule
Profit
Single Tier
-37.26%
Single Tier - 20% Discount -26.73%
Single Tier - 40% Discount -13.46%
Single Tier - 60% Discount 3.45%
NL - Linear
-19.22%
NL - 2 Lo and 4 Hi
-21.21%
NL - 2 Even and 4 Even
-0.31%
NL - 2 Hi and 4 Lo
6.96%
NL - 3 Hi and 5 Lo
-13.76%
No. of Orders
-30.8%
-15.17%
4.94%
29.83%
-15.53%
-17.11%
-0.63%
5.56%
-12.8%
Q=1
-23.25%
-4.6%
19.82%
50.7%
-9.9%
-11.66%
-0.89%
2.21%
-7.25%
Q=2
-63.2%
-50.68%
-30.6%
-2.32%
-41.5%
-41.03%
-0.33%
18.08%
-51.89%
Order Sizes
Q=3
-46.92%
-34.45%
-17.1%
17.76%
-9.3%
-30.21%
-0.27%
17.9%
59.79%
Q=4
-67.16%
-60.21%
-54.81%
-38.53%
-47.23%
-30.51%
0.14%
8.99%
-35.88%
Q=5
-50.67%
-42.42%
-26.66%
-12.41%
-14.29%
-22.13%
-0.76%
10.35%
20.43%
Table 15: Counterfactual Analysis 1 - Market Maturity and Quantity Discounts
I examine two different counterfactual scenarios, one a new market and the other a mature market. The new market is
assumed to have all new patients. The mature market is assumed to have 60% refill, 30% switcher, and 10% new. In the
table below, I present the ratio of the optimal contract value’s profitability versus the current plan’s profitability, which is
Profit in the table below. Initial corresponds to the initial price discount, in dollars, from the current list price. Discount
corresponds to the variable D, or the incremental discount, in dollars.
Price Schedule
Single Tier
NL - Linear
NL - 2 Hi 4 Lo
NL - 2 Lo 4 Hi
NL - 3 Hi 5 Lo
NL - 2 Even 4 Even
Profit
2.233
2.275
2.313
2.188
2.215
2.222
New Market
Initial
12
10
10
10
10
12
Discount
0
1
0.5
0.5
0.5
0.5
54
Profit
1.185
1.466
1.559
1.35
1.316
1.313
Mature Market
Initial
10
4
0
6
6
6
Discount
0
3.5
3.5
3
1.5
3.5
Figure 6: Heat Maps of Profitability from Counterfactual Analysis 1 - Part 1
I show the results, in terms of heat maps, of the counterfactual results. The value of interest is the ratio of profitability
between the simulated price schedule and the current price schedule. The first set is a key, which gives an indication of which
heat map corresponds to which price schedule. The second set is the heat maps of the 0 refill scenario, where blue levels
are higher relative levels of profitability and red levels are lower levels of relative profitability. The final set of heat maps
correspond to the economic environment with 60% refills, 30% switchers, and 10% new patients.
Relative Profitability
5
Model
10
Model
2.0
3
1.5
2
1.0
Nonlinear Discount
1
0.5
Model
Model
Model
0.0
3
−0.5
2
−1.0
1
−1.5
−2.0
5
10
55
Initial Price Discount
5
10
Figure 7: Heat Maps of Profitability from Counterfactual Analysis 1 - Part 2
I show the results, in terms of heat maps, of the counterfactual results. The value of interest is the ratio of profitability
between the simulated price schedule and the current price schedule. The first set is a key, which gives an indication of which
heat map corresponds to which price schedule. The second set of heat maps correspond to the economic environment with
60% refills, 30% switchers, and 10% new patients., where blue levels are higher relative levels of profitability and red levels
are lower levels of relative profitability.
Relative Profitability
5
Model
10
Model
1.5
3
1.0
2
0.5
Nonlinear Discount
1
0.0
Model
Model
Model
−0.5
3
−1.0
2
−1.5
1
−2.0
5
10
56
Initial Price Discount
5
10