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 man-

ufacturers compete by o�ering 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 o�ered. 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 dis-

tortion. 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

di�erent price schedules by adjusting the threshold levels and per-unit discounts. I �nd one system

that increases �rm pro�tability by 7% without changing the initial and �nal tier per-unit price of the

�rm's products. The best single-price plan only increases pro�tability by 3.5% in comparison. Also,

I examine the pro�tability implications of volume discounts in two di�erent counterfactual scenarios;

one in which 60% of the patients require a re�ll of their previous prescription and the other with 0%

re�ll patients. I �nd volume discounts provide the greatest pro�tability gains over a single-tier pricing

system in the setting with 60% re�ll patients. In the other scenario, some of the optimal nonlinear

price schedules actually result in less pro�tability 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 pro�tability.

The choice of pricing has a direct, sometimes immediate, impact on the behavior of buyers in the market.

However, understanding the potential bene�ts 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 o�ered 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 ful�ll an order must take into consideration not only the nonlinear price incentives o�ered 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 in�uence of

preferred client status, sales force e�ectiveness, and a measure of price elasticity for the retailer. The

speci�c empirical application is observing prescription choices made by a doctor in an international

market, where the doctor chooses between manufacturers that each o�er 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 in�uence pro�tability 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 speci�c

international division. Preliminary evidence from the �rst year after implementation shows a 9% increase

in revenue under the new price schedule, as opposed to the year-over-year 15% decline in previous periods.1

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 speci�c 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 o�ce, noting if certain patients are coming in for a re�ll 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 o�ers, 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" o�er or a more

complex loyalty program that o�ers 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

1Although 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 bene�t based on their choices. The presence of these

nonlinear gains make agent dynamics an important part of modeling behavior.

In B2B transactions, manufacturers o�er �rms a menu of prices for their products. Conditioning

on speci�c volume targets, such as total revenue in a quarter or overall order size, the cost per unit

is lower for the purchasing �rm. In Table 1, I present two examples from the empirical application.2

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 in�uence 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 coordi-

nation 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 pro�tability and

better coordinate the ordering process between manufacturers and retailers. Nonlinear pricing is shown

to induce self-selection of customers into di�erent quantity-allocation levels (Gerstner and Hess 1987,

Spence 1980). The addition of multiple products o�ered by a monopolist using nonlinear pricing has

been addressed as well in Adams and Yellen (1976), McAfee, McMillan, and Whinston (1989), Arm-

strong (1996), Rochet and Choné (1998). However, my application is based on a set of �rms o�ering

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 pro�tability if a set of conditions hold: (i) demand is

elastic, (ii) heterogeneity exists in the preferences for the products of di�erent manufacturers, (iii) the

mix of products contain correlated preferences, and (iv) shopping costs are su�ciently high. In the em-

pirical application, all these conditions hold and I �nd support for the theoretical claims of Armstrong

and Vickers, by showing how di�erent price schedules increase pro�tability.

The aforementioned works indicate that nonlinear pricing can in�uence consumer demand for a prod-

uct; however, the marketing literature also discusses this pricing strategy as a method of channel coor-

dination. 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)

2Due to con�dentiality 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 di�erent 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 o�ers.

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, Netzer, and Ansari (2012), 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 e�ort accordingly. Finally, Kopalle et al. (2012) examine how both frequency

rewards and customer tier bene�ts 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 e�ect.

Descriptive analysis of the focal �rm's data show that doctors shift their prescription choices based on

the nonlinear price incentives o�ered by the focal manufacturer. I �nd that doctors distort their ordering

behavior at the quantity thresholds. This information not only aides in identi�cation 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 pro�tability.

The estimation of a model with forward-looking agents that make quantity-allocation decisions across

multiple brands and multiple product o�erings 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 di�cult 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 di�erent brand/product combinations, which makes calculating all

the choice-speci�c 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 in�uenced 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 unidenti�ed separately from the intercept term.

I use the estimated structural primitives to explore the relative performance of di�erent 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 pro�tability by 7%, compared to 3% with the single-price option. Second, not all price schedules

provide adequate incentives to obtain greater pro�tability; 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 su�cient 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� o�ering a smaller discount on a larger order size,

and the multiplicative e�ect 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 in�ux 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 re�ll 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 re�ll 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 pro�tability.

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 o�ered 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 di�erent 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 identi�cation 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 com-

petitive nonlinear pricing on the manufacturer and quantity choices made by the retailer. Speci�cally, I

examine the daily prescription habits of a doctor prescribing a commonly used medical device in an inter-

national market. In this section, I detail the cost minimization problem faced by the doctor, the factors

that in�uence the prescription decision, and industry speci�c 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 o�ce and their individual characteristics. Speci�cally, if the patient has been seen by the

o�ce 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 re�ll

of their prescription or intends to switch to a di�erent manufacturer due to a mismatch in the product

itself. The class of products under observation are horizontally di�erentiated, 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 di�erent manufacturer's

product. The patient incurs no cost in switching between manufacturers and the side e�ects are common

across all manufacturers; namely, mild discomfort and irritation. Each patient has a set of symptoms

that requires a speci�c 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 pro�tability.

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 o�er 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 o�ered price schedules are �xed for long periods of

time, sometimes remaining the same for years. Although manufacturers may o�er infrequent, temporary

price promotions on a select number of products to select doctors, the prices remain otherwise �xed. The

8

only di�erence in the list price o�ered 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 o�ers a

di�erent �at discount on prices charged by the manufacturer.

The prescribed product must be customized to the speci�cations 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 o�ered by each manufacturer, I limit the

estimation to four separate product categories, where each product is o�ered 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 t during day τ . The total

number of patients a doctor sees in a day is T . For each patient, the doctor decides qjkt, which repre-

sents the product j from manufacturer k that is prescribed to patient t. The doctor's contemporaneous

prescription choice is impacted by three forces: the prescription choices already made in the day, the ex-

pected demand of products for patients that are arriving later in day τ , and the nonlinear price schedules

o�ered 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 o�er incentives for the doctor to accumulate

the prescriptions and order at the end of each business day. The mechanism that doctors internalize the

prices o�ered by the manufacturers is through the use of a price schedule. The price schedule o�ered by

a manufacturer takes the form of Ck(Qk), the per unit prices that the doctor realizes is a function of the

total quantity ordered at the end of day τ , which is expressed as Qk and Qk =∑Jj=1 qjk or the sum of

all products j prescribed by the doctor to manufacturer k, qjk. The quantity that determines the per

unit price is based on the total order volume, Qk, and not the brand/product speci�c volume, qjk. This

function is a step function, where the realized per-unit price for manufacturer k's product j is c′

jk, and

conditional on both the total amount ordered during the day and the quantity thresholds established by

9

the manufacturer, Q′

k. The prime superscript denotes the current price/quantity level the doctor has

achieved through the prescriptions written. More formally, the step function is3:

Ck(Qk) =

c1jk

c2jk

c3jk

if Qk < Q1k

if Q1k ≤ Qk < Q2

k

if Q2k ≤ Qk

(1)

A manufacturer sets the levels of the quantity thresholds and the per-unit prices such that c3jk ≤

c2jk ≤ c1jk. Each of the superscripts, 1 through 3, corresponds to a di�erent 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 speci�c quantity-threshold levels and prices a

manufacturer o�ers for its products is not in the scope of this model, because these change infrequently.

However, I analyze di�erent 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 j o�ered by manufacturer k that doctor i has ordered up to patient

t, Qkt =∑Jj=1 qjkt. I collect the price schedules for all manufacturers, Ck 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 t, Qkt, the

number of patients that have not yet arrived in the day, Nt = T − t, and the information about the

current and future patient preferences, Pt..PT , are all state variables, which I collect into a vector

st = {Qkτt, Nt, Pt, Pt+1, .., PT }.

3.1 Actions and Per-Period Utility

When a doctor meets with patient t, the doctor observes his current state, st, and chooses to order qjkt

from a single manufacturer k. The doctor then updates the total quantity purchased from manufacturer

k, Qkt = Qkt + qjkt. 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-

3If 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 o�er other services to an agent to gain some of its business,

Xkt. Further, the agent considers not only its own cost sensitivity, β, in his choices, but also his belief on

the e�cacy of the product αjk. εjkt 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 speci�c 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 speci�c utility for patient t < T is

ujkt = αjk + δXkt + εjkt (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 . The total quantity allocated to each manufacturer

resets at the end of each day after the �nal patient T is prescribed. Therefore, the state transitions of

Qkt is:

Qkt+1 =

Qkt + qjkt

0

if t < T

if t = T

(3)

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, Pt contains his product needs jt, if he is a new/re�ll/ or

manufacturer switcher rt, and, if applicable, the manufacturer he was prescribed previously kt. If the

patient is a new patient, then the doctor has the ability to prescribe any manufacturer k's product. If the

patient is a re�ll patient, then the patient receives only 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, Nt, the number of prescriptions the doctor has to write during the day, until

the terminal patient T arrives, updates deterministically.

Nτ =

Nt

Nt − 1

0

if t = 0

if t < T

if t=T

(4)

The number of patients arriving into the o�ce 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 o�ce 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 re�ll 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

re�ll. The expected patient preferences evolve deterministically. σjk is the total amount of manufacturer

k's product j that has been ordered in the quarter. As a result, the prescriptions written today in�uence

the doctor's choices in the future:

ξ'jk =

σjk+1(∑

k σjk)+qjkt

σjk

(∑

k σjk)+qikt

σjk

(∑

k σjk)

if k = K and j = J

if k 6= K and j = J

if j 6= J

(5)

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 speci�c 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, θ = {αk, β, δ}, and the state

space observed by the doctor st, the value function of optimal choices is formulated. I start with the

12

value function at the terminal period, T :

VT (sT , φ, θ) =maxqjkT

αjk + δXikT + εjkT − β

∑j∈J

∑k∈K

(QjkT−1 + qjkT )Cjk (QkT−1 + qjkT )

(6)

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 pre-

sented in equation 2,∑j∈J

∑k∈K (Qjkt + qjkT )Cjk (Qkt + qjkT ) represents the total costs of all the

prescriptions written up to time T . 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 man-

ufacturer o�ers, that a threshold value Q1k exists where by the per-unit cost shifts from c1jk to c

2jk. If the

allocation of qjkT causes QkT > Q1k, that choice carries the speci�c bene�t of

∑j∈J

(c1jk − c2jk

)QjkT−1

in cost savings to the doctor on all the other choices already made through time T − 1. 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 speci�c price schedules.

For any time t < T , the doctor takes expectations of both the patient preferences for the di�erent

products o�ered by the manufacturers and the error term in the next period. The resulting value function

is:

Vt(st, φ, θ) =maxqjkt

ujkt +∑st+1

ˆε

Vt+1(st+1,|st, φ, θ)f(ε)f(st+1|st)dεdst+1

(7)

s.t. qjkt ∈ Pt

Given the demand shocks realized when the doctor meets with a speci�c patient t, the value in any

future period is stochastic. 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 di�erent 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 di�erent at each time period t. As the doctor orders from

a manufacturer at each time point t, the total costs for all the choices are updated based on the total

quantity allocated at time t 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 In�uence 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 pref-

erences 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 and B. Manufacturer

A is the high cost product that has a deep discount at the quantity threshold. Manufacturer B is the

low cost product, with a smaller discount at the threshold value. 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 di�erent 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 manufac-

turer. The second scenario assumes that the last patient of the day requires a re�ll from manufacturer A,

(P3 = A). The third scenario assumes that patients two and three both require re�lls from manufacturer

A, (P2 = A,P3 = A). The fourth scenario assumes that patient 3 requires manufacturer A and patient

2 requires manufacturer B, (P2 = B,P3 = A). Finally, the �fth scenario assumes that patient three

requires manufacturer B, (P3 = B).

For simplicity, 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 β = .5

14

QA = 0 QA = 1 QA = 2 QA = 3

Scenario 1 (No Re�ll) 0.9973 0.0010 0.0017 0

Scenario 2 (P3=A) 0 0.2283 0.7529 0.0187

Scenario 3 (P2=A,P3=A) 0 0 0.9526 0.0474

Scenario 4 (P2 = B,P3 = A) 0 0.3775 0.6225 0

Scenario 5 (P3 = B) 0.9988 0.0007 0.0006 0

When there are no re�ll patients for manufacturer A the probability of prescribing any amount of A's

product is virtually 0. However, once a patient requires manufacturer A's product, then the probability

of ordering two units from manufacturer A 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. Due to the discontinuity in the per unit price at two units in the price schedule for

manufacturer A 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 re�ll 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 β = .3 and β = .7.

Simulation of β = .3

QA = 0 QA = 1 QA = 2 QA = 3

Scenario 1 (No Re�ll) 0.9435 0.0233 0.0314 0.0017

Scenario 2 (P3=A) 0 0.2549 0.6882 0.0569

Scenario 3 (P2=A,P3=A) 0 0 0.8581 0.1419

Scenario 4 (P2 = B,P3 = A) 0 0.4256 0.5744 0

Scenario 5 (P3 = B) 0.9732 0.016 0.0108 0

Simulation of β = .7

QA = 0 QA = 1 QA = 2 QA = 3

Scenario 1 (No Re�ll) 0.9999 0 0.0001 0

Scenario 2 (P3=A) 0 0.1977 0.7963 0.006

Scenario 3 (P2=A,P3=A) 0 0 0.9852 0.0148

Scenario 4 (P2 = B,P3 = A) 0 0.3318 0.6682 0

Scenario 5 (P3=B) 1.0000 0 0 0

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 re�ll 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 su�cient 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 o�ce 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 insu�cient 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

o�ered 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 e�ective 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 o�ered 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 speci�c 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 o�ered 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

summarizes the results based on doctors accumulating prescriptions across two days.4

If doctors are accumulating orders across business days, two patterns in transition probabilities should

be present. First, the probability of Qt > 0 should be highest when Qt−1 = 0. On the other hand, the

probability of Qt = 0 should be higher when Qt−1 > 0. I do not observe either e�ect. The pattern shown

in the empirically estimated transition probabilities is the opposite. The probability of Qt = 0 is highest

when Qt−1 = 0. Further, the probability that Qt > 0 increases as Qt−1 increases. Therefore, I �nd no

indication from the transition probabilities that doctors are accumulating orders across days.

4.2.2 Are Doctors In�uenced by Price?

Knowing that doctors order on a daily basis is only part of the story. I next explore whether they are

in�uenced by the costs they incur from their prescription choice. �Met with \#\#\#. She has �t a

4Checking 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 o�ce. This summary indicates that the doctor is

concerned with the current cost position of the focal manufacturer's product and, if the price o�ered 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. I create the marginal probability of purchasing the

discounted product, P (A), 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 o�ered 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 di�erence-in-di�erence (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 signi�cant 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. Doctors did alter their

19

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 e�cient visits or there may be a promotional e�ect that

caused the increase in the allocation of prescriptions to the focal �rm.5

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 e�ect or a

promotional e�ect. If sales-force agents use the change in price to promote the product more e�ciently,

then detailing e�orts 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 e�ect, in Model 3, I run a regression of the order size by a doctor i at time

τ as the dependent variable. Covariates that in�uence 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 e�ect, in Model 4, I test for the presence of a halo e�ect. If the promotion

causes the brand to be top of mind, then there should be a relationship between an order placed at τ − 1

and τ during the promotional window. 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 signi�cant. No statistical relationship

exists between the sales-force e�ect and a promotional e�ect during this time. The choice to increase the

amount ordered and the e�ect of bundling is then strongly associated with the change in price by the

focal manufacturer. 6

5Additional analysis suggests the number of orders did not change. Instead, only the amount per-order changed as a

result of this pricing study6I 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 e�ort 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. However, this evidence is anecdotal

at best. 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 o�ce 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 o�ce, π2, times the joint probability of both patients receiving the brand's

product, Pk(t1 = k, t2 = k). tn is the patient number, and k is the brand choice.

21

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 o�ce, π1 or π2, respectively, gives an estimate of the total probability of seeing an

order of size one of the brand,π1 [Pk(τ1 = k)] + π2 [Pk(τ1 6= k, τ2 = k) + Pk(τ1 = k, τ2 6= k)] = Pk(Q = 1),

where Q is the total quantity ordered.

By minimizing the distance between the constructed order-size frequency, Pk(Q = n), and the actual

order-size frequency, Pk(Q = n), 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

o�ce 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 di�erence 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 in�uenced by price, but they also make their

quantity-allocation decisions based on nonlinear price incentives.9 Because the histogram of order sizes

captures the desired behavior of doctors aligning the volume of prescriptions written on a daily basis with

the nonlinear incentives o�ered by the focal manufacturer, I use the distribution of order sizes as a set of

moments to identify cost sensitivity.

7With 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.8The 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.9I 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 con-

structing a utility framework to model sequential brand choice. Doctors are not only in�uenced 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 bene�t 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 Identi�cation

5.1 Econometric Assumptions

The intended use of this model is to quantify the e�ect 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

re�ll types. Further, I assume doctors are heterogeneous in the manufacturer choice in the past, which

in�uences 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-speci�c characteristics, as a result I cannot identify any

patient-speci�c e�ects outside the patient's simulated state space. This assumption allows for a clean

IID error term, because each patient visit within a day τ is unique and unrelated to any other patient

visiting on day τ . 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 τ in�uence the patient

preferences in the future. If a doctor prescribes brand k to patients, the returning patients at some future

time point will request brand k due to the nature of re�lls 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 speci�c 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 o�er 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 speci�c 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

speci�city of the products and space constraints. Therefore, the daily ordering process is limited to the

number of patients that arrive to the o�ce 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 o�ce 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 λd to estimate the distribution of patients arrivals.10

10Another speci�cation allows each doctor to have a di�erent arrival rate, which results in the simulation of each individual

24

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-speci�c 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:

P (N = n) =1

D

D∑d

DaysdDays

× TP (n|λd) (8)

Where D represents the total number of doctors in the study, Daysd is the total number of days the

doctor entered data for the journal study, and TP (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, re�ll 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 o�ce.

Similarly, I assume the distribution of new, re�ll, 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 di�erence 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 re�ll and switch-

ing patients, there is an initial conditions problem that must be overcome. I compute the average number

of patients arriving in the o�ce in a given quarter by multiplying the average number of patients arriving

to the o�ce 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 re�ll cycle is six months. Tak-

ing this amount and dividing by the total a doctor should have seen for that product type produces as

doctor having a di�erent λ. I use this speci�cation, 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 re�ll states. If the patient is a new

patient, the potential to allocate one of 16 di�erent brand/product combinations exists. A re�ll 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 S, 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 θ, I compute the �tted, value function, in the terminal state for

the NT , given the choices the doctor made during the day. The terminal value function is represented by

W 0NT

(ST ), which is calculated for each simulated set of states from the set of all possible states, ST ∈S.

Next, I assume a �exible linear relationship exists between the states S and the terminal value function,

W 0NT

(SNT), which is g(SNT

).

W 0NT

(SNT) = g(SNT

)ν+ιNT(9)

Where ν is a vector of linear coe�cients and ιNTis 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 S by W (S) = S′ν0. 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′|S,kW

0NT−1(S

′)]

(10)

Where k is the manufacturer choice and S′ is the new state conditional on the previous state and

manufacturer choice for patient NT −1. Using this new approximation, I estimate a new linear regression

presented in equation 11 and obtain 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 di�erent potential for nonlinear savings, I need to compute this series of value function for NT =

{2, 3, 4, 5, 6}. I take uniform draws from the product requirements, re�ll state, and previous products

o�ered for the patient states used in value function formulation. I also take uniform draws of the doctor-

speci�c states: buying group status, preferred client tier, and sales force visit.

5.4 Simulated Method of Moments Estimation

A full likelihood speci�cation 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 in�uence

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 MN , which are the sample counterparts to population moments.

msn 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:

b = argminb

[Mn −

1

S

S∑s=1

m

]′

WN

[Mn −

1

S

S∑s=1

m

](11)

WN is a positive de�nite matrix of weights for each of the selected moments. To approximate the

optimal weighing matrix, I employ the in�uence function method from Erickson and Whited (2000). In

27

brief, W is the inverse of the variance matrix of the actual moments. To achieve an estimate of variance

matrix, I estimate the di�erence between the data moment value MN and the moment value for each unit

of observation. The inner product of the in�uence function vectors approximates the variance matrix,

which is then inverted to obtain, the optimal weighing matrix WN . I use a modi�ed 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 re�ll 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 untilNT 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 Identi�cation

This section summarizes the method used to identify the structural parameters by using a novel, informa-

tive 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 identi�ed

using the available data sources. I explain my moment selection for the intercept terms �rst, then the

focal �rm speci�c 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-re�ll patients. I choose to use the patient share of the non-re�ll patients as the the doctor has

in�uence 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 speci�c e�ect of marketing mix variables, my moment selection are a set of

regression coe�cients. Speci�cally, 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 e�ect 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 identi�es the e�ect 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 in�uences 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 brie�y discuss the estimated coe�cients 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 speci�cation, and comment on the di�erence. Second, I provide

a measure of the distortion e�ect 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 pro�tability. 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 signi�cant. 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 signi�cant. 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 in�uence 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 di�erence can

be attributed to the sample of doctors in the empirical analysis. The regression coe�cient 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

di�erence 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 di�erent 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 o�ce during the day.

Next, I examine the e�ect 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 Q represents the current quantity of the order. For ease of

presentation, I present the following ratio, 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 e�ect is an

important factor in developing a new nonlinear price schedule, presented in the next section.

6.1 Pro�tability Improvement under Focal Firm Constraints

At its core, the framework and methodology presented is useful in aiding marketing managers in manu-

facturing �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 Qi allocated to manufacturer k. χj is the variable cost of production to make a unit of product j.

qij is the total amount of product j doctor i orders from the manufacturer. Finally, FC 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 pro�tability for manufacturer k on

an order from doctor i on day τ is

πiτ =∑j∈J

(Cij(Qiτ )− χj

)qijτ − FC (12)

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 ful�lling 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 di�erent 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 su�ciently interesting to study the change in pro�tability, order frequency, and

order sizes. These three items characterize how the characteristics of the price schedule in�uence both

channel coordination and pro�tability.

32

To generate the price schedules, I take the following steps. I use the initial, c1j , and �nal tier price, c3j ,

of each product as benchmarks. I then subtract c1j − c3j = ωj , where ωj is the total dollar value discount

allowable in this exercise. I discretize this value into �fths, so dj =ωj

5 , and use these values and the

initial price c1j to construct the di�erent 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 di�erent price. I examine the percentage change of

pro�tability, order frequency, and order sizes from each di�erent 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 pro�tability

over the current plan, Single Tier - 60% Discount and NL - 2 Hi and 4 Lo. The nonlinear price schedule

provides the largest increase in pro�tability, 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 pre-

scriptions 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 incre-

mental 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 pro�tability 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 sacri�ces 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 pro�tability 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 re�ll 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 re�ll 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 speci�cs 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, c1. Then

a nonlinear discount is applied if a certain quantity threshold is met, D. I �x D in each evaluation, but

have di�erent quantities of D 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. Then, at Q2, when the order quantity

equals two prescriptions, the per-unit price would be c1−D. At Q3, 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 o�ers 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 o�ered; at four units a more modest discount is o�ered. 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. Schedule 5 is

another Hi Lo price schedule. 11 . 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 pro�tability

formulation in equation 14 and present the results of this analysis as a ratio of the simulated contract

pro�tability against the focal manufacturer current plan's pro�tability.

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 re�ll and switcher customers between the two

counterfactual exercises. On one extreme, I assume both the percentage of re�ll 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 re�ll 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% re�ll 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 pro�tability levels in Figures 7 and 8. Comparing the relative pro�t

levels, the current plan would do poorly in a new market scenario. None of the optimal plans do worse than

2.18 times the pro�tability achieved by using the current plan. Given that this economic environment

is completely di�erent 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 pro�t level of 2.313 versus the single tier relative pro�tability of 2.233. Even in a

system where there is no potential for re�ll 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

pro�tability. To show the regions of pro�tability, Figure 7 presents a heat map of the relative pro�tability

levels for each price schedule. Areas of deeper blue indicate higher relative pro�tability and red are areas

of lower relative pro�tability. In all cases, the regions of the highest pro�tability are clustered in the same

general area.

11I have also tested using di�erent threshold levels, but due to the limitation of only seeing six patients a day, the volume

thresholds provided above always provide greater pro�tability.

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 pro�tability. 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 di�erent 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 pro�tability 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 re�ll 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 di�erent view. The presence of a large number of

re�lls 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 o�set 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

su�ciently 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 incen-

tives. 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 bene�ts o�ered 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 identi�cation of the parameter associated

with the cost sensitivity of a doctor. I further show additional moments of interest to estimate the struc-

tural 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 in�uence of nonlinear

pricing on pro�tability. 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 re�ll patient in the doctor's

38

ordering process causes a re-assessment of the quantity gains. The more patients that come in for re�lls 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 pro�table than a

single-tier price system in the presence of a higher percentage of re�ll patients. Conversely, in newer

markets without the �ow of re�ll 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, di�erent

�rms are known to specialize in di�erent products, which causes heterogeneity in the market preferences

for the di�erent product types. Finally, as mentioned previously, this model can be extended to other

nonlinear bene�t 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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42

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 haveselected two schedules to present, the one for the focal firm, Brand A, and a second firm, Brand B. Due to confidentialityreasons, I cannot present the actual dollar values. So, I have divided all the values by the initial price for each brand/productcombination. Brand A has discounts in two levels while Brand B has only one. In addition, Brand A offers larger discountsthan Brand B does.

Firm A 1-3 Boxes 4-7 Boxes 8 or more Boxes(1 Patient) (2-3 Patients) (4+ Patients)

Product 1 1 0.93 0.884Product 2 1 0.979 0.947Product 3 1 0.917 0.85Product 4 1 0.942 0.865

Firm B 1-3 Boxes 4-7 Boxes 8 or more Boxes(1 Patient) (2-3 Patients) (4+ Patients)

Product 1 1 1 0.964Product 2 1 1 0.913Product 3 1 1 0.913Product 4 1 1 1

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 accumulatingorders across days, rather than ordering on a daily basis. The left hand side is the order quantity on day τ − 1 and thecolumns are the order sizes of day τ . The percentages in the middle are the probability of transitioning from one quantityamount to the next, conditional on the order quantity size ordered at τ − 1

Current Order SizeLagged Order Size 0 1 2 3 4 50 0.905 0.064 0.023 0.005 0.002 0.0011 0.835 0.108 0.041 0.01 0.004 0.0012 0.805 0.105 0.062 0.016 0.007 0.0033 0.729 0.148 0.078 0.024 0.014 0.0034 0.709 0.134 0.096 0.026 0.02 0.0095 0.625 0.195 0.119 0.018 0.018 0.012

43

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 CommentComment 1 Use Brand 3 right now not much Brand 2 because of price.Comment 2 Need more of an incentive to move over.Comment 3 She has a few of her patients with our product. Would fit more if the price was right.Comment 4 BLANK said that due to price and design, he is going to make this his number one Product 3!!Comment 5 Not interested in going to Product 2. Mostly Brand 3 now based on price.Comment 6 Believes in Product 4 modality but too high priced.Comment 7 Pricing is quite high.Comment 8 Not doing much in Product 4 due to pricing but is willing to promote if can provide great price.Comment 9 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 alteredpricing 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 controlgroups: before, during and after the price change.

Control Group Test GroupBefore During After Before During After

Average Quantity Per Order 0.421 0.432 0.399 0.725 0.840 0.775Standard Deviation 0.882 0.989 0.933 1.357 1.484 1.353Minimum 0 0 0 0 0 0Median 0.300 0.310 0.250 0.512 0.680 0.585Maximum 3.068 3.833 2.50 2.255 2.667 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). Theprobability of binning is defined as the probability of ordering any other product offered by the firm, conditional on orderingthe discounted product.

Before During AfterP (A) P (B|A) P (A) P (B|A) P (A) P (B|A)

Test Group 9.31% 39.03% 9.94% 39.88% 8.53% 36.56%Control Group 2.56% 21.07% 2.55% 19.97% 2.26% 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 observesthe future patients’ states. As choices are made throughout the day, the terminal period’s value is updated and alters thecontemporaneous choice the doctor has to make.

The Day Begins

The Day Ends

First Patient Arrives

Doctor Observes Patient’s State and Expected Value based on Current Choice

Initial Quantities Created based on

Patient States

Cre

ate

s E

xp

ecte

d

Utilitie

s

Doctor Prescribes a Product for Patient 1

Doctor Updates Order for Brand k Brand k’s Quantity

Increases by 1

Nth Patient Arrives

Doctor Prescribes a Product for Patient N

Doctor Updates Order for Brand k

Brand k’s Quantity Increases by 1

Orders are Placed to All Brands K and Final

Cost is Realized

Observes Information Arriving Patients

Observes Patient States and Creates Expectation of Final Costs based on them

•New or Returning Patient •If Returning – Refill State, Product

Need and Previous Brand •If New – No information is known

Up

da

tes

Ex

pe

cted

U

tilities

Doctor Observes Patient’s State and Final Costs based on Current Choice

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 asubset of 85 doctors, the price was reduced for a single product, all other prices were kept constant. I examine the impactof 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 testgroup purchased more of other products (Model 2). X1 is a dummy variable for the doctor being in the test group. X2 is adummy variable for if the observation is during the test period when the price is discounted. Finally, X1*X2 is the interactionbetween 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 Model 2 - All Other ProductsEstimate SE Sig. Estimate SE Sig.

Intercept .448 .005 ∗ ∗ ∗ 1.083 .007 ∗ ∗ ∗X1 .303 .020 ∗ ∗ ∗ -.014 .030X2 -.009 .007 .08 .010 ∗ ∗ ∗X1*X2 .054 .028 ∗ .081 .042 ∗R Squared .005 .001

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 salesand bundling. Both models are regression models where, Ydt is the quantity ordered by doctor d at time t. Model 3 is testingthe influence of sales force visits (X3), measured as a dummy variable if the doctor had been visited by a sales force agentin the past 5 business days. Two other variables of interest are if the doctor is included in the test group(X1) and if theobserved day t is during the pricing study (X2). I then interact all the variables to determine if sales force activity duringthe promotion for doctors receiving the promotion have a significant effect on increasing the probability of purchase. Model2 uses the same variables Ydt, X1, and X2. I then include variable to measure the effectiveness of a promotional effect. Tocapture this behavior, I use a lagged term of an order (X4). All the interactions between the variables are included as well todetermine if the promotional effect is strongest for doctors receiving the discount during the promotional period. The tablesummarizes the estimated parameter, the standard error (SE), and statistical significance (Sig.). If Sig. is ∗ ∗ ∗, then it issignificant to the 99% level.

Model 3 - Sales Force Effect Model 4 - Promotional EffectEstimate SE Sig. Estimate SE Sig.

Intercept .218 .002 ∗ ∗ ∗ .198 .002 ∗ ∗ ∗X1 .149 .007 ∗ ∗ ∗ .150 .008 ∗ ∗ ∗X2 -.015 .002 ∗ ∗ ∗ -.010 .002 ∗ ∗ ∗X1*X2 -.007 .010 -.008 .011X3 .063 .015 ∗ ∗ ∗X1*X3 -.042 .044X2*X3 -.004 .020X1*X2*X3 -.027 .058X4 .097 .003 ∗ ∗ ∗X1*X4 -.043 .009 ∗ ∗ ∗X2*X4 -.017 .004 ∗ ∗ ∗X1*X2*X4 .007 .012R Squared .003 .010

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 summarizethe total amount of units ordered from the focal firm on a daily basis. The distribution of orders have peaks at the quantitylevels that correspond with nonlinear incentives in the price schedule. The frequency value, for ease of reading, is presentedas the total number of orders of a given size divided by the total number of business days.

ID 0 Patients 1 Patient 2 Patients 3 Patients 4 Patients 5 Patients980 53.42% 17.95% 21.37% 5.56% 1.28% 0.43%1530 72.22% 6.84% 15.81% 2.99% 1.71% 0.43%2397 73.93% 5.98% 15.38% 0.85% 2.99% 0.85%557 70.51% 7.26% 14.96% 2.14% 3.42% 0.85%141 73.50% 8.55% 14.10% 2.14% 1.71% 0.00%1204 75.21% 9.83% 13.68% 0.43% 0.85% 0.00%12 74.36% 8.12% 13.25% 2.14% 1.28% 0.85%1458 73.93% 5.98% 13.25% 1.71% 4.27% 0.43%1428 74.36% 12.82% 8.97% 0.85% 2.56% 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 of1,254 doctors for 252 business days.

Mean S. D. Minimum Median Maximum0 Boxes (0 Patients) 85.45% .1 23.9% 88.67% 94.87%1-2 Boxes (1 Patient) 9.41% .0524 1.28% 8.12% 30.34%3-4 Boxes (2 Patients) 3.8% .0359 0% 2.56% 23.5%5-6 Boxes (3 Patients) .84% .0145 0% .427% 14.96%7-8 Boxes (4 Patients) .34% .008 0% 0% 8.97%9-10 Boxes (5 Patients) .11% .003 0% 0% 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 observethe total number of days recorded, the total number of patients that arrived into the office, and the total number of boxesordered.

Mean S. D. Minimum Median MaximumDays Recorded 8.14 4.08 3 7 21Total Number of Patients 16.1 10.62 5 12 72Average Number of Patients 2.06 1.04 1 1.6 5.25

47

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 optionto better show the distribution of order sizes for Brand A’s products. In each case, note the disproportionate number oforders clustered around the threshold values of 2 and 4.

Freq

0.00

0.05

0.10

0.15

0.20

1 2 3 4 5

0.00

0.05

0.10

0.15

1 2 3 4 5

0.000.020.040.060.080.100.120.14

1 2 3 4 5

Freq

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

1 2 3 4 5

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

1 2 3 4 5

0.00

0.02

0.04

0.06

0.08

0.10

0.12

1 2 3 4 5

Order Size

Freq

0.00

0.02

0.04

0.06

0.08

0.10

0.12

1 2 3 4 5Order Size

0.00

0.02

0.04

0.06

0.08

0.10

0.12

1 2 3 4 5Order Size

0.00

0.02

0.04

0.06

0.08

0.10

0.12

1 2 3 4 5

48

Figure 3: Recovery of Conditional Probabilities of Binning From the Aggregate Observed Distributionof 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 atthe overall market share. The second model (Flexible), in red, is where the conditional probabilities of ordering additionalquantities are semi-parametrically estimated from the observed data. I, also, present a table that summarizes these estimatedconditional probabilities as well as the continuation probabilities computed from the empirical distribution of order sizes.

−1.0

−0.5

0.0

0.5

0 1 2 3 4Order Size

% D

iffer

ence

Bet

wee

n E

stim

ated

and

Acu

tal F

requ

enci

es

Model

Flexible

Simple

Comparison of Estimated vs. Actual Frequencies of Order Sizes

Continuation Probabilities

PO>0|0 PO>1|1 PO>2|2 PO>3|3 PO>4|4 PO>5|514.5% 35.2% 25.8% 36.4% 31.7% 31.4%

Recovered conditional probabilities

P1 P2|1 P3|2 P4|3 P5|48.5% 55.35% 39.38% 87.69% 60.78%

49

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 intoa doctor’s office during the day using Study A’s data. I start with the distribution of average number of patients arriving intothe office, λ, shown in the first graph. The second graph presents the estimated Truncated Poisson distribution associatedwith 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 themethod proposed in equation 10.

Average Number of Patients Arriving in a Day per Doctor

Den

sity

0.0

0.2

0.4

0.6

0.8

1 2 3 4 5 6

Number of Patients per Order

Den

sity

0.000.050.100.150.200.250.300.35

0 1 2 3 4 5 6

Number of Patients Arriving in a Day

Den

sity

0.00

0.05

0.10

0.15

0.20

0.25

0 1 2 3 4 5 6

50

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 doctorsfor 252 business days. I view the daily orders, order sizes, and products prescribed by each doctor. In addition, I have detailson 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 doctorsin the table below.

Mean St.Dev. MedianOrders per Doctor 34.15 23.48 26.00Average Order Size per Doctor 1.40 0.27 1.33Total Amount Ordered per Doctor 51.46 46.80 35.00Total Sales Force Visits per Doctor 3.48 3.85 2.00Tier A Doctors 0.20Tier B Doctors 0.43

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 dummyvariables for the preferred client tiers and another variable for the total number of sales force visits in the quarter. The tablesummarizes the estimated parameter, the standard error (SE), and statistical significance (Sig.). If Sig. is ∗ ∗ ∗, then it issignificant to the 99% level.

Variable Estimate SE Sig.(Intercept) 1.816 0.016 ***Tier A Dummy 1.201 0.027 ***Tier B Dummy 0.49 0.021 ***Number of Sales Force Visits (Quarterly) 0.056 0.007 ***R Squared .314

51

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 fromfocal 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. Inline with the literature, I present both the actual and the moments recovered from the structural model. . The second tablecontains the estimates and standard errors of the estimated structural parameters: aA the brand intercept for brand A, aBthe brand intercept for brand B, aC the brand intercept for brand C, β the measure of price sensitivity, the utility gainedfor 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 FitMeasure Actual Structural

Market Shares Brand A 0.1339 0.1413(in Patient Brand B 0.2979 0.297Visits) Brand C 0.3562 0.3554log Tier A 1.2 1.2Sales Tier B 0.49 0.49Regression SF Visits 0.056 0.053Histogram of 0 Patients 0.854 0.858Daily Order 1 Patient 0.094 0.098Size (in 2 Patients 0.038 0.0325Patients) 3 Patients 0.0084 0.0063

4 Patients 0.0034 0.0033

Estimated ParametersGroup Parameters Estimates SEs Sig.Intercepts aA -0.012 0.008

aB 0.903 0.045 ***aC -2.556 0.154 ***

Cost β -0.428 0.101 ***Focal Client Tier A Dummy 2.494 0.015 ***Firm Client Tier B Dummy 0.937 0.068 ***Effects Sales Force 1-Day Dummy 10.739 0.222 ***

52

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 priceschedule, where there is no incentive for purchasing additional units. The second is a linear discount schedule. For eachadditional unit purchased, a smaller per unit discount is achieved. The third is a Hi - Lo price schedule, where the smallerquantity is given a larger discount at the threshold and the larger quantity is given a smaller discount. The final is a Lo - Hiprice schedule, which is the opposite of the previous price schedule. C1 is the initial price point and d is the discount.

C1

NL – Linear Price

Quantity

d

C1

NL – Lo Hi Price

Quantity

d

C1

Single - Tier Price

Quantity

C1

NL – Hi Lo Price

Quantity

d

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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 theircurrent plan. Single-Tier is a plan with just one price per product; the discount is the percentage of the dollar value discountallowable by the focal firm. NL is a plan with nonlinearities present in the price schedule. NL-Linear is a linear discountingbased on order size. The of the NL plans have a number, which corresponds to the quantity size where the break pointoccurs, and a description of the discount given, either high, even or low. The column, Profit, showcases the change in profitfrom using the new plan versus the current plan. No. of Orders corresponds to the percentage change in how many ordersare 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 Order SizesProfit No. of Orders Q=1 Q=2 Q=3 Q=4 Q=5

Single Tier -37.26% -30.8% -23.25% -63.2% -46.92% -67.16% -50.67%Single Tier - 20% Discount -26.73% -15.17% -4.6% -50.68% -34.45% -60.21% -42.42%Single Tier - 40% Discount -13.46% 4.94% 19.82% -30.6% -17.1% -54.81% -26.66%Single Tier - 60% Discount 3.45% 29.83% 50.7% -2.32% 17.76% -38.53% -12.41%NL - Linear -19.22% -15.53% -9.9% -41.5% -9.3% -47.23% -14.29%NL - 2 Lo and 4 Hi -21.21% -17.11% -11.66% -41.03% -30.21% -30.51% -22.13%NL - 2 Even and 4 Even -0.31% -0.63% -0.89% -0.33% -0.27% 0.14% -0.76%NL - 2 Hi and 4 Lo 6.96% 5.56% 2.21% 18.08% 17.9% 8.99% 10.35%NL - 3 Hi and 5 Lo -13.76% -12.8% -7.25% -51.89% 59.79% -35.88% 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 isassumed to have all new patients. The mature market is assumed to have 60% refill, 30% switcher, and 10% new. In thetable below, I present the ratio of the optimal contract value’s profitability versus the current plan’s profitability, which isProfit in the table below. Initial corresponds to the initial price discount, in dollars, from the current list price. Discountcorresponds to the variable D, or the incremental discount, in dollars.

Price Schedule New Market Mature MarketProfit Initial Discount Profit Initial Discount

Single Tier 2.233 12 0 1.185 10 0NL - Linear 2.275 10 1 1.466 4 3.5NL - 2 Hi 4 Lo 2.313 10 0.5 1.559 0 3.5NL - 2 Lo 4 Hi 2.188 10 0.5 1.35 6 3NL - 3 Hi 5 Lo 2.215 10 0.5 1.316 6 1.5NL - 2 Even 4 Even 2.222 12 0.5 1.313 6 3.5

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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 profitabilitybetween the simulated price schedule and the current price schedule. The first set is a key, which gives an indication of whichheat map corresponds to which price schedule. The second set is the heat maps of the 0 refill scenario, where blue levelsare higher relative levels of profitability and red levels are lower levels of relative profitability. The final set of heat mapscorrespond to the economic environment with 60% refills, 30% switchers, and 10% new patients.

Relative Profitability

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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 profitabilitybetween the simulated price schedule and the current price schedule. The first set is a key, which gives an indication of whichheat map corresponds to which price schedule. The second set of heat maps correspond to the economic environment with60% refills, 30% switchers, and 10% new patients., where blue levels are higher relative levels of profitability and red levelsare lower levels of relative profitability.

Relative Profitability

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