Stackelberg's Game in Moser Baer -Cost Estimation - Mathematics for business decisions

MATHEMATICS FOR
BUSINESS DECISION
Submitted on September 28, 2018
Mayank Agrawal

Page | 1
A REPORT ON:
STACKELBERG’S GAME IN MOSER BAER
Submitted to: Dr. Nivedita Haldar
In partial fulfilment of the requirement of the
Mathematics for Business Decision
Course
By:
Mayank Agrawal
On:
September 28, 2018

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Letter of Transmittal
September 28, 2018
Dr. Nivedita Haldar
Mathematics for Business Decisions
Department of Business Administration
Jindal Global Business School
Subject: Case Analysis Report
Dear Ma’am
As per the guidelines provided by you during the classroom discussion, we hereby submit a
report on the case titled ‘Stackelberg’s Game in Moser Baer’. This report has been prepared
after going through the entire case and carefully scrutinizing the facts and assumptions and
using the six-step problem solving model of Penrose, Rasberry and Myres. The report
contains the relevant facts in the introduction, problem statement and list of possible
alternatives and best action plan to be implemented.
Kindly evaluate the same and provide your valuable inputs.
Best Regards,
Mayank Agrawal
Jindal Global Business School

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TABLE OF CONTENTS:
LIST OF ILLUSTRATIONS

Page | 4
OBJECTIVE:
To find the optimal wholesale price and the retail price for Moser Baer Compact Disks (CDs)
in a simple duopoly consisting one manufacturer and one retailer.
We have chosen to work with demand function of Moser Baer Compact Discs.
Industry: Information Technology
Company: Moser Baer
Product: Compact Discs
MRP: 44
The Data for the price vs demand is as follows:
Month p q
Jun-2017 20 52
Jul-2017 22 50
Aug-2017 24 48
Sep-2017 26 46
Oct-2017 28 44
Nov-2017 30 42
Dec-2017 32 40
Jan-2018 34 38
Feb-2018 36 36
Mar-2018 38 34
Apr-18 40 32
May-18 42 30
Jun-18 44 28
Table 1.1
Where price, p is price in rupees and demand, q is in unit in thousands.

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DEFINITIONS:
What is Supply Chain?
“A supply chain is the network of all the individuals, organizations, resources, activities and
technology involved in the creation and sale of a product, from the delivery of source
materials from the supplier to the manufacturer, through to its eventual delivery to the end
user. The supply chain segment involved with getting the finished product from the
manufacturer to the consumer is known as the distribution channel.
Supply chain management (SCM) is the oversight of materials, information, and finances as
they move in a process from supplier to manufacturer to wholesaler to retailer to consumer.
The three main flows of the supply chain are the product flow, the information flow and the
finances flow. SCM involves coordinating and integrating these flows both within and among
companies.”
Supply Chain (n.d.)
What is the importance of price in oligopoly market?
“We define oligopoly as the form of market organization in which there are few sellers of a
homogeneous or differentiated product. If the product is homogeneous, we have a pure
oligopoly. If the product is differentiated, we have a differentiated oligopoly. Oligopoly is the
most prevalent form of market organization in the manufacturing sector of most nations,
including India. Some oligopolistic industries in India automobiles, primary aluminium, steel,
electrical equipment, glass, breakfast cereals,cigarettes, and many others.
Since an oligopolist knows that its own actions will have a significant impact on the other
oligopolists in the industry, each oligopolist must consider the possible reaction of

Page | 6
competitors in deciding its pricing policies, the degree of product differentiation to introduce,
the level of advertising to be undertaken, the amount of service to provide, etc. Since
competitors can react in many different ways (depending on the nature of the industry, the
type of product, etc.) We do not have a single oligopoly model but many-each based on
the particular behavioural response of competitors to the actions of the first. Because of this
interdependence, managerial decision making is much more complex under oligopoly than
under other forms of market structure.”
Pricing (n.d.)
What is duopoly?
“A duopoly is a form of oligopoly occurring when two companies (or countries) control all
or most of the market for a product or service.
There are two kinds of duopolies. In the first, the Cournot duopoly, competition between the
two companies is based on the quantity of products supplied. The duopoly members
essentially agree to split the market. The price each company receives for the product is based
on the quantity of items produced, and the two companies react to each other's production
changes until an equilibrium is achieved.
In a Bertrand duopoly, the two companies compete on price. Because
consumers will purchase the cheaper of two identical products, this leads to a zero-profit
price as the two competitors attempt to attract more customers (and thus more profit) through
price cuts. The threat of price undercutting means that Bertrand equilibrium prices and profits
are generally lower (and quantities higher) than in Cournot duopolies.”
Duopoly (n.d.)

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MATERIAL FLOW:
FIG 1.1
ANALYSIS:
Let’s take price of Moser Baer Compact Disks (CDs) ranging from Rs 20 to 44.
Let’s take a step of INR 2 i.e. an increase of INR 2 across different stores in the market.
Demand at Market:
Let the linear demand function be
q = a – bp
where,
q = quantity sold
p = retail price

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a = market potential
b = price elasticity
Estimation of Demand:
To establish the demand function we will do the following calculations:
∑ 𝑞 = 𝑛𝑎 − 𝑏 ∑ 𝑝
And
∑ 𝑝𝑞 = 𝑎 ∑ 𝑝 − 𝑏 ∑ 𝑝2
Let us now prepare a table consisting of retail price(p), quantity sold(q), Total
revenue(pq) and p2.
Month p q pq p2
Jun-2017 20 52 1040 400
Jul-2017 22 50 1100 484
Aug-2017 24 48 1152 576
Sep-2017 26 46 1196 676
Oct-2017 28 44 1232 784
Nov-2017 30 42 1260 900
Dec-2017 32 40 1280 1024
Jan-2018 34 38 1292 1156
Feb-2018 36 36 1296 1296
Mar-2018 38 34 1292 1444
Apr-18 40 32 1280 1600
May-18 42 30 1260 1764
Jun-18 44 28 1232 1936
∑ 416 520 15912 14040
Table 1.2

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Therefore,
∑ 𝑞 = 𝑛𝑎 − 𝑏 ∑ 𝑝
520 = 13 a – 416 b --------(1)
and
∑ 𝑝𝑞 = 𝑎 ∑ 𝑝 − 𝑏 ∑ 𝑝2
15,912 = 416 a – 14,040 b ---------(2)
From equation 1 & 2 we can get the value of a & b,
And those will be a=72 and b=1.
Hence the demand function is
q = 72 – p (i)
Verification of Demand:
Let us plot the straight line graph of Price V/s Quantity.
FIG 1.2
We know that the equation of the straight line is
y = mx + c
From the graph, we get
y = -x + 72
Therefore,
m = -1, intercept
and c = 72, constant
y = -x + 72
0
10
20
30
40
50
60
15 20 25 30 35 40 45 50
Quantity
Price
Demand V/s Price Curve

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Hence, our equation derived from the curve and the values from the table match and are one
and the same.
THE PROBLEM:
We will consider a simple duopoly market where there is only one manufacturer and one
retailer.
The problem is to find the optimal wholesale price and retail price. We will be using
Stackelberg Game Theory to decide the optimum wholesale and retail price.
STACKELBERG GAME:
“The Stackelberg leadership model is a strategic game in economics in which the leader firm
moves first and then the follower firms move sequentially. It is named after the German
economist Heinrich Freiherr von Stackelberg who published Market Structure and
Equilibrium (Masrktform und Gleichgewicht) in 1934 which described the model.”
In game theory terms, the players of this game are a leader and a follower and they compete
on quantity and price. The Stackelberg leader is sometimes referred to as the Market Leader.
Firms may engage in Stackelberg competition if one has some sort of advantage enabling it to
move first. More generally, the leader must have commitment power. Moving observably
first is the most obvious means of commitment: once the leader has made its move, it cannot
undo it - it is committed to that action. Moving first may be possible if the leader was the
incumbent monopoly of the industry and the follower is a new entrant. Holding excess
capacity is another means of commitment.
The game is played in a sequential manner. Specifically, the manufacturer is the leader in the
game, which means it initiates the game by making the first move (e.g., order quantity to

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suppliers), while the suppliers are the followers. The manufacturer knows that the suppliers
will decide by reacting to its decisions, so it takes this into account while making its own
decisions. Consequently, the game is modelled as a single leader − multiple follower
Stackelberg game. As in any variant of game theory and its applications, the central
assumption is that players are rational and claimed that this assumption is often a common
knowledge. In the proposed game, the manufacturer, being the leader of this non-cooperative
Stackelberg game, is assumed to be rational in nature and his/her objective in a non-
cooperative Stackelberg game is to discover the best strategy in order maximize its profit
given all the possible move of its followers. Rationality is also expected from the suppliers as
long as they never choose dominated strategies. Thus, as long as the manufacturer and the
suppliers satisfy the rationality principle and are willing to remain in the game, the proposed
game is applicable; that is, they can determine their strategies that maximize their profit
functions. Otherwise, optimal decisions may not be guaranteed when the manufacturer or
suppliers or both leave the game.
Key Points:
STACKELBERG GAME:
• The decision-making is sequential.
• Either the manufacturer or the retailer acts as the market leader and first decides on
the wholesale price (w) or the retail price (p)
• Then the other acts as the follower and then decides on p or w accordingly
• If the manufacturer acts as the leader the game is called Manufacturer-Stackelberg
(MS) game
• Else if the retailer acts as the leader the game is called Retailer-Stackelberg (RS) game
The assumptions we will use to solve the problem:
• No stock-out or inventory at the retailer’s end
• The retailer anticipates the demand q and orders exactly q quantity and sells exactly q
quantity
• No stock-out or inventory at the manufacturer’s end
• The manufacturer produces exactly q quantity and sells exactly q quantities to the
retailer
• The manufacturer possess enough capacity to fulfil the order

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What is Manufacturer Stackelberg Model?
The game is played in a sequential manner. Specifically, the manufacturer is the
leader in the game, which means it initiates the game by making the first move (i.e.
order quantity to suppliers), while the suppliers are the followers. The manufacturer
knows that the suppliers will make their decisions by reacting to its decisions, so it
takes this into account while making its own decisions. Accordingly, the game is
modelled as a single leader-multiple follower Stackelberg game.
What is Retailer Stackelberg Model?
In the context of a retailer Stackelberg supply chain, we examine how a direct channel
added by a manufacturer can influence the decisions of the retailer and the
manufacturer, and the impact it will have on their profits. We obtain equilibrium
prices for a retailer Stackelberg dual-channel supply chain. We find that, as compared
to a single retail channel supply chain, a dual-channel supply chain can enhance the
profits of the manufacturer and the supply chain. The retailer, however, will benefit
from the direct channel when the maximum sales in the retail channel are high. We
further propose a retailer's margin contract that can coordinate the dual-channel
supply chain and ensure that both the retailer and the manufacturer will be more
profitable.
The cost function for the manufacturer can be given as:
Cq = A +vq
Cq = (20 + 3q) (in thousand rupees)…………….. Cost of manufacturer
(ii)
where,
A = Total Fixed cost = 20, and
v = per unit variable cost = 3
Also Let,

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w = wholesale price, and
p = Retail Price
𝑝𝑞 = Revenue of the retailer
𝑤𝑞 = Cost of the retailer
𝑤𝑞 = Revenue of the manufacturer
Manufacturer’s revenue/profit = 𝜫 𝑴 = 𝒘𝒒 − 𝑪 𝒒 (iii)
Using equation (ii) in (iii), we get
𝜫 𝑴 = 𝒘𝒒 − (𝟐𝟎 + 𝟑𝒒) = (𝒘 − 𝟑)𝒒 − 𝟐𝟎 (iv)
Using equation (i) in (iv), we get:
𝜫 𝑴 = (𝒘 − 𝟑)(𝟕𝟐 − 𝒑) − 𝟐0 (v)
Also, Retailer’s Revenue/profit = 𝜫 𝑹 = 𝒑𝒒 – 𝒘𝒒 (vi)
Using equation (i) In equation (vi), we get:
𝜫 𝑹 = (𝒑 − 𝒘)𝒒 = (𝒑 − 𝒘)(𝟕𝟐 − 𝒑)
𝜫 𝑹 = (𝒑 − 𝒘)(𝟕𝟐 − 𝒑)
(vii)
OPTIMIZATION MODELS:
MS GAME (Manufacturer Stackelberg):
Manufacturer is the market leader and retailer follows the manufacturer. The wholesale
price(w) is set by the manufacturer and based on that the retailer will set the retail price(p).
We will try to optimize the wholesale price(w) first and the retail price(p) based on the
optimized wholesale price.
Maximizing the manufacturers profit:
Using equation (v) to maximize, we get
𝐦𝐚𝐱
𝒘
∏ = (𝒘 − 𝟑)(𝟕𝟐 − 𝒑) − 𝟐𝟎
𝑴
Where for a given w, retailer will find optimal retail price at
𝒑 = 𝒂𝒓𝒈𝒎𝒂𝒙 ∏ (𝒑 − 𝒘)(𝟕𝟐 − 𝒑)
𝑹
Hence to find the point of maxima, we find the partial derivative with respect to p, i.e.

Page | 14
𝜕𝛱 𝑅
𝜕𝑝
=
𝜕(𝑝 − 𝑤)(72 − 𝑝)
𝜕𝑝
=
(𝑝−𝑤)𝜕(72−𝑝)
𝜕𝑝
+
(72−𝑝)𝜕(𝑝−𝑤)
𝜕𝑝
= (𝑝 − 𝑤) ∗ (−1) + (72 − 𝑝) ∗ 1
= −2𝑝 + 𝑤 + 72
Thus,
𝝏𝜫 𝑹
𝝏𝒑
= −𝟐𝒑 + 𝒘 + 𝟕𝟐
For maxima,
𝜕𝛱 𝑅
𝜕𝑝
= 0
Hence, −2𝑝 + 𝑤 + 72 = 0
𝒑(𝒘) =
𝐰+𝟕𝟐
𝟐
………… Reaction function of retailer.
(viii)
Therefore,
𝛱 𝑀 = (𝑤 − 3)(72 − 𝑝) − 20 (ix)
𝛱 𝑀 =
(𝑤 − 3)(144 − (𝑤 + 72))
2
− 20
𝛱 𝑀 =
(𝑤 − 3)(72 − 𝑤)
2
− 20
To maximize the manufacturer profit, we differentiate the profit function with w
Therefore,
𝑑𝛱 𝑀
𝑑𝑤
=
𝑑(((𝑤 − 3)(72 − 𝑤))/2) − 20)
𝑑𝑤
𝑑𝛱 𝑀
𝑑𝑤
=
(72 − 𝑤)𝑑((𝑤 − 3)
2 ∗ 𝑑𝑤
+
(𝑤 − 3)𝑑(72 − 𝑤)
2 ∗ 𝑑𝑤
𝑑𝛱 𝑀
𝑑𝑤
=
(72 − 𝑤)
2
+
(𝑤 − 3)(−1)
2
𝑑𝛱 𝑀
𝑑𝑤
=
(72 − 𝑤 − 𝑤 + 3)
2
𝑑𝛱 𝑀
𝑑𝑤
=
(72 − 2𝑤 + 3)
2

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Optimum wholesale price is achieved when,
𝑑𝛱 𝑀
𝑑𝑤
= 0
0 =
(72 − 2𝑤∗
+ 3)
2
0 = (72 − 2𝑤∗
+ 3)
𝑤∗
=
(72 + 3)
2
𝑤∗
= 75/2
𝑤∗
= 37.5
Also optimum Retail price at 𝑤∗
= 37.5 is
𝒑(𝒘) =
𝐰+𝟕𝟐
𝟐
(From equation(viii))
𝑝∗
=
(37.5 + 72)
2
𝒑∗
= 𝟓𝟒. 𝟕𝟓
Optimum Demanded quantity in our case is
𝑞∗
= 72 − 𝑝∗
(From equation(viii))
𝑞∗
= 72 − 54.75
𝒒∗
= 𝟏𝟕. 𝟐𝟓
Thus, the optimum profit of the manufacturer is
Max 𝛱 𝑀 = (𝑤 − 3)(72 − 𝑝) − 20 (from
equation (ix))
𝛱 𝑀
∗
= (37.5 − 3)(72 − 54.75) − 20
𝛱 𝑀
∗
= (34.5 ∗ 17.25) − 20
𝛱 𝑀
∗
= 575.125
And the maximum profit of the retailer is
𝛱 𝑅 = (𝑝 − 𝑤)(72 − 𝑝) (from equation
(vii))
𝛱 𝑅
∗
= (54.75 − 37.5)(72 − 54.75)
𝛱 𝑅
∗
= 17.25 ∗ 17.25
𝛱 𝑅
∗
= 297.562

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RS GAME(Retailer Stackelberg):
The retailer is the market leader and sets the retail price first and the manufacturer is the
follower, follows the retail price to set the wholesale price. We will try to establish the MRP
in this model.
Maximizing the retailers profit
max
𝑤
∏ = (𝑝 − 𝑤)(72 − 𝑝)
𝑅
Where for a given p , the retailer will find optimal wholesale price
𝑤 = 𝑎𝑟𝑔𝑚𝑎𝑥 ∏ = (𝑤 − 3)(72 − 𝑝) − 20
𝑀
To maximize the manufacturer’s profit we find partial derivative with respect to w, i.e.
𝜕𝛱 𝑀
𝜕𝑤
=
𝜕((𝑤 − 3)(72 − 𝑝) − 20)
𝜕𝑤
=
(72 − 𝑝)𝜕((𝑤 − 3)
𝜕𝑤
+
(𝑤 − 3)𝜕(72 − 𝑝)
𝜕𝑤
= (72 − 𝑝)
Therefore, MRP =
𝜕𝛱 𝑀
𝜕𝑤
= 0 and hence p = 72, Therefor MRP is 72.
Result Validation:
The table shows the data for retail price, quantity demanded, manufacturer’s profit and
retailer’s profit for different wholesale price. (MS Game)
Table 1.3
Wholesale
Price (w)
Retail
Price(p)
Quantity
(q)
Manufacturer
Cost ( c )
Profit
Manufacturer
(Π 𝑀)
Profit
Retailer (ΠR)
35 53.5 18.5 75.5 572 342.25
35.5 53.75 18.25 74.75 573.125 333.0625
36 54 18 74 574 324
36.5 54.25 17.75 73.25 574.625 315.0625
37 54.5 17.5 72.5 575 306.25
37.5 54.75 17.25 71.75 575.125 297.5625
38 55 17 71 575 289
38.5 55.25 16.75 70.25 574.625 280.5625
39 55.5 16.5 69.5 574 272.25
39.5 55.75 16.25 68.75 573.125 264.0625
40 56 16 68 572 256

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FIG 1.3
From the graph between the wholesale price and the manufacturer’s profit, it is clear that
the profit is optimum at a wholesale price of 37.5 and that is the price optimum price
established through MS game.
FIG 1.4
Also the graph between retail price and retailer’s profit for different wholesale price is
declining in nature. However as in MS game the manufacturer sets the optimum wholesale
0
50
100
150
200
250
300
350
400
53 53.5 54 54.5 55 55.5 56 56.5
Retailer'sProfit(ΠR)
Retailer's Price (p)
571.5
572
572.5
573
573.5
574
574.5
575
575.5
34 35 36 37 38 39 40 41
Manufacturer'sProfit(Π𝑀)
Wholesale Price(w)

Page | 18
price i.e. 37.5 in our case , varying the retail price at fixed wholesale price ,we can establish
that the profit of the retailer is maximized at Rs 54.75 that was derived in MS game.
Wholesale
Price (w)
Retail
Price(p)
Quantity
(q)
Profit Retailer
(ΠR)
37.5
46 26 221
48 24 252
50 22 275
52 20 290
54 18 297
54.75 17.25 297.5625
55 17 297.5
57 15 292.5
59 13 279.5
61 11 258.5
63 9 229.5
Table 1.4
FIG 1.5
210
220
230
240
250
260
270
280
290
300
310
45 47 49 51 53 55 57 59 61 63 65
RetailerProfit(ΠR)
Retail Price (p)

Page | 19
Hence, we are able to set the wholesale price and the retail price through MS game as
manufacturer as the leader of the market and retailer as the follower such that both the
profits of the manufacturer and retailer are at optimum level.
LIST OF REFERENCES:
1. Supply Chain (n.d.) retrived from
https://whatis.techtarget.com/definition/supply-chain
2. Pricing (n.d.) retrived from
http://www.jbdon.com/pricing-under-monopolistic-and-oligopolistic-competition.html
3. Duopoly (n.d.) retrived from
https://investinganswers.com/financial-dictionary/economics/duopoly-1315

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