Economies of scale to exploit quantity discount

Economies of Scale to
Exploit Quantity Discount

Economies of Scale to Exploit Quantity Discounts
There are two types of quantity discounts
 Lot Size based quantity discount:
 Volume Based quantity discount:
 What motivates for discounts?
 All-unit quantity discounts
 Marginal unit quantity discounts
Based on quantity
ordered in a single lot
• Improved coordination to increase total supply chain profits
• Extraction of surplus by suppliers through price discrimination
Based on total
quantity over a given
period (regardless of
number of lots)

Coordination in Supply Chain
A supply chain is coordinated if the decisions the retailer and supplier make maximize total
supply chain profits.
Supplier
Retailer
Distributor
All stages try to
maximize their own
profit, independently
This may results in
lack of coordination
Action that maximizes
retailers profit may not
maximize total supply
chain profit
What should manufacturer do to
maximize total supply chain profit?

 Quantity discounts (for commodity products):
When there is a large number of competitors in the market, the market sets the
price of that commodity and demand is fixed. (Ex. milk)
 Quantity discounts (for products for which the firm has market power):
When there are few competitors in the market, demand varies with price charged
by the retailer. (Ex. Herbal products)
❖ Two Part Tariff
❖ Volume based quantity discount
Coordination in Supply Chain
Manufacturer charges it’s entire profit as an up-
front franchise fee (ff) from the retailer and sets
its wholesale price as CR = CM
Based on total quantity over a given period
(regardless of number of lots)

Quantity discount for commodity products
The Impact of Locally Optimal Lot Sizes on Supply chain:
Problem No.1 Given Data: Demand (D) = 10,000 bottles/month
Data Related To Retailer
SR= Rs.100/Lot
IR = 0.2
CR = Rs.3
Data Related To Manufacturer
SM= Rs.250/Lot
IM = 0.2
CM = Rs.2/Unit
● Evaluate the optimal lot size for retailer.
● What is the Annual fulfillment and holding cost incurred by the
manufacturer as a result of retailer’s ordering policy?
Fixed order
placement,
transportation and
receiving cost
Price charged by
the manufacturer
Fixed order filling
cost
Production cost

Without Coordination
Solution:
𝑄 𝑅 =
2𝐷𝑆 𝑅
𝐼 𝑅 𝐶 𝑅
𝑄 𝑅 =
2 ∗ 1,20,000 ∗ 100
0.2 ∗ 3
𝑄 𝑅 = 𝟔, 𝟑𝟐𝟓 𝐮𝐧𝐢𝐭𝐬
𝐴𝑛𝑛𝑢𝑎𝑙 cos𝑡 𝑜𝑓 Re𝑡𝑎𝑖𝑙𝑒𝑟 =
𝐷
𝑄 𝑅
∗ 𝑆 𝑅 +
𝑄 𝑅
2
∗ 𝐼 𝑅 ∗ CR
1,20,000
6325
∗ 100 +
6325
2
∗ 0.2 ∗ 3
Re𝑡𝑎𝑖𝑙𝑒𝑟′
𝑠 𝑂𝑝𝑡𝑖𝑚𝑎𝑙 𝐿𝑜𝑡 𝑆𝑖𝑧𝑒:

Without Coordination
Solution cont.. 𝐴𝑛𝑛𝑢𝑎𝑙 𝑐𝑜𝑠𝑡 𝑜𝑓 𝑅𝑒𝑡𝑎𝑖𝑙𝑒𝑟 = 𝑹𝒔. 𝟑, 𝟕𝟗𝟓
𝐼𝑓 𝑅𝑒𝑡𝑎𝑖𝑙𝑒𝑟 𝑂𝑟𝑑𝑒𝑟𝑠 𝑖𝑛 𝑙𝑜𝑡 𝑠𝑖𝑧𝑒 𝑜𝑓 𝑄𝑅 = 6,325:
𝐴𝑛𝑛𝑢𝑎𝑙 𝑐𝑜𝑠𝑡 𝑜𝑓 Manufacturer =
𝐷
𝑄 𝑅
∗ 𝑆 𝑀 +
𝑄 𝑅
2
∗ 𝐼 𝑀 ∗ 𝐶M
𝐴𝑛𝑛𝑢𝑎𝑙 𝑐𝑜𝑠𝑡 𝑜𝑓 𝑀𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑒𝑟 =
1,20,000
6325
∗ 250 +
6325
2
∗ 0.2 ∗ 2
𝐴𝑛𝑛𝑢𝑎𝑙 𝑐𝑜𝑠𝑡 𝑜𝑓 𝑀𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑒𝑟 = 𝐑𝐬. 𝟔, 𝟎𝟎𝟖
𝐴𝑛𝑛𝑢𝑎𝑙 Suply Chain 𝑐𝑜𝑠𝑡 = 3,795 + 6,008 = 𝐑𝐬. 𝟗, 𝟖𝟎𝟑

Solution cont..
 In the above Example Retailer picks the lot size of 6,325 with an objective of minimizing only its own
cost.
 From a supply chain perspective, the optimal lot size should account for the fact that both the retailer
and the manufacturer incur costs associated with each replenishment lot.
 The Total supply chain cost using a lot size Q is obtained as follows:
𝐴𝑛𝑛𝑢𝑎𝑙 Cos𝑡 𝑜𝑓 𝑆𝑢𝑝𝑝𝑙𝑦𝐶ℎ𝑎𝑖𝑛 =
𝐷
𝑄
∗ 𝑆 𝑅 +
𝑄
2
∗ 𝐼 𝑅 ∗ 𝐶 𝑅 +
𝐷
𝑄
∗ 𝑆 𝑀 +
𝑄
2
∗ 𝐼 𝑀 ∗ CM
𝐹𝑜𝑟 𝑜𝑝𝑡𝑖𝑚𝑎𝑙 𝑙𝑜𝑡 𝑠𝑖𝑧𝑒 𝑄
∗
𝑑(𝑇𝑜𝑡𝑎𝑙 sup𝑝𝑙𝑦 𝑐ℎ𝑎𝑖𝑛 cos𝑡
𝑑𝑄
= 0

Solution cont..
-
𝐷
𝑄2 ∗ 𝑆 𝑅 +
𝐼 𝑅∗𝐶 𝑅
2
−
𝐷
𝑄2 ∗ 𝑆 𝑀 +
𝐼 𝑀∗CM
2
=
0
𝐷
𝑄2
∗ 𝑆 𝑅 + 𝑆M =
𝐼 𝑅 ∗ 𝐶 𝑅
2
+
𝐼 𝑀 ∗ CM
2
𝑄
∗
=
2𝐷(𝑆 𝑅 + 𝑆 𝑀
𝐼 𝑅 ∗ 𝐶 𝑅 + (𝐼 𝑀 ∗ 𝐶 𝑀
⇒
⇒
⇒

Solution cont..
𝑄
∗
=
𝐼 𝑅 ∗ 𝐶 𝑅 + (𝐼 𝑀 ∗ 𝐶 𝑀
𝑄
∗
=
2 ∗ 1,20,000 ∗ (100 + 250
0.2 ∗ 100 + (0.2 ∗ 250
𝑄
∗
= 𝟗, 𝟏𝟔𝟓 units
𝐼𝑓 𝑅𝑒𝑡𝑎𝑖𝑙𝑒𝑟 𝑂𝑟𝑒𝑑𝑒𝑟𝑠 𝑖𝑛 𝑙𝑜𝑡 𝑠𝑖𝑧𝑒 𝑜𝑓 𝑄
∗
= 9,165 units
𝐷
𝑄
∗ ∗ 𝑆 𝑅 +
𝑄
∗
2
1,20,000
9165
∗ 100 +
9165
2
∗ 0.2 ∗ 3 = Rs. 𝟒𝟎𝟓𝟗

𝐴𝑛𝑛𝑢𝑎𝑙 cos𝑡 𝑜𝑓 Manufacturer =
𝐷
𝑄
∗ ∗ 𝑆 𝑀 +
𝑄
∗
2
1,20,000
9165
∗ 250 +
9165
2
∗ 0.2 ∗ 2
𝐴𝑛𝑛𝑢𝑎𝑙 cos𝑡 𝑜𝑓 Manufacturer = Rs. 𝟓, 𝟏𝟎𝟔
𝐴𝑛𝑛𝑢𝑎𝑙 Supply chain cos𝑡 = Rs. 4059 + Rs. 5,106 = Rs. 𝟗, 𝟏𝟔𝟓
Solution cont..

Summary:
When QR= 6,325
Without coordination
When Q*= 9,165
With coordination
Raise or Down in
Cost (After coordination)
Annual cost of
Retailer (Rs.)
3795 4059 Raise of Rs.264
Annual cost of
Manufacturer (Rs.)
6008 5106 Down of Rs. 902
Total supply chain
cost (Rs.)
9803 9165 Down of Rs. 638
So, manufacturer must offer retailer a suitable incentive to raise
it’s lot size to Q*=9,165 (as the total cost of retailer is raising by
Rs. 264 as he orders in lots of 9165)

Designing a suitable Lot size based quantity discount
Problem No.2: (Consider the data from previous problem)
Design a suitable quantity discount that gets retailer to order in lots of 9,165 units when it
aims to minimize only its own total costs.
Solution:
 CR = Rs.3/unit (CR: Price charged to retailer)
 Manufacturer should reduce the material cost by Rs 264/year (as retailer’s total cost
increased by Rs. 264/year when he orders in lots of 9165) for the sales of 1,20,000
units/year.
= 3 −
264
1,20,000
⇒ Rs. 2.9978/unit

Quantity ordered by Retailer Unit price
If Q < 9165 units Rs. 3
If Q ≥ 9165 units Rs. 2.9978
Pricing Scheme:
Important points:
• For commodity products for which price is set by the market, manufacturer with large fixed cost per lot
can use lot size based quantity discount to maximize total supply chain profit.
• Lot size based discount, however, increase cycle inventory in the supply chain.

Problem: If manufacturer lowers it’s fixed cost per order from Rs. 250 to Rs. 100 & SM=
Rs.100/ order (no coordination in supply chain)
when QR= 6,325∶ 𝐴𝑛𝑛𝑢𝑎𝑙 𝑐𝑜𝑠𝑡 𝑜𝑓 𝑅𝑒𝑡𝑎𝑖𝑙𝑒𝑟 = 𝑅𝑠. 3,795
Impact of lowering fixed cost per lot
𝐷
𝑄 𝑅
∗ 𝑆 𝑀 +
𝑄 𝑅
2
1,20,000
6325
∗ 100 +
6325
2
∗ 0.2 ∗ 2 = 𝑅𝑠. 𝟑𝟏𝟔𝟐
𝐴𝑛𝑛𝑢𝑎𝑙 Supply chain cos𝑡 = Rs. 3795 + Rs. 3162 = Rs. 𝟔𝟗𝟓𝟕
From previous
example and not
affected by
change in SM

Problem: If manufacturer lowers it’s fixed cost per order from Rs. 250 to Rs. 100 i.e. SM=
Rs.100/ order (with coordination in supply chain)
𝑂𝑝𝑡𝑖𝑚𝑎𝑙 𝐿𝑜𝑡 𝑠𝑖𝑧𝑒 𝑄
∗
=
𝐼 𝑅 ∗ 𝐶 𝑅 + (𝐼 𝑀 ∗ 𝐶 𝑀
𝑄
∗
=
2 ∗ 1,20,000 ∗ (100 + 100
0.2 ∗ 100 + (0.2 ∗ 100
= 𝟔, 𝟗𝟐𝟖 𝐮𝐧𝐢𝐭𝐬
𝐷
𝑄
∗ ∗ 𝑆 𝑅 +
𝑄
∗
2
1,20,000
6928
∗ 100 +
6928
2
∗ 0.2 ∗ 3 = Rs. 𝟑𝟖𝟏𝟏

Problem cont..
𝐷
𝑄
∗ ∗ 𝑆 𝑀 +
𝑄
∗
2
𝐴𝑛𝑛𝑢𝑎𝑙 cos𝑡 𝑜𝑓 𝑀𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑒𝑟 =
1,20,000
6928
∗ 100 +
6928
2
∗ 0.2 ∗ 2 = Rs. 𝟑𝟏𝟏𝟖
𝐴𝑛𝑛𝑢𝑎𝑙 Supply chain cos𝑡 = Rs. 3811 + Rs. 3118 = Rs. 𝟔𝟗𝟐𝟗

Summary:
No
coordination
(When SM=250)
Coordination
(When SM=250 )
No
coordination
(When SM=100)
Coordination
(When SM=100)
QR= 6325 Q*= 9165 QR= 6325 Q*= 6928
Total Supply
Chain Cost
Rs.9803 Rs. 9165 Rs.6957 Rs. 6929
All quantity discount can be removed if Sm is lowered
to Rs 100.

 Here price at which the retailer sells the product influences demand.
Problem: Let annual demand faced by retailer is given by Demand Curve: (3,60,000 - 60,000p)
Case 1. Policy: (When No coordination in supply chain)
Quantity discount for products for which firm has market
power
p = price at which retailer
sells products
• What should the manufacturer charge (CR) to the retailer?
• What should the retailer charge (p) to the customer?

Solution: Profit at Retailer (ProfR) = (p - CR) (3,60,000 – 60,000p)
Profit at Manufacturer (ProfM) = (CR - CM) (3,60,000 – 60,000p)
Price p at which Retailer maximizes its profit is obtained by
power
𝑑(Pr𝑜𝑓 𝑅
𝑑𝑝
= 0
⇒
𝑑[(𝑝 − 𝐶 𝑅 (3,60,000 − 60000𝑝 ]
𝑑𝑝
= 0
⇒ 3,60,000 − 60,000𝑝 + (𝑝 − 𝐶 𝑅 (−60,000 = 0
⇒ 𝑝 = 3 +
𝐶 𝑅
2

Solution cont..
ProfM=
ProfM=
To maximize ProfM
So
𝐶 𝑅 − 𝐶 𝑀 (3,60,000 – 60,000(3 +
𝐶 𝑅
2
𝐶 𝑅 − 2 (1,80,000 – 30,000𝐶 𝑅
𝑑(Pr𝑜𝑓 𝑀
𝑑𝐶 𝑅
= 0
1,80,000 − 30,000𝐶 𝑅 + (CR − 2 (−30,000 = 0
𝐶 𝑅 = Rs. 4
⇒
⇒
Where CM is
production cost
CM=Rs 2 per unit
𝑝 = 3 +
𝐶 𝑅
2
⇒ 𝑝 = 3 +
4
2
⇒ 𝑝 = Rs. 5
power

Summary: When decisions are made independently it is optimal (No Coordination)
Price charged by manufacturer (CR) Rs.4
Price charged by retailer (p) Rs. 5
𝑇𝑜𝑡𝑎𝑙 𝑀𝑎𝑟𝑘𝑒𝑡 𝐷𝑒𝑚𝑎𝑛𝑑 = 3,60,000 – 60,000p
3,60,000 − 60,000 ∗ 5 = 𝟔𝟎, 𝟎𝟎𝟎 𝒖𝒏𝒊𝒕𝒔
Pr𝑜𝑓 𝑅 = (p − CR) (3,60,000 – 60,000p) ⇒ 5 − 4 (3,60,000 − 60,000 ∗ 5 = 𝑅𝑠. 𝟔𝟎, 𝟎𝟎𝟎
Pr𝑜𝑓 𝑀 = 𝐶 𝑅 − 2 (1,80,000 – 30,000𝐶 𝑅 ⇒ (4 − 2 (1,80,000 − 30,000 ∗ 4 = 𝑅𝑠. 𝟏, 𝟐𝟎, 𝟎𝟎𝟎
𝑇𝑜𝑡𝑎𝑙 𝑆𝑢𝑝𝑝𝑙𝑦 𝐶ℎ𝑎𝑖𝑛 𝑝𝑟𝑜𝑓𝑖𝑡 = 60,000 + 1,20,000 = 𝑅𝑠. 𝟏, 𝟖𝟎, 𝟎𝟎𝟎
power

Case 2. When There is coordination in supply chain: (Two stages coordinate their pricing decision to
maximize the supply chin profit
For optimal retail price
Pr𝑜𝑓SC = (p − CM (3,60,000 − 60,000p
𝑑(Pr𝑜𝑓 𝑆𝐶
𝑑𝑝
= 0
3,60,000 − 60,000𝑝 + (𝑝 − CM (−60,000 = 0 𝑝 = 𝐑𝐬. 𝟒/𝐮𝐧𝐢𝐭⇒⇒
power
𝑇𝑜𝑡𝑎𝑙 𝑀𝑎𝑟𝑘𝑒𝑡 𝐷𝑒𝑚𝑎𝑛𝑑 = 3,60,000 – 60,000p ⇒ 3,60,000 − 60,000 ∗ 4 = 𝟏, 𝟐𝟎, 𝟎𝟎𝟎 𝒖𝒏𝒊𝒕𝒔
𝑇𝑜𝑡𝑎𝑙 𝑠𝑢𝑝𝑝𝑙𝑦 𝑐ℎ𝑎𝑖𝑛 𝑝𝑟𝑜𝑓𝑖𝑡 (Pr𝑜𝑓 𝑆𝐶 = (p − CM (3,60,000 − 60,000p ⇒ (4 − 2 (3,60,000 − 60,000 ∗ 4
=Rs.2,40,000
CM=Rs 2/unit

power
Summary:
 From summary table it is clear that when each stage of supply chain is setting its price independently
(i.e. no coordination) there is a loss of Rs. 6000 in supply chain profit.
 This phenomenon is called Double Marginalization.
 Double marginalization : Supply chain margin is divided into two stages but each stage makes its
pricing decision considering only its own local profit and this results in loss in profit.
Without
Coordination
With
Coordination
Loss due to lack
of coordination
Total supply chain
profit ProfitSC Rs. 1,80,000 Rs. 2,40,000 Rs. 6000

 New pricing schemes to achieve coordinated solution and maximize supply chain profit (Even if
decisions are made independently)
power
I. Two Part Tariff
II. Volume based quantity discount

Manufacturer charges it’s entire profit as an up-
front franchise fee (ff) and sets its wholesale
price as CR = CM
 Manufacturer can construct a two part tariff by which the retailer is charged an up-front franchise fee (ff )
𝑓𝑓 = Pr𝑜𝑓 𝑆𝐶 − Pr𝑜𝑓 𝑅
From previous example
ProfSC = Rs. 2,40,000 (with coordination)
ProfR = Rs.60,000 (without coordination)
𝑓𝑓 = 2,40,000 − 60,000
𝑓𝑓 = Rs. 1,80,000
Two part tariff constructed by manufacturer
𝑓𝑓 = Rs. 1,80,000
Material cost CR = CM =Rs. 2/unit
Two Part Tariff

 Retail pricing decision is based on maximizing profit.
Optimal Retail price
 Retailer gets the maximum profit when
Now
Two Part Tariff (Analysis)
Pr𝑜𝑓 𝑅 = (𝑝 − 𝐶 𝑅 (360000 − 60000𝑝 − 𝑓𝑓
Pr𝑜𝑓 𝑅 = (𝑝 − 𝐶M (360000 − 60000𝑝 − 𝑓𝑓
Since
CR = CM
𝑝 = 3 +
𝐶 𝑀
2
As Derived in
previous example
𝑝 = 3 +
2
2
= Rs. 4/unit
Since
CM=Rs. 2/unit
𝑇𝑜𝑡𝑎𝑙 𝑀𝑎𝑟𝑘𝑒𝑡 𝐷𝑒𝑚𝑎𝑛𝑑 = 360000 − 60000𝑝 ⇒ 3,60,000 − 60,000 ∗ 4 = 𝟏, 𝟐𝟎, 𝟎𝟎𝟎 𝑢𝑛𝑖𝑡𝑠
Pr𝑜𝑓 𝑅 = (𝑝 − 𝐶M (360000 − 60000𝑝 − 𝑓𝑓 ⇒ 4 − 2 360000 − 60000 ∗ 4 − 1,80,000 = Rs. 𝟔𝟎, 𝟎𝟎𝟎
Pr𝑜𝑓M = 𝟏, 𝟖𝟎, 𝟎𝟎𝟎
Charged by manufacturer as an up-front
(Franchise fee)
𝑑(Pr𝑜𝑓 𝑅
𝑑𝑝
= 0 ⇒

Pr𝑜𝑓SC = 60,000 + 1,80,000 = Rs. 2,40,000
Without coordination Two part tariff
Rs.1,80,000 Rs. 2,40,000Pr𝑜𝑓SC
Same as what we got when
supply chain is coordinated
Two Part Tariff (Analysis)

Problem: Total demand Dcoord = 1,20,000 units, Retail price (p) = Rs. 4/unit
 Manufacturer want to design a volume based quantity discount scheme that gets the retailer to buy
(sell) 1,20,000 units/year.
 Pricing scheme must be such that
Volume based quantity discount
From previous example when supply chain
stages are coordinated
Pr𝑜𝑓R = At least Rs. 60,000
Pr𝑜𝑓M = At least Rs. 1,20,000 From previous example
when supply chain stages
was not coordinated

 Analysis: Several schemes can be designed, one such scheme is when
D < 1,20,000 CR = Rs.4/unit
CR = Rs.3.50/unitD𝑐𝑜𝑜𝑟𝑑 ≥ 1,20,000
Pr𝑜𝑓 𝑅 = (𝑝 − 𝐶 𝑅 (360000 − 60000𝑝 ⇒ (4 − 3.5 (3,60,000 − 60,000 ∗ 4 = 𝑅𝑠. 60,000
Pr𝑜𝑓 𝑀 = (𝑝 − 𝐶M (360000 − 60000𝑝 ⇒ (3.5 − 2 (1,20,000 = 𝑅𝑠. 1,80,000
𝑇𝑜𝑡𝑎𝑙 𝑆𝑢𝑝𝑝𝑙𝑦 𝐶ℎ𝑎𝑖𝑛 𝑝𝑟𝑜𝑓𝑖𝑡 (𝑃𝑟𝑜𝑓SC = 60,000 + 1,80,000 = 𝑅𝑠. 2,40,000
Non coordinated Volume based
ProfSC Rs. 1,80,000 Rs. 2,40,000
Volume based quantity discount
Same as what we got when
supply chain is coordinated

Lot-size-based vs. volume based quantity discount
Lot-size-based quantity discount Volume based quantity discount
1. It is based on quantity purchased per lot. 1. It is based on rate of purchase or volume
purchased on average per specified time
period.
2. It tend to increase cycle inventory in the
supply chain
2. It is compatible with small lots that reduces
cycle inventory
3. It makes sense only when the manufacturer
incurs high fixed cost per period.
3. In all other instances it is better to have
volume based quantity discount.

Hockey Stick Phenomenon:
 In volume based discount orders from the retailer peak toward the end of a financial horizon, this is
referred to as the hockey stick phenomenon.
 As demand from retailer increases dramatically toward the end of a period, similar to the way a hockey
stick bends upward toward the end of the stick.
 Solution to hockey stick phenomenon : Base the volume discounts on a rolling horizon.
Volume based Quantity discount
For example, each week the manufacturer may offer retailer the volume
discount based on sales over the last 12 weeks.

Price Discrimination to Maximize Supplier Profits
Price discrimination is the practice whereby a firm charges differential prices to different segment of
customers to maximize profits.
 The goal of supplier is to price so as to maximize it’s profit, by dividing its customer into segments.
Example: In Airlines, passengers travelling on the same plane often pay different
prices for their seats.

Problem: A contract manufacturer has identified two customer segments for its production capacity.
Production cost : c = Rs. 10/unit
(i) What price should the contract manufacturer charge each segment if its goal is to maximize profits?
(ii) If the contract manufacturer were to charge a single price over both segments, what should it be?
(iii) How much increase in profits does differential pricing provide?
Demand curve
First segment of customers 𝑑1 = 5000 − 20𝑃1
Second segment of customers 𝑑2 = 5000 − 40𝑃2

Total profit made
(with capacity constraint)
If there is no capacity constraint , for segment i the supplier attempts to maximize
To find the optimal price for each segment
𝑀𝑎𝑥
𝑖=1
𝑘
𝑃𝑖 − 𝑐 (𝐴𝑖 − 𝐵𝑖 𝑃𝑖
Profit made by
supplier per unit
𝑃𝑖 − 𝑐 (𝐴𝑖 − 𝐵𝑖 𝑃𝑖
]𝑑[(𝑃𝑖 − 𝑐 (𝐴𝑖 − 𝐵𝑖 𝑃𝑖
𝑑𝑃𝑖
= 0
𝐴𝑖 − 𝐵𝑖 𝑃𝑖 + (𝑃𝑖 − c (−𝐵𝑖 = 0 ⇒ 𝑃𝑖 =
𝐴𝑖
2𝐵𝑖
+
𝑐
2
Optimal price for
each segment
Demand curve
for segment i

Solution:
(i) Without capacity constraints, the differential prices to be charged each segment are given by
Equation
Demand from two segments:
𝑃𝑖 =
𝐴𝑖
2𝐵𝑖
+
𝑐
2
𝑃1 =
𝐴1
2𝐵1
+
𝑐
2
𝑃2 =
𝐴2
2𝐵2
+
𝑐
2
⇒
⇒ 𝑃1 =
5000
2 ∗ 20
+
10
2
𝑃2 =
5000
2 ∗ 40
+
10
2
⇒
⇒
𝑃1 = Rs. 130
𝑃2 = Rs. 67.50
𝑑1 = 5000 − 20P1 ⇒ 5000 − 20 ∗ 130 ⇒ 𝑑1 = 2,400 units
𝑑2 = 5000 − 40𝑃2 ⇒ 5000 − 40 ∗ 67.50 ⇒ 𝑑2 = 2,300 units

Total profit of supplier :
(ii) If the contract manufacturer charges the same price p to both segments, he is attempting to maximize
Optimal price
𝑑1 ∗ 𝑃1 + (𝑑2 ∗ 𝑃2 − [𝑐(𝑑1 + 𝑑2
2400 ∗ 130 + (2300 ∗ 67.50 − [10(2400 + 2300
𝑇𝑜𝑡𝑎𝑙 𝑝𝑟𝑜𝑓𝑖𝑡 = 𝑅𝑠. 4,20,250
𝑃 − 10 5000 − 20𝑃 + (𝑃 − 10 (5000 − 40𝑃
= 𝑃 − 10 10000 − 60𝑃
𝑃 =
10000
2 ∗ 60
+
10
2
⇒ 𝑃 = Rs. 88.33
Solution cont..

Demand for two customers
Total profit =
(iii) Increase in profit due to differential pricing
𝑑1 = 5000 − 20 ∗ 88.33 ⇒ 𝑑1 = 3,323.40
𝑑2 = 5000 − 40 ∗ 88.33 ⇒ 𝑑2 = 1,466.80
(𝑃 − 𝑐 (𝑑1 + 𝑑2
= (88.33 − 10 (3323.40 + 1466.80
= 𝑅𝑠. 3,68,166.67
= 4,20,250 − 3,68,166.67
= Rs. 52083.33
Conclusion: Setting a fixed price for all segment of customers or for all units will not maximize profits for
manufacturer.

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