Module 3. Product differentiation
Horizontal differentiation
Javier Ballesteros Muñoz
jballesterosm@faculty.ie.edu
Industrial Organization & Strategy
November, 2025
IE University 3. Product Differentiation November, 2025 1/ 23
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Module 3. Product differentiation Horizontal differentiation, Diapositivas de Gestión Industrial
Module 3. Product differentiation Horizontal differentiation
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Module 3. Product differentiation
Horizontal differentiation
Javier Ballesteros Muñoz
Industrial Organization & Strategy
November, 2025
Device Usage Policy in the Classroom
No electronic devices are allowed during class this includes phones, laptops, and tablets. Nothing should be on the table except a sheet of paper and a pen/pencil for notes. Phones must be kept silent and out of sight. Respect your peers’ learning environment – avoid distractions of any kind. Repeated misuse will negatively affect your Class Participation grade.
Introduction
Contents
- Introduction
- Quantity competition without localization
- Price competition without localization
Introduction
Introduction
One of the crucial assumptions behind the Bertrand paradox is that firms produce a homogeneous product (perfect substitutes). Price is the only variable of interest to consumers, and no firm can raise its price above marginal cost without losing its entire market share.
In practice, such an assumption is unlikely to be satisfied. Some consumers... will prefer buying the firm’s brand even at a small premium because it is available at a closer store, can be delivered sooner will remain faithful to the high-price firm because they are unaware of the existence of other brands will be concerned that alternative brands do not have the same quality or will not satisfy their preferences as well
Introduction
Introduction – Horizontal Differentiation
Within horizontal differentiation Competition without localization: characteristics of the product are exogenous. Spatial competition: firms choose the characteristics of their products
The main difference in the assumptions wrt previous models is that we consider the case where firms produce differentiated products. Horizontal differentiation Products as a set of measurable characteristics Consumers hace preferences over characteristics Product space: set of all possible products Address of a product: location in the product space
Quantity competition without localization
Contents
- Introduction
- Quantity competition without localization
- Price competition without localization
Quantity competition without localization
Quantity competition wo localization (II)
Inverse demand for each product, where d ∈ (0, 1) P 1 (q 1 , q 2 ) = a − q 1 − dq 2 P 2 (q 1 , q 2 ) = a − q 2 − dq 1 What happens if... d = 0? d = 1? Firm i profit πi(qi, qj ) = qi(Pi(qi, qj ) − c) = qi(a − qi − dqj − c) Our goal : derive the Nash equilibrium of game played between the two firms (qC 1 , qC 2 ), i.e. each firm is maximizing its profit given the actions of the others. q 1 C = arg max q 1 π 1 (q 1 , qC 2 )
Quantity competition without localization
Quantity competition wo localization (III)
In order to determine the NE:
- Derive each firm’s optimal choice given its conjecture of what the rival does, that is, the firm’s best response.
Ri(qj ) = arg max qi≥ 0
πi(qi, qj ) = arg max qi≥ 0
qi(a − qi − dqj − c)
FOC: a − 2 qi − dqj − c = 0 ⇒ qi =
a − dqj − c 2
= Ri(qj )
- Find a mutually consistent combination of actions. In equilibrium: qC 1 = R 1 (q 2 C ) and qC 2 = R 2 (qC 1 )
q 1 C = qC 2 =
a − c 2 + d
= qC
Quantity competition without localization
Class Participation
Two firms produce differentiated products. Inverse demand functions:
P 1 = 120 − q 1 − 0. 5 q 2 and P 2 = 120 − q 2 − 0. 5 q 1
Constant marginal cost: c = 20.
Tasks: (^1) Write firm 1’s profit function and derive its best-response R 1 (q 2 ). (^2) Write firm 2’s best-response R 2 (q 1 ). (^3) Solve for the Nash equilibrium quantities q 1 C = qC 2 = qC^. (^4) Compute equilibrium price pC^ and profit πC^ for each firm. (^5) Compare outcomes for d = 0, d = 0. 5 , and d = 1.
Quantity competition without localization
Class Participation – solution
Firm 1’s profit: π 1 = q 1 (120 − q 1 − 0. 5 q 2 − 20) FOC: 120 − 20 − 2 q 1 − 0. 5 q 2 = 0 ⇒ q 1 = 50 − 0. 25 q 2 By symmetry: q 2 = 50 − 0. 25 q 1 Solve simultaneously: q 1 = 50 − 0 .25(50 − 0. 25 q 1 ) ⇒ q 1 = 50 − 12 .5 + 0. 0625 q 1 ⇒ 0. 9375 q 1 = 37. 5 ⇒ qC 1 = qC 2 = 40
Equilibrium price: pC^ = 120 − 40 − 0 .5(40) = 60 Profit per firm: πC^ = 40 (60 − 20) = 1, 600 Comparative statics (per firm): d qC^ pC^ πC 0 50. 00 70. 00 2 , 500
- 5 40. 00 60. 00 1 , 600 1 33. 33 53. 33 1 , 111. 11 As d increases: outputs fall, prices fall, and profits fall.
Quantity competition without localization
Class Participation
Two firms, quantity competition with differentiated products:
P 1 = 78 − q 1 − 0. 5 q 2 , P 2 = 78 − q 2 − 0. 75 q 1
Constant marginal cost: c = 20. Firms choose quantities simultaneously.
Tasks: (^1) Write the profit functions π 1 , π 2. (^2) Derive best responses R 1 (q 2 ), R 2 (q 1 ). (^3) Solve for the Cournot–Nash equilibrium q 1 C , q 2 C. (^4) Compute equilibrium prices pC 1 , pC 2 and profits πC 1 , πC 2. (^5) Briefly explain which firm produces more and why.
Quantity competition without localization
Class Participation – solution
Profits and best responses: π 1 = q 1 (78 − q 1 − 0. 5 q 2 − 20), FOC: 58 − 2 q 1 − 0. 5 q 2 = 0 ⇒ q 1 = 29 − 0. 25 q 2
π 2 = q 2 (78 − q 2 − 0. 75 q 1 − 20), FOC: 58 − 2 q 2 − 0. 75 q 1 = 0 ⇒ q 2 = 29 − 0. 375 q 1 Equilibrium quantities:
q 1 = 29 − 0 .25(29 − 0. 375 q 1 ) ⇒ 2932 q 1 =^874 ⇒ qC 1 = 24, qC 2 = 29 − 0. 375 · 24 = 20
Equilibrium prices:
pC 1 = 78 − 24 − 0 .5(20) = 44, pC 2 = 78 − 20 − 0 .75(24) = 40 Profits: πC 1 = 24 (44 − 20) = 576, πC 2 = 20 (40 − 20) = 400 Interpretation: Firm 1’s demand is less sensitive to the rival ( 0. 5 < 0. 75 ), so it faces weaker competitive pressure ⇒ produces more and earns higher profits.
Price competition without localization
Price competition wo localization (I)
Bertrand model with differentiated products : firms cannot choose the characteristics of their products (it is exogenous).
Assumptions N symmetric firms that produce differentiated products. Same constant marginal cost c. There are no capacity constraints. Firms choose prices simultaneously and non-cooperatively. Firm i chooses pi.
We consider a duopoly (N = 2).
Price competition without localization
Price competition wo localization (II)
Given the inverse demand for each product, where d ∈ (0, 1)
P 1 (q 1 , q 2 ) = a − q 1 − dq 2 P 2 (q 1 , q 2 ) = a − q 2 − dq 1
We find the demand for each product, where α, β, γ > 0
D 1 (p 1 , p 2 ) = α − βp 1 + γp 2 D 2 (p 1 , p 2 ) = α − βp 2 + γp 1
Exercise Compute α, β, γ in terms of a and d given that P 1 and P 2 are the inverse demand of D 1 and D 2.