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5 Characteristics of perfect competition: Large number of buyers or sellers, product
homogeneity, firms are price takers, there is perfect information available to consumers,
there are no barriers to entry.
In most markets, oligopolistic firms sell heterogenous, or differentiated products, rather
than homogeneous ones. Generally, by the principle of differentiation, firms wish to
differentiate themselves from other firms to gain market power or profit margin (“market
niche” in business term). Of course, entry can dissipate this market power.
Product Differentiation can assume many forms: quality, location, complementary service
bundled with product, advertising or branding…
Product differentiation can also be horizontal or vertical: Horizontal differentiation is where
consumers have divergent views of what is the best version i.e. colour, taste, location. Or if
for equal prices, consumers do not agree on which product is the preferred one, products
are horizontally differentiated.
Vertical differentiation is where consumers generally agree on what constitutes higher
quality. Or if for equal prices, all consumers prefer one over the other product, products are
vertically differentiated.
Two common approaches to the specification of consumers’ preferences when products are
horizontally differentiated: “address” or “non-address”.
In non-address models, the way the products are differentiated is not defined and we do
not explicitly specify consumer preferences with respect to the characteristics of the goods.
Consumers express preferences over a pre-determined set of all possible (differentiated)
goods, and have a taste for variety (the consumer cares if they are close substitutes or not).
In address models, the elements of differentiation are explicitly introduced into preferences
of the consumer. Consumers derive from the characteristics or attributes of the products.
Section 1: Non-address models: Monopolistic competition (perfect competition but added in
product differentiation): Consumers’ preferences over the set of differentiated products are
symmetric and can be aggregated by the preferences of a single “representative consumer”.
Very large set of possible differentiated products. Free entry of new brand-producing firms.
The purpose is to abstract from strategic interaction between products to study product
diversity offered by a market economy. Widely used in the analysis of international trade to
explain bilateral trade of similar products at the origin.
Dixit-Stiglitz-Spence Model: Representative Consumer: i=1,2 , … , N firms, each firm
produces at most one differentiated product, with quantity q i and price pi
N
“Representative consumer” utility is given by: U ( q 1 , q 2 , .. , q N )=∑ qi =q 1 + q2 ...+q N
ρ ρ ρ ρ
i=1
ρ
with ρ<1. o.e. U =N . ( q )
Product substitutability is captured by ρ : As products become less substitutable, consumers
find the products to be more differentiated ( ρ → 0¿ .
As products become closely substitutable, consumers find the products to be more
homogeneous ( ρ →1 ¿
, N
The representative consumer maximises U subject to the budget constraint: ∑ pi q i ≤ I
i=1
where I is the income of the representative consumer. Npq=I o.e.
Dixit-Stiglitz-Spence Model: Firms and Equilibrium Definition: Firms have identical
{
technologies with increasing returns to scale: T C i ( qi ) =
F +c qi if qi >0
0 if q i=0
In equilibrium, we have: Each consumer takes his income and prices as given and maximises
his utility; Each firm behaves as a monopoly over its brand and chooses pi to maximise its
profit; Given the free entry of firms, each firm makes zero profits:
N N
∑ π i =0 → ∑ ( F +c qi ) =I
i=1 i=1
Total cost of n firms = total revenue of N firms
For representative consumer, at optimal consumption level, the elasticity of demand for
1
product i is given by: ε i=
1−ρ
p i−c 1 ¿ c
For firm i, it sets the monopoly price such that: = → pi=
pi εi ρ
¿ ¿ ¿ 1−ρ ¿ ¿ ρ F
Zero profit condition implies: π i ( p i ) =( pi −c ) qi −F= c q i −F=0→ q i = .
ρ 1−ρ c
Finally, as all firms make zero profits, the resource constraint implies:
(1−ρ ) I
N . ( F+ c q ¿i )=I → N =
F
For when the market is not at equilibrium, the consumer’s optimisation problem becomes:
¿ I
q=
Np
The utility of the representative consumer is then: U =N . ( q )ρ
The profit of each firm is: π=( p−c ) .q−F . Total profit of firms ¿ N . π
Implications: The equilibrium number of firms depend on the extent of scale economies (F)
and the elasticity of substitution (between products) ( ρ ¿ .
In general, there are two effects operating in opposite directions: Nonappropriability of
social surplus: This effect says that a firm cannot generally capture the whole consumer
surplus associated with the introduction of a good (due to the lack of market power – there
is still competition due to the substitutability of product). The positive externality on
consumers implies that firms tend to introduce socially too few products.
Business stealing: By introducing a product, a firm steals consumers from other firms. The
rivals who have a positive profit margin, lose income from these diverted consumers. This
negative externality on other firms implies that firms tend to introduce too many products.
So when a new entrant enters the market, the utility of a representative consumer goes up
(nonappropriability of CS), but the new entrant can still earn abnormal profit (market
stealing effect)
,Section 2: Address Model – Horizontal Differentiation: Oligopolistic competition with
differentiated products.
Monopolistic competition firms cannot interact with each other. Oligopolistic they can.
Location model of product differentiation: Product differentiation: Differences between
products modelled as differences in a product’s location in “product space”, or the spectrum
of characteristics.
Consumer heterogeneity: Differences in consumer preferences modelled by their location
on that same product space.
Hotelling linear model is used to study price competition with differentiated products, as
well as firms’ choice of product in duopoly.
Salop circular model is used to study entry and product diversity when there are no barriers
to entry other than fixed cost of production.
Hotelling Model: Consider a “linear city” of length 1, along which all consumers reside.
Consumers are uniformly distributed on the interval [0, 1] and have mass 1.
Two (duopoly) firms located at I 1 , I 2 on the interval. Without loss of generality, assume
I 1< I 2. Firms incur constant marginal costs c.
The consumers have unit demands (i.e. each consumes one or zero units of the good), and
incur transportation costs which are increasing functions of the distance needed to travel to
the firm.
- Consumer located at x and buying from firm 1 gets utility:
U −p 1−τ (|x−I 1|)
Constant surplus from consumption price of firm 1 transportation cost to firm 1
In particular, we consider two variants of the transportation cost:
Linear transportation cost: τ (| x−I 1|) =t | x−I 1|
Quadratic transportation cost: τ (| x−I 1|) =t ( x−I 1 )
2
Location Game with Fixed Price: First, we take the firms’ prices as given (due to regulated
prices) and look for the Nash equilibrium in location.
Assume that p1= p2= p . Recall that I 1< I 2:
Consumers maximises U −p−τ (|x−I 1|) → consumers shop from the closest store.
( I +I
)
The profit of each firm is given by: π 1=( p−c ) 1 2 → π 1 is increasing in I1 (incentive to
2
move to the right)
(
π 2=( p−c ) 1−
2 )
I1 + I 2
→ π 2 is decreasing in I1 (incentive to move to the left)
The unique equilibrium in simultaneous location game is I 1=I 2 =0.5
The absence of price competition leads to minimal differentiation, that is, firms will
maximise market share by locating where they can best meet consumers’ preferences
(demand effect – drive to differentiate less).
Price Competition with Fixed Location: location as given (due to design patent) and look for
the Bertrand Nash equilibrium.
Assume that firms locations are fixed at I 1=0 , I 2=1 and U is sufficiently large that each
consumer buys one unit of good:
, Then consumer located at x gets utility: U x ={¿ firm1 ¿ U −p 2−τ (|1−x|) if he buys ¿ firm 2 ¿
Assuming linear transportation cost τ (| x−I i|) =t |x−I i|. Then the
consumer that is indifferent to buy from firm 1 or 2 is located at:
p −p 1
^x ( p 1 , p2 )= 2 1 +
2t 2
All consumers located at x < ^x strictly prefer to purchase from firm 1.
All consumers located at x > ^x strictly prefer to purchase from firm 2.
Then the demand for each firm is given by:
1 p p 1 p p
q 1= ^x = + 2 − 1 , q 2=1− x^ = + 1 − 2
2 2 t 2t 2 2t 2t
t
Solving for Bertrand Nash equilibrium, we have: p1= p2=c+ t , π 1=π 2=
2
The products appear differentiated more for the consumer when the transportation cost is
higher (and prices are not fixed). When the transportation cost increases, firms compete
less intensively for the consumers, giving firms market power to increase its prices and
profits (strategic effect – firm has incentive to differentiate itself to enjoy market power).
Hotelling Model with Quadratic Cost: Firms here can choose both prices and locations.
The linear cost model is not very tractable if firms are located too close to each other. Firms
always have an incentive to offer substitutes to generate more demand, which may lead to
instability in competition.
Consider a two stage game where: In the first stage of the game, firms simultaneously
decide which location to pick (or, equivalently, the design of their product).
In the second stage, after observing the location of their competitors, they simultaneously
set prices.
The transportation cost is quadratic, that is, τ ¿
The solution concept is subgame perfect equilibria.
¿ ¿
First, we solve for the optimal pricing p1 ( I 1 , I 2 ) , p 2 (I 1 , I 2 ) for each location pair I 1< I 2.
Assume that U is large enough that each consumer buys 1 unit of good (market is covered).
Then consumer located at x gets utility:
{
2
U x = U− p1−t ( x−I 1) if he buys firm 1 ¿ U− p2 −t ( I 2−x ) if he buys ¿ firm 2 ¿
2
¿
Then the consumer that is indifferent to buy from firm 1 or 2 is located at:
I +I p − p1
^x ( p 1 , p2 , I 1 , I 2 )= 1 2 + 2
2 2t ( I 2−I 1)
All consumers located at x < ^x strictly prefer to purchase from firm 1
All consumers located at x > ^x strictly prefer to purchase from firm 2
Then the demand function for each firm is given by:
I 1+ I 2 p2 − p 1
q 1= ^x = +
2 2t (I 2−I 1)
2−( I 1 + I 2 ) p1 −p 2
q 2=1− x^ = +
2 2t (I 2−I 1)
Solving for Bertrand Nash equilibrium we have:
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