Estimating Market Power in Differentiated Product
Markets
Metin Cakir
Purdue University
December 6, 2010
Metin Cakir (Purdue)
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December 6, 2010
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Outline
Outline
Estimating structural market equilibrium models
Supply side: Review of Cournot and Bertrand competition
Demand side: Homogeneous vs. differentiated product markets
Review of classical demand system models
Random utility models: Discrete choice modeling
An empirical application to ready to eat cereal industry
Nevo, 2001
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Supply Side
Cournot Competition
Quantity Setting Firms
Cournot Model:
Max πi = P(Q)qi − c(qi )
qi
where Q is aggregate quantity and qi is quantity produced by firm i.
The FOC is:
∂πi
∂qi
0
= P + qi ∂P(Q)
∂qi − c (qi ) = 0
where
∂P(Q)
∂qi
=
∂P(Q) ∂Q
∂Q ∂qi
∂P(Q)
∂qi
=
∂P(Q)
∂Q
∂P(Q)
∂qi
=
∂qj
∂P(Q)
1+
∂q
| ∂Q
{z }|
{z i }
P 0 (Q)
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∂qi
∂qi
+
∂qj
∂qi
θC
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Supply Side
Cournot Competition
Quantity Setting Firms
At the optimum
P = MC − P 0 (Q)θC qi
where P 0 (Q) is the slope of inverse demand,
θC =
∂Q
∂qi
=1+
∂qj
∂qi
is the conjectural variation (not elasticity).
∂q
θC = 1 ⇒ Cournot-Nash. ( ∂qji = 0)
∂q
θC = 0 ⇒ Perfect Comp. ( ∂qij = −1)
∂q
θC = N ⇒ Cartel (Symmetry) ( ∂qji = 1)
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Supply Side
Bertrand Competition
Price Setting Firms
Bertrand Model: “Bertrand competition” refers to a model of oligopoly
in which two or more firms compete by simultaneously setting prices.
A price setting game can be formalized as:
Players: n > 2 players (firms), i, . . . , n
Actions: Each firm simultaneously sets price pi ∈ Pi = [0, ∞]
Payoffs: πi (pi , p−i ) = pi D(pi , p−i ) − Ci (D(pi , p−i ))
Under assumption of profit maximization
∗ ) such that
A Bertrand–Nash equilibrium is a vector of prices (pi∗ , p−i
for each player i,
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Supply Side
Bertrand Competition
Price Setting Firms
Bertrand equilibrium with differentiated products is simply the
solution to the system of first-order conditions implied by each
firms profit-maximizing pricing decision.
Let output of each firm denoted as:
Firm i’s profit maximization problem is given as:
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Supply Side
Bertrand Competition
Price Setting Firms
i
Pi = MC − qi / dq
dpi
where
dqi
dpi
=
∂qi ∂pi
∂pi ∂pi
dqi
dpi
=
∂qi
∂pi
+
+
∂qi ∂pj
∂pj ∂pi
∂qi B
∂pj θ
i
θB = 0 ⇒ Bertrand-Nash. (pi = MC − qi / ∂q
∂pi )
θB = −∞ ⇒ Perfect Comp. pi = MC
i
θB = 1 ⇒ Cartel (Symmetry) (pi = MC − qi /( ∂q
∂pi +
∂qi
∂pj ))
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Demand Side
Homogeneous Product Markets
NEIO models are typical examples of structural market equilibrium
models for homogeneous product markets.
Empirical applications use macro (aggregate) data
Demand side: Typically a single product demand curve is
estimated
Supply side: A model of Cournot competition is employed
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Demand Side
Differentiated Product Markets
Could refer to vertical or horizontal differentiation
Empirical applications use brand and/or individual level data
Demand side: Requires a demand system estimation of
competing brands
Supply side: A model of Bertrand competition is employed
Wide range of applicability of these models in IO and Marketing:
Analysis of competition
Market power
Price discrimination
Advertisement
Merger analysis
Analysis of consumer behavior
Sensitivity to price
Valuation of product attributes
Brand loyalty
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Demand Side
Demand system models: The dimensionality problem
Suppose we have observations of the same 50 brands of ice–cream
from 50 cities, 2500 data points. We also have data on 4 product and 6
consumer characteristics. We seek to estimate bulk ice–cream
demand elasticity.
We can take a classical demand system approach, i.e. AIDS
model. This would require:
Estimating a total of 2500 own and cross–price elasticities, using
2500 data points
Even with symmetry, homogeneity and adding up restrictions
estimation would be impractical
Instead we can work in product characteristics space rather than
product space (discrete choice models):
Redefine consumer’s utility as a function of product characteristics
Redefine consumer’s problem as a probability of buying a
particular product rather than how much to buy
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Demand Side
Random Utility Models
Random Utility Models, RUM
RUMs describe the relation of explanatory variables to the outcome of
a choice, without reference to exactly how the choice is made.
Agent i, i = 1, ..., n is assumed to make a choice among
j = 1, 2, ..., J possible alternatives.
Agent i chooses the alternative that provides the greatest utility,
Uij > Uik ∀j , k
Researcher observes the choice and its attributes, xij , but not the
utility
Therefore, the utility is assumed to be represented additively as
the sum of a partially observable part, Vij , and an unobservable
part, ij , such that
Usually we’ll assume Vij = xij β
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Demand Side
Random Utility Models
Random Utility Models, RUM
Let f (i ) denote the joint density of random vector i0 = (i1 , . . . , iJ ).
The probability that agent i chooses alternative j is:
This cumulative distribution is the probability of each random term
ik − ij is below observed quantity Vij − Vik . Since the joint density of
random errors is f (i ), we can rewrite as:
where I is indicator function equal to 1 if the term in parentheses is
true.
The model collapses to logit if f (i ) is i.i.d. extreme value
The model collapses to probit if f (i ) is multivariate normal
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Demand Side
Multinomial Logit
Multinomial Logit
Logit model is derived by assuming error terms follow a Type I extreme
value distribution. Specifically, assume that the errors are independent
across i and j with a density function:
f (ij ) = exp(−ij )exp(−e −ij )),
and the cumulative function
F(ij ) = exp(−e −ij ))
By this assumption the logit probability of individual i choosing
alternative j is
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Demand Side
Multinomial Logit
Estimation
Likelihood function:
Q Q
L̄ (β) = ni=1 Jj=1 (Pij )Yij
where Yij =1 if person i selects j and zero otherwise. The loglikelihood
is:
P P
P
L (β) = ni=1 Jj=1 Yij (xij β − log[ Jj=1 exp(xij β)])
where Vij = xij β. The score is
Lβ (β) =
P P
i
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j (Yij
− Pij )xij0 )
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Demand Side
Multinomial Logit
Independence of Irrelevant Alternatives, IIA
MNL model is subject to IIA problem. Take 2 alternatives k and l
Pj
Pj
Pik
−1
l=1 exp(Vil ][exp(Vil )/
l=1 exp(Vil ]
Pil = [exp(Vik )/
That is, the ratio of probabilities are unaffected from any other outside
alternatives. Because they cancel at the ratio.
This imposes restrictions on substitution patterns
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Demand Side
Multinomial Logit
Independence of Irrelevant Alternatives, IIA
The reason for IIA problem is i.i.d errors not the logit itself
The problem is that consumer characteristics are independent of
the observed product
The solution to this problem can be interacting consumer
characteristics with product characteristics
Nested Logit
Mixed Logit
Multinomial Probit
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Empirical Application
Nevo, 2001, Econometrica
RTE industry is a classic example of a concentrated
differentiated-products industry:
high concentration
high price-cost margins
large advertising to sales ratio
aggressive introduction of new products
The paper analyzes price competition by estimating the true economic
price-cost margins (PCM) and distinguishing between three sources:
firm’s ability to differentiate its brands from those of its competitors
portfolio effect (ability to charge more by producing imperfect
substitutes)
price collusion
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Empirical Application
Nevo, 2001, Econometrica
The steps of estimation are:
estimate the demand function (random coefficient logit model)
as a function of observed product characteristics, unobserved
product characteristics, and unknown parameters
Compute PCM implied by three industry structures by using
demand elasticities
single product firms
multi-brand oligopolistic firms
multi-brand monopolistic firm
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Empirical Application
Supply side
Empirical Framework: Supply Side
Suppose there are F firms, each of which produces some subset, Γj , of
the j = 1, . . . , J different brands.
define
Ω∗ = 1 if ∃f : (r, j) ⊂ Γf , Ω∗ = 0
otherwise.
r (p)
Ωjr = Ω∗ ∗ Sjr where Sjr = − ∂s∂p
j
then
s(p) − Ω(p − mc) = 0
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Empirical Application
Supply side
Empirical Framework: Supply Side
Notice the resemblance to the simple Bertrand Model
∂qi
θB = 0 ⇒ Bertrand-Nash. (Pi = MC − qi / ∂P
)
i
θB = −∞ ⇒ Perfect Comp. Pi = MC
∂qi
θB = 1 ⇒ Cartel (Symmetry) (Pi = MC − qi /( ∂P
+
i
∂qi
∂Pj ))
Ownership matrix allows to estimate different market structures.
If none of the products are within the same subset it is
single-product Bertrand-Nash
If some of the products are within the same subset it is
multi-product Bertrand-Nash
If all of the products are within the same subset it is collusion
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Empirical Application
Demand side
Empirical Framework: Demand Side
Multinomial Logit:
The estimation of the demand curves using random coefficient logit
model is not an easy task. To better understand the procedure we first
focus on estimation via multinomial logit.
Let the utility of consumer i from choosing alternative j in town t is
given by the equation:
where i = 1, . . . , N, j = 1, . . . , 50, t = 1, . . . , 50
pjt observed price of product j in town t, xjt is 4 dimensional
observed product characteristics, ξjt is the unobserved product
characteristics, ijt is mean zero error term
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Empirical Application
Demand side
Empirical Framework: Demand Side
Multinomial Logit:
Since the coefficients are the same for all consumers we can obtain an
aggregate utility function
Assuming jt is i.i.d extreme value, the market share of product j is
given by
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Empirical Application
Demand side
Empirical Framework: Demand Side
Multinomial Logit: Elasticities of Demand
sjt =
x β−αp +ξ
jt
jt
e jt
P
xkt β−αpkt +ξkt
1+ 50
1 e
We seek to obtain
sjt =
∂sjt
∂pkt
∂sjt
∂pkt .
Let Gj = e xjt β−αpjt +ξjt so that
Gj
P
1+ 50
1 Gk
=
∂Gj
∂pkt
P
1+ 50
1
Gk
+
−Gj
P
2
(1+ 50
1 Gk )
+
−Gj
P
2
(1+ 50
1 Gk )
∂Gk
∂pkt
If k , j then
∂sjt
∂pkt
∂sjt
∂pkt
=
0
P
1+ 50
1 Gk
=α
Gj
P
(1+ 50
1 Gk )
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(−αGk )
G
Pk
(1+ 50
1 Gk )
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Empirical Application
Demand side
Empirical Framework: Demand Side
Multinomial Logit: Elasticities of Demand
∂sjt
∂pkt
=α
Gj
P
(1+ 50
1 Gk )
G
Pk
(1+ 50
1 Gk )
Products with higher shares are close substitutes regardless of
content
similarly it can be shown that if If k = j then
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Empirical Application
Demand side
Empirical Framework: Demand Side
Random Coefficient Logit
uijt = xj β∗i − α∗i pjt + ξj + ∆ξjt + ijt
individual heterogeneity is captured as deviations from mean value:
α∗i = α + ΠDi + Σvi
β∗i = β + ΠDi + Σvi
vi ∼ N(0, IK +1 )
Note that coefficients are functions of demographic variables and
unobserved component vi
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Empirical Application
Demand side
Empirical Framework: Demand Side
Random Coefficient Logit
Let θ = (θ1 , θ2 ) where θ1 = (α, β) and θ2 = (vec(Π), vec(Σ)) then the
utility can be rewritten as deviations from mean
uijt = δjt (xj , pjt , ξj , ∆ξjt ; θ1 ) + µijt (xj , pjt , vi , Di ; θ2 ) + ijt
δjt = xj βi − αi pjt + ξj + ∆ξjt
Component of utility from alternative j that is same across all
consumers
0
µijt = [pjt , xj ] ∗ (ΠDi + Σvi )
component of utility from alternative j that is individual specific
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Empirical Application
Demand side
Empirical Framework: Demand Side
Random Coefficient Logit: Market Shares
Under the assumption of Type I extreme value for ijt the market share
sijt is given by:
sijt =
δ +µ
e jt ijt
P
δjt +µijt
1+ 50
1 e
R R
∗
ν D dP (D, v, ) =
dP ∗ (v)dP ∗ (D)
δjt +µijt
sjt (x, pt , δt ; θ2 ) =
R R
δ +µ
e jt ijt
ν
D
P
1+ 50
1 e
Integral can be evaluated by simulation to obtain shares. It adds up the
market shares of different types of consumers based on distribution of
types.
Nevo (2000a): “A Practitioner’s Guide to estimation of Random
Coefficients Logit Models of Demand”, Journal of Economics and
Management Strategy, 9, 513-549.
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Empirical Application
Demand side
Empirical Framework: Demand Side
Random Coefficient Logit: Elasticities of Demand
Own–price elasticity, j = k ,
p R R
ηjkt = − sjtjt ν D αi sijt (1 − sijt )dP ∗ (v)dP ∗ (D)
Cross–price elasticity, j , k ,
R R
ηjkt = psktjt ν D αi sijt sikt dP ∗ (v)dP ∗ (D)
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