Topic

1
Topic 3 Estimating Supply and Demand Systems
with Differentiated Products Fall
2003
Introduction This topic is in some sense the
heart and soul of the NEIO. It is about
estimating supply and demand systems in
industries with differentiated products. In a
lot of ways, the articles were going to be
talking about are related to the NEIO articles
that we talked about in the case of homogeneous
products. Basic idea of estimating parameters of
the demand equation (price elasticities) and
parameters of the cost function, and then
embedding them in an equilibrium concept to
generate predictions about prices and quantities
is the same.
2
In that way, this topic is essentially NEIO for
differentiated products. The catch is that,
unlike the setting with homogeneous products, the
estimation techniques can be pretty complicated.
Estimating a demand function for a good that has
multiple attributes and many potential
substitutes with slightly different
characteristics gets pretty messy. When I took
IO back in prehistoric times, the techniques
were not known. So I am self-taught, and by using
the word taught, I do not mean to imply that I
have learned everything there is to know about
these models. There is still a lot to learn. My
experience suggests that actually doing your own
study with this methodology is the only way to
really learn it.
3
Uses of Demand Systems Major tool for comparative
static analysis. Examples include mergers, tax
changes, the effect of interruptions in supply.
The analysis usually assumes a mode of
competition (typically Bertrand price
competition). The analysis either has cost data
or more typically infers what the costs must be
based on actual data if indeed the mode of
competition assumption is correct. The analysis
can simulate the effects of price changes on
demand and welfare. This is important in
regulated markets The analysis can assess the
benefit of the introduction of new goods
4
Estimating Demand The Product Space
Approach This approach is self-explanatory.
Demand is a function of own price elasticity and
cross-price elasticities. Problems ps
endogenous (not necessarily a problem, since we
can instrument) degrees of freedom- we need NxN
observations , e.g. with 100 car models, we need
10,000 observations at the minimum
5
The Multilevel Budgeting Approach (Deaton and
Muelbauer) helps alleviate the too many
parameters problem. The approach imposes
several restrictions that significantly reduce
the number of parameters to be estimated. A
popular model here is the AIDS model. Well
discuss this model in topic IV (merger analysis).
6
The Characteristic Space Approach The idea is
that the product is just a bundle of
characteristics. The advantage is that demand
will be a function of the products price and a
small number of characteristics. Note that basic
industrial organization models include the notion
of product location, which is a characteristic.
Examples include Hotelling on a line, Salop on a
circle, the vertical differentiation model. New
Empirical Industrial Organization models build on
the basics in these models. While the analyses
have been greatly facilitated by high-speed
computers, many of the basic ideas date back to
Lancaster and McFadden.
7

• Problems with characteristic space analysis
• What are the appropriate characteristics? Were
  going to be talking a lot about automobiles later
  on, so lets think about it in those terms for a
  little while. Appropriate characteristics for
  automobiles might include the engine size, safety
  characteristics, whether the car has an
  air-conditioner, etc. But an automobile has
  hundreds of characteristics. How do you decide
  which are the appropriate ones?
• There is a need for unobserved characteristics
  in estimation. This introduces computational
  problems
• How are new goods and totally new characteristics
  treated?
• The richness of the data and functional forms.
  Often we only have product level data. In such
  a case, we need to estimate the whole
  distribution of preferences from aggregate
  shares, prices, and characteristics. This is
  asking a great deal from the data.

8
Summary of Product Space Approach
vs. Characteristics Space
Approach The best system for a given problem
depends on the issues one wants to address and
the type of data that are available. A small
number of goods, lots of data, and no important
observable characteristics suggests that one
should use the product space approach.
(Estimating the demand for soft drinks, i.e.,
Coke vs. Pepsi.) A large number of goods, and
data on important characteristics suggests that
one should use the characteristic space approach.
(Estimating the demand for automobiles.) A
sufficiently flexible functional form and proper
treatment of simultaneity should generate
reasonable elasticities at points near the center
of the data. Often this is sufficient for our
needs.
9
Discrete Choice Models of Product Differentiation

• Data Requirements Aggregate market shares,
  average prices, and characteristics of products.
• Specify supply and demand system to estimate from
  aggregate price and quantity data. (Newer
  techniques use individual or micro/micro data.
  If youre lucky enough to have such data, I can
  point you to the relevant literature.)
• Embed this system (elasticity, cost) in
  equilibrium concept (such as a Bertrand Nash
  equilibrium) to draw inferences about conduct
  (price-cost margins).
• N.B. - these two steps are identical to NEIO.
• The estimated system can be used make predictions
  about impacts of changes in tariffs, supply
  restrictions due to sanctions, environmental
  regulations, etc.

10
Discrete choice models solve the dimensionality
problem by projecting the N products onto the
space defined by a small number of product
characteristics In the case of automobiles,
characteristics might be size, weight, hp, mpg,
whether the car has air bags, etc. Obtain
aggregate quantity of good j demanded, start by
specifying an individual consumer indirect
utility from consuming good j (consumer i from
product j)
p is the price, x are observed characteristics, ?
is an unobserved product characteristic mean, ?
represents unobserved consumer heterogeneity, and
? represents parameters to be estimated.
11
Consumer i buys product j if
Let
Then the market share of good j is
(
)
(
)
?

x
q
n
n

s
p
x
f
d
,
,

j
v ? Aj

where f(?) is the density of ? in the population.
12

• Comments
• Note that market shares are as good as the
  quantity demanded if we know the size of the
  market.
• To compute the integral, we need to give a
  functional form to utility and say something
  about the distribution of ?. Then we need to
  transform the market share expressions into
  expressions we can actually estimate.
• There are two predominant techniques
• Logit (Nested Logit) Model
• Full Random Coefficients Model

13
Consider a demand equation that relates observed
market shares, sj to market shares predicted by
the model sj sj sj(x,p,?,?) (0) RHS of
(0) is market share function. The presence of ?
raises a difficult econometric problem. The RHS
of (0) contains both prices and product level
errors. Since the unobserved product
characteristics (?) are correlated with prices,
instrumental variables are needed. However ?
enters (0) in a non-linear fashion, making it
difficult to use traditional IV
estimation. Solution Estimating from mean
utility levels
14
Define the mean utility of product j as
follows First suppose that the distribution of
consumer unobservables is known so that market
shares depend only on mean utility levels, so
that sj sj(?). At the true vales of ?, and
market shares, these equations must hold exactly.
That is, since the mean utility levels contain
the aggregate error, ?, the model should fit the
data exactly. The exact fit of the model
conditional on mean utility levels (?) can be
exploited in an estimation procedure Berry shows
that under weak regularity conditions, s s(?)
can be inverted to produce ?s-1(s) and that he
market share function is one-to-one. Hence every
vector of market shares can be explained by one
and only one vector of utility means.
15
The unique calculated vector ?(s) can be used in
a simple estimation procedure. When the density
of ? is known, so that the market share function
depends only on ?, the mean utility levels can be
treated as a known, nonlinear transformation of
the market shares, s. For the true values of
(?,?), ?j(s)xj? - ? pj ?j (0) We can treat
(0) as an estimation equation and use standard
instrumental variable techniques to estimate the
unknown parameters. The fact that is a
transformation of the original data on market
shares is not important.
16

• Example Logit
• Let utility have the following functional form

The last two terms of the above equation are
error terms ?j is the average value of product
j's unobserved characteristics and ?ij is the
same for all consumers and represents the
distribution of consumer preferences around this
mean. The term ?ij introduces heterogeneity and
its distribution determines the substitution
patterns among products. Hence ?ij is the only
element of ?. If (i)?ij is orthogonal to the
rest of the model, (ii) ?ij is distributed iid
extreme value (f(?) exp(-exp(-?)),
17
(1)
Recall that
If we normalize the mean utility of the outside
good (k0) to be 0, and take logs of equation
(1), we get ? as a function of market
shares (2) s0 is the share of the
outside good.
?j
18

To derive (2), let Then ln(sj)?j-
ln(D) ln(s0)?0 - ln(D) - ln(D) (since ?00
by normalization) Combining yields ln(sj) -
ln(s0) ?j
19
Hence the observed ratio of market shares is the
dependent variable. Note that (2) could be
estimated using 2SLS, if we had instruments for
price. Characteristics of other products can be
used as instruments. This is because they are
excluded from the utility function, but they are
correlated with prices via the markups in the
first order conditions. The presence of the
outside good is important otherwise, if all the
prices went up, people would still buy the same
number of cars.
20
The Supply Side Assume that the firms compete
in prices. For simplicity, assume that each firm
sells a single product. This doesnt matter,
because we can easily extend this to
multi-product pricing. It can easily be shown
that the first order condition for firm j
is pj1/?(1-sj) mcj Let Then the supply
equation to estimate is pj 1/?(1-sj) wj?
?j
21
Problem Additive separability of product
characteristics and consumer characteristics
imply that substitution patterns follow market
shares (i.e. consumers substitute towards other
popular products). If Audis and Yugos have the
same market share, a change in the price of a BMW
will have the same impact on each of these two
products, and these two products will have the
same cross-price elasticities. Hence the logit
model itself is only appropriate when its
reasonable to assume that all products are
equally substitutable. For example, the logit
model might be appropriate for the cellular
industry
22
Nested Logit Model In order to overcome the
unreasonable substitution patterns inherent with
the Logit, several authors have employed a
nested logit model. See Goldberg (1995),
Verboven (1996), and Fershtman and Gandal (1998).
The nested logit model gets around this problem
somewhat by creating nests of goods (e.g. luxury
and economy cars) within which the substitution
patterns are like logit across the nests,
however, the substitution patterns are different.
As Berry (1994) notes, the nested logit is
appropriate when the substitution effects among
products depend primarily on predetermined
classes of products. This assumption seems
reasonable in the case of automobiles indeed
industry groups employ a standard classification
system that puts each car in one of the following
groups subcompact, compact, midsize, large, and
luxury/sport, according to its characteristics.
23
The difference between the logit and the nested
logit is that there is an additional variable,
denoted ?, which is common to all products in
group g and has a distribution that depends on ?,
0lt ? lt1.
In this case, the probability of choosing product
j in group g is
where
(3),
24
Gg denotes the set of automobiles of type g, and
? is an additional parameter to be estimated it
measures the degree of substitution among the
products in the classes or groups. If ?0, the
cross elasticities among products do not depend
on the classification in this case, the simple
multinomial logit model (Eq. (1)) is appropriate.
When ?gt0, there is a higher degree of
substitution among cars that belong to the same
group than among cars from different groups. If
? approaches one, the cross elasticity between
any two cars that belong to different groups
approaches zero
25
D01, s01/?g

• sbarg (share of group g) / ?g
• () sj sbarg sbarj/g (where the latter term is
  the share of product j in group g)
• ln(sj) ?j/(1-?)- ?ln(Dg)- ln(Q) (where Q )
• ln(s0)-ln(Q), which implies
• ln(sj)-ln(s0) ?j/(1-?)- ?ln(Dg)
• ln(sbarg ) (1-?)ln(Dg)- ln(Q), ln(s0)-ln(Q),
  which implies
• (2) ln(Dg) ln(sbarg )-ln(s0)/(1-?).
• Plugging (2) into (1) and substituting () yields
  (3) on next page

26

As shown in Berry (1994), Eq. (3) can be inverted
to yield the following equation
(4)
-
where sj/g is the share of product j in group g.
S0 is the proportion of consumers that choose
not to purchase a new car, that is, the
proportion of consumers that choose the outside
good. Since the price and the group share are
endogenous, we can obtain consistent estimates of
?, ? and ? from an instrumental variable
regression. Characteristics of other products
can be used as instruments. (Why?)
27
Specifying the supply side Assume that the firms
compete in prices. This is really straight
forward we just must account for multi-product
firms. Assume there are F firms, each of which
produce a subset ?f of the N total products.
Then firm fs total profits are
The firm maximizes with respect to p (because x
and ? are considered exogenous). FOC are
28
We can stack the N expressions for each product,
noting that
if products r and j are produced by different
firms. With assumptions about the functional form
of the marginal cost function, this expression
can be estimated along with the demand equations.
Instruments come from demand characteristics
(xj) and cost characteristics (wj). Letting
Z(x,w), estimation is based on the assumption
that E(error/Z)0.
29
Back to the Nested Logit Since
Oligopoly pricing FOC conditions
are
s
-
)
1
(
(5)
u
g
w



p
,

å
å
j
j
s
s
-
-

/
)
1
(
/
1-
a
M
q
Q
q
j
k
g
k
Î
Î
f
k
f
k
g
g
where fg represents the set of products that
firm is selling in group g, Qg is the total
number of sales in group g, and M is the
potential market. The last term on the right
hand side is endogenous, suggesting that
instrumental variables are also needed in order
to estimate the pricing equation.
30
The model to be estimated consists of the demand
and the pricing equations Can estimate this two
equation system using the general method of
moments (GMM). GMM estimation is attractive for
the following reasons (i) the unobserved demand
characteristics, ?j , and the unobserved cost
characteristics, ?j, might be correlated (ii) ?
and ? appear in both equations (iii) the
equations are not linear in ? and ?. (iv) other
methods require structure on the correlation
between the error terms of the demand and
oligopoly pricing equation. Finally, GMM, which
is an iterative procedure (like MLE), is
preferable to 3SLS, which essentially involves a
single iteration.
31
Estimating the Full Random Coefficients
Model Consider a standard discrete choice model
in which consumer utility is given by where uij
represents idiosyncratic shocks to consumer is
preferences that has some distribution that
depends on parameters ?. It is convenient to
define the mean utility of alternative j to be
so that the utility to i from alternative j can
be written We refer to the ? as the linear
parameters and the ? as the non linear parameters.
32
All that matters for the computation of these
predicted probabilities are the distributional
parameters and the mean utilities, not its
individual components at this point. In the case
of the logit model, these predicted shares would
be given by
for each good j where the mean utility of the
outside alternative has been normalized to 0.
Note that these shares are computed at the
initial guess. The key here is that these
predicted market shares can be computed for any
guess of the mean utilities (at least any
reasonable guess).
33
Now we wish to set the observed market shares
equal to those predicted by the model. However,
unless the initial guess is very lucky, it
generally will be the case that
Thus, we need to update our guess for the mean
utilities. BLP suggest the contraction mapping
So we would take the initial guess of mean
utilities, compute the predicted market shares
and update to a new guess via
34
We would then iterate until the contraction
mapping converges. Once convergence has been
reached, we have found the value of the mean
utilities that solves the system of equations
Let the value of the mean utilities that solve
these equations be denoted and note that
these mean utilities are implicitly functions of
the observed shares and distributional
parameters, that is, .
Now we have the value of the mean utilities that
set the observed shares equal to the predicted
shares. However, we know that
35
But we want to find the value of that minimizes
the GMM objective function given This GMM
problem reduces to the problem of solving Where
Z is the matrix of exogenous right hand side
variables and instrumental variables and W is a
symmetric positive definite weighting matrix,
which is a consistent estimate of the asymptotic
variance-covariance matrix of the moment
conditions. This has a closed form solution for
the linear parameters given by
36
We then compute the implied values of the
unobservables, i.e., We then evaluate the GMM
objective function We would then update the
value of the distributional parameters using an
optimization algorithm and repeat this procedure
until the GMM objective function is minimized.
37
Summary
38
Berry Levinsohn, and Pakes EMA 1995)
U( ) ?j ? ij

• is a parameter, xjk is the kth element of xj, ?
  is random variable describing consumer is taste
  for characteristic k. So, this would be my taste
  for mpg versus your taste, which may be a
  function of our taste for style, our incomes,
  etc.
• The specification is particularly tractable if ?
  ij is i.i.d. with the extreme value distribution.
• Key point ? ij depends on the interaction
  between consumer preferences and product
  characteristics.

39
The coefficient on the kth product
characteristics is E(?ik)0, V(?ik)1 Now
consumers who purchase good j say a Cadillac
probably have lower than mean valuation for mpg,
so that if the price of a Cadillac changes they
will substitute towards cars with similar
characteristics, and not, even if they have the
same market shares, towards a Yugo.
40
Since E(?ik2)1, the mean and variance of the
marginal utilities associated with characteristic
k are bk and sk2 respectively. Now, we cannot
analytically calculate the integral (see slide
7). To account for the fact that ?j is
correlated with pj we would want to do IV, but
its a big problem since the expression would be a
nonlinear function of the error term. As
mentioned, Berry shows how you can invert the
market-share equation to get something that is
linear in the unobserved product
characteristics. Let Berry shows that s s(?)
can be inverted to produce ?s-1(s)
41
Fortunately, the ? function is unique. This
can be estimated using instrumental variables
techniques (NL2SLS or GMM). (You calculate ? by
numerically aggregating over observations) Note
that you can use the distribution of consumer
incomes across markets to enhance the estimation
of ?. Note that BLP (and others) who use these
models still assume that the error term ?ij is
Weibull in order to ease the numerical
difficulties of having many products in the
demand system. Note that BLP use a slightly
different functional form than that appearing in
equation (1), since they have additional
information on the distribution of income across
households.
42
Comments Results may not be robust to changes in
the set of instruments (which are just
characteristics of other products) if no cost
shifters. The model is identified off of a
(restrictive) functional form Exogenous product
characteristics From an empirical perspective,
the model does not account for company or
segment effects.
43
BLP data and results Data set includes nearly
all models sold during 1971-1990 period in
U.S. Unbalanced panel with 2217 observations (997
distinct models). Product characteristics
horsepower/weight, size, fuel efficiency (MP or
MPG), air conditioning. Table III Logit and OLS
(But no nested logit) Table IV (BLP estimation
(I) CRS (II) allowing for IRS Table VI Cross
Price Elasticities Table VIII Price-Cost
Markups
44
The importance of instruments
45
CRS IRS
Term on price different than (1)
46
(P-MC)/P 15.8 21.4 26.1 29.3