A Likelihood Approach to Estimating Market Equilibrium Models 1 Michaela Draganska Stanford University Graduate School of Business Stanford, CA 94305-5015 draganska [email protected]Dipak Jain Kellogg School of Management Northwestern University Evanston, IL 60208-2001 [email protected]
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A Likelihood Approach to Estimating Market Equilibrium Models 1
Michaela DraganskaStanford University
Graduate School of BusinessStanford, CA 94305-5015
This paper develops a new likelihood-based method for the simultaneous estimation ofstructural demand-and-supply models for markets with differentiated products. We specifyan individual-level discrete choice model of demand and derive the supply side assumingmanufacturers compete in prices. The proposed estimation method considers price en-dogeneity through simultaneous estimation of demand and supply, allows for consumerheterogeneity, and incorporates a pricing rule consistent with economic theory.
The basic idea behind the proposed estimation procedure is to simulate prices andchoice probabilities by solving for the market equilibrium. By repeating this many times,we obtain an empirical distribution of equilibrium prices and probabilities. The empiricaldistribution is then smoothed and used in a likelihood procedure to estimate the parametersof the model. The advantage of this method is that it avoids the need to perform atransformation of variables. If consumers’ tastes are independent across market periods,our approach yields maximum-likelihood estimates; otherwise it yields consistent but notfully efficient partial likelihood estimates.
Key Words: price endogeneity, competitive strategy, maximum likelihood.
1 Introduction
In recent years marketers have become increasingly interested in estimating structural market
equilibrium models, where demand is derived from utility maximization on the part of con-
sumers, and the supply side is obtained by assuming that firms maximize profits given the
characteristics of the market. Because competitive environment (i.e., market structure) and
policy variables (i.e., marketing mix) are specified explicitly, we can identify separate demand,
cost, and competitive effects. Estimating a market equilibrium model enables us to analyze
questions pertaining to firms’ strategies in the marketplace through “what-if” type analyses by
taking into account all interdependencies between the demand and supply sides of the market.
The simultaneous estimation of demand and supply is also motivated by the so-called
endogeneity problem. In short, endogeneity arises because marketing variables not only affect
consumer choice, but because consumer choice also affects marketing mix decisions. It has
been well documented that ignoring endogeneity leads to biased coefficient estimates of the
marketing mix variables and therefore to suboptimal decisions (Besanko, Gupta and Jain 1998,
Villas-Boas and Winer 1999).
It is often argued that the use of individual-level data solves the endogeneity problem, since
individuals are price takers. However, even though price is exogenous in a microeconomic sense,
there still might be important correlations between the price and the error term in the demand
equation, thus leading to econometric endogeneity (Kennan 1989). Product attributes that
are unobservable to the researcher such as coupon availability, national advertising and shelf
space allocation have an impact on consumer utility as well as on price setting decisions by
firms (Villas-Boas and Winer 1999, Besanko, Dube and Gupta 2003). Prices should thus be
viewed as endogenous independent of the aggregation level of the data used in the analysis.
In this research, we focus on developing a new likelihood based method for the estimation of
structural demand-and-supply models. Our demand model falls into the broad class of discrete
1
choice models of markets for differentiated products (Anderson, de Palma and Thisse 1992).
The supply model is derived from the profit maximization behavior of the firms, assuming
Bertrand-Nash competition in prices between manufacturers. Market equilibrium is determined
jointly by the demand and supply specifications, and our estimation procedure accordingly
considers the equilibrium equations simultaneously.
Once the presence of unobserved product attributes is acknowledged, it is no longer possi-
ble to estimate a discrete choice model using traditional maximum likelihood methods because
in this case prices will be correlated with the unobservables due to the strategic price-setting
behavior of firms (Berry 1994). Therefore, choice probabilities depend on the unobserved prod-
uct attributes not only directly but also indirectly via prices. Hence, one cannot integrate the
unobserved product attributes out of the choice probabilities without taking this latter depen-
dency into account. Berry (1994) proposed a technique for the estimation of discrete choice
models using instrumental variables to account for the endogeneity of prices. His approach
is easy to implement and has been widely applied to the analysis of aggregate data (Berry,
Levinsohn and Pakes 1995, Besanko et al. 1998, Nevo 2001).
Marketing researchers, however, have long recognized the advantages of data describing
the purchase behavior of individual consumers. Such disaggregate scanner panel data provide
detailed information that can be used to learn about their preferences. For example, they enable
us to understand the source of behaviors such as variety seeking or deal proneness. Given the
richness of scanner panel data, a large literature has evolved that uses them to estimate discrete
choice models of consumer behavior (Guadagni and Little 1983, Kamakura and Russel 1989,
Chintagunta, Jain and Vilcassim 1991, Gonul and Srinivasan 1993, Fader and Hardie 1996).
These models have focused on estimating the demand side and have not considered the possible
presence of endogeneity. Recently Goolsbee and Petrin (2003) and Chintagunta, Dube and Goh
(2003) apply variants of Berry’s (1994) method to estimate consumer demand using individual-
2
level choice data. These approaches are useful when the main interest lies in obtaining precise
demand-side estimates because they provide a way to account for price endogeneity without
the need of making assumptions about supply-side behavior. If conducting policy experiments
is our goal, however, then estimating an equilibrium model is preferable, since it enables us to
take advantage of the cross-equation dependencies of the structural parameters.
An equilibrium model provides a mapping from unobserved product attributes and cost
shocks to market outcomes, i.e., prices and choice probabilities. Extending traditional MLE
methods to include a supply side in addition to a consumer choice model is not straightforward
because it requires that the researcher is able to write down the joint distribution of these
equilibrium outcomes. Assuming that this distribution is known runs counter to the notion of
an equilibrium (Berry 1994). Hence, the joint distribution of these equilibrium outcomes needs
to be derived from the distribution of the unobservables. Performing this transformation of
variables proves to be very difficult due to the highly nonlinear nature of the model. Villas-
Boas and Winer (1999) circumvent this problem by estimating a reduced-form pricing rule that
relates current prices to lagged prices. In a subsequent article, Villas-Boas and Zhao (2001)
specify a structural supply-side model derived from manufacturers’ and retailers’ optimization
problem and estimate the equilibrium model directly using maximum likelihood. This direct
approach to estimating the Jacobian, however, prevents them from incorporating consumer
heterogeneity. Recently, Yang, Chen and Allenby (2003) have proposed a Bayesian approach
to resolve the issue.
In this article, we propose a likelihood based approach to the estimation of a structural
demand-and-supply model using individual-level choice data. The basic idea behind the pro-
posed estimation procedure is to simulate prices and probabilities by randomly drawing the
shocks from an assumed joint distribution, and then solve for the equilibrium. By repeating
this many times, we obtain an empirical distribution of equilibrium prices and probabilities.
3
The empirical distribution is then smoothed and used in a maximum-likelihood procedure to
estimate the parameters of the model. The advantage of this method is that it avoids the need
to perform a transformation of variables and thus enables us to estimate the model when the
evaluation of the Jacobian seems infeasible.
In computing the likelihood of the data, we treat market periods as independent from each
other. This implicitly assumes that there is no persistence in the preferences of consumers
across market periods.2 If markets are geographical regions rather than time periods, then
this assumption is warranted. Furthermore, the psychology literature suggests that consumers’
preferences change over time, often depending on contextual effects that are unobserved by the
econometrician (Petty and Cacioppo 1986, Burnkrant and Unnava 1995). To the extent that
this leads to independence over time, our procedure yields maximum-likelihood estimates of
the model parameters. If, on the other hand, there is a correlation in consumers’ preferences
over time, then our procedure yields a so-called partial likelihood (Wooldridge 2002), and the
resulting estimates are consistent but not fully efficient.
The remainder of the paper is organized as follows. Section 2 develops the equilibrium
model. In Section 3, the estimation procedure is described along with the details of the im-
plementation. In Section 4, we apply the estimation method to two frequently purchased
consumer products, yogurt and laundry detergent. We demonstrate the accuracy of the pro-
posed procedure in a Monte Carlo study presented in Section 5. In Section 6 we conclude with
a summary and directions for future research.
2 Model Formulation
2.1 Demand Specification
Brands are indexed by j = 0, . . . , J , and market periods by t = 1, . . . , T . Let household
types be indexed by n = 1, . . . , N , where a type denotes a set of households with identical
4
demographic characteristics. There are mn individuals of type n. To capture unobserved
consumer heterogeneity, we use a latent class approach and specify random coefficients with
an L-point distribution (Kamakura and Russel 1989).3 This specification is appealing in terms
of interpretability for marketing purposes and has been applied both in the economics and
marketing literature (Berry, Carnall and Spiller 1997, Besanko et al. 2003). Let the latent
market segments be indexed by l = 1, . . . , L. The share of segment l in the population is
λl ≥ 0, where∑L
l=1 λl = 1.
Consumer behavior is governed by the following utility function:
un0t = εn0t,
unjt = xnjtβl − αlpjt + ξjt + εnjt,
where {εn0t, . . . , εnJt} are iid extreme value distributed, xnjt are observed characteristics of an
alternative or decision-maker, and pjt denotes the price of alternative j in period t. βl and αl are
the respective response parameters. We allow for household-specific variation in these response
parameters to capture consumer heterogeneity. The demand shocks {ξ1t, . . . , ξJt} are common
across consumers and represent product characteristics that are unobserved by the researcher,
but are taken into account by the firms in their pricing decision. While some unobserved
product characteristics, such as quality and brand image, can be captured through the inclusion
of brand-specific constants, ξjt reflects time-varying factors like coupon availability, shelf space,
and national advertising.
Brand 0 is the outside good (i.e., no-purchase alternative). Including an outside good
allows for category expansion effects of marketing actions. We assume that the outside good
is non-strategic, i.e., its price is not set as a best response to the inside goods.
Utility maximization and the assumptions on the error term imply that the probability of
5
household n purchasing brand j in market period t, Dnjt, is given by
Dnjt =L∑
l=1
λlDnljt =L∑
l=1
λlexp(xnjtβl − αlpjt + ξjt)
1 +∑J
k=1 exp(xnktβl − αlpkt + ξkt)(1)
and the probability of the outside good being chosen is
Dn0t =L∑
l=1
λlDnl0t =L∑
l=1
λl1
1 +∑J
k=1 exp(xnktβl − αlpkt + ξkt). (2)
2.2 Supply Specification
The supply side is characterized by Bertrand-Nash behavior on part of oligopolistic firms (Berry
et al. 1995, Besanko et al. 1998). We assume that retailers pass through the manufacturers’
decisions, which is likely to hold for categories that do not have strategic impact on store traffic
or are a primary driver of retailers’ profits. Under this assumption we do not need to explicitly
include a retailer in the supply-side model.
The production function has constant returns to scale. Marginal costs for firm j in period
t are denoted by cjt. In market period t firm j maximizes profits,
maxpjt
Πjt = (pjt − cjt)N∑
n=1
mnDnjt, (3)
where∑N
n=1 mnDnjt is the expected demand for product j in period t. Expected demand is
thus given by the weighted sum of the choice probabilities for all consumer types in the market.
The first order condition for this problem is given by
N∑
n=1
mn∂Dnjt
∂pjt(pjt − cjt) +
N∑
n=1
mnDnjt = 0. (4)
Given our demand model, the above equation can be rewritten as
pjt = cjt +∑N
n=1 mn∑L
l=1 λlDnljt∑Nn=1 mn
∑Ll=1 λlαlDnljt(1−Dnljt)
. (5)
We infer marginal cost from the data using the relationship
cjt = wtγj + ηjt, (6)
6
where wt are observable variables, e.g. input prices, and ηjt denotes cost characteristics that
are unobserved by the researcher. Substituting (6) in (5) yields
pjt = wtγj +∑N
n=1 mn∑L
l=1 λlDnljt∑Nn=1 mn
∑Ll=1 λlαlDnljt(1−Dnljt)
+ ηjt, j = 1, . . . , J. (7)
2.3 Market Equilibrium
Considering the demand equations (1) and supply equations (7) jointly, the market equilibrium
where Jac is the (2J × 2J) Jacobian of the inverse transformation
∂u1∂p1t
. . . ∂u1∂pJt
∂u1∂D1t
. . . ∂u1∂DJt
...... . . .
......
...∂uJ∂p1t
. . . ∂uJ∂pJt
∂uJ∂D1t
. . . ∂uJ∂DJt
∂v1∂p1t
. . . ∂v1∂pJt
∂v1∂D1t
. . . ∂vJ∂DJt
...... . . .
......
...∂vJ∂p1t
. . . ∂vJ∂pJt
∂vJ∂D1t
. . . ∂vJ∂DJt
.
8
The problem is that the equilibrium equations generally cannot be solved to obtain the in-
verse transformations {uj(·)}j and {vj(·)}j . Moreover, even if we could obtain {uj(·)}j and
{vj(·)}j , e.g., using numerical methods, then we would still have to compute the Jacobian
of this (unknown!) inverse transformation. Due to the highly nonlinear model specification,
this is a daunting task. In a recent article, Yang et al. (2003) propose a Bayesian approach
to estimating market equilibrium models. While the authors simplify the transformation of
variables considerably by transforming the supply shocks into prices conditional on demand
shocks, the computation of the Jacobian still needs to be done using numerical methods.4
Our approach is different. We avoid performing the transformation of variables altogether
and instead obtain equilibrium prices and probabilities using simulation. Recall that, for given
values of the exogenous variables, the joint distribution of demand and supply shocks induces
a distribution of prices and probabilities. We exploit this by numerically solving the model
repeatedly for simulated demand and supply shocks. For each draw of the demand and supply
shocks, we obtain the corresponding equilibrium prices and probabilities. Then we compute
the joint distribution of prices and probabilities.
There is, however, no guarantee that the empirical distribution of prices and probabilities
obtained through the simulation will be smooth, which is a property we need for the optimiza-
tion. We therefore employ nonparametric techniques to estimate the joint density of prices and
probabilities, and then evaluate it at the actual data to obtain a smooth, well-behaved likeli-
hood function. The parameter estimates are obtained by maximizing this likelihood function
using an iterative optimization procedure.
Estimation Algorithm
Based on the previous discussion, there are three main components to the proposed estimation
procedure:
(i) simulation of equilibrium prices and probabilities,
9
(ii) estimation of the joint density of prices and probabilities in order to smooth the likeli-
hood function, and
(iii) maximization of the loglikelihood to obtain the parameters of the model.
The estimation algorithm proceeds as follows. Let s = 1, . . . , S index the simulations per
time period t.
Step 1: Draw {ξjt}j,t, and {ηjt}j,t S times.
Step 2: Choose a starting value for θ.
Step 3: Set s = 1.
Step 4: Set t = 1.
Step 5: Using the sth draw solve
pjt = wtγj
+
∑Nn=1 mn
∑l λl
exp(xnjtβl−αlpjt+ξjt)
1+PK
k=1 exp(xnktβl−αlpkt+ξkt)∑Nn=1 mn
∑l λlαl
exp(xnjtβl−αlpjt+ξjt)
1+PK
k=1 exp(xnktβl−αlpkt+ξkt)
{1− exp(xnjtβl−αlpjt+ξjt)
1+PK
k=1 exp(xnktβl−αlpkt+ξkt)
}
+ ηjt (11)
to obtain {pjt}j . Using {pjt}j calculate Dnjt =∑
l λlexp(xnjtβl−αlpjt+ξjt)
1+PJ
k=1 exp(xnktβl−αlpkt+ξkt).
Step 6: Increase s by 1. If s ≤ S, go back to step 5.
Step 7: Estimate the joint density of the calculated prices and probabilities, ϕ({pjt}j , {Dnjt}n,j), at theactual prices and probabilities to get the period t contribution to the loglikelihood.
Step 8: Increase t by 1. If t ≤ T , then go back to step 5.
Step 9: Update θ to maximize the loglikelihood. If convergence is reached, terminate. Else go back to
step 3.
We now discuss the details of the estimation algorithm step by step.
3.1 Simulation of Equilibrium Prices and Probabilities
The first component of the estimation procedure is the simulation of equilibrium prices and
probabilities. We assume that demand and supply shocks are normally distributed,
Another important factor affecting the performance of the estimation procedure is the
choice of bandwidth (see Section 3). The bandwidth determines the smoothness of the joint
density of equilibrium prices and probabilities, i.e., the likelihood function. Too small a band-
width leads to a likelihood that is not well-behaved and hence makes finding a global maximum
very difficult. Too large a bandwidth, however, may cause the likelihood function to differ
greatly from the true underlying density of equilibrium prices and probabilities. We examined
the sensitivity of the price estimate to the choice of this parameter by looking at bandwidths
that are 1/4, 1/2, 2, and 4 times the normal reference rule bandwidth. Table 7 summarizes
the results for the price coefficient for two different sets of parameters. One set of parameters
22
was generated as above based on preliminary estimates in the peanut butter category (true
value for price is −0.21), the other set of parameters corresponds to the parameter estimates
in the laundry detergent category (true value for price −0.77). It appears that the bandwidth
obtained by the normal reference rule (equation (13)) performs well. Moreover, the precision
of the estimates is not overly sensitive to the choice of the smoothing parameter.
Table 7: Comparison of Monte Carlo results for different bandwidths (price coefficient only).NR is the bandwidth computed from the normal reference rule.
To summarize, our Monte Carlo simulations demonstrate the ability of the proposed estima-
tion procedure to reliably recover the true parameters of an equilibrium model. In particular,
the parameter of interest, namely the price coefficient, is estimated with a very high degree of
precision. The conducted robustness checks indicate that our methodology is fairly robust to
modifications of the distributional assumptions as well as bandwidth selection.
6 Concluding Remarks
In this article we develop a new likelihood-based methodology for the estimation of structural
demand-and-supply models using disaggregate data. Marketing researchers have established a
long tradition of estimating random utility models of consumer demand using maximum likeli-
hood methods. Tying a traditional individual-level choice model such as a logit or probit with
a supply side specification is a non-trivial task. Simply assuming a joint distribution of prices
23
and probabilities is inconsistent with the equilibrium notion. Furthermore, the nonlinearity
of brand choice models makes writing down the joint distribution of equilibrium prices and
probabilities implied by the unobserved demand and supply shocks very challenging.
We solve these problems by simulating equilibrium prices and probabilities and then us-
ing the empirical likelihood of these prices and probabilities to obtain the parameters of the
model. Estimating the demand and supply equations jointly deals with the problem of price
endogeneity and ensures that we obtain reliable estimates of the price response parameter.
Moreover, the estimated structural equilibrium model can be used to perform “what-if” type
analyses (Draganska and Jain 2003).
We apply the proposed algorithm to both real-world scanner data and to simulated data in
order to assess the properties of the estimation method and highlight its merits and limitations.
Overall, the new procedure performs very well. It yields estimates of plausible magnitude when
applied to individual level choice data in several product categories. The conducted Monte
Carlo experiments demonstrate both the accuracy of our method and its robustness.
One of the attractive features of our approach relative to previous research considering
endogeneity in individual-level models (Villas-Boas and Winer 1999, Villas-Boas and Zhao
2001) is the ability to model explicitly the heterogeneity structure of the population. We specify
and estimate a latent class model to incorporate unobserved heterogeneity across households.
In its current form, however, our method cannot readily take into account the panel structure of
the household-level data. That is, if there is a correlation in the tastes of individual households,
our procedure yields a partial likelihood and the estimated standard errors need to be corrected.
Extending the proposed methodology to explicitly incorporate the dependencies in households
choices over time is an important area for future research.14
On the supply side, one might think about the reasonability of the assumed Nash behavior in
prices. Our method does not require any particular assumption about the strategic interactions
24
between firms. A conjectural variation approach or a menu approach to test for different
behavioral assumptions could be employed to reveal the nature of competition in the market.
This is critical because misspecification of the supply side translates into a misspecified system,
thus leading to inconsistent parameter estimates. Future research could also focus on enriching
the supply side by explicitly incorporating the channel structure (Villas-Boas 2001, Sudhir
2001).
In the current analysis we only consider the endogeneity of prices to illustrate the proposed
methodology. Recent studies have suggested, however, that other strategic instruments such as
advertising (Vilcassim, Kadiyali and Chintagunta 1999) and product line length (Draganska
and Jain 2003) should also be considered endogenous. One fruitful venue for future study
would therefore be to apply the estimation procedure developed in this paper to the analysis
of other marketing mix instruments.
In sum, the present research is a first step towards the estimation of a market equilibrium
model with a disaggregate discrete choice model on the demand side and an oligopoly model
on the supply side. The proposed estimation procedure explicitly accounts for the price endo-
geneity problem. It further bears the potential of combining the advantages of simultaneous
estimation of market models with recent developments in incorporating richer heterogeneity
structures and more flexible error specifications in disaggregate models.
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Notes
1The authors wish to thank Arie Beresteanu, Ulrich Doraszelski, Jean-Pierre Dube, Gau-tam Gowrisankaran, Charles Manski, Mike Mazzeo, Brian Viard and participants at the 1999Marketing Science conference in Syracuse for their helpful comments and suggestions. MariuszRabus provided expert research assistance for this project.
2An anonymous referee drew our attention to the fact that our assumption is somewhatsimilar to what Yang and Allenby (2000) call ‘observation’ heterogeneity. Yang and Allenby(2000) define this term in the context of a latent class model as a specification in which thelatent class probabilities depend on observable covariates. This contrasts with ‘household’ or’structural’ heterogeneity, which entails dependence over time.
3The main drawback of a continuous distribution of consumer heterogeneity is its compu-tational complexity, since we need to numerically evaluate multidimensional integrals. Whilethis is also true in standard models (e.g., Berry et al. (1995)), our estimation algorithm isalready computational intense, so we prefer to work with a discrete distribution.
4For a lucid discussion of this approach, see Dube (2003).
5In small samples, most kernel density estimators are biased. Our Monte Carlo results indi-cate that this does not impair the ability of our procedure to recover the structural parametersof the equilibrium model. If unbiasedness is desired, one can use so-called higher-order kernels,which are computationally more demanding.
6Another possibility to obtain a smooth likelihood function has been explored by Ackerbergand Gowrisankaran (2001). The authors make the auxiliary assumption of normal measurementerror that allows them to express the likelihood function in terms of the normal density. Asimilar assumption has also been employed by Viard, Polson and Gron (2002) who estimate anequilibrium model using Bayesian methods (Markov Chain Monte Carlo techniques). Theseapproaches may be problematic if the underlying density of the endogenous variables differssignificantly from a normal density.
7For a thorough treatment the interested reader is referred to Yatchew (1998).
8Details on the estimation procedure available from the authors upon request.
9In the laundry detergent category, we use data for 107 weeks.
10Yoplait only offers single-serving size yogurt. Dannon also carries 16oz and 32oz of plainand vanilla yogurt in addition to single-serving size. It is often argued that these two particularflavors are used for cooking purposes and constitute a different market.
11Details are available from the authors upon request.
12We are grateful to an anonymous referee for bringing this point to our attention.
30
13These numbers are computed from the standard logit specification with no heterogeneity.
14In a recent article, Yang et al. (2003) propose a Bayesian approach to this problem.