1 Objective 1. Study of the SPML estimation method

Planned activities: Simulations for the model parametersPromised results: Simulation results

In this phase of the project we intended to produce results regarding the perfor-

mance of the semi-parametric maximum likelihood estimator (SPML) proposed by Sán-

dor (2012) �that is a practical version of the estimator developed by Ai (1997)� in

comparison with the reference estimator of these models that is the generalized method

of moments (GMM) proposed by Berry, Lensohn si Pakes (1995, henceforth BLP). These

results are intended to complement our results reported in Sándor (2012). We recall that

that the main objective is to demonstrate that the SPML estimator of the BLP model

(described in Sándor 2013, downloadable from the web page of the project) has �nite

sample properties better than the GMM estimator. The performance is measured in the

mean squared error of the estimators, which is computed by Monte Carlo simulations.

The results of the Monte Carlo study are presented in the table below. For the exact

de�nition of the parameters presented in the leftmost column we refer to Sándor (2013).

In the upper block of the table, in each market the prices are determined as the Nash

equilibrium of a pricing game between two �rms, of which the �rst has 2 and the second

has 3 out of the 5 products in the market. The marginal cost is speci�ed as linear as a

function of the cost characteristics. For the demand parameters we use true values for

�, �� and �1 di¤erent from those in Section 5.2 because these values allow the numerical

computation of the Nash equilibrium of prices. In the lower block of the table the prices

depend exponentially on the price-shifters. In both cases the unobserved characteristics

have the normal distribution.

We notice from the table that the GMM estimators can have large bias and in general

have large standard deviation; consequently, most of their RMSEs are again above 1:5.

The performance of the SPML estimators is di¤erent in the two cases considered. In the

�rst case, when the price equilibrium is misspeci�ed, the SPML estimators are virtually

unbiased with low standard deviations. The SPML estimator is clearly better than

the GMM estimators. In the second case, when the pricing function is misspeci�ed,

the SPML estimators have a small bias, and most of their RMSEs are below 0:5. The

SPML estimator is again clearly better than the GMM estimators. We can conclude

that SPML performs well under misspeci�cation of the pricing equation. Regarding the

question whether the additional supply side information is responsible for the substantial

di¤erence in the performance of GMM and SPML, we can claim that it does not appear

to be so in the case when the price equilibrium is misspeci�ed. In the case when the

pricing function is misspeci�ed the performance di¤erence between the two estimators

1

is still large, but slightly smaller than in the other cases.

Table 1. Performance of GMM and SPML estimatorsT = 50 GMM GMM-OI SPMLTrue Bias StD RMSE Bias StD RMSE Bias StD RMSE

Prices are determined as a Nash equilibrium� = -5.0 0.752 1.649 1.812 0.295 2.743 2.758 -0.097 0.244 0.262�� = 1.0 -0.045 0.615 0.616 0.129 0.871 0.881 0.006 0.068 0.069�1 = -2.0 1.437 3.156 3.468 1.035 2.224 2.453 0.030 0.348 0.349�2 = 2.0 -1.392 1.514 2.056 -1.057 1.375 1.735 -0.066 0.244 0.252�3 = 2.5 -1.570 2.230 2.727 -1.267 1.909 2.291 -0.055 0.296 0.301�1 = 2.5 -0.113 2.321 2.323 0.370 3.318 3.338 -0.046 0.262 0.266�2 = 2.0 0.032 1.442 1.442 0.158 1.498 1.506 0.017 0.242 0.243�3 = 2.0 0.251 1.649 1.668 0.064 1.109 1.111 0.081 0.151 0.172

Prices are exponential functions of cost characteristics� = -3.0 1.444 1.202 1.879 0.914 1.863 2.075 0.292 0.435 0.524�� = 1.5 -0.601 0.864 1.052 -0.221 1.299 1.318 -0.179 0.187 0.259�1 = -5.0 -0.717 1.483 1.648 0.014 1.807 1.807 -0.009 0.466 0.466�2 = 2.0 -0.658 1.028 1.220 -0.737 1.010 1.250 -0.288 0.368 0.467�3 = 2.5 -0.902 1.329 1.606 -0.957 1.520 1.796 -0.344 0.406 0.532�1 = 2.5 0.036 2.351 2.351 0.075 2.889 2.890 -0.167 0.395 0.429�2 = 2.0 -0.346 1.351 1.395 0.322 1.715 1.745 -0.061 0.276 0.283�3 = 2.0 -0.394 1.750 1.794 0.305 2.037 2.060 0.020 0.367 0.367

The above table contains the promised results. These results are very promising given

the importance of the problem and the substantial di¤erence between the performances

of the two estimators.

References

[1] C. Ai, A semiparametric maximum likelihood estimator, Econometrica 65, 933�963,

1997.

[2] Steven Berry, James Levinsohn and Ariel Pakes, Automobile Prices in Market Equi-

librium,�Econometrica, 63, 841�890, 1995.

[3] M. Reynaert and F. Verboven, Improving the Performance of Random Coe¢ cients

Demand Models: the Role of Optimal Instruments, Working Paper, 2013.

[4] Z. Sándor, Semiparametric Maximum Likelihood Estimation of Di¤erentiated Prod-

uct Demand Systems, Working Paper, 2012.

[5] Z. Sándor, Semiparametric Maximum Likelihood Estimation of Di¤erentiated Prod-

uct Demand Systems, Working Paper, 2013.

2

2 Objective 2. Study of price equilibrium and the�xed point algorithm

Planned activities: Study of literature on price equilibriumPromised results: Literature review of price equilibrium

2.1 Introduction

First we present the general problem and mention some important special examples. In

order to solve the problem we propose two theories, as follows. In Section 2 we present

the literature that treats the price equilibrium problem as a vector equilibrium problem.

In Section 3 we present the literature that treats the price equilibrium problem as a

correlated equilibrium problem.

Suppose that there are J products in a market denoted by 1; :::; J . For j 2 f1; :::; Jg,let pj, sj and cj denote the price, market share and marginal cost of product j, respec-

tively. Suppose that for each j, sj : (0;1)J ! (0; 1) is a continuously di¤erentiable

function of p = (p1; :::; pJ)0, where x0 denotes the transposed of vector x. We assume

thatPJ

j=1 sj (p) < 1 for any p 2 (0;1)J , that is, there is an outside product withstrictly positive market share.

Suppose further that the J products are produced by F �rms and each �rm f 2f1; :::; Fg sells a subset Gf of the J products. Let pf be the column-vector of own

prices, that is, of those components pj of p for which j 2 Gf ;1 let p�f be the column-vector of the rest of the components of p. Each �rm f determines its own vector of

prices pf such that its pro�t

�f (pf ;p�f ) =Xj2Gf

(pj � cj)sj (p)

is maximized, given p�f . The resulting price vector is a Nash equilibrium.

If a Nash equilibrium exists, then it satis�es that

@�f (pf ;p�f )

@pf= 0;

which is equivalent to

sj(p) +Xh2Gf

(ph �mch)@sh@pj

(p) = 0: (1)

1In general, we use the notation that boldface characters are column-vectors and if they have sub-script f then they refer to the column vector corresponding to the products of �rm f .

3

In matrix form this is the same as saying that p solves the system of equations

p = c+�(p)�1s(p); (2)

where the matrix �(p) is block-diagonal and can be written as

�(p) =

264 �1(p) 0. . .

0 �F (p)

375 ;and �f (p) = �

@sf (p)

@p0ffor f = 1; :::; F . Equation (2) determines a �xed-point equation,

which may be used to show that a unique Nash equilibrium exists.

2.1.1 Special case: The random coe¢ cient logit

The most commonly used version of the model has the market share

sj (p) =

ZRK+1

exp��pj + xj� + �j

�1 +

PJh=1 exp (�ph + xh� + �h)

dF (�; �) ; (3)

where xj 2 RK , �j 2 R, for j = 1; :::; J , and F is a cumulative distribution function.

The interpretation of �; � is that they are random coe¢ cients of pj and xj.

Conjecture 1 If the marginal distribution of � has support included in (�1; 0), thenthere is a unique Nash equilibrium.

Proposition 2 If the support of the marginal distribution of � includes some intervalof positive values, then there may be multiple Nash equilibria.

Allon et al. (2010) show uniqueness of price equilibrium, if the coe¢ cient of price

is non-random (in fact is a product-dependent parameter) and the distribution of � is

discrete. The key of their proof is a condition by which they exogenously bound the

prices. Their results are challenging because they also prove as a side result that the

pricing game is supermodular, which is not true for the version of the model for which

both �, � are non-random (Sándor 2001).

2.1.2 Special case: The simple logit

In this case in (3) both �, � are non-random and

sj (p) =exp

��pj + xj� + �j

�1 +

PJh=1 exp (�ph + xh� + �h)

:

Then � < 0 is a su¢ cient condition for the existence of a unique equilibrium (Konovalov

and Sándor 2010). The proof can be based on either Kellogg�s (1976) �xed point result

or the Gale and Nikaido (1965) univalence theorem.

4

2.1.3 Special case: Single-product �rms

In this case each Gf is a singleton. The most important contribution is Caplin and

Nalebu¤ (1991), who prove existence of equilibrium for a class of models that include

(3) in the case where � is non-random and � is random with a log-concave density.

Their proof is based on Prékopa-Borel results on the quasi-concavity of probabilities.2

For a logit model with � non-random and � random with a discrete distribution,

Allon et al. (2011) and Aksoy-Pierson et al. (2013) prove that, if the market share is less

than 1=2 and the prices are bounded by some value, then there is a Nash equilibrium.

Further, if the market share is less than 1=3, then there is a unique Nash equilibrium

of prices. The proof of existence is based on reducing the interval of candidate price

equilibria and establishing the concavity of the pro�t function on the reduced price

interval.

Another line of research had been by establishing supermodularity conditions for

the pricing game and applying Tarski�s �xed point theorem (e.g., Milgrom and Roberts

1990). This has turned out to be too restrictive for the model in (3) because according to

Sándor (2001) in the simple logit model the pricing game cannot be either supermodular

or log-supermodular.

Mizuno (2003) extended the Caplin and Nalebu¤ (1991) results by adding some

weak supermodularity conditions for uniqueness. The key condition used is that the

best response function of any �rm is increasing.

References

[1] G. Allon, A.Federgruen, M. Pierson, Price competition under multinomial logit

demand functions with random coe¢ cients, Working paper, Dartmouth College,

Hanover, (2010)

[2] G. Allon, A.Federgruen, M. Pierson, Price competition under multinomial logit

demand functions with random coe¢ cients, Working paper, Dartmouth College,

Hanover, (2011)

[3] M. Aksoy-Pierson, G. Allon, A.Federgruen, Price competition under mixed multino-

mial logit demand functions, Management Science, 59 (2013) 1817-1835.

[4] A. Caplin and B. Nalebu¤, Aggregation and imperfect competition: on the existence

of equilibrium, Econometrica 59, 25�59 (1991)

2Quasi-concavity fails when �rms produce multiple products. For example, Hanson and Martin(1996) provide an example of a model with simple logit market shares in which the pro�t function of a3-product �rm is not quasi-concave.

5

[5] D. Gale, H. Nikaido, The Jacobian matrix and global univalence of mapping. Math

Ann 159, 81�93 (1965)

[6] W. Hanson, K. Martin, Optimizing multinomial logit pro�t functions. Manage Sci

42, 992�1003 (1996)

[7] R.B. Kellogg Uniqueness in the Schauder �xed point theorem. Proc Am Math Soc

60, 207�210 (1976)

[8] A. Konovalov and Z. Sándor, On price equilibrium with multi-product �rms, Econ

Theory 44, 271�292 (2010)

[9] P. Milgrom and J. Roberts, Rationalizability, Learning, and Equilibrium in Games

with Strategic Complementarities, Econometrica, 58, 1255�77.(1990)

[10] T. Mizuno, On the existence of a unique price equilibrium for models of product

di¤erentiation. Int J Ind. Organ 21, 761�793 (2003)

[11] J. Nash. Noncooperative games. Ann Math 54, 286-295, 1951.

[12] Z. Sándor, Computation, e¢ ciency and endogeneity in discrete choice models,

Labyrint Publication, The Netherlands, (2001)

2.2 Price equilibrium as vector equilibrium

The price equilibrium problem is a Nash equilibrium problem, which is a special case of

a vector equilibrium problem. Let (D; f) be a game with n players, where Dj are the

strategy sets of the players, D = D1�� � ��Dn and Fj : D ! R are the pro�t functions.The strategy (x1; x2; :::; xn) 2 D Nash equilibrium point if

Fj (xj; x�j) � Fj (yj; x�j) for any yj 2 Dj:

If f (x; y) = (f1 (x; y) ; :::; fn (x; y)) where

fj (x; y) = Fj (xj; x�j)� Fj (yj; x�j)

for any x = (x1; x2; :::; xn) ; y = (y1; :::; yn) 2 D and C = Rn+, then the Nash equilibriumproblem coincides with the next vector equilibrium problem:

Find x 2 D such that f (x; y) 2 C; 8y 2 D:

In order to establish necessary conditions for the existence of the solution, Ansari and

Flores Fazan (2005) use the so-called recession method; in the proof Gong (2006) employs

6

the separation theorem for convex sets; Long, Huang and Teo (2008) the Kakutani�

Fan�Glicksberg �xed point theorem. Lin (2005) introduces several notions of maximal

pseudomonotonicity, which he used for obtaining existence results. Fu and Wang (2013)

examine the strong vector equilibrium problems with domination structures.

Gong (2007) studies symmetric strong vector quasi-equilibrium problems, and Yang

and Pu (2013) analyze the system of strong vector equilibrium problems.

References

[1] Ansari Q.H., Flores-Bazán F. , Recession methods for generalized vector equilibrium

problems, J. Math. Anal. Appl. 321, 132-146 (2006).

[2] Fu, J., Wang, S.: Generalized strong vector quasi-equilibrium problem with domi-

nation structure. J. Glob. Optim. 55, 839-847 (2013)

[3] Gong, X.H.: Strong vector equilibrium problems. J. Glob. Optim. 36, 339-349 (2006)

[4] Gong, X. H.: Symmetric strong vector quasi-equilibrium problems. Math. Meth.

Oper. Res. 65, 305�314 (2007)

[5] Lin, L.J.: Existence Results for primal and dual generalized vector equilibrium prob-

lems with applications to generalized semi-in�nite programming. J. Glob. Optim. 33,

579-595 (2005)

[6] Long, X.J., Huang, N.J., Teo, K.L., Existence and stability of solutions for gener-

alized strong vector quasi-equilibrium problem. Math. Comp. Modelling 47, 445�451

(2008)

[7] Yang, Z., Pu, Y.J, On existence and essential components for solution set for system

of strong vector quasi-equilibrium problems. J Glob Optim 55, 253�259 (2013)

2.3 Price equilibrium as correlated equilibrium

Let G denote a �nite noncooperative game, let n be the number of players. Each player

k � n has a �nite set of strategies, Sk, with jSkj � 2. The utility or payo¤ function ofplayer k is a function uk. The set

S =nYk=1

Sk

7

is called the set of all joint strategy pro�les, andN = jSj denote the number of outcomes.Further, let

S�k =

nYq=1;q 6=k

Sq:

Let uk (s) be the payo¤of player k when the joint strategy s is played, and let uk (dk; s�k)

denote the payo¤ to player k when he chooses strategy dk 2 Sk and the others adhereto s:

The game G is non-trivial if there exist a player k with s 2 S and dk 2 Sk such thatuk (s) 6= uk (dk; s�k) :Correlated equilibrium is a generalization of Nash equilibrium that allows the prob-

abilities to be arbitrarily correlated in the strategy spaces. For the �rst time it was

formulated by Aumann (1974), as follows:

A correlated equilibrium of G is a distribution p on S such that for all players

k and all strategies i; j 2 Sk the following is true:

Conditional on the k-th component of strategy pro�le drawn from p being

i, the expected utility for player k of playing i is not smaller than that of

playing j, i.e. Xs2S�k

ukispis �Xs2S�k

ukjspjs: (4)

The condition that p is a joint distribution on S implies an N � 1-dimensionalsimplex, henceforth denoted by �, consisting of all probability distributions on joint

strategies. The set of all correlated equilibrium distributions determined by (4) is a

convex polytope, henceforth denoted by C, which is a proper subset of �, if the game

is non-trivial. The polytope C is of full dimension if it has dimension N � 1; the sameas �.

The set I of all joint probability distributions that are independent between players is

de�ned by a system of nonlinear constraints, that is, I is the set of all joint distributions

p on S for which there exists a marginal probability distribution xk on Sk such that

p =Qnk=1 xk.

The geometric relationship between the Nash equilibrium and the correlated equi-

librium is formulated in Nau et al (2003). In any �nite, non-trivial game, the Nash

equilibria are on the boundary of the correlated equilibrium. If the polytope is of full

dimension, the Nash equilibria are on its relative boundary. Speci�cally:

(i) In any �nite, non-trivial bimatrix game, if (u; v) is a Nash equilibrium

strategy, then the joint distribution p, where pij = uivj (i = 1;m; j = 1; n)

is a correlated equilibrium.

8

(ii) In any �nite, non-trivial bimatrix game, if the joint distribution p is a

correlated equilibrium and there exists a marginal distribution (u; v) such

that pij = uivj (i = 1;m; j = 1; n), then (u; v) is Nash equilibrium strategy.

Example 3 The chicken games has 2 players (think of them as very competitive driversspeeding from di¤erent streets to an intersection), each with two strategies: S1 = S2 =

fStop;Gog. The utilities, tabulated below, re�ect the situation (in the format u1(s); u2(s),where the strategies of player 1 are the rows and of player 2 the columns):

Stop GoStop 6,6 2,7Go 7,2 0,0

In a 2 � 2 game, � is a 3-dimensional tetrahedron, I is a 2-dimensional saddle andC is a 3-dimensional convex polytope (see Figure 1).

In the chicken game the correlated equilibrium is a distribution

p =

�p11 p12p21 p22

�that satis�es the following inequalities:8>>>>>><>>>>>>:

6p11 + 2p12 � 7p11 + 0p127p21 + 0p22 � 6p21 + 2p226p11 + 2p21 � 7p11 + 0p217p12 + 0p22 � 6p12 + 2p22p11 + p12 + p21 + p22 = 1

pij � 0; i; j = 1; 2

(5)

The following �ve distributions are correlated equilibria in the chicken game

SG =

�0 10 0

�; GS =

�0 01 0

�;MNash =

�4=9 2=92=9 1=9

�

p1 =

�2=4 1=41=4 0

�; p2 =

�0 2=52=5 1=5

�The �rst two are pure Nash equilibria, the third is a mixed strategy Nash equilibrium.

The fourth and �fth are only correlated equilibria. Figure 1 shows the geometry of this

game.

The problem of determining a Nash equilibrium is a combinatorial problem; it im-

plies the identi�cation of a support for each player. None of the known algorithms for

determining the supports of the strategies are known to be e¢ cient (in the sense of

determining the supports in a polynomial number of steps). This problem is studied by

Jiang et al. (2013); Papadimitriu et al. (2008).

9

Figure 1: The geometry of Example 3: The polytope is the symplex of correlatedequilibria (the set C); the sadle is the set of distributions independent between players(the set I); and the three intersection points (SG, GS, MNash) are Nash equilibria.

Recently, in the literature there have appeared a number of generalizations of the

notion of correlated equilibrium (Forgo, 2011). Stein et al. (2011) generalized this

concept for games in which each correlated equilibrium has a continuous utility function.

It is interesting to note that the logic for determining the distribution of correlated

equilibria for these games is similar to the estimation algorithm for the BLP model.

The operator equilibrium problems can help the study of existence of correlated equi-

librium points. Domokos and Kolumbán (2002) introduced and studied a class of oper-

ator variational inequalities. The importance of these operator variational inequalities

stems not only from the fact that they include scalar and vector variational inequalities

as special cases, but they are also interesting as a problem on its own. Inspired by this

work, Kum and Kim (2005, 2007) extend the problem of operator variational inequali-

ties from the single-valued to the multi-valued case. The operator equilibrium problems

were studied by Kazmi and Raouf (2005), Kum and Kim (2008).

References

[1] R. J. Aumann. Subjectivity and correlation in randomized strategies. Journal of

Mathematical Economics, 1: 67-96, 1974.

[2] A. Domokos, J. Kolumbán, Variational inequalities with operator solutions, J. Global

Optim. 23 (2002) 99-110.

[3] F. Forgo. Generalized correlated equilibrium for two-person games in extensive form

with perfect information. CEJOR 19, 201-213, 2011.

10

[4] A. X. Jiang, K. Leyton-Brown. Polynomial-time computation of exact correlated

equilibrium in compact games. Games and Economic Behavior, 2013.

[5] K.R. Kazmi, A. Raouf, A class of operator equilibrium problems, J. Math. Anal.

Appl. 308 (2005) 554-564.

[6] S. Kum, W.K. Kim, Generalized vector variational and quasi-variational inequalities

with operator solutions, J. Global Optim. 32 (2005) 581-595.

[7] S. Kum, W.K. Kim, Applications of generalized variational and quasi-variational

inequalities with operator solutions in a TVS, J. Optim. Theory Appl. 133 (2007)

65-75.

[8] S. Kum, W.K. Kim, On generalized operator quasi-equilibrium problems, J. Math.

Anal. Appl. 345 (2008) 559-565.

[9] R. Nau, S. G. Canovas, P. Hansen. On the geometry of Nash equilibria and correlated

equilibria. Int. J. Game Theory 32, 443-453, 2003.

[10] C. H. Papadimitriou, T. Roughgarden. Computing correlated equilibria in multi-

player games. J. ACM 55(3), 14, 2008.

[11] N. D. Stein, P. A. Parrilo, A. Ozdaglar. Correlated equilibria in continuous games:

Characterization and computation. Games and Economic Behavior 71, 436-455, 2011.

[12] N. D. Stein, A. Ozdaglar, P. A. Parrilo. Structure of extreme correlated equilibria:

a zero-sum example and its implications. LIDS Technical Report 2929, January 27,

2011.

11

3 Objective 2. Study of price equilibrium and the�xed point algorithm

Planned activities: Study of literature on the �xed point algorithmPromised results: Literature review of the �xed point algorithm

From an algorithmic point of view the BLP estimation algorithm contains two loops:

an exterior loop for estimating the model parameters and an interior loop for deter-

mining the unobservables (i.e., the product characteristics that are unobservable to the

researcher). These unobservables (�) are the �xed points of the following discrete dy-

namical system:

�k+1 = �k + ln (~s)� ln (s (�k)) ; (6)

where s is the vectorial function of the market shares with component j given by

sj (�) =1

N

NXi=1

exp f�pj; xj; �j; �i

�1 +

JPr=1

exp f (pr; xr; �r; �i)

; (7)

and ~s the observed market share vector.

Reynaerts et al. (2012) and Dubé et al. (2012) study the �xed point algorithm given

in (6).3 These authors draw the conclusion that the numerical errors from the interior

loop have a substantial e¤ect on the errors of the estimates obtained in the exterior loop.

For example, in order to obtain the estimates with precision 10�6, the unobservables

need to be approximated in the interior loop with precision 10�14 (Dubé et al. 2012).

This fact increases the computing time of the interior loop substantially.

In order to eliminate this drawback Reynaerts et al. (2012) propose four other

algorithms for the inner loop, which were developed for solving nonlinear systems of

equation: the Newton-Raphson method, the Quasi-Newton � Broyden method, the

spectral method and the quadratic method. They �nd that the latter two algorithms

outperform (6) in a signi�cant number of situations. Our numerical results obtained

in MATLAB also con�rm this �nding. More precisely, we have found that if the sum

of the market shares of all products is less than 0:5, then the quadratic method and

the spectral method indeed have a better performance regarding the computing time

and the number of iterations. If the sum of the market shares is close to 1, then the

arguments in the exponentials become large and this slows down the computations. In

this case we have found that the spectral method has the best performance.

3We have also veri�ed the numerical �ndings of the Reynaerts et al. (2012) paper through a MAT-LAB implementation of the algorithms proposed.

12

From a theoretic point of view the Quasi-Newton �Broyden method is remarkable

because in addition to the solution it also determines the Jacobian of the dynamical

system iteratively. This feature can help us study the stability of the �xed point �.

For the analysis of nonlinear �xed point problems an initial step is �nding the �xed

points of the system and the analysis of the stability around the �xed points. The nature

of the stability around the �xed point can be determined by analyzing the eigenvalues

of the Jacobian matrix. In order to determine the stability of a �xed point �, we intend

to analyze the location of the eigenvalues of the Jacobian matrix in the complex plane

with respect to the unit circle. If the eigenvalues of the Jacobian matrix are either

inside or outside the unit circle, the corresponding �xed point is called hyperbolic. A

hyperbolic �xed point is called saddle if one part of the corresponding eigenvalues are

inside and the rest outside the unit circle. A hyperbolic �xed point is called sink node

if all the corresponding eigenvalues are inside the unit circle. A hyperbolic �xed point

is called source node if all the corresponding eigenvalues are outside the unit circle. If

at least one eigenvalue is on the unit circle, then the corresponding �xed point is called

nonhyperbolic (Nayfeh and Balachandran, 2004).

Applying the Hartman-Grobman theorem for the �xed points of mappings, we can

say that linearizing the mappingH (�) = �+ln (~s)�ln (s (�)) is su¢ cient for determiningthe stability of the �xed point �. This way, if all the eigenvalues of the Jacobian matrix

are in the interior of the unit circle, we say that the �xed point � is asymptotically stable,

while if at least one eigenvalue is outside the unit circle, we say that the �xed point �

is unstable. If none of the eigenvalues of the Jacobian is outside the unit circle, the

linear analysis is no more su¢ cient for determining the stability of the nonhyperbolic

�xed point, and therefore, it is necessary to introduce nonlinear terms (Nayfeh and

Balachandran, 2004).

Berry and Pakes (2007) prove that if the function s satis�es some monotonicity,

continuity, linearity and substitutability conditions, then for any vector of market shares

~s in which the sum of the components is strictly less than one, the dynamical system (6)

has a unique �xed point. Berry et al. (1995) prove that this is a sink node (contraction).

We have veri�ed this fact for some speci�c numerical examples by determining the

eigenvalues of the Jacobian approximated by the Quasi-Newton �Broyden method.

The nonlinear dynamic analysis has a very important role for providing new infor-

mation regarding the behavior of complex dynamical systems. Some of the methods

employed are: spectral Fourier analysis, drawing of histograms, reconstruction of the

state space, construction of Poincaré mappings, computation of Lyapunov coe¢ cients,

construction of recurrence mappings and of wavelet mappings. Out of these methods

the reconstruction of the attractors of the dynamics system is one of the most interesting

13

and mostly used. These methods are described and illustrated in the book by Nayfeh

and Balachandran (2004).

This type of analysis has a special importance in the case of the pure characteristics

model proposed by Berry and Pakes (2007). This model has some appealing features

compared to the BLP model.4 One appealing feature is that the substitution structure

of the products is less restricted in the pure characteristics model; another appealing

feature is that by increasing the number of products in the BLP model, the maximum

utility of consumers increases without bound, a feature that does not characterize the

pure characteristics model. Song demonstrates the advantages of this latter model from

an empirical point of view by analyzing consumer welfare in the personal computer

market (Song 2007) and by computing the quality adjusted price index for computers

between 1996 and 2000 (Song 2010). Song (2008) provides a computational note on a

composite algorithm for determining the �xed point �.

Such a composite algorithm is needed due to the fact that, from a computational

point of view, the pure characteristics model is more di¢ cult than BLP. This is because,

on one hand, the market share function is not di¤erentiable everywhere with respect to

the unobserved characteristics, and on the other hand, it is not known if the vector of

unobservables � is a sink node or not. Consequently, the �xed point analysis described

above and the application of the algorithms provided by Reynaerts et al. (2012) can

facilitate substantially the econometric analysis of this model.

References

[1] Jo Reynaerts, Ravi Varadhan, John C. Nash, Enhancing the Convergence Properties

of the BLP (1995) Contraction Mapping, Discussion Paper, Vives, 2012.

[2] Jean-Pierre Dubé, Jeremy Fox and Che-Lin Su, Improving the numerical perfor-

mance of static and dynamic aggregate discrete choice random coe¢ cients demand

estimation, Econometrica, Vol. 80, No. 5, 2231�2267, 2012.

[3] Steven Berry and Ariel Pakes, The Pure Characteristics Demand Model, Interna-

tional Economic Review, Vol. 48, No. 4, November 2007.

[4] Minjae Song, Measuring Consumer Welfare in the CPU Market: An Application of

the Pure Characteristics Demand Model, RAND Journal of Economics, 38, 429�446,

2007.4This model is described in detail in the working paper Sándor (2013).

14

[5] Minjae Song, Estimating the Pure Charasteristics Demand Model: A Computational

Note, Simon School Working Paper No. FR 08-28, 2008.

[6] Minjae Song, The Quality Adjusted Price Index in the Pure Characteristics Demand

Model, Journal of Business and Economic Statistics, 28, 190-199, 2010.

[7] Jong-Shi Pang, Che-Lin Su, and Yu-Ching Leey, Estimation of Pure Characteristics

Demand Models with Pricing, Work Paper, 2013.

[8] Steven Berry, James Levinsohn and Ariel Pakes, Automobile Prices in Market Equi-

librium,�Econometrica, 63, 841�890, 1995.

[9] A. H. Nayfeh, B. Balachandran, Applied Nonlinear Dynamics. Analytical, Compu-

tational and Experimental Methods, Wiley-VCH Verlag, 2004;

15

4 Objective 3. Study of brand value and warrantytheory

Planned activities: Study of literature on brand valuePromised results: Literature review of brand value

The objective of our research is to provide solutions to those methodological problems

related to brand equity measurement presented later in this literature review.

4.1 General Presentation

Brands stand out of the other marketing mix elements owing to the fact that they are

capable of incorporating the positive e¤ects of all marketing activities, and by this they

become e¤ective signals of quality for the experience and credence attributes (Erdem et

al. 2006, Goldfarb et al. 2009).

The concept of brand equity has gained in popularity since the 1980s. The �eld has

undergone signi�cant development, and an increasing number of empirical models (Yoo

and Donthu 2000, Netemeyer et al. 2004, Erdem et al. 2006) succeed the conceptual

models (Aaker 1991, Keller 1993). The research priorities indicated by the Market-

ing Science Institute (MSI 2010) for the 2010-2012 periods include research related to

brand management. The latest comprehensive literature review related to brand equity

measurement appeared in 2010 (Christodoulides and Chernatony 2010).

In order to distinguish between consumer-based brand equity and brand equity ex-

pressed in �nancial terms, the literature uses consumer-based brand equity (Keller 1993)

instead of brand equity, the latter appears without a distinctive epithet referring to brand

equity expressed in �nancial terms (Ailawadi et al. 2003, Srinivasan et al. 2005) and in

other cases is called brand value (Raggio and Leone 2006).

We will use the following formulations: brand equity refers to the preference of

consumers towards the brand; brand value refers to the �nancial performance of the

measured brand.

The greatest number of studies is concerned with brand equity measurement (Ka-

makura and Russel 1993, Erdem et al. 2006); there is a smaller number of studies

focusing on brand value estimation (Ailawadi et al. 2003, Simon and Sullivan 1993);

there are only a few that estimate both brand equity and brand value by linking (Srini-

vasan et al. 2005, Goldfarb et al. 2009). A great number of studies are based on survey

based data (Yoo and Donthu 2000, Netemeyer et al. 2004, Srinivasan et al. 2005, Erdem

et al. 2006), others are using market level scanner data (Kamakura and Russel 1993,

Ailawadi et al. 2003, Goldfarb et al. 2009).

16

The concept of consumer-based brand equity has become a central marketing con-

cept due to the increasing scienti�c and business interest in brands, since the approach

according to which brands constitute one of the most valuable intangible assets of com-

panies is becoming increasingly widespread (Kapferer 2008). Brands are able to stay

on the market in the long term until products transform or disappear (Kapferer 2008);

that is why it is worth investing in developing brands.

Owing to Aaker�s publications at the beginning of the nineties (1991), there has

been a growing interest in the problem of �nancial returns generated by valuable brands

(Fehle et al. 2008). Companies of high brand equity can expect signi�cant market

share increase if they cut prices, while their share decrease would be insigni�cant if they

increased their prices (Ailawadi et al. 2003). Several empirical researches report about

a positive relationship between agency-based brand equity measures (BAV, Interbrand

and Equitrend) and �nancial returns as well as stock returns (Barth et al. 1998, Fehle

et al. 2008).

The most commonly known and frequently referred-to brand equity measure cate-

gorization belongs to Keller and Lehman (2001). In their system, we can speak about

three large categories:

- Customer mind-set measures

- Product market measures

- Financial measures

Customer mind-set measures mostly assess awareness, associations, loyalty and per-

ceived quality (Aaker 1991, Keller 1993, Yoo and Donthu 1997, Christodoulides et al.

2006,). Their advantages are that they assess brand equity sources, can predict brand

equity changes and predict a brand�s potential (Ailawadi et al. 2003).

Product market measures assess brand equity in the brand�s market performance.

The most common product- market measure is price premium (Randall et al. 1998,

Aaker 1991, Agarwal and Rao 1996, Sethuraman 2000). Further product- market mea-

sures use market share (Chaudhuri and Holbrook 2001) or revenue premium (Ailawadi,

Lehman and Neslin 2003) to de�ne brand equity. The advantage of these measures is

that they can measure the result of the process by which the brand name adds value to

the product; that is, they quantify the performance due to the brand name. Their de�-

ciency lies in data providing and analyzing methods. Revenue premium measures often

refer to hypothetical situations, conjoint analyses are costly and they do not make con-

tinuous measures possible due to the di¢ culties of data collection and the complicated

statistical methods used in the analyses (Ailawadi et al. 2003).

Financial market measures assess the value of a brand as a �nancial asset, establish-

ing a �nancial value of a brand. Measures often use the discounted cash �ow model to

17

assess �nancial value (Interbrand). The advantage of the �nancial value is that it can

quantify future cash �ow (Ailawadi et al. 2003).

4.2 Theory - Modeling

Two outstanding theories in brand equity literature de�ne brand equity, the Resource

Based View (RBV) (Hooley et al. 1998, Slotegraaf and Pauwels 2008) and the Signaling

Theory (Erdem et al. 2006). The RBV theory de�nes the brand as intangible asset or

market-based asset (Bharadwaj et al. 1993, Srivastava et al. 1998), while the signaling

strategy de�nes brands as e¤ective signals (Erdem and Swait 1988). Besides the above

mentioned theories, the consumer-based brand equity literature (Keller 1993) also refers

to the cognitive psychological approach. However, according to the authors�present

knowledge, cognitive psychology has no developed theory applied to marketing.

One of the most-referred-to de�nitions of brand equity was given by Farquhar (1989).

According to it, brand equity is the added value endowed by the brand to the product.

This de�nition stood at the basis of several further instruments measuring brand equity

(Kamakura and Russel 1993, Srinivasan et al. 2005).

In Keller�s interpretation (1993), brand equity is �the di¤erential e¤ect of brand

knowledge on consumer response to the marketing of the brand�given by the di¤erence

between consumer response to the marketing of the branded and unbranded product.

Consumers give a more favorable response to marketing mix in the case of brands with

high brand equity than in those with low equity. As a consequence, relative marketing

costs decrease as the e¢ ciency of marketing activities increase.

�The productivity of brand assets comes from both the demand and supply sides.�

Goldfarb et al. (2009).

Demand side e¤ects - Brand equity

A brand, through indicating quality, decreases the time for consumer decision, par-

ticularly when a consumer has no knowledge of the product category (Pashigian and

Bowen 1994, Tsao et al. 2006, Ramello 2006). Consumers possess an image of product

attributes based on previous experiences or marketing communication activities, and

they, when recognizing a certain brand, are able to make a quick decision whether to

purchase a particular product or not.

The demand side e¤ect of the brand was operationalized in the economic theory

based literature as the brand preference measured with the intercept term in the utility

function (Kamakura and Russel 1993). Jedidi et el. (1999) captures the main e¤ects

of advertising and promotions on brand intercepts interpreted as brand equity. Brand

preference is also extracted from brand intercept in the random utility of the multinomial

logit in Chintagunta (1994). In Chintagunta et al. (2005) and Aribarg and Arora (2008)

18

intrinsic brand value is also measured with the intercept term of the utility function. are

also measuring brand equity with the intercept term. In Goldfarb et al. (2009) brand

equity is conceptualized as the unobserved component of utility of products (Berry et

al. 1995).

4.2.1 Supply side e¤ect �brand value

A company�s protection against competitive attacks increases as the more di¤erentiated

brands result in lower price elasticity (Boulding et al. 1994), the company is more pro-

tected against competititve attacks (Srivastava and Shocker 1991), can apply premium

pricing (Farquhar 1998), and can achieve a more successful brand extension (Keller

2003). Simon and Sullivan (1993) con�rmed that stock exchange evolution contains in-

formation referring to brand equity as well. There is also a positive relationship between

new products and stock return, which is a strong relationship only when a company has

introduced a great number of new developed products into the market (Chaney et al.

1991). Companies of high brand equity can expect signi�cant market share increase if

they cut prices, while their share decrease would be insigni�cant if they increased their

prices (Ailawadi et al. 2003).

We are building a random logit model based on previous empirical �ndings (Ka-

makura and Russel 1993, Goldfarb et al. 2009) and methodological advancement (Berry

et al. 1995). We propose a model that will account for both demand and supply side

e¤ects of brands measured using market level scanner data representing real consumer

choices. The estimation will be realized by using the SPML (semiparametric maximum

likelihood) and SPB (semiparamteric bayesian) estimators which holds some important

advantages over previous estimation methods.

4.3 Most Important Issues of Brand Equity Literature

One of the unsolved problems of brand equity measurements is related to the proper

operationalization of the base brand.

According to the widely accepted (Keller 1993) de�nition measurement of brand

equity presupposes a comparison between the measured brand and a base brand or a

factual and a counterfactual (Goldfarb et al. 2009). Di¤erent solutions were o¤ered:

private label (Ailawadi et al. 2003), �ctive brand (Ferjani 2009), or the brand with the

smallest market share.

Srinivasan et al. (2005) de�nes brand equity with the help of the incremental choice

probability, that is, brand equity is the di¤erence between the choice probability of a

certain brand and that of the base product. In their model, the base product is neither a

private label, nor a �ctive brand. At the individual consumer level, the model compares

19

a certain brand to one in the sample in whose case the di¤erence between product-

related or awareness-related associations and objective measures is the smallest. Jourdan

(2002) also measures brand equity as the di¤erence between objective and subjective

preferences, but their data collection referring to objective measures was not based on

expert survey. Jourdan (2002) uses one sample instead of two, and, throughout the data

collection built on the conjoint method, the members of the sample �rst had to assess

attributes without knowing the brand name, later they re-assessed the attributes, this

time in knowledge of the brand name.

Goldfarb et al. (2009) builds a counterfactual model by introducing the brand with-

out its experiential attributes, and calculates the di¤erence between the factual equilib-

rium and counterfactual equilibrium pro�t.

Brand equity measurement is fueled by the increasing interest in the need for mea-

surement of e¢ ciency of di¤erent marketing activities. Because brands are capable of

incorporating all the positive e¤ects of all marketing activities brand equity could pro-

vide an aggregate indicator of marketing activities (Srinivasan et al. 2005, Sriram et

al. 2007). The process of value creation is illustrated with the help of the BVC (Brand

Value Chain) by Keller and Lehmann (2003). BVC describes the process in which brand

equity sources (awareness, associations) are created as an impact of marketing invest-

ments (advertisements, product development) and the way they increase the company�s

market performance (price premium, market share), which eventually is able to create

and increase the shareholder value.

It is a common practice to use market level data on advertising expenditures, pro-

motion frequency (Jedidi et al. 1999, Ailawadi et al. 2003, Sriram et al. 2007, Goldfarb

et al. 2009) in the literature closest to our approach. Beside advertising and promotion

Sriram et al. (2007) studies also the in�uence of product innovation on brand equity.

4.4 The most important methodological approaches

Consumer-based brand equity research predominantly uses structural equations for es-

timation (Erdem and Swait 1998, Yoo and Donthu 2000, Netemeyer et al. 2004, Erdem

et al. 2006). Some studies do not operationalize brand equity but only its dimensions

(Netemeyer et al. 2004), others do not estimate brand equity as a second order latent

variable, but they include the construct as a dependent variable (Yoo and Donthu 2001).

Some studies estimate the causal measurement model in isolation and do not report on

appropriate �t indicators.

Structural equation models are very popular in brand equity literature, however we

follow a di¤erent econometric stream of literature based on the BLP model (Goldfarb et

al. 2009) that makes possible measurement of brand e¤ect on both demand and supply

20

side while using market level scanner data.

In brand equity literature Kamakura and Russel (1993) are the �rst to address the

problem of brand valuation by using a random utility framework. The probability of

choice is estimated in a multinomial logit model by using market level scanner data.

Brand value is modeled by the utility component intrinsic to the brand, which re�ects

consumer�s perceptions and needs after controlling for price and advertising expenditure.

This conceptualization is related to the logic of consumer mindset measures, but it

has to be estimated, opposed to survey based brand equity models where these per-

ceptions are measured variables or are conceptualized as latent variables measured by

re�ective or causal/formative indicators. Kamakura and Russel (1993) are calculating

brand value as indices; the values of measured brands are scaled to sum up zero.

Although they are using the brand value concept, actually they are measuring brand

equity, meaning they are estimating consumer level utility, but not providing �nancial

estimates of brand value.

One innovative feature of their model is that they are breaking down brand value

into brand tangible and intangible value where the brands�intangible value represents

that part of the brand value which is not explained by physical attributes. Because

they are using experience attributes in measuring the brand tangible value Goldfarb

et al. (2009) assumes that they are likely to overestimate brand value. According to

Christodoulides and Chernatony (2010) Brand Value provides a diagnostic ability of

the brands competitive position, while the Brand Intangible value isolates the utility

associated with intangible factors.

Kamakura and Russel (1993) proposed brand intercept as a brand equity measure.

Sriram et al (2007) based on recent methodological advances (Berry et al. 1995) tested

this measurement method using store level data on toothpaste and dish detergents. They

found that the brand intercept e¤ectively captures high equity of measured popular

brands. They also validated brand intercept as a brand equity measure by analyzing its

correlation with other brand equity measures (price premium, sales premium, revenue

premium) �nding in every case positive correlations.

Ferjani et al. (2009) built a conjoint based brand-attribute interaction model, and

tested its �t while analyzing �ve other speci�cations. They found strong evidence for

the empirical superiority of the brand-attribute interaction model, and the Kamakura

and Russel (1993) model speci�cation performed second best in this comparison. Ac-

cording to Borkovsky et al. (2013) the addition of conjoint analysis enables a detailed

understanding of the role of speci�c search and experience attributes in driving brand

value.

The Goldfarb et al. (2009) model can be considered one big step forward in brand

21

equity measurement. Was built on the Ailawadi et al. (2003) model however they

measure brand value in pro�t terms while Ailawadi et al. (2003) operationalize brand

value as revenue premium realized over a private label. The revenue premium assures a

better measurement in comparison to others (price premium, volume premium) since it

gives a more comprehensive picture. There might be cases when a brand assures price

premium as opposed to a private label, but private label sales may exceed brand sales,

which can result in a negative income. However pro�t premium is a more appealing

measurement method of brand value over revenue premium as o¤ers information about

pro�tability while revenue premium does not.

In contrast to Ailawadi et al. (2003) Goldfarb et al. (2009) calculates the di¤er-

ence between equilibrium pro�t earned by the brand and the pro�t equilibrium in a

counterfactual situation where the brand enters the estimation without its experiential

attributes.

The Goldfarb et al. (2009) model measures brand e¤ect on both demand (brand

equity) and supply side (brand value). The brand positive e¤ect on consumers (demand

side e¤ect) is de�nes as brand equity, and positive e¤ect of brand on the �nancial

performance (supply side e¤ect) is de�ned as brand value (operationalized as pro�t

premium).

In Goldfarb et al. (2009) brand equity is conceptualized as the unobserved utility (see

Berry et al. 2005) component opposed to Kamakura and Russel (1993) who measures

brand equity with the intercept term of the utility function.

Srinivasan et al. (2005) de�ned brand choice probability taking a multiattribute

model as a starting point. Srinivasan et al. (2005) de�nes brand equity with the help

of the incremental choice probability, that is, brand equity is the di¤erence between

the choice probability of a certain brand and that of the base product. In their model,

the base product is neither a private label, nor a �ctive brand. At the individual

consumer level, the model compares a certain brand to one in the sample in whose case

the di¤erence between product-related or awareness-related associations and objective

measures is the smallest. Finally, brand equity in �nancial terms is given by the product

of brand choice probability and its contribution margin.

Christodoulides and Chernatony (2010) considers it a de�ciency that Srinivasan et

al. (2005) did not decompose the non-attributes component, but this is not possible

in the Srinivasan et al. (2005) model since they calculated this component as the

di¤erence between brand preference and multiattribute preference measured on the basis

of product-related attributes.

A further de�ciency of the Srinivasan et al. (2005) model is that the objective

brand measure is based on a survey of experts, and the objectivity of expert opinions

22

should be treated with reservation as a brand name may also have an impact on experts.

We might assume that mobile telephone test results might have been a more reliable

objective measure.

In the Srivastava and Shocker (1991) brand equity model, similarly to the previous

one, brand equity is made up of two elements: brand strength and the brand�s �nancial

value. The Srivastava and Shocker (1991) model also combines consumer mind-set

measures with product market measures, brand strength comes into existence based

on consumer mind-set measures, while the second component, brand value de�nes the

�nancial bene�t for the �rm.

Jourdan (2002), similarly to Srinivasan et al. (2005), used a multiattribute model

as a starting point in devising his brand equity model (Christodoulides and Chernatony

2008). Jourdan (2002) developed his own brand equity model from Srinivasan and Park�s

(1994) model. Jourdan (2002) also measured brand equity as the di¤erence between

objective and subjective preferences, but their data collection referring to objective

measures was not based on expert survey. Jourdan (2002) used one sample instead of

two, and, throughout the data collection built on the conjoint method, the members of

the sample �rst had to assess attributes without knowing the brand name, later they

re-assessed the attributes, this time in knowledge of the brand name.

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Barth, M. E., Clement, M. B., Foster, G. and Kasznik, R. (1998): Brand Values and

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Bharadwaj, S. P., Varadarajan, P., and Fahy, J. (1993): Sustainable Competitive Ad-

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of Marketing, 57 (4), 83�99.

Boulding, W., Lee, E. and Staelin, R. (1994): Mastering the Mix: Do Advertising,

Promotion and Sales Force Activities Lead to Di¤erentiation? Journal of Marketing

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Chaney, P. K., Devinney, T. M. andWiner, R. S. (1991): The Impact of New Product

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Chintagunta, P. K., Dubé, J-P., and Goh, K. Y. (2005): Beyond the Endogeneity

Bias: The E¤ect of Unmeasured Brand Characteristics on Household Level Brand Choice

Models, Management Science, 2005, 51(5), pp. 832.

Christodoulides G. and de Chernatony, L. (2010): Consumer-Based Brand Equity

Conceptualization and Measurement. A Literature Review. International Journal of

Market Research, 52 (1), 43-66.

Erdem, T. and Swait, J. (1998): Brand Equity as a Signaling Phenomenon. Journal

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25

5 Objective 3. Study of brand value and warrantytheory

Planned activities: Study of literature on warranty theoryPromised results: Literature review of warranty theory

5.1 General description

An empirical application of the research project has got the aim to determine how

warranties in�uence the demand of a product, or more speci�cally, to measure the degree

to which warranties are valued as signals of product quality. The length of an automobile

manufacturer�s powertrain warranty has been a prominent aspect of modern automobile

advertisements. The trend is a movement from a �pure manufacturing�paradigm to

a business model in which a central role is assigned to the service components. For

example, the aftersales services of the products represented 8% of U.S. GDP in 2006

(Guajardo et al. 2012).

The automobile industry serves as a preferred setting for the empirical study of the

e¤ects of warranty. Standard&Poor report that in this industry �product quality and

design are becoming less of an issue in di¤erentiating foreign and domestic manufac-

turers.�Thus services represent an important di¤erentiating factor for automakers, in

particular, if we measure the service dimension of a brand by the length of its war-

ranty. The length of warranty de�nes the period in which repair services are provided

by the OEM. Firms have been active on adjusting the length of their warranties in

the last decade. They face an important trade-o¤ when de�ning their warranty period:

while longer warranties may potentially increase demand, they also generate signi�cant

costs, which represent 2-4% of yearly revenue of U.S.-based automakers. Choi and Ishii

(2010) enumerates several examples from the U.S. automobile market how the extended

powertrain warranty of a type increased its market share (Hyundai, Dodge) or how

the warranty reduction declined the share (Volkswagen). These examples suggest (but

do not conclude) the importance of warranty coverage on consumer demand for new

automobiles.

Besides warranty duration service competition is a major topic of interest in oper-

ation management (OM). Theoretical models and related empirical research examined

competition when consumer demand depends on price and service levels (So 2000, Bern-

stein and Federgruen 2007, Allon et al. 2011 etc.). In the economics literature theoretical

models of product di¤erentiation have prompted empirical studies, especially in the au-

tomobile industry where di¤erent aspects of �rm competition and consumer demand

26

have been studied (e.g., Berry et al. 1995 and 2004, Train and Winston 2007).

A list of empirical models of demand that include consumer response to warranties

include the following.

1. Chu and Chintagunta (2009). They use BLP for modeling warranties, di¤erent

models for automakers and retailers.

2. Chu and Chintagunta (2011). They empirically test all four warranty theories in

U.S. Computer server and automobile market. Do not �nd evidence for signaling and

incentives theories.

3. Guajardo et al. (2012). They formulate and estimate a structural model to

measure the impact of service attributes on consumer demand in the U.S. automobile

industry. They use OLS, IVLOGIT and Random coe¢ cient model.

4. Emons (1989). Emons gives a comprehensive overview of the warranty theories.

5. Choi and Ishii (2010). Choi and Ishii seek empirical evidence on the role of

warranties as signals of unobservable quality. They adapt the linear random utility

model of consumer automobile demand to investigate the extent to which warranties

a¤ect consumer choice and the extent to which this estimated warranty e¤ect is dues to

risk aversion and signaling motives.

6. Train and Winston (2007). Provide a method that facilitates identi�cation and

estimation when consumer level data are available. They also measure the e¤ect of

brand loyalty.

5.2 Theoretical-modeling issues

The following four theories have been proposed in the literature on economic rationales

for warranty provision.

1. Insurance theory. Warranties provide insurance to customers and work as a

risk-sharing mechanism, because oblige the manufacturer or seller to compensate the

buyer in the event of product failure. According to Chu and Chintagunta (2011) the

underlying key assumption of the theory is that customers are risk-averse and the prob-

ability of product failure is higher than zero.

2. Sorting theory. Warranties work as a means for second-degree price discrim-

ination among customers with di¤erent risk preferences. It explores how �rms design

and price a line of products distinguished by di¤erent quality and warranty levels to

extract the maximum surplus from each consumer type.

The key assumption according to Chu and Chintagunta (2011) is the presence of

consumer heterogeneity. In our opinion risk-aversion heterogeneity of the customers is

also an assumption, and this theory seems to be an adoption of the insurance theory

27

mainly used for market segmentation and positioning. We note that Emons (1989) does

not discuss this theory.

3. Signalling theory. By Chu and Chintagunta (2011) the key assumption is

information asymmetry in the sense that sellers have better knowledge about the product

quality than buyers, thus sellers need to signal product quality through warranties, just

like they signal quality through advertising (Nelson 1974). Choi and Ishii (2010) �nd a

stronger role of signaling than risk aversion in the US car market.

4. Incentive theory. Warranties work as an incentive mechanism for �rms to

reveal and improve product quality. The key assumption is risk endogeneity in the sense

that sellers�actions can a¤ect product performance. Chu and Chintagunta (2011) test

empirically two implications of the insurance theory:

- The same customers buy longer warranties when product reliability decreases.

- More risk-averse customers buy longer warranties.

Choi and Ishii (2010) apply conditional and mixed logit models to household level

data. Their model speci�cation is similar to the one in Goldberg (1995), except that

they use the mixed logit instead of the nested logit in order to relax the �independence

of irrelevant alternatives (IIA)�assumption.

Guajardo et al (2012) use three types of models and draw conclusions not only

from the results directly but also indirectly from di¤erences obtained from the models.

Their Random coe¢ cients model allows for customer heterogeneity and accounts for

the endogeneity of prices and warranties. OLS �if customer heterogeneity is ignored.

IVLOGIT �using instrumental variable technique.

5.3 The main research questions

These theoretical themes lead to the following empirically testable research questions.

1. The validity of the insurance theory. Chu and Chintagunta (2011) test empir-

ically all the four theories. Choi and Ishii (2010).

2. The validity of the sorting theory. Chu and Chintagunta (2011)

3. The validity of the signalling theory. Chu and Chintagunta (2011) �not proved

empirically. Choi and Ishii (2010) separate the e¤ect of insurance and signaling. They

�nd that longer warranties appear to be valued most for automobile models on which

consumers have little a priori information and for those o¤ered by brands with which

consumers have no prior purchase experience.

4. The validity of the incentive theory. Chu and Chintagunta (2011)

5. The e¤ect on demand of the interaction between service attributes and product

quality. Guajardo et al (2012)

28

6. The moral hazard and adverse selection. Emons (1989). Moral hazard prob-

lems arise when there is imperfect information concerning the actions of those who

purchase insurance. The more insurance individuals have, the less incentive they have

to take appropriate actions. Adverse selection problems arise when there is incomplete

information about the attributes of those who purchase insurance. Changing the terms

of insurances a¤ects the mix of those who apply.

7. Warranties as a marketing device Emons (1989): monopolistic consumer dis-

crimination, search and price-warranty distributions.

8. Choi and Ishii (2010) derives the marginal e¤ects of both the binary and non-

binary explanatory variables in the conditional and mixed logit models. These results

also demonstrate the strong role of brand loyalty, a �nding also found in Train and

Winston (2007). They measure marginal e¤ects for selected brand dummies, where the

excluded brand is Suzuki. These results suggest the presence of important, heteroge-

neous brand reputations.

9. Choi and Ishii (2010) also analyze the relationship between warranties and

brands, where brand reputation is regarded to be an important signal. They examine if

longer warranty o¤sets the di¤erences in the estimated brand dummy, their best proxy

for brand reputation. They compare Hyundai with Honda and Lexus with BMW, in each

case a newer brand with lower reputation with the segment leader in the small/medium

categories and in the luxury category. In both comparisons, the newer brand o¤ers a

powertrain warranty that is longer by 2 years. They consider three scenarios:

- both models are unrated by CR (i.e., consumer reports), so there is no past

purchase experience; thus the di¤erence in product quality is strictly the di¤erence in

brand reputation;

- the category leader type is rated by CR; this is the most common, as consumers

usually have some third party information about the category leader�s make but perhaps

not any based on personal experience;

- the category leader type is rated by CR and has previously been purchased.

The results suggests that Hyundai�s two-year longer warranty in 1998 was inadequate

to compensate for Honda�s greater reputation, and Lexus would need to o¤er a 4-year

longer warranty to o¤set its brand disadvantage to BMW.

According to the authors these results suggests -but do not imply- that automobile

manufacturers engage in interesting trade-o¤s involving their possible product quality

signals.

29

5.4 The main methodological approaches

According to Choi and Ishii (2010) one reason for the low number of empirical papers

examining warranties as a form of signaling is the di¢ culty in obtaining the necessary

data that combine the appropriate product, manufacturer, and consumer information.

In the table below we summarize the variables introduced in the models with their

de�nitions and the data sources for each.

30

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31