# Estimating Coke and Pepsi's Price and Advertising Strategies

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**Estimating Coke and Pepsi’s Price and Advertising Strategies**

Amos Golan*

Larry S. Karp**

Jeffrey M. Perloff**

March 1999

*

American University

**

University of California, Berkeley, and Giannini Foundation

We benefitted greatly from George Judge’s comments about econometrics and

Leo Simon’s comments on game theory. We are very grateful to Jean Jacques

Laffont, Quang Vuong, and especially Farid Gasmi for generously providing us

with the data used in this study and for advice. We received useful suggestions

from participants at the "Industrial Organization and Food-Processing Industry"

conference at the Institute D’Economie Industrielle at the Université des

Sciences Sociales de Toulouse and anonymous referees and the associate editor.

We thank Gary Casterline and Dana Keil for substantial help with computer

programs.

Contact:

Jeffrey M. Perloff (510/642-9574; 510/643-8911 fax)

Department of Agricultural and Resource Economics

207 Giannini Hall

University of California

Berkeley, California 94720

[email protected]

2

**Table of Contents**

1. INTRODUCTION

1

2. OLIGOPOLY GAME

3

2.1 Strategies

4

2.2 Econometric Implications

5

3. GENERALIZED-MAXIMUM-ENTROPY ESTIMATION APPROACH

7

3.1 Background: Classical Maximum Entropy Formulation

7

3.2 Incorporating Sample Information

9

3.3 Incorporating the Non-Sample (Game-Theoretic) Information

11

3.4 Properties of the Estimators and Normalized Entropy

14

4. COLAS

15

4.1 Cola Estimates

17

4.2 Tests

20

4.3 Lerner Measures

22

4.4 Effects of the Exogenous Variables

23

5. CONCLUSIONS

24

REFERENCES

26

Appendix 1: The GME-Nash Estimator

30

**Abstract**

A semi-parametric, information-based estimator is used to estimate strategies in prices

and advertising for Coca-Cola and Pepsi-Cola. Separate strategies for each firm are estimated

with and without restrictions from game theory. These information/entropy estimators are

consistent and efficient. These estimates are used to test theories about the strategies of firms

and to see how changes in incomes or factor prices affect these strategies.

KEYWORDS: strategies, noncooperative games, oligopoly, generalized maximum

entropy, beverages

JEL: C13, C35, C72, L13, L66

**1. INTRODUCTION**

This paper presents two methods for estimating oligopoly strategies. The first method

allows strategies to depend on variables that affect demand and cost. The second method

adds restrictions from game theory. We use these methods to estimate the pricing and adver-

tising strategies of Coca-Cola and Pepsi-Cola.

Unlike most previous empirical studies of oligopoly behavior, we do not assume that

firms use a single pure strategy nor do we make the sort of ad hoc assumptions used in

conjectural variations models.1 Both our approaches recognize that firms may use either

pure or mixed (perhaps more accurately, distributional) strategies.

In our application to Coca-Cola and Pepsi-Cola, we assume that the firms’ decision

variables are prices and advertising. We divide each firm’s continuous price-advertising

action space into a grid over prices and advertising. Then we estimate the vector of probabil-

ities — the mixed or pure strategies — that a firm chooses an action (a rectangle in the price-

advertising grid). We use our estimates to calculate the Lerner index of market structure and

examine how changes in exogenous variables affect strategies.

The main advantages of using our method are that we can flexibly estimate firms’

strategies subject to restrictions implied by game theory and test hypotheses based on these

estimated strategies. The restrictions we impose are consistent with a variety of assumptions

regarding the information that firms have when making their decisions and with either pure or

mixed strategies.

1 Bresnahan (1989) and Perloff (1992) survey conjectural variations and other structural

and reduced-form "new empirical industrial organization" studies.

2

For example, suppose that a firm’s marginal cost in a period is a random variable

observed by the firm but not by the econometrician. Given the realization of marginal cost,

the firm chooses either a pure or a mixed strategy, which results in an action: a price-

advertising pair. The econometrician observes only the firm’s action and not the marginal

cost. As a consequence, the econometrician cannot distinguish between pure or mixed strate-

gies. If both firms in a market use pure strategies and each observes its rival’s marginal cost,

each firm can anticipate its rival’s action in each period. Alteratively, firms might use pure

strategies and know the distribution but not the realization of their rival’s cost. Due to the

randomness of the marginal cost, it appears to both the rival and the econometrician that a

firm is using a mixed strategy. The equilibrium depends on whether firms’ private informa-

tion is correlated.

All of these possibilities — firms have only public information, firms observe each

other’s private information but the econometrician does not, or a firm only knows that its

private information is correlated or uncorrelated with its rival’s — lead to restrictions of the

same form. For expositional simplicity, we concentrate on the situation where firms have

private, uncorrelated information about their own - but not their rival’s - marginal costs (or

some other payoff-relevant variable). Firms choose either pure or mixed strategies.

There have been few previous studies that estimated mixed or pure strategies based on

a game-theoretic model. These studies (Bjorn and Vuong 1985, Bresnahan and Reiss 1991,

and Kooreman 1994) involve discrete action spaces. For example, Bjorn and Vuong and

Kooreman estimate mixed strategies in a game involving spouses’ joint labor market

participation decisions using a maximum likelihood (ML) technique. Our approach differs

3

from these studies in three important ways. First, they assume that there is no exogenous

uncertainty. Second, they allow each agent a choice of only two possible actions. Third, in

order to use a ML approach, they assume a specific error distribution and likelihood function.

Despite the limited number of actions, their ML estimation problems are complex.

Our problem requires that we include a large number of possible actions in order to

analyze oligopoly behavior and allow for mixed strategies. Doing so using a ML approach

would be difficult if not impossible. Instead, we use a generalized-maximum-entropy (GME)

estimator. An important advantage of our GME estimator is its computational simplicity.

Using GME, we can estimate a model with a large number of possible actions and impose

inequality and equality restrictions implied by the equilibrium conditions of the game. In

addition to this practical advantage, the GME estimator does not require the same strong,

explicit distributional assumptions used in standard ML approaches. A special case of our

GME estimator is identical to the ML multinomial logit estimator (when the ML multinomial

logit has a unique solution), which indicates that those restrictions to the GME model is

identical to the distributional assumption of the standard approach.

In the next section, we present a game-theoretic model of firms’ behavior. In the third

section, we describe a GME approach to estimating this game. The fourth section contains

estimates of the strategies of Coke and Pepsi. In the final section, we discuss our results and

possible extensions.

**2. OLIGOPOLY GAME**

Our objective is to determine the strategies of oligopolistic firms using time-series data

on prices, advertising, quantities, and variables that affect cost or demand, such as input

4

prices or seasonal dummies. We assume that two firms,

*i*and

*j*, play a static game in each

period of the sample.

The econometrician observes payoff-relevant public information, such as demand and

cost shifters,

*z*, but does not observe private information known only to the firms. Firm

*i*

(and possibly Firm

*j*), but not the econometrician, observes Firm

*i*’s marginal cost or some

other payoff-relevant random variable ?i(

*t*) in period

*t*= 1, ...,

*T*. Where possible, we sup-

press the time variable

*t*for notational simplicity. The set of

*K*possible realizations,

{? , ? , ..., ? }, is the same every period for both firms. The distributions are constant over

1

2

K

time but may differ across firms. The firms, but not the econometrician, know these distribu-

tions. To simplify the description of the problem, we assume that ?i and ?j are private,

uncorrelated information.

*2.1 Strategies*

The set of

*n*possible actions (price-advertising pairs) for Firm

*i*is {

*x*i,

*x*i, ...,

*x*i }. We

1

2

n

now describe the problem where the random state of nature is private information and

uncorrelated across firms.

The profit of Firm

*i*in a particular time period is ?i (

*z*) = ?i(

*x*i,

*x*j, ?i,

*z*), where

*r*is

rsk

r

s

k

the action chosen by Firm

*i*and

*s*is the action chosen by Firm

*j*. In state

*k*, Firm

*i*’s strategy

is ?i(

*z*) = (?i (

*z*), ?i (

*z*), ..., ?i (

*z*)), where ?i (

*z*) is the probability that Firm

*i*chooses

k

k1

k2

kn

kr

action

*x*given private information ?i and public information

*z*. If Firm

*i*uses a pure

r

k

strategy, ?i (

*z*) is one for a particular

*r*and zero otherwise.

kr

Firm

*j*does not observe Firm

*i*’s private information, so it does not know the condi-

tional probability ?i (

*z*). Firm

*j*knows, however, the distribution of Firm

*i*’s private informa-

kr

5

tion. The Nash assumption is that Firm

*j*knows the unconditional probability of Firm

*i*using

action

*r*. This probability is the expectation over Firm

*i*’s private information: ?i(

*z*) = E

r

k

?i (

*z*), where E is the expectations operator. Similarly Firm

*i*knows the unconditional

kr

k

probability ?j(

*z*) of Firm

*j*.

s

In state

*k*, Firm

*i*chooses ? (

*z*) to maximize expected profits, ? ?j(

*z*)? (

*z*), where

k

s

s

rsk

the expectation is taken over its rival’s actions. If

*Y*i(

*z*) is Firm

*i*’s maximum expected profits

k

given ?i and

*z*, then Firm

*i*’s expected loss from using action

*x*is

k

r

*i*

*i*

(2.1)

*L*

(

*z*) ?

?

*j*(

*z*) ?

*i*(

*z*)

*Y*(

*z*) ? 0 ,

*r k*

*s*

*r s k*

*k*

*s*

which is non-positive. If it is optimal for Firm

*i*to use action

*r*with positive probability, the

expected loss of using that action must be 0. Hence, optimality requires that

(2.2)

*i*

*L*

(

*z*) ?

*i*(

*z*)

0 .

*r k*

*r k*

The equilibrium to this game may not be unique. Our estimation method selects the pure or

mixed strategy equilibrium that is most consistent with the data.

*2.2 Econometric Implications*

Our objective is to estimate the firms’ strategies subject to the constraints implied by

optimization, Equations 2.1 and 2.2. We cannot use these constraints directly, however,

because they involve private information ?i. By taking expectations, we eliminate these

k

unobserved variables and obtain usable restrictions.

6

We define

*Y*i(

*z*) ? E

*Y*i(

*z*) and ?i (

*z*) ? E ?i (

*z*). Taking expectations with respect to

k

k

rs

k

rsk

*k*of Equations 2.1 and 2.2 and using the previous definitions, we obtain

(2.3)

?

*j*(

*z*) ?

*i*(

*z*)

*Y i*(

*z*) ? 0 ,

*s*

*r s*

*s*

?

?

(2.4)

?

?

*j*(

*z*) ?

*i*(

*z*)

*Y i*(

*z*) ?

*i*(

*z*)

?

*i*(

*z*)

0 ,

*s*

*r s*

?

*r*

*r*

?

*s*

?

where ?i ? cov(

*L*i , ?i ) ? 0. For each Firm

*i*= 1, 2, we can estimate the unobservable

r

rk

rk

strategies ?i(

*z*) subject to the conditions implied by Firm

*i*’s optimization problem, Equations

2.3 and 2.4.2

Firms may use approximately optimal decisions due to bounded rationality, or there

may be measurement error. Therefore, we treat Equation 2.4 as a stochastic restriction and

include additive errors in estimation. Equation 2.4, however, already has an additive function,

?(

*z*), which we cannot distinguish from the additive error in 2.4. Thus, ?(

*z*) is the only "error

term" we include in this equation.

If we tried to estimate this model (Equations 2.3 - 2.4) using traditional techniques, we

would run into two problems. First, imposing the various equality and inequality restrictions

2 If ?i and ?j are correlated or observed by both firms, the restrictions are slightly more

complicated. If information is correlated, it would be reasonable to suppose that Firm

*i*’s

beliefs about

*j*’s actions depend on the realization of ?i, so that ?j is replaced by ?j . If

s

ks

information is observed by both firms, Firm

*i*’s beliefs would also be conditioned on the

realization of ?j. In both cases, we can take expectations with respect to the private informa-

tion and obtain equations analogous to 2.3 and 2.4. However, with either generalization, we

would have an additional additive term in 2.3, say ?, and the definition of ? would be

changed. The signs of both ? and ? would be indeterminate. In our empirical application to

the cola market, all the estimated ? are positive, which is consistent with the model in the text

where ?i and ?j are uncorrelated.

7

from our game-theoretic model would be very difficult if not impossible with standard

techniques. Second, as the problem is ill posed in small samples (there may be more param-

eters than observations), we would have to impose additional assumptions to make the

problem well posed. To avoid these and other estimation and inference problems, we propose

an alternative approach.

**3. GENERALIZED-MAXIMUM-ENTROPY ESTIMATION APPROACH**

We use generalized maximum entropy (GME) to estimate the firms’ strategies. In this

section, we start by briefly describing the traditional maximum entropy (ME) estimation

procedure. Then, we present the GME formulation as a method of recovering information

from the data consistent with our game. Our GME method is closely related to the GME

multinomial choice approach in Golan, Judge, and Perloff (1996 — henceforth GJP). Unlike

ML estimators, the GME approach does not require explicit distributional assumptions,

performs well with small samples, and can incorporate inequality restrictions.

*3.1 Background: Classical Maximum Entropy Formulation*

The traditional entropy formulation is described in Shannon (1948), Jaynes (1957a;

1957b), Kullback (1959), Gokhale and Kullback (1978), Levine (1980), Jaynes (1984), Shore

and Johnson (1980), Denzau, Gibbons, and Greenberg (1989), Skilling (1989), Csiszár (1991),

Soofi (1992, 1994) and Golan, Judge, and Miller (1996). In this approach, Shannon’s (1948)

entropy is used to measure the uncertainty (state of knowledge) we have about the occurrence

of a collection of events. Letting

*x*be a random variable with possible outcomes

*x*,

*s*= 1, 2,

s