© 2009 jiaoju ge
TRANSCRIPT
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BARGAINING BETWEEN COOPERATIVES AND PROCESSORS: A MODELING AND EMPIRICAL STUDY IN FLORIDA DAIRY INDUSTRY
By
JIAOJU GE
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2009
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© 2009 Jiaoju Ge
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To the most important loved ones in my life: my husband, Qiyong; my son, Michael; and my daughter, Sarah.
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ACKNOWLEDGMENTS
First, I express my deepest appreciation to all my committee, Dr. Richard L. Kilmer, Dr.
Alfonso Flores-Lagunes, Dr. Lisa House, Dr. Thomas Spreen and Dr. Lawrence Kenny.
Foremost, I would like to gratefully express my sincere gratitude to my committee chair,
Dr. Richard L. Kilmer, for his generous guidance and encouragement. I was very fortunate to
work with him and learned much invaluable knowledge from him. I would like to express my
deepest appreciation to him for his thorough review of the manuscript and patience to
successfully complete my study.
I am also very grateful to Dr. Alfonso Flores-Lagunes for guiding me through the
econometric knowledge of disequilibrium model and the method of Expectation Maximization
algorithm. I sincerely thank him for providing me with invaluable comments and suggestions for
me to complete this dissertation.
I am very appreciative of the support I received from Dr. Lisa House and thank her for
providing me the comments and suggestions.
Many thanks must also go to the other members of my supervisory committee, Dr. Thomas
Spreen and Dr. Lawrence Kenny. Thank them all for their support and guidance, ,comments and
suggestions.
I would also like to thank Dr. Charles Moss, Dr. Ronald Ward for their generous help and
suggestions, thank for my fellow graduate students in the Food and Resource Economics
Department, Ledia Guci and Xinxin Zhang, for their support and suggestions, and all other
fellow students for all the unforgettable memories.
I thank my families, especially my mom for her support for me to finishing my
dissertation. Finally, the greatest thank you goes to my husband, Qiyong Xu, for his
understanding, encouragement and love.
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TABLE OF CONTENTS
page
ACKNOWLEDGMENTS.................................................................................................................... 4
LIST OF TABLES................................................................................................................................ 8
LIST OF FIGURES .............................................................................................................................. 9
ABSTRACT ........................................................................................................................................ 10
CHAPTER
1 INTRODUCTION, PROBLEM STATEMENT AND OBJECTIVES ................................... 11
1.1 Introduction ........................................................................................................................... 11 1.2 Problem Statement ................................................................................................................ 12 1.3 Objectives .............................................................................................................................. 12 1.4 Overview of the Study .......................................................................................................... 13
2 LITERATURE REVIEW ........................................................................................................... 14
2.1 Introduction ........................................................................................................................... 14 2.2 Bargaining Theory ................................................................................................................ 14
2.2.1 Introduction................................................................................................................. 14 2.2.2 Bilateral Bargaining ................................................................................................... 16
2.2.2.1 Axiomatic approach model ............................................................................. 16 2.2.2.2 Strategic approach model ................................................................................ 19 2.2.2.3 Summary of bilateral bargaining theory......................................................... 34
2.2.3 Multilateral Bargaining .............................................................................................. 35 2.2.3.1 Axiomatic approach ......................................................................................... 35 2.2.3.2 Strategic approach ........................................................................................... 36 2.2.3.3 Solution approximation of strategic approach to axiomatic approach ......... 37 2.2.3.4 Summary of multilateral bargaining theory ................................................... 39
2.2.4 Bargaining Theory Summary .................................................................................... 40 2.3 Distribution Channels Bargaining Theory........................................................................... 41
2.3.1 Introduction................................................................................................................. 41 2.3.2 Supply Chain Bargaining ........................................................................................... 41 2.3.3 Cooperative Bargaining ............................................................................................. 43 2.3.4 Distribution Channel Bargaining Theory Summary ................................................ 47
2.4 Bargaining Model Applications ........................................................................................... 47 2.4.1 Model Applications .................................................................................................... 47 2.4.2 Bargaining Model Applications Summary ............................................................... 52
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2.5 Summary, Conclusions and Discussions ............................................................................. 53 2.5.1 Summary ..................................................................................................................... 53 2.5.2 Conclusions and Discussions..................................................................................... 54
2.6 Relevant Information For Model Developing Next Chapter ............................................. 56
3 THE THEORETICAL MODEL ................................................................................................ 59
3.1 Building Model ..................................................................................................................... 59 3.1.1 Bilateral Model with Risk of Breakdown ................................................................. 59 3.1.2 Bilateral Model with Risk of Breakdown and Outside Options .............................. 60 3.1.3 Bilateral Model with Risk of Breakdown and Outside Options between A
Cooperative and Processor .............................................................................................. 61 3.2 Model Solution ...................................................................................................................... 62
3.2.1 Case 1: Outside Option Prices are Variable ............................................................. 63 3.2.2 Case 2: Outside Option Prices are Fixed .................................................................. 72 3.2.3 Case 3: One Outside Option Price is fixed and The Other Outside Option
Price is Variable ............................................................................................................... 74 3.2.4 Summary of Model Solution ..................................................................................... 75
4 THE EMPIRICAL MODEL....................................................................................................... 77
4.1 Florida Milk Industry ............................................................................................................ 77 4.2 The Econometric Model ....................................................................................................... 79 4.3 Discussion of The Exogenous Variables ............................................................................. 82 4.4 Data Description ................................................................................................................... 85 4.5 Model Estimation .................................................................................................................. 86 4.6 Summary................................................................................................................................ 91
5 RESULTS .................................................................................................................................... 93
5.1 Summary Statistics................................................................................................................ 93 5.2 Parameter Estimates .............................................................................................................. 94 5.3 Bargaining Power Parameter ................................................................................................ 98 5.4 Variance Covariance ........................................................................................................... 101
6 SUMMARY, CONCLUSIONS AND IMPLICATIONS ....................................................... 103
6.1 Summary.............................................................................................................................. 103 6.2 Conclusions ......................................................................................................................... 105 6.3 Implications ......................................................................................................................... 107 6.4 Further Researches .............................................................................................................. 107
APPENDIX
A CALCULATION OF THE NEGOTIATED QUANTITY Q ................................................. 109
B DATA ........................................................................................................................................ 122
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C GAUSS PROGRAM ................................................................................................................. 130
D TAO TESTING ......................................................................................................................... 134
LIST OF REFERENCES ................................................................................................................. 138
BIOGRAPHICAL SKETCH ........................................................................................................... 142
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LIST OF TABLES
Table page 4-1 Florida Milk Processor Plants ............................................................................................... 78
4-2 Monthly SDC Non-Florida Cooperative members .............................................................. 79
5-1 Summary Statistics of Data ................................................................................................... 93
5-2 Correlation Matrix of Data .................................................................................................... 94
5-3 Coefficient Estimates ............................................................................................................. 96
5-4 Monthly Bargaining Power Coefficient Estimates for SDC and Processors ..................... 98
5-5 Yearly Bargaining Power Coefficient Estimates for SDC and Processors....................... 100
5-6 Test Result for Zero Covariance ......................................................................................... 102
B-1 Data(1) .................................................................................................................................. 122
B-2 Data(2) .................................................................................................................................. 126
B-1 Testing Results for Maximum Bargaining Power for SDC in Year 2008 ........................ 134
B-2 Testing Results for Minimum Bargaining Power for SDC in Year 2001......................... 135
B-3 Testing Results for Maximum Bargaining Power for SDC in April................................. 136
B-4 Testing Results for Minimum Bargaining Power for SDC in November ........................ 137
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LIST OF FIGURES
Figure page 5-1 SDC’s Monthly Bargaining Power Coefficient and Over-order Premium....................... 101
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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy
BARGAINING BETWEEN COOPERATIVES AND PROCESSORS: A MODELING AND
EMPIRICAL STUDY IN FLORIDA DAIRY INDUSTRY
By
Jiaoju Ge
December 2009 Chair: Richard L. Kilmer Major: Food and Resources Economics
The Florida dairy market has a few fluid milk processors and many dairy farmers. The
dairy farmers are represented in negotiation with the processors by a cooperative. This
dissertation builds a theoretical model for bargaining between the processors and a cooperative
with outside options and risk of breakdown. The model is applied to the Florida dairy market to
examine price negotiations between Florida milk processors and a dairy cooperative. Time series
data was collected for the period of October 1998 to May 2009. An expectation maximization
(EM) algorithm along with Maximum Likelihood Estimation was used to analyze the
econometric disequilibrium model empirically in Gauss statistical software.
The results show that the class I price set by the Federal Milk Marketing Order is the major
factor influencing the processors’ demand reservation price. Negotiated quantity and production
seasonality affect the cooperative’s supply reservation price. The cooperative appears to be more
patient and has higher average bargaining power (0.7055) than the bargaining power for
processors (0.2945). The highest (lowest) bargaining power for the cooperative (processors)
occured in 2008 and the lowest (highest) bargaining power for the cooperative (processors)
occurred in both 2001 and 2004.
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CHAPTER 1 INTRODUCTION, PROBLEM STATEMENT AND OBJECTIVES
1.1 Introduction
Food processors are generally large and few in number due to economies of size (Durham
and Sexton, 1992). However, large numbers of independent farmers are small which put
processors at an advantage over individual farmers when contracts are negotiated. Thus,
marketing cooperatives were established to give individual farmers a better bargaining position
with processors (Jesse, et al., 1982). Florida’s milk marketing cooperative was established to
represent Florida dairy farmers and negotiate the price of fluid milk with Florida dairy
processors.
The objective of the milk marketing cooperative is to maximize the profit of cooperative
members (farmers) (Iyer and Villas-boas, 2003, Nagarajan and Bassok, 2002, Oczkowski, 1999,
Oczkowski, 1991, Oczkowski, 2006). For a given quantity of milk supplied by farmers, the price
paid by processors decides the net revenue for farmers. Thus, a milk marketing cooperative
would prefer to negotiate with processors and receive a high price.
This negotiated fluid milk price includes (1) the class 1 (fluid milk) price and (2) the over-
order premium. The class 1 price is set monthly by the federal milk marketing order which
means it is exogenous to the bargaining process. This implies that the higher the over-order
premium, the higher the negotiated price. Therefore, dairy farmers and their cooperative would
prefer a high over-order premium and the processors would prefer a low over-order premium.
Thus, the interesting question becomes what factors affect the negotiated price (class 1 price plus
over-order premium). While the milk cooperative and milk processor bargain over the over-order
premium, some other milk cooperatives and processors outside of the state of Florida exist and
they might provide a better offer for either the Florida milk cooperative or Florida milk
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processor. Thus, when outside offers are accepted, the negotiation process between the Florida
milk cooperative and the Florida milk processor breaks down.
In Florida, there exist several milk processors. However, when it comes to negotiating with
the milk cooperative for the fluid milk price, the biggest milk processor generally represents all
milk processors. Thus, the Florida milk cooperative and the Florida milk processor can be treated
as a monopoly and monopsony organization. Nash indicates that “The economic situation of
monopoly versus monopsony may be regarded as a bargaining problem.” (Nash 1950).
1.2 Problem Statement
The problem to be examined in this study is what factors affect the negotiated price
between milk cooperatives and milk processors when outside options are available for both sides.
1.3 Objectives
The general objective of this study is to understand the factors that influence the bargaining
process between the dairy cooperatives and the milk processors. Specific objectives are listed
below:
1. To do a comprehensive literature review on bargaining theory, its applications, and estimating disequilibrium models;
2. To develop a theoretic bargaining model with two-side outside options and risk of breakdown;
3. To develop a disequilibrium econometric model and collect the data needed;
4. To specify and estimate the model including the demand reservation price equation, the supply reservation price equation, the negotiated price equation, and the bargaining power parameter equation in order to examine how they are affected by different factors;
5. To examine and interpret the factors that affect the demand reservation price, the supply reservation price, the negotiated price and the bargaining power parameter;
6. To summarize the results and draw conclusions and implications from the results.
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1.4 Overview of the Study
Chapter 2 is the literature review chapter on bargaining theory and applications which
provide all the information needed for developing a theoretical bargaining model. There are three
categories of literature including (1) bargaining theory, (2) distribution channel theory, and (3)
bargaining model applications.
A general bargaining model is developed in chapter 3 with outside options and risk of
breakdown. Three cases are discussed including (1) the case of variable outside option
bargaining price, (2) the case of fixed outside option price, and (3) the case where one player
accepts a fixed outside option price and the other player’s outside option price is variable.
An empirical disequilibrium econometric model is developed in chapter 4 for bargaining
between a milk cooperative and Florida milk processors. Associated econometric literature will
be reviewed. The Expectation Maximization method is used to estimate the empirical model. The
data needed will be identified.
The model will be estimated in chapter 5 and the results of the econometric model will be
presented. Chapter 6 provides a summary, conclusions and implications.
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CHAPTER 2 LITERATURE REVIEW
2.1 Introduction
In Florida, the milk cooperative and the milk processors can be treated as a monopsony
and monopoly structure during bargaining. The price negotiation process between milk
cooperatives and milk processors is complicated. Nash indicates that “The economic situation of
monopoly versus monopsony may be regarded as bargaining problem.” (Nash, 1950). Thus this
price determination process is complicated and can be captured through the negotiation process
by using bargaining theory.
The purpose of this chapter is to review the literature associated with the bargaining
problem and its application to bargaining between a milk cooperative and the Florida milk
processors. Three major areas of past literature will be reviewed and discussed. The first area is
the development and the history of bargaining theory. The second area is the distribution channel
bargaining theory, which is concentrated on cooperative theory development in the bargaining
area. The third area is the application of bargaining theory including the empirical econometric
works of bargaining models.
2.2 Bargaining Theory
“Bargaining is any process through which the players on their own try to reach an
agreement” (Muthoo, 1999). In my research, bargaining refers to the price negotiation process
between milk cooperatives and milk processors.
2.2.1 Introduction
Bargaining theory has caught people’s eyes ever since 1950, when John Nash published
his famous work containing a bilateral bargaining solution by using the axiomatic approach
(Nash, 1950). The axiomatic approach can be defined as the approach of analyzing bargaining
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behavior to have a unique solution based on a series of axiomatic assumptions. This bargaining
solution is called a symmetric Nash bargaining solution. Then, Roth (Roth, 1979) followed the
same axiomatic approach, but generalized the symmetric Nash bargaining solution to the
asymmetric Nash bargaining solution, which is called the generalized Nash bargaining solution.
However, this axiomatic approach has overlooked the complexity of the negotiation
process, which might have an unexpected effect on the bargaining solution. This motivated the
appearance of the other approach which is called the strategic approach. In 1982, Rubinstein first
proposed a two person alternating offer model (Rubinstein, 1982) by using the strategic
approach. The strategic approach can be thought of as an approach to find the bargaining
solution taking into consideration the strategies each player would use at each negotiation stage.
Roth (Roth, 1985) then added the risk component to Rubinstein’s strategic alternating model.
By 1986, people paid attention to the relationship between the above two approaches. One
article (Binmore, et al., 1986) examined this relationship and suggested that the solution of the
strategic approach approximates the Nash bargaining solution if the negotiation time goes to zero
in each bargaining period.
In the 1990s, more articles studied bargaining theory by adding new components into the
bilateral bargaining model, such as risk and outside options (Muthoo, 1999, Muthoo, 1995) to the
bilateral bargaining model. Furthermore, additional work has been done on a n-player bargaining
model (Chae and Yang, 1988, Jun, 1987, Nagarajan and Bassok, 2002, Schneider, 2005, Suh and
Wen, 2006). In this case, some authors tried to reduce the n-player game to a multi-stage game
with bilateral bargaining at each stage (Nagarajan and Bassok, 2002) where the first stage is to
form a coalition on the supply side and a coalition on the demand side and the second stage is the
bilateral bargaining between the supply side coalition and the demand side coalition. Other
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authors reduced the game to a series of bilateral bargaining sessions (Chae and Yang, 1994, Chae
and Yang, 1988, Jun, 1987, Suh and Wen, 2006). The direct n-player game with risk and outside
option components is still developing.
2.2.2 Bilateral Bargaining
Napel (Napel, 2002)pointed out that economically, bilateral bargaining is a situation
where two agents have a common interest in cooperating to get joint payoffs, but conflicting
interests in the process of how to reach the cooperation and how to divide the payoffs. In my
research, bilateral bargaining refers to milk price bargaining between two monopolies (one milk
processor and one milk cooperative) for the purpose of cooperating to obtain a market solution
and dividing the joint profits. The development of bilateral bargaining has been the major field in
bargaining theory development history, probably because of the interesting monopoly power
between two parties. Both the axiomatic approach and the strategic approach have the theory
developed in bilateral bargaining.
2.2.2.1 Axiomatic approach model
The standard economic bargaining model uses the axiomatic Nash (Nash, 1950)
equilibrium concept. For any bargaining game, as Rubinstein et al.(Rubinstein, et al., 1992)
pointed out, this unique Nash bargaining solution can be represented by the following function
N, defined as
both for and ),())(max(arg),( 212211 iduSuudududSN ii ≥∈−−= (2-1)
where N is the Nash bargaining solution; S is the feasible bargaining solution set; d is the
disagreement point; arg max represents the maximization through logical discussion; u1 and d1
are utility representation for party 1 at the agreement point and the disagreement point,
respectively; u2 and d2 are the utility representation for party 2 at the agreement point and the
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disagreement point, respectively. This Nash bargaining solution is a unique solution satisfying
the following four axioms:
1.(IAT) Invariance to positive affine transformations: The solution is invariant to all independent person-by-person, positive affine transformations of utilities.
2.(SYM) Symmetry: If the set S is symmetric with respect to the main diagonal and if d1 = d2, then the solution should assign equal utilities to both players.
3.(PAR) Pareto Optimality: There is no point in S which Pareto-dominates the solution outcome.
4.(IIA) Independence of irrelevant alternatives.(Muthoo, 1999)
The first axiom of IAT implies that the bargaining outcome does not rely on the calibration
of player’s utilities (Binmore, 1992), or in other words that the bargaining outcome is based on
the player’s preferences, not the utility representation of them (Muthoo, 1999). This axiom seems
reasonable from Muthoo’s view. For example, if the utility representations for both players are
multiplied by a scale, the new bargaining outcome should be the old bargaining outcome
multiplied by a scale .
The second axiom SYM concerns a symmetric bargaining problem and suggests that both
players should get the same bargaining outcome. This axiom is feasible as long as both parties
choose to cooperate. For example, under a strike, a labor bargaining agreement is not reached
implying that both parties are not cooperating, then the outcome for both parties is not
symmetric.
The third axiom PAR explains that there is no other bargaining outcome that is preferred
by both players to this unique Nash bargaining solution because both parties are playing the
game in a rational way. Finally, the last axiom IIA says that if both parties agree on the payoff
set S while another payoff set T is feasible, they should never agree on T (Binmore, 1992). This
axiom indicates that the disagreement point for both parties should be the same.
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Similarly, Muthoo (Muthoo, 1999) used the following optimization model and showed us
the same conclusion such that the Nash bargaining solution is the unique solution to the
following maximization problem based on the above four axioms as
))(( max)( ,
BBAAuududuN
bA
−−=Θ∈
(2-2)
where , and ),( BBAABA duduSuu ≥≥∈≡Θ , N is the maximization problem; uA, uB, S, dA
and dB are defined the same as above. Now, we can solve the problem mathematically rather than
logically as discussed above in equation (2-1).
However, Nash (1950) suggests that the symmetry assumption needs to be reconsidered.
Since the symmetric axiom might not be feasible for all games, asymmetric Nash economic
bargaining models were developed. Binmore ((Binmore 1992), p.180-195) discussed the
asymmetric Nash bargaining solution by introducing the bargaining power parameter for both
players. The same result is shown by Muthoo. Muthoo (1999) explains that the bargaining
solution may be affected by many other factors, such as the tactics used by the bargainers, or the
negotiation procedure, which results in asymmetric power for the participating parties. Thus, for
a given value which is defined as bargaining power, a generalized or an asymmetric Nash
bargaining solution is a unique solution of the maximization problem
ττ −−−= 1
),()()( max BBAAUU
duduNBA
(2-3)
as long as all but the symmetric axiom is satisfied (Muthoo, 1999). When 5.0=τ , this
asymmetric Nash’s bargaining solution is equivalent to the symmetric Nash bargaining solution.
Regarding the economics of bargaining, the symmetric Nash bargaining solution, when
joint profits are maximized, results in a single transacted price which is half way between the
demand reservation price and the supply reservation price if both parties are assumed to seek
maximized profits ((Oczkowski 2004), p. 6). Roth (Roth 1979) shows that the difference
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between the symmetric and the asymmetric Nash bargaining solution is that the negotiated price
for the asymmetric Nash equilibrium falls between the demand reservation price and the supply
reservation price instead of at the half way point. This is determined by the relative bargaining
power of the parties.
2.2.2.2 Strategic approach model
The axiomatic approach discussed above has certain disadvantages when exploring the
details of the bargaining process. Thus, another approach called the strategic approach appeared
in the early 1980s. It was introduced by Rubinstein (Rubinstein, 1982).
2.2.2.2.1 Basic strategic approach alternating offer model: In 1982, Rubinstein wrote
an article on the pure game theory of bargaining for a two player game. It allowed the Nash
equilibrium to be combined with the complexity of the negotiation process. The bilateral
bargaining model in Rubinstein’s article assumes that one player makes an offer, and then the
other player accepts the offer reaching the agreement or rejects the offer and makes another
offer. Negotiation continues until the agreement is reached. Rationality and complete
information of preferences are also assumed for both players.
Rubinstein’s model described two players who are bargaining on the partition of a pie
which is represented by s, a number in the unit interval. That is, the portion of the pie player 1
receives is 1s and the portion of the pie player 2 receives is 12 1 ss −= . Rubinstein defined t as the
bargaining period. The outcome function of this alternating bilateral bargaining model is defined
as
∞=∞∞<
=T(f,g)T(f,g)gfTgfD
gfP ),,0(
)),,(),,((),( (2-4)
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where ),( gfP is the outcome function; f is the set of all strategies of the player who starts the
bargaining by making an offer; g is the set of all strategies of the player who in the first move has
to respond to the offer; ),( gfD is the partition induced by ),( gf ; and ),( gfT is the length of
the sequence of offers. Thus, the outcome ),( ts represents reaching the agreement s in
bargaining period t; and ),0( ∞ suggests a permanent disagreement. Then, the ordered pair
),(∧∧
gf is a Nash equilibrium if there is no f such that ),(),(∧∧∧
> gfPgfP or no g such
that ),(),(∧∧∧
> gfPgfP . In other words, there is no outcome that is preferred by either player 1 or
2 except the equilibrium offer.
Rubinstein’s model shows how the results of the existing perfect Nash equilibrium are
different from the axiomatic approach model by looking at two different cases. First, bargaining
costs are fixed for each player in each bargaining period and second, a case where the discount
factor is fixed for each player. For the case of the fixed bargaining cost models, each player i has
a number of fixed bargaining costs ic such that ),(),( 21 tsts ≥ if and only if
),()( 21 tcstcs iiii −≥⋅− where s and s are the portion of the pie received by player 1 and 2 and
1t and 2t represent the negotiation time period, then the perfect equilibrium (P.E.) is:
1.If 21 cc > , 2c is the only P.E..
2. If 21 cc = , every 11 ≤≤ xc is a P.E..
3.If 21 cc < , 1 is the only P.E.. (Rubinstein, 1982)
where x is an offer associated with each strategy that the player chooses to play; player 1 accepts
the offer 2c when 21 cc > ; player 1 accepts the offer between 1c and the entire negotiated pie
when 21 cc = ; and player 1 accepts the offer of the entire negotiated pie when 21 cc < .
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For the case of the fixed discount factor models Rubinstein shows that, each player i has
a number of discount factors 10 ≤< iδ such that the outcome ),(),( 21 tsts ≥ if and only
if 21 tii
tii ss δδ ≥ , then if at least one of the discount factors is strictly less than 1 and at least one of
them is strictly positive, the only P.E. is )1/()1( 212 δδδ −− .
Later, Muthoo (Muthoo, 1999) explains this alternating Rubinstein strategic approach
model with more mathematical detail and shows the results at the limit. Two players, A and B,
bargain over a partition of a cake size of π (π >0). At time 0, A makes an offer. Then if B
accepts the offer, the agreement is reached; if B rejects the offer, B will make a counteroffer at
time period ∆ (∆>0), and A will make a decision on either to accept or reject. If A rejects the
offer, player A makes a counter offer to B at time period ∆2 . This process continues until an
agreement is reached. Muthoo defines ix as a share of the cake for player i when the agreement is
reached, where π≤≤ ix0 . Then, given the discount rate )0( >ii rr for player i, the payoff for
player i is )exp( ∆−⋅ trx ii where ,...2,1,0=t . For simplicity, the discount factor )exp( ∆−≡ triiδ is
defined. Notice that 10 << iδ . Thus, player i’s payoff at the agreement point is iix δ . Notice that
if perpetual disagreement occurs, and then each player’s payoff is zero.
Muthoo indicates that this subgame perfect equilibrium outcome, denoted by *ix , should
satisfy the following two properties:
1.(No Delay). Whenever a player has to make an offer, her equilibrium offer is accepted by the other offer.
2.(Stationarity).In equilibrium, a player makes the same offer whenever she has to make an offer. (Muthoo, 1999)
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Given these two properties, the subgame perfect equilibrium of this alternating offer game is a
unique payoff set, ),( **BA xx , for player A and B, respectively. Muthoo shows that the equilibrium
can be solved from the following equations
**BBA xx δπ =− (2-5)
**AAB xx δπ =− (2-6)
where
πδδδ
BA
BAx
−−
=1
1* (2-7)
πδδδ
BA
ABx
−−
=1
1* . (2-8)
In the limit as 0→∆ ,
BA
B
BA
B
BA
B
BA
B
BA
BA
rrr
rrr
rrr
rrrx
+=
∆+∆
=∆+−−
∆−−→
∆+−−∆−−
=−−
=
)())(1(1)1(1
))(exp(1)exp(1
11*
δδδ
(2-9)
BA
A
BA
A
BA
A
BA
A
BA
AB
rrr
rrr
rrr
rrrx
+=
∆+∆
=∆+−−
∆−−→
∆+−−∆−−
=−−
=
)())(1(1)1(1
))(exp(1)exp(1
11*
δδδ
. (2-10)
Equations (2-9) and (2-10) show that the unique equilibrium outcome set ),( **BA xx depends on the
relative discount rate between two players. When, BA rr / gets bigger, the equilibrium outcome
for player A ( *Ax ) decreases and the equilibrium outcome for player B ( *
Bx ) increases. This
means that the equilibrium outcome for the player who faces a higher discount rate is smaller
than the equilibrium outcome for the player who faces a lower discount rate. Notice that as
0→∆ , the discount factor for both players goes to unity.
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From the result of Rubinstein’s model (equation 2-4) and Muthoo’s result (equation (2-7)
and (2-8)), they reach the same conclusion such that the unique subgame perfect equilibrium
exists and the outcome for player 1 and 2 is )1/()1( 212 δδδ −− , or )1/()1( 211 δδδ −− multiplied
by the bargaining cake π , respectively.
Based on the above basic strategic approach alternating offer bargaining model, other
perspectives of the model are discussed. Some authors discuss the risk of breakdown of the
model, such as Roth (Roth, 1985), Binmore, etal. (Binmore, et al., 1986), and Muthoo (Muthoo,
1999). Others discuss the outside option of the model, such as Muthoo (Muthoo, 1995), Ponsati
and Sakovics (Ponsati and Sakovics, 1998), and Muthoo (Muthoo, 1999). Even later, this
bilateral alternating offer bargaining model has been extended to the multilateral bargaining
level, which will be reviewed in the multilateral bargaining section.
2.2.2.2.2 Strategic approach alternating offer model with risk of breakdown: For the
risk component of the bargaining model, Roth (Roth, 1985) had a note on risk aversion at perfect
equilibrium in the Rubinstein bilateral alternating offer bargaining model. Roth pointed out that
the more risk averse the party is, the smaller share at equilibrium the party has. Roth explains the
effect of risk by replacing one player i with a risk averse player *i in the original alternating
offer fixed discount factor model, but keeps the other player unchanged. Recall that it was
discussed above, *x is the equilibrium share (for the first move player) of the cake. Now, we
introduce )2 ,2 ;1 ,1, ( ),,( **=jijix representing the share but measured by commodities rather
than utility so that the share ),( jix can be compared directly. Thus, if player 1 is replaced by *1 ,
then the equilibrium share for the first move player 1 or *1 can be represented
by )2,1(x or )2,1( *x , respectively. Similarly, if player 2 is replaced by *2 , then the equilibrium
24
share for the second move player 2 or *2 can be represented by ( )2,1(1 x− ) or ( )2,1(1 *x− ),
respectively. Roth proved the theorem such that
)2,1()2,1()2,1( ** xxx ≤≤ . (2-11)
For the part of )2,1()2,1( * xx ≤ , it suggests that for the same bargaining game, if player 1 is
replaced with a risk averse player *1 , the equilibrium share is smaller for the risk averse player *1 .
For the part of )2,1()2,1( *xx ≤ , it implies that )2,1(1)2,1(1 *xx −≥− , which says that for the same
bargaining game, if player 2 is replaced with a risk averse player *2 (player 1 is no longer risk
averse), the equilibrium share is smaller for the risk averse player *2 . Thus, Roth concludes that
the more risk averse player gets the smaller share than the less risk averse player does. Roth
discussed the concept of risk such that for both the axiomatic approach models and the strategic
approach models, the risk is the bargainers’ subjective probability of failing to reach an
agreement. In other words, if a breakdown in the negotiation process occurs, then an agreement
is not reached.
Later, Binmore, Rubinstein and Wolinsky (Binmore, et al., 1986) present a strategic
model (Rubinstein’s alternating offer bilateral model discussed above) and add the risk of
breakdown(failing to reach an agreement). They use probability to represent the risk of
breakdown and showe the utility maximization problem mathematically. Binmore, Rubinstein
and Wolinsky assume that the time of the breakdown is exponentially distributed with a
parameter λ such that the risk of breakdown in each bargaining period can be presented by a
positive probability
)exp(1)( ∆−−=∆= λpp (2-12)
where for each bargaining period of length∆ , there is p probability that the bargaining process
will breakdown. Binmore, Rubinstein and Wolinsky assume that the outcome is b at the point of
25
breakdown. Recall that in Rubinstein’s basic model discussed above, x is the outcome when the
agreement is reached. Thus, Binmore, Rubinstein and Wolinsky suggest that the outcome for this
bargaining game is shown below as
bpxpO tt ))1(1()1( −−+−= (2-13)
where O represents expected offer; t is defined the same as the time period. Binmore, Rubinstein
and Wolinsky also point out that if ordered preference is assumed, then the utility outcome of
this bargaining model can be represented as
)())1(1()()1( bupxupEU it
it −−+−= (2-14)
where EU represents expected utility; )(xui and )(bui are defined as the utility pair at the
agreement point and the disagreement point, respectively. The above results imply that the
outcome for each player in the strategic model with risk of breakdown is the expected utility.
Then some years later, Muthoo (Muthoo, 1999) showed the same strategic bilateral
model with risk of breakdown by defining the probability a different way. Muthoo explained that
the potential cause of the risk of breakdown is that players might just walk away from the
negotiation table or the intervention by a third party occurs. The model discussed by Muthoo
shows that the equilibrium payoffs for both players are heavily affected by the degree of risk
aversion when negotiation breaks down.
The model is based on Rubinstein’ alternating offer bargaining model discussed above.
Player A and B bargain over the partition of a cake size π . At any time period ∆t , the probability
that the negotiation breaks down is p (where 0<p<1), and then 1-p represents the probability that
the bargaining proceeds to time period ∆+ )1( t . Muthoo assumes that as 0→∆ , 0→p (Recall
that )exp(1 ∆−−= tp so that p goes to zero when ∆goes to zero). Let ix be defined as the payoff
(share of the cake) for the first move player when the agreement is reached, then ix−1 is the
26
payoff for the second move player. Let ib be the payoff for both players at the breakdown point,
then Muthoo called this payoff pair ),( BA bb as the impasse point. Muthoo suggests that the
unique subgame perfect equilibrium with risk averse players is the solution to the following two-
equation system
)()1()( **BBBAB xUppbxU −+=−π (2-15)
)()1()( **AAABA xUppbxU −+=−π (2-16)
where BU and AU are the utility outcomes from this bargaining game; *Ax−π and *
Bx−π are the
outcome cake share; p is the overall probability that negotiation breaks down; 1-p is the overall
probability when the agreement is reached; Bb and Ab are the payoffs at the disagreement point;
*Bx and *
Ax are the payoff cake shares when the agreement is reached for player B and A,
respectively.
As we can see, all the articles mentioned above regarding the strategic approach
alternating offer model with a risk component, define risk as the probability of the negotiations
breaking down. Further more, when the agreement is reached, there exists a unique subgame
perfect equilibrium payoff set ),( **BA xx . When the disagreement occurs, the payoff set for both
players can be represented by ),( BA bb . Thus, the final outcome from this type of bargaining
game is the combination of the payoff at the agreement point and disagreement point considering
the risk.
2.2.2.2.3 Strategic approach alternating offer model with outside options: For any
bargaining game, players can have outside opportunities that are relevant for the negotiation
outcome. However, the strategic approach model with outside options was not developed until
the nineties. In 1984, Shaked and Sutton (Shaked and Sutton, 1984) clearly established the
27
conceptual difference between a “threat point” and an outside option where a threat point can
exist between two players. One question is can only one player have an outside option or can
both players have outside options? Shaked (Shaked, 1994) analyzed a bilateral model where only
one player can opt out and recognized that the assumption that only one side can opt out is not
without loss of generality, which implies that outside options are possible for both players for the
bilateral bargaining game.
In 1995, Muthoo (Muthoo, 1995) studied a bargaining strategic model with two-sided
outside options. The model assumes that player B can search for outside options while
temporally leaving the negotiation table. Thus, the bargaining results can be either player B
chooses the outside option or player B chooses to go back to the negotiation table and set an
agreement with player A. The conclusion suggests that this kind of outside option does not
affect the equilibrium outcome, and the unique equilibrium can still be found approximately by
applying the Nash bargaining solution.
In 1998, Ponsati and Sakovics (Ponsati and Sakovics, 1998) had a note on Rubinstein’s
alternating offer with both players having outside options. The model assumes that (1) players 1
and 2 bargain over a fixed surplus, which is normalized to one, (2) negotiation time is numbered
by the natural numbers and players make alternating offers in each period starting with player 2
making the offer first, (3) both players have outside options that yield payoff ix (where i=1,2;
ix≤0 and 1≤∑ ix ). When both players opt out, they will get outside payoffs, (4) if both players
decide to come back to the negotiation table, when the agreement is reached, the payoff set
)1,( yy − is being discounted for both players by a discount factor tiδ . That is the payoffs for
player 1 and 2 are yt1δ and )1(2 yt −δ , and (5) if the outside options are taken at the negotiation
time period r, then the payoffs for player 1 and 2 are 11 xrδ and 22 xrδ , respectively.
28
Assumptions 1 and 2 are the standard assumptions of the basic Rubinstein alternating
offer model. Assumption 3 suggests that when the payoff from the outside option is greater than
the bargaining surplus (recall that is normalized to 1), the bargaining does not even begin, which
makes the whole bargaining discussion nonsense. That’s the reason the assumption 1≤∑ ix is
made. Similarly, if the outside option is less than 0, no players will ever choose to opt out. Then
the model becomes the standard model again. That’s why ix≤0 is made. Assumptions 4 and 5
are the discount factor, which implies that continued bargaining over periods will result in the
reduction of payoffs due to the presence of the discount factor.
Ponsati and Sakovics do not prove the result mathematically, but more in a logical way.
The article suggests that for any 110 21 ≤−≤≤ xx , no matter when the proposers can opt out
(i.e., in the first round or each negotiation period), a subgame perfect equilibrium is the outcome
set ),1( 22 xx− . This implies that the payoff from the offer that the proposer makes is greater than
the payoff from the outside option for the responder, then the responder will take the offer and
the agreement is reached. And this is true for all periods. Since player 2 is assumed to make the
first offer, the equilibrium payoff for player 2 is 2x . Then, at the agreement point, the payoff for
player 1 will be 21 x− .
In 1999, Muthoo ((Muthoo, 1999), p.100-105) shows mathematically how the model with
outside options but no risk of breakdown works. He describes the model in the following way.
Two players, A and B, bargain over the partition of a cake sizeπ . When one player makes an
offer, the other player has three choices which include “(i) accept the offer, (ii) reject the offer
and make a counter offer ∆ time units later, and (iii) reject the offer and opt out ((Muthoo, 1999),
p.100-101).” It suggests that if the agreement is reached at the time period ∆t , the equilibrium
29
payoff set is ),( **BA xx where BAitrxx iii , ),exp(* =∆−= . Recall that ii tr δ≡∆− )exp( is the
discount factor. If the players choose to opt out to the outside options, then the players obtain the
payoff )exp( ∆− triiω . Notice that the set ),( BA ωω is the payoff pair at the outside option point,
where )(),( πωπω << BA and πωω <+ BA . This is almost the same assumption as the Ponsati
and Sakovics’ article made but allows the outside point payoff to go negative. Then, Muthoo
(Muthoo, 1999) shows that the unique subgame perfect equilibrium for this outside option model
is ),( **BA xx , where
)( and )( if
)( and )1
1( if )-(1
)1
1( and )( if
)1
1( and )
11
( if )1
1(
BBAA
BBAABAB
BBAA
BBAA
*
−>−>−
−≤−−
>+
−−
>−≤−
−−
≤−−
≤−−
=
ABB
ABA
B
BA
ABB
BA
A
BA
B
BA
B
Ax
ωπδωωπδωωπ
ωπδωπδδδ
δωπδωδ
πδδδ
δωωπδωωπ
πδδδ
δωπδδδ
δωπδδδ
(2-17)
and
)( and )( if
)( and )1
1( if
)1
1( and )( if )-(1
)1
1( and )
11
( if )1
1(
BBAA
BBAA
BBAAABB
BBAA
*
−>−>−
−≤−−
>−
−−
>−≤+
−−
≤−−
≤−−
=
ABA
ABA
BA
BA
AB
BA
A
BA
B
BA
A
Bx
ωπδωωπδωωπ
ωπδωπδδδ
δωωπ
πδδδ
δωωπδωπδωδ
πδδδ
δωπδδδ
δωπδδδ
(2-18)
This equilibrium payoff set implies that if the outside option payoffs for player A and B are less
(or greater) than the discounted payoffs from the bargaining agreement, respectively, then both
player A and B will choose to negotiate until the agreement is reached (or to opt out). The
30
payoffs for A and B are πδδδ
BA
B
−−
11 (or Bωπ − ) and π
δδδ
BA
A
−−
11 (or Aωπ − ), respectively. If
player B’s outside option payoff is greater than the payoff when the agreement is reached and
player A’s outside option payoff is less than the payoff when player A takes player B’s outside
offer, then player B will make an outside option payoff offer to player A and A will take that
offer to get the payoff Bωπ − and B’s payoff will be )-(1 ABB πδωδ + . Similarly, if the situation
is reversed for player A and B, the result will be similarly reversed.
In the limit, as 0→∆ , recall that
BA
B
BA
B
rrr+
→−−
δδδ
11 and
BA
A
BA
A
rrr+
→−−
δδδ
11 , then Muthoo (Muthoo, 1999) shows that the
unique subgame perfect equilibrium ),( **BA xx converges to ),( ****
BA xx , which is called “the
outside option principle”
)( and )( if
)( and )( if
)( and )( if )(
BAA
BA
BA
**
+≤
+>
+>
+≤−
+≤
+≤
+
=
πωπωω
πωπωωπ
πωπωπ
BA
A
BA
B
BA
A
BA
BB
BA
A
BA
B
BA
B
A
rrr
rrr
rrr
rrr
rrr
rrr
rrr
x (2-19)
****AB xx −= π (2-20)
Equation (2-19) and (2-20) suggest that when the negotiation time period goes to zero, the payoff
is only affected by the discount rate if both players choose to negotiate until the agreement is
reached.
In conclusion, all the strategic models paid more attention to what strategies are being
used which might affect the final bargaining results. The axiomatic approach model has a simple
31
equation to maximize, and then the bargaining problem can be solved when the utility
presentation is established. The axiomatic approach is very easy to apply in a real world
problem. However, the strategic approach models do not have any equations to solve for
bargaining directly, and the result of what could be depends on what strategies are being used.
This approach is difficult to apply. Thus, people are thinking that there might be a way to
combine the advantages of the two different approach models. The question is can the solution of
the strategic approach model converge to the solution (Nash equilibrium) of the axiomatic
approach model? If so, under what conditions?
2.2.2.2.4 Solution approximation of strategic approach to axiomatic approach: In
1986, Binmore et al. (Binmore, et al., 1986) examined the relationship between the axiomatic
approach and the strategic approach to bargaining theory. They explained that compared to the
axiomatic approach, the strategic approach of non-cooperative bargaining games deals with the
complexity of the negotiation process and might have two different incentives for parties to
bargain which include the parties’ impatience to settle an agreement and parties’ fear of
breakdown (i.e., the component of risk). Binmore, Rubinstein et al. conclude that Nash’s
bargaining solution can be seen as the approximation of the equilibria of the strategic approach
models.
Binmore et al. assume rationality and complete information of preferences for all parties,
the same as Rubinstein’s strategic approach alternating offer model. Following the strategic
model with the exogenous risk of breakdown discussed above, Binmore, Rubinstein et al.
showed that the approximate solution for this model is the Nash bargaining solution of
)]()()][()([max arg 2211xbUxUbUxU −− (2-21)
32
where 1U and 2U are the utilities for player 1 and 2 respectively at the agreement point; )(1 bU
and )(2 bU are the utilities for player 1 and 2 respectively at the breakdown point; x is the
solution payoff at the agreement point; and b the payoff when negotiation breaks down. It proves
that as 0→∆ , the solution *x of the equation (2-14) )())1(1()()1( bupxup it
it −−+− converges
to the Nash bargaining solution of the equation (2-21) )]()()][()([max arg 2211xbUxUbUxU −− .
Overall, Binmore, Rubinstein et al. show that the Nash bargaining solution can be applied
to some economic bargaining situations by using the strategic approach models involved with the
utility representations and the consideration of either parties’ impatience or parties’ fear of
breakdown during the negotiation process.
In 1999, Muthoo ((Muthoo, 1999), p.113-116) first showed a model with risk of
breakdown which could include negotiators walking away from the negotiation table, and
intervention by a third party. Then, Muthoo showed another model with both the risk of
breakdown and outside options. These models have already been discussed in this chapter.
Muthoo suggested that the results from both models can converge to the generalized Nash
bargaining solution as long as the time period between two negotiations goes to zero ( 0→∆ ).
Following Muthoo’s ((Muthoo 1999), p.78-118) strategic approach alternating bilateral
model with risk discussed above, the author shows that the solution of that model can converge
to the Nash bargaining solution under some conditions. Define the risk of breakdown in the same
way as the overall probability that negotiation breaks down, which is )exp(1 ∆−−= λp . Recall
that λ is the probability that negotiation breaks down in each period. The author suggests that in
the limit as 0→∆ , the payoff pair ))(),(( **BBAA xUxU (equations (2-15) and (2-16)) of the above
model with a risk component converges to the following Nash bargaining solution of
33
))((max, BBAAUU
bUbUBA
−− (2-22)
where the terms are defined the same as equations (2-15) and (2-16). Notice that p goes to zero
as 0→∆ (equation (2-12)) which means that when the risk of breakdown gets smaller, the
approximation of the solution of the strategic approach model to the Nash bargaining solution is
applicable.
Then, Muthoo extends this solution approximation to the generalized Nash bargaining
solution. To extend the strategic approach bilateral alternate offer model with risk of breakdown,
the author assumes that player A and B have different views on the risk of breakdown, which is
represented by Ap and Bp , respectively. Moreover, the timing of the agreement is considered so
that there exists discount factors Aδ and Bδ for player A and B respectively. Thus, by considering
an outside option as the constraints for player A and B’s decision, when BB b≥ω and AA b≥ω ,
the solution pair ),( **BA VV from the following equation system
))1(( *1*BBBBBA VpbpV δφ −+= − (2-23)
))1(( *1*AAAAAB VpbpV δφ −+= − (2-24)
converges to the generalized Nash bargaining solution of
AA
BABBAAUU
dUdU σσ −−− 1
,)()(max (2-25)
subject to AABA dUUU ≥Ω∈ ,),( and BB dU ≥ , where *AV , *
BV represent the utility for player A
and B at the agreement point, respectively. All other terms are defined the same as those of the
strategic approach bilateral alternate offer model with risk of breakdown discussed above. AU ,
Ad represent the utility for player A at the agreement point and the disagreement point,
respectively; BU , Bd represent the utility for player B at the agreement point and the
34
disagreement point, respectively; τσ =A is defined as the bargaining power for player A and
Aσ−1 is the bargaining power parameter for player B. The variables can be defined as
BBAAABABA UUUUUUU ωωππ ≥≥−=≤≤=Ω and ,,0:),( (2-26)
)2/()( BABA rrr +++= λλσ (2-27)
),(),(λ
λλ
λ++
=B
B
A
ABA r
br
bdd (2-28)
where λ (λ >0) is the negotiation breakdown rate calculated as the number of disagreements
divided by the total number of negotiations.
However, there are some components that we need to pay attention to which include (1) if
ib is zero, then id becomes zero, (2)λ is actually the calculated probability of breakdown, (3) the
total payoff from the outside options )( BA ωω + is less than the total bargaining cake π ,
otherwise bargaining will not begin in the first place, (4) the author does not assume that the
breakdown point is always the same as the opt out point (not always iib ω= ), outside option
payoff only enters as a constraint, and (5) at the limit as 0→∆ , the present value of expected
payoff only slightly differs from the actual payoff at time ∆ which implies that the discount
factor does not actually affect the payoff that much.
2.2.2.3 Summary of bilateral bargaining theory
As a summary of bilateral bargaining theory, two different approaches were discussed
which include the axiomatic approach and the strategic approach. The axiomatic approach model
is based on a series of assumptions and the result is called the Nash bargaining solution. For most
of the cases where symmetry is not satisfied, the solution is considered as a generalized Nash
bargaining solution. The strategic approach models are more complicated depending on what
strategy players choose during the negotiation process. The basic strategic approach model was
35
developed by Rubinstein in 1982. Then the model was extended to the basic model with risk of
breakdown or outside options. Furthermore, as it is discussed above, the solution of the strategic
approach models converge to the Nash bargaining solution at the limit as the negotiation time
period goes to zero. This allows the application of these models to be easier where pure
mathematics of the Nash bargaining solution can be used.
2.2.3 Multilateral Bargaining
Recently, some theoretical work was done in the field to extend the above bilateral
bargaining model. Compared to the above extended outside option models, some of them extend
the model to the outside option but do not allow the negotiator to come back to the negotiation
table once that player chooses to opt out, such as Sloof (Sloof, 2004), and Li (Li, et al., 2005).
But most of the literature has paid more attention to developing the multilateral bargaining
theory based on bilateral bargaining theory. These models can be divided into two categories
which include the axiomatic approach models and the strategic approach models. The strategic
approach multilateral bargaining models are presented in two ways which include (1) reduce the
multilateral bargaining to a series of bilateral bargaining models and (2) form coalitions among
the negotiators on the supply and the demand side, then bilateral bargaining theory is used
between these two sides.
2.2.3.1 Axiomatic approach
To extend the axiomatic approach bilateral bargaining model, Krishna and Serrano
(Krishna and Serrano, 1996) and Schneider (Schneider, 2005) all pointed out that the Nash
bargaining solution from the axiomatic approach is very easy to extend to the n-player case,
where the solution must maximize the product of all players’ utilities and also satisfy Nash’s four
axioms ( IAT,SYM,PAR,IIA) discussed above.
36
For the case of extending Nash’s bargaining solution to n players, Schneider (Schneider,
2005) showed us the bargaining power associated with the n player case model with application
to international relations . It defines power as “actors are powerful if they possess some capacity
to withstand pressure or, to put it another way around, to force other actors to give in”(Schneider,
2005). Schneider suggests that the bargaining power can shift Nash’s bargaining solution when
one player has a better outside option. The n-player situation model is
icii
n
iOQuOu ))()((max
1−Π
=Θ∈ (2-29)
where Θ represents all feasible offer sets for all players; ∑=
=n
ii Cc
1
; ic is capacity for each player
representing bargaining power; C is the overall bargaining power parameter; )(Oui represents the
outcome utility for player i; and )(Qui is the utility at a disagreement point for player i.
Schneider concludes that the larger the country’s capabilities, the more negotiation power the
country has over the international negotiations. It is obvious that to extend the bilateral model to
a multilateral model for the axiomatic approach is easy.
2.2.3.2 Strategic approach
To extend the Rubinstein alternating offer model to the n players’ case, several different
model forms have been analyzed based on different negotiation processes. Some early articles
tried to reconcile the Nash bargaining solution and the Rubinstein alternating offer model for the
n player negotiating situation. For example, Krishna and Serrano (Krishna and Serrano, 1996)
had an article about multilateral bargaining which attempted to extend Rubinstein ‘s alternating
offer model to a n player model and links the solution to Nash’s bargaining solution.
Unfortunately this n player game has many perfect equilibria, which can’t be represented by a
unique Nash bargaining solution. To extend Rubinstein’s alternating offer model directly to the
37
multilateral level is unfeasible. Thus, past literature is focused on either reducing multilateral to
bilateral or forming a coalition before bilateral negotiation, in order to have the result of the
model approximate the multilateral Nash bargaining solution.
2.2.3.3 Solution approximation of strategic approach to axiomatic approach
Some of the passed articles assume that there could be coalitions forming before the
negotiation process, and then the n player multilateral game can be reduced. However, Bennett
(Bennett, 1997) points out that some problems are associated with a coalition. For instance, the
distribution of the gain within each coalition and the opportunity cost of participating in a
particular coalition are problems that must be addressed.
Bloch and Gomes (Bloch and Gomes, 2006) studied a model by assuming multiple players
in the negotiation process with two step coalitional games. Step one is the contracting phase,
which is to form a coalition among some players. Step two is the action stage, which means that
each existing coalition is playing a simultaneous bargaining game together, and inside each
coalition, there exists a unique Nash bargaining solution. However, this article restricts the
strategies to the Markovian strategies which result in a markov perfect equilibrium. This
equilibrium could be efficient or inefficient depending on whether outside options are
independent or non-independent of the actions of other players.
Some other studies assume that this n player bargaining situation can be reduced to n-1
bilateral bargaining sessions where the bilateral strategic model is used to analyze the subgames.
For example, Suh and Wen (Suh and Wen, 2006) point out that as the discount factor goes to one
for all players, the subgame perfect equilibrium for every subgame within a n-1 bilateral
bargaining game converges to the Nash bargaining solution. Suh and Wen assume the model has
n players who negotiate how to split a fixed pie through n-1 bilateral subgames. Further more,
38
they assume that there are two types of procedures in each bilateral bargaining Rubinstein
alternating offer game which include the offer procedure and the demand procedure.
In the offer procedure, Suh and Wen assume that one player (the proposer) makes an offer
and if the other player (responder) accepts the offer, then the responder will receive their share
and exit the game. Then, the proposer will continue to bargain with the next player by continuing
the same process. The results show that there is a unique subgame pefect equilibrium that is
efficient and converges to the Nash equilibrium when the discount factor goes to one. The
solution for this procedure is
))(()( jjjjjij XYsuXu += δ (2-30)
))(()( ij
iiiji XYsuXu += δ (2-31)
where, for player j and i respectively, ju and iu represent the utility at the agreement point; iX
and jX are players i and j’s offer at the agreement point; jδ and iδ are the discount factors; Y is
the total share offered to n-2 players; )( jjj XYs + is the share for player j when j offers jX and
accepts iX ; )( jj
i XYs + is the share for player i when i offers iX and accepts jX .
The demand procedure is slightly different from the offer procedure at which the
proposer will exit the game instead of the responder when the offer from the proposer is not
accepted by the responder, and all other procedures are the same. Thus, similar results can be
drawn from the procedure where the utilities are
)())(( jjjijjj XuXYsu δ=+ (2-32)
)())(( iiijj
ii XuXYsu δ=+ (2-33)
where all terms are defined the same as equations (2-30) and (2-31)
39
Suh and Wen then prove that the solution of the above two subgame perfect equilibrium
outcomes converge to the symmetric Nash bargaining solution of the following problem, as the
players’ common discount factor goes to one.
)( max1
ii
n
ixuΠ
=
, subject to ∑=
≤n
iix
11 and nixi ,...,1,0 =∀≥ (2-34)
where ix represents the offer played by player i.
Suh and Wen conclude that the Nash bargaining solution can be used to solve this
complex n player multilateral bargaining situation when the common discount factor goes to one.
Moreover, if players have different discount factors, when the length of each bargaining period
goes to zero (i.e., )0→∆ , the subgame perfect equilibrium outcomes of the above multilateral
bargaining model, which when reduced to n-1 bilateral bargaining models, converges to the
asymmetric Nash bargaining solution.
2.2.3.4 Summary of multilateral bargaining theory
As a summary of multilateral bargaining theory, it’s still developing. Carraro et al.
(Carraro, et al., 2005) have a systematic discussion of bargaining theory development until 2005
in their World Bank policy research working paper. The paper includes all major developments
of bargaining theory and has almost the same conclusion on the multiple player non-cooperative
bargaining models such that the direct extension to an n player model from Rubinstein’s two
player alternating offer model is less successful due to the existence of multiple subgame perfect
equilibria, for instance, Shaked’s past work (Shaked, 1986). Shaked also concludes that various
solutions have been proposed to isolate a unique equilibrium and an appealing way is to modify
the structure of the game. Other authors, Jun (Jun, 1987), Chae and Yang (Chae and Yang, 1994,
Chae and Yang, 1988), and Krishna and Serrano (Krishna and Serrano, 1996) follow Shaked’s
40
suggestion to solve the multi-player game by reducing the game to a series of bilateral
negotiation games.
2.2.4 Bargaining Theory Summary
In summary, bargaining theory has been developing since 1950 when Nash’s axiomatic
bargaining solution was first introduced. Symmetric (Nash, 1950) and asymmetric (Binmore,
1992) axiomatic models have been developed. A more complex strategic model ( (Rubinstein,
1982) alternating offer model) was introduced in 1982. All these models were concentrated on
bilateral two player games. Later, Roth (Roth, 1985), Binmore et al. (Binmore, et al., 1986), and
Muthoo (Muthoo, 1999)) extended the strategic model with risk of breakdown. In the 1990’s and
2000’s, Shaked (Shaked, 1994), Muthoo (Muthoo, 1995), Ponsati and Sakovics (Ponsati and
Sakovics, 1998), Muthoo(Muthoo, 1999), Sloof (Sloof, 2004), and Li et al.(Li, et al., 2005)
extended Rubinstein’s basic alternating model to the model with outside options. Shaked
(Shaked, 1994) pointed out that the assumption of “one side opt out” lost generality. But
compared with the axiomatic approach by Nash, all strategic models are difficult to apply. Thus,
Binmore et al.(Binmore, et al., 1986), and Muthoo (Muthoo, 1999) showed that the strategic
model’s solution approximated the Nash bargaining solution at the limit when the negotiation
time period goes to zero ( 0→∆ ).
However, in reality, most issues are composed of multiple players instead of only two
players. More and more authors try to extend the two player alternating model to an n player
game. To extend the axiomatic approach models to solve for the n-player Nash bargaining
solution is straight forward (Krishna and Serrano, 1996), (Schneider, 2005); however, the direct
application of Rubinstein’s strategic model is less successful in the empirical world because of
the existence of multiple subgame perfect equilibria (Krishna and Serrano, 1996). To solve the
multiple equilibrium problems, most authors change the structure of the multi-player game. Jun
41
(Jun, 1987), Chae and Yang (Chae and Yang, 1988), Chae and Yang (Chae and Yang, 1994),
Krishna and Serrano (Krishna and Serrano, 1996), and Suh and Wen (Suh and Wen, 2006) tried
to reduce the multilateral bargaining to a series of bilateral bargaining sessions. Suh and Wen
(Suh and Wen, 2006) conclude that at the limit, when negotiation time goes to zero, the solution
of this series of bilateral models converge to the asymmetric Nash bargaining solution.
Other work by Bloch and Gomes (Bloch and Gomes, 2006) tried to reduce the game to
multi-stage games such that coalitions are formed at the first stage, and the second stage is to
negotiate as the bilateral alternating offer bargaining model, and the third stage is to discuss the
profit distribution inside the coalition. Bloch and Gomes conclude that inside the coalitions, the
convergence of the Nash bargaining solution can be proved, but the over all game has a Markov
equilibrium. The theory of multilateral bargaining is still developing. As we can see above,
people are still trying to figure out a way to solve the basic multiple player problem. No literature
on multiple-player models with risk of breakdown and outside options has been found.
2.3 Distribution Channels Bargaining Theory
2.3.1 Introduction
A cooperative is one distribution channel participant. Once finishing the review of
bargaining theory, the question becomes has bargaining theory developed in the area of
distribution channels? So I extend the literature of bargaining theory to the specific field of
distribution channels, which includes supply chain bargaining and cooperative bargaining.
2.3.2 Supply Chain Bargaining
Literature on the bargaining within distribution channels or supply chains can be very
complex. Some of the literature paid attention to distributional bargaining through bilateral
bargaining models (Iyer and Villas-boas, 2003). Iyer and Villas-Boas define a distribution
channel as a manufacturer producing a product and a retailer selling this product to end
42
consumers. They examine the channel relationship and the relative bargaining power between
manufacturers and retailers by using two different well-known bargaining processes which
include Nash’s bargaining solution and Rubinstein’s alternating offer bargaining model. The
same equilibrium solution for the demand from the above two models with different definitions
of the bargaining power parameter are analyzed. By the end, Iyer and Villas-Boas explained that
the two player case can be extended to allow more retail competition; however, the authors left
this for further research.
Other authors analyzed the supply chain bargaining through multistage game analysis.
Nagarajan and Bassok (Nagarajan and Bassok, 2002) examined a supply chain contracting
problem with one single buyer (assembler) and n suppliers. Two assumptions the authors made
were that all players were risk neutral and that the disagreement points were normalized to zero.
The two major issues solved were profit allocation among coalition members and at equilibrium
how the stable supplier coalitions were affected by their relative bargaining power. Three stages
of the negotiating game include stage 1 as the suppliers form coalitions, stage 2 as these
coalitions compete for a better position in the negotiation sequence, and stage 3 as the buyer
(assembler) negotiates with the coalitions in sequence. The game was solved backwards from
stage 3 to stage 1. In stage 3, the Nash bargaining solution was used for the multilateral (a
sequential bilateral) negotiation process as
βα )()(max, nAxx
xxnA
(2-35)
where Ax represents the share of the negotiated pie for the assembler; nx is the total share of the
negotiated pie for all n suppliers inside the coalition; α and β represent the bargaining power
for the assembler and the supplier coalition. The outcome solution of this stage for the assembler
43
and the thi supplier is CnA Π=απ and Ci
i Π= − βαπ 1 (where CΠ is the allocated pie),
respectively.
Then, in stage 2, a dynamic coalition formation was used to solve the problem of these
supplier coalitions competing for the negotiation sequence by using the Nash bargaining
solution. Nagarajan and Bassok show that at every Nash equilibrium, the profits for every
supplier and the assembler are Cn Π− βα 1 and )1( 1βα −−Π nC n , respectively, where all terms are
defined the same as above.
Finally, back to stage 1, the stability of supplier coalitions is discussed where unique
stablity occurs when the assembler’s bargaining power parameter α is less than 1/n (n is the
number of suppliers). Over all, this article provides a way to solve the multi player bargaining
problem by restructuring the game to a multi-stage game and still use the Nash bargaining
solution to solve the problem. However, Nagarajan and Bassok indicated that the risk averse
players could be very hard, if not impossible, to obtain closed form solutions for profit
allocations.
Over all, there is only a few articles applying bargaining theory on the supply chain. From
the two articles reviewed on supply chain bargaining, (1) profit maximization is assumed to be
the goal for all negotiators, (2) the Nash bargaining solution is used to solve the problem, and (3)
the multi player bargaining game has been reduced to a bilateral game so that the problem can be
solved.
2.3.3 Cooperative Bargaining
Cooperative bargaining in U.S. agricultural fruit and vegetable industry has a long history
dating from the 1960’s. In 1962, Helmberger and Hoos (Helmberger and Hoos, 1962) developed
a theory of cooperation by recognizing a cooperative enterprise as a decision-making entity, a
44
broader interpreted firm. Within an organizational framework, Helmberger and Hoos analyzed
the performance of cooperatives by assuming that the goal of the cooperative organization was to
maximize the price Pm for any amount of M which the member firms choose to supply, but
subject to the constraint that all costs including fixed costs, F, are met. The result showed that the
cooperative surplus was maximized by a maximum Pm.
In 1963, Helmberger and Hoos (Helmberger and Hoos, 1963) discussed the economic
theory of bargaining in agriculture. Regarding the theoretical approaches to bargaining,
Helmberger and Hoos indicate that bilateral monopoly theory was the simplest type of market
structure that implies a bargaining process to determine the outcome. They assumed that joint
profit of the monopolist and monopsonist was maximized at a given quantity with an
undetermined price. Helmberger and Hoos indicated that cooperative bargaining can be an
effective tool for certain markets and for certain farm products such as processing fruits and
vegetables, sugar beets, and fluid milk.
In 1967, Johnson (Johnson, 1967) appraised collective bargaining in the U.S. milk
industry. Johnson introduced the use of bargaining theory to analyze U.S. milk cooperatives and
concludes that to achieve income gains from bargaining, cooperative unity and strength was the
key.
In 1974, Ladd (Ladd, 1974) pointed out the lack of well-developed models to understand
cooperative behavior, then developed an operational method to analyze the behavior of a
cooperative selling raw material to processors and the bargaining price. Ladd assumed that the
cooperative's objectives could be either the maximization of the raw material price received by
members or the maximization of quantity marketed through the cooperative.
45
In 2006, Oczkowski (Oczkowski, 2006) developed the generalized Nash bargaining
solution for a bargaining cooperative selling its output to a single buyer. Oczkowski analyzed the
case of the bilateral monopoly where a cooperative bargains with a single trader over the price
and quantity for a good. The theory used in the article was the generalized Nash bargaining
model. The generalized framework was consistent with various strategic models and the standard
axiomatic foundation of the Nash program.
Oczkowski assumed that the cooperative acts only as a bargaining agent, and the
cooperative only exists for the members and hence makes no profit (ΠC=0). Also, all of the
cooperative surplus (CS) is returned to members (CS=PMY), where Y is output and PM is the per-
unit price returned to members. The cooperative faces fixed bargaining costs (B) which members
incur if negotiations reach agreement. Oczkowski also assumed that the Nash disagreement point
is a zero payoff for both the cooperative and processor. Based on all the above assumptions, the
cooperative’s profit function (ΠC), the cooperative’s surplus function (CS), and the members’
profit function ( Mπ ) are constructed. By using the static approach, when cooperatives and
processors bargained for price and quantity, three objectives can be made.
The first objective is to maximize members’ profits ( Mπ ). The Nash bargaining program
is used to find the optimal output Y and output price PY by
ττττ ππ −− −−== 11
),(P)()()()( max
YYYY
PARYATCPYPMF (2-36)
where F represents the objective function that needs to be maximized; Mπ , Y, and PY are defined
the same as above; τ )10( ≤≤ τ and τ−1 are the bargaining power for cooperative and processor,
respectively ( 5.0=τ implies equal bargaining power); Pπ is the profit for the processor; ATC is
the average total cost for cooperatives; and AR is the average revenue for processors. The
solution of this maximization problem can be obtained as
46
ATCARPY ⋅−+⋅= )1( ττ (2-37)
)/()(Y
ARY
ATCATCARY∂∂
−∂
∂−= (2-38)
where all terms are defined the same as above. The results show that the bargained price falls
between an upper limit determined by the processor’s average revenue and a lower limit
determined by the cooperative’s average total cost and transacted quantity occurs at the
intersection of the marginal cost and marginal revenue curves.
The second objective is to maximize the cooperative’s surplus (CS). The Nash bargaining
equation to find the optimal Y and PY is
ττττ π −− −−== 11
),(P)()()()( max
YYYY
PARYABPYPCSF (2-39)
where all terms are defined the same as above. The third objective is to maximize average
returns ( MP ) for cooperative members. The Nash bargaining equation is to find the optimal Y
and PY is
ττττ π −− −−== 11
),(P)()()()( max
YYYMY
PARYABPYPPF (2-40)
where AB is the average bargaining cost for cooperatives. However, Oczkowski shows that when
either maximizing the members’ surplus or maximizing the price, the Nash solution will result in
highest quantity with lowest price, or lowest quantity with highest price, respectively.
In summary of cooperative bargaining theory, we found that a cooperative acts as an
organization or as a firm, which has possibly three different goals which include maximize the
member’s surplus, maximize the member’s price of the commodity, or maximize the member’s
profit assuming the processor is a profit maximization firm. Furthermore, the above literature
suggests that cooperative bargaining is a bilateral bargaining problem where the Nash bargaining
47
solution can be used to solve a profit maximization problem for both players to find out the
negotiated price and quantity of the product.
2.3.4 Distribution Channel Bargaining Theory Summary
Whether it is a cooperative, representing the producers or a manufacturer acting as a firm,
taking profit maximization as their goal to negotiate with the processors or other buyers, a Nash
bargaining model with bilateral bargaining or a multi player bargaining (which can be reduced to
a series of bilateral bargaining) is used to solve the negotiation problem, no matter what
strategies are adapted.
2.4 Bargaining Model Applications
Theoretically, from the literature review on bargaining theory, the multi player
bargaining game can be reduced to a series of bilateral bargaining games. At the limit, the
generalized Nash bargaining solution can be used for each bilateral game. However, it’s also
important to know what the applications are.
2.4.1 Model Applications
The application of bargaining theory has existed for a long time. However, most of the past
work is related to labor issues, such as labor bargaining (Bloom, 1980), (Neumayer and Soysa,
2006), or related to insurance premium bargaining (Maude-Griffin, et al., 2004). Only a few
articles, showed in the following content, relate to cooperative bargaining.
In 1969, Babb et al.(Babb, et al., 1969) analyzed factors which affected the bargaining
process and outcome of negotiations for the tomato processing industry in Indiana and Ohio
during the 1966 growing season through ranking the survey results from tomato processors,
grower representatives, and growers. They concluded that processors and grower representatives
(cooperatives) underestimated supply response and consumers' response to higher prices.
Significant differences between grower and processor attitudes toward bargaining were
48
identified. Processors were primarily concerned about quality factors and growers with price.
This article is an early cooperative bargaining work; however, it does not use a Nash bargaining
framework.
In 1998, Folwell et al. (Folwell, et al., 1998) used the general economic theory of
bargaining to analyze the bargaining process and its role in price discovery within the Pacific
Northwest asparagus industry. The theory is based on the equilibrium of quantity and
disequilibrium of price for the inverse supply and demand equations as
),( dddd ZQPP = Buyer’s Inverse Demand (2-41)
),( ssss ZQPP = Seller’s Inverse Supply (2-42)
*QQQ sd == Quantity Equilibrium (2-43)
ds PPP αα +−= )1(* Price Equilibrium under Bargaining (2-44)
where dP and sP represent the demand and supply price for the buyer and the seller, respectively;
dQ and sQ are the quantity transacted by the buyer and the seller, respectively; dZ and sZ are
exogenous variables that affect the demand and the supply, respectively; *Q is the equilibrium
quantity; *P is the negotiated price; and α−1 and α represent the relative bargaining power
coefficients for the buyer and the seller, respectively.
The empirical model is based on two-rounds of bargaining to set up the price for asparagus
with processors and growers. At the beginning, a growers’ representative (a cooperative)
provides the first round price and processors either accept or reject the offer. Then the
cooperative will offer the second price. If processors accept, the bargaining game ends; however,
if processors reject the offer, the price will be determined by the arbitration board with members
from both growers and processors. Folwell, Mittelhammer et al. estimated their model
49
econometrically with ordinary least squares for the first round bargaining and Heckman’s two-
stage estimation procedure (probit model) for the second round of bargaining.
The results indicated that (1) basic supply and demand forces exerted substantial influences
on the bargaining process, (2) expected levels of supply played a large role in the level of prices
offered and also influenced whether the bargaining process required more than one round to
complete, and (3) past prices also influence current offers. This article estimated the bargaining
price econometrically by also considering the process of bargaining as all strategic approach
models do.
In 1991 and 1999, Oczkowski conducted some analysis on bilateral monopoly bargaining
by using the Nash bargaining solution and econometric techniques. In his 1991 article,
Oczkowski (Oczkowski, 1991) developed a disequilibrium econometric model for single markets
with fixed quantity transacted or contracted and varied negotiated price. The maximum
likelihood method was used to estimate the bargaining power of the Australian tobacco-leaf
market.
The 1999 article by Oczkowski (Oczkowski, 1999) is an extension of his 1991 article. It
allows fix quantity to be varied over different contracts, and allows the strictly bilateral
monopoly market to have outside traders. A disequilibrium econometric model was used and the
bargaining power was estimated by the maximum likelihood (ML) method in the coking coal
trade market between Japan and Australia. The advantage to use this model and ML is that only
the data on price and quantity transacted was needed plus some exogenous shifters for the
demand and supply equations.
Oczkowski follows the assumption that both seller and buyer will trade if their expected
profit is greater than 0. Then, the maximization of joint profit was assumed to have the
50
asymmetric Nash equilibrium solution. Under the above assumptions, joint profit is maximized
for this pure monopoly market as
ττ ππ −−− 1
PQ,]0),([]0),([ max PQPQ ds (2-45)
where Q and P represent the negotiated quantity and price; sπ and dπ are the profit for the
supplier (cooperative) and the buyer (processor), respectively; and τ and τ−1 represent the
bargaining power for the cooperative and the processor, respectively.
For the case with possible outside options, in addition to the above assumptions, the
breakdown point is assumed to be the outside option point instead of the zero profit point, and
then the objective function becomes
ττ −⋅Π−⋅Π⋅Π−⋅Π 1
PQ,)](([)()]([()( max QPQPQQPQPQ d
dds
ss (2-46)
where )([( QPQ ss
⋅Π and )(([ QPQ dd
⋅Π are the profits for the cooperative and the processor at the
outside option point, respectively; and all other terms are defined the same as above.
Oczkowski indicates that for the above cases (equation (2-45) and (2-46)), the same
solution can be obtained as
)()1()( QPQPP sd ⋅−+⋅= ττ (2-47)
/)(/)(/)()( QQPQQPQPQPQ dssd ∂∂−∂∂−= . (2-48)
Econometrically, the article shows that the solution can be written as the following
disequilibrium model
dtt
dt
dt uQXgP += ),,(α (2-49)
stt
st
st uQXhP += ),,(β (2-50)
Pt
stt
dttt uPPP +⋅−+⋅= )1( ττ (2-51)
51
),( θτ btt Xk= (2-52)
qt
st
dttt uPPQ +−= )(δ (2-53)
/),,(/),,(/1 ttdttt
stt QQXgQQXh ∂∂−∂∂= αβδ (2-54)
where for the processor and the cooperative, respectively, dtP and s
tP are the demand price and
the supply price; tP and tQ are the negotiated price and quantity; dtX , s
tX and btX are the
exogenous shifters for the demand, the supply and the bargaining power equation, respectively;
g, h, and k are functional forms; and α , β , θ and δ are parameters; all u are residuals; and τ
and τ−1 are bargaining power parameters for the cooperative and the processor, respectively.
This econometric model is estimated using the maximum likelihood method
dt
stP
P
tts
td
t dPPdQPPPfLt
t
∫ ∫∞
∞−∏= ),,,( (2-55)
where f is the joint density function assumed to be a bivariate normal distribution. Double log
functional form was assumed for both the supply and the demand equation. A general linear
function form was assumed for the negotiated price equation.
Overall, this article is a good application example of bargaining theory. It illustrated the
utility of employing limited dependent variable techniques to estimate the bargaining power in
bilateral monopoly markets. It suggests that the more patient bargainer generally has more
bargaining power.
In 2008, there is one article done in the milk bargaining area (Prasertsri and Kilmer, 2008).
Prasertsri and Kilmer use the generalized Nash bargaining solution (bilateral case) to develop an
econometric model and estimate the negotiated price and bargaining power between the milk
processor and the milk cooperative.
52
Instead of assuming profit maximization, both the cooperative’s utility and processor’s
utility are jointly maximized using the following generalized Nash bargaining model as
αα −−− 1)()(max bppbccp
UUUUN
(2-56)
with the econometric solution of
ijbcj
jbcjbpj
jbpNi eDppDppp
iiii+−−−−+−++= ∑∑
==
)]1(1[)1( 1
12
211
12
21 αααααα (2-57)
where cU and bcU are utilities for the cooperative at the agreement and breakdown point,
respectively; pU and bpU represent utilities for the processor at the agreement and breakdown
point, respectively; α and α−1 are bargaining power parameters for the cooperative and
processor, respectively; Np is the negotiated price; bpp and bcp are the cost and the minimum
price that the processor pays to the cooperative when agreement is reached or breaks down; and
D is a monthly dummy variable. The model is estimated by the maximum likelihood method and
concludes that the cooperatives’ bargaining strength exceeds that of the processor for all
negotiated periods.
2.4.2 Bargaining Model Applications Summary
In summary, all the past works have been shown such that the generalized Nash bargaining
solution can be used to analyze bilateral negotiation between any two traders over some
commodity, or used to analyze a multi player game by reducing it to a series of bilateral
bargaining games. Some empirical work has been done in the field related to both bargaining and
econometrics, such as Folwell, Mittelhammer et al.(Folwell, et al., 1998), Oczkowski
(Oczkowski, 1999), and Prasertsri and Kilmer (Prasertsri and Kilmer, 2008). However, all three
empirical econometric models are dealing with bilateral bargaining. None of them has risk of
breakdown associated.
53
2.5 Summary, Conclusions and Discussions
2.5.1 Summary
The bargaining theory has been developing ever since 1950 when John Nash (Nash, 1950)
presented an axiomatic approach to solve the bilateral bargaining problem. This approach is
called the symmetric Nash bargaining solution based on four axioms which include Invariance to
positive affine transformations (IAT), Symmetry (SYM), Pareto optimality (PAR) and
Independence of irrelevant alternatives (IIA). Then, Roth (Roth, 1979), and Muthoo(Muthoo,
1999), relaxed the symmetry assumption and developed the generalized (asymmetric) Nash
bargaining solution. Another approach called the strategic approach model, was introduced by
Rubinstein in 1982 (the basic alternating bilateral offer model) (Rubinstein, 1982). Further, Roth
(Roth, 1985), Binmore, Rubinstein et al.(Binmore, et al., 1986), and Muthoo (Muthoo, 1999)
extended this basic bilateral alternative offer model to a model with the component of risk of
breakdown. Muthoo (Muthoo, 1995), Ponsati and Sakovics (Ponsati and Sakovics, 1998) and
Muthoo (Muthoo, 1999) further extended the bilateral alternative offer basic model to a model
with possible outside options. Finally, Binmore et al.(Binmore, et al., 1986) and Muthoo
(Muthoo, 1999) showed that the solution to the strategic approach model approximated the Nash
bargaining solution at the limit when the negotiation time period goes to zero.
Later, bargaining theory development extended from the bilateral two player model to the
multi player model. The axiomatic Nash bargaining solution is easy to extend to an n player
bargaining model, (Krishna and Serrano, 1996) and (Schneider, 2005). However, authors tried to
extend the strategic bilateral model to an n player bargaining model in two ways. First, Jun (Jun,
1987), Chae and Yang (Chae and Yang, 1994, Chae and Yang, 1988), Krishna and
Serrano(Krishna and Serrano, 1996), Suh and Wen (Suh and Wen, 2006) reduce the multiple
54
player game to a series bilateral games. Second, Bloch and Gomes(Bloch and Gomes, 2006)
formed a coalition first, and then reduce the game to a bilateral game.
Since cooperatives are one of the participants in the distribution channel, the reviews on
distribution channel bargaining has been done too. Regarding distribution channel bargaining,
Iyer and Villas-boas (Iyer and Villas-boas, 2003) and Nagarajan and Bassok (Nagarajan and
Bassok, 2002) agree that profit maximization is the goal for all participants including
manufacturers, retailers and assemblers. Furthermore, the Nash bargaining solution was used for
both the bilateral bargaining models (Iyer and Villas-boas, 2003) and multi player bargaining
models (Nagarajan and Bassok, 2002). As a special participant in agricultural, the cooperative
and its bargaining game with processors has a long history being discussed back to the 1960s.
Maximizing the commodity price, maximizing the cooperative’s surplus and maximizing profit
are all discussed as the goal of cooperatives (Helmberger and Hoos, 1962), (Helmberger and
Hoos, 1963),(Oczkowski, 2006).
The empirical work of bargaining theory majorly focused on agricultural products
(Folwell, et al., 1998), (Prasertsri and Kilmer, 2008), and some international trade commodities
(Oczkowski, 1999). These three articles applied bargaining theory in the real world bilateral
negotiation problems and used the Nash bargaining solution to solve the problems. Then, the
econometric technique (Maximum Likelihood Estimator) was used to estimate the solution
models. The empirical work has analyzed the bargaining power through the negotiated price
estimation.
2.5.2 Conclusions and Discussions
Over all, the axiomatic approach model is based on a series of axioms and less on the
consideration of strategies adopted by all bargaining game participants. The results of the
strategic approach model heavily depend on what strategies and who plays the strategy first.
55
However, the results of axiomatic approach models ( i,e., Nash bargaining solution) are easier to
apply in the real world than the results of the strategic approach models. At the limit, when the
negotiation time period goes to zero, the solution of the strategic approach model converges to
the Nash bargaining solution. To solve any bilateral bargaining problem, it’s obvious that the
Nash bargaining solution can be applied at the limit. If the negotiation time period does not go to
zero ( 0→∆ is not satisfied), the solution of strategic approach model does not converge to the
Nash Bargaining solution.
When it comes to the multiple player game, the problem can be reduced to a series of
bilateral bargaining games, or to form a coalition first and then to adopt the bilateral bargaining.
Therefore, regardless of bilateral or multi player games, no matter what approach is used, the
generalized Nash bargaining solution of bilateral bargaining can be used to solve these
negotiation problems when negotiation time goes to zero.
When the negotiated game extends to the game with outside options or risk of breakdown,
the theory on bilateral bargaining is abundant; however, the bargaining theory on multi player
games with outside options and risk of breakdown is still developing.
For a cooperative, profit maximization should be considered as the goal when negotiating
with a processor over any commodity rather than maximize price or maximize a cooperative’s
surplus because the last two situations only occur when the cooperative or the processor has
absolute bargaining power. And if one side has absolute bargaining power, the bargaining game
will not begin in the first place. The major negotiated topic is the commodity price in this
bilateral game for a given quantity. Empirically, the generalized Nash bargaining solution can be
estimated econometrically, which suggests that the negotiated price is the price somewhere
between the maximum price that the buyer is willing to pay and the minimum price that the
56
seller is willing to sell. It depends on the bargaining power of each side. With a multiple player
game ( e.g., one cooperative and multiple processors), it appears that whether the risk of
breakdown or the outside option exists or not, the best way to solve the game is to reduce the
game to a series bilateral bargaining games and adopt the generalized Nash bargaining solution
of bilateral bargaining theory. For this case, outside options can be assumed as the only break
point of the negotiation. Thus, the utility pair for the generalized Nash bargaining solution would
be the profit at the agreement point and the profit when outside options are chosen for both
players.
The resulting model of the Nash bargaining solution can be estimated by some
econometric techniques. Thus, to compare the results of all different econometric techniques
could be one area of further research. Since the lack of abundance of work in the multiple player
bargaining game area, more research could be done in this related area such as outside options
and players risk attitude.
2.6 Relevant Information For Model Developing Next Chapter
The bargaining problem was first introduced by Nash (Nash, 1950), who analyzed the
bilateral bargaining game by the axiomatic approach based on four axioms (IAT, SYM, PAR and
IIA). It results in a famous Nash bargaining solution, a unique solution of the following
maximization problem (Muthoo, 1999)
))(( max BBAA dudu −− (2-58)
where Au and Bu are the utilities at the agreement point for player A and B, respectively; Ad and
Bd represent the utility pair at the disagreement point for player A and B, respectively.
Later, Binmore (Binmore, 1992) and Muthoo (Muthoo, 1999) discussed the generalized
(asymmetric) Nash bargaining solution by abandoning the non-generality axiom assumption of
57
symmetry (SYM). The asymmetric Nash bargaining solution is a unique solution of the
maximization problem
ττ −−− 1)()( max BBAA dudu (2-59)
where τ )10( << τ is the bargaining power parameter such that τ represents the bargaining
power for player A and ( τ−1 ) is the bargaining power for player B, and all other terms are
defined the same as the above.
Further, for the same bilateral game discussed above, a strategic approach was introduced
by Rubinstein (Rubinstein, 1982) called the alternating offer bilateral bargaining model. Then,
Binmore et al. (Binmore, et al., 1986) extended this basic bilateral model to the model with risk
of breakdown, where the utility outcome can be represented as
)())1(1()()1( bUpxUp it
it −−+− (2-60)
where )exp(1 ∆−−= λp represents the probability of the negotiation breakdown; ∆ is the length
of the negotiation time period (1,2,3,…t); )(xU i and )(bU i represent the utility for player i at the
agreement point and the breakdown point, respectively; x and b are the offers accepted by player
i at the agreement point and disagreement point, respectively. Binmore indicates that as the time
period of bargaining∆ goes to zero, the solution equation (2-60) (the outcome of strategic model
with risk of breakdown) converges to the unique Nash bargaining solution.
Further, the bilateral bargaining theory was extended to the multi player bargaining level.
When the axiomatic approach is applied in the multi player game, it is very easy to extend the
Nash bargaining solution (Krishna and Serrano, 1996),(Schneider, 2005) to
icii
n
iOQuOu ))()((max
1−Π
=Θ∈ where ∑
=
=n
ii Cc
1
(2-61)
58
where all terms are defined the same as the main body literature. However, as Krishna and
Serrano, and Schneider point out that to extend the strategic approach bilateral bargaining game
to a multi player game is less successful due to the existence of multiple sub-game perfect
equilibria. Thus, to solve the multiple player bargaining game, one way to solve it is to reduce
the game to a series bilateral bargaining games as shown by Jun (Jun, 1987), Chae and Yang
(Chae and Yang, 1988), Krishna and Serrano (Krishna and Serrano, 1996) and Suh and Wen
(Suh and Wen, 2006). Each bilateral game is solved by the Nash bargaining solution.
More over, when it comes to cooperative bargaining, the theory has been developed ever
since the 1960s, such as Helmberger and Hoos (Helmberger and Hoos, 1962), Helmberger and
Hoos (Helmberger and Hoos, 1963) and Ladd (Ladd, 1974). The goal of the cooperatives and the
factors affecting bargaining were discussed. At that early time, maximize price or quantity is
assumed to be cooperative’s objective. However, as more and more articles in distribution
channel bargaining was published by Folwell, Mittelhammer et al.(Folwell, et al., 1998),
Oczkowski (Oczkowski, 1991), (Oczkowski, 1999), Nagarajan and Bassok (Nagarajan and
Bassok, 2002), and Iyer and Villas-boas (Iyer and Villas-boas, 2003) profit maximization
became a more appealing objective for cooperative bargaining.
59
CHAPTER 3 THE THEORETICAL MODEL
3.1 Building Model
Following the literature review of chapter two from the studies of Binmore and Rubinstein
et al.(Binmore, et al., 1986) and Muthoo (Muthoo, 1999), a standard bilateral bargaining game is
used where two players, A and B, bargain over a fixed pie for a commodity. Assume that player
A makes an offer first, and then player B makes a decision either to accept the offer or reject the
offer and make a counter offer. For each bargaining period ∆ , assume that the probability of
negotiation breakdown is represented by the exponentially distributed Variable
)exp(1 ∆−−= γrP where γ is a distributional parameter. This process continues until the
agreement is reached or perpetual disagreement happens, which will be explained later.
3.1.1 Bilateral Model with Risk of Breakdown
Thus, according to Binmore et al. (Binmore, et al., 1986), the outcome solution of this
bilateral bargaining model with risk of breakdown converges to the following generalized Nash
bargaining solution of
ττ −1RA )()(U max R
BU (3-1)
where
ArArRA dpUpU +−= )1( (3-2)
BrBrRB dpUpU +−= )1( (3-3)
where RAU and R
BU are the utility outcome for player A and B, respectively; τ and 1-τ represent
the bargaining power for player A and B, respectively; rp is the probability that negotiation
breaks down; AU , Ad and BU , Bd are the utilities at the agreement and the disagreement points
for players A and B, respectively where Ad and Bd could be any value less or equal to zero.
60
Further, let’s discuss the possible disagreement point in detail. Muthoo (Muthoo, 1999)
and Oczkowski (Oczkowski, 1999) point out that an outside option point can be considered as
the disagreement point when negotiation breakdown happens. Now let’s assume that when
player B makes a decision, they have one more option such that player B can reject the offer and
opt out for an outside option. Similarly, if player B makes a counter offer, player A can choose to
make a counter offer or choose to opt out. This process continues until the agreement is reached
or the negotiation breaks down. If both players have a choice to opt out, they opt out because
they reject the offer and still have the demand or the supply needs. If they do nothing, the
expected utility from this bargaining disagreement would be zero. If they choose to opt out and
do something, then they can get the expected utility from an outside trade which must be greater
than zero but less than the expected utility from the success of the negotiation; otherwise, they
will not opt out. Thus, let’s assume that the breakdown point for this model is the outside option
point.
3.1.2 Bilateral Model with Risk of Breakdown and Outside Options
Following Binmore et al. (Binmore, et al., 1986), assuming that the breakdown points are
the outside option points, let AW , and BW be the utilities that player A and B receive when the
player opts out, respectively. Further, assume that 0,0 >> BA WW . Then, the disagreement point
utility pair of two players become AW and BW . Thus, the solution of the bilateral bargaining
model with both risk of breakdown and two-side outside options becomes
τROB
τROA )(U)(U −1 max (3-4)
where
ArArROA WpUpU +−= )1( (3-5)
BrBrROB WpUpU +−= )1( (3-6)
61
and all terms are defined the same as the above.
The next question becomes how to define the utilities for the bargaining game between a
milk cooperative and milk processor. As discussed in the literature review in chapter two, the
Nash bargaining solution can be used to solve cooperative bargaining (Iyer and Villas-boas,
2003), (Nagarajan and Bassok, 2002). Regarding the utility pair at the agreement point and the
breakdown point, Helmberger and Hoos (Helmberger and Hoos, 1962), (Helmberger and Hoos,
1963) used the price of the commodity or the quantity of the commodity. Oczkowski
(Oczkowski, 2006) suggests that besides price and quantity, profit can be used to replace utility.
In addition, Iyer and Villas-boas (Iyer and Villas-boas, 2003), and Nagarajan and Bassok
(Nagarajan and Bassok, 2002) use profit instead of utilities for all players entering into the Nash
bargaining solution.
3.1.3 Bilateral Model with Risk of Breakdown and Outside Options between A Cooperative and Processor
Thus, for the bilateral bargaining game between the Florida milk cooperative and the
Florida milk processor, profit is used to replace the utilities in the Nash bargaining solution of
equation (3-4) (Iyer and Villas-boas, 2003, Nagarajan and Bassok, 2002, Oczkowski, 1999,
Oczkowski, 1991, Oczkowski, 2006). Let’s assume that both the cooperative and the processor
maximize their profits and define A as the cooperative and B as the processor. Following the
above authors, using profit instead of utility for both players, the Nash bargaining solution for
this bilateral bargaining model with risk of breakdown and outside options becomes
ττ ππ −1ROA )()( max RO
B (3-7)
where
WArArROA pp πππ +−= )1( (3-8)
WBrBrROB pp πππ +−= )1( (3-9)
62
and, ROAπ , RO
Bπ are expected profits for the milk cooperative and milk processor, respectively;
Aπ , Bπ are profits for the milk cooperative and milk processor at the agreement point,
respectively; WAπ , WBπ are the profits for the milk cooperative and milk processor when both
choose to opt out; and all other terms are defined the same as above.
The outside option prices can be fixed or variable. First, the outside option prices for the
milk cooperative and milk processor can be fixed when they decide to opt out. They will choose
to accept the outsider’s offer without negotiations which makes the outside offer price fixed.
Second, the outside option prices are variable if the milk cooperative and milk processor choose
to opt out and they have to search for the possible outside option and negotiate the price with
outsiders which implies that the outside option price varies due to the presence of another
bargaining process. Third, the outside option price can be fixed for the milk marketing
cooperative (milk processor) and variable for the milk processor (milk marketing cooperative)
when the milk marketing cooperative (milk processor) accepts the outsider’s offer and the milk
processor (milk marketing cooperative) has to search for outside options and bargains over the
commodity price and quantity. Thus, the model can be solved for all above three cases.
3.2 Model Solution
Following Oczkowski (Oczkowski, 2006) to solve the bilateral bargaining model with risk
of breakdown and outside options, the profit functions for the cooperative and the processor are
defined as
)(QCCPQM PBC −−=π (3-10)
PQQRP −= )(π (3-11)
where CMπ is the cooperative members’ profit; P is the negotiated price of the commodity; Q is
the output of the commodity; BC is the bargaining cost; )(QCP is the production cost of the
63
commodity; Pπ represents the profit of the processor; and )(QR is the commodity associated net
revenue for the processor.
The latest cooperative bargaining theory is only found in the following works. Oczkowski
(Oczkowski, 1999) defined the outside option price as variable along the supply and the demand
curve but fixed for any quantity of the bargaining product. Later, Oczkowski (Oczkowski, 2006)
had a theoretical bilateral bargaining model on cooperative and processor bargaining but without
the consideration of outside options and risk of breakdown. Besides to what Oczkowski has
done, I assume that the outside option prices could be fixed or variable for any quantity of the
bargaining product and there exists three possible cases including (1) both outside option prices
are variable, (2) both outside option prices are fixed, and (3) one outside option price is fixed and
the other outside option price is variable.
3.2.1 Case 1: Outside Option Prices are Variable
Differing from what Oczkowski (Oczkowski, 1999) defined as the outside option price
being variable along the supply and the demand curve but fixed for any quantity of the
bargaining product, I assume the outside option prices are bargaining prices which are variable
for any quantity of the bargaining product. In this case, when the negotiation breaks down and
both players need to go and search for outside options, they need to start the bargaining process
again but with outsiders. This means that the outside prices are endogenous variables. Then, the
following objective function needs to be maximized with respect to four endogenous variables
( P
O
C
O ,PP,Q,P ) as
τWBrBr
τWArAr
,PP,Q,Pπpπ-pπpπ-p F
PO
CO
−++= 1
)(])1[(])1[( max (3-12)
where
)(QCPQ CPA −=π (3-13)
64
)(QCQP COC
OWA −=π (3-14)
PCPCB BPQQR −−= )(π (3-15)
POP
OPOWB BQPQR −−= )(π (3-16)
where P is the negotiated price between the cooperative and the processor; Q is the negotiated
quantity; CPC is the total cost (bargaining cost and production cost) for the cooperative when the
cooperative contracts with the processor; COP is defined as the negotiated price between the
cooperative and its outside option; COC is the total cost (bargaining cost and production cost) for
the cooperative when the cooperative contracts with an outside option; PCR is processor’s
commodity associated net revenue when contracting with the cooperative; PCB is bargaining cost
for the processor when contracting with cooperatives; POR is processor’s commodity associated
revenue when contracting with an outside option; POP is the negotiated price between the
processor and its outside option; POB is bargaining cost for the processor when contracting with
an outside option; and all other terms are defined the same as above.
Thus, the negotiated price between the Florida milk cooperative and Florida milk processor
can be found by solving this profit maximization problem with respect to ,, COPP Q and P
OP . The
first order condition (FOC) of equation (3-12) with respect to P is
0])1([][])[[1(]][)1[(][ 11 =−−−+−=∂∂ −−− QpQp
PF
rROB
ROA
ROBr
ROA
ττττ ππτππτ . (3-17)
Multiplying both sides by τπ −1][ ROA (see equation 3-7), then
0])1([]][)[1(]][)1[( 1 =−−−+− −− QpQp rROB
ROA
ROBr
ττ ππτπτ (3-18)
and
65
0]][)[1(][ 1 =−− −− ττ ππτπτ ROB
ROA
ROB . (3-19)
Multiplying both sides by τπ ][ ROB (see equation 3-7), then
0])[1(][ =−− ROA
ROB πτπτ . (3-20)
Then, substitute ROAπ and RO
Bπ (see equation (3-8), (3-9), (3-13) through (3-16)) into the
above equation (3-20) and solve for P as
0))](())()(1)[(1(
)])(())()(1[(
=−+−−−−
−−+−−−
QCQPpQCPQpBQPQRpBPQQRp
COC
OrCPr
POP
OPOrPCPCr
τ
τ (3-21)
0))](())()(1)[(1(
)])(())()(1[(
])1)[(1(])1[(
=−+−−−−
−−+−−+
−−−−−
QCQPpQCpQPBQRpBQRp
PQpPQp
COC
OrCPr
POPOPOrPCPCr
rr
τ
τ
ττ
(3-22)
))](())()(1)[(1(
)])(())()(1[(])1)[((
QCQPpQCpQPBQRpBQRpPQp
COC
OrCPr
POPOPOrPCPCrr
−+−−−+
−−+−−−=−−
τ
τ (3-23)
)])((1
)()[1(
)])((1
))(([
QQCP
pp
QQC
PQ
BQRp
pQ
BQRP
COCO
r
rCP
PO
POPO
r
rPCPC
−−
−−+
−−
−+
−=
τ
τ. (3-24)
Simplifying the negotiated price P (equation 3-24) as
)](1
)[1(
)](1
)[(
COC
Or
rCP
POPOPO
r
rPCPC
ATCPp
pATC
PABARp
pABARP
−−
−−+
−−−
+−=
τ
τ (3-25)
where PCAR = Q
RPC is the processor’s commodity associated net revenue when contracting with
cooperatives; PCAB = Q
BPC is the processor’s bargaining cost when contracting with
66
cooperatives; POAR = Q
RPO is the processor’s commodity net associated revenue when
contracting with an outside option; POAB = Q
BPO is the processor’s bargaining cost when
contracting with an outside option; CPATC = Q
CCP is the average total cost (bargaining cost and
production cost) for the cooperative when the cooperative contracts with the processor; COATC =
QCCO is the average total cost (bargaining cost and production cost) for the cooperative when the
cooperative contracts with an outside option.
The solution found in equation (3-25) is different from what Oczkowski (Oczkowski,
1999, Oczkowski, 2006) found in both his articles such that the probability of breakdown and
outside option prices are inside the solution (equation (3-25)). The reason is that Oczkowski’s
1999 article assumed outside option price varies along the supply curve and the demand curve
but is fixed for any quantity. This dissertation assumes that outside option prices vary along the
average net revenue and average cost curve and is variable for any quantity.
However, notice that if the probability of breakdown goes to zero, which means that my
model simplifies to Oczkowski’s model in 2006 (Oczkowski, 2006), the above solution
(equation 3-25) of negotiated price becomes
CPPCPCY ATCABARP )1()( ττ −+−= (3-26)
which is identical to the Oczkowski’s solution (Oczkowski, 2006) which is the result that should
be the same because the model is the same when 0=rp (i.e., no risk of breakdown and no
outside options).
The first order condition of equation (3-12) with respect to COP is
67
0]][[][ 11 ==∂∂ −− ττ ππτ RO
BrROAC
O
QpPF . (3-27)
And the solution is
0=τ (3-28)
or
0=rp . (3-29)
0=τ means that when the cooperative has no bargaining power at all, they will not choose to
trade with the processor and the cooperative will opt out. This is the same result that Oczkowski
(Oczkowski, 2006) found; however, compared to the mathematical derivations shown here, only
a word explanation was presented by Oczkowski. 0=rp implies that there is no chance that
both players would opt out.
The first order condition of equation (3-12) with respect to POP is
0]][)[1(][ =−−=∂∂ −ττ πτπ RO
BrROAP
O
QpPF . (3-30)
And the solution is
1=τ (3-31)
or
0=rp . (3-32)
1=τ means that when the cooperative has absolute power, then the processor will not choose to
trade with the cooperative and they will opt out. This is identical to what Oczkowski
(Oczkowski, 2006) found but without the mathematical derivations along with the explanations
shown here.
68
Let’s discuss the solutions with respect to COP and P
OP (equations (3-28), (3-29), (3-31),
and (3-32)) before solving the first order condition with respect to Q. There are four possible
solutions:
(1). 0=τ (equation (3-28)) for maximizing with respect to COP and 1=τ (equation (3-31)) for
maximizing with respect to POP ,
(2). 0=τ (equation (3-28)) for maximizing with respect to COP and 0=rp (equation (3-32))
for maximizing with respect to POP ,
(3). 0=rp (equation (3-29)) for maximizing with respect to COP and 1=τ (equation (3-31))
for maximizing with respect to POP ,
(4). 0=rp (equation (3-29), equation (3-32)) for maximizing with respect to COP and P
OP .
Possible solution (1) 0=τ and 1=τ conflict, thus, this is not a solution. Possible solution
(2) 0=τ implies that the processor has absolute bargaining power over the cooperative, thus the
bargaining process would not begin in the first place. In the mean time, 0=rp suggests that
there is no chance that both players will go to an outside option. However, since the bargaining
process does not actually begin, 0=τ and 0=rp is not a bargaining solution. Similarly,
possible solution (3) 0=rp and 1=τ is not a bargaining solution. Possible solution (4) 0=rp
is the only solution for maximizing the expected profits with respect to COP and P
OP . This is a
solution because both players have to go outside and search for a possible option when
negotiation breakdown occurs. Furthermore, they need to bargain with the outside options over
the price and quantity again. 0=rp suggests that the costs of searching outside options,
bargaining, and timing exceed the benefit of breakdown so that the best decision for both players
is to continue negotiation until agreement is reached.
69
Then, when 0=rp , let’s solve the first order condition of equation (3-12) with respect to
Q,
0][])[[1(][][ 11 =∂∂
−+∂∂
=∂∂ −−−
QQQF RO
BROB
ROA
ROB
ROARO
Aπππτπππτ ττττ . (3-33)
Multiplying both sides by τπ −1][ ROA , then
0]][)[1(][ 1 =∂∂
−+∂∂ −−
ROBRO
BROA
ROB
ROA πππτππτ ττ . (3-34)
Multiplying both sides by τπ ][ ROB , then
0][)1(][ =∂∂
−+∂∂ RO
A
ROBRO
B
ROA
QQππτππτ . (3-35)
Then, solve for Q
ROA
∂∂π and
Q
ROB
∂∂π
QATCQATCP
QCP
CP
ROA
∂∂
−−=∂∂π (3-36)
)(Q
ABQ
ARQPABARQ
PCPCPCPC
ROB
∂∂
−∂
∂+−−=
∂∂π . (3-37)
Then, substitute ROAπ , RO
Bπ , equations (3-36) and (3-37) into equation (3-35) to get
0][)]()[1(
][)(
=−∂
∂−
∂∂
+−−−+
−−∂
∂−−
CPPCPC
PCPC
PCPCCP
CP
ATCPQQ
ABQ
ARQPABAR
PABARQQ
ATCQATCP
τ
τ. (3-38)
Simplify equation (3-38) to get
70
0)()()()(
)()()()(
)()()(
=−∂
∂−
∂∂
−−−−−
−∂
∂−
∂∂
+−−−+
−−∂
∂−−−−
CPPCPC
CPPCPC
CPPCPC
CPPCPC
PCPCCP
PCPCCP
ATCPQQ
ABQ
ARQATCPQPABAR
ATCPQQ
ABQ
ARQATCPQPABAR
PABARQQ
ATCQPABARQATCP
ττ
ττ
. (3-39)
Then, divide both sides of equation (3-39) by Q
0)()()()(
))(()(
=−∂
∂−
∂∂
−−∂
∂−
∂∂
+
−−−+−−∂
∂−
CPPCPC
CPPCPC
CPPCPCPCPCCP
ATCPQQ
ABQ
ARATCPQQ
ABQ
AR
ATCPPABARPABARQQ
ATC
τ
τ. (3-40)
When 0=rp , the negotiated price (P) equation (3-25) can be simplified as
))(1()( CPPCPC ATCABARP ττ −+−= . (3-41)
Then, substitute equation (3-42) into equation (3-41) to get
0]))(1()([)(
]))(1()([)(
]))(1()([ ))])(1()(([
))])(1()(([
=−−+−∂
∂−
∂∂
−
−−+−∂
∂−
∂∂
+
−−+−•−+−−−+
−+−−−∂
∂−
CPCPPCPCPCPC
CPCPPCPCPCPC
CPCPPCPC
CPPCPCPCPC
CPPCPCPCPCCP
ATCATCABARQQ
ABQ
AR
ATCATCABARQQ
ABQ
ARATCATCABAR
ATCABARABAR
ATCABARABARQQ
ATC
τττ
ττ
ττττ
τττ
. (3-42)
Then, simplify it to get
0)()(
)()(
)())(1(
))(1(
=−−∂
∂−
∂∂
−
−−∂
∂−
∂∂
+
−−−−−+
−−−∂
∂−
CPPCPCPCPC
CPPCPCPCPC
CPPCPCCPPCPC
CPPCPCCP
ATCABARQQ
ABQ
AR
ATCABARQQ
ABQ
ARATCABARATCABAR
ATCABARQQ
ATC
ττ
τ
ττ
ττ
. (3-43)
Divide both sides of equation (3-43) by )( CPPCPC ATCABAR −−τ to get
71
0)()(
))(1()1(
=∂
∂−
∂∂
−∂
∂−
∂∂
+
−−−+−∂
∂−
ABQ
ARQQ
ABQ
AR
ATCABARQQ
ATC
PCPCPCPC
CPPCPCCP
τ
ττ. (3-44)
Then simplify to get
0))(1(
))(1()1(
=∂
∂−
∂∂
−+
−−−+−∂
∂−
ABQ
AR
ATCABARQQ
ATC
PCPC
CPPCPCCP
τ
ττ. (3-45)
Divide both sides of equation (3-45) by )1( τ− to get
0)()( =∂
∂−
∂∂
+−−+∂
∂− Q
QAB
QARATCABARQ
QATC PCPC
CPPCPCCP . (3-46)
Then solving for Q
CPPCPCPCPCCP ATCABAR
QAB
QAR
QATC
Q −−=∂
∂+
∂∂
−∂
∂)( . (3-47)
Thus, the negotiated quantity equation can be presented as
QATC
QAB
QAR
ABARATCQ
CPPCPC
PCPCCP
∂∂
−∂
∂−
∂∂
−−=
)(
)(. (3-48)
where terms are defined the same as equation (3-25). This result (equation 3-48) is equivalent to
what Ozkcowski (Oczkowski, 2006) found but we assume that bargaining cost exists for the
processor ( 0≠PCAB ).
In summary, the solutions for the bilateral bargaining model with outside options and risk
of breakdown with variable outside option prices (equation 3-12) are equations (3-29, 3-32, 3-41
and 3-48) which are listed below
(3-29, 3-32) PO
CO PP ∀∀ , with 0=rp
72
(3-41) ))(1()( CPPCPC ATCABARP ττ −+−= .
(3-48)
QATC
QAB
QAR
ABARATCQ
CPPCPC
PCPCCP
∂∂
−∂
∂−
∂∂
−−=
)(
)(.
The above results imply that the probability of opting out or in other words, the risk of
breakdown is zero which suggests that outside options are not binding and do not affect the
negotiated price and the negotiated quantity. A possible reason might be the existing high cost
(outside option search cost and bargaining cost). The negotiated price is between the processor’s
average revenue and the cooperative’s average total cost, and the final negotiated price depends
on the bargaining power of each side. The negotiated quantity is also affected by the processor’s
average revenue and the cooperative’s average total cost.
3.2.2 Case 2: Outside Option Prices are Fixed
In this case, when the negotiation breaks down, both players go outside options and accept
the prices that outsiders offer. Thus, the objective function of the model needs to be maximized
only with respect to the negotiated price P and the negotiated quantity Q shown as
τWBrBr
τWArAr(P,Q)
πpπ-pπpπ-p F −++= 1])1[(])1[( max (3-49)
where
)]([)( QATCPQQCPQ CPCPA −=−=π (3-50)
)]([)( QATCPQQCQP COC
OCOC
OWA −=−=π (3-51)
])()([)( PQABQARQBPQQR PCPCPCPCB −−=−−=π (3-52)
])()([)( POPOPOPO
POPOWB PQABQARQBQPQR −−=−−=π (3-53)
Solve for the negotiated price P by using the first order condition of equation (3-49) with
respect to P (same as case I, equation 3-25) shown below as
73
)](1
)[1(
)](1
)[(
COC
Or
rCP
POPOPO
r
rPCPC
ATCPp
pATC
PABARp
pABARP
−−
−−+
−−−
+−=
τ
τ. (3-54)
Solve for the negotiated quantity Q by using the first order condition of equation (3-49)
with respect to Q (Appendix A)
]
1[
])11
1
]1
)(
))()(1(2
))((2
)()1(
)(
))()(1(2
22
2
22
2
22
22
)ATC(Pp)PAB(ARp
)ATCAB)(ARp(p)Q
ABQ
AR(
)ATC(Pp(p)PAB)(ARp(p
)ATCAB(AR)p)[(Q
ABQ
AR(
)ATC(P p)ATCAB)(ARp(
)P(Q)AB(Q)(AR p)[Q
ATC(
ATCPpATCABARATCPpp
PABARATCPpATCABARp
PABARpPABARATCABARpp
Q
COC
OrP
OPOPOr
CPPCPCrrPOPO
COC
OrrP
OPOPOrr
CPPCPCrPCPC
COC
OrCPPCPCr
POPOPOr
CP
COC
Or
CPPCPCCOC
Orr
POPOPOCO
COr
CPPCPCr
POPOPOr
POPOPOCPPCPCrr
−+−−+
−−−∂
∂−
∂∂
+
−−+−−−+
−−−∂
∂−
∂∂
+
−−−−−−
−−−∂
∂−−
−−−−−
−−−−
−−−−
−−−
−−−−−−
=
. (3-55)
In summary, the solution for the bilateral bargaining model with outside options and risk
of breakdown with fixed outside option prices (equation (3-49)) are equations (3-54) and (3-55)
for the negotiated price and quantity, which suggests that the negotiated price and quantity is not
only affected by the bargaining power parameter τ , but also affected by the risk of breakdown
when outside options are chosen. No other authors have found this result before. In other words,
outside options do affect the negotiated price and the negotiated quantity through the risk of
breakdown.
74
Notice that when 0=rp , the negotiated price and negotiated quantity become
CPPCPC ATCABARP )1()( ττ −+−= (3-56)
QATC
)Q
ABQ
AR(
ABARATC Q
CPPCPC
PCPCCP
∂∂
−∂
∂−
∂∂
−−=
)( (3-57)
which is same as the solution of case I when outside option prices vary.
3.2.3 Case 3: One Outside Option Price is fixed and The Other Outside Option Price is Variable
The third possible case is that negotiation breakdown occurs because one player decides
to accept the offer provided by an outsider, and then the other player has to search for an outside
option to start the negotiation process all over again. Thus, the outside option price for the player
who accepts the outsider’s offer is fixed, but the outside option price for the player who has to
search and negotiate with the outsider is variable.
For simplicity, let’s assume that the processor has a fixed outside option price POP and the
cooperative has a variable outside option price COP . Then, the following objective function needs
to be maximized with respect to three endogenous variables ( P
OP,Q,P ) as
τWBrBr
τWArAr
P,Q,Pπpπ-pπpπ-p F
PO
−++= 1
)(])1[(])1[( max (3-58)
where terms are defined the same as in case 1.
To solve the problem, let’s follow the same procedure in case 1 to find the first order
conditions with respect to P, Q and COP . The solution is the same as shown in equation (3-41),
equation (3-48) and equations (3-28), (3-29) for P, Q and COP , respectively.
When 0=τ (equation 3-28), it means that the processor has absolute bargaining power.
The cooperative will not choose to bargaining because they lose at the beginning. Therefore, no
75
bargaining problem exists which indicates that this is not a real solution. Then, 0=rp (equation
3-29) is the solution. It means that both players will choose to continue negotiating until the
agreement is reached. So the solution for case 3 is the same as that for case 1 which is shown in
equation (3-41) for P and equation (3-48) for Q as
COP∀ with 0=rp (3-59)
CPPCPC ATCABARP )1()( ττ −+−= (3-60)
QATC
)Q
ABQ
AR(
ABARATC Q
CPPCPC
PCPCCP
∂∂
−∂
∂−
∂∂
−−=
)( (3-61)
3.2.4 Summary of Model Solution
The model presented in this chapter differs from the model presented by Oczkowski in
1999 (Oczkowski, 1999). Instead of assuming that the outside option price is variable along the
supply and the demand curve but fixed for any quantity of the bargaining product, I assume that
the outside options can be fixed or variable for any existing quantity so that the model was
developed and solved in three different cases (never seen before in other literatures). Further, the
model developed in this chapter differs from the model presented by Oczkowski in 2006
(Oczkowski, 2006) in the way that outside options and risk of breakdown are included in the
bilateral model. Notice that, when 0=rp , the bilateral bargaining model with risk of breakdown
and outside options simplifies to the same solution of what Oczkowski (Oczkowski, 2006) has.
In summary, case 1 (both prices are variable) and case 3 (one price fixed and one price
variable) have same solutions for negotiated price and negotiated quantity such that no players
would choose to opt out where 0=rp . They will continue to negotiate until the agreement is
reached where the negotiated price (equation 3-(60)) and the negotiated quantity (equation (3-
76
61)) are the same for these two cases. This suggests that an outside option does not affect the
negotiated price and the negotiated quantity directly, but might instead affect the negotiated price
and the negotiated quantity through the bargaining power parameter.
Furthermore, the result of case 2 is completely new and different from the result of case 1
and case 3. The solution has a risk of breakdown variable ( rp ) and both outside option prices
inside the solution of the negotiated price (equation (3-41)) and the negotiated quantity (equation
(3-48)). This solution implies that option prices do affect the negotiated price and the negotiated
quantity directly. When outside option prices are fixed for both players, if the profit from the
outside option is higher than that from the insider, players will opt out where negotiation
breakdown occurs ( 0≠rp ), and if the profit from outside option is less than what both players
get from each other, both players will choose to continue the negotiation process until an
agreement is reached ( 0=rp ).
77
CHAPTER 4 THE EMPIRICAL MODEL
4.1 Florida Milk Industry
In the Florida dairy industry, two milk cooperatives, Florida Dairy Farmers' Association
(FDFA) and Tampa Independent Dairy Farmers' Association, Inc. (TIDFA) were established in
1956 and in 1967, respectively (http://www.southeastmilk.org/v2/about/). In 1998 with the
merger of FDFA and TIDFA, Southeast Milk, Inc. (SMI) was created as a new cooperative with
members from FDFA and TIDFA. SMI is a cooperative with approximately 280 members and
markets approximately 3 billion pounds of milk annually
(http://www.southeastmilk.org/v2/about/). SMI represents most Florida dairy farmers plus
approximately one half of the Georgia dairy farmers and some members in Alabama and
Tennessee. SMI is a member of the Southeast Dairy Cooperative, Inc. (SDC). SDC negotiates
with Florida milk processers. There were fourteen milk processors in Florida from the year 1998
to the year 2009 (Table 4-1).
The fluid milk price and quantity are determined through the negotiation process between
SDC and the leading processor. The negotiated price is adopted by all Florida processors. The
actual milk supply contract between SDC and the Florida processors is a yearly contract which is
annually renewable and can be cancelled given a 60 day notice. However, under the contract, the
quantity and price is negotiated once a month. The quantity is variable throughout the month as
processors increase or decrease their demand.
This market environment is a monopoly on the supply side (SDC) and oligopsony on the
demand side. Iskow and Sexton (Iskow and Sexton 1992) indentified this type of the market
structure first in the fruit and vegetable markets. Then Sexton (Sexton 1993) indicated that the
78
market approximated a bilateral monopoly during negotiations. Thus, the bargaining between
SDC and milk processors in the Florida dairy market approximates bilateral bargaining.
Table 4-1 Florida Milk Processor Plants Number Plant Name City Name ZipCode 1 Gustafsons L.L.C. Green Cove Springs 32043 2 Hato Patrero Farms, Inc. Clewiston 33440 3 M&B Dairy Products, Inc. Tampa 33637 4 McArthur Dairy, Inc. Miami 33325
5 Publix Supermarkets, Inc. Deerfield Beach 33442 Lakeland 33802
6 Sunbelt Dairy and Food Company Tampa 33686 7 Sunshine State Dairy Farms, LLC Plant City 33567
8 T. G. Lee Foods, Inc. Orange City 32763 Orlando 32802
9 Velda Farms, LLC Miami. 33164 Winter Haven 33881
10 Ryan Foods Company Jacksonville 32209
11 Superbrand Dairy Products, Inc Miami 33167 Plant City 33567
12 Morningstar Foods, Inc. Jacksonville 32209 13 WhiteWave Foods Jacksonville 32209 14 Winn-Dixie Stores, Inc. Plant City 33567 source: http://future.aae.wisc.edu/tab/production.html#43, date visited, July 3rd, 2009
Every month, SDC is the only supplier of milk to those processors whether it’s a deficit
month or surplus month. During the deficit months, SDC will have more cooperative members
from outside the state of Florida to provide milk to processors (table 4-2). The numbers inside
the table represent months. When SDC has surplus months, the number of cooperative members
decrease (table 4-2) and SDC sells surplus milk to manufacturing plants who make non fluid
milk products.
The negotiation process has broken down only once over the past twenty five years which
suggests that the probability of negotiations breaking down and SDC and the processors opting-
out can be assumed to be zero ( 0=rp ).
79
Table 4-2. Monthly SDC Non-Florida Cooperative members
Year
Cobblestone Milk Cooperative, Maryland/Virginia Milk Cooperative, Virginia
Lone Star Milk Producer, Inc, Texas
Select Milk Producers, New Mexico
White Eagle Cooperative Association, Indiana
Land-O-Lakes, Pennsylvania
Michigan Milk Producers, Michigan
2000 8 2001 8 2002 2003 4-12 1, 7-11 8-9 2004 1-6,9-12 9 1, 3, 8-12
2005 1-12 7-12 3-4, 6, 8-12 8 8-9
2006 1-12 1-12 1, 7-12 2007 1-12 1-12 1, 6, 8-11 9-11 8 2008 1-12 1-12 1 source: http://www.fmmatlanta.com/Annual%20Statistics.htm, date visited, October 20th, 2009
4.2 The Econometric Model
Assuming 0=rp , the bilateral solution of the negotiated price (P) and the negotiated
quantity (Q) for all three cases (fixed outside option prices, variable outside option prices, and
one fixed/one variable outside option prices) mentioned in chapter 3 can be presented as
CPPCPC ATCABARP )1()( ττ −+−= (4-1)
QATC
)Q
ABQ
AR(
ABARATC Q
CPPCPC
PCPCCP
∂∂
−∂
∂−
∂∂
−−=
)( (4-2)
where PCPC ABAR − is the processors’ average net revenue (AR) associated with fluid milk
( PCAR ) (Blair, et al., 1989) which does not include the bargaining cost for the processor ( PCAB ),
and CPATC is the associated average total cost (ATC) for SDC members. Recall that τ is the
bargaining power for SDC and τ−1 is the bargaining power for processors. Following
Oczkowski (Oczkowski, 1999), AR is processor’s demand reservation price which is
QQRARP D )(
== and ATC is the supply reservation price which is defined as
80
QQCATCP S )(
== , where R(Q) and C(Q) represent the net revenue function for the processor
and the cost function for SDC. Thus, equation (4-1) and (4-2) can be expressed as
SD PPP )1( ττ −+= (4-3)
QP
QP
PP Q SD
DS
∂∂
−∂∂
−= (4-4)
The four endogenous variables are the negotiated unit price (P), the demand reservation price
( DP ), the supply reservation price ( SP ), and the negotiated quantity (Q). τ and τ−1 are the
bargaining power parameters for SDC and the milk processor.
Following Oczkowski (Oczkowski, 1999), the bilateral generalized Nash bargaining
econometric model (equations (4-3) and (4-4)) with risk of breakdown and outside options for
the Florida dairy market can be presented as
Dt
Dt
Dt XfP εα += ),( (4-5)
St
St
St XgP εβ += ),( (4-6)
Pt
Stt
Dtt PPP εττ +−+= )1( (4-7)
),( θτ τtt Xh= (4-8)
Qt
Dt
Sttt PP Q εγ +−= )( (4-9)
)]),(),(/[(1Q
XgQXf S
tDt
t ∂∂
−∂
∂=
βαγ (4-10)
where DtP , S
tP , tP and tQ are endogenous variables representing the demand reservation price for
the processor, the supply reservation price for SDC, the negotiated price and the negotiated
quantity; f, g and h represent functional forms; α , β , τ , θ and γ are estimated parameters; τ
81
and τ−1 are the bargaining parameters for SDC and milk processors; DtX , S
tX , and τtX are
exogenous variables which affect the demand reservation price ( DtP ), the supply reservation
price ( StP ) and the bargaining power parameter (τ ), respectively; D
tε , Stε , P
tε and Qtε are error
terms with constant variance 2Dσ , 2
Sσ , 2Pσ , and 2
Qσ , respectively.
Following the argument of Oczkowski (Oczkowski, 1999), theoretically, log-linear,
double-log and interaction permit all directional changes so that they are potentially fitted for
empirical analysis. The double log function is assumed in our research for the processor’s
demand reservation price (equation (4-11)) and SDC’s supply reservation price (equation (4-
12)). A logistic function is used to ensure that the bargaining power parameter τ is between zero
and one for equation (4-8) (equation (4-14)). Thus, the econometric model can be rewritten as
Dt
K
i
Ditit
Dt XQP εααα +++= ∑
=
1
210 )ln()ln()ln( (4-11)
St
K
i
Sitit
St XQP εβββ +++= ∑
=
2
210 )ln()ln()ln( (4-12)
Pt
Stt
Dttt PPP εττ +−+= )1( (4-13)
∑=
++=
3
10 )exp(1
1K
iiti
t
X τθθτ (4-14)
Qt
Dt
Sttt PP Q εγ +−= )( (4-15)
)/(1 11
QQtβα
γ −= (4-16)
where 1K , 2K and 3K represent the number of exogenous variables in equations (4-11), (4-12)
and (4-14), respectively, and all other terms are defined in equations (4-5) to (4-10).
82
4.3 Discussion of The Exogenous Variables
By rewriting the processor’s demand reservation price equation to include the exogenous
variables and the expected signs of the parameters, equation (4-11) becomes
Dttttt
Dt WPLCQP εααααα ++−+−= )ln()ln()1ln()ln()ln( 43210 (4-17)
where DtP is the demand reservation price for the processor; α are the parameters associated
with the exogenous variables; tQ is the negotiated quantity; tC1 is the Florida Federal Milk
Marketing Order (FMMO) #6 Class 1 price; tL is the labor cost index; and tWP is the wholesale
price index for dairy products; and all other terms are defined the same as in equations (4-5) to
(4-10).
Regarding the possible variables affecting the processor’s demand reservation price ( DtP ),
the more fluid milk the processors buy from SDC, the lower the price processors are willing to
pay. This indicates a negative relationship between negotiated quantity ( tQ ) and the processor’s
demand reservation price ( DtP ). For the dairy industry, there exists Federal Milk Marketing
Orders (FMMO) who set the Class 1 (fluid milk) price every month for each individual Federal
Milk Marketing Order area. Responding to FMMO policy, the class I price ( tC1 ) is the minimum
price that the fluid milk processors have to pay. The higher the class I price, the higher the
demand reservation is for processors1.
Both the cost of processing fluid milk and the price of selling the containerized fluid milk
to wholesalers needs to be considered. Among all the variable costs, labor accounts for 31.4% of
the cost of processing fluid milk (Dalton, Griner and Halloran, 2002). Thus, following Prato 1 Oczkowski Oczkowski, E. "An Econometric Analysis of the Bilateral Monopoly Model." Economic Modelling 16, no. 1(1999): 11. suggested that besides the factors affecting cost for SDC and net revenue for processors, other price variables may be considered.
83
(Prato, 1973), the total labor compensation index ( tL ) is used to examine the relationship
between labor cost and the processor’s demand reservation price ( DtP ). The higher the labor
input cost, the lower the processors demand reservation price which suggests a negative
relationship between them. Similarly, if the processors can sell their processed fluid milk to
wholesalers at a higher price ( tWP ), the higher the demand reservation price ( DtP ) will be.
To include the exogenous variables and the expected signs of the parameters in the
SDC’s supply reservation price, equation (4-12) becomes
Stttttt
St TCSFCCQP εββββββ ++−+++= )ln()ln()ln()1ln()ln()ln( 543210
(4-18)
where StP is the supply reservation price for SDC; β are the parameters associated with the
exogenous variables; tQ is the negotiated quantity; tC1 is the Florida FMMO #6 Class 1 price;
tFC is the feed cost index; tS represents seasonality; tTC is the transportation cost index; and
all other terms are defined the same in equations (4-5) to (4-10).
Considering the supply reservation price ( StP ) for SDC who represents dairy farmers, it is
partly associated with milk production ( tQ ).When the cost of milk production ( tQ ) increases,
StP goes up which implies a positive relationship between S
tP and tQ . Since the class 1 price is
the minimum price that processors must pay SDC for buying fluid milk. The Class 1 price ( tC1 )
will be used as a policy related exogenous variable. When the Florida Class 1 price is high, it
means that the minimum price is high. This suggests a positive relationship between StP and
tC1 .
The exogenous variables that go into the supply reservation price equation (equation (4-
14)) should include some major cost variables. Feed cost should be included ( tFC ). When tFC
84
is high, StP has to go up. Milk production per cow is chosen to represent seasonality ( tS ). SMI
states that the Florida’s summer heat reduces milk production
(http://www.southeastmilk.org/v2/about/). This increases production cost per cow which results
in a higher StP . This suggests a negative relationship between S
tP and tS . SMI states that
transportation cost is the major cost of selling fluid milk
(http://www.southeastmilk.org/v2/about/) to Florida processors ( tTC ).When cost increases, StP
goes up which implies a positive between tTC and StP .
The exogenous variables and the expected signs of the parameters in the bargaining power
equation (4-14) can be rewritten as
)*exp(11
10 tt OOPθθ
τ−+
= (4-19)
where tOOP is the over order premium for cooperatives (SDC).
For the bargaining power parameter, Oczkowski (Oczkowski, 1999) and Binmore, Rubinstein et
al. (Binmore, et al., 1986) suggest that the impatience of either buyer or seller has an impact on
the bargaining power parameter. The more patience the player is, the stronger bargaining power
that player has. The over order premium ( tOOP ) is calculated as the difference between the
negotiated price and class 1 price. When SDC expects to have a higher tOOP , SDC should be
more patient when negotiate with the processor. Therefore, the expected sign for the parameter
associated with the over order premium ( tOOP ) inside the logistic function should be negative
which implies that when the over order premium ( tOOP ) increases, SDC’s bargaining power
( tτ ) increases due to the functional form in equation (4-19).
85
4.4 Data Description
The data used for the above model are monthly time series data based on the monthly
negotiation process between SDC and processor(s). It ranges from October 1998 to May 2009
(APPENDIX B). All data are available at the websites of the Agricultural Marketing Service
(AMS), the Bureau of Labor Statistics (BLS), and the National Agricultural Statistics Services
(NASS). The data set includes:
(1) Negotiated price ( tP in dollars per hundred weight) is the monthly cooperative announced fluid milk price in Miami, Florida (http://www.ams.usda.gov/AMSv1.0/ams.fetchTemplateData.do?template=TemplateF&navID=IndustryMarketingandPromotion&leftNav=IndustryMarketingandPromotion&page=AnnouncedCooperativeClassIPrices&description=Announced+Cooperative+Class+I+Prices&acct=dmktord);
(2) Negotiated quantity ( tQ in million pounds) is the monthly in area fluid milk sales for FMMO #6 in Florida (http://www.ams.usda.gov/AMSv1.0/ams.fetchTemplateData.do?startIndex=1&template=TemplateV&navID=IndustryMarketingandPromotion&leftNav=IndustryMarketingandPromotion&page=MilkMarketingandUtilizationIndividualOrdersYeartoDate&acct=dmktord);
(3) Class I price ( tC1 in dollars per hundred weight) is the monthly FMMO #6 minimum fluid milk price paid to farmers (http://www.ams.usda.gov/AMSv1.0/ams.fetchTemplateData.do?template=TemplateF&navID=IndustryMarketingandPromotion&leftNav=IndustryMarketingandPromotion&page=AnnouncedCooperativeClassIPrices&description=Announced+Cooperative+Class+I+Prices&acct=dmktord);
(4) Index of feed price paid by farmers ( tFC ) is the seasonally unadjusted index calculated from the survey of 48 contiguous states of the USA with the base period of 1990 to 1992 (http://usda.mannlib.cornell.edu/MannUsda/viewDocumentInfo.do?documentID=1002);
(5) Seasonality ( tS in pounds per cow) is the monthly milk production per cow in Florida (http://future.aae.wisc.edu/data/monthly_values/by_area/98?tab=production&grid=true&area=Florida);
(6) Transportation cost index ( tTC ) is the seasonally unadjusted Producer Price Index (PPI) for freight/truck transportation published by BLS every month based on June 1993 (http://www.bls.gov/schedule/archives/ppi_nr.htm);
86
(7) Labor cost index ( tL ) is the seasonally unadjusted total employment cost index of total compensation in the manufacturing industry based on 1989 which is used to represent the labor cost of processing fluid milk (http://www.bls.gov/schedule/archives/eci_nr.htm#2009);
(8) Fluid milk wholesale level price index ( tWP ) is the seasonally unadjusted PPI index for dairy products with the base year of 1982 (http://www.bls.gov/schedule/archives/ppi_nr.htm);
(9) The over-order-premium ( tOOP in dollars per hundred weight) is calculated by using the negotiated price ( tP ) minus the FMMO #6 Class I price ( tC1 ).
The data set that has been collected includes all exogenous variables and two endogenous
variables ( tP and tQ ). However, the other two endogenous variables ( DtP and S
tP ) are
uncollectable because they are the reservation prices for all the past negotiation period, and just
as the willingness to pay or sell prices for the past. Therefore, the collected data set is
incomplete. How to estimate the model with incomplete data set is a difficulty. This problem will
be addressed and solved in the next section.
4.5 Model Estimation
Quandt (Quandt, 1988) argues that disequilibrium models have two classes which are (1)
models where the observed demand is less or greater than the observed supply of a particular
good (the demand price is not equal to the supply price) and (2) models where the demand and
supply information is not available. The bargaining econometric model presented above showed
that the negotiated price lies between the demand reservation price (demand information) and the
supply reservation price (supply information) depending on the value of the bargaining power
parameter τ (equation (4-13)). The data for DtP and S
tP is not available which suggests that the
econometric bilateral bargaining model between Florida milk processors and SDC is a
disequilibrium model.
87
Quandt ((Quandt, 1988) suggested to estimate the disequilibrium model using maximum
likelihood estimation (MLE). The MLE method was pioneered by Fisher between 1912 and
1922 (Aldrich, 1997). Then, Oczkowski (Oczkowski, 1999) followed Quandt’s suggestion and
applied MLE in the disequilibrium model presented in his article and showed that the likelihood
function can be written as
Dt
StP
P
ttS
tD
t dPPdQPPPfLt
t
∫ ∫∞
∞−Π= ),,,( (4-20)
where ),,,( ttS
tD
t QPPPf is the joint density function of DtP , S
tP , tP , and tQ which are the
endogenous variables assuming a multivariate normal distribution. The likelihood function
presented here has double integrals inside as it was shown by Dempster et al (Dempster, et al.,
1977). This because the data set is incomplete where no data available for DtP and S
tP , the
probability density function (pdf) is then calculated by integrated the variables DtP and S
tP out.
Let’s define
4] [ ×= nQt
Pt
St
Dt εεεεε (4-21)
where Pt
St
Dt εεε ,, and Q
tε can be defined from equation (4-11) to equation (4-16), respectively, as
∑=
−−−=1
210 )ln()ln()ln(
K
i
Ditit
Dt
Dt XQP αααε (4-22)
∑=
−−−=2
210 )ln()ln()ln(
K
i
Sitit
St
St XQP βββε (4-23)
Stt
Dttt
Pt PPP )1( ττε −−−= (4-24)
)( Dt
Sttt
Qt PPQ −−= γε (4-25)
Then, by assumption, ε is distributed normally with zero mean and a variance-covariance
matrix Σ as
88
),0(~ ΣNε (4-26)
where the variance-covariance matrix ( Σ ) among equations can be defined as
442
PQSQ
PQ2
SP
SQSP2
DS
DQDPDS2
×
=Σ
QDQ
PDP
S
D
σσσσ
σσσσ
σσσσ
σσσσ
(4-27)
where 2Dσ , 2
Sσ , 2Pσ , and 2
Qσ are the constant variances of the error terms Dtε , S
tε , Ptε and Q
tε ,
respectively; PQSQSPDP σσσσσσ and , , , , , DQDS are covariance between errors Dtε and S
tε , Dtε
and Ptε , D
tε and Qtε , S
tε and Ptε , S
tε and Qtε , P
tε and Qtε , respectively. Notice that all the variance
and covariance terms defined here are parameters to be estimated.
Thus, the joint normal density function can be written as
)21exp()2(),,,( 12/12/ εεπ −−− Σ−Σ= Tk
tttt QPATCARf (4-28)
where 4=k represents the number of error terms; ε and Σ are defined above in equations (4-21)
to (4-25) and equation (4-27), respectively. By substituting the joint normal density function
(equation (4-28)) into the likelihood function (equation (4-20)), the likelihood function of the
econometric model presented here can be then written as
Dt
StP
P Tk dPPdLt
t
∫ ∫∞
∞−
−−− Σ−ΣΠ= )]21exp()2[( 12/12/ εεπ . (4-29)
Recall that the data for DtP and S
tP is not available. So to maximize this likelihood
function (equation (4-29)) is difficult. This issue is solved by using the Expectation
Maximization (EM) algorithm (Dempster, et al., 1977). This method was used by Oczkowski
(Oczkowski, 1999) with the assumption that the covariance among all equations was zero. The
covariance assumption is not made in this dissertation. As suggested by Dempster , Laird and
89
Rubin (Dempster, et al., 1977), the EM algorithm includes two major steps which are expectation
and maximization. The first step is expectation which will find the expectation of the unobserved
variables ( DtP and S
tP ) based on the specified model (equation (4-11) through (4-12)). The
second step is the maximization step which says that once the unobserved data is calculated by
using the expectation, the data set will be considered complete and then the regular log-
likelihood function will be maximized to get the optimum level parameter estimates which then
are used to calculate the expectation step again for the unobserved variables followed by another
maximization step. This process continues until the optimum level parameter estimates for two
consecutive EM algorithm runs converge.
To explain the EM algorithm mathematically, for the empirical model, the supply
reservation price for SDC ( DtP ) and the demand reservation price for processors ( S
tP ) are
unobservable. Following the assumption that all errors have zero means, the mean of DtP and
StP can be calculated from equation (4-11) and (4-12) to get
])ln()ln(exp[_1
210 ∑
=
++=K
i
Ditit
Dt XQPMean ααα (4-30)
])ln()ln(exp[_2
210 ∑
=
++=K
i
Sitit
St XQPMean βββ (4-31)
where α and β are estimated parameters associated with the exogenous variables which are
defined in equation (4-11) to equation (4-16). Once the means for DtP and S
tP are computed, for
a given variance covariance matrix between DtP and S
tP , the data for DtP and S
tP are simulated
from the bilateral distribution which is assumed.
Once the expectation step is done as shown by Dempster, et al. (Dempster, et al., 1977),
the second step is to maximize the likelihood function using a complete data set including
90
DtP and S
tP simulated in the expectation step. Following Greene (Greene, 2008) but in matrix
form, the likelihood function that will be maximized for the normal distribution in the
maximization step is
)]21exp()2[( 12/12/ εεπ −−− Σ−ΣΠ= TkL
. (4-32)
Then, the log-likelihood function to be maximized becomes
∑=
−Σ−Σ−−=n
itt
nknL1
1' )(21ln
2)2ln(
2ln εεπ (4-33)
where k is the number of normally distributed variables; n is the number of observations; Σ is
the variance covariance matrix defined in equation (4-27); and ε is a matrix of all errors for all
observations and all equations defined in equations (4-21) to (4-25). To maximize equation (4-
33), the necessary condition is that all parameters should be solved by setting the first order
derivatives with respect to all parameters equal to zero and the sufficient condition is that the
Hessian matrix (second order derivatives) has to be negative definite. The first order conditions
(FOC) are
0'2
ln 1 =Σ=∂∂ −εµ
xnL (4-34)
0)'(21
2ln 11 =ΣΣ+
Σ−=
Σ∂∂ −− εεnL (4-35)
where x is the exogenous variables arranged horizontally for all equations with errors. The FOC
are solved to get the maximum likelihood estimator (MLE) for the variance covariance matrix as
εε '1n
MLE =Σ∧
. (4-36)
91
Following Greene ((Greene, 2008), p533), substitute the solution (equation (4-36)) of the
variance-covariance matrix Σ into the log-likelihood function (equation (4-33)). Then the log-
likelihood function becomes the concentrated log-likelihood function
εεπ '1ln2
)2ln(22
lnn
nnknkL −−−= . (4-37)
Notice that the first and second terms are constant. Thus, the MLE of all coefficients for all
variables can be found through minimizing εε 'ln . Based on the law of large numbers, MLE has
the following asymptotic properties (Greene, 2008): asymptotic normal, consistent and
asymptotic efficient because the variance-covariance has achieved the Cramer-Rao Lower
bound. So the second step of the EM algorithm for maximization finishes here. If the last two
parameter estimates from the EM algorithm do not converge, the process starts over from the
expectation step to maximization step and continues until the last two parameter estimates
converge.
By using the EM algorithm on the expectation step (equation (4-30) and (4-31)) and the
maximization step (maximizing equation (4-37) by MLE), the parameter estimates are consistent
parameter estimates for the likelihood function presented in equation (4-29). In summary, by
assuming error terms that are distributed normal, the EM algorithm and the MLE are used to
estimate the empirical model for Florida dairy market. The statistical software GAUSS is used
for programming (APPENDIX C).
4.6 Summary
Since milk price bargaining in the Florida dairy market actually occurs between one
cooperative and one processor, the theoretical bilateral bargaining model can be applied to this
market. Moreover, during the past twenty five plus years which implies more than three hundred
negotiation periods, negotiations broke down only once. This suggests a very low probability for
92
both players to opt out and contract with out of state cooperatives or processors. Thus, by
assuming the probability of breakdown equals to zero, the common solution of Chapter 3 (
equations (4-1) and (4-2)) can be analyzed econometrically for the Florida dairy market to
examine the bargaining power of both players (equation (4-1) and (4-2)). The data was collected
as time series data from October 1998 to May 2009. However, the data set is incomplete due to
the unobservable supply reservation price for SDC and the demand reservation price for
processors. The incomplete data set problem is solved by using the EM algorithm and MLE to
estimate the model by programming inside GAUSS software (APPENDIX C).
93
CHAPTER 5 RESULTS
5.1 Summary Statistics
Following all the variables defined in chapter 4, the summary statistics of the data are
shown in Table 5-1. Negotiated price ( tP ) has the mean of $21.35 per hundredweight with a
minimum of $17.16 and a maximum of $31.37. The negotiated quantity ( tQ ) has the average of
209.82 million pounds with the minimum at 169 million pounds and the maximum at 242.10
million pounds.
Table 5-1. Summary Statistics of Data Variables Mean Standard Deviation Minimum Maximum
tP 21.35a 3.46 17.16a 31.37a
tQ 209.82b 15.48 169.00b 242.10b
tC1 18.33a 3.23 13.94a 26.78a
tFC 125.95c 29.66 96.00c 217.00c
tS 1346.47d 181.11 1020.00d 1760.00d
tTC 133.69c 14.07 112.40c 164.74c
tL 168.44c 17.38 138.90c 193.45c
tWP 151.24c 16.66 130.10c 189.50c
tOOP 3.02a 0.68 1.94a 4.85a
a Dollars per hundredweight. b Million pounds per month. c Index with various base yearS. d Pounds per cow per month.
The data correlations in Table 5-2 shows that a majority of the exogenous variables are less
than 0.5, and only two are 0.9 and above. The negotiated price ( tP ) and negotiated quantity ( tQ )
are endogenous variables. However, the correlation between the wholesale dairy price index
( tWP ) and the Class 1 price ( tC1 ) is 0.9 and between the labor index ( tL ) and transportation cost
index ( tTC ) is 0.97.
94
Table 5-2. Correlation Matrix of Data tP tQ tC1 tFC tS tTC tL tWP tOOP
tP 1.00
tQ 0.11 1.00
tC1 0.98 0.07 1.00
tFC 0.68 0.27 0.57 1.00
tS 0.09 0.74 0.05 0.36 1.00
tTC 0.61 0.30 0.50 0.86 0.32 1.00
tL 0.52 0.29 0.42 0.78 0.35 0.97 1.00
tWP 0.94 0.15 0.90 0.78 0.17 0.78 0.72 1.00
tOOP 0.43 0.26 0.25 0.74 0.23 0.73 0.62 0.53 1.00 The data collected is time series data with data for two endogenous variables are
simulated. As Cecchetti et al. (Cecchetti, et al., 2006) pointed out that the stationary residuals are
necessary for the reliability of the estimated parameters. First of all, all variables (Table 5-1) are
tested stationary individually by using Phillips and Perron (Phillips and Perron, 1988) test where
the null hypothesis is with unit root and the alternative hypothesis is stationary data. The results
suggest that among all 9 variables, only four of them are stationary at 5% significant level.
Therefore, the residuals are further tested. The test results reject the null hypothesis at 1%
significant level for all residuals which suggest that statistically, all residuals are stationary and
implies that the parameter estimated shown in the next section are reliable.
5.2 Parameter Estimates
The econometric model is
Dttttt
Dt WPLCQP εααααα ˆ)ln(ˆ)ln(ˆ)1ln(ˆ)ln(ˆˆ)ln( 43210 ++−+−= (5-1)
Stttttt
St TCSFCCQP εββββββ ˆ)ln(ˆ)ln(ˆ)ln(ˆ)1ln(ˆ)ln(ˆˆ)ln( 543210 ++−+++= (5-2)
Pt
Stt
Dttt PPP εττ ˆ)ˆ1(ˆ +−+= (5-3)
95
)*ˆˆexp(11ˆ
10 tt OOPθθ
τ−+
= (5-4)
Qt
Dt
Sttt PP Q εγ ˆ)(ˆ +−= (5-5)
)ˆˆ
/(1ˆ 11
QQtβα
γ −−
= (5-6)
where DtP , S
tP , tP and tQ are endogenous variables representing the processor’s demand
reservation price, the SDC’s supply reservation price, the negotiated price and the negotiated
quantity; α , β , τ , θ and γ are estimated coefficients; tC1 is the Florida Class 1 price; tL is the
labor cost index; tWP is the wholesale price index for dairy products; tFC is the feed cost index;
tS is the monthly average Florida production per cow which represents seasonality; tTC is the
transportation cost index; τ and τ1− are the bargaining power coefficients for SDC and the milk
processor; tOOP is the over order premium for SDC; Dtε , S
tε , Ptε and Q
tε are estimated standard
errors with constant variance 2Dσ , 2
Sσ , 2Pσ , and 2
Qσ , respectively.
Serial correlation and heteroscedasticity are potential problems that need to be dealt with
to ensure consistent estimators. Generally, time series data might have serial correlation but no
heteroscedasticity. The Durbin-Watson d test was used to test for serial correlation. Positive
serial correlation was found for equations 5-1, 5-2, 5-3 and 5-5. However, the MLE coefficient
estimates are asymptotically consistent (Levine, 1983). In addition, by using MLE to estimate the
model, the correction for serial correlation is intractable (Oczkowski, 1999). So, no corrections
were done.
The results show that all signs of the coefficients are the same as theoretically expected
except the sign of the parameter for transportation cost ( tTC ) (Table 5-3). However, the
96
coefficient estimate for tTC is insignificant which indicates that the estimated coefficient (-
2.5099) is not different from zero, statistically.
Table 5-3. Coefficient Estimates equations Variables Coefficient Estimates standard error
DtP Constant 1.3316 1.2011
tQ -0.0072 0.1612 tC1 0.6974*** 0.2548 tL -0.0308 0.1732 tWP 0.0165 0.3895
StP Constant -30.0392*** 0.9317
tQ 14.1532*** 1.5892 tC1 1.9391 4.6594 tFC 1.1535 3.1874 tS -5.8438* 3.4020 tTC -2.5099 3.0431 τ Constant 0.4814 0.6499 tOOP -0.4547* 0.2473
tP DtP 0.7055***(1) 0.0603
StP 0.2945***(1) 0.0603
tQ Dt
St PP − -14.8174***(1) 1.0931
*** 0.01 significance ** 0.05 significance * 0.1 significance (1) The coefficients for tP and tQ equations are the average coefficients
In the demand reservation price equation ( D
tP ), the estimated coefficients (Table 5-3) are
insignificant (statistically not different from zero) except the coefficient (0.6974) for the class I
price ( tC1 ). This suggests that when tC1 increases by one percent, the processors demand
reservation price goes up by 0.6974%. This result is reasonable because for fluid milk
processors, the class I price is the minimum price that they have to pay. This result implies that
government policy has a huge impact on the demand reservation price.
97
In the supply reservation price equation ( StP ), the estimated coefficients for the Constant,
negotiated quantity ( tQ ) and seasonality ( tS ) are significant. All other coefficients are
insignificant which implies that they are statistically not different from zero. The coefficient of
negotiated quantity ( tQ ) is 14.1532 which suggests that when the negotiated quantity increases
by one percent, the supply reservation price for SDC goes up by 14.1532%. The coefficient of
seasonality ( tS ) is -5.8438, which indicates that when the production per cow deceases by one
percent, the supply reservation price will increase by 5.8438%. The results suggest that quantity
is very important for SDC. When SDC has a surplus quantity, they have to sell them to the
manufacture plants. If the quantity is in deficit, SDC has to have more members from outside of
Florida to be able to meet the demand of the processors. The similar results also founded by
Folwell (Folwell, et al., 1998) for the analysis in tomato processing industry.
In the τ equation, the coefficient (-0.4547) for the over order premium ( tOOP ) is
negative and statistically significant. Recall that the bargaining power parameter (τ ) for SDC is
a logistic function of tOOP . This indicates that if tOOP between the Class I price (C1) and the
announced fluid milk price goes up, SDC is more patient during the negotiation, thus the
bargaining power for SDC goes up. This means that the bargaining power for processors goes
down by the same magnitude.
In the negotiated price equation ( tP ), both average coefficients for the demand
reservation price ( DtP ) and supply reservation price ( S
tP ) are statistically significant. If DtP
increases by one unit, tP will increase by 0.7055 units. One unit increase in StP will increase tP
by 0.2945 units.
98
In the negotiated quantity equation ( tQ ), the coefficient of the difference between the
supply reservation price and the demand reservation price ( Dt
St PP − ) is statistically significant
which shows that if Dt
St PP − gets larger by one unit, in other words, if the rang of the supply
reservation price and the demand reservation price gets larger, tQ will decrease by 14.8174
unites.
5.3 Bargaining Power Parameter
Table 5-4. Monthly Bargaining Power Coefficient Estimates for SDC and Processors
Months SDC Bargaining Power Estimates
Processors Bargaining Power Estimates
Differences Standard Errors(1)
January 0.6990*** 0.3010*** 0.3980 0.0511 Febuary 0.7158*** 0.2842*** 0.4316 0.0541 March 0.7099*** 0.2901*** 0.4198 0.0600 April 0.7210*** 0.2790*** 0.4420 0.0583 May 0.7094*** 0.2906*** 0.4188 0.0662 June 0.7032*** 0.2968*** 0.4064 0.0743 July 0.7065*** 0.2935*** 0.4130 0.0698 August 0.7100*** 0.2900*** 0.4200 0.0690 September 0.7101*** 0.2899*** 0.4202 0.0652 October 0.6903*** 0.3097*** 0.3806 0.0675 November 0.6889*** 0.3111*** 0.3778 0.0654 December 0.7031*** 0.2969*** 0.4062 0.0540 mean 0.7055*** 0.2945*** 0.4110 0.0603 *** 0.01 significance different from 0 (1) Standard errors are the same for both SDC’s bargaining power and processors’ bargaining power.
All monthly bargaining power coefficient estimates for both SDC and processors are
statistically significant and different from 0 (Table 5-4). Further, all monthly estimates are
statistically different from 0.5 which suggests that SDC and the processors have unequal
bargaining power during negotiations. The average bargaining power for SDC is 0.7055 where
the monthly average varies from 0.6889 in November to 0.7210 in April (Table 5-4). The
average bargaining power for processors is 0.2945 where the monthly average varies from
99
0.2790 in April to the highest 0.3111 in November. The overall average bargaining power
difference between SDC and the processors is 0.4110 which suggests that the overall bargaining
power for SDC is higher than the bargaining power for processors. The biggest bargaining power
difference between SDC and processors occurred in April (0.4420) and the lowest appeared in
November (0.3778).
Although we did not expect any seasonality among months, a statistical one tail t test was
used to examine whether SDC (processors) has the highest (lowest) bargaining power in April,
and whether SDC (processors) has the lowest (highest) bargaining power in November. The
results (Appendix B) show that statistically, there is no evidence to reject the null hypothesis
such that there is no difference between the bargaining power parameter for SDC (processors) in
April (November) and the bargaining power in the remaining 11 months. This indicates that
statistically, there is no difference among all monthly bargaining power coefficients for SDC.
The same result for processors. This implies that there is no seasonality among the months.
All yearly bargaining power coefficient estimates for both SDC and the processors are
statistically significant and different from 0 (Tables 5-5). In addition, all yearly estimates are
statistically different from 0.5 which suggests that SDC and processors have unequal bargaining
power over years. Compared to the monthly estimates, the yearly bargaining power for SDC
varies more from 0.6316 in 2001 to 0.8146 in 2008, while that for processors varies from 0.1854
in 2008 to 0.3684 in 2001. The difference among the yearly bargaining power estimates between
SDC and the processors ranges from 0.6292 in 2008 to 0.2631 in 2001.
In addition, a statistical one tail t test was used to analyze whether the SDC (processors)
has the highest (lowest) bargaining power in 2008, and whether SDC (processors) has the lowest
(highest) bargaining power in 2001. The results (Appendix D) show that statistically, there is
100
evidence to reject the null hypothesis such that the yearly average bargaining power in 2008 is
different from that in other years and in favor of the alternative hypothesis such that SDC
(processors) has the highest (lowest) yearly average bargaining power in 2008. Furthermore, the
null hypothesis of the yearly bargaining power in 2001 is not different from that in 2004;
however, the null hypotheses is rejected for all other years and indicates that SDC (processors)
has the lowest (highest) yearly average bargaining power in 2001. This implies that for the year
of 2001 and 2004, SDC (processors) has the lowest (highest) bargaining power.
Table 5-5. Yearly Bargaining Power Coefficient Estimates for SDC and Processors
Year SDC Bargaining Power Estimates
Processors Bargaining Power Estimates
Differences Standard Errors(1)
1998 0.6783*** 0.3217*** 0.3566 0.0124 1999 0.6630*** 0.3370*** 0.3260 0.0395 2000 0.6847*** 0.3153*** 0.3694 0.0265 2001 0.6316*** 0.3684*** 0.2631 0.0227 2002 0.7111*** 0.2889*** 0.4222 0.0271 2003 0.6960*** 0.3040*** 0.3920 0.0420 2004 0.6462*** 0.3538*** 0.2924 0.0327 2005 0.6855*** 0.3145*** 0.3710 0.0255 2006 0.7489*** 0.2511*** 0.4978 0.0250 2007 0.7665*** 0.2335*** 0.5331 0.0170 2008 0.8146*** 0.1854*** 0.6292 0.0351 2009 0.7394*** 0.2606*** 0.4788 0.0249 mean 0.7055*** 0.2945*** 0.4110 0.0603 *** 0.01 significant level different from 0 (1) Standard errors are the same for both SDC’s bargaining power and processors’ bargaining power
Graphically, the bargaining power for SDC has the same trend as the over-order premium
for SDC in the entire negotiation period from October 1998 to May 2009. Recall that for
cooperatives, the negotiated price is the sum of the Class I price and the over-order premium.
The class I price is a fixed minimum price that processors have to pay. So a higher bargaining
power for SDC implies more over-order premium for SDC. Figure 5-1 shows this positive
relationship for the bargaining power coefficient τ and over-order premium for cooperatives.
101
They are highly correlated with lowest point in 2001 and highest point in 2008 and an upward
trend started in 2004 through 2008 and started down in 2009.
Year
99 01 03 05 07 09 98 00 02 04 06 08 10
τ
0.5
0.6
0.7
0.8
0.9
1.0
Ove
r Ord
er P
rem
ium
($
per
hun
dred
wei
ght)
0
1
2
3
4
5
6τOver Order Premium
Figure 5-1. SDC’s Monthly Bargaining Power Coefficient and Over-order Premium
5.4 Variance Covariance
Let’s look at the estimated variance covariance matrix from the MLE even though they are
inconsistent due to the presence of serial correlation. The estimated variance covariance matrix
( Σ ) among equations is
4*42
PQSQ
PQ2
SP
SQSP2
DS
DQDPDS2
ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ
ˆ
=Σ
QDQ
PDP
S
D
σσσσ
σσσσ
σσσσ
σσσσ
(5-7)
102
where 2ˆ Dσ , 2ˆ Sσ , 2ˆ Pσ , and 2ˆQσ are the constant variance of the error terms Dtε , S
tε , Ptε and Q
tε ,
respectively; PQSQSPDP σσσσσσ ˆ and ,ˆ ,ˆ ,ˆ ,ˆ ,ˆ DQDS are covariance between errors Dtε and S
tε , Dtε
and Ptε , D
tε and Qtε , S
tε and Ptε , S
tε and Qtε , P
tε and Qtε , respectively. Following the definition in
equation 5-7, testing the null hypothesis that all covariances are zero by using the likelihood ratio
test
)()ln(ln2 2 kxLL UR →−− (5-8)
where RL is the likelihood value for the restricted model by restricting the covariance terms to
equal zero; UL is the likelihood value for the unrestricted model (covariance are not restricted to
zero) which is the model that has been presented and estimated in chapter 4; the statistic
)ln(ln2 UR LL −− follows a chi-square distribution with k (the number of restrictions) degrees of
freedom (Greene, 2008). Thus, the null hypothesis that all covariances equal zero can be written
as
Ho: 0ˆ ˆ ˆ ˆ ˆ ˆ DQDS ====== PQSQSPDP σσσσσσ
Ha: not all co-variances are zero
Table 5-6. Test Result for Zero Covariance ln RL ln UL )ln(ln2 UR LL −− 01.0 )6(2 atx
-1018 -717 601 16.8119
The result (Table 5-6) indicates that the test statistic (601) is larger then the associated
chi-square value (16.8119). The null hypothesis is rejected which implies that there is only a 1%
probability that the null hypothesis of all zero covariances is true. This result indicates that one
or more non-zero covariances among equation DtP , S
tP ,P and Q do exist.
103
CHAPTER 6 SUMMARY, CONCLUSIONS AND IMPLICATIONS
6.1 Summary
In chapter 2, a comprehensive literature review was done on three major categories. The
first category is bargaining theory including bilateral bargaining and multilateral bargaining
theory review. The second category is distributional channels bargaining including supply chain
and especially the cooperatives bargaining theory review. The third category is the empirical
applications associated with bargaining theory.
In summary, bargaining theory was developed by two different approaches: the axiomatic
approach and the strategic approach. The axiomatic approach is based on four axioms which
include Invariance to positive affine transformations (IAT), Symmetry (SYM), Pareto optimality
(PAR) and Independence of irrelevant alternatives (IIA). This approach develops the bargaining
model for two monopoly/monoposony players, and then easily extends to a multilateral model.
The solution for the axiomatic approach model is the generalized Nash bargaining solution for
both the bilateral and multilateral models.
The strategic approach had difficulties extending the model from bilateral to multilateral
with outside options and risk of breakdown. Two different methods were used to solve the
difficulties. One is to reduce the multilateral strategic game to a multistage game by forming
coalitions in the first stage and bilateral bargaining as the second stage. The other method is to
reduce the game to a series of bilateral games. The solution to the strategic approach model
approximated the generalized Nash bargaining solution for both the bilateral and multilateral
models at the limit when the negotiation time period goes to zero.
In chapter 3, a theoretical strategic bargaining model was developed for bilateral
bargaining with outside options for both sides under the assumption that when negotiation
104
breakdown occurs, both players will go to outside options. Three cases were discussed when the
outside option price was fixed or variable for any existing quantity. When the probability of
negotiation breakdown goes to zero, the model was solved by using the generalized Nash
bargaining solution and the common solution of all three cases indicates that the negotiated price
is between the processor’s average net revenue and SDC’s average total cost depending on the
bargaining power for each side. The negotiated quantity is also a function of the processor’s
average net revenue and SDC’s average total cost. The interesting result is that under the
assumption that both SDC and processor can opt out when breakdown occurs, outside options
only affects the solution of the case where one outside option price is variable and the other one
is fixed.
An empirical econometric model was developed for the Florida dairy market.
Econometrically, the common solution to the Nash model was developed into a six equation
disequilibrium econometric model. The time series data for the econometric model, from
October 1998 to May 2009, is incomplete due to the unavailable supply reservation price
(average total cost) for SDC and the unavailable demand reservation price (average net revenue)
for processor(s). The estimation difficulty of incomplete data was solved by using the
expectation maximization (EM) algorithm along with maximum likelihood estimation (MLE)
inside the EM algorithm where the concentrated log-likelihood function was estimated.
The results were then presented in chapter 5. The data was tested for serial correlation and
heteroscedasticity. The results show that serial correlation does exist for all four error terms, but
no evidence of heteroscedasticity was found. This implies that the coefficient estimates are
consistent. The coefficient estimates for all exogenous variables and the bargaining power
coefficient τ were presented and discussed.
105
6.2 Conclusions
Regardless of bilateral or multi player games, no matter what approach is used, the
generalized Nash bargaining solution of bilateral bargaining can be used to solve these
negotiation problems when negotiation time goes to zero. Therefore, the strategic approach was
adopted to develop the theoretical bilateral bargaining model with outside options and risk of
breakdown under the assumption that both sides are maximizing their profits. The common
solution of the Nash model was chosen for empirical use to examine the bargaining power for
dairy cooperatives and Florida fluid milk processors.
The negotiated price is influenced the most by the class I price (C1) set by the FMMO #6.
The market intervention by the FMMO significantly influences the demand reservation price set
by the processors. The coefficient of C1 is much larger than the coefficients of any other
variables even through their signs are correct. Further, the coefficient estimate for the negotiated
quantity turns out to be statistically insignificant which implies that negotiated quantity is
actually not a major issue for processors. Processors need the milk and they know that there is a
minimum price they must pay. This suggests a certain a mount of impatience by the processors.
On the supply side, the negotiated quantity and seasonality are the most significant factors
for SDC. The negotiated quantity is important because SDC must find milk in deficit months and
must sell milk to manufacturing plants during surplus months. This influences their supply
reservation price. Furthermore, seasonality in milk production influences the cost to produce.
Hot weather decreases milk production per cow and expenses per hundredweight of milk are
increased. This causes the average total cost per cow to increase which increases SDC’s supply
reservation price.
The results for the bargaining power coefficient τ show that statistically the monthly and
yearly average bargaining power coefficient estimates are different from 0.5, and the average
106
bargaining power for SDC (0.7055) is higher than the average bargaining power for processors
(0.2945). The reasons for this were explained by Prasertsri and Kilmer (Prasertsri and Kilmer,
2008). First, Florida milk “processors are more impatient than SDC because they have buyers
who need a continuous supply of dairy products.” Second, if negotiation breaks down, Florida
milk processors must go outside of the state to find milk suppliers whose milk will not be as
fresh as the milk provided by SDC considering the time for transportation. Third, the cost of
delivering milk to Florida processors from non-Florida milk suppliers is high which will result in
a higher price to processors.
All monthly and yearly bargaining power coefficients are statistically different from zero.
This suggests that the model captures well the bilateral bargaining between SDC and Florida
milk processors. The bargaining is bilateral bargaining which means that both sides have some
bargaining power but neither side has absolute power (bargaining power coefficient estimates are
different from zero). If one side had absolute power, then the bargaining process would not begin
in the first place.
Furthermore, the monthly bargaining power coefficient estimates are statistically the same
for each month for both SDC and processors. However, the yearly average bargaining power
coefficient estimates vary. SDC (processors) has the lowest (highest) yearly bargaining power in
2001 and year 2004, and the highest (lowest) yearly bargaining power in 2008. This occurred
because the over order premium trended upward from 2004 to 2008.
Theoretically, when the bargaining power for SDC is high, the over-order premium above
the class I price should be high which gives more revenue to dairy farmers. The estimated results
from the empirical model capture and present this theoretical explanation well such that the
bargaining power coefficient for SDC and the over-order premium trends in the same direction
107
with the lowest value for both of them occurring in 2001 and highest values for both of them
appearing in 2008.
Over all, the model indicates that on average, SDC has higher bargaining power than
Florida milk processors which suggests that SDC is competitive with processors during the
negotiation process. Class I price is the most important factor for processors when buying milk
from SDC. Negotiated quantity is more important for SDC when it comes to bargaining.
6.3 Implications
Increasing Class 1 price increases the demand reservation price for processors which raises
the negotiated price. Thus, dairy policy, a price floor policy (class 1 price) set by the Federal
Milk Marketing Order (FMMO), has a strong impact on negotiated prices. Government
intervention matters for processors.
Higher bargaining power for SDC but statistically, non-variable monthly bargaining power
parameters for both SDC and processors indicate that both SDC and processors are effective
negotiators. However, compared to monthly bargaining power, the variable yearly bargaining
power suggests a dynamic bargaining environment.
On the supply side, the supply reservation price is more volatile than the demand
reservation price. This suggests that SDC must put more time into determining its reservation
price than the processors. The quantity turns out to be a very important factor for SDC. Further,
SDC has the ability to deal with the surplus (selling the product to the manufacturing plants) or
deficit (needing more milk) situation in quantity. This suggests that they are more patient during
negotiation which results in higher bargaining power.
6.4 Further Research
Thus, future research can be done in the following areas: (1) extend the bargaining power
equation to include risk preferences; (2) the method can be used in other commodity markets; (3)
108
building more complex dynamic strategic approaches for bilateral markets and how to
econometrically estimate the dynamic bargaining power coefficients over time; (4) develop multi
player games for oligopoly market structure; (5) research related to examining whether the
negotiated price would be different if there were no price floor policy or other similar price
policies.
109
APPENDIX A CALCULATION OF THE NEGOTIATED QUANTITY Q
τWBrBr
τWArAr ]πp)π-p[(]πp)π-p[( F −++= 1
)Q(P,11 max (A-1)
where
)]([)( QATCPQQCPQ CPCPA −=−=π (A-2)
)]([)( QATCPQQCQP COC
OCOC
OWA −=−=π (A-3)
])()([)( PQABQARQBPQQR PCPCPCPCB −−=−−=π (A-4)
])()([)( POPOPOPO
POPOWB PQABQARQBQPQR −−=−−=π (A-5)
Solve for the negotiated quantity Q by using the first order condition of equation (A-1)
with respect to Q
0][])[[1(][][ 11 =∂∂
−+∂∂
=∂∂ −−−
QQQF RO
BROB
ROA
ROB
ROARO
Aπ
ππτππ
πτ ττττ . (A-6)
Multiplying both sides by τπ −1][ ROA , then
0]][)[1(][ 1 =∂∂
−+∂∂ −−
ROBRO
BROA
ROB
ROA π
ππτππ
τ ττ (A-7)
Multiplying both sides by τπ ][ ROB , then
0][)1(][. =∂∂
−+∂∂ RO
A
ROBRO
B
ROA
QQπ
πτπ
πτ . (A-8)
Then, solve for Q
ROA
∂∂π and
Q
ROB
∂∂π
])(
)1())(1[(
QATC
QpATCPp
QATC
QpATCPpQ
COrCO
COr
CPrCPr
ROA
∂∂
−−+
∂∂
−−−−=∂∂π
(A-9)
110
)]()([
)]()1())(1[(
QAB
QARQpPABARp
QAB
QARQpPABARp
Q
POPOr
POPOPOr
PCPCrPCPCr
ROB
∂∂
−∂
∂+−−+
∂∂
−∂
∂−+−−−=
∂∂π
(A-10)
Then, substitute ROBπ
and equation (A-9) into the first part of equation (A-8) to get
)])()((
))()(()1[(
])(
)1())(1[(][
POPOPOr
PCPCr
COrCO
COr
CPrCPr
ROB
ROA
PQABQARQpPQABQARQp
QATCQpATCPp
QATCQpATCPp
Q
−−+
−−−•∂
∂−−+
∂∂
−−−−=∂∂ τππτ
(A-11)
Extend all terms of equation (A-11) to get
))()((
))()()(1(
))()(()(
))()(()1)((
))()(()1(
))()(()1(
))()(())(1(
))()(()()1(][
22
2
2
2
22
2
POPOPO
COr
PCPCrCO
r
POPOPOCO
COr
PCPCrCOC
Or
POPOPOr
CPr
PCPCCP
r
POPOPOrCPr
PCPCCPrROB
ROA
PQABQARQ
ATCQpτ
PQABQARpQ
ATCQp
PQABQARQATCPpPQABQARQpATCPp
PQABQARpQ
ATCQp
PQABQARQ
ATCQp
PQABQARQpATCPp
PQABQARQATCPpQ
−−∂
∂−
−−−∂
∂−
−−−+
−−−−+
−−∂
∂−−
−−∂
∂−−
−−−−+
−−−−=∂∂
τ
τ
τ
τ
τ
τ
τππτ
(A-12)
Then, substitute ROAπ and equation (A-10) into the second part of equation (A-8) to get
111
))]((.))(()1[(
)]()(
)()1())(1)[(1(
][)1(
QATCPQpQATCPQpQ
ABQ
ARQpPABARp
QAB
QARQpPABARp
Q
COC
OrCPr
POPOr
POPOPOr
PCPCrPCPCr
ROA
ROB
−+−−•
∂∂
−∂
∂+−−+
∂∂
−∂
∂−+−−−−=
∂∂
−
τ
ππτ
(A-13)
Extend all terms of equation (A-13) to get
))()((
))()(1)((
))(()(
))(()1)((
))(()()1(
))()(()1(
))(())(1(
))(()()1(
))()((
))()(1)((
))(()(
))(()1)((
))(()()1(
))()(()1(
))(())(1(
))(()()1(
][)1(
22
2
2
2
22
2
22
2
2
2
22
2
QATCPQ
ABQ
ARQp
QATCPpQ
ABQ
ARQp
QATCPQPABARpQATCPQpPABARp
QATCPpQ
ABQ
ARQp
QATCPQ
ABQ
ARQp
QATCPQpPABARpQATCPQPABARp
QATCPQ
ABQ
ARQp
QATCPpQ
ABQ
ARQp
QATCPQPABARpQATCPQpPABARp
QATCPpQ
ABQ
ARQp
QATCPQ
ABQ
ARQp
QATCPQpPABARpQATCPQPABARp
Q
COC
OPOPO
r
CPrPOPO
r
COC
OP
OPOPOr
CPrP
OPOPOr
COC
OrPCPC
r
CPPCPC
r
COC
OrPCPCr
CPPCPCr
COC
OPOPO
r
CPrPOPO
r
COC
OP
OPOPOr
CPrP
OPOPOr
COC
OrPCPC
r
CPPCPC
r
COC
OrPCPCr
CPPCPCr
ROA
ROB
−∂
∂−
∂∂
−
−−∂
∂−
∂∂
−
−−−−
−−−−−
−∂
∂−
∂∂
−−
−∂
∂−
∂∂
−−
−−−−−
−−−−−
−∂
∂−
∂∂
+
−−∂
∂−
∂∂
+
−−−+
−−−−+
−∂
∂−
∂∂
−+
−∂
∂−
∂∂
−+
−−−−+
−−−−=
∂∂
−
τ
τ
τ
τ
τ
τ
τ
τ
ππτ
(A-14)
112
Substitute equation (A-12) and (A-14) into equation (A-8) to get
0))()((
))()(1)((
))(()(
))(()1)((
))(()()1(
))()(()1(
))(())(1(
))()((
))()(1)((
))(()(
))(()1)((
))(()()1(
))()(()1(
))(())(1(
))(()()1(
))()((
))()()(1(
))()(()(
))()(()1)((
))()((.)1(
))()(()1(
))()(())(1(
22
2
2
2
22
22
2
2
2
22
2
22
2
2
2
22
=−∂
∂−
∂∂
−
−−∂
∂−
∂∂
−
−−−−
−−−−−
−∂
∂−
∂∂
−−
−∂
∂−
∂∂
−−
−−−−−
−∂
∂−
∂∂
+
−−∂
∂−
∂∂
+
−−−+
−−−−+
−∂
∂−
∂∂
−+
−∂
∂−
∂∂
−+
−−−−+
−−−−+
−−∂
∂−
−−−∂
∂−
−−−+
−−−−+
−−∂
∂−−
−−∂
∂−−
−−−−+
QATCPQ
ABQ
ARQp
QATCPpQ
ABQ
ARQp
QATCPQPABARpQATCPQpPABARp
QATCPpQ
ABQ
ARQp
QATCPQ
ABQ
ARQp
QATCPQpPABARp
QATCPQ
ABQ
ARQp
QATCPpQ
ABQ
ARQp
QATCPQPABARpQATCPQpPABARp
QATCPpQ
ABQ
ARQp
QATCPQ
ABQ
ARQp
QATCPQpPABARpQATCPQPABARp
PQABQARQ
ATCQpτ
PQABQARpQ
ATCQp
PQABQARQATCPpPQABQARQpATCPp
PQABQARpQ
ATCQp
PQABQARQ
ATCQp
PQABQARQpATCPp
COC
OPOPO
r
CPrPOPO
r
COC
OP
OPOPOr
CPrP
OPOPOr
COC
OrPCPC
r
CPPCPC
r
COC
OrPCPCr
COC
OPOPO
r
CPrPOPO
r
COC
OP
OPOPOr
CPrP
OPOPOr
COC
OrPCPC
r
CPPCPC
r
COC
OrPCPCr
CPPCPCr
POPOPO
COr
PCPCrCO
r
POPOPOCO
COr
PCPCrCOC
Or
POPOPOr
CPr
PCPCCP
r
POPOPOrCPr
τ
τ
τ
τ
τ
τ
τ
τ
τ
τ
τ
τ
τ
(A-15)
113
Divide both sides of equation (16) by Q to get
0))()((
))()(1)((
))()((
))()(1)((
))(()()1(
))()(()1(
))(())(1(
))()((
))()(1)((
))()((
))()(1)((
))(()()1(
))()(()1(
))(())(1(
))()(()1(
))()((
))()()(1(.
))()()((
))()()(1)((
))()(()1(
))()(()1(
))()(())(1(
2
2
2
2
2
2
2
2
2
2
=−∂
∂−
∂∂
−
−−∂
∂−
∂∂
−
−−−−
−−−−−
−∂
∂−
∂∂
−−
−∂
∂−
∂∂
−−
−−−−−
−∂
∂−
∂∂
+
−−∂
∂−
∂∂
+
−−−+
−−−−+
−∂
∂−
∂∂
−+
−∂
∂−
∂∂
−+
−−−−+
−−−−+
−−∂
∂−
−−−∂
∂−
−−−+
−−−−+
−−∂
∂−−
−−∂
∂−−
−−−−+
QATCPQ
ABQ
ARQp
QATCPpQ
ABQ
ARQp
QATCPPABARpQATCPpPABARp
QATCPpQ
ABQ
ARQp
QATCPQ
ABQ
ARQp
QATCPpPABARp
QATCPQ
ABQ
ARQp
QATCPpQ
ABQ
ARQp
QATCPPABARpQATCPpPABARp
QATCPpQ
ABQ
ARQp
QATCPQ
ABQ
ARQp
QATCPpPABARpQATCPPABARp
PQABQARQ
ATCQpτ
PQABQARpQ
ATCQp
PQABQARATCPpPQABQARpATCPp
PQABQARpQ
ATCQp
PQABQARQ
ATCQp
PQABQARpATCPp
COC
OPOPO
r
CPrPOPO
r
COC
OP
OPOPOr
CPrP
OPOPOr
COC
OrPCPC
r
CPPCPC
r
COC
OrPCPCr
COC
OPOPO
r
CPrPOPO
r
COC
OP
OPOPOr
CPrP
OPOPOr
COC
OrPCPC
r
CPPCPC
r
COC
OrPCPCr
CPPCPCr
POPOPO
COr
PCPCrCO
r
POPOPOCO
COr
PCPCrCOC
Or
POPOPOr
CPr
PCPCCP
r
POPOPOrCPr
τ
τ
τ
τ
τ
τ
τ
τ
τ
τ
τ
τ
τ
(A-16)
114
Collect all the terms with “Q”, all the terms without Q and move the one without “Q” to the right
hand side of equation (A-16) to get
2
2
2
(1 ) ( ( ) ( ) )
(1 ) ( ( ) ( ) )
(1 )( ( ) ( ) )
( ( ) ( ) )
(1 ) ( )( ( ))
(1 ) (
CPr PC PC
PCPr r PO PO O
COr r PC PC
PCOr PO PO O
PC PCr CP
r
ATCp Q AR Q AB Q PQ
ATCp Q p AR Q AB Q PQ
ATCp Q p AR Q AB Q PQ
ATCτ p Q AR Q AB Q PQ
AR ABp Q P ATC QQ Q
p Q
τ
τ
τ
∂− − − −
∂∂
− − − −∂
∂− − − −
∂∂
− − −∂∂ ∂
+ − − −∂ ∂
∂+ −
2
2
) ( ( ))
( )(1 )( ( ))
( )( ( ))
(1 ) ( )( ( ))
(1 ) ( ) ( ( ))
(
CPC PCr O CO
PO POr r CP
CPO POr O CO
PC PCr CP
CPC PCr r O CO
POr
AR AB p P ATC QQ Q
AR ABp Q p P ATC QQ Q
AR ABp Q P ATC QQ Q
AR ABp Q P ATC QQ Q
AR ABp Q p P ATC QQ Q
ARp Q
τ
τ
τ
∂− −
∂ ∂∂ ∂
+ − − −∂ ∂
∂ ∂+ − −
∂ ∂∂ ∂
− − − −∂ ∂
∂ ∂− − − −
∂ ∂∂
−∂
2
)(1 )( ( ))
( )( ( ))
POr CP
CPO POr O CO
AB p P ATC QQ Q
AR ABp Q P ATC QQ Q
τ
∂− − −
∂∂ ∂
− − −∂ ∂
(A-17)
115
2
2
(1 )( ) ( ( ) ( ) )
( )(1 )( ( ) ( ) )
( )( ( ) ( ) )
(1 ) ( )( ( ))
(1 )( ) ( ( ))
(
Pr CP r PO PO O
Cr O CO r PC PC
C Pr O CO PO PO O
r PC PC CP
Cr PC PC r O CO
r PO
p P ATC p AR Q AB Q P
p P ATC p AR Q AB Q P
p P ATC AR Q AB Q P
p AR AB P P ATC Q
p AR AB P p P ATC Q
p AR AB
τ
τ
τ
=
− − − − −
− − − − −
− − − −
− − − − −
− − − − −
− −2
2
)(1 )( ( ))
( )( ( ))
(1 )( ) ( ( ))
( )(1 )( ( ))
( )( ( ))
PPO O r CP
P Cr PO PO O O CO
Cr PC PC r O CO
Pr PO PO O r CP
P Cr PO PO O O CO
P p P ATC Q
p AR AB P P ATC Q
p AR AB P p P ATC Q
p AR AB P p P ATC Q
p AR AB P P ATC Q
τ
τ
τ
− − −
− − − −
+ − − − −
+ − − − −
+ − − −
(A-17 continued)
Let’s collect the terms of the left hand side of equation (A-17) and define it as “Left”, and
))]((
))()(1( ))((
))()(1()[(
))](()1(
))(()1(
))(()1(
))(()1)[((
)])()((
))()()(1( ))()(()1(
))()(()1()[(
2
2
2
2
2
2
QATCPpQATCPpp
QATCPp
QATCPppQ
ABQ
ARQATCPpp
QATCPpQATCPpp
QATCPpQ
ABQ
ARPQABQARpτ
PQABQARppPQABQARpp
PQABQARpQ
ATCQleft
COC
Or
CPrr
COC
Or
CPrrPOPO
COC
Orr
CPr
COC
Orr
CPrPCPC
POPOPOr
PCPCrr
POPOPOrr
PCPCrCP
−−
−−−−+
−−+∂
∂−
∂∂
+
−−−
−−−
−−+
−−∂
∂−
∂∂
+
−−−
−−−−−−−−
−−−−∂
∂•=
τ
τ
τ
τ
ττ
τ
(A-18)
116
))](( )1(
))()(1()1)[((
))](()1)(1(
))(()1)(1)[((
)])()((
))()()(1()[(
2
2
QATCPp
QATCPppQ
ABQ
ARQATCPpp
QATCPpQ
ABQ
ARPQABQARpτ
PQABQARpQ
ATCQleft
COC
Or
CPrrPOPO
COC
Orr
CPrPCPC
POPOPOr
PCPCrCP
−−+
−−−∂
∂−
∂∂
+
−−−+
−−−∂
∂−
∂∂
+
−−−
−−−−∂
∂•=
τ
τ
τ
τ
τ
(A-19)
Let’s collect the terms of the right hand side of equation (A-17) and define it as “right”, and
])[))(((
])1()[))(((
])1()[)()()((
)))(()(1(
2
2
rP
OPOPOCOC
O
rPCPCCP
rrPCPCCOC
O
POPOPOCPrr
pPABARQATCPpPABARQATCP
ppPQABQARATCPPABARQATCPppright
−−−−+
−−−−−+
−−−−−+
−−−−−=
(A-20)
Solving for )(QATCP CP− by substituting the negotiated price equation (3-25) to get
)(1
)1(
)()](1
)[(
)](1
[
)(1
)](1
)[(
)()](1
)[1(
)](1
)[(
)(
COC
Or
r
CPP
OPOPOr
rPCPC
COC
Or
rCP
COC
Or
rPOPOPO
r
rPCPC
CPCOC
Or
rCP
POPOPO
r
rPCPC
CP
ATCPp
p
ATCPABARp
pABAR
ATCPp
pATC
ATCPp
pPABARp
pABAR
QATCATCPp
pATC
PABARp
pABAR
QATCP
−−
−−
−−−−
+−=
−−
−−
−−
−−−−
+−=
−−−
−−+
−−−
+−=
−
τ
ττ
τ
τ
τ
τ
(A-21)
Solving for PABAR PCPC −− by substituting the negotiated price equation (3-25) to get
117
)](1
)[1(
)(1
))(1(
)](1
)[1(
)](1
)[(
COC
Or
rCP
POPOPO
r
rPCPC
COC
Or
rCP
POPOPO
r
rPCPCPCPC
PCPC
ATCPp
pATC
PABARp
pABAR
ATCPp
pATC
PABARp
pABARABAR
PABAR
−−
−−−
−−−
−−−=
−−
−−−
−−−
+−−−=
−−
τ
ττ
τ
τ
(A-22)
Then, substitute equation (A-21) and (A-22) into equation (A-19) to get
))](( )1(
)](1
)1(
)())(1
)(()[1()1)((
))(()1)(1(
)](1
)1(
)())(1
)(([)1)(1)((
))()((
))](1
)(1(
)(1
))(1)[(1()(
2
2
QATCPp
ATCPp
p
ATCPABARp
p
ABARppQ
ABQ
ARQATCPpp
ATCPp
p
ATCPABARp
p
ABARpQ
ABQ
ARPQABQARpτ
ATCPp
pATC
PABARp
p
ABARpQ
ATCQleft
COC
Or
COC
Or
r
CPP
OPOPOr
r
PCPCrrPOPO
COC
Orr
COC
Or
r
CPP
OPOPOr
r
PCPCrPCPC
POPOPOr
COC
Or
rCP
POPOPO
r
r
PCPCrCP
−−+
−−
−−
−−−−
+
−−−∂
∂−
∂∂
+
−−−+
−−
−−
−−−−
+
−−−∂
∂−
∂∂
+
−−−
−−
−−−
−−−
−
−−−−∂
∂•=
τ
τ
τ
ττ
τ
τ
τ
ττ
τ
τ
ττ
(A-23)
Simplifying equation (A-23) to get
118
]11
11[
])1111
11
]111
1
22
2
)ATC(Pτ)p()PAB(ARτ)p(
)ATCAB)(ARp(τ)p()Q
ABQ
AR(
)ATC(Pp(τ)p()PAB)(ARp(τ)p(
)ATCAB(AR)pτ)(()[Q
ABQ
AR(
)ATC(Pτ) pτ()ATCAB)(ARpτ)(τ(
)P(Q)AB(Q)(ARτ)τ p()[Q
ATC(Q
left
COC
OrP
OPOPOr
CPPCPCrrPOPO
COC
OrrP
OPOPOrr
CPPCPCrPCPC
COC
OrCPPCPCr
POPOPOr
CP
−−+−−−+
−−−−∂
∂−
∂∂
+
−−−+−−−−+
−−−−∂
∂−
∂∂
+
−−−−−−−−
−−−−∂
∂•=
ττ
τ
ττ
τ (A-24)
]
1[
])11
1
]1
1
22
2
)ATC(Pp)PAB(ARp
)ATCAB)(ARp(p)Q
ABQ
AR(
)ATC(Pp(p)PAB)(ARp(p
)ATCAB(AR)p)[(Q
ABQ
AR(
)ATC(P p)ATCAB)(ARp(
)P(Q)AB(Q)(AR p)[Q
ATC(τ)τ(Q
left
COC
OrP
OPOPOr
CPPCPCrrPOPO
COC
OrrP
OPOPOrr
CPPCPCrPCPC
COC
OrCPPCPCr
POPOPOr
CP
−+−−+
−−−∂
∂−
∂∂
+
−−+−−−+
−−−∂
∂−
∂∂
+
−−−−−−
−−−∂
∂•−•=
(A-25)
Then, substitute equation (22) and (23) into equation (21) to get
119
])[))(((
])1()][(1
)[1(
)(1
))(1)((1
)1(
)()](1
)[(
)]1()[)()()((1
)1(
)()](1
)[(
])1()][(1
)[1(
)(1
))(1)((
2
2
rP
OPOPOCOC
O
rCOC
Or
rCP
POPOPO
r
r
PCPCCOC
Or
r
CPP
OPOPOr
rPCPC
rrP
OPOPOCOC
Or
r
CPP
OPOPOr
rPCPC
rrCOC
Or
rCP
POPOPO
r
rPCPCCO
CO
pPABARQATCP
pATCPp
pATC
PABARp
p
ABARATCPp
p
ATCPABARp
pABAR
ppPQABQARATCPp
p
ATCPABARp
pABAR
ppATCPp
pATC
PABARp
pABARATCP
right
−−−−
+−−−−
−−−
−−−
−
−−−−
−−
−−−−
+−
+−−−−−−
−−
−−−−
+−
+−−−−
−−−
−−−
−−−−
=
τ
τ
ττ
ττ
τ
ττ
τ
ττ
(A-26)
Simplifying equation (A-26) to get
120
))((
)()1(
)( )()1(
))()(1()1(
))(()1(
)(
))()(1()1(
))()(1()1(
))()(1(
)()1)(1(
))()()()()(1(
)()(
)]1()[)()()((
))(1(
))((
))()(1()1(
2
222
2
2
2
222
2
22
2
22
22
2
POPOPOCO
COr
COC
Or
POPOPOCO
COr
CPPCPCCOC
Orr
COC
OP
OPOPOr
POPOPOr
CPPCPCP
OPOPOrr
COC
OCPPCPCrr
POPOPOCPPCPCrr
CPPCPCr
rP
OPOPOCOC
O
rP
OPOPO
rrP
OPOPOCPPCPC
COC
Or
POPOPOCO
COr
CPPCPCCOC
Orr
PABARATCPpATCPp
PABARATCPpATCABARATCPpp
ATCPPABARpPABARp
ATCABARPABARppATCPATCABARpp
PABARATCABARppATCABARp
pPQABQARATCPpPABAR
ppPQABQARATCABARATCPp
PABARATCPpATCABARATCPpp
right
−−−−
−−+
−−−−−
−−−−−+
−−−−−
−−+
−−−−−−−
−−−−−−
−−−−−+
−−−−−
−−−−+
−−−
−−−−−−
−−−
−−−+
−−−−−−
=
τ
ττ
τ
ττ
τ
ττ
ττ
τ
ττ
τ
τ
τ
τ
τ
τ
(A-27)
Simplifying equation (A-27) to get
22
2
22
22
)()1(
))()(1()1(2
))(()1(2
)()1)(1(
)()1(
))()(1()1(2
COC
Or
CPPCPCCOC
Orr
POPOPOCO
COr
CPPCPCr
POPOPOr
POPOPOCPPCPCrr
ATCPpATCABARATCPpp
PABARATCPpATCABARp
PABARpPABARATCABARpp
right
−−−
−−−−−−
−−−−−
−−−−−
−−−−
−−−−−−−
=
ττ
ττ
ττ
ττ
ττ
ττ
(A-28)
121
)(
))()(1(2
))((2
)()1(
)(
))()(1(2)1(
22
2
22
22
COC
Or
CPPCPCCOC
Orr
POPOPOCO
COr
CPPCPCr
POPOPOr
POPOPOCPPCPCrr
ATCPpATCABARATCPpp
PABARATCPpATCABARp
PABARpPABARATCABARpp
right
−−
−−−−−
−−−−
−−−−
−−−
−−−−−−
•−= ττ
(A-29)
Solving the negotiated price of Q by using “right” (equation A-29, the right hand side of
the equation (A-17)) divided by “left” (equation A-25, the left hand side of the equation (A-17))
leftrightQ = (A-30)
where “right” and “left” are calculated in equation (A-29) and (A-25). Then, the terms
)1( ττ − can be cancelled out, and Q becomes
]
1[
])11
1
]1
)(
))()(1(2
))((2
)()1(
)(
))()(1(2
22
2
22
2
22
22
)ATC(Pp)PAB(ARp
)ATCAB)(ARp(p)Q
ABQ
AR(
)ATC(Pp(p)PAB)(ARp(p
)ATCAB(AR)p)[(Q
ABQ
AR(
)ATC(P p)ATCAB)(ARp(
)P(Q)AB(Q)(AR p)[Q
ATC(
ATCPpATCABARATCPpp
PABARATCPpATCABARp
PABARpPABARATCABARpp
Q
COC
OrP
OPOPOr
CPPCPCrrPOPO
COC
OrrP
OPOPOrr
CPPCPCrPCPC
COC
OrCPPCPCr
POPOPOr
CP
COC
Or
CPPCPCCOC
Orr
POPOPOCO
COr
CPPCPCr
POPOPOr
POPOPOCPPCPCrr
−+−−+
−−−∂
∂−
∂∂
+
−−+−−−+
−−−∂
∂−
∂∂
+
−−−−−−
−−−∂
∂−−
−−−−−
−−−−
−−−−
−−−
−−−−−−
=
(A-31)
122
APPENDIX B DATA
Table B-1. Data(1)a
Year Month
Negotiated Price ( tP ) ($ per hundredweight)
Negotiated Quantity ( tQ ) (million pounds)
FMMO #6 Class I price ( tC1 ) ($ per hundredweight))
Index of Feed Cost Paid by Farmers ( tFC ) (US)
1998 10 21.99 189.7 19.17 100 11 21.99 191.7 19.28 103 12 22.79 217.3 20.22 104 1999 1 23.39 224.3 21.02 104 2 23.89 208.4 21.52 103 3 22.82 230.5 20.45 101 4 17.32 219.4 14.45 102 5 18.17 207.4 15.80 101 6 18.36 198.3 15.99 100 7 18.36 205.6 15.44 97 8 18.87 182.0 15.60 97 9 20.54 180.6 17.77 98 10 21.99 186.8 19.97 97 11 22.46 205.4 20.44 97 12 18.64 221.2 15.67 98 2000 1 18.32 225.0 15.20 98 2 18.06 221.0 15.01 101 3 18.2 236.0 15.14 102 4 18.37 212.0 15.23 102 5 18.67 215.0 15.78 105 6 18.64 203.0 16.00 104 7 19.15 199.0 16.76 100 8 18.89 202.0 16.25 96 9 18.78 187.0 16.14 99 10 18.83 199.0 16.19 101 11 18.76 209.0 16.12 103 12 18.82 219.0 16.43 108 2001 1 20.68 233.0 18.29 112 2 18.88 209.0 16.24 108 3 19.26 240.0 16.95 107 4 20.13 211.0 17.74 106 5 20.79 213.0 18.51 106 6 21.23 203.0 19.29 107 7 21.73 197.0 19.64 108 8 21.79 199.0 19.70 111
123
Table B-1. Continueda
Year Month
Negotiated Price ( tP ) ($ per hundredweight)
Negotiated Quantity ( tQ ) (million pounds)
FMMO #6 Class I price ( tC1 ) ($ per hundredweight))
Index of Feed Cost Paid by Farmers ( tFC ) (US)
2002 9 21.95 179.0 19.86 110 10 22.32 201.0 20.23 109 11 22.15 203.0 20.06 109 12 18.84 204.0 16.28 109 2002 1 18.76 220.0 16.26 108 2 18.76 202.0 16.25 107 3 18.76 218.0 15.92 109 4 18.76 211.0 15.77 109 5 18.80 207.0 15.56 108 6 18.57 186.0 15.33 110 7 18.16 194.0 14.92 114 8 18.02 200.0 14.78 116 9 18.00 173.0 14.76 118 10 17.69 185.0 14.45 116 11 18.04 191.0 14.90 114 12 17.96 207.0 14.82 114 2003 1 18.00 228.0 14.86 114 2 17.67 202.0 14.53 114 3 17.33 219.0 14.11 114 4 17.16 209.0 13.94 114 5 17.23 207.0 14.01 115 6 17.26 186.0 14.04 114 7 17.29 196.0 14.07 111 8 18.27 186.0 15.27 107 9 20.51 177.0 18.01 112 10 20.80 193.0 18.57 112 11 20.80 190.0 18.67 118 12 20.64 218.0 18.14 117 2004 1 18.90 227.0 16.15 117 2 18.90 204.0 15.89 121 3 18.90 225.0 16.24 124 4 20.54 217.0 17.94 131 5 26.05 197.0 23.95 135 6 27.53 189.0 25.43 129 7 24.35 203.0 22.25 128 8 21.02 190.0 18.92 119 9 20.84 169.0 18.24 116 10 21.18 190.0 19.08 111
124
Table B-1. Continueda
Year Month
Negotiated Price ( tP ) ($ per hundredweight)
Negotiated Quantity ( tQ ) (million pounds)
FMMO #6 Class I price ( tC1 ) ($ per hundredweight))
Index of Feed Cost Paid by Farmers ( tFC ) (US)
2004 11 20.90 200.0 18.59 109 12 20.98 222.0 18.73 108 2005 1 23.29 229.0 21.04 112 2 20.86 209.0 18.18 110 3 22.41 231.0 19.73 115 4 21.11 224.0 18.43 116 5 21.78 208.0 19.10 118 6 20.60 199.0 17.92 121 7 20.87 203.0 18.19 122 8 21.42 218.0 18.74 122 9 21.08 201.0 18.00 119 10 21.57 200.0 18.57 117 11 21.86 218.0 18.86 115 12 21.11 231.0 17.87 118 2006 1 20.66 238.0 17.68 121 2 20.66 216.0 17.68 121 3 19.85 242.0 16.79 123 4 19.08 221.0 15.52 123 5 18.83 218.0 15.27 123 6 18.69 208.0 15.05 122 7 19.28 207.0 15.64 123 8 18.91 215.0 15.27 120 9 18.87 205.0 15.15 120 10 20.36 211.0 16.72 125 11 20.26 221.0 16.70 133 12 20.39 225.0 16.73 138 2007 1 21.51 239.1 17.89 140 2 21.32 218.2 17.69 148 3 22.10 242.1 18.55 150 4 22.93 219.9 19.30 148 5 23.85 214.7 20.22 145 6 25.85 204.7 22.14 147 7 28.77 198.7 25.21 147 8 29.47 213.6 26.06 145 9 29.62 199.2 26.21 147 10 29.80 216.6 25.89 151 11 29.74 217.0 25.75 156 12 28.42 219.7 24.34 160
125
Table B-1. Continueda
Year Month
Negotiated Price ( tP ) ($ per hundredweight)
Negotiated Quantity ( tQ ) (million pounds)
FMMO #6 Class I price ( tC1 ) ($ per hundredweight))
Index of Feed Cost Paid by Farmers ( tFC ) (US)
2008 1 29.12 234.0 25.27 168 2 28.50 220.6 23.98 176 3 25.77 233.0 21.00 183 4 27.76 218.7 22.91 186 5 27.13 227.2 22.62 199 6 28.77 196.6 24.18 203 7 31.37 204.2 26.78 217 8 29.22 211.1 24.47 215 9 28.4 205.2 23.65 209 10 25.32 220.7 21.53 196 11 27.04 210.1 23.33 191 12 24.90 223.6 21.43 185 2009 1 25.05 234.0 21.74 189 2 20.60 218.0 16.72 187 3 18.66 234.0 15.43 185 4 19.51 222.0 16.36 184 5 20.20 212.0 16.97 192 a) Data Source found in the 4.4 Data Description section of Chapter 4
126
Table B-2. Data(2)a
Year Month
Seasonality ( tS ) (Milk production per cow in Florida, lbs/cow)
Transportation Cost ( tTC ) (PPI Freight transportation)
Labor Index ( tL )
Whole Sale level Price ( tWP ) ( finished dairy products PPI)
Over Order Premium ( tOOP )
1998 10 1030 112.5 138.9 148.0 2.8 11 1120 112.4 138.9 148.6 2.7 12 1325 112.6 138.9 148.5 2.6 1999 1 1415 113.2 139.9 149.0 2.4 2 1365 113.4 139.9 145.1 2.4 3 1540 113.9 139.9 142.6 2.4 4 1435 114.4 140.9 132.1 2.9 5 1400 114.4 140.9 132.9 2.4 6 1290 114.8 140.9 135.5 2.4 7 1190 114.8 142.1 136.4 2.9 8 1045 115.4 142.1 139.9 3.3 9 1020 115.7 142.1 143.9 2.8 10 1025 115.4 143.6 144.1 2.0 11 1115 115.3 143.6 142.5 2.0 12 1320 115.8 143.6 132.7 3.0 2000 1 1455 116.5 146.0 130.9 3.1 2 1410 116.8 146.0 130.1 3.1 3 1565 118.1 146.0 130.5 3.1 4 1470 118.2 147.5 131.7 3.1 5 1465 118.8 147.5 133.1 2.9 6 1325 119.4 147.5 134.4 2.6 7 1215 118.8 148.7 136.3 2.4 8 1120 120.1 148.7 134.9 2.6 9 1030 120.6 148.7 135.6 2.6 10 1120 121.4 149.3 134.6 2.6 11 1170 121.6 149.3 135.6 2.6 12 1360 121.5 149.3 136.8 2.4 2001 1 1440 121.9 151.3 136.8 2.4 2 1380 122.5 151.3 136.1 2.6 3 1580 122.6 151.3 138.6 2.3 4 1475 122.7 152.6 141.3 2.4 5 1515 123.0 152.6 146.4 2.3 6 1330 123.2 152.6 150.1 1.9 7 1235 123.3 153.3 150.9 2.1 8 1115 123.4 153.3 152.0 2.1 9 1030 123.6 153.3 153.5 2.1 10 1120 123.8 154.6 150.6 2.1 11 1170 124.0 154.6 145.4 2.1 12 1320 123.2 154.6 140.3 2.6
127
Table B-2. Continueda
Year Month
Seasonality ( tS ) (Milk production per cow in Florida, lbs/cow)
Transportation Cost ( tTC ) (PPI Freight transportation)
Labor Index ( tL )
Whole Sale level Price ( tWP ) ( finished dairy products PPI)
Over Order Premium ( tOOP )
2002 1 1410 123.4 156.6 140.9 2.5 2 1335 123.3 156.6 139.8 2.5 3 1515 123.2 156.6 138.1 2.8 4 1455 123.8 158.1 137.7 3.0 5 1450 123.8 158.1 136.2 3.2 6 1345 124.3 158.1 135.2 3.2 7 1250 124.2 159.1 134.0 3.2 8 1190 124.6 159.1 134.5 3.2 9 1045 125.0 159.1 133.9 3.2 10 1055 125.4 160.5 136.6 3.2 11 1100 125.9 160.5 134.3 3.1 12 1265 125.9 160.5 135.3 3.1 2003 1 1360 126.5 164.0 134.8 3.1 2 1290 126.8 164.0 133.6 3.1 3 1435 127.3 164.0 132.5 3.2 4 1415 127.4 165.4 133.7 3.2 5 1385 127.3 165.4 134.1 3.2 6 1290 127.5 165.4 134.1 3.2 7 1215 127.8 166.5 139.3 3.2 8 1120 128.3 166.5 143.6 3.0 9 1020 128.7 166.5 147.5 2.5 10 1105 128.6 167.1 147.6 2.2 11 1180 128.8 167.1 145.7 2.1 12 1355 128.8 167.1 143.5 2.5 2004 1 1435 129.1 171.7 141.5 2.8 2 1415 130.3 171.7 142.2 3.0 3 1550 130.5 171.7 147.4 2.7 4 1515 130.7 173.2 162.7 2.6 5 1525 131.6 173.2 173.4 2.1 6 1395 132.3 173.2 169.8 2.1 7 1340 132.9 174.9 159.7 2.1 8 1225 133.4 174.9 155.1 2.1 9 1025 133.9 174.9 154.6 2.6 10 1165 134.9 175.4 154.3 2.1 11 1250 135.8 175.4 154.5 2.3 12 1450 135.6 175.4 157.4 2.3 2005 1 1520 136.7 178.2 157.7 2.3 2 1475 136.8 178.2 154.8 2.7 3 1640 137.7 178.2 155.1 2.7
128
Table B-2. Continueda
Year Month
Seasonality ( tS ) (Milk production per cow in Florida, lbs/cow)
Transportation Cost ( tTC ) (PPI Freight transportation)
Labor Index ( tL )
Whole Sale level Price ( tWP ) ( finished dairy products PPI)
Over Order Premium ( tOOP )
2005 4 1570 138.7 179.6 155.8 2.7 5 1585 139.2 179.6 153.8 2.7 6 1440 139.9 179.6 152.7 2.7 7 1315 140.6 180.7 155.2 2.7 8 1235 140.6 180.7 153.7 2.7 9 1105 142.9 180.7 155.3 3.1 10 1135 144.5 181.3 155.7 3.0 11 1220 144.3 181.3 153.5 3.0 12 1360 143.1 181.3 153.7 3.2 2006 1 1485 143.4 181.5 152.6 3.0 2 1425 143.1 181.5 149.2 3.0 3 1640 143.7 181.5 145.9 3.1 4 1540 144.8 183.1 144.4 3.6 5 1520 146.8 183.1 143.3 3.6 6 1415 147.1 183.1 144.1 3.6 7 1350 146.9 183.8 143.8 3.6 8 1180 147.9 183.8 145.3 3.6 9 1110 148.1 183.8 148.2 3.7 10 1185 146.4 184.6 149.1 3.6 11 1245 146.3 184.6 151.4 3.6 12 1390 145.9 184.6 152.4 3.7 2007 1 1470 146.8 184.9 154.7 3.6 2 1415 146.1 184.9 156.8 3.6 3 1620 146.8 184.9 159.8 3.6 4 1590 148.1 186.6 162.3 3.6 5 1585 148.8 186.6 169.6 3.6 6 1470 148.8 186.6 180.3 3.7 7 1400 148.9 187.1 186.8 3.6 8 1230 149.0 187.1 186.5 3.4 9 1170 149.1 187.1 189.5 3.4 10 1160 149.5 188.2 187.8 3.9 11 1240 151.1 188.2 188.4 4.0 12 1410 151.6 188.2 188.7 4.1 2008 1 1550 151.9 189.8 187.5 3.9 2 1530 153.0 189.8 184.6 4.5 3 1655 154.3 189.8 181.2 4.8 4 1568 157.3 190.5 181.9 4.9 5 1625 160.2 190.5 180.8 4.5 6 1440 163.8 190.5 187.1 4.6
129
Table B-2. Continueda
Year Month
Seasonality ( tS ) (Milk production per cow in Florida, lbs/cow)
Transportation Cost ( tTC ) (PPI Freight transportation)
Labor Index ( tL )
Whole Sale level Price ( tWP ) ( finished dairy products PPI)
Over Order Premium ( tOOP )
2008 7 1375 164.7 191.5 189.3 4.6 8 1240 163.7 191.5 187.0 4.8 9 1160 160.7 191.5 183.2 4.8 10 1225 159.7 192.0 181.6 3.8 11 1320 155.5 192.0 178.3 3.7 12 1520 151.7 192.0 174.3 3.5 2009 1 1660 151.7 193.1 162.1 3.3 2 1540 151.9 193.1 155.1 3.9 3 1760 149.8 193.1 153.1 3.2 4 1710 150.8 193.4 153.8 3.2 5 1700 150.6 193.4 153.1 3.2 a) Data Source found in the 4.4 Data Description section of Chapter 4
130
APPENDIX C GAUSS PROGRAM
new; clear all; rndseed 3434567; /* read data and variable names in from the disc*/; data = xlsreadm("SepData.xls", "d2:p129",1, 0); datanames=xlsreadSA("SepData.xls", "d1:p1", 1, ""); names=datanames'; print data[.,1]; print "Variable Names=" names; /* caculate the means */; means=meanc(data); print "Means=" means; /* calculate the standard deviation */ stds = stdc(data); print "Standard Deviations=" stds; /* caculate variance-covariance matrix */; VarCovs = vcx(data[.,2:12]); print " Variance Covariance Matrix=" VarCovs; /* calculate the correlation matrix */; Cor=corrvc(VarCovs); print "Correlation Matrix=" Cor; detcor=det(cor); vif=inv(cor); print "Vif (variance inflation factors)=" diag(vif); /* define model terms */; n=128; P=data[.,1]; Q=data[.,2]; XD=data[.,3]~data[.,10]~data[.,11]; XS=data[.,3]~data[.,4]~data[.,8:9]; XT=data[.,12]; NP=12; /* 4 demand + 7 supply +1 bargaining */; print " Demand price EQ exogenous variables=" Q~XD; print " Supply price EQ exogenous variables=" Q~XS; print " Tao EQ exogenous variables=" XT;
131
//************* EM algorithm loop Starts here***************// diff=1; count=1; /******************Initial Values**********/; x0=-0.004, 0.82, -0.002, 0.057, 0.9, 3.548, -16.03, -5.761, 1.68, 4.63, 23, 0.01, 0.51; print "initial values=" x0; do until diff<=0.1; /***********simulation by using expectation of PD, PS****** */; /* calculate the mean of PD and PS2 */; lnPD_means = ln(Q)*x0[1,1]+ln(XD)*x0[2:4,1]+ones(128,1)*x0[5,1]; lnPS_means = ln(Q)*x0[6,1]+((ln(XS[.,1:2]))~(ln(XS[.,3:4])))*x0[7:10,1]+ones(128,1)*x0[11,1]; mu = meanc(lnPD_means~lnPS_means); /* print "Demand and Supply equation means=" mu; */ /* define the variance covariance matrix of PD and PS */; vc = (120~0)|(0~130); /* print "Variance Covariance Matrix of Demand and Supply Equations=" vc; */ /* simulating all needed PD and PS */ trial=128^2; PDPS=zeros(256, trial); j=1; for j(1, trial, 1); library rndmn; mvnvars=rndmn(mu,vc, 128); @ calling the biviraite normal generator procedure @ /* print "Psuedo biviraite normal PD and PS=" mvnvars; */ PDPS[1:128,j] = mvnvars[.,1]; PDPS[129:256,j] = mvnvars[.,2]; endfor; PDall=PDPS[1:128,.]; PSall=PDPS[129:256,.]; /* taking the average of all trials */ PDsum=sumr(PDall); 1 Originally, some difficulties occurred for finding initial values. Then, by making the assumption for PD and PS, a simple OLS provide some possible initial values. With the knowledge that PD is above P and PS is below P, some magnitude changes in the OLS initial values were made.
2 Besides double log functional form for PD and PS, other functional forms (linear log, linear) have been tried. The results are non sensible.
132
PSsum=sumr(PSall); lnPD=PDsum./trial; lnPS=PSsum./trial; PD=exp(lnPD); PS=exp(lnPS); // print "PD and PS=" PD~PS; /*****************maximization*************/; /***** define log-likelihood function *****/; proc ll(x0); local BD, BS, BDQ, BSQ, BT, BDC, BSC, BTC, e, tao, gama, test; /* define parameters as the element of b */ BDQ = x0[1,1]; BD = x0[2:4,1]; BDC= x0[5,1]; BSQ = x0[6,1]; BS = x0[7:10,1]; BSC= x0[11,1]; BT = x0[12,1]; BTC= x0[13,1]; /* define variance covariance matrix */ e=zeros(128,4); e[.,1] = lnPD- ln(Q)*BDQ-ln(XD)*BD-ones(128,1)*BDC; e[.,2] = lnPS- ln(Q)*BSQ-((ln(XS[.,1:2]))~(ln(XS[.,3:4])))*BS-ones(128,1)*BSC; tao=1/(1+exp(XT*BT+ones(128,1)*BTC)); // print "Tao=" tao; e[.,3]= P-tao.*PD-(1.-tao).*PS; gama = Q./(BDQ-BSQ); e[.,4]= Q-gama.*(PS-PD); /* testing sigularity */ test=det(e'e); if test==0; print "sigular"; endif; retp(ln(det(e'e))); endp; /* calling the library and setting globals */; library optmum; optset; _opgtol=0.0001; x,fmin,g,retcode = optprt(optmum(&ll,x0)); bn=x; diff=maxc(abs(bn-x0)); count=count+1; print "count=" count;
133
x0=bn; endo; print "count=" count; print "final estimates=" bn; /* standard errors of parameters */ print "standard erroes of parameters =" sqrt(diag(inv(_opfhess))); /* variance covariance matrix of equation errors */ er=zeros(128,4); er[.,1] = lnPD- ln(Q)*x[1]-ln(XD)*x[2:4]-ones(128,1)*x[5]; er[.,2] = lnPS- ln(Q)*x[6]-((ln(XS[.,1:2]))~(ln(XS[.,3:4])))*x[7:10]-ones(128,1)*x[11]; taoc=1/(1+exp(XT*x[12]+ones(128,1)*x[13])); er[.,3]= P-taoc.*PD-(1.-taoc).*PS; gamac = Q./(x[1]-x[6]); er[.,4]= Q-gamac.*(PS-PD); varc=((er'er)/128); print "variance covariance matrix of all equation errors=" varc; print "bargaining power parameter tao=" taoc; print "Gama =" gamac; print "mean Gama =" meanc(gamac);
134
APPENDIX D TAO TESTING
The one tail t test is used to test whether the average bargaining power coefficient in year
2008 is the highest for SDC and the lowest for processors (Table B-1). The hypothesis is:
Ho: 02008 =− yearµµ
Ha: 02008 >− yearµµ
Table B-1. Testing Results for Maximum Bargaining Power for SDC in Year 20081 Year P-value Conclusion 2009 0.0003 reject Ho at 5% significant level 2007 0.0003 reject Ho at 5% significant level 2006 0.0000 reject Ho at 5% significant level 2005 0.0000 reject Ho at 5% significant level 2004 0.0000 reject Ho at 5% significant level 2003 0.0000 reject Ho at 5% significant level 2002 0.0000 reject Ho at 5% significant level 2001 0.0000 reject Ho at 5% significant level 2000 0.0000 reject Ho at 5% significant level 1999 0.0000 reject Ho at 5% significant level 1998 0.0000 reject Ho at 5% significant level
1 Before the one tail t test, an equal variance test has been done for all t tests. The results show that the variances are statistically equal at 5% significant level except the test for year 2007.
135
The one tail t test is used to test whether the average bargaining power coefficient in year
2001 is the lowest for SDC and the highest for processors (Table B-2). The hypothesis is:
Ho: 02001 =− yearµµ
Ha: 02001 <− yearµµ
Table B-2. Testing Results for Minimum Bargaining Power for SDC in Year 20011 Year P-value Conclusion 2009 0.0000 reject Ho at 5% significant level 2008 0.0000 reject Ho at 5% significant level 2007 0.0000 reject Ho at 5% significant level 2006 0.0000 reject Ho at 5% significant level 2005 0.0000 reject Ho at 5% significant level 2004 0.1082 fail to reject Ho at 5% significant level 2003 0.0001 reject Ho at 5% significant level 2002 0.0000 reject Ho at 5% significant level 2000 0.0000 reject Ho at 5% significant level 1999 0.013 reject Ho at 5% significant level 1998 0.0025 reject Ho at 5% significant level
1 Before the one tail t test, an equal variance test has been done for all t tests. The results show that the variances are statistically equal at 5% significant level except the test for year 2003.
136
The one tail t test is used to test whether the average bargaining power coefficient in April
is the highest for SDC and the lowest for processors (Table B-3). The hypothesis is:
Ho: 0=− MonthApril µµ
Ha: 0≠− MonthApril µµ
Table B-3. Testing Results for Maximum Bargaining Power for SDC in April1 Month P-value Conclusion 1 0.1695 fail to reject Ho at 5% significant level 2 0.4133 fail to reject Ho at 5% significant level 3 0.3236 fail to reject Ho at 5% significant level 5 0.3251 fail to reject Ho at 5% significant level 6 0.2686 fail to reject Ho at 5% significant level 7 0.3009 fail to reject Ho at 5% significant level 8 0.3446 fail to reject Ho at 5% significant level 9 0.3413 fail to reject Ho at 5% significant level 10 0.1282 fail to reject Ho at 5% significant level 11 0.1141 fail to reject Ho at 5% significant level 12 0.2255 fail to reject Ho at 5% significant level
1 Before the two tail t test, an equal variance test has been done for all t tests. The results show that the variances are statistically equal at 5% significant level.
137
The one tail t test is used to test whether the average bargaining power coefficient in
November is the lowest for SDC and the highest for processors (Table B-4). The hypothesis is:
Ho: 0=− MonthNovember µµ
Ha: 0≠− MonthNovember µµ
Table B-4. Testing Results for Minimum Bargaining Power for SDC in November1 Month P-value Conclusion 1 0.3445 fail to reject Ho at 5% significant level 2 0.1567 fail to reject Ho 3 0.2169 fail to reject Ho 4 0.1141 fail to reject Ho 5 0.2321 fail to reject Ho 6 0.3227 fail to reject Ho 7 0.2789 fail to reject Ho 8 0.2408 fail to reject Ho 9 0.2336 fail to reject Ho 10 0.4809 fail to reject Ho 12 0.2927 fail to reject Ho
1 Before the two tail t test, an equal variance test has been done for all t tests. The results show that the variances are statistically equal at 5% significant level.
138
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BIOGRAPHICAL SKETCH
Jiaoju Ge was born in Hubei province, P.R. China. After she obtained her Bachelor of
Science degree in investment economics from Sichuan University in 1999, she worked for LG
Electronics for 3 years as a marketing product manager.
In August 2006, Jiaoju Ge was awarded the Master of Science degree from the Food and
Resource Economics in University of Florida under the direction of Dr. Allen Wysocki. Then she
was admitted to the PhD program in the Food and Resource Economics Department specialized
in marketing, econometrics and international trade and expected to graduate on December 2009.
She was married to Qiyong Xu on May 21st, 2002 and they have two kids, Michael and Sarah.