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UNIVERSITY OF CALIFORNIA, IRVINE
Essays on Corruption and Governance
DISSERTATION
Submitted in partial satisfaction of the requirements for the degree of
DOCTOR OF PHILOSOPHY
in Economics
by
Amjad Toukan
Dissertation Committee:
Stergios Skaperdas, Chair Michelle Garfmkel
Priya Ranjan Donald Saari
2007
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UMI Number: 3271334
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© 2007 Amjad Toukan
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The Dissertation of Amjad Toukan is approved and is acceptable in quality and form for publication on microfilm:
Committee Chair
University of California, Irvine 2007
ii
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DEDICATION
To
My Mother, Father, Aiman, Yassar, Nancy my beloved wife, and our newMohammad:
The true inspiration in my life
iii
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TABLE OF CONTENTS
Page
LIST OF FIGURES vi
LIST OF TABLES vii
ACKNOWLEDGEMENTS viii
CURRICULUM VITAE ix
ABSTRACT OF THE DISSERTATION xi
Introduction 1
Chapter 1: Privately Held or Publicly Owned? Large Shareholders and
Corporate Control 5
1.1 Introduction 5
1.2 The model 10
1.3 Equilibrium choices where the partners compete in
an asymmetric contest for control 15
1.4 Equilibrium choices where the partners compete with
the powers of persuasion 21
1.5 Concluding remarks 27
Chapter 2: Risk of Expropriation and the Rybczynski Theorem 33
2.1 Introduction 33
2.2 The model 37
2.3 Data description 45
2.4 Empirical Results 48
2.5 Concluding remarks 56
iv
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Chapter 3: Contests with a Generalized Difference Form 58
3.1 Introduction 58
3.2 Persuasion Function as an Alternative to the
Tullock Functional Form 63
3.3 The symmetric case 65
3.4 Non-cooperative equilibrium with asymmetric cost
functions and asymmetric contestable rents 69
3.5 Non-cooperative equilibrium with asymmetric
evidence production 72
3.6 Non-cooperative pure strategy equilibria with N agents 75
3.7 Persuasion with a fixed number of agents N 77
3.8 Non-cooperative equilibrium and the extent of rent
dissipation 78
3.9 Non-cooperative equilibrium - asymmetric case 80
3.10 Conclusion 86
Appendix 87
BIBLIOGRAPHY 93
v
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LIST OF FIGURES
Page
Figure 1.1 The game between shareholders and managers 11
Figure 1.2 Optimal value to owners/ managers (the partners
compete in an asymmetric contest for control) 29
Figure 1.3 Optimal ownership structure to owners/ managers
(the partners compete in an asymmetric contest for control) 30
Figure 1.4 Optimal value to owners/ managers (the partners
compete with the powers of persuasion) 31
Figure 1.5 Optimal ownership structure to owners/ managers (the partners
compete with the powers of persuasion) 32
Figure 2.1 Equilibrium analysis 40
Figure 3.1 Player 1 ’s reaction curve 67
Figure 3.2 Non-cooperative pure strategy equilibrium 69
c cFigure 3.3 Non-cooperative pure strategy equilibrium - —L < 72X\ X 2
Figure 3.4 Non-cooperative pure strategy equilibrium -(1-6) <6 75
Figure 3.5 Non-cooperative pure strategy equilibrium - asymmetric
example 1 84
Figure 3.6 Non-cooperative persuasion equilibrium - asymmetric
example 2 84
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LIST OF TABLES
Table 2.1 Comparison table for Japan and Venezuela
Table 2.2 Estimating Rybczynski effects for a sample of
22 OECD countries
Table 2.3 Estimating Rybczynski effects for a sample of
16 developing countries
Page
35
51
53
vii
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ACKNOWLEDGEMENTS
I am indebted to Professor Stergios Skaperdas for his constant guidance,
encouragement and support. Professor Skaperdas was not only my advisor, he was
also my mentor who gave his endless intellectual and moral support that played an
invaluable role in the timely completion of this project. I am also very grateful to
Professor Donald Saari, Professor Michelle Garfinkel and Professor Priya Ranjan for
their valuable feedback and thought-provoking suggestions at different stages in the
development of this work.
I am very grateful to the Department of Economics at UCI, the Institute for
Mathematical Behavioral Sciences (IMBS) at UCI for their generous financial
support.
viii
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CURRICULUM VITAE
Amjad Toukan
EDUCATION
2007 Ph.D. in Economics, University of California, Irvine
2001 MBA, University of California, Irvine
1988 M.Sc. in Electrical Engineering, California State University, Fullerton
1986 B.Sc. in Electrical Engineering, University of California, Irvine
FIELDS OF SPECIALIZATION
Applied Microeconomic Theory, Public Choice/Political Economy,
International Trade
HONORS, SCHOLARSHIPS AND FELLOWSHIPS
Regents Fellowship, Summer 2005, Department of Economics, UC-Irvine
Regents Fellowship, Summer 2004, Department of Economics, UC-Irvine
Summer Fellowship, Summer 2002 and 2003, Institute for Mathematical and
Behavioral Sciences, UC-Irvine
Invited Panelist to the Teaching Assistant Professional Development Program,
2003-2004
Dean’s Letter for Outstanding Teaching Evaluations, Fall 2002, Spring 2003,
Winter 2005
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Sarah Scaife Foundation Scholarship to attend Public Choice Outreach
Conference at George Mason University, Fairfax, Virginia (2004)
TEACHING EXPERIENCE
Summer 2005: Teaching Associate, Department of Economics, UC-Irvine
Summer 2004: Teaching Associate, Department of Economics, UC-Irvine
Spring 2002: Teaching Associate, School of Social Science, UC-Irvine
Fall 2000 to present: Teaching Assistant, School of Social Science, UC-Irvine
PROFESSIONAL ACTIVITIES
Conference Participation: Public Choice Society Meeting, (2004); Public
Choice
Outreach Conference (2004)
Referee: Economic Theory
Conference Organization: Assisted Prof. Donald Saari in organizing the
graduate student conference on social choice and behavioral sciences, 2004
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ABSTRACT OF THE DISSERTATION
Essays on Corruption and Governance
by
Amjad Toukan
Doctor of Philosophy in Economics
University of California, Irvine, 2007
Professor Stergios Skaperdas, Chair
Nearly 2500 years ago, the old Indian treatise entitled “Arthashastra” had recognized
the impact of corruption on the conduct of the economy. Corruption is not just an
economic problem, however; it is also associated with bad governance (governance
being defined as the way in which both public and private institutions perform their
functions in a country). In my dissertation, I focus on the principal-agent model of
corruption. The agency relationship links at least two actors and is the basic unit of
analysis. The first chapter evaluates a corporation’s decision to go public, draws the
distinction between large and dispersed shareholders and examines how the
differences in their incentives to monitor the managers affect the shape of ownership
structure in public firms. In the second chapter, I find evidence contradicting the
predictions of the Rybczynski theorem using a sample of 28 manufacturing industries
in 16 developing countries over eight years. This contradiction is examined using a
modification of the Heckscher-Ohlin model to allow for international variability in
corruption and risk of expropriation. The final chapter explores the properties and
implications of a general class of “difference-form” contests that has been derived for
settings in which rent-seeking involves persuasion. Such class of contests could be
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employed to analyze the impact of corruption in governance such as the decision
making in the courtroom, the decision making within bureaucracies, the interactions
among interest groups among others.
xii
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Introduction
Nearly 2500 years ago, the old Indian treatise entitled “Arthashastra” had recognized
the impact of corruption on the conduct of the economy. Chanakya in Arthashastra
urged the king’s administrators to control the state income and expenses in order to
avoid embezzlement of state funds. Corruption is not just an economic problem,
however; it is also associated with bad governance (governance being defined as the
way in which both public and private institutions perform their functions in a
country). Ineffective formal governance institutions lead to the creation of informal
institutions to substitute for the functions that the formal ones are unable to perform.
Corruption at high levels of government has even a more profound impact on the
degree of informality: it forms barriers to entry by creating a less competitive
business environment and adds to business risks by increasing the unpredictability of
government policies [Johnson, Kaufmann, and Shleifer (1997)].
Following Becker and Stigler [1974], Banfield [1975], Rose-Ackerman [1975,
1978], and Klitgaard [1988, 1991] my research will focus on the principal-agent
model of corruption. The agency relationship links at least two actors and is the basic
unit of analysis. Whenever there is a potential conflict of interest between the
principals and the principals' agents, the principals are induced to limit the extent to
which the agents may seek to further their own interests rather than those of the
principals. Agency costs are incurred when agents do not maximize principals’
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objectives and when principals spend time and money to monitor agents and to
influence their actions.
The first chapter “Privately Held or Publicly Owned? Large Shareholders
and Corporate Control” examines the decision to go public in the presence of large
and dispersed shareholders. In contrast to much of the existing literature, I make the
distinction between large and dispersed shareholders, and examine the differences in
their incentives to monitor the managers. My analysis takes a game theoretic
approach, modeling the conflict between managers and shareholders as a contest.
According to Anderton (2001), one of the fundamental building blocks of a unifying
micro-theory of conflict economics is the contest success function (CSF), which
specifies how the appropriative efforts of agents lead to an appropriative outcome.
Results predicted by the use of two families of CSFs (In the first family of the CSF, a
contestant’s winning probability depends on the ratio of fighting efforts. In the second
family of the CSF, called “ difference-form” success functions, a contestant’s
probability of winning depends upon the difference of fighting efforts) are consistent
with the existing literature for example, similar to La Porta, Lopez-de-Silanes,
Shleifer and Vishny (1998), Shleifer and Wolfenzon (2002) and Burkart, Panunzi and
Shleifer (2003). In particular, I obtain a negative relationship between the
concentration of ownership shares in public companies and the legal protection of
outside shareholders. As the legal protection of outside shareholders improves,
entrepreneurs choose to decrease their share of ownership in the public firm while
increasing that of dispersed shareholders. The share of ownership sold to large
shareholders is non-monotonic in the legal protection of outside shareholders.
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In the second chapter “Risk of Expropriation and the Rybczynski
Theorem,” I find evidence contradicting the predictions of the Rybczynski theorem
using a sample of 28 manufacturing industries in 16 developing countries over eight
years. This contradiction is examined using a modification of the Heckscher-Ohlin
model to allow for international variability in corruption and risk of expropriation.
Our model predicts that countries (with similar capital stock per worker) with higher
incidence of corruption and higher risk of expropriation have a lower ratio of capital-
intensive output to labor-intensive output. This implies that the spread of corruption
and the weak enforcement of property rights can have adverse effects on a country’s
development predicted by the ladder-of-development or product-cycle hypothesis: a
country's output mix depends on its stage of development, with countries moving
from agriculture to labor-intensive manufactures to high-tech manufacturing and
services as their aggregate labor productivity increases.
The third chapter “Contests with a Generalized Difference Form” (co
authored with Stergios Skaperdas), explores the properties and implications of a
general class of “difference-form” contests that has been derived for settings in which
rent-seeking involves persuasion. Our study characterizes equilibria and analyzes the
relationship between the extent of rent dissipation and the underlying contest
characteristics. Our results differ from those of the traditional ratio model. For
instance, in the pure-strategy equilibrium, it is possible that one or both contestants
expend zero effort. Applications of such outcomes include lobbying, election
campaigns, industrial disputes and lawsuits where one-sided submission and two-
sided peace between the parties can occur as a Cournot equilibrium. Also in contrast
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to the traditional ratio model, we find that the extent of rent dissipation is non
monotonic in the number of contestants N while in the traditional ratio model it
strictly increasing in N.
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Chapter 1
Privately Held or Publicly Owned? Large
Shareholders and Corporate Control
1.1 Introduction
Economists since Adam Smith have warned that a separation between ownership and
management opens the possibility of insider abuse (Enron and WorldCom scandals
are recent examples of insider abuse). The separation between ownership and
management is the principal-agent problem that occurs between shareholders and
managers and the agency costs incurred as a result of it. In our analysis we will
examine the decision to go public in the presence of both large and dispersed
shareholders and we will focus on the agency costs that are incurred due to (1) the
monitoring costs incurred by large shareholder’s in trying to keep managers’
objectives aligned with their own to maximize the value of the firm and (2) managers’
furthering their own interest rather than maximizing the value of the firm.
According to La Porta, et. al. (2000), monitoring by shareholders includes
more than just measuring or observing the behavior of the managers. It includes
efforts by shareholders to ‘control’ the behavior of the managers through budget
restrictions, compensation policies, operating rules, and other methods. The way
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managers further their own interest (specifically expropriate shareholders) can take a
variety of forms as well. In some instances, the managers simply steal the profits. In
other instances, managers sell the output, the assets, or the additional securities in the
firm they control to another firm they own at below market-prices. Such transfer
pricing, asset stripping, and investor dilution, though often legal, have largely the
same effect as theft. In still other instances, expropriation takes the form of diversion
of corporate opportunities from the firm, installing possibly unqualified family
members in managerial positions, or overpaying executives.
Our paper joins the literature on corporate governance in the area of investor
protection. Castillo and Skaperdas (2005) examine how the legal protection of
outside shareholders and the appropriative costs that they induce influence the
incentives for private firms to go public. They model the conflict between the
owners/managers and outside shareholders as a contest to secure part of the value of
the public firm. Their findings indicate that owners are more likely to go public when
outside shareholders are better protected. In the case of going public, Castillo and
Skaperdas (2005) obtain a non-monotone relationship between the legal protection of
outside shareholders and the size of the ownership share retained by the
owners/managers. Building on their work, our paper considers not only the decision
to go public, but also the initial firm’s optimizing ownership structure in terms of the
composition of concentrated (or large) and dispersed (or small) shareholders.
We present a model of entrepreneurs who consider the possibility of taking
their company public. Absent an outright buyer, the entrepreneurs choose among two
options: they can sell a share of the company in the stock market and create a publicly
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held firm run by the original founders or they can keep the company private. In going
public, the entrepreneurs choose the future ownership structure in a manner that
maximizes their expected payoff. In contrast to much of the existing literature, we
make the distinction between large and small shareholders, and examine the
differences in their incentives to monitor the managers. Dispersed shareholders,
having a high opportunity cost of monitoring the managers, free ride on the efforts of
large shareholders.1 Supposing that the initial owners factor in the differential
monitoring effects of large and dispersed shareholders represents a point of departure
of the analysis. This point of departure provides some insight into the large cross-
sectional differences across countries in the ownership concentration in publicly
traded firms and the possible origins of these differences. According to La Porta et.
al. (1998), the patterns of ownership vary across countries with the highest
concentration of ownership found in the French-civil-law countries and the lowest
found in a sample of East Asian countries where company law has been significantly
influenced by the United States.
1 Shleifer and Vishny (1986) show that in a corporation with many small owners, it may not pay any
one of them to monitor the performance o f the management. They build on Grossman and Hart (1980)
argument that opposite to what is often suggested, the free rider problem cannot be avoided by the use
o f the takeover bid mechanism. Outsiders without a share in a diffusely held firm would never take
over a diffusely held firm in order to improve it. The reason is that outsiders’ improvement plan would
be understood by atomistic incumbent shareholders and they will demand the value o f the
improvement in return for their shares or else they stay on. If the outsider can gain only on the shares
they already own (which are few if any) but have to pay all the monitoring and takeover costs, the deal
may not be worth their while. For the same reason, small shareholders do not have a big enough stake
in the firm to absorb the costs o f watching the management.
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So who are the large shareholders and what characterizes their ownership
share in public firms? According to Shleifer and Vishny (1986), large shareholdings
are extremely widespread and very substantial where present. In a sample of 456 of
the Fortune 500 companies, the authors find that 354 have at least one shareholder
owning 5 percent or more of the firm. In only 15 cases does the largest shareholder
own less than 3 percent of the firm. The average holding of the largest shareholder
among the 456 firms is 15.4 percent. In their sample, large shareholders are families
represented on boards of directors (149 cases), pension and profit-sharing plans (90
cases), financial firms such as banks, insurance companies, or investment funds (117
cases) and the final category consists of firms and family holding companies with
large stakes that do not have board seats (100 cases).
Jensen and Meckling (1976), show that, in contrast to small shareholders,
large shareholders receive a discount on the price of their shares equal in value to the
cost of the efforts they exert in monitoring the managers. Specifically, they argue that
prospective large shareholders recognize that the owners/managers’ interests will
diverge somewhat from theirs; hence the highest price that they are willing to pay for
shares will reflect the monitoring costs and the effect of the divergence between the
manager’s interest and theirs.
Similar to Burkart and Panunzi (2004), we assume that large shareholders and
managers are distinct parties. Being a Board Member or even its Chairman is quite
different from being the CEO of the firm, and their interests are likely to differ. We
depart from others in assuming that there is no collusion between large shareholders
and managers in expropriating dispersed shareholders. Rather we assume that the
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interests of the large and the dispersed shareholders are perfectly congruent. That is,
shareholders, whether large or small want the managers to maximize the firms’ value.
Our analysis takes a game theoretic approach, modeling the conflict between
managers and shareholders as a contest. According to Anderton (2001), one of the
fundamental building blocks of a unifying micro-theory of conflict economics is the
contest success function (CSF), which specifies how the appropriative efforts of
agents lead to an appropriative outcome. To date, two families of CSFs have been
developed. In one family, the conflict outcome depends on the ratio of fighting
efforts; in the other family it depends upon the difference of fighting efforts
(Hirshleifer 1995). Our main analysis incorporates the first family of the CSF where
the ratio of the efforts expended by the owners/managers and large shareholders
determines how much is appropriated by each (Clark and Riis, 1997). As a
robustness check we compare the results obtained in our main analysis with the
results obtained by the use of the other family of the CSF in which the difference of
the efforts expended by the owners/managers and large shareholders determines how
much is appropriated by each (Skaperdas and Vaidya, 2005).
Results predicted by the use of both types of CSFs are consistent with the
existing literature for example, La Porta, Lopez-de-Silanes, Shleifer and Vishny
(1997), Shleifer and Wolfenzon (2002) and Pagano and Roell (1998) show that the
better the legal protection of outside shareholders the more valuable the public firm
and the more likely that the owners of a private firm will take their company public.
Consistent with the results of La Porta, Lopez-de-Silanes, Shleifer and Vishny
(1998), Shleifer and Wolfenzon (2002) and Burkart, Panunzi and Shleifer (2003), we
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also obtain a negative relationship between the concentration of ownership shares in
public companies and the legal protection of outside shareholders. As the legal
protection of outside shareholders improves, entrepreneurs choose to decrease their
share of ownership in the public firm while increasing that of dispersed shareholders.
The share of ownership sold to large shareholders is non-monotonic in the legal
protection of outside shareholders.
The paper is organized as follows. Section 2 outlines the model. Section 3
solves the model utilizing Tullock’s ratio-form contest success function, examines the
owner’s decision to go public and analyzes the effect of a change in the efficiency of
the legal system on the shape of the firm’s ownership structure. Section 4 solves the
model utilizing Skaperdas and Vaidya’s difference-form contest success function,
examines the owner’s decision to go public and analyzes the effect of a change in the
efficiency of the legal system on the shape of the firm’s ownership structure. Section
5 concludes.
1.2 The Model
We consider a three-stage model, where in the first stage the firm is privately owned
- that is, by its original founders, the entrepreneurs. In stage 1, the owners decide
whether to take their company public. We assume that, if they go public, they will
stay on as managers of the public firm due to their special expertise in running the
firm. A decision to go public incorporates what fraction /3 of the shares to sell to
large shareholders, what fraction a to keep, what fraction 1 - a - fi to sell to
dispersed shareholders and what fraction y of the equity sales proceeds to reinvest
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back into the public firm. Dispersed shareholders exert no effort in protecting their
investment in the public firm. Instead they free ride on the efforts of large
shareholders in monitoring the managers. The sequence of actions in this game is
depicted in figure 1.1 below.
Stage 1;Private Finn
/ \Private Finn Public Finn
iStage 2:
i r i 1 - a - f i & a
Dispersed Large Owners/Shareholders Shareholders Managers
Large shareholders andowners/managers competefor corporate control
Figure 1.1: The game between shareholders and managers
In stage three of the game, owners/managers and large shareholders compete
for corporate control. We assume that owners/managers are forward-looking,
choosing the optimal ownership structure in stage two of the game in a manner that
maximizes their expected payoff in stage three.
According to Zingales (1995), entrepreneurs must weigh the benefits and the
costs of going public before deciding to proceed with an Initial Public Offering (IPO)
On the costs side, there are the registration and underwriting costs, the underpricing
costs (Ritter (1987)), the annual disclosure costs, and the agency problems generated
by a separation between ownership and management (Jensen and Meckling (1976)).
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On the benefit side, there are benefits of diversification, expanding the possibility of
financing beyond the initial entrepreneurs’ limited wealth, less costly access to the
capital markets, an increased liquidity of the company’s shares, and some outside
monitoring (Holmstrom and Triole (1993)).
In our analysis we assume that the gross value of the public firm is given by
Vp (yS) , which is increasing in the amount of equity sales proceeds (yS) reinvested
back into the firm: Vp (yS) > 0 . Vf represents the market value of the private firm
and is exogenously determined. The share of the gross value of the public firm
expropriated by the owners/managers is a function, q(em,els) , depending on two
kinds of effort: em representing costly efforts exerted by the owners/managers to
expropriate part of the value of the public firm, and els representing costly efforts
exerted by large shareholders to protect their investment in the firm. Assume,
q(em, els) e [0,1], which is increasing in em and decreasing in els. 1 - q(em, els),
represents the share received by shareholders (including the owners/managers who
keep a share of the firm).
Examples of the efforts exerted by large shareholders in monitoring the
managers include auditing, formal control systems, budget restrictions, and the
establishment of incentive compensation systems, which serve to align the manager’s
interests more closely with those of shareholders. Examples of the efforts exerted by
the managers in expropriating shareholders include managers paying themselves large
salaries and generous perquisites, diverting company resources for corporate empire
building and for private benefits, and stealing business opportunities from the
company.
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The net payoff to the owners/managers depends on the share of the firm they
can expropriate q{em,els) and the share they never relinquished a as follows:
The net value to large shareholders who own a fraction, f) , of the firm is given by:
The net value to dispersed shareholders is then given as the residual value of the
public ownership of the firm ((1 - a - fi):
Recall that dispersed shareholders, by definition are free-riders, and thus exert no
effort in the competition for corporate control. Therefore eds- 0 .
Third stage choice of efforts:
Large shareholders and owners/managers choose their equilibrium efforts
simultaneously and in a manner that maximizes their total payoffs in stage three of
the game. Given values of the owners/managers share in the public firm ( a ) , large
shareholders share ( f i ) and the value of the public firm to both large and dispersed
shareholders ( S ), the owners/managers choose em to maximize their payoff Vm
shown in (1). Similarly large shareholders choose eb to maximize their payoff Vls
shown in (2). Assuming interior optima, e*n and e,s, these solutions are defined
implicitly by the respective first order conditions as functions of a , p and yS .
Substituting the equilibrium efforts e*m and e*s into equations (1) and (2) above we
K ,(em,eb;a,f i , jS) = (q(em,els) + a( 1 - q(em,eb))VP(yS) - em (1)
(2)
v ds ( e m ,eb;a,fi ,}S) = Q . - a - f i ) ( l - q(em, els ))VP (}S) (3)
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get the equilibrium payoffs to the owners/managers V* and to shareholders Vs*. e„
and e*s depend on a , ft and yS , V* and /7<j are also functions of a , f) and y S .
Expected Payoff to Owners/Managers:
The expected payoff to the owners/managers from taking their private firm public
(Vo) equals the equilibrium payoff to owners/managers ( V*) plus the amount that
owners/managers ( l - y )S decide to keep from the equity sales proceeds.
V'0{a ,p ,S ) = V:(a,p,y5) + ( \ - r )S (4)
The amount (S) that shareholders are willing to pay for their share in the public firm
should equal the equilibrium payoff to shareholders ( Vs*) or the amount that
shareholders expect to receive from their share of ownership in the public firm, so
that:
5 = V*(a,fl ,yS) = V*ds(a,f3,yS) + v;s {a ,p ,y S ) (5)
Due to their added efforts in monitoring the management, large shareholders pay a
lower price for the same amount of shares than dispersed shareholders (Shleifer and
Vishny, 1986). In our model large shareholders receive a discount on the price of
their shares equal in value to the cost of the efforts they exert in monitoring the
managers.
Choosing the optimal ownership structure to owners/managers:
Owners/managers choose the proportion of shares to sell to large shareholders ( /J ),
the proportion to keep ( a ) , which together imply the proportion to sell to dispersed
shareholders (1 - a - f i ) , and the fraction ( y ) of the equity sales proceeds to keep that
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maximize their expected payoff from taking their company public. Specifically, they
solve:
To proceed we consider specific functional forms for the contest success
function q(em,els) and for the gross value of the public firm Vp (yS). We will utilize
two different types of contest success functions (Clark and Riis (1997) and Skaperdas
and Vaidya (2005)) in the contest between managers and shareholders as illustrated in
sections III and IV below.
1.3 Equilibrium choices where the partners compete in an
asymmetric contest for control
In the third stage of the game, the competition between large shareholders and
owners/managers is modeled as a contest in which the participants exert costly efforts
to increase their probability of winning part of the value of the public firm [Clark and
Riis 1997]. What is unique about this specification is that it supposes that, even when
the two partners expend identical efforts, one of the two partners will enjoy a greater
share of the value of the firm.
The parameter o represents the effectiveness of the conflict technology or the
degree to which greater appropriative effort translates into conflict success. 6
represents the efficiency of the judiciary and law enforcement system in a country
Max(V:{a,f},YS) + { l - r )V;(a,p,y))a,p,y (6)
where o > 0. (7)
15
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and it varies between 0 and 1. An increase in 6 towards 1 would indicate stronger
law enforcement or a more efficient legal system which would favor shareholders.
Conversely a movement of 6 toward 0, would indicate weaker law enforcement or a
less efficient legal system. Suppose, for example, that 6 <1/2, if both parties devoted
an equal amount of effort to the contest, the outcome would favor the original
owners/managers.
Large shareholders and owners/managers each choose their equilibrium
efforts simultaneously and in a manner that maximizes their respective net payoffs in
stage three of the game. Given values of the owners/managers share in the public
firm ( a ) , large shareholders share ([)) and the value of the public firm to both large
and dispersed shareholders ( S ) we solve for the owners/managers equilibrium efforts
by differentiating equation (1) above with respect to em and setting it equal to zero:
« \ - 0 ) e am' +9 el*)1
In order to solve for large shareholders’ equilibrium efforts, we differentiate
equation (3) above with respect to els and we set it equal to zero:
C T g q - g x y *((1 - 0 ) e ° : + d e i y
PVP(yS) = 1 (9)
Combining equations (8) and (9) above, we get:
e,s* = (10)1 — a
Plugging in equation (10) into equations (8) and (9) above, we can solve for
the owners’/managers’ and large shareholders equilibrium efforts as given by
equations (11) and (12) below:
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^ ( l - S X l - a X J L ) "_1 - a
Pem* = ------------------- ^ — VP{yS), (11)[(i - e ) + e ( 7^ y f
1 - a
1~ a —Vp(yS), (12)[(i - e ) + e ( - ^ - y f
1 - a
Substituting (11) and (12) into (1) and (2) above we get the equilibrium
payoffs to the owners/managers and to shareholders respectively:
[a - 9)+ae m i -0)+e(.-~-yy - m - «x^-rV'm{a,p,yS) =-------- ------------------------------- --------------------- — Vp(yS) •••(13)
[ ( \ - 0 ) + d ( J ^ y f1 - a
(1 - a ) d m - 0 ) + 0 < J - y ] - <70(\ - <9)(1 - a ) ( - ? - y +l V*i.a,p,yS) = ------------------------- ^ ^ ----- Vp(y>)...(l4)
[(i - 0) + 0( - ^ - y f1 - a
Choosing the optimal ownership structure to owners/managers:
We now turn to the owners/managers choice of ownership structure. This choice
factors in the effects the choices will have on els and e*m and thus on V* and V*.
Owners/managers solve:
Max (V* (a,p,yS) + ( l - r )V *(a ,p ,y ) ) (15)a,P,r
Due to the complexity in solving the maximization problem above, we will
illustrate by an example as shown below:
Example:
17
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We assume that that the value of the public firm is determined through a Cobb-
Douglas-like form:
Vp(yS) = K ( y S y , whereK > 0 and ¥ is between 0 and 1 (16)
We interpret 'F , as a measure of the productivity of a public firm, which
would increase with improvements in the efficiency of the judiciary and law
enforcement system. According to Hall and Jones (1997), differences in levels of
economic success across countries are driven primarily by the institutions and
government policies (or infrastructure) that frame the economic environment in which
people produce and transact. Societies with secure physical and intellectual property
rights that encourage production are successful. Societies in which the economic
environment encourages the diversion of output instead of its production produce
much less output per worker. Diversion encompasses a wide range of activities,
including theft, corruption, litigation, and expropriation. In our analysis we will
allow for increases in VF that are commensurate with the improvements in the
efficiency of the judiciary and law enforcement system.
Plugging in (16) into (14) above, we get:
1 (1 - a)6p[{\ - 6 ) + ] - cx<9(l - <9)(1 ' ,S = r^(a,fi,yS) = K l-'i‘r 1 [ --------------------------^ ------- — -------------------- ]'-'p (17)
[(i -9)+e(T^ r f1 - a
To determine the optimal share to sell to large shareholders (5 , the optimal
share to keep a* and the optimal share to sell to dispersed shareholders (1 - a - [1) ,
we plug in (16) and (17) into the maximization problem (15) above. Due to the
difficulty in obtaining a closed form solution, we have provided a numerical solution
18
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to the maximization problem as shown in figures 1.2 and 1.3 attached. Numerical
solution results show the optimal ownership structure that maximizes the expected
payoff to owners/managers for the different values of 6 (efficiency of the judiciary
and law enforcement system) and are discussed in the sections below.
Analysis of figure 1.2 attached:
Numerical solution results show that owners/managers get a higher return from
investing an additional dollar in the public firm than they would from retaining it
(except for investing in the public firm, we assume that the owners/managers earn
zero return on their money), so they reinvest the full amount of the equity sale
proceeds back into the public firm (y* =1).
In our model, we have assumed that the legal protection of outside
shareholders complements the efforts exerted by large shareholders in monitoring the
managers, so as the efficiency of the judiciary and law enforcement system improves
( Q t ), equity valuation by outside shareholders increase and the value of public firms
increase. Results obtained from our numerical solution are consistent with the
existing literature as we show that as the efficiency of the judiciary and law
enforcement system improves (0 T ), the optimal value to the owners/managers from
taking their firm public increases and the likelihood that entrepreneurs will take their
privately held firms public increases. An example with Vf 2 as the market value of
the private firm and Vq being the optimal value to the owners/managers from taking
their firm public is shown in figure 1.2 attached.
Analysis of figure 1.3 attached:
2 The value o f the private firm is exogenously determined in our model.
19
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In deciding to take their company public, three sources of value contribute to the
owners/managers payoff: expropriating part of the value of the public firm; retaining
part of the proceeds received from selling equity; and the value received from being
shareholders.
In a forward looking game, the owners/managers will decide on the optimal
ownership structure in stage two of the game in a manner that maximizes their
expected payoff from taking their firm public in stage three. The owners/managers
weigh the costs and benefits of expropriation of outside investors that comes with
control. Such private benefits of control, as described by Jensen and Meckling
(1976), do come at the expense of profits accruing to the outside investors (including
the owners/managers who keep a share of the firm). In legal regimes with weak
investor protection (low values of 6 ), the owners/managers can steal a firm’s profits
perfectly efficiently and no rational outsider would finance such a firm. In such legal
regimes, the cost of raising capital to entrepreneurs is high and the family firm
emerges as the value maximizing outcome where the original owners retain both
ownership and control.
As investor protection improves or the expropriation technology becomes less
efficient, the owners/managers expropriate less, and their private benefits of control
diminish. In such situations entrepreneurs obtain outside finance on better terms. In
addition to the legal protection of outside shareholders, the presence of large
shareholders restricts managers’ excessive spending and is in the best interest of
dispersed shareholders. Pagano and Roell (1998) show that the optimal ownership
structure chosen by the entrepreneur generally involves some measure of dispersion
20
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(more than one external investor). It may also involve a certain degree of monitoring
by a large external shareholder, because in order to obtain equity capital more
cheaply, the initial owner needs to restrain his own future tendency to stray. So in
legal regimes with intermediate investor protection (intermediate values of 6 ), the
optimal ownership structure chosen by entrepreneurs involves keeping the majority of
firm ownership while selling the rest of the company shares to large shareholders.
The family controlled firm emerges as the equilibrium outcome.
When legal protection of outside investors is very good (high values of 6),
there is no need for monitoring in equilibrium, and the widely held professionally
managed firm emerges as the equilibrium outcome.
Our results are consistent with La Porta, Lopez-de-Silanes, Shleifer and
Vishny (1998), Shleifer and Wolfenzon (2002) and Burkart, Panunzi and Shleifer
(2003) as we show that the concentration of ownership of shares in public companies
is negatively related to the legal protection of outside shareholders. In contrast with
Castillo and Skaperdas (2005), we obtain a negative and monotone relationship
between the legal protection of outside shareholders and the size of the ownership
share retained by the owners/managers.
1.4 Equilibrium choices where the partners compete with
the powers of persuasion
In stage three of the game, the competition between large shareholders and
owners/managers involves owners/managers and large shareholders devoting costly
21
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resources to influence the opinion of a third party: Board of Directors, Securities and
Exchange Commission or the courts in their favor [Skaperdas and Vaidya 2005].
Briefly, the functional form is derived using the following process: Managers and
large shareholders expend resources em and e]s on gathering information and
evidence. The evidence and information produced are presented to the Board of
Directors, Securities and Exchange Commission or the courts. Based on the evidence
and information presented, the Board of Directors, Securities and Exchange
Commission or the courts make an inference about the truth. They update their prior
beliefs in light of the evidence and information presented and the posterior beliefs
thus produced determine the probability of their judgment being in favor or against
the claimant.
In addition to the effect that resources have in collecting information and
uncovering evidence, the efficiency of the judiciary and law enforcement system
should have an effect on the ease or difficulty with which each side can collect
information or uncover evidence in favor of its cause. In our analysis, the higher the
efficiency of the judiciary and law enforcement system the easier it is for
shareholders to collect information or uncover evidence in their favor and the harder
it is for managers to collect information or uncover evidence in theirs.
As before, the share of the gross value of the public firm received by the
owners/managers is a function of the two kinds of effort:
q(em,es)=0.5-<l>[del , O c r r c l (18)
with em representing costly efforts exerted by the owners/managers to influence the
opinion of the Board of Directors, Securities and Exchange Commission or the courts
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in their favor, and els denotes costly efforts exerted by large shareholders to influence
the opinion of the Board of Directors, Securities and Exchange Commission or the
courts in theirs. 1 - q(em, els) represents the share of the gross value of the public
firm received by shareholders (including the owners/managers who keep a share of
the firm).
As before the parameter 6 reflects the efficiency of the judiciary and law
enforcement system in a country and it varies between 0 and 1. An increase in 6
towards 1 would indicate stronger law enforcement or a more efficient legal system.
Conversely a movement of 6 toward 0, would indicate weaker law enforcement or a
less efficient legal system. The function ea is positive, increasing and strictly
concave. The parameter (j) is taken as exogenous in our model and it varies between
0 and 0.5. In a more fully articulated model, it would be an increasing function of the
likelihood of conviction of the managers of the firm given that there is legal evidence
against them.
Solving for Equilibrium:
Large shareholders and owners/managers choose their equilibrium efforts
simultaneously and in a manner that maximizes their total payoffs in stage three of
the game. Given values of the owners/managers share in the public firm ( a ), large
shareholders share ( f t ) and the value of the public firm to both large and dispersed
shareholders ( S ) we solve for the owners/managers equilibrium efforts by
differentiating equation (1) above with respect to em and setting it equal to zero. The
owners’/managers’ equilibrium efforts are given by equation (19) below:
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em = ( ( ! - « ) ( ! - # ) ?* O '^ ) (|“ff)5 O 9)
In order to solve for large shareholders’ equilibrium efforts, we differentiate
equation (3) above with respect to e,s and we set it equal to zero. Large shareholders’
equilibrium efforts are given by equation (20) below:
e;=(pe<|>aVp)< \ (20)
Substituting equations (19) and (20) into equations (1) and (2) above we get
the equilibrium payoffs to the owners/managers and to shareholders respectively:
V'm =Vp +0.5(a - l )Vp + ( a - 1 ) ^ 9 ) ^ (J3 V
- ((1 - a )(l - (21)
r;=O.5(l-a)F,+(l-aX00)W iP o r ) ^ V ^ c t ) < t > ( \ - V
- ( p 0 t < r ) & V & (22)
Choosing the optimal ownership structure to owners/managers:
Owners/managers choose the proportion of shares to sell to large shareholders ( f t ),
the proportion to keep ( a ) , the proportion to sell to dispersed shareholders
(1 - a - (5 ) and the fraction ( y ) of the equity sales proceeds to keep in a manner that
maximizes their expected payoff from taking their company public.
Max(V;t(a,j3,yS) + (\-y)V:(a , j3 ,y)) (23)a,p,y
Due to the complexity in solving the maximization problem above, we will
illustrate by an example as shown below:
Example:
24
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We assume that that the value of the public firm is determined through a Cobb-
Douglas-like form:
Vp(yS) = K ( y S f 5 whereK > 0 (24)
Substituting (24) into (22) above and in the special case of o - 0.5 , we get:
S = v;{a,p,yS) =----------------- ----------- W ~ a y K ' y ------ ...(25)[ 4 - 2 ( l - a ) p e 2f - K y + 2 ( l - a ) 2f - ( l - d ) - K 2r + J32<l>2e 2K 2r f
To determine the optimal share to sell to large shareholders , the optimal
share to owners/managers a* and the optimal share to sell to dispersed shareholders
(1 - a -/?)* , we plug in (24) and (25) into the maximization problem (23) above.
Due to the difficulty in obtaining a closed form solution, we have provided a
numerical solution to the maximization problem as shown in figures 1.4 and 1.5
attached. Numerical solution results show the optimal ownership structure that
maximizes the expected payoff to owners/managers for the different values of 6
(efficiency of the judiciary and law enforcement system) and are discussed in the
sections below.
The choice of the constant K depends in part on the value of ^ and both
values of ^ and K should be chosen in a manner that would be consistent with what
is observed in the real world. For our analysis we have chosen K to equal 20 and (f
to equal 0.25. Future work is required to get an empirical estimation for both K and
</>.
Analysis of figure 1.4 attached:
In our model, we have assumed that the legal protection of outside shareholders
complements the efforts exerted by large shareholders in monitoring the managers, so
25
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as the efficiency of the judiciary and law enforcement system improves (9 T ), equity
valuation by outside shareholders increase and the value of public firms increase.
Results obtained from our numerical solution are consistent with the existing
literature as we show that as the efficiency of the judiciary and law enforcement
system improves ( 0 T ), the optimal value to the owners/managers from taking their
firm public increases and the likelihood that entrepreneurs will take their privately
held firms public increases. An example with V( 3 as the market value of the private
firm and Vq being the optimal value to the owners/managers from taking their firm
public is shown in figure 1.4 attached.
Analysis of figure 1.5 attached:
Numerical solution results show that owners/managers get a higher return from
investing an additional dollar in the public firm than they would from retaining it
(except for investing in the public firm, we assume that the owners/managers earn
zero return on their money), so they reinvest the full amount of the equity sale
proceeds back into the public firm ( y* =1).
Upon deciding to take their privately held company public, the
owners/managers have to find the particular combination of concentrated and
dispersed ownership that maximizes their wealth. Three sources of value contribute
to the owners/managers payoff: expropriating part of the value of the public firm;
retaining part of the proceeds received from selling equity; and the value received
from being shareholders.
3 The value o f the private firm is exogenously determined in our model.
26
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In legal regimes with weakest shareholder protection, relatively low values of
6 , the value of the private firm exceeds its value being public (Fig. 1.4) and the
original owners decide against taking their privately held firm public. Ownership
stays with the original founders and the family owned corporation emerges as the
equilibrium outcome.
In legal regimes with intermediate shareholder protection, values of 6
midway between 0 and 1, the value of the public firm exceeds its value being private
(Fig. 1.4) and the original owners decide to take their privately held firm public.
Majority ownership stays with the original founders and the family controlled
corporation emerges as the equilibrium outcome (a* > 0.5).
In legal regimes that successfully limit the expropriation of shareholders, 6
approaching 1, the value of the public firm exceeds its value being private (Fig. 1.4)
and the original owners decide to take their privately held firm public. The widely
held corporation emerges as the equilibrium outcome.
1.5 Concluding Remarks
Our results are consistent to a large extent with the existing literature on Corporate
Governance as Burkart, Panunzi and Shleifer (2003) and Shleifer and Wolfenzon
(2002) show that firms are more valuable, shareholder expropriation is lower and
ownership concentration is lower, with better protection of shareholders. Our results
are also consistent with Burkart, Panunzi and Shleifer (2003) as they show that in
legal regimes that successfully limit the expropriation of minority shareholders, the
widely held professionally managed corporation emerges as the equilibrium outcome.
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In legal regimes with intermediate protection, management is delegated to a
professional, but the family stays on as large shareholders to monitor the manager. In
legal regimes with the weakest protection, the founder designates his heir to manage
and ownership remains inside the family. In Western Europe for example, many
publicly traded firms are family controlled through a majority ownership while in
emerging markets such as the Middle East, both ownership and control tend to stay
with the family. In the United States, separation of ownership and control occurs at
an early stage where the original founder and his family retain only marginal
ownership.
Even though our findings are consistent to a large extent with the existing
empirical literature, measuring the correct level of legal protection of outside
shareholders ( 6 ) for different countries would be vital in interpreting empirical
results. Conflicts between shareholders and managers are not the only situations
where agency costs are incurred. We can think of the company’s overall value as a
pie that is divided among a number of claimants. These include management,
shareholders, company workforce, banks, creditors and the government. Work is
currently underway to analyze the situation where the owners raise cash from both
equity and debt financing to fund the firm’s investment decisions.
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Optimal Value to Owners/Managers
0.6
0.4P u b l i c F i r m s
0.2
0.1 0.2 0.8 0.9 10.3 0.4 0.5 0.6 0.7
Theta
Figure 1.2: The partners compete in an asymmetric contest for control
(Vo*) Optimal Value toOwners/Managers
(Vf) Value of PrivateFirm
29
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Optimal Ownership Structure to Owners/Managers1 -
0.9 |I
I0.7 4
0.6 -
0.5
0.4 -f—
0.3
0.2 4—
[
0.1
o L0.1
p*
0.2 0.3
1
0.4 0.7 0.8 0.9 10.5 0.6
Theta
Figure 1.3 : The partners compete in an asymmetric contest for control
Owners/Managers(Alpha*)
Large Shareholders (Beta*)
Dispersed Shareholders (1- Alpha*-Beta*)
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Optimal Value to Owners/Managers
1.2 -
1 -
0.8
0.4 ■
0.2 -
/P u b l i c F i r m s
0 V 0 V <5 o - O ' O ' O ' O ' O ' O ' O ' O ' O ' O ' O ' O ' O ' O ' O ' O ' O ' O ' O ' O ' O '
Theta
Figure 1.4: The partners compete with the powers of persuasion
(Vo*) Optimal Value to Owners/Managers
(Vf) Value of Private Firm
31
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Optimal Ownership Structure to Owners/Managers
Owners/Managers
■— Large Shareholders
Dispersed Shareholders
0.2
1 - a * -|3
- 0 .2 J
Theta
Figure 1.5: The partners compete with the powers of persuasion
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Chapter 2
Risk of Expropriation and the Rybczynski Theorem
2.1 Introduction
Based on the Heckscher-Ohlin (HO) Model of international trade (see for example
Bhagwati, Srinivasan and Panagariya 1998), we can derive growth paths for
production and trade in terms of a country’s capital-labor ratio. As a country grows, it
will accumulate more capital relative to the world leading to an increase in the output
of its capital-intensive goods relative to the output of its labor-intensive goods. This
is the ladder-of-development or product-cycle hypothesis: a country's output mix
depends on its stage of development, with countries moving from agriculture to labor-
intensive manufactures to high-tech manufacturing and services as their aggregate
labor productivity increases [James Harrigan and Egon Zakrajsek, 2000]. Rybczynski
theorem formally states that “If a factor endowment in a country rises (falls), and if
prices of the outputs remain the same, then the output of the good that uses that factor
intensively will rise (fall) while the output of the other good will fall (rise).”
Harrigan (1995) and Bernstein and Weinstein (1998) used the Heckscher-
Ohlin (HO) general equilibrium model with factor price equalization to provide
empirical confirmation of Rybczynski theorem. Harrigan (1995) use data on
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manufacturing output and factor endowments for 20 OECD countries from 1970 to
1985 and show that under well-known conditions, there will be a linear Rybczynski
relationship between sectoral outputs and factor endowments across countries.
Bernstein and Weinstein (1998) use data on production patterns and factor
endowments at the regional level for both OECD countries and Japan to verify the
Rybczynski theorem. In a recent paper, Xu [2002] focuses on developing countries,
finding capital abundance to be statistically significant in determining production
patterns in 18 of 28 examined industries. In his panel data regressions controlling for
time and country fixed effects as well as industry skill level (proxied by industry
average wage rate relative to the US), the value-added shares of 11 of the 12
relatively labor-intensive industries increase with country capital abundance, with
five of them statistically significant, and the value-added shares of 10 of the 16
relatively capital intensive industries decrease with country capital abundance, with
four of the 10 statistically significant. This finding contradicts the predictions of
Rybczynski theorem and presents the author with a puzzle.
The starting point of departure of this paper is to ask how corruption and risk
of expropriation affect the Rybczynski predictions. A preliminary analysis,
comparing data (1980-1983) for two countries with similar capital stock per worker
(K/L), reveals that the ratio of the output of capital intensive goods to the output of
labor intensive goods is higher for countries that have less corruption and higher
bureaucratic efficiency [The relevant data on corruption and the bureaucratic
efficiency index were taken from Mauro’s 1995]. Both indices, (recorded by country
representatives in each country) are subjective. The bureaucratic efficiency index is
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the average of the judiciary system, red tape and corruption indices. The preliminary
analysis comparing data for Japan and Venezuela is shown in table 2.1 below:
Country Year K/L C/L BEI Country Year K/L C/L BEI
JPN 1981 23,345 1.72 9.08 VEN 1981 21,635 0.45 5.42
JPN 1982 24,614 1.70 9.08 VEN 1982 21,603 0.52 5.42
JPN 1983 25,785 1.70 9.08 VEN 1983 21,606 0.48 5.42K /L = capital stock per workerC/L = (value added o f capital intensive goods) / (value added o f labor intensive goods)BEI = Bureaucratic efficiency index
Table 2.1: Comparison table for Japan and Venezuela
In this paper, we use the Heckscher-Ohlin (HO) general equilibrium model to
test empirically the hypothesis that Rybczynski predictions are less likely to hold in
countries with higher risk of expropriation and higher incidence of corruption. Risk
of expropriation and incidence of corruption by corrupt government officials are
higher in the case of capital-intensive projects due to the larger size of the prize
(larger size of the pie). So in countries where the risk of expropriation and the
incidence of corruption are high, investors prefer to invest in labor-intensive projects
instead of capital-intensive projects.
Many economists argue that it is easier for a corrupt government official to
expropriate large non-standard capital-intensive projects as opposed to smaller,
standardized labor-intensive projects. Kaufmann [1998] suggests that bribing and
rent-seeking exact a significant economic cost. Corrupt bureaucrats tend to favor
non-standard, complex, and expensive capital-intensive projects that make it easier to
skim significant sums. Coolidge and Rose-Ackerman [1997], argue that kleptocrats
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will differentially support projects that generate large corrupt payoffs. And if
revelations of corruption would destabilize the regime, the kleptocrat will favor
projects where payoffs can be easily hidden. Examples of such projects are
specialized capital-intensive projects with one of kind designs such that no one would
be able to locate a reliable cost benchmark. Coolidge and Rose-Ackerman [1997]
also point out that lacking credible commitment mechanisms, such as independent
law enforcement institutions, the corrupt autocrat may have difficulty convincing
investors to make capital investments since they may fear expropriation or
confiscatory tax and regulatory systems. The only investors willing to commit funds
may be those with a short term, get rich-quick attitude.
In conducting panel data regressions (16 developing countries over a period of
eight years 1984-1991), controlling for time and country fixed effects as well as
incidence of corruption and risk of expropriation, I find a negative and not
statistically significant relationship between capital stock per worker and the ratio of
the value-added share of the relatively capital-intensive industries to the value-added
share of the relatively labor-intensive industries. In the case of developing countries,
lack of statistical significance between capital stock per worker and the ratio of value
added share of capital intensive goods to the value added share of labor intensive
goods contradicts the predictions of the Rybczynski Theorem. A positive and
statistically significant relationship obtains between the Corruption and Expropriation
Index (the higher the Corruption and Expropriation Index the higher the incidence of
corruption and risk of expropriation) and the ratio of the value-added share of the
relatively capital-intensive industries to the value-added share of the relatively labor-
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intensive industries. This supports the notion that the risk of expropriation and the
incidence of corruption are important factors in determining of the shape investment
structure in a country. In countries where the risk of expropriation and the incidence
of corruption are high, investors prefer to invest in labor-intensive projects instead of
capital-intensive projects.
In conducting panel data regressions (22 developed OECD countries over a
period of eight years 1984-1991), controlling for time and country fixed effects as
well as incidence of corruption and risk of expropriation, I find a positive and
statistically significant relationship between both independent variables (capital stock
per worker and the Corruption and Expropriation Index) and the ratio of the value-
added share of the relatively capital-intensive industries to the value-added share of
the relatively labor-intensive industries.
In the next section I will suggest a modification to the Heckscher-Ohlin model
where I will allow for international variability in corruption and risk of expropriation.
The modification to the Heckscher-Ohlin model is intended to explain cross country
differences in the ratio of capital-intensive output to labor intensive output and is not
intended to explain the contradiction in the predictions of Rybczynski theorem.
Results predicted by the model show that a negative relationship exists between the
risk of expropriation by corrupt government officials and the ratio of capital-intensive
output to labor-intensive output.
2.2 The Model
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We assume that we have two countries. Each country produces two final goods, X
(capital intensive) and Y (labor intensive) using two factors of production labor and
capital. We will consider two approaches to solving our model where the same result
of higher taxation of the capital intensive good relative to the labor intensive good
obtains.
In our first approach, capital-intensive output and labor-intensive output are
subject to an expropriation tax t imposed by corrupt government officials. Risk of
expropriation by corrupt government officials is higher in the case of capital-intensive
projects due to the larger size of the prize (larger size of the pie), so we assume that
capital-intensive output is taxed at a higher rate than labor intensive output. Capital-
intensive firms maximize the following profit function:
M a x ( \ - t x )Px f x (Kx ,Lx ) - r K x - w L x (1)KX ’LX
Labor-intensive firms maximize the following profit function:
Max(\ - ty )Py f y (Ky , Ly) ~ rK y ~ WLy (2)K y ,Ly
where t x > t Y (good X is effectively taxed at a higher rate than good Y)
In our second approach, we examine the specific case of a Cobb Douglas
production function, and from our argument in section I above we assume that it is
easier for corrupt government officials to expropriate the capital endowment of firms
rather than their labor endowment. Capital-intensive firms maximize the following
profit function:
Max Px [(1 - t)(Kx ) f [Lx ]*-“ - rKx - wLx
= M a x { \ - t ) aPx [Kx ]a[Lx t a ~ r K x - w L x (3)KX’LX
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Labor-intensive firms maximize the following profit function:
MaxPY[ ( \ - t ) (K Y)Y[LYt P - rKY - w L YK y , L y
= MOX( 1 ~ t ) PPy [Ky ] P [Ly ]'~fi ~ rK y ~ WLy (4)K y , Ly
a > ft , to guarantee that good X has a higher capital intensity than good Y
(1 - 1)“ > (1 - t)/f (good X is effectively taxed at a higher rate than good Y)
From the two approaches illustrated above, we can show that taxing the output
of individual firms effectively translates into a reduction in the relative price of the
labor intensive good X to the price of the labor intensive good Y. The effective
reduction in the relative price of good X to the price of good Y, results in an increase
in the capital to labor ratio for both goods X and Y as shown in Figure (2.1) below.
In figure 2.1 below we have equilibrium initially, with the goods-price ratio
Py Wexchanging (-J—) X for Y and with the factor-price ratio (—) at ABCD. FactorPx r
K Kproportions in X and Y are indicated by points B and C respectively with —— > ——
Lx Ly
pat all factor-price ratios. An increase in the effective goods-price ratio (— ) results
Px
in the upward shift of good X isoquant. The new equilibrium factor-price ratio is at
EFGH and it is tangential to the new (—— ,y^P--)X isoquant and the Y isoquant.0 — tx ~)Px
wThe above implies that we must have a higher factor-price ratio (—) and thus higher
r
K ratios in both sectors.L
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Facto
r K
For any finite factor-price ratio, there must be full employment of both factors
in equilibrium. Furthermore, the overall factor-endowment ratio must be a weighted
average of the factor ratios in the two sectors, X and Y. This is shown in the identity:
( f ) = ( ^ X ~ L) + ( | L) ( ^ ) (5)J-j X Y
Where,
L "I- Ly — L
K X + K Y = K
and —^ and — are weights adding up to unity.
O Factor L
Figure 2.1: Equilibrium analysis
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The effective reduction in the relative price of good X to the price of good Y
P K( — i ), results in an increase in the capital to labor ratio for both goods ( —— T ) and
PY L x
Y
A change in the effective goods-price ratio does not alter the amount of capital
Kand labor available in the economy and hence the ratio -j— stays the same. In order
for equation (5) to hold, we must have a decrease in the weight relative to theL
weight — . In other words Lx has to decrease and LY has to increase. From above, L
we know that —- is increasing and since LY is increasing then K Y must be LY
increasing and K x must be decreasing.
So an effective reduction in the relative price of good X to the price of good Y
p( — -I), results in a decrease in K x and Lx and an increase in K y and Ly. This
Py
implies that the output of capital-intensive goods decreases and the output of labor-
intensive goods increases.
The same result can be reached using an algebraic representation as shown in
the derivation below. Capital-intensive firms solve the following optimization
problem.
Max Px f x (K x , Lx ) — rK x — wLxKx ,PX
The first-order conditions are
41
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d ° f x ( K x , L x ) _ u .
X SLX
p dfx (Kx ,Lx ) X SKX
dfx ( K x , Lx )MPT...
(6)w _ dLx _ MPLX
7 ~ 8fx (Kx ,L x r MPKx
dKx
Labor-intensive firms solve the following profit maximizing problem:
MaxPY f Y(KY,Ly) ~ rKY - wLy
the first-order conditions are
p dfY( K Y, L Y) _1 y ---------------------------------- W
dLy
dfY( K y , L Y)Y 6Ky
dfY( K Y, LY)
w _ dLy _ MPLy ^r 8fy (Ky , Ly ) MPKy
SKy
In a competitive equilibrium, each output price must equal its marginal cost,
which under the assumption of constant returns to scale equals the average cost.
Therefore we have the equations of production equilibrium:
L xP* = r ~ ^ +W^ <8)
P r E jL + w h L (9)Y Y
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where X and Y are the outputs of the capital-intensive and labor-intensive goods
P rrespectively. Now, we define the relative prices n = — and Q = — . Dividing
PY w
equation (8) by equation (9) above, we have a functional relationship between relative
output-prices and relative factor-prices:
Q K X | Lx X XK = -
ClKy Ly
Y Y
To see whether the relative goods-price ratio is increasing or decreasing in the
relative factor-price ratio, we take the logarithms of both sides and differentiate. This
yields:
Ly Lx1 dn Ky K x
71X1 (h + K-m + K)K X Ky
Therefore a higher Q T (— T) corresponds to a higher n t (— t ) if andW Py
K Konly if —— > —— which is always true since good X is more capital-intensive than
Lx Ly
good Y.
The above also implies that an effective reduction in the relative price of good
X to the price of good Y ( —X J ,) wjH result in an increase in — AnPY Q r
Wincrease in — T will result in an increase in the marginal product of labor relative to r
the marginal product of capital for both capital-intensive and labor-intensive goods as
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can be seen from equations (6) and (7) above. An increase in the marginal product of
labor relative to the marginal product of capital for both goods X and Y implies that
K Kthe capital to labor ratio will increase for both goods ( —— t ) and ( —— T ).
Lx LY
Adopting the same argument we used under the geometric representation above, an
increase in the capital to labor ratio for both goods X and Y will result in a decrease
in the output of capital-intensive goods and an increase in the output of labor-
intensive goods.
Our model developed above is useful in explaining cross-country differences
in the ratio of capital-intensive output to labor-intensive output. Our model predicts
that countries (with similar capital stock per worker) with higher incidence of
corruption and higher risk of expropriation have a lower ratio of capital-intensive
output to labor-intensive output. This implies that the spread of corruption and the
weak enforcement of property rights can have adverse effects on a country’s
development predicted by the ladder-of-development or product-cycle hypothesis: a
country’s output mix depends on its stage of development, with countries moving
from agriculture to labor-intensive manufactures to high-tech manufacturing and
services as their aggregate labor productivity increases [James Harrigan and Egon
Zakrajsek, 2000]. In section IV below, we use panel data collected for 22 developed
and 16 developing countries to show that an increase in the incidence of corruption
and the risk of expropriation can render the Rybczynski predictions statistically not
significant. We start by providing an illustration of the Rybczynski theorem.
The Rybczynski theorem predicts that holding product prices constant, an
increase in the relative supply of capital increases the relative output of the capital-
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intensive good so much that the output of the labor-intensive good decline. As can be
Kshown from equation (5) above, an increase in the relative supply of capital ( — T )L
will lead to an increase in the weight -L- relative to the weight —E . In other wordsL L
L x has to increase and Ly has to decrease.
K KIf prices are constant then ( ——) and ( —- ) remain unchanged and from the
Lx Ly
goods-output ratio shown in equation (10) below, an increase in Lx relative to LY
will lead to an increase in the relative output of the capital-intensive good to the
labor-intensive good.
Y , / * ( — ->!)— = ( - ^ - ) -------- - A ( 1 0 )Y T KY Ly f A r , i)
L y
Equation (10) above defines the relationship predicted by Rybczynski theorem
which formally states that “If a factor endowment in a country rises (falls), and if
prices of the outputs remain the same, then the output of the good that uses that factor
intensively will rise (fall) while the output of the other good will fall (rise).”
2.3 Data Description
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In this section we provide a description for the panel data collected for 22 developed
1 2(OECD) and 16 developing countries used in the empirical estimation of the model.
Subsection III.l provides a description of the corruption and risk of expropriation
index and section III.2 provides a description of the remaining variables.
2.3.1. IRIS-3 File of International Country Risk Guide (ICRG) Data - The data files
provided by the IRIS-3 file contain annual values for indicators of the quality of
governance, 1982-1997, constructed by Stephen Knack and the IRIS Center,
University of Maryland, from monthly ICRG data provided by The PRS Group.
Currently, IRIS-3 provides annual ratings for the following indicators: Corruption in
government, rule of law (law and order tradition), bureaucratic quality, ethnic
tensions, repudiation of contracts by government, and risk of expropriation. To
measure the incidence of corruption and risk of expropriation in a country, I took the
simple average of the risk of expropriation and corruption indices. The IRIS Center’s
definitions of these indices are reported below:
1- Risk of Expropriation: This variable evaluates the risk of outright
confiscation and forced nationalization of property. Lower ratings are given to
countries where expropriation of private foreign investment is a likely event (Knack,
Stephen [principal investigator(s)] / PRS Group [distributor], 1982-1997).
2- Corruption: Lower scores indicate that high government officials are likely
to demand special payments and that illegal payments are generally expected
throughout lower levels of government in the form of bribes connected with import
' List o f developed (OECD) countries: Australia, Austria, Canada, Denmark, Finland,France, Germany, Greece, Ireland, Italy, Japan, Korea, M exico, Netherlands, N ew Zealand, Norway, Portugal, Spain, Sweden, Turkey, UK, and U SA .2 List o f developing countries: Argentina, B olivia, Chile, Colombia, Ecuador, Guatemala, Honduras, India, Iran, Kenya, M alawi, M orocco, Panama, Peru, Philippines, and V enezuela.
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and export licenses, exchange controls, tax assessment, police protection, or loans
(Knack, Stephen [principal investigator(s)]/'PRS Group [distributor], 1982-1997).
From the definitions above, it can be seen that the two indices are closely
related. There may be measurement error in each individual index, and taking the
simple average of the two yields a better estimate of the determinants of the
confiscatory tax imposed by corrupt government officials. I will label the average of
the two indices the Expropriation Index (El), where the higher the El the lower the
incidence of corruption and risk of expropriation.
2.3.2. Value added data for 28 three-digit ISIC manufacturing aggregates were drawn
from the UNIDO INDSTAT3 database, available from the United Nations. The
dependent variable (value-added share of the relatively capital-intensive industries to
the value-added share of the relatively labor-intensive industries) used in the panel
data regressions below, was constructed by dividing the total value added output of
the 14 most capital-intensive industries by the total value added output of the 14 most
labor-intensive industries. Classification of labor-intensive and capital-intensive
industries wwas done in accordance with Schott [2001], where sectors were ordered
in terms of increasing capital intensity according to maximum observed value added
per worker.
Capital Stock per worker data in 1985 prices were collected from version 5.6
of the Penn World Table. The Penn World Table displays a set of national accounts
economic time series covering many countries. Its expenditure entries are
denominated in a common set of prices in a common currency so that real quantity
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comparisons can be made, both between countries and over time. The table also
provides information about relative prices within and between countries, as well as
demographic data and capital stock estimates.
Gross enrollment ratios for primary, secondary and tertiary levels of education
were collected from the UNESCO Institute for Statistics. Total enrollment in a
specific level of education, regardless of age, is expressed as a percentage of the
official school-age population corresponding to the same level of education in a given
school-year.
Finally, oil production data were collected from the CIA World Factbook.
The data are used to construct a dummy variable which takes on a value of 1 for a
country if it is one of the top 30 oil producing nations or a 0 if it is not.
2.4 Empirical Results
In this section we use panel data for 22 developed (OECD) and 16 developing
countries to test the relationship between capital stock per worker and the ratio of the
value-added share of the relatively capital-intensive industries to the value-added
share of the relatively labor-intensive industries. Subsection 2.4.1 focuses on OECD
countries where I find a positive and statistically significant relationship between
capital stock per worker and the ratio of the value-added share of the relatively
capital-intensive industries to the value-added share of the relatively labor-intensive
industries. The relationship remains positive and statistically significant even after
controlling for the expropriation index, education index and the oil dummy variable.
Subsection 2.4.2 focuses on developing countries where I find a negative and not
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statistically significant relationship between capital stock per worker and the ratio of
the value-added share of the relatively capital-intensive industries to the value-added
share of the relatively labor-intensive industries. The relationship remains negative
and not statistically significant even after controlling for the expropriation index,
education index and the oil dummy variable.
To test my hypothesis I use panel data regressions. Below is the regression
equation for my estimation:
LN(C, / Z.) = a + fi lLN(Ki /L ,) + fi2LN(EI i) + fi-iLN(OILi) + fiALN(EDUi) + u, ,
where C / L represents the ratio of the value-added share of the relatively capital-
intensive industries to the value-added share of the relatively labor-intensive
industries, K / L represents capital stock per worker, El represents the risk of
expropriation index, OIL represents the oil production index and EDU represents
the level of education index.
2.4.1. OECD Countries - In Table 2.2 below, we report our results from estimating
the Rybczynski effects for a sample of 22 OECD countries using both fixed-effects
and random-effects methods. The results are similar between the two methods. The
Hausman test supports the hypothesis of no correlation between the independent
variables and the country specific effects, so the random-effects estimator is valid.
Table 2.2 shows that there is a positive and statistically significant relationship
between capital stock per worker and the dependent variable (ratio of the value-added
share of the relatively capital-intensive industries to the value-added share of the
relatively labor-intensive industries), both in the random-effects and fixed-effects
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methods. Controlling for the risk of expropriation (expropriation index), I find a
positive and statistically significant relationship between both independent variables
(capital stock per worker and the expropriation index) and the dependent variable
(ratio of the value-added share of the relatively capital-intensive industries to the
value-added share of the relatively labor-intensive industries). Overall the R-square
increases from 0.346 to 0.415 for the random effects model and from 0.346 to 0.427
for the fixed effects model. The increase in the R-square implies that the
expropriation index plays an important role in explaining variations in the ratio of the
value-added share of the relatively capital-intensive industries to the value-added
share of the relatively labor-intensive industries.
Controlling for the level of education (education index), oil production (oil
dummy variable) and risk of expropriation (expropriation index) we find a not
statistically significant relationships between two independent variables (the
education index and the oil dummy variable) and the dependent variable (ratio of the
value-added share of the relatively capital-intensive industries to the value-added
share of the relatively labor-intensive industries), while the relationships between the
remaining two independent variables (capital stock per worker and the expropriation
index) and the dependent variable (ratio of the value-added share of the relatively
capital-intensive industries to the value-added share of the relatively labor-intensive
industries) remain positive and statistically significant.
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Table 2.2Estimating Rybczynski effects for a sample of 22 OECD Countries
Dependent variable: LN(value-added share o f the relatively capital-intensive industries / value-added share o f the relatively labor-intensive industries)
Independentvariable (1) (2) (3) (4) (5) (6) (7) (8)
Constant -0 .618 -0 .317 -0 .502 -0 .172 -0.502 -0 .172 -0 .613 -0 .328LN(Capital stock per worker)
0 .260(0 .071)***
0 .192(0 .078)**
0.183(0 .077)**
0.1124(0 .084)
0 .180(0.077)**
0.1124(0 .084)
0 .186(0 .103)*
0.101(0 .113)
LN(Expropriation Index)
0 .240(0 .091)***
0.224(0 .091)**
0.239(0 .091)***
0.224(0 .091)**
0 .220(0 .097)**
0 .206(0 .097)**
Oil(OIL)
0.0567(0 .077)
0 .056(0 .079)
--------
LN(Education)
0 .056(0 .258)
0 .116(0 .266)
R 2 0 .346 0 .346 0.415 0 .427 0.401 0 .427 0 .374 0 .406
Number o f observations
175 175 175 175 175 175 165 165
Estimationmethod
Randomeffects
Fixedeffects
Randomeffects
Fixedeffects
Randomeffects
Fixedeffects
Randomeffects
Fixedeffects
Hausman Test (Prob.Rejecting Ho)
0.0392 0.5886 0.7281 0 .9739
*** = significant at the 0.01 level, * = significant at the 0.10 level** = significant at the 0.05 level, Standard error in parenthesisHo (Hausman Test): Difference in coefficients not systematic
Overall the R-square drops from 0.415 to 0.374 for the random effects model and
from 0.427 to 0.406 for the fixed effects model.
2.4.2. Developing Countries — In Table 2.3 below, we report our results from
estimating Rybczynski effects for a sample of 16 developing countries using both
fixed-effects and random-effects methods. The results are similar between the two
methods. The Hausman test supports the hypothesis of no correlation between the
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independent variables and the country specific effects in almost all cases, so the
random-effects estimator is valid. Table 2.3 shows that there is a negative and not
statistically significant relationship between capital stock per worker and the ratio of
the value-added share of the relatively capital-intensive industries to the value-added
share of the relatively labor-intensive industries, both in the random-effects and fixed-
effects methods. Controlling for the risk of expropriation (expropriation index), I find
a positive and statistically significant relationship between the expropriation index
and the dependent variable (ratio of the value-added share of the relatively capital-
intensive industries to the value-added share of the relatively labor-intensive
industries) while a negative and not statistically significant relationship maintains
between capital stock per worker and the dependent variable. Overall R-square
increases from 0.006 to 0.011 for the random effects model and from 0.006 to 0.018
for the fixed effects model. The significant increase in R-square implies that the
expropriation index plays a very important role in explaining the variation in the ratio
of the value-added share of the relatively capital-intensive industries to the value-
added share of the relatively labor-intensive industries. The low level of R-square
with capital stock per worker as the only independent variable implies that capital
stock per worker plays a minor role in explaining variations in the ratio of the value-
added share of the relatively capital-intensive industries to the value-added share of
the relatively labor-intensive industries.
Controlling for the level of education (education index), oil production (oil
dummy variable) and risk of expropriation (expropriation index) we find not
statistically significant relationships between three independent variables (capital
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stock per worker, the education index and the oil dummy variable) and the dependent
variable (ratio of the value-added share of the relatively capital-intensive industries to
the value-added share of the relatively labor-intensive
Table 2.3Estimating Rybczynski effects for a sample o f 16 developing countries
Dependent variable: LN(value-added share o f the relatively capital-intensive industries / value-added share o f the relatively labor-intensive industries)
Independentvariable (1) (2) (3) (4) (5) (6) (7) (8)
Constant 0 .514 0.558 0.292 -0 .208 0.146 -0 .208 0.262 0 .329LN(Capital stock per worker)
-0 .030(0 .069)
-0.041(0 .190)
-0 .006(0 .071)
0 .116(0 .197)
0 .040(0 .081)
0 .116(0 .197)
0.051(0 .103)
0 .155(0 .228)
LN(Expropriation Index)
0.201(0 .099)**
0.265(0 .108)**
0 .210(0 .099)**
0 .265(0 .108)**
0 .154(0 .115)
0 .230(0 .127)*
Oil(OIL)
-0 .097(0 .081)
. --------------------- -0 .101(0 .086)
---------------------
LN(Education)
-0 .067(0 .341)
-0 .370(0 .442)
R 2 0 .006 0 .006 0.011 0.018 0.023 0.018 0 .030 0 .059
Number o f observations
127 127 127 127 127 127 106 106
Estimationmethod
Randomeffects
Fixedeffects
Randomeffects
Fixedeffects
Randomeffects
Fixedeffects
Randomeffects
Fixedeffects
Hausman Test (Prob.Rejecting Ho)
0.9492 0.2472 0.3239 0.4065
*** = significant at the 0.01 level, * = significant at the 0.10 level** = significant at the 0.05 level, Standard error in parenthesisHo (Hausman Test): Difference in coefficients not systematic
industries) while the relationship between the expropriation index and the dependent
variable remains positive and statistically significant in the fixed-effects method.
Overall R-square increases from 0.018 to 0.059 for the random effects model and
from 0.011 to 0.030 for the fixed effects model. The increase in R-square implies that
the education level plays an important role in explaining variations in the ratio of the
value-added share of the relatively capital-intensive industries to the value-added
share of the relatively labor-intensive industries.
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Results from the panel data regressions are consistent with the theoretical
results obtained in the analytical section of the paper. Controlling for the level of
education (education index), oil production (oil dummy variable) and risk of
expropriation (expropriation index), a not statistically significant relationship obtains
between capital stock per worker and the dependent variable (value-added share of
the relatively capital-intensive industries to the value-added share of the relatively
labor-intensive industries) for a sample of 16 developing countries while a positive
and statistically significant relationship obtains between capital stock per worker and
the dependent variable (value-added share of the relatively capital-intensive industries
to the value- added share of the relatively labor-intensive industries) for a sample of
22 developed (OECD) countries. The not statistically significant relationship
obtained for the 16 developing countries implies a statistically not significant rate of
change in the ratio of capital-intensive output to labor intensive output caused by a
change in the capital stock per worker, while the statistically significant relationship
obtained for the 22 developed countries implies a positive and significant rate of
change in the ratio of capital intensive output to labor intensive output caused by a
change in the capital stock per worker.
The positive and statistically significant relationship between the
expropriation index and the dependent variable (ratio of the value-added share of the
relatively capital- intensive industries to the value-added share of the relatively labor-
intensive industries) for both developed and developing countries implies that risk of
expropriation in a country plays a significant role in explaining the variation in the
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ratio of the value-added share of the relatively capital-intensive industries to the
value-added share of the relatively labor-intensive industries.
Finally we conduct robustness checks regarding our measures of industry
classification and our measures of education level. To check our measure of industry
classification, we ran panel data regressions using the eight most factor intensive
industries instead of the 14 most factor intensive industries used in our analysis
above.
Our results did not change qualitatively. For developing countries, the
relationship between capital stock per worker and the dependent variable (ratio of the
value-added share of the relatively capital-intensive industries to the value-added
share of the relatively labor-intensive industries) remained negative and statistically
not significant while the relationship between the expropriation index and the
dependent variable (ratio of the value-added share of the relatively capital-intensive
industries to the value-added share of the relatively labor-intensive industries),
remained positive and statistically significant. For OECD countries, the relationship
between both independent variables (capital stock per worker and the expropriation
index) and the dependent variable (ratio of the value-added share of the relatively
capital-intensive industries to the value-added share of the relatively labor-intensive
industries) remained positive and statistically significant.
To check our measure of education level, we ran panel data regressions using
separate gross enrolment ratios for each of the primary, secondary and tertiary levels
of education. Our results did not change qualitatively. For developing countries, the
relationship between capital stock per worker and the dependent variable (ratio of the
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value-added share of the relatively capital-intensive industries to the value-added
share of the relatively labor-intensive industries) remained negative and statistically
not significant while the relationship between the expropriation index and the
dependent variable (ratio of the value-added share of the relatively capital-intensive
industries to the value-added share of the relatively labor-intensive industries),
remained positive and statistically significant. For OECD countries, the relationship
between both independent variables (capital stock per worker and the expropriation
index) and the dependent variable (ratio of the value-added share of the relatively
capital-intensive industries to the value-added share of the relatively labor-intensive
industries) remained positive and statistically significant. So we can conclude that
our panel data regressions results are robust to changes in both our measures of
industry classification and education level.
2.5 Concluding Remarks
From our model developed above, we were able to provide an explanation for cross
country differences in the ratio of capital-intensive output to labor intensive output
(for countries with similar capital stock per worker). We were also able to provide an
explanation as to why certain developing countries (higher incidence of corruption
and higher risk of expropriation) are slower in climbing the ladder of comparative
advantage that a country will climb as it accumulates capital relative to the world. In
the second part of the paper we used a sample of 16 developing countries, 28
manufacturing industries and eight years, to find evidence contradicting the
predictions of Rybczynski theorem. Our empirical findings suggest that Rybczynski
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predictions could seize to hold in countries with high incidence of corruption and
high risk of expropriation. It is important to mention that the model developed in this
paper is intended to explain cross country differences in the ratio of capital-intensive
output to labor intensive output and is not intended to explain the contradiction in the
predictions of Rybczynski theorem arri ved at in the empirical part of the paper.
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Chapter 3
Contests with a Generalized Difference Form*
3.1 Introduction
The literature on rent- seeking games has expanded quite rapidly in the past two
decades. According to Anderton (2001), one of the necessary building blocks of a
unifying micro-theory of conflict economics is the contest success function (CSF),
which specifies how the appropriative efforts of agents lead to an appropriative
outcome. To date, two families of CSFs have been developed. The first family of the
CSF comes from the Tullock (1980) rent-seeking game in which a contestant’s
winning probability depends on the ratio of fighting efforts. In the second family of
success functions, called “ difference-form” success functions, a contestant’s
probability of winning depends upon the difference of fighting efforts (Hirshleifer
1995).
The game-theoretic rent-seeking model studied by Tullock (1980) marked a
starting point for numerous studies on the subject. Perez-Castrillo and Verdier (1992)
stressed the importance of the shape of the players’ reaction curve in order to
* This chapter is co-authored with Stergios Skaperdas.
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understand the impact of the technology4 of rent-seeking on the structure of the
outcome of the game. Their findings indicate that, in the case when the rent-seeking
technology displays constant or decreasing returns to scale5, the reaction curve of the
agent is continuous in the bets of the other agents, increasing at first when the outside
competition is weak and decreasing continuously as the outside competition
increases. In the case of increasing returns to scale, a sharp discontinuity in the
reaction curve obtains. This discontinuity is essentially due to the nonconvexity of
the profit function of the agent. Based on the type of technology of rent-seeking, the
authors also characterize the type of pure strategy equilibria that can result in the
game. With constant or decreasing returns to scale and for a fixed number of agents,
there exists a unique Nash equilibrium which is symmetric. With increasing returns
to scale, if the number of agents is not too large, there also exists a unique symmetric
Nash equilibrium. However, if the number of agents is too large, then there exists a
multiplicity of equilibria, which are asymmetric with some agents devoting the same
amount of resources to rent-seeking and the remaining agents remain inactive.
Nitzan (1994) surveys alternative ways of modeling rent seeking contests,
focusing primarily on the relationship between the extent of rent dissipation6 and the
underlying contest characteristics: for example the number of players, the degree of
4 Following Tullock (1980) contestant i ’s probability o f winning a contested prize isN
P j = e ' ! { e \ + , w hen contestants) = 1, ,N expend “effort” C - > 0 . A ccording to Perez-7=1j * i
Castrillo and Verdier (1992), r > 0 characterizes the returns o f scale o f the technology o f rent- seeking.5 When r < 1 the technology o f rent-seeking may be considered with decreasing returns o f scale while when r > 1 the technology is with increasing returns.6 Rent dissipation is defined as the ratio between total rent-seeking outlays in equilibrium and the value o f the contested rent.
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asymmetry between the players, and the nature and source of the rent. He showed that
the extent of rent dissipation is increasing in the number of rent seekers and in the
marginal return to lobbying outlays.
Hirshleifer (1989), studied difference-form contests with two contestants,
obtaining results that are different from those of the Tullock game. Hirshleifer
pointed out that a crucial flaw of the traditional ratio model is that neither one-sided
submission nor two-sided peace between the parties can ever occur as a Cournot
equilibrium. In contrast, both of these outcomes are entirely consistent with a model
in which success is a function of the difference between the parties' resource
commitments. Che and Gale (2000) characterized equilibria for all parameter values
for a particular class of difference-form contest success function, namely the
piecewise linear function. In their work, they find similarities between general
difference-form contests and all-pay auctions.
Skaperdas and Vaidya (2005) propose a general class of “difference-form”
contests for settings in which rent-seeking involves persuasion. Examples of such
settings include litigation, advertising, lobbying, electoral campaigning or
argumentation in policy debates where contending parties expend resources to
persuade an audience of the correctness of their view. They examine how the
probability of persuading the audience depends on the resources expended by the
parties, so that persuasion can be modeled as a contest. In the present work we
attempt to explore the properties and implications of the functional form proposed by
Skaperdas and Vaidya (2005) by providing a complete characterization of the
players’ reaction functions and the pure strategy equilibria. We also discuss the
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relationship between the extent of rent dissipation and the underlying contest
characteristics: the nature of the contested rent, the number of players, the cost per
unit of persuasion activity, and the force of the evidence.
Our results for the case of two players show that when the persuasion function
is symmetric, the reaction curve of each agent is independent of the efforts of the
other agent and the non-cooperative (persuasion) equilibrium is symmetric in the
equilibrium efforts expended by both agents. Moving to more complicated cases with
asymmetric cost functions and asymmetric contestable rents, the non-cooperative
persuasion equilibrium is asymmetric in the equilibrium efforts expended by both
agents. The agent with the relatively lower cost of persuasion per unit of her
valuation of the contestable rent expends greater effort in equilibrium. The increase
in the agent’s equilibrium effort is due to the concavity of the evidence production
function and to the decrease in her marginal cost relative to the marginal cost of the
other agent. With asymmetric evidence production functions, the non-cooperative
persuasion equilibrium is also asymmetric in the equilibrium efforts expended by
both agents, with the agent on the side of the truth expending greater persuasion effort
in equilibrium. In this case, the agent’s greater equilibrium persuasion effort is due to
the increase in her relative advantage in the production of evidence at the margin.
We also characterize the type of pure strategy equilibria that may result in the
game. First we discuss the case where the total number of agents potentially
interested in persuasion is fixed to some number, which can exceed 2, and the
persuasion function is symmetric. If the number of agents is such that the positive
profits condition is satisfied then there exists a unique Nash Equilibrium which is
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symmetric. However, if the number of agents is too large and the positive profits
condition is no longer satisfied, then there exists a multiplicity of equilibria, which
are asymmetric. Some agents devote the same amount of resources to persuasion and
the remaining agents choose not to participate. In contrast to the literature surveyed
by Nitzan (1994), we show that the extent of rent dissipation is non-monotonic in the
number of rent seekers N. We also show that the extent of rent dissipation is non
monotonic in the contestable rent and in the cost per unit of persuasion activity while
increasing in the force of the evidence presented to a third party audience. The
greater the force of the evidence presented to the third party audience, the greater the
return to the resource investment by both contestants and the higher is the amount of
persuasion effort expended by all parties in equilibrium.
Finally we examine the case when the persuasion function is asymmetric. Our
results show that the reaction curve of our agent is determined by the amount of effort
expended by the other agent and by the degree of asymmetry between the likelihood
ratios of judgment held by the third party audience. The reaction curve of each agent
is continuous in the efforts of the other agent with the reaction curve of the favored
agent increasing continuously as the outside competition increases while the reaction
curve of the other agent decreases continuously as the outside competition increases.
The paper is organized as follows. Section 2 describes an alternative to the
Tullock functional form proposed by Skaperdas and Vaidya (2005). Section 3
outlines the model under symmetry and characterizes the non-cooperative pure
strategy equilibrium with two agents. Sections 4 through 8 focus on the symmetric
7 One agent is favored in terms o f the force o f the evidence presented to the third party audience and in terms o f the negative bias in judgm ent b y the third party audience.
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case and characterize: the non-cooperative pure-strategy equilibrium with asymmetric
cost functions and asymmetric contestable rents, the non-cooperative persuasion
equilibrium with asymmetric evidence production functions, the non-cooperative pure
strategy equilibria with N agents, persuasion with a fixed number of agents N, and the
relationship between the extent of rent dissipation and the underlying contest
characteristics. Section 9 examines the non-cooperative persuasion equilibrium with
asymmetry. Section 10 concludes.
3.2 Persuasion Function as an Alternative to the Tullock
Functional Form
To lay out the building blocks of persuasion, Skaperdas and Vaidya (2005) examine
an evidence production process. Two players, player 1 and player 2, compete to
gather and present evidence so as to influence the verdict of a third party audience in
their favor. With discrete evidence production, Player 1 can either produce evidence
in her favor denoted by Ex, or offer no evidence, denoted by { }. Similarly, Player 2
can either produce evidence in her favor, denoted by E2, or offer no evidence, { }.
The production of such evidence is not deterministic. The amount of resources
enhances the probability of finding a favorable piece of evidence. The authors let
h(r}) denote the probability that player 1 will find evidence in her favor. This
probability is increasing in rx, the resources expended on finding that evidence.
Similarly h(r2) denotes the probability that player 2 will find evidence in her favor,
with that probability also increasing in the resources r2 expended by the player. Thus
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in terms of evidence there are four possible states of the world that can be faced by
the third party audience: (EX,E2),(EX,{ }),({ },E2),and({ },{ }) occurring with the
following probabilities: h(rx )h{r2), h(rx )[1 - h(r2)], [1 - h{rx )]h(r2), and
[1 - h(rx )][1 - h(r2)] respectively. Given the posterior probability of player 1 winning
(and of player 2 losing) that will be induced by each realized combination of evidence
and given the function h(.), the ex ante probability of player 1 winning (and of player
2 losing) can be straightforwardly calculated:
where n represents the third party audience’s posterior probability of winning for
player 1 and n represents the third party audience’s prior. T and 6 are defined
according to the restrictions below:
P\ Oi,r2) = Krx)h(r2 (Ex, E2) + h(rx)[1 - h(r2)]x*(Ex,{ })
+ [1 - h ( r xm r 2) n \ { },E2) + [1 - h{rx)}[\ - h{r2)}n ({ },{ })(1)
Equation (1) above can be rearranged as:
Pl(rx,r2) = n + n[(T- 1)h(rx)-(1 - S)h(r2)] + [n (Ex,E2) + (1 - S - T)jc]h(r,)h(r2) (2)
n (<(), (j)) = k \ (3)
n (ff),E2) = 8n for some 8 e (0,1);
whereT > 1
Skaperdas and Vaidya (2005) also examine equation (2) above in the case
with symmetry when a = l - S = T - 1 , 7r*(j>,fi) = 7t*(Ex,E2) = 7r, and n
probability that player 1 wins then takes the following simple form:
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P\(rv ri) ~ n +om[h(r]) - h ( r 2)] 1 > a > 0, (4)
where rx, r2, and h{.) are as defined above, a represents the force of the evidence
(the higher is the force of the evidence the higher is the contestant’s probability of
winning the contest).
3.3 The symmetric case
Using the symmetric form of the persuasion function that was derived in Skaperdas
and Vaidya (2005) and explained in section 2 above, we consider the basic rent-
seeking contest with 2 contending parties confronting the opportunity of winning a
fixed prize, the contestable rent, X . The contending parties expend resources to
persuade a third party audience of the correctness of their view. The probability of
persuading the audience depends on the resources expended by the parties so that the
probability that agent 1 wins takes the simple form:
where the parameter a and the probability function h(r) are as defined in equation
(4) above. We also make the following assumptioms:
(5)
h :U c: 91 —» 91 is C2 ,i.e. h is continuous with h (r) < 0 for all r e U .
The expected profit of agent 1 can then be written as:
V' (r,) = ! i + a i [A(r,) - H r , )] }X - cr, (6)
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where c represents the cost per unit of persuasion activity and X , as previously
defined, represents the contestable rent. The agent’s decision problem then becomes
to maximize her expected profit V 1 taken r2 as given and under the constraint r > 0 .
The first order condition, at an interior optimum yields:
n = < i > Y A (7)a X
We note that r* represents the optimal effort chosen by agent 1. Proposition
1 below characterizes her best response function:
Proposition 1
Two cases are possible:
For 0 < r2 < G(c ,a ,X) , r* is strictly positive for and determined by the first order
condition (7).
For r2 > G(c,a ,X) , r*= 0.
where G(c, a, X ) =hTx[h{r\)Jt - ----- —— r,*] and h \ r ) < 0.a a X
Proof: See Appendix.
Proposition 1 states, player 1 will choose to invest in persuasion activities
determined by the first order condition (7) only if the persuasion effort exerted by
player 2, r2, is less than a certain level given by G(c ,a ,X ). If player 2 chooses a
greater level of effort, r2 > G (c ,a ,X ) , player 1 will choose not to participate in
persuasion activities and her best response function r* equals zero. The point
G(c,a ,X) marks the point of complete dissipation of the rents.
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From Proposition 1, we can see the shape of the reaction function r *, as
illustrated in Figure 3.1 below. As Figure 3.1 shows, the best response of player 1 r*
falls discontinuously from a positive value determined by the first order condition (7)
to zero as the effort exerted by the other player r2, passes through the threshold value
of G(c,a ,X) .
K
F ig , 3. J Player 1 rs reaction curve
Proposition 2
A multiplicity of Nash equilibria are possible in this case. One possible Nash
equilibrium is such that both agents invest the same amount of persuasion effort r* in
persuasion given by equation (7) above. A second possible Nash equilibrium is such
that both agents invest zero persuasion effort in persuasion.
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A necessary condition for the existence of the first Nash equilibrium (where
« Xboth agents invest the same amount of persuasion effort r ) is that < — and
2c
* Xr2 < — . The equilibrium expected profit of agent 1 involved in persuasion is then:
2c
K = l \ x + ~ X h ( r ; ) - ~ X h(r;)]-cr; (8)
V ' X x - c r (9)
while agent 2’s expected profit takes the following form:
v;=[±x+jXh(r;)-~xh(r;)]-Cr; m
V2* = l x - c r * (11)
Proof: See Appendix.
Figure 3.2 below maps a possible Nash equilibrium of the 2-player game
described in proposition 2 above. We can see from the figure that the total cost of
persuasion effort expended in equilibrium 2 cr* cannot exceed the amount of the
contestable rent X .
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*
X2c ■ 2
F ig , 3 .2 Non-cooperative pure strategy equilibrium
The analysis in sections 4 though 8 below will focus on the symmetric case
discussed above. Section 9 will examine the non-cooperative persuasion equilibrium
with asymmetry when 1 - 6 ^ T -1 and n *(<!>, (/>) * n*(Ev E2) .
3.4 Non-cooperative equilibrium with asymmetric cost
functions and asymmetric contestable rents
We consider the case with two agents under asymmetric cost structures. Agent 1 ’s
cost function is C, (r,) = c,r\ while agent 2’s cost function is C2 (r2) = c2r2 with
Cj ^ c'2 • We will also consider that agent 1 ’s interpretation of the contestable rent X
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differs from agent 2’s and are given as X ] and X 2 respectively. The expected profit
of agent 1 may be then written as:
>/V,) = <i + f [A(r,)-A(r2)]}.r, - c , ,„ (12)
where agent’s 1 decision problem is to maximize her expected profit V 1 taking r2 as
given and under the constraint r > 0 . The first order condition yields:
r,'=(A')-,(J^ -) , 03)a X x
We note that represents the optimal effort chosen by agent 1. Similarly for
agent 2:
(14)a X 2
Proposition 3
A multiplicity of Nash equilibria are possible in this case. One possible Nash
equilibrium is such that the 2 agents invest different amounts of persuasion effort
given by their optimal efforts in equations (13) and (14) above. A second possible
Nash equilibrium is such that one agent exerts zero effort while the second agent
exerts positive effort given by her first order condition in equations (13) or (14)
above. The third possible Nash equilibrium is such that both agents invest zero
persuasion effort.
Necessary conditions for the existence o f such equilibria are
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] which ensures
V'x , F2* > 0. The equilibrium expected profits to each agent involved in persuasion
are:
Proof: See Appendix.
Figure 3.3 below maps one possible Nash equilibrium of the 2-player game
described in proposition 3 above. The non-cooperative persuasion equilibrium is
asymmetric in the equilibrium efforts expended by both agents. The agent with the
relatively lower cost of persuasion per unit of her own interpretation of the
contestable rent expends greater persuasion effort in equilibrium.
V* = ^ - [ l + a h(r*) - a h{r *)] - c,r/ (15)
(16)
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c cF ig , 3 ,3 Non-cooperative pure strategy equilibria -
3.5 Non-cooperative equilibrium - with asymmetric
evidence production functions
We examine a setting where two players, player 1 and player 2, compete to gather
and present evidence so as to influence the verdict of a third party audience in their
favor. When player 1 is on the side of the truth, evidence production by player 1 and
player 2 can be determined by the following functions:
e, = 6 h{rx) (17)
e 2 = 0 -6 )A (r2) (18)
where,
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- < 9 < 1 (19)2
where e, and e2 represent evidence production for player 1 and player 2 respectively.
The function /?(.) is as defined in (5) above while the parameter 6 captures the fact
that the truth does matter in the production of evidence. Given, for example, that
6 >1/2, if both parties devoted an equal amount of effort to gather and present
evidence to the court, the outcome would favor player 1.
The expected profit to player 1 may be then written as:
V'' (r, ) = { + j [e Hr, ) - (1 - 6 ) h(n)] }X - cr,, (20)
where player 1 ’s decision problem is to maximize her expected payoff F 1 taking
r2 > 0 as given and under the constraint rx > 0. The first order condition yields:
(21)aO X
We note that r * represents the optimal effort chosen by player 1. Similarly
for player 2:
(22)
Proposition 4
A multiplicity of Nash equilibria are possible in this case. One possible Nash
equilibrium is such that the 2 agents invest different amounts of persuasion efforts
given by their optimal efforts in equations (21) and (22) above. A second possible
Nash equilibrium is such that one agent exerts zero effort while the second agent
exerts positive effort given by her first order condition in equations (21) or (22)
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above. The third possible Nash equilibrium is such that both agents invest zero
persuasion effort in persuasion. The parameter 6 captures the fact that the truth does
matter in the production of evidence with player 1 being favored in this case.
Necessary conditions for the existence of such equilibrium are
r; <[gh(r; ) - ( \ - e ) h ( r;) + - \ [ C \ and r2* < [ 0 - e)ACr2")- 6>A(V> + —a 2c a 2c
which ensures F,*, V*2 > 0. The equilibrium expected profits to each agent involved
in persuasion are:
K =[^x+^exh(r;)~(\~e)x h(r;)\-cr; (23)
K = l ^ X + 0 X H r ; ) - ^ ( l - 0 ) X h ( r ; ) ] - cr; (24)
Proof: See Appendix.
Figure 3.4 below maps a possible Nash equilibrium of the 2-player game
described in proposition 4 above. The non-cooperative persuasion equilibrium is
asymmetric in the equilibrium efforts expended by both players with the player on the
side of the truth expending greater persuasion effort in equilibrium. The increase in
the equilibrium persuasion effort exerted by the player is due to the increase in her
marginal benefit relative to the marginal benefit of the other player.
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1 CY. X[(1 - & ) k ( r ' ) - d h ( ^ ) + — ][—
a 2 c
F ig , 3 ,4 Non-cooperative pin e strategy equilibria - (l - &) < &
3.6 Non-cooperative pure strategy equilibria with N agents
With N contending parties confronting the opportunity of winning a fixed prize X ,
we conjecture that the probability that agent i wins depends on the difference between
her effort, rt , and the mean value of the efforts expended by all other agents, r . ,
j ^ i and it takes the following form:
7=1i*jN ~ 1
(25)
h{.) is increasing in r and the other variables and functions are similarly
defined to those in (5). The expected profit of agent i may be then written as:
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Where the agent’s decision problem is to maximize her expected profit V
taking the effort exerted by other agents r . , j i as given and under the constraint
r . > 0 . The first order condition yields the optimal persuasion effort chosen by agent
Proposition 5 below characterizes agent i’s best response function:
Proposition 5
Two cases are possible:
For r < G ( N , c , a , X ) , then r* is strictly positive for ry >0 and determined by the
first order condition (27).
For r > G( N, c ,a , X) , then r*= 0.
Proof: See Appendix.
Proposition 6
Nash equilibria with N agents are such that all agents invest the same amount
of persuasion effort r in persuasion. One possible Nash equilibrium is such that all
i :
(27)
where G(N, c ,a , X) =h l [h(r*)+-
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agents invest the same amount of effort given by equation (27) above. The other
possible Nash equilibrium is such that all agents invest zero effort.
XA necessary condition for the existence of such an equilibrium is r < ----- .
N c
The equilibrium expected profit to each agent involved in persuasion is:
V* = [ ^ + ^ h ( r * ) - — h(r*)]-cr* (28)N N N
V* = - - c r * (29)N
Proof: See Appendix.
Similar to section 3 above, the total cost of persuasion effort expended in
equilibrium N cr* cannot exceed the amount of contestable rent X .
3.7 Persuasion with a fixed number of agents N
In this section we discuss how the total number of agents N(> 2) potentially
interested in persuasion can affect the symmetry of the Nash equilibrium in pure
strategies.
Proposition 7
Two cases are possible.
a. If — > N(h (-^-^-) > 0 , then there is a unique Nash equilibrium in which all thec a X
N agents participate in persuasion activity with each of them investing
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N cr - (h y l (----- ) . The equilibrium expected profit for each agent is equal toa X
v‘ =^--(c)(h'r'A> o.N a X
b. If — < N(h y ' ( - ^ - ) , let N* be the highest number of agents such that c a X
— > N(h')~x ( - ^ - ) > 0. Then if N* > 1, the Nash equilibria in pure strategies are c a X
asymmetric. There exists an equilibrium with N* agents participating in persuasion
activity and N - N * non-participating agents in which each of the participating
* - _ t N Cagents devotes r - (h y (----- ) of resources to persuasion and receives a profita X
* X . N cV (c)(h ) (----- ) > 0. Each of the non-participating agents invests nothingN a X
and has a zero payoff.
3.8 Non-cooperative equilibrium and the extent of rent
dissipation
The existing rent-seeking literature is concerned with the existence and
characterization of Nash-equilibria and, in particular, with the relationship between
total rent-seeking outlays in equilibrium and the value of the contested rent. The ratio
D between these two values is called the extent of rent dissipation. This ratio is
important, as it measures the resources squandered on the contested rent from its
value [Nitzan (1994)]. Our results show that the extent of rent dissipation is non-
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monotonic in the number of agents N, while in the case of the Tullock function it is
strictly increasing in N.
Proposition 8
When the probabilistic contest success functions are symmetric and of the
form shown in (27) and if an interior Nash equilibrium in pure strategies exists, then
N c . N cthe extent of rent dissipation D = ( )(h ) ( ) < 1. The extent of rent
X cc X
N cdissipation can be decomposed into two multiplicative parts:---- which represents
X
, N cthe total cost of persuasion activity per unit of the contestable rent and (h ) (----- )
a X
which represents the symmetric equilibrium effort (see equation (27)). The extent of
rent dissipation is non-monotonic in the contestable rent X , in the number of rent
seekers N and in the cost per unit of persuasion activity c while increasing in the
parameter a . The extent of rent dissipation is increasing in the number of agents N
and in the cost per unit of persuasion activity c when the symmetric equilibrium effort
is decreasing at a slower rate than the rate of increase in the total cost of persuasion
activity per unit of the contestable rent — and — respectively. Conversely the extentN c
of rent dissipation is decreasing in the contestable rent X when the symmetric
equilibrium effort is decreasing at a faster rate than the rate of decrease in the total
cost of persuasion activity per unit of the contestable rent - — . The parameter aX
represents the force of the evidence (the higher is a the higher is the contestant’s
probability of winning the contest). So the higher the a , the greater the return to the
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resource investment by both contestants and the higher is the amount of persuasion
effort expended by all parties in equilibrium.
Proof: See Appendix.
In the appendix, we show that when the rate of decrease in the symmetric
cost of persuasion activity per unit of the contestable rent — then the extent of rent
dissipation D is increasing in the number of agents N and decreasing otherwise.
From a more technical viewpoint, A is a measure of the degree of concavity of the
probability function h( .) . It measures the speed at which the marginal probability of
finding evidence in the agent’s favor is decreasing. When the marginal probability of
finding evidence in the agent’s favor is decreasing at a slower rate than the rate of
increase in the total cost of persuasion activity per unit of the contestable rent ,
then the extent of rent dissipation, D , is increasing in the number of agents N ,
otherwise, it is decreasing in N . A similar relationship can be established between
the cost per unit of persuasion activity c and the extent of rent dissipation D while a
converse relationship can be established between the contestable rent X and the
extent of rent dissipation D.
3.9 Non-cooperative equilibrium - asymmetric case
dj h ' y1 N cdN a X )
equilibrium effort A - - is less than the rate of increase in the total
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With asymmetry, the probability that agent 1 wins takes the following functional
form:
p x(rx,r2) = n + n [(T -1 )h{rx) - (1 - 6 )h(r2)]
+ [x* (Ex, E2) + (1 - S - T)x]h(rx )h(r2),
while agent 2’s probability takes the following form:
p 2^ , r 2) = \ - n + n [ ( l - m r 2) - ( r - m r , ) ]
+ [(<5 + r - \ )x - x \ E x,E2)]h(rx)h(r2),
where T, 6, n and h(.) are as defined in (2) and (3) above. The expected profit of
agents 1 and 2 may be then written as:
V1 (rx) = {x + 4 r -1 )h(rx) - (1 - S)h(r2)]
+ [x (Ex,E2) + { \ - 5 - T ) x ] h ( r x)h(r2) } X - crx,
V 2(r2) = { l - x + [(1 - S)h(r2) - (T -1 )h(rx)]
- [x (Ex,E2) + ( 1-S - F) x]h{rx)h(r2)}X - cr2,
Here each agent i’s decision problem is to maximize her respective expected profit Vj
taking the resources expended by the other agent as given and under the constraint
rt >0.
The best response of agent 1 to a given effort r2 exerted by agent 2 is:
r; = ( h ' y1 (------------------- ;--------- >, (34)W r-i)i+[/(£1,£2>+(i-j-r>p(/'!)]
Similarly for player 2, we have
r; = ( h ' y x(------------------- ; ), (35)[^(1 - S ) X - [x (Ex, E2) + (1 - 5 - T)x]Xh(rx)]
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Note that the term [ft* (.£',, /:,) + (1 - 8 - F)/r] is positive providing agent 1
with the advantage in the persuasion contest. We can rewrite the term
[ft* (El ,E1) + ( l - S - T);r] as:
[* ({ },{ } ) - * ( { } ,£ ,) ] - [ * (£ „ { } ) - * (£ „ £ ,) ] (36)
which is positive if the following condition holds:
| * • ( £ „ £ , ) - * • ( £ „ { }) !< | *■•({ } ,£ , ) - » ( { },{ }) | (37)
The above condition is always satisfied since in the early stage of the contest
(when no evidence is presented), the marginal probability increase (that is the extra
contribution) of the evidence E2 presented by agent 2 when no evidence { } has been
presented by agent 1, {ft*({ },E2) ~ ft '({ M }» , is greater than the marginal
probability increase of the same evidence E2 presented by agent 2 when evidence Ex
has already been presented by agent 1, {ft* (Ex, E2) - n* (El , { })}. Proposition 9
below characterizes the agents best response functions:
Proposition 9
The following cases are possible:
r,* is strictly positive and increasing in r2 for r2 >0 and determined by the first-order
condition (34).
For r} < G( c ,X ,n , S , r ) , then r2 is strictly positive and decreasing in r, for r, >0 and
determined by the first order condition (35).
For rx > G(c, X , n , S,T), then r2 = 0.
where G(c,X, f t ,5,T) =h 1 [—----—— ---------------] and h (r)< 0ft (Ex,E2) + ( l - S - T ) f t
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Nash equilibria are defined at the intersection of r* and r*_
Proof: See Appendix.
From proposition 9 above, we can see that agent 2 will choose to invest in
persuasion activities determined by the first order condition (35) only if the
persuasion effort rx exerted by agent 1 is less than a certain level given by
G( c , X ,n , S , T ) . For persuasion effort r, greater than G(c,X,n ,6 ,T) , agent 2’s
marginal return to persuasion becomes negative and she will choose not to participate
in persuasion activities with her best response function r*2 being equal to zero.
Proposition 9 defines the Nash equilibria of the 2-player game and it also
defines the shape of the reaction functions r* and r2 • In figure 3.5 below, the Nash
equilibrium of the game occurs at the point of intersection of the two reaction curves
r* and r2 which is indicated by the point of intersection of the two lightly shaded
curves in the middle of the graph. In figure 3.6 below the Nash equilibrium of the
game also occurs at the point of intersection of the two reaction curves >{ and r2*
• cwhich is indicated by the point of intersection (h ) ' (--------------) at the bottom of the[;r(r - \ ) X
graph.
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r2
Ibts
C r■i7T (Sj.jSj) + (1- 5-r)jr'
Fig. 3,5 Non-cooperative pure strategy' equilibrium - asymmeric example 1
r2
Ib*
T*
C■1n (filt5a)+ ( l-5 - r )? r '
Fig. 3.6 Non-cooperative pure strategy equilibrium - asymmetric example 2
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Proposition 10
Nash equilibria with 2 agents under asymmetry are such that both agents
invest different amounts of effort r in persuasion. One possible set of Nash
equilibria is such that both agents invest different amount of effort satisfying equation
(38) below. The other possible set of Nash equilibria is such that one agent exerts
zero effort while the second agent exerts positive effort given by the first order
condition (36) above.
i = (38)r2 \ ( T - i ) + [ x \ E , , E 2 ) + ( l - S - r>]ft(r: )
Necessary conditions for the existence of such equilibrium are
, - ir 1 c h(r*) cr, < h [------- 1-----------------;— ------------------- r. 1,1 - S 7C(1-S)X h (r,) 7t(\ - S ) X
, -ir 1 - f t c h(rj) c „ .... .r, < n [------------ 1--------------------r-~----------------- r71. The equilibrium expected
S r (T - l) 7r (T - \ )X h (r2) n ( r ~ \ ) X
profit to each agent involved in persuasion is:
V,‘ = x X + ^ p ± - K ( \ - d ) X h ( r " 2) - c r ; (39)h'{rx )
r ; = ( l - x ) X + ? / ^ ± - x ( r - l ) X h ( r ; ) - c r ; (40)h'(r2)
Proof: See Appendix.
Our results show that the reaction curve of agent 1 is determined by the
amount of effort expended by agent 2 and by the degree of asymmetry between the
likelihood ratios of judgment held by the third party audience. The reaction curve of
each agent is continuous in the efforts of the other agent with the reaction curve of
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agent 1 (the agent with the advantage as shown in equations (36) and (37) above)
increasing continuously in the persuasion efforts of agent 2 while the reaction curve
of agent 2 decreases continuously in the persuasion efforts of agent 2.
3.10 Conclusion
In this paper, we have offered a review of the general properties and implications of
the new game-theoretic rent-seeking model proposed by Skaperdas and Vaidya
(2005). We also offered a comparison with the traditional ratio model studied by
Tullock (1980). As argued by Hirshleifer (1989), a crucial flaw of the traditional
ratio model is that neither one-sided submission nor two-sided peace between the
parties can ever occur as a Cournot equilibrium. In contrast, both of these outcomes
are entirely consistent with the new game-theoretic rent-seeking model proposed by
Skaperdas and Vaidya (2005) in which success is a function of the difference between
the parties' resource commitments. We argue that the new proposed functional form
is suitable for a wider range of applications than the traditional ratio model proposed
by Tullock. Examples of such applications include lobbying, military combats,
election campaigns, industrial disputes and lawsuits where one-sided submission and
two-sided peace between the parties can occur as a Cournot equilibrium.
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Appendix
Proof of proposition 1:
The first order condition equation states:
r\ = 1 > a > 0 (A .l)a X
Agents will expend positive effort only in the case when they can achieve positive
expected profits:
rV,*)>0 o { I j r + | j r A ( r 1, ) - | j r A ( r J) - c ^ } > 0 (A.2)
r2 <*"' [*('i’ ) + - ---- ^77'f] (A.3)a a X
G(c,a,X)=h~l [ h ( r ; ) + - - ^ - r ; ] (A.4)a a X
Proof of proposition 2:
The equilibrium expected profit of agent 1 involved in persuasion is:
V'* =[\ X + ~ \ X h ^ ~ c i (A'5)
V * = ^ X - c r (A.6)
similarly agent 2’s equilibrium expected profit takes the following form:
K + j X K r ) - j X K r ) ] - c r (A.7)
V"2 X x - c r (A.8)
Equilibria can exist only if the equilibrium profits of both agents are positive or if
V* > 0 (A.9)
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V*>0 (A. 10)
This is equivalent to the conditions that:
» Xr, < — (A.l 1)
2c
ri < ~ (A. 12)2 c
Proof of proposition 3:
A Nash equilibrium with 2 agents under asymmetric cost functions and
asymmetric contestable rents is such that both agents invest different amounts of
effort r* in persuasion with
r; = 1 > a > 0 (A. 13)a X x
1 > a > 0 (A. 14)a X 2
and
r - = (*■)-' ( S i - ) r ; (A.15)c2X,
The equilibrium equation above characterizes the shape of all possible Nash
equilibria between players 1 and 2 and is determined by the magnitude of
. w r ' ( \ )Ar = --------- —— = (h') — ~) ■ From Hao and Zheng (1998), we know that the' 2 C 2 X <
a X 2
inverse of an increasing (decreasing) monotone function is also increasing
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(decreasing) which implies that (/f )"'(•) is a decreasing function since /?"(.) < 0.
* *V C C V c c
This implies that -v > 1 for —- < —— and -V < 1 for — > ——.X, X 2 r2 X, X 2
Proof of proposition 4:
See proof of proposition 3 above.
Proof of proposition 5:
The first order condition equation states:
N rn = ( h Y ( — ), a >0 (A. 16)
a X
Agents will expend positive effort only in the case when they can achieve positive
expected profits:
7=1
V(r*)> 0 o - + — } - c r * } > 0 (A.17)lN N ' N N - 1, X a X , , a X ,\*j
r~-‘ < h-1 [h(r*)+— - — rf*] (A. 18)a a X
o G(N, c ,a , X) =h~l [h(r*) + - -----——r*] (A. 19)a a X
Proof of proposition 6:
With N agents there can only be symmetric equilibria. This equilibrium can exist
only if the equilibrium profit is positive or if
Y ^ h i r )7=1
{ — ------------’N ' N ' v ' iV N - 1
This is equivalent to the condition that
V = [— + (— )/,(r * ) - ( — X— ------- )] -cr* > 0 • (A.20)
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Xr < -----N c
(A.21)
Proof of proposition 8:
If an interior Nash equilibrium in pure strategies exists, then the extent of rent
N c i _t N cdissipation is D = (-----)(h ) (-) . A necessary condition for the existence of anX ex, X
interior Nash equilibrium in pure strategies is (h ' y1 ( - ^ - ) < which implies thata X N c
the extent of rent dissipation is:
D < (— )(— ) (A.22)X N c
<=> Z) < 1 (A.23)
The change in D with respect to N can be shown by the representation below:
f = < > y < ^ ) + ( ^ ) { - ^ « A y ' ) ( ^ ) } (a .24,dN X a X X d N a X
From equation (A. 16) and from proposition 4 above, it can be easily shown that the
extent of rent dissipation D is increasing in the number of agents N when
>{h'Y1 > - N ((h ' y 1 ) ( - ^ r ) , and decreasing whenN c a X d N a X
t c c*0 <{h )_1 (----- ) < - N ------((h ) 1)(----- ). Similarly we can show that the extent of
a X d N a X
rent dissipation D is increasing in the cost per unit of persuasion activity c when
^ ->(h')~x ( ^ C) > - N — ((h ' y 1)(—— ), and decreasing whenN c a X dc a X
0 < ( h y ] ( - ^ - ) < - N —— ((h ) ' . It can be also shown that the extent of renta X dc a X
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dissipation D is decreasing in the contestable rent X when
X „ 8 • j N c _->{h ) (----- )>X (\h ) )(----- ) , and increasing whenN c a X 8 X a X
a X 8 X a X
Proof of proposition 9:
The first order condition equation states:
r* =( h ' y ' ( --------------------;-------- ), (A.25)W r - l ) J + [^ (E\,E2) + ( l -S - r )7r]Xh(r2)]
Agent 1 will expend positive effort only in the case when:
r* > 0 <=> n ( T - \ ) X + [7t\E\,E2) + ( \ - 8 - Y ) n } X h { r 2) > 0 (A.26)
, _ l r *(T -1 )<=> r2 <h ----------------------------- ]n (E\,E2) + ( l - S - T ) 7 r
(A.27)
Proof of proposition 10:
With two agents there can only be asymmetric equilibria. This equilibrium can exist
only if the equilibrium profits are positive or if
V; = n X + p ± - 7 i { \ - 8 ) X h ( r ; ) - c r ; (A.28)h \rx )
and
v ; = (1 -7C)X + - 7t(T -1) X h(r; ) - cr2 (A.29)h \r2)
This is equivalent to the conditions that
j - l r C K r i ) Cr ,<h [------------1---------------------------------------- r21 (A.30)S r (T - l) n ( T - \ ) X h( r2) tt{T - \ ) X 2
and
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1 h{r•*)1 - S 7 c ( \ - S ) X h (r[ ) - S ) X
•ir\ 1
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(A.31)
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