Corruption and Gouvernance Problem

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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Committee Chair

University of California, Irvine 2007

ii


DEDICATION

To My Mother, Father, Aiman, Yassar, Nancy my beloved wife, and our new Mohammad: The true inspiration in my life

iii


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


Chapter 3: Contests with a Generalized Difference Form 3.1 Introduction

58 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


LIST OF FIGURES Page

Figure 1.1

The game between shareholders and managers

Figure 1.2

Optimal value to owners/ managers (the partners

11

compete in an asymmetric contest for control) Figure 1.3

29

Optimal ownership structure to owners/ managers (the partners compete in an asymmetric contest for control)

Figure 1.4

Optimal value to owners/ managers (the partners compete with the powers of persuasion)

Figure 1.5

30

31

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

Figure 3.3

Non-cooperative pure strategy equilibrium - —L < X\ X 2

72

Figure 3.4

Non-cooperative pure strategy equilibrium -(1-6) <6

75

Figure 3.5

Non-cooperative pure strategy equilibrium - asymmetric

c

c

example 1 Figure 3.6

84

Non-cooperative persuasion equilibrium - asymmetric example 2


84

LIST OF TABLES Page

Table 2.1

Comparison table for Japan and Venezuela

Table 2.2

Estimating Rybczynski effects for a sample of 51

22 OECD countries Table 2.3

35

Estimating Rybczynski effects for a sample of 53

16 developing countries

vii


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


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

ix


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


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 o f 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

xi


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


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 o f government has even a more profound impact on the degree o f 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’

1


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 o f 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 o f 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 o f 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 o f 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 o f 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 o f 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.

2


In the second chapter “Risk of Expropriation and the Rybczynski Theorem,” I find evidence contradicting the predictions of the Rybczynski theorem using a sample o f 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 o f corruption and higher risk of expropriation have a lower ratio of capitalintensive 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 twosided peace between the parties can occur as a Cournot equilibrium. Also in contrast

3


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.

4


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 o f 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

5


managers further their own interest (specifically expropriate shareholders) can take a variety o f 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 o f 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

6


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 o f large and dispersed shareholders represents a point of departure o f the analysis. This point of departure provides some insight into the large crosssectional 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 o f 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 o f 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.

7


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

8


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 o f La Porta, Lopez-de-Silanes, Shleifer and Vishny (1998), Shleifer and Wolfenzon (2002) and Burkart, Panunzi and Shleifer (2003), we

9


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 o f ownership in the public firm while increasing that of dispersed shareholders. The share o f ownership sold to large shareholders is non-monotonic in the legal protection o f 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

10


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.

Private Finn Stage 1;

Stage 2:

/ \ Private F inn Public F inn i i r i 1- a - f i & a Dispersed Large Owners/ Shareholders Shareholders M anagers Large shareholders and owners/managers compete for 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)). 11


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 o f 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 o f 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. 12


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: K ,(em,eb ;a ,f i,j S) = (q(em,els) + a( 1 - q(em, eb))VP(yS) - em

(1)

The net value to large shareholders who own a fraction, f) , of the firm is given by: (2)

The net value to dispersed shareholders is then given as the residual value of the public ownership of the firm ((1 - a - fi): v ds ( e m

,eb;a,fi,}S) = Q . - a - f i ) ( l - q(em, els))VP(}S)

(3)

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

13


get the equilibrium payoffs to the owners/managers V* and to shareholders Vs*. e„ and e*s depend on a , ft and y S , V* and /7
(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

14


maximize their expected payoff from taking their company public. Specifically, they solve: Max(V:{a,f},YS) + { l - r )V;(a,p,y))

(6)

a,p,y

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 o f 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 o f 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 o f the value of the firm. where

o > 0.

(7)

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

15


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 o f 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)em a ' +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 * PV P(yS) = 1 ((1 - 0 ) e ° : + d e i y

(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:

16


^ (l-S X l-a X J L )" _1 - a ^ — VP{yS), em* = ------------------P [(i - e ) + e ( 7^ y f 1- a

(11)

1~ a —Vp(yS), [(i - e ) + e ( - ^ - y f 1- a

(12)

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^-r

V'm{a,p,yS) =-------- ------------------------------- --------------------[(\-

0)

— Vp(yS) •••(13)

+d ( J ^ y f

1- 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 f 1- 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*mand 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


We assume that that the value of the public firm is determined through a CobbDouglas-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 V F 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 ) + S = r^(a,fi,yS) = K l-'i‘r 1^ [ --------------------------^

] - cx<9(l - <9)(1 ------- — --------------------

,

'

]'-'p

(17)

[(i -9)+e(T^ r f 1- 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


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


In deciding to take their company public, three sources of value contribute to the owners/managers payoff: expropriating part o f 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


(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 o f the game, the competition between large shareholders and owners/managers involves owners/managers and large shareholders devoting costly

21


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 o f 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-[del

, Ocrrcl

(18)

with em representing costly efforts exerted by the owners/managers to influence the opinion o f the Board o f Directors, Securities and Exchange Commission or the courts

22


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 o f 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:

23


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<|>aV p)<^ \

(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

-(p0t< 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 )) a,p,y

(23)

Due to the complexity in solving the maximization problem above, we will illustrate by an example as shown below: Example:

24


We assume that that the value of the public firm is determined through a CobbDouglas-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 + J322e 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 o f the constant K depends in part on the value of ^ and both values o f ^ 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


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 o f 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


In legal regimes with weakest shareholder protection, relatively low values of 6 , the value o f 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.

27


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.

28


Optimal Value to Owners/Managers

(Vo*) Optimal Value to Owners/Managers

0.6

(Vf) Value of Private Firm

0.4 P u b lic Firm s

0.2

0.1

0.2

0.3

0.4

0.6

0.5

0.7

0.8

0.9

1

Theta

Figure 1.2: The partners compete in an asymmetric contest for control

29


Optimal Ownership Structure to Owners/Managers 10.9 |

I I

0.7 4

0.6

Owners/Managers (Alpha*)

-

Large Shareholders (Beta*)

0.5

Dispersed Shareholders (1Alpha*-Beta*)

0.4 -f—

p*

0.3

0.2 4—

1

[

0.1

oL 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Theta

Figure 1.3 : The partners compete in an asymmetric contest for control

30


Optimal Value to Owners/Managers 1.2

-

1

-

0.8 (Vo*) Optimal Value to Owners/Managers (Vf) Value of Private Firm

/

0.4 ■

0.2

P u b lic F irm 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

31


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

32


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 o f 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 laborintensive 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 o f 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 HeckscherOhlin (HO) general equilibrium model with factor price equalization to provide empirical confirmation of Rybczynski theorem. Harrigan (1995) use data on

33


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 o f 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

34


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

Y ear

K/L

C/L

JPN

1981

23,345

1.72

JPN

1982

24,614

JPN

1983

25,785

BEI

Country

Year

K/L

C/L

BEI

9.08

VEN

1981

21,635

0.45

5.42

1.70

9.08

VEN

1982

21,603

0.52

5.42

1.70

9.08

VEN

1983

21,606

0.48

5.42

K /L = capital stock per worker C /L = (value added o f capital intensive goods) / (value added o f labor intensive good s) B E I = Bureaucratic efficien cy 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 o f 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 o f 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

35


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 o f the relatively labor-intensive industries. In the case of developing countries, lack o f 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-

36


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 valueadded share o f 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 o f expropriation by corrupt government officials and the ratio of capital-intensive output to labor-intensive output.

2.2

The Model

37


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. Capitalintensive firms maximize the following profit function: M a x ( \ - t x )Px f x ( K x , L x ) - r K x - w L x KX’LX

(1)

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 KX’LX

38


(3)

Labor-intensive firms maximize the following profit function: MaxPY[ ( \ - t ) ( K Y) Y [ L Yt P - rKY - w L Y Ky,Ly

= MOX( 1 ~ t ) PPy [Ky ] P[Ly ]'~fi ~ r K 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 o f 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

W

exchanging (-J—) X for Y and with the factor-price ratio (—) at ABCD. Factor Px r K K proportions in X and Y are indicated by points B and C respectively with ——> —— Lx

Ly

p

at 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 w The above implies that we must have a higher factor-price ratio (—) and thus higher r K L

ratios in both sectors.

39


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 o f the factor ratios in the two sectors, X and Y. This is shown in the identity: ( f ) = ( ^ X ~ L) + ( | L) ( ^ ) J -j

X

(5)

Y

Where, L

"I- Ly —L

K X + K Y =K

Factor K

and —^ and — are weights adding up to unity.

O

Factor L

Figure 2.1: Equilibrium analysis

40


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 Lx

Y

A change in the effective goods-price ratio does not alter the amount of capital K and 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

L

relative to the

weight — . In other words L x has to decrease and LY has to increase. From above, L we know that —- is increasing and since L Y 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 laborintensive 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 —wLx Kx ,PX The first-order conditions are

41


d

° f x ( K

X

x

, L x

)

_

u .

SLX

p

dfx ( K x , L x ) X

SKX dfx ( K x , L x )

w_

MPT... _ MPLX

dLx

7 ~ 8fx ( K x , L x r dKx

(6)

MPK x

Labor-intensive firms solve the following profit maximizing problem: MaxPYf Y( K Y, L y) ~ rKY - wLy the first-order conditions are p dfY( K Y, L Y) _ 1 y ----------------------------------

W

dLy dfY( K y , L Y)

Y

6K y dfY( K Y, L Y)

w_ r

dLy

_ MPLy

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: Lx

P* = r ~ ^ +W^

<8)

P

(9)

r E jL + w h L

Y

Y

42


where X and Y are the outputs of the capital-intensive and labor-intensive goods P r respectively. 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 K

= -

X ClKy

X 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:

1 dn

Ly

Lx

Ky

Kx

71X1 (h+K-m +K) KX Ky Therefore a higher Q T (— T) corresponds to a higher n t (— t ) if and Py

W

K

K

Lx

Ly

only if ——> ——which is always true since good X is more capital-intensive than

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 — PY Q

r

An

W increase 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

43


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 K the 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 laborintensive 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 o f 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-

44


intensive good so much that the output of the labor-intensive good decline. As can be K shown 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 words L L L x has to increase and Ly has to decrease. K K If prices are constant then ( ——) and ( —- ) remain unchanged and from the Lx Ly goods-output ratio shown in equation (10) below, an increase in L x relative to L Y will lead to an increase in the relative output of the capital-intensive good to the labor-intensive good.

Y —

Y

,

/ * ( — ->!)

= ( - ^ - ) -------- - A

T

(1 0 )

K

Y Ly f ALry , i) 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 o f 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

45


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 o f 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 o f 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 o f 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 d eveloped (O E C D ) countries: Australia, Austria, Canada, Denmark, Finland, France, Germany, G reece, Ireland, Italy, Japan, Korea, M exico, Netherlands, N e w Zealand, N orw ay, Portugal, Spain, Sw eden, Turkey, U K , and U S A . 2 List o f d evelop ing countries: Argentina, B olivia, C hile, Colom bia, Ecuador, Guatemala, Honduras, India, Iran, K enya, M alaw i, M orocco, Panama, Peru, Philippines, and V enezuela.

46


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 o f 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 o f 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

47


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 o f 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

48


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 + filL N ( K i /L ,) + fi2LN(EIi) + fi-iLN(OILi) + fiALN(ED Ui ) + u, , where C / L represents the ratio of the value-added share of the relatively capitalintensive 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 o f 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 o f the relatively capital-intensive industries to the value-added share of the relatively labor-intensive industries), both in the random-effects and fixed-effects

49


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 o f 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 o f 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 o f 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.

50


Table 2.2 Estimating Rybczynski effects for a sample o f 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) Independent variable Constant LN (Capital stock per worker)

(1) -0 .6 1 8 0 .2 6 0 (0 .0 7 1 ) ***

(2) -0 .3 1 7 0 .1 9 2 (0 .0 7 8 ) **

LN (Expropr iation Index)

(3) -0 .5 0 2 0 .1 8 3 (0 .0 7 7 ) **

(4) -0 .1 7 2 0 .1 1 2 4 (0 .0 8 4 )

(5) -0 .5 0 2 0 .1 8 0 (0 .0 7 7 ) **

(6) -0 .1 7 2 0 .1 1 2 4 (0 .0 8 4 )

(7) -0 .6 1 3 0 .1 8 6 (0 .1 0 3 ) *

(8) -0 .3 2 8 0.101 (0 .1 1 3 )

0 .2 4 0 (0 .0 9 1 ) ***

0 .2 2 4 (0 .0 9 1 ) **

0 .2 3 9 (0 .0 9 1 ) ***

0 .2 2 4 (0 .0 9 1 ) **

0 .2 2 0 (0 .0 9 7 ) **

0 .2 0 6 (0 .0 9 7 ) **

0 .0 5 6 7 (0 .0 7 7 )

0 .0 5 6 (0 .0 7 9 )

-------0 .1 1 6 (0 .2 6 6 )

Oil (OIL) LN (Education)

R2

0 .3 4 6

0 .3 4 6

0 .4 1 5

0 .4 2 7

0.401

0 .4 2 7

0 .0 5 6 (0 .2 5 8 ) 0 .3 7 4

Number o f observations Estimation method

175

175

175

175

175

175

165

165

Random

Fixed

Random

Fixed

Random

Fixed

Random

Fixed

effects

effects

effects

effects

effects

effects

effects

e ffects

Hausman Test (Prob. Rejecting Ho)

0 .0 3 9 2

0 .5 8 8 6

0.7281

0 .4 0 6

0 .9 7 3 9

*** = significant at the 0.01 level, * = significant at the 0.10 level ** = significant at the 0.05 level, Standard error in parenthesis Ho (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

51


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 o f the relatively labor-intensive industries, both in the random-effects and fixedeffects 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 capitalintensive 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 o f the relatively capital-intensive industries to the valueadded 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 valueadded share o f 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

52


stock per worker, the education index and the oil dummy variable) and the dependent variable (ratio o f the value-added share of the relatively capital-intensive industries to the value-added share of the relatively labor-intensive

Table 2.3 Estimating 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) Independent variable Constant LN (Capital stock per worker)

(1) 0 .5 1 4 -0 .0 3 0 (0 .0 6 9 )

(2) 0 .5 5 8 -0.041 (0 .1 9 0 )

LN (Expropr iation Index)

(3) 0 .2 9 2 -0 .0 0 6 (0 .0 7 1 )

(4) -0 .2 0 8

(5) 0 .1 4 6

0 .1 1 6 (0 .1 9 7 )

0.201 (0 .0 9 9 ) **

0 .2 6 5 (0 .1 0 8 ) **

Oil (OIL)

0 .0 4 0 (0 .0 8 1 )

(6) -0 .2 0 8 0 .1 1 6 (0 .1 9 7 )

(7) 0 .2 6 2 0.051 (0 .1 0 3 )

(8) 0 .3 2 9 0 .1 5 5 (0 .2 2 8 )

0 .2 1 0 (0 .0 9 9 ) **

0 .2 6 5 (0 .1 0 8 ) **

0 .1 5 4 (0 .1 1 5 )

0 .2 3 0 (0 .1 2 7 ) *

-0 .0 9 7 (0 .0 8 1)

R2

0 .0 0 6

0 .0 0 6

0.011

0 .0 1 8

0 .0 2 3

0 .0 1 8

-0 .1 0 1 (0 .0 8 6 ) -0 .0 6 7 (0 .3 4 1 ) 0 .0 3 0

Number o f observations Estimation method Hausman Test (Prob. Rejecting Ho)

127

127

127

127

127

127

106

106

Random effects

Fixed effects

Random effects

Fixed effects

Random effects

Fixed effects

Random effects

Fixed effects

.

---------------------

LN (Education)

0 .9 4 9 2

0 .2 4 7 2

0 .3 2 3 9

---------------------

-0 .3 7 0 (0 .4 4 2 ) 0 .0 5 9

0 .4 0 6 5

*** = significant at the 0.01 level, * = significant at the 0.10 level ** = significant at the 0.05 level, Standard error in parenthesis Ho (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. 53


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 o f 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 laborintensive 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

54


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 o f 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

55


value-added share of the relatively capital-intensive industries to the value-added share o f 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 o f 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

56


predictions could seize to hold in countries with high incidence of corruption and high risk o f 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.

57


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.

58


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 o f 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 is N P

j

=

e '

! { e \

, w h en con testa n ts) = 1 ,

+

,N expend “effort”

C

-

>

0 . A ccord in g

to P erez-

7=1 j* i

Castrillo and Verdier (1992), r > 0 characterizes the returns o f scale o f the technology o f rentseeking. 5When 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.

59


asymmetry between the players, and the nature and source of the rent. He showed that the extent o f 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 o f the correctness of their view. They examine how the probability o f 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

60


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 o f 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 o f 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

61


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 o f 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 o f 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 term s o f the force o f the ev id en ce presented to the third party a u d ien ce and in term s o f the n egative b ias in ju d gm en t b y the third party audience.

62


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

63


in terms of evidence there are four possible states of the world that can be faced by the third party audience: (EX,E 2),(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 o f 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 o f player 2 losing) can be straightforwardly calculated: P\ Oi,r2) = Kr x)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)

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: 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:

64


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 o f 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 rentseeking contest with 2 contending parties confronting the opportunity o f 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:

(5)

where the parameter a and the probability function h(r) are as defined in equation (4) above. We also make the following assumptioms: h :U c: 91 —» 91 is C 2 ,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)

65


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 aX

(7)

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 aX 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 o f complete dissipation of the rents.

66


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 o f 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.

67


A necessary condition for the existence of the first Nash equilibrium (where both agents invest the same amount of persuasion effort r ) is that

«

X < — and 2c

* X r2 < — . 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'Xx-cr

(9)

while agent 2 ’s expected profit takes the following form:

v; = [ ± x + j X h ( 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 .

68


*

X

2c

F ig, 3 .2

■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 o f the contestable rent X

69


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,) =
(12)

[A(r,)-A(r2)]}.r, - c ,, „

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)

aXx

We note that

represents the optimal effort chosen by agent 1. Similarly for

agent 2: (14)

aX2 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

70


] which ensures

V'x , F2* > 0. The equilibrium expected profits to each agent involved in persuasion are: V* = ^ - [ l + a h(r*) - a h{r *)] - c,r/

(15)

(16) 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 o f persuasion per unit of her own interpretation of the contestable rent expends greater persuasion effort in equilibrium.

71


F ig, 3 ,3 Non-cooperative pure strategy equilibria -

c

c

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: (17)

e, = 6 h{rx) e2

(18)

= 0 - 6 ) A ( r 2)

where,

72


-<9< 1 2

(19)

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)

73


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^ \ a 2c

and r2* < [ 0 - e)ACr2")- 6>A(V> + — 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.

74


1 CY. X [(1 - & ) k ( r ' ) - d h ( ^ ) + — ][— a 2c

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=1

i*j

(25)

N~ 1

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: 75


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 i:

(27)

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.

where G ( N , c , a , X ) =h l [h(r*)+-

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

76


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. X A necessary condition for the existence of such an equilibrium is r < ----- . Nc The equilibrium expected profit to each agent involved in persuasion is: V* = [ ^ + ^ h ( r * ) - — N N N

h(r*)]-cr*

(28)

V* = - - c r * N

(29)

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 c

(-^-^-) > 0 , then there is a unique Nash equilibrium in which all the aX

N agents participate in persuasion activity with each of them investing

77


Nc r - (h y l (----- ) . The equilibrium expected profit for each agent is equal to aX

v‘ =^--(c)(h'r'A> o. N

aX

b. If — < N(h y ' ( - ^ - ) , let N* be the highest number of agents such that c aX — > N(h')~x( - ^ - ) > 0. Then if N* > 1, the Nash equilibria in pure strategies are c aX asymmetric. There exists an equilibrium with N* agents participating in persuasion activity and N - N * non-participating agents in which each of the participating * NC agents devotes r - (h y (----- ) of resources to persuasion and receives a profit aX _t

V

*

X N

. Nc (c)(h ) (----- ) >0. Each of the non-participating agents invests nothing aX

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-

78


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 the extent o f rent dissipation D = (

Nc . Nc )(h ) ( ) <1. The extent of rent X cc X

Nc dissipation can be decomposed into two multiplicative p arts:---- which represents X , Nc the total cost o f persuasion activity per unit of the contestable rent and (h ) (----- ) aX

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 o f the contestable rent — and — respectively. Conversely the extent N c o f 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 a X represents the force o f 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

79


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

equilibrium effort A - -

djh'y1 Nc ) dN aX

is less 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 and decreasing otherwise. From a more technical viewpoint, A is a measure of the degree of concavity o f 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 o f 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 o f 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 o f rent dissipation D.

3.9 Non-cooperative equilibrium - asymmetric case

80


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: V 1(rx) = {x + 4 r -1 )h(rx) - (1 - S)h(r2)] + [x (E x, 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 (E x, E2) + ( 1 - S - F ) x ] h { r x)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 ' y 1(------------------- ;---------

>,

(34)

r; = ( h ' y x(------------------- ; ), [^(1 - S ) X - [x (Ex, E 2) + (1 - 5 - T)x]Xh(rx)]

(35)

W r-i)i+ [/(£ 1,£2>+(i-j-r>p(/'!)]

Similarly for player 2, we have

81


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 o f 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)< 0 ft (Ex, E2) + ( l - S - T ) f t

82


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 o f 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 o f 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* • c which is indicated by the point of intersection (h ) ' (--------------) at the bottom of the [;r(r - \ ) X graph.

83


r2

I b

ts

C

■i

7T (Sj.jSj) + (1- 5-r)jr'

r

Fig. 3,5 Non-cooperative pure strategy' equilibrium - asymmeric example 1

r

2

I b

*

T*

■1

C

n (filt5 a) + ( l- 5 - r ) ? r '

Fig. 3.6 Non-cooperative pure strategy equilibrium - asymmetric example 2

84


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

r2

=

\ ( T - i ) + [ x \ E , , E 2) + ( l - S - r>]ft(r: )

(38)

Necessary conditions for the existence of such equilibrium are ,-ir 1 c h(r*) c r, < h [------- 1-----------------;— ------------------- r. 1, 1 - S 7C(1-S)X h (r,) 7t(\ - S ) X , -ir 1 - f t c h(rj) c „ .... . [------------ 1--------------------r-~----------------- r7 1. The equilibrium expected S r ( T - l) 7 r ( T - \ ) X h (r2 ) n ( r ~ \ ) X

r, < n

profit to each agent involved in persuasion is: V,‘ = x X + ^ p ± - K ( \ - d ) X h ( r " 2) - c r ; h'{rx )

(39)

r ; = ( l - x ) X +? / ^ ± - x ( r - l ) X h ( r ; ) - c r ; h'(r2)

(40)

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 o f 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

85


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.

86


Appendix Proof of proposition 1: The first order condition equation states:

r\ =

1> a > 0

aX

(A .l)

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’ ) + -a----a^77'f] X

(A.3)

G( c, a, X) =h~l [ h ( r ; ) + - - ^ - r ; ] a aX

(A.4)

Proof o f proposition 2: The equilibrium expected profit of agent 1 involved in persuasion is:

V'* =

[\ X+

~ \ Xh^

(A'5)

~c i

V*=^X-cr

(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)

87


V*>0

(A. 10)

This is equivalent to the conditions that: » X r, < — 2c

(A.l 1)

ri < ~ 2c

(A. 12)

Proof o f 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; =

aXx

a X2

1> a > 0

(A. 13)

1> a > 0

(A. 14)

and r - = (*■)-' ( S i - ) r ; c2X,

(A.15)

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') '2

— ~) ■ From Hao and Zheng (1998), we know that the C2 X <

a X2 inverse o f an increasing (decreasing) monotone function is also increasing

88


(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 o f proposition 4: See proof of proposition 3 above. Proof of proposition 5: The first order condition equation states: Nr n = ( h Y ( — ), aX

a >0

(A. 16)

Agents will expend positive effort only in the case when they can achieve positive expected profits:

V(r*)> 0 o -

o

,X a X ,, +— lN N '

7=1 a X ,\*j } - c r *}> 0 N N- 1

(A.17)

r~-‘ < h-1[h(r*)+— - — rf*] a aX

(A. 18)

G ( N , c , a , X ) =h~l [h(r*) + - -----——r*] a aX

(A. 19)

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 ir ) 7=1

V = [— + (— )/,(r * ) - ( — X— ------- )] -c r* > 0 { — -----------’N ' N ' v ' iV N -1

•

This is equivalent to the condition that

89


(A.20)

X r < ----Nc

(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 c dissipation is D = (-----)(h ) (-) . A necessary condition for the existence of an X ex, X interior Nash equilibrium in pure strategies is ( h ' y 1( - ^ - ) < which implies that aX Nc the extent of rent dissipation is: D < (— )(— ) X Nc

(A.22)

<=> Z) < 1

(A.23)

The change in D with respect to N can be shown by the representation below: f = < > y < ^ ) +(^ ){-^ « A y ')(^ )} dN X aX X dN aX

(a .24,

From equation (A. 16) and from proposition 4 above, it can be easily shown that the extent o f rent dissipation D is increasing in the number of agents N when

Nc

>{h'Y1

t

aX

>

- N ((h ' y 1) ( - ^ r ) , and decreasing when dN aX

c

c*

0 <{h )_1(----- ) < - N ------((h ) 1)(----- ). Similarly we can show that the extent of aX dN aX rent dissipation D is increasing in the cost per unit of persuasion activity c when ^ ->(h')~x( ^ C) > - N — ((h ' y 1)(—— ), and decreasing when Nc aX dc aX

0 < ( h y ]( - ^ - ) < - N ——((h ) ' aX dc

aX

. It can be also shown that the extent of rent

90


dissipation D is decreasing in the contestable rent X when X „ 8 • j Nc _ (\h ) )(----- ) , and increasing when ->{h ) (----- ) >X Nc aX 8X aX

aX

8X

aX

Proof o f proposition 9: The first order condition equation states: r* = ( h ' y ' ( --------------------;-------), W r - l ) J + [^ (E\,E2) + ( l - S - r ) 7 r ] X h ( r 2)]

(A.25)

Agent 1 will expend positive effort only in the case when: r* > 0

<=> ,_lr

<=> r2
n ( T - \ ) X + [7t\E\,E2) + ( \ - 8 - Y ) n } X h { r 2) > 0

* (T -1 ) ----------------------------- ] n (E\,E2) + ( l - S - T ) 7 r

(A.26) (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 ; h \r x )

(A.28)

and v ; = (1 -7C)X +

h \r 2 )

- 7t(T -1 ) X h(r; ) - cr2

(A.29)

This is equivalent to the conditions that C K ri ) C r , < hj - l r[------------1---------------------------------------r2 1 S r ( T - l) n ( T - \ ) X h ( r 2 ) tt{T - \ ) X 2

and

91


(A.30)

1

h{r•*)

1- S

7 c ( \ - S ) X h (r[ )

-S)X

•i r\ 1

92


(A.31)

B ib lio g ra p h y

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