UNIVERSITY OF CALIFORNIA, IRVINE

On the Dynamics of Property Rights DISSERTATION submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in Economics by Vimal Kumar Dissertation Committee: Professor Michelle Garfinkel, Co-Chair Professor Stergios Skaperdas, Co-Chair Professor Priya Ranjan Professor Dan Bogart

2008

c 2008 Vimal Kumar

The dissertation of Vimal Kumar is approved and is acceptable in quality and form for publication on microfilm and in digital formats:

Committee Co-Chair

Committee Co-Chair

University of California, Irvine 2008 ii

DEDICATION

To my grandfather, Dr. Bindeshwar Singh, for teaching me the value of asking questions, To my mother, Poonam Singh, for her love and support, To my father, Dr. Baijnath Singh, for teaching me the way to tackle problems.

iii

Contents

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . .

vi

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

ACKNOWLEDGEMENTS

. . . . . . . . . . . . . . . . . . . . . .

ix

CURRICULUM VITAE . . . . . . . . . . . . . . . . . . . . . . . .

xi

ABSTRACT OF THE DISSERTATION . . . . . . . . . . . . . . . xiii

1 Production, Appropriation and Emergence of Property Rights

1

1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.2

The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

iv

1.3

Equilibrium Analysis . . . . . . . . . . . . . . . . . . . . . . . 12

1.3.1

The Static Optimization Problem . . . . . . . . . . . . 12

1.3.2

The Dynamic Allocation Problem . . . . . . . . . . . . 15

1.3.3

The Open-Loop Equilibrium . . . . . . . . . . . . . . . 20

1.3.4

The Feedback Equilibrium . . . . . . . . . . . . . . . . 27

1.4

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.5

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2 Innovation, Standardization and Income Distribution in the Presence of Complete Property rights

2.1

41

The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.1.1

Consumers . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.1.2

Producers . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.1.3

Labor Market Clearing Conditions . . . . . . . . . . . 57

v

2.1.4

Product Development . . . . . . . . . . . . . . . . . . . 57

2.2

Steady-State Analysis . . . . . . . . . . . . . . . . . . . . . . . 60

2.3

Long Run Determinants of the Growth Rate and the Relative Wage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.4

2.3.1

Effects of an Increase in Labor Supply . . . . . . . . . 68

2.3.2

Effects of an Increase in Productivity Parameter . . . . 70

2.3.3

Effects of an Increase in Standardization Time . . . . . 70

Extension: Product Cycle without the Scale effect . . . . . . . 72

2.4.1

Comparative Statistics . . . . . . . . . . . . . . . . . . 78

2.5

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

2.6

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

vi

List of Figures

1.1

Differential Games . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2

Open loop Equilibrium . . . . . . . . . . . . . . . . . . . . . . 26

1.3

Feed-Back System . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.1

Timeline in The Economy. . . . . . . . . . . . . . . . . . . . . 62

2.2

Equilibrium Conditions and the Effects of a Change in the Labor Supply . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2.3

Equilibrium Conditions and Effects of Change in Research Productivity Parameter. . . . . . . . . . . . . . . . . . . . . . 71

vii

List of Tables

2.1

Effects of changes in parameters (With Scale Effect) . . . . . . 72

2.2

Effects of changes in parameters (In absence of Scale Effect) . 80

viii

ACKNOWLEDGEMENTS I would like to express my deepest gratitude to my advisors Professors Stergios Skaperdas and Michelle Garfinkel for their support, patience, and encouragement throughout my research. I have been amazingly fortunate to have two advisors who gave me the freedom to explore my own research, and at the same time the guidance to recover when my steps faltered. Professor Skaperdas taught me how to question thoughts, and Professor Garfinkel, how to express ideas. Their advice was essential to the completion of this dissertation.

I am grateful to my committee members, Professors Priya Ranjan and Dan Bogart, for reading my papers, commenting on my views and helping me understand and enrich my ideas. A special thanks to Professor Ranjan who is responsible for helping me complete the second paper as well as the challenging research that lies behind it.

I would like to thank School of Social Science, University of California Irvine for making my graduate years financially secure.

This dissertation has also been considerably improved because of comments from and discussion with Busik Chou, Hyeok Ki Min, Chen Ng, Gregor Franz, Iris Franz and especially Vivek Pai. I am very grateful for their help and friendship. ix

My parents, Dr. Baijnath Singh and Poonam Singh, receive my deepest gratitude and love for their dedication and many years of support during my studies. A special thanks to my uncle Sanjiv, and my brothers Vijay and Vikas.

Finally, I thank my fianc´ee Pallavi Anand. This dissertation would not have been written without her patience and understanding.

x

CURRICULUM VITAE Vimal Kumar

EDUCATION Doctor of Philosophy in Economics

2008

University of California, Irvine

Irvine, California

Master of technology in Mechanical Engineering

2003

Indian Institute of Technology, Bombay

Mumbai, India

Bachelor of technology in Mechanical Engineering Indian Institute of Technology, Bombay

2003 Mumbai, India

TEACHING EXPERIENCE Instructor

Summer 2006

University of California, Irvine Summer Program Teaching Assistant

Irvine, California 2004–2008

University of California, Irvine

Irvine, California

Teaching Assistant

2002–2003

Indian Institute of Technology, Bombay

xi

Mumbai, India

SELECTED HONORS AND AWARDS Social Science Merit Fellowship

2003–2008

University of California, Irvine Summer Program Social Science Summer Fellowship University of California, Irvine Summer Program

Irvine, California 2004, 2006–2008 Irvine, California

National Talent Search Scholarship (NTSE)

1995

National Council of Educational Research and Training

India

SELECTED WORKS

Production, Appropriation and The Dynamic Emergence of Property rights. Growth and Inequality in Closed and Open Economies: The Role of the Product Cycle. (with Priya Ranjan) Practicing Presidential Leadership: A Model of Presidents’ Positive Power in U.S. Lawmaking. (with Matthew Beckmann) Opportunism in Polarization: Presidential Success in U.S. Senate, 1953-2006. (With Matthew Beckmann) On The Economics of Organized Crime. (with Stergios Skaperdas), Prepared for inclusion in Criminal Law and Economics. (Nuno Garoupa, ed.)

xii

ABSTRACT OF THE DISSERTATION On the Dynamics of Property Rights By Vimal Kumar Doctor of Philosophy in Economics University of California, Irvine, 2008 Professor Michelle Garfinkel, Co-Chair Professor Stergios Skaperdas, Co-Chair

This dissertation contains two stand-alone research projects within the context of dynamics of property rights.

Chapter 2 analyzes the dynamic emergence of property rights in a decentralized economy devoid of an exogenous enforcement mechanism. Imperfection in property rights enforcement gives rise to appropriative activities, which take away resources from productive activities, and thus, hampers the performance of the economy. Therefore, agents in the economy strategically invest in definition and enforcement of property rights to limit the detrimental appropriative competitions for the use of resources. Using a differential game framework, this paper obtains the open-loop and the Markov-perfect equilibrium level of property rights enforcement in an economy. The exact level depends on the economy’s characteristics, such as fractionalization, xiii

value of affected assets, productivity of the tools employed to build the institution of property rights, future discount rate, as well as the economy’s norms, culture and traditions.

Chapter 3 uses insight from the product cycle model with skilled and unskilled labor to show the effect of innovation and product standardization on the distribution of income in a closed economy. The rate of innovation in the economy is endogenized. The paper shows that an increase in the size of unskilled worker will decrease its relative wage, while an increase in the size of skilled workers which is required for both production and research might or might not increase its relative wage. The relative wage of skilled workers will increase with increase in their population only when the research effect dominates the population effect. The replacement of endogenous growth by semi-endogenous growth in the model wipes out the scale effect and suggests that an increase in skilled workers’ population decreases its relative wage in a closed economy. Further, the paper using the insight from product cycle literature, shows how an endogenous growth that is skill neutral can have the same impact on income distribution as international trade and skill-biased technical change.

xiv

Chapter 1

Production, Appropriation and Emergence of Property Rights The first principle of economics is that every agent is actuated only by self-interest. The workings of this principle may be viewed under two aspects, according as the agent acts without, or with, the consent of others affected by his actions. In wide senses, the first species of action may be called war; the second contract. Edgeworth (1881) The efforts of men are utilized in two different ways: they are directed to the production or transformation of economic goods, or else to the appropriation of goods produced by others. 1 As

Vilfredo Pareto.1

quoted in Hirshleifer (2001).

1

The fundamental purpose of property rights, and their fundamental accomplishment, is that they eliminate destructive competition for control of economic resources. Well-defined and wellprotected property rights replace competition by violence with competition by peaceful means. Alchian (2006)

1.1

Introduction

Economists have long recognized the role of property rights in determining the allocation of resources and the distribution of output, and thus, in shaping the incentive structures for successful economic performance of the economy. Acemoglu et al. (2005) points to work by John Locke, Adam Smith, John Stuart Mill, among many others, on the topic. However, economists, barring a few, tend to abstract from such issues by assuming the presence of perfectly enforced property right in models.

The notable exceptions are Coase (1960) and Demsetz (1967), who study the role of property rights in resource allocation. Grossman and Hart (1986) and Hart and Moore (1990) extend their studies to provide an elaborate and precise model on the assignment of property rights in economic organizations. While the assignment of property rights is one important issue, so is the 2

role of property rights in determining the distribution of output. Alchian (1965) has defined property rights over an asset as the ability to enjoy the outcome from the use of that asset. Barzel (1997) christened the way property rights are assigned as a “legal property right regime” and the way output is distributed as a “economic property right regime”. Thus, economic rights are the end which economic agents seek while legal rights are means to achieve that end. The present paper takes the assignment of property right as given, and deals solely with the role of property rights in resource allocation and output distribution.

Economic studies which explicitly model the institutions of property rights tend to assume some fixed structure of property rights. Those that consider endogenous determination of these institutions assume they change discretely. In particular, when it is beneficial, economies jump from one regime to another regime, in a costless manner (Gradstein, 2004), or by incurring one time fixed cost (Tornell, 1997). In reality, a regime change is neither perfect nor fixed, but rather evolves continuously over time. Thus, another aspect of property rights that is left untouched by previous models is the costly transition and maintenance of the institution of property rights. A report by the World Bank (1997), The State in a Changing World, emphasizes not only the need to have secure property rights but also the need to pay attention to the evolving nature of property rights for the development of third world countries. In view of the importance of the subject, this paper develops a model of evolution of property rights in a decentralized economy. 3

A decentralized economy is defined as an economy in which a state actor who can enforce the property right is absent. While the choice of such an economy for the model is stark, the description is very close to reality in many developing and underdeveloped countries, as well as in many informal and unorganized economic sectors. In places where the state actor cannot and does not enforce property rights, individuals and groups in these economies attempt to establish institutions. In some cases such private institutions are able to enforce property rights. Ensminger (1992) describes one such case study of Orma tribe in Kenya. Greif (2006) presents another case study of Maghribi traders’ coalition in medieval Europe who developed a private order to enforce property rights. Instead of taking an outside enforcer of property rights as given, the present model discusses the evolution of property rights in an homogeneous economy.

When do property rights evolve? Demsetz (1967), Cheung (1970), Pejovich (1972), among many others, have proposed theories of property right emergence. All of them agree that a well defined property right system evolves when the benefit of having secured property rights outweighs the cost of having it. Demsetz (1967), for example, suggests “that property rights arise when it becomes economic for those affected by externalities to internalize benefits and costs.” Demsetz and others support their theory from case studies. They neither provide an analytical model nor mention variables which affect the benefits and costs of institution of property rights. This paper explicitly describes factors affecting the benefit and costs and develops a 4

functional relationship among them.

The paper tries to answer questions such as why, when and how the institution of property rights emerges. Furthermore, it considers: its role in resources allocation and output distribution, and its continuous evolution set this paper apart from the existing literature.

The framework in the paper comes from “state of nature models” of conflict and appropriation (Skaperdas, 1992; Grossman and Kim, 1995; Garfinkel and Skaperdas, 1996). Instead of assuming a market where property rights are perfectly secured, these models begin the analysis from anarchy.2 In anarchy, property rights are not presumed to be enforced. The absence of enforcement gives rise to various appropriation activities which hamper economic performance, as these activities take resources away from productive activities. The balance between productive activities leading to higher wealth, along with conflict to decide who gets the wealth, plays the central role in these models. An economic agent, who is self interested and rational, balances on the margin two alternative ways of generating income: peaceful production or forceful appropriation of goods produced by others. The balance plays a central role in these models.

Even in the absence of property rights enforcement, economic activity cannot 2 Hirshleifer

(1995) defines anarchy as a system in which economic agents can seize and defend resources without regulations from the above.

5

grind to a halt; otherwise too much potential of the economy would go unrealized. Individuals and groups in these economies have an incentive to develop private institutions to provide the needed mechanism for economic activities to take place (Dixit, 2004). Specifically, contributions from economic agents improve the level of property right enforcement, which in turn dampens the severity of appropriative activities in the economy.3 While private provision of such collective good is subject to the free rider problem, in dynamic settings the shadow of the future somewhat mitigates these problems.

Following Demsetz (1967), the institution of property rights is treated as a public good that is produced by economic agents. It is useful to discuss two polar scenarios: a commiment scenario which presupposes a full commitment by agents at the beginning of the game to follow the agreed upon contribution towards the institution of property rights, and the noncooperative scenario, in which economic agents choose their contribution to further their own interest given other agents contributions and the level of property right in the economy. The Nash-equilibrium is found in both scenarios. It is important to emphasize that the Nash equilibrium in the first scenario may not be subgame perfect. However, the equilibrium in the second scenario is self-enforcing, which means that it is subgame perfect. Subgame perfection means that at any stage in the game, it is in the best strategy of each agent to 3 In

a recent paper, Evia et al. (2007) provide estimates of the costs of conflict and examine its relationship with economic activities. Further, using Bolivian recent history they discuss how levels of conflict, economic performance, and property rights might be related.

6

follow the equilibrium strategy. The presence of an equilibrium strategy does not rule out any pre-game play, but only makes the equilibrium strategies renegotiation proof. The second scenario appears more intuitive than the committed scenario. However, both scenarios are desribed in detail. Consideration of cooperative scenario in this paper can be, in part, justified on the basis of its possible application in some close knit economies and in providing benchmark for the second scenario.

1.2

The Model

Consider an economy populated by n ≥ 2 infinitely-lived agents. These agents can be individuals or groups. If groups, any intra-group conflicts and the free-rider problem are assumed to be resolved and each group acts as a unitary actor. At time t, an agent i ∈ N = {1, . . . , n} allocates her endowment Ri (t) among production (ei (t)), appropriation (ai (t)) to seize part of other’s produce, and investment (gi ) to strengthen the institution of property rights in the economy. Her resource constraint is, then, given by the following:

ei (t) + ai (t) + gi (t) = Ri (t).

7

(1.1)

I assume agents are identical and the endowed resources is time-independent such that Ri (t) = R, ∀i ∈ {1, . . . , n}, ∀t ≥ 0..

The production technology, which transforms agent i’s effort into consumable goods yi , is linear and given in the following equation.

yi (t) = A · ei (t),

(1.2)

where A is the total factor productivity in the economy.

After individuals allocate their endowments, production takes place. Subsequently, individuals try to seize a part of other agents’ product. The fraction of each agent’s output that is subject of appropriation depends on the level of property-rights enforcement θ(t) ∈ [0, 1] in the economy. The total amount of output, subject to appropriation (X(t)) is given in the following equation:

X(t) = (1 − θ(t))

n X

A · ej (t).

(1.3)

j=1

Agent i’s share of the contestable output from the common pool depends on her appropriative effort vis-a-vis the appropriative effort of all other agents.

8

The share of agent i is given by following appropriation technology:4

Pi (t) =

 

Pnai (t) j=1 aj (t)

Pn



1 n

Otherwise.

j=1

aj > 0

(1.4)

Agent i’s total consumption at any time t, ci (t), depends on her current level of productive effort, appropriative efforts and the level of property right enforcement (θ(t)) in the economy.5

ci (t) = θ(t)A · ei (t) + Pi (t)(1 − θ(t))

n X

A · ej (t)

(1.5)

j=1

= θ(t)A · (R − ai (t) − gi (t)) + Pi (t)(1 − θ(t))

n X

A · (R − aj (t) − gj (t)) .

j=1

The first term in the expression above represents the part of agent i’s production not subject to appropriation by others, while the second term gives the share appropriated by agent i from the contestable pool.

All agents share identical preferences over consumption goods, and they max4 The

appropriation technology has an alternative interpretation in that it gives agent i’s probability of winning all goods in the appropriative pool. 5 Dinopoulos and Syropoulos (1998) use a similar index θ to measure degree of institutional security in intellectual property

9

imize their utility over an infinite horizon. Agent i’s instantaneous utility at time t is given as:6

ui (t) = log(ci (t)).

(1.6)

ui (·) is concave and increasing in agent i’s consumption. The preferences of agent i for consumption are aggregated over time by integrating the discounted sum of instantaneous utilities: Z Ui (t) =



e−ρ(τ −t) ui (τ )dτ.

(1.7)

t

The parameter ρ is the rate of time preference in the economy, and is assumed to be strictly positive.

Having imperfect property right enforcement has negative consequences as it induces appropriative activities, though improved property rights help avoid this negative consequence. However, the evolution of property rights is not taken as exogenous: the degree of property right enforcement at any given time depends on previous actions of economic agents in the economy, as they invest in defining and improving property-rights enforcement. I assume a linear property right production function that is additively separable in different agents’ efforts, {gj }nj=1 . It is also assumed that the institution of property rights depreciates at constant rate δ. The equation of motion for θ 6 The

use of a logarithmic utility function in place of a more general CRRA function form is to keep the calculation and analysis simple.

10

is given by the following equation:

˙ θ(t) = B

n X

gj (t) − δθ(t),

B > 0,

(1.8)

j=1

where B is the factor productivity of agents to build the institution of property rights in the economy.

Each agent, i, in maximizing her lifetime utility chooses how many resources to devote to production, appropriation, and to strengthen property rights. The optimization problem for agent i at time t is: ∞

Z max ∞

{ei ,ai ,gi }t

subject to :

e−ρ(τ −t) log(ci (τ ))dτ

(1.9)

t

˙ =B θ(t)

n X

gj (τ ) − δθ(t),

j=1

θ(t) = θt ,

θ(τ ) ∈ [0, 1],

ei (t) + ai (t) + gi (t) = R,

where ci (t) is given in equation (1.5).

Since at any moment of time agent i’s endowment of resources, R, is exogenous, only two choices need to be made. One of these choices is investment gi to improve the level of property right enforcement in the future. Such investments have no direct effect on the current level of property right en11

forcement. The other choice is the allocation of R between appropriation efforts, ai , and productive effort, ei . This allocation does not affect present or future level of property rights. Thus, one can consider the individual’s allocation of resource in two parts; the static allocation problem which takes {gj }n1 as given, and the dynamic problem to choose gi , given the solution of the static problem.

1.3

1.3.1

Equilibrium Analysis

The Static Optimization Problem

Agent i’s static optimization problem at time t given {gj }j6=i can be expressed as:7

max log(ci ), ei ,ai

subject to :

ei + ai = R − gi .

(1.10)

Since log(ci ) is an increasing function of ci , whatever values of ai and ei maximize ci maximize log(ci ). Using the expression for ci from equation (1.5), and taking R − gj , ∀j, as given, the maximization problem of agent i 7 Even

though the model is dynamic, time notation is suppressed where possible to avoid notational cluttering.

12

in period t can be written as " max ai

θ(t) [A · (R − ai (t) − gi (t))] + (1 − θ(t)) Pi (t)

n X

# A · (R − aj (t) − gj (t))

j=1

(1.11)

Agents i takes as given other agents choice of appropriation efforts, {aj }j6=i , and chooses ai to maximize her payoff. The appropriation technology shown in equation (1.4) ensures that all agents make a positive appropriation effort as long as X(t) > 0. For if all but one makes zero effort for appropriating the common pool, that one remaining agent needs to make only an infinitesimally small effort to capture the entire common pool. Hence, P i ai (t) > 0. Therefore, in a homogeneous economy, all agents expend positive effort, aj > 0∀j ∈ N . Agent i’s optimizing appropriation effort level satisfies the following first-order condition: P n X ai j6=i ai P P − θA − n (1 − θ)A + (1 − θ) A · (R − gi − ai ) = 0 ( nj=1 aj (t))2 j=1 aj (t) j=1 (1.12)

At the margin, an increase in appropriation effort by individual (ai ) implies decreased production of the good, yi . This effect is reflected in first two terms. The first term represents the marginal decrease in the secured share of production; the second term represents the marginal decrease in agent i’s share of appropriated goods from common pool X(t) because her increased appropriation effort decreases the size of production, thus decreasing her 13

contribution to the common pool. The third term represents the marginal increase in the fraction that agent i captures from the common pool due to increased appropriation effort.

The first-order conditions in appropriation efforts for all agents of N , if satisfied as an equality, yield the following result:8 Proposition 1. There exists a unique, symmetric pure-strategy Nash equilibrium in appropriation efforts provided agents’ contribution towards improving property rights are sufficiently close. The equilibrium appropriation effort profile can be expresed as n

a∗i

= a



X gj (n − 1) (R − ), = (1 − θ) n n j=1

∀i ∈ N.

(1.13)

Proof: See Appendix Corollary 1.1. The optimal value of consumption for agent i, Ci , in this symmetric equilibrium is given by: " Ci = A

#      1 n−1 1−θ 1−θ X + θ R− θ+ gi − gj (1.14) . n n n2 n2 j6=i

8 The

economy is populated by identical agents having the same endowments. As such, satisfying each first-order conditions simultaneously as a strict equality requires only that there is not too much variation among individuals’ choice of gj s, ∀j.

14

Ci is obtained using a∗j = a∗ , ∀j ∈ N, in equation (1.5).

As described earlier, efforts gi (t), ∀i, work towards improving the level of property rights in the future. It follows immediately, from the above expression, that an increase in gi induces two opposite effects. It increases θ, thus increasing the future value of total available consumable good for agent i, Ci ; however, it also decreases the current Ci . As mentioned, an infinitely lived agent i will dynamically optimize the effort levels for production, appropriation, and strengthening future property rights.

1.3.2

The Dynamic Allocation Problem

Using the result from the static allocation problem, agent i’s optimization problem at time t expressed in equation (1.9) can be restated as

Z max ∞

{gi }t



e−ρ(τ −t) log(Ci (τ ))dτ

∀i ∈ N

t

subject to

θ˙ = B

n X

gj (τ ) − δθ,

j=1

θ(0) = θ0 ,

15

θ(τ ) ∈ [0, 1].

(1.15)

where Ci (t) is obtained as a solution of the static allocation problem, and is expressed in equation (1.14).

In control-theoretic problems, the felicity equation and the state of motion is typically exogenously given. Here, however, both the felicity equation and the equation of motion for θ are determined endogenously. They depend on the strategies of all individuals in the economy. Such problems are modeled as differential games.9 In the setting, agents make their choice of gj in a noncooperative manner. In a differential game, the players interact repeatedly through time. However, the differential game is not a simple repetition of the original game. Instead, there is a state variable θ which continuously changes. Since, other agents’ choice variable affects agent i’s optimization problem, she must take into account the other agents’ choice of control variable {gj }j6=i in choosing her control variable gi . As this is true for all agents j ∈ N , each agent needs to choose her control variable so as to maximize her payoff for every possible choice of other player’s control variable. All agents j, ∀j ∈ N , choose their control variable simultaneously. Accordingly, in order to make optimal choice, agents need to guess what others are doing and going to do in the future. After observing the real choices, some agents might like to revise their choice of control variable. When there exists no incentive for any agent to revise his choice of control variable, then the choices are said to be in a Nash equilibrium. If the expression Ji denotes agent i’s objective 9 For

excellent introduction, see Kamien and Schwartz (1991) Sec.23, and for details see Dockner et al. (2000)

16

function, then Z Ji ({g1 }, . . . , {gi }, . . . , {gn }) =



e−ρ(τ −t) log(Ci (τ ))dτ

∀i ∈ N,

t

where Ci (τ ) is a function of gj (τ ), ∀j ∈ N , and {gj } in the above expression means {gj }∞ t . Then, symbolically, the Nash equilibrium can be given as: Ji ({g1∗ }, . . . , {gi∗ }, . . . , {gn∗ }) ≥ Ji ({g1∗ }, . . . , {gi }, . . . , {gn∗ }) ∀i ∈ N (1.16) where superscript “∗ ” denotes the equilibrium strategy.

As in any game-theoretic problem, the information structure available in the present problem plays a very important role in determining the equilibrium strategies of agents. The two most commonly employed assumptions regulating the information structures in differential games are: 1) each agent is aware of the initial condition of the state variable, θ0 2) each agent observes the current state variable, θ(t). The corresponding strategies are called “open-loop” and “feedback”, respectively.

Since open-loop strategies are conditioned on the initial value of the state variable, they imply that each player has committed to her entire course of action in the beginning of game and will not revise her strategy at any

17

point of time (figure 1.1(a)). In the present problem, the open loop game corresponds to a cooperative scenario where agents are aware of the initial level of property rights enforcement; and based on this value either they commit to the lifetime stream of choice variable or they fail to observe the evolution of property rights. Both interpretations, that agents either commit to entire sequence of actions through time in the beginning of game or do not observe the evolution of property rights, are relatively stringent and can rarely be achieved in a dynamic setting.10

Alternatively, I could assume all agents observe the current level of property rights enforcement, and then have option to revise their action throughout the game. The resulting feedback strategy is characterized by the requirement that the choice variable is a function of both time and the state of the system (figure 1.1(b)). In addition to considering the open-loop strategy for the benchmark, a special case of feed-back strategy is considered in the paper that is the stationary Markov feedback strategy. The term stationary indicates that the feedback strategy depends on time only through the state variable θ(t). The Markov perfection of the strategy implies that all the information about the state variable is captured into its current value. Feedback strategies have the property of being subgame perfect.

gi = gi (θ(t)). 10 This

(1.17)

is because of two facts. First, in a game theoretic setting, rival players do not precommit. Second, they do not ignore the outcome of their strategic interaction on the evolution of game.

18

Efforts form agents {j≠i} towards the institution of property rights.

Effort gi from Agent i towards the institution of property rights

Dynamic System θ=f(θ, gi,{gj(t)}{j≠i})

Change in State Variable

θ The Level of Property Rights

(a) Open Loop System Efforts form agents {j≠i} towards the institution of property rights.

Effort gi from Agent i towards the institution of property rights

Dynamic System θ=f(θ, gi,{gj(θ)}{j≠i})

Change in State Variable

Feedback from current value of State Variable.

(b) Close Loop/ Feed Back System

Figure 1.1: Differential Games

19

θ The Level of Property Rights

To grasp concepts, first the open loop equilibrium is discussed in detail, and then Markov perfect feedback equilibrium is characterized.

1.3.3

The Open-Loop Equilibrium

When agents commit to an action plan at the beginning of the game and stick to that plan forever, the open loop solution characterizes the equilibrium behavior. Each agent takes the other agents’ control variable as the function of time only. The current value Hamiltonian function for agent i is:11 " Hi = log A " + λi B

n X





1 n−1 1−θ + θ R− θ+ n n n2 #



 gi −

1−θ n2

X

#! gj

j6=i

gj − δθ

j=1

where λi is the shadow price that agent i sees associated to θ.12 Necessary conditions for optimality include satisfaction of the equation of motion, constraints, and the following first-order conditions.

11 I

suppress the time subscripts to avoid notational cluttering. the state of motion, given in equation (1.8), it is clear that θ ≥ 0 is always satisfied as θ falls to 0, that is, θ cannot fall below 0. In order to restrict the equilibrium θ below 1, it is necessary to introduce a multiplier with the constraint θ ≤ 1 and form a Lagrangian. However, to keep analysis simple, it is assumed that B is sufficiently small, so θ = 1 is never achieved. 12 From

20

The first-order condition with respect to gi is given by: ∂Hi = ∂gi

1 n

 − θ + 1−θ 2 n   1−θ + n−1 θ R − θ + gi − 2 n n

1−θ n2

P

j6=i gj

+ λi B. (1.18)

The first-order condition with respect to the state variable θ (the adjoint equation) is ∂Hi = ∂θ

1 n

+

n−1 R− 1 n  n−1 θ R− n

1 n2

gi + n12  θ + 1−θ gi − n2





P

j6=i gj P 1−θ j6=i n2

gj

− λi δ = ρλi − λ˙ i(. 1.19)

It can be shown that, for an economy populated by identical agents, the denominator in the first-order condition is always positive. Solving the firstorder conditions with respect to gi , for all i = 1, 2...n, equation (1.18), for a symmetric result (λi = λ, gi = g ∀i ∈ N ) yields:



1 n

  > 0 ⇒g=R    1−θ θ + n2  + λB = 0 ⇒ g ∈ (0, R) ,  + n−1 θ (R − g)  n   < 0 ⇒ g = 0.

At any interior optimum, the following holds:

21

(1.20)

λB =

1 n

θ + 1−θ (1 + (n2 − 1)θ) n2  = . n (1 + (n − 1)θ) (R − g) + n−1 θ (R − g) n

(1.21)

The above relation gives an expression for the shadow price of property rights, λ, in terms of choice variable, gj = g, ∀j. Using this expression the equation of motion can be expressed in terms of λ: n2 θ + 1 − θ 1 · − δθ. θ˙ = BnR − 1 + (n − 1)θ λ

(1.22)

The first-order condition for θ given in equation (1.19) implies: λ˙ = (ρ + δ)λ −

n−1 . 1 + (n − 1)θ

(1.23)

The above two equations along with the initial condition, θ(t) = θt describes the system completely. Using these two equations, the initial condition and the transversality condition determine the time path of g ∗ , θ∗ and λ∗ . For an interior solution, these equations can be analyzed using the phase diagram as shown in figure (1.2).

The schedule λ˙ = 0 shows combinations of the state variable θ and its value λ for which the value of the property rights λ remains momentarily unchanged. The negative slope of this schedule can be understood as follows. An increase 22

in the value of property rights raises the rate of return from having the property rights at the level θ. Then economy will experience zero instantaneous rate of change in the value of property rights, λ, only if the increase in rate of return is completely absorbed in corresponding increase in benefit received by the agent i for having the property right θ.13 . The utility of agent i is concave in the level of property rights, so higher dividend implies lower level of property rights in the economy.

The schedule θ˙ = 0 plane is positively sloped in the θ-λ. This can be explained with the help of the equation of motion in θ. As the level of property rights, θ, increases, the total disintegration in the institution of property right increases (δ > 0). To maintain the same level of property rights, more efforts from agents are required. Agents put more effort only if the value of property right increases which gives positively sloped θ˙ = 0 schedule.

Now one can use figure (1.2) to describe the equilibrium dynamics. Suppose that at the beginning of the game, t, the level of property right in the economy is given as θt . The figure shows three of infinitely many possible trajectories that begins from points with initial condition θt . Along all these trajectories all first-order conditions are satisfied, however the initial value/shadow price of property right differ. Trajectory 1, which assigns the highest shadow price, 13 The

benefit from improving property rights from θ to θ + dθ can be defined as rate of utility gain of agent i at the property right θ multiplied by dθ, which is ∂ui ∂θ · dθ. In tradition literature where the state variable is tied to profit, the benefit is called dividend.

23

leads in the figure, to a very high level of property rights with a very high shadow price. A high shadow price implies higher contribution towards improvement of property rights (equation 1.21), which chokes off flow of efforts towards production increasing the scarcity of consumption good in economy unnecessarily. As the initial valuation of property rights is too high to be consistent with perfect foresight, one can rule out this type of trajectories as candidates for equilibrium. Trajectory 2, which assigns the lowest shadow price with the figure, leads to 0 property rights. Trajectories like this one can also be ruled out based on the fact that, with perfect foresight, agents would never contribute towards improving property rights to achieve 0 level of property rights. Only the saddle path, denoted by trajectory 0, reflects expectations that can be fulfilled everywhere. The saddle path describes the unique dynamic equilibrium. The equilibrium has a property that the value of current effort level to maintain this level of property rights is equivalent to the current value of future gain that accrue to agent i for having the property right as this level.14

As it is discussed earlier, λ˙ = 0 curve is downward slopping and θ˙ = 0 curve is upward slopping. There are two conditions under which these two curves do not intersect. Under one condition, the λ˙ = 0 lies completely below the θ˙ = 0 curve. In this case, the economy ends up with no property rights (θ = 0). This is because at all points of curve θ˙ = 0, λ˙ is positive, which 14 The

value of current effort level can be measured in terms of current consumption forgone

24

renders the cost of property right very high, which, in turn, makes any contribution towards property right improvement very costly. In this case, the level of property rights achieved in equilibrium is 0. The opposite is true when θ˙ = 0 lies completely under the curve λ˙ = 0. At all points of θ˙ = 0, λ˙ is negative making the maintenance and improvement of property rights virtually costless which results in perfect property rights (θ = 1) in equilibrium. Proposition (2) sums up the discussion on the open loop equilibrium and gives the level of property rights in all three conditions, one internal and two boundary conditions, discussed above. Proposition 2. There exists a unique and dynamically stable open-loop solution that results in a steady state level of property right enforcement given by:

θ =

   0   

ρ+δ BnR− n−1 (n+1)(ρ+δ)+δ

     1

:

B ρ+δ

:

1 Rn(n−1)

:

B ρ+δ

<

>

1 Rn(n−1)

h

B ρ+δ



n (n−1)R

+



h

n (n−1)R

δ (ρ+δ)nR

+ i

δ (ρ+δ)nR

i

(1.24)

What happens if political fragmentation, n, in economy increases? One can consider two scenarios. In the first scenario, more and more group share the resource available in economy T . In this case, resources available to an agent (a group) can be given as R =

T . n

In the second scenario, the econ-

omy has more agents, however per capita resources in the economy remains unchanged: where T = nR, where T increases with n.

25

Figure 1.2: Open loop Equilibrium Corollary 2.1. The level of property right in the economy with a fixed amount of resources (T ) either decreases or first increases/remain constant and then decreases with increasing political fragmentation (n), and asymptotically reaches to zero.

Proof: See Appendix Corollary 2.2. For sufficiently high n, the level of property rights in the economy is non decreasing in number of agents, as more and more agents having the amount R joins in the economy, and asymptotically reaches to the level, θ∞ .  BR = min ,1 ρ+δ 

θ∞

26

(1.25)

Proof: See Appendix

The open-loop solution provides the level of property right enforcement that can be achieved if agents either do not observe the evolution of the state variable or they commit in the beginning of the game to ignore the effect of change in the state on their strategy. Of course, such conditions cannot be enforced as all agents have an incentive to free ride and deviate based on the observation of the state variable.

1.3.4

The Feedback Equilibrium

The equilibrium concept of feedback strategy is more intuitive and appealing in the present problem, as agents cannot gain by unilaterally deviating from their equilibrium strategy. Here, agents optimize their actions in all subgames. These subgames can be understood as a new game which starts after each agent’s action have caused the level of property right to evolve from its initial state to a new state. The continuation of the game with a new level of property rights can be considered as a subgame of the original game. A feedback strategy permits agents to take the best possible action in each subgame. A feedback strategy is, therefore, optimal not only in the beginning of the game but throughout the game. Although feedback strategies appear very appealing, they are very difficult to compute. In order to simplify the analysis, two assumptions are made: 27

1. All information emanating from the observation of the state variable θ is available through its current value. (Markov Perfect Property) 2. The feedback strategies depend on time only through state variable. (Stationary Property)

Feedback strategies are difficult to calculate because finding agent i’s strategy requires that all other agent’s optimal strategies be known which, in turn, requires player i’s optimal strategies be known. In order to optimize, as in the case of open loop strategies, agents need to guess what others are doing and are going to do. However, in case of feedback strategies, agent i’s guess of other agents’ strategies are a function of θ, the state variable, which leads to the presence of an interaction term in agent i’s adjoint equation. The interaction term, in turn, makes the computation of the shadow price difficult. To find the optimal feedback strategy for agent i, I set up current value Hamiltonian.

#!      1−θ 1−θ X 1 n−1 + θ R− θ+ gi − gj (θ) = log A n n n2 n2 j6=i " ! # X + λi B gi + gj (θ) − δθ (1.26) "

Hi

j6=i

where λi is the shadow price that agent i see associated to θ. Necessary conditions for optimality include satisfaction of the equation of motion, con28

straints, and following first-order conditions.

The first-order condition with respect to gi is given by: ∂Hi = ∂gi

1 n

 − θ + 1−θ 2 n   1−θ gi − + n−1 θ R − θ + 2 n n

1−θ n2

P

j6=i

gj (θ)

+ λi B(1.27)

It implies the following



1 n

  > 0 ⇒g=R    1−θ θ + n2  + λB = 0 ⇒ g ∈ (0, R)  + n−1 θ (R − g)  n   < 0 ⇒g=0

(1.28)

Using equation (1.27) and looking at symmetric solutions I get

λB =

1 n

θ + 1−θ (1 + (n2 − 1)θ) n2  = . n (1 + (n − 1)θ) (R − g) + n−1 θ (R − g) n

(1.29)

Equation (1.29) gives an expression for the shadow price of property rights, λ, in terms of choice variable, gj = g, ∀j. Using this expression the equation of motion can be expressed in terms of λ: n2 θ + 1 − θ 1 θ˙ = BnR − · − δθ. 1 + (n − 1)θ λ

29

(1.30)

The first-order condition with respect to state variable θ (the adjoint equation) is given by ∂Hi = ∂θ

n−1 R n

1 0 gi + n12 j6=i gj (θ) − 1−θ j6=i gj (θ) n2 n2    P n−1 θ R − θ + 1−θ gi − 1−θ j6=i gj n n2 n2

− 1− 1 n

+



P

P

! + λi B

X

gj0 (θ) − δ

= ρλi − λ˙ i .

(1.31)

j6=i

It implies the following: λ˙ = (ρ + δ) λ −

(1 − θ)(n − 1)g 0 (θ) n−1 − B(n − 1)g 0 (θ)λ + 1 + (n − 1)θ n (1 + (n − 1)θ) [R − g(θ)] (1.32)

One can rewrite the adjoint equation using first-order condition with respect to the choice variable given in equation (1.29) as: g 0 (θ) n−1 n(n − 1)θ · λ˙ = (ρ + δ) λ − (1.33) − 1 + (n − 1)θ (1 + (n − 1)θ) [R − g(θ)] | {z }| {z } Same as Open loop Strategies

Interaction Term

The expression for the interaction term in the terms of θ, n, and λ can be obtained from differentiating the first-order condition in choice variable. Using

30

equation of motion, one can obtain:15 (ρ + δ) λ − F (n, θ) i λ0 (θ) = h 1 BnR − δθ − λ(θ) where F (n, θ) =

n−1 1+(n−1)θ



n(n−1)θ (1+(n−1)θ)

h

(n2 −1) (1+(n2 −1)θ)



(n−1) (1+(n−1)θ)

(1.34)

i

The expression in (1.34), the equation of motion for θ given in equation (1.30) and the initial condition, θ(0) = θ0 describes the evolution of property rights in the economy completely. Based on equations (1.34 and 1.30), I present the phase portrait of λ0 (θ) in figure 1.3(a) and the Markov perfect strategies and equilibria in figure 1.3(b). The flow paths (formed by arrows) in figure (1.3) can, at least locally, be interpreted as the graphs of λ(θ) associated with possible symmetric feedback strategies g(θ).

It is important to note that for the feed back system, I do not give the ˙ This is because canonical equations for the system in terms of λ˙ and θ. ˙ as the expression for λ˙ of this system can only be expressed in terms of θ, ˙ Accordingly, λ˙ and θ˙ are not independent of each other. λ˙ = λ0 (θ)θ.

By definition, in equilibrium, the shadow price λ and the level of property right θ in the economy do not change. So, both λ˙ and θ˙ should be equal to zero in equilibrium. As I discussed, λ˙ is always zero whenever θ˙ is zero. 15 See

the appendix for the derivation of the equation.

31

⋅ θ=0 ⋅ λ= 0

λ′ = ∞

λ′ = 0

(a) Phase portrait of λ0 (θ) in feed-back system

⋅ θ=0 ⋅ λ= 0

λ′ = ∞

λ′ = 0

(b) Feed-Back Equilibrium

Figure 1.3: Feed-Back System 32

So all strategies which cross the θ˙ curve provide equilibria. However, not all of these equilibria are stable. Since, in this feedback system, I possibly encounter multiple equilibria (see figure (1.3)), I use the stability condition to characterize the equilibria rather than use phase diagrams to analyze the system.

The stability criterion for the system can be obtained from the equation of motion expressed in terms of the shadow price λ given in equation (1.30). θ˙ is positive above the curve θ˙ = 0, shown in figure (1.3), and negative below. If λ(θ) be a feedback value of property rights corresponding to a feedback strategy g(θ), and {λ∗ , θ∗ } be an equilibrium such that λ∗ = λ∗ (θ), the equilibrium is stable if d f (λ∗ , θ∗ ) < 0 dθ where f (·), the equation of motion, is equal to BnR −

(1.35) n2 θ+1−θ 1+(n−1)θ

·

1 λ

− δθ.

Differentiating f (·) with respect to θ, I obtain the stability condition:  0   1 λ (θ) n2 − n 2 ∗ |˙ 1 + (n − 1)θ − <δ λ∗ (θ) · (1 + (n − 1)θ∗ ) λ(θ) θ=0 1 + (n − 1)θ∗ ) (1.36)

At any stable equilibrium point, the slope of shadow value of property rights (λ0 (θ)) actuated by an equilibrium strategy should be less than the slope of the curve θ˙ = 0, otherwise, the stability condition given in equation (1.36)

33

would not hold. From the figure 1.3(a), it is apparent that points on θ˙ = 0 close to the origin are stable as the slopes of the flow paths of λ(θ) strategies crossing the θ˙ = 0 curve is less than the slope of the curve θ˙ = 0, while points closer to θ = 1 are unstable. There is a point on the curve θ˙ = 0 where equilibria change from stable to unstable. This point is unique and lies at the position where λ0 (θ) is tangent to θ˙ = 0. I call this point θ¯ which can be obtained analytically, and is given in the following equation (see derivation in appendix): Bn(n − 1)R − (ρ + δ) θ¯ = 2(n − 1)δ + (n2 − 1)ρ

(1.37)

Proposition 3. There exist multiple equilibria in the feedback system. Any ¯ is feasible as the equilibrium level of property rights. θ ∈ [0, θ]

For increasing political fragmentation in the economy, I can derive corollaries analogous to that derived for open loop system. Corollary 3.1. The property rights enforcement in the economy with fixed amount of resources either decrease or first increase and then decreases with increasing political fragmentation (n), and asymptotically becomes zero.

Proof: Similar to Corollary 2.1. Corollary 3.2. The feasible range of property right in the economy expands as more and more identical agents having the same amount of resources R 34

joins in the economy, and asymptotically becomes [0, θ¯∞ ], where θ¯∞ is given in the following expression:

θ¯∞



 BR = min ,1 ρ

(1.38)

Proof: Similar to Corollary 2.2.

1.4

Conclusion

The main contribution of this paper is to develop and analyze an analytical model that explains the evolution of property rights in a decentralized economy. The model captures the basic elements affecting the benefits and costs associated with the institution of property rights and describes the strategic interaction among agents involved in productive and appropriative activities. The paper uses a dynamic-game framework and characterizes an open loop strategy and a stationary Markov feedback strategy in the evolution of property rights without relying on the guessing method. For the open loop strategy, I show a unique and stable equilibrium which depends on the economy’s characteristics such as the productivity factor in institution building (B), institutional rate of depreciation, and also on individual characteristics 35

such as individuals’ endowment of resources, discount rate for future. For stationary Markov feedback strategies, I show that a range of equilibrium level of property rights are feasible in an economy.

36

1.5

Appendix

Proof of Proposition 1

For an internal solution, equation (1.12) gives: P n X ai j6=i ai (1 − θ)A + Pn (1 − θ) A · (R − gi − ai ) − θA − Pn ( j=1 aj (t))2 j=1 aj (t) j=1 = 0 ∀i ∈ N

(1.39)

Taking the first-order condition for agents i, and j, and dividing one by another, one can obtain: θ+ θ+

Pn ai (1 k=1 ak (t) a Pn j (1 k=1 ak (t)

− θ)

P k6=i ak = P − θ) k6=j ak

(1.40)

The above equation is true for any i, j pair belonging to N . This is possible only when a∗i = a∗j , ∀i, j ∈ N . Using this fact in the first-order condition for agent i, one can obtain the expression for a∗i given in Proposition 1.

Proof of Correlation 2.1

From differentiating equation (1.24) with respect to n and taking n · R = T as constant: 37

  

[(n+1)(ρ+δ)+δ]

∂θ =  ∂n  0

ρ+δ −(ρ+δ) (n−1)2

ρ+δ [BT − n−1 ]

:

(n+1)(ρ+δ)+δ

1 Rn(n−1)



B ρ+δ



h

n (n−1)R

+

δ (ρ+δ)nR

i

: Otherwise (1.41)

Also, ∂θ =0 n→∞ ∂n

lim θ(n) = 0

lim

n→∞

(1.42)

Proof of Correlation 2.2

From equation (1.24)

θ =

   0    B(1+ 1 )R−

ρ+δ (n−1)2 δ (ρ+δ)+ n+1 n

     1

:

B ρ+δ

:

1 Rn(n−1)

:

B ρ+δ

<

>

1 Rn(n−1)

h

lim θ(n) =

n→∞

38

B ρ+δ



n (n−1)R

+



BR ρ+δ

h

n (n−1)R

δ (ρ+δ)nR

+ i

δ (ρ+δ)nR

i

(1.43)

(1.44)

Derivation of equation (1.34)

Taking logs of both sides and differentiating the first-order condition with respect to choice variable gi gives:

(n2 − 1) (n − 1) g 0 (θ) λ0 (θ) = − + , λ(θ) (1 + (n2 − 1)θ) (1 + (n − 1)θ) R − g(θ)

(1.45)

which can be rearranged to obtain an expression for the interaction term: g 0 (θ) = R − g(θ)

λ0 (θ) λ(θ)



h

(n2 −1) (1+(n2 −1)θ)



(n−1) (1+(n−1)θ)

i

˙ one can rewrite the adjoint Using this expression and the identity λ˙ = λ0 (θ)θ, equation as  λ (θ) θ˙ + 0

n(n − 1)θ 1 (1 + (n − 1)θ) λ(θ)



n−1 1 + (n − 1)θ   n(n − 1)θ (n2 − 1) (n − 1) + − (1 + (n − 1)θ) (1 + (n2 − 1)θ) (1 + (n − 1)θ) = (ρ + δ) λ −

Using the expression for θ˙ in terms of λ given in equation (1.30) in LHS of

39

the above equation, one can obtain:   1 n−1 λ (θ) BnR − δθ − = (ρ + δ) λ − (1.46) λ(θ) 1 + (n − 1)θ n2 (n − 1)2 θ + (1 + (n2 − 1)θ) (1 + (n − 1)θ)2 0

Derivation of equation (1.37)

Using the expression for λ0 (θ) given in equation (1.34) in the stability condition given in equation (1.36), I obtain: h 1 λ(θ)

(ρ + δ)λ∗ (θ) −

n−1 1+(n−1)θ∗

(n2 − n)θ∗

i <δ

(1.47)

Since λ∗ lies on the curve θ˙ = 0, I can obtain an expression for λ∗ in terms of θ∗ and other system parameters by equating the equation of motion to 0. Using this expression for λ∗ (θ) in the above equation, I obtain the critical value θ¯ that shown in equation (1.37).

40

Chapter 2

Innovation, Standardization and Income Distribution in the Presence of Complete Property rights

The pattern of income distribution in developed as well as in developing countries has changed considerably in recent past. Katz and Autor (1999) report 29 percent increase in the gap between the 90th percentile of earner and the 10 percentile of earner from the late 1970s to the mid 1990s in the United States. The gap increased by 27 percent in the same period in the United 41

Kingdom and by 9 percent in the Canada. Feenstra and Hanson (1997) report a similarly high increase in wage inequality for Mexico, a developing country. Observations such as these have motivated a large body of literature that aims to understand the determination of an economy’s distribution of income at a given point in time and in the long run. These explanation can be categorized broadly in two groups based on two different theories used to explain the problem at hand. The first emphasizes international trade with other countries (Wood, 1995; Dinopoulos and Segerstrom, 1999). The second centers on skill-biased-technical-change (STBC) hypothesis (Berman et al., 1998; Acemoglu, 2002).

The rise in wage inequality has coincided with the gradual removal of trade barriers. So, it was natural, at least initially, to hypothesize that increase in international trade leads to the rise in wage inequality. This hypothesis found firm theoretical grounding in the work of Stolper and Samuelson (1941). Specifically the Stolper-Samuelson theorem, when applied to the HeckscherOhlin model, postulates that a decrease in the relative price of a good reduces the real return to the factor used intensively in its production. Usually with a larger relative supply of skilled woker, a developed country engaged in trade with a developing country specializes in the production of goods which use skilled workers intensively. Since international trade leads to decrease in the cost of imported good, the removal of trade barriers pushes the relative wage of skilled workers in developed countries upward. Borjas and Ramey (1994) and many others report empirical evidence correlating the decrease in 42

relative wage of unskilled workers with net imports of durable goods.

The view that international trade leads to wage inequality has not gone unchallenged. Some empirical papers have suggested that inequality is rising not only in developed economies but also in developing economies (Feenstra and Hanson, 1997). Even when trade barriers were lowered, the domestic relative prices of imported good in developed countries remained roughly constant (Lawrence and Slaughter, 1993). Since the Stolper Samuelson theorem predicts a change in relative wage of unskilled worker in opposite directions for a developed and a developing country and the theorem’s prediction about the change in relative wage works through the changes in product prices, this evidence casts serious doubts on the international trade based explanation for increasing wage inequality. Krugman and Lawrence (1994) argued very forcefully that, even though US trade with the rest of the world has increased manifold in the past, the living standards in the economy are still determined by domestic factors. In their view, technological change plays a major role in explaining the current wage inequality.

The second type of explanation for the rise in wage inequality is based on “skill-biased-technological-change (SBTC)” hypothesis. This type of explanation is largely shaped by the observation of skill biased technological changes taking place in the economy along with the rise in wage inequality. As the name suggests, SBTC indicates a particular type of technological change in the economy that is biased in favor of one particular type of worker. 43

Typically, the bias is for skilled workers as it is assumed that technical advances eliminate or reduce the need for unskilled workers. This leads to a change in labor composition in favor of skilled workers. A common feature of models based on the SBTC hypothesis is that, as technical progress occurs, the relative marginal productivity of different inputs change.

SBTC based explanations are also not free from criticism. The main criticism comes from the fact that a rise in relative wage makes use of unskilled labor relatively cheap. Thus, an incentive exists to develop technologies which favors unskilled workers rather than skilled workers. Still why do STBC favor skilled workers? Very few SBTC based models try to answer this question. Acemoglu (2002) claims and shows empirically that a positive supply shock can lead to a technical change that is biased towards a particular factor of production.

The debate over what leads to rise in wage inequality and thus influences the income distribution is hardly settled, Dinopoulos and Segerstrom (1999) describe a product cycle based Schumpeterian growth model to argue that the role of international trade in rising wage inequality has been underestimated. This underestimation is because a traditional Heckscher-Ohlin trade model, which focuses on trade driven by differences in relative factor endowments between countries, is hardly suitable to explain the international trade that is driven by differences in knowledge between countries. According to them, knowledge-difference based trade can explain observed rise in wage inequality. 44

The new knowledge based trade explanations, and the SBTC hypothesis suggest that a differences in level of knowledge/technology has emerged as the main explanation for the rise in wage inequality.

Building on the established trend in the literarure, I provide a technologybased explanation to understand the distribution of income. This explanation is different from existing ones. My intention is not to point out weaknesses of existing models, but to augment them. My model, based on a product cycle with standardization, suggests that a product cycle with standardization has the same impact on the distribution of income as international trade or skillbiased technical change.

The term ‘product cycle’ was first used by Vernon (1966) to describe a phenomena that most new goods are manufactured first in the countries where they were originally discovered and developed, and later in countries where production costs are lower, once the products have been standardized. Vernon’s implicit assumption was that, in the beginning of cycle the production function is not clearly specified such that production can only take place under the supervision of skilled engineers. As time progresses, the manufacturer gradually gains knowledge on how to produce the good without such assistance, and gradually production becomes less skill intensive.

Vernon’s (1966) description of the “product cycle hypothesis” led to a large body of empirical research pointing to a richer implication than what Vernon 45

envisioned in his paper. Apart from suggesting the existence of the product cycle in the shift of production of standardized goods to countries where production costs are lower, the empirical findings also suggested the presence of an entire product cycle inside an economy. Hekman (1980) provides evidence on the shift in production of textiles from mature industrial region (New England) to the low wage worker abundant south in United States after 1880.

Vernon’s (1966) work also pushed theorists to formalize the theory behind the cycle. The first notable formalization came from Krugman (1979). In his model, a developed country (industrialized north) innovates and produces new goods, and a developing country (south) produces old goods. Since agents in both economies have a ‘love of variety’, there is trade between the north and the south. In his model, a new good becomes an old good over time with a lag specified exogenously. Since the south can imitate old goods and produce them more cheaply, southern manufacturers drive northern manufacturers of old goods out of the market. Grossman and Helpman (1991) developed a model of product cycle based on endogenous growth theory. In endogenous growth models, whenever the discounted present value of the expected profits exceeds the current cost of development resources (skilled labor), entrepreneurs spend resources to bring new products to the market. The cost of developing a new product decreases in real terms as the number of already developed products increases in the economy. The reasoning is that available products represent disembodied knowledge in the 46

economy. As disembodied knowledge in the economy increases, development costs decrease.

Krugman (1979), Grossman and Helpman (1991), and most if not all other existing analyses of product cycles study the various implications of international trade. All these models have only one type of factor of production. Hence they are not suitable to study the distribution of income in a closed economy. These models abstract from Vernon’s description of continuous standardization and suppose the same production function remains in place throughout the whole product life cycle. In a recent paper, Antras (2005) describes a model that is much closer to Vernon’s description of standardization. He uses a standardization process to describe the change in factor requirements to produce the same good. Although, his model is based on two factors of production (skilled and unskilled workers), both factors of production are assumed to be paid the same wage, thereby making the model unsuitable to study the distribution of income. The main focus of Antras (2005) is to describe how an endogenous product cycle can arise due to incomplete contracts.

I develop a model of endogenous product cycle with process standardization to explain the income distribution in a closed economy.1 Unlike the tradi1 My

paper is much closer in spirit to Ranjan (2005). My model differs from his model in three ways. First, he uses a random standardization process. Second, His model assumes perfectly competitive market. Third, in his model productivity growth is driven by publicly funded R&D, while in this paper, I use private funding

47

tional product cycle literature, in my model production of standardized goods does not move to developing countries. Instead, the standardization of production frees up skilled workers tied in the production of existing goods. The freed skilled workers contribute to endogenous growth and contribute to the creation of new products. Over time, technological innovation itself requires fewer resources which implies an increasing number of newly developed goods over any given interval of time. The production of newly developed products are skill intensive. Given a fixed ratio of skilled and unskilled workers in the economy, this leads to an equilibrium distribution of income in a closed economy.

My paper makes two contribution to the literature. First, it describes a endogenous product cycle with production process standardization. Second, it presents a product cycle based mechanism to describe the equilibrium income distribution in a closed economy.

2.1

The Model

In this section, I present my model of endogenous product cycle. The model builds upon Grossman-Helpman (1991), yet differs in two important ways. First, my model incorporates two types of labor, skilled and unskilled. This for R&D

48

distribution is important to consider intra economy income distribution. Second, my formulation allows for the standardization of production function in the product cycle that represents product development over the life cycle.

2.1.1

Consumers

Consider an economy populated by two types of infinitely lived workers, skilled (h) and unskilled (l), with populations equal to H and L, respectively. At time t, there are n(t) differentiated goods available in the economy. A worker of type k ∈ {h, l}, has a time-separable intertemporal lifetime utility function, Uk (t), with a common discount rate, ρ. Worker k’s lifetime utility depends on her instantaneous sub-utility function, uk (τ ), which, in turn, depends on her instantaneous consumptions, Ckj (τ ) of (j ∈ n(τ )) products, of n(τ ) differentiated goods, available at time τ .



Z

e−ρτ log[uk (τ )]dτ

Uk (t) = Zt uk (τ ) = [

1

Ckj α (τ )dj] α ,

(2.1) α ∈ (0, 1)

(2.2)

j∈n(τ )

The assumption of CES (constant elasticity of substitution) implies that consumers have a ‘love of variety’. It also implies the elasticity of substitution between any two products is constant and equal to σ = 49

1 1−α

> 1.

Consumer k faces an optimization problem which can be divided in two stages. She solves her optimization problem using the backward induction technique. In the first stage, consumer k ∈ {h, l} chooses the path of her expenditures, Ek (t) to maximize Uk . While maximizing Uk , she needs to satisfy her intertemporal budget constraint. The budget constraint depends on her wages {wk (τ )}∞ τ =t , assets holding at time t, Aj (t), and instantaneous ˙ )}∞ , prevailing in the capital market. The cumulative interest rates, {R(τ τ =t interest factor from time 0 to time t that a worker faces in the capital market Rt ˙ is given by R(t) = 0 R(t)

Assuming a consumer k can lend and borrow freely in the capital market, her budget constraint is Z

∞ −[R(τ )−R(t)]

e

Z Ek (τ )dτ =

t



e−[R(τ )−R(t)] wk (τ )dτ + Aj (t)

(2.3)

t

In the second stage she chooses the consumption of good i at time t so as to maximize uk (t) given prices of all available goods and expenditure Ek (t), Z where Ek (t) = pj (t)Ckj (t)dj. Optimization at the second stage implies: j∈n(τ )

Ckj (t) = R

p−σ j (t) j 0 ∈n(t)

50

pj1−σ (t)dj 0 0

Ek (t)

(2.4)

The instantaneous demand function of good i in the economy is

Yj (t) = L · Clj (t) + H · Chj (t) = R = λp−σ j (t),

where λ = R

p−σ j (t) j 0 ∈n(t)

0 p1−σ j 0 dj

E(t)

(2.5)

E(t) 1 1−σ 0 and σ = 1−α p dj j 0 ∈n(t) j 0

E(t) = L · El (t) + H · Eh (t), is economy’s total expenditure at time t.

Consumption of good j for worker k, obtained in the second stage and given by {Ckj }j∈n(τ ) , leads to an indirect utility function uk (τ ) that is weakly separable in the level of k’s expenditure, Ek (τ ), and in a function of prices of differentiated goods, {pj (t)}j∈n(τ ) . It implies that uk (τ ) can be written as uk [p(τ ), E(τ )] = Ek (τ )f (p(τ )). I can rewrite the lifetime utility function, equation (2.1), as Z Uk (t) =



e−ρτ [log Ek (.) + log f (p(.))]dτ

(2.6)

t

The Lagrangian expression using equation (2.6) and lifetime budget constraint (2.3) is given by Z

L =



e−ρτ [log E(τ ) + log f (p(τ ))]dτ t Z ∞  −[R(τ )−R(t)] −µt e (Ek (τ ) − W (τ ))dτ − Aj (t) t

where µt denotes the Lagrangian multiplier on the budget constraint. The first-order condition for maximizing Uk (t) with respect to Ek (τ ) can be writ51

ten as e−ρτ

1 − µt e−[R(τ )−R(t)] = 0 Ek (τ )

(2.7)

Taking logs on both of the sides and differentiating with respect to t gives E˙k = R˙ − ρ, Ek

k ∈ (L, H)

(2.8)

The economy’s total expenditure in period t is E(t) = L · El (t) + H · Ek (t) which gives LE˙ l + H E˙ h E˙ = E LEl + HEh By using componendo and dividendo method on equation (2.8), I get E˙ = R˙ − ρ E

(2.9)

The above conditions implies that the individual’s (skilled or unskilled), and the economy’s expenditure, all grow at the same instantaneous rate, equal to the instantaneous interest rate corrected by future discount rate.

52

2.1.2

Producers

The number of potential products is infinite. To begin production of one of the potential differentiated goods, the producer needs to learn how to produce that good. All new producers incur a development cost to start production. A new producer does not want to develop an already existing type, as this leads to Bertrand competition between two identical products. In Bertrand competition, competitors have to set price of the good equal to the marginal cost. Since the learning process is costly, a new producer of an already existing product would never able to recover the development cost of the good.

Once a new producer is developed, production takes place under the constant return to scale technology. In the beginning, only skilled laborers are capable of producing the new good. Following the ‘product cycle’ literature, any new product goes through a standardization process, and once standardized the production shifts to unskilled laborers. For convenience, I call the good a “new good” when produced solely by skilled workers, and an “old good” when production uses unskilled hand. For good j, the production function is given as Yj :   hj When good j is new Yj =  l When good j is old j

(2.10)

where hj is the number of skilled workers employed to produce good j when

53

it is new and lj is the number of skilled workers employed to produce good j when it is old.

The process of standardization in this model is taken as exogenous and discrete in nature. After producing a new good for an exogenously given period of time, say T , the producer accumulates enough information regarding the production process, such that it can be undertaken by unskilled workers.2

Consumers’ CES type of preference lead to an iso-elastic aggregate demand 2 One

can envision a more general production function such as a Cobb-Douglas function with continuous standardization process given as

Yj

= ζh1−z ljz , j

0 ≤ z ≤ 1,

where ζ = z −z (1 − z)−(1−z) . Since limz→1 z −z (1 − z)1−z = 1 and limz→0 z −z (1 − z)1−z = 1, the production function is continuous in z. z as a function of time (τ ) captures the standardization process. Standardization implies that the output elasticity of unskilled workers increases as the product becomes older, and that of skilled workers decreases. The basic characteristics of continuous standardization can be given as z(0) = 1, z 0 (τ ) < 0, and lim z(τ ) = 0. τ →θ

Antras (2005) envisions one such standardization process in which productdevelopment intensity of the good is inversely related to product maturity. For τ this, he proposes a exponential standardization process, z = e− θ . My model can accommodate this special standardization, or a more general standardization process. However, to keep calculation and analysis simple I use a discrete and tractable standardization process.

54

curve for a unique differentiated good j, ∀j ∈ n(t). 1

Yj (t) = λ [pj (t)]− (1−α) ,

0<α<1

(2.11)

Where λ is a parameter given in equation (2.5) that the producer takes as given. Such producers maximizes profit by setting a price pj (t) that is a fixed mark up over marginal cost of production.

Since the production function is linear, the marginal cost of production is equal to unit cost of production. At time t, the cost of production of good j, cj (t), can be given in terms of unit factor prices, wage of skilled wh (t) and unskilled workers wl (t).   wh (t) for new goods cj (t) =  w (t) for old goods l

(2.12)

A monopolist has the unique ability to produce good j. The demand she faces for good j at time t, Yj (t), is given in equation (2.11). To maximize her profit, she solves the following optimization problem:

max [pj (t) − cj (t)] · Yj (t)

(2.13)

pj (t)

It is straightforward to check that to maximize her profits, the producer of

55

good j set the price as pj (t) =

cj (t) α

as the optimal price for good j. The price

pj (t) is a fixed mark-up over marginal cost, cj (t).

pj (t) =

 

wh (t) α

for new goods



wl (t) α

for old goods

(2.14)

If I denote the number of new goods at time t by nN (t) and number of old goods at time t by nO (t), then the instantaneous profit for the producer of good j at time t can be given as

(2.15) πj (t) = max [pj (t) − cj (t)] · Yj (t) = (1 − α) · pj (t) · Yi (t) pj (t)  h i wh (t) 1−σ  α    (1 − α) n (t)h wh (t) i1−σ +n (t)h wl (t) i1−σ E(t), for a new product j O N α α h i = wl (t) 1−σ  α   h h i i E(t), for an old product j,  (1 − α) wh (t) 1−σ wl (t) 1−σ nN (t)

α

+nO (t)

α

where the expression for Yi (t) is given in the equation (2.5). Let the relwh (t) ative wage of a skilled worker at time t be denoted by ω(t) = , the wl (t) instantaneous profit for a good i at time t can be rewritten as:

πj (t) =

   πN (t) = (1 − α) nN (t)

+

  πO (t) = (1 − α) n (t) N

+

n(t)

n(t)

1 nO (t) [ω(t)]σ−1 n(t) σ−1

[ω(t)]

nO (t) [ω(t)]1−σ n(t)

E(t) , n(t)

for a new product j

E(t) , n(t)

for an old product j. (2.16)

where n(t) = nN (t) + nO (t) is the total number of goods in the economy at t.

56

2.1.3

Labor Market Clearing Conditions

At any time t, as given in equation (2.10), one skilled worker produces one unit of a new good and one unskilled worker produces one unit of an old good. The derived demand for labor for each differentiated good is simply equal to the demand of that good. Total demand for skilled labor, Hp (t), (unskilled labor, Lp (t)), engaged in productive activities at time t can be obtained by integrating the demand for skilled (unskilled) labor over all new (old) goods available in the economy at time t.

Z Lp (t) =

lj (t)dj.

(2.17)

j∈nO (t)

Z Hp (t) =

hj (t)dj. j∈nN (t)

nN (t) and nO (t) are number of new goods and old goods available in the economy at time t.

2.1.4

Product Development

Following the endogenous growth literature, particularly Romer (1986, 1990), Grossman and Helpman (1991), I assume that the resources dedicated to

57

research lead to two types of outputs. First, a direct output that is the ability to produce a new differentiated product from the pool of infinitely feasible products. It gives the developer a monopoly over the production of the new good and earns her a stream of monopoly profits. Second is an indirect and unintended output. The development of each new good leads to the addition of general knowledge available in the economy. The underlying assumption is that such knowledge has widespread scientific applicability, and it increases the productivity of any such development efforts in the future.

Following Grossman and Helpman (1991), if K denotes the level of disembodied knowledge capital in the economy and ad denotes a fixed productivity parameter in the product development sector, the resources required to come ad up with a new product could be given as units of skilled labor.3 The total K available number of products in the economy can be used as proxy for the disembodied knowledge capital. If Hd , where Hd + Hp = H, is the number of high skilled laborers involved in the development work, the rate of development n˙ can be given as

n˙ =

n · Hd K · Hd = ad ad

3 Grossman

(2.18)

and Helpman (1991) describe the requirement to come up with a new product in a very similar fashion. Since, their model has only one type of factor of production, the same factor is used for the development. Here, I shy away from using both factors of production as this would unnecessarily complicate the model without adding any extra insight.

58

The cost to develop a new product at time t, V (t), can be given as

V (t) =

ad wh (t) n(t)

(2.19)

A newly developed product is a ‘new good’ for period T . After this period the good become an ‘old good’. The discounted value of cumulative profit for the innovator of good j can be given as Vj (t).4 Z



e−[R(τ )−R(t)] πj (τ )dτ

Vj (t) = t

Z

t+T −[R(τ )−R(t)]

=

e

Z



e−[R(τ )−R(t)] πO (τ )dτ (2.20)

πN (τ )dτ +

t

t+T

where subscript N denotes a new and O denotes an old product, and T is the time span that a newly innovated product takes for standardization.

The model allows for free entry. Accordingly, the discounted value of the cumulative profit for an individual producer Vj (t) should equal her development cost at time t, V (t). ad wh (t) = n(t)

Z

t+T −[R(τ )−R(t)]

e

Z



πN (τ )dτ +

t

e−[R(τ )−R(t)] πO (τ )dτ

(2.21)

t+T

Differentiating equation (2.21) with respect to time t while noting ad is a 4 It

is clear from equation (2.16) that πj (τ ) is independent of j, and depends only on the production process (old or new) employed at time τ to produce j.

59

constant, I can write, 

 w˙h n˙ ˙ · V (t) − πN (t) − e−[R(t+T )−R(t)] [πO (t + T ) − πN (t + T )] − V (t) = R(t) wh n (2.22)

If I use the expression for V (t) given in equation (2.19), equation (2.22) implies, w˙h n˙ πN (t) + e−[R(t+T )−R(t)] [πO (t + T ) − πN (t + T )] +( − ) R˙ = ad wh (t)/n(t) wh n (2.23)

The equations derived above can completely determine the evolution of the economy from any initial conditions. Provided E(0) is consistent with long term convergence, the economy attains a steady state.

2.2

Steady-State Analysis

I am interested in showing and characterizing long term rate of product development and the distribution of income in the economy. In the steady state, a fixed fraction of skilled workers participates in product development making

Hd ad

constant. I denote the growth rate of number of products ( nn˙ ) by

g in the steady state.

60

In the model, there is no monetary authority. So, I am free to give an arbitrary value to one of variables in the model. Following Grossman and Helpman (1991), I fix the economy’s expenditure at every time equal to the number of products available at that time, E(t) = n(t). Using this normalization and equation (2.9), the instantaneous interest in the economy can be written as n˙ R˙ = ρ + n

(2.24)

Since, total expenditure in the economy, E(t), is equal to the number of products in the economy, in the steady state, the economy’s expenditure grows at constant rate g. Given the fixed supply of skilled and unskilled labor, the growth rate for both wages would also be the same as g. The equation (2.24) implies: R˙ = g + ρ

(2.25)

The equation (2.25) gives the instantaneous interest rate as sum of the discount rate and the growth rate of the number of varieties. In the previous section, equation (2.23) also gives instantaneous interest rate in the economy. Equating these two instantaneous interest rates provides the no-arbitrage condition in research and development sector. To calculate the value of instantaneous interest rate in the economy using equation (2.23), one needs to   N (t) N (t) know the proportion of old and new goods nn(t) and nn(t) in the economy.

61

In the steady state, the proportion of new and old goods can be expressed in terms of g, which is the growth rate of number of products in the economy ( nn˙ ), and T , which is the time span after which production of a newly innovated good shifts from skilled workers to unskilled workers.

Old Products, nO(t)

n(t-T) t-T

dn(t-s) = gn(t)e-gsds ds

s

n(t)

t Current Time

New Products, nN(t)

Figure 2.1: Timeline in The Economy.

t

= current time.

n(t)

= the number for products innovated in the economy up to time t.

nO (t)

= n(t − T ) =n(t)e−gT = the number of old products in the economy at time t.

nN (t)

= n(t)(1 − e−gT ) the number of new products at time t. (Grows at the rate of g).

dn(t − s) = g · n(t)e−g·s ds = the number of products with maturity s < T (innovated between t − s to t − (s + ds) time).

Notice that in the steady state, the ratios

nN (t) n(t)

and

nO (t) n(t)

are independent

of time t. I have taken E(t) = n(t) as the num´eraire in the economy. The number of products in the economy, n(t), grows at rate g in the steady 62

state, therefore, E(t) also grows at rate g in the steady state. Further, to achieve the steady state, wk , k ∈ {h, l}, must grow at the same rate as the growth rate of Ek . From equation (2.9), E˙ E

=

w˙ h wh

=

w˙ l wl

E˙ E

=

E˙ k , ∀k Ek

∈ {h, l}, which implies

= g. Also notice that the expressions for πN (t) and πO (t) given

in equation (2.16) depend only on

nN (t) nO (t) , n(t) , n(t)

ω(t) and

E(t) ; n(t)

all of these are

constant in the steady state, implying πN (t) and πO (t) are independent of time in the steady state.

In the steady state, as the growth rate in the wage is equal to the growth rate in the number of products. The profit from producing a good is independent of time t, as

n wh

is constant in the steady state. I can rewrite the expression

for R˙ given in equation (2.23) as πN + e−[R(t+T )−R(t)] [πO − πN ] ˙ R = ad wh /n

(2.26)

˙ given above and in equation (2.25), πN and Using the expressions for R, πO , given in equation (2.16), the steady state equilibrium values of ratios nN (t) n(t)

and

nO (t) n(t)

derived earlier, and numeraire E(t) = n(t), I can give the

no-arbitrage condition in research and development sector as:   n 1 + e−(ρ+g)T [ω σ−1 − 1] ρ + g = (1 − α) ad · wh 1 + e−gT [ω σ−1 − 1]

(2.27)

To explore the equilibrium condition in research and development sector,

63

one needs to know the steady state value of

n . wh

It can be obtained from

labor market clearing condition in the equilibrium. No one in the economy is unemployed. It means that both skilled and unskilled labor markets clear. Equation (2.17) gives the expression for labor involved in productive activities. I can obtain the expression for labor involved in development activities from equation (2.18). The labor market clearing condition can be written as5 .

H = Hp + Hd

(2.28)

L = Lp

where Hp and Lp are given in equation (2.17). Using the expression for Yj (t) given in equation (2.5), Hp and Lp can be expressed in terms of g, ω and system parameters such as T and ρ.

Hp =

α 1 − e−gT n(t) wh (t) 1 − e−gT + e−gT ω σ−1

(2.29)

Lp =

α e−gT ω σ−1 n(t) wl (t) 1 − e−gT + e−gT ω σ−1

(2.30)

Using equation (2.18), I obtain the number of skilled workers engaged in 5

Labor supply is inelastic in the economy, therefore,Lp (t) is always equal to L. Further, Hd (t) is constant in the steady state. So, Hp (t) = Hp

64

development activity.

Hd = g · ad

(2.31)

Using equations (2.29), (2.30), and (2.31), one can obtain the expression for n(t) in the steady state. By multiplying equation (2.29) with wh (t) and wh (t) equation (2.30) with wl (t), adding them up, then, dividing the sum by wh (t), and using equation (2.31) for Hd , and equation (2.28) for Hp as a function of H and Hd , I obtain:   1 n(t) L = H − ad g + wh (t) α ω

Since σ =

1 , 1−α

(2.32)

using equations (2.32) and (2.27), the equilibrium condition

in the research and development sector can be rewritten as:    1 L 1 + e−(ρ+g)T [ω σ−1 − 1] ρ+g = H − ad g + ; (σ − 1)ad ω 1 + e−gT [ω σ−1 − 1] (2.33)

Now, similar to the equilibrium condition for research and development sector, I can write an equilibrium condition for labor market clearing in terms of variables relative wage (ω), growth rate in number of products, g, and parameters, standardization period, T , elasticity of substitution σ and productivity parameter in research sector, ad . Dividing equation (2.29) by equation (2.30) 65

and rearranging, I get the equilibrium condition in labor market.  ω = (egT − 1)

2.3

L H − ad g

 σ1 (2.34)

Long Run Determinants of the Growth Rate and the Relative Wage

In the previous section, I obtained two conditions, labor market clearing condition (equation 2.34) and research no-arbitrage condition (equation 2.33). A pair of {g, ω} that satisfies both conditions gives the steady state growth rate g ∗ and relative wage ω ∗ in the economy. I am interested in the determinants of these long run values of growth rate and relative wage.

In figure (2.2), the M-M curve represents the labor market clearing condition given in equation (2.34). This curve is upward slopping and represents the combinations of steady-state rates of growth and relative wage that are consistent with both the labor market clearing conditions in the economy. An increase in g takes skilled workers away from productive activities to research activities, thereby creating a scarcity of skilled workers in production, which in turn, raises the skilled wage for labor market to clear.

The equation representing the equilibrium condition in research and devel66

opment sector is not as straightforward as the other equilibrium condition. To get better understanding of this equation, consider a special case: ρ = 0, implying that consumers are very patient; they do not discount the future consumption. McGrattan (1994) estimates various parameters involved in growth models and suggests the value of discount rate ρ equals 0.0075. With this condition (ρ = 0) the research and development equilibrium condition given in equation (2.33) reduces to the following form:

ω =

L σad g − H

(2.35)

The curve representing equation (2.35) in {g, ω} plane is downward slopping and represents the combinations of steady state growth rate and relative wage that are consistent with the equilibrium condition in research and development sector, given ρ = 0. The curve describing this equilibrium condition in {g, ω} plane is denoted by R-R in figure (2.2). An increase in ω makes research work expensive, thereby it decreases the number of skilled worker employed in research activities, which in turn, decrease the growth rate of the number of products in the economy. As ρ is very low in real world economies, for a practical purpose the equilibrium condition in research and development sector (equation 2.33) behaves in the same fashion in {g, ω} as the curve R-R (representing equation 2.35) does.

67

ω

R'

M H↑ or L↓ M'

R H↑ or L↑

ω*' ω* M R'

M'

R

g*

g*'

g

Figure 2.2: Equilibrium Conditions and the Effects of a Change in the Labor Supply

2.3.1

Effects of an Increase in Labor Supply

An increase in H, as shown by equations (2.33 and 2.34) and in figure (2.2) shifts M-M curve down and R-R curve up, leading to an increase in the 0

equilibrium value of g ∗ . The increased value of g ∗ is denoted as g ∗ in the figure (2.2). From the same figure, it is seen that the increase in H has an ambiguous effect on the equilibrium value of ω. An increase in H increases the supply of skilled workers participating in productive activities causing their relative wage to fall, which in turn, shifts the M-M curve down. At the same time, an increase in H leads to two different effects on the R-R curve. First, an increase in H increases the demand for all goods in the economy 68

increasing the profit per differentiated variety, as a consequence research and development activities in the economy increase. This effect is known as “scale effect”.6 Second, an increase in H also increases the number of innovations fueling the demand for skilled worker to manufacture newly developed new goods. I call this effect the “innovation effect”. The scale effect and the innovation effect work in the same direction and shift the R-R curve up. The magnitudes of effects of an increased H on M-M and R-R curves depend on the various parameters in the economy making the overall effect of a change in H on the equilibrium value of ω ∗ ambiguous.

An increase in L as shown by equation (2.34) shifts both M-M and R-R curves up, implying an increase in the equilibrium value of relative wage, ω ∗ and an ambiguous effect on the equilibrium value of growth rate, g ∗ . An increase in L increases the supply of unskilled workers involved in the production of old goods causing their relative wage,

1 , ω

to fall. This effect leads to an upward

shift in M-M curve. Also, an increase in L, like an increase in H, generates a scale effect implying an increased profit for a differentiated good. The scale effect increases the return from a new innovation, and thus, moves the R-R curve up. 6I

discuss this effect in detail in the next section

69

2.3.2

Effects of an Increase in Productivity Parameter

An increase in the productivity parameter in research and development sector, ad , increases the cost of a new innovation that creates a disincentive to participate in research and development sector. This disincentive, in turn, leads to a downward shift in R-R curve. Similarly, an increase in ad necessitates an increased number of skilled workers per innovation that implies a decreased supply of skilled workers in production sector. This shift, in turn, raises the wage for a skilled worker and leads to an upward shift in M-M curve. Overall, an increase in ad decreases the equilibrium value of economy wide growth rate g ∗ . The overall effect of an exogenous increase in ad on the equilibrium value of relative wage ω ∗ is ambiguous as the change in the productivity parameter affects the two equilibrium condition for the determination of ω ∗ in an opposite way.

2.3.3

Effects of an Increase in Standardization Time

When I take the discount rate as 0, the standardization period T does not appear in the equilibrium condition in research and development sector (equation 2.35). However, an increase in T shifts the M-M curve up. A higher T means that skilled workers’ service is required for a longer period to produce the same good, and by the same token, unskilled worker’s service is required

70

ω

ad↑

ω* ω*'

ad↑

g*'

g*

g

Figure 2.3: Equilibrium Conditions and Effects of Change in Research Productivity Parameter. for lesser time period. Therefore, an increase in T shifts M-M curve up and does not affect R-R curve. The analysis implies that an increase in T , when future discount rate is low, increases the equilibrium value of relative wage, ω and decreases the equilibrium value of growth rate g ∗ .

Table 2.1 shows the effects of changes in parameters on curves representing equilibrium conditions and equilibrium values of long term growth rate and relative wage in the economy. Dash, ‘-’, denotes ambiguous effect and “0” implies no change.

71

Effects on Equilibrium Values

Curves Parameters

M-M

R-R

g∗

ω∗

H↑ L↑ ad ↑ T↑

↓ ↑ ↑ ↑

↑ ↑ ↓ 0

↑ ↓ ↓

↑ ↑

Table 2.1: Effects of changes in parameters (With Scale Effect)

2.4

Extension: Product Cycle without the Scale effect

The source of economic growth, in the model described above, is knowledge creation. Knowledge is very unique as it is non-rival in nature: the use of a piece of knowledge by one economic agent does not preclude the simultaneous use of the same piece by another agent. In knowledge-based growth models, the growth rate of the economy is directly related to the amount of knowledge created in the economy. When the rate of knowledge creation is linearly dependent on the available knowledge in the economy and labor employed in research, as described in the earlier model, any increase in labor supply raises the resources devoted to knowledge creation linearly and leads to an exponential increase in per capita growth rate. This effect is known as scale effect. It implies an accelerating per capita income growth in presence of population growth. Jones (1995) was first to point out the lack of evidence in support of such accelerated growth in presence of population growth. Sub72

sequently, endogenous models with scale effect came under heavy criticism. Later theorists have attempted to remove the scale effect from endogenous growth models.7 I follow the original prescription by Jones (1995) to remove the unintended scale effect from my model.

In the previous model, I suggested that the number of products in the economy can be a proxy for the disembodied knowledge (K) in the economy. Jones (1995) shows that this particular choice of the proxy leads to the scale effect in models based on Grossman and Helpman (1991). To remove the effect, Jones suggested an alternative formulation which differs from GrossmanHelpman’s formulation in two important ways. First, the productivity of a worker is negatively affected by the total number of skilled labor devoted to research and development. This relationship can be attributed to the possibility of duplication in research. Second, the knowledge in the economy is a concave function of the total number of goods available in the economy. With the assumption that the technology of research exhibits constant returns to scale with respect to the number of labor devoted to research at the firm level, Jones gives the following formulation for the product development at the firm level: 1 hd nφ Hdλ−1 , ad

(2.36)

where 0 < φ < 1 and 0 < λ < 1, and hd is unit of labor employed by a firm, ad , 7 See

Jones (1999) for an excellent survey of such models.

73

a labor productivity parameter, n, number of good available in the economy, and Hd is the total skilled workers in the research and development sector. In equilibrium, by aggregating over all firm in research and development, one P gets hd = Hd which gives the total number of new products developed in the economy at date t is given by n. ˙

n˙ =

nφ · Hdλ . ad

(2.37)

The rate of innovation is given by g = nn˙ . The growth rate in g can be written as H˙ d g˙ = −(1 − φ)g + λ g Hd

(2.38)

Unlike the model in the previous section, here the growth rate, g, is not endogenous and depends on population growth rate and other exogenously given parameters. If there is no population growth in the economy, there will be no growth in this model. Therefore, I introduce an exogenous population growth with rate η in the economy. The populations, of skilled workers and unskilled workers, grow at the rate η. In the steady state the fraction of skilled worker employed in the research and development work would be a constant that implies

H˙ d Hd

=

H˙ H

= η that, in turn, implies a constant g in the

74

steady state.

g =

λη 1−φ

(2.39)

Population growth also affects the optimal condition on the consumer side given in equation (2.9) as the evolution of economy-wide expenditure will no longer follow the same path as the evolution of any individuals’ expenditure. The economy’s total expenditure in period t is E(t) = L(t).El (t)+H(t).Ek (t). This gives: E˙k E˙ = η+ = η + R˙ − ρ, E Ek where where, η =

L˙ L

=

H˙ , H

(2.40)

andk ∈ {h, l}. Total spending in the economy

grows at the rate equal to individuals’ spending growth rate corrected by the population growth rate.

I define the economy-wide labor productivity, ment sector as

m ad

n˙ , Hd

in research and develop-

expression for which is given in the following equation: m n˙ nφ · Hdλ−1 = = , ad Hd ad

(2.41)

m is a scaled labor productivity in the economy which follows the growth path of labor productivity in the R & D sector, and its steady state growth

75

rate can be given by m ˙ = φg − (1 − λ)η = ψg, m

where g =

λη 1−φ , and ψ = 1 − . 1−φ λ (2.42)

The growth rate of labor productivity is positive only when λ > 1 − φ.

As in the previous model, there is no monetary authority in this model. So, I am free to give an arbitrary value to one of variables in the model. For convenience I follow Lai (1998) and normalize the labor productivity in the R & D sector, m(t) by making it equal to a skilled worker’s wage, wh (t). This particular specification implies that the wage paid to a skilled worker is always proportional to labor productivity in R & D sector. It also implies that an innovator incurs a fixed cost equal to ad to come up with a new h (t) product as innovation cost for a new good is ad wm(t) . In other word, this

particular normalization makes the cost of a new design the numeraire in the economy. This normalization also implies that wages and expenditures for skilled and unskilled workers grow at the same rate as labor productivity in the R & D sector, while the economy wide expenditure grows at the same rate as number of products in the economy.8

The new specification for product development would imply that new equi8 Rate

of growth in economy wide expenditure is equal to the sum of the rate ˙ m ˙ of growth in population and an individual’s expenditure. E E = m + η = ψg + η, Using expressions given in equation (2.42), ψg + η can be shown equal to g

76

librim condition in R & D sector differs from the condition given in equation (2.23). The new equilibrium condition would be: πN + e−[R(t+T )−R(t)] [πO − πN ] R˙ = ad

(2.43)

In the steady state, the growth rate of number of products ( nn˙ ) is given by g. In the steady state, the economy’s expenditure grows at constant rate g. Then, equation (2.40) implies: R˙ = g + ρ − η

(2.44)

˙ given in equation (2.44), πN and πO , given in equaUsing expressions for R, tion (2.16), and steady state equilibrium values of ratios

nN (t) n(t)

and

nO (t) , n(t)

derived earlier, I can rewrite the above equation as:   (1 − α) 1 + e−(g+ρ−η)T [ω σ−1 − 1] E(t) g+ρ−η = ad 1 + e−gT [ω σ−1 − 1] n(t)

(2.45)

The above equation gives the equilibrium condition in the research and development market. It is important to note that equation (2.45) is not a no-arbitrage condition, as the economy’s growth rate g is no longer an endogenous variable. Instead, g is determined by population growth rate η. The relationship between g and η is given in equation (2.39). Expression for E(t) n(t)

is given in Appendix (equation 2.51). If I substitute the value of

E(t) n(t)

from equation (2.51) in equation (2.45). The resulting expression gives the number of skilled workers engaged in and R & D activities. Using equations 77

(2.45) and (2.29), I obtain the expression for Hp (t) = H(t) − Hd (t), in terms of H(t), g, T and ω.9   α[g + ρ − η] 1 − e−gT H(t) Hp (t) = α[g + ρ − η][1 − e−gT ] + (1 − α)g [1 + e−(g+ρ−η)T [ω σ−1 − 1]] (2.46)

Now, I can write the equilibrium condition for labor market clearing in terms of variables relative wage (ω), growth rate in number of products, g, and parameters, maturity period, elasticity of substitution σ. Dividing equation (2.46), by equation (2.49), given in appendix, and rearranging, I get the equilibrium condition in labor market.

ω

∗σ

 =

gT

e

where g ∗ =

2.4.1

  (1 − α)  gT  L(t) g −(ρ−η)T ∗(σ−1) +1 + e +e ω −1 α [g + ρ − η] H(t) (2.47) λη . 1−φ

Comparative Statistics

In the steady state of this model, the number of varieties grows at the constant rate g ∗ =

λη 1−φ

which is exogenous to the model and relative wage

remains at the level ω ∗ which is given in equation (2.47). Unlike the previous model, there is only one endogenous parameter, relative wage ω, in this 9 See

appendix for formulation

78

model.

Effects of an Increase in Labor Supply

An increase in H(t) at time t, keeping all other parameters constant, lowers the steady state relative wage ω ∗ in the economy and does not affect the growth rate g ∗ in the economy.10 Equation (2.47) is an identity in the steady state. From equation (2.46),

Hp (t) H(t)

is independent of H(t) in the steady state.

Therefore, an increase in H(t) does not affect the proportion of skilled workers employed in research and development sector. However, it raises the ratio of skilled workers employed in productive activities with respect to unskilled worker, which in turn lowers the equilibrium relative wage. Similarly, an increase in L(t), keeping all parameters constant, increases the steady-state relative wage and does not affect the steady state growth rate g ∗ in the economy.

Effects of an Increase in Standardization Period

An increase in T does not affect the steady state growth rate and increase the relative wage ω ∗ . A higher T implies longer service from skilled workers which increases the relative wage in the steady state. 10 See

the derivation in Appendix

79

Effects on Equilibrium Values Parameters

g∗

ω∗

H↑ L↑ ad ↑ T↑

0 0 0 0

↓ ↑ 0 ↑

Table 2.2: Effects of changes in parameters (In absence of Scale Effect) The effect of economy wide parameters on ω∗ and g ∗ is given in table (2.2).

2.5

Conclusion

Grossman and Helpman (1991) show that relative wage between North and South varies directly with the size of the labor in the these region, and the long-run rate of innovation in North and imitation in South are determined endogenously. By assuming an innovation process in the North similar to Grossman-Helpman and replacing the process of imitation by South in Grossman-Helpman model by product standardization in the North (No trade), I find that an increase in the size of unskilled worker will decrease its relative wage. An increase in the size of skilled workers which is required for both production and research might or might not increase its relative wage. The relative wage of skilled workers will increase with increase in their population only when the research effect dominates the population effect. The

80

replacement of Grossman Helpman type innovation by Jones (1995) type of innovation in the model wipes out the scale effect and suggests that an increase in skilled workers’ population decreases its relative wage in a closed economy. Further, the paper using the insight from product cycle literature, shows how an endogenous growth that is skill neutral can have the same impact on income distribution as international trade and skill-biased technical change.

81

2.6

Appendix

Using the expression for Yj (t) given in equation (2.5), Hp and Lp can be expressed in terms of g, ω and system parameters such as T and ρ.

Hp (t) = H(t) − Hd (t) =

Lp (t) = L(t) =

α 1 − e−gT E(t) wh (t) 1 − e−gT + e−gT ω σ−1

α e−gT ω σ−1 E(t) wl (t) 1 − e−gT + e−gT ω σ−1

(2.48)

(2.49)

E(t) in the wh (t) steady state. By multiplying equation (2.48) with wh (t) and equation (2.49) Using equations (2.48), (2.49), one can obtain the expression for

with wl (t), adding them up and using equation (2.31) for Hd (t), I obtain:   1 E(t) L(t) = H(t) − Hd (t) + wh (t) α ω Using wh (t) = m(t), m(t) = ad Hdn˙(t) from equation (2.41), and

(2.50) n˙ n

= g in

equation (2.50), I obtain:   E(t) ad g L(t) H(t) − Hd (t) + = n(t) Hd (t)α ω Using equations (2.45) and (2.51), I can write: L α [g + ρ − η] 1 − e−gT H − Hd + = Hd ω (1 − α) g 1 + e−(g+ρ−η)T [ω σ−1 − 1]

82

(2.51)

(2.52)

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