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and therefore may not be sustainable as an equilibrium Inefficiency may set inplayers may follow first-best policies of accumulation at higher levels of wealth and first-best policies ma

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Social Conflict, Growth and Inequality

by

Jess BenhabibDepartment of EconomicsNew York University

New York, NY 10003

andAido RustichiniDepartment of EconomicsNorthwestern University

Evanston, IL 60208

August 1991 Revised April1992

Social Conflict, Growth and Inequality

byJess Benhabib

New York University

and Aldo RustichiniNorthwestern University

AbstractDespite the predictions of the neoclassical theory of economic growth,

we observe that poor countries have invested at lower rates and have not grownfaster than rich countries To explain these empirical regularities we

provide a game-theoretic model of conflict between social groups over thedistribution of income Among all possible equilibria, we concentrate onthose which are on the constrained Pareto frontier We study how the level ofwealth and the degree of inequality affects growth We show how lower wealthleads to lower growth and even to stagnation when the incentives to domesticaccumulation are weakened by redistributive considerations

JEL Classification numbers

Key Words: Dynamic Games

010, C73

Department of EconomicsNew York University

269 Mercer Street, 7th FloorNew York, New York 10003 USA

1 Introduction.

Neoclassical growth theory predicts that poor countries because of the law

of diminishing returns, should grow at faster rates than rich countries

strengthened by the diffusion of technology and the opportunites for "catch-up"

flat and possibly hump-shaped, not downward sloping

and Wolff [1988), Figure 2; and Easterly [1991).) To explain this discrepancy

on endogenous growth theory has introduced economy-wide externalities threshold

Here we pursueeffects and other mechanisms that overcome diminishing returns

between the levels of wealth, social and political conflict, and the incentives

As such our work is related to that of Persson and Tabellinifor accumulation

and Alesina and Rodrik 1991

[1991

We have in mind a situation where organized social groups can capture, or

1 Some of the non-violent redistributive mechanisms that are used indeveloping countries include nationalization; bursts of inflationary finance tosustain the incomes of government bureaucracies and the military; the squeezing

of the agricultural sectors in favor of politically powerful urban classesthrou~ exchange rate policies price controls and monopolistic marketing boards;legislation and other measures that alter the bargaining power of labor (eitherpositively or negatively); the allocation of highly desirable government and

country these groups may represent, among others, organized labor industrial and

the military, the bureaucracy, o~ racial, ethnic and

business associations,

tribal groups.2 Such redistributive and expropriative activities undertaken by

accUJDulate, which

or threshold effects in the production technology (See for example Romer 1986]

or Azariadis and Drazen [1990}.)

The recent empirical literature on the convergence- hypothesis (see Barro

[1991} Levine and Rene1t [1991}} suggests that the lower growth rates observed

reestablished, Indeed investment rates in physical and human capital (primaryschooling) are negatively correlated with income levels (see Fisher 1991], Table

3) Furthermore investment rates show a robust negative correlation with various

and large scale bureaucratic corruption tolerated and condoned by the government.

[1974]; Laothamatas [1992]; Malon and Sourri11e [1975]; O'Donnell [1973],[1988]; Peralta-Ramos [1992]; Sachs [1989]; Veliz [1980], chapter 13.; and thevarious essays in Goldman and Wilson [1984], and in Nelson [1990]

2 The role of the enforcement of property rights by the state to internalizesocial gains and promote growth has been discussed by D North [1981], [1991] in

a historical context For a wide-ranging historical analysis of the role ofrational collective action by social groups in the political arena, see Tilly[1978] The effects of rent-seeking behavior by organized groups on the economicefficiency of mature economies has been studied by Olson [1982] See also Becker[1983], Romer [1990] and Brock and Magee [1978]

Venieris and Gupta [1986]) and there is a negative relationship between measures

then that poorer countries are more prone to political instability have lowerinvestment rates and consequently may not have realized their growth potential

to catch-up with rich countries

The historical evidence is in line with the evidence from cross-national

studies as well

and imperceptible growth, the richer nations of Europe, together with the US and

prices.4 This is higher than the 1988 per capita GNP, at 1985 prices of about

set.S These observations reflect the well-known Landes-Kuznets thesis, which

recently has been reconfirmed by Maddison [1983.].' Kuznets summarized this

thesis in his Nobel prize speech in 1971 "The less developed areas that accountfor the largest part of the world population today are at much lower per capita

industrialization.- We must however be cautious in drawing comparisons between

3 Countries like Taiwan and Korea on the other hand have had strong growthperformance despite their low initial income levels However, the eliminationand suppression of landlord classes under Japanese occupation and strong armtactics towards business and labor unions to implement liberalization in the1960's and 1970's may have been critical elements See Amsden [1988], Jones andSakong [1980], Datta-Chauduri [1990] and Westphal [1990]

4 The countries are Austria, Belgium, Denmark, France, Germany, Japan,Netherlands, Norway, Sweden, Switzerland, UK and USA The UK led with a percapita GNP of about $1311

S Without adjusting exchange rates for purchasing power differences in 1988

half the countries in the world had GNP levels below $974 See the WorldDevelopment Report [1990]

Compared to

the present and the past world of a hundred and seventy years ago

the present, 19th century European governments were significantly more repressive

of social classes,

As wealth levels increased, redistributive pressures were in part accomodated by

welfare worldwide

above, we use a simple dynamic game framework in which each player independently

chooses a consumption level and the residual output, if any, becomes the capital

1980],

(See also Torne11 and Velasco

Majumdar and Sundaram [1991], and many others.

We consider equilibrium paths of accumulation in which players receive[1990].)

thoseutilities that are at least as high as

appropriating higher immediate consumption levels and suffering some retaliation

(For a related framework of analysis, see Karcet and Marimon [1990];later on

Chari and Kehoe [1990]; Kaita1a and Pohjo1a [1990].) We focus, however, on thosesubgame-perfect equilibria which are second-best, that is on a subset of subgame-

perfect equilibria which lie on the constrained Pareto frontier Within this set

steady state income levels.

We also consider cases which produce classical "growthleads to lower growth

Even though first-best policies lead to growth, along second-best traps

incentive constraints:

appropriation and to high consumption levels by other players and therefore may

not be sustainable as an equilibrium

Inefficiency may set inplayers may follow first-best policies of accumulation

at higher levels of wealth and first-best policies may have to be abandoned asthe incentives for appropriation grow and redistributive pressures increase

possibility that inefficiencies are associated with stable and wealthy economies

in which organized groups have had the time to mature and to exert redistributive

pressures has been suggested by Mancur Olson

section 7 below

There may be good evolutionary or institutional reasons to focus on second

For our purposesbest equilibria which lie on the constrained Pareto frontier

however, there is an additional and compelling reason to study symmetric, that

In section 3 we show that when incentive

is egalitarian, second best equilibria

constraints are binding, the fastest growing sub game perfect equilibrium is the

if incentive constraints are

(egalitarian) second best equilibrium sets an upper bound to the growth rates at

non-low wealth levels

Our model therefore

symmetric or inegalitarian second best, must be even lower

implies that for any given level of wealth, there is a trade-off between growthand inequality, where inequality is measured by the disparities of consumption

High rates of accumulation in

rates and welfare

because the disadvantaged groups can undertake redistributive actions or exert

The political

attainable if income

consensus necessary for efficient growth may not be

country data, Venieris and Gupta [1988] established the negative effect of income

And more recently Persson and Tabellini [1991

inequality on investment rates

affects growth rates

Our paper is organized as follows The next section sets up the problem

that among all equilibria, the symmetric (egalitarian) second best is the fastest

Section 4 works out a simple and illustrative second best problemgrowing one

where incentive constraints retard growth but accumulation rates do not depend

incentive constraints along the symmetric (egalitarian) equilibrium Section 5 provides some general conditions under which a political "growth trap occurs

Again a numerical without having to explicitly compute the "second best"

example is provided Section 6 computes an explicit example of a growth trap with a discontinuous value function Section 7 illustrates the "Olson" case

that is the case where first best policies are optimal at low stock levels butcannot be sustained at high stock levels Section 8 contains some final remarksand a discussion of the role of the state in economic growth

2 The Second Best Problem.

sequences must satisfy f(kt) - Ctl - Ct2 ~ kt+l' and c: ~ 0, t = 0, 1, .; i =

1, 2 In our game, histories at time t are sequences of consumption pairs

~ = (cl,cl" ,Ct ,Ct an strateg1es are maps rom 1stor1es 0 consump 1ons

For a given initial stock k, the second best value is defined by

vsb(k) = sup :E~ pt[olUl(Ctl) + °2U2(c;)] (2.1)

t

where the supremum is taken over the sequences (Ctl, Ct2)t~O of subgame perfect

equilibrium outcomes Here °1' °2 ~ O

The purpose of this section is to prove that the second best is achievedover a smaller set of SPE To avoid ambiguities, we describe in detail how theallocation of consumDtion is regulated It will be useful to distinguish betweenattempted consumption and consumption (the first is the consumption a player istrying to get, the second is what the allocation rule gives him) For a givencapital stock k and two attempted consumptions cl and c2' the allocatedconsumption is

8

Cl + C2 ~ £(k) or Cl ~ £(k)/2

if

Cl

Cl+CZ~ f(k) and Cl ~ f(k)f2 ~ C2if

Al (C1J CZJ k) - f(k) -C2

C1' Cz ~ £(k)/2

Note that if Cz $ f(k)/2, then Al(Clt cz, k)

-and similarly for~.

min ( Cl' f(k) -C2

fast consUInution stratev.ies,

constitute an equilibrium for all values of the capital stock k

cl(k) -cz(k) - f(k)

Note in fact that in this case the allocation rule gives Al(Cl' cz k) - Az(Cz

Also note that if the second player attemptsCl' k) - f(k)/2 to both players

the capital stock in the next period is

to consume f(k) , for any choice of Cl

So by reducing cl the first player can only reduce his payoff

zero

While we adopted a symmetric specification for the allocation rule, this

To assign asymmetric appropriation powers to the playerscan easily be modified

we could have assumed that one of the players can obtain up to say 3/4 of the

What sustains fast consumption as

of the opposing player to 1/4 of the output

an equilibrium is that any attempt to save by a player is defeated because the

Positive savings may be possible ifopposing player then exhausts the residual

the fast consumption rates of one or both players are bounded, that is if

where Cjf(k} - cJ > 0 is the bound for the consumption rate of the j ~ playerConditions under which fast consumption rates are still equilibrium strategies

when positive savings rates are possible have been studied (in a continuous timeframework) by Benhabib and Radner [1992] for the case of linear utility, and byRustichini [1992] with non-linear utility.

Under the allocation rule described above, the worst SPE is easilydescribed This allows us to utilize the worst equilibrium in order to sustainany other SPE with trigger strategies Under more general allocation rules theworst SPE is difficult to characterize Note, of course, that one may alsoarbitrarily choose a simple SPE, for example the interior Markovian one underwhich the sum of attempted consumptions never exceeds output, and then study theset of SPE that are enforcable with trigger strategies using that Markovian SPE

as a threat (For such an example see the end of section 4.)

As noted above, it is clear that the pair (Cl' cz) is a SPE, since theutility functions of the players are strictly increasing The value of thisequilibrium to player i is given by:

vi(k) = L ~ fJtUi(Ai(Cl(~)' cz(~)'~», i =1,2

t

where ko = k, kt+l = f(kt) - Al(Cl(kt) , cz(kt) , kt) - AZ(Cl(kt) , cz(kt) , kt), t ~ O

Of course if f(O) - 0, the above summation reduces to Ui(f(k)/2) A triggerstrategy pair is described by an agreed consumption path (Ctl, Ctz)t~O and the

threat of a shift to a fast consumption equilibrium after the first defection isdetected The individual rationalitv constraint for player i on an outcome path

is the condition:

10

LPt.U(c;) ~ Vi (k) t.

satisfies this inequality

Consider now a trigger strategy equilibrium For any capital stock k and

value for a player of deviating optimally, that is:

denote by C1D(k, c) the consumption giving the optimal deviation In the games

possible there

We state and prove it for completeness

The following lemma is clear

1 2Let (~'~)t.~o be the outcome of a SPE, ; say Then the trigger

Lemma 2.1

alternative choice is cz (~ Ct) - C In the equilibrium ~ such a choice would

give him a payoff of U2(cD) plus the equilibrium value of the subgame starting

at (~ c; CD). In the equilibrium of this subgame the individual rationality

constraint is satisfied, so

It follows that the supremum in the definition of second best is the same

This reduction allows us also

as the supremum over trigger strategy equilibria

to prove that the second best value is in fact achieved We turn to this now

Let °1' °2 ~ 0 be weights attached to the players From what we have seen,second best is the solution of the problem

In the following we shall refer to this as the second best vroblem

We assume now that the production function f and the discount factor p

1. A solution to the second best problem exists

The function v.b is uppersemicontinuous.7

2.

7 Section 6 provides an example of a discontinuous value function Ingeneral, the conditions to apply dynamic programming methods and the contractionmapping, which assure the continuitl of the value function, may not hold for ourproblem In particular, Blackwell s discounting condition, T(v+a) ~ T(v) + pamay be violated because the constraint set for V+a becomes larger, allowingconsumption levels that would be ruled out under v.

of sequences {(~, Ctl, Ctz)t?;o} such that (1) and (2) above are satisfied In aproperly chosen weighted space this set is compact (because it is a closed subset

of an order interval) Now existence follows immediately from the continuity of

the funct1.on (Ct' Ct )t?;O -+ L.,{3 [alU1 (Ct) + azUz(ct)] For the second statement,

note that the correspondence defining the set of admissible paths has a closedgraph, and since the image space is compact, it is also uppersemicontinuous Now

For an extension of the above results to renegotiation proof equilibria insome special cases see Benhabib and Rustichini [1991]

3 Why Do We Study Second Best Eauilibria?

It may be reasonable to expect that institutions and bargaining mechanismsthat implement second best outcomes will evolve in societies where the strategicbehavior of social groups has a strong influence on the production and on thedistribution of output In this section however we motivate our choice of secondbest equilibria as the focus of analysis by another consideration We areprimarily interested in studying growth rates of economies when incentiveconstraints are binding, at least on some part of the equilibrium trajectory

In such situations we show that the symmetric (egalitarian) second bestequilibrium is the one that affords the fastest growth rate over all subgameperfect equilibria and therefore represents an upper-bound to growth rates Whenincentive constraints do not bind, on the other hand, second best equilibria maynot lead to the fastest growth For example, it is obvious that if first-best

14

outcomes can be sustained by trigger strategies, less efficient equilibria with

particular focus is on those equilibria which are incentive constrained either

initially, and maybe forever, at low levels of wealth (the classical case), orultimately, as wealth reaches higher levels (the Olson case)

Before we proceed with a formal proof of a more general case, let us notice

that the above claim easily follows if we only allow comparisons among symmetric

equilibria To achieve a faster growth, consumption must be reduced, and so, for

players On the other hand, since we had a symmetric second best, the value for

defection were initially equal, now the second is greater than the first

4 A Simple Example of Second Best EQuilibrium with no Wealth Dependence.

We will start by exploring a simple case of a second best equilibrium indetail to illustrate how growth rates may differ between first and second bestequilibria This first example is simple because growth rates on equilibrium paths will turn out to be independent of the levels of wealth, that is of the

capital stock More interesting and complex cases will be studied later

Each of the two identical players in this example have an instantaneous

utility function given by

(l-C)-lCl-.

U(c

where 0<.8<1 and 0 < E < 1 Production is linear, and is given by

y - ak

1 for the problem given by (2.2) above) where both players get equal consumption

equilibrium can be described by a dynamic program:

-) ~ 0

and where we have imposed the restrictions

negative consumption levels and to assure a well-defined value function, For any

where

~ ~ 0, the value function is given by v(k) - sy

(4.5)

is derived here for arbitrary ~ ~ 0, not only

We note for further use that s

.

function later on

strategies will be enacted subsequently Optimal defection value is thereforegiven by

(1- E) -l~l-CMax .

O:S~:S(l-A)Y

vD(k, c(k»

following a defection, all output is consumed in equal shares by the two players

The optimal d~fection policy for consumption is given by

~(k, ~y) K(l-~)y ~oY

~ :S 1/2

r ~

where vD(k ~y) - sDY.

they generate for each player must dominate the values of defection at each point

on the equilibrium path that is v(k) ~vD(k,c(k» for all k on the equilibrium path. As we illustrate in later examples however v(k) and vD(k) can intersect

so that first-best outcomes can be enforced from some k'st but not others This

.state or "wealth" dependence of equilibria was explored in Benhabib and Radner

[1987

A symmetric second-best equilibrium incentive

constraints will be given by the solution to the following problem:

vab(k) - Max (l-E)-lCl-c + v b(ak - 2c)

2

subject to v8b(k) ~vD(k.c) Alternatively, if q is a Lagrange multiplier, the

problem can be defined as

vab(k) Max y (l-c)-lcl-f + 8v.b(ak-2c) + a(v8b(k) - vD(k c»

2

o~c~-(4.8)

We now characterize the solution to (4.8):

U(c) - (l-C)-lCl-C

Then the symmetric second-best consumption

8 We note that when~ is determined from s(~.) - 8D(~J we also obtain~

~ 1/2 which is required to hold in the analysis above ~~rthermore we have

~.:S z - K(l+M)-l which illplies ~ ~ K(l-~.) - ~D

Figure 1 below illustrates the second-best solution The solution is to

policyc.b - ~.y with the value of defecting from it.

This requires equating s.

- s(~.) - SD(~.) In other words, consumption rates must be increased and

accumulation slowed down up to the point where defection is no longer attractive.

VD in Figure 1 is the value of defecting against a player following first best

strategies

Firore 1

The following numerical values illustrate the effects of incentivecompatibility constraints on economic growth along the symmetric equilibrium for

the proposition above We set a - 3.3, p - 0.325 (implausibly high discounting

of course) and E - 5 For these values vD(k, c(k» > v(k) for all k> 0, where

We compute A

sustained, the capital stock would perpetually grow (since we have an (a-k)

technology) at 15\ On the second-best path however the economy grows at

Of course parameters were chosen to make this-0.0015' that is it contracts

stark point Slightly different parameter values would allow positive growth

course in some cases the first-best may be enforceable as an equilibrium from all

the parameters above, it is easy to check that q - 0.6032, and that y - 2cm(k)

~ 0 and y - c(k) - cD(k) ~ 0 for all k ~O as assumed in the computations

An alternative example with a more reasonable discount factor is given by -1.058, ~ - 0.95, e - 0.1 This economy is barely productive (ap - 1.005) and

equilibriw In the latter, the close to linear preferences lead to rapid

contraction at a disastrous rate of about 99'

trigger strategy that is a trigger strategy where players exhaust the stock if

a defection occurs We can also compute the best symmetric equilibrium that is sustained by a weaker trigger strategy that 1 a strategy where players revert

The Markov

to an interior stationary Markov equilibrium if a defection occurs

equilibrium solves for player 1 the problem given by

v~(k} Max(l -

E}-lC: Cl

M

+ 8v1(ak - Cl - cz(k»

21

easily solved, with 8trategies where players revert to the stationary Markov

above, each player consumes ~ (ak) with ~8H - 0.435 This yields a

contraction rate of about 43' much higher than the contraction rate of 0.0015'

The point is that even along

enforceable equilibrium will be to the first-best.

symmetric aecond-best equilibria sustained by grta atrategiea growth may not be

possible Of course it is easy to construct examples where positive growth

occurs, at different rates, for fir8t-best and second-best equilibria, as well

strategies

s Wealth DeDendent Growth

section for example to the form y - ak + b b > 0 it is no longer possible to

In particular for ~ - ~ v(k)find a constant ~ to equate v(k) and VD(k, c)

and vP(k, c) may intersect at some k If v(k) ~ vD(k c) for ~ - i and k ~ k.

conditions below k where v(k) < VD(k, c(k», it may be possible to construct

first best policies are followed once k is attained This was demonstrated in Benhabib and Radner [1992J (see also Benhabib and Ferri [1987]) In thi section

we will derive conditions under which the second-best growth rates will be wealthdependent in particular we will find conditions under Which first-best growth

rates are sustainable from high stocks while growth is not at all possible fromlow stocks because of incentive compatibility constraints The 1ntu1 t1on for the

increased and accumulation slowed down to prevent defection When stocks are

low, consumption must be increased so much to prevent defection that growth is

no longer possible Examples will follow

The general proposition below will allow us to show how growth rates areaffected by wealth levels

(1) v(k) < vP(k, c(k»;

(11) f(k) - 2c :S k,

2c ~ k for any c which is the first period consumption rate of a symmetric

In particular this is true for symmetric second best equilibria

Proof: Assume that f(k) - 2c > k; then clearly c< c. But then, if v(k) isthe value of the equilibrium,

v»(k, c) :S v(k)

~ U(c) + pV(f(k) - 2c)

(5.3)

holds by the fact that U(c') + pV(f(k) - 2c') < vP(k, c') for every c e (c(k).

c); this interval is non empty because of the assumptions (i) and (ii) We have

derived a contradiction We note that if a c as defined above does not exist,

condition (i1 can be taken to be trivially satisfied .

We slightly alter the example of the previous section by adding a constant to the

where s is given by (4.5) as before

The optimal defection policy against an opposing player consuming c - ~y

+ '1:S y/2 is

9 A sufficient interiority condition for c - iy + ij $ y/2 for all k ~ 0 is

easily computed to be pa ~ 1 This condition will be satisfied in all ourexamples below

a-1 a

(5.7)

We now assign parameter values to the example above as follows: a - 1.058

With these values, for k ~ 1.2 we have v(k) >

b - 0.025, ~ - .95,10 - 2.

v(k) < v»(k c(k».

v»(k, c(k», while for k~ 1.1, It is easily shown that for

at the rate of about 2.5' Thua for k ~ 1.2 thi growth rate can be sustained

above propoaition apply For k - 0.4 (- 0.9) c defined in the proposition is

given by 0.10015 (0.05055) and y - 2c - k < 0 Therefore even the second-bestequilibrium cannot generate growth for k e [0.4.0.9] Of course, we can check

that for k ~ 0.4 we have y - 2c(k) > 0 and y - c(k) -cD(k) > 0

The above example and proposition allow us to starkly establish how growth

rates can depend on wealth because incentive constraints can be strongly binding

10 Since sa > 1, C - iy + ij < y/2 for all k ~ 0, as pointed out in the previous footnote.

25