C.V. STARR CENTER FOR APPLIED ECONOMICS pdf

ECONOMIC RESEARCH REPORTS
Individual copies of papers may be obtained free of charge by writing to.
The C. v: Starr Center for Applied Economics
New York University
269 Mercer Street, yd Floor
New York. NY 10003-6687
. edu/working/
Social Conflict, Growth and Inequality
by
Jess Benhabib
Department of Economics
New York University
New York, NY 10003
and
Aido Rustichini
Department of Economics
Northwestern University
Evanston, IL 60208
August 1991
Revised April1992
Social Conflict, Growth and Inequality
by
Jess Benhabib
New York University
and
Aldo Rustichini
Northwestern University
Abstract
Despite the predictions of the neoclassical theory of economic growth,
we observe that poor countries have invested at lower rates and have not grown
faster than rich countries. To explain these empirical regularities we

provide a game-theoretic model of conflict between social groups over the
distribution of income. Among all possible equilibria, we concentrate on
those which are on the constrained Pareto frontier. We study how the level of
wealth and the degree of inequality affects growth. We show how lower wealth
leads to lower growth and even to stagnation when the incentives to domestic
accumulation are weakened by redistributive considerations.
JEL Classification numbers
Key Words: Dynamic Games
010, C73
Send Correspondence to:
Professor Jess Benhabib
Department of Economics
New York University
269 Mercer Street, 7th Floor
New York, New York 10003 USA
1
1
Introduction.
Neoclassical growth theory predicts that poor countries. because of the law
of diminishing returns, should grow at faster rates than rich countries.
inverse relation between wealth levels and growth rates should further be
strengthened by the diffusion of technology and the opportunites for "catch-up"
despite concerted efforts at faster development,
we observe that
Yet,
countries have invested at lower rates, exhibited more intense social conflict
and political instability,
and consequently have not grown faster than
The empirical relationship between income levels and growth rates is
countries.

Baumol
(See De Long [1988
flat and possibly hump-shaped, not downward sloping.
and Wolff [1988), Figure 2; and Easterly [1991).)
To explain this discrepancy
be~een the data and the predictions of the neoclassical model the literature
on endogenous growth theory has introduced economy-wide externalities. threshold
Here we pursue
effects and other mechanisms that overcome diminishing returns
an alternative game theoretic course that emphasizes the interrelationships
between the levels of wealth, social and political conflict, and the incentives
As such our work is related to that of Persson and Tabellini
for accumulation
and Alesina and Rodrik
1991
[1991
We have in mind a situation where organized social groups can capture, or
attempt to capture, a larger share of the output either by means of direct
appropriation, or by manipulating the political system to implement favorable
transfers, regulations and other redistributive policies.l
Depending on the
1 Some of the non-violent redistributive mechanisms that are used in
developing countries include nationalization; bursts of inflationary finance to
sustain the incomes of government bureaucracies and the military; the squeezing
of the agricultural sectors in favor of politically powerful urban classes
throu~ exchange rate policies. price controls and monopolistic marketing boards;
legislation and other measures that alter the bargaining power of labor (either
positively or negatively); the allocation of highly desirable government and
civil service jobs and university admissions to favored ethnic and tribal groups;
2

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
social
disincentives
create
significant
to
groups
can
furthermore can be stronger at lower levels of wealth than at higher ones, so
that poorer countries grow more slowly or even stagnate at lower levels of
wealth.
We obtain these results in our model without having to rely on the
alternative, and probably complementary framework which requires non-convexities
or threshold effects in the production technology
1986]
(See for example Romer
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
in poorer countries are essentially due to lower rates of accumulation in
physical and human capital.

the
When factor accumulation is taken into account
predicted negative relation between growth rates and initial income levels is
reestablished,
Indeed investment rates in physical and human capital (primary
schooling) are negatively correlated with income levels (see Fisher
1991], Table
3) .
Furthermore investment rates show a robust negative correlation with various
measures of political instability (see Barro 1991], Levine and Rene1t [1991],
and large scale bureaucratic corruption tolerated and condoned by the government.
For some detailed accounts of various redistributive mechanisms see Bardhan
[1984], [1988]; Bates [1983], [1988]; Ca11aiby [1990]; Dornbush and Edwards
[1990]; Findlay [1989]; Frieden
{ 1991]; Gould [1980]; Horowitz [1985]; Krueger
[1974]; Laothamatas [1992]; Malon and Sourri11e [1975]; O'Donnell [1973],
[1988]; Peralta-Ramos [1992]; Sachs [1989]; Veliz [1980], chapter 13.; and the
various essays in Goldman and Wilson [1984], and in Nelson [1990].
2 The role of the enforcement of property rights by the state to internalize
social 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 of
rational collective action by social groups in the political arena, see Tilly
[1978]. The effects of rent-seeking behavior by organized groups on the economic
efficiency of mature economies has been studied by Olson [1982]. See also Becker
[1983], Romer [1990] and Brock and Magee [1978].
3
Venieris and Gupta [1986]) and there is a negative relationship between measures
of political instability and levels of income [see Gupta [1990], Zimmerman
(1980], or Londregan and Poole (1990})
This cross-country evidence suggests

then that poorer countries are more prone to political instability
have lower
investment 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
Maddison's [1982] estimates show that after centuries of
studies as well.
and imperceptible growth, the richer nations of Europe, together with the US and
Japan, had acheived an average GNP per capita in 1820 of about $974 in 1985
prices.4
This is higher than the 1988 per capita GNP, at 1985 prices
of about
a quarter (35 out of 138) of the countries in the Summers-Heston [1991
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 account
for the largest part of the world population today are at much lower per capita
countries
before
their
levels
than
the
developed just
product
were
industrialization

We must however be cautious in drawing comparisons between
3 Countries like Taiwan and Korea on the other hand have had strong growth
performance despite their low initial income levels. However, the elimination
and suppression of landlord classes under Japanese occupation and strong arm
tactics towards business and labor unions to implement liberalization in the
1960's and 1970's may have been critical elements. See Amsden [1988], Jones and
Sakong [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 per
capita 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 World
Development Report [1990].
6 See Landes [1969) or Kuznets [1974. p. 179).
chapter 7).
[1971, chapter 1], [1966,
4
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
often enforcing limited franchise as well as suppressive
of social classes,
policies towards organized labor in order to sustain growth and accumulation.
As wealth levels increased, redistributive pressures were in part accomodated by
(See Maddison [1984])
Of
the significant expansion of the welfare state
course we must also consider that today great progress in communications and
information technology has not only vastly enhanced the possibilities for
technology diffusion but also created much higher expectations of income and

welfare worldwide
To capture the empirical relationship between wealth and growth discussed
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
Stationary equilibria in
or the productive resource in the following period
1980],
such games have been studied by Lancaster [1973], Levhari and Mirman
(See also Torne11 and Velasco
Majumdar and Sundaram [1991], and many others.
We consider equilibrium paths of accumulation in which players receive
[1990].)
that they could obtain by
those
utilities
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 those
subgame-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
we analyze the effects of wealth (or the stock of capital) on growth and on
In particular we study cases where lower wealth
steady state income levels.
We also consider cases which produce classical "growth
leads to lower growth.

5
Even though first-best policies lead to growth,
along second-best
traps
equilibria growth may not be possible from low levels of wealth because of
the accumulation of wealth by one player can lead to
incentive constraints:
appropriation and to high consumption levels by other players. and therefore may
not be sustainable as an equilibrium
Another possibility is for incentive constraints to bind at high wealth
Capital may be too precious at low levels
levels and not at low ones
Inefficiency may set in
players may follow first-best policies of accumulation
at higher levels of wealth and first-best policies may have to be abandoned as
the 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
1982]
We illustrate this case in
pressures has been suggested by Mancur Olson
section 7 below
There may be good evolutionary or institutional reasons to focus on second
For our purposes
best 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

symmetric (egalitarian) second best
For ins tance
of
the symmetric
low levels of
wealth,
then
the
binding
growth rate
at
(egalitarian) second best equilibrium sets an upper bound to the growth rates at
Growth rates on all other equilibria, including the 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 growth
and inequality, where inequality is measured by the disparities of consumption
6
High rates of accumulation in
levels
(see section 3)
rates and welfare
economies with pronounced and persistent inequalities may not be sustainable
because the disadvantaged groups can undertake redistributive actions or exert
The political
redistributive pressures that discourage domestic investment
attainable
if
income

consensus necessary for efficient growth may not be
Recent empirical work has confirmed the inverse
inequality is too severe
relationship of income inequality with investment and growth
Using cross
country data, Venieris and Gupta [1988] established the negative effect of income
And more recently. Persson and Tabellini [1991
inequality on investment rates
have shown that income inequality adversely
also using cross country data,
affects growth rates
Our paper is organized as follows
The next section sets up the problem
in a general framework and provides an existence result
Section 3 establishes
that among all equilibria, the symmetric (egalitarian) second best is the fastest
Section 4 works out a simple and illustrative second best problem
growing one
where incentive constraints retard growth but accumulation rates do not depend
illustrate how growth is influenced by the
on wealth
Numerical
examples
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 but
cannot be sustained at high stock levels
Section 8 contains some final remarks
and a discussion of the role of the state in economic growth
7
2. The Second Best Problem.
We consider two players characterized by two concave and strictly
increasing utility functions Ui, i = 1, 2, and a common discount factor p E
(0,1). kt represents the capital stock at time t. The production function is
concave, increasing and f(O) ~ O. The feasible paths of the consumption
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
1 2 1 2
) d " f h" "
t t "
~ = (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 achieved
over a smaller set of SPE. To avoid ambiguities, we describe in detail how the
allocation of consumDtion is regulated. It will be useful to distinguish between
attempted consumption and consumption (the first is the consumption a player is
trying to get, the second is what the allocation rule gives him). For a given
capital stock k and two attempted consumptions cl and c2' the allocated

consumption is
8
"
I, (
Cl + C2 ~ £(k) or Cl ~ £(k)/2
if
Cl
Cl+CZ~ f(k)
and
Cl ~ f(k)f2 ~ C2
if
Al (C1J CZJ k) -
f(k) -C2
C1' Cz ~ £(k)/2
f(k)/2
if
Note that if Cz $ f(k)/2, then Al(Clt cz, k) -
and similarly for~.
min ( Cl' f(k) -C2
Since the utility function of both players is strictly increasing, the
fast consUInution stratev.ies,
which we call
following pair
of
strategies,
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 attempts
Cl' 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 players
can easily be modified
we could have assumed that one of the players can obtain up to. say 3/4 of the
output under fast consumption strategies (Cl' C2)' and confine the consumption
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 if
opposing player then exhausts the residual
the fast consumption rates of one or both players are bounded, that is if
where Cj
f(k} - cJ > 0
is the bound for the consumption rate of the j ~ player
Conditions under which fast consumption rates are still equilibrium strategies
9
.
when positive savings rates are possible have been studied (in a continuous time
framework) by Benhabib and Radner [1992] for the case of linear utility, and by
Rustichini [1992] with non-linear utility.
Under the allocation rule described above, the worst SPE is easily
described. This allows us to utilize the worst equilibrium in order to sustain
any other SPE with trigger strategies. Under more general allocation rules the
worst SPE is difficult to characterize. Note, of course, that one may also
arbitrarily choose a simple SPE, for example the interior Markovian one under

which the sum of attempted consumptions never exceeds output, and then study the
set 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 the
utility functions of the players are strictly increasing. The value of this
equilibrium 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 trigger
strategy 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 is
detected. The individual rationalitv constraint for player i on an outcome path
is the condition:
10
v
LPt.U(c;) ~ Vi (k)
t.
Clearly, in a SPE,
the equilibrium outcome of the equilibrium of any subgame
satisfies this inequality
Consider now a trigger strategy equilibrium For any capital stock k and
equilibrium consumption
of the other player, the value of defection is the
c
value for a player of deviating optimally, that is:
where
wD(k,c) .
U(f(O)/2) . (2.2)
1=i

max
o~ c' ~f(k)-c
and
kl- f(k) - c - ~(ktCtC/)t ~+1 - f(~) - Ai (Cl (~). C2(~)'~) - Az(Ci (~). C2(~)'~)'
t~l
(2.3)
Note that this optimization problem can be expressed without the maximization
D D -
- by simply adding the constraint Vi (k,c) ~vi(k).operator in defining v;
We
denote by C1D(k, c) the consumption giving the optimal deviation
In the games
we consider such optimal consumption exists and is unique, so no ambiguity is
11.
possible there
We state and prove it for completeness.
The following lemma is clear.
1 2
Let (~'~)t.~o be the outcome of a SPE, ;. say Then the trigger
Lemma 2.1.
strategy pair with this agreed cons~tion path, ~' is an SPE
For any history~. we denote vi(~) the value to the i~ player of
Proof.
starting with hto
We only need to consider equilibrium
the equilibrium in
~
( 1 1
histories ~-1. ~1.C2'
1 a

C~-l' C~.l) .
Let ~ be the capital stock
We claim
that ct,2 is an optimal choice for player 2 next period,
in F.'.
The best
D 1 D
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
and our claim follows
.
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
12
Let °1' °2 ~ 0 be weights attached to the players
From what we have seen,
second best is the solution of the problem
SU~ ~ t
{(Ctl, C ) } ".8 [alU1 (l! 1
t t~O t -to
+ °2U2<C;:
v.h(k) -
1 2 1

subject to f(~) - Ct - Ct ~ ""'t+l
j . 1,1
1 -1,2, ;
In the following we shall refer to this as the second best vroblem
We assume now that the production function
f
p
and the discount factor
8atiafy
11m f'(k).8 < 1.
k-+~
(AO)
Then we have
Lemma 2. 2
1.
A solution to the second best problem exists.
The function v.b is uppersemicontinuous.7
2.
Proof. For every capital stock
k the set of admissible paths is the
7 Section 6 provides an example of a discontinuous value function. In
general, the conditions to apply dynamic programming methods and the contraction
mapping, which assure the continuitl of the value function, may not hold for our
problem. In particular, Blackwell s discounting condition, T(v+a) ~ T(v) + pa
may be violated because the constraint set for V+a becomes larger, allowing
consumption levels that would be ruled out under v.
13
of sequences {(~, Ctl, Ctz)t?;o} such that (1) and (2) above are satisfied. In a
properly 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

. 1 Z ~ t 1 Z
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 closed
graph, and since the image space is compact, it is also uppersemicontinuous. Now
apply the Maximum Theorem. .
For an extension of the above results to renegotiation proof equilibria in
some 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 mechanisms
that implement second best outcomes will evolve in societies where the strategic
behavior of social groups has a strong influence on the production and on the
distribution of output. In this section however we motivate our choice of second
best equilibria as the focus of analysis by another consideration. We are
primarily interested in studying growth rates of economies when incentive
constraints are binding, at least on some part of the equilibrium trajectory.
In such situations we show that the symmetric (egalitarian) second best
equilibrium is the one that affords the fastest growth rate over all subgame
perfect equilibria and therefore represents an upper-bound to growth rates. When
incentive constraints do not bind, on the other hand, second best equilibria may
not 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
higher growth rates or higher steady
states can also be sustained
particular focus is on those equilibria which are incentive constrained
either
initially, and maybe forever, at low levels of wealth (the classical case), or
ultimately, 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
a fixed amount of capital stock,
the value of defection increases for both
players.
On the other hand, since we had a symmetric second best, the value for
both players must decrease
if the value of continuation and the value of
defection were initially equal, now the second is greater than the first
therefore we cannot have an equilibrium with a faster growth rate.
The next
proposition
extends
claia
comparisons
to
include
non-symmetric
our
that
equilibr1a.
ProDosition 3.1.
t t
For a given k, let (cl,c2)t~O be a second best equilibrium
outcome, starting from k, which is symmetric and such that
vi(k) - v»(k, Ci')'
for 1-1,2
If k1 - f(k) - cl - C2t and k1' is the next period capital stock for
any other sub game perfect equilibrium, then k1' S k1
Proof

See Appendix Al
15
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 in
detail to illustrate how growth rates may differ between first and second best
equilibria.
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
where k~O is the capital stock, and
'We will
is a non-negative constanta
consider in this example symmetric (egalitarian) equilibria (setting °1 - ~
1 for the problem given by (2.2) above). where both players get equal consumption
levels In this case the total utility of each player along the first-best
equilibrium can be described by a dynamic program:
v(k) - Max
o~c~I

2
1-c + pV(y - 2c)
(4.3)
~ 1
rr:;
The consumption function which solves this program is given by c - iy, where
16
(I~.)
-
) ~ 0
a > 1, ,81/'.(1-' )/« < 1
to avoid
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
.
for the first-best ~
We will use this fact in deriving the second-best value
function later on
When a player defects against first-best play by his opponent
he must
choose his consumption in the current period taking into account that trigger
strategies will be enacted subsequently.
Optimal defection value is therefore
given by

(1- E) -l~l-C
Max .
O:S~:S(l-A)Y
vD(k, c(k» -
where c(k) - ~y
This value reflects the trigger strategy equilibrium for which
following a defection, all output is consumed in equal shares by the two players
The optimal d~fection policy for consumption is given by
17
~(k, ~y) . K(l-~)y . ~oY
~ :S 1/2.whenever the other player's consumption policy 1s ~y. and where
r
~
K-
<1.
.
(Of course when the other player chooses ~ -~. he is
The value of optimal defection isfollowing the first-best strategy.
where
vD(k. ~y) - sDY.
~ . (l-~)l-c(l-()-l[Ml + P«1-M)a/2)1 . (l-~)l-c(l-E)-lK-C
(4.7)
For first best policies that constitute an equilibrium, the values that
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

with
compatibility
A
symmetric equilibrium
incentivesecond-best
constraints will be given by the solution to the following problem:
vab(k) -
Max (l-E)-lCl-c + v b(ak - 2c)
o~c~I .
2
subject to v8b(k) ~vD(k.c) Alternatively, if q is a Lagrange multiplier, the
18
problem can be defined as
vab(k) .
Max y (l-c)-lcl-f + .8v.b(ak-2c) + a(v8b(k) - vD(k. c»
o~c~-
2
(4.8)
We now characterize the solution to (4.8):
U(c) - (l-C)-lCl-C
Pronosition 4.1.
and
y - ak
where 0 < t' < 1,
Let
Then the symmetric second-best consumption
policy is given by
.
CD - ~y
. .

.(.A,) ~ ao(.A,)
(4.9&)
1f
(a)
& &
s(A-) < 8o(A-)
(4.9b)
(b)
Cab - ~.Y
if
K 1
z ~-
l+K 2
Kin U I ~ e [i. z] .
where
~.
.(~) . 80(~)} ~ ~.
and
-
,-1
+~
,l/f
~
8
~1K-
Proof
See Appendix A.2
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.

19
Figure 1 below illustrates the second-best solution.
The solution is to
find ~. which equates the value for each player of following the consumption
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
the
illustrate
of
effects
incentive
compatibility 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
v(k) corresponds to the first-best values with policies c - ~y.
.

We compute A
0.326, ),. - 0.349
These magnitudes imply that if the first best could be
sustained,
the capital stock would perpetually grow (since we have an (a-k)
20
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
along the second best equilibrium
Of
but at a slower rate than the first best.
course in some cases the first-best may be enforceable as an equilibrium from all
stocks so that incentive constraints will not bind
Finally, we note that for
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
the utility function is close to linear
On the first best equilibrium the
growth rate i. 5.2'. but again growth becomes impossible on the second best
equilibriw
In the latter,
the close to linear preferences lead to rapid
contraction. at a disastrous rate of about 99'
In the above examples the second-best equilibrium is sustained by a grim

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»
where in the symmetric case the solution ti.fie.
cl(k) - c2(k), and
c2(k) is
also a best response for the second player
The trigger-strategy problem is
21
easily solved, with 8trategies where players revert to the stationary Markov
equilibriua after a defection
On such an equilibrium, using the parameters
above,
~ (ak)
with
~8H - 0.435.
each player consumes
This yields a
contraction rate of about 43' much higher than the contraction rate of 0.0015'
for the second-best equilibrium under grim trigger strategies
This is not
surprising since

the
"grimmer"
the
trigger
the
strategy,
the closer
The point is that even alongenforceable 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
as for equilibria obtained by trigger strategies that are weaker than
strategies
s. Wealth DeDendent Growth
If we slightly alter the linear production function of
the previous
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.
first-best policies will be sustainable a8 equilibria for k ~~.
Fro. initial
conditions below k where v(k) < VD(k, c(k», it may be possible to construct
.switching. equilibria (which are not necessarily second-best), along
growth occurs at a rate slower than first-best rates until k is reached and
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 wealth
dependent
in particular we will find conditions under Which first-best growth
22