This paper explores some implications of the comparison between feedback Nash and Stackelberg equilibria for growth and welfare in a `voracity' model. We show that as compared to the Nash equilibrium, the Stackelberg equilibrium involves a lower growth rate while it leaves both the leaders and the followers better o, i.e., the Stackelberg equilibrium is Pareto superior to the Nash equilibrium.Dynamic game, Growth, Welfare, Feedback Nash equilibrium, Feedback Stackelberg equilibrium

Kenji Fujiwara

English

Research Papers in Economics

 
DISCUSSIONç PAPERç SERIES 
 
 
 
Discussion paper No. 77 
 
 
 
 
Voracity, growth and welfare 
 
 
 
 
 
Kenji Fujiwara 
  
School of Economics, Kwansei Gakuin University 
 
 
September, 2011 
 
 
 
 
 
 
 
 
 
 
SCHOOL OF ECONOMICS 
KWANSEIç GAKUINç UNIVERSITY 
 
1-155 Uegahara Ichiban-cho 
Nishinomiya 662-8501, Japan Voracity, growth and welfare
Kenji Fujiwara
School of Economics, Kwansei Gakuin University, 662-8501
Nishinomiya, Hyogo, Japan
Abstract
This paper explores some implications of the comparison between feedback Nash
and Stackelberg equilibria for growth and welfare in a `voracity' model. We show
that as compared to the Nash equilibrium, the Stackelberg equilibrium involves a
lower growth rate while it leaves both the leaders and the followers better o, i.e.,
the Stackelberg equilibrium is Pareto superior to the Nash equilibrium.
JEL classication: C73, O41
Keywords: Dynamic game, Growth, Welfare, Feedback Nash equilibrium, Feedback
Stackelberg equilibrium
Tel: +81-798-54-7066 Fax: +81-798-51-0944. E-mail: kenjifujiwara@kwansei.ac.jp.
11 Introduction
The last decades have witnessed rapid growth of an individual country and the world.
Reecting this observation, there has been a large literature of theories and evidences of
economic growth.1 Most of the previous works in this eld have commonly adopted an
assumption that there is no strategic interaction among agents. Relaxing this assump-
tion, Tornell and Velasco (1992) make clear some interesting implications of a feedback
Nash equilibrium of an AK model of endogenous growth, nding a possibility that a
technological progress can reduce both the growth rate and welfare. This is because the
technological progress boosts over-extraction of each consumer, and hence accelerates the
tragedy of the commons. Tornell and Velasco (1992) call this perverse eect of a tech-
nological progress a voracity eect. This result indicates that conventional wisdom that
rests on the assumption of no strategic interactions may be invalid in a strategic setting.
This paper seeks more about growth and welfare in a Tornell-Velasco (1992) model.2
In particular, we pay special attention to the role of a leadership by deriving a feedback
Stackelberg equilibrium of the Tornell-Velasco model. There exists a certain literature
that extends the Tornell-Velasco model. Introducing a private asset, Tornell and Lane
(1995) and Tornell (1998) make richer arguments on the voracity eect. Taking into
account endogenous labor supply, Mino (2006) demonstrates that the growth rate is
aected by the interplay between the voracity eect and the scale eect, the former of
which has a negative impact on growth and the latter of which has a positive impact. None
of these works examines a leader-follower model by focusing on the Nash equilibrium.
However, the heterogeneity of agents is profoundly observed and their action is far
from simultaneous, which requires us to allow for a hierarchical play. To our knowledge,
Shimomura (1991) is the rst to formally characterize the feedback Nash and Stackel-
berg equilibria in a capitalistic game a la Lancaster (1973). While his focus is on the
characterization of equilibria, we discuss the implications of the presence of a leadership
for growth and welfare. After deriving the feedback Stackelberg equilibrium where there
are an arbitrary number of leaders and followers, we show three main results. First, the
1See, for instance, Barro and Sala-i-Martin (2004) and Acemoglu (2007) for a comprehensive review.
2There is another strand of literature that introduces strategic interactions into an endogenous growth
model, e.g., Vencatachellum (1998), Shibata (2002), and Dockner and Nishimura (2004, 2005). These
papers are, however, mainly interested in the comparison among ecient, open-loop Nash, and feedback
Nash equilibria, which diers much from this paper in scope.
2voracity eect is larger in the Stackelberg equilibrium than in the Nash equilibrium. Sec-
ond, the Stackelberg equilibrium involves a lower growth rate than the Nash equilibrium.
Third, the Stackelberg equilibrium leaves both the leaders and the followers better o
relative to the Nash equilibrium.
This paper is organized as follows. Section 2 briey reviews the Tornell-Velasco (1992)
model and the feedback Nash equilibrium. Section 3 turns to the feedback Stackelberg
equilibrium. Section 4 compares these two equilibria and proves our main results. Section
5 concludes.
2 A model
The model is an extension of an AK model of Tornell and Velasco (1992). There are
m  1 leaders and n  1 followers, both of which extract a renewable resource for their
consumption. Thus, the problem of player j is formulated as
max
cj
Z 1
0
e
 rt c
1 
j
1   
dt;  2 (0;1)
s.t. _ k = ak   cj  
X
i6=j
ci; a > 0:
where cj is consumption of player j, r > 0 is a discount rate, and k is a stock of a
renewable resource. Tornell and Velasco (1992) compute the feedback Nash equilibrium
in linear strategies in this model, according to which the equilibrium consumption and
welfare per player are obtained as follows.3
c(k) =
"
r   (1   )a
1   (m + n)(1   )
#
k (1)
V (k0) =
k
1 
0
1   
"
r   (1   )a
1   (m + n)(1   )
# 
; (2)
where V () is a value function of each player. Following Tornell and Velasco (1992), let
us make
Assumption 1. r   (1   )a < 0 and 1   (m + n)(1   ) < 0.
Under this assumption, equilibrium consumption in (1) is positive, and welfare in (2) is
bounded.
3Subscript j signifying players is dropped since we focus on the symmetric equilibrium.
33 A feedback Stackelberg equilibrium
This section turns to the leader-follower model in which each of m leaders announce
a linear feedback strategy !ik before the n followers move. The game is solved with
backward induction and the problem of a representative follower j is given by
max
cj
Z 1
0
e
 rt c
1 
j
1   
dt
s.t. _ k = ak   cj  
m X
i=1
!ik  
n X
l6=j
cl:
To solve this problem, let us construct a Hamilton-Jacobi-Bellman equation of player j:
rV
j(k) = max
cj
8
<
:
c
1 
j
1   
+ V
j
k (k)
0
@ak   cj  
m X
i=1
!ik  
n X
l6=j
cl
1
A
9
=
;; (3)
where V
j
k ()  dV j()=dk. The rst-order condition for maximizing the right-hand side is
cj =
h
V
j
k (k)
i  1
 : (4)
Guessing that V j(k) = Ak1 =(1   ), (4) becomes cj = A 1=k. Substituting these into
(3), we have an identity in k:
rAk1 
1   
=
A
 1
 k1 
1   
+ Ak
1 
 
a   nA
  1
  
m X
i=1
!i
!
:
The undetermined coecient A that satises this identity is obtained as
A =
"
r   (1   )(a  
Pm
i=1 !i)
1   n(1   )
# 
; (5)
and the equilibrium consumption of the followers is
cj(k) =
"
r   (1   )(a  
Pm
i=1 !i)
1   n(1   )
#
k: (6)
We now turn to the problem of leaders. Given the announced strategy of leader i,
!ik, and the followers' strategy (6), the resource dynamics is rewritten as
_ k = ak  
m X
i=1
!ik   n 
r   (1   )(a  
Pm
i=1 !i)
1   n(1   )
=
a   nr  
Pm
i=1 !i
1   n(1   )
;
4the solution of which is explicitly computed:
k(t) = k0e
a nr 
Pm
i=1
!i
1 n(1 ) t:
Substituting this into the strategy of leader i and its discounted stream of utility, we have
Z 1
0
e
 rt(!ik)1 
1   
dt
=
Z 1
0
e
 rt!
1 
i k
1 
0
1   
e
(1 )(a nr 
Pm
i=1
!i)
1 n(1 ) tdt
=
!
1 
i k
1 
0
1   
Z 1
0
e
 r[1 n(1 )]+(1 )(a nr 
Pm
i=1
!i)
1 n(1 ) tdt
= k
1 
0 
1   n(1   )
1   

!
1 
i
r   (1   )(a  
Pm
k=1 !i)
; (7)
which is to be maximized by player i.
Each leader chooses !i to maximize (7), which involves the rst-order condition:
(1   )!
 
i [r   (1   )(a  
Pm
k=1 !k)   !i]
[r   (1   )(a  
Pm
k=1 !k)]
2 = 0:
From this equation, and focusing on a non-zero strategy, any leader i chooses !i =
r  (1 )(a  
Pm
k=1 !k), which implies that all the leaders choose the same strategy. In
the symmetric equilibrium where !i = !k, we have
ci(k) = !ik =
r   (1   )a
1   m(1   )
k: (8)
Substituting this into (6), the follower's strategy becomes
cj(k) =
r   (1   )(a   m!i)
1   n(1   )
k =
r   (1   )a
[1   n(1   )][1   m(1   )]
k: (9)
As in the Nash case, we introduce the following restrictions on the parameters.
Assumption 2. 1   m(1   ) < 0 and 1   n(1   ) > 0, i.e., m > 1=(1   ) > n.
The inequality 1   m(1   ) < 0 is used to ensure the positivity of !i in (8) and the
inequality 1   n(1   ) > 0 is analogously adopted to guarantee cj(k) > 0.
54 Comparison of Nash and Stackelberg equilibria
Having derived the Nash and Stackelberg equilibria in linear feedback strategies, we
readily compare how the two regimes aect the growth rate and welfare.
4.1 Growth rates
Making use of (1), the growth rate in the feedback Nash equilibrium is
g
N 
  _ k
k
!N
= a   (m + n)
r   (1   )a
1   (m + n)(1   )
=
a   r(m + n)
1   (m + n)(1   )
; (10)
where superscript N refers to the Nash equilibrium. In a similar way, Eqs. (8) and (9)
allow us to compute the growth rate in the feedback Stackelberg equilibrium:
g
S 
  _ k
k
!S
= a   m
r   (1   )a
1   m(1   )
  n
r   (1   )a
[1   n(1   )][1   m(1   )]
=
a   r[m + n   mn(1   )]
[1   m(1   )][1   n(1   )]
: (11)
where superscript S stands for the Stackelberg equilibrium. Given Assumptions 1 and
2, we can conrm the voracity eect of a technological progress both in the Nash and in
the Stackelberg equilibrium, namely, @gN=@a < 0 and @gS=@a < 0. In addition, we can
prove:
Lemma 1. The voracity eect is stronger in the feedback Stackelberg equilibrium than
in the feedback Nash equilibrium.
Proof. Under Assumptions 1 and 2, the eect of an increase in a on the growth rate in
each equilibrium is
@gN
@a
=
1
1   (m + n)(1   )
< 0
@gS
@a
=
1
[1   m(1   )][1   n(1   )]
< 0:
Subtracting the former from the latter yields
@gS
@a
 
@gN
@a
=  
mn(1   )2
[1   m(1   )][1   n(1   )][1   (m + n)(1   )]
< 0:
6that is, it follows that @gS=@a < @gN=@a < 0. Thus, the detrimental eect of a rise in a
on the growth rate is stronger in the Stackelberg case than in the Nash case. Q.E.D.
The reason behind this result is obvious in view of the property of the strategic com-
plement between the strategy of leaders and followers. In the Stackelberg equilibrium,
both class of players opt for higher consumption than in the Nash equilibrium. A tech-
nological progress in the form of a rise in a accelerates this tendency for the tragedy of
the commons and hence its detrimental eect on the growth rate is also enhanced. Let
us turn to the comparison between gN and gS. Since subtracting gN from gS yields
g
S   g
N =
mn(1   )[r   (1   )a]
[1   m(1   )][1   n(1   )][1   (m + n)(1   )]
< 0;
by noting Assumptions 1 and 2, we have established:
Proposition 1. The growth rate is higher in the feedback Nash equilibrium than in the
feedback Stackelberg equilibrium.
(Figure 1 around here)
The intuition behind this result is well understood by using Figure 1. This gure
depicts the relationship between !i and !j in a two-player case. From (6) and the
assumption that 1   n(1   ) > 0, there is a strategic complement between !i and !j.
In the Nash game, the equilibrium is determined by the intersection of the two reaction
curves and given by N. If, on the other hand, player i is a leader, it chooses !i such that
its iso-welfare curve is tangent to the reaction curve of the follower (player j). Therefore,
the Stackelberg equilibrium is obtained by S. As is clear from the gure, both players
consume more in the Stackelberg equilibrium than in the Nash equilibrium. That is,
the Stackelberg equilibrium makes the tragedy of the commons stronger and hence the
associated growth rate is smaller than the growth rate in the Nash equilibrium.
4.2 Welfare
While Proposition 1 is concerned with how the presence of a leadership inuences the
equilibrium growth rate, we now consider welfare aspects of it. For this purpose, we
7compare the payo level of the follower in the two regimes. Substituting (8) into (5), the
follower's welfare in the Stackelberg equilibrium is
V
j(k0) =
k
1 
0
1   
"
r   (1   )(a   m!i)
1   n(1   )
# 
=
k
1 
0
1   
(
r   (1   )a
[1   m(1   )][1   n(1   )]
) 
:
(12)
Therefore, (2) and (12) yield the dierence in the equilibrium payo of the follower in
the Stackelberg and Nash equilibria:
V
j(k0)   V (k0) =
k
1 
0
1   
"
r   (1   )a
1   (m + n)(1   )
#  2
4
(
[1   m(1   )][1   n(1   )]
1   (m + n)(1   )
)
  1
3
5 > 0:
Thus, the Stackelberg equilibrium leaves the follower better o than the Nash equilib-
rium. Noting that the Stackelberg leader always enjoys a higher payo than in the Nash
equilibrium, we can state:
Proposition 2. Both the leader and follower are better o in the feedback Stackelberg
equilibrium than in the feedback Nash equilibrium.
Figure 1 gives a diagrammatic representation Proposition 2. In the gure, it is clear
that both the leader (player i) and the follower (player j) at S achieve higher welfare than
at N. The reason for this nding is as follows. In the present model, the leaders have
an incentive to extract the common resource to enjoy a higher utility by moving rst.
In response to this behavior of the leaders, the followers also increase their consumption
than the Nash level. This bilateral increase in consumption enhances welfare of both all
players as compared to the Nash case because they commit to increased consumption in
every point of time.
Finally, we address why the leader-follower game enhances a voracity but improves
welfare of both classes of players. To seek the reason, let us consider what happens
if we have a technological progress in the form of increased a. This encourages capital
accumulation as a direct through the capital accumulation equation. Taking into account
the resulting increase in k, all players expand their consumption, which applies to both
the Nash game and the Stackelberg game, and leads to the voracity eect on growth.
Comparing the two games, the Stackelberg leaders commit to more consumption than
8the Nash level by taking advantage of their leadership. Observing this commitment
of the leaders, the followers also consume more than the Nash level because otherwise
their utility decrease as a result of the leaders' aggressive behavior. Accordingly, capital
accumulation is slower, namely, the voracity eect is stronger in the Stackelberg game
than in the Nash game. What is striking that this stronger voracity raises welfare of not
only the leaders but also the followers. This is because the leaders' precommitment to
more consumption encourages the followers to consume more in every point of time, and
improves the followers' utility. That is, the strategic complement property of the model
causes a Pareto superiority of the Stackelberg equilibrium over the Nash equilibrium
although the former involves a stronger voracity than the latter.
5 Concluding remarks
We have extended a dynamic game model of endogenous growth to accommodate the
presence of leaderships to address how it aects the growth rate and welfare. It has been
established that the growth rate is lower but welfare of both classes of players is higher
in the feedback Stackelberg equilibrium than in the feedback Nash equilibrium. While
the above result has a certain novelty, it is admittedly proved in the simplest version of
the Tornell-Velasco (1992) model. It is an interesting extension to allow for private assets
as in the second model of Tornell and Velasco (1992) and endogenous labor supply as in
Mino (2001).
Acknowledgements
I gratefully acknowledge helpful comments from Ryo Arawatari, Yoichi Gokan, Yunfang
Hu, Shin Imoto, Kazuo Mino, Keiichi Morimoto, Jumpei Tanaka, and an anonymous
referee. Any remaining error is my own responsibility.
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10!j
!i
O
N
S
!i(!j)
!j(!i)
Figure 1: Nash and Stackelberg equilibria
11

A dierential R&D game: implications for knowledge-based growth models.

Capital accumulation games with a non-concave production function.

Economic growth.

Introduction to modern economic growth.

Strategic growth.

Strategic interactions in a growth model with public infrastructure.

The dynamic ineciency of capitalism.

The feedback equilibria of a dierential game of capitalism.

The tragedy of the commons and economic growth: why does capital ow from poor to rich countries?

The voracity eect.

Voracity and growth in discrete time.

Voracity vs. scale eect in a growing economy without secure property rights.

Voracity, growth and welfare

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Voracity, growth and welfare

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