GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.38

A Dynamic Game Model of Endogenous
Growth with Consumption Externalities

Katsuhiko Hori and Akihisa Shibata

March 2009GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, JapanA Dynamic Game Model of Endogenous
Growth with Consumption Externalities∗
Katsuhiko Hori†

Akihisa Shibata‡February 16, 2009

Abstract
This paper introduces consumption externalities into an endogenous growthmodel of general capital accumulation and characterizes balanced growthequilibria. Contrary to the common argument in previous studies, we showthat the growth rate in a feedback Nash equilibrium can be higher than that inan open-loop Nash equilibrium if agents strongly admire others’ consumption.

Keywords: diﬀerential game, consumption externalities, endogenous growth,
open-loop Nash equilibrium, feedback Nash equilibrium∗

This work is supported by the Global Centre of Excellence (GCOE) program.
GCOE Researcher, Institute of Economic Research, Kyoto University, Yoshida, Sakyo-ku,
Kyoto 606-8501, Japan‡
Professor, Institute of Economic Research, Kyoto University, Yoshida, Sakyo-ku, Kyoto 6068501, Japan
†

1

1

Introduction

In the literature on commons games, it is usually argued that the lack of agents’commitment to their future actions leads to the so-called tragedy of the commons;that is, if agents condition their actions on the basis of the aggregate stock of thecommons, rather than commit to their initial decisions, then it could worsen theoverconsumption and/or underinvestment of the commons.1 For example, Fershtman and Nitzan (Ref. 2) develop a dynamic model of the voluntary provision of
public goods and show that the contribution of conditioning agents to the collective contributions aggravates the degree of the free-rider problem2 . Moreover, using endogenous growth models, Tornell and Velasco (Ref. 4) and Shibata (Ref. 5)
show that the balanced growth rate without commitment is lower than that withcommitment.3 This paper shows that this relationship between the growth rates
with and without commitment is vulnerable; that is, if preferences include consumption externalities, the growth rate without commitment can be higher thanthat with commitment.
The existence of consumption externalities is emphasized in an early workby Veblen (Ref. 14) and validated as a determinant of aggregate consumption
by Duesenberry (Ref. 15). Moreover, many recent studies analyze the eﬀects ofconsumption externalities. For example, to reconcile the equity premium puzzle, Abel (Ref. 16), Constantinides (Ref. 17), and Galı́ (Ref. 18) incorporate theconsumption externalities into consumption-based asset pricing models. Liu andTurnovsky (Ref. 19) examine the eﬀects on the overconsumption phenomena in a
growth model with consumption externalities.4 Along these lines of research, weconstruct a simple dynamic game model of common capital accumulation with
consumption externalities and analyze how the presence of these externalities
modifies the conventional results on the growth rates with and without commitment.

2

The Model

There are N homogenous agents in our economy. They jointly produce a good byusing common capital and divide it into consumption and common capital accumulation. The lifetime utility function of each agent is assumed to be additively
1

Gordon (Ref. 1) is the first study that presents an example of the tragedy of the commons.
See also Levhari and Mirman (Ref. 3).
3
See also Benhabib and Radner (Ref. 6), Cozzi (Ref. 7), Dockner and Sorger (Ref. 8), Sorger(Ref. 9), Vencatachellum (Ref. 10), Vencatachellum (Ref. 11), Dockner and Nishimura (Ref. 12),and Luckraz (Ref. 13).
4
See Ljungqvist and Uhlig (Ref. 20), Turnovsky and Monteiro (Ref. 21), and Mino (Ref. 22)
for other recent examples.
22

separable in time. The agent’s subjective discount rate is denoted by ρ and theelasticity of intertemporal substitution is represented by η. We specify the lifetime utility function as follows:5∫ ∞η ( −α )1− 1η
ci c̄iŪi =exp(−ρt)dt for i = 1, . . . , N,
(1)
0 η−1
∑
where ci : R+ → R+ is countinuously diﬀerentiable; c̄i = j,i c j /(N − 1); ρ, η ∈R++ ; and α ∈ (−∞, 1). Here, ci is the consumption of agent i and c̄i is the levelof average consumption of other agents, which represents the externalities fromother agents’ consumption. Parameter α represents the magnitude of the external
eﬀects of consumption. Following Dupor and Liu (Ref. 23), we define “jealousy”
and “admiration” as follows.
Definition 2.1. We say that consumption externalities indicate
1. “jealousy” if α > 0 while “admiration” if α < 0.
2. “keeping up with the Joneses” (KUJ) if α(1 − η) > 0 and “running awayfrom the Joneses” (RAJ) if α(1 − η) < 0.
The definitions of jealousy and admiration are intuitively interpreted as follows. If the utility of an agent decreases as others’ consumption rises, we can statethat his preference exhibits jealousy. On the other hand, if the utility increases asothers’ consumption falls, we state that the preference exhibits admiration. Similarly, if the marginal utility increases as others’ consumption rises, we state thathis preference exhibits KUJ, and, if not we state that the preference exhibits RAJ.
The production technology is defined as of the Ak form, and therefore, thedynamics of the common capital stock arek̇ = Ak −N∑

ci , given k0 ,

(2)

i=1

where k : R+ → R+ , k0 ∈ R++ , and A ∈ R++ . Here, k denotes the stock of common
capital, k0 is its initial value, and A is a constant productivity parameter. If eachagent chooses a symmetric strategy, we have c = ci for all i = 1, · · · , N. In thiscase, from (2), the growth rate, g, can be written ascg= A−N .
kThroughout this paper, we assume the following condition.
5

This specification follows Galı́ (Ref. 18).

3

(3)

Assumption 2.1. It holds that

(
)
ρ
1−α
ψ < < min 1,
,
A
N

(4)

where ψ = (1 − α)(1 − 1/η).
Here, we impose Assumption 2.1 to ensure that the problem is well defined
and that the rate of balanced growth is positive.6

3Open-loop Nash Equilibrium

We first consider a situation where agents commit to their announced actions.
Definition 3.1 (Open-loop Nash equilibrium). An N-tuple of consumption paths,(c1 , . . . , cN ), is called an open-loop Nash equilibrium if, for each i = 1, . . . , N, ci
maximizes (1) subject to (2).
An open-loop Nash equilibrium is a plausible equation concept in the situation
where agents can not observe the value of common capital at each time7 , andthus, condition their strategies only on the basis of the initial value of common
capital, and precommit themselves to their future consumption paths. To solve
the problem, we define the following current value Hamiltonian:


N
∑


η ( −α )1− 1ηoci c̄iHi =+ qi Ak −ci  ,η−1i=1
where qoi is the costate variable of the common capital. The optimal conditions
for this problem are given as()− 1η −α 1− 1η
ci c̄i= qoi ,
Aqoi = ρqoi − q̇oi ,

(5)
(6)

lim qoi k exp(−ρt) = 0.

(7)

and
t→∞
6

By Assumption 2.1, the value function of each agent is well defined. Some literature studies
the case that it does not exist in finite values and refers to the several concepts of optimality.
See, for example, studies such as Stern (Ref. 24); Seierstad and Sydsæter (Ref. 25); Dockner,
Jorgensen, van Long, and Sorger (Ref. 26).
7
For example, if agents have to pay extremely high value of cost to observe the current amountof instracture accumulated in the economy, they would give up to observe the current amount of
infrastracture and adopt the open-loop Nash equilibrium.

4In equilibrium, it follows from (5) and (6) that c = c̄i = ci for all i = 1, · · · , N, andthus, the Euler equation is given asċ A − ρ
=
.
c 1−ψ

(8)

Therefore, we have the following result.
Proposition 3.1. The growth rate in the open-loop Nash equilibrium is given by
go =

A−ρ
.
1−ψ

(9)

Proof. From (3), (5), and (6), we obtain the balanced growth rate in this open-loop
Nash equilibrium.

Note that from conditions (5) to (7), we have a unique open-loop Nash equilibrium in this model. It is also easy to show that the open-loop Nash equilibrium
is Pareto eﬃcient. This eﬃciency result is similar to that derived by Chiarella,Kemp, van Long, and Okuguchi (Ref. 27), who show, contrary to the traditionalview, that an open-loop Nash equilibrium can be socially eﬃcient in various situations.

4

Feedback Nash Equilibrium

We next derive the case where each agent cannot commit to his future actions.
This case can be analyzed by applying the feedback Nash equilibrium concept,
which allows agents to choose and expect optimal consumption paths that dependon the current stock of common capital at each time.
Definition 4.1 (Feedback Nash equilibrium). An N-tuple of consumption paths,(c1 , . . . , cN ), is called a feedback Nash equilibrium if, for each i = 1, . . . , N, ci
maximizes (1) subject to (2) and c j = c j (k) for j , i.
Note that the feedback Nash equilibrium implies that agents can monitor thestock of common capital, and they condition their future consumption paths on thevalue of common capital at each time. We solve the feedback Nash equilibriumby using dynamic programming. The Hamilton-Jacobi-Bellman equation of thisproblem is



N
∑


 η (
1
)
1−
f
−α
η
+ qi Ak −ci c̄i (k)ci  ,ρUi (k) = max 
(10)
ci
η−1
i=1
5

∑
where c̄i (k) = j,i c j (k)/(N − 1); qif = dUi /dk; and Ui (k), defined by below,
denotes the value function of agent i.
∫ ∞
) 1
η (
Ui (k) = max
ci c̄i (k)−α 1− η exp (−ρ(s − t)) ds.
(11)
t η−1
From (10), the first-order condition is obtained as
(

−1

)

ci η c̄i (k)−α 1− η = qif (k).
1

(12)

In what follows, we consider the equilibrium in linear and nonlinear strategies.

4.1

Equilibrium in Linear Strategies

In this subsection, we consider equilibrium where the agents asopt the following
symmetric linear strategy.
c(k) = βk + γ,

(13)where β and γ are constants. In this situation, we have the following lemma:
Lemma 4.1. If agents adopt the linear strategy defined in (13), it holds that
ωBGP =

ρ − ψA
1 − α − ψN

U(k) =

c(k)ψ
ψωBGP

for all t ∈ R+ ,

(14)

and
for all k ∈ R+ ,(15)

where ωBGP denotes the ratio of consumption to common capital in the case thatthe agents adopt the linear strategy.
Proof. From (12) it holds that in equilibrium,
U(k) =

1c(k)ψ + δ,
βψ

(16)

where δ denotes an integral constant. By using (12) and (16), (11) can be written
as
(
)
ρ
η
k
ψ
c(k) + δ =−N+Ac(k)ψ .
βψη−1
βk + γ6

Therefore, for the above equality to hold for any value of k, it must hold that
δ=γ=0

(17)

and
β=

ρ − ψA
.
1 − α − ψN

(18)

Thus, by substituting (17) and (18) into (13) and (16), we have (14) and (15),respectively.

From Lemma 4.1, we have the following proposition.
Proposition 4.1. If agents adopt the linear strategy defined in (13), the growthrate in the feedback Nash equilibrium is positive and given asf= N (ωSS − ωBGP ) ,
gBGP

(19)

f
denotes the growth rate of the economy and ωSS = A/N.
where gBGP

Proof. If agents adopt (13), Lemma 4.1 holds. Therefore, substituting (14) into
(3), we getk̇ ċN (ψA − ρ)
= = A+= N (ωSS − ωBGP ) ,k c
1 − α − ψN
where ωSS > ωBGP by Assumption 2.1. Next we check that (19) satisfies thetransversality condition. From (15) and (19), we get
( [
])
cψ0
ψ A(1 − α) − ρN
U(k) =
exp
t .
ψωBGP
1 − α − ψN
Therefore, we have
(
)
cψ0
f
lim U(k) exp (−ρt) = lim
exp (α − 1)gBGPt =0
t→∞t→∞ ψωBGP
fsince α < 1 and gBGP
> 0 by the assumptions.

7

(20)


4.2

Equilibrium in Nonlinear Strategies

This subsection considers a more general case where agents adopt symmetric nonlinear strategies.8
Lemma 4.2. In a symmetric feedback Nash equilibrium, it holds thatA

k = Zc A−ρ (1−ψ) +

c
ωBGP

,

(21)

where Z ∈ R denotes an integral constant.
Proof. In a symmetric equilibrium, it follows from (12) that (10) can be written
as
(
) ψ
η
ρU =
(22)
− N q f ψ−1 + q f Ak.
η−1
Here, note that from (4), the coeﬃcient of the first term on the right-hand sideof the above equation is positive. Therefore, since (22) is D’Alembert’s (or Lagrange’s) equation, the solution of (22) satisfies the following equation:− A

k = Zq f A−ρ +1
1 − α − ψN f ψ−1q,ρ − ψAAgain, using (12), we get (21).

The sign of Z in Lemma 4.2 plays a crucial role in characterizing the dynamics
of the economy, as will be clear in the statement in the last part of this section.
Lemma 4.2 gives the dynamics of the economy as follows.
Proposition 4.2. The dynamics of consumption in a symmetric feedback Nashequilibrium are characterized by (3) andċN(A − ρ)(ωSS − ω)=,c (1 − α − ψN) (ωUB − ω)

(23)

where ω = c/k andωUB =(1 − ψ)A.
1 − α − ψN

8See, for example, Tsutsui and Mino (Ref. 28), Dockner and Sorger (Ref. 8), Sorger (Ref. 9),
Vencatachellum (Ref. 11), and Itaya and Shimomura (Ref. 29).

8

Proof. Since ω = ωUB is not feasible since it violates the diﬀerentiability of c,
we assume that ω , ωUB in the following analyses. From Lemma 4.2, in thesymmetric feedback Nash equilibrium, we have (21). Rearranging (21) yields()
Ac
Z = k−c A−ρ (ψ−1) .
ωBGP
Taking the logarithm of both sides of the above equation and diﬀerentiating themwith respect to t, we have
] 1 ċk̇ [c= (1 − α − ψN) + (1 − ψ)A
.
k
k
A−ρc
Therefore, by substituting (3) into the above equation, we have (23).

Note that (23) is a necessary condition for the symmetric feedback Nash equilibrium, since it is derived from the first-order condition (12).
Before we state Proposition 4.3, we state the following lemma.
Lemma 4.3. In the symmetric feedback Nash equilibrium, it holds thatω̇ =N (ωBGP − ω) (ω − ωSS ) ω.
ωUB − ω(24)

Proof. Since ω = ωUB is not feasible, we assume here that ω , ωUB . Subtracting(3) from (23), we haveω̇ ċ k̇ N (ωBGP − ω) (ω − ωSS )
= − =
.
ω c k
ωUB − ωThe above equation completes the proof.



Therefore, the dynamics of the economy are summarized as that of ω. We now
state the following proposition.
Proposition 4.3. Denoting the growth rate of consumption in the feedback Nashequilibrium by gc , we have the following:
1. In the case that ω0 = ωBGP , the growth rate in the feedback Nash equilibrium is given by (19).
2. In the case that ω0 > ωBGP , the feedback Nash equilibria are only the steadystates if 1 − α > N and they converge to steady states if 1 − α < N: gcf → 0
as t → ∞.

9

3. In the case that ω < ωBGP , the growth rate in the feedback Nash equilibrium
converges to that in the open-loop Nash equilibrium (9): gcf → go as t → ∞,where ω0 denotes the initial ratio of c to k: ω = ω0 at t = 0.
Proof.
1. Case 1: ω0 = ωBGP
Substituting ω0 = ωBGP into (3), we have the result.
2. Case 2: ω0 > ωBGP
See Appendix.
3. Case 3: ω0 < ωBGP
It follows from (24) that ω̇ < 0 for all ω < ωBGP and that ω̇ = 0 at ω = 0.
Since this implies ω → 0 as t → ∞, Proposition 4.2 gives
N(A − ρ)ωSS
A−ρ
ċ
=
=
,
t→∞ c
(1 − α − ψN)ωUB 1 − ψlimwhere the right-hand side is the same as the growth rate in the open-loop
Nash equilibrium. Therefore, it satisfies the transversality condition under
Assumption 2.1.
Figure 1 is the phase diagram in the case that 1 − α < N. The solid curves in
Figure 1 depict the loci of capital and consumption satisfying (21) corresponding
to various values of ω0 in Proposition 4.2. The BGP line plots the locus of capitaland consumption in the case that ω0 = ωBGP , and the curves above and below the
BGP line depict the loci in the case that ω0 > ωBGP and ω0 < ωBGP , respectively.
The SS line depicts the k̇ = 0 (ω = ωSS ) locus, and thus, k̇ > 0 when a path islocated below the line and k̇ < 0 when it lies in the area above the line. The slopeof the SS line is larger than that of the BGP line by Assumption 2.1. It should also
be noted that the SS line in the case that ω0 > ωBGP lies in the area below the UB
line. The UB line plots ω = ωUB , at which the slope of the peaks of the curves
drawn from (21) become infinite.
The case that ω0 = ωBGP corresponds to the balanced growth path. Notethat a balanced growth is realized only when ω0 = ωBGP . Here, it should be
noted that the growth rate in this case coincides with (19), and thus, it satisfies thetransversality condition.
In the case that ω0 > ωBGP , since the paths in the area above the UB line
move up in the north-west direction along the curves drawn from (21) and reachthe vertical axis in a finite period, the equilibrium paths drawn from (21) must lie
between the UB and BGP lines. Moreover, since the paths between the UB andSS line move down in the south-west direction and the paths between the SS and
10

ct
UB
SS
BGP

O

α = −0.5, η = 4/3, ρ = 0.5, A = 1, N = 2
BGP: ct = ωBGP kt , SS: ct = ωSS kt , UB: ct = ωUB kt
Figure 1: Dynamics for feedback Nash equilibrium

11

kt

BGP line move up in the north-east direction along the curves, the equilibrium
paths converge to the SS line. This implies there are infinitely many feedback
Nash equilibria which converge to corresponding steady states.9
In the case that ω0 < ωBGP , the paths must be below the BGP line and move
up in the north-east direction along the curves. Along these curves, the growth
rates converge to the balanced growth path in the open-loop Nash equilibrium.
Therefore, we find that there also exist infinitely many nonbalanced growth pathsin the feedback Nash equilibrium .10

5Comparison of the Growth Rates in Open-loop
and Feedback Nash Equilibria

5.1 Linear Strategies
As Tornell and Velasco (Ref. 4) and Shibata (Ref. 5) illlustrate, in an economy
without consumption externalities, the balanced growth rate in the feedback Nashequilibrium is lower than that in the open-loop Nash equilibrium. In this section,
we show that the existence of consumption externalities may destroy this relationship between the two equilibrium growth rates.
Theorem 5.1. Under Assumption 2.1, it holds that
f
> go
gBGP

if and only if
α < 1 − N.

(25)

Proof. To compare the growth rates in the open-loop and feedback Nash equilibria, presented in respectively, (9) and (19), we calculate their diﬀerence:fgBGP− go =

ωBGP(1 − α − N) .
1−ψ

(26)

Since the first multiplying term is positive by Assumption 2.1, the growth rate in
the feedback Nash equilibrium becomes higher than that in the open-loop Nashequilibrium if and only if (25) holds.
9Note that these equilibrium paths are not Markov perfect since the domain of each path does
not span the entire state space. However, we can easily extend the strategy to be made Markov
perfect. For example, by extending the strategy to (21) with ω > ωBGP if k > kmax , where kmax
denotes a maximum value of k satisfying (21), it can be Markov perfect. See also Clemhout and
Wan Jr. (Ref. 30) and Shibata (Ref. 5).
10
See also Shibata (Ref. 5).

12Note that under (25), we can always choose the value of η to satisfy (4); that is,
we can prove that in this model, there exists a situation in which (26) is positive.
Since N > 1, (25) states that the growth rate in the feedback Nash equilibrium ishigher than that in the open-loop Nash equilibrium if preferences exhibit admiration to other agents’ consumption. This result shows that although the growthrate in the feedback Nash equilibrium is lower than that in the open-loop Nashequilibrium as generally argued, the presence of strong admiration reverses theconclusion.
Intuitively, the presence of jealousy implies that the utility of agents decreases
with an increase of the others’ consumption. This suggests that the presence of
jealousy is a factor raising their current consumption and reducing their contribution to the accumulation of common capital. On the other hand, the presenceof admiration implies that their utility increases with increase of the others’ consumption, suggesting that admiration is a factor raising contribution to the accumulation of common capital. It shows that the growth rate in the feedback Nashequilibrium, where agents can change their consumption on the basis of others’
consumption, may be higher than that in the open-loop Nash equilibrium if thedegree of admiration is strong enough. Alternatively, the presence of admiration
can also be interpreted as the other source of the free-rider problem, since theycan enjoy utility from others’ consumption. In this sense, it can be said that thegrowth rate without commitment becomes higher than that with commitment if
the free ride on others’ consumption is stronger than that on the common capital
accumulation.
Finally, we make the following remark.
Remark 5.1. The relative magnitude of the two growth rates is irrelevant towhether preferences exhibit KUJ (α(1 − η) > 0) or RAJ (α(1 − η) < 0).
Remark 5.1 follows from the fact that (25) does not include the intertemporalelasticity of substitution.

5.2

Nonlinear Strategies

Finally, we compare the open-loop Nash equilibrium and the nonlinear feedback
Nash equilibrium. It is obvious that the growth rate in the open-loop Nash equilibrium is higher than that in the feedback Nash equilibrium in the long run in thecase that Z < 0, where consumption and capital stock converge to a steady state.
In the case that Z > 0, the growth rate of common capital converges to A sinceω converges to 0. We have the following theorem on the rate of consumption
growth.

13

Theorem 5.2. There exist infinitely many nonlinear feedback Nash equilibria suchthat
gcf > gofor all t ∈ R+

if and only if α < 1 − N and ω0 < ωBGP .
Proof. Rearranging (23), we havegcf =A − ρ (1 − ψ)A − (1 − ψ)Nω.
1 − ψ (1 − ψ)A − (1 − α − ψN)ωThe above equation implies that gcf > go if (1 − ψ)N < (1 − α − ψN), that is,
α < 1 − N.
Therefore, we find the same condition for the ordinal relationship between
growth rates to be destroyed as in the case that agents adopt linear feedback strategies, (25), even if they adopt nonlinear ones.

6

Conclusion

This paper introduced consumption externalities into a dynamic game growthmodel of common capital accumulation. Contrary to the general argument thatthe growth rate in a feedback Nash equilibrium is lower than that in an open-loop
Nash equilibrium, we showed that in our model, the growth rate in the feedback
Nash equilibrium under both linear and nonlinear strategies could be higher thanthat in the open-loop Nash equilibrium if agents strongly admire others’ consumption, while a conventional relationship is maintained between the two growth rates
if the agents envy others’ consumption.

14

Appendix
Proof of Proposition 4.3 in the Case that ω0 > ωBGP
We first consider the case that 1 − α > N. Noting that ωSS > ωUB > ωBGP in this case, weinvestigate (24) by classifying four cases.
1. ω0 > ωSS
In this case, it must hold thatω̇ ≥ ω̇0 > 0,
and thus, it follows from (3) thatk̇
≤ A − Nω0 < 0.
k
Therefore, the value of k reaches 0 in finite time. Since this implies that (21) cannot
hold for any c ∈ R+ at some finite time, we do not classify this case as feedback
Nash equilibrium.
2. ωSS = ω0
In this case, we haveω̇ = k̇ = ċ = 0

for all t ∈ R+ .

Since this case obviously satisfies the transversality condition and (21) for all t ∈R+ , it must be classified as a feedback Nash equilibrium.
3. ωSS > ω0 > ωUB
In this case, it follows thatω̇ <N (ωUB − ωBGP ) (ωSS − ω0 ) ωUB< 0.
ωUB − ω0

Therefore, the value of ω reaches ωUB in finite time, where the right-hand side of(24) diverges and (21) is not satisfied.
4. ωUB > ω0 > ωBGP
In this case, it holds thatω̇ >N (ω0 − ωBGP ) (ωSS − ωUB ) ωBGP
> 0.
ωUB − ω0

Therefore, the value of ω reaches ωUB in finite time, where the right-hand side of(24) diverges and (21) does not hold. Thus, it cannot be classified as a feedback
Nash equilibrium.
Therefore, in the case that 1 − α > N, we have only the steady states as feedback Nashequilibria.
We next consider the case that 1 − α < N. In this case, ωUB > ωSS > ωBGP by
Assumption 2.1. We investigate (24) by classifying two cases.

15

1. ω0 > ωUB
In this case, it holds thatω̇ ≥ ω̇0 > 0,
and thus, it follows from (3) thatk̇
≤ A − Nω0 < 0.
k
Therefore, the value of k reaches 0 in finite time. Since this implies that (21) cannot
hold for any c ∈ R+ at some finite time, we exclude this case as a feedback Nashequilibrium.
2. ωUB > ω0 > ωBGP
In this case,ω̇ < 0 if ω0 > ωSSandω̇ > 0 if ω0 < ωSS .
Therefore, ω → ωSS as t → ∞, whereω̇ = k̇ = ċ = 0.
The transversality condition is obviously satisfied in this case.

16

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