A Note on Existence of the Business-as-Usual Equilibria in a
Carbon Commons:
An External Appendix for
“Business-as-usual in a Global Carbon Commons”
by R. Harrison and R. Lagunoff
In this note, we establish existence of Business-as-usual equilibria under certain
parameter configurations. To do this, we lay out the corresponding dynamic stochastic
game for the model specified in Section 2. We then show that for certain configurations, the stochastic game satisfies the conditions of a Theorem of Horst (2005). That
theorem establishes sufficient conditions for the existence of a pure strategy Markov
Perfect equilibrium in which the strategies are almost-everywhere differentiable functions of the state. Our notion of the BAU equilibrium for the global carbon commons
fits this definition precisely.
To get existence, we modify the model slightly by requiring that the parameter A
in the carbon dynamic (Equation (3) in the paper) vary stochastically over time.
We begin by assuming log payoffs in output. Define the payoff function for each
country i by
Ui (ω, θ, c) = θit ci + (1 − θit ) log (ω − C)
Clearly, Ui is smooth, strictly concave in ci , and continuous in (ω, θ) and c.
Next, let ei denote the extraction rate, i.e., ci = ei ω, and let ēi = maxθ estatic
(θ)
i
where estatic
(θ)
is
i’s
optimal
extraction
rate
in
an
equilibrium
in
the
one-shot
static
i
(θ) is constructed explicitly as
game (with δ = 0). From the paper, estatic
i
estatic
(θ)
i
θi
=
1 − θi
1+
X
j
θj
1 − θj
!−1
(1)
Consequently, ēi is the maximum statically optimal rate for country i and is, hence,
an upper bound for any BAU equilibrium extraction rate for i. Using (1), one can
verify that
n−1
z }| {
static
ēi = ei
(θi , θ, . . . , θ)
(2)
In other words, the maximum statically optimal rate for i is determined at its own
highest extraction elasticity and the lowest for all other countries.
1
Let ei the best response to
1
max
ei
1−δ
!
θ log ei + (1 − θ) log (1 − ei −
X
ēj )
j6=i
Then ei is a lower bound for any BAU equilibrium extraction rate that i would choose
since ei is chosen assuming that every other country is choosing its statically optimal
(hence maximal) extraction rate every period.
Any BAU equilibrium extraction rate e∗i must therefore satisfy ei ≤ e∗i (ωt , θt ) ≤ ēi
for all states (ωt , θt ) ∈ IR+ × [θ, θ]n .
Since the bounds ei and ēi do not depend on the carbon stock ω, we can effectively
bound the state space by the maximal steady state induced over all extraction profiles
in [ei , ēi ]n . Call the maximal steady state ω. We restrict attention to compact stocks
in the compact set Ω ≡ [F, ω]. Let Θ = [θ, θ]n . Our state space then is Ω × Θ which
is obviously compact and convex.
The feasible action correspondence for each country i is then given by Di :
(ω, θ) 7→→ [ei ω, ēi ω] over all (ω, θ) ∈ Ω×Θ. Notice that Di is upper hemi-continuous,
nonempty, compact, and convex valued. Since Di is identical over all countries, we
drop the subscript and let Dn = ×i D. We will sometimes write Dn (Ω × Θ) to express
the range space of Dn .
Next, we assume that parameter A varies stochastically. The process {At } is
assumed IID with each At determined by a nonatomic distribution Λ that admits a
twice differentiable density λ on support [1, Ā].
The transition probability on next period’s carbon stocks ω 0 is then given by the
Markov distribution



 Λ(ω 0 /(ω − C − b)γ ) if ω 0 ≥ F
0
χ(ω | ω, θ, C) =


 0
if ω 0 < F
Note that this distribution dependsPonly onP
the aggregate consumption C and is norm
continuous on Ω × Θ and on C ∈ [ j ej ω, j ēj ω]. The transition function mapping
states and actions to distributions on future states is given by
0
0
0
Z
Q(ω , θ | ω, θ, C) = χ(ω | ω, θ, C)
θ0
π(θ0 | θ)dθ0
θ
Assuming that π is twice diffentiable, it follows that for each (ω 0 , θ0 ) ∈ Ω × Θ,
2
Q(ω 0 , θ0 | ·) is norm-continuous on Ω × Θ × Dn (Ω × Θ), and admits an almosteverywhere twice differentiable density q(ω 0 , θ0 | ω, θ, C).1
Finally, let
U=
sup
Ui (ω, θ, c).
i,ω∈Ω,θ∈Θ,c∈Dn
Notice that U < ∞
To summarize, the stochastic game is represented by G(Dn , Ω × Θ, q, U ) where
• Ω × Θ is a compact, convex state space,
• Dn is an upper hemi-continuous, compact and convex valued mapping from
states to extraction profiles.
• q is a norm-continuous, transition density determined as the product of densities
λ and π on ω 0 and θ0 , respectively.
• U = (Ui ) with Ui : Ω × Θ × Dn representing i’s differentiable, strictly concave
(in ci ) payoff.
To verify existence, we verify that G(Dn , Ω × Θ, q, U ) satisfies the conditions of
Horst (2005, Theorem 2.3) for an open set of parameter configuration of λ, π, b, γ,
F , Ā, θ, θ and n, and for “robust” distributions λ and π.
In particular, we need to show that there exists β < 1 such that for each country
i and each state (ω, θ) ∈ Ω × Θ,
2
X
X
δ
||∂ 2 Ui /∂ci ∂cj ||
||∂ q(·| ω, θ, C)/∂C 2 ||
sup
+
U
sup
< β
(3)
2 U /∂c2 |
2 U /∂c2 |
|∂
1
−
δ
|∂
c∈Dn
c∈Dn
i
i
i
i
i
j6=i
Horst refers to (3) as the Moderate Social Influence (MSI) assumption. MSI resembles
a dominant diagonal assumption by which a country’s own carbon consumption has
a larger effect on its long run marginal payoff than the carbon decisions of others.
We now establish the following result.
Claim. Let G(Dn , Ω×Θ, q, U ) be a stochastic game with λ and π (in the construction
of q) uniform on their respective supports, [1, Ā] and [θ, θ]n . Holding fixed all other
parameters, there exists A0 < ∞ and θ0 < θ such that for any Ā ∈ [A0 , ∞) and any
θ ∈ [θ0 , θ), a BAU equilibrium exists for the given Ā and θ.
1
The density may have an atom at ω 0 = F .
3
Proof of the Claim. Using the definition for U above, it is easy to verify that there
is a βu < 1 such that for all i and for all states (ω, θ) ∈ Ω × Θ,
X
j6=i
sup
c∈Dn
||∂ 2 Ui /∂ci ∂cj ||
< βu
|∂ 2 Ui /∂c2i |
To see this observe that in the log payoff case,
X
j6=i
sup
c∈Dn ,ω,θ
(n − 1)(1 − θ)ei
||∂ 2 Ui /∂ci ∂cj ||
≤
≡ βu
2
2
|∂ Ui /∂ci |
θ(1 − nei ) + (1 − θ)ei
Then one can verify that βu < 1 if θ is close enough to θ, given that ēi is determined
by Equations (1) and (2) above.
Next we construct q, in other words construct λ, π, b, γ and F , so that
2
X
||∂ q(·| ω, θ, C)/∂C 2 ||
δ
< 1 − βu
U
sup
2 U /∂c2 |
1−δ
|∂
c∈D
i
i
i
Clearly this holds for all ω, θ and C such that A(ω − C − b)γ < F . Now take λ
and π to be uniform on their respective supports. Then
||∂ 2 q(·| ω, θ, C)/∂C 2 ||
U
sup
≤
|∂ 2 Ui /∂c2i |
c∈Dn
i
ω 0 c2i (ω − nci )2
n
sup
(Ā − 1)(θ − θ) ci ∈D,ω,ω0 ∈Ω,θi ∈Θ (ω − nci − b)2−γ (θi (ω − nci )2 + (1 − θi )c2i )
X
Given compactness of all the sets over which the sup is taken, the right-hand side
term is finite. By taking Ā sufficiently large, the right-hand side can be made smaller
than (1 − δ)(1 − βu )/δ). Consequently, Horst’s Moderate Social Influence condition
holds, and so a BAU equilibrium exists for these parameters. This concludes the
argument.
References
[1] Harrison, R. and R. Lagunoff (2015), “Business-as-usual in a Global Carbon
Commons,” Georgetown University Working Paper.
[2] Horst, U. (2005), “Stationary Equilibria in Discounted Stochastic Games with
Weakly Interacting Players,” Games and Economic Behavior, 51: 83-108.
4