Transcript c discounte

The LinC strategy for cooperation
in the international emissions game
Jobst Heitzig (RD IV)
Kai Lessmann (RD III), Yong Zou (RD IV)
Work in progress, presented at PIK Research Days, 14 Dec 2010
Summary
Although…
•
•
a globally optimal GHG emissions path has large payoffs
and this outcome can be made profitable for all world regions,
(e.g. by emissions trading or direct monetary transfers)
…the free-rider incentive makes cooperation quite difficult.
(a player can hope the others will solve the problem without him)
We show that (under certain conditions)
this problem can be solved by using
a strategy of linear compensation (LinC)
realising a dynamic redistribution of permits.
Jobst Heitzig (RD IV) The LinC strategy for cooperation
2
Outline
• Framework, costs, benefits
• Profitability doesn’t deter free-riding
• Dynamic permits and the LinC strategy
• A road-map for stable cooperation
• How egalitarian can we get?
Jobst Heitzig (RD IV) The LinC strategy for cooperation
3
Framework of our analysis
• World consists of N regions interpreted as “players” (e.g. 12)
• Consider benefits and costs of emissions reductions
relative to some baseline emissions (e.g. “business as usual”)
• Benefits for region i of global emissions reductions in year t
= economic damages avoided after t, at a discount rate (e.g. 2%)
(e.g. damages = 2.7% of GDP for a doubling of CO2 concentrations)
• Non-decreasing marginal costs of reducing regional emissions
by 1 Gton (e.g. quadratic, Ellerman and Decaux 1998)
• Some “optimal path” of regional emissions
maximizes the total global payoff
• Further details mostly unimportant for this analysis
(though not in reality)
Jobst Heitzig (RD IV) The LinC strategy for cooperation
4
Main question and pessimistic findings so far
Can there be a stable agreement to realize the global optimum?
Finus, Ierland, Dellink (2006): (with simple cost-benefit model “STACO”)
• Global optimum not profitable for some regions
• China has to reduce very much due to its low marginal costs
 gets less payoff than under business as usual
• All possible long-time coalitions are unstable
• Assumption: A group of regions builds a coalition once and for all
and maximizes their joint payoff
• Finding: Always some region wants to leave (be a free-rider) or join
• Monetary transfers can not
remove free-rider incentives
in their model
Jobst Heitzig (RD IV) The LinC strategy for cooperation
5
Profitability doesn’t deter free-riding
How to make the optimal path profitable for all regions?
 Efficient emissions trading
• Reductions can be done at home
or bought from a cheaper region
 marginal costs get equal for all regions
 Global “cap” can be allocated flexibly
 Many different allocations of the optimal global path
are profitable for all regions
So: Just estimate the globally optimal path,
then negotiate a regional allocation of these permits?
• If other’s compliance cannot be enforced,
free-riding remains attractive!
Jobst Heitzig (RD IV) The LinC strategy for cooperation
6
Reacting on each other can deter free-riding
• Finus et al. assume a coalition builds once
and maximizes their joint payoff over 100 years
• More realistic: Decisions to cooperate
can be made and changed at all times
Better model: an iterated game played in periods
Game theory: Cooperation might evolve easier
if players can react on each other’s actions
Jobst Heitzig (RD IV) The LinC strategy for cooperation
7
A better model of the emissions game
• Game theory: Cooperation might evolve easier
if players can react on each other’s actions
Our model:
In each period t,
each region i
chooses net emissions ei (t)
= emissions at home
– reductions payed elsewhere
• usually > 0, but might be < 0
• hopefully equal to the target allocation, but maybe different
• Task: Find a joint strategy for choosing ei (t) that
• realizes the global optimum and deters free-riding
• will not be renegotiated even if deviations did happen someday
Jobst Heitzig (RD IV) The LinC strategy for cooperation
8
The basic idea in a simple analogy
• Alice, Berta, and Celia each farm some part of their back-yard
Alice
Berta
Celia
Harvest 1:
Berta falls short
Harvest 2:
She has higher liabilities
and fulfils them
Jobst Heitzig (RD IV) The LinC strategy for cooperation
Next spring:
Back to normal
9
A strategy based on dynamic permits
• Assume a target allocation ei*(t) of the optimal path
has been agreed upon
• We suggest each region i uses this strategy to choose ei (t):
Linear Compensation (LinC)
In period t+1, compute excesses
dj (t) = max { ej (t) – aj (t), 0 }
and a new permit allocation
aj (t+1) = ej*(t+1) + [ dmean (t) – dj (t) ] · c
In period 1, permits equal targets: aj (1) = ej*(1)
Strategy: Always emit as much as permitted: ei (t) = ai (t)
Jobst Heitzig (RD IV) The LinC strategy for cooperation
10
Effect of dynamic permits
Linear Compensation (LinC)
In period t+1, compute excesses
and a new permit allocation:
dj (t) = max { ej (t) – aj (t), 0 }
aj (t+1) = ej*(t+1) + [ dmean (t) – dj (t) ] · c
new permits = target + (mean excess – own excess) · factor
• If all follow this exactly, excesses d are zero and
both permits a and net emissions e equal the target values e*
• Otherwise, permits are redistributed temporarily
so that who emitted too much in t is allowed less in t +1
– smoothly and proportionately, keeping the global target
Jobst Heitzig (RD IV) The LinC strategy for cooperation
11
LinC is stable in more than one ways
Linear Compensation (LinC)
new permits = target + (mean excess – own excess) · factor
If total target i ei*(t) = globally optimal emissions path,
we can prove that in every period t …
• no region (or group of regions) has an incentive
to deviate from their permits (LinC is subgame-perfect, even groupwise)
• ignoring past excesses or switching to a different strategy
can never profit all regions (LinC is weakly and strongly renegotiation-proof)
• Both holds no matter whether excesses happened before t or not!
Still, excesses may happen (e.g. due to error), but …
Jobst Heitzig (RD IV) The LinC strategy for cooperation
12
LinC is stable in more than one ways (cont.)
Linear Compensation of excesses (LinC)
new perm. = targ. + (mean – own exc.) · c
region A
• No incentives to emit more
• No incentives to renegotiate
region B
If excesses do happen anyhow,
• permits are redistributed for one period
(LinC has short memory)
• new permits are still globally optimal
(LinC remains Pareto-efficient)
1 2 3 4 5 6 7 8
target
permits
net emissions
• deviators have incentives to soon “make up” for earlier excesses
• small errors don’t add up or lead away from the target path
(similar to trembling hand perfectness, also supported by simulations)
Jobst Heitzig (RD IV) The LinC strategy for cooperation
13
A road-map for stable cooperation
1. Determine…
•
•
a discount rate to use in estimation (2%? 1%?)
the shortest possible period length (2 years?)
in which emissions can be monitored reliably
2. Estimate the globally optimal emissions path
3. Negotiate an allocation of this into regional targets ei*(t)
•
use any criteria such as per capita permits,
per capita payoffs, historical responsibilities, …
4. Agree to use the LinC strategy to avoid free-rider incentives
and set the compensation factor c sufficiently high
5.
Be sufficiently “rational” and comply with this strategy
Jobst Heitzig (RD IV) The LinC strategy for cooperation
14
How egalitarian can we get?
• LinC can stabilize any target allocation of the global optimum
• e.g., one that achieves equal per capita payoffs
• A weaker strategy with bounded compensations (BLinC)
can only completely stabilize some target allocations
• e.g. distribute 1/3 of the payoff by population, the rest by GDP
(in simple cost/benefit model “STACO”)
but with other targets, stability is decreased
• e.g. achieving equal per capita payoffs
in purchasing power units (PPP)
• is only stable unless USA+Japan+Europe
jointly choose large excesses
Jobst Heitzig (RD IV) The LinC strategy for cooperation
15
Work done and to do
Done
• Game theoretic analysis of LinC almost finished (see our draft)
• First numerical simulations with simplified model
• Stability analysis of BLinC in the STACO model (as shown)
To do
•
•
•
•
•
•
What allocations are stable with bounded redistributions?
Use a better cost-benefit model for simulations (Suggestions?)
Assess the influence of uncertainty and number of players
More thorough comparison with existing literature
Try non-linear compensations to improve efficiency & stability
Simulate errors, learning, irrational behaviour, …
Jobst Heitzig (RD IV) The LinC strategy for cooperation
16
Take-home message
A simple dynamic strategy
using linear redistribution of permits
can stabilize an international agreement
to follow a globally optimal emissions path.
Thank you for your attention!
Jobst Heitzig
RD IV, Transdisciplinary Concepts and Methods
[email protected]
Jobst Heitzig (RD IV) The LinC strategy for cooperation
17
Appendix / Back-up slides
•
•
•
•
•
•
•
•
First simulations
Related work
LinC is subgame-perfect and renegotiation-proof (proof sketch)
Choice of compensation factor c
The problem of extreme allocations
Strategy BLinC and stability with bounded allocations
Repairing two shortcomings of STACO
Some allocations
Jobst Heitzig (RD IV) The LinC strategy for cooperation
18
First simulations
(rescaled)
(rescaled)

emissions

permits
Development of permits and emissions in several scenarios
(simplified cost-benefit model, but results qualitatively similar)
If all comply but make
If blue region always abates
small errors (std.dev. ):
only 90% of what is expected:
target path
blue’s permits quickly
settle on a smaller level
rest soon can emit more
blue soon emits less
Jobst Heitzig (RD IV) The LinC strategy for cooperation
target path
19
First simulations (cont.)
Development of permits and emissions in several scenarios
(simplified cost-benefit model, but results qualitatively similar)
If all comply 90% of the time
If blue region stays on its
& totally fail 10% of the time:
baseline (business-as-usual):
(rescaled)
baseline
emissions
(rescaled)
permits
baseline
target path
target
baseline
rest also goes back
to baseline quickly
target
Jobst Heitzig (RD IV) The LinC strategy for cooperation
20
Related work (Selection)
• Finus, Ierland, Dellink (2006, see above)
• “One-shot” “cooperative” game of coalition formation
 Coalition maximizes their payoff, no reactions to other’s actions
 Not joining pays  No stable long-term coalitions
• Barrett (1994, 1999), Asheim et al. (2006)
• Iterated “non-cooperative” game of emissions (as in our case)
• But only two actions: Cooperate or Defect (Prisoner’s Dilemma)
 No smooth reactions possible, play leaves the optimal path
 No weakly renegotiation-proof strategy
• Weikard, Dellink, Ierland (2010)
• Mixed model: Finite-horizon iterated non-cooperative game,
but in each period a cooperative game of coalition formation
 Somewhat more optimistic results, but no global cooperation
Jobst Heitzig (RD IV) The LinC strategy for cooperation
21
LinC is subgame-perfect, even groupwise
Linear compensation (LinC)
new permits = target – (own excess – mean excess) · c
No group of at most k regions has an incentive to net-emit…
• … more than their permits
• Game theory: It suffices to check that it doesn’t pay to
deviate once and then return to compliance
 Assume their joint excesses in t are x
In t, they jointly gain at most
x · average costs
In t +1, they jointly loose at least x · (1 – k/12) · c · average costs
Just choose c large enough
• … or less than their permits
• since marginal costs increase faster than marginal benefits
Jobst Heitzig (RD IV) The LinC strategy for cooperation
22
LinC is weakly and strongly renegotiation-proof
Linear compensation (LinC)
new permits = target + (mean excess – own excess) · factor
Formal definition of terms: Farrell & Maskin (1989), Bergin & MacLeod (1989)
Ignoring past excesses (pretenting a different past) can never profit all regions
• This would lead to a different allocation of the same total target
 At least one region’s allocation would have to decrease
 The region would have higher costs but the same benefits as before
Switching to a different strategy can never profit all regions
• Whatever happened before t, following LinC from t on
realizes the maximum long-term global payoff possible from t on
 Switching to a different strategy from t on
cannot realize a higher long-term global payoff from t on
 If some region gains from the switch, another must loose from it
Jobst Heitzig (RD IV) The LinC strategy for cooperation
23
Choice of compensation factor c
Linear compensation (LinC)
new permits aj (t+1) = ej*(t+1) + [ dmean (t) – dj (t) ] · c
Compensation factor c =  –p · 12/(12-k)
 = discount factor (0.98?), p = period length in years (2?)
k = sought degree of group-wise stability (5?)
• This way, no group of at most k regions has an
incentive to jointly exceed their permits
• If some regions emitted more in t,
it benefits all other regions in t +1
 No incentive to ignore excesses by others
Jobst Heitzig (RD IV) The LinC strategy for cooperation
24
The problem of extreme allocations
• Assume for some period t,
many individual allocations ei (t) exceed the baseline emissions
Emissions market might get inefficient, and prices unpredictable
Our assumption of equal cost curves might get invalid
Our strategic analysis might get invalid
 Modify the strategy so that all individual allocations
stay below some bounds (e.g. maximum of baseline and target)
Requires some modification of the compensation scheme
Some stability properties now depend on target and bound!
(Groupwise) subgame-perfection now requires that
neither the bounds nor the individual targets get too small
(see below)
Jobst Heitzig (RD IV) The LinC strategy for cooperation
25
Compensation scheme with bounded allocations
Bounded linear compensation (BLinC)
Excesses dj (t) = max { ej (t) – aj (t), 0 }
New allocation:
aj (t+1) = min { ej*(t+1) + [ dmean (t) – dj (t) ] · c, ajmax } if dj (t) = 0
aj (t+1) =
ej*(t+1) + [ dmean (t) – dj (t) ] · c / f (t)
if dj (t) > 0
f (t) is chosen so that total allocated permits = global target
• If excesses are not too large, this equals LinC
aj (t+1) = ej*(t+1) + [ dmean (t) – dj (t) ] · c
• Otherwise, permits are capped at their upper bound and the
compensation by those with excesses is rescaled accordingly
Jobst Heitzig (RD IV) The LinC strategy for cooperation
26
Stability with bounded allocations
 For subgame-perfection with bounded allocations,
it suffices if for all periods, regions i, and possible excesses x,
target – a min
e
i
i
C(Eopt) – C(Eopt + x) – bi x < p C(Eopt) –––––––––––
Ebase – Eopt
max. profit from excess x in t
C(E)
Ebase, Eopt
bi
eitarget
aimin
p
loss from minimal permits in t +1
global abatement costs given global emissions E
baseline and optimal global emissions (Eopt < Ebase)
marginal abatement benefits for region i
initially negotiated target for region i (= ei*, sum equals Eopt)
minimal possible permit allocation for region i
discounting factor for one period
Condition can be fulfilled with equal per capita payoffs in PPP
(STACO, 2-year periods, 2% discounting, aimin = Eopt – Ebase + eibase
so that the worst punishment for i is to pay for all global abatements)
Jobst Heitzig (RD IV) The LinC strategy for cooperation
27
Stability with bounded allocations (cont.)
 For groupwise subgame-perfection with bounded allocations,
it suffices if for all periods, groups of regions G, and excesses x,
target – A min
E
G
G
C(Eopt) – C(Eopt + x) – BG x < p C(Eopt) ––––––––––––
Ebase – Eopt
max. profit from excess x in t
BG
EGtarget
AGmin
loss from minimal permits in t +1
marginal joint abatement benefits for the group
initially negotiated joint target for the group
minimal possible joint permit allocation for the group
Condition fulfilled not with equal per capita payoffs in PPP
but with payoffs proportional to GDP (1995)
(STACO, 2-year periods, 2% discounting,
AGmin = Eopt – j  G max{ejbase, ejtarget}
so that the worst punishment for G is to pay for all global abatements
and for the excesses allowed to EEX, China, Brazil and ROW)
Jobst Heitzig (RD IV) The LinC strategy for cooperation
28
Details of cost-benefit model STACO
Used in Finus, Ierland, Dellink (2006)
• GHG emissions over 100 years in 12 world regions
• Benefits of emissions reductions in year t
= economic damages avoided after t, at discount rate 2%
(assumed damages = 2.7% of GDP for a doubling of CO2 concentrations)
• Approx. constant marginal benefits of not emitting 1 Gt CO2
(proportional to estimated regional GDP; Fankhauser 1995 and Tol 1997)
• Quadratically increasing marginal costs
(Ellerman and Decaux 1998)
• Not quantitatively realistic
due do simplifications,
but sufficient for
illustration here
Jobst Heitzig (RD IV) The LinC strategy for cooperation
29
Repairing two shortcomings of STACO
1. Benefits of emissions reductions in year 2109
= economic damages avoided between 2109 and 2110
 Almost all damages caused by emissions in 2109
occur later than 2110 but are not counted in STACO!
 Benefits are hugely underestimated
 Solution: Count all avoided damages, no matter how late
(based on estimated GDP and discounted at 2%)
 Result: Optimal path is worth almost twice as much!
2. The difference between optimal and baseline emissions
was assumed to be constant over time
 More realistic: Allow that emissions reductions vary in time
 Result: Optimal reductions increase
Jobst Heitzig (RD IV) The LinC strategy for cooperation
30
Emissions for equal per capita payoffs in PPP
• net emissions negative for Japan, large for EEX, China, Brazil, ROW
(Modified STACO, 100 years discounted at 2% yearly, two-year periods)
450
baseline emissions
optimal actual emissions
target net emissions
400
350
300
250
200
150
100
50
0
USA Japan
-50
Jobst Heitzig (RD IV) The LinC strategy for cooperation
EU
OOE
EET
FSU
EEX
China India
DAE
Brazil ROW
31
Emissions for complete groupwise stability
with bounded allocations
• one third of global payoff distributed by population, rest by GDP
(Modified STACO, 100 years discounted at 1% yearly, two-year periods)
450
baseline emissions
optimal actual emissions
target net emissions
400
350
300
250
200
150
100
50
0
USA Japan
-50
Jobst Heitzig (RD IV) The LinC strategy for cooperation
EU
OOE
EET
FSU
EEX China India
DAE
Brazil ROW
32