Analyzing Carbon Trade Permits in Optimization Framework

Department of Mathematics and Physics

Laboratory of Applied Mathematics

ANALYZING CARBON TRADE PERMITS IN

OPTIMIZATION FRAMEWORK

The topic of this master's thesis wasapproved by the department councilof

Department of Mathematics and Physics onDecember,

12 ^th

^{, 2008.}

The examiners of the thesis were Prof. Markku Lukka and Ph.D. MattiHeiliö.

The thesis was supervised by Ph.D MattiHeiliö.

In Lappeenranta,May,

22 ^nd

^{, 2009.}

Goodluck Mika Mlay

Ruskonlahdenkatu 13-15 E2

53850 Lappeenranta

p. +358449480915

goodluck.mlay@lut.fi

Lapeenranta University of Technology

Department of Mathematics and Physics

Goodluck Mika Mlay

Analyzing Carbon trade permits in Optimization framework

Thesis for the Degree of Master of Science inTechnology

2009

77pages, 22 gures,5 tables,1 appendix

Examiner: Prof. Markku Lukka and PhD. MattiHeiliö

Keywords: CarbonTrade,Emissionpermit,Kyototreaty,Optimization,Lagrangian,

Sequential Quadratic Programming,DierentialEvolution.

The threatscaused by globalwarming motivatedierent stakeholders todeal with

and control them. This Master's thesis focuses on analyzing carbon trade permits

in optimization framework. The studied model determines optimal emission and

uncertainty levelswhich minimizethe totalcost. Researchquestionsare formulated

and answered by using dierent optimization tools. The model is developed and

calibratedbyusing availableconsistentdata inthe areaof carbonemissiontechnol-

ogy and control. Data and some basic modeling assumptions were extracted from

reportsand existingliteratures. The datacollectedfromthecountriesintheKyoto

treaty are used toestimate the cost functions. Theory and methods of constrained

optimization are briey presented. A two-level optimization problem (individual

and between the parties) is analyzed by using several optimization methods. The

combined cost optimization between the parties leads into multivariate model and

calls for advancedtechniques. Lagrangian, SequentialQuadratic Programming and

DierentialEvolution(DE)algorithmare referredto. Theroleof inherentmeasure-

mentuncertainty inthe monitoringof emissionsisdiscussed. We brieyinvestigate

anapproachwhereemissionuncertaintywouldbedescribedinstochasticframework.

MATLAB software has been used to provide visualizations including the relation-

shipbetweendecisionvariablesandobjectivefunctionvalues. Interpretationsinthe

context of carbon trading were briey presented. Suggestions for future work are

giveninstochasticmodeling,emissiontradingand coupledanalysisofenergy prices

I amhighly indebted tothe Laboratoryof Applied Mathematics and Physics of

Lappeenranta University of Technology for great support ithas provided tome

throughout my studies. Anexceedingly cooperation I received from stateam

make mefeel athome.

I would like toexpress myspecial gratitude tomy supervisor of the thesis Ph.D.

MattiHeiliöfor valuablecomments, assistance and guidance; and Prof. Markku

Lukka for examiningmy thesis. Their guidance without limitationsmakemy work

done.

I would like toextend my sinceregratitude to Lic. Phil. Sirkku Parviainen forher

special advice in the area of optimization.

I alsowantto thank allmy friends includingfew tomention for their

contributions; especiallyJere Heikkinen and Tapio Leppälampi and Matylda

Jabªo«ska, who helped meintechnical supports, forwithout themthis workwould

have been too dicult.

I nallythank my familyfor how has stand with me shoulder by shoulder,

encouraging meeven inthe time of diculty and reminding methat one day, 'it

willbe possible'.

'Kiitos Kaikesta'

Lappeenranta,June,

6 ^th

^,2009.

Goodluck Mika Mlay.

1 Convex and concave behavior revealed by rst order derivative of

goal function,

G ⁱ (x i )

^{in Equation (3.30) expressed in terms variable}

x i

a i

= u i

^and parameters,

α i = γ i = 0.25

^{) . . . . . . . . . . . . . . . . 22}

2 Convex and concave behavior revealed by rst order derivative of

goal function,

G ⁱ (x i )

^{in Equation (3.30) expressed in terms variable}

x i

a i

= u i

^and parameters,

α i = γ i = 0.2 < 0.25

^{) . . . . . . . . . . . . . 23}

3 Convex and concave behavior revealed by rst order derivative of

goal function,

G ⁱ (x i )

^{in Equation (3.30) expressed in terms variable}

x i

a i

= u i

^and parameters,

α i = γ i = 0.3 > 0.25

^{) . . . . . . . . . . . . . 24}

4 The generalscheme of Evolutionary algorithmasow-chart. . . 34

5 The number of generationsagainst permits in USRegion . . . 37

6 Parameter value,

x 1

^and

R 1

^{plottedagainst numberof generation in}

US Region . . . 37

7 Objective function value,

G ¹ (x 1 , R 1 )

^{againstnumberof generation . . 38}

8 Objective function value, against parameter,

x ₁

^{. . . . . . . . . . . . 38}

9 Objective function value, against parameter,

R ₁

^{. . . . . . . . . . . . 40}

10 Costs for reducing emissions against reported emissionsfor US with

reference toEquation (3.46) . . . 50

11 Costsformonitoringuncertaintyagainstreportedrelativeuncertainty

for USwith reference to Equation(3.47) and data from table 1. . . . 51

12 Costsagainstregions'reportedemissionsholdtogetherwithreference

todata from table 1 . . . 52

13 Costsagainstregions'reportedrelativeuncertaintyholdtogetherwith

reference todata fromtable 1 . . . 53

14 Marginal Abatement Cost (MAC) against Emission for US with ref-

erenceto data fromtable 2. . . 54

15 MarginalAbatementCost(MAC)againstEmissionforUSwhenper-

mits are traded with referenceto data fromtable 3 . . . 55

uationswhennopermit tradeand whenpermit tradeisallowed with

reference to data fromtables 2 and 3 . . . 56

17 MarginalAbatement Costs (MACs) againstEmissions for CANZfor

situations when no permit trade and when permit trade is allowed

with referenceto data fromtables 2 and 3 . . . 57

18 3-D surface shown how Emissions, uncertainty and total cost for re-

ducingemission are relatedin USregion with referenceto data from

table 1 . . . 58

19 The histogram showing normal distribution for emissions in US and

OECDE . . . 60

20 The histogram showing normal distribution for emissionsin JAPAN

and CANZ . . . 60

21 The histogramshowing normaldistribution for emissionsin EEFSU 61

22 The histogramshowing normal distribution fortotal emissionsin all

Regionsdescribed in table2 . . . 62

1 Projections withreferenceto2010of Kyoto targettogetherwith cost

parameters of emissionsand uncertainty reductions. Source:[3] . . . . 27

2 Optimalvalues of both emissions,

x i

^and uncertainty,

R i

^{in the situ-}

ation beforetrade i.e,

y i = 0

^{. . . . . . . . . . . . . . . . . . . . . . . 29}

3 The optimal values of emissions and relative uncertainty when the

permits were traded. . . 29

4 Theoptimalvaluesofemissions,emissionpermitsand relativeuncer-

tainty aftertransaction using SQP method . . . 31

5 The optimal values of emissions, relative uncertainty and permits

found by usingDierentialEvolution algorithm5.1 . . . 36

List of Algorithms

5.1 DE/rand/l/bin (DEGS) . . . 35

1 Introduction 1

1.1 Background of the Thesis . . . 1

1.2 Objective of the Thesis . . . 3

1.3 Research Questions . . . 4

1.4 Structure of the Thesis . . . 5

2 Constrained Optimization 6 2.1 Lagrangian for constrainedproblem . . . 6

2.2 Optimality conditions. . . 7

2.2.1 Necessary conditions for optimality . . . 7

2.2.2 Sucient conditionsfor Optimality . . . 9

2.2.3 Saddle pointconditions . . . 10

3 Methodology and Modeling 10 3.1 Introduction of Relativeuncertainty to the Model . . . 13

3.2 Analysisof localminima . . . 15

3.3 Choiceof cost function . . . 19

3.4 Estimatesof cost parameters . . . 25

4 Data 26 5 Solution to the Model 26 5.1 ModelResults . . . 27

5.2 Sequential QuadraticProgramming (SQP) . . . 28

5.3 Implementation of solutionusing SQP . . . 31

5.4 DierentialEvolution(DE) algorithm . . . 32

5.5 Implementation of solutionusing DE . . . 34

5.6 Employment of Kuhn Tucker conditions to the Model . . . 41

5.6.1 KT sucient second orderconditions . . . 43

5.7 Evaluationof the ModelResults . . . 48

7 Conclusion and Future work 64

Appendix: The MATLAB Codes for SQP results presented in table4 69

In recent years our planet, earth has faced drastic climate changes. There has

been global warming that makes our planet hotter than one century ago. Several

environmental risks such as drought, ooding, spread of tropical diseases, destruc-

tion ofcostlinedue tohigh oceantides, meltingoficecovering peaksofmountains,

raging forest re, hurricanes and storms have brought worrisome to the mass of

people. Datashows that, over last100years, the temperatureof airnearby earth's

surface hasrisen by almost1degreeCelsiusscaleand for20warmestyears,19have

occurred since 1980and threehottest years ever observed, have occurred inthelast

eightyears[19]. Thewarmingofplanetearthhasbeencausedbyhighaccumulation

of green house gases (GHG) which absorb more heat fromthe sun. Most of green-

house gases are related to human activities such as heavy industries and intensive

agriculture. Dierenttreatieshavebeen signedby industrialdevelopedand fastde-

velopingcountries tocurb the situation. Among those treaties,Kyoto Protocolhas

become successful to addressand give solutionof reducing global warmingthreats.

This Protocol embraces exible mechanisms of alleviating global warming such as

Emission Trading, Clean Development Mechanism (CDM) and Joint Implementa-

tion [20].

Inthissection,threemainpointswillbeputinthecentralofourdiscussion. The

rst one will be the background of the thesis, where by a reader will nd exactly

what motivated the author to the topic and the tools author has been using to

analyze research data in the context of carbon trading. The second point will be

the objective of the thesis where by main goals and researchquestions are outlined

and how they will be solved explicitly. The last point is the structure of the thesis

in brief, where all sections are outlined shortly. After this short introduction the

background of the thesis willbe discussed in the following subsection.

1.1 Background of the Thesis

Drastic climate change due to high concentrations of greenhouse gases (GHG)

in the atmosphere is one of the most severe environmental risks. Most of green

housegasesare causedbyhumanactivitiessuchasindustries andagriculture. Most

of these gases are categorized into six types [5], which are carbondioxide

(CO 2 )

^,

methane

(CH ₄ )

^{,nitrousoxide}

(N ₂ O)

^,hydrouorocarbons (HFCs),Peruorocarbons (PFCs) and sulfur hexauoride

(SF ₆ )

^.

several international conventions. The United Nations Framework Convention on

Climate Change [6] that was conducted at Rio de Janeiro, Brazilin 1992 aimed at

reinforcingher Parties tostabilizegreenhousegas concentrationsinthe atmosphere

atalevelthatwould preventdangerousanthropogenicinterferencewith theclimate

system. Laterunderthesameconvention,Kyoto ProtocolwasestablishedatKyoto,

JapaninDecember,

11 ^th

^{,1997. TheprotocolcameintoeectinFebruary,}

16 ^th

^,2005

afterRussianratication. Itspeciestoitsmembersanemissionlevelwhichhasnot

tobeexceeded withinthecommitmentperiodof2008-2012. Under Kyoto Protocol,

industrialized countries [7] agreed principally to cut down emission on average of

about 5.2 percent below 1990 levels.

Article 5 of Kyoto protocol[8],requires her Parties toconduct the nationalsys-

tem of estimating anthropogenic emission and prepare the inventory for reporting

emissions by sources and their removal by sinks. The guidelines for such national

system havebeen specied inInternational Panel onClimateChange [9]. However

the protocolseeks fromher members to reportthe emissiontoher Convention Sec-

retariatand complytotheir levelsof endowment butit doesnotregard uncertainty

whichis associatedwith reported emission.

Allemissionestimatesaccordingto[10]containuncertaintyduetoerrorsinmea-

surement instruments, natural variability of emission generating process and bias

expert judgements. However [3]pointed that uncertainties are generally caused by

emissionfactors and activity data reecting GHGrelatedactivities. Uncertainty in

emission factors arise from lack of sucient knowledge about processes generating

emissions, lack of relevant measurements and thus inappropriate generalizations.

Depending on the source generating emission, [11] and [12] reported that uncer-

tainty of

CO ₂

^{fromenergy sources issmall, around 5percent. Otherpollutantsare}

reported to have much more uncertainty, usually more than 20 percent as pointed

in [13]. These include

N 2 O

^from agricultural soil, PFCs and

SF 6

^{from aluminium}

production,

CH 4

^{from landlls and}

N 2 O

^{from road trac. The} implementation of Kyoto Protocolneedhighqualityemissioninventories toensurethatpartiescomply

to their commitments and right reducing measures of emissions are taken. Uncer-

tainty reporting is a crucial in the context of emission trading. Kyoto protocol in

response, from its article 17 introduced emission trading to facilitate achievement

of national agreed reduction targets. Annex B countries to the protocol allowed to

selltheir excess emissionreductions as itis potentialfor cost-eciency.

by [1] and [2]. Both assumed that uncertainty has to be associated with emissions

to oversee the compliance with Kyoto targets. In order to meet these targets one

has toinvest inemissionreductionormonitoring uncertaintyorby buying permits.

Reducing emission levels createcosts investment intechnology process improve-

mentsand renovations of process installations. Reduction of uncertainty isanother

source ofcosts. Improvingmeasurementaccuracy and extendingthe scope of emis-

sion monitoring is possible only by investment into improved technology. Hence

smaller uncertainty meanshigh cost level.

Several optimization methods have been used to solve the optimal value such

as emission levels of dierent Parties or Regions under consideration. Classical

Lagrangian, Sequential Quadratic Programming (SQP) and Dierential evolution

methods have been applied widely to analyze to solve the optimal values and to

analyzerelationshipexistingbetweendierentdecisionvariablesinthemodel. Inthe

next subsection whichisthe objectiveof thethesis, thesemethodswillbediscussed

briey on howthey are going tosolve our optimization problems.

1.2 Objective of the Thesis

Primary objective of this master'sthesis is to analyze two schemes in Emission

trading. The rst scheme is when there is no transaction, i.e. no trade of carbon

permits but each Party carry its own initiative to cut down emissions and relative

uncertainty. The secondscheme of our analysiswillbeuponthe situationwhereby

the trade of permits isdone, by carbonpermits being exchanged between potential

buyers andsellers. Inboth schemeswe aregoingtondand comparetotalcosts for

cutting down both emissions and relativeuncertainty.

The secondary objectivesof our work are toanalyze:

•

^{the nature of our optimal solution.}

•

^{eect of} introducingrelativeuncertainty tothe cost function.

•

^emissionuncertainty in stochastic framework.

Our researchquestionshavebeen extractedfromobjectivesand willbeanswered

immediatelytothe next coming sections. The research questions are as follows:

Research Question 1

Isthereanysignicantimpactbetweenthe situationswhencarbontradepermits

are traded ornot traded?

Toanswerthis questioncostfunctionsare introduced. Thesefunctionsarebasedon

the datacollected fromthe countries ofKyototreaty (confertable1). Weare going

tousethemethodofLagrangemultiplier,SequentialQuadraticProgramming(SQP)

andDierentialEvolution(DE)tocalculatetheoptimalpointsofbothemissionsand

relative uncertainty and compare the total cost for reducing emissions and relative

uncertainty incase thereisno transactionsandwhen carbonpermitsare boughtor

traded.

Research Question 2

Does our solutionattains localor globalminimum?

The nature of the solution will be determined by using rst order conditions

(KKT)and sucientsecond order conditions.

Research Question 3

Does inclusion of relative uncertainty to the cost function has an impact in

analyzing the cost of reducing uncertainty?

We are going to replace absolute uncertainty with relative uncertainty and nd

the relationship between emissionsand relative uncertainty.

Research Question 4

and relative uncertainty?

We are going to study analternativeformulationin modelingof total emissions

when the uncertainty intervals are replaced by probability distributions.

After describing how are we going to answer our research questions, the next

subsection willdescribebriey how the thesis is structured.

1.4 Structure of the Thesis

The thesis work comprises of seven sections. It begins with introductionsection

in brief describing the background, objective and research questions related toour

master's thesis. In this sectionsome importantpoints suchas Lagrangian, Sequen-

tial Quadratic Programming (SQP), Dierential Evolution optimization methods,

localand global minimum points and costs of reducing both emission and relative

uncertainty are briey reviewed.

Analysis of necessary and sucient conditions for optimality of general con-

strained minimization problem have been presented in section 2. In this section,

Lagrangian of constrainedproblem, activeconstraints,constraint qualicationcon-

ditionsandKuhnTuckerconditionsforoptimalityhavebeendeeply examined. The

end of this section is marked by briefexplanation of saddle point conditions which

are more restrictivethan Kuhn Tucker conditions.

Methodology and modeling have been presented in section 3. The section de-

scribeshowthemodelhasbeenchosen,parameterestimatesofthemodel, introduc-

tion ofrelativeuncertaintytothemodelandanalysis oflocalminima. Assumptions

used in estimating the cost parameters of the model have been clearly described.

Dataandsource fromwhichthedata have extractedare presented brieyinsection

4. Section 5 entails extensively several methods have been used to come up with

the solution of our optimization problem. Classical Lagrangian for optimization,

Sequential Quadratic Programming (SQP) and Dierential Evolution(DE) meth-

ods have been adequately described and used to solve parameters such as optimal

emissionleveland relativeemissionuncertainty tothe situationswhere permitsare

not transacted and when permits are allowed to be bought or sold within the par-

ticipants to the carbon trade. In addition to the optimal number of permits each

tionshipbetween decisionvariables andobjective functionvaluehas been explicitly

presented. The section is ending by evaluating the model results by giving brief

interpretations inthe context of Carbon Trading.

Section 6is concerned about stochastic emissionuncertainty, where by uncertainty

relatedto emissionlevels are explicitlyanalyzedin stochastic framework. The nal

section isconclusion and future workfollowed by references.

2 Constrained Optimization

In this section we are going to analyze necessary and sucient conditions for

optimality of general constrained minimization problem(CP). We are going to ex-

aminedeeplytheLagrangianofconstrainedproblem,'activeconstraints',constraint

qualicationconditions and Kuhn Tuckerconditions for optimality.

2.1 Lagrangian for constrained problem

Let usconsider the followinggeneralconstrained minimizationproblem(CP) as

it ispointed in[17]:

(

^CP

)

^min

f (x) = f (x 1 , ..., x n )

^(2.1)

s.t

c i (x) = 0, i = 1, ..., m e

^(2.2)

c i (x) ≥ 0, i = m e + 1, ..., m

^(2.3)

The constraints

c(x) = (c i (x), ..., c m (x)) ^T

^{in(2.2) and (2.3) is a column vector. We}

are going to deneimportantterminologies inthe context of constrained optimiza-

tion as follows:

Denition 1. (Feasible set)

The feasible set

Ω ⊂ R ⁿ

^{is the set of all points which satisfy all given}constraints.

According to [16], p.308, active and inactive set of constraints in constrained opti-

mizationcan beexplained briey as follows:

Denition 2. (Active set)

indices

E

^{together with indices of inequality} constraints

i

^{such that}

c i (x) = 0

^{, that}

is,

A (x) = E ∪ { i ∈ I| c i (x) = 0 }

^.

Wecansaythattheinequalityconstraint

i ∈ I

^{issaidtobeactive atafeasiblepoint}

x

^if

c i (x) = 0

^{and inactive if thestrict inequality}

c i (x) > 0

^{issatised. Furthermore}

allequality constraints are active ateveryfeasible point

x

^.

The Lagrangian function of the constrained problem(CP) can be writtenas:

L(x, λ) = f(x) − λ ^T c(x) = f(x) −

m

X

i=1

λ i c i (x)

^(2.4)

where

λ = (λ 1 , ..., λ m ) ^T

^{is a vector of Lagrange} multipliers. The rst order partial derivative with respect to

x

^gives:

∇ ^x L(x ^∗ , λ ^∗ ) = ∇ f (x ^∗ ) −

m

X

i=1

λ i

∗ ∇ c i (x ^∗ )

^(2.5)

2.2 Optimality conditions

In this subsection we are going to discuss necessary and sucient conditions for

optimality. Before looking for optimality conditions of constrained optimization

problem we have to check if linear independence constraint qualication (LICQ)

holds. This has been explainedin [16], p. 320 that:

Denition 3. Given

x

^{isthe feasible point and}

A (x)

^{is theset of active} constraints dened in Denition2 then the linear independenceconstraint qualication (LICQ)

it issaid to hold if the gradients of active constraints are linearly independent.

A feasible point

x

^{mentionedin Denition3 is known as regular point.}

2.2.1 Necessary conditions for optimality

Thenecessary conditionsfor

x ^∗

^{tobealocalminimizerof}constrainedproblem(CP) are also called the rst order conditions. According to [16], p. 321 the necessary

conditions for optimality are stated inthe following theorem:

Theorem 2.1. (First-Order Necessary Conditions)

Suppose

x ^∗

^{is the local minimizer of} constrained problem (CP) in (2.1), (2.2) and

(2.3) and both objective function

f

^{and the} constraints

c i

^are continuously dieren- tiable and that linear independence constraint qualication (LICQ) holds thenthere

exist Lagrange multipliers

λ ^∗ = (λ 1

∗ , ..., λ m

∗ )

^{such that the following conditions are}

satised at

(x ^∗ , λ ^∗ )

^:

•

^Feasibility:

c i (x ^∗ ) = 0 i = 1, ..., m e

c i (x ^∗ ) ≥ 0 i = m e + 1, ..., m

(2.6)

•

Stationarity

∇ ^x L(x ^∗ , λ ^∗ ) = 0 or ∇ f (x ^∗ ) =

m

X

i=1

λ ^∗ _i ∇ c i (x ^∗ )

^(2.7)

•

Complementarity

λ i

∗ c i (x ^∗ ) = 0 i = m e + 1, ..., m

^(2.8)

•

^Dual feasibility

λ ^∗ _i ≥ 0 i = m e + 1, ..., m

^(2.9)

Theseconditionsfromtheorem2.1areknownasKuhn-Tucker(KT)orsometimes

KarushKuhnTucker(KKT)conditions. Thepoint

x

^{thatsatisestheKTconditions}

is called KT-point or KKT- point. It is pointed in [18] that the conditions in (2.9)

are for dualfeasibility thatmeans that theLagrangemultipliersthat correspond to

active constraintcan bezero.

and (2.8), mean that the gradient

∇ f (x ^∗ )

^{is the linear} combination of gradients of the active constraints at

x ^∗

^{. According to [16] the conditions in (2.8) are called}

complementaryin the sense that either constraint

i

^{is active or}

λ ^∗ _i = 0

^{or possibly}

both. It is obvious that the lagrange multiplierscorrespond to inactive constraints

are zero. This leads us todene the special case for complementarityas follows:

Denition 4. (Strict Complementarity).

Given that

A (x ^∗ )

^{is the set of active} constraints at optimal point

x ^∗

^and

I

^{is an}

indexof inactive constraints and

λ ^∗

^{satisfy the KT} conditions,wecansay thatstrict complementary conditionsholdsif exactlyoneof

λ ^∗ _i

^and

c i (x ^∗ )

^{iszero foreach index}

i ∈ I

^{. In other words we can say that}

λ ^∗ _i > 0

^{for each}

i ∈ I ∩ A (x ^∗ )

^.

2.2.2 Sucient conditions for Optimality

Theseareconditionswhichwillguaranteeoursolutiontothelocalminimumsolu-

tion forconstrained minimizationproblem. They includeinboth rst order(KKT)

and second order conditions. These conditions are summarized in the following

theorem as itis pointed in[17]:

Theorem 2.2. (Sucient Conditions).

x ^∗

^{islocalminimum point on} constrainedproblem(CP) ifthere existLagrangemul- tipliers

λ ^∗ = (λ ₁ ^∗ , ..., λ m

∗ )

^{such that the followingconditionshold:}

1. KKT-conditions of Theorem 2.1.

2. Second order conditions:

For every non-zero vector

y ∈ R ⁿ

^{such that;}

y ^T ∇ c i (x ^∗ ) = 0

^{for allequality}constraints

i = 1, ..., m e

^.

y ^T ∇ c i (x ^∗ ) = 0

^{for allinequality} constraints with

λ i

∗ > 0

^,

y ^T ∇ ^{x 2} L(x ^∗ , λ ^∗ )y > 0

^.

The condition

y ^T ∇ ^{x 2} L(x ^∗ , λ ^∗ )y > 0

^{pointed in Theorem 2.2 implies that the sym-}

metric matrix

∇ ^{x 2} L(x ^∗ , λ ^∗ )

^{is positivedenite.}

These are more restrictive conditions than those which have been mentioned in

Theorem2.2andtheydeterminewhethertheconstrainedproblem(CP)hasattained

global minimum point or not. These conditions are presented in [17] and being

summarized inthe following theorem;

Theorem 2.3. (Saddle point conditions).

If

(x ^∗ , λ ^∗ )

^{is a saddle point of the Lagrangian of (CP), i.e. if}

λ i

∗ ≥ 0

^for,

i =

m e + 1, ..., m

^and

L(x ^∗ , λ) ≤ L(x ^∗ , λ ^∗ ) ≤ L(x, λ ^∗ )

^{for all}

x ∈ R ⁿ

^{and all}

λ ∈ R ^m

with

λ i ≥ 0

^for

i = m e + 1, ..., m

^{, then}

x ^∗

^{is the global minimum point.}

3 Methodology and Modeling

Our optimization model willfocus to minimizecosts of reducing reported emission

and its associated uncertainties. In our case we shall incorporate in both absolute

and relative uncertainties.

The necessary set of variablescan be dened asfollows. Let

i

^{=Partiesorcountries}participatingunderKyotoProtocol,for

i = 1, 2, 3, ..., N

^.

x i

^{=reported emission inevery party}

i

^.

i

^{=volume of absolute}uncertainty emission.

C i (x i )

^{=total costs forParty}

i

^{of keeping reported emissionon level,}

x i

^.

F ⁱ ( i )

^{=total costs forParty}

i

^{of keeping absolute} uncertainty onthe level

i

K i

^{=Kyoto target of emissionto eachparty}

i

^and

y i

^{=number of emission permits accrued by Party}

i

^{(might be positive for net}

purchaser,or negative for net supplier of permit).

Each Party faces a two step optimization problem. The rst step optimization

problemisforeachPartytocarryitsownindividualtasktodecidewhether toabate

emissions ortoinvest inmonitoring the volume of absoluteuncertainty emissions.

The secondstep optimizationisforthe Party (orcountry)to decidewhether ornot

to exchange the number of emissionpermits with other Parties [1].

For individual optimization,the least cost of reducing reported emission,

C i (x i )

^as

well asmonitoring the volume of absoluteuncertainty,

F ⁱ ( i )

^{, is given by:}

f i (y i ) = min

x i , i

[C i (x i ) + F ⁱ ( i )]

^(3.1)

s.t

x i + i ≤ K i + y i

^(3.2)

Both cost functions

C i (x i )

^and

F ⁱ ( i )

^{are assumed to be positive, decreasing and}

convex in

x i

^and

i

respectively. Furthermore these functions are assumed to be continuously dierentiable [3]. The convexity of

f i (y i )

^{is assured since it is a mini-}

mum sum of two convex functions

C i (x i )

^and

F ⁱ ( i )

^.

This is according tothe following lemmain [17] as;

Lemma 3.1. (Sum of convex functions).

If

f (x)

^and

g(x)

^{are both convex functions, hence}

f (x) + g(x)

^{is also convex.}

The marginal costs

C _i ⁰ (x i )

^and

F i ⁰ ( i )

^{are both negative in}

x i

^and

i

respectively so as tobe positivein reducing

x i

^and

i

^{. The theorem proved in[4] willbehelpful to}

examine how objective value of optimization problem changes as the result of the

change of its parametersand is stated here under asfollows:

Theorem 3.2. (Envelope theorem) :

Suppose

M (a) = maxf (x, a)

^{gives the maximized value of objective function}

f

^{as a}

function of parameter

a

^then

M (a) = f (x(a), a)

^and

M (a)

^{changes as parameter}

a

changes, namely

dM (a)

da = ∂f (x ^∗ , a)

∂a | ^{x ∗ =x(a)}

^{at optimal point}

x ^∗

By substitutingequation (3.2) into (3.1) and eliminating

x i

^{we have;}

f i (y i ) = min

ⁱ [(C i (K i + y i − i )) + F ⁱ ( i )]

^(3.3)

Then by applying theorem 3.2 inequation (3.3) weget;

f _i ⁰ (y i ) = ∂

∂y i

min i

[(C i (K i + y i − i )) + F ⁱ ( i )] = C _i ⁰ (x i ∗ )

^(3.4)

If substituting(3.2) into (3.1) and eliminating

i

^{we have;}

f i (y i ) = min

x i

[C i (x i ) + F ⁱ (K i + y i − x i )]

^(3.5)

The use ofenvelope theorem to(3.5) gives;

f _i ⁰ (y i ) = ∂

∂y i

min i

[C i (x i ) + ( F ⁱ (K i + y i − x i ))] = F i ⁰ ( i ∗ )

^(3.6)

where by

x ^∗ _i

^and

^∗ _i

^{are optimal solutions of subproblem (3.1) and (3.2), i.e. the}

optimalreported emissionand optimalvolume of absoluteuncertainty respectively.

Asequation(3.4)is equalto(3.6)thenweobtainthe optimalitycondition

C _i ⁰ (x ^∗ _i ) = F i ⁰ ( ^∗ _i )

^{, that imply that the marginal costs for cutting down reported emissions to}

x ^∗ _i

^{is equal to marginal cost of reducing the volume of absolute} uncertainty to

^∗ _i

at optimal conditions. By minimizing (3.1) subject to linear constraint (3.2) and

setting Lagrangian

λ i

^{we obtainthe Lagrangian function;}

`(x i , i , λ i ) = C i (x i ) + F ⁱ ( i ) − λ i (x i + i − K i − y i )

^(3.7)

Then the rst order partial derivativesof (3.7) gives;

∂`

∂x i

= C _i ⁰ (x i ) − λ i = 0

^(3.8)

∂`

∂ i

= F i ⁰ ( i ) − λ i = 0

^(3.9)

∂`

∂λ i

= x i + i − K i − y i = 0

^(3.10)

Solving (3.8) and (3.9) we obtain:

λ i = C _i ⁰ (x ^∗ _i ) = F i ⁰ ( ^∗ _i )

^(3.11)

from which

λ i

^{is Lagrangian multiplier and is} interpreted as the shadow price, i.e.

the willingness of Party

i

^{to pay for emitting one more unit of reported emission,}

x i

^{or volume of absolute} uncertainty,

i

^by considerably relaxing constraint (3.2) by one unit. At optimal conditions, shadow price is equal to the marginal costs of

reducingreportedemissionaswellasreducingvolumeofabsoluteuncertainty. Since

the marketwill not be inequilibrium, the shadow prices

λ i

^willdier considerably between buyers and sellers of permits reecting potential for trade. This will au-

tomaticallylead to secondoptimization problemtond permit distributionamong

theparticipantssoastoequalizetheshadowpriceamongthem. Theaggregate cost

of reaching the Kyoto targetsis dened as the sum of individual costs asfollows;

Suppose the aggregate cost functionis given by:

F (y 1 , ..., y N ) =

N

X

i=1

f i (y i )

^(3.12)

min y i

F (y 1 , ..., y N )

^(3.13)

s.t

N

X

i=1

y i = 0

^(3.14)

By setting Lagrangian multiplier

µ

^{to (3.13) and (3.14) and solving the rst order}

condition we have;

L(y 1 , ..., y N , µ) =

N

X

i=1

f i (y i ) − µ

N

X

i=1

y i

^(3.15)

∂L

∂y i

= f _i ⁰ (y i ) − µ = 0

^(3.16)

∂L

∂µ = −

N

X

i=1

y i = 0

^(3.17)

Solving equation (3.16)we obtain the rst order condition:

f _i ⁰ (y i ) = µ, ∀ i

^(3.18)

Condition (3.18) imply that the marginal cost of permits,

y i

^{, shall in} equilibrium equal toa specic level

µ

^toall participants. It is obvious that by combining (3.4) and (3.6) we deduce that:

f _i ⁰ (y i ) = C _i ⁰ (x ^∗ _i ) = F i ⁰ ( ^∗ _i )

^(3.19)

Thisshowsthattheonlynecessaryconditiontobringthepermitmarketintoequilib-

rium isforpermitprice equaltobothmarginalcostsforreducing reportedemission

and volume of absoluteuncertainty.

3.1 Introduction of Relative uncertainty to the Model

Weaimatinvestigatingtheeectsofintroducingtherelativityuncertaintytothe

cost function (3.1). We claim that this approach is suitablefor analyzingthe costs

of reducing uncertainty that involved to the inventory of non-

CO 2

^{GHG such as}

methane,

CH 4

^{, nitrousoxide,}

(N 2 O)

^{andtheir aggregatein}combinationwith

CO 2

^.

sourceandthe knowledgeaboutprocessesgeneratingemission,thenitwasreported

to [11] and [12] that

CO 2

^{emissionfactors fromenergy related sourcesis 5percent.}

It is reported in [3]that, other GHG depending on the emission source, have more

uncertainties,forexample

N 2 O

^fromagriculturalsourcesisup to100percent,while

N ₂ O

^{fromcombustion isup to200 percent.}

Let

H ⁱ (R i )

^{be the costs for reducing relative}uncertainty,

R i

^{. It isassumed that}

relative uncertainty is given by

R i = i /x i

^{. The total abatement costs is the sum}

of

H ⁱ (R i )

^{and emission reduction costs,}

C i (x i )

^{. Under Kyoto protocol, as pointed}

in [3] we are needed to express uncertainties in absolute terms. That is emission

level

x i

^{plus absolute} uncertainty

= x i .R i

^{shall not exceed the Kyoto target,}

K i

increased or decreased by a certainspecic levelof Permit,

y i

^{(see condition (3.2)).}

Let

Z i (x i , R i )

^{representsthetotalcostforabatingbothemissions,}

x i

^andrelative

uncertainty,

R i

^suchthat:

Z i (x i , R i ) = C i (x i ) + H ⁱ (R i )

^(3.20)

Thenafter introducing the relative uncertainty into trade system then equation

(3.1)and (3.2) become;

G ⁱ (x i , R i ) = min

x i ,R i

Z i (x i , R i )

^(3.21)

s.t

x i + x i .R i ≤ K i + y i

^(3.22)

The approach showing in (3.21) and (3.22) strongly reect the dependence be-

tween both emissionsand their associated uncertainty. As ithas been noticed, the

constraint(3.22)isnon-linearincontrastto(3.2). Itisassumedthat bothfunctions

C i (x i )

^and

H ⁱ (R i )

^{display the usual economic} properties: that is they are convex,

decreasingandcontinuousdierentiable. Thisimpliesthatboth

x i

^and

R i

^shouldbe

positivetoreect therealityand itisassumed that

K i + y i

^{shouldbestrict positive}

and Parties do not sell more permit than their Kyoto compliances. The Lagrange

function of (3.21) and (3.22)is:

L (x i , R i , λ i ) = C i (x i ) + H ⁱ (R i ) − λ i (x i + x i .R i − K i − y i ) ,

^(3.23)

∂L

∂x i

= C _i ⁰ (x i ) − λ i − R i .λ i = 0

^(3.24)

∂L

∂R i

= H ⁰ i (R i ) − λ i .x i = 0

^(3.25)

∂L

∂λ i

= x i + x i .R i − K i − y i = 0

^(3.26)

Solving (3.24) and (3.25) foroptimalitywe get

λ i = C _i ⁰ (x ^∗ _i )

1 + R ^∗ _i = H ⁰ i (R ^∗ _i )

x ^∗ _i ,

^(3.27)

where by

x ^∗ _i

^and

R _i ^∗

^{are optimal levels of emissions and relative} uncertainties and Lagrangemultiplier,

λ i

^is interpreted asthe permit shadow price.

With rst order conditions of (3.23) itis assumed that the cost function

H ⁰ i (R i )

is independent of emission level

x i

^{, that is relative} uncertainty does not change in case of change in emissions. However from (3.27) we notice that the marginal

costratio

C _i ⁰ (x i )

H ⁰ i (R i )

^{depends onbothoptimallevelofemission,}

x ^∗ _i

^{andoptimalrelative}

uncertainty

R ^∗ _i

^{whileincaseof}independentemissionandabsoluteuncertaintystated in (3.11)reveals that the ratio of marginalcosts is 1.

Thecostfunction

Z i (x i , R i )

^{in(3.21)istheminimumsumoftwoconvexfunctions}

subject to the non-linear constraint (3.22) with respect to the variables

x i

^and

R i

from the factthat

i = x i .R i

^.

3.2 Analysis of local minima

Tohavemore insightof theminimumof Lagrangefunction, thenthe secondderiva-

tive of

Z i (x i , R i )

^{has to be analyzed so as to check the existence of several local}

minima. Takingintoconsiderationthatcountries neednot toover-complytoKyoto

targets,we take a case of equality constraint (3.22). We express our goal function

(3.21) todepend only on

x i

^{. Now constraint(3.22) becomes:}

x i + x i .R i − K i − y i = 0

^(3.28)

By making

R i

^{the subject from (3.28)we obtain:}

R i = K i + y i − x i

x i

(3.29)

Substituting (3.29)into(3.21) weget:

G ⁱ (x i ) = C i (x i ) + H ⁱ

K i + y i − x i

x i

(3.30)

The rst order derivative of (3.30)gives:

d G ⁱ dx i

= C _i ⁰ (x i ) +

− x i − (K i + y i − x i ) x i 2

. H ⁰ i

K i + y i − x i

x i

(3.31)

= C _i ⁰ (x i ) − (K i + y i ) x i 2 . H ⁰ i

K i + y i − x i

x i

(3.32)

Setting the rst derivativeto zero and from (3.29),it follows that:

K i + y i

x i 2 = C _i ⁰ (x i )

H ⁰ i (R i )

^(3.33)

The second order derivativewith respect to

x i

^becomes:

d ² G ⁱ

dx i 2 = C _i ⁰⁰ (x i ) + 2 x i (K i + y i ) x i 4 H i ⁰

K i + y i − x i

x i

+

K i + y i

x i 2

2

H ⁰⁰ i

K i + y i − x i

x i

= C _i ⁰⁰ (x i ) + 2 (K i + y i )

x i 3 H i ⁰ (R i ) +

K i + y i

x i 2

2

H ⁰⁰ i (R i )

(3.34)

From condition (3.33) which was valued at

d G ⁱ dx i

= 0

^{we make both}

H ⁰ i (R i )

^and

x i

the subject:

H ⁰ i (R i ) = x i 2 C _i ⁰ (x i ) K i + y i

(3.35)

x i = s

H ⁰ i (R i )

C _i ⁰ (x i ) (K i + y i )

^(3.36)

d ² G ⁱ dx i 2 | ^{dG i}

dxi =0 = C _i ⁰⁰ (x i ) +

C _i ⁰ (x i ) H ⁰ i (R i )

2

H ⁰⁰ i (R i ) + 2 (K i + y i ) x i 3

x i 2 C _i ⁰ (x i ) K i + y i

= C _i ⁰⁰ (x i ) +

C _i ⁰ (x i ) H ⁰ i (R i )

2

H ⁰⁰ i (R i ) + 2 C _i ⁰ (x i ) x i

(3.37)

By substituting(3.36) into (3.37)we have:

d ² G ⁱ dx i 2 | ^{dG i}

dxi =0 = C _i ⁰⁰ (x i ) +

C _i ⁰ (x i ) H ⁰ i (R i )

2

H ⁰⁰ i (R i ) + 2

s C _i ⁰ (x i ) H i ⁰ (R i )

C _i ⁰ (x i )

√ K i + y i

(3.38)

Thersttwotermsof(3.38)arepositivesincethecostfunction

C i (x i )

^and

H ⁱ (R i )

are convex. The third term can be negative since both marginal costs

C _i ⁰ (x i )

^and

H ⁰ i (R i )

^{are negative. Nowdependingonmagnitudeof}

C _i ⁰ (x i )

^and

H i ⁰ (R i )

^,

d ² G ⁱ

dx i 2 | ^{dG i}

dxi =0

can be negative. This implies that the problem (3.21), (3.22) can be non-convex

and could haveseveral localminima.

Without the loss of generality we can derive the goal function that depends on

relative uncertainty,

R i

^{and its}corresponding secondorder derivative asfollows:

Making

x i

^{the subject from(3.28) we have:}

x i = K i + y i

1 + R i

(3.39)

Substituting (3.39)into(3.21) weget:

P ⁱ (R i ) = C i

K i + y i

1 + R i

+ H ⁱ (R i )

^(3.40)

The rst order derivative of (3.40)gives:

d P ⁱ dR i

= − (K i + y i ) (1 + R i ) ² C _i ⁰

K i + y i

1 + R i

+ H ⁰ i (R i )

^(3.41)

Setting the rst order derivativeto zero and making use of (3.39) we nd that:

H i ⁰ (R i )

C _i ⁰ (x i ) = K i + y i

(1 + R i ) ²

(3.42)

d ² P ⁱ

dR i 2 = 2(1 + R i )(K i + y i )

(1 + R i ) ⁴ C _i ⁰ (x i ) − (K i + y i )

(1 + R i ) ² . − (K i + y i )

(1 + R i ) ² C _i ⁰⁰ (x i ) + H ⁰⁰ i (R i )

= H ⁰⁰ i (R i ) +

K i + y i

(1 + R i ) ² 2

C _i ⁰⁰ (x i ) + 2 (K i + y i ) (1 + R i ) ³ C _i ⁰ (x i )

(3.43)

Solving

d ² P ⁱ dR i 2

^at

d P ⁱ dR i

= 0

^{we manipulate (3.42),that willgive:}

(1 + R i ) = s

C _i ⁰ (x i ) H ⁰ i (R i ) . p

K i + y i

= ⇒ (1 + R i ) ³ = C _i ⁰ (x i )

H ⁰ i (R i ) (K i + y i ) s

C _i ⁰ (x i ) H ⁰ i (R i ) . p

K i + y i

(3.44)

Substituting(3.44) into(3.43) andmakethe use of (3.42)we nallyget the expres-

sion:

d ² P ⁱ dR i 2 | ^{dP i}

dRi =0 = H ⁰⁰ i (R i ) +

H i ⁰ (R i ) C _i ⁰ (x i )

2

C _i ⁰⁰ (x i ) + 2

s H ⁰ i (R i )

C _i ⁰ (x i ) . H ⁰ i (R i )

√ K i + y i

(3.45)

Since both

H ⁱ (R i )

^and

C i (x i )

^{are assumed tobe convex then the rst two terms}

to the right of (3.45) are positive. That is

H ⁰⁰ i (R i )

^and

C _i ⁰⁰ (x i )

^{are strictly greater}

than zero. The third term is negative when both

H ⁰ i (R i )

^and

C _i ⁰ (x i )

^{are negative.}

Depending on magnitude of marginal costs,

H ⁰ i (R i )

^and

C _i ⁰ (x i )

^{we can signal}

d ² P ⁱ dR i 2

at

d P ⁱ dR i

= 0

^{to be either positive or negative. This implies that problem(3.21) can}

be non-convex and could have several localminima.

The analysis of second order is very important in the context of carbon permit

market aswe can not achieveglobal minimum cost solution. The marketmight be

locked tolocalminima.

In order to reach local minimum conditions we considered a convex function for

emissionreduction tobe:

C i (x i ) =



 

 

 

 

b i (x i − a i ) ²

^for

x i ∈ [0, a i ] 0

^for

x i > a i

(3.46)

where

a i

^{is initial emission or} 'Business-As-Usual' (BAU). If

x i = a i

^{, no cost or}

emission regulation is taken into account to reduce emission. This reects baseline

emissionand isalsoknown asbusiness asusual (BAU)asitispointed in[3]. In the

same mannerwe formulatecost function for reducing the relative uncertainty as;

H ⁱ (R i ) =



 

 

 

 

d i (R i − R 0,i ) ²

^for

R i ∈ [0, R 0,i ] 0

^for

R i > R 0,i

(3.47)

whereby

R 0,i

^{isinitialvolume ofrelative}uncertainty. If

R i = R 0,i

^{,itmeansthatno}

cost is incurred to reduce relative uncertainty and

R 0,i

^{will reect the baseline for}

relative uncertainty.

These cost functions have been proposed by several authors [1], [2], and [3].

Quadraticcurve reectsthe typicalwellknown featureof increasingmarginalcosts.

The parameter values inthe modelare derived from available datain the countries

of Kyoto treaty (refer table 1).

Our claim is that although (3.46) and (3.47) are convex (downward) functions,

they cannotachievetheleastcostsolution. Inotherwordsthereexistlocalmaxima

as an indication of these functions to exhibit non-convexity to some points. To

support our claim let us consider both functions

H ⁱ (R i )

^and

C i (x i )

^{by writing}

(3.30) interms of (3.46) and (3.47) toget:

T ⁱ (x i ) = b i (x i − a i ) ² + d i

K i + y i − x i

x i − R 0,i

2

(3.48)

Setting tozero the rst orderderivative of (3.48)gives:

d T ⁱ dx i

= 2b i (x i − a i )+2d i

K i + y i − x i

x i − R 0,i − x i − (K i + y i − x i ) x i 2

= 0

^(3.49)

Manipulating (3.49) by dividing itby

2

^and multiplyingby

x i 3

we get:

b i (x i − a i ) x i 3

− d i (K i + y i − x i − x i R 0,i ) (K i + y i ) = 0

= ⇒ b i (x i − a i ) x i 3

− d i (K i + y i ) ² + d i (1 + R 0,i ) (K i + y i ) x i = 0

= ⇒ (x i − a i ) x i 3 + d i

b i

(1 + R 0,i ) (K i + y i ) x i − d i

b i

(K i + y i ) ² = 0

(3.50)

By normalizing

x i

^upon

a i

^{wend that}

x i

a i

= u i

^fromwhich:

x i = a i u i

^(3.51)

Substituting (3.51)into(3.50) weget:

0 = a i 4 u i 4

− a i 4 u i 3 + d i

b i

(1 + R 0,i ) (K i + y i ) a i u i − d i

b i

(K i + y i ) ²

⇒ 0 = u i 4

− u i 3 + d i

a i 3 b i

(1 + R 0,i ) (K i + y i ) u i − d i

b i a i 4 (K i + y i ) ²

⇒ 0 = u i 4

− u i 3 + d i

a i 3 b i

(1 + R _0,i ) (K i + y i ) u i − d i

a i 3 b i

(1 + R _0,i ) (K i + y i ) K i + y i

a i (1 + R 0,i )

(3.52)

Nowwe lettwodimensionless parameters

α i

^and

γ i

^represent:

α i = d i

a i 3 b i

(1 + R 0,i ) (K i + y i )

^(3.53)

γ i = K i + y i

a i (1 + R 0,i )

^(3.54)

Substituting Equations(3.53) and (3.54) into (3.52)we have:

0 = u i 4

− u i 3 + α i u i − α i γ i

⇒ 0 = u i 3 (u i − 1) + α i (u i − γ i )

^{, for}

u i > 0

^,

α i > 0

^and

γ i > 0

(3.55)

To have minima solutions,letus write Equation (3.55)as:

α i (u i − γ i ) − u i 3 (1 − u i ) = 0

Wehavetoconsidertheexpression

α i (u i − γ i )

^{asthetangentlineto}

u i 3 (1 − u i )

ⁱⁿ

ordertodetermineparameters

α i

^and

γ i

^{whichmighthelpustodrawthe conclusion}

about the minima. The function

f ⁰⁰ (u i ) = u i 3 (1 − u i )

^{has two point of inexion}

from the fact that

f ⁰⁰ (u i ) = 6u i − 12u i 2 = 0

^{gives us}

u i = 0

^or

u i = 0.5

^{. When}

u i < 1/2

^{implies that}

u i 3 (1 − u i )

^{isconvexandif}

u i > 1/2

^{theexpression isconcave.}

Our focus will be when

u i = 0.5

^{. The slope at this point,}

f ⁰ (u i ) | ^{u i} =0.5 = 3u i 2 − 4u i 3 | ^{u i} =0.5 = 0.25

^{. That is}

f(0.5) = 0.5 ³ (1 − 0.5) = 1/16

^{and clearly the linepasses}

through

(1/2, 1/16)

^{. Furthermore we can nd that:}

α i (u i − γ i ) = 1/4u i − 1/16

^(3.57)

α i = γ i = 0.25

^(3.58)

Equation (3.56) exhibits only one positive solution

u i = 1/2

^for

α i = 0.25

^and

γ i = 0.25

^{. For values of}

α i

^and

γ i

^{less than}

0.25

^{, Equation (3.56) exhibits more}

thanonesolution. Thisisshowedingure1whereby

f (u i ) = α i (u i − γ i ) − u i 3 (1 − u i )

^,

g(u i ) = α i (u i − γ i )

^and

h(u i ) = u ³ _i (1 − u i )

^.

Figure 2 represents functions

f (u i )

^,

g(u i )

^and

h(u i )

^with

α i = γ i = 0.2 < 0.25

^.

In fact

f(u i )

^{exhibits more than one positive solution at this range of}

α i = γ i = 0.2 < 0.25

^{which are}

u i = 0.7236, 0.4472

^and

0.2764

^{and itisconcavewhen}

u i < 0.5

and convex when

u i > 0.5

^,while

h(u i )

^{isconvex when}

u i < 0.5

^{and isconcavewhen}

u i > 0.5

^.

On the other hand, gure 3 shows functions

f (u i )

^,

g(u i )

^and

h(u i )

^with

α i = γ i = 0.3 > 0.25

^{. Function}

f (u i )

^{exhibits apositivereal solutionat}

u i = 0.5477

^and

itisconcavewhen

u i < 0.5

^{andconvexwhen}

u i > 0.5

^{,whilefunction}

h(u i )

^isconvex

when

u i < 0.5

^{and it isconcave when}

u i > 0.5

^.

Itispointedin[3]thatparameter

γ i

^{inthecontextofcarbonmarketis}interpreted as the ratio between Kyoto emission plus traded permit tothe BAU emission level

0 0.2 0.4 0.6 0.8 1

−0.1

−0.05 0 0.05 0.1 0.15 0.2 0.25

u

f(u _i ) g(u _i ) h(u _i )

Figure 1: Convex and concave behavior revealed by rst order derivative of goal

function,

G ⁱ (x i )

^{in Equation (3.30)expressed in terms variable}

x i

a i

= u i

^{and param-}

eters,

α i = γ i = 0.25

⁾

plus absolute uncertainty.Thus;

γ i = K i + y i

a i (1 + R i )

= K i

1 + _K ^{y i} _i a i (1 + R i )

= K i

a i

(1 + τ i ) (1 + R i ) ^{− 1}

(3.59)

By binomialexpansionwe nd that;

γ i = K i

a i

(1 + τ i ) 1 − R i + R i 2 + . . .

≈ K i

a i

(1 + τ i − R i − R i τ i + . . .)

≈ K i

a (1 + τ i − R i )

(3.60)

0 0.2 0.4 0.6 0.8 1

−0.05 0 0.05 0.1 0.15 0.2

u

f(u i ) g(u i ) h(u i )

Figure 2: Convex and concave behavior revealed by rst order derivative of goal

function,

G ⁱ (x i )

^{in Equation (3.30)expressed in terms variable}

x i

a i

= u i

^{and param-}

eters,

α i = γ i = 0.2 < 0.25

⁾

0 0.2 0.4 0.6 0.8 1

−0.1

−0.05 0 0.05 0.1 0.15 0.2 0.25

u

f(u i ) g(u i ) h(u i )

Figure 3: Convex and concave behavior revealed by rst order derivative of goal

function,

G ⁱ (x i )

^{in Equation (3.30)expressed in terms variable}

x i

a i

= u i

^{and param-}

eters,

α i = γ i = 0.3 > 0.25

⁾

where by

K i

a i

is aratio of agreedKyoto Protocolto Business asUsualemission and

τ i = y i

K i

is the ratio of traded emission permits to Kyoto Protocol targets. If for

sure

γ i < 0.25

^is approximately as saying more than 75 percent of BAU emission levelhas been reduced.

3.4 Estimates of cost parameters

Duetothe factthatthe informationaboutcosts forreducing relativeuncertainty is

limitedthenparameter

d i

^{fromEquation(3.47)canbeobtainedbyassumingthatthe}

costs of relativeuncertainty reduction atany level

R _i ¹

^{relative toinitial}uncertainty

R 0,i

^{aredependentoncostsofemissionreductionaccordingtothefollowingformula:}

∂C i (x i )

∂x i | ^{x i} =x ^{i 1} = ∂ H ⁱ (R i )

∂R i | R i =R ^{i 1} . 1 a i

(3.61)

with

x i 1

a i

= R i 1

R _0,i

^(3.62)

This formulation originated from [1] that marginalcost of absolute uncertainty

reduction

F i ⁰ ( i )

^{at any level relative to the initial} uncertainty

_0,i

^{is the same as}

marginalcostofemissionreduction

C _i ⁰ (x i )

^{atthesamepercentage ofBAU(Business}

As Usual) level. That is :

∂ F ⁱ

∂ i | ^{i = i 1} = ∂C i

∂x i | ^{x i =x i 1}

^(3.63)

with

i 1

0,i

= x i 1

a i

, wherethe cost functionfor absolute uncertainty

F ⁱ ( i )

^{is downside}

function.

Nowfrom Equations (3.46)and (3.47) we nd that;

C _i ⁰ (x i ) | ^{x i =x i 1} = 1

a i H ⁰ i (R i ) | R i =R i 1

⇒ 2b i x i 1

− a i

= 2 d i

a i

R i 1

− R 0,i

⇒ b i x i 1

− a i

= d i

a i

R i 1

− R _0,i

(3.64)