Carbon Emissions Under Cap-And-trade and Carbon Tax Regulations

7/26/2019 Carbon Emissions Under Cap-And-trade and Carbon Tax Regulations

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Production lot-sizing and carbon emissions under cap-and-trade andcarbon tax regulations

Ping He a , Wei Zhang a , Xiaoyan Xu a, Yiwen Bian b , *

a School of Management, Zhejiang University, Hangzhou, Zhejiang 310058, PR Chinab SHU-UTS SILC (Sydney Institute of Language & Commerce) Business School, Shanghai University, Shanghai 201800, PR China

a r t i c l e i n f o

Article history:

Received 20 November 2013

Received in revised form

28 August 2014

Accepted 29 August 2014

Available online 6 September 2014

Keywords:

Lot-sizing

Carbon emission regulation

Economic order quantity (EOQ)

a b s t r a c t

Cap-and-trade and carbon tax are two emission regulations widely used to curb the carbon emissions

generated from rms. Based on economic order quantity (EOQ) model, this paper examines the pro-

duction lot-sizing issues of a rm under these two regulations, respectively. The optimal lot-size and

emissions under the two regulations are achieved. We then investigate the impacts of production and

regulation parameters on the optimal lot-size and emissions. Furthermore, we compare the rm's

optimal carbon emissions under the two regulations. It is found that under the cap-and-trade regulation,

the rm's decisions of the optimal emissions as well as permits trading depend on the differentiated

permits trading prices. If setup incurs the same cost as holding incurs per unit of generated emissions,

both regulations always lead to the same optimal emissions (which is also equal to that without emission

regulation). Otherwise, neither regulation always leads to lower emissions than the other does.

2014 Elsevier Ltd. All rights reserved.

1. Introduction

There is an increasing consensus that the carbon emission

generated fromrms' activities is one of the main causes of global

climate change. To curb the carbon emissions, many countries and

regions enact various regulations on rms' activities. Cap-and-

trade (or emissions trading)and carbon tax are two most popular

regulations implemented in the world. Under the cap-and-trade

regulation,rms initially receive a free amount of permits (cap)

over a planning horizon (e.g., one year), and are allowed to trade

the permits with other rms or government agencies through

special markets (e.g., carbon market). The European Union's

Emissions Trading System (EU ETS) is the rst and biggest inter-

national scheme for permits trade. Up to 2010, the EU ETS covers

11,000 power stations and industrial plants in 30 countries

(European Commission, 2013), and involves over 50% of all emis-

sions in the European Union (Benjaafar et al., 2013). Advocated as

an alternative cost-effective instrument for reducing emissions,

carbon tax regulation is much easier to implement than cap-and-

trade regulation is. Under carbon tax regulation, rms are

charged for their carbon emissions at a constant tax rate level. A

growing number of scholars (Avi-yonah and Uhlmann, 2009),

politicians and economists (Inglis and Laffer, 2008) and business

leaders (Pontin, 2010) advocatecarbon tax regimes rather than cap-and-trade.

As we know, carbon emissions are generated in almost all ac-

tivities ofrms, e.g., procurement, production, inventory holding,

order processing, transportation and some others (Hua et al., 2011;

Chen et al., 2013). Generally, carbon emissions from different ac-

tivities are generated in different ways. For example, emissions

from procurement are generated only when a procurement activity

is implemented, usually irrelevant to the procured quantity; while

emissions from inventory holding depend on the inventory quan-

tity and inventorytime. In production process, if the production lot-

size is too small (which is advocated by Just-In-Time production

theory), lots of emissions are generated from frequent setups;

otherwise, if the production lot-size is too large, lots of emissions

are generated from inventory. In the presence of emission regula-

tions, emission-related costs arise in terms of buying additional

permits (under cap-and-trade regulation) or paying tax (under

carbon tax regulation). These emission-related costs can be sub-

stantial (Drake et al., 2010), which induces carbon-intensive rms

to take the emission-related costs into consideration when deter-

mining the production lot-size.

This paper addresses the issues of the production lot-sizing of a

rm under cap-and-trade and carbon tax regulations based on EOQ

model. Under each regulation, the optimal lot-size and emissions of

the rm are characterized, and the impacts of production param-

eters and regulation parameters on the optimal lot-size and* Corresponding author. Tel.:86 21 69980028; fax:86 21 69980017.E-mail addresses:[email protected],[email protected](Y. Bian).

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2014 Elsevier Ltd. All rights reserved.

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emissions are also investigated, respectively. Due to their different

mechanisms, cap-and-trade and carbon tax regulations lead to

different forms of emissions costs, and have different impacts on

rms' operational decisions. The comparison of these two regula-

tions may provide governments the guidance on determining the

cap (under cap-and-trade regulation) or the tax rate level.

In the cap-and-trade regulation considered in this paper, the

permits buying and selling prices of the rm can be different. To our

bestknowledge, most of the existing studies on operational decisions

under the cap-and-trade regulation treat these two prices as the

same. The only exception isGong and Zhou (2013), who investigate

the impactof emission tradingon a manufacturer's technologychoice

and production planning by using differentiated permits buying and

selling prices.Since the emission trade takes place in a carbon market

and a rm can buy permits from or sell permits to agencies, the

permits buying price and selling price of a rm could be different.

FollowingGong and Zhou (2013), we differentiate these two prices

and assume that the rm's permits buying price is not smaller than

the rm's permits selling price in the cap-and-trade regulation. The

rationale for this price differentiation is well-documented in Gong

and Zhou (2013). First, the trading prices of permits actually repre-

sent the cost and the revenue of buying and selling a unit of permits,

respectively, which include transaction costs. Transaction costs inemissions trading can be signicant and have been studied both

empirically and theoretically (e.g.,Stavins, 1995; Woerdman, 2001).

Second, the bideask price spreads, often seen in various trading

markets, are anothercause of non-identical selling and buying prices.

For instance, the ask and bid prices for ECX EUA (European Union

Allowances: carbon credits issued under the EU ETS to CO2-emitting

installations) futures for December 2010 are V15.48 and V14.20 per

metric ton (12:00 p.m., Aug. 14, 2010, Hong Kong Time), respectively

(http://www.ecx.eu/market-data). This implies that the rm's per-

mits buying price may be higher than the selling price in practice.

Furthermore, we hold that if the buying price is smaller than the

selling price, rms might raise prot by purely buying and selling

carbon permits. This speculation in turn weakensthe effect of carbon

trading regulation on reducingrms' emissions. It is noteworthy that,the production lot-sizing issue with identical permits buying and

selling prices is a special case of what is discussed in our paper.

The rest of this paper is organized as follows. In Section 2,

related literature is reviewed. In Sections 3 and 4, the lot-sizing

decisions under cap-and-trade and carbon tax regulations are

explored, respectively. In Section 5, the two regulations are

compared with respect to the optimal emissions. Section 6 con-

cludes this paper.

2. Literature review

In recent years, the research on cap-and-trade and carbon tax

regulations has received extensive attentions both in empirical and

theoretical studies.The rst stream mainly discusses the concepts, advantages and

disadvantages of cap-and-trade and carbon tax regulations at

strategic levels based on empirical studies.Ekins and Barker (2001)

provide a detailed survey of the literature on carbon tax and

emissions trading as well as their implementations. They conclude

that there is a general agreement that market-based instruments of

carbon control will achieve a given level of emission reductions at

lower cost. As indicatedby Harrison and Smith (2009), the cap-and-

trade regulation is business-friendly and can produce more jobs.

However, carbon tax regulation is simpler and easier to implement

than cap-and-trade regulation is, and the tax increases the revenue

of government which can be used as the investment of carbon

abatement (Baranzini et al., 2000). Theoretically, both cap-and-

trade and carbon tax can achieve cost-effective emission

reductions (Stavins, 2008), and there is a broad equivalence be-

tween emissions trading scheme and carbon tax regulation under

some assumptions (Pezzey, 1992; Farrow, 1995).

The second stream examines the operational decisions ofrms

under emission regulations. Letmathe and Balakrishnan (2005)

study the production mix and production quantities of a rm un-

der several different environmental constraints, e.g., threshold

values, penalties and taxes, and/or emissions trading. From the

perspective of carbon abatement efciency,Mandell (2008)shows

that utilizing the two regulations (i.e., cap-and-trade and carbon

tax) can be superior to adopting only one regulation (either cap-

and-trade or a carbon tax). Benjaafar et al. (2013)introduce a se-

ries of simple and general models to illustrate howcarbon footprint

could be incorporated into operational decisions, where many ob-

servations and insights are obtained. Drake et al. (2010) study a

two-stage decision problem of a rm under the two regulations

(cap-and-trade and carbon tax). In the rst stage, the rm chooses

capacities under two technologies, dirty and clean. With the

given technology, the rm in the second stage chooses production

quantities to maximize its own prot.Hoen et al. (2014) examine

the effect of different emission regulations (including voluntary

targets) on transportation mode selection for a carbon-aware

company (either by choice or enforced by regulation) under sto-chastic demand.Jaber et al. (2013)study the coordination in a two-

level supply chain in the EU ETS, where greenhouse gas emissions

are generated in the manufacturing processes. Jin et al. (2013)

investigate the impact of carbon policies on supply chain design

and logistics of a major retailer, where three carbon policies are

considered: carbon emission tax, inexible cap and cap-and-trade.

The third stream is related to the estimation of emission costs and

carbon accounting under carbon emission regulations. Tsai et al.

(2012a) develop a mixed Activity-Based Costing (ABC) decision

model for green airline eet planning under emissions trading

scheme.Sthls et al. (2011) investigate the impacts of international

commoditytrade on carbon ows of forestindustry in Finland, using a

quantitative analysis method. The carbon ows are embodied in the

tradedforest.They show that in Finland, thedirect impactof theforestindustry is only a minor fraction of the total CO 2emissions related to

production, and almost all of the emissions are caused due to pro-

duction of exports. Stechemesser and Guenther (2012) systematically

review theliterature related to carbon accounting. Onecan refer toTsai

et al. (2011, 2012b, 2013)and Mozner (2013) for other similar studies.

Close to our work, Van der Veen and Venugopal (2014) incor-

porate the cost of energy usage into EOQ model, and nd that the

economic and environmental performance of a rm can be synergy

or trade-off, depending on the values of specic parameters of the

emission regulations. Huaet al.(2011) investigate howrmsreactin

inventory management under carbon emission regulation based on

EOQ model. They derive the optimal order quantity, and examine

the impacts of regulation parameters on the optimal decisions,

carbon emissions and total costs. However, in their work, only thecap-and-trade regulation is discussed, and the permits buying and

selling prices are assumed to be equal. Bonney and Jaber (2011)

incorporate transportation cost and waste into EOQ model, and

develop an environmental economic order quantity model, which

results in a larger optimal ordering lot-size than that under the

standard EOQ model.Arslan and Turkay (2010)revise the standard

EOQ model by incorporating sustainability constraints. Various

sustainability constraints, such as carbon tax, cap and trade, direct

cap andcarbon offset, areconsidered.They show that in most cases,

the optimal ordering quantity with the presence of sustainability

constraints is larger than that without the constraints. It is note-

worthy that, in the above mentioned studies, the permits buying

and selling prices are all assumed to be the same. One exception is

Chen et al. (2013). They investigate the EOQ model under various

P. He et al. / Journal of Cleaner Production 103 (2015) 241e248242
http://www.ecx.eu/market-datahttp://www.ecx.eu/market-data


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carbon constraints including carbon tax, carbon offset, and cap and

price (i.e. cap-and-trade). For each carbon constraint, they show

whether and howthe ordering quantity can be adjusted to decrease

rms'costs. Although theymention two prices (award and penalize)

under cap and price constraint, they do not examine the optimal

decisions ofrms when these two prices are different.

Our paper differentiates itself from the existing studies in that it

treats the permits trading prices to be two different variables and

assumes that the permit buying price is not smaller than the permit

selling price. Based on this assumption, some novel insights for

rms' optimal decisions can be obtained.

3. Cap-and-trade regulation

This section examines the optimal production lot-sizing and the

corresponding optimal carbon emissions under cap-and-trade

regulation based on EOQ model.

Consider a carbon-intensive rm which produces a product to

satisfy the market demand under cap-and-trade regulation. The

rm rst receives a free quantity of permits (i.e. cap C) on its

emissions. If necessary, the rm can buy more or sell the granted

permits through an outside market with unit buying price b andunit selling prices (sb), respectively. Similar toHua et al. (2011)andChen et al. (2013), the variability of buying/selling price over

time is not considered in this paper. The annual market demand is

xed atD. Once the rm starts a production run, a setup cost Kand

related emissionseKare incurred. The unit production cost and the

corresponding emissions are denoted by c and ec, respectively.

Assume that the production is instantaneous. Each unit of product

kept in inventory incurs an annual holding cost h and annual

emissionseh. The three types of emissions eK,ecand eh, also called

emission intensities (Drake et al., 2010), relate the production lot-

size decision to the emissions cost. The objective of the rm is to

determine the optimal lot-size Q to minimize the sum of produc-

tion cost, setup cost, holding cost and emissions cost.

Denote by Ethe total annual emissions of the rm. For given lot-

sizeQ, the annual setup times is D/Q, thus the emissions generated

during setups iseKD/Q. The average inventory during a year is Q/2,

thus the emissions generatedby holding inventory is ehQ/2. Besides

these two parts, the total annual emissions of the rm also include

the emissions generated during the production process. Therefore,

the total annual emissions of the rm is EecDeKD/QehQ/2.Hereafter, the corresponding emissions under the optimal lot-size

are termed as the optimal emissions.

Let TCc be the minimal total annual cost under cap-and-trade

regulation, which can be solved by the following model:

TCc minQ>0n

cDKD=QhQ=2 bECsCEo

;

(1)

where cD KD/Q hQ/2 is the total cost caused by production,setups and inventory holding, and b(E C) s(C E)(u max(u,0)) is the emissions cost (revenue) resulted frombuying or selling permits.

The following monotonic properties of the optimal total cost can

be directly obtained from Model(1)(throughout this paper, we use

increasing and decreasing in the non-strict sense to mean non-

decreasing and non-increasing, respectively): (i) TCc is

increasing inc,K,h,ec,eK,eh, respectively; and (ii) TCc is increasing

in b but decreasing in s. Therefore, in the subsequent sections, we

do not discuss the above properties, but focus on the properties of

the optimal decisions, i.e., the optimal lot-size and emissions.

For ease of analysis, we rst solve the optimal emissions and

then the optimal lot-size. It is clear that the amount of emissions

(ecD) associated with production for the xed demand is constant.

Thus, we focus on the remaining part of the emissions. LetbEEecDeKD=QehQ=2. It is easy to verify thatbE ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2eKehDp . By solving the equationbEeKD=QehQ=2, wehave two lot-sizes in terms of a function ofbE, i.e.Q1 bE ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibE2 2eKehDq eh and Q2

bE ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibE2 2eKehDq eh:

Let fbE be the minimal total cost of setups and inventoryholding under givenbE, i.e.fbE min

eKD=Qeh Q=2bEfKD=QhQ=2g.By substitutingQ1 andQ2 intofbE, we have.

fbE K=eKh=ehbE.2 jK=eKh=ehj ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibE2 2eKehDq 2:

Note that if K/eK h/eh, f

bE achieves its minimum at Q Q1;

otherwisef

bEachieves its minimum at QQ2.K/eK(h/eh) can beinterpreted as the average setup (holding) cost for each unit ofgenerated emissions. For given emissionsbE, K/eK h/eh indicatesthat setup incurs a lower cost than holding does for each unit of

generated emissions, which leads to more setups with smaller lot-

size. In contrast, K/eKh/eh leads to fewer setups with larger lot-size. Particularly, ifK/eK< h/ehor K/eK> h/eh, it can be easily veri-

ed that v2fbE=vbE2 >0, and thus fbE is strictly convex inbE. IfK/eKh/eh, it is clear that vfbE=vbE>0 and v2fbE=vbE2 0, thusfbEis linearly increasing inbE.

LetbCCecD. Based on the above process, Model(1) can betransformed as:

TCc min

nmin

bEffiffiffiffiffiffiffiffiffiffiffiffi2eKehDp

;bEbCTCc1

bE

;

minbEffiffiffiffiffiffiffiffiffiffiffiffi2eKehDp ;bEbCTCc2bEo; (2)where TCc1bE cDfbE bbEbbC andTCc2bE cDfbE sbEsbCare the costs for the case with buyingpermits bEbC and selling permits bEbC, respectively. Note thatthe overall emissions arebEecD.

We next develop the optimal lot-size and the optimal emissions

based on Model(2). Dene the following thresholds:

Cdargmin

bE>0

nfbEbbEo and Cdargmin

bE>0

nfbE sbEo:

Since b s, it isclearthat CCbased on the convexity offbE. NotethatCCifbs. Denote bybEc the optimal solution to Model(2).Based on the convexity offbE, we have the following result.Lemma 1. The thresholds C and C satisfy:

CK=eKh=eh 2bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2eKehDp

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

K=eKbh=ehbp ; and C

K=eKh=eh 2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2eKehDp

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

K=eKsh=ehsp ;

with CC

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2eKehD

p , and

bE

csatises that

bE

c C when

bCC.

P. He et al. / Journal of Cleaner Production 103 (2015) 241e248 243


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TheProof of Lemma 1(and all subsequent results) can be found

in the Appendix part. Lemma 1 shows the values ofbEc underdifferentbC. In particular, the buying and selling prices (of thepermits) determine the lower bound Candthe upperbound CofbEc,respectively. A larger difference between b and s implies a wider

span betweenCand C.

Now, we characterize the optimal solution to Model(1). Denote

by Qc

andEc

the optimal lot-size and emissions under the cap-and-

trade regulation, respectively. Based on Lemma 1, we have the

following theorem:

Theorem 1. Under the cap-and-trade regulation,

(i) when C< ecDC, the optimal emissions are Ec ecDC, andthe optimal lot-size is Q

c ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2KbeKD=hbehp

;

(ii) when ecDCCecDC, the optimal emissions are equalto the cap, i.e., E

c C, and the optimal lot-size satises:(a) if K/eK h/eh, then Q c CecDffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

CecD2 2eKehDq

=eh,(b) if K/eK h/eh, then Q c CecD

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiCecD2 2eKehDq =eh;(iii) when C> ecDC, the optimal emissions are Ec ecDC andthe optimal lot-size is Q

c ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2KseKD=hsehp .Based on Theorem 1, we know that the emissions E

care

increasing (fromecDCtoecDC) in the capC. Recall inLemma 1that ifbs, thenCC, which implies that the optimal lot-size andemissions are independent of cap C. The rationale is that the im-

pacts of emissionsEunder different levels of cap Con the total cost

are the same when b s (the emissions trading cost can beexpressed as the sum of two separate linear functions ofEand C,

respectively). Note that the optimal emissions are increasing in ec,

but independent of the unit production cost c.

Theorem 1 also shows the optimal decision for emissions

trading. It is optimal to buy C

ecD

C unit permits when

C< ecDC, to buy and sell nothing when ecDCCecDC,and to sell CecDC unit emissions when C> ecDC. Theresulted permits range (ecDCto ecDC) is ultimately the rangeof the actual amount of emissions. The determination of the two

thresholds on the rm's emissions depends on the permits

buying price and selling price, respectively. The lower threshold

decreases in the permit buying price, and the upper threshold

decreases in the permit selling price. The larger the difference

between permits buying price and selling price is, the larger the

range between these two thresholds is. It is noteworthy that the

difference between the permit buying price and selling price in

this paper can be regarded as the transaction cost.

Particularly, if these twoprices are thesame,as assumed in many

existing studies, thermwill alwaysbuy or sell some permits unless

the initial cap equals a special value, i.e. CecDCecDC. Thismeans, the optimal emissions of the rm will always be a xed

amount, which is determined by the permits trading prices but not

the rm's received free permits.

Based on Theorem 1, we derive the impacts of regulation pa-

rameters (i.e.,C,s and b) on the optimal emissionsEc, as shown in

Proposition 1.

Proposition 1. Under the cap-and-trade regulation,

(i) if K/eK h/eh, then the optimal emissionsare independent of C, sand b, and E

cecD


p ;

(ii) if K/eKs h/eh, then the optimal emissions are increasing in C,

but decreasing in s and b, respectively.

Proposition 1(i) indicates that if setup incurs the same cost as

holding does for each unit of generated emissions, then the cap-

and-trade regulation has no direct impact on the rm's optimal

amount of emissions. The reason for this is that, the optimal lot-

size also minimizes the carbon emissions, so the rm should

keep the lot-size without engaging in carbon trade. However, if

setup incurs a lower or higher cost than holding does for each

unit of generated emissions, in the presence of smaller cap,

higher buying or selling prices, the rm will have more in-

centives to decrease its emissions, as indicated in Proposition

1(ii).

The following proposition characterizes the impacts of regu-

lation parameters on the optimal lot-sizeQc. Denote by Q* theoptimal lot-size in the classic EOQ model without the emissions

regulation, then Q*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2KD=h

p . We have the following results

about the comparison of the optimal lot-sizes.

Proposition 2. By comparing Qc

with Q*

, wend:

(i) if K/eK h/eh, then Qc is independent of C, s and b, andQ

c

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2eKD=eh

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2KD=h

p Q*;

(ii) if K/eK < h/eh, then Qc>Q

*

, and Qc

is decreasing in C but

increasing in s and b,respectively;(iii) if K/eK > h/eh, then Q

c


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TCt minQ>0fcDKD=QhQ=2 tEg; (3)

where tEis emissions cost (i.e. the tax).

The following monotonic properties of the optimal total cost

can also be directly obtained from Model (3): (i) TCt is increasing

in c, K, h , e c, eK, e h, respectively; and (ii) TCt is increasing in t.

Comparing the emissions cost under Model (3)with that under

Model(1), it is clear that carbon tax regulation is equivalent to cap-and-trade regulation with C0 andbt, whiles can be any valuesmaller than b because there is no permits to be sold in carbon tax

regulation (or equivalently C 0).Thus in thissection we directly givethe optimal decisions of the rm under the carbon tax regulation.

Denote by Qt

and Et

the optimal lot-size and the optimal

emissions under the carbon tax regulation, respectively. By setting

C0 and b t, the values and properties of the optimal lot-sizeand emissions under the carbon tax regulation are given in Theo-

rem 2. Theorem 2 can be directly obtained based on Theorem 1 and

Propositions 13, and thus its proof is omitted for brevity.

Theorem 2. Under the carbon tax regulation, the optimal lot-size

and emissions satisfy:

Qt ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2KteKD=h tehq ; EtecDK=eKh=eh 2t


p2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

K=eK th=ehtp ; and

(i) if K/eK h/eh, both the optimal lot-size and emissions are in-dependent of t, with Q

t


p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2KD=hp Qc,E

t EcecDffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2eKehDp

;

(ii) if K/eK s h/eh, the optimal emissions are decreasing in t.

Particularly, if K/eK< h/eh, the optimal lot-size is increasing int,

while if K/eK> h/eh,the optimal lot-size is decreasing in t;

(iii) the optimal lot-size is increasing in K but decreasing in h.

Particularly,if K/eKh/eh,the optimal emissions are increasingin K but decreasing in h; if K/eKh/eh, the optimal emissionsare decreasing in K but increasing in h.

Theorem 2 shows the optimal lot-size and emissions under

the carbon tax regulation, and summarizes the impacts of pro-

duction and regulation parameters on the optimal decisions. By

comparing Theorem 2(iii) with Proposition 3, we nd that the

impacts of production parameters (i.e. K and h) on the optimal

lot-size and emissions are the same under the both regulations.

Similar to the discussion of the cap-and-trade regulation, if setup

incurs the same cost as holding does for each unit of generated

emissions, then the rm's optimal lot-size under the carbon tax

regulation is also equal to the solution to the classic EOQ model

without the emission regulations.Based onLemma 1,Theorems 1 and 2, we can derive a condi-

tion under which the optimal decisions under the two regulations

are the same. Based on Lemma 1andTheorem 2, we nd that if

t b, Et CecD; if t s, Et CecD; if t b s,E

t CecDCecD. Hence, by further comparing Theorem 1with Theorem 2, we know that when t b s, both theoptimal lot-size and emissions under the carbon tax regulation

are the same as those under the cap-and-trade regulation,

respectively, which are irrelevant to any capC.

5. Comparison of the optimal emissions

Due to their different mechanisms, cap-and-trade and carbon

tax regulations may have different impacts on

rms' performance,

especially on the optimal emissions. Based on Theorems1 and 2, we

have the following results about the optimal emissions of the rm

under cap-and-trade and carbon tax regulations.

Theorem 3. Under cap-and-trade and carbon tax regulations,

(i) if K/eKh/eh,both the two regulations have no impact on theoptimal emissions, and E

tE

cecD ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2eKehDp ;(ii) if K/eKs h/eh,the optimal emissions under two regulations Et

and Ec

satisfy:

(a) if t > b,EtEc when CE

cfor all C.

Theorem 3indicates that, the difference between the optimal

emissions of the rm under two regulations depends on not only

the regulation parameters (i.e., C, s, b, t) but also the intrinsic

production parameters of the rm (i.e.,K,h,eh,eK).

Note that the buying price of permits under cap-and-trade

regulation determines the lower bound of

bE

cand thus of E

c

(see Lemma 1). If the tax rate level is higher than the buying

price of permits, the emissions should be even lower than thelower bound ofE

c. If the tax rate level is lower than the selling

price of permits, the emissions should be even higher than the

upper bound ofEc. If the tax rate level is between the range of

the buying and selling prices of permits, the relative magnitude

of the optimal emissions under these two regulations depends on

the initial capC. Therefore, none of these two regulations leads to

lower emissions than the other does all the time.

To further elaborateTheorem 3, a numerical example is con-

ducted. To this end, let D90,000,c20, K200,h2, ec2,eK 25, e h 0.5, s 3 and b 7. The optimal emissions of therm with different caps (under cap-and-trade regulation) and

with different tax rate levels (under carbon tax regulation) are

depicted inFig. 1.

The straight lines inFig. 1denote the optimal emissions of the

rmEt under the carbon tax regulation with different tax ratelevels. The polygonal line denotes the optimal emissions of the

rm Ec under the cap-and-trade regulation. The caps corre-sponding to the two ex points on the polygonal line are CecDand CecD, respectively. Note that in this example, K/eK 200/258, andh/eh2/0.54, thusK/eKs h/eh. As shown inFig. 1, ift2, i.e. ts,Et is greater thanEc over all levels of capC. Ift8,i.e. t b, Et is less than Ec over all levels of cap C. If t 5, i.e.s < t < b,E

tis greater thanE

cwhen the capCis less thanE

t, andE

t

is less than Ec

when the capCis greater than Et. These results are

consistent with the conclusions inTheorem 3(ii).

Fig. 1. The optimal emissions under the two regulations.

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6. Conclusions

This paper addresses the production lot-sizing issues of a rm

under cap-and-trade and carbon tax regulations based on EOQ

model. We characterize the optimal lot-size and corresponding

(optimal) emissions under each regulation, and compare the

performance of the rm with respect to the optimal carbon

emissions under the two regulations.

Our results show that, under the cap-and-trade regulation,

the rm may buy some permits for production, or sell some

surplus permits, or buy and sell no permits at all, depending on

the value of initial cap. If setup incurs the same cost as holding

incurs per unit of generated emissions, both regulations always

lead to the same optimal emissions (which is also equal to that

without emissions regulation). Otherwise, neither regulation al-

ways leads to lower emissions than the other does.

In this work, we only consider the lot-size and emissions

decisions in one particular period, and thus assume that the

permits buying and selling prices do not change over time. This

modelling and analysis is appropriate for those carbon-intensive

rms with small or modest carbon emissions, especially for those

producing single-period products such as food and electricity,

which are more likely to buy or sell permits on the spot market.Firms with large carbon emissions are likely to make permits

trading actions by taking actual production into account, multi-

ple period analysis may be more appropriate. On the other hand,

we focus our research on a single rm's decision making and do

not consider multiple competitive rms. These aspects are the

main limitations of our work, which can be further examined in

future research. Furthermore, more research can also be done in

the following ways. First, the decision problem can be investi-

gated based on other models, e.g. newsvendor model. Stochastic

demand can be more realistic and may generate some other in-

sights. Second, the decision-making of the government can be

incorporated. The rm and the government take part in a deci-

sion game, with the objective of minimizing the total cost of the

rm and maximizing the social welfare, respectively.

Acknowledgements

This research was supported by the National Natural Science

Foundation of China (nos. 71001094, 71101085, 71201153,

71371176), NSFC Major Program (nos. 71090401/71090400), and

Innovation Program of Shanghai Municipal Education Commis-

sion (no. 12ZS099). The authors thank the editor and three ref-

erees for helpful comments and suggestions on the earlier

manuscript.

Appendices

Proof of Lemma 1

It is clear that CdargminbE>0ffbE bbEg can be found bycomputing the corresponding emissions when the lot-size is the

solution to the following problem.

minQ>0fKD=QhQ=2 beKD=QehQ=2g;

whereKD/QhQ/2 is the sum of setup and inventory holding cost,andbEeKD=QehQ=2. It is clear that the optimal solution for theabove problem is

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2KbeKD=hbeh

p . Hence the correspond-

ing emissions satisfy

CK=eKh=eh 2bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2eKehDp


K=eKbh=ehbp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2ekehDp K=eKb h=ehb


K=eKbh=ehbp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2eKehDp :

Similarly, we have

CK=eKh=eh 2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2eKehDp


K=eKsh=ehsp


2ekehDp K=eKs h=ehs

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK=eKsh=ehsp


p :

Recall thatCCfrom the convexity off(z). Consequently, we haveCC ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2eKehDp .

Dene

bE

c

1dargmin

bEffiffiffiffiffiffiffiffiffiffiffiffi2eKehDp

;bEbCnf

bE

b

bE

o and

bE

c

2

argminbEffiffiffiffiffiffiffiffiffiffiffiffi2eKehDp ;bEbCnfbEsbEo:Since CdargminbE>0ffbE bbEg and CdargminbE>0ffbE sbEg,we havebEc1maxC;bC andbEc2minbC; C based on the con-vexity offbE. Consequently, we have

(i) whenbCC,bEc1bCandbEc2C. SincebCis a feasible solutionto minbEffiffiffiffiffiffiffiffiffiffiffiffi2eKehDp ;bEbCffbE sbEg, we haveTCc cDfC sCsbCcDfC sCecDC.

Proof of Theorem 1

(i) and (iii) can be easily veried from Lemma 1. Details are

omitted for brevity. We next prove (ii). When

ecDCCecDC, we have C

bCC, and thus

E

c

ecDbEc ecDbC C, and the feasible lot-sizes areCecD

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiCecD2 2eKehD

q =eh. Substituting these lot-sizes

in Model(1), we have the total costs are.

cDK=eKh=ehCecD=2h=ehK=eK


q 2:

Consequently, if K/ek h/eh,Q

c CecD ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

CecD2 2eKehDq

=eh, and if K/ek h/eh,Q

c CecD


q =eh.

It can be veri

ed that

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Cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

C2 2eKehDq

ehK=eKh=eh 2bjK=eKh=ehj

2K=eKbffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2KbeKD

hbeh

s ;

C ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC2

2eKehDq eh K=eKh=eh 2sjK=eKh=ehj2K=eKsffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2KseKDhseh

s :

Hence, we have that ifK/ekh/eh,0@C ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC2 2eKehDq 1A,eh ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2KbeKD=hbehq

andC

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC

2 2eKehDq

ehffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2KseKD=hsehq

;

and ifK/ekh/eh,0@C ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC2 2eKehDq 1A,eh ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2KbeKD=hbehq ;

andC

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC

2 2eKehDq

ehffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2KseKD=hsehq

:

So we have that if K/eK h/eh,Q

c Ec ecD ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Ec ecD2 2eKehDq

=eh, otherwise

(K/eKh/eh),Qc

Ec ecD ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEc ecD2 2eKehDq =eh. Proof of Proposition 1

IfK/eK h/eh, thenLemma 1indicates that CCffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2eKehDp

.

Hence, the optimal emissions are a constant, i .e.,

EcecD


p . So (i) holds.

(ii) can be proved from the following process. Theorem 1

directly indicates that Ec

is increasing in C. Specically, Ec

is

increasing from ecDC to ecDC. Recall thatCdargminbE>0ffbE bbEg and CdargminbE>0ffbE sbEg. Conse-quently, the convexity of f(z) indicates that C(C) is decreasing in

b(s). So we have thatEc

is decreasing ins and b; respectively.

Proof of Proposition 2

Recall that EcecD


p if K/eK h/eh. Since

Qc Ec ecD

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEc ecD2 2eKehD

q =eh (from the Proof of

Theorem 1), we have that Qc


p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2KD=hp Q* ifK/eKh/eh.

We next prove (ii) and (iii). Dene

g1zdzffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

z2 2eKehDq

and g2zdzffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

z2 2eKehDq

:

It is clear thatg1(z)d 2eKehD/g2(z). Sinceg2(z) is increasing inz,

we have g1(z) is decreasing in z. This indicates that

Ec ecD ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Ec ecD2 2eKehDq

=eh is decreasing in Ec ecD,

while Ec ecD ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Ec ecD2 2eKehDq

=eh is increasing inE

c ecD.Proposition 1(ii) shows thatEc is increasing inCwhile it isdecreasing in s and b, respectively. So we have that ifK/eK< h/eh,

thenQc

is decreasing inC, but is increasing ins andb, respectively,

and ifK/eK> h/eh, thenQc

is increasing inC, but is decreasing in s

andb, respectively.

Proof of Proposition 3

We rst show the monotonic properties of the optimal emis-

sions because we have shown the impact of the optimal emissions

on the optimal lot-size in theProof of Proposition 2.

Denote

Rv K=eKh=eh 2vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2eKehDp


K=eK vh=ehvp

ffiffiffiffiffiffiffi

2Dp

2

KehheK 2veKeh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKveKh veh

p :

It is clear that CRband CRs. SincevRvveK


2Dp

2

K2h 3veh veKh 2veh2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

KveK3h vehq >0

and

vRvveh


2Dp

2

h2K 3veK vehK 2veK2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKveKh veh3

q >0;we have bothCand Care increasing ineKandeh, respectively. Note

that both Cand Care clearly independent of. ec.

The impacts ofKand h on Cand Ccan be veried as follows. R(v)

can be rewritten as.

Rvffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2eKehDp

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK=eK v

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih=ehv

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih=ehvpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK=eK v

p !


2eKehDp

2 r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK=eK v

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih=ehv

p !;wherer(t)t1/t. It is clear thatr(t) is decreasing when t1, butis increasing whent1. Consequently, we have that: ifK/eKh/eh,R(v) is increasing inKbut is decreasing inhfor allv, and ifK/eKh/eh, R(v) is decreasing in decreasing in Kbut is increasing in h for all v.

CRb and CRs indicate that the cost parameters K and hhave the same impacts on Cand C.

Now, we have that (i) bothCand Care increasing in eKand eh,

respectively, but are independent of ec; and (ii) if K/eK h/eh,both Cand Care increasing in Kbut is decreasing in h , and ifK/

eK h/eh, is decreasing in Kbut is increasing in h .FromTheorem 1, we have

Ec

8>:ecDC; C< ecDCC; ecDCCecDCecDC; C> ecDC

Hence, Ec

is increasing in ec, C and C, respectively, for all C. The

above impacts of parameters on C and C indicate that: (i) the

optimal emissions are increasing in ec, eK and eh, respectively,

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and (ii) ifK/eK h/eh, the optimal emissions are increasing in Kbut is decreasing in h, and ifK/eK h/eh, the optimal emissionsare decreasing in decreasing in Kbut is increasing in h. Conse-

quently,Proposition 3(ii, iii) hold.

From the Proof of Theorem 1 and Proof of Proposition 2, we

have that if K/eK h/eh,Q

c Ec ecD ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE

c ecD2 2eKehDq =eh is decreasing inEc ecD, and if K/ek h/eh,Q

c Ec ecD ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Ec ecD2 2eKehDq

=eh is increasing inE

c ecD. Consequently, the impacts of K and h on the optimalemissions E

cindicate that the optimal lot-size is increasing in K

but is decreasing in. h .

Proof of Theorem 3

Part (i) can be directly veried from Proposition 1(i) and

Theorem 2(i). We next discuss the optimal emissions under the

two regulations whenK/ek s h/eh. Recall that the emissions under

the two regulations are

Ec

8>:ecDC; C< ecDCC; ecDCCecDCecDC; C> ecDC

and

Et ecDK=eKh=eh

2t ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2eKehDp2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

K=eK th=ehtp : (A.1)

It can be veried thatEt ecDargminbE>0ffbE tbEg. Recall that

fbEis strictly convex ifK/eKs h/eh. Consequently, (1) ift > b, thenE

t ecDC. Based on the above three points, part (ii) holds fromEquation(A.1).

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Carbon Emissions Under Cap-And-trade and Carbon Tax Regulations

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