F o i s i e S c h o o l o f B u s i n e s s | 1 0 0 I n s t i t u t e R d . | W o r c e s t e r , M A 0 1 6 0 9 5 0 8 - 8 3 1 - 5 2 1 8 | w w w . w p i . e d u / + C S B

2014

Pricing of Emission Permits in Internal Markets: A Bayesian Markov Chain Monte

Carlo Approach Working Paper WP5-2014

Joseph Sarkis and Dileep G. Dhavale

Pricing of Emission Permits in Internal Markets: A Bayesian Markov Chain

Monte Carlo Approach

Joseph Sarkis Robert A. Foise School of Business

Worcester Polytechnic Institute 100 Institute Road | Worcester, MA 01609-2289

Phone: +508.831.4831 email: jsarkis@wpi.edu

Dileep G. Dhavale* Graduate School of Management

Clark University 950 Main Street | Worcester, MA 01610

Phone: +508.793.7781 email: ddhavale@clarku.edu

*Please direct correspondence to Dileep Dhavale

SEPTEMBER 2014

FOR INQUIRY PLEASE REFERENCE THIS WORKING PAPER AS WP5-2014

1

Pricing of Emission Permits in Internal Markets: A Bayesian Markov Chain Monte Carlo Approach

Abstract

Global climate change has caused significant risk to world populations and their livelihoods. Climate change has been undeniably linked to man-made, anthropogenic, actions many arising from industrial activities. To help manage and mitigate climate change concerns, regulators and industry have sought ways to manage greenhouse gas emissions to reduce anthropogenic effects on the world’s climate. Policy makers have sought to manage these efforts through various market-based mechanisms of which ‘carbon trading’ is one. The external emissions trading market prices can help in planning purposes. But organizations have come to realize that external pricing may not actually provide an accurate costing, risk, and accounting measure. Many organizations have turned to internal emissions trading schemes (IETS) to help them manage their emissions. One of the difficulties facing managers is to determine an accurate or acceptable price or cost for emissions. To help address this management accounting issue, one that the research and practitioner literature has largely ignored, we introduce and develop a Bayesian framework to determine effective internal pricing measures and thresholds. To achieve this, a Monte Carlo Markov Chain technique using a Gibbs sampler algorithm and a standard investment appraisal model is employed. Our findings show that higher required rate of return requires higher prices for the carbon credits. More uncertainty in the cash flows results in a wider range for variation in the prices. Carbon credit prices tend to increase over the life of the carbon-credit producing asset. Avenues for future research into this important managerial issue are presented.

Keywords: Internal rate of return, carbon emission credits, Bayesian analysis, sustainable manufacturing, discounted cash flow, Gibbs sampler

1.0 Introduction

Climate change has been on the agenda of organizations for over two decades (Kassinis and Vefeas,

2006; Sarkis et al., 2010). Various regulatory schemes have been introduced to internalize the

environmental externalities that lead to climate change and to help mitigate the impact of climate change.

Market-based trading mechanisms are quite popular in achieving this internalization goal (Vincent, 2013).

Understanding these external demands and requirements and integrating them into internal risk and cost

management decisions and activities has now become an accounting imperative.

2

Internalizing environmental externalities requires for-profit organizations to reconsider their

capital investments strategies. These investments will require effective pricing and cost estimation given

the significant uncertainties associated with market-based-regulatory mechanisms for control of

greenhouse gases (GHG).1 Although many of these mechanisms focus on external markets, internal

emissions trading schemes (IETS) have also been developed and proposed to address actual and perceived

risks and social and regulatory costs (Sarkis and Tamarkin, 2005; Horisch, 2013). The difficulty in many

of these internal systems is that the prices for emissions permits vary greatly and are difficult to

determine. Many times the internal costs are determined by external forces (e.g. the external emissions

market prices) but sometimes they are artificially set by IETS managers who determine the caps and

allocations across the organization.

To help address these concerns, we introduce a methodology to help organizations determine an

economically optimal and fair price for carbon trading between the organization’s subunits in an IETS.

This price can be compared to the external market price and can aid organizations in making decisions on

whether or not to invest in certain carbon emissions abatement technologies and processes. An internal

market for trading of emissions permits or carbon credits allows subunits in the organization to sell

carbon credit they do not need to meet their own goals. These carbon credits are purchased by subunits

that exceed the allowed emission goals. It is for these markets that this paper develops a pricing method

based on discounted cash flows and expected rates of return. We also analyze different uncertainties that

affect the prices and the corresponding cash flows.

One of our contributions in this paper deals with integration of Bayesian decision-making

framework in the evaluation of environmental externalities that impact investment decisions in GHG 1 There are several greenhouse gases. The main ones are carbon dioxide, methane, nitrous oxide (N2O), Freon refrigerant and its later versions (CFC-12 and HCFC-22), and sulfur hexafluoride. These gases have different radiative impact thus have different global warming potential (GWP). Based on the GWP and how long the gases stay in the atmosphere, environmental scientists have designed one common measure of impact for all emitted GHG. All emissions are measured in number of equivalent metric tons (tonnes) of CO2. This is referred to as one carbon credit or one emission permit. These credits are traded in open or internal markets. We present a method in the paper to estimate the price per credit in the internal market.

3

reducing technologies. We do so by considering the uncertainties and risk inherent in carbon credits

markets and their impact on carbon credit prices. The methodology developed in this paper provides the

decision makers with useful data which helps them to set equitable carbon credit prices that can help the

organization and its subunits in meeting environmental goals in an economical fashion. We also examine

the effect of increased uncertainty and volatility as measured by the standard deviation on pricing

structure in the internal markets.

This paper proceeds in the following fashion: after providing some background to the issues at

hand, we position this paper within the environmental decision making and investment literature. A brief

description of Bayesian statistical model and Gibbs sampler, which are used in pricing and return on

investments analysis, is provided. The model is applied to various scenarios of an illustrative example,

based on real-world data, which provides useful insights and generalized guidance to decision makers.

Practice-, management- and policy-oriented insights are identified. Some theoretical issues are

highlighted. A summary of the results and conclusions are then presented with future research avenues

identified.

.

2.0 Background

To aid in understanding the process for estimating internal pricing of carbon emissions and trading, we

first provide some background on emissions trading and various policies. Included in this overview are

some current practices for price setting and trading mechanisms. Initially an overview of carbon markets

is provided. This overview will include some history and characteristics of both internal and external

markets.

2.1 External and Internal Carbon markets

Internalization of external environmental costs and policies can occur at multiple levels: corporate,

regional, national and international (Engel, 2005; Krause, 2011; Xue et al., 2014). Important in these

practices is the setting of prices. The cost related to environmental concerns are dependent on the

4

regulatory regime and mechanism, the type and boundaries (geographical as well as political) for market

for emissions credits, the number of participants involved, and the costs of technology. These are some of

the major factors, among others, that influence costs and pricing of carbon credits.

Market based regulatory mechanisms typically include determination of carbon pricing by the

carbon-trading markets (Hepburn, 2010; Speck, 2013; Wietzman, 2013). For many years, use of the

natural environment had been considered as a free good; the cost of avoiding or repairing damage to the

environment was considered a dispersed social cost, which nobody had to pay. Thus, organizations have

tended not to include these costs in their normal operating expenses. Due to a changing political and

social normative landscape and increased demands of social responsibilities on the part of organizations,

there is a strong impetus to internalize these social costs as normal and usual costs of operations and need

to manage them in a socially responsible and beneficial way. From an IETS perspective, this would

require setting the price(s) that are considered fair and equitable by subunits of an organization as well as

induce the type of behavior from subunits managers that will help the organization meet its environmental

obligation in an optimal fashion. Pricing and accounting for carbon credits need to be actively managed

and controlled. Since, as we will see, due to significant uncertainties that exist about many facets of

carbon trading policies, internal pricing presents a challenging problem.

Before focusing on internal cost management and pricing of GHG emissions a brief overview

history of the market mechanism known as ‘cap-and-trade’ is provided.2 One of the earliest efforts of

cap-and-trade involved the trading of emissions permits or allowances in sulfur dioxide emissions by U.S.

energy generating companies. It was introduced by the U.S. Environmental Protection Agency to help

mitigate acid rain concerns (Shmalensee, et al., 1998). The Kyoto protocol then supported the use of

carbon trading to reduce GHG emissions (Böhringer and Löschel, 2003). The European Union (Convery

and Redmond, 2007), Northeast United States (Ranson and Stavins, 2014), California (London et al.,

2013), New Zealand, Australia, and China in selected cities such as Hong Kong, have also introduced

carbon trading markets. Voluntary markets for industrial trading have also existed, such as the Chicago 2 For an in-depth review please see Calel (2013).

5

Climate Exchange (CCX), but have not gained as much traction (Boulatoff, et al, 2013; Yang, 2005). A

broad variety of extra-organization carbon markets have existed in the past and currently exist.

IETS by corporations have also been increasingly popular, voluntary mechanisms to manage

industrial GHG emissions (Sarkis and Tamarkin, 2005; Horisch, 2013). IETS have had an almost two

decade history in response to the Kyoto Protocol and market based regulatory mechanisms to reduce

GHG emissions. For example, British Petroleum (BP) and Shell Corporation each had internal emissions

trading programs (Pinkse and Kolk, 2007). Given the wide variety of global emissions regulations and

variations in trading schemes and organizations, corporations, especially multinational ones, have

responded in varied ways. IETS may be used to generate a response that is flexible enough to these

various external forces.

Similar to external markets, for IETS a cap is set on emissions and permits allocated across

subunits of an organization (Horisch, 2013). The subunits can buy and sell emissions permits in an

internal market, and thus the term ‘cap-and-trade’. The cap level can be manipulated for each subunit to

manage the overall emissions of the organization. Distribution across the organization can be determined

based on historical levels or characteristics of the operations. Some subunits can reduce their emissions

more effectively/efficiently because the technology or processes they use are amenable to easy or

inexpensive modification. Initially, these subunits would be targeted for investment in abatement

technology, while the other organizational units will be required to buy the credits to cover the excess

emissions above their respective caps (Akhurst, et al., 2003; Horisch, 2013; Lee 2013).

2.2 Carbon Pricing and Practices

Central to organizational carbon emissions management is the costing or pricing of these

emissions. A great deal of uncertainty exists in pricing and operations of trading markets for carbon

credits. Reasons for uncertainty, for both internal and external markets, are economic and political

(Paltsev et al., 2008; Dietz and Fankhauser, 2010). Economic uncertainties result from limited market

exposure and experience, and regional rather than global markets. These inefficiencies and nor-

transferability of credits from one market or region to the others allow different prices to exist in different

6

markets (Ranson and Stavins, 2014; Weitzman, 2013). These external differences can influence internal

prices. Spread of technology can also impact trading markets; easily available and inexpensive

technology can cause great fluctuations in carbon prices.

Carbon credit pricing in external markets is the result of many forces. The market prices are

influenced by efforts such as auctions which will set initial prices or through a policy about setting caps or

taxes, lower caps will mean higher prices and vice versa (Keohane, 2009). Trading then allows the

market to determine subsequent prices. Researchers and governments have made estimates of the ‘social’

cost of emissions to get a non-market estimate. Many of these estimates are made through integrated

assessment models (IAM’s), which have their own limitations (Ackerman, et al., 2009). These IAM’s

have been utilized by the U.S. government, for example, it has estimated that on average for 2015 the

social cost per ton of carbon is $37 per tonne (US Government, 2013). Although higher than current

market trading prices, this governmental valuation has been considered a conservative estimate by watch

groups and activists who felt that a number of social impacts were not considered in the evaluation

(Howard, 2014). Since governmental and regulatory agencies have used IAM’s to determine social costs,

it has caused organizations to pay close attention to them lest they may be used to set external market

prices and tax rates by regulators.

Internal carbon prices are becoming an increasingly common business tool and are used within

firms for planning purposes, (Economist, 2013). A study by the Carbon Disclosure Project (CDP) found

that 29 American companies used an internal carbon price. The number of German companies doing so is

about 20 (Horisch, 2013) and increasing. The U.S. corporate prices range from $6-7 per ton of CO2

equivalent at Microsoft to $60 per ton of CO2 equivalent at Exxon Mobil. It was generally found that the

companies with long productive lives and those affected by regulatory policies (such as oil companies)

tend to use higher prices (Economist, 2013). Usually, the internal ETS prices have been higher than

external market prices, possibly because the true cost of compliance may be higher or the organizations

may want to accelerate compliance. It is not known whether any methods or proprietary information was

used to arrive at these estimates. Based on significant variations seen in the internal pricing of carbon

7

credits in different organizations, it is likely that the organizations may have used widely varying methods

to estimate the price. Our search of published literature indicates that there are no techniques or

methodologies available for installing pricing mechanisms in internal market for emission permits.

In this paper, we propose just such a technique, which is described in the following sections. The

technique involves a Bayesian decision-making framework where the organization is able to integrate the

prior knowledge and expertise it has about the carbon credit pricing with current observations to come up

with revised price applicable for trading in the internal market. The pricing of the carbon credits is based

on the well-known investment appraisal method of internal rate of return which is embedded in the

Bayesian framework. Bayesian modeling allows emulation of uncertainties and volatilities in carbon

prices seen in the external markets. The results of the model provide us with detailed information about

the prices that ought to be charged to attain a certain rate of return on the investment in GHG emission

reducing assets.

2.3 Setting the Stage for a Bayesian Model for IETS Pricing

In this paper we focus on the internal carbon credit markets run by organizations for the benefit of

their subunits to help them comply with GHG emission requirements so the organizations as a whole is

also in compliance with emission control regime currently in force. It is not economically optimal to

require all subunits of an organization to proceed at the same pace in achieving the environmental goals.

Depending on the processes that are responsible for emitting GHG in subunits’ operations and the amount

of GHG involved, it may be less expensive to reduce GHG in certain subunits than in others. Since the

regulatory authorities generally track the emissions from the entire organization and not its smaller

subunits, it behooves the organization to invest in emission reduction equipment in those subunits where

the impact per dollar spent will be the greatest.3 However, following this strategy, though optimal from

the organization’s perspective, creates moral hazard for the subunits that are not required to invest in

emission-abating equipment at the current time. Additionally, it would create a situation where some

subunits bear the entire cost of the organization’s GHG abatement program, making the managers of 3 Large stand-alone subunits, such as a power plant, are tracked individually.

8

those subunits aggrieved, since their performance measures can be adversely affected by the additional

pollution control costs that the other subunits do not have.

There are several ways to handle this situation. An organization may “tax” (i.e., allocate emission

alleviation costs) its subunits based on sales, cost of energy used or some other allocation base. It could

require each subunit responsible to reduce the emissions by some percentage. These and other similar

policies may work in many situations in others they may create suboptimal decisions by managers of the

subunits due to incongruence of goals.

In this paper we propose an internal market for trading of carbon credits, which will allow

subunits in the organization to sell carbon credits they do not need to meet their own goals. These carbon

credits are purchased by subunits that exceed the allowed emission goals. We have devised a method of

determining the pricing of these units.

In this method, it is assumed that the organization has experience from previous years about

prices and thresholds; it may use that prior knowledge; alternatively, it may decide to guide the prices in a

different direction than historical trends due to changing regulatory requirements and use the new desired

prices as “prior knowledge”. The method is able to incorporate this information as well as fluctuation in

prices that have been observed in the open markets as prior knowledge. We further assume that the

organization is not totally altruistic and may expect some return on its investment in GHG reducing

equipment. The rate of return could be equal to what the organization expects in its normal capital

budgeting decisions or it could be set lower to make the compliance with environmental goals easier.

To describe briefly the proposed method, we start with the “prior knowledge” described in the

above paragraph, to that we add the information about uncertain nature of cash flows expected from sale

of carbon credits within the internal market, the organization’s expected rate of return; then using Bayes

Theorem, we determine the revised probability distribution of carbon credit cash flows, and carbon credit

prices for the internal market. The organization may pick the mean of the revised distribution as the price

of a carbon credit. Alternatively, it could pick some other value from the distribution, say, 75th percentile.

By choosing a 75th percentile value, the organization will be implicitly rewarding the subunits that have

9

made the investment in GHG reducing equipment. Any price picked that is lower than the mean will

penalize the subunits that invest in emission-reducing equipment.

3.0 Application of Bayesian Framework to determine the pricing of Carbon Credits

To recapitulate, a Bayesian model is developed to amalgamate the knowledge and experience

decision makers and experts in an organization have accumulated about the internal dynamics of carbon

credits usage by different subunits, the organization’s objective and goals in reduction of emissions of

GHG, as well as the external restrictions and limitations currently faced and expected to be in force in the

future. An organization may wish to reward a certain type of action by the managers in charge of its

subunits related to GHG reduction, and thus may decide to purposely raise the price of carbon credits (to

make a quick adoption of emission control technology lucrative). These types of inducements, or

alternatively penalties for early adoption, may be built into this study’s valuation scheme by appropriately

specifying the prior and hyper prior distributions of carbon credit cash flows and their volatility.

The model, which determines the profitability of an investment for environmentally sustainable

technology, in our case example energy efficient manufacturing to reduce energy usage emissions4,

begins with the conventional NPV model. We consider incremental after-tax cash flows and use an after-

tax discount rate, 𝑟, in computation of NPV as follows:

𝑁𝑃𝑉 = −𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡+ ∑ 𝑒𝑛𝑒𝑟𝑔𝑦 𝑠𝑎𝑣𝑖𝑛𝑔𝑠𝑖+𝑑𝑒𝑝𝑟𝑒𝑐𝑖𝑎𝑡𝑖𝑜𝑛 𝑡𝑎𝑥 𝑏𝑒𝑛𝑒𝑓𝑖𝑡𝑠𝑖−𝑚𝑎𝑖𝑛𝑡𝑒𝑛𝑎𝑛𝑐𝑒 𝑎𝑛𝑑 𝑟𝑒𝑝𝑎𝑖𝑟𝑖+𝑐𝑎𝑟𝑏𝑜𝑛 𝑐𝑟𝑒𝑑𝑖𝑡𝑖(1+𝑟)𝑖

𝑁𝑖=1 (1)

where subscript 𝑖 indicates cash flows in the 𝑖𝑡ℎ year; and 𝑁 is economic life in years of the investment.

The internal rate of return is that value of the discount rate, which will make 𝑁𝑃𝑉 = 0. Expression (1)

4 The eligibility of whether a technology or activity is considered eligible for emissions reductions may depend on the ‘scope’ of emissions covered by the IETS. Scope 1 are the emissions from sources under the jurisdiction of the company, Scope 2 are offsite emissions from the purchase of electricity, and Scope 3 are offsite emissions from the company’s supply chain or from products sold by the company. This example falls more into the Scope 2 emissions related to energy savings. Many companies utilize Scope 1 and Scope 2 emissions since they are more easily measurable and controlled (Peters, 2010; Neville and Whisnant, 2014).

10

does not have a closed-form solution for r. However, efficient search algorithms using trial and error

method are available to find the value of the discount rate at which NPV equals zero.

In an IETS, only the subunits of the organization can participate in selling and buying of carbon

credits; only carbon credits generated within the organization by a subunit can be sold to other subunits.

These internal market transactions have no impact on the organization’s operating income, however the

subunit selling the emission permits will show increased revenue and one that is purchasing will show

increased expense.5

Since the internal markets are insulated from conventional market mechanisms (arms-length

transactions, and diametrically-opposite interests of the parties involved where each is trying to maximize

its utility), the price or value of carbon credit for sale between the subunits needs be determined by some

other mechanism. As described earlier, a significant amount of uncertainty and volatility exists in open

emission trading markets around the world. In our model, we emulate this stochastic nature of carbon

credit prices by assuming the prior distribution of prices (hence the corresponding cash flows) have

normal or multivariate normal distributions with increasing standard deviation at various rates, over the

life of the emission control equipment6.

Prices of carbon credits exhibit significant amounts of risk and improbability in open emissions

trading markets. In this study, we want to match the impact of those characteristics on pricing of the

credits in internal market; hence we have purposefully kept all other cash flows (the initial investment,

depreciation tax benefits, savings due to energy efficiency, and maintenance and repair expenses)

deterministic while varying only the carbon credit cash flows7.

The model allows an organization and its decision makers to take into account the information

they already have about external markets run by regulatory bodies, the internal objective and goals in 5 Because they exactly offset each other, the intra-subunit transactions in the internal market have no impact on the bottom line of the organization. However, the organization and the individual subunits that install pollution control and other abatement systems will incur depreciable costs. 6 Although we assume the normal distribution family for illustration purposes, the methodology is flexible enough to allow for any type of distribution including categorical distributions. 7 These deterministic cash flows can be probabilistic, if required. The methodology does not put any constraints on the nature of cash flows.

11

meeting GHG emissions and current progress in achieving them. The organization could also take into

account the ability of individual subunits in contributing to overall performance. This information is

reflected in hyper prior and prior distribution. The Bayesian framework does not require that such

information be developed with great accuracy. The information, decision makers’ beliefs about the most

appropriate prices of the carbon credits based on all relevant information and organization’s goals, is

described as a probability density functions (pdf). Bayesian modeling allows a great deal of flexibility in

how the pdf is specified. It could be any standard, analytical, continuous or discrete pdf or non-standard

i.e. a categorical probability mass function. We use normal or multivariate normal as a prior with a beta

distribution as a hyper priors in the illustrative example. These two-parameter distributions afford an

organization the ability to correctly reflect the average value of carbon credit prices as well as any

volatility that may anticipated over the life of the asset that is producing the carbon credits.

In our example, we consider investment in an energy efficient manufacturing system that will

help reduce GHG emissions as it consumes less electrical power to operate. This reduction in power use

reduces emissions of GHG and creates carbon credit allowances for the subunit in the organization that

installs it. If the subunit has satisfied the internal organizational goals in reducing the emissions then the

subunit is in a position to offer these carbon credits for sale to the other subunits, if needed, to meet their

internal goals. In a fortunate situation when all subunits meet their own goals and the organization is in

full compliance with environmental regulations and regulatory goals, it may be able to sell the carbon

credits in an external emission trading markets8 or bank it for future use.

To summarize the technique developed in this paper, we are determining the price of the carbon

credit in the internal market using a Bayes’ Theorem and Markov Chain Monte Carlo (MCMC)

8 In such an instance, the price of a carbon credit will be decided by the prevailing market price and not based on price determined by the model presented in the paper. The model presented here is applicable only to the organization’s internal market.

12

technique. All cash flows are deterministic except the cash flows related carbon credits because the price

fluctuations.9

Note that at this point the necessary information needed to compute the NPV in Equation 1 exists,

except for the discount rate 𝑟. The organization may set the discount rate equal to the hurdle rate or any

other rate it chooses. We consider only the incremental cash flows that result from incremental investment

in GHG reducing equipment. Thus, the organization has the ability of setting a separate, possibly more

lenient rate of return just for environmental compliance equipment. Once the discount rate is known the

NPV can be calculated using the traditional method if the carbon credit prices are known. That

information is not known and in this paper we propose a method to find the prices implicit in Equation 1

using the Bayesian framework and MCMC method, specifically a Gibbs sampler.

The Bayesian framework is set up as follows: The prior knowledge about the prices is expressed

as a prior pdf. We also use a hyper prior to inject more uncertainty about the prices since emissions

trading markets have traditionally exhibited substantial gyrations. The sampling distribution is the

distribution of NPV which is assumed to be normal with zero mean and small variance. This assumption

imposes a condition on the carbon credit prices; their posterior or revised values should be such that the

NPV of all cash flows should be equal to zero. Posterior prices or the implicit prices of carbon credits the

model finds are those prices that make the rate of return (discount rate) required by the organization

exactly equal to the internal rate of return.

3.1 The Bayesian Framework

The Bayes formula updates the existing information with experimental observations (or similar

new information that may become available) to arrive at revised information about an event or a set of

events. The prior existing knowledge about the events is expressed as a prior pdf, 𝑓(𝜽) . Experimental or

9 We use the terms cash flows resulting from carbon credits and price of carbon credits interchangeably. Since carbon credit cash flows = price per credit x number of carbon credits, and the number of credits are constant in the model. What is stochastic is the price and hence the corresponding cash flow. The number of carbon credits in the illustrative problem is 2,963.4 each year for 10 years.

13

new information gathered is quantified by a sampling distribution or likelihood function, 𝑓(𝒚|𝜽) where 𝒚

is a vector of observed or sampled values and 𝜽 represents a vector (or a set of vectors) of parameters of

the prior distributions. The Bayes formula updates prior knowledge about some given event(s) using new

information contained in the observations to come up with revised knowledge, which is expressed in the

form of a posterior conditional distribution, 𝑓(𝜽|𝒚).

𝑓(𝜽|𝒚) = 𝑓(𝒚|𝜽)𝑓(𝜽) 𝑓(𝒚)

(2)

The quantity in the denominator is a normalizing factor, which is not needed for further analysis in this

paper, so

𝑓(𝜽|𝒚) ∝ 𝑓(𝒚|𝜽)𝑓(𝜽) (3)

When an organization has additional information about the prior pdf, such as the distribution of one or

more of its parameters then a hyper prior distribution, 𝑓(𝜃|∅), is employed to incorporate this additional

information in the revision process. The Bayes formula then becomes:

𝑓(𝜽;∅|𝒚) = 𝑓(𝒚|𝜃)𝑓(𝜽|∅) 𝑓(∅)𝑓(𝒚)

(4)

where ∅ is the hyper prior parameter vector and provides more information about the prior parameter

vector 𝜃 .

In the illustrative example, the prior distributions are for the carbon credit cash flows of the

project, hence θ=C where 𝑪 = {𝑐1, 𝑐2, … 𝑐10} are annual cash flows resulting from sale of carbon credit

in the internal market by a subunit.

It is assumed that that the carbon credit cash flows, 𝑐𝑖′𝑠, occurring over the ten years are

independently (but not identically) distributed.10

𝑐𝑖~ 𝑛𝑜𝑟𝑚𝑎𝑙�𝑚𝑖 ,𝜎𝑖2� (𝑓𝑜𝑟 𝑖 = 1,2, … ,10) (5)

10 We also explore the case when the cash flows taking place over the investment horizon, instead of being independently distributed, are distributed according to a multivariate normal. This case is discussed in a later section.

14

In lieu of a normal distribution, the decision maker may use any standard continuous or discrete

distribution if prior knowledge warrants it; for example a uniform distribution would be appropriate as a

prior if the cash flows are expected to be equally likely over a range of values. It is also possible to use a

categorical frequency distribution as a prior distribution.

If there is substantial amount of uncertainty about the means of the normal distribution of the cash

flows, 𝑪, this additional vagueness may be taken into account by using a hyper prior distribution defined

on the means. A hyper prior is not always needed to describe fully an organization’s prior knowledge

about events; in the example a hyper prior is included for illustrative purposes.

It is assumed that mean values of the cash flows, 𝑚𝑖 ′ 𝑠, (the vector 𝑴) are not completely known.

There is some uncertainty about the exact values of the means. We assume that the decision makers

expect the actual mean values of the cash flow lie somewhere in the range of 80% to120% of the

estimated mean value, 𝑒𝑖 of each year’s cash flow. Furthermore they believe that the shoulder of the

distribution (within ± one standard deviation from the mean) has more probability mass than outside of it.

This prior knowledge is quantified in the example by a beta distribution with both the right and left shape

parameters equal to 1.5.11 This beta distribution’s density function is exactly shaped like a half circle over

the range 0 to 1 with a center at 0.5. Its probability mass within one standard deviation off the mean, 0.5

±0.25, is about 61%. The probability of the beta random variable falling below 0.1 or above 0.9 is slightly

greater that 10%.12

𝑚𝑖 ~�0.8 + 0.4 𝐵𝑒𝑡𝑎(𝑥: 1.5, 1.5)�𝑒𝑖 = (0.8 + 1.0186 𝑥0.5(1 − 𝑥)0.5)𝑒𝑖, 𝑥 ∈ [0, 1] (6)

The variance of the cash flow for year 𝑖, 𝜎𝑖2, is changed over the investment horizon to explore

the impact of volatility in the cash flows on the pricing of carbon credits. The standard deviation of the

cash flow during the first year is $1,000 in all scenarios. In the first scenarios explored, the standard

11 Many types of distributions satisfy this requirement including other beta distributions with different parameters. The decision makers decide which distribution best represents the prior knowledge. 12 The methodology presented is capable of handling separate types of prior or hyper prior distribution for each annual cash flow. In the example discussed, for sake of simplicity in illustration, all prior distributions are assumed normal (or multivariate normal) and all hyper prior distributions are beta (1.5, 1.5).

15

deviation increases by 8% over the previous year. Thus, the second year’s standard deviation is $1080

and so on. By the end of the 10th year the standard deviation would almost double to $2,000. The three

other scenarios use 17%, 26% and 36% annual increases such that by the final year of the horizon the

increase in the standard deviation is about 4-, 8-, and 16-fold, respectively.

At this point let us consider Equation 1. There are four different annual cash flows related to the

additional investment in energy efficient manufacturing system. Three of them are assumed to be

deterministic and the fourth is a stochastic cash flow from the carbon credits. It may be noted that the

stochastic cash flow, described above, is based on the prior knowledge of the organization and it will

change when the organization includes experimental observations and applies the Bayesian formula. The

right hand side of the Equation 1 needs a value for the discount rate 𝑟. The organization may set this rate

equal to any rate-of-return measure it uses to make capital investment decisions for pollution control

investments.

Taking a financially conservative stance, the organization may expect the same rate of return on

the investment it makes in energy-efficient, carbon-credit-producing investment as any other

investment.13 For illustration purposes, 𝑟 is set equal to 15%. When the organization sets the required rate

of return, it indirectly sets the price for the carbon credits. The price would be such that the cash flows

earn the rate of return exactly equal to the rate of return required by the organization. In other words,

carbon credit pricing would provide an internal rate of return equal to the desired rate of return. This

requirement provides us the “experimental observations” (the sampling distribution) of the Bayes

Theorem. We require that observed net present value in Equation 1 must be equal to 0. Since the carbon

credit cash flows are probabilistic, the requirement is stated as14:

13 If the organization has a more lenient standard for investment in GHG reduction equipment and uses a lower rate of return, the proposed method can use the lower rate since only the incremental cash flows related to the investment are considered. The example in a later section illustrates how incremental cash flows are computed. 14 The proposed method is flexible enough to determine pricing at any specified net present value (other than just 0 for internal rate of return) for a given discount rate. We use the concept of internal rate of return here to simplify the description of the method. Let us say that NPV=10,000 is required then Equation 7 would be stated as: 𝑁𝑃𝑉~𝑛𝑜𝑟𝑚𝑎𝑙(10,000,𝜎02).

16

𝑁𝑃𝑉~𝑛𝑜𝑟𝑚𝑎𝑙(0,𝜎02) (7)

where 𝜎02 is a small amount of assigned variance. Now using Bayes’ Theorem, we combine the

prior knowledge the organization has about the expected cash flows from carbon credits, and the

corresponding prices, and experimental observations to obtain the posterior or revised information.

The posterior distribution is a joint conditional distribution which provides information about the

cash flows (and thus carbon credit prices) in the form of a pdf. The following section describes pdf’s of

the posterior distributions, which are to be used in a Gibbs sampler described in the next section.

3.2 Posterior joint conditional probability function

The posterior distribution 𝑓(𝑪;𝑴;𝑥|𝑁𝑃𝑉) cannot always be expressed as a closed-form, known,

mathematical function. Such a function, if available, would allow us to make inferential and predictive

statements about statistical characteristics and perform conventional tests concerning the cash flows or

prices of the carbon credit or emission permits. Additional statistical tests can be carried out to determine

confidence intervals and estimates about the carbon credit prices. Due to the lack of the closed-form

mathematical solution, we will use a Gibbs sampler to extract the needed information. A separate section,

appearing below, provides a brief description of how a Gibbs sampler works. The expressions for

posterior distributions with different prior distributions used in this paper are provided below:

The general expression for a posterior distribution for our case is:

𝑓(𝑪;𝑴; 𝑥|𝑁𝑃𝑉) ∝ 𝑓(𝑁𝑃𝑉|𝑪) ∙ 𝑓(𝑴|𝑥) ∙ 𝑓(𝑥) (8)

Expression (8) can be rewritten taking into account the actual density functions. Note that the sampling

distribution 𝑓(𝑁𝑃𝑉|𝑪) is a normal distribution with mean 0 and variance 𝜎0 2 and the hyper prior is

𝑏𝑒𝑡𝑎(1.5, 1.5) rescaled over the range [0.8𝑒𝑖 , 1.2𝑒𝑖]. We use two prior distributions, the first one is a set

of independent but not identical normal distributions, and the other one is a multivariate normal

distribution.

1) When carbon credit cash flows are independently distributed normal distributions, we get the joint

conditional posterior distribution shown in expression (9):

17

𝑓(𝑪;𝑴; 𝑥|𝑁𝑃𝑉) ∝ � 1𝜎0√2𝜋

exp (−12

(𝑁𝑃𝑉)2 )� �∏ � 1𝜎𝑖√2𝜋

exp (−12

(𝑐𝑖 − 𝑚𝑖)2 )�10𝑖=1 � �∏ (0.8 +10

𝑖=1

1.0186 𝑥0.5(1− 𝑥)0.5)𝑒𝑖� (9)

2) When the carbon credit cash flows are dependent on each other, they are distributed according to a

multivariate normal distribution, and the joint conditional posterior distribution in expression (10) is

found:

𝑓(𝑪;𝑴;𝑥|𝑁𝑃𝑉) ∝ � 1𝜎0√2𝜋

exp �−12

(𝑁𝑃𝑉)2 �� � 1�(2𝜋)10|𝚺|

exp �− 12

(𝑪 −𝑴)𝑇 𝚺−1(𝑪 −𝑴)��

�∏ (0.8 + 1.0186 𝑥0.5(1 − 𝑥)0.5)𝑒𝑖10𝑖=1 � (10)

The 𝜮 is the covariance matrix obtained from the correlations and variances of carbon credit cash flows;

exponent -1 represents the inverse of the matrix and |𝚺| is the determinant of the matrix.

3.3 The Gibbs Sampler

An inspection of the above joint conditional posterior distributions makes it clear that they do not have a

closed-form analytical expression. If the sampling and the prior distributions do not belong to a family of

conjugate distributions or if there is more than one parameter involved in a prior distribution (which

means the posterior is a joint conditional distribution), it is generally not possible to derive an analytical-

form of the marginal posterior distribution. Without such an expression for the distribution, it is not

possible to develop inferential and predictive statements about statistical properties of the posterior

distribution. The posterior distributions in closed analytic forms are available only for a few special cases

when both the prior and posterior distributions are members of the same family of distributions for a

specific likelihood or sampling distribution. Gamma distributions are conjugates when a Poisson

distribution is the sampling distribution. Prior gamma distribution’s parameters will be revised based on

the experimental results from Poisson distribution. The revised parameters give us the posterior

distribution which is also gamma. Such conjugate distributions are few and generally not available most

statistical models.

18

The Bayesian framework offers powerful statistical tools, which are useful only if the posterior

distribution turns out to be one of the standard distributions whose density function is a known closed

form mathematical expression. Furthermore, the posterior distribution is a conditional distribution. If

there is more than one prior parameter involved then the resulting posterior distribution is a joint

conditional distribution, which complicates the analyses of the revised information from the posterior

distribution. Due to these limitations, Bayesian analysis could not traditionally be applied to complex

statistical models. However with recent advances in application of the Gibbs sampler (a special case of

Markov Chain Monte Carlo technique), a marginal posterior distribution for each prior random variable

of interest for complex statistical models can now be obtained.

Geman and Geman (1984) describe the Gibbs sampler technique. Gelafand and Smith (1990)

opened the possibility of applying Gibbs sampler technique to many complex Bayesian models. The

Gibbs sampler obtains the marginal conditional (also called the full conditional) distribution for each prior

or hyper prior variable without actually obtaining the joint posterior probability density functions. In the

illustrative example, there are twenty prior, (𝑪,𝑴) , and one hyper prior (𝑥) random variables. The Gibbs

sampler generates data points that have a steady-state, ergodic Markov Chain property. Successive values

of a random variable generated by the Gibbs sampler depend only on the random variable’s immediate

prior value. The distribution of the values generated by the sampler duplicates the intended distribution

with a sufficient sample size (Congdon, 2007).

4.0 An Illustrative Example

We now present an example to describe how the method developed in this paper can estimate the

pricing of carbon credits. We borrow an example developed by Dhavale and Sarkis (2014), the relevant

segments of which are described below.

The example describes a hypothetical manufacturing facility. The data from real world factory

was unobtainable due to concerns about the confidentiality, and proprietary nature of the processes used.

19

Additionally, no one site or one company could provide all details necessary. So the needed data were

gathered from publically available and government sources.

Since the hypothetical organization is interested in pricing the carbon credits, the illustrative

example described below considers only the incremental cash flows associated with the installation and

operation of special equipment and ancillaries related to improving energy efficiency of a manufacturing

system. It does not consider the total cash flows needed to install and operate the entire system.

As a result of energy efficiency, two cash inflows are generated. One set is from the decreased

use of electrical power or other energy source, and the other from carbon credits that the company

generates due to reduced GHG emissions. The organization allows the subunit making the investment in

energy efficient equipment to sell the credits in the internal market to other subunits, which may need

them to meet internal company goals regarding GHG emissions. In this way, the company can meet its

environmental obligations without requiring each subunit to reduce the emissions by proportionately

equal amounts. The cost of reaching the environmental goals is distributed to all subunits by an internal

market mechanism rather than an allocations scheme.

It is assumed that the initial investment in the manufacturing facility is $40 million. Five million

out of which is the incremental cost of ancillary equipment needed to increase the energy efficiency. This

estimate is based on results of the study by U. S. Department of Energy (U.S. Department of Energy,

Industrial Technologies Program, 2011). According to the study, the cost of efficient equipment is

estimated as an additional 10 to 20% of the cost the whole system; in the illustrative example, a

conservative value of about 12% is chosen. Thus, the incremental cost to acquire the energy efficient

equipment will be about $5 million. The study estimates a savings of about 30% in energy use. A

manufacturing facility has a variety of machine tools, it is estimated that manufacturing equipment in the

facility costs on an average $4,000 for each kilowatt (kW) of installed power capacity. The estimated

power usage of the manufacturing facility then would be $40,000,000 / ($4,000/kW) = 10,000 kW. If this

facility operates 2,000 hours a year, it would consume 20 million Kilowatt hours (kWh) of energy (ibid.).

The incremental investment of $5 million in the energy efficient equipment results in annual energy

20

savings of 30% of 20 million kWh, or 6 million kWh. The average commercial rate in the U.S. for

electrical energy is about $0.10 per kWh in 2012 (U. S. Energy Information Administration, Electricity,

2013). Thus dollar savings in the electric bill of the manufacturing facility amounts to $600,000 per year.

Historical data indicates that electrical energy costs have been increasing at a rate of about 2% a year

(ibid.), thus the savings will increase at the same rate. The savings for year 2 will be $600,000 x 1.02 =

$612,000 and so on. The example assumes the useful life is 10 years and considers the cash flows taking

place during that time period.

The U. S. tax code allows full depreciation of investment within 6 to 8 years for pollution control

equipment. We use 7 years in our illustrative example and use the prescribed 150% declining balance

method and a statutory federal tax rate of 35%. The tax benefits generated in the first seven years for a $5

million incremental investment are: $382,900; $341,775; $256,375; $246,050; $246,050; $246,050;

$31,150. No tax benefits are allowed for the remaining three years.

The maintenance and repair costs are assumed to be 1% of the equipment cost in the first year and

increases at a rate of 2% of the prior year’s cost. Thus, the cost during the first year is $50,000 and for the

second year it is $51,000, etc.

The cash flows described above are deterministic in nature. The stochastic models for cash flows

resulting from carbon credits were described earlier in this paper. Additional details about the cash flows

are provided below.

It is estimated that one kWh of electrical energy production and transmission results in 0.4939

kilograms of CO2-equivalent-emitted GHG (Department of Environment Food and Rural Affairs,

Government of U. K., 2012). As described earlier, the energy savings due to installation of sustainable

and efficient manufacturing facility is 6 million kWh. This efficient facility due to reduction in use of

energy would reduce the emission of greenhouse gases by 6,000,000 X 0.4939, or 2,963,400 kilograms

CO2 equivalent. These are the carbon credits the subunit can sell in the internal market. The carbon

credits are denoted in metric tons. In our example, the carbon credits amounts to 2,963.4 tonnes (TCO2e ).

21

We estimate that the average value of carbon credits in various international markets (e.g.

Australia, China, California, and the EU) is about $7 to $12 per unit; the illustrative example uses $10.

This is the value used in year 1. It is expected that as effects of climate change become more pronounced,

the political pressure on regulators will increase resulting in stringent regulations and much tighter GHG

emission targets and caps. To account for this tightening, the price of a unit of carbon credit is assumed to

increase by 10% each year. Thus the cash flow will be (2,963.4 x $10=) $29,634 for the first year,

$32,597 the second year and so forth for the ten-year investment horizon; the cash flow for the 10th year is

$69,875. This is prior information and it is incorporated in the model as expected cash flows from carbon

credits, 𝑒𝑖′𝑠.

To account for increased uncertainty associated with the carbon credit prices (thus cash flow)

taking place farther into the future, the standard deviation (SD) of the cash flow is increased over the

years, starting with SD equal to $1,000 in year one. The SD is increased in by 8, 17, 26, and 36 percent

each year to study the impact of variability on the prices.

For the multivariate distribution, we assume the organization believes that less correlation exists

as the temporal distance between 𝑐𝑖′𝑠 increases. This is part of the prior knowledge available to the

Bayesian framework. For example, 𝑐4 is highly correlated to the adjacent cash flows 𝑐3 and 𝑐5; a

correlation of 0.9 is assumed. The correlations are assumed to be less as the time distance gets longer. In

this illustration, a decrease in correlation of 0.1 for a distance of one year is assumed. Thus the correlation

between 𝑐4 and 𝑐1 is 0.7 and that of 𝑐4 with 𝑐10 is 0.4. The highest correlation assumed is 0.9 and the

lowest is 0.1. A covariance matrix is determined from the correlation matrix.

4.1 Setup of Simulation Experiment

We use the Gibbs sampler to obtain the posterior distribution of carbon credit prices for and IETS

market for each of the ten years. Data developed by the sampler provides detailed information about the

distribution including the graph of its pdf. To save space only a few characteristics of interest are shown

22

in the following tables. It is clear from earlier discussion that the posterior distributions do not belong to

any known family of distributions. The information available from the distribution of prices will help the

organization with task of setting the prices of internal trading of carbon credits.

We consider the two distributions as prior distributions. In one case, the prices for each of the ten

years have an independent normal distribution with increasing mean and standard deviation. In the other

case, the prior is assumed to be multivariate normal and the cash flows for each year are correlated with

other years. The two parameters, mean and variance (or covariance in multivariate case), of these

distributions allow us to independently specify those values. Volatility or uncertainty is ascribed to carbon

credit prices through its variance.

5.0 Results and Discussion

Results of the Gibbs sampler, under different test conditions, are provided in Tables 1 to 4. In each

experiment, a given distribution and a given SD, was run to obtain 2,000,000 realizations. To attain a

steady state and to eliminate transient readings, the first 4,000 realizations were discarded. After that, to

eliminate autocorrelation, every 40th realization was selected for inclusion in the sample. As a result of

these precautions, there were 49,900 usable data points for each experiment.

Tables 1 through 4 about here

To obtain a reference value, we computed the carbon credit price per TCO2e for the illustrative

example assuming carbon credit cash flows that were “almost deterministic”. It was assumed that the

carbon credit cash flows were equal to the expected values mentioned earlier (year 1 cash flow equal to

$29,634, year 2: $32,597 and so on); and the standard deviation was reduced to 0.001.15 We then ran the

Gibbs sampler on this configuration and results are shown in the second column of all tables. These

15 The standard deviation is reduced to the negligible amount making the pdf almost a vertical line and the random variable almost a constant.

23

posterior prices match the prior prices to two decimal places.16 These would be the prices the subunit

selling the carbon credit would charge if there was not any uncertainty or variation about the prices. These

prices are the same for normal and multivariate normal distribution when rounded up to two decimal

places.

In Tables 1 through 4 show the prices under different rates of increase in standard deviation and

different required rates of return. The first year’s standard deviation in all scenarios is $1,000. It increases

by 8% per year over the previous year in the third segment of the tables so that by 10th year the standard

deviation of prior cash flow distribution is almost doubled to about $2,000. The increases in standard

deviation reflect the uncertainty about the future direction of the prices over the horizon. This uncertainty

in prices is the result of several factors. Prices of the carbon credits are affected by supply and demand.

The regulatory authorities can control the supply to a certain extent by changing the regulation governing

generation, use, banking, and sale of carbon credits in an open market. A longer horizon means that there

is an inherent uncertainty associated with prices farther into future because of greater lack of credible

information about future events as compared to events in the near future. We consider 3 additional

scenarios where the SD of expected cash flows increases by 17%, 26% and 36%, which result in

approximate increases in SD of 4, 8 and 16 fold by the 10th year.

It may be noted that Table 1 and 2 assume a rate of return on the investment of 15%. Tables 3 and

4 are similar to Tables 1 and 2 except the rate of return is changed to 12% for comparison. Table 1

assumes that the prior distribution of carbon credit prices is normal with mean whose range is determined

by the beta hyper prior and SD that is changed according the scheme described in the above paragraph.

One may note that in Table 1 under all scenarios the mean prices of carbon credits in the internal market

are expected to increase over the ten-year period.

It is observed that as the rate of standard deviation increase gets larger, the price means during the

initial years are lower and later years higher. In other words, with the larger standard deviations, resulting 16 The values are different if more than two decimal places are taken into account. Revision of the prior prices is minimal because of the small standard deviation of the prior distribution. This indirectly confirms that our statistical modeling is correct.

24

in higher uncertainty, the range of change in the mean prices increases. For example, for the 8% increase

in SD the range of mean price change is $21.90 ($75.39 - $53.49), when the SD increase is 36% a year

the range is $267.04 ($282.55 - $15.51). Higher ranges in price increase will require increased diligence

in proper planning for emission control expenditures, and acquisition or generation of carbon credits by

the subunit managers.

Another observation from these results is that the distribution of prices is tightly clustered around

the mean in the early years irrespective of SD variability. For example, with the 8% SD increase in year 1,

the middle 95% spread of the distribution is bound by $52.84 (2.5 percentile point) to $54.16 (97.5

percentile point) or a spread of $0.67. This type of assurance that the price fluctuation is very limited in

the early years is very helpful to managers since it removes the risk of fluctuating carbon prices, which

offers a significant advantage in planning for GHG related aspects. The middle 95% range is also

relatively small for 36% increase in SD; it is only $2.89. However, this middle 95% range increases over

investment horizon for each year in all scenarios in Table 1 (as well as in the other tables) until in the year

10 it becomes $177.09 for 36% SD, adding significant challenge and difficulty to the planning and

decision making.17

Table 2 is similar to Table 1 except the prior distribution for the cash flows is multivariate normal

with correlated cash flows. A similar observation and conclusion can be drawn about the price behavior

from Table 2 as were obtained for Table 1. It is possible to draw more specific conclusions about the

pricing of the credits when they are independently distributed as opposed to when they are correlated.

We caution that the amounts and characteristics mentioned in the above discussion are specific to

data in our illustrative example; however, the trends observed are generalizable. Our purpose in providing

the illustrative example is to explain the methodology developed in this paper to obtain prices for internal

markets under the Bayesian framework.

17 Instead of the middle 95% range of the distribution, one may also compare the SD’s (shown in the tables) of the distributions and arrive at the same conclusions. From a practical perspective, however, the middle ranges are easier to explain to a practicing manager than the concept of SD.

25

Tables 3 and 4 are developed for a 12% rate of return. Similar general conclusions can be drawn

from Tables 3 and 4 as for Table 1. It is interesting to compare the pricing for the same scenarios under

the two required rates of return. Let us focus on multivariate normal prior and SD increase of 26%

(Tables 2 and 4). Focusing on the mean price column, it may be observed that with a 12% return the mean

prices are lower for each of the ten years compared to the 15% return. For example, the 12% mean price

for year 1 is $13.48 per carbon credit (per TCO2e) compared to 15% mean price, which is $23.22. The

10th year’s respective mean prices are $50.38 and $146.93. The reason for a higher price for a higher

return rate is quite straightforward and expected. To generate higher returns, cash inflows have to be

higher. To obtain higher cash flows, higher prices have to be charged for the carbon credits.

The organization will have to decide what price should prevail in the internal market for each of

the next ten years. If the rate or return required is 15%, and let us say the prior belief supports a normal

distribution for the cash flows with SD increasing at 8% a year, then in the first year one carbon credit

should be priced at $53.49, the second year $55.27 and so on, for the tenth year the price is $75.39. These

are the means of the posterior distributions of the prices for each of those years. Pricing the carbon credits

at the mean values will give a rate of return of 15% on the incremental investment.

With this information, the organization may decide to select a price lower than the mean price to

accelerate the adoption of energy efficient systems. Obviously, such a decision would imply that it is

ready to accept a return lower than 15% on pollution control equipment for the social good and/or to

comply with the regulatory requirements.

In this example, we focused on only one GHG reducing investment. The organization may have

several such investments. To compute the carbon credit price for the internal market, all these investments

should be considered together and all cash flows combined. The method developed in this paper then can

be applied to the combined total investments and cash flows. It is clear that there will be one carbon credit

price in the internal market for a given year for all investments generating the credits.

26

5.1 Implications of the Results and Limitations

The results indicate that it is important for an organization running its own IETS for carbon

credits to pay particular attention to the required rate of return and uncertainty of future cash flows. It

appears that higher uncertainty (as measured by SD) invites greater increases in future prices, and

furthermore those prices fluctuate over larger and larger ranges over time. Unreasonably low rates of

return, according to our model, means lower prices for carbon credits, which takes away the incentives to

invest in GHG reducing technology, resulting in the organization’s failure to comply with environmental

regulation. Alternatively, unreasonably high prices may induce subunits to invest in the technologies

whether they make economic sense from an organization-wide perspective or not. The organization, in

this situation, will be in compliance with environmental regulation but quite possibly with a great deal of

excessive and unnecessary cost.

Given the sensitivities at different time periods, organizations and managers would need to

determine how much uncertainty they are willing to take when managing emissions reduction projects

and determining investments. The level of uncertainties and knowledge will tend to increase over time.

This new knowledge and expectations can be integrated into the model where the distributions can be

reevaluated and the pricing system is refined. The special characteristics of the projects under evaluation,

e.g., longer term, heavy capital assets investments, legal requirement, etc., should be considered and

whether the current pricing schemes using this technique is appropriate should be evaluated.

The use of this technique may provide insights as to whether managers have been using the

appropriate pricing scheme for the internal market if one is already in existence. If the existing prices fall

within the ranges given by this model and given the assumptions and parameters are acceptable, then they

are within acceptable return requirements. If the prices are outside these ranges, consideration needs to be

given to resetting the prices. Information related to the types of investments, cash flows, and prices

accumulated from this model over time can be utilized as “prior knowledge” for the next generation of the

models.

27

In addition to previous data and monitoring aspects, future predictions can be made. Managers

using this model can help identify potential, emergent issues related to price fluctuations. Given that

mean prices can increase greatly, integrating these expected prices into other decision models (e.g.

optimization models) may be another direction to help managers make decisions. Our paper provides an

initial, and important attempt at developing a much-needed pricing mechanism for internal markets. More

and diverse techniques are needed so organizations have a set of analytical tools to obtain guidance in

determining the pricing for internal markets.

Limitations of the technique include need to develop pdf’s or frequency distributions

encompassing prior beliefs. It may not be always possible to quantify fully all prior information about the

price changes and variations in them as a pdf. In addition to the risk that is inherent in assumptions

dealing with events in the longer terms, carbon credit markets face a strong external and unpredictable

influence in the form of diktats of regulatory bodies, which are difficult to predict and incorporate as risk

in the prior pdf’s. Gibbs sampler and the underlying statistical theory are not as well-known as some other

statistical techniques. The method uses rate of return in determining prices. Other metrics may be

appropriate depending on the specific circumstances faced by an organization. Even when the rate of

return is appropriate whether to use the hurdle rate for general capital budgeting decisions for GHG

reduction technologies is a decision that would have to be made.

6.0 Summary, Conclusion, and Future Research

In this paper, we devise a methodologically sound technique to estimate the price for carbon

credits in an IETS run by an organization for its subunits. Increasingly organizations view IETS markets

as a fair, equitable and economically optimal way of allocating funds available for GHG reduction

amongst its subunits. These IETS have been applied in a broad variety of industries. IETS lack market

forces (arms-length transactions, conflicting economic interests) to create the pricing mechanism found in

the external markets. In this paper, we propose such a pricing mechanism for internal markets.

28

The carbon credit price determined by our model is based on the values of parameters defined by

the organization. The price is implicit in cash flows, their uncertain behavior and the rate of return. In the

model developed in this paper, a technique is introduced that incorporates well-known methods such as

Bayes’ Theorem, discounted cash flows and internal rate of return to come up with pricing for carbon

credits.

The existing external markets have shown a great deal of variability in prices. It is very likely that

future prices of the credits will rise as more stringent cap and trade policies are put into force to address

issues of climate change. This will put an upward pressure on prices in internal markets as organizations

face stricter emission standards. Given this background, an organization must carefully price the carbon

credits so that subunits investing in GHG reducing technology have incentive to do so at the same time

avoid the risk of creating moral hazard.

The results of the paper indicate that higher prices will provide a higher rate of return on the GHG

reducing technology for the subunits that invest in it. Higher expected uncertainty in the future requires

higher prices. This may indicate that higher risks of uncertainties are rewarded by higher returns. In

addition, fluctuation or variability in the prices will increases over the life of carbon-credits-generating

asset, adding an additional element of risk in setting the prices. The price spread is much narrower in the

distribution of prices in the first year compared to the last. In a nutshell, based on what the organization

currently expects or knows about the dynamics of its internal market, we can, using the model developed

in this paper, obtain a probability density function for internal price for carbon credits. This density

function will have smaller spreads in early years and higher spreads in later years, making price setting

for later years more difficult. More uncertain cash flows and higher return on investment both will require

higher prices. The ranges and distributions that are generated from these data can be integrated across the

organization such that limits and trends can be identified by managers for planning and setting of prices

and/or provide incentives to act in certain ways in response to natural environmental pressures.

As far as we can tell, this is the first paper that develops a model for pricing for internal emission

markets. We rely on time-tested accounting and economic techniques to develop our pricing method. The

29

pricing could be based on a metric other than the required rate of return. Based on the goals of the

organization for GHG emission reduction, pricing mechanisms could be developed that uses

mathematical programming, game theory, or economic information theory. Very little information is

available about how organizations operate the internal markets or decide the prices, since all of it is still in

its infancy. Empirical research in those areas would add to the knowledge about practical and operating

concerns and constraints faced by those organizations. Revenue to one subunit in an internal market

transaction is an expense to other subunit; whether inclusion of these items on the segment performance

report of the respective subunit would provide a better measure of their overall performance needs to be

examined.

30

References

Ackerman, F., DeCanio, S. J., Howarth, R. B., & Sheeran, K. (2009). Limitations of integrated

assessment models of climate change. Climatic change, 95(3-4), 297-315.

Akhurst, M., Morgheim, J. and Lewis, R. (2003), ‘‘Greenhouse gas emissions trading in BP’’,

Energy Policy, Vol. 31 No. 7, pp. 657-663.

Böhringer, C., & Löschel, A. (2003). Market power and hot air in international emissions trading: the

impacts of US withdrawal from the Kyoto Protocol. Applied Economics, 35(6), 651-663.

Boulatoff, C., Boyer, C., & Ciccone, S. J. (2013). Voluntary Environmental Regulation and Firm Performance: The Chicago Climate Exchange. The Journal of Alternative Investments, 15(3), 114-122.

Calel, R. (2013). Carbon markets: a historical overview. Wiley Interdisciplinary Reviews: Climate Change, 4(2), 107-119.

Congdon, P. 2007. Bayesian Statistical Modelling second edition. John Wiley and Sons, 2007.

Convery, F. J., & Redmond, L. (2007). Market and price developments in the European Union emissions

trading scheme. Review of Environmental Economics and Policy, 1(1), 88-111.

Department of Environment Food and Rural Affairs, Government of U. K. (2012) Guidelines to Defra /

DECC‟s GHG Conversion Factors for Company Reporting (Table 5, page 17)

https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/69568/pb13792-emission-

factor-methodology-paper-120706.pdf

Dhavale, D. and J. Sarkis. (2014). Illustrative example of a sustainable and energy efficient

manufacturing system. Working Paper, Worcester Polytechnic Institute, Worcester, MA.

Dietz, S., & Fankhauser, S. (2010). Environmental prices, uncertainty, and learning. Oxford Review of Economic Policy, 26(2), 270-284.

Economist (2013), Carbon copy; Companies and emissions. The Economist. 409.8866 (Dec 14, 2013):

70 (Accessed on-line http://www.economist.com/news/business/21591601-some-firms-are-preparing-

carbon-price-would-make-big-difference-carbon-copy: August, 2014).

Engel, K. H. (2005). Mitigating global climate change in the United States: a regional approach. NYU

Envtl. LJ, 14, 54.

Gelfand A. & Smith AFM (1990). Sampling-based approaches to calculating marginal densities. J

American Statistical Association 85(410), 398–409.

31

Geman S, Geman D (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of

images. IEEE Transactions on Pattern Analysis and Machine Intelligence 6, 721–741.

Hepburn, C. (2010). Environmental policy, government, and the market. Oxford review of economic

policy, 26(2), 117-136.

Hörisch, J. (2013). Combating climate change through organisational innovation: an empirical analysis of

internal emission trading schemes. Corporate Governance, 13(5), 569-582.

Howard, P. H. (2014). Omitted Damages: What’s Missing From the Social Cost of Carbon. The Cost of

Carbon Pollution Project. Available at http://costofcarbon.org/reports/entry/omitted-damages-whats-

missing-from-the-social-cost-of-carbon. (Last Accessed, August, 2014).

Kassinis, G., & Vafeas, N. (2006). Stakeholder pressures and environmental performance. Academy of

Management Journal, 49(1), 145-159.

Keohane, N. O. (2009). Cap and trade, rehabilitated: Using tradable permits to control US greenhouse gases. Review of Environmental Economics and policy, ren021.

Krause, R. M. (2011). Policy innovation, intergovernmental relations, and the adoption of climate

protection initiatives by US cities. Journal of Urban Affairs, 33(1), 45-60.

Lee, S. Y. (2013). Existing and anticipated technology strategies for reducing greenhouse gas emissions

in Korea’s petrochemical and steel industries. Journal of Cleaner Production, 40, 83-92.

London, J., Karner, A., Sze, J., Rowan, D., Gambirazzio, G., & Niemeier, D. (2013). Racing climate

change: Collaboration and conflict in California's global climate change policy arena. Global

Environmental Change, 23(4), 791-799.

Melville, N. P., & Whisnant, R. (2014). Energy and Carbon Management Systems. Journal of Industrial

Ecology. Forthcoming. DOI: 10.1111/jiec.12135.

National Center for Environmental Economics , Office of Policy, U.S. Environmental Protection Agency

(2010) Guidelines for preparing economic Analyses

http://yosemite.epa.gov/ee/epa/eed.nsf/webpages/Guidelines.html

Paltsev, S., Reilly, J. M., Jacoby, H. D., Gurgel, A. C., Metcalf, G. E., Sokolov, A. P., & Holak, J. F.

(2008). Assessment of US GHG cap-and-trade proposals. Climate Policy, 8(4), 395-420.

32

Peters, G. P. (2010). Carbon footprints and embodied carbon at multiple scales. Current Opinion in

Environmental Sustainability, 2(4), 245-250.

Pinkse, J., & Kolk, A. (2007). Multinational Corporations and Emissions Trading:: Strategic Responses to New Institutional Constraints. European Management Journal, 25(6), 441-452.

Ranson, M., & Stavins, R. (2014). Linkage of Greenhouse Gas Emissions Trading Systems: Learning

from Experience (No. w19824). National Bureau of Economic Research.

Sarkis, J., Gonzalez-Torre, P., & Adenso-Diaz, B. (2010). Stakeholder pressure and the adoption of

environmental practices: The mediating effect of training. Journal of Operations Management, 28(2), 163-

176.

Sarkis, J., & Tamarkin, M. (2005). Real options analysis for “green trading”: the case of greenhouse

gases. The Engineering Economist, 50(3), 273-294.

Speck, S. (2013). Carbon taxation: two decades of experience and future prospects. Carbon Management,

4(2), 171-183.

U.S. Department of Energy, Industrial Technologies Program, 2011) Energy intensive processes portfolio:

addressing key energy challenges across U.S. industry

https://www1.eere.energy.gov/manufacturing/intensiveprocesses/pdfs/eip_report.pdf

U. S. Energy Information Administration, Electricity (2013) Table 1.2. Summary Statistics for the United

States, 2002 – 2012 http://www.eia.gov/electricity/annual/html/epa_01_02.html

U.S. Government, (2013), Technical Update of the Social Cost of Carbon for Regulatory Impact Analysis

Under Executive Order 12866. Interagency Working Group on Social Cost of Carbon, United States

Government, Whitehouse, Washington, DC. (Accessed on-line August 2013:

http://www.whitehouse.gov/sites/default/files/omb/assets/inforeg/technical-update-social-cost-of-carbon-

for-regulator-impact-analysis.pdf).

Vincent, D. P. (2013). Internalizing Externalities: An Economic and Legal Analysis of an International

Carbon Tax Regime. Oregon Law Review, 92(1).

Weitzman, M. L. (2013). Can Harmonized National Carbon Taxes Internalize the Global Warming

Externality? (No. w19637). National Bureau of Economic Research.

Xue, B., Mitchell, B., Geng, Y., Ren, W., Müller, K., Ma, Z., Oliveira, J., Fujita, T., & Tobias, M. (2014).

A review on China's pollutant emissions reduction assessment. Ecological Indicators, 38, 272-278.

33

Yang, T. (2005). Problem of Maintaining Emission Caps in Carbon Trading Programs without Federal

Government Involvement: A Brief Examination of the Chicago Climate Exchange and the Northeast

Regional Greenhouse Gas Initiative, The. Fordham Envtl. L. Rev., 17, 271.

34

Table 1: Posterior Distribution of Prices, Rate of Return 15%

(Normal Prior, Beta Hyper Prior Distributions)

*Prices in dollars per tonne of CO2 equivalent

Year

Almost

Deterministic

Prior Distribution

SD Rate of Increase 8%

Prior Distribution

SD Rate of Increase 17%

Prior Distribution

SD Rate of Increase 26%

Prior Distribution

SD Rate of Increase 36%

Mean price*

$

Mean price*

$

SD of

Price

2.5 %

point

97.5 %

Point

Mean price*

$

SD of

Price

2.5 %

point

97.5 %

Point

Mean price*

$

SD of

Price

2.5 %

point

97.5 %

Point

Mean price*

$

SD of

Price

2.5 %

point

97.5 %

Point 1 10.00 53.49 1.09 52.84 54.16 33.36 1.18 32.74 34.08 21.59 1.02 20.83 22.37 15.51 0.89 13.42 16.31 2 11.00 55.27 0.87 54.57 55.98 38.71 1.69 37.90 39.48 26.54 2.42 25.65 27.51 19.03 3.18 16.01 19.99 3 12.10 57.06 1.06 56.29 57.81 44.91 2.52 43.94 45.79 33.06 4.26 31.76 34.12 24.10 6.26 19.80 25.29 4 13.31 59.32 1.15 58.48 60.13 52.20 2.90 51.02 53.18 41.61 5.32 39.79 42.92 31.49 8.42 26.10 33.32 5 14.64 61.48 1.06 60.57 62.39 60.67 2.84 59.26 61.89 52.88 5.78 50.69 54.63 42.55 10.43 35.53 45.12 6 16.11 63.88 0.97 62.87 64.86 70.59 2.82 68.97 72.05 68.10 6.40 65.43 70.36 59.46 12.63 49.13 64.59 7 17.72 66.48 1.00 65.47 67.52 82.27 3.12 80.52 83.96 88.48 7.17 84.77 91.45 85.14 14.85 69.21 92.56 8 19.49 69.21 1.21 68.06 70.32 95.87 4.02 93.78 98.00 115.75 8.58 111.80 120.03 125.40 17.24 105.52 152.16 9 21.43 72.21 1.44 70.97 73.50 111.83 5.08 109.64 114.36 152.70 11.69 147.74 159.01 187.42 22.95 171.36 224.94

10 23.58 75.39 1.67 74.10 76.77 130.76 6.70 128.23 133.87 202.67 17.68 195.72 210.06 282.55 39.82 123.34 300.43

35

Table 2: Posterior Distribution of Prices, Rate of Return 15%

(Multivariate Normal Prior, Beta Hyper Prior Distribution)

*Prices in dollars per tonne of CO2 equivalent

Year Almost Deterministic

Prior Distribution SD Rate of Increase 8%

Prior Distribution SD Rate of Increase 17%

Prior Distribution SD Rate of Increase 26%

Prior Distribution SD Rate of Increase 36%

Mean price*

$

Mean price*

$

SD of

Price

2.5 %

Point

97.5 %

Point

Mean price*

$

SD of

Price

2.5 %

point

97.5 %

Point

Mean price*

$

SD of

Price

2.5 %

point

97.5 %

Point

Mean price*

$

SD of

Price

2.5 %

point

97.5 %

Point 1 10.00 41.27 0.23 40.80 41.71 30.56 0.26 30.05 31.06 23.22 0.29 22.65 23.78 18.11 0.36 17.34 18.12 2 11.00 48.69 0.21 48.29 49.10 38.23 0.26 37.69 38.74 29.96 0.33 29.32 30.60 23.46 0.43 22.56 23.48 3 12.10 55.65 0.19 55.27 56.02 46.77 0.26 46.26 47.31 38.40 0.36 37.69 39.11 30.84 0.52 29.78 30.86 4 13.31 62.02 0.17 61.69 62.36 56.15 0.26 55.65 56.66 48.86 0.39 48.09 49.61 40.93 0.60 39.68 40.97 5 14.64 67.59 0.18 67.22 67.93 66.14 0.27 65.63 66.65 61.48 0.41 60.64 62.26 54.53 0.69 53.11 54.53 6 16.11 72.25 0.22 71.81 72.65 76.47 0.30 75.86 77.04 76.36 0.46 75.45 77.24 72.48 0.78 70.90 72.52 7 17.72 75.72 0.28 75.18 76.26 86.72 0.39 85.95 87.50 93.37 0.56 92.26 94.45 95.60 0.92 93.74 95.63 8 19.49 77.82 0.36 77.14 78.52 96.17 0.54 95.13 97.22 111.86 0.78 110.35 113.38 124.28 1.22 121.82 124.28 9 21.43 78.25 0.44 77.41 79.13 103.93 0.74 102.48 105.39 130.56 1.18 128.23 132.89 157.72 1.90 153.98 157.76

10 23.58 76.74 0.54 75.69 77.82 108.63 1.01 106.63 110.58 146.93 1.79 143.45 150.44 192.92 3.18 186.68 192.95

36

Table 3: Posterior Distribution of Prices, Rate of Return 12%

(Normal Prior, Beta Hyper Prior Distributions)

*Prices in dollars per tonne of CO2 equivalent

Year

Almost

Deterministic

Prior Distribution

SD Rate of Increase 8%

Prior Distribution

SD Rate of Increase 17%

Prior Distribution

SD Rate of Increase 26%

Prior Distribution

SD Rate of Increase 36%

Mean price*

$

Mean price*

$

SD of

Price

2.5 %

point

97.5 %

Point

Mean price*

$

SD of

Price

2.5 %

point

97.5 %

Point

Mean price*

$

SD of

Price

2.5 %

point

97.5 %

Point

Mean price*

$

SD of

Price

2.5 %

point

97.5 %

Point 1 10.00 18.87 0.44 18.22 19.51 15.29 0.43 14.60 16.01 13.30 0.44 12.48 13.99 12.14 0.73 9.68 13.03 2 11.00 20.37 0.47 19.67 21.06 17.22 0.65 16.40 18.04 15.08 0.87 14.04 15.93 13.70 1.18 12.05 14.71 3 12.10 21.97 0.56 21.23 22.72 19.48 0.92 18.52 20.40 17.32 1.34 15.99 18.39 15.58 1.87 13.09 17.00 4 13.31 23.75 0.60 22.91 24.58 22.04 1.04 20.92 23.16 19.99 1.62 18.34 21.41 18.09 2.35 15.43 19.97 5 14.64 25.66 0.54 24.82 26.51 25.02 1.05 23.73 26.27 23.31 1.80 21.31 25.28 21.41 2.92 17.07 24.03 6 16.11 27.72 0.58 26.77 28.66 28.36 1.12 26.85 29.89 27.54 1.99 25.22 29.72 25.57 3.52 20.81 30.56 7 17.72 30.02 0.63 29.01 31.06 32.30 1.26 30.60 33.95 32.88 2.38 29.89 36.11 32.11 4.32 24.81 42.92 8 19.49 32.48 0.80 31.35 33.61 36.92 1.57 34.82 38.84 39.75 2.82 36.17 43.43 40.90 4.79 30.27 54.46 9 21.43 35.26 0.78 34.05 36.48 42.22 1.89 39.65 44.58 48.56 3.38 44.04 53.05 53.99 6.48 41.47 70.73

10 23.58 38.17 0.00 23.58 23.58 48.32 2.30 45.66 51.19 60.81 4.97 55.41 66.85 75.05 9.93 35.47 88.51

37

Table 4: Posterior Distribution of Prices, Rate of Return 12%

(Multivariate Normal Prior, Beta Hyper Prior Distribution)

*Prices in dollars per tonne of CO2 equivalent

Year Almost Deterministic

Prior Distribution SD Rate of Increase 8%

Prior Distribution SD Rate of Increase 17%

Prior Distribution SD Rate of Increase 26%

Prior Distribution SD Rate of Increase 36%

Mean price*

$

Mean price*

$

SD of

Price

2.5 %

Point

97.5 %

Point

Mean price*

$

SD of

Price

2.5 %

point

97.5 %

Point

Mean price*

$

SD of

Price

2.5 %

point

97.5 %

Point

Mean price*

$

SD of

Price

2.5 %

point

97.5 %

Point 1 10.00 17.12 0.27 16.59 17.64 15.04 0.36 14.26 15.68 13.48 0.56 12.16 14.36 12.11 0.85 10.11 13.38 2 11.00 19.48 0.26 18.95 19.97 17.41 0.38 16.57 18.09 15.67 0.62 14.22 16.65 14.03 0.94 11.84 15.50 3 12.10 21.86 0.25 21.34 22.32 20.06 0.40 19.16 20.76 18.29 0.68 16.69 19.38 16.43 1.06 14.02 18.11 4 13.31 24.27 0.25 23.74 24.72 22.98 0.43 22.01 23.71 21.40 0.76 19.60 22.61 19.52 1.18 16.82 21.43 5 14.64 26.66 0.26 26.09 27.12 26.16 0.46 25.13 26.94 25.07 0.83 23.12 26.40 23.43 1.31 20.45 25.59 6 16.11 29.01 0.30 28.37 29.54 29.55 0.51 28.39 30.41 29.33 0.91 27.18 30.81 28.38 1.45 25.17 30.83 7 17.72 31.30 0.35 30.54 31.94 33.10 0.59 31.76 34.12 34.18 1.04 31.76 35.90 34.55 1.62 31.02 37.36 8 19.49 33.48 0.43 32.57 34.25 36.68 0.73 35.09 37.96 39.52 1.23 36.75 41.64 42.08 1.92 38.00 45.56 9 21.43 35.50 0.52 34.42 36.48 40.09 0.91 38.13 41.74 45.05 1.58 41.71 47.95 50.85 2.49 45.72 55.51

10 23.58 37.32 0.62 36.04 38.50 43.13 1.18 40.66 45.32 50.38 2.11 46.06 54.36 60.20 3.61 53.05 67.22