NBSIR 77-1199 /

The Effect of "Resource

Impact Factors" on Energy

Conservation Standards for

Buildings

Stephen F. Weber

Building Economics Section

Center for Building Technology

Institute for Applied Technology

National Bureau of Standards

Washington, D. C. 20234

Sponsored by:

The Office of Conservation and Environment

Federal Energy Administration

February 1977

Final Report

NBSIR 77-1199

THE EFFECT OF "RESOURCE

IMPACT FACTORS'" ON ENERGY

CONSERVATION STANDARDS FOR

BUILDINGS

Stephen F. Weber

Building Economics Section

Center for Building Technology

Institute for Applied Technology

National Bureau of Standards

Washington, D. C. 20234

Sponsored by:

The Office of Conservation and Environment

Federal Energy Administration

February 1977

Final Report

U.S. DEPARTMENT OF COMMERCE. Juanita M. Kreps, Secretary

Dr. Betsy Ancker-Johnson. Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

o

PREFACE

This study was funded by the Federal Energy Administration (FEA)

as part of a program being carried out by the National Bureau of Standards

to develop energy conservation performance standards for the design of

buildings. The legislative mandate for this program is in PL 94-385,

Title III, "Energy Conservation Standards for New Buildings Act of 1976,"

which calls for the development and implementation of performance standards

for new buildings to achieve the maximum practicable improvements in energy

efficiency. The performance standards being developed under this program

are expected to take energy prices into account. This study addresses

the question of whether the energy prices used in determining the standards

should be the actual market prices paid or a price which has been adjusted

to reflect the social value of energy resources. FEA is developing such

adjustment factors under the name "Resource Impact Factors" (RIF's). Moreover, the report assesses the effect of using RIF's on the rate of energy

consumption allowed by the standard and on the economic efficiency of the

standard

The Building Economics Section of the Center for Building Technology,

Institute for Applied Technology, National Bureau of Standards, has prepared

this report to provide those involved in the development of energy

conservation standards with some guidance in determining what factors are

appropriate to be included in RIF's, and how the use of such RIF's would

affect the standards being developed.
' * - 'V .v 'J'.-

in

V
Dr. Harold E. Marshall, Chief of the Building Economics Section, and

Stephen R. Petersen deserve special thanks for the time they spent discussing these issues, reading and commenting on the earlier drafts of

this report, and providing the author with invaluable insights and ideas.

The author is also grateful for the helpful comments and suggestions of

the reviewers, Robert R. Jones, William L. Carroll, Dr. Douglas R. Shier,

and Dr. Carl 0. Muehlhause. Thanks are also due to Noreen Rigopoulos

and Linda Sacchet, who typed the several drafts of this report, as well

as to Reid Hartsell, the general technical representati ve for FEA. Responsibility for whatever errors and shortcomings may remain rests solely with

the author.
ABSTRACT

This report addresses the question of the proper price for energy

to be used in the development of optimum (i.e., cost-effective) energy

conservation performance standards for buildings. This study finds that

the appropriate price for energy is its social value, which should be

determined through the development and application of Resource Impact

Factors (RIF's). Some guidelines are provided for the formulation and

development of RIF's. A simple life-cycle cost minimization model for

determining the optimum conservation standard is employed to show how the

use of RIF's would generally lower the maximum allowable energy consumption

specified in the standard. Indeed, it is found that the higher the RIF

value, the lower the energy consumption allowed by the standard, although

this effect steadily diminishes as the RIF value increases. When a more

general cost model with less restrictive assumptions is employed, the

same inverse relationship appears between the energy consumption allowed

by the standard and the RIF value. Finally, a geometric measure is

derived for the net gain in economic efficiency that would result from

using RIF's in developing energy conservation performance standards.

1 v
EXECUTIVE SUMMARY

This report focuses on the role which the price of energy is expected

to play in the development of optimum (i.e., cost-effective) energy conservation performance standards for new buildings. The maximum rate of

energy consumption allowed under the optimum standard is based on a comparison of the value of the energy saved relative to the cost of complying

with the standard over the life cycle of the building. Thus, the price

used to value the energy saved has a significant effect on the resulting

standard. If the standard is to be set at a level which is socially optimum,

then it must be based on the social value of energy. Because of a range

of price distorting factors in energy markets, however, it is unlikely

that the actual prices paid for various energy types accurately reflect

the true social values of these resources. The Federal Energy Administration (FEA) is developing a system of indices which could be used to adjust

actual prices so that the true social values of energy resources would be

better represented. One of the purposes of this report is to offer some

guidelines for the development of these indices which are called "Resource

Impact Factors" (RIF's). The other major purpose of this report is to

assess the effects of using a system of RIF's in the development of energy

conservation standards. To accomplish this latter goal, a comparison is

made between a standard that is optimum from the private point of view

(using the actual unadjusted prices paid for energy) and one that is optimum

from the social point of view (using prices adjusted with RIF values).

The comparision is made in terms of two implications of the standards:

(1) the relative amounts of energy saved; and (2) the relative economic

efficiencies of the standards.

v
The guidelines offered for developing RIF's fall into two categories.

The first concerns the explicit method of formulation. One method of

formulation, as a quantity multiplier, results in impact numbers which

would serve well to compare alternative energy types. Because these impact

numbers are not denominated in dollar terms, however, they would be

inappropriate for determining the economic balance between energy and

nonenergy resources. To achieve such a balance, a common basis is needed

to allow comparison between the value of the energy saved and the cost of

the resources expended to save it. The other method of formulation, as

a price multiplier, provides this common denominator because the RIF value

converts the actual price paid for the energy to its corresponding social

price. Since this social price is denominated in dollars, it can be compared

with the cost of the nonenergy resources.

The second category of guidelines relates to the factors which RIF's

should take into account. These are the same factors that cause a divergence

between the actual price and the social value of energy. These factors

include the presence of unit taxes or of monopoly power, environmental

effects of energy production or consumption, the desire for national

economic independence, and the existence of price controls such as those

on natural gas.

To assess the likely effect on energy consumption resulting from the

use of RIF's, several forms of a life-cycle cost minimization model are

employed. The present value of total life-cycle heating-related costs

is expressed as a function of the thermal resistance of the building envelope. This cost expression is minimized to find the economically

vi
optimum (i.e., most cost-effective) level of thermal resistance. Because

of the exactly inverse relationship between resistance and the rate of

heat flow conducted through the building envelope, this model also yields

a formula for the economically optimum rate of thermal transmission, which

forms the basis for the optimum performance standard itself. Since the

actual price paid for energy is used in this formula, the resulting rate

of thermal transmission is considered privately optimum. Moreover, it

is possible to introduce price multiplier RIF's into this formula in

order to determine the socially optimum rate of transmission. Thus, the

socially optimum rate (using RIF's) can be compared with the privately

optimum rate (without RIF's). The comparison is made for the alternative

forms of the model which result from varying the assumptions regarding the

cost of resistance and effective degree days.

In general, it is found that the introduction of RIF's would have

a restraining influence on the energy consumption ceiling allowed under

the conservation performance standard, assuming the most probable range

for the price multiplier RIF values: RIF > 1 (that is, the social value

is greater than the actual price paid for energy). Using the example

of natural gas, a rough estimate can be made of the additional energy

savings resulting from the use of RIF's. A RIF value of 2.82 results

from a comparison of the regulated interstate wellhead price of $1 . 42/MCF

with the expected cost of synthetic gas from Western coal of about $4.00.

Introducing this 2.82 figure into the expression for the socially optimum

standard shows that the energy consumption ceiling allowed under the

standard would be 40.5% lower with RIF's than without them. The model
further shows that in general the higher the RIF value introduced (that

is, the higher the ratio of social value to actual price) for an energy

type, the lower the energy consumption ceiling allowed by the standard,

although this effect steadily diminishes as the RIF value increases.

When the more general forms of the cost model are employed with less restrictive assumptions regarding degree days and the cost of resistance,

the same inverse relationship appears between the energy consumption ceiling

and the RIF value.

Regarding the efficiency effect of RIF's, it turns out that the

introduction of RIF's increases the economic efficiency of the resulting

energy conservation standard. A standard developed without RIF's is

in effect based on the actual private prices paid for energy, whereas to

assure a socially optimum standard, the appropriate prices to use are the

social values of the energy types (that is, the product of the private

prices times the corresponding RIF values). Using the private rather than

the social prices of energy will likely result in a standard that is not

socially optimum. This deviation of the privately optimum from the socially

optimum standard can be termed economically inefficient in the sense that

an opportunity for a net economic gain to society is being foregone.

That is, the use of a RIF value greater than one will generally lead to an

additional reduction in energy consumption whose social value exceeds that

of the additional thermal resistance required to achieve it. This net

gain in economic efficiency (the social value of the additional energy

savings less the cost of achieving them) which results from the use of RIF's

is depicted graphically and a geometric measure of its magnitude is illustrated in the Appendix.

vi i i
One important area requiring further research concerns the actual

development of empirical estimates for RIF's. Those factors preventing

the actual price from reflecting the social value need to be modeled

more realistically and accurately. These models must be applied to

derive empirical estimates for each type of price-distorting factor.

Then, a method must be found to combine these individual factors into

an overall RIF value.

The other important area for further research is that of the relationship of RIF's to building (rather than component) performance standards.

This type of analysis would take into account the many energy-flow

interactions which occur among the various components of a building,

rather than study each component in isolation. Researchers at NBS

are currently involved in developing such a model for determining

optimum building performance standards. Once this model has been completed,

the effect of using RIF's in determining socially optimum building

performance standards can be assessed.

IX
TABLE OF CONTENTS

PAGE

PREFACE ii

ABSTRACT iv

EXECUTIVE SUMMARY v LIST OF FIGURES xii

LIST OF TABLES xiii

1.0 INTRODUCTION 1 1.1 PURPOSE 1

1 .2 ENERGY CONSERVATION STANDARDS FOR BUILDINGS 2

1 .2.1 TYPES OF STANDARDS 2

1.2.2 LEGISLATION 3

1.2.3 ASHRAE STANDARD 90-75 6

1.2. 3.1 CONTRAST WITH LEGISLATION 6

1.2. 3.

2

RESOURCE UTILIZATION FACTORS 7

1 .3 APPROACH AND ORGANIZATION 9

2.0 CONCEPTUAL FRAMEWORK FOR AN OPTIMUM STANDARD 11

2.1 OPTIMUM STANDARDS FROM THE PRIVATE PERSPECTIVE 12

2.1.1 LIFE-CYCLE COST MINIMIZATION 12

2. 1.1.1 ALTERNATIVE CHOICES FOR THE SAME COMPONENT 12

2. 1.1.

2

ALTERNATIVE COMPONENTS 13

2. 1.1.

3

ALTERNATIVE BUILDINGS 14

2.1.2 PRICE DIFFERENTATION AMONG ENERGY TYPES 15

2.2 OPTIMUM STANDARDS FROM THE SOCIAL PERSPECTIVE 16

2.2.1 PRIVATE PRICES DO NOT REFLECT SOCIAL ENERGY VALUES 16

2. 2. 1.1 UNIT TAXES AND MONOPOLY 19

2. 2. 1.2 ENVIRONMENTAL CONSIDERATIONS 22

2. 2. 1.3 NATIONAL ECONOMIC INDEPENDENCE 23

2.2,1

.4 PRICE CONTROLS 25

2.2.2 THE NEED TO ADJUST PRICES TO REFLECT SOCIAL VALUES 26

3.0 RIF's IN AN OPTIMUM STANDARD DETERMINATION MODEL 28

3.1 ALTERNATIVE FORMULATIONS FOR RIF's 28

3.1.1 QUANTITY MULTIPLIER FORMULATION 28

3.1.2 PRICE MULTIPLIER FORMULATION 31

x
PAGE

3.2 AN OPTIMUM STANDARD DETERMINATION MODEL 32

3.2.1 DERIVATION OF THE MODEL 33

3.2.2 SIGNIFICANCE OF THE MODEL 35

3.3 APPLICATION OF PRICE MULTIPLIER RIF'S 36

3.4 LIMITATIONS OF THE ANALYSIS 43

4.0 THE OPTIMUM STANDARDS MODEL GENERALIZED 45

4.1 ALTERNATIVE COST FUNCTIONS 46

4.1.1 CASE I: A LINEAR COST FUNCTION 47

4.1.2 CASE II: A QUADRATIC COST FUNCTION 50

4.1.3 CASE III: HIGHER-ORDER COST FUNCTIONS 54

4.1.4 COMPARISON OF ALTERNATIVE COST FUNCTIONS 55

4.2 DEGREE DAYS AS A FUNCTION OF RESISTANCE 60

4.2.1 DEGREE-DAY TEMPERATURE BASE AND RESISTANCE 61

4.2.2 DEGREE DAYS AND THE DEGREE-DAY BASE 62

4.2.3 CASE IV: A QUADRATIC DEGREE-DAY FUNCTION 65

4.3 CASE V: THE GENERAL OPTIMUM STANDARDS MODEL 68

5.0 SUMMARY, CONCLUSIONS AND FURTHER RESEARCH . 73

5.1 SUMMARY . 73

5.2 CONCLUSIONS 75

5.2.1 PROPER FORMULATION OF RIF'S 76

5.2.2 THE ENERGY CONSUMPTION EFFECT 76

5.2.3 THE ECONOMIC EFFICIENCY EFFECT 79

5.3 SUGGESTIONS FOR FURTHER RESEARCH 80

BIBLIOGRAPHY 82

APPENDIX: ECONOMIC EFFICIENCY EFFECTS OF RIF'S A-l

xi
LIST OF FIGURES

Figure Page

2.1 Competitive Market for a Resource 18

2.2 Effects of a Unit Tax 21

2.3 Social Costs of Coal-Fired Electric Power 24

2.4 Regulation of the Natural Gas Price 26

3.1 Plot of the Socially Optimum Standard (in units of U ) as

a Function of the RIF Value, for a Given Value of the

Privately Optimum Standard (U

Q

) 4 1

4.1 Degree Days as a Function of Thermal Resistance for a Typical House in Washington, D.C 66

A.l Trade-off Curve between Energy Consumption and Thermal

Resistance for Given Climate and Indoor Temperature . . A-2

A. 2 Privately Optimum Energy Consumption Rate (A

Q

). A-4

A. 3 Socially Optimum Energy Consumption Rate (A ) and the Net

Efficiency Gain from Using RIF's (N-M) A-6

xi i
LIST OF TABLES

Table Page

3.1 Hypothetical Quantity Multiplier RIF's 29

3.2 Hypothetical Price Multiplier RIF's 32

3.3 Possible RIF Values with Corresponding Socially Optimum

Standards 39

3.4 Selected RIF Values with Corresponding First Derivatives

of U* 42

4.1 Ratios of Socially Optimum to Privately Optimum Standards

for Various RIF Values and Cost Functions 56

4.2 Ratios of Extra Energy Savings Due to RIF's in Colder

Versus Warmer Climates, for Various Cost Functions .... 58

4.3 Effective Balance Points for Various Levels of Thermal

Resistance: Hypothetical Example 63

4.4 Degree Days Corresponding to Various Balance Points for

Six Major U.S. Cities 64

ij 4.5 Restrictions on the First and Second Derivatives of D(R)

and C(R) 69

A.l Conversion Factors from Customary to Metric (SI) Units . . . A-9

xi i i
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..

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1.0 INTRODUCTION

1.1 PURPOSE

The focus of this report is on the role of energy prices in the

development of optimum (i.e., cost-effective) energy conservation standards

for buildings. It is expected that the level of energy prices will influence

the amount of energy allowed under such an optimum standard. This study

assesses the effect on the conservation standard of using "adjusted" prices

rather than actual market prices for energy. The method of adjustment

would be to employ indices being developed by the Federal Energy Administration (FEA). These indices are called "Resource Impact Factors"

(RIF's) and are intended to reflect the true social value of energy

resources

This report compares an energy conservation standard developed using

actual market prices of energy with one developed using adjusted social

prices of energy. Thus, a standard that is optimum in private profitability

terms is compared with one that is optimum from a social perspective. The

comparison is made in terms of two implications of the standards: (1) the

relative amounts of energy saved, and (2) the relative economic efficiencies

of the standards.

In addition, the report will discuss the manner in which RIF's

should be specified in order to be useful in the development of optimum

energy conservation standards for buildings. Moreover, there will be

some discussion of the need for RIF's and of the various factors which

will require quantification before usable RIF's are developed by FEA.

1
These factors are the main causes for the failure of actual market

prices to reflect the true social values of energy resources.

1.2 ENERGY CONSERVATION STANDARDS FOR BUILDINGS

1.2.1 Types of Standard: ;

There are three fundamental types of energy conservation standards

which could be developed for buildings. The first is a prescriptive

standard which might specify the kind and amount of insulating materials

to be used in the walls and attic of a particular type of building.

Another kind of standard is a component performance standard, which

would only require that a component be able to meet a particular goal

or target specification without regard to how that goal is achieved. An

example of such a standard would be a requirement that the attic have an

overall coefficient of heat transmission (or U value) of less than

.06 Btu/hr./sq. ft./°F J The third type is a building performance

standard, which would specify an overall goal for the design energy consumption of the building, thus allowing the designer to make cost-reducing

tradeoffs between building components in order to conform to the standard.

For example, a building performance standard might require that the

amount of energy needed to satisfy the annual heating and cooling load

be less than 60,000 Btu per square foot of floor space. This is referred

to as the allowable "annual energy budget" for heating and cooling.

Another building performance standard might impose an upper limit on

the annual energy consumption for domestic hot water.

^Because of the present accepted practice in this country for building

technology, customary U.S. units of measurement are used throughout this

report. Relevant conversion factors from customary to metric (SI) units

can be found in Table A.l at the end of the Appendix.

2
1.2.2 Legislation

On August 14, 1976, the President signed into law the Energy

Conservation and Production Act (P.L. 94-385). Title III of this law.

Energy Conservation Standards for New Buildings Act of 1976, requires

the Secretary of Housing and Urban Development to consult with the

Federal Energy Administration, the National Bureau of Standards, and

the General Services Administration in order to develop proposed performance standards for new residential and commercial buildings. Key

elements of this law are: (1) the explicit specification of performance

rather than prescriptive standards; (2) the requirement that the standards

developed be analyzed in terms of their economic costs and benefits;

(3) the stated goal of achieving the maximum practicable improvements

in energy efficiency through reasonable conservation features; (4) the

requirement that the standards take into account the climatic variations

among different regions of the country; (5) the avowed purpose of encouraging greater use of nondepletabl e sources of energy; and (6) that the

standard apply to the heating, cooling, ventilating, and domestic hot

water loads of buildings.

The fact that performance standards are to be developed means that

designers will be free to choose the most economical means of meeting

the standards, given regional variations in the costs of building materials

and labor. From the legislative hearings, it appears that building performance standards are the ultimate goal of the Act. These building

performance standards will be developed by combining the results of anlaysis

of individual components and their energy interdependencies. In most

->0
applications, manuals of accepted practice regarding individual building

components will provide the guidelines for conformity with the standards.

The second key element of the Act, the need to consider costs and

benefits, points out the importance of developing an economically

balanced conservation standard. This requirement seems to suggest

that the value of the total energy saved by the standard should at

least cover the extra costs imposed by the standards. A further guideline for developing the standard may be implicit in this requirement.

That is, that the standard be set so as to achieve the greatest possible

net economic gain, taking into account both savings and costs. Such

a standard could be considered economically optimum.

The goal of maximum practicable gains in energy efficiency requires

that the standard provide for the most energy savings possible within

the limits of practicability. This suggests that the operating constraint

preventing an extremely stringent standard is the increased cost required

for compliance. According to this interpretation, the standard would

be made increasingly stringent as long as the additional energy savings

still justify the extra costs. In this way, cost-effective

energy conservation features would be introduced into the design of

buildings. Likewise, such a cost-effective standard would directly

address the expressed concern of Congress over excessive long-term

operating costs due to inadequate energy conservation measures in new

buildings.

The concern for cost effectiveness is also apparent in the fourth

element of the Act, the requirement to consider climatic variations in

developing the standard. The reason for considering these variations is

4
so that the standard can be tailored to the specific heating and cooling

loads of each region. The dollar-value benefit of the energy saved by

one additional inch of attic insulation is greater in more severe climates.

Thus, these higher levels of benefits will tend to justify the extra

cost of complying with more stringent performance standards developed

for more severe climates.

The fifth key element of the Act, the desire to encourage greater

use of nondepletable resources, is especially relevant to this report.

We will show how RIF's can be used in the development of performance

standards which are optimum from the social point of view.

Properly designed RIF's will take into account the policy preferences of

society, which could easily include the goal of fostering the development and use of nondepletable resources.

The sixth key element concerns which energy loads will have standards.

Those loads specifically mentioned in the Act are for heating, cooling,

ventilating, and domestic hot water services. It is likely, however,

that standards will be developed for illumination loads as well. Of

these energy loads, heating and cooling are the ones most affected by

the design of commercial and residential buildings. Hence, a wider

range of trade-offs are possible between heating and cooling loads

and building design decisions. For this reason, we have chosen to concentrate our analysis on the development of energy conservation performance standards for the heating and cooling of buildings.

5
1.2.3 ASHRAE Standard 90-75

An energy conservation standard for buildings has already been

developed and is being proposed by the American Society of Heating,

Refrigerating, and Air-Conditioning Engineers (ASHRAE). This is ASHRAE

Standard 90-75 and is entitled, "Energy Conservation in New Building

Design." This standard serves as the basis for energy conservation

regulation in several states as well as for the formulation of energy

conservation provisions of three model codes. For purposes of comparison

it is worth noting how it differs from the standard called for by the

Energy Conservation and Production Act.

1.2.3. 1 Contrast with Legislation

The major point of difference between these two standards regards

the method of establishing the appropriate degree of stringency. ASHRAE

90-75 did not attempt to achieve the maximum practicable improvements in

energy efficiency as called for by the Energy Conservation and Production

Act. Indeed, the ASHRAE standard specifically states that it is presenting

only minimum requi rements J Moreover, the degree of stringency embodied

in the ASHRAE standard does not seem to be based on economic considerations.

By contrast, as we have seen, the language of the Energy Act suggests

that the standards developed should be based on a balancing of costs and

benefits.

Another less significant point of difference between these two

standards lies in the relative emphasis placed on component versus building

performance. Most of the text of the ASHRAE standard uses the component

performance approach. Sections 4 through 9 specify performance

standards for the various building components such as the exterior

^ ASHRAE 90-75: Energy Conservation in New Building Design (New York:

ASHRAE, Inc., 1975), Section 4.2.1.

6
envelope, heating and cooling equipment, service water heating, and

lighting. Only in the rather brief Section 10 is a systems analysis

approach suggested as an acceptable alternative to the component standards.

In other words, deviations from the specific component standards are

allowed as long as the entire building's annual energy performance is

shown to be superior to that of a comparable building which does conform

to the component standards. The Energy Conservation and Production Act,

on the other hand, has building performance standards as its ultimate

goal, although these will be based on the analysis of individual components and will be mainly complied with through manuals of accepted

practice.

1.2. 3. 2 Resource Utilization Factors (RUF's)

The proposed twelfth and final section of ASHRAE 90-75, "Annual Fuel and

Energy Resource Determination," provides a method for calculating the

equivalent resource quantities required to supply the energy needs of a

building. The procedure involves the use of conversion coefficients,

1 called Resource Utilization Factors (RUF's), to be applied to the

required energy output delivered to the building site. These RUF's are

based on the total energy losses that result from processing, refining,

transporting, converting, and delivering the energy from the point of

extraction to the building site.

The rationale behind the development of RUF's was to treat separately

those factors which are more readily quantifiable. This is because they

are basically technical in nature. The less manageable factors, such as

-

RUF's are not to be confused with RIF's.

7
environmental, economic, and national security concerns, would be

accounted for by developing a system of RIF's, an analogous kind of

energy quantity multiplier. The format for annual fuel and energy

calculations presented in Section 12 of ASHRAE 90-75 suggests that the

RIF's are to be applied after and in addition to the RUF's,

so that the total resource impact of each energy load can be determined.

This distinction between RUF's and RIF's has several

limitations. In the first place, RUF's do not include all of the

energy losses which occur in making energy available to deliver a

service to a building. The energy expended in extracting the resource

from the ground is ignored, as is the energy lost in the final conversion at the building. At a more fundamental level, the use of RUF's

is similar to applying the technique known as Net Energy Analysis.

According to this technique, resources to be employed in the production

of energy for consumer use should be so allocated as to maximize "net

energy." In this context net energy means the amount of energy that

remains for consumer use after the energy costs of finding, developing,

extracting, upgrading, and delivering the energy have been paid. These

energy costs are measured by the energy content of the resources (i.e., by

the amount of energy required to produce them) which are employed in

carrying out these processes. The basic shortcoming of this type of

analysis is that it implies that all nonenergy resources are relevant

Vor further discussion of Section 12 of ASHRAE Standard 90-75

and its distinction between RIF's and RUF's, see Robert R. Jones, "Resource

Impact Factor (RIF) Approach to Optimal Use of Energy Resources," ASHRAE

Journal , Vol . 18, No. 10 (October 1976) pp. 15-18. Also, see Harry H. Phipps, "The RUF Concept as an Enerqy Measurement Tool," ASHRAE Journal,

Vol. 18, No. 5 (May 1976) pp. 28-30.

8
only in the measure of the energy required to produce them. Thus, many

important resources are either totally or partially ignored.

1

Another difficulty with this distinction between RIF's and RUF's

is that it involves a certain amount of double counting. This occurs

because all of the elements included in RUF's also affect the economic

cost (which is included in RIF's) of making the energy available to serve

the building's needs. That is, the effective cost of delivering a million

Btu's to a building project site could be reduced by an increase in refining

efficiency (i.e., a decrease in energy losses at the refining stage).

1.3 APPROACH AND ORGANIZATION

In the next section we will develop the conceptual framework for

determining the optimum energy conservation standard for the heating

and cooling loads of buildings. The concept of life-cycle cost minimization will be discussed from the private and then the social points

of view. The reasons why actual market prices may fail to reflect the

social values of energy resources will be explained and illustrated.

These reasons will serve to suggest ways to adjust energy prices so as

to reflect social values.

In Section 3.0, RIF's will be applied to a simple model for determining optimum standards for the heating load characteristics of a

building component. Alternative formulations for RIF's will first be

For a discussion of the inadequacies of Net Energy Analysis, see David A. Huettner, "Net Energy Analysis: An Economic Assessment,"

Science , Vol . 192 (April 9, 1976), pp. 101-104. For a brief overview of

several approaches to energy analysis see Kenneth C. Hoffman, "Some

Comments on Energy Accounting," paper presented at panel discussion

Energy, Environment and Economics, 68th Annual Meeting, American Institute

of Chemical Engineers, November 1975.

9
presented and then the cost minimization model will be developed. Then,

a family of RIF values will be introduced into the model, the resulting

optimum standards will be analyzed, and the limitations of the analysis

will be discussed. In Section 4.0 several alternative forms of the cost

model will be developed by varying the assumptions regarding climatic

loads and the cost of energy conservation techniques. The effects of

introducing a family of RIF's into these more general forms of the model

will be analyzed.

Section 5.0 will present a general summary of the report and of

its major findings. These conclusions concern: (1) the appropriate

formulation of RIF's for use in developing an energy conservation

standard; (2) the effects of using RIF's on the degree of stringency

(allowable energy consumption rate) specified by the standard; and

(3) the net gain in economic efficiency from using RIF's rather than

unadjusted market prices for energy in the development of the standard.

Finally, this gain in economic efficiency resulting from the use of

RIF's will be graphically illustrated and analyzed in the Appendix.

10
2.0 CONCEPTUAL FRAMEWORK FOR AN OPTIMUM STANDARD

As we have seen in our discussion of the Energy Conservation and

Production Act, the language of Title III calls for development

of a conservation standard which is designed to achieve the maximum

practicable improvements in energy efficiency. Moreover, the Act

emphasizes the need to analyze the standard in terms of its costs

and benefits. Thus, we have chosen to concentrate our analysis on

the effects of RIF's on an optimum standard, that is, a standard which

achieves the greatest possible net benefits (dollar savings minus

costs) J Moreover, for purposes of illustration we will focus on

setting a standard for the conservation of energy used.onlv to heat

and cool buildings.

There are two possible points of view to take in measuring net

benefits: (1) that of a private investor or potential homebuyer; and

(2) that of society or the public at large. In rare circumstances

these two points of view lead to the same optimum. Often, what

is optimum for a private consumer or producer may not be

optimum for society as a whole. As we shall see, the latter is

usually the case when dealing with energy. We will define more

precisely what is meant by an optimum standard first from the

private viewpoint and then from the social viewpoint.

If, on the other hand, the standard to be developed is not based

on a comparison of costs and benefits, then there would be no way to

introduce RIF's into the development process.

11
2.1

OPTIMUM STANDARDS FROM THE PRIVATE PERSPECTIVE

2.1.1 Life-Cycle Cost Minimization

To select an optimum energy conservation standard, one must consider

all costs occurring over the life of the building which relate to the

supply of heating and cooling services. * These would include such

first-cost items as the insulation used in the building envelope, the

heating and cooling equipment and the choice of storm windows. The

recurring costs would depend on the annual fuel consumption required to

achieve a given level of comfort. These recurring future costs must be

discounted in order to be comparable with the first-cost items. Then,

all these cost items are summed to get the total present value of the

cost of heating and cooling the building over its life. Finally, this

present value figure is minimized with respect to the entire range of

choices for insulation levels, heating and cooling equipment, and storm

windows. That is, the least-cost design combination of these components

is selected.

2. 1.1.1 Alternative Choices for the Same Component

The simplest application of life-cycle cost minimization is for the

case of a single building component with alternative levels of energy

conservation. An example would be various levels of insulation in the

Whis discussion of minimization of life-cycle heating and cooling

costs is intended only as a brief overview. For a more detailed development see Stephen R. Petersen, Retrofitting Existing Housing for Energy

Conservation: An Economic Analysis (Building Science Series 64; Washington

,

D.C.: U.S. Department of Commerce, 1974), and "Economic Optimization

in the Energy Conservation Design of Single-Family Housing," ASHRAE

Transactions, Vol . 82, Part I, 1976.

12
attic. As the level of insulation is increased, the energy required to

heat and cool the building decreases, but at a decreasing rate. This

means that each successive unit of insulation yields less benefits in

terms of saved energy than the preceding units. If the cost of each

unit of insulation increases, remains constant, or at least does not

decrease at a rate faster than that of the benefits, then that level of

insulation which yields incremental benefits equal to its incremental

costs will be the level which also minimizes the life-cycle costs of

supplying heating and cooling services to the building.

2. 1.1. 2 Alternative Components

If the above minimization procedure is applied to all the components

of a building, then the optimum energy conservation standard for that

building will generally be found. One exception to this result is due

to possible interdependencies between the various building components.

Thus, for example, the efficiency of the heating equipment will have

some effect on the optimal level of insulation in the walls and attic.

When such interdependencies are significant, all possible combinations

of levels of equipment efficiency and of insulation should ideally be

examined to find the one with minimum life-cycle costs

J

An optimum building design developed with this minimum-cost method

has the convenient characteristic that no more energy savings can be achieved by

shifting resources from one component to another. In other words,

greater energy savings are available only by increasing construction

^Currently, research is being conducted at the National Bureau of

Standards to model these various interdependencies.

13
costs and the extra dollar savings would not justify those additional

construction costs. This means that an economic balance has been achieved

among the components in that the total energy-related construction

budget cannot be reallocated differently among the components to reduce

energy consumption.

2. 1.1. 3 Alternative Buildings

An energy conservation standard that is to be optimum must also

achieve an economic balance among all buildings governed by the standard.

This requires that an alternative standard could not achieve lower

energy expenditures for the same national energy-related construction

budget by setting stricter specifications for some buildings and more lenient

ones for others. Otherwise the alternative standard would be superior.

An important implication of this optimality requirement is that the

climatic conditions of each building must be taken into account.

Buildings located in severe climates have greater energy loads than

elsewhere, and consequently the benefits available from an additional

expenditure on energy-saving construction techniques are greater for

such buildings. A standard which treats all climates alike by requiring

the same overall coefficient of heat transmission (or U value), could

always be improved by raising the maximum allowable U value for mild

climates and lowering it for severe climates -- without increasing the

nation's energy-related construction budget at all. Thus, the lifecycle cost minimization procedure will lead to a more stringent standard

(in the sense of a lower allowable U value) for buildings in severe

climates than for similar buildings in mild climates."*

^It should be noted that if a cost-effective standard were stated in terms

of an annual energy budget rather than a U value, the more severe climate would

be permitted a higher annual energy budget., This is because the additional insulation called for under a cost-effective standard will not completely compensate

for the greater heating load imposed by the severe climate. This result is

demonstrated quantitatively (pp. 43-50).
2.1.2 Price Differentiation among Energy Types

Just as for differentiation among climatic conditions, a similar

argument can be made for distinguishing among the effective prices'*'

paid for different energy types. A building which uses a more costly

energy form will yield greater benefits from an extra expenditure on

energy-saving techniques because the value of the saved energy is

greater. If a standard were developed which ignored the differences in

the prices of energy types, then there would be room for improvement

in benefits without even adding to the overall cost of compliance. One

could simply design a new standard that would shift some of the energysaving construction resources from buildings which use low-cost energy

to those which use high-cost energy. The net dollar savings from the

latter standard would be greater than those from the original standard

which did not differentiate among the prices of energy types.

Thus, if the standard is to achieve the greatest possible net economic

benefits (i.e., be truly optimum), it must take account of differences

in energy prices. There is, moreover, a further relationship between

optimality and energy prices. The type of energy prices used in the

analysis determines the nature of the optimality achieved. The conceptual framework presented thus far has focused on the development of a

standard which is optimum from the private building owner's point of view.

^Effective price here means the number of dollars required to be

spent to get one unit of usable heat, cooling power, or light added to

the conditioned space.

15
That is, the standard was said to be based on the principle of minimizing

the present value of all the privately incurred energy-related costs of

the building over its entire life cycle. We now will modify this principle

so that it will lead to a standard which is optimum from society's

viewpoint.

2.2 OPTIMUM STANDARDS FROM THE SOCIAL PERSPECTIVE

The purpose of developing a national energy conservation standard

is to encourage practices in the design of new buildings which are

optimum for the nation. Because of certain distortions in the markets

for energy resources, what is optimum for the nation may

not be optimum for the individual prospective homeowner. That is, a

standard which minimizes the life-cycle energy-related costs actually

incurred by the homeowner will not necessarily minimize those costs when they

are measured from society's point of view. This is because energy resources are valued differently by individuals and by society.

2.2.1 Private Prices do not Reflect Social Energy Values

Traditional economic theory of perfectly competitive markets

shows that under very restrictive assumptions the prices paid for

* resources are equal to their true social valuesJ According to this

theory, resource users are motivated to employ them up to the point at

which the value produced by the last unit of the resource equals the

For example, see Richard H. Leftwich, The Price System and Resource

A1 location (3rd ed.; New York: Holt, Rinehart, and Winston, 1966),

Chapter 15, or Ian M.D. Little and James A. Mirrlees, Manual of Industrial

Project Analysis in Developing Cou ntries, Vol . II: Social Cost Benefit

Analysis (Paris development Centre of the Organization for Economic Cooperation and Development, 1968), Chapter II.

16
price paid for it. Similarly, the suppliers of the resource will make

it available up to the point at which the cost of supplying the last

unit will just be covered by the price received for it. In this way

resources are allocated most efficiently since no more surplus can be

squeezed out of them. That is, to use a resource beyond the point

dictated by its price would mean that the additional units employed

would produce outputs worth less than the cost of the resources themselves.

Thus, the resource price is the key which leads users to employ and

suppliers to make available the optimum amount.

This ideal world of perfect competition in the supply and use of a

resource is illustrated graphically in Figure 2.1. The line S represents

the quantities of the resource that suppliers would be willing to make

available at various selling prices in a given market and during a

specific time period. The upward slope is used to show that as more of the resource is produced, the cost of producing each successive unit generally increases,

which in turn requires a higher price to justify production. The line U,

on the other hand, represents the quantities of the resource that users

would be willing to buy at various prices in the same market and during

the same time period. The downward slope indicates that as more of the

resource is used, the value generated by each successive unit employed

declines, which in turn reduces the price which users are willing to pay

for additional units. At prices above P*, suppliers make more units

available than users want and the resulting surplus drives the price

down to P*. At prices below P*, suppliers make fewer units available

17
Pri ce

Figure 2.1

Competitive Market for a Resource

18
than users want and the resulting shortage drives the price up to P*.

At P* there is neither shortage nor surplus and Q* is supplied and used.

Users and suppliers are willing to continue buying and selling this

equlibrium quantity at this equilibrium price. In addition to

providing for an equilibrium in the sense that supply and demand are in

balance, P* and Q* are optimum in the sense that the correct amount of

the resource is being produced and used. That is, units of the

resource used in excess of Q* would be employed to produce output whose

value is less than the cost of supplying the resources (U is below S to

the right of Q*). On the other hand, if resource units fewer than Q* are

used, they would be employed to produce output valued above the cost of

supplying the resources (U is above S to the left of Q*).

To operate so efficiently, however, the conditions of the perfectly

competitive model must be satisfied for the market being considered. These

conditions guarantee that the resulting equilibrium price that is visible to

all participants in the market will indicate to the suppliers, on the one

hand, the true value of additional units of the resource, and to the users,

on the other hand, the true social cost of producing additional units. If

the price is in any way prevented from performing this vital function, then the

resource will not be supplied nor employed at optimum levels. There are a

number of ways in which the price can fail to provide this essential

information to the users and suppliers of the resource. We will now focus

on several of the most important ones for energy resources.

2. 2. 1.1 Unit Taxes and Monopoly

One possible source of divergence between actual prices and social values

or costs of energy resources is a tax per unit produced or soldJ Examples

useful collection of articles on energy taxes and their implications

can be found in Gerard M. Brannon (ed.). Studies in Energy Tax Policy

(Cambridge, Mass.: Ballinger Publishing Company, 1975).

19
would be the utility taxes on the number of therms of gas or kilowatthours of electricity consumed, and the former oil import tax per barrel

included in the per gallon price of home heating oil. Such taxes generally

force the market equilibrium price to differ from the true social cost

of supplying the resource.

1

This difference is illustrated in Figure 2.2,

which is the same as Figure 2.1 except that the line Sy has been added

above and parallel to line S. The line Sy represents actual costs of

production plus the amount of the unit tax. This upward shift in the

suppliers' requirements changes both the equilibrium quantity (from Q* to

Qy) and price (from P* to Py). The effect on the price means that the

actual market price (Py) visible to and paid by the users exceeds the

price (P*) which would exist if there were no tax and by an even greater

amount exceeds the price (P

c

) necessary to cover the actual costs of

production. This implies that users do not correctly perceive the

true social cost of producing the resource, are forced to pay a higher

3 price, and consequently under-util ize the resource.

A similar situation of distorted market prices exists when there

are so few firms competing in the market that the suppliers have a

certain degree of monopoly power. In this case, suppliers in an attempt

to maximize profits, will tend to restrict the quantity available such

that the price paid per unit is greater than the true cost of producing

1 This is true unless the purpose of the tax is to

offset neqative environmental effects or excess profits, or to fulfill some other policy objective of the aqencv responsible for imposing the tax.

2 The difference between Py and P~ is given by the amount of the per

unit tax.

3 The same type of analysis applies to the case of subsidies, which

can be treated as negative taxes. In this case, however, a subsidy per

unit of production would cause the supply line to fal

1

by the amount of

the subsidy rather than to rise. As a result, users are able to purchase

the resource at a price below its true cost of production, and consequently

the resource is over-uti 1 i zed .
Price

Figure 2.2

Effects of a Unit Tax

21
those units.

1 Here again, then, the decisions of resource users are not

guided by the true cost of the resource.

2. 2. 1.2 Environmental Considerations

Another source of possible divergence between the actual price paid

for a resource and the social cost of making it available is a group of

factors called externalities or environmental effects. These effects

are present whenever a particular production or consumption process

generates a cost or benefit for persons not directly involved in the

process. As a result the price established by the suppliers and users

of the resource will generally not reflect these external (or indirect)

benefits and costs. In the case of energy resources most of the external

3 effects take the form of costs rather than benefits. As an illustration, consider the generation of electricity from coal. To the

extent that environmental control legislation allows some residual air

pollution, the price users pay for electric power will not reflect all

of society's pollution costs of producing the power, but only the direct

4

costs incurred by the utility. This example is shown graphically in

See Edwin Mansfield, Microeconomics: Theory and Applications

(New York: W. W. Norton & Company, 1970), Chapter 9, for a detailed

explanation of the theory of pricing and output choice for monopolists.

2 For two complete collections of articles on the economic theory

of externalities, see Robert J. Staaf and Francis X. Tannian, Externalities : Theoretical Dimensions of Political Economy (New York: Dunellen Publishing

Company, Inc. n.d.)> and Steven A. Y . Lin (ed . ) , Theory and Measurement of

Economic Externalities (New York: Academic Press, 1976).

For a very detailed overview of alternative energy technologies

and their environmental impacts, see Oklahoma University Science and

Public Policy Program, Energy Alternatives: A Comparative Analysis (Springfield,

V a: U.S. Dept, of Commerce, National Technical Information Service, 1975).

4

Of course, if a tax were imposed which exactly offset these external

costs, then the price (including the tax) paid would reflect all social costs

of production.

22
Figure 2.3. The line Sp indicates the costs incurred by utilities which

must be covered by the price actually paid by users. The line S

s is the

sum of both the private costs and the external costs (i.e., the total

social costs) caused by the air pollution. In the absence of an

adequate pollution tax, the suppliers and users will agree on amount

Q and exchange price P for each unit of electric power. This private

price, however, is deceptively low in that it ignores the social costs

incurred by the pollution. In fact, the total social costs of producing

the last unit of power in this case would be P

g and the amount by which

that last unit is underpriced is P -P.

2. 2. 1.3 National Economic Independence

In support of various energy policies an appeal has been made in

recent years to the need for national economic independence J The appeal

is based on concern over U.S. vulnerability to political influence through

oil embargoes and unreasonable price rises. To the extent that energy

independence is a national policy objective, it can be reasonably argued

that each unit of energy consumed has an extra social cost in terms of

increasing our requirement for oil imports and thus our vulnerability to

embargoes. Indeed, when considering energy conservation standards that

affect the energy use of buildings for many years to come, the argument

becomes even more cogent. The more stringent the standard, the less our

commitment to energy imports during the next 30 to 40 year life of

buildings now being designed.

^ Uzi Arad, Barry J. Smernoff, and Haim ben-Shaher, American Security

and the Interna ti onal Energy Situation , 4 vols. (Crotori-on-Hudson , N . V .

;

Hudson Institute, 1975) provides an overview of the energy independence issue.

23
Price

Figure 2.3

Social Costs of Coal-Fired Electric Power

24
This national independence goal has implications for the need to

adjust energy prices similar to those of externalities discussed above.

Just as the external cost of pollution causes the social price of

electricity to be above the privately determined market price, so the

cost to the nation's economic independence (if such is a public policy

goal) makes the social cost of energy consumption exceed its private cost.

2.2. 1 .4 Price Control

s

Deviations between the prices paid for resources and their social

value can also come about through price controls. In this situation,

the price actually paid is set by legislation or regulation and not by

the forces of the market operating through the costs of supply and the

benefits from use. The objective of such regulation may be to control

excess profits in an industry with little competition or to make a

vital resource affordable for low-income households. Thus, even in the

absence of externalities, monopoly power, and taxes, the administered

price will not necessarily be equal to the social value of the resource.

This is illustrated for the case of natural gas in Figure 2.4. Because

the regulated price, P , is set so far below the optimum price, P*, the

suppliers of natural gas are willing to make only Q

r units available

rather than the optimum quantity, Q*. With so few units available, there

are many unsatisfied high-value uses to which the gas could have been put.

This is indicated by the fact that at Q , the premium users would be willing

to pay for additional units over and above the regulated price, would be

as much as P -P . s r

2.2.2 The Need to Adjust Prices to Reflect Social Values

The preceding discussion covers some of the ways in which the actual

prices paid for energy resources may fail to reflect their full

25
Price

Fiqure 2.4

Regulation of the Natural Gas Price

26
social values. As such it offers some insight into the difficulty of

using actual energy prices as a basis for energy-rel ated policies such

as the conservation standard for building design. Moreover, the items

discussed provide at least a partial checklist of factors which should

be considered in the development and actual quantification of RIF's.

Before RIF's can be quantified, however, a further consideration will

have to be made regarding possibly conflicting objectives which may

exist among government agencies. For example, a goal of keeping the

cost of vital home heating fuels within the budgets of low-income

households may conflict with a goal of national energy independence.

If RIF's are to reflect a unified national energy policy, such conflicts

must first be resolved.

27
3.0

RIF's IN AN OPTIMUM STANDARD DETERMINATION MODEL

In this section we will apply a range of RIF's to a life-cycle cost

minimization model for the determination of a socially optimum conservation

standard. Two alternative ways of formulating RIF's will be presented

and then the optimum standard determination model will be explained. These

formulations will be used to introduce a family of RIF values explicitly

into the model to compare the resulting energy standard with and without

RIF's.

3.1 ALTERNATIVE FORMULATIONS FOR RIF's

RIF's may be formulated to be applied to energy resources in

either of two fundamental ways. The first way would have RIF's formulated to be multiplied by the physical quantities of the energy

resource used. According to the other formulation, RIF's would be

applied directly to the actual price of the energy resource. Each of

these formulations will be discussed in turn.

3.1.1 Quantity Multiplier Formulation

Quantity multiplier RIF's would be formulated as coefficients to be

multiplied times the particular energy load in question.^ The annual energy

load would be denominated in units of energy output (10^ Btu's) required

to meet the final energy consumption needs of the building. In the case

of space heating, for example, this energy load would be the heating

^ T h i s quantity formulation is suggested in the definition of RIF under

review by the ASHRAE Section 12 Panel: "A multiplier applied to the fuel

and energy resources required by a building project to permit evaluation

of the effect on non-renewable resources resulting from the selection of

fuel and energy sources, giving consideration to time, location, economic

environmental and national interest issues."

28
value of the energy output into the conditioned space that is necessary

to maintain a desired comfort level. To specify the RIF coefficient

one would have to define a unit of measurement for scaling the relative

impacts of using the various energy types. Such a measurement unit

could be arbitrarily specified as a "resource impact unit." 1 These

impact units would attempt to measure such factors as the environmental

and national security issues. Then the RIF for a particular energy type

would be a coefficient dimensioned as the number of resource impact

units per unit of output for that energy type. Table 3.1 presents some

|l hypothetical quantity multiplier RIF's to illustrate how they might

operate.

Table 3.1

Hypothetical Quantity Multiplier RIF's

Units Needed9 Heating Load (10^ Btu) RIF Impact

Energy Type

(1) (2) (3) (4)=(2)x(3)

Nat. Gas (MCF) 60 36 1.3 46.8

Oil (Gal.) 429 36 1.2 43.2

Electric (kWh) 10,548 36 2.0 72.0

a The number of physical units of input of the energy type needed to

satisfy the given annual heating load of 36 x 106 Btu. The heating

system output of 36 x 10° Btu was converted to corresponding input

requirements by assuming the following energy contents and conversion

efficiencies:

(1) Natural Gas: 10^ Btu/MCF and 60% efficiency

(2) Oil: 140,000 Btu/Gal. and 60% efficiency; and

(3) Electric: 3413 Btu/kWh and 100% efficiency.

Alternatively, the impact unit could be stated in terms of the

number of equivalent Btu's, along the lines suggested for Resource Utilization Factors in the proposed ASHRAE Standard 90-75, Section 12. As

indicated in Section 1.0 above, however, the Btu equivalence approach

amounts to Net Energy Analysis with its noted shortcomings.

29
Column 1 shows the number of physical units of each alternative

C energy type required to satisfy the annual heating load of 36 x 10 Btu

specified for a particular building (e.g., 429 gallons of fuel oil).

Column 2 gives the energy load in terms of the heating value of the

energy output needed to maintain a desired degree of comfort (36 x 10^

Btu). Column 3 presents the RIF value corresponding to each energy

C

resource type, measured in terms of impact units per 10 Btu of energy

output needed to meet the heating load (e.g., 1.2 impact units per 10^

Btu of heating output from oil). Finally, Column 4 shows the product of

Columns 2 and 3, which represents the number of impact units required to

meet the heating load by means of each energy type (e.g., 43.2 impact

uni ts for oil )

.

This quantity multiplier formulation is useful in that the resulting

impact numbers permit comparison of the relative merits of using one

energy type rather than another. Using these impact numbers, standards

could be developed so as to achieve a relative balance among energy

types with respect to the stringency of the standards. Thus, for a given

energy conservation construction budget, a system of such standards could

be designed so as to minimize the sum of the impact numbers for space

heating.

The fundamental shortcoming of this formulation lies in its inability

to establish the appropriate construction budget. Because the impact

numbers are not denominated in dollars (the measurement unit of the

construction costs), the formulation cannot be used to establish the

appropriate economic balance between the amount of one energy type and

the application of energy conservation features in a building design.

30
This quantity multiplier does not allow one to weigh the importance of

using all types of energy resources as compared with energy-conserving

resources. Consequently, it cannot be used to determine how much energy

is worth being saved. Of course, this quantity multiplier could be

converted into a value multiplier if a dollar value could be established

for the resource impact unit chosen. Then, the energy and non-energy

resources would have a common basis for comparison. In this case,

however, the RIF would no longer be a quantity multiplier, but rather a

value multiplier. This is equivalent to the price multiplier formulation

which we will now discuss.

3.1.2 Price Multiplier Formulation

The other formulation possible for RIF's is as a multiplier to be

applied to the actual price of each energy type. This multiplier would

represent the ratio of the estimated social price to the actual price of

the resource. As such, this factor would convert the private price or

the actual price paid for the energy to its corresponding social price.

In short, it would act as an adjustment factor which corrects existing

actual prices for the various price-distorting effects (external effects,

the need for economic independence, unit taxes, monopoly power, and price

controls) which were discussed in Subsection 2.2.1. Table 3.2 offers

some hypothetical price multiplier RIF's to illustrate this kind of

formulation.

31
Table 3.2

Hypothetical Price Multiplier RIF's

Energy Type

Actual Price

($/Unit)

Effective Price3

($/106 Btu)

RIF Social ,-Price

($/10

D Btu)

0) (2) (3) (4)=(2)x(3)

Nat. Gas (MCF) 2.00 3.33 2.0 6.66

Oil (Gal.) .40 4.76 1.5 7.14

Electric (kWh) .03 8.79 1.1 9.67

a Energy contents and conversion efficiencies for each type are as follows:

(1) Natural Gas: 10 Btu/MCF and 60% efficiency;

(2) Oil: 140,000 Btu/Gal. and 60% efficiency; and

(3) Electric: 3,413 Btu/kWh and 100% efficiency.

Column 1 gives the approximate prices actually paid by homeowners

for each physical unit of the energy types and Column 2 uses energy

contents and efficiencies to convert these to effective prices paid for

one million Btu's delivered to the conditioned space. Column 3 lists

hypothetical price multiplier RIF's for each type of energy and Column 4

gives the products of Columns 2 and 3 which represent the adjusted

social prices of the energy types.

3.2 AN OPTIMUM STANDARD DETERMINATION MODEL

In this section we shall briefly summarize the life-cycle cost

minimization model developed by Stephen Petersen to determine the

economically optimum levels of thermal resistance in building components

J

^Stephen R. Petersen, "Economic Optimization in the Energy Conservation

Design of Single-Family Housing," ASHRAE Transactions , Vol . 82, Part I, 1976.

32
Thermal resistance (R) is defined as the exact reciprocal of the overall

coefficient of thermal transmission (U). The resistance of components

of the building envelope is one of the primary determinants of heating

and cooling energy used in buildings, especially of the residential

and small commercial type. This model will be used to show how climate

factors and energy prices can be considered directly in determining the

extent to which energy conservation modifications should be made to the

design of the building envelope. By determining the optimum level of

resistance (R

Q

) for a particular component such as the attic floor, the

model also determines the optimum coefficient of thermal transmission

(U ), because of the exactly inverse relationship between R and U. In

speaking of the optimum standard we shall henceforth be referring to

this optimum coefficient of thermal transmission (U

Q

). Thus, the more

stringent the standard (i.e., the more energy conserving), the lower

will be U .

3.2.1 Derivation of the Model

Considering only the heating load, the life-cycle costs (LCC) attributable to heat losses through one square foot of attic floor area for

given heating equipment and climatic conditions can be represented by the

following expression:

LCC = a + b • R + (UPW • 24 • DD • P Q

) (1/R). (3.1)

The notation is to be interpreted in the following manner:

a = The fixed cost component of thermal resistance ($/sq. ft.),

b = The cost per additional unit of thermal resistance ($/sq. ft./R).

R = The number of units of thermal resistance on each square foot of

attic floor area.

33
where e is the annual rate of fuel price escalation in real terms,

d is the real discount rate, and N is the number of years in the

useful life of the building. For example, with a 10% real discount

rate, a 3% rate of real fuel price escalation, and a 40 year life,

UPW = 13.65. This uniform present worth ( UPW ) factor operates

as a fixed coefficient to convert annual energy costs to their

equivalent present value.

DD

365

(B - T. ), for all T. < B, where is the average

temperature for the ith day and B is the base temperature (usually

65° F), chosen as the cutoff point below which heat is needed from

the heating system to keep the conditioned space at a target

temperature (usually 70°). This summation measure is referred to

as heating degree days and is commonly used to approximate the

annual heating load imposed by a given climate. Multiplication

by 24 simply converts degree days to degree hours.

P 0

= the effective price actually paid for one Btu of heating energy

output delivered to the conditioned space, taking account of

conversion efficiencies.

34
This expression for life-cycle costs can be minimized by partial

differentiation with respect to R, setting the first derivative equal

to zero and solving to find the optimum level of thermal resistance,

Kofi = [UPW • 24 • DD • P e

• (1/b)

f

5

. (3.2)

Since the rate of thermal transmission, U, is defined as the exact

reciprocal of R, we have the optimum coefficient of thermal transmission,

U

Q

, for a square foot of attic floor area:

Although this formulation of the model is quite simple, it provides the

most important elements needed to determine the optimum component performance standard for attics.

3.2.2 Significance of the Model

Several features of this expression for the optimum component

performance standard are worthy of note. In the first place the

optimum standard (U ) is an increasing function of the marginal

cost of adding resistance units (b), although it increases as the

square root of this cost. This means that a doubling of the price

of insulation would result in only a 41.4% increase in the maximum

heat transmission rate allowed by the optimum standard, other things

being equal J This positive relationship has the intuitive appeal

^Letting u' represent the standard after the doubling of the

marginal cost of resistance, we have U' 3 ooo = /2 • U = 1.414 U , which

means an increase of 41.4%.

UPW • 24 • DD

(3.3)

35
that the more costly insulation becomes, the more lenient will

be the standard by allowing higher rates of thermal transmission,

and consequently calling for lesser amounts of insulation.

Another significant aspect of this expression is that there is

an inverse relationship between the number of degree days (DD) and

the standard (U

Q

). This implies that buildings in more severe

climates should be subject to a more stringent standard requiring a

lower rate of thermal transmission. It should be noted that

this inverse relationship is not linear but rather is based on the square

root of degree days. Thus, a house in a climate with twice as many

degree days would be subject to a standard requiring a 29.3% lower

rate of thermal transmission.^

The final noteworthy feature of this expression for the optimum

component performance standard is that, just as with degree days, U

Q

also varies inversely as the square root of the price of energy (P

e

).

That is, a doubling of the price of energy calls for a 29,3% decrease

in U Q

, and consequently the same 29.3% decrease in the maximum energy

consumption allowed by the optimum standard. The negative direction of

this relationship makes sense in that an increased price of energy should

lead to decreased consumption of it through the substitution of additional

insulation. This last feature has a direct bearing on the effect of

applying RIF's to the model.

3.3 APPLICATION OF PRICE MULTIPLIER RIF's

As it stands, the expression for U Q developed above is based on

the goal of cost minimization from the private point of view. To

^Letting U' represent the standard for the climate with twice as many degree days, we have U

' = U / /2 = 0.707 U , which means a decrease

of 29.3%. oo o
adjust the expression to reflect the social viewpoint, it is necessary

to replace the effective price paid for energy (P ) with the social

price of energy J Using the price multiplier formulation shown in

Table 3.2 above, which gives RIF's as the ratio of the social price

to the effective price, all that needs to be done is to multiply the

symbol P g

in the expression for U

Q by the appropriate RIF of the fuel

2

type being considered. This will result in the following new expression for the optimum performance standard from the social point

if of view (U ):

* b

UPW • 24 • DD • P • RIF

e (3.4)

Thus, the relationship of the socially optimum standard to the privately

optimum standard can be established as follows:

It might also be necessary to replace the marginal cost paid for

insulation (b) by its appropriate social cost counterpart, if there is reason to believe that a significant divergence exists between its

actual cost and the social cost. If such social cost estimates are developed, then they should be used in place of b in the formula.

2 This formulation allows us to establish clearly the effect of

RIF's on the optimum standard. It is conceivable, however, that the

estimated social value of energy may not continue to be a constant

(or even constantly growing) multiple of the actual price paid for

energy over time. In this case, the proper approach to actually

determining the socially optimum standard would be to replace the product UPW • P • RIF in expression (3.4) with the summation,

N (P ) 6

21 „ , where (P 1 represents the estimated social value of the

n=l (l+d)

n s n

energy type for the nth year.

37
, or UPW • 24 • DD • P,

U = U

o o RIF (3.5)

So, in general, it can be said that for a given privately optimum

standard the socially optimum standard varies inversely with the square

root of the RIF value.

In order to analyze this relationship properly, the range of

RIF values must be established. A RIF value would always be positive

but could be less than, equal to, or greater than one, depending on

the relative importance of the various elements to be included in the

derivation of RIF's. For example, if the price effects of unit taxes

and monopoly power predominate, then the social price of the energy type

could be lower than the actual price, making RIF less than one. If

the actual price paid for one type of energy exactly reflected its

social value, then the corresponding RIF would equal one. Finally,

if factors such as external effects and price controls predominate,

then RIF's will be greater than one.

Table 3.3 shows the relationship between particular RIF values

and the resulting socially optimum standard expressed as a proportion

of the privately optimum standard. As can be seen, the value of the

RIF determines whether or not the use of RIF's leads to greater or

less energy consumption. When the RIF is less than one ( i . e . , the

social price is less than the actual price paid), then the introduction

of RIF's would have the effect of raising the level of energy consumption

38
Table 3.3

Possible RIF Values with Corresponding Socially Optimum Standards

RIF

*

Uo [1/RIF]

.50 1.414

.60 1.291

.70 1.195

.80 1.118

.90 1.054

1.00 1.000

1.10 0.953

1.20 0.913

1.30 0.877

1.40 0.845

1.50 0.816

1.60 0.791

1.70 0.767

1.80 0.745

1.90 0.725

2.00 0.707

2.25 0.667

2.50 0.632

2.75 0.603

3.00 0.577

4.00 0.500

5.00 0.447

10.00 0.316

100.00 0.100

Uo

(Equivalent to not

using RTF's)

39
allowed by the standard. This would mean less energy conservation

with RIF's than without them. If the RIF equals one, then the introduction of RIF's would have no effect on energy consumption. In the

more probable event that RIF's are greater than one, the effect of using

them is to lower the allowable energy consumption and, thus, to increase

the energy savings attributable to the standard. For example, the

introduction of a RIF equal to 1.5 (meaning the social price is 50%

higher than the actual price paid) would lower the maximum energy consumption allowed by the standard by 18.4%, to only 81.6% of what it

would be without the RIF.

An observation can be made from Table 3.3 which holds generally,

regardless of the value of the RIF applied. That is, as the RIF value

increases, the corresponding energy consumption rate allowed under the

socially optimum standard decreases relative to the privately optimum

standard, and absolutely. This inverse relationship is also apparent

from Figure 3.1 which merely plots the points given in Table 3.3. The

general pattern of the points is downward sloping to the right which

•k indicates that increases in the RIF value lead to decreases in U

Q as

a proportion of U

Q

. A more rigorous mathematical demonstration of this

observed inverse relationship requires evaluation of the sign of the

first derivative of UQ

with respect to RIF. Thus, partial differentiation

of expression (3.5) leads to the following:

*

3 R I

[-0.5 RIF"

1,5

] (3.6)

40
Social 1y

Optimum Standard

(in

units

ofU

rt )

41

Figure

3.1

Plot

of

the

Socially

Optimum

Standard

(in

units

of

Uj

asa

Function

of

the

RIF

Value,

fora

Given

Value

of

the

Privately

OptimunT

Standard

(U)
Since the value of RIF is always positive, this expression must be

negative for the entire range of possible RIF values. This negative

value of the first derivative means that the inverse relationship

(downward slope) between RIF values and U

Q

holds everywhere.

Table 3.4 presents the calculated values of this first derivative

measured in units of U for selected RIF values.

o

Table 3.4

Selected RIF Values with Corresponding First Derivatives of U

Q 9U*RIF */8RIF = 0

-1 5

[-0.5 RIF

l,D

] • uo

.50 -1.414 • Uo

.75 -0.770 • U

o 1.00 -0.500 • uo 1.25 -0.358 • Uo 1.50 -0.272 • U

o 2.00 -0.177 • Uo 3.00 -0.096 •

As indicated above, the signs of all the calculated derivatives are

negative. Moreover, an additional pattern emerges from Table 3.4.

Note that as the RIF values increase, the calculated derivatives decrease

in absolute value but increase in actual value. This means that, as

one moves to the right (increasing RIF's) on the curve plotted in Figure

3.1, the slope of the tangent to that curve gets steadily flatter. This

can be demonstrated mathematically by inspecting the sign of the second

derivative of U

Q

with respect to RIF. Differentiating expression (3.6)

we have

42
(3.7) = U [0.75 RIF’

2 ' 5

].

3RIF^ 0

Since this second derivative is positive for all positive values of

RIF, it means that the first derivative, which is negative, is always

increasing (i.e., decreasing in absolute value). Thus, as the RIF value

is increased by increments of equal size, the effect of each increase

in reducing the energy consumption rate allowable by the socially optimum

standard steadily decreases. The important implication of this result

is that there are diminishing returns in terms of the energy savings

attributable to the socially optimum standard from successive equal

increments in the value of RIF's.

3.4 LIMITATIONS OF THE ANALYSIS

The foregoing analysis showed the energy-saving effects of using

RIF's in determining the socially optimum performance standard for

a building component such as the attic floor. Generally, it was found

that the higher the RIF value, the lower would be the maximum energy

consumption rate allowed by the socially optimum standard. It was

also found that in the more likely situation of RIF values greater than

one, the energy saved by the socially optimum standard is greater than

that saved by the privately optimum standard. The analysis of the

economic efficiency effects of using RIF's will be treated in the

Appendix.

The model used in the above analysis has two general limitations:

(1) it is suitable for studying component rather than building performance

standards; and (2) it is based on some restrictive assumptions which

43
may be violated under actual conditions. A model for determining

optimum building performance standards is currently being developed

at the National Bureau of Standards. When completed, the model will

incorporate the many energy-flow interactions among the various building

components. This will allow analysis of the economic trade-offs between

interrelated components. Once this model is fully developed, the

effect of using RIF's for determining building performance standards

could be assessed.

Two restrictive assumptions underlying the model presented above

limit its general applicability. The first regards the cost of each

additional unit of resistance. It was assumed that the cost of resistance

was a linear function of the level of resistance. That is, the marginal

cost of each successive unit of resistance was assumed to be constant,

regardless of the total level of resistance. This marginal cost

component was represented by the constant parameter, b. It seems likely,

however, that within the relevant range the marginal cost of resistance

increases as more units are added. Secondly, it was assumed that the

temperature base for calculating degree days was the same regardless

of the level of resistance. This meant that DD could be treated as a

fixed parameter, independent of the level of resistance. It will be

shown that, as more resistance is added, the degree-day temperature base

is likely to fall, resulting in fewer effective degree days for a given

climate. In the next section a model is developed which takes account

of these more realistic conditions.

44
4.0 THE OPTIMUM STANDARDS MODEL GENERALIZED

In this section we modify the optimum standards selection model

of Section 3.0 by making alternative assumptions regarding the manner

in which resistance costs and the number of degree days depend on the

level of thermal resistance. We begin by maintaining our original

assumption that the number of degree days is invariant and independent

of the resistance level, while allowing resistance costs to assume

several alternative functional forms. These modifications regarding

resistance costs lead to the development of two new cases of the model

which have different formulations for the rate of thermal transmission

allowed by the optimum standard. These new formulations are each

analyzed and compared in terms of the effect of using RIF's in the

determination of the socially optimum standard. In the second subsection we consider the manner in which the number of degree days are likely

vary as a function of the level of thermal resistance. This leads to

a modification of the original model with degree days specified as a

quadratic function of the resistance level. The implications of this

formulation for the effect of using RIF's are discussed. In the final

subsection, both cost and degree days are formulated as general functions

of resistance without specifying particular functional forms. By making

some reasonable assumptions about the first and second derivatives of

these functions, some general conclusions are drawn regarding the relationship between the magnitude of the RIF values and the level of

the socially optimum standard.

45
To facilitate the presentation of these alternate forms of the model,

a slightly different, more condensed system of notation will be used

throughout this section. As in Section 3.0, the objective of the model

is to determine which level of thermal resistance is consistent

with some minimum required indoor temperature for the least life-cycle

cost. This optimum level of resistance is found by minimizing life-cycle

heating-related costs (L) expressed as a function of thermal resistance

(R). The model and explanations of the new notation are as follows:

Minimize

with respect to R L = K • D • (1/R) + C, (4.1)

where

L = life-cycle heating -related costs of a unit of attic floor

area,

K = UPW • P • 24,

e

D = the number of effective degree days, which in one variation

of the model is expressed as a function dependent on the

level of thermal resistance,

R = the thermal resistance of the attic floor, and

C = the total construction costs of a unit of attic floor area expressed as a function of thermal resistance.

4.1 ALTERNATIVE COST FUNCTIONS

As stated above, the first group of model variations maintains the

original assumption of degree days being fixed and independent of resistance, while allowing alternative assumptions regarding the manner

in which construction costs (C) depend on the level of thermal resistance.

In Section 3.0 we made the assumption that the per unit area construction

46
costs of the attic portion of the building envelope are a linear function

of the level of resistance. This assumption implies that each additional

(or incremental) unit of resistance costs the same regardless of the

linearity assumption to allow for the case of rising incremental costs

first by a quadratic function and then by cubic and higher-order functions.

For the sake of completeness, we begin by restating the original model

of Section 3.0 using the new format given by expression (4.1) above.

4.1.1 Case I: A Linear Cost Function

Under the assumptions of linear costs and fixed degree days, the

expression for life-cycle cost becomes

Differentiating with respect to R and setting the derivative equal

to zero yields the following expression for the optimum resistance,

level of total resistance. In this subsection we will relax that

L = K • D • (1/R) + a + b . R. (4.2)

R

o

K • D

b

(4.3)

This implies that the standard is given by

Uob (4.4)

K • D

In the terminology of economists, this is equivalent to assuming

that the marginal cost of resistance is constant.

47
Substituting F • K for K, where F is the RIF value, we can establish

* the relationship between the socially optimum standard U

Q

, and the

privately optimum one, U

Q

:

b

*2

1 F • K • D

• o

3

II F

As was pointed out in Section 3.0, this means that as the RIF value

increases, the rate of thermal transmission allowed by the socially

optimum standard decreases (i.e., greater levels of thermal resistance

could be needed to meet the standard), but at a steadily diminishing

rate.

With this formulation it is also possible to compare the relative

impacts of employing RIF's in different climates. For the entire

range of heating loads represented by the number of degree days (D),

the introduction of a given RIF value has the same percentage impact

on the rate of thermal transmission allowed by the optimum standards.

This is because the two different values of U

Q

in equation (4.5) are

being multiplied by the same value of F

2

, so that the percentage

change is the same. By the same token, this implies that the absolute

value of the impact on UQ

is greater for warmer climates than for colder

ones, because the base values of U Q are greater in the warmer climate.

This result is reversed, however, when one considers the relative

impact of RIF's on the annual amount (rather than the rate per degree

day) of energy consumption allowed by a socially optimum standard. A

numerical example illustrates the point. Suppose the values of the

parameters b, K, and D in expression (4.4) were such that U

Q

equalled

48
0.10 in a particular region with a relatively warm climate. A colder

climate with four times as many degree days would then require a

U Q

equal to 0.10/v/4 (or one half as high a rate of transmission),

for the same values of b and K. Let us denote the extra annual

energy savings per square foot of attic attributable to the use of

RIF's in the warmer climate by AS, and those in the colder climate

w

by aS

c

- Then we have

AS

C

= (1-F' %

)

• (24) • (4D) • (0.10//4), and

ASw

= (1-F^2

)

• (24) • (D) • (0.10).

(4.6)

Forming the ratio of the colder climate savings to the warmer climate

savings, and cancelling like terms in the numerator and denominator,

we have

AS

= 4//4 = /4 = 2. (4.7)

Thus in comparing two different climates, one with four times as

many degree days as the other, the extra annual energy savings

attributable to using RIF's are twi ce as great for the colder

climate than for the wanner one. It is important to note that this

result holds for all RIF values greater than oneJ This result can

be stated in a more general form. If a colder climate has n times

^If F=1 , then aS

c

= AS

w

= 0 and the ratio becomes meaningless.

If F < 1, then AS

C and aS w are both negative, which implies that application of the RIF would allow higher rather than lower levels of

energy consumption.

49
as many degree days as a warmer climate, the extra annual energy

savings attributable to using a positive RIF to develop the standard

for the colder climate would be /n times as much as ’that for the

warmer climate.

4.1.2 Case II: A Quadratic Cost Function

We now abandon the linear cost assumption in order to investigate

the case in which unit area construction costs of the attic floor are

formulated as a quadratic function of thermal resistance. The general

form of a quadratic cost function is as follows:

C = a + b • R + c • R^, (4.8)

where a, b, and c represent fixed parameters. This formulation

implies that the marginal or incremental cost of adding a unit of

thermal resistance is no longer constant but steadily changes as

more resistance is added. Since the more probable situation is that

the marginal cost increases (rather than decreases), the parameter

c is assumed to be positive. Thus, the marginal cost rises at a

constant rate equal to 2c J Moreover, to rule out the unlikely

situation of a negative marginal cost, we assume b _> 0. In terms

of geometry, these restrictions on b and c mean that we are confining

our analysis to the upward sloping portion of a U-shaped quadratic

curve, that is, one which is concave upward.

^By taking the first derivative of (4.8) we find that marginal

cost (MC) equals b + 2c • R. Thus, MC rises with increasing R if

c is positive, falls with increasing R if c is negative. Of course,

if c = 0, then MC = b regardless of the value of R, which is equivalent

to the linear cost case.
Under the assumption of fixed degree days and quadratic costs,

the expression for life-cycle cost takes on the following form:

L = K • D • (1/R) + a + b • R + c • R

2

. (4.9)

Differentiating this expression with respect to R and setting the

result equal to zero yields the following cubic equation:

2 c R

2 + b o

R

o

- K D = 0, (4.10)

The expression for the optimum level of thermal resistance, R

q

, can

be found by applying the formula for the solutions to a cubic equation.^

Unfortunately, this results in an expression too cumbersome to be of

any use in interpreting the relationship between RIF's and the resulting optimum resistance.

For the special case of b = 0, on the other hand, the explicit

solution for R

q can be easily derived and interpreted. The implication

of assuming b = 0 is that marginal cost has no fixed component, but

simply starts at zero and increases by 2c for every unit of resistance

added. Under this assumption, the following formula for Rq can be

derived from expression (4.10):

R

o

K • D 2c

1/3

(4.11)

^Murray R. Spiegel, Mathematical Handbook of Formulas and Tables

(Schaum's Outline Series; New York: McGraw-Hill, Inc., 1968), p. 32.

2

The geometric interpretation of b = 0 is that the MC straight

line goes through the origin.

51
This implies that the maximum rate of thermal transmission allowed

by the optimum standard is given by

U 2c

K • D

1/3 (4.12)

Substituting F • K for K, where F is the RIF value, we can establish

the following relationship between the transmission rate allowed

by the socially optimum standard, U

Q

, and that by the privately

optimum standard, U q

:

*

U o

2c

1/3

1 1/3

= U . 1 K • D

- F

L.

0

F (4.13)

As with the linear cost case, this means that as the RIF value

increases, U

0

decreases, but at a steadily diminishing rate. There is,

however, a significant difference between the two cases. Note that for

* this quadratic cost case U Q

varies inversely as the cube root of F, whereas

* 1 for the linear cost case, U q

varies inversely as the square root of F.

For example, the effect of a RIF value of 2 is to reduce the maximum rate

of transmission allowed under the quadratic cost standard by only 20.6%,

whereas we saw in Section 3.0 that the effect of the same RIF value on

the rate allowed under the linear cost standard would be a reduction of

29.3%. Thus, the higher-order cost function allows a smaller impact for

RIF's on the rate of thermal transmission.

As with the linear cost case, it is possible to compare the

relative impact of RIF's on the annual energy consumption allowed

by the standard in different climates. A numerical example helps

1 See expression (4.5) above.

52
to illustrate the relative impacts. Suppose the values of the parameters

c, K, and D in expression (4.13) were such that U

Q equalled 0.10

in a particular region with a relatively warm climate. Then, a much

colder climate with four times as many degree days would require a

UQ

equal to 0.10/4^ (or about 0.63 as high a rate of thermal

transmission), for the same values of c and K. As before we can

denote the extra annual energy savings per square foot of attic attributable to the use of RIF's in the warmer and colder climates by

aSw and aS

c

, respectively. Then we have

AS = (1 - F'

1/3

)

• (24) • (4D) • (0.10/4

1/3

), and

AS

W

= (1 - F' 1/3)

• (24) • (D) • (0.10).

(4.14)

Thus, the ratio of the colder to the warmer climate savings becomes

~~ = 4/4

1/3 = 42/3 a 2.52. (4.15)

ASw

Thus, we can conclude that for a climate with four times as many degree

days as another, the extra annual energy savings attributable to

RIF's would be about 2.52 times as great. Again note that this

relationship holds for all RIF values greater than oneJ In general,

this relationship means that for a colder climate with n times as

many degree days as a warmer climate, the extra annual energy savings

2/3

due to using a positive RIF value would be n times as much as for

the warmer climate.

^See footnote 1 on page 49 above.

53
4.1.3 Case III: Higher-Order Cost Functions

We now consider the implications of using cubic and even higherorder functions to represent the relationship between total construction

costs and the level of thermal resistance. The general form of the

functions we will analyze is

C = a + c • R

q

, (4.16)

where the coefficient c is positive and the exponent, q, is a parameter

(usually an integer) which specifies the order of the function. For

q _> 3, this form means the marginal cost of resistance not only rises

but rises at an increasing rate as R increases. In general, for given

values of a and c, both total construction costs, C, and marginal

costs rise more quickly, the greater the value of q (the higher the

order of the function).

Under the assumptions of fixed degree days and the cost function

of expression (4.16), life-cycle costs become

L = K • D • ( 1/R) + a + c • R

q

. (4.17)

Following the same procedure used for Cases I and II, we can derive

the following expression for the maximum rate of thermal transmission

allowed by the socially optimum standard:

*

U o

1 1

q

• c]

q+1

• 1

” q+1

= U • 1 K • D F u o F

1 q+i

(4.18)

This relationship means that in general the higher the order

of the cost function, the smaller will be the impact of a given

RIF value in reducing the maximum rate of thermal transmission allowed

54
by the standard. For example, we saw in Case II with a quadratic

cost function that a RIF value of 2 would reduce the maximum rate

by 20.6%. With a cubic cost function (q=3), however, this reduction

would be a mere 15.9%.

For the case of a cubic cost function, the relative impact of using

RIF's on annual energy consumption in different climates can be established.

It turns out that if a colder climate has n times as many degree days as

a warmer climate, the ratio of extra energy savings in the colder climate

3/4

to those in the warmer climate is equal to n . For example, a factor

of four times as many degree days implies 2.83 times as much energy

savings.

4.1.4 Comparison of Alternative Cost Functions

It is now appropriate to compare the alternative cost functions

presented. They will first be compared in terms of the effects of

various RIF values on the maximum rate of thermal transmission allowed

* by the socially optimum standard. Table 4.1 gives the ratio, U

o /U

Q

,

for a range of RIF values and various cost functions. The ratios in

the column entitled "linear" are the same as those in Table 3.3 of

Section 3.0.

Several patterns emerge from an analysis of these ratios. In the

first place, for all cost functions the ratio decreases as the

RIF value increases, but does so at a diminishing rateJ Secondly,

^ Thi s can also be shown in general b^ evaluating the signs of the

first and second partial derivatives of Uq with respect to F in expression

(4.18). As expected, the first derivative is negative, indicating the

inverse relationship, and the second derivative is positive, verifying

the diminishing effect.

55
Table 4.1

Ratios of Socially Optimum to Privately Optimum Standards

for Various RIF Values and Cost Functions 3

RIF

Val ue

Cost Function

Linear

(q=l)

Quadratic

(q=2)

Cubic

(q=3)

Quartic

(q=4)

1.00 1.000 1.000 1.000 1.000

1.25 0.894 0.928 0.946 0.956

1.50 0.817 0.874 0.904 0.922

1.75 0.756 0.830 0.869 0.894

2.00 0.707 0.794 0.841 0.871

2.25 0.667 0.763 0.817 0.850

2.50 0.633 0.737 0.795 0.833

2.75 0.603 0.714 0.777 0.817

3.00 0.577 0.693 0.760 0.803

a These ratios are calculated by the following relationship derived

from expression (4.18):

-1 U*/U = F q+1

o o

56
for any given RIF value as one moves from lower to higher- order cost

functions (reading from left to right across one row of the table), the

ratios get larger and closer to unity. This implies that as q increases,

the effect of RIF's in lowering the maximum rate of transmission decreases. Thirdly, a unit change in the RIF value has less of an

effect on the ratios, the higher the order of the cost function. For

example, an increase in the RIF value from 2.00 to 2.25 yields a change

of 0.040 for the linear cost case, while the same increase in the

RIF leads to a change of only 0.031 for the quadratic cost case.

Another type of comparison can be made regarding the alternative

cost functions. We can analyze the relative impact of using RIF's

on annual energy consumption in different climates for all cost functions.

If a colder climate has n times as many degree days as a warmer climate,

the ratio of extra energy sa vings in the colder versus the warmer

climate turns out to equal n^ , where q is the order of the cost

function. Table 4.2 presents these ratios for a range of values

of n and the various cost functions. For example, in a climate with

twice as many degree days* RIF's would have a 41.4% greater impact on

annual energy consumption with a linear cost function, but they would

have a 58.7% greater impact with a quadratic cost function.

There are two general conclusions that can be drawn from these

ratios. First, note that for any given value of n, the ratios increase

as the order of the cost function (q) increases. This means that

the energy-saving impact of having more degree days is greater for

57
Table 4.2

Ratios of Extra Energy Savings Due to RIF's in Colder

Versus Warmer Climates, for Various Cost Functions 5

Degree-Day

Multiple

(n)

Cost Function

Li near (q=D

Quadratic

(q=2)

Cubic

(q = 3)

Quartic

(q=4)

1.00 1.000 1.000 1.000 1.000

1.50 1.225 1 .310 1.355 1.383

2.00 1.414 1.587 1.682 1.741

2.50 1.581 1.842 1 .988 2.081

3.00 1.732 2.080 2.280 2.408

4.00 2.000 2.520 2.828 3.031

5.00 2.236 2.924 3.344 3.624

6.00 2.450 3.302 3.834 4.193

a The ratios of

relationship:

the

AS

c ASw

extra savings

_9_

q+1

= n M

are calculated using the following

58
higher-order cost functions . 1 The second point is that for all values

of q the ratios increase as n increases. This simply means that the

greater the difference between the colder and warmer climates, the

greater would be the relative energy-saving impact of using RIF's.

Moreover, the effect of a one unit change in n decreases as n increases.

This means that the same difference in degree days leads to extra

annual energy savings due to RIF's which are greater when comparing

two relatively warm climates than when comparing two relatively cold ones.

This concludes our analysis of alternative cost functions. There

are a host of other forms possible (e.g., trigonometric, logarithmic,

exponential). Most of these would not lead to an explicit expression

for U

Q

. Moreover, the family of polynomial forms which have been

analyzed adequately represents the probable way in which unit area

construction costs of the attic floor are expected to vary with the level of

thermal resistance. There is a shortcoming, however, which all these

functional forms share. They assume that costs vary continuously. Because

glass fiber batts or blankets come in a limited number of thicknesses, it

would be more accurate to have these construction costs vary discretely as

more resistance is added. For the loose fill type of insulation, on

the other hand, the continuous assumption is quite realistic. Moreover,

even for the case of limited sized batting, the continuous assumption

is valid for establishing the general relationships between resistance

^ote that this effect of increasing q gradually diminishes and approaches

the limiting value of n as q gets arbitrarily large. That is,

i

• q+i nm n M = n. q

-> oo

59
costs, energy costs, degree days and RIF's on the one hand, and the

socially optimum energy conservation standard, on the other. When the

actual calculations are made to develop the standard, the discrete nature

of these costs can be taken into account.

4.2 DEGREE DAYS AS A FUNCTION OF RESISTANCE

Up to this point we have maintained our original assumption that

the number of degree days (D) associated with the heating requirements

of a building is invariant with respect to the level of resistance (R).

In fact, it is expected that as more units of thermal resistance are

added to the building envelope, the number of degree days effectively

contributing to the heating load will decrease. It is further expected

that as R is increased, the rate at which D falls' will diminish. These

expectations suggest that degree days could be adequately represented

by a quadratic function of resistance.

1

The reasoning leading to these expectations regarding the pattern

of variation of degree days is best explained by distinguishing between

the two links in the chain of causality from R to D: (1) an increase

in resistance will cause the effective degree-day temperature base to

decrease, (2) which, in turn, will lower the number of degree days in

the heating season. Each of these links will be explained and illustrated

in turn.

Vor the sake of completeness, the life-cycle model was also analyzed

for the case of degree days assumed to be a linear function of resistance.

This assumption leads to an expression for R

Q

which would have a higher

value than that which results from the fixed degree-day assumption. This

result has the intuitive appeal that when the extra benefit of lowering

the number of degree days is attributed to resistance, it makes sense

to select a higher level of resistance than would otherwise have been chosen.

60
4.2.1 Degree-Day Temperature Base and Resistance

With regard to the first link, the effective degree-day temperature

base is given by that critical temperature below which a supply of heat

to the conditioned space is needed in order to maintain a given indoor

temperature. This critical temperature is sometimes referred to as the

balance point (B) and is generally less than that specified by the

desired indoor winter temperature, because solar radiation and the normal

activities of a household (lighting, hot water use, appliances, and body

warmth) replace some of the heat being lost through the building envelope.

Given an indoor temperature requirement and rates of solar gain and incidental internal heat generation, the lower the rate at which heat is

being lost through the building envelope, the lower the balance point.

This means that there is an inverse relationship between the balance

point and the level of thermal resistance up to a limit determined by

the rate of air infiltration in the building. The impact on the balance

point of adding one unit of resistance, however, gradually diminishes as

more units are added because the direct effect of more resistance on

the rate of heat loss also diminishes. This implies that unit increases

in resistance lead to reductions in the balance point but at a decreasing

rate.

This inverse relationship between resistance and the balance point

can be illustrated with a numerical example. Suppose the occupants

of a building with 3000 sq. ft. of envelope area having an R-value of 5

set the thermostat at 70° F. If the heat loss due to air infiltration

is 200 Btu's per hour per degree of indoor-outdoor temperature difference.

61
then an assumed balance point of 65° F implies that spontaneous heat

gains from solar radiation and household activities equal 4000 Btu's per

hour. That is, if the outdoor temperature were exactly 65° F, then the

4000 Btu's of spontaneous heat gains would just offset the heat losses

due to conductance and infiltration so that the desired indoor temperature

of 70° F could be maintained without heat from the furnace. Now by

successively adding 5 more units of resistance to the original resistance

level of R-5, we can calculate the effective balance point for each new

level of resistance. The results of these calculations are presented

in Table 4.3. It is clear that as R is increased, B decreases. Moreover,

note that the rate of decrease of B diminishes. For example, increasing

R from 5 to 10 causes B to drop 3 degrees, whereas increasing R from

10 to 15 causes a drop of only 2 degrees. In fact, B approaches a theoretical limit of 50° F, corresponding to an infinite level of resistance

(no heat losses due to conduction).

4.2.2 Degree Days and the Degree-Day Base

The second link in the chain of causality between resistance and degree

days deals with the way in which these reductions in the balance point

affect the actual number of degree days in particular climatic areas. The

pattern of this relationship is illustrated in Table 4.4 with data for

several representative cities. Note that for all the cities as the balance

point drops the number of degree days decreases but at a slightly diminishing

rate. For example, in Washington the one-degree shift from 65° to 64°

would cause a 213 degree-day decrease, whereas the one-degree drop from 64°

to 63° would cause a 208 degree-day decrease. This pattern continues until

the one-degree drop from 51° to 50° causes only a 150 degree-day decrease.

62
Table 4.3

Effective Balance Points for Various Levels of Thermal Resistance:

Hypothetical Example8

Thermal Resistance(R) Balance Point (°F)

5 65.0

10 62.0

15 60.0

20 58.6

25 57.5

30 56.7

35 56.0

40 55.5

45 55.0

50 54.6

55 54.3

60 54.0

00 (U= 0) 50.0

a The following assumptions underlie these calculations: (1) the

envelope area (A) is 3000 sq. ft.; (2) the desired indoor temperature

(T) is 70°F; (3) the spontaneous heat gain (H) from solar radiation

and household activities is 4000 Btu's/hour; and (4) heat loss (I)

due to air infiltration is 200 Btu/hour/degree of indoor-outdoor

temperature difference. These parameter values are substituted into

the following formula for calculating the balance point corresponding

to each resistance level:

B = T - [imr .

63
Table 4.4

Degree Days Corresponding to Various Balance Points

for Six Major U.S. Cities 3

City

it ( U

F) Chicago New York Washington San Francisco Dallas Los Angeles

65 6509 - 5217 4667 3510 2552 1524

64 6250 4976 4454 3192 2391 1310

63 5998 4741 4246 2885 2236 1112

62 5752 4512 4044 2591 2087 938

61 5512 4289 3849 2310 1942 781

60 5278 4071 3659 2042 1803 639

59 5049 3859 3472 1789 1670 514

58 4825 3653 3290 1554 1541 402

57 4605 3453 3112 1331 1416 308

56 4389 3258 2939 1127 1296 224

55 4178 3067 2769 945 1180 151

54 3971 2880 2604 769 1069 97

53 3768 2698 2444 603 961 68

52 3566 2519 2288 455 857 44

51 3371 2343 2135 320 755 26

50 3178 2172 1985 193 656 10

a These data were made available through a private communication from Richard Erth,

York Division of Borg-Warner Corporation, York, Pa. 17405. Monthly data were

chosen to be representative of a ten-year historical pattern. Daily temperatures

are based on the average of 24 hourly readings, rather than the midpoint of the

high and low temperatures

.

64
4.2.3 Case IV: A Quadratic Degree-Day Function

Now it is possible to bring these two links together to establish

the relationship between degree days and the level of thermal resistance.

By combining the data from Table 4.4 on Washington with that in Table

4. 3, we can plot a curve representing degree days as a function of resistance.

This curve is shown in Figure 4.1. As is apparent from the shape of the

curve, degree days decrease as the level of resistance increases. Moreover,

the rate of decrease in degree days gradually diminishes. The pattern of

the relationship shown in Figure 4.1 can be represented reasonably well by

expressing degree days as a quadratic function of thermal resistance:

D=d + e- R + f-R2

, (4.19)

where d, the intercept is positive, f must also be positive to guarantee

upward concave curvature, and e is negative because the slope is negative.

Now we can apply this assumption of quadratic degree days and maintain

our original linear cost function to obtain the following expression for

life-cycle cost:

L = K • (d + e • R + f • R

2

)

• (1/R) + a + b • R. (4.20)

Differentiating L with respect to R, solving for the optimum level of

resistance, and inverting yields the following expression for the optimum

rate of thermal transmission:

65
Degree Days

66

Degree

Days

as

a

Function

of

Thermal

Resistance

for

a

Typical

House

in

Washington,

D.C.
I (4.21)

When RIF's are introduced by substituting F • K for K in this expression,

we obtain the maximum rate of transmission allowed by the socially optimum

standard:

* b/F + f • K

d • K

(4.22)

Unfortunately, expression (4.22) does not permit the separation of F

which would be required in order to express the socially optimum transit mission rate, UQ

, as a function relative to the privately optimum rate,

U . The best that can be done under these circumstances is to assess the

* general direction of the relationship between F and U

Q

by evaluating the

sign of the first derivative. Thus, we have

*

8F

- 1/2

"b/F + f • K‘ ~h b

*

1 d • K d • K

F

2 (4.23)

Since it is assumed that additional units of resistance have a positive

cost, we have b > 0. Moreover, since degree days are assumed to decrease

at a decreasing rate, with increases in resistance, we have d and f both

positive. Finally, since the only meaningful values of F and K are all

positive, we know that the derivative is less than zero, which means that

•k in general U

Q

is a decreasing function of F. That is, the higher the value

of the RIF, the lower will be the maximum rate of thermal transmission

allowed by the socially optimum conservation standard. This is the same

67
general conclusion we have seen with the previous formulations of the

model. The higher RIF value implies a higher social value being imputed

to energy, which in turn justifies more investment in energy conservation

so as to lower the rate of energy consumption. On the other hand, whether

*

U Qdecreases at a diminishing rate as it does in Cases I through III

can only be determined by evaluating the sign of the second derivative.

Unfortunately, it turns out that this sign cannot be unambiguously determined without actually specifying the magnitudes of the variables in

expression (4.23).

4.3 Case V: THE GENERAL OPTIMUM STANDARDS MODEL

So far in Cases I through IV we have assumed alternative specific

functional forms to represent the manner in which the level of thermal

resistance may affect the building envelope construction costs on the

one hand, and the number of degree days on the other hand. Rather than

assuming particular algebraic forms for the cost and degree-day functions,

we now employ general functional forms to represent these relationships.

That is, we simply state that the number of degree days is a function of

the level of resistance, D(R), and similarly that cost is a function of

resistance, C(R). Thus, the expression for life-cycle cost becomes

L = K • D(R) • (1/R) + C(R) . (4.24)

Moreover, it is reasonable to impose certain restrictions on the first and

second derivatives of the functions D(R) and C(R). These restrictions are

stated and briefly interpreted in Table 4.5.
Table 4.5

Restrictions on the First and Second Derivatives

of D(R) and C(R)

Restriction Interpretation

1. D'(R) < 0 [D is a decreasing function of R1 2. D" (R) > 0 [D decreases at a decreasing ratel

3. C'(R) > 0 [C is an increasing function of R]

4. C" (R) > 0 [C increases at an increasing or constant rate!

The rationale behind restrictions 1 and 2 regarding degree days was

already discussed under Case IV (Subsection 4.2). Restriction 3 simply

means that the addition of one unit of resistance increases rather than

decreases the total amount spent on resistance (i.e., the marginal cost

is positive). Restriction 4 implies that as more units of resistance

are added, each successive unit costs more or the same as the previous one.

When the derivative of expression (4.24) with respect to R is set

equal to zero, we obtain

K • [R

q

• D'(R

0

) - D(R

q

)] + R

Zq

C' (R

Q

) = 0. (4.25)

As before, the RIF value can be introduced into this expression by substituting F • K in place of K. Although the resulting expression does

not permit the explicit solution for the socially optimum level of

* resistance Rq

, it is possible to determine the general direction of the

* * effect of RIF's upon R

q

and in turn upon its inverse, U

Q

. By employing

69
the implicit function rule of calculus,^ we can find the derivative

•k of R

q

with respect to F:

dR

( dF

K[D(R*) - R* • D'(R*)]

K • R* • D" ( R*

) + 2R*

0 0 0

C ' (R

Q

) + (Rq ) 2

« C'(Rq)

(4.26)

Using the restrictions specified in Table 4.5 and the fact that F, K,

D(R), and R

q are all positive, it can be seen that the entire expression

is greater than zero. This means that the higher the RIF value introduced into the standard setting process, the greater the level of thermal

resistance required to meet that standard.

This result can be restated in terms of the effect of RIF's on the

maximum rate of thermal transmission allowed by the socially optimum

standard. What is needed is the sign of the derivative of U Q

with respect

to F.

Since in general U is the reciprocal of R, we have

U* = 1/Rq. (4.27)

^The implicit function rule states that for an implicit function

of two variables G(x,y) = 0,

dX = _ 9G/ax

dx " 8G/3y

’

See Jack R. Britton, Calcul us (New York: Holt, Rinehart and Winston, 1961)

p. 414.

70
Differentiating, we have

dU

o *

2

-4 = - i/r

0

< °- dR

o

By the chain rule, we know that

* * i dU dU0 dR„

0 0 0

dF * dR

dF

L 0

(4.28)

(4.29)

From expression (4.28) the bracketed term on the left is negative and

from our discussion of expression (4.26) the term on the right is

positive. Thus, the product is negative and we can conclude that the

use of higher RIF values will lead to the selection of a lower maximum

rate of thermal transmission allowed by the socially optimum standard.

It would be possible to investigate the second derivative of Rq

with respect to F to discover whether the effect of RIF's diminishes

or increases as higher RIF values are introduced. Unfortunately , however,

the expression for the second derivative contains third-order derivatives

of the degree-day and cost functions. Without actual empirical work,

it would be meaningless to attempt to make assumptions regarding the signs

^The chain rule for derivatives simply states that

dy^ = dy . ch

dx du * dx

See Spiegel, Mathematical Handbook of Formulas and Tables , p. 53.

71
of these third-order derivatives. Hence, the sign of the second derivative

* of R

q with respect to F could not be established.

Comparison of these results of the general model with those of the

earlier, more specific model formulations reveals the same basic conclusion

the higher the RIF value used in deriving the socially optimum standard,

the lower will be the corresponding maximum rate of thermal transmission

allowed by that standard, ceteribus paribus . On the other hand, while

for Cases I through III (fixed degree days) we were able to conclude

that this effect of higher RIF values tends to diminish, we were unable

to reach such a conclusion for Cases IV (quadratic degree days) and V (the

general model).

72
5.0 SUMMARY, CONCLUSIONS AND FURTHER RESEARCH

This final section offers a brief summary of the report and presents

the conclusions regarding the likely effects RIF's would have on the

energy conservation performance standard. The section closes with

several suggestions for further research.

5.1 SUMMARY

Our purpose has been to analyze the effects of using RIF's in the

development of socially optimum energy conservation performance standards

for buildings. We have discussed the various types of standards and the

Energy Conservation and Production Act of 1976, which calls for the

development of such a conservation performance standard. The language

of this Act led us to concentrate our analysis on the effects of RIF's

on an optimum standard, that is, a standard which reduces energy use to

the extent that is economically justifiable while maintaining the habitability

requirements of building occupants. The costs considered in such a

standard include all expenditures occurring over the life of the building

which relate to the energy uses governed by the standard --both first

costs, such as for insulation or storm windows, and recurring annual

energy costs. These cost items are put on a time-equivalent basis by

discounting the future items to their present values. The process of

minimizing these life-cycle costs implies a trade-off between expenditures

for energy conservation and expenditures for energy. Thus, the price of

energy used in the analysis is critical in determining the optimum

standard.

73
Section 2.0 showed how the actual price paid for energy may not

necessarily reflect its true value to society. Some of the possible

reasons for a divergence between the actual price and the social value

of energy were presented. These reasons include the presence of unit

taxes or of monopoly power, environmental effects due to production or

consumption, the desire for national economic independence, and the

existence of price controls such as those on natural gas. Because of

this divergence between actual energy prices and their social values, it

is necessary to adjust the actual prices before using them to determine

the socially optimum conservation standard. RIF's provide a method of

making such an adjustment.

Section 3.0 began by discussing two alternative methods of formulating

RIF's: (1) as a quantity multiplier; or (2) as a price or value multiplier.

Then, a life-cycle cost minimization model was presented for determining

the economically optimum level of thermal resistance in a building component such as the attic floor. Because of the exactly inverse relationship

between resistance and the rate of thermal transmission, this model also

determines the optimum rate of transmission which forms the basis for

the optimum component performance standard itself. RIF's were introduced

into this model so that the socially optimum standard (using RIF's)

could be compared with the privately optimum standard (without RIF's).

The relationship between the social and private standards was derived

and analyzed. This analysis has several limitations. In the first

place, it is addressed to component performance standards rather than

74
building performance standards. As noted below in the suggestions for

further research, the proper analysis of the effect of RIF's on building

performance standards will be possible only after the energy-flow interactions

between building components are adequately modeled. A second limitation

is that this analysis focuses on a particular component, insulation in

the building envelope. On the other hand, the same type of inverse

relationship found to exist between the social value of energy and the

optimum level of thermal resistance is expected to hold for the performance

characteristics of other building components such as space heating and

cooling equipment, illumination, and water heaters. Finally, for the

particular component analyzed, the restrictive assumptions made regarding

envelope construction costs and degree days may be violated under actual

conditions.

Section 4.0 responded to this last limitation of the analysis by

generalizing the cost minimization model. Four additional , more realistic

sets of assumptions were developed and the corresponding formulas for

the optimum standard were derived. Then, RIF's were introduced into

these models and the effect of using RIF's was analyzed in terms of the

amount of energy saved. The additional energy savings due to RIF's were

also studied in relation to the number of degree days in the climate

region.

5.2 CONCLUSIONS

The major conclusions of this report fall into three areas: (1)

the appropriate method of formulating RIF's: (2) the potential effect
RIF's would have on energy consumption in buildings; and (3) the effect

of using RIF's on the economic efficiency of the selected energy conservation

standard. These conclusions are presented in turn.

5.2.1 Proper Formulation of RIF's

We conclude that the appropriate method of formulating RIF's is as

a price multiplier rather than as a quantity multiplier. The application

of a quantity multiplier RIF results in impact numbers which would serve

well to compare alternative energy types. Because these impact numbers

are not denominated in dollars, however, they would be unsuitable for

determining the economic balance between energy and nonenergy resources.

To achieve such a balance, a common basis for comparison is needed. The

price multiplier formulation provides this common denominator because

the RIF value converts the actual price paid for the energy to its

corresponding social price. The resulting social price remains denominated

in dollars and thus, is comparable with the values of nonenergy resources.

Thus, with price multiplier RIF's an economic trade-off can be established

between the proper amount of a particular type of energy and the appropriate

level of insulation.

5.2.2 The Energy Consumption Effect

In general, it was found that the introduction of RIF's would have

a restraining influence on the energy consumption of new buildings. The

most probable situation is that price multiplier RIF values would be

greater than one (that is, the social value of energy is most likely

greater than the price paid for energy). This is because those factors

76
which cause the social price to be higher than the actual price (environmental

effects, price controls, and the national desire for economic independence)

are likely to outweigh the other factors (unit taxes and monopoly power).

We can indicate the order of magnitude of the additional energy

savings that might result from the use of RIF's in determining the

standard for the case of natural gas. The current regulated price for

interstate natural gas at the wellhead is $1 - 42/MCF . On the other hand,

a rough approximation of the social cost of using natural gas can be derived

from the fact that synthetic gas from Western coal is expected to cost

about $4.00/MCfJ Thus, the corresponding RIF value would be 4.00/1.42 -

2.82. Introducing this figure into expression (3.5) for the socially

optimum standard, we find that the use of RIF's would lower the maximum

rate of thermal transmission and hence, the annual energy consumption

for space heating allowed by the standard by 40.5% from what they would

have been without RIF's. A smaller reduction (29.2%) in the optimum

rate of transmission results from the same RIF value of 2.82 when the cost

function is assumed to be quadratic rather than linear. Finally, with

a cubic cost function, the same RIF value leads to only a 22.8% reduction

in the maximum rate of thermal transmission allowed by the socially

optimum standard.

It was also found that the energy consumption effect of introducing

RIF's depends on the severity of the climatic heating load. That is,

the greater the number of degree days in an area, the greater will be

the additional energy saved due to the introduction of RIF's. More

^See Edward F. Renshaw, "A Cost Sharing Approach to the Conservation

of Natural Gas," Public Utilities Fortnightly, Vol . 98, No. 1 (July 1, 1976),

pp. 37-39.

77
specifically, for the linear cost model it was found that doubling the

number of degree days leads to a 41.4% increase in the amount energy

saved from RIF's. On the other hand, when envelope construction costs

are assumed to be a quadratic function of the level of thermal resistance,

it turns out that doubling the number of degree days leads to a 58.7%

increase in RIF-induced energy savings. In general, this impact of

degree days on the energy savings due to RIF's, steadily increases as

the order of the cost function increases.

The ability to use a particular RIF value to find the corresponding

percentage reduction in the energy consumption allowed by the standard

is limited to Cases I through III (assuming fixed degree days) of the

optimum standards determination model. For the other cases we are able

to reach conclusions only regarding the general direction of the relationship

between RIF values and the allowable energy consumption level. That is,

for both the case in which degree days are assumed to be quadratic (Case

IV) and the case in which degree days and costs were written as general

functions of thermal resistance (Case V), we can conclude that the

higher the RIF value used in determining the socially optimum standard,

the lower ( i . e . , more restrictive) will be the maximum allowable energy

consumption level specified in that standard. This makes intuitive

sense because a higher RIF value implies a higher social value of energy,

given the actual private price paid for it. It seems reasonable for

society to recommend lower consumption levels for those resources which

are valued more dearly.

78
4 5.2.3 The Economic Efficiency Effect

Our conclusion is that the use of RIF's will improve the economic

efficiency (from the national standpoint) of an energy conservation

standard based on the actual private prices paid for energy. As we have

seen in Section 2.0, in order to reflect the national viewpoint, a

standard should be based on the social values of energy types, which are

equivalent to the product of the private prices times the corresponding

price multiplier RIF values. Using the private rather than the social

prices of energy results in an incorrect choice of the standard --

one that is not socially optimum. This deviation of the privately

optimum from the socially optimum standard can be termed economically

inefficient because an opportunity for a net economic gain to society is

being foregone. That is, the use of a RIF value greater than one will

lead to an additional reduction in energy consumption (over and above

that of a privately optimum standard) whose social value exceeds that of

the additional thermal resistance required to achieve it. This net gain

in economic efficiency (value of additional energy savings less the cost

of achieving them) which results from the use of RIF's can be depicted

graphically.

The geometric measure of this gain in economic efficiency is developed

and illustrated in the Appendix. In essence this measure is denominated

in the same units as the annual energy consumption for space heating

attributable to a square foot of the building envelope. The change in

envelope construction costs due to the use of RIF's is converted to

79
equivalent units of annual energy consumption by dividing by the social

price of energy and the uniform present worth factor.

1 This equivalent

value of construction costs can then be combined with the change in

annual energy consumption due to the use of RIF's. In the case of a

RIF value greater than one, the additional construction costs are subtracted from the additional annual energy savings to arrive at the net

efficiency gain. In the case of a RIF value less than one, the additional

energy consumption is subtracted from the additional construction cost

savings to yield the net gain in economic efficiency.

5.3 SUGGESTIONS FOR FURTHER RESEARCH

There are two major areas related to RIF's in which further research

is needed. The first and most obvious need is for the actual development

of empirical estimates for RIF's. This report has suggested that the

most useful formulation for RIF's is as price multipliers equivalent to

the ratio of the social price of the energy type to its actual price.

Section 2 briefly outlined the nature of those factors which cause the

actual price paid for energy to deviate from its true social value.

What remains to be done is: (1) to develop workable models which adequately

reflect how each of those factors operate in reality; (2) to apply those

models to derive empirical estimates for the magnitude and direction of

each type of price-distorting factor; (3) to employ these empirical

estimates to arrive at an overall numerical value for the social price

of each energy type; and (4) to divide this social value by its corresponding

actual price to obtain the price multiplier RIF value.

^ Th i s makes annual energy consumption the numeraire.

80
The second area which requires further research is that of the

relationship of RIF's to building performance standards. The analysis

of Sections 3.0 and 4.0 was carried out using models to develop component

performance standards for building envelope characteristics as they

relate to space heating requirements. A full understanding of the

relationship between RIF's and energy conservation performance standards

will only be possible after the model for determining optimal building

performance standards, which is currently being developed at NBS, has

been completed. This model will incorporate the many energy-flow interactions

which occur among the various components of a building. Thus, it will

be possible to analyze the economic trade-offs between interrelated

components. Moreover, the model will be able to take explicit account

of the changes in the degree-day base which result from energy conservation

techniques. This building performance standard model will be able to

determine the socially optimum standard by basing the economic optimization

routine on the social values of energy types. Thus, the effects of using

RIF's will be determined by comparing the standard based on social values

with one based on the actual private prices paid for the energy resources.

81
BIRL IOGRAPHY

1. Arad, Uzi ; Smernoff, Barry J.; and ben-Shaber, Haim. Amer i can Security

and the Internat ional Energy Situation. 4 Vols. Croton-on-Hudson , N.Y.;

Hudson Institute, April 1975.

2 . ASHRAE Standard 90-75: _Ejoe

r

q.y Conserv ati

o

n i n New Bu i 1 di ng Design . New York: American Society of Heating, Refrigerating arid Air Conditioning

Engineers, Inc. (ASHRAE), 1975.

3. Brannon, Gerard M. (Ed.) Studies i n Ene rgy Tax Pol icy. Cambridge, Mass.:

Ballinger Publishing Company, 1975.

4. Britton, Jack R. Calcu lus . New York: Holt, Rinehart and Winston, 1961.

5. Erth, Richard. Private communication concerning the relationship between

degree days and the balance point for selected U.S. cities. York Division

of Borg-Warner Corporation, York, Pa. 17405.

6. Hoffman, Kenneth C. "Some Comments on Energy Accounting," Paper presented

at panel discussion on Energy, Environment and Economics, 68th Annual

Meeting, American Institute of Chemical Engineers, November 1975.

(Available as BNL 20567 from the National Technical Information Service.)

7. House of Representatives 12169, "Energy Conservation and Production Act."

Public Law 94-385 (August 14, 1976).

8. Huettner, David A. "Net Energy Analysis: An Economic Assessment."

Science , Vol . 192 (April 9, 1976), pp. 101-104.

9. Jones, Robert R. "Resource Impact Factor (RIF) Approach to Optimal Use of

Energy Resources." ASHRAE Journal , Vol. 18, No. 10 (October 1976), pp. 15-18.

10.

Leftwich, Richard H. The Price System and Resource Allocation . 3rd ed.

New York: Holt, Rinehart, and Winston, 1966.

11.

Lin, Steven A . Y . ( Ed . ) Th eory and Measurement of Economic Externalities . New York: Academic Press, 1976.

12- Little, Ian M.D., and Mirrlees, James A. Manual of Industrial Project

Analysis in Developing Countries. Vol. II: Social Cost Benefit Analysis . Paris: Development Centre of the Organization of Economic Co-Operation

and Development, 1968.

13 . Mansfield, Edwin. Microeconomics: Theory and Applications . New York:

W.W. Norton & Company, 1970.

14- Oklahoma University Science and Public Policy Program. Energy Alternatives : A Comparative Analysis . Springfield, V a.: U.S. Dept, of Commerce, National

Technical Information Service, 1975.

15- Petersen, Stephen R. "Economic Optimization in the Energy Conservation

Design of Single-Family Housing." ASHRAE Transactions , Vol. 82, Part I, 1976.

16. Petersen, Stephen R. Retrofitting Existing Housing for Energy Conservation:

^LL. Econ omic Analysis . Building Sc i ence Ser ios 64. Wash mg to rf," D.C.:

U.S. Department of Commerce, 1974.

82
17. Phipps, Harry H. "The RUF Concept as an Energy Measurement Tool."

ASHRAE Journal , Vol . 18, No. 5 (May 1976), pp. 28-30.

18. Renshaw, Edward F. "A Cost Sharing Approach to the Conservation of

Natural Gas." Public Utilities Fortnightly, Vol. 98, No. 1 (July 1 , 1976), pp. 37-39.

19. Spiegel, Murray R. Mathematical Handb ook of Formulas and Tables . Schaum's Outline Series. New York: McGraw-Hill, Inc., 1968.

20. Staaf, Robert J., and Tannian, Francis X. (Eds.) External ities : Theoretical Dimensions o f Po litical Economy . New York: Dunellen

Publishing Company, Inc'. , N . D

8J

APPENDIX

Economic Efficiency Effects of RIF's

As was mentioned in Section 5.0, the use of RIF's leads to a conservation

standard with greater economic efficiency than the standard developed without

RIF's. This is because the social value of the extra energy saved by using

RIF's exceeds the additional cost required to save it. The purpose of this

appendix is to present a geometric measure of this net gain in economic

efficiency due to the use of RIF's.

Figure A.l illustrates the physical trade-off possibilities between

the level of thermal resistance (R) and the corresponding annual energy

consumption (A) required to replace the heat lost through one square foot

of attic floor surface for a given climate and desired indoor temperature

level. The curve, TT', is drawn as an exact rectangular hyperbola because

A is inversely proportional to R according to the following expression:

A = (DD 24)/R, (A.l)

where DD is a parameter representing the number of degree days which is

determined by the given climate and indoor temperature. For convenience, DD

is taken to be a fixed parameter, as was assumed for the model developed

in Section 3.0. The alternative assumption of quadratic degree days discussed

as Case IV in Section 4.0 would not change the basic analysis, but only

slightly alter the position, slope, and curvature of the relationship. It

should be noted that an infinite number of combinations of R and A are available which satisfy the indoor temperature requirements and climatic conditions

A-l
Annual

Energy

Consumption

(A)

Fi qure A.

1

Trade-off Curve between Energy Consumption and Thermal

Resistance for Given Climate and Indoor Temperature
specified. That is, one can achieve the desired indoor temperature by

using energy at a high annual rate with little thermal resistance, or by

using less energy with more resistance. Which of the many combinations

of R and A should be chosen depends on the cost of energy relative to

that of thermal resistance.

Figure A. 2 has the same trade-off curve, TT 1 , as shown above but

introduces the concept of relative costs using the straight line, LL

1 J

This is called an isocost line because it represents all those possible

combinations of A and R which lead to the same total present value of

life-cycle heating-related costs, given the price of energy (P ) and of

resistance (b) and the uniform present worth factor (UPW). The algebraic

formula for this line is given by

L = (P . UPW) • A + b • R, (A. 2)

where L is a parameter representing the total present value of life-cycle

2 heating-related costs. This equation can be rewritten to express A

as an explicit function of R so that the intercept and slope can be

clearly seen:

A = L/(P • UPW) - [b/(P

e

• UPW) ] • R. (A. 3)

By varying the value of the parameter L, the A-axis intercept is changed

so that the isocost line can be lowered or raised. The particular isocost

line drawn in Figure A. 2 is just tangent to the trade-off curve TT'

,

^If thermal resistance costs are assumed to be a quadratic or cubic rather

than a linear function of the level of resistance, then LL' would be concave downward rather than a straight line.

2 This formulation assumes there is no fixed cost component for thermal

resistance. The inclusion of such fixed costs would unnecessari ly complicate

the argument without altering the conclusion.

A-3
Annual

Energy

Fiqure A.

2

Privately Optimum Energy Consumption Rate (A

q

)

A-4
which means that the value of L chosen is the lowest possible life-cycle

cost which will still satisfy the indoor temperature requirements and

climatic conditions given by TT ' . Hence, the values, A

q and R , given by the

point of tangency can be considered optimum for the particular ratio of

relative prices given by the slope, b/(P • UPW). Since the actual

price paid for energy, P^, was used rather than the social price,

P 0

• RIF, it can be said that A

q represents the rate of annual energy

consumption which would be embodied in the privately optimum conservation

standard.

In Figure A. 3, two more isocost lines have been added, both of which

have the same slope, b/(P • RIF • UPW), based on the social price of

energy. The first line MM 1 is drawn to be tangent to the trade-off curve

•k TT' so that the value A

q at the point of tangency is interpreted as the

social-ly optimum rate of annual energy consumption embodied in the standard.

The second line NN

1 is drawn parallel to MM* by using the same slope,

but it is made to go through the original point of tangency of isocost

line LL

1 with trade-off curve TT'. This construction allows us to compare

the life-cycle heating-related cost of the socially optimum standard

with that of the privately optimum standard. The former is given by

M and the latter by N, both found on the A axis. These cost figures are

both denominated in units of annual energy consumption and represent the

sum of the energy use plus the thermal resistance cost, with the latter

being converted to equivalent energy units by division by the relative social

price of energy.

^This procedure means that annual energy consumption serves as the

numeraire.
Annual

Energy

Figure A.

3

Socially Optimum Energy Consumption Rate (A ) and the Net Efficiency Gain from Using RIF's (N-M)

A-6
The proof that N represents the life-cycle heating-related cost under

the privately optimum standard is demonstrated in the following steps:

1. A

q

= the energy use under the standard.

2. R = the resistance used under the standard.

o

I

I 3. To sum these we can convert the units of R to their equivalent

o

M energy units using the relative price ratio based on the social

price of energy. Thus, life-cycle cost measured in energy units

equals A

q

+ [b/(P

g

• RIF • UPW) J

• R .

4. Now the tangent of the angle a in Figure A. 3 is equivalent to the

absolute value of the slope of line NN ' , which we know to be b/(P • RIF UPW).

5. But by defirtition, tan a

= (N-A )/R . o o

6. Thus, N = A + (tan a )

• R = + [b/(P • RIF • UPW)] • R , o o o e o which combined with Step 3 shows that N represents life-cycle

cost denominated in equivalent units of annual energy consumption.

Similarly, M can be shown to represent the life-cycle heating-related cost

i under the socially optimum standard. Thus, we can conclude that the net

gain in economic efficiency is given by N - M, which represents the

amount by which the social value of the extra energy saved by using RIF's

exceeds the additional social cost required to save it.

It should be noted that MM' and NN' were drawn under the assumption

that the RIF value is greater than one. If the RIF value for a particular

i fuel type happens to be less than one, then these lines would have a slope

steeper than that of LL' . The same procedure should be followed for

measuring the net gain in efficiency that would result from using RIF's.

As already noted, this analysis was carried out using a straight

isocost line which is based on the assumption of linear resistance costs.

A-7
If quadratic or cubic costs are assumed, the isocost line becomes concave

downward. Nevertheless, the optimum still occurs at the point of tangency

between the isocost curve and the trade-off curve. In this case, the analysis

of the net gain in efficiency would be based on the straight tangent line

which goes through the socially optimum combination of A and R, rather than

on the isocost curve itself. This is because the absolute value of the

slope of that straight tangent line is given by the ratio of the marginal

cost of thermal resistance (at the optimum point) to the social value of the

energy saved.

A-3
Table A.l

Conversion Factors from Customary to Metric (SI) Units

Physical Characteristic To Convert From To Multiply By

Length ft m 3.048 x 10" 1

Area ft

2 2 m 9.290 x 10" 2

Temperature °F °C t

c

= (

t

f

~32 ) / 1 .8

Temperature difference °F °C 5.556 x 10' 1

Energy Btu J 1.055 x 10

3

U-val ue Btu/hr-ft2 -°F W/m

2 °C 5.678

Thermal Resistance hr*ft

2

*°F/Btu m

2 °C/W 1.761 x 10" 1

A-9
:
NBS-114A (REV. 7-73)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION ()|< REPORT NO.

NBSIR 77-1109

2. Gov’t Accession

No.

3. Recipient’s Accession N<

4. TITLE AND SUBTITLE

The Effect of "Resource Impact Factors" on Energy Conservation

Standards for Buildings

5. Publication Date

6. Performing Organization Code

7. AUTHOR(S)

Stephen F. Weber

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No.

11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

Federal Energy Administration

1200 Pennsylvania Ave. , N.W.

Washington, D.C. 20461

13. Type of Report & Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most si&iificant information. If document includes a significant

bibliography or literature survey, mention it here.)

This report addresses the question of the proper price for energy in the development

of cost-effective energy conservation performance standards for buildings. This

study finds that the appropriate price for energy is its social value, which should

be determined through the development and application of Resource Impact Factors

(RIF's). Some guidelines are provided for the formulation and development of RIF's.

A simple life-cycle cost minimization model for determining the optimum conservation

standard is employed to show how the use of RIF's would generally lower the maximum

allowable energy consumption specified in the standard. Indeed, it is found that the

higher the RIF value, the lower the energy consumption allowed by the standard,

although this effect steadily diminishes as the RIF value increases. When a more

general cost model with less restrictive assumptions is employed, the same inverse

relationship appears between the energy consumption allowed by the standard and the

RIF value. Finally, a geometric measure is derived for the net gain in economic efficiency that would result from using RIF's in developing energy conservation performance

standards

.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons

)

Building economics; economics; economic efficiency: energy; energy conservation:

life-cycle cost; optimization; performance standards; resources; resource impact

factors; social optimum; standards

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