u

C.2

NBS BUILDING SCIENCE SERIES 114

The Effect of "Resource Impact Factors"

on Energy Conservation Standards

for Buildings

U.S. DEPARTMENT OF COMMERCE • NATIONAL BUREAU OF STANDARDS
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards' was established by an act of Congress March 3, 1901. The

Bureau's overall goal is to strengthen and advance the Nation's science and technology and

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and the Institute for Computer Sciences and Technology.

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2 Some divisions within the center are located at Boulder, Colorado, 80303.

The National Bureau of Standards was reorganized, effective April 9, 1978.
National Bureau et Stan OCT 3 1 1978

NBS BUILDING SCIENCE SERIES 114

The Effect of "Resource Impact Factors"

on Energy Conservation Standards

for Buildings

Stephen F. Weber

Center for Building Technology

National Engineering Laboratory

National Bureau of Standards

Washington, D.C. 20234

Sponsored by the

Department of Energy

Washington, D.C. 20545

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

Dr. Sidney Harman, Under Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued September 1978
Library of Congress Cataloging in Publication Data

Weber, Stephen F. The effect of "resource impact factors" on energy conservation standards for buildings.

(National Bureau of Standards building science series ; 114)

"Supersedes NBSIR 77-1199."

Supt. of Docs, no.: Cl3.29/2:114

1. Buildings--energy conservation. I. United States. Dept. of Energy. II. Title. III. Series: United States. National Bureau of Standards. Building science series ; 114. TA435.U58 no. 114 [TJ163.5.B84] 690'.21s 696 78-606072

National Bureau of Standards Building Science Series 114

Nat. Bur. Stand. (U.S.), Bldg. Sci. Ser. 114, 55 pages (Sept. 1978) CODEN: BSSNBV

Supersedes NBSIR 77-1199

U.S. GOVERNMENT PRINTING OFFICE WASHINGTON: 1978

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

Stock No. O03-0O3-O1952-6 Price $2.30 (Add 25 percent additional for other than U.S. mailing).

>

i
PREFACE

The first phase of this research was funded by the Federal Energy Administration (FEA), and the review and revision process leading to this final report was funded by the Energy Research and Development

Administration. The functions of both of these former

agencies now reside in the Department of Energy

(DOE). This work is in support of the DOE/NBS

research program entitled "Building Energy Conservation Criteria," whose broad objective is to facilitate the development and promulgation of energy conservation performance standards for the design and evaluation of buildings. The performance criteria being inves- tigated under this program are expected to take energy values into account. This study addresses the question of whether the energy values used in developing standards based on these criteria should be

the actual market prices paid or a value which has been adjusted to reflect the total value to the nation of

the energy saved. This total value of energy can be

represented by a system of multiplicative factors called "Resource Impact Factors" (RIF's) applied to the market prices of energy. This report assesses the

effect of using RIF's on the design energy consumption to be incorporated into performance standards and on the economic efficiency of such standards. The Applied Economics Program of the Center for Building Technology, National Engineering

Laboratory, 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, how such RIF's can be

integrated into the actual standards development

process, and how the use of such RIF's would affect the standards being developed.

Dr. Harold E. Marshall, Chief of the Applied

Economics Program, and Stephen R. Petersen deserve

special thanks for the time they spent discussing these

issues, reading and commenting on the several drafts of this report, and providing the author with invalu- able insights and ideas. The author is also grateful for the helpful comments and suggestions of the NBS

reviewers, Robert R. Jones, William L. Carroll, John

D. Ryan, Jim L. Heldenbrand, Dr. Douglas R. Shier,

and Dr. Carl O. Muehlhause, as well as of all those who participated in the public review of the earlier version of this report, NBSIR 77-1199 (R). Thanks are

also due to Noreen Rigopoulos, Linda Sacchet, and

Carol Thompson who typed the several drafts of this report, as well as to Reid Hartsell and James F. Shivar, the general technical representatives for FEA

and DOE. Responsibility for any errors and

shortcomings that remain rests solely with the author.

Cover: Energy conservation standards based on Resource Impact Factors could reduce the lifetime energy consumption of new buildings.

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iii
ABSTRACT

This report addresses the question of the proper value

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 value 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 devel- opment of RIF's. A 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 con- sumption 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. Moreover, the additional energy savings resulting

from using RIF's are shown to increase as the severity

of the climate increases. When two important

restrictive assumptions of the model are relaxed, these same relationships between energy consumption and

the RIF value on the one hand, and between the extra energy savings due to RIF's and the climate on the other hand, are both maintained. Finally, geometric

and algebraic measures are derived for the net gain in economic efficiency that would result from using RIF's in developing energy conservation performance

standards.

Key words: Building economics; economics; economic

efficiency; energy; energy conservation; life-cycle cost; optimization; performance standards; resources; 'resource impact factors; social optimum; standards.
CONTENTS

Page

PREFACE iii ABSTRACT iv EXECUTIVE SUMMARY 1

1. INTRODUCTION 5

1.1 PURPOSE 5 1.2 ENERGY CONSERVATION STANDARDS FOR BUILDINGS 6

1.2.1 Types of Standards 6

1.2.2 Legislation 6

1.2.3 ASHRAE Standard 90-75 7

1.3 APPROACH AND ORGANIZATION 8

2. CONCEPTUAL FRAMEWORK FOR AN OPTIMUM STANDARD 1

2. 1 OPTIMUM STANDARDS FROM THE PRIVATE PERSPECTIVE 12

2.1.1 Life-Cycle Cost Minimization 12

2.1.2 Value Differentiation among Energy Types 13

2.2 OPTIMUM STANDARDS FROM THE SOCIAL PERSPECTIVE 13

3. RIF's IN AN OPTIMUM STANDARD DETERMINATION MODEL 19

3.1 ALTERNATIVE FORMULATIONS FOR RIF's 19

3.1.1 Quantity Multiplier Formulation 20

3.1.2 Price Multiplier Formulation 21

3.2 AN OPTIMUM STANDARD DETERMINATION MODEL 21

3.2.1 Derivation of the Model 22

3.2.2 Significance of the Model 22

3.3 APPLICATION OF PRICE MULTIPLIER RIF's 23

3.4 LIMITATIONS OF THE ANALYSIS 24

4. THE OPTIMUM STANDARDS MODEL GENERALIZED 27

4.1 ALTERNATIVE COST FUNCTIONS 28

4.1.1 Case I: A Linear Cost Function 28

4.1.2 Case II: A Quadratic Cost Function 29

4.1.3 Case III: Higher-Order Cost Functions 30

4.1.4 Comparison of Alternative Cost Functions 31

4.2 DEGREE DAYS AS A FUNCTION OF RESISTANCE 32

4.2.1 Degree-Day Temperature Base and Resistance 33

4.2.2 Degree Days and the Degree-Day Base 34

4.2.3 Case IV: A Quadratic Degree-Day Function 34

4.3 CASE V: THE GENERAL OPTIMUM STANDARDS MODEL 36

5. SUMMARY, CONCLUSIONS AND FURTHER RESEARCH 39

5.1 SUMMARY 39

5.2 CONCLUSIONS 40

5.2.1 Proper Formulation of RIF's 40

5.2.2 The Energy Consumption Effect 40

5.2.3 The Economic Efficiency Effect 41

5.3 SUGGESTIONS FOR FURTHER RESEARCH 42 REFERENCES 43 APPENDIX: ECONOMIC EFFICIENCY EFFECTS OF RIF'S 44 A.l A GEOMETRIC MEASURE OF ECONOMIC EFFICIENCY 44 A.2 AN ALGEBRAIC MEASURE OF ECONOMIC EFFICIENCY 47

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Figures

Page

Fig. 2.1 Competitive Market for a Resource 14

Fig. 2.2 Effects of a Unit Tax 15 Fig. 2.3 Social Costs of Coal-Fired Electric Power 16

Fig. 2.4 Regulation of the Natural Gas Price 17 Fig. 3.1 Plot of the Ratio of Socially to Privately Optimum

Standards as a Function of the RIF Value 25

Fig. 4. 1 Degree Days as a Function of Thermal Resistance for

a Typical House in Washington, D.C 35

Fig. A. 1 Trade-off Curve between Energy Consumption and Thermal

Resistance for Given Climate and Indoor Temperature 44

Fig. A.2 Privately Optimum Energy Consumption Rate (A 0) 45

Fig. A. 3 Socially Optimum Energy Consumption Rate {Af} and the Net Efficiency Gain from Using RIF's (TV-A/) 46

Tables

Page

Table 3.1 Hypothetical Quantity Multiplier RIF's....: 20 Table 3.2 Hypothetical Price Multiplier RIF's 21 Table 3.3 Possible RIF Values with Corresponding Ratios of

Socially to Privately Optimum Standards 24 Table 3.4 Selected RIF Values with Corresponding First Derivatives of U* 24 Table 4. 1 Ratios of Socially Optimum to Privately Optimum

Standards for Various RIF Values and Cost Functions 31 Table 4.2 Ratios of Extra Energy Savings Due to RIF's in Colder Versus Warmer Climates, for Various Cost Functions 32 Table 4.3 Effective Balance Points for Various Levels of

Thermal Resistance: Hypothetical Example 33 Table 4.4 Degree Days Corresponding to Various Balance Points

for Six Major U.S. Cities 34 Table 4.5 Restrictions on the First and Second Derivatives

of D(R) and C(R) 36 Table A. 1 Conversion Factors from Customary to Metric (SI) Units 48

Facing page: Standards for energy conservation would

apply to new high-rise apartment buildings.

vi

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EXECUTIVE SUMMARY

This report focuses on the role which the value of energy is expected to play in the development of optimum (i.e., cost-effective) energy conservation per- formance 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 method 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. For

developing optimum energy conservation standards, the social values of energy can be represented by

applying a system of indices called "Resource Impact

Factors" (RIF's) to the actual energy prices. One of

the purposes of this report is to offer some guidelines

for the development of such a system. 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 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. 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 value. Since this social value 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 pro- duction or consumption, the desire for national economic independence, the existence of price controls such as those on natural gas, and inadequate

information about the long-run scarcity of exhaustible energy resources. 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 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 con- sumption ceiling allowed under the conservation per- formance 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, rough

estimates can be made of the additional energy savings

resulting from the use of RIF's. A comparison of domestic with imported gas prices is used to derive

high and low RIF values. Introducing the resulting

RIF values of 2.9 and 1.5 into the expression for the

socially optimum standard, we find that the energy consumption ceiling allowed under the standard would be 41.3 percent and 18.4 percent lower, respectively, with RIF's than without them. The model also 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. Moreover, the additional energy

savings resulting from using RIF's are shown to increase as the severity of the climate increases. When

the restrictive assumptions regarding degree days and

the cost of thermal resistance are relaxed, these same

relationships between energy consumption and the RIF value on the one hand, and between the extra energy savings due to RIF's and the climate on the other hand, are both maintained.

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 ap- propriate 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 the

cost of the additional thermal resistance required to

2
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 geometric and

algebraic measures of its magnitude are developed in the Appendix.

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. With Department of Energy

funding, researchers at NBS are currently developing

such a model for determining optimum building perfor- mance criteria. Once this model has been completed,

the effect of using RIF's in determining socially

optimum building performance standards can be

assessed.

267-424 O - 78 - 2
Facing page: Polystyrene sheathing applied to these townhouses helps reduce heat flow through exterior

walls.
1. INTRODUCTION

1.1 PURPOSE

The focus of this report is on the role of energy values

in the development of optimum (i.e., cost-effective) energy conservation standards for buildings. It is expected that the value of energy used in the stand- ards development process will influence the energy consumption allowed under such an optimum standard. This study assesses the effect on the conservation

standard of using "adjusted" values rather than actual market prices for energy. The purpose of the

adjustment is to ensure that the energy values on which the standard is based accurately reflect the true

social values of the energy resources to be used in the

building. In developing an optimum standard, the

social values of energy can be represented by applying

a system of multiplicative factors called "Resource

Impact Factors" (RIF's) to the actual prices paid for energy. This report compares an energy conservation standard developed using actual market prices of energy with

5
one developed using social values 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 efficien- cies 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 can be developed. 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 Standards

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 per- formance 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 0.06

Btu/hr./sq.ft./°F. The third type is a building perfor- mance standard, which would specify an overall goal

for the design energy consumption of the building, thus allowing the designer to make cost-reducing

trade-offs between building components in order to conform to the standard. For example, a building per- formance 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.

1 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.

1.2.2 Legislation

On August 14, 1976, the Energy Conservation and Pro- duction Act (P.L. 94-385) was signed into law. Title

III of this law, the Energy Conservation Standards for New Buildings Act of 1976, called for the Secretary of Housing and Urban Development (HUD) to develop energy conservation performance standards for new

residential and commercial buildings within three

years. In his April 20, 1977 Energy Message to the

Joint Session of Congress, President Carter

accelerated this program schedule by one year. On

August 4, 1977, Section 304 of the Department of

Energy Organization Act (P.L. 95-91) transferred

responsibility for the development and promulgation

of these standards to this new Department. The

responsibility for implementing the standards has remained with HUD.

Key elements of this energy conservation standards

legislation are: (1) the explicit specification ofperfor- mance 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 im- provements 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

nondepletable sources of energy; and (6) that the stand- ard 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 standards of the building performance

type are the ultimate goal of the legislation. These

building performance standards are expected to be

developed by combining the results of analysis of

individual components and their energy

interdependencies.

The second key element of the legislation, 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 standard. A further guideline for developing the standard, or

future improved versions of it, 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.

6
The goal of maximum practicable gains in energy effi- ciency 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 inade- quate energy conservation measures in new buildings.

The concern for cost effectiveness is also apparent in the fourth element of the legislation, the requirement

to consider climatic variations in developing the standard. The reason for considering these variations is 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 legislation, 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 is 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.

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 AirConditioning 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.

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 effi- ciency as called for by the Energy Conservation and

Production Act. Indeed, the ASHRAE standard

specifically states that it is presenting only minimum

requirements. 2 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 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 stand- ards 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 per- formance standards as its ultimate goal, although these are expected to be based initially on the analysis of

individual components and may be complied with

mainly through manuals of accepted practice.

Resource Utilization Factors and Resource Impact

Factors: The proposed section 12 of ASHRAE 90-75, "Annual Fuel and Energy Resource Determination,"

provides a method for calculating the equivalent

2 ASHRAE 90-75: Energy Conservation in New Building

Design (New York: ASHRAE, Inc., 1975), Section 4.2.1.

7
resource quantities required to supply the energy

needs of a building. 3 The procedure involves the use of conversion coefficients, 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 as

distinct from RIF's is to treat separately those factors which are more readily quantifiable. This is because

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

and national security concerns, are to 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.4

This approach has several shortcomings. In the first place, RUF's do not take into account the energy expended in extracting the resource from the ground.

Secondly, this use of RUF's is similar to relying solely on 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. Exclusive reliance on Net Energy

Analysis incorrectly implies that all nonenergy

resources are relevant only in the measure of the energy required to produce them. Thus, many

3 Section 12 has undergone two open reviews, and Panel 12 of the Standard 90 Project Committee has recommended

that it be added to the Standard. For further details on the

status and content of Section 12, see William J. Coad,

"Section 12: Toward a More Effective Use of Energy

Resources," ASHRAE Journal, Vol. 19, No. 5 (May 1977), pp. 32-33.

4 For 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 Energy Measurement

Tool," ASHRAE Journal, Vol. 18, No. 5 (May 1976), pp. 28-

30.

important resources are either totally or partially

ignored. 5

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. For example, an increase in refining efficiency resulting from a decrease in energy losses at the refining stage would

cause a reduction in the corresponding RUF value. At

the same time, however, this change in efficiency

would also be reflected in a lowering of the effective cost of delivering energy to a building project site.

In this report, we suggest an alternative method of

formulating RIF's—one which combines all those

factors not included in the market price of an energy

resource into a single multiplicative coefficient. This

coefficient can be used to adjust the market price to

reflect the true social value of the energy resource.

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, 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

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 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.

For a discussion of the inappropriateness of relying solely on 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 the panel discussion on Energy, Environment

and Economics, 68th Annual Meeting, American Institute of Chemical Engineers, November 1975.

8
Section 5 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.
Facing page: The price paidfor energy at the building site may differ from its total value to the nation. Resource Impact Factors can help conservation standards for new

buildings reflect the total value of energy.
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72

2. CONCEPTUAL FRAMEWORK

FOR AN OPTIMUM

STANDARD

As we have seen in our discussion of the relevant

legislation, the ultimate goal is to achieve the maximum practicable improvements in energy efficien- cy. Moreover, the legislation 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).

6 Moreover, for purposes of illustration we will focus on setting a standard for the conservation of energy used only to heat and cool buildings.

6

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.

267-424 O - 78 - 3
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 op- timum standard first from the private viewpoint and

then from the social viewpoint.

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.

7 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.

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 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

7 This 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.

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.

Alternative components: If the above minimization

procedure is applied to all the components of a building, then the optimum energy conservation stand- ard 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 optimum 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.

8 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 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.

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 stand- ard 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.

Currently, research is being conducted at the National Bureau of Standards to model these various

interdependencies.

12
Thus, the life-cycle 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.

9

2.1.2 Value 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.

10 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 values of the 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 energy-saving construction resources from buildings

which use low-cost energy to those which use highcost energy. The net dollar savings from the latter standard would be greater than those from the original standard which did not differentiate among the values of the 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 the values of the energy

types. There is, moreover, a further relationship

between optimality and the value of energy. The way

in which energy resources are valued 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. 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.

9 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.

10 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.

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.

Traditional economic theory of perfectly competitive

markets shows that under very restrictive assumptions

the prices paid for resources are equal to their true

social values." 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 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 demand of a resource is illustrated graphically in figure 2.1. The line 5 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. 12 The line D, on the other hand,

represents the quantities of the resource that users would be willing to buy at various prices in the same

11 For example, see Richard H. Leftwich, The Price System and Resource Allocation (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 Countries, Vol. II: Social Cost Benefit

Analysis (Paris: Development Centre of the Organization

for Economic Cooperation and Development, 1968), Chapter II.

12 This increasing cost assumption generally holds in the

short run when existing capacity cannot be expanded. In the long run, if capacity is expanded, unit costs may

increase, decrease, or remain constant.

13
Q*

Quantity

FIGURE 2.1 COMPETITIVE MARKET FOR A RESOURCE

market and during the same time period. The

downward slope indicates that as more of the resource

is demanded, 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 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 demanded. Users and suppliers are

willing to continue buying and selling this equilibrium

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 demanded in excess of Q* would be employed to produce output whose value is less than the cost of

supplying the resources (D is below 5 to the right of

Q*). On the other hand, if resource units fewer than Q* are demanded, they would be employed to produce output valued above the cost of supplying the resources (D is above 5 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 demanders, 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 demanders and suppliers of the resource. We will now

focus on several of the most important ones for energy

resources. Unit taxes and monopoly: One possible source of diver- gence between actual prices and social values or costs of energy resources is a tax per unit produced or

sold.

13 Examples would be the utility taxes on the number of therms of gas or kilowatt-hours 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

13 A 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).

14
Q t Q*

Quantity

FIGURE 2.2 EFFECTS OF A UNIT TAX

equilibrium price to differ from the true social cost of

supplying the resource.

14 This difference is illustrated

in figure 2.2, which is the same as figure 2. 1 except

that the line 5, has been added above the parallel to

line S. The line S, 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 Q,) and price (from P* to P,). The effect on the price means that the actual market price (P,) 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 (Pc) necessary to cover the actual costs of

production. 15 This implies that users do not correctly

perceive the true social cost of producing the

resource, are forced to pay a higher price, and

consequently under-utilize the resource.

16

14 This is true unless the purpose of the tax is to offset negative environmental effects or excess profits, or to

fulfill some other policy objective of the agency

responsible for imposing the tax.

15 The difference between P, and Pc is given by the amount

of the per unit tax.

16 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 fall 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-utilized.

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 those units.

17 Here

again, then, the decisions of resource users are not guided by the true cost of the resource. 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.

18 These effects are present whenever a

particular production or consumption process

generates a cost or benefit for persons not directly

17 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.

18 For an excellent survey of the literature on environmental economics, see Anthony C. Fisher and Frederick M.

Peterson, "The Environment in Economics: A Survey,"

Journal of Economic Literature Vol. 14, No. 1 (March

1976), pp. 1-33. Also, for two convenient 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).

15
I Q

Quantity

FIGURE 2.3 SOCIAL COSTS OF COAL-FIRED ELECTRIC POWER

involved in the process. As a result the price

established by the supply and demand of the resource

will generally not reflect these external (or indirect) benefits and costs. In the case of energy resources most of the external effects take the form of costs rather than benefits.

19 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 costs incurred by the utility.

20 This example is shown

graphically in 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 55 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,

19 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, Va: U.S. Dept. of Commerce, National Technical

Information Service, 1975).

20 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.

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 Ps and the amount by which that last unit is underpriced is Ps-P.

National economic independence: In support of various energy policies an appeal has been made in recent years to the need for national economic indepen- dence. 21 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.

This national independence goal has implications for the need to adjust energy prices similar to those of

21 Uzi Arad, Barry J. Smernoff, and Haim ben-Shaher, American Security and the International Energy Situation, 4

vols. (Croton-on-Hudson, N.Y.: Hudson Institute, 1975) provides an overview of the energy independence issue.

16
Qr Q*

Quantity

FIGURE 2.4 REGULATION OF THE NATURAL GAS PRICE

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.

Price controls: 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, Pr, is set so

far below the optimum price, P*, the suppliers of

natural gas are willing to make only Qr 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 Qr , the premium users would be willing to pay for additional units over and

above the regulated price, would be as much as Ps -Pr .

Other factors: There are other factors which may

cause market prices of energy to be different from the corresponding social values. Because of inadequate in- formation about the long-run scarcity of exhaustible energy resources, present supply and demand forces may lead to equilibrium prices which do not reflect true values. This situation could result in non-optimal

rates of resource exploitation.

A similar problem arises because private rates of

discount are likely to be greater than the social rate.

22

Consequently, private producers will tend to undervalue the future earnings from resources saved

for later exploitation. This leads to rates of extraction which are too rapid from society's viewpoint. 23 The

same result occurs when two or more producers are

exploiting a common resource pool, each producer's

share of which depends on his individual rate of

extraction.

24

22 William J. Baumol, "On the Social Rate of Discount," American Economic Review, Vol. 58 (September 1968), pp. 788-802.

23 Robert M. Solow, "The Economics of Resources or the Resources of Economics," American Economic Review, Vol. 64, No. 2 (May 1974), pp. 1-14.

24 For further discussion of various aspects of common

property resources, see Edwin T. Haefele (ed.), The

Governance of Common Property Resources (Baltimore: The

Johns Hopkins University Press, 1974).

17
The preceding discussion covers some of the ways in which the actual prices paid for energy resources may

fail to reflect their full social values. As such it offers some insight into the difficulty of using actual energy

prices as a basis for energy-related 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 develop- ment 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 lowincome 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.

Facing page: The economic model presented in this section determines the optimum amount of insulation, given its cost and the social value of the energy saved.

18
3. 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.

19

267-424 0-78-4
3.1.1 Quantity Multiplier Formulation

Quantity multiplier RIF's could 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

6

Btu's) required to satisfy the needs of the building. In the case of space heating, for example, this energy

load would be the heating value of the energy output

into the conditioned space that is necessary to maintain a desired comfort level.

25 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."

26 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 hypothetical

quantity multiplier RIF's to illustrate how they might

operate.

Column 1 shows the number of physical units of each

alternative energy type required to satisfy the annual

heating load of 36 X 10

6 Btu specified for a particular

building (e.g., 429 gallons of fuel oil). Column 2 gives

. 25 Another way to formulate quantity multiplier RIF's would be as multipliers applied to the energy content of

the fuel and energy delivered to the building site. In this case, energy losses occurring at the building site would not be included in the RIF value itself, but rather would be

included in the estimate of the energy content of the delivered fuel or energy to which the RIF value is applied.

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

the number of equivalent Btu's.

the energy load in terms of the heating value of the energy output needed to maintain a desired degree of comfort (36 X 106 Btu). Column 3 presents the RIF

value corresponding to each energy resource type, measured in terms of impact units per 10

6 Btu of energy output needed to meet the heating load (e.g., 1.2 impact units per 106 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 units 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. 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

Table 3.1 HYPOTHETICAL QUANTITY MULTIPLIER RIF S

Units Needed3 Heating Load RIF Impact

(10

6 Btu)

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

J The number of physical units of input of the energy type needed to satisfy the given annual heating load of 36 X 10

6 Btu. The heating system output of 36 X 10

6 Btu was converted to corresponding input requirements by assuming the following energy contents and conversion efficiencies:

(1) Natural Gas: 10

6 Btu/MCF and 60% efficiency

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

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

20
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 for- mulation 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.

27 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. 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 efficien- cies 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

27 For a good introduction to this type of price multiplier (sometimes called a shadow price) applied to benefit-cost

analysis, see Peter G. Sassone, "Shadow Pricing in CBA:

Mathematical Programming and Economic Theory," The

Engineering Economist, Vol. 22, No. 3 (Spring 1977), pp. 219-233.

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.

28 Thermal resistance (R) is defined as the exact

reciprocal of the overall coefficient of thermal trans- mission (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 (R0) for a particular component such as the attic floor, the model also determines the optimum

coefficient of thermal transmission (U0), 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 (U0). Thus, the more stringent the stand- ard (i.e., the more energy conserving), the lower will be U0

.

Stephen R. Petersen, "Economic Optimization in the Energy Conservation Design of Single-Family Housing,"

ASHRAE Transactions, Vol. 82, Part I, 1976.

Table 3.2 HYPOTHETICAL PRICE MULTIPLIER RIF'S

Actual Price Effective Price2 RIF Social Price ($/Unit) ($/10

6Btu) ($/106 Btu)

Energy Type (1) (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

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

(1) Natural Gas: 10

6 Btu/MCF and 60% efficiency;

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

(3) Electric: 3,413 BtuAWh and i00% efficiency.

21
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:

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 op- timum level of thermal resistance, R0

.

Ro = [UPW • 24 DD Pe (l/b)) m. (3.2)

LCC a + b R

+ (UPW 24 • DD Pc) (\/R). (3.1)

The notation is to be interpreted in the following manner:

a = The fixed cost component of thermal

resistance ($/ft

2

).

b — The cost per additional unit of thermal

resistance ($/ft

2 /R).

R = The number of units of thermal resistance on

each square foot of attic floor area

(hr-ft

2 -°F/Btu).

UPW

N

2

i

l + e

l + d

l+i

1+rf

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 percent real discount

rate, a 3 percent 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.

365 DD = £ (B-Ti), for all T, <5, where T is the average

i = l

temperature for the /th 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 = the effective price actually paid for one Btu

of heating energy output delivered to the conditioned space, taking account of

conversion efficiencies.

Since the rate of thermal transmission, U, is defined as the exact reciprocal of R, we have the optimum

coefficient of thermal transmission, U0

, for a square

foot of attic floor area:

UPW 24 DD

-1Pe J (3.3)

Although this formulation of the model is quite simple,

it provides the most important elements needed to determine the optimum component performance stand- ard 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 (U0) 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 percent

increase in the maximum heat transmission rate allowed by the optimum standard, other things being

equal.

29 This positive relationship has the intuitive appeal 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 (U0

). 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 percent lower

rate of thermal transmission. 30

"Letting U'0 represent the standard after the doubling of the marginal cost of resistance, we have U„ = V~I U„ = 1.414 Uo, which means an increase of 41.4 percent.

30 Letting U'0 represent the standard for the climate with twice as many degree days, we have U'o = U0 / V~I = 0.707 Uo, which means a decrease of 29.3 percent.

22
The final noteworthy feature of this expression for the optimum component performance standard is that, just as with degree days, U0 also varies inversely as the square root of the price of energy (Pe

). That is, a doubling of the price of energy calls for a 29.3 percent decrease in U0, and consequently the same 29.3 percent 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 con- sumption 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.

mance standard from the social point of view (£/*>:

UPW . 24 • DD • Pe R1F (3.4)

Thus, the relationship of the socially optimum stand- ard to the privately optimum standard can be

established as follows:

U*

U*

\ b

1/2

1

1/ 2 UPW • 24 • DD Pe RIF .

, or

RIF (3.5)

3.3 APPLICATION OF PRICE MULTIPLIER

RIF'S

As it stands, the expression for U0 developed above is based on the goal of cost minimization from the

private point of view. To adjust the expression to

reflect the social viewpoint, it is necessary to replace

the effective price paid for energy (Pe) with the social price of energy.

31 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 Pe in the expression for U0 by the appropriate RIF of the

fuel type being considered. 32 This will result in the following new expression for the optimum perforIt 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. A preliminary

estimate might be made by applying RIF's to the energy

content of insulating materials. Some work has been

initiated in this area of energy content of selected building

materials and for the construction sector as a whole. See B. M. Hannon, et al, Energy Use for Building Construction.

Progress Report, March 1, 1976-August 1, 1976 (Center for Advanced Computation, Illinois University, Urbana,

August 1976).

32 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-/>f-RIF in expression (3 4) with the summation,

" (pX

nil (\+d)n '

where (P^„ represents the estimated social value of the energy type for the «th year.

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 corre- sponding 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 stand- ard 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 allowed by the standard. This would mean less energy conservation with RIF's than without them. If the RIF equals one, then the intro- duction of RIF's would have no effect on energy con- sumption. 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 percent higher than

the actual price paid) would lower the maximum

energy consumption allowed by the standard by 18.4 percent, to only 81.6 percent 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

23
Table 3.3 POSSIBLE RIF VALUES WITH CORRESPONDING RATIOS OF SOCIALLY TO PRIVATELY OPTIMUM

STANDARDS

RIF U*/U0 = [1/RIF] 172

.50 1.414

.60 1.291

.70 1.195

.80 1.118

.90 1.054

1.00 3 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.UL) U. /U / 2.25 0.667

2.50 0.632

2.75 0.603

3.00 0.577

4.00 0.500

Equivalent to not using RIF's.

applied. That is, as the RIF value increases, the corre- sponding 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 indicates that increases in the RIF

value lead to decreases in £/*as a proportion of U0

. A

more rigorous mathematical demonstration of this observed inverse relationship requires evaluation of

the sign of the first derivative of U* with respect to RIF. Thus, partial differentiation of expression (3.5) leads to the following: = [-0.5 RIF" 1 ' 5

]

• U0

. (3.6)

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* holds everywhere.

Table 3.4 presents the calculated values of this first derivative measured in units of U0 for selected RIF

values. As indicated in table 3.4, the signs of all the

Table 3.4 SELECTED RIF VALUES WITH

CORRESPONDING FIRST DERIVATIVES OF U0*

RIF af/*/3RlF=

[-0.5 RIF 1

]

• U0

•50 -1.414 • f/0 .75 -0.770 • U0 1-00 -0.500- U0 1.25 -0.358 • U0 1.50 -0.272 • U0 2.00 -o.m U0 3.00 -0.096 • U0

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* with respect to RIF.

Differentiating expression (3.6) we have

jgfa = [0.75 RIF"2 ' 5

]

• U0

. (3.7)

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 stand- ard is greater than that saved by the privately optimum standard. The analysis of the economic efficien- cy effects of using RIF's will be treated in the appendix.

24
I I I I I

I I I L

0 0.2 0.4 0 6 0.8 1 0 1.2 1 4 1 6 1.8 2.0 2.2 2.4 2 6 2 8 3.0

Resource Impact Factor (RIF) FIGURE 3.1 PLOT OF THE RATIO OF SOCIALLY TO PRIVATELY OPTIMUM

STANDARDS AS A FUNCTION OF THE RIF VALUE

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 may be

violated under actual conditions. A model for deter- mining optimum building performance criteria is currently being developed at the National Bureau of Standards with funding from the Department of Energy. 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.

25
Facing page: Because of their smaller exposed envelope

area, townhouses generally require less energy for heating

and cooling than do single-family detached houses.
4. THE OPTIMUM STANDARDS

MODEL GENERALIZED

In this section we modify the optimum standards

selection model of section 3 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 to 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 27
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 relation- ship between the magnitude of the RIF values and the

level of the socially optimum standard. 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, 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 heatingrelated 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 (\/R) + C, (4.1)

where

L = life-cycle heating-related costs of a unit of attic floor area, K = UPW • Pe 24, 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 we made the assumption that the per unit area construction 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 level of total resistance.

33 In

this subsection we will relax that linearity assumption

33 In the terminology of economists, this is equivalent to assuming that the marginal cost of resistance is constant.

to allow for the case of rising incremental costs first by a quadratic function and then by cubic and higherorder functions. For the sake of completeness, we

begin by restating the original model of section 3 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

L = K D (\/R) + a + b R.

Differentiating with respect to R and setting the

derivative equal to zero yields the following

expression for the optimum resistance, R0:

(4.2)

KD 1/2

This implies that the standard is given by

U„ = K • D

1/2

(4.3)

(4.4)

Substituting F K for K, where F is the RIF value, we

can establish the relationship between the socially optimum standard U* and the privately optimum one, U0

:

U* FKD U, (4.5)

As was pointed out in section 3, 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 would 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 in- troduction 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 U0 in equation (4.5) are being

multiplied by the same value of F~

xn

, so that the percentage change is the same. By the same token, this implies that the absolute value of the impact of t/*is greater for warmer climates than for colder ones, because the base values of U0 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 con- sumption allowed by a socially optimum standard. A

numerical example illustrates the point. Suppose the

28
values of the parameters b, K, and D in expression

(4.4) were such that U0 equalled 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 U0 equal to 0. 10/V4^(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 by AS,.. Then we have

A5C (\-Fm) (24) • (AD) (0.10/V4), and

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.

35 Moreover, to rule out the unlikely situation of a negative marginal cost, we assume b^Q. In terms of geometry, these restrictions on b and c mean that we

are confining our analysis to the upward sloping

portion of a t/-shaped quadratic curve, that is, one which is concave upward.

Under the assumption of fixed degree days and

quadratic costs, the expression for life-cycle cost takes on the following form:

&SW = (\-Fv2

)- (24) -(D)- (0.10). (4.6) K D (1/7?) + a + b R + c R

2

. (4.9)

Forming the ratio of the colder climate savings to the warmer climate savings, and cancelling like terms in the numerator and denominator, we have

f§ =4/V4 = V4

AS„

(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

twice as great for the colder climate than for the warmer one. It is important to note that this result holds for all RIF values greater than one.

34 This result can be stated in a more general form. If a colder

climate has n times 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 VTTtimes 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:

a + b R + c R

2

, (4.8)

where a, b, and c represent fixed parameters. This for- mulation 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

If F=l, then ASC = AS„=0 and the ratio becomes

meaningless. If F< 1, then ASf and AS„, are both negative, which implies that application of the RIF would allow

higher rather than lower levels of energy consumption. In

this latter case, the decrease in energy savings in the colder

climate would be twice as much as the decrease in savings

in the warmer climate.

Differentiating this expression with respect to R and

setting the result equal to zero yields the following

cubic equation:

2 c Ri + b Ri - K D = 0. (4.10)

The expression for the optimum level of thermal

resistance, R0

, can be found by applying the formula

for the solutions to a cubic equation. 36 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 6=0, on the other hand, the

explicit solution for R0 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. 37 Under, the assumption, the following formula for R0 can be derived from

expression (4.10): \KD 1 1/3

2c

(4.11)

This implies that the maximum rate of thermal trans- mission allowed by the optimum standard is given by

2c KD

1/3 (4.12)

35 By taking the first derivative of (4.8) we find the 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.

30 Murray R. Spiegel, Mathematical Handbook of Formulas and Tables (Schaum's Odtline Series; New York: McGrawHill, Inc., 1968), p. 32.

37 The geometric interpretation of 6 = 0 is that the MC

straight line goes through the origin.

24
Substituting FK 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* and that by the privately optimum standard, U-.

U*

r 2c i 1/3 KDj

ill 1/3 F \ (4.13)

As with the linear cost case, this means that as the RIF value increases, U* 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* varies inversely as the cube

root of F, whereas for the linear cost case, U* varies inversely as the square root of F.

38 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 percent, whereas we saw in section 3 that the effect of the same RIF value on the

rate allowed under the linear cost standard would be a reduction of 29.3 percent. 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 con- sumption allowed by the standard in different climates. A numerical example helps to illustrate the relative impacts. Suppose the values of the parameters c, K,

and D in expression (4.13) were such that U0 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 U0 equal to

0. 10/4 1/3 (or about 0.63 as high a rate of thermal trans- mission), 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 A.SW and A5C, respectively. Then we have

A5C = (l-rui

) (24) • (4D) • (0.10/4 173), and

AS„ = (l-r m) (24) • (D) • (0.10). (4.14)

Thus, the ratio of the colder to the warmer climate savings becomes

AS

4/41/3 = 42/3 2 .52. (4.15)

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 one.

39 In general,

1 See expression (4.5) above.

' See footnote 34 above.

this relationship means that for a colder climate with n times as many degree days as a warmer climate, the

extra annual energy savings due to using a positive

RIF value would be n

2/i times as much as for the warmer climate.

4.1.3 Case III: Higher-Order Cost Functions

We now consider the implications of using cubic and

even higher-order functions to represent the relation- ship between total construction costs and the level of thermal resistance. The general form of the functions we will analyze is

C = a + c Rq (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.

40 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 (\/R) + a + c Rg

. (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:

rr* _ r

<?- c ii/<<? + ' r 1 D riii/<<? + « °~ik d\ 'm = un

(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 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 percent. With a cubic cost function (q = 3), however, this reduction would be a mere 15.9 percent. 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

It might be noted that this functional form could be used

to analyze the case in which total costs are a less than

linear function of thermal resistance (e.g., q = 0.5). The

economic interpretation of this situation is that the marginal cost of resistance decreases as more is added. 30
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 to those in the warmer

climate is equal to n

i/A

. 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, Uyi/0

, 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.

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 rate.

41 Secondly, 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

This can also be shown in general by evaluating the signs of the first and second partial derivatives of U* with

respect to Fin expression (4.18). As expected, the first derivative is negative, indicating the inverse relationship, and the second derivative is positive, verifying the diminishing effect.

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 con- sumption 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 savings in the colder versus the warmer climate turns out to equal n

ql{l> +

»>, 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 percent greater impact on

annual energy consumption with a linear cost

function, but they would have a 58.7 percent 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 energysaving impact of having more degree days is greater

for higher-order cost functions.42 The second point is that for all values for q the ratios increase as n

42 Note that this effect of increasing q gradually diminishes and approaches the limiting value of n as q gets arbitrarily

large. That is, lim»*/( ' + '>=«.

Table 4.1 RATIOS OF SOCIALLY OPTIMUM TO PRI- VATELY OPTIMUM STANDARDS FOR VARIOUS RIF VALUES AND COST FUNCTIONS a

RIF Cost Function

Value

Linear Quadratic Cubic Quartic

(q=\) (q=2) (q = 3) (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

"These ratios are calculated by the following relationship derived from

expression (4.18): f/*„/t/0 =.F-i/<<7+ i)

31
Table 4.2 RATIOS OF EXTRA ENERGY SAVINGS DUE TO RIF'S

IN COLDER VERSUS WARMER CLIMATES, FOR

VARIOUS COST FUNCTIONS a

Cost Function

iviuiupic Linear Quadratic Cubic Quartic

(n\ \n) W= 2) (9 = 3) (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

' The ratios of the extra savings are calculated using the following relationship: AS, gl(q + i) AS„ = "

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*

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 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.43

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.

43 For 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 Ra 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.

32
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 l 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, 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 would 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

Table 4.3 EFFECTIVE BALANCE POINTS FOR VARIOUS LEVELS OF THERMAL RESISTANCE:

HYPOTHETICAL EXAMPLE a

Thermal Resistance (/?) 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 (7) is 70°F; (3) the spontaneous heat gain (H) from solar radiation and household activities

is 4000 Btu's/hour; and (4) heat loss (/) 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 ~ \a7rTi\

33
10 causes B todrop 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

area. 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 degreeday 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.

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. The data from table 4.4 can be combined with that in table 4.3 to plot curves

representing degree days as functions of resistance. One such curve is shown in figure 4. 1 for Washington,

D.C. 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 relation- ship 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 • R

2

, (4.19) where d, the intercept is positive, /must also be

positive to guarantee upward concave curvature, and

e is negative because the slope is negative. 44

44 Regression analysis of this quadratic relationship reveals

that for the six cities of table 4.4, all the coefficients have

the expected signs and are significantly different from zero

at the 99 percent level of confidence. Moreover, the R

2 values for all six cities exceed 0.96.

Table 4.4 DEGREE DAYS CORRESPONDING TO VARIOUS

BALANCE POINTS FOR SIX MAJOR U.S. CITIES

Balance City

Point (°F)

Chicago N.Y. Wash. San Fran. St. Louis Mpls.

65 6191 4578 4164 3394 5225 8405

64 5928 4344 3942 3058 5002 8145

63 5667 4115 3725 2733 4784 7889

62 5413 3888 3515 2422 4571 7640

61 5170 3668 3310 2126 4365 7397

60 4930 3455 3110 1848 4165 7155

59 4698 3246 2917 1598 3970 6916

58 4475 3047 2726 1370 3782 6681

57 4256 2853 2542 1162 3598 6449

56 4043 2663 2360 977 3417 6221

55 3836 2482 2185 802 3239 5995

54 3636 2304 2015 645 3065 5772

53 3442 2135 1857 507 2892 5556

52 3255 1975 1709 387 2721 5342

51 3072 1824 1566 290 2554 5134

50 2899 1683 1430 217 2392 4933

° These data are taken from computer calculations performed at the National Bureau

of Standards by Edward A. Arens and William L. Carroll using Test Reference Year

weather tapes. These tapes contain a particular year's climate records selected for each

location from a population of 27 years of records.. The selection procedure is described by Eugene Stamper, "Weather Data," ASHRAE Journal, Vol. 19, No. 2 (February 1977),

p. 47.

34
4200 1- #

4000 -

2000 -

10 20 30 40

Thermal Resistance (R)

50 60

FIGURE 4.1 DEGREE DAYS AS A FUNCTION OF THERMAL RESISTANCE FOR

A TYPICAL HOUSE IN WASHINGTON, D.C.

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 + R + f + a +

R

2

) O/R)

R. (4.20)

Differentiating L with respect to R, solving for the op- timum level of resistance, and inverting yields the following expression for the optimum rate of thermal

transmission:

. \b+f-K] -[ d-K

1/2 (4.21)

When RIF's are introduced by substituting FK for K

in this expression, we obtain the maximum rate of

transmission allowed by the socially optimum stand

ard: b/F+f-K

U*

d-K

1/2 (4.22)

Unfortunately, expression (4.22) does not permit the separation of F which would be required in order to express the socially optimum transmission rate, U* as a function relative to the privately optimum rate, U0

. The best that can be done under these circumstances is to assess the general direction of the relationship

between Fand U*by evaluating the sign of the first derivative. Thus, we have

dU*

dF

-1/2 b/F + / K

d-K

1/ 2 [d-K ' \fF \ (4.23)

35
Since it is assumed that additional units of resistance have a positive cost, we have Z>>0. Moreover, since degree days are assumed to decrease at a decreasing

rate, with increases in resistance, we have d and /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 in general U* 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 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* decreases 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) (l/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. The rationale behind restrictions 1 and 2 regarding degree days was

already discussed under Case IV (subsec. 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.

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 R.

2. D"(R) > 0 D decreases at a decreasing rate.

3. C(R) > 0 C is an increasing function of R.

4. C"(R) > 0 C increases at an increasing or

constant rate.

When the derivative of expression (4.24) with respect

to R is set equal to zero, we obtain K • [R0 D\Ra) - D(R0)] + R\ C{R0

) = 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 R* it is possible to determine the general direction of

the effect of RIF's upon R* and in turn upon its inverse, U* By employing the implicit function rule of

calculus,

45 we can find the derivative of R* with

respect to F: dR%

dF

K\D{R^-R*-D\Rt)]

F-KR*B"(R*) + 2R*-C(Rf) + (Rtf-C"{R?)

'

(4.26)

Using the restrictions specified in table 4.5 and the fact that F, K, D{R), and R * 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* with respect

to F.

Since in general U is the reciprocal of R, we have

U* = l/R* (4.27)

Differentiating, we have

dU*

dR* = -\/R* 2 < 0. (4.28)

The implicit function rule states that for an implicit function of two variables G(x,y), dG/dX

dx dG/dy'

See Jack R. Britton, Calculus (New York: Holt, Rinehart and Winston, 1961) p. 414. 36
1

By the chain rule, we know that

dU*

dF

\dU*\ \dR*\

[dR*\ [dF\ (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 R* 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 of these third-order derivatives. Hence, the sign

of the second derivative of R* 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, other

things being equal. On the other hand, while for Cases

I through III (fixed degree days) we were able to 1 conclude that this effect of higher RIF values tends to I diminish, we were unable to reach such a conclusion

for Cases IV (quadratic degree days) and V (the general model).

46 The chain rule for derivatives simply states that

dx ~ du dx'

See Spiegel, Mathematical Handbook of Formulas and

Tables, p. 53.

37
Facing page: Incorporating energy-conservingfeatures into new buildings is usually less costly than adding them to

existing buildings such as these townhouses.
5. 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 econom

ically justifiable while maintaining the habitability re- quirements 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

39
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 value of energy

used in the analysis is critical in determining the optimum standard.

Section 2 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, the existence of

price controls such as those on natural gas, and inade- quate information about the long-run scarcity of ex- haustible energy resources. Because of this likely diver- gence between actual energy prices and their social values, it is necessary to adjust the actual prices before

using them to determine the socially optimum conser- vation standard. RIF's provide a method of making

such an adjustment.

Section 3 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

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 direct 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 responded to this last limitation of the

analysis by generalizing the cost minimization model.

Four alternative sets of assumptions were developed

and the corresponding formulas for the optimum stand- ard 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 deter- mining 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 con- sumption 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 which cause the social

40
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 natural gas used to heat buildings

in the U.S. is either produced domestically or is imported. Almost all of the imported gas purchased by

the major interstate pipeline companies comes from

Canadian and Mexican sources, although small amounts of liquified natural gas (LNG) have begun to be imported. Because of the current price controls on

domestically produced gas, these interstate pipeline

companies pay considerably less for domestic than for imported gas. For the year ending September 1977, the weighted average price for gas purchased from

domestic producers was S0.67/MCF, while that for Canadian and Mexican gas was S1.94/MCF.47 Thus,

two very different prices exist in separate markets for the same energy resource. This price differential is expected to be at least partially maintained at the retail level. As a result, if an energy conservation standard were based on the retail price charged by a local distributor using only domestic gas, the value of the energy saved would be understated. This is because

whatever gas is saved through conservation makes

possible an equal reduction in costly gas imports. To

correct for this undervaluation, a RIF could be used

equal to the ratio of the two retail prices. Assuming

for the moment that retail gas prices remain

proportional to the wellhead prices paid at the source,

the same RIF value would be given by the ratio of the two wellhead prices. Thus, RIF= 1.94/0.67 = 2.9. At

the other extreme, retail prices could be assumed to exceed their respective wellhead prices by a given

dollar amount needed to cover the average costs of

transport and distribution. This average mark-up can

be roughly estimated at S1.78/MCF.48 Adding this amount to both wellhead prices and forming a ratio, we obtain a lower estimate for the RIF value, 1.5.

47 U.S. Department of Energy, Energy Information

Administration, "September 1977 Interstate Gas Prices and

Financial Data," Energy Data Reports, January 13, 1978, Table 1, p. 4.

48 The average retail price of natural gas for twenty major

Standard Metropolitan Statistical Areas in October 1977 was about S2.57/MCF, as reported in U.S. Department of

Labor, Bureau of Labor Statistics, "Retail Prices and

Indexes of Fuels and Utilities: Residential Usage,"

November 1977, p. 6. The average wellhead price

including both domestic and imported gas was S0.79/MCF

for the year ending September 1977, as reported in U.S. Department of Energy, Energy Information

Administration, "September 1977 Interstate Gas Prices and

Financial Data," Energy Data Reports. January 13, 1978, Table 1, p. 4.

Introducing these two RIF values 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

between 18.4 percent and 41.3 percent from what they

would have been without RIF's. Smaller reductions,

12.6 percent and 29.9 percent, in the optimum rate of

transmission result from the same RIF values of 1.5 and 2.9, respectively, when the cost function is assumed to be quadratic rather than linear. Finally,

with a cubic cost function, the same RIF values lead

to reductions of between 9.6 percent and 23.4 percent

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 specifically, for the linear cost model it was found that doubling the number of degree days

leads to a 41.4 percent 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 percent 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 con- sumption 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 deter- mining 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.

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)

41
of an energy conservation standard based on the

actual private prices paid for energy. As we have seen

in section 2, 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 opti- mum. 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 (i.e., the value of

additional energy savings less the cost of achieving

them) which results from the use of RIF's is illustrated geometrically and derived algebraically in the appendix. The geometric 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 equivalent units of annual energy consumption by

dividing by the social price of energy and the uniform

present worth factor.

49 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.

An algebraic measure of the net gain in economic effi- ciency from using RIF's is also derived in the appendix. This measure is the amount by which the

life-cycle heating-related cost under the privately optimum standard exceeds that under the socially optimum standard. The energy consumed under both

standards is evaluated at its social value. It is shown

that for all RIF values (other than unity), there is a positive net gain in economic efficiency from using

RIF's. That is, present value life-cycle social costs are always lower with RIF's than without them. In

addition, for a typical house in a moderate climate this cost reduction is estimated to be $58.20 for a RIF

value of 1.5, and $569.10 for a RIF value of 2.9.

This makes annual energy consumption the numeraire.

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 operated in reality; (2) to apply those models to derive empirical estimates for the magnitude and direction of each type of pricedistorting 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.

50

The second area which requires further research is that of the relationship of RIF's to building perfor- mance standards. The analysis of sections 3 and 4 was

carried out using models to develop component perfor- mance 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 energy performance, which is currently being

developed at NBS, has been completed. This model is expected to incorporate the many energy-flow

interactions which occur among the various components of a building. If this effort is successful, it will be possible to analyze the economic trade-offs between interrelated components. Moreover, the model should be able to take explicit account of the changes in the degree-day base which result from

energy conservation techniques. This model should be

able to determine the socially or privately optimum

energy performance of buildings by basing the economic optimization routine on the social or private values of energy types, respectively. Thus, the effects of using RIF's in developing a building performance

standard would be determined by directly comparing

a standard based on the socially optimum performance

with one based on the privately optimum perfor- mance.

One attempt to develop a system of energy resource shadow prices similar in concept to RIF's is being made at Brookhaven National Laboratory. See Harold Abilock, et aL, A Study to Assess the Application of Shadow Pricing

Techniques to National Energy Resource Planning, Year 1 Final Report (Upton, NY: Brookhaven National Laboratory, June 1976).

42
REFERENCES

1. Abilock, Harold, et al. A Study to Assess the Application ofShadow Pricing Techniques to National Energy

Resource Planning, Year 1 Final Report. Upton, N.Y.: Brookhaven National Laboratory, June 1976.

2. Arad, Uzi; Smernoff, Barry J.; and ben-Shaber, Haim.

American Security and the International Energy

Situation. 4 Vols. Croton-on-Hudson, N.Y.; Hudson

Institute, April 1975.

3. ASHRAE Standard 90-75: Energy Conservation in New

Building Design. New York: American Society of

Heating, Refrigerating and Air Conditioning Engineers,

Inc. (ASHRAE), 1975.

4. Brannon, Gerard M. (Ed.) Studies in Energy Tax Policy. Cambridge, Mass.: Ballinger Publishing Company,

1975.

5. Britton, Jack R. Calculus. New York: Holt, Rinehart and Winston, 1961.

6. Coad, William J. "Section 12: Toward a More

Effective Use of Energy Resources." ASHRAE Journal, Vol. 19, No. 5 (May 1977), pp. 32-33.

7. Fisher, Anthony C, and Peterson, Frederick M. "The

Environment in Economics: A Survey." Journal of Economic Literature. Vol. 14, No. 1 (March 1976), pp.

1-33.

8. Haefele, Edwin T. (Ed.) The Governance of Common

Property Resources. Baltimore: The Johns Hopkins

University Press, 1974.

9. Hannon, B. M., et al. Energy Use for Building

Construction. Progress Report, March 1, 1976-August

1, 1976 (Center for Advanced Computation, Illinois University, Urbana, August 1976).

10. Hoffman, Kenneth C. "Some Comments on Energy

Accounting," Paper presented at the panel discussion on Energy, Environment and Economics, 68th Annual

Meeting, American Institute of Chemical Engineers,

November 1975. (Available as BNL 2567 from the National Technical Information Service.)

11. House of Representatives 12169, "Energy Conservation and Production Act." Public Law 94-385 (August 14, 1976).

12. Huettner, David A. "Net Energy Analysis: An

Economic Assessment." Science, Vol. 192 (April 9, 1976), pp. 101-104.

13. Jones, Robert R. "Resource Impact Factor (RIF)

Approach to Optimal Use of Energy Resources." ASHRAE Journal, Vol. 18, No. 18 (October 1976), pp.

15-18.

14. Leftwich, Richard H. The Price System and Resource

Allocation. 3rd ed. New York: Holt, Rinehart, and

Winston, 1966.

15. Lin, Steven A. Y. (Ed.) Theory and Measurement of Economic Externalities. New York: Academic Press, 1976.

16. Little, Ian M. D., and Mirrlees, James A. Manual of

Industrial Project Analysis in Development Countries. Vol. II: Social Cost Benefit Analysis. Paris: Development Centre of the Organisation of Economic

Co-Operation and Development, 1968.

17. Mansfield, Edwin. Microeconomics: Theory and

Applications. New York: W.W. Norton & Company,

1970.

18. Oklahoma University Science and Public Policy

Program. Energy Alternatives: A Comparative Analysis. Springfield, Va.: U.S. Dept. of Commerce, National Technical Information Service, 1975.

19. Petersen, Stephen R. "Economic Optimization in the Energy Conservation Design of Single-Family

Housing." ASHRAE Transactions, Vol. 82, Part I, 1976.

20. Petersen, Stephen R. Retrofitting Existing Housing for Energy Conservation: An Economic Analysis. Building Science Series 64. Washington, D.C.: U.S. Department

of Commerce, 1974.

21. Phipps, Harry H. "The RUF Concept as an Energy

Measurement Tool." ASHRAE Journal, Vol. 18, No. 5 (May 1976), pp. 28-30.

22. Sassone, Peter G. "Shadow Pricing in CBA:

Mathematical Programming and Economic Theory."

The Engineering Economist, Vol. 22, No. 3 (Spring

1977), pp. 219-233.

23. Solow, Robert M. "The Economics of Resources or the Resources of Economics." American Economic Review, Vol. 64, No.2 (May 1974), pp 1-14. 24. Spiegel, Murray R. Mathematical Handbook of Formulas and Tables. Schaum's Outline Series. New York: McGraw-Hill, Inc., 1968.

25. Staaf, Robert J., and Tannian, Francis X. (Eds.)

Externalities: Theoretical Dimensions of Political Economy. New York: Dunellen Publishing Company,

Inc., No date.

26. Stamper, Eugene. "Weather Data." ASHRAE Journal, Vol. 19, No. 2 (February 1977), p. 47.

27. U.S. Department of Energy, Energy Information

Administration, "September 1977 Interstate Gas Prices and Financial Data," Energy Data Reports, January 13, 1978.

28. U.S. Department of Labor, Bureau of Labor Statistics, "Retail Prices and Indexes of Fuels and Utilities: Residential Usage," November 1977.

43
APPENDIX

ECONOMIC EFFICIENCY

EFFECTS OF RIF'S

As was mentioned in section 5, the use of RIF's leads

to a conservation standard with greater economic effi- ciency 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 both geometric and algebraic measures of this net gain in economic efficiency due to the use of

RIF's.

A.l A GEOMETRIC MEASURE OF

ECONOMIC EFFICIENCY

Figure A. 1 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, 7T', 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. The alternative assumption of

quadratic degree days discussed as Case IV in section 4 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 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 the energy relative to that of thermal

resistance.

T

T

Thermal Resistance (R) FIGURE A.l TRADE-OFF CURVE BETWEEN ENERGY CONSUMPTION AND

THERMAL RESISTANCE FOR GIVEN CLIMATE AND INDOOR TEMPERATURE

44
T

Thermal Resistance (R)

FIGURE A.2 PRIVATELY OPTIMUM ENERGY CONSUMPTION RATE (A)

Figure A.2 has the same trade-off curve, TT', as shown above but introduces the concept of relative costs using the straight line, LL''. 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 (Pe) and of resistance (b) and the uniform present worth factor (UPW). The algebraic

formula for this line is given by

L = (Pe UPW) • A + b R, (A.2)

where L is a parameter representing the total present value of life-cycle heating-related costs.

2 This equation

1 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 unnecessarily complicate the argument without

altering the conclusion.

can be rewritten to express A as an explicit function of R so that the intercept and slope can be clearly seen:

A = L/(Pe UPW) - [b/(Pe 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', 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 A0 and R0 , given by the point of tangency can be considered optimum for the particular ratio of relative prices given by

the slope, b/(Pe -UPW). Since the actual price paid for energy, Pe

, was used rather than the social price,

•Pe-RIF, it can be said that A0 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, 6/(ZVRIFUPW),

based on the social price of energy. The first line MM

45
T

R 0 Rq L' M' N'

Thermal Resistance (R)

FIGURE A.3 SOCIALLY OPTIMUM ENERGY CONSUMPTION RATE (A AND

THE NET EFFICIENCY GAIN FROM USING RIF'S (N-M)

is drawn to be tangent to the trade-off curve TT so

that the value ^*at the point of tangency is interpreted as the socially optimum rate of annual energy consumption embodied in the standard. The

second line AW 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' 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.

3

3 This procedure means that annual energy consumption

serves as the numeraire.

The proof that TV represents the life-cycle heatingrelated cost under the privately optimum standard is demonstrated in the following steps:

1. A0 = the energy use under the standard.

2. R0 = the resistance used under the standard.

3. To sum these we can convert the units of R0 to

their equivalent energy units using the relative price ratio based on the social price of energy. Thus, life-cycle cost measured in energy units equals A0 + [b/(Pe -Kl¥-\JVWj\-R0 .

4. Now the tangent of the angle a in figure A.3 is equivalent to the absolute value of the slope of

line AW, which we know to be 6/(Pe.RIF-UPW).

5. But by definition, tan a = (N-A0)/R0

.

6. Thus, N=A0 + (tan a)-R0 =A0 +

[f>/(iVRIF-UPW)]vR0) which combined with

Step 3 shows that TV represents life-cycle cost denominated in equivalent units of annual energy consumption.

46
Similarly, M can be shown to represent the life-cycle

heating-related cost 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 AW were drawn

under the assumption that the RIF value is greater than one. If the RIF value for a particular 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. 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.2 AN ALGEBRAIC MEASURE OF

ECONOMIC EFFICIENCY

An algebraic measure of the net gain in economic effi- ciency resulting from the use of RIF's can also be

derived. 4 This measure is based on the same concept

underlying the geometric measure developed above. That is, the net gain in efficiency is equal to the amount by which the life-cycle heating-related cost under the privately optimum standard exceeds that under the socially optimum standard. The energy consumed under both standards is evaluated at its social value, i.e., the market price times the RIF.

Using the abbreviated notation of expression (4.2), for both standards, the life-cycle heating-related cost, adjusted for the social value of energy, can be

represented as follows:

L —- F K • D • (1/R) + a + b • R. (A.4)

4 The author is indebted to Dr. Douglas Shier for suggesting

this algebraic measure of efficiency.

Thus, for the privately optimum standard life-cycle cost (Z-J, expression (4.3) is substituted for R.

L0 = F K • D \K D (1/6)]

1/2 + a + b \K D (\/b)) ]/2

. (A 5 >

Simplifying,

L0 = a + (1 + F) [K D b\ m. (A.6)

Similarly, for the socially optimum standard life-cycle cost (L$, expression (4.5) is substituted for R:

L* = F K D [F K D {\/b)\ m

+ a + b [F K D (l/b)] W2

. (A' 7 >

Simplifying,

L* = [a + 2F]/2

} [K D b] m. (A.8)

Now the net gain in economic efficiency is equal to the savings in life-cycle costs, as given by

L„ - L* = [1 + F - 2 Fm] [K D • bf2

. (A.9)

It is interesting to note that in the case where the

social and private values of energy are equal, we have

F= 1 and expression (A.9) reduces to zero, as expected. Moreover, it can also be shown that the net gain in efficiency is always positive, regardless of the RIF value used. 5 This means that even if the social value of a resource is less than its market price (i.e. RIF< 1), a positive gain in economic efficiency would

result from using the RIF in the development of the standard. To get an idea of the order of magnitude of this net gain in economic efficiency, the following reasonable

values are assumed for the parameters in expression

(A.9):K = UPW • 24 • Pe = (13.65) • (24) • (4.5/1,000,000);

D = 5000; and

b = 0.02.

Under these assumptions, the net gain given by

expression (A.9) becomes 0.0194 for a RIF value of

1.5 and 0.1897 for a RIF value of 2.9. This means that

All the elements in the second set of brackets are positive. The first set of brackets can also be shown to be positive since (l+F)2 >4For (1-F)

2 >0.

47
for each square foot of envelope area in a new

building, the use of the RIF values of 1.5 and 2.9 would generate reductions in the present value life- cycle heating-related costs of 1.94 cents and 18.97

cents, respectively. Thus, for a typical single-family

detached house with 3000 square feet of envelope

area, the cost savings (or gains in economic efficiency)

would be $58.20 or $569.10 for the two RIF values assumed.

Table A.l CONVERSION FACTORS FROM CUSTOMARY TO

METRIC (SI) UNITS

Physical

Characteristic To Convert From To Multiply by

Length

Area

Temperature

Temperature difference Energy

{/-value

Thermal Resistance

ft

ft2 m 3.048 X10"1

9.290 X10'2 °F°F

Btu

Btu/hr-ft

2 -°F

hr-ft

2 -°F/Btu

W/m2o C

m

2o C/W

tc=(tp-32)/1.8

5.556X10'1

1.055 X103

5.678 1.761X10-'

48

A U.S. GOVERNMENT PRINTING OFFICE : 1978 O—267-424
U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO.

NBS BSS 114

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

The Effect of "Resource Impact Factors" on Energy Conservation

Standards for Buildings

5. Publication Date September 1978

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)

Department of Energy

Washington, D.C. 20545

13. Type of Report & Period

Covered

Final

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

Library of Congress Catalog Card Number: 78-606072

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

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 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.

Moreover, the additional energy savings resulting from using RIF's are shown to

increase as the severity of the climate increases. When two important restrictive

assumptions of the model are relaxed, these same relationships between energy con- sumption and the RIF value on the one hand, and between the extra energy savings

due to RIF's and the climate on the other hand, are both maintained. Finally,

geometric and algebraic measures are 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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computer sciences. Papers cover a broad range of subjects, with major emphasis on measurement methodology, and

the basic technology underlying standardization. Also in- cluded from time to time are survey articles on topics closely

related to the Bureau's technical and scientific programs. As

a special service to subscribers each issue contains complete

citations to all recent NBS publications in NBS and non- NBS media. Issued six times a year. Annual subscription: domestic $17.00; foreign $21.25. Single copy, $3.00 domestic;

$3.75 foreign. Note: The Journal was formerly published in two sections: Section A "Physics and Chemistry" and Section B "Mathematical Sciences." DIMENSIONS/NBS

This monthly magazine is published to inform scientists, engineers, businessmen, industry, teachers, students, and

consumers of the latest advances in science and technology,

with primary emphasis on the work at NBS. The magazine

highlights and reviews such issues as energy research, fire protection, building technology, metric conversion, pollution

abatement, health and safety, and consumer product per- formance. In addition, it reports the results of Bureau pro- grams in measurement standards and techniques, properties of matter and materials, engineering standards and services, instrumentation, and automatic data processing.

Annual subscription: Domestic, $11.00; Foreign $13.75

NONPERIODICALS

Monographs—Major contributions to the technical liter- ature on various subjects related to the Bureau's scientific and technical activities. Handbooks—Recommended codes of engineering and industrial practice (including safety codes) developed in cooperation with interested industries, professional organizations,

and regulatory bodies.

Special Publications—Include proceedings of conferences sponsored by NBS, NBS annual reports, and other special

publications appropriate to this grouping such as wall charts, pocket cards, and bibliographies.

Applied Mathematics Series—Mathematical tables, manuals, and studies of special interest to physicists, engineers,

chemists, biologists, mathematicians, computer programmers,

and others engaged in scientific and technical work.

National Standard Reference Data Series—Provides quantitative data on the physical and chemical properties of

materials, compiled from the world's literature and critically evaluated. Developed under a world-wide program co- ordinated by NBS. Program under authority of National

Standard Data Act (Public Law 90-396).

NOTE: At present the principal publication outlet for these data is the Journal of Physical and Chemical Reference Data (JPCRD) published quarterly for NBS by the American Chemical Society (ACS) and the American Institute of

Physics (AIP). Subscriptions, reprints, and supplements

available from ACS, 1155 Sixteenth St. N.W., Wash., D.C.

20056.

Building Science Series—Disseminates technical information developed at the Bureau on building materials, components,

systems, and whole structures. The series presents research

results, test methods, and performance criteria related to the

structural and environmental functions and the durability and safety characteristics of building elements and systems. Technical Notes—Studies or reports which are complete in themselves but restrictive in their treatment of a subject.

Analogous to monographs but not so comprehensive in scope or definitive in treatment of the subject area. Often

serve as a vehicle for final reports of work performed at NBS under the sponsorship of other government agencies.

Voluntary Product Standards—Developed under procedures

published by the Department of Commerce in Part 10, Title 15, of the Code of Federal Regulations. The purpose

of the standards is to establish nationally recognized require- ments for products, and to provide all concerned interests with a basis for common understanding of the characteristics of the products. NBS administers this program as a supple- ment to the activities of the private sector standardizing

organizations.

Consumer Information Series—Practical information, based on NBS research and experience, covering areas of interest to the consumer. Easily understandable language and

illustrations provide useful background knowledge for shopping in today's technological marketplace.

Order above NBS publications from: Superintendent of Documents, Government Printing Office, Washington, D.C.

20402. Order following NBS publications—NBSlR's and FIPS from

the National Technical Information Services, Springfield, Va. 22161.

Federal Information Processing Standards Publications (FIPS PUB)—Publications in this series collectively consti- tute the Federal Information Processing Standards Register.

Register serves as the official source of information in the Federal Government regarding standards issued by NBS

pursuant to the Federal Property and Administrative Services Act of 1949 as amended, Public Law 89-306 (79 Stat. 1127), and as implemented by Executive Order 11717

(38 FR 12315, dated May 11, 1973) and Part 6 of Title 15 CFR (Code of Federal Regulations).

NBS Interagency Reports (NBSER)—A special series of

interim or final reports on work performed by NBS for outside sponsors (both government and non-government).

In general, initial distribution is handled by the sponsor;

public distribution is by the National Technical Information

Services (Springfield, Va. 22161) in paper copy or microfiche form.

BIBLIOGRAPHIC SUBSCRIPTION SERVICES

The following current-awareness and literature-survey bibli- ographies are issued periodically by the Bureau:

Cryogenic Data Center Current Awareness Service. A litera- ture survey issued biweekly. Annual subscription: Domestic, $25.00; Foreign, $30.00.

Liquified Natural Gas. A literature survey issued quarterly.

Annual subscription: $20.00.

Superconducting Devices and Materials. A literature survey

issued quarterly. Annual subscription: $30.00. Send subscription orders and remittances for the preceding bibliographic

services to National Bureau of Standards, Cryogenic Data

Center (275.02) Boulder, Colorado 80302.
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