Notes on the Economics of Household Energy Consumption and 

Technology Choice 

Alan H. Sanstad* 

Lawrence Berkeley National Laboratory 

Abstract 

The Office of Management and Budget (OMB) and the Department of Energy (DOE) have 

initiated a joint effort to examine the issue of consumer welfare impacts of appliance energy 

efficiency standards, and to extend and discuss enhancements to the methodology by which these 

impacts are defined and estimated in the regulatory process. DOE's economic analysis of 

efficiency standards generally takes a life-cycle cost investment perspective focused on the tradeoff between initial and operating costs for efficient equipment. In this perspective, the time 

value-of-money is represented by the cost of capital. In a more general framework, additional 

trade-offs exist between investment and consumption, and consumer choice over the planning 

horizon also reflects preferences for future consumption. In this framework, these preferences 

combine with the cost-of-capital as drivers of consumer choice. This document presents a first 

version of a mathematical framework for analyzing the similarities and differences between these 

two decision modeling approaches, and thus starts to address several theoretical economic issues 

raised by OMB. It is anticipated that further elaboration of this framework may support empirical 

analysis to develop practical quantitative tools for improved assessment of the effects of 

appliance standards. 

* Staff Scientist, Lawrence Berkeley National Laboratory. #1 Cyclotron Rd., Berkeley, CA 94720. 

Voice: 510-486-6433; Fax: 510-486-6996; e-mail: ahsanstad@lbl.gov. 
   

1. Introduction 

U. S. federal appliance energy efficiency standards are established using a set of criteria 

pertaining to their effects on industry, consumers, environmental quality, and other factors. 

These notes are part of a joint effort by the Office of Management and Budget (OMB) and the 

Department of Energy (DOE) to examine the issue of consumer welfare impacts of efficiency 

standards, and to extend and enhance the methodology by which these impacts are defined and 

estimated in the regulatory process. 

DOE's economic analysis of efficiency standards generally takes a life-cycle cost 

investment perspective focused on the trade-off between initial and operating costs for efficient 

equipment. In this perspective, the time value-of-money is represented by the cost of capital. In 

a more general framework, additional trade-offs may exist between investment and consumption, 

and consumer choice over the planning horizon can reflect preferences for future consumption. 

In this framework, these preferences combine with the cost-of-capital as drivers of consumer 

choice. This document presents a first version of a mathematical framework for analyzing the 

similarities and differences between these two choice modeling approaches, and thus starts to 

address several theoretical economic issues raised by OMB. These notes have been prepared to 

facilitate discussion and investigation of analytical metrics for assessing welfare effects, initially 

from a theoretical perspective. Terminology and basic concepts in engineering and economic 

approaches to modeling household or consumer energy demand are reviewed, and a simple 

theoretical economic model of consumer energy efficiency and fuel choice is introduced and 

discussed. 

Going forward, this theoretical material may be useful in supporting empirical analysis 

to define and implement quantitative welfare estimates that relate life-cycle cost and other 

aspects of consumer appliance choices. This document reflects the philosophy that a clearlyarticulated theoretical framework can be useful in dealing with the potential challenges and 

complexities of identifying and obtaining data for such estimates and integrating it into practical 

quantitative tools. 

There is a long history of debate regarding consumer welfare effects of appliance 

standards, but the literature on this debate, as such, is not reviewed here. Indeed, this first version 

does not explicitly discuss standards per se. Instead, its aim is to facilitate discussion of the 

issues, provide a modeling starting point that can be discussed, debated, extended and improved, 

and to inform subsequent quantitative analysis. The departure point is the specific topic of 

metrics for assessing these consumer welfare effects. In the appliance standards regulatory 

methodology, direct consumer impacts are estimated by projecting life-cycle cost changes 

resulting from standards. In this paradigm, these direct cost outcomes are implicitly the measure 

of welfare effects. The key economic inputs to these calculations are purchase prices of 

appliances, energy prices, and discount rates. By contrast, the conventional microeconomic 

conception of consumer welfare is based on models of consumer utility maximization; while 

1 
   

such models can be constructed in applications to yield welfare metrics denominated in dollars, 

the factors underlying these metrics include, in addition to equipment and energy prices and 

discount rates, so-called “behavioral parameters,” such as substitution elasticities, that 

empirically characterize consumers’ choices of energy and technology as predicted by utility 

maximization. 

For applied quantitative analysis, all of these inputs and parameters – costs, prices, 

elasticities, etc. – must be empirically measured and/ or estimated, and issues such as data 

availability and quality, measurement error, and functional forms, must be addressed. The 

philosophy reflected in this first version of these notes is that these issues can tend to obscure the 

role of underlying principles; moreover, without a clear statement of theoretical and modeling 

assumptions, practical choices made in applied work – such as of functional forms for consumer 

utility - can have substantive consequences that may not be readily apparent (or desirable). In the 

present project, examples are the roles of utility discounting and intertemporal substitution per se

in understanding consumer energy choices, as opposed to the appropriate numerical values of the 

rate-of-time-preference or substitution elasticity. In addition, a clear theoretical foundation can 

be invaluable in focusing and facilitating applied analysis. For these reasons, this draft deals 

strictly with theory; it establishes terminology and basic background concepts, and presents and 

discusses an initial simple theoretical model of consumer energy efficiency choice. 

This is a working document in the sense that it is expected to evolve via revisions and 

additions from the stakeholders as the joint project proceeds. The exposition is intended to be 

self-contained (although familiarity with elementary optimization theory is assumed), and 

therefore deliberately begins from basic first principles of both engineering and microeconomic 

approaches to analyzing efficiency standards from the standpoint of consumer choice. For this 

reason, stakeholders are very likely to find some of the content already familiar – particularly 

Section 2. However, it is hoped that this summary of background material will help to clearly 

identify underlying assumptions and to define a frame of reference for analyzing consumer 

efficiency choices incorporating both engineering and economic perspectives and techniques. 

2. Basic engineering and economic choice models 

2.a The life-cycle cost model 

Appliance efficiency standards are based in part on the observation that households 

derive value or utility not from the direct consumption of fuels – electricity, national gas, etc. – 

but rather from the energy services that are produced when these fuels are used in conjunction 

with energy-using equipment such as refrigerators, air-conditioners, and water heaters. Thus, 

refrigeration, air-conditioning, and water heating are examples of energy services. Within a 

given end-use energy service category, different unit models require different levels of fuel input 

to produce a given level of energy service, that is, have different engineering energy efficiencies. 

2 
   

Therefore, in principle, a specific energy service output level can be produced by different 

combinations of fuel and equipment; put differently, there exists a fuel-efficiency trade-off. 

Appliance standards act on this relationship by requiring a specific minimum efficiency – or 

equivalently, all else being equal, a maximum fuel demand - for a given equipment type and 

service level. 

The regulatory process assumes that, in terms of prices faced by households in retail 

markets, the fuel-efficiency trade-off canonically corresponds to a cost trade-off, with more 

efficient equipment being more expensive to purchase initially while less expensive to operate. 

When combined with the assumption that both the level and the characteristics of the underlying 

energy service are held constant across fuel/ efficiency combinations, and initial costs, operating 

characteristics, and future fuel prices are assumed known with certainty, the problem of 

minimizing the cost of obtaining energy services is quite naturally captured in a deterministic 

discrete-time engineering-economic, i.e., discounted cash flow or life-cycle cost (LCC) model.1

 

In the case of a choice between two discrete efficiency levels, the LCC model is as 

follows. Suppose that for a given end-use energy service category, two units of equipment, 

labeled 1 and 2 respectively, have initial costs (purchase prices) and and require energy 

(fuel) inputs of and per period to produce an exogenously-given service level, with 

 and . An example would be two refrigerators of equal volumes and features (i.e., 

of the same product class), one with higher purchase price but lower annual energy consumption 

in kilowatt hours under equivalent operating conditions. Further assume that the two units have 

the same anticipated operating lifetime of T periods, that an initial fuel price

P1 P2

E1

E1 >

E2

P P 1 < 2 E2

0 p is given and that 

a future sequence of per-period fuel prices is assumed, . Finally, assume that a fixed 

per-period “discount rate” is given. Then the expected LCC of purchasing and operating unit i , 

, is 

1 2 p p, ,K

r

i = 1, 2

0 ( ) 1

T

t i

i i t

t

p E LCC P

= r

= +∑ +, (2.1) 

where t i p E

t

LCC

 is the operating cost of unit i in period t . In this set-up, cost minimization means 

simply choosing the unit with the lowest LCC. Denoting the operating cost of the unit in 

period as , this criterion can be stated as: Choose the more efficient unit (#2) if and 

only if , i.e., 

th i

OC p E it t i ≡

2 1 < LCC

                                                           

1

 The choice of discrete rather than continuous time for this exposition reflects the convention used in the regulatory 

process and in much of the investment literature; it makes no substantive difference to the results. 

3 
   

( ) ( )

2

2 1

0 0 1 1

T T

t

t

t t

OC OC P P

= = r r

+ < + ∑ ∑ + +

1t

t , (2.2) 

or 

( )

1

2 1

0 1

T

t

t

t

OC OC P P

= r

− − <∑ +

2t

1

. (2.3) 

That is, the more efficient unit should be chosen if and only if the present value of the operating 

cost savings exceeds the incremental purchase price. This frames the energy efficiency choice as 

an investment decision using the net-present value criterion: The more efficient unit should be 

chosen if the initial “investment” of P P 2 − is exceeded by the discounted sum of the stream of 

“returns” . This highlights the critical role of the discount rate . To explore this 

point further, assume that the fuel price is constant across periods, 

OC OC 1t − 2t r

t p = p for all t , so that the 

(undiscounted) operating costs are also constant across periods, OC OC it = i for all . Recalling 

the summation formula 

t

( ) ( ( )

0

1 1 1 1

1

T

T

t

t

r

r r

−

=

= −+ ∑ + )

1 2

, (2.4) 

and letting and , the previous LCC inequality can be re-arranged 

as 

Δ= − PPP 2 Δ= − OC OC OC 1

1

1 (1 ) T

OC

r

P r −

Δ ⎛

> ⎜ Δ −+ ⎝ ⎠

⎞

⎟. (2.5) 

In keeping with the investment interpretation, the expression OC

P

Δ

Δ is the incremental return, in 

operating cost savings, for an incremental initial investment in energy efficiency. The right-hand 

side of this inequality approximates the discount rate . Although an infinite horizon in this 

type of model is not practically relevant, it is nonetheless instructive – for mathematical 

simplicity – to imagine the limiting case as T increases. In the limit, the inequality approaches 

r

OC

r

P

Δ

>

Δ , (2.6) 

and equality – i.e., equal LCCs of the two technologies – would hold if ΔOC P r Δ = . In other 

words, in this limiting case, the ratio ΔOC PΔ is the internal rate-of-return (IRR) of the 

efficiency investment, and the previous inequality is the standard criterion “invest if the IRR 

exceeds the discount rate.” 

4 
   

Beyond this two-technology example, in practice – e.g., in appliance standards regulatory 

analysis - a set of technologies is usually posited, ranked by efficiency, with the purchase prices 

increasing, and operating costs decreasing, with higher efficiency – the familiar “cost curve.” 

Notwithstanding the fundamentally discrete nature of technology types or options, it will be 

useful to consider a continuous generalization (or abstraction), which we introduce here and 

consider in more detail in the next section. Suppose that, for a given end-use energy service 

category, efficiency is represented by a continuous variable ε , the purchase price of technologies 

is represented by an increasing function P(ε ) , and the per-period energy input requirement by a 

decreasing function E( ) ε , where as in the above discrete example an exogenous service level is 

assumed. Then the operating cost in period t is OCt t (ε ) = p E(ε ) , and we can write the lifecycle cost as a function of ε as 

() () ( )

0 ( ) 1

T

t

t

t

p E LCC P

r

ε

ε ε

=

= +∑ +. (2.7) 

Further assuming differentiability of P(ε ) and E(ε ), the condition for life-cycle cost 

minimization is 

0 dLCC

dε = , (2.8) 

or 

0 ( ) 1

T

t

t

t

dP p dE d

d r

ε

ε =

= −∑ +. (2.9) 

As before, if we assume constant energy price p and therefore constant operating cost OC(ε ) , 

then (using the summation formula above) the equation becomes 

1

1 1 T

dOC dP

r

d d ε ε ( r)−

⎛ − = ⎜

⎝ ⎠ − +

⎞

⎟, (2.10) 

a continuous analogue to the discrete, two-technology condition ΔOC P r Δ = . 

2.b Static utility maximization 

In this section, the basic two-good microeconomic model of consumer choice is briefly 

reviewed. In complete textbook expositions of this theory (for example, Varian 1992), a number 

of technical properties of utility functions and consumer optimization are defined and analyzed; 

here we note only the following essentials, in a two-good model. A hypothetical consumer is 

5 
   

assumed to choose strictly positive quantities of goods or services 1x and 2 x with prices 1 p and 

2 p , respectively, by maximizing a utility function U x ,x ( 1 2 ) subject to the budget constraint that 

total expenditures on the two goods not exceed an amount y . The function represents 

the consumers’ preferences regarding 

U , ()

1x and 2 x individually and jointly, and is assumed to be 

strictly increasing, twice continuously differentiable, and strictly concave in its arguments; in 

terms of partial derivatives, for i =1,2 , 

2

2

22 2

2 2

1 2 12

0

0

i

i

U , x

U

x

UU U . x x xx

∂

> 

∂

∂

<

∂

∂∂ ∂

>

∂ ∂ ∂∂

 (2.11) 

The derivative U xi ∂ ∂ is known as “marginal utility.” The first inequality states that marginal 

utility is positive, i.e., utility increases with increased consumption. The second inequality states 

the property of “diminishing marginal utility,” i.e., although utility increases, it does so at a 

declining rate. 

We assume that the consumer possesses perfect information on these goods, services, and 

prices, and solves the choice problem under conditions of certainty. This terminology warrants 

clarification: In this context, “imperfect information” and “uncertainty” are distinct although 

related concepts. A standard example of decision-making under uncertainty is of choice among 

lotteries – i.e., bets – with outcomes defined by random variables the probability distributions of 

which are known to the consumer. In this case, the usual assumption is that the expected value 

of utility is maximized. This differs from the colloquial idea of having “imperfect information” 

in the sense of being uninformed or perhaps simply mistaken. 

Mathematically, the consumer’s behavior is described as solving the following 

optimization problem (where “max” means “maximize,” and “s. t.” means “subject to”): 

1 2

11 2 2

U(x ,x )

s.t. p x p x y

max

+ ≤ (2.12)

 

The Lagrangian for this problem is 

L ( ) x1 2 , , (, ) x U x x y px px λ λ = +−− 1 2 ( 11 2 2 ),

 (2.13)

and the first-order necessary conditions for optimality are 

6 
   

1

1

2

2

11 2 2

U

p

x

U

p

x

p x px y

λ

λ

.

∂ = ∂

∂ = ∂

+ ≤

 (2.14)

Solving the first two equations for λ , equating the results, and re-arranging yields the condition 

1 2

1 2

U U

p p

x x

∂ ∂ = ∂ ∂ . (2.15) 

To gain some intuition for this, first suppose that 1 2 p p = =1. Then the condition is simply 

1 2

U U

x x

∂ ∂ = ∂ ∂ , (2.16) 

that is, a necessary condition for utility maximization is that the levels of consumption of each of 

the two goods are such that marginal increases in utility from marginal increases in their 

consumption are equal. Were this not the case – say 

1 2

U U

x x

∂ ∂

>

∂ ∂

 - then an infinitesimal increase in 

the consumption of 1x and offsetting decrease in the consumption of 2 x would increase utility 

while staying within the budget constraint, contradicting the assumption of optimality. In the 

general case of 1 2 p ≠ p , this logic is amended to take into account the unequal marginal costs, 

i.e., prices, of consuming an extra amount of either good, so that the ratio of marginal utilities at 

optimality equals the relative prices. It can be shown that what this in turn implies for the optimal 

relative quantities of 1x and 2 x that are consumer depends on the concavity assumption; an 

example is discussed in detail in the next section. 

3. A simple model of intertemporal energy service choice 

 3.a Modeling context 

The previous section introduced the concept of energy services, such as refrigeration, air 

conditioning, and water heating, which are produced by combining energy (fuel) with energyusing equipment such as household appliances. The term “produced” is used here deliberately, 

for it places household energy and efficiency choices in the context of the economic theory of 

“household production” (Deaton and Muellbauer 1980, LaFrance 2001). This theory builds on an 

analogy to the theory of the firm by observing that households implicitly “produce” their own 

goods and services by combining various “raw” inputs, as in the present example, and that in a 

7 
   

microeconomic sense these goods and services are the actual inputs to utility. The model 

presented below adopts this approach. The key goal is to explicitly address the household’s 

energy service choice, rather than specifying it exogenously as in the life-cycle cost approach. 

In the previous section, we described the life-cycle cost minimization problem in the case 

of two discrete technologies of differing efficiencies. Modeling consumer choice among a set of 

discrete options is the focus of what is known as “qualitative choice analysis” (QCA), a 

methodology which has been applied to energy technology choice in many studies. (See for 

example Train 1986.) Although QCA is in certain ways the natural approach to studying this 

type of consumer choice, it is also probabilistic and data-intensive, and designed specifically for 

numerical econometric analysis. Because these notes are initially concerned with underlying 

principles independent of statistical issues or the role of specific functional forms or parameter 

values, the following discussion instead assumes continuous relationships among energy 

efficiency, technology cost, and energy service, elaborating on the assumptions of the continuous 

life-cycle cost minimization model described in the previous section. 

3.b Model assumptions 

We assume that energy E is an input to a technology that produces an energy 

service S according to the relationship S E = ε , whereε is a continuous index of service output 

per energy input. Thus ε represents the technology’s technical efficiency: Increasingε

corresponds to an increased service output per unit of energy input or, equivalently, a reduced 

energy input requirement per unit of service. The cost of purchasing the technology with 

efficiency level ε is assumed to be a twice continuously-differentiable function P(ε ) defined 

for ε in the interval E = [0,ε max ](where max ε is a maximum-feasible efficiency level) and 

satisfying P(0 0 ) = , P( ) ε > 0 for ε > 0, 0 dP

dε

≥ , and 0 dP

dε

> , 

2

2 0 d P

dε

> for ε > 0, i.e., P(ε ) is 

positive, strictly increasing, and strictly convex for positive ε . 

The basic static utility maximization model sketched in Section 2 can be extended in 

various ways to represent “dynamic” or “intertemporal” choice problems, i.e., those that involve 

a time dimension in the sense that decisions regarding future as well as present consumption are 

made, or planned, jointly. Such extensions can include time horizons of arbitrary length, for 

example the lifetime T of energy-using equipment in the LCC model. In order to introduce as 

simply as possible the essentials of intertemporal choice from a microeconomic perspective as 

they apply to the energy and technology choice problem, however, we confine ourselves here to 

a two-period model in which the “present” corresponds to period 1, and “future” to period 2. We 

simplify further by initially focusing only on the intertemporal choice of energy services, rather 

8 
   

than that of energy services as well as other household goods. Finally, we assume that the 

household or consumer has perfect information not only about present-period but also future 

prices and energy technologies, and makes the intertemporal choice under conditions of 

certainty. In the context of intertemporal choice, of course, these assumptions can be considered 

much stronger than their static analogues (as in Section 2b), for here they imply perfect foresight. 

Quantities of energy and energy services in periods 1 and 2 will be denoted and , 

respectively. 

E ,S 1 1 E ,S 2 2

The behavioral assumption is discounted utility maximization. This has a different 

meaning than the discounting of cash flows in the LCC model (for example); here, it means that 

anticipated future utility is given less “weight” than present utility. We assume that two-period 

utility is additive separable, and introduce an intra-period, single-good utility function U() so 

that, in terms of the notation of the previous section, we can write 

( )() ( )

( )

2

12 1 1

S

U S ,S S

ρ

= +

+

U = , (3.1) 

where U( ) is strictly increasing and concave - 0

i

d

dS

>

U , 

2

2 0

i

d

dS

<

U - and ρ is a positive rate of 

time preference that represents the above-mentioned “weighting” of future utility. 

We assume that an initial wealth endowment W , first period income 1 y , and expected 

second period income 2 y are given exogenously. We also assume a fixed, exogenous “discount 

rate” that is a riskless rate for either borrowing or saving, implicitly through a capital market. 

One “dollar” in the present can be invested or saved to yield 1

r

+ r dollars in the future; 

conversely, one dollar can be borrowed in the present at the cost of repaying 1 dollars in the 

future. The household’s two-period budget constraint then takes the form 

+ r

( ) ( ) ( )

2 2 2

1 1 1 1 1

p E P pE W y

r r

ε + + ≤ + + y

+ +

, (3.2) 

which states that the present value of expenditures on fuel and technology must not exceed the 

present value of wealth and income. This present-value formulation derives from the existence of 

a capital market allowing the household to shift wealth or income, and therefore expenditures 

and consumption, between present and future. Note that this interpretation makes sense even if 

the discount rate is zero; the point is that the consumer is not restricted to spending wealth or 

income only in the period in which they become available. (Put differently, the budget constraint 

for the two-period problem does not consist of two single-period budget constraints.) Instead, in 

the first period, the household faces a choice of allocating between consumption and saving. 

 3.c Model solution and interpretation 

9 
   

The household chooses levels of energy services in periods 1 and 2 to solve the following 

problem: 

( ) ( )

( )

( ) ( ) ( )

1 2

2

1

2 2 2

1 1 1

1 1

2 2

1

1 1

S S,

S

S

p E s.t. P p E W y

r r

S E

S E.

max ρ

ε

ε

ε

 +

+

 + + ≤ + + y

+ +

=

=

U

U

 (3.3) 

Note that in keeping with the perspective that energy services are the inputs to utility, the 

service demands in the two periods are the decision variables in this problem. However, this 

raises the following issue: While the equations relating services to energy can be used to 

substitute for the energy terms in the budget constraint, the definition of the optimization appears 

to leave unresolved the problem of choosing the optimal value of energy efficiency ε . 

To deal with this, we use the household production theory idea of “two-stage” or “twotiered” decision-making: In the lower tier, the optimal combination of fuel and efficiency is 

determined conditional on the allocation of service consumption between the two periods; at the 

top tier, this intertemporal allocation is decided using the lower-tier results as inputs. In this 

model, the lower-tier optimality criterion is cost-minimization. Under our assumptions, given 

values of services, the cost of “producing” these services is the sum of fuel 

expenditure in period 1, discounted fuel expenditure in period 2, and the cost of the technology 

as a function of the efficiency level, that is: 

1 2 S ,S C S ,S ( 1 2 )

( ) () ( )

2 2

1 2 1 1 1

p E C S ,S P p E

r

= ++ ε

+

, (3.4) 

where 1

1

S E

ε = and 2

2

S E

ε = . The lower-tier problem is thus 

( ) 1 2 ( )

2 2

1 1

1 1

2 2

E ,E , 1

p E min P p E

r

s.t. E S

E S.

ε

ε

ε

ε

 ++

+

 =

 =

 (3.5) 

Note that this is a special case of the continuous version of life-cycle cost minimization 

described in Section 2, with the operating cost function implicitly defined here by 

OC p S t tt ( ) ε = ε . It is shown in the Appendix that the solution to this problem is a unique 

10 
   

efficiency level ε

∗ , energy consumption quantities E S ,E S 11 2 2 ε ε ∗ ∗ ∗ = = ∗ , and Lagrange 

multipliers 1 , 2 τ τ ∗ ∗ . 

The optimal values 1 2 τ ,τ ∗ ∗ of the Lagrange multipliers in the cost-minimization problem 

are called the shadow prices of the energy services in the two periods. Under our assumptions, 

they are the marginal costs of the energy services. Returning to the two-stage framework, these 

shadow prices allow us to re-write the top-tier utility maximization problem as 

( ) ( )

( )

( ) ( )

1 2

2

1

2 2 2

1 1 1

1

1 1

S S,

S

S

S y s.t. S W y .

r r

max ρ

τ

τ

∗

∗

 +

+

 + ≤ + +

+ +

U

U

 (3.6) 

In other words, we have subsumed or embedded the solution to the conditional energy service 

cost minimization problem, and the top-tier problem is now to allocate energy service 

consumption intertemporally, i.e., between periods 1 and 2. The Lagrangian for this problem is 

( ) () ( )

( ) ( ) ( )

2 2 2

1 2 1 1 1 1 1 1

S y SS S W y S

r r

τ λ λτ

ρ

∗

∗ ⎛ ⎞

, , = + + + + − − ⎜ ⎟

+ + ⎝ ⎠

U

L U 2

1

S

+

 (3.7) 

and the first-order conditions are 

( ) ( )

( ) ( )

1

1

2

2

2 2 2

1 1 1

1

1 1

1 1

d

dS

d

dS r

S y S Wy

r r

λτ

τ λ

ρ

τ

τ

∗

∗

∗

∗

=

= + +

+ ≤ + +

+ +

U

U

.

 (3.8) 

As in the previous section, we can solve for λ and combine the first two equations to obtain the 

condition 

( )

( )

1

1 2 2

1

1

d d r

dS dS

τ

τ ρ

∗

∗

+ = +

U U . (3.9) 

This equation describes the optimal trade-off – i.e., between present and future consumption of 

energy services – in this intertemporal problem. Comparing with Equation 2.15, we see that the 

static optimality relationship is now augmented by the introduction of the rates of discounting 

and time preference: In addition to the effect of the prices 1 τ ∗ and 2 τ ∗

, the extent to which the 

11 
   

utility-maximizing household will favor either present or future consumption is also influenced 

by the relative magnitudes of r and ρ , as reflected in the ratio 1

1

r

ρ

+

+ . 

We mentioned in the previous section the importance of concavity in interpreting the 

optimality condition in the static model. Analogously, in this two-period model the concavity of 

U plays a critical role, which we now describe. To explain this, suppose first that r = = ρ 0 , i.e., 

that the household does not discount future consumption and that there is no cost for borrowing 

nor a “return” for saving. In this case, the “dynamic” problem is completely equivalent to the 

static; the optimality condition equates the ratio of marginal utilities to that of prices. Now 

consider the effect of incrementally increasing r and ρ so that both are positive, and reoptimizing. If r > ρ , then the right-hand side of Equation 3.9 will increase to maintain equality; 

if r < ρ , it will decrease. Thus, the relative magnitudes of 1 d dS U and 2 d dS U must in turn 

adjust. What does this imply for the adjustment in the consumption levels of and ? Because 

the concavity assumption implies that 

1 S 2 S

2 2 0 i S < , it follows that 2 dU d i dU dS will increase 

(decrease) if and only if decreases (increases). Suppose that r i S > ρ ; then to maintain 

optimality 1 d dS U must increase relative to 2 d d U S , and therefore must decrease relative to 

. (Either must decrease or must increase, or both.) Conversely, by exactly analogous 

reasoning, it follows that if 

1 S

2 S 1 S 2 S

r < ρ after the incremental change, then must increase relative to 

. 

1 S

2 S

Let us interpret these conclusions in terms of the consumption/saving choice noted above. 

The consumer derives utility from the consumption of energy services, but must decide how to 

allocate this utility between present and future. As in the static problem, the relative period 1 

and period 2 prices favor either present or future consumption. However, the assumption of a 

non-zero rate-of-time preference ρ means that, all else being equal, the consumer prefers present 

to future consumption. Moreover, by virtue of the capital market, the consumer also has the 

option to borrow in order to increase present consumption at the expense of future, or to save and 

thereby have additional resources to finance consumption in the future. The discussion in the 

previous paragraph says that the consumption vs. saving dimension of this choice problem is 

governed by the ratio 1

1

r

ρ

+

+

. If r > ρ , then the value of marginal incremental future consumption 

available from deferring present consumption in order to save and subsequently 

increase exceeds the loss of utility from lower consumption of . Conversely, if 2 S 1 S r < ρ , then 

at the margin present consumption is favored. 

3.d Discussion 

12 
   

We now summarize the main elements of this section. We introduced a deterministic 

two-period model of energy service choice by a discounted utility maximizing consumer subject 

to an intertemporal budget constraint. Adopting a two-stage budgeting approach from household 

production theory, we showed that the consumer’s optimal solution incorporates life-cycle cost 

minimization conditional upon two-period energy service demands. The roles of the rates of 

discount and time preference (in addition to prices) in determining the consumer’s allocation of 

energy service consumption between present and future were discussed, including the 

importance of concavity in the utility function. 

With respect to the background and objectives presented in the Introduction, we would 

emphasize the following points. First, in this simple model, life-cycle cost minimization and 

intertemporal utility maximization are not alternative assumptions, but consistent and interrelated; this result should be robust to at least some extensions (including multi-period time 

horizons). Second, the distinction between the “discount rate” and the rate of time preference is 

critical, and highlights the problems of using only observed or implicit discount or “hurdle” rates 

in an engineering-economic framework to make inferences about consumers’ intertemporal 

choices related to energy efficiency and technology. Although the details will vary, this 

conclusion will also be robust in more complex models (including those applying different 

behavioral assumptions than exponential discounting of utility). Third, even this simple model 

begins to indicate the type and importance of microeconomic “behavioral parameters,” such as 

the rate of time preference. 

3.e Elaborating the model 

As stated in the Introduction, these notes have been prepared as part of a joint OMB-DOE 

discussion on consumer welfare effects of appliance efficiency standards, and are aimed at 

facilitating discussion of issues and contributing to further work on technical methodology. Such 

work can proceed in several (complementary) directions. Specifically with regard to theoretical 

extensions of the model and results described above, next steps could include the following. 

First, in addition to the rate-of-time-preference, the so-called “intertemporal elasticity of 

substitution” is a critical behavioral parameter in dynamic choice models, and its effects on 

optimal choice in the model should be analyzed. Second, to investigate how energy service 

choice is combined with, and affected by, the consumer’s preferences for other goods and 

services, a non-energy composite good can be introduced. Third, extensions to longer time 

horizons are important – in particular, the lifetime of energy technology as represented in the 

life-cycle cost model. Fourth, even before considering empirical applications, it will be 

important to introduce specific functional forms for utility in order to gain understanding of how 

these may affect the results. Fifth, more complex representations of energy service “production 

technologies” – e.g., incorporating possible nonlinearities – should be explored. As these 

examples indicate, there are multiple possibilities for further development. 

13 
   

5. References 

Deaton, Angus, and John Muellbauer. 1980. Economics and consumer behavior. 

Cambridge, England: Cambridge University Press. 

LaFrance, Jeffrey T. 2001. “Duality for the Household: Theory and Application.” 

Chapter 18 in B. Gardner and G. Rausser, eds., Handbook of Agricultural Economics. New 

York: Elsevier Science. 

Train, Kenneth E. 1986. Qualitative Choice Analysis: Theory, Econometrics, and an 

Application to Automobile Demand. Cambridge, Massachusetts: The MIT Press. 

Varian, Hal R. 1992. Microeconomic Analysis. New York: W. W. Norton. 

14 
   

15 

Appendix 

The existence of a solution to the cost-minimization problem in Section 3 can be shown as 

follows. The Lagrangian for this problem is 

( ) () ( ) ( )( 2 2

1 2 12 1 1 11 1 2 2 2 1

p E E ,E , , , P p E S E S E

r

ε ττ ε = ++ + −+ − τ ε τ ε ) +L

and the first-order optimality conditions are 

( )

11 2 2

1 1

2

2

1 1

2 2

0

0

0

1

dP E E

d

p

p

r

S E

S E.

τ τ

ε

τ ε

τ ε

ε

ε

− − =

− =

− = +

=

=

This is a system of five equations in five unknowns; by first solving for in the fourth and 

fifth equations and then for 

E ,E1 2

1 2 τ ,τ in the second and third, we can reduce the system to a single 

equation in the unknown ε : 

( )

2 2 2

1 1 1

dP p S p S d r

ε

ε

= +

+ . (∗)

Because we have deliberately avoided introducing specific functional forms, we cannot 

solve this equation explicitly for ε ; instead, we reason as follows. We assume (trivially) that 

( )

2 2

1 1 0

1

p S p S

r

+

+

> , and by the assumptions on P(ε ) , 

0

0 dP

d ε ε =

= . Differentiating, 

2

2 2

2 2 0 d dP d P dP

dd d d

ε εε

εε ε ε

⎡ ⎤ = + > ⎢ ⎥ ⎣ ⎦

for ε > 0, where the inequality again follows from the assumptions on P(ε ) (strictly increasing 

and concave, with continuous second derivative). If we make the reasonable assumption that the 

quantity ( )

2 2

1 1 1

p S p S

r

+

+

 lies in the range of P(ε ) , it follows that there is a unique solution ε

∗ to 

Equation ∗ (by the Intermediate Value Theorem and the strict monotonicity of P( ) ε ).