Energy efficiency

Energy Efficiency: A Study in Derived Demand
Michael Alexander1
International Innovative Institute

ABSTRACT:

Examination of the economics of energy efficiency can be unduly complicated by the
assumption that the demand for energy is a real demand. However, this paper takes the
approach that the demand for energy is not a fundamental demand, but rather a derived
demand – derived from the demand for energy services. This approach can greatly
simplify the analysis of energy efficiency. Note that although the paper is expressed
primarily in terms of the demand for electricity, the same underlying analysis would
apply to automotive demand for gasoline, water for irrigation, or any other derived
demand. This paper focuses on the Takeback Effect (sometimes called the “Rebound
Effect”) as an application of the derived demand approach.

Section I of this paper shows some of the problems with using the demand for energy as
the basis of analysis, and discusses how looking at the demand for energy services can
simplify the analysis. Section II discusses the relationship between the demand for
energy services and the demand for energy and shows how one can convert from one to
the other. Section III deals with the limitations of traditional supply and demand analysis
in analyzing the real world.

The Appendix examines the use of the concept of “negawatts” to produce similar results.

SECTION I: THE PROBLEMS WITH USING THE DEMAND FOR
ENERGY

In a traditional economic setting, the equilibrium price and quantity of a good can be
determined by a supply and demand curve.

1
The author may be contacted at <urban@leprechaun.com>

page 1 of 15


Figure 1: Traditional Supply and Demand

In such an analysis, the Consumer Surplus can be thought of as the area between the
demand curve and the price line (P0).

Figure 2: Consumer Surplus

page 2 of 15


(Note that producer surplus is the area between the supply curve and the price. However,
in utility regulation, the producer surplus is usually regulated by a fixed rate of return,
and therefore is not relevant to most analyses of energy efficiency involving regulated
industries. For that reason, it is omitted from the discussion in this paper.)

However, when examining the effects of conservation and the demand for electricity,
however, it is clear that this framework requires refinement. A simple example will
suffice:

Consider a customer who keeps his home heated and air-conditioned to 72 degrees year
round, paying a price of 10¢/KWH for electricity. After enrolling in a new government
program, the customer’s heating and air conditioning units have their efficiency increased
by 10%. (The cost in this example is exogenous to the consumer, and occurs as a
one-time cost, not an on-going marginal cost of electricity) Since the consumer is getting
the same amount of heating and cooling at for a lower electricity use, one would expect
that their demand curve would shift in. Let us assume, for simplicity sake, that in this
case the supply of electricity is completely elastic.

Since the customer is living in the same 72o house, and paying 10% less in total, logic
would dictate that he is better off. However, without changing the assumptions
underlying our original analysis we would come to the conclusion that the consumer was
not better off, but rather worse off.

Figure 3: Misleading Effect of a 10% improvement in Energy Efficiency

In the prior situation, his consumer surplus was area CPB, now that is reduced to area
CPA, which means that he is somehow area CAB worse off, which is not logical.

page 3 of 15


One solution is to assume that energy efficiency is beneficial if there is a sufficient
decrease in the price.

Figure 4: Demand Declining While Facing an Upward sloping Supply Curve

As we see in figure 4, our theoretical consumer has added to their surplus by area BCED
and lost area ADF. Therefore, under this transition, they are better off if and only if BCE
is larger than ADF. This is approach gets us half way there; the customer can be better
off, but not all the way there; the customer is not definitely better off.

In order to deal with this dilemma, Brennoni suggested that “Because energy efficiency
increase the amount of energy service one gets, the value consumers gets from initial
levels of consumption will increase, not decrease. Thus the effect of energy efficiency
investments is not to shift the demand curve down… but to pivot it.”

page 4 of 15


Figure 5: Brannon’s Rotated Demand Curve (from his Figure 4)

Unfortunately, the fundamental problem remains that if the price is unchanged at C, then
the consumer surplus change is ACD (the new Surplus) minus BCE (the old surplus), and
again, even though the customer gets the same temperature, the consumer surplus could
actually shrink if BCE > ACD. It is also not clear intuitively why the shift would occur,
other than in some specialized cases of “crowding out.”2

The problem with the preceding examples is that they are all set in terms of the demand
for electricity.ii But, people do not want electricity, per se. They want the services of
electricity. iii Therefore, the demand for electricity is a derived demand, not a primary
demand.iv We need to look at the area under the demand for energy services curve if we
want to see what is happening, in terms of consumer surplus.

The first thing to note is that the demand curve for energy services does not shift with
changes in energy efficiency. The pleasure that a person derives from a 72o house (or X
lumens of light in the house, etc.) is the same. Therefore, energy efficiency does not shift
the demand curve for energy services at all. What does change is to decrease the cost of
obtaining those energy services. Looked at in this way, instead of seeing a 10% increase
in efficiency as a decline in the demand for energy, we can see it as a 10% decrease in the
price of those services. So, the cost free 10% improvement in efficiency at a fixed cost
for electricity gives us Figure 3A.

2
I shall discuss crowding out later in the paper.
page 5 of 15


Figure 3A: Effect of Energy Efficiency in terms of Demand for Energy Services3

Now we see that the effect on consumer surplus of the added efficiency in our simplified
example is a clear increase of area PGBA (note that G=90% of P, since we posited a 10%
efficiency increase. Further, we have the difference between the amount demanded at A
and the amount demanded at B (which is what we expected), which is the intuitive
explanation of the takeback effectv (sometimes called the “Rebound Effect”). The actual
takeback would be determined by the transformation function which converts energy
services to energy demand. Note that the size of the takeback effect will depend upon
the change in energy efficiency (the larger the increase in efficiency, the larger the
takeback effect) and the slope of the demand curve, with a highly inelastic demand
resulting in a small takeback effect, and a highly elastic demand resulting in a large
takeback effect.

A more complex formulation of the supply curve does not create a problem. Customers
would still be demanding energy services, but they get the same services at a cheaper
rate. It might be as simple as the 10% reduction seen above, or it might be more
complicated, with varying slopes depending upon the degrees of savings, and even an
initial increase in cost to pay for the more efficient equipment.

3
In this paper, I will not be using “Heating and Cooling Degree days” the same way as NOAA (National
Oceanic and Atmospheric Administration) does, but rather simply to indicate a number of degrees of
temperature adjustment for duration of time. In reality the customer’s utility is probably a function of
temperature, not change in the temperature. It is also worth noting, in passing, that the cost of heating and
the cost of cooling are probably not the same.
page 6 of 15


Figure 7: Effect of Increasing Energy Efficiency

Note that in this example, there is an initial cost (A-B) to running the more efficient
equipment4 as well as a decreased marginal cost of the energy service (reflected in the
change in slope between the two supply curves). Therefore, whether the price of the
services increases or decreases (and therefore whether there is a net increase or decrease
in consumer surplus, will depend on whether the Demand curve crosses the two supply
curves to the left of D (the intersection of the “inefficient” and “efficient” supply curves)
or to the right of D.

SECTION II: CONVERTING BETWEEN THE DEMAND FOR ENERGY
SERVICES AND THE DEMAND FOR ENERGY

While customers demand energy services, utilities provide energy. Therefore, it is
necessary to look at how to convert from one to the other. In general, this is relatively
straightforward if one has good engineering estimates of the energy consumption of the
appliances involved in converting energy consumption to energy services. The easiest
example would be an automobile which gets x miles per gallon. One could then look at
the demand for gasoline, multiply it by x and get the demand for miles driven. Similarly,
one could convert the cost of gas to the cost of miles, simply by dividing it by x.

While the equations may be more complicated, if we assume that we know the function
for the production of energy services (which may require conversations with an engineer,
especially if things get complicated), then we know:
4
As mentioned before, the supply curve for regulated utilities is seldom, if ever, the marginal cost of
energy, so the reasons that the supply curves take on the shifts and reformations that they do is not a
concern for this paper.
page 7 of 15


EQ-1: f(energy consumption) = energy services,

and

EQ-2: f-1(energy services) = energy consumption.

Similarly, if the price of energy is linear5 then:

EQ-3: f-1(supply cost of energy)=supply cost of energy services

and

EQ-4: f(supply cost of energy services) =supply cost of energy.

Therefore, if one knows the demand curve for energy, D(p), then one knows the Demand
for energy services f(D(p)). If one knows the Supply curve for energy S(p), then one
knows the Supply curve for energy services f-1(S(p)).

The relationship between the elasticity of demand (εenergy) for energy and the elasticity of
demand for energy services (εservices) is relatively easy to establish.

Short-term elasticity for energy (εenergy) is defined as the percent change in Quantity of
energy services divided by the percent change in price of energy.

EQ-5: εenergy = (∆Q/Q)/(∆P/P)

Subsituting EQ-1 into EQ-5

EQ-6: εservices = (∆f(Q)/f(Q))/(∆f-1(P)/f-1(P))

If f is a linear function (or linear at the margin), EQ-6 will simplify to:

EQ-7 εenergy = εservices

Short Term vs. Long Term Elasticity

By definition, short term elasticity covers the change in quantity demanded when there is
no change in technology. Long term elasticity covers the change in quantity demanded
5
Increasing and decreasing block prices for energy are not consistent with this analysis, but will not be
considered in this paper.
page 8 of 15


after a technological response has taken place. This distinction is not relevant to the
demand for energy services, which are unchanged with technology. In effect, long-term
elasticity is the sum of short-run elasticity and technological elasticity (the study of the
adoption of technological change, endogenously and exogenously supplied is not
examined in this paper. Therefore all discussions of elasticity in this paper concern
short-run elasticity only.)

A Definition of the Takeback Effect

What does this mean in terms of the takeback effect? The takeback effect is defined as
the difference between the amount of energy used after an improvement in conservation
and the savings one would obtain if customers use the same amount of energy services.
In other words, if one were to improve the insulation of a home by 10% one would expect
to see a 10% change in the energy used for home heating and air conditioning if the
customer did not change their heating/cooling behavior. However, because the price of
heating and cooling a home is now 10% less (since the same energy consumption will
result in more heating and cooling), as the result of the price effect customers would
likely increase the amount by which they heated and cooled their homes, which will
diminish the energy savings experienced. It has been postulated that that takeback effect
could exceed the savings from the energy conservation program.

In a simple example with completely elastic supply curves, consider the 10% increase in
energy conservation:

$/Heating and cooling degree Day

A
P0 S0

B
.9*P0 S1

C D

Heating and cooling degree Days

Figure 8: Takeback Effect

page 9 of 15


Because of the lower price, however, the consumption of energy services has increased
from C to D in terms of energy services. In terms of energy itself, the increase in demand
(prior to applying the increase in energy efficiency) would be f-1(D) - f-1(C). After
applying the increase in energy efficiency, the takeback would be 0.9*[f-1 (D) - f-1(C)].

Note that by the definition of elasticity:

EQ-7 D-C= ∆Q

Substituting this into EQ-5, we get:

EQ-8: ε = ((D-C)/Q)/(∆P/P)

Solving for (D-C)/Q by multiplying both sides by [(∆P/P)], we get:

EQ-9: (D-C)/Q= ε*(∆P/P)

In other words, in terms of energy services, the takeback effect as a percentage of
demand (with Q≈(D+C)/2 is simply elasticity times the percent change in price. Note
that the percent change in the price of energy services is 1 minus the increase in
efficiency.

As an example, if the elasticity of demand for electricity is about 0.25, a 10% increase in
efficiency will result in a (0.25*0.10=0.03) 3% takeback effect. Therefore, if a study,
such as those surveyed by Gottronvi reports a significantly higher takeback effect, one
should suspect that something other than price effects may be going on.

If we assume that f(), is roughly linear, then in terms of energy demand, the takeback
effect as a percentage of demand is also equal to the same elasticity times the percent
change in price. Hence, the higher the price elasticity the higher the takeback effect. In
fact, the takeback effect will exceed 100% if

EQ-10 1 < ε*(∆P/P)

or multiplying both sides by (P/∆P), the takeback effect will exceed 100% if

EQ-11 (P/∆P) < ε

Note that, if the supply curves are not complely elastic, the same result holds, although
(∆P/P) may be more difficult to calculate, especially in a situation, such as that shown in
Figure 7 above.

page 10 of 15


SECTION III: PROBLEMS WITH TRADITIONAL SUPPLY AND
DEMAND ANALYSIS

While Sections I and II dealt with traditional supply and demand analysis, the bread and
butter of the micro-economist, there are a number of other possible models of decision
making for consumers and they might affect the final results. Consider the following
alternatives to the traditional demand curve.

• Budget Decision making

A consumer may put together a budget for consumption. Therefore, the consumer rule is
“I will spend roughly $x for electricity each month.” In this case, the consumer’s total
outlay for energy will be the same, and an improvement in the energy efficiency of an
appliance will be wiped out with a corresponding increase in energy use either on the
same appliance (i.e. more comfortable temperatures), or with an increase in the use of
other appliances in the home. The result would be a 100% takeback effect.

• Conservation thinking

There is reason to believe that customer’s utility functions include a belief that they are
saving energy. (And based on imperfect knowledge, that belief may not be accurate.)
Therefore, a consumer may say something to the effect of “Since I have better insulated
my home, I have done my bit for the environment. Now I can set my thermostat to a
more comfortable temperature, etc.” To the degree that the feeling of conserving is not
based in actual data, the exact effect may be unknown.

• Higher conservation costs drive out cheaper options

As an example, consider a person whose windows leak. If that person could install low
cost single pane windows, they probably would. However, if the market only offers
higher priced dual pane windows, for some customers, the cost of the upgrade might
exceed the value for some customers. Looking at Figure 7-a, we see that this could
happen on a system wide basis if the demand for energy services crosses the original and
increased efficiency supply lines to the left of their intersection point

page 11 of 15


Figure 7a: Effect of Increasing Energy Efficiency

As a result, that customer will use less efficient windows than they might have. As a
result one will see a rotated demand curve for electricity looking like the one in Figure 5
above:

Figure 9: More Expensive Conservation Measures Crowd Out Conservation at Some
Points

page 12 of 15


This means that for some customers, the amount of energy they use might actually be
above the

APPENDIX: NEGAWATTS AND THE TRADITIONAL ANALYSIS

The proponents of the “Negawatt” view of energy efficiency would call increases in
energy efficiency an increase in supply rather than a decrease in demand. In order to
graph this in a traditional Supply and demand curve, one would leave the demand curve
unchanged, but generate a supply curve S2, which is the sum of the tangible energy (from
S1 the supply curve of “tangible” electricity) and the number of negawatts produced.
One would then look at the intersection of the demand curve and S2.

Figure A-1: The “Negawatt” View

In the Negawatt view of the universe, the actual reduction in energy demand is the
quantity demanded at D minus that demanded at F consumers are made better off by
energy conservation, since when S0 shifts out to S1, their surplus rises from ABD to
ACE. Producers see a new surplus, instead of BGE, they now see, CGE, which might be
an increase or a decrease. Although it is clear that total surplus has increased by DEF, so
(costless) energy efficiency is a benefit both to consumers and to society. The actual
amount of energy demanded in this analysis was J before the increase in efficiency. K is
the amount of “tangible” and negawatts of energy, of which H are real and need to be
generated. If one looks at J, the amount of energy plus negawatts generated to meet the
old equilibrium level, one can see that L is the amount of tangible electricity generated,
and J-L is the amount of negawatts “consumed”. H-L would be the take-back effect. It
would not be possible, with a non-increasing demand curve and a non-decreasing supply
curve for H-L to exceed J-L, so at no point can the takeback effect exceed the savings
from a demand curve. The exceptions would occur if the traditional supply/demand rules
do not apply as mentioned in section III.

page 13 of 15


In the model above, can we estimate the takeback effect, given that we know the price
reduction B-C? Had there been no change in the quantity of electricity demanded, the
total demanded would be J. That demand would be made up of L megawatts and J-L
negawatts. However, because the price of electricity is lowered (held down), by the
introduction of conservation, the new quantity demanded is K (Note that K-J can be
determined using the price elasticity: η*J*(B-C)/B). The sum of the megawatts and
negawatts demanded is where the demand curve crosses the Supply curve, in this case K.
The price for the quantity K demanded is C. At price C, H units of real electricity
(megawatts) will be consumed and K-H negawatts will be consumed. The takeback
effect is the increase in real megawatts, therefore, the takeback effect, in this case is H-L.

Figure A-1 may be complicated if one assumes that there is an initial cost (M) to adopting
these energy conservation measures.

The Negawatt version of the model, unfortunately, cannot be graphed if supply is
completely elastic, although the analysis can be reconstructed without particular
difficulty.

The concept of negawatts, unfortunately, is not directly compatible with the concept of
derived demand. For instance, if we assume a basic linear supply curve:

EQ-12: P=mQ + G

Improving efficiency by (1-x)6 would result in the negawatt world in an equation of the
form

EQ-13: Pn = x*mQ + G

In other word, the price for Q=0 would be the same (G), and would diverge from there.
However, in terms of energy services, an increase in efficiency of (1-y)7 would result in
an instantaneous decrease in the cost of energy services, resulting in an equation of the
form

EQ-13a: Pd = y*mQ + y*G

In other words, while a graphical representation might intersect at a single point, at all
other points Pn ≠ Pd.

REFERENCES

6
(0<x<1)
7
(0<y<1)
page 14 of 15

i
Brennan, Timothy, “Energy Efficiency: Efficiency or Monopsony”, presented at the Center for Research in Regulated
Industries Western Conference, June 18, 2009
ii
Bently, W.G., C. E. Cosgrove, and P. W. Stallcup, "Integrating Econometric and End-Use Models: A Realistic Approach
to Conservation Programs," Proceedings of the Third EPRI Load-Forecasting Symposium March 25-27, 1981, July 1982,
pp. 44-76
iii
Crew, Michael and Spiegel, Menahem, “Price Caps, Demand Side Management and Public Goods”, presented at the
CRRI Western Conference, June 18, 2009
iv
Becker, G.S. "A Theory of the Allocation of Time", Economic Journal, Vol 75, No. 299 (September 1965) pp 493-517.
v
Alexander, Michael, The Effect of Changes in Technology of Derived Demand The Takeback Effect in Energy
Conservation Programs, Doctoral Dissertation, University of Minnesota, December 1997, available upon request to the
author <michael.alexander@sce.com>.
vi
Gottron, Frank, “Energy Efficiency and the Rebound Effect: Does Increasing Efficiency Decrease Demand?” CRS Report
for Congress, Congressional Research Service, Library of Congress, Order Code RS20981 (July 30, 2001), available at
https://www.policyarchive.org/bitstream/handle/10207/3492/RS20981_20010730.pdf?sequence=1

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