M.J. Kaiser r Energy Economics 22 2000 463]495 478
Ž . The cumulative distribution function H x describes the percent of the popula-
t
tion that spends x or less of their income on electricity consumption. The equity Ž
. measure E
E t
, f is defined in terms of this quantity:
2
x
Ž .
Ž . Ž .
E E
t , f s H x s
h y
d y,
H
2 t
t
and is a function
7
of the evaluation parameter x. This is a more accurate and comprehensive assessment of equity as compared to the moment method. Tariff t
1
Ž .
Ž .
is thus preferred to tariff t and is said to be more equitable if E E
t , f E
E t
, f ;
2 2
1 2
2
Ž .
Ž .
i.e. H 0.1 H 0.1 .
t t
1 2
4. The flat-rate tariff
A flat tariff rate is a one]parameter tariff that provides a benchmark for comparison with more complicated tariff designs. A flat tariff rate is the simplest
possible tariff and is still the most widely used throughout the world, especially for residential rates. A flat tariff is given by
Ž .
t s t
ADRrkWh ,
1
or in other words, a flat rate of t is imposed for any quantity of electricity
1
consumed. This neither encourages nor discourages consumption. A consumer who Ž
. uses E kWh electricity per month is charged t E
ADR . The total revenue
i 1
i
Ž .
R R
t , f generated by the tariff is thus given by:
Ž .
R R
t , f s Ýt E s t ÝE s t E
1 i
1 i
1 T
,
where E represents the total electricity consumption of the sample S S
. It is clear
T
that under a flat-rate tariff, total revenue does not depend explicitly on the shape of the distribution curve, but only on the total amount of consumption in the
sample. A plot of revenue vs. quantity electricity consumed for a typical residential consumer is a linear function that passes through the origin with slope t .
1
A number of conclusions result from the application of the flat-rate tariff. The first two results hold in general and follow from the revenue expression.
Ž .
1. R
R t ,f
is an increasing linear function of t and is independent of the form of f
. Ž .
2. The average value E g is an increasing linear function of t. For a given sample S
S , E is fixed and revenue is a linear function of t, and so
T
7
The choice of the value x s 10 is arbitrary but appeared to be a natural cut-off when examining Ž
. the output data of the analysis although in principle any value can be used . In fact, due to the
Ž . Ž .
complicated nature of the functional and its dependence on the underlying distribution h x , H x
t t
may induce ambiguous preferences as the cut-off value x changes.
M.J. Kaiser r Energy Economics 22 2000 463]495 479
Table 5 Flat tariff rate and related parameters
Ž .
Ž . Ž
. Tariff t
R R
t , f
E g H
0.10
t
Ž .
Ž .
ADRrkWh ADR
10 596 957
5.4 91.7
12 756 349
6.5 87.2
14 855 741
7.6 80.5
16 955 132
8.7 70.3
18 1 074 524
9.7 64.2
20 1 193 915
10.8 58.1
22 1 313 307
11.9 50.6
24 1 432 693
13.0 40.7
Ž .
as the magnitude of the flat-rate increases decreases , the revenue generated Ž
. will increase decrease in proportion to the tariff. In the short-run, therefore,
and assuming no change in the consumer response, total revenues will vary linearly with the tariff change. In the long-run, however, as individuals adjust
their consumption patterns in accord with the new tariff, the revenue change is dependent on the price elasticity of demand of the population.
In the short-run, the percentage of a household’s income spent on electricity
Ž .
Ž .
will rise fall with a tariff increase decrease . This can be seen as follows. The random variable g s C r I is described by the density function h, and in the
discrete case, the expected value of g is given by:
Ž .
Ž .
Ž .
Ýg Ý C rI
Ý E t rI t
Ý E rI
i i
i i
1 i
1 i
i
g s
s s
s ,
n n
n n
a linear function of t since E and I are fixed in the sample data.
i i
The next two conclusions are empirical results based on the specific form of the Ž
. collected data refer to Table 5 :
3. The percentage of the population that spends 10 or less of their total income Ž
Ž ..
Ž .
on electricity consumption H
0.1 is a decreasing approximately linear
t
function of t. Ž
. Ž
. Ž
. 4. Total revenue R
R t
, f and H 0.1 are approximately negatively correlated.
t
Ž .
Ž .
Ž .
As total revenue increases, the equity E E
t , f s E
E t
, f as measured by H 0.10 ,
2 t
decreases. From Fig. 4 it is clear that the equity measure is an approximately linear .
decreasing function of t. Since revenue R R
t ,f and t are proportional, this is
Ž .
equivalent to stating that total revenue is approximately a linear decreasing Ž
. Ž
. function of H 0.10 as shown in Fig. 5. A plot of revenue R
R t ,f
vs. equity E E
t , f
t
represents an ‘efficient frontier’; i.e. it is not possible to increase revenue without a subsequent decrease in equity, and vice-versa. Note that the shape of the efficient
Ž . frontier depends on the specific value of x used in H x .
t
M.J. Kaiser
r Energy
Economics 22
2000 463
] 495
480
Ž .
Fig. 4. Graph of the equity measure E E
t , f as a function of the tariff rate t for a flat-rate tariff.
M.J. Kaiser
r Energy
Economics 22
2000 463
] 495
481 Ž
. Ž
. Fig. 5. Graph of the total revenue R
R t ,f
vs. the equity measure E E
t ,f .
M.J. Kaiser r Energy Economics 22 2000 463]495 482
5. The two-tier tariff
