M.J. Kaiser r Energy Economics 22 2000 463]495 482
5. The two-tier tariff
A general model for the two-tier tariff rate is given as: Ž
. Ž
. t
ADRrkWh if E F E kWh
1 i
t s
½
Ž .
Ž .
t ADRrkWh
if E E kWh .
2 i
Ž .
For the first E kWh electricity consumed per month, the price is t ADRrkWh ,
1
Ž .
and for the balance of the monthly consumption E y E
, the price is
i
Ž .
t ADRrkWh . This is a form of lifeline pricing which involves charging a low,
2
subsidized price for a fixed energy quota and a higher price for consumption above that level. A plot of revenue vs. quantity electricity consumed is a piecewise linear
function that changes slope from t to t at the value given by E . The tariff t can
1 2
Ž be represented by the parameters t , t and E written in vector form as t s t ,
1 2
1
. t
, E , and so the two-tier rate structure is a three-parameter tariff design; e.g. in
2
Ž .
Armenia from January 1997 to September 1997, t s 13, 20, 100 . Similar to the flat-rate analysis, a class of tariff rates dependent upon the
parameters t , t and E can be compared, and an optimal tariff in terms of
1 2
revenue and equity can be determined. The major distinction between the flat-rate and the two-tier rate is that the tariff is no longer a function of a single variable;
tariff design is a function of three parameters and by design gives more flexibility in terms of distributional impact.
5.1. Parametric analysis An approximate range for the tariff parameters was a priori assigned and based
on the nominal settings, t s 13, t s 20, E s 100:
1 2
4 t
:s 10, 13, 16
1
4 t
:s 17, 20, 23
2
4 E
:s 75, 100, 125
3
Ž .
Associated with each of the 3 s 27 possible tariff combinations 10, 17, 75 ,
Ž . Ž
. Ž .
4 Ž
. 10, 17, 100 , 10, 17, 125 , 13, 17, 75 , . . . is a value for the revenue R
R t
, f and Ž
. equity E
E s H
0.1 based on the collected sample data. For each tariff rate
t
Ž .
Ž Ž . Ž .
Ž . Ž ..
combination t, the revenue R R
t ,f , moment vector E g , s g , sk g , ku g , and
Ž .
equity measure H 0.1 is computed directly from the sample data and is summa-
t
rized in Table 6. An example illustrates a typical computation. Ž
.4 The population sample provides energy consumption and income data
E , I ,
i i
Ž .
and for a given tariff, t s t , t , E is used to compute the cost of consumption
1 2
C and g s C rI .
i i
i i
The values of C and g are functions of t, and the density function of g, h , is
t
used to determine the equity measure:
M.J. Kaiser
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483 Table 6
Two]tier tariff rates and related parameters Ž
. Ž
. Tariff t
R t , f
Moment vector H
0.10 Histogram h
t t
Ž .
Ž .
Ž Ž . Ž .
Ž . Ž ..
t t E
ADR E g
, s g , sk g , ku g
1, 2,
Ž .
Ž .
Ž .
Ž .
0, 0.05 0.05, 0.10
0.10, 0.15 0.15, 1
Ž .
Ž .
10, 17, 75 911 302
7.2, 4.9, 2.1, 5.9 30
58 7
5 87
Ž .
Ž .
10, 17, 100 878 825
6.8, 4.5, 2.2, 7.2 33
55 5
7 89
Ž .
Ž .
10, 17, 125 660 020
4.7, 3.4, 3.4, 21.5 67
27 5
1 94
Ž .
Ž .
10, 20, 75 1 046 022
8.0, 5.4, 2.0, 4.9 28
54 9
9 82
Ž .
Ž .
10, 20, 100 999 625
7.4, 4.9, 2.1, 5.8 28
58 5
9 86
Ž .
Ž .
10, 20, 125 956 240
7.0, 4.5, 2.1, 6.9 32
57 5
6 89
Ž .
Ž .
10, 23,75 1 180 741
8.7, 5.9, 1.9,4.2 24
52 7
7 76
Ž .
Ž .
10, 23, 100 1 120 425
8.0, 5.2, 2.0, 5.0 23
57 8
12 80
Ž .
Ž .
10, 23, 123 1 064 024
7.5, 4.7,2.0,5.9 27
58 5
10 85
Ž .
Ž .
13, 17, 75 955 671
8.1, 5.5, 2.4, 8.8 27
51 12
10 78
Ž .
Ž .
13, 17, 100 937 112
7.8, 5.3, 2.5, 9.8 27
51 10
12 78
Ž .
Ž .
13, 17, 125 919 758
7.7, 5.2, 2.6, 10.7 26
51 8
15 77
Ž .
Ž .
13, 20, 75 1 090 390
8.8, 6.0, 2.2, 6.9 21
50 20
9 71
Ž .
Ž .
13, 20, 100 1 057 912
8.5, 5.6, 2.3, 8.1 21
50 19
11 71
Ž .
Ž .
13, 20, 125 1 027 542
8.1, 5.4, 2.4, 5.2 26
52 12
10 78
Ž .
Ž .
13, 23, 75 1 225 109
9.6, 6.5, 2.1, 5.6 20
43 27
10 63
Ž .
Ž .
13, 23, 100 1 178 712
9.1, 6, 2.2, 6.7 20
50 20
10 70
Ž .
Ž .
13, 23, 125 1 135 327
8.6, 5.6, 2.3, 7.9 20
51 16
13 71
Ž .
Ž .
16, 17, 75 1 000 039
8.9, 6.3, 2.7, 11.8 24
50 16
10 74
Ž .
Ž .
16, 17, 100 995 399
8.9, 6.2, 2.7, 12.2 22
50 17
11 72
Ž .
Ž .
16, 17, 125 991 060
8.8, 6.2, 2.7, 12.4 22
50 18
10 72
Ž .
Ž .
16, 20, 75 1 134 758
9.7, 6.7, 2.5, 9.4 18
44 26
12 62
Ž .
Ž .
16, 20, 100 1 116 199
9.5, 6.5, 2.5, 10.3 19
48 23
10 67
Ž .
Ž .
16, 20, 125 1 098 845
9.3, 6.3,2.6,11.1 20
50 20
10 70
Ž .
Ž .
16, 23, 75 1 269 477
10.5, 7.1, 2.3, 7.6 15
41 29
15 56
Ž .
Ž .
16, 23, 100 1 236 999
10.1, 6.8, 2.4, 8.8 19
44 27
10 63
Ž .
Ž .
16, 23, 125 1 206 630
9.8, 6.5, 2.5, 9.8 18
45 27
10 63
M.J. Kaiser
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484
Ž . Ž
. Fig. 6. The histogram h g for the tariff rate t s 10, 17, 75 .
t
M.J. Kaiser
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485 Ž .
Ž .
Fig. 7. The histogram h g for the tariff rate t s 10, 17, 100 .
t
M.J. Kaiser
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486
Ž . Ž
. Fig. 8. The histogram h g for the tariff rate t s 10, 17, 125 .
t
M.J. Kaiser r Energy Economics 22 2000 463]495 487
0.1
Ž .
Ž .
Ž . E
E t
, f s H 0.1 s h
y d y.
H
t t
Ž .
For the tariff t s 10, 17, 75 , the revenue is calculated from the consumption Ž
. sample data and yields R
R t ,f s
911302 ADR and distribution h as shown in Fig.
t
6. The histogram subsequently yields the moment vector: Ž Ž .
Ž . Ž .
Ž .. Ž
. E g
, s g , sk g , ku g s
7.2, 4.9, 2.1, 5.9 , Ž
. Ž
. and H 0.1 s 0.32 q 0.55 s 0.87. For the tariff t s 10, 17, 100 , the revenue
t
Ž .
generated is R R
t ,f s 878825 ADR, and the distribution h is shown in Fig 7. The
t
moment vector of the distribution yields Ž
. 6.8, 4.5, 2.2, 7.2 ,
Ž .
and H 0.1 s 0.34 q 0.55 s 0.89. Observe that as revenue declines equity in-
t
creases, while the average amount of income spent on electricity consumption decreases. The value of g changes in response to the change D E s 100 y 75 s 25
kW, and the amount of change depends upon the form of f and the magnitude of Ž
. D
E . The distribution h for the tariff t s 10, 17, 125 is depicted in Fig. 8.
t
5.2. Model analysis The main results of the non-linear tariff are summarized as follows:
Ž .
Ž . 1. Total revenue R
R t ,f
is an increasing function of E g . Ž
. 2. Total revenue R
R t
, f decreases approximately linearly as a function of one Ž
. parameter e.g. t , t , or E while holding the other two parameters fixed .
1 2
In terms of the sample data specific to this study: Ž
. Ž
. 3. The efficient frontier of the R
R t
, f ]E E
t , f graph is composed of the Pareto-
Ž .
U
optimal efficient or non-dominated tariffs t :
U
Ž .
t s
16, 23, 75
1 U
Ž .
t s
10, 23, 75
2 U
Ž .
t s
10, 23, 100
3 U
Ž .
t s
10, 23, 125
4 U
Ž .
t s
10, 17, 125
5 U
U
Ž 4. The tariff rates t and t represent the extreme ranges of the constraints t ,
1 5
1
. t
, E :
2
M.J. Kaiser r Energy Economics 22 2000 463]495 488
U
Ž .
t s high
, high, low
1 U
Ž .
t s low
, low, high
5
Ž
U
. and yield, as expected, maximum revenue]minimum equity t
and minimum
1
Ž
U
. revenue]maximum equity t
measures.
5 A
Ž .
5. The Armenian tariff t s 13, 20, 100 is non-Pareto-efficient. Ž
. 6. The mean and variance decreases within each tariff class t , t , E
as E
1 2
Ž .
increases t
and t held fixed , while the skewness remains essentially
1 2
constant and the kurtosis measure increases. .
Ž . In the flat-rate tariff, revenue R
R t
, f and E g was directly proportional to t, and thus, directly proportional to one another. In the two-tier tariff, revenue is now
Ž .
Ž . a function of three parameters, but R
R t ,f
and E g still depict a general increasing trend. This is illustrated in Fig. 9 and can be seen in Table 6 within each
Ž .
Ž tariff class t ,t ,? as a function of E . Note that as E increases from 75 to 100
1 2
. Ž
. Ž .
to 125 kWh , revenue R R
t ,f decreases along with the value of E g . The slope of
the data in each tariff rate class provides a measure of the sensitivity of the total Ž
. revenue to changes in g; e.g. for 10, 17,? ,
Ž .
D R R
t , f
911 302 y 660 020
5
s s
1.0 = 10 Dg
7.2 y 4.7 Ž
. while for 10, 20,? ,
Ž .
D R R
t , f
1 046 022 y 956 240
4
s s
8.9 = 10 Dg
8.0 y 7.0 These sensitivity measures can be used to compare the various tariff rates.
Ž .
Ž . The relationship between R
R t
, f and sk g for the 27 tariff rate combinations is depicted in Fig. 10.
In Fig. 11 the 27 tariff combinations in Table 6 are plotted relative to the Ž
. revenue and equity they generate columns 2 and 5 in Table 6 . The efficient
frontier is illustrated in Fig. 12 and depicts the Pareto-optimal tariff rates. A Pareto-optimal tariff rate t
U
is defined such that no other tariff rate is at least as good as t
U
on every objective and strictly better than t
U
on at least one objective. In this setting the objectives are revenue and equity, and the tariff rates t
U
,
i
i s 1, . . . ,5, are the Pareto-optimal tariffs. The same general trend between rev-
Ž .
Ž .
enue R R
t ,f and equity E
E t
, f is observed as the flat rate but the structure is complicated by the non-linear nature of the tariff. As the revenue increases, the
equity measure decreases, but there are a number of combinations that are Ž
non-efficient and can be improved either with an increase in the revenue or .
A
Ž .
equity . The Armenian tariff t s 13, 20, 100 , for example, falls in this classifica-
U
Ž .
tion and is thus a non-optimal tariff. Observe that the tariff t s 10, 23, 125
4
increases the equity measure as compared to t
A
by nearly 15 while not changing
M.J. Kaiser
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Economics 22
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] 495
489 Ž
. Ž .
Ž .
Fig. 9. Graph of the total revenue R R
t , f vs. the mean value E g for the 27 tariff combinations t s t , t , E
t s
10,
1 2
1
4 13, 16; t s 17, 20, 23; E s 75, 100, 125 .
2
M.J. Kaiser
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490
Ž .
Ž . Ž
. Fig. 10. Graph of the total revenue R
R t ,f
vs. the skewness sk g for the 27 tariff combinations t s t , t , E t
s 10,
1 2
1
4 13, 16; t s 17, 20, 23; E s 75, 100, 125 .
2
M.J. Kaiser
r Energy
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2000 463
] 495
491 Ž
. Ž
. Ž
. Fig. 11. Graph of the total revenue R
R t
, f vs. the equity H 0.1 for the 27 tariff combinations t s t ,t , E t
s 10, 13, 16;
t 1
2 1
4 t
s 17, 20, 23; E s 75, 100, 125 .
2
M.J. Kaiser r Energy Economics 22 2000 463]495 492
U
Ž .
the revenue generated. The tariff t s 10, 23, 75 is observed to increase revenue
2
by 12 and equity by 3. This result follows from the fact that t
U
and t
U
are
2 4
elements of the efficient frontier, while t
A
is not. 5.3. Optimization model formulation
The rates t
U
and t
U
represent the extreme ranges of the constraints and can be
1 5
solved for in terms of an optimization model. In particular, t
U
and t
U
tariff designs
1 5
represent the solution to the constrained optimization problems: Ž
. max g t , f
s.t. 10 F t F 16
1
Ž .
17 F t F 23
2
Ž .
75 F E F 125 Ž
. Ž
. Ž
. Ž
. for g t , f s R
R t
, f and g t , f s E E
t , f , respectively. The Pareto-optimal tariffs
t
U
, t
U
, t
U
can also be solved for directly using a linear programming formulation,
2 3
4
and for the two objective case, a visual solution is straightforward. The framework of optimization theory, however, allows the problem class to be generalized in
significant directions.
6. Conclusions
Kategori