Policy Paper
shoomoosh
LIHEAPreconsidered.pdf
Energy Policy 31 (2003) 1441–1458
LIHEAP reconsidered
Mark J. Kaiser*, Allan G. Pulsipher
Center for Energy Studies, Louisiana State University, One East Fraternity Circle, Baton Rouge, LA 70803-0301, USA
Abstract
The Low-Income Home Energy Assistance Program (LIHEAP) is a federal block grant program established in 1981 to help low-
income households meet a portion of their home energy costs. The manner in which LIHEAP funds are allocated to states, however,
has been a contentious issue since the inception of the program. In 1984, the Health and Human Services developed a new formula
to increase equity among the states by incorporating state cooling requirements in an equal weighting scheme with state heating
requirements. In addition to the new distribution formula, various provisions were also included in the LIHEAP re-authorization
amendment that specified when and how the 1984 formula could be employed. These provisions have turned out to be so
constraining, however, that they have effectively disabled the 1984 formula. The purpose of this paper is to introduce realistic policy
alternatives to the current LIHEAP allocation mechanism and to examine the impact of each alternative. Three options are
discussed ranging from the complete elimination of the hold-harmless (HH) provisions to a proposal that maintains the primary HH
provision but reduces the trigger level when it is enabled. A simple allotment block distribution based on mixing the two competing
funding formulas is also considered. The models presented in this paper represent the first time that policy alternatives to the
LIHEAP allocation mechanism has been examined within an analytic framework.
r 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction
The Low-Income Home Energy Assistance Program (LIHEAP) was authorized by Congress through Title XXVI of the Omnibus Budget Reconciliation Act of 1981, and currently authorized through the end of FY 2004 by the Coats Human Services Re-authorization Act of 1998, P.L.105-285, enacted on October 27, 1998, www.acf.dhhs.gov.
The Low-Income Home Energy Assistance Act of 1981 established the Low-Income Home Energy Assis- tance block grant program to help eligible households meet home energy costs. States can provide assistance to low-income households through various program com- ponents, including home heating and cooling assistance, energy crisis assistance, and home weatherization (General Accounting Office, US Congress, 1986):
* Home energy assistance consists of helping low- income households pay heating and cooling costs. Grantees provide assistance in the form of cash, vouchers, coupons, and two-party checks to eligible households, and may make payments to landlords or
home energy suppliers on behalf of eligible house- holds.
* Energy crisis assistance includes funding for weather- related, supply shortages, and other household energy-related emergencies. States provide cash, shelter, emergency supplies, or supplemental heating sources to households without heat or in imminent danger of having their fuel supplies terminated.
* Weatherization assistance includes funding for low- cost residential weatherization or other energy- related home repair.
LIHEAP is limited to households that include recipients of aid to families with dependent children, supplemen- tary security income, food stamps, or certain veteran’s benefits. Households with incomes less than 150% of the poverty level, or less than 60% of the state’s median income, whichever is greater, also qualify under the statute.
The LIHEAP is a block grant program, meaning that the state can—within the statutory requirements— choose their method of administration, eligibility criteria, benefit levels, and funding levels for program activities. A number of legislative amendments over the years have placed some restrictions on state program discretion; for example, no more than 25% of LIHEAP
ARTICLE IN PRESS
*Corresponding author. Tel.: +1-225-578-4554; fax: +1-225-578-
4541.
E-mail address: [email protected] (M.J. Kaiser).
0301-4215/03/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved.
PII: S 0 3 0 1 - 4 2 1 5 ( 0 2 ) 0 0 2 0 0 - 8
funds can be used for weatherization and no more than 10% can be spent on state administration costs. States can transfer a maximum of 10% of their funds to other block grants and cannot carry more than 15% of their allotment to the succeeding fiscal year.
The General Accounting Office by legislative mandate is required to evaluate LIHEAP, and as requested by various government units, have produced a number of reports on the program, e.g., General Accounting Office, US Congress (1986, 1987, 1990). A handful of academic papers on federal energy assistance programs are scattered throughout the literature (Colton, 1990; Higgins and Lutzenhiser, 1995; Kennedy, 1987; Kings- ley, 1992; Landsberg and Dukert, 1981), including a few Congressional Research Service reports (Abbey, 2001; Gish, 2000; Stoltzfus, 2002) and numerous LIHEAP reports to Congress (Low Income Home Energy Assistance Programs, 1983, 1985, 1994, 2000).
The federal government distributes funds to states for LIHEAP using a legislated formula. The LIHEAP formula was originally legislated in 1981 and then revised in 1984 in large measure due to political pressure from many warm-weather states that maintained that the 1981 formula over-allocated funds to cold-weather states. Pursuant to the re-authorization amendment for LIHEAP in 1984, the Health and Human Services developed a new formula to increase the overall equity among the states. The LIHEAP formula was revised to incorporate state cooling requirements in an equal weighting scheme with state heating requirements, but various additional conditions were also included in the amendment that specified when and how the 1984 formula could be employed. To ‘‘turn-on’’ the 1984 formula for instance, LIHEAP’s regular appropriation is required to exceed a minimum threshold level of $1.975 billion, but because program commitment to LIHEAP has fallen over the years, the revised allocation mechanism has only rarely been applied—specifically, twice over the past 17 years, and last during FY 1986.
The primary intent of the 1984 legislation to increase the overall equity of the distribution of LIHEAP funds has thus failed miserably, and so there is a need to examine alternative allotment methods to bring the 1984 formula ‘‘back into’’ the allocation mechanism as intended by the legislation. The primary purpose of this paper is to suggest viable policy alternatives to the current legislative framework and to develop an analytic model to quantify the impact of each alternative. The present work represents the first time that alternative allocation mechanisms to LIHEAP have been modeled and presented publicly. This model not only provides legislators insight on competing policy proposals, but also hopefully, provides a forum and stimulus for future debate.
The outline of the paper is as follows. The current LIHEAP statute is discussed in Section 2, followed by a
summary of LIHEAP funding levels in Section 3 and a general description in Section 4 of three realistic policy alternatives. In Sections 5–7 the policy alternatives are examined. In Section 5, a comparison of the 1981 and 1984 formula allocation formulas are discussed by state and region, and in Section 6, the effect of a trigger-level reduction on state allotments and system measures are outlined. In Section 7 a block allotment procedure is described which combines the 1981 and 1984 formula. In Section 8 conclusions complete the paper. To maintain focus on the policy proposals, the theoretical foundation of the allocation mechanisms are maintained in separate appendices.
2. The LIHEAP statute
The LIHEAP statute currently assigns each state and the District of Columbia an allotment percentage based on the value of the regular appropriation $D for the fiscal year. One of three cases can occur depending on the value of D:
Case I : Dp$1:975B Case II : $1:975BoDo$2:25B Case III : DX$2:25B
Case I: Dp$1:975B: When the regular appropriation in a fiscal year is less than or equal to $1.975B, the 1981 formula is employed to determine state allocation percentages. In other words, the 1981 formula repre- sents a default condition if regular appropriation does not exceed the trigger F ¼ $1:975B: The 1981 allocation percentages shown in Table 1 are now over 20-years old and the arbitrary basis of their ‘‘derivation’’ forgotten by the legislative community. Further, and more importantly, since regular appropriations have fallen below the trigger level every year for the past 15 years (and counting), the high frequency of usage of the 1981 formula has helped to create a sense of ‘‘acceptance’’ and ‘‘validity’’ for its application, which is anything but satisfactory.
Three main ‘‘problems’’ exist with the 1981 formula:
(1) The 1981 formula represented the outcome of a political process as opposed to being based on good science.
(2) The 1981 formula was poorly designed, extremely complex, and arbitrarily ‘‘derived’’ in a manner that strongly favored cold-weather climates.
(3) The 1981 formula is a ‘‘static’’ formula that fixed the allocation percentages of states based on pre- 1980 data for state climate, fuel price, demographic, and expenditure patterns.
The 1981 formula is unusually easy to criticize and very difficult to defend. It is important to recognize
ARTICLE IN PRESS M.J. Kaiser, A.G. Pulsipher / Energy Policy 31 (2003) 1441–14581442
that the 1981 formula was designed in response to conditions specific to the times that emphasized the rapidly increasing fuel oil prices in the Northeast during the late 1970s. The formula allocation caused great consternation on the part of many warm-weather
states 1
and the continued application of the 1981 formula remains one of the better-kept ‘‘secrets’’ in Washington.
The primary shortcomings of the 1981 formula are easy to recognize: The 1981 formula does not nor cannot reflect the current low-income demographic, consump- tion, and fuel price mix in the country (since the data on which the formula is based is over 20-years old), and further, the formula itself cannot be ‘‘updated’’ to reflect more recent data (since the allocation mechanism is a static design). The most egregious aspect of the 1981 formula is that the allocation mechanism does not distribute funds based on the need for home energy assistance. Instead, the 1981 formula defines need in a mysterious and complicated fashion primarily in terms of the home heating needs of cold-climate states.
A complex and convoluted formula is not necessarily a ‘‘bad’’ formula, but unfortunately—and more to the point—it is not clear what the 1981 state allocation percentages mean or if they identify the need for home energy assistance. Four separate formulas are entangled with the 1981 formula allotment—two House of Representatives formula alternatives, the FY 1980 state allotment, and the Windfall Profit Tax Act formula— commingled with pro-rata reductions, weak hold-harm- less (HH) provisions, and arbitrary formula compar- isons. And as each one of the four formulas is biased toward cold-climate states, no matter what procedural elements are invoked, a biased
2 cold-climate formula
remains a biased cold-climate formula. Pursuant to the re-authorization amendment for
LIHEAP in 1984, the 1981 formula was completely revised in the ‘‘1984 formula’’, but for the 1984 formula to be ‘‘turned-on’’ the regular appropriation must exceed $1.975B.
Case II: $1:975BoDo$2:25B: When the regular appropriation exceeds $1.975B in a fiscal year, then an allocation mechanism based on the 1984 formula coupled with two HH provisions is adopted. The first HH condition, or primary HH provision, is enabled when D > $1:975B: When DX$2:25B a second HH provision kicks in.
The 1984 formula, or ‘‘new formula’’, was established to provide a mechanism to distribute LIHEAP funds based solely on the home energy needs of low-income
ARTICLE IN PRESS
Table 1
LIHEAP allocation percentages according to the 1981 and 1984
formula
State 1981 Formula
(%)
1984 Formula
(%)
Statutory
floor ($M)
Alabama 0.86 1.68 16.99
Alaska 0.55 0.36 10.84
Arizona 0.42 1.25 8.21
Arkansas 0.66 1.21 12.96
California 4.61 6.00 91.12
Colorado 1.61 1.15 31.77
Connecticut 2.10 1.40 41.45
Delaware 0.28 0.37 5.50
D.C. 0.33 0.27 6.44
Florida 1.36 4.16 26.88
Georgia 1.08 2.68 21.25
Hawaii 0.11 0.12 2.14
Idaho 0.63 0.28 12.39
Illinois 5.81 5.30 114.72
Indiana 2.63 2.21 51.94
Iowa 1.86 1.22 36.81
Kansas 0.86 1.07 16.91
Kentucky 1.37 1.73 27.03
Louisiana 0.88 1.79 17.37
Maine 1.36 0.55 26.85
Maryland 1.61 2.26 31.74
Massachusetts 4.20 2.91 82.91
Michigan 5.51 4.82 108.92
Minnesota 3.97 1.68 78.47
Mississippi 0.74 1.87 14.56
Missouri 2.32 2.41 45.82
Montana 0.74 0.35 14.54
Nebraska 0.92 0.55 18.21
Nevada 0.20 0.51 3.86
New Hampshire 0.79 0.45 15.69
New Jersey 3.90 3.28 76.97
New Mexico 0.52 0.54 10.28
New York 12.72 8.68 251.34
North Carolina 1.90 3.14 37.45
North Dakota 0.80 0.21 15.79
Ohio 5.14 4.86 101.49
Oklahoma 0.79 1.31 15.61
Oregon 1.25 0.95 24.62
Pennsylvania 6.84 5.15 134.99
Rhode Island 0.69 0.47 13.65
South Carolina 0.68 1.44 13.49
South Dakota 0.65 0.29 12.83
Tennessee 1.39 2.08 27.38
Texas 2.26 6.60 44.71
Utah 0.75 0.54 14.76
Vermont 0.60 0.23 11.76
Virginia 1.96 3.05 38.66
Washington 2.05 1.50 40.50
West Virginia 0.91 0.96 17.89
Wisconsin 3.58 1.93 70.63
Wyoming 0.30 0.18 5.91 1 ‘‘It has only been through the making of legislative history with
regard to this and previous appropriations bills for this program that
the prejudice favoring cold-weather energy bill assistance and opposing
hot-weather energy bill assistance has developedy it is counter to the
American tradition of fair play. I would urge the Congress to re-
examine its conscience and design a fairer solution’’. Additional view
of Bill Alexander, Low Income Energy Assistance, House of
Representatives Report No. 96-1244, August 21, 1980. 2 Granted, the home heating needs of cold-climate states were
particularly severe in the early 1980s but the incorporation of these
same factors today is not defendable.
M.J. Kaiser, A.G. Pulsipher / Energy Policy 31 (2003) 1441–1458 1443
households—where ‘‘home energy’’ was interpreted to mean the heating and cooling requirements of house- holds. According to Section 2604(a)(2) of P.L. 97-35 as amended by P.L.98-558,
‘‘y For fiscal year 1985 and thereafter, a state’s allotment percentage is the percentage which expen- ditures for home energy by low-income households in that state bears to such expenditures in all statesy’’
The 1984 formula was a significant improvement over the 1981 formula in terms of its transparency and equity and in many regards can be considered the ‘‘ideal’’ mechanism to distribute LIHEAP funds:
(1) The 1984 formula determines state allocation percentages according to a simple, easy-to-under- stand, and science-based mechanism using estimates of state expenditures for the Btu requirements of low-income households for each fuel source con- tributing to home heating and cooling needs.
(2) The 1984 formula is extremely well designed and easy-to-defend and is based on the ‘‘best-available’’ statistical data.
(3) The 1984 formula is a dynamic allocation that is updated annually to normalize for changes in weather patterns (heating and cooling degree-days) and fuel prices (coal, electricity, fuel oil, liquid petroleum gases, and natural gas) at the state level.
The 1984 formula allocations are shown in Table 1 alongside the 1981 allotments, but unlike the 1981 percentage allocation values, the 1984 formula values do not represent the final state allotment percentages. The final allotment percentages can only be calculated after the regular appropriation D is known for the fiscal year and the impact of the HH and give-back (GB) provisions are incorporated within the allocation mechanism.
The HH condition establishes a statutory floor for each state determined by the 1981 formula allocation and the appropriation level F ¼ $1:975B as shown in Table 1. If a state does not achieve its statutory floor under the appropriation D and 1984 formula allotment, then it is held ‘‘harmless’’ at its floor. The funds necessary to maintain states at their floor is mandated by a GB provision which specifies that all states above their floor will have their allotments reduced propor- tionally until the funds required to make-up the deficit is achieved. Since the funds necessary to make-up the deficit come from states that benefit under the new allocation mechanism, these states effective allotment percentages are reduced from their 1984 allotment.
Case III: DX$2:25B: When the regular appropriation for a given fiscal year is equal to or exceeds $2.25B, then another HH and GB condition is triggered, referred to as the secondary HH-GB provision or the LTOP (less than 1%) provision. The LTOP provision stipulates that
any state that receives less than 1% of the total allotment funding at the appropriation level D cannot have a smaller allotment percentage than its allotment percentage at $2.14B. The impact of the secondary HH- GB provision acts to maintain a set of states at their allotment percentage achieved at $2.14B. Since a set of states are maintained at an allotment percentage that is higher than dictated by the primary HH-GB provision, the complementary set of states which are not affected by the LTOP provision provide the funds required to satisfy this condition.
The inclusion of the secondary HH-GB provision is interesting because it indicates the ‘‘mind-set’’ of the legislators at the time the LIHEAP statute was revised. It was expected, or at least anticipated, that regular appropriations for LIHEAP would increase in the years ahead and might conceivably even exceed $2.25B! As it turns out, however, due to factors such as the oil price crash of 1986 and the Emergency Deficit Control Act, funding levels for LIHEAP in the years following the 1984 re-authorization plummeted.
3. LIHEAP funding levels
The LIHEAP statute provides for two types of program funds: regular block grant and emergency. Regular funds are authorized under Section 2602(b) of the LIHEAP statute and are currently allocated according to the mechanism described in Section 2. Emergency funds are authorized under 2602(e) of the LIHEAP statute and are authorized and distributed at
ARTICLE IN PRESS
Table 2
LIHEAP funding history, 1981–2002
FY Regular appropriation ($B) Emergency ($M)
1981 1.850
1982 1.875
1983 1.975
1984 2.075
1985 2.100
1986 2.010
1987 1.825
1988 1.532
1989 1.383
1990 1.393 50
1991 1.415 195
1992 1.500
1993 1.346
1994 1.439 298
1995 1.319 100
1996 0.900 180
1997 1.000 215
1998 1.000 160
1999 1.100 180
2000 1.100 744
2001 1.400 456
2002 1.700
M.J. Kaiser, A.G. Pulsipher / Energy Policy 31 (2003) 1441–14581444
the discretion of the President and Secretary of the Health and Human Services at any point in time in the fiscal year. The funding history of LIHEAP is shown in Table 2. The average value of LIHEAP regular appropriation from 1981–2002 is EðDÞ ¼ $1:51B with a standard deviation of sðDÞ ¼ $0:37B: Observe that since the LIHEAP program was reauthorized in 1984, funding levels have exceeded the trigger level $1.975B only during FY 1985 and FY 1986.
4. Policy proposals
The principal condition that has prevented the application of the 1984 formula is immediately clear. The level of the regular funding appropriated for LIHEAP over the years has not achieved the legislated trigger, and so by default, the 1981 formula—despite its obvious shortcomings—has become the effective alloca- tion mechanism for federal funding.
To address the failure of the 1984 legislation, three suggestions follow:
* Eliminate the trigger level, * Reduce the trigger level or * Devise an alternative allocation mechanism.
These suggestions are formalized as the following options:
Option A: Eliminate the HH-GB provisions and apply the 1984 formula allocation at a zero trigger level.
Option B: Reduce the trigger level of the HH-GB provision and eliminate the LTOP provision.
Option C: Apply an allocation mechanism based on a linear combination of the 1981 and 1984 formula values.
A brief description of each option is now provided followed in Sections 5–7 by models that explore the impact of each alternative.
Option A: This is the most dramatic approach, the simplest, and arguably the most equitable. Unfortu- nately, Option A is also unlikely to garner the political support necessary to achieve passage.
The purpose of LIHEAP is to assist low-income households meet the costs of home energy. This purpose is clear and unambiguous and a very good allocation mechanism already exists in the 1984 formula to satisfy this objective—and in fact it is doubtful that a ‘‘better’’ formula can be developed unless the legislative mandate of LIHEAP radically changes—but for the past 15 years the 1984 formula could not be employed since the trigger level to turn it ‘‘on’’ has never been attained. Hence, by eliminating the HH provisions and setting the trigger point to zero, the 1984 formula will serve as the new default formula.
The basic premise of Option A is that any use of the 1981 formula perpetrates a distribution of funds that is inequitable since the allocation is not based on the home
energy needs of low-income households. The manner in which LIHEAP funds should be distributed to states should mimic the state’s home energy expenditures
3 paid
by low-income households. Interestingly, this also appears to be the Bush Administration position.
4
Although Option A is believed to be the fairest, simplest and best way to revise LIHEAP, it may not be politically viable or develop a strong enough coalition of support, especially in the Senate since a majority of states realize a smaller allotment percentage under the 1984 formula. The fight in Congress over Option A will be especially difficult since Senators are not likely to vote their state a lower allotment percentage to ‘‘right the wrongs’’ of an archaic and cryptic formula. Be that as it may, if legislators can be convinced to move on LIHEAP—perhaps for the sake of good government, good conscience, or daresay, a good formula—viability may hinge on maintaining some form of the HH condition as opposed to the complete elimination of the statutory floor. One simple means to accomplish this task is to simply re-set the trigger level of F downward as proposed in Option B.
Option B: Any revision to LIHEAP should be realistic and based on a politically viable strategy.
The value of the trigger point F currently represents a level that has not, nor likely will, be surpassed in the future, and hence its adjustment downward is necessary to ensure that a more equitable allocation mechanism be applied to future LIHEAP funding. The value of the trigger point was set by legislative mandate and over time has lost its relation to D: At one time (15-years ago) the values of D and F were roughly comparable, while today F appears as an upper bound that is unlikely to be exceeded.
There is also some precedence for trigger re-adjust- ment. When the HH-GB provision was initially devised in 1984, F was set at the FY 1984 appropriation level of $2.075B. In 1986, when the regular appropriation fell below the required trigger, the value of F was reduced to the FY 1983 level of $1.975 to bring the 1984 formula back ‘‘in’’ the allocation mechanism.
5 Unfortunately,
FY 1986 represented the last year in which program appropriations exceeded $1.975B, and subsequent to this time no further action has been taken to address this issue and/or re-adjust F:
ARTICLE IN PRESS
3 Home energy expenditures paid by low-income households is also
relatively easy to estimate, and so there is no technical difficulties
associated with the computation. 4 ‘‘y The legislatively established formula currently used to
distribute LIHEAP block grant funds to states is based on 20-year
old population and winter heating cost data. The Administration is
interested in options that would make block grant allocations more
equitable by basing the formula on current home energy expenditures
paid by low-income households’’. Bush Administration LIHEAP
Budget for Fiscal Year 2003. 5 Apparently LIHEAP re-authorization legislation was still fresh in
legislators minds.
M.J. Kaiser, A.G. Pulsipher / Energy Policy 31 (2003) 1441–1458 1445
The ideal equitable allocation mechanism would be based on the 1984 formula with no provisions attached (Option A), but because of the historical precedence to couple the 1984 formula with HH and GB provisions, and the need to satisfy a majority of votes in the Senate, Option B is likely to be viewed as compromise legislation.
Option C: An alternative approach to maintain the application of the 1981 formula that does not rely explicitly on re-adjusting the trigger level F is to set-aside a certain percentage ðrÞ of the regular appro- priation to be used with the 1981 formula, rD; and then to apply the 1984 formula to the remaining funds, ð1 � rÞD: This has an advantage over Option B since regardless of the appropriation level the 1984 formula would always be applied to the same portion of the regular appropriation.
6 Option C ensures application of
the 1984 formula with no ‘‘downside risk’’ as in Option B, but it also foregoes the ‘‘upside potential’’ when D > F: On methodological grounds a formula mix is not especially appealing, but as will be demonstrated in Section 7, a linear combination of the 1981 and 1984 formula is equivalent to Option B under a suitable correspondence.
A summary of the three options is depicted in Table 3 along with a brief description of their advantages and disadvantages.
5. A comparison of the 1981 and 1984 formula allotments
The 1981 formula is for all practical purposes a ‘‘perpetual’’ formula since the funding levels required to ‘‘turn-on’’ the new allocation mechanism has only rarely been achieved in the past and is unlikely to be attained in the future. The HH and GB provisions associated with the allocation mechanism are therefore relaxed and the trigger level required to turn-on the 1984 formula is set equal to zero. A direct comparison is performed between the old and new formula allotment percentages by state and region.
5.1. Comparison by state
If the vectors f and g denote the 1981 and 1984 state allotment percentages:
f ¼ ðf1; y; f51Þ; 0ofio1; X
fi ¼ 1;
g ¼ ðg1; y; g51Þ; 0ogio1; X
gi ¼ 1;
then the set of states that would benefit from a formula change is denoted
fBg ¼ fi j fiogig;
while the set of states that would be harmed is denoted
fCg ¼ fi j fi > gig:
The sets fBg and fCg partition the 50 states and District of Columbia (D.C.) according to states that ‘‘benefit’’ and are ‘‘harmed’’ by the formula change. If a state’s 1984 formula allotment percentage is greater than its
ARTICLE IN PRESS
Table 3
A comparison of three policy options
Policy Description Advantages Disadvantages
Option A Eliminate the primary and Simple Senate fight
secondary HH-GB provisions Fair Large percentage change in state allotment
and nullify the trigger level Rational
mechanism
Satisfies original intent of LIHEAP
Bush Administration position
House passage likely
Option B Maintain the primary HH-GB Maintains form of Maintains convoluted formula
provision but reduce the statute Not a rational mechanism for distribution
trigger level when it is enabled Enables 1984 Level of fairness not guaranteed
formula
Compromise formula
Senate support more likely
Option C Apply an allotment based Simple Formula mix
on a linear combination Easy to understand Not a rational mechanism for distribution
of the 1981 and 1984 No downside risk Foregoes upside potential
formula values Correspondence with
Option B
6 Under Option B, if F is re-set at a value that is ‘‘too high’’ (say
$1.5B) and D again drifts downward away from F; then the same inequity will return to the distribution.
M.J. Kaiser, A.G. Pulsipher / Energy Policy 31 (2003) 1441–14581446
1981 allotment percentage, fiogi; then state i is said to benefit from the new policy.
In Table 4, 23 states are shown to benefit under Option A while 28 states would be harmed. States that benefit realize a percentage change in their allotment ranging from 4% (New Mexico, Missouri) to 200%
(Arizona, Florida) while states that are harmed range from �6% (Ohio) to �74% (North Dakota). A graphical representation of allotment percentage changes is depicted in Fig. 1. The average percentage change for states in fBg and fCg is 81% and �35%; respectively. Eight states in fBg realize an allocation percentage change greater than 100% (FL, AZ, TX, NV, MS, GA, SC, LA), while seven states in fCg realize a change in allocation percentage greater than 50% (ND, VT, MN, ID, SD, ME, MT).
Under the 1984 formula allotment, the set of states that benefit from the new policy would realize a 20.4% increase in total allotment, which would be transferred from the states in fCg:7 The total amount of money transferred depends on the value of D; e.g., if D ¼ $1B; then $204M would be transferred under Option A from fCg to fBg:
All states are impacted by the new policy. The expected annual difference in state allocations is com- puted in Table 4 based on the 1981–2002 average annual regular appropriations for LIHEAP, EðDÞ ¼ $1:51B; multiplied by the difference in allotment percentages, gi � fi: The ‘‘big winners’’ under Option A include Texas ($65M), Florida ($42M), Georgia ($24M), California ($20M), and North Carolina ($18M), while states that lose money include New York ð�$61MÞ; Minnesota ð�$35MÞ; Pennsylvania ð�$25MÞ; Wisconsin ð�$25MÞ; and Massachusetts ð�$19MÞ: See Fig. 2 for a graph of the expected annual difference in funding under Option A. The average annual difference in funding levels in fBg and fCg is $13.4M and �$11:0M; respectively. The top five states in fCg represent nearly 60% of the total funds that would be reallocated under Option A to the top five states in fBg: The remaining funds are transferred among the 41 other states in fBg and fCg:
To provide a rough measure of the degree of inequity that has been promulgated over the past decade through the continued use of the 1981 formula, refer to the last column in Table 4. The values depicted are based on the
ARTICLE IN PRESS
-100 -50 0 50 100 150 200 250
S ta
te s
Percentage Change (%)
{C} {B}
Fig. 1. State LIHEAP allotment percentage change: 1981 vs. 1984
formula.
Table 4
The impact of using the 1984 formula allocation
State Percentage
change (%)
Expected
annual
difference
($M)
10-year
cumulative
impact ($M)
Alabama 95.02 12.34 112.80
Alaska �34.34 �2.85 �26.02 Arizona 200.42 12.59 115.07
Arkansas 84.80 8.40 76.82
California 29.93 20.85 190.62
Colorado �28.38 �6.89 �63.03 Connecticut �33.41 �10.59 �96.79 Delaware 33.73 1.42 12.97
D.C. �17.24 �0.85 �7.76 Florida 205.14 42.15 385.35
Georgia 148.79 24.17 221.00
Hawaii 7.21 0.12 1.08
Idaho �55.66 �5.27 �48.22 Illinois �8.77 �7.70 �70.35 Indiana �15.88 �6.31 �57.65 Iowa �34.33 �9.66 �88.34 Kansas 24.40 3.15 28.84
Kentucky 26.51 5.48 50.08
Louisiana 103.62 13.76 125.77
Maine �54.82 �11.26 �102.89 Maryland 40.82 9.91 90.55
Massachusetts �30.64 �19.42 �177.56 Michigan �12.63 �10.52 �96.20 Minnesota �57.74 �34.64 �316.65 Mississippi 153.94 17.14 156.69
Missouri 3.98 1.40 12.76
Montana �52.58 �5.84 �53.43 Nebraska �40.55 �5.64 �51.60 Nevada 159.36 4.70 42.97
New Hampshire �42.59 �5.15 �47.11 New Jersey �15.97 �9.40 �85.89 New Mexico 3.62 0.28 2.60
New York �31.87 �61.23 �559.73 North Carolina 65.31 18.70 170.97
North Dakota �73.56 �8.88 �81.18 Ohio �5.56 �4.31 �39.42 Oklahoma 65.51 7.82 71.50
Oregon �23.96 �4.51 �41.23 Pennsylvania �24.64 �25.43 �232.51 Rhode Island �32.56 �3.40 �31.06 South Carolina 110.44 11.39 104.13
South Dakota �55.35 �5.43 �49.62 Tennessee 49.99 10.47 95.68
Texas 191.33 65.41 597.96
Utah �27.92 �3.15 �28.81 Vermont �62.14 �5.59 �51.09 Virginia 55.88 16.52 151.00
Washington �27.08 �8.38 �76.65 West Virginia 5.56 0.76 6.95
Wisconsin �45.98 �24.83 �226.98 Wyoming �39.63 �1.79 �16.37
7 Capital redistribution is a zero-sum game.
M.J. Kaiser, A.G. Pulsipher / Energy Policy 31 (2003) 1441–1458 1447
LIHEAP cumulative allotments from 1992–2002, $13.80B, multiplied by the allocation differential, gi � fi: The values shown give an indication of what states can expect to gain and/or lose over the next decade should Option A be implemented.
5.2. Comparison by region
Regional comparisons are especially insightful since they indicate the inequity that the 1984 legislation was originally intended to address. Assign each state to one of four census regions: Northeast, North Central, South, and West, as shown in Table 5. The percentage of the total appropriation by region according to the 1981 and 1984 formula allocation is shown in Table 6.
States that benefit from Option A are drawn predominately from the South and to a lesser extent the West and North Central regions. No state in the Northeast benefits from a formula change, and hence it is not surprising that the Northeast forms a strong and vocal coalition against any changes to LIHEAP’s status quo (Pataki, 2001). Relative to the 1984 formula, the Northeast and North Central states receive a 17.6% over-allotment of funds, which are effectively being withdrawn from the South. The Southern states mean- while have expressed dissatisfaction with the mechanism that promotes this distribution although their criticism remains rather soft (Patton and Hodges, 2001) con- sidering the duration and extent of the inequity— although political considerations may also be at work here. The Western region as a class does not realize a noticeable difference in allotment percentages under the two formulas, although individual states may realize significant variation, e.g., Arizona (200%), California (30%), Nevada (159%), Idaho ð�56%Þ; Oregon ð�24%Þ and Washington ð�27%Þ:
5.3. The cumulative allocation
A description of the cumulative percentage allotment provides further insight into the allocation mechanism.
Under the 1981 formula, nine states receive 53% of LIHEAP funds (NY, PA, IL, MI, OH, CA, MA, MN, NJ); 19 states receive 75% of total funds (WI, IN, MO, TX, CT, WA, VA, NC, IA, CO), and 33 states receive 90% of funds. Under the 1984 formula, 10 states receive 52% of LIHEAP funds (NY, TX, CA, IL, PA, OH, MI, FL, NJ, NC); 20 states receive 75% of the funds (VA, MS, GA, MO, MD, IN, TN, WI, MI, LA); and 30 states receive 90% of the funds. The placement of states under the two formulas shift slightly, and from a global perspective the allotments are quite similar. Under the 1981 formula allocation, 25 states receive less than 1% of the total funding, while with the 1984 formula, 19 states receive less than a 1% allocation. A cumulative representation of the two allocations can be compared by first ordering the elements in descending rank and then perform a cumulative summation. The result in Fig. 3 shows the curves to be nearly identical. The 1984 cumulative curve falls ever so below the 1981 curve indicating an allocation that distributes funds in aggregate slightly more evenly across states.
ARTICLE IN PRESS
-80 -60 -40 -20 0 20 40 60 80
S ta
te s
Expected Annual Difference ($M)
{C} {B}
Fig. 2. Expected difference in annual LIHEAP allotments: 1981 vs.
1984 formula.
Table 6
Percentage of the total appropriation by census region
Region 1981
Allotment
(%)
1984
Allotment
(%)
Difference
(%)
Northeast 33.2 23.2 �10.0 North
Central
34.1 26.5 �7.5
South 19.0 36.6 17.6
West 13.7 13.7 —
Table 5
State assignment via census region
Northeast North Central South West
Connecticut Illinois Alabama a
Alaska
Maine Indiana Arkansas a
Arizona a
Massachusetts Iowa Delaware a
California a
New Hampshire Kansas a
D.C. Colorado
New Jersey Michigan Florida a
Hawaii a
New York Minnesota Georgia a
Idaho
Pennsylvania Missouri a
Kentucky a
Montana
Rhode Island Nebraska Louisiana a
Nevada a
Vermont North Dakota Maryland a
New Mexico a
Ohio Mississippi a
Oregon
South Dakota North Carolina a
Utah
Wisconsin Oklahoma a
Washington
South Carolina a
Wyoming
Tennessee a
Texas a
Virginia a
West Virginia a
a Indicate states that benefit from the application of the 1984
formula allotment.
M.J. Kaiser, A.G. Pulsipher / Energy Policy 31 (2003) 1441–14581448
6. The effect of a trigger-level reduction
Application of the 1981 formula to distribute funds is considered fundamentally unfair since it does not satisfy the intent of LIHEAP nor does it allocate federal dollars in a rational manner. Application of the 1984 formula was originally tied to a HH provision to protect states from receiving less money under the new formula. Unfortunately, regular appropriations for LIHEAP fell after the 1984 legislation was passed and has come in under the threshold 15 of the last 17 years, and so by default, the problematic 1981 formula continues to be used to distribute federal dollars.
The primary problem with the LIHEAP allocation mechanism is due to the fact that the trigger level F was set nearly two decades ago at a value commensurate to mid-1980 appropriation levels, while actual (real) appropriations began to slide shortly after the 1984 legislation was passed. Today, the trigger level F ¼ $1:975B is not representative of current LIHEAP appropriation levels and future appropriation is not expected to reach F ¼ $1:975B—let alone surpass F to ‘‘turn-on’’ the 1984 formula. To increase the chance of allocating some funds with the 1984 formula, it is necessary to lower the value of the trigger. Ultimately, the value of F will need to be negotiated, and so it is useful to understand how the selection of F impacts global measures and individual state allotments. The basic idea of Option B is to set F at a level below the expected value of D:
6.1. Theoretical structure
The LIHEAP allocation mechanism described in Section 2 is first simplified by eliminating the secondary HH provision, and for analytical purposes, the value of F is considered variable. The LIHEAP allocation mechanism is constructed according to Theorem 1 and its limiting relations are proved in Theorem 2.
Theorem 1. The LIHEAP allocation mechanism under
the primary HH-GB provision yields state allotments
according to the following recipe:
AiðF; DÞ ¼
Dfi; iAfSg; DpF; Ffi; iAfHg; D > F;
Ffi þ Ei; iAfNg; D > F;
8>< >:
where fSg ¼ fHg,fNg; fHg ¼ fi j Ffi=Dgi > 1g; fNg ¼ fi j Ffi=Dgip1g; Ei ¼ gðDgi � FfiÞ; g ¼ X =Q; X ¼ D � F; and Q ¼
P iAfNg ðDgi � FfiÞ:
Proof. See Appendix A.
Theorem 2. The LIHEAP allocation mechanism under
the primary HH-GB yields the following limiting rela-
tions:
lim D=F-1
aiðF; DÞ ¼ fi;
lim D=F-N
aiðF; DÞ ¼ gi:
Proof. See Appendix A.
The LIHEAP allocation mechanism described in Theorem 1 and system measures derived from the mechanism can be expressed as a function of the ratio D=F as shown in Theorems 3 and 4.
Theorem 3. For D > F; the LIHEAP allocation mechan- ism under the primary HH-GB provision yields state
allotment percentages
aiðF; DÞ ¼ F D
fi; iAfHg; F D
fi þ Ei D ; iAfNg
(
that are functions of the ratio F=D:
Proof. See Appendix B.
Theorem 4. The LIHEAP system measures
sðF; DÞ; iðF; DÞ; and rðF; DÞ can be expressed in terms of the ratio D=F as sðD=FÞ; iðD=FÞ; and rðD=FÞ:
ARTICLE IN PRESS
0
20
40
60
80
100
120
1 4 7
1 0
1 3
1 6
1 9
2 2
2 5
2 8
3 1
3 4
3 7
4 0
4 3
4 6
4 9
Number of States
C u
m u
la ti
v e P
e rc
e n
ta g
e
1981 allotment 1984 allotment
Fig. 3. The 1981 and 1984 cumulative percentage LIHEAP allotment.
M.J. Kaiser, A.G. Pulsipher / Energy Policy 31 (2003) 1441–1458 1449
Proof. See Appendix B.
6.2. Universal tableaux
Application of the theoretical structure is now presented. Theorem 1 describes how to construct state allotments and Theorem 2 presents the limiting cases when D ¼ F and DbF: The result of Theorems 3 and 4 demonstrate that the LIHEAP allocation percen- tages, aiðF; DÞ; and aggregate measures sðF; DÞ; iðF; DÞ; and rðF; DÞ—which are a function of F and D—can be expressed in terms of one variable, the ratio D=F: This is not only a very convenient result but also quite useful since rather than enumerate all the various combinations of F and D sepa- rately, e.g., F ¼ 1; D ¼ ð1; 1:25; 1:5; 1:75; 2:0; yÞ; F ¼ 1:5; D ¼ ð1:5; 1:75; 2:0; 2:25; 2:5; 2:75; 3:0; yÞ; etc. one ‘‘universal tableaux’’ suffices to present all relevant allocation percentages and aggregate measures.
The aggregate measures employed are empirically derived and are defined formally as follows:
H ¼ P
iAH Ffi Amount of money allocated to H under the imposition of the HH provision ($)
G ¼ P
iAH Dgi Amount of money allocated to H if the HH provision was eliminated ($)
N ¼ P
iAN Ffi þ X Amount of money allocated to N under the imposition of the HH provision ($)
X ¼ D � F Money available for distribu- tion after all state statutory floors are maintained ($)
M ¼ P
iAN Dgi Amount of money allocated to N if the HH provision was eliminated ($)
sðD=FÞ ¼ ðH � GÞ=D Percentage of money trans- ferred from N to H relative to the total appropriation D; 0psðD=FÞp1
rðD=FÞ ¼ ðH � GÞ=M Percentage reduction in the funding allocation to N due to the imposition of the HH-GB provision, 0prðD=FÞp1
iðD=FÞ ¼ ðH � GÞ=G Percentage increase in the fund- ing allocation to H due to the imposition of the HH-GB pro- vision, 0piðD=FÞp1
pðD=FÞ ¼ H=D Percentage of the total funds D allocated to H Due to the imposition of the HH-GB pro- vision, 0ppðD=FÞp1
qðD=FÞ ¼ G=D Percentage of the total funds D allocated to H if the HH-GB
provision was eliminated, 0pqðD=FÞp1
H The number of states in H N The number of states in N
Application of the tableaux follows three steps: Step 1: Assume D: Step 2: Select F: Step 3: Compute D=F:
* If D=Fp1; then read the row of Table 7 (or column of Table 8) for D=F ¼ 1:
* If 1oD=Fp3; then select the row (or column) for D=F closest to the ratio value.
* If D=F > 3; then read the row (or column) for D=F ¼ 3:
The tableaux is easy to use and the results are bound through the limiting relations described in Theorem 2. A few examples illustrate the application of the tableaux.
Example. (a) If D ¼ $1:3B and F ¼ $0:5B; then D=F ¼ 2:6 and the row for D=F ¼ 2:5 should be used in Table 7 and the column for D=F ¼ 2:5 should be used in Table 8.
(b) If D ¼ $1:5B and F ¼ $1B; then use the row and column in Tables 7 and 8 corresponding to D=F ¼ 1:5:
ARTICLE IN PRESS
Table 7
The impact of the primary HH-GB provision on LIHEAP parameters
as a function of D=F for the appropriation level D ($B) and trigger level F ($B)
D=F sðD=FÞ rðD=FÞ iðD=FÞ pðD=FÞ qðD=FÞ #fHg #fNg
1.00 20.5 42.4 39.5 72.3 51.8 28 23
1.05 17.0 35.3 32.9 68.8 51.8 28 23
1.10 14.1 24.1 33.8 55.7 41.7 28 23
1.15 11.7 18.5 31.7 48.5 36.8 25 26
1.20 9.7 14.2 31.0 41.1 31.3 23 28
1.25 8.1 11.7 26.0 39.2 31.1 22 29
1.30 6.6 9.5 21.2 37.7 31.1 22 29
1.35 5.3 7.1 21.2 30.3 25.0 20 31
1.40 4.3 5.4 19.5 26.0 21.8 17 34
1.45 3.4 4.2 17.8 22.2 18.9 16 35
1.50 2.8 3.1 28.9 12.6 9.7 13 38
1.55 2.5 2.7 36.8 9.2 6.8 11 40
1.60 2.2 2.4 32.5 9.0 6.8 11 40
1.65 1.9 2.1 28.5 8.7 6.8 11 40
1.70 1.7 1.8 27.8 7.7 6.0 9 42
1.75 1.5 1.6 24.2 7.5 6.0 9 42
1.80 1.3 1.3 22.6 6.8 5.6 8 43
1.85 1.1 1.1 19.3 6.7 5.6 8 43
1.90 1.0 1.0 26.1 4.6 3.6 7 44
1.95 0.8 0.9 22.9 4.5 3.6 7 44
2.00 0.7 0.8 19.8 4.4 3.6 7 44
2.25 0.3 0.3 11.3 2.7 2.4 4 47
2.50 0.1 0.1 27.7 0.5 0.4 2 49
2.75 0.1 0.1 37.5 0.3 0.2 1 50
3.00 0.1 0.1 26.0 0.3 0.2 1 50
M.J. Kaiser, A.G. Pulsipher / Energy Policy 31 (2003) 1441–14581450
(c) If D ¼ $0:8B and F ¼ $0:75B; then D=F ¼ 1:067 and the row for D=F ¼ 1:05 should be used in Table 7 and the column entry for D=F ¼ 1:1 used in Table 8.
The impact of the primary HH-GB provision on LIHEAP parameters sðD=FÞ; rðD=FÞ; iðD=FÞ; pðD=FÞ;
qðD=FÞ; #fHg and #fNg presented in Table 7 are graphed in Figs. 4–6. In Table 8 the final LIHEAP allocation percentages under the primary HH-GB provision as a function of D=F is depicted. To determine state allotment dollars for a given value of D and F; compute D=F; retrieve the appropriate final allotment
ARTICLE IN PRESS
Table 8
Final LIHEAP allocation percentages under the primary HH-GB provision as a function of D=F
State 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.5 3.0
Alabama 0.86 1.13 1.32 1.45 1.54 1.59 1.61 1.63 1.64 1.65 1.66 1.67 1.68
Alaska 0.55 0.50 a
0.46 a
0.42 a
0.39 a
0.37 a
0.36 0.36 0.36 0.36 0.36 0.36 0.36
Arizona 0.42 0.72 0.92 1.04 1.13 1.17 1.19 1.21 1.22 1.23 1.23 1.25 1.25
Arkansas 0.66 0.84 0.97 1.06 1.12 1.15 1.17 1.18 1.19 1.20 1.21 1.21 1.21
California 4.61 4.90 5.20 5.45 5.65 5.77 5.82 5.87 5.90 5.92 5.94 5.99 5.99
Colorado 1.61 1.46 a
1.34 a
1.24 a
1.15 1.15 1.14 1.14 1.14 1.15 1.15 1.15 1.15
Connecticut 2.10 1.91 a
1.75 a
1.61 a
1.50 a
1.40 1.39 1.39 1.39 1.39 1.39 1.40 1.40
Delaware 0.28 0.30 0.32 0.34 0.35 0.36 0.36 0.36 0.37 0.37 0.37 0.37 0.37
D.C. 0.33 0.30 a
0.27 a
0.27 0.26 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27
Florida 1.36 2.38 3.04 3.46 3.74 3.90 3.97 4.02 4.06 4.08 4.10 4.15 4.15
Georgia 1.08 1.64 2.02 2.27 2.43 2.52 2.57 2.60 2.62 2.64 2.65 2.68 2.68
Hawaii 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.12 0.12 0.12 0.12
Idaho 0.63 0.57 a
0.52 a
0.48 a
0.45 a
0.42 a
0.39 a
0.37 a
0.35 a
0.33 a
0.31 a
0.28 0.28
Illinois 5.81 5.29 a
5.13 5.11 5.15 5.19 5.21 5.23 5.24 5.25 5.26 5.29 5.30
Indiana 2.63 2.39 a
2.20 2.17 2.17 2.18 2.18 2.19 2.19 2.20 2.20 2.21 2.21
Iowa 1.86 1.69 a
1.55 a
1.43 a
1.33 a
1.24 a
1.22 1.22 1.22 1.22 1.22 1.22 1.22
Kansas 0.86 0.89 0.94 0.97 1.01 1.03 1.04 1.04 1.05 1.05 1.06 1.06 1.06
Kentucky 1.37 1.44 1.51 1.58 1.63 1.67 1.68 1.70 1.70 1.71 1.77 1.73 1.73
Louisiana 0.88 1.19 1.40 1.54 1.64 1.70 1.72 1.74 1.75 1.76 1.77 1.79 1.79
Maine 1.36 1.24 a
1.13 a
1.05 a
0.97 a
0.91 a
0.85 a
0.80 a
0.76 a
0.72 a
0.68 a
0.61 0.61
Maryland 1.61 1.78 1.92 2.04 2.12 2.17 2.19 2.21 2.22 2.23 2.24 2.26 2.26
Massachusetts 4.20 3.82 a
3.50 a
3.23 a
3.00 a
2.90 2.90 2.89 2.90 2.90 2.90 2.91 2.91
Michigan 5.51 5.01 a
4.74 4.69 4.70 4.73 4.74 4.76 4.77 4.78 4.79 4.81 4.82
Minnesota 3.97 3.61 a
3.31 a
3.06 a
2.84 a
2.65 a
2.48 a
2.34 a
2.21 a
2.09 a
1.99 a
1.68 1.68
Mississippi 0.74 1.14 1.41 1.58 1.70 1.76 1.79 1.82 1.83 1.84 1.85 1.87 1.87
Missouri 2.32 2.23 2.24 2.27 2.31 2.35 2.36 2.37 2.38 2.39 2.39 2.41 2.41
Montana 0.74 0.67 a
0.61 a
0.57 a
0.53 a
0.49 a
0.46 a
0.43 a
0.41 a
0.39 a
0.37 a
0.35 0.35
Nebraska 0.92 0.84 a
0.77 a
0.71 a
0.66 a
0.61 a
0.58 a
0.55 0.55 0.55 0.55 0.55 0.55
Nevada 0.20 0.31 0.38 0.43 0.46 0.48 0.49 0.49 0.50 0.50 0.50 0.51 0.51
New Hampshire 0.79 0.72 a
0.66 a
0.61 a
0.57 a
0.53 a
0.50 a
0.47 a
0.45 0.45 0.45 0.45 0.45
New Jersey 3.90 3.54 a
3.26 3.21 3.21 3.22 3.23 3.24 3.24 3.25 3.26 3.27 3.27
New Mexico 0.52 0.50 0.50 0.51 0.52 0.52 0.53 0.53 0.53 0.53 0.54 0.54 0.54
New York 12.72 11.57 a
10.60 a
9.79 a
9.09 a
8.66 8.63 8.63 8.63 8.63 8.63 8.66 8.67
North Carolina 1.90 2.28 2.56 2.76 2.90 2.99 3.03 3.06 3.08 3.09 3.10 3.13 3.13
North Dakota 0.80 0.73 a
0.67 a
0.62 a
0.57 a
0.53 a
0.50 a
0.47 a
0.44 a
0.42 a
0.40 a
0.32 a
0.27 a
Ohio 5.14 4.74 a
4.64 4.65 4.70 4.74 4.76 4.78 4.80 4.81 4.82 4.85 4.85
Oklahoma 0.79 0.95 1.07 1.15 1.21 1.25 1.26 1.28 1.28 1.29 1.30 1.31 1.31
Oregon 1.25 1.13 a
1.04 a
0.96 a
0.94 0.94 0.94 0.94 0.94 0.94 0.94 0.95 0.95
Pennsylvania 6.84 6.21 a
5.70 a
5.26 a
5.12 5.10 5.10 5.11 5.11 5.12 5.13 5.15 5.15
Rhode Island 0.69 0.63 a
0.58 a
0.53 a
0.49 a
0.47 0.46 0.46 0.46 0.46 0.46 0.47 0.47
South Carolina 0.68 0.94 1.12 1.24 1.31 1.36 1.38 1.40 1.41 1.42 1.42 1.44 1.44
South Dakota 0.65 0.59 a
0.54 a
0.50 a
0.46 a
0.43 a
0.41 a
0.38 a
0.36 a
0.34 a
0.32 a
0.29 0.29
Tennessee 1.39 1.58 1.74 1.85 1.94 1.99 2.01 2.03 2.04 2.05 2.06 2.08 2.08
Texas 2.26 3.84 4.86 5.52 5.95 6.20 6.31 6.39 6.45 6.49 6.52 6.58 6.59
Utah 0.75 0.68 a
0.62 a
0.58 a
0.54 0.54 0.53 0.53 0.54 0.54 0.54 0.54 0.54
Vermont 0.60 0.54 a
0.50 a
0.46 a
0.43 a
0.40 a
0.37 a
0.35 a
0.33 a
0.31 a
0.30 a
0.24 a
0.23 a
Virginia 1.96 2.28 2.53 2.71 2.84 2.92 2.95 2.98 3.00 3.01 3.02 3.05 3.05
Washington 2.05 1.86 a
1.71 a
1.58 a
1.49 1.49 1.48 1.48 1.49 1.49 1.49 1.49 1.49
West Virginia 0.91 0.88 0.88 0.90 0.92 0.93 0.93 0.94 0.95 0.95 0.95 0.95 0.96
Wisconsin 3.58 3.25 a
2.98 a
2.75 a
2.55 a
2.38 a
2.24 a
2.10 a
1.99 a
1.93 1.93 1.93 1.93
Wyoming 0.30 0.27 a
0.25 a
0.23 a
0.21 a
0.20 a
0.19 a
0.18 0.18 0.18 0.18 0.18 0.18
a Designates membership in H:
M.J. Kaiser, A.G. Pulsipher / Energy Policy 31 (2003) 1441–1458 1451
percentage depicted in Table 8, and then multiply by the value of D:
AiðD=FÞ ¼ DaiðD=FÞ:
The functionals sðD=FÞ; pðD=FÞ; and qðD=FÞ are defined in terms of the regular appropriation D; while rðD=FÞ is defined in terms of the money allocated to the states that fall within the set fNg: When D=F ¼ 1 þ E; the 28 states in fHg control 72% of the total funds
allocated to the program, and as D increases, pðF; DÞ ¼ H=D declines since the number of states within fHg will decrease and with it the amount of money that fHg controls. This is shown in Fig. 4. At D=F ¼ 1:15; roughly half of the states control half of the total appropriation. The value of qðF; DÞ ¼ G=D represents the proportion of funds that the states fHg would control without the HH-GB provision. The difference between pðF; DÞ and qðF; DÞ—equivalent to sðF; DÞ— represents the distortion of the 1984 formula in percentage terms relative to the total appropriation. In Fig. 5 the impact of the primary HH-GB provision on the functional rðD=FÞ represents the percentage reduc- tion in the funding allocation to fNg due to the HH-GB provision. The counting functions #fHg and #fNg count the number of elements of fHg and fNg and is depicted in Fig. 6. The counting functionals are mirror images of one another since #fHg þ #fNg ¼ 51; and as D=F increases, #fHg-0 and fNg-fSg:
6.3. Regional distribution
The state allocation percentages depicted in Table 8 are aggregated according to region and presented
ARTICLE IN PRESS
0 10 20 30 40 50 60 70 80
1 1.
1 1.
2 1.
3 1.
4 1.
5 1.
6 1.
7 1.
8 1.
9 2
D/F
P e rc
e n
ta g
e o
f D
( %
)
s (D/F) p (D/F) q (D/F)
Fig. 4. Impact of the primary HH-GB provision on LIHEAP functionals sðD=FÞ; pðD=FÞ; and qðD=FÞ:
0
10
20
30
40
50
1 1.
1 1.
2 1.
3 1.
4 1.
5 1.
6 1.
7 1.
8 1.
9 2
D/F
P e rc
e n
ta g
e (
% )
r (D/F)
Fig. 5. Impact of the primary HH-GB provision on LIHEAP functional rðD=FÞ:
0
10
20
30
40
50
60
1
1 .1
5
1 .3
1 .4
5
1 .6
1 .7
5
1 .9
2 .0
5
2 .2
2 .3
5
2 .5
2 .6
5
2 .8
2 .9
5
D/F
N u
m b
e r
o f
S ta
te s
#{H} #{N}
Fig. 6. Impact of the primary HH-GB provision on the LIHEAP
counting functionals #fHg and #fNg:
M.J. Kaiser, A.G. Pulsipher / Energy Policy 31 (2003) 1441–14581452
graphically in Fig. 7 as a function of D=F: The limits of the distribution for the 1981 formula allotment ðD=Fp1Þ and the 1984 allotment ðD=FX1:5Þ represent the endpoints of the graph and were described previously in Table 6. As the ratio D=F increases federal allocation percentage in the Northeast and North Central regions are reallocated to the South, while the Western region maintains an essentially constant allot- ment percentage over the entire range of D=F: As D=F exceeds 1.5 the allotment percentages stabilize as the impact of the HH-GB provision significantly declines. Recall from Table 7 that when D=F ¼ 1:5 less than 3% of the regular appropriation is required to satisfy the HH provision.
7. Allotment block procedure
An alternative approach to distribute funds that does not depend on selecting a specific value of F is to ‘‘set- aside’’ a given percentage of the regular appropriation to be used with the 1981 formula and then to apply the 1984 formula to the remaining funds. This mechanism does not hold states harmless in the sense described by the LIHEAP statute or in the trigger-level re-adjust- ment, but can be correlated with these allotment procedures using an approximating algebraic correspon- dence.
If the percentage of funds allocated through the 1981 formula is denoted by r; then a state’s allotment dollars for a given fiscal year is given by
AiðrÞ ¼ rDfi þ ð1 � rÞDgi
and its allotment percentage is ai ¼ Ai=D or
aiðrÞ ¼ rfi þ ð1 � rÞgi; 0oro1:
The allotment percentage is a linear combination of fi and gi and so aðrÞ ¼ ða1; y; a51Þ is an allotment
percentage sinceX aiðrÞ ¼
X ðrfi þ ð1 � rÞgiÞ
¼ r X
fi þ ð1 � rÞ X
gi ¼ 1:
The value of aðrÞ as a function of r is shown in Table 9.
ARTICLE IN PRESS
0
5
10
15
20
25
30
35
40
1 1.
05 1. 1
1. 15 1.
2 1.
25 1. 3
1. 35 1.
4 1.
45 1. 5
D/F
P e rc
e n
ta g
e (
% )
Northeast
North Central
South
West
Fig. 7. Regional distributional of funds.
Table 9
Weighted linear combination of the 1981 and 1984 formula, aiðrÞ ¼ rfi þ ð1 � rÞgi
State r
0.2 0.4 0.5 0.6 0.8
Alabama 1.51 1.35 1.27 1.19 1.02
Alaska 0.40 0.44 0.45 0.47 0.51
Arizona 1.08 0.92 0.83 0.75 0.58
Arkansas 1.11 0.99 0.93 0.88 0.77
California 5.72 5.44 5.30 5.17 4.89
Colorado 1.24 1.33 1.38 1.43 1.52
Connecticut 1.54 1.68 1.75 1.82 1.96
Delaware 0.35 0.33 0.33 0.32 0.30
D.C. 0.28 0.29 0.30 0.30 0.31
Florida 3.59 3.04 2.76 2.48 1.92
Georgia 2.36 2.04 1.88 1.72 1.40
Hawaii 0.11 0.11 0.11 0.11 0.11
Idaho 0.35 0.42 0.45 0.49 0.56
Illinois 5.40 5.50 5.55 5.60 5.71
Indiana 2.30 2.38 2.42 2.46 2.55
Iowa 1.35 1.48 1.54 1.61 1.74
Kansas 1.02 0.98 0.96 0.94 0.90
Kentucky 1.66 1.59 1.55 1.51 1.44
Louisiana 1.61 1.43 1.33 1.24 1.06
Maine 0.76 0.91 0.99 1.06 1.21
Maryland 2.13 2.00 1.93 1.87 1.74
Massachusetts 3.17 3.43 3.55 3.68 3.94
Michigan 4.96 5.10 5.17 5.24 5.38
Minnesota 2.14 2.60 2.83 3.06 3.51
Mississippi 1.65 1.42 1.30 1.19 0.96
Missouri 2.39 2.38 2.37 2.36 2.34
Montana 0.43 0.50 0.54 0.58 0.66
Nebraska 0.62 0.70 0.73 0.77 0.85
Nevada 0.44 0.38 0.35 0.32 0.26
New Hampshire 0.52 0.59 0.62 0.66 0.73
New Jersey 3.40 3.52 3.59 3.65 3.77
New Mexico 0.54 0.53 0.53 0.53 0.52
New York 9.48 10.29 10.70 11.10 11.91
North Carolina 2.89 2.64 2.52 2.39 2.14
North Dakota 0.33 0.45 0.51 0.56 0.68
Ohio 4.91 4.97 5.00 5.02 5.08
Oklahoma 1.20 1.10 1.05 1.00 0.89
Oregon 1.01 1.07 1.10 1.13 1.19
Pennsylvania 5.49 5.82 5.99 6.16 6.50
Rhode Island 0.51 0.56 0.58 0.60 0.65
South Carolina 1.29 1.14 1.06 0.98 0.83
South Dakota 0.36 0.43 0.47 0.51 0.58
Tennessee 1.94 1.80 1.73 1.66 1.53
Texas 5.73 4.86 4.43 4.00 3.13
Utah 0.58 0.62 0.64 0.66 0.71
Vermont 0.30 0.37 0.41 0.45 0.52
Virginia 2.83 2.61 2.50 2.39 2.18
Washington 1.61 1.72 1.77 1.83 1.94
West Virginia 0.95 0.94 0.93 0.93 0.92
Wisconsin 2.26 2.59 2.75 2.92 3.25
Wyoming 0.20 0.23 0.24 0.25 0.28
M.J. Kaiser, A.G. Pulsipher / Energy Policy 31 (2003) 1441–1458 1453
7.1. The correspondence between aðD=FÞ and aðrÞ
A useful exercise is to correlate the values of aðrÞ and aðD=FÞ; that is, for a given value of D=F determine the value of r such that aðrÞ achieves an allocation percentage ‘‘equivalent’’ to aðD=FÞ: The limits of the relation are obvious:
lim D=F-N
aiðD=FÞ ¼ gi ¼ lim r-0
aiðrÞ;
lim D=F-1
aiðD=FÞ ¼ fi ¼ lim r-1
aiðrÞ
and so we are interested in the correspondence between aðD=FÞ and aðrÞ for 0oro1: A correspondence using a least-squares criteria is derived in Theorem 4.
Theorem 5. For a given value of D=F; the final allocation percentages specified by aðD=FÞ correspond to aðrÞ ¼ rf þ ð1 � rÞg for the least-squares estimator
r ¼ rðD=FÞ ¼ P
aiðD=FÞðfi � giÞ � P
giðfi � giÞP ðfi � giÞ
2
Proof. See Appendix C.
The correspondence between aðD=FÞ and aðrÞ is determined through the value of rðD=FÞ shown in Fig. 8 and provides a useful means to interpret the HH condition in terms of a simple (linear) allocation mechanism. For example, when D=F ¼ 1:3; the value of r that ‘‘equates’’ aðD=FÞ in the least-squares sense with aðrÞ is given by r ¼ 0:25; or in other words, when D=F ¼ 1:3; the allocation mechanism determined through Option B is roughly equivalent to taking 25% of the 1981 formula values plus 75% of the 1984 formula values g: The main use of Fig. 8 is as a tool to establish the relation between competing alternative policies:
* For a given value of D=F; the user can determine how the average state HH allotment aðD=FÞ distributes funds in terms of the allocation mechanism aðrÞ ¼ rf þ ð1 � rÞg:
* Since the value of aðrÞ and aðD=FÞ are equivalent under the derived correspondence, the fairness criteria employed for aðD=FÞ translates to aðrÞ under rðD=FÞ:
* A decision maker that can determine a range of acceptable percentage values for the contribution of f to the final allotment mechanism can use the functional rðD=FÞ to determine acceptable D=F ratios.
7.2. Induced hold-harmless allotment percentages
To illustrate one application of the correspondence the allotment percentage induced through a reduced trigger level is computed.
The correspondence functional rðD=FÞ describes the percentage of the HH provision funding that is allocated according to the 1981 formula. Hence, using the correspondence we can determine the effective weight of the 1981 formula as a function of the trigger level. Given rðD=FÞ and the historic LIHEAP funding levels from 1981–2002, the ratio D=F is computed each year based on the regular appropriation D and trigger level F; rðD=FÞ is computed, and then the average value of rðD=FÞ reported:
/rðD=FÞS ¼ P2002
t¼1981 rðD=F; tÞ 21
:
The value /rðD=FÞS describes the expected contribu- tion of the 1981 formula allotment to the final allotment percentages for a given value of F:
In Fig. 9 the contribution of the 1981 formula to the final allotment percentages as a function of the trigger level F is shown. Since the current trigger level is F ¼ $1:975B; Fig. 9 illustrates that 98% of the LIHEAP funds has effectively been allocated based on the 1981 formula, with 2% of the funds allocated using the 1984 formula. If F ¼ $1B; then based on historic levels of LIHEAP funding, 28% of federal dollars would have been allocated using the 1981 formula with the remaining 72% allocated with the 1984 formula.
ARTICLE IN PRESS
0
0.2
0.4
0.6
0.8
1
1.2
1 1.
1 1.
2 1.
3 1.
4 1.
5 1.
6 1.
7 1.
8 1.
9 2
D/F
r (%
)
Fig. 8. The correspondence between aðD=FÞ and aðrÞ:
0 0.2
0.4 0.6 0.8
1 1.2
1.975 1.75 1.5 1.25 1 0.75 0.5 0.25
F ($B)
r (%
)
Fig. 9. Induced allotment percentage of the HH-GB provision based
on LIHEAP funding levels, 1981–2002.
M.J. Kaiser, A.G. Pulsipher / Energy Policy 31 (2003) 1441–14581454
7.3. Re-interpreting the primary hold-harmless provision
The LIHEAP allocation percentages under the primary HH provision (Option B) can be interpreted in a manner analogous to the allotment block proce- dure. Recall that for states iAfNg;
AiðF; DÞ ¼ Ffi þ Ei ¼ Ffi þ gðDgi � FfiÞ
¼ ð1 � gÞFfi þ gDgi
or
aiðF; DÞ ¼ ð1 � gÞkfi þ ggi;
where k ¼ F=D and g ¼ X =Q: For a given value of D > F; ko1; and so the term kfi represents a reduction in the 1981 formula percentage. As D increases, k approaches zero while the term g approaches one (recall Theorem 1), indicating that the contribution to aiðF; DÞ from fi will approach zero. The LIHEAP allocation percentages in Option B is thus similar in structure to the allotment block procedure, but while the parameter r is a decision parameter, the value of k and g is dependent on D=F and is not controllable unless the ratio D=F is legislated.
8. Simplified legislative road map
The main issues surrounding the LIHEAP debate are summarized as follows:
* The 1981 distribution formula is not an appropriate baseline.
* Corporate scandals are sexy and good publicity issues, while government ‘‘scandals’’ resist debate.
* Impetus to change LIHEAP will always be an uphill struggle since the votes for change simply do not add up for any of the alternatives described.
* Full ‘‘disclosure’’ and transparency may help lead to change.
* Shame is a good motivator for change. * Arguments should be based on essential facts and a
clear storyline.
A legislative ‘‘road map’’ of sorts is suggested to help push LIHEAP along in the right direction. Since a simple count of votes is not expected to add up in favor of changing the LIHEAP allocation mechanism, the story presented must be moral tale and based on established facts. Shame is used as the primary motivator for change.
Fact 1: The high trigger level established within the 1984 re-authorization legislation has disabled the application of the 1984 formula.
Shame/result: The 1981 formula is still being used to distribute LIHEAP funding to this day (inequitable program).
Fact 2: The 1981 formula allotments hold little meaning 20 years after their creation and should not be used to distribute federal dollars.
Shame/result: LIHEAP allotments to states are not distributed in a rational manner (poorly executed program).
Fact 3: The primary intent of LIHEAP is to assist low-income households meet their home energy needs.
Shame/result: LIHEAP funds are not being distrib- uted in accord with the program intent (irrational program).
9. Conclusions
The purpose of this paper was to propose policy alternatives for distributing LIHEAP funds to states and to quantify the impact of each alternative. Three policy alternatives were examined.
In the first alternative, the LIHEAP trigger level is nullified and the HH-GB provisions attached to the allocation are eliminated. The 1984 formula is applied in ‘‘pure’’ form unencumbered by the weight of additional provisions or conditions. This policy alternative paral- lels the Bush Administration position expressed in the FY 2003 federal budget and is the simplest and probably the best of the three options considered, although it is not expected to develop the political support necessary to achieve passage.
In the second alternative, the primary HH-GB provision of the LIHEAP statute is maintained but the trigger level when it is enabled is reduced. Regular appropriations for LIHEAP generally do not exceed the current trigger F ¼ $1:975B; and in fact, have only exceeded this level twice in the past 17 years—the last time in FY 1986. At one time (15-years ago) the values of D and F were roughly comparable, while today F appears as a large (mostly unattainable) upper bound. Reducing the current trigger level increases the like- lihood that the 1984 formula will be applied, and since this alternative does not involve the complete disman- tling of the legislative statute, it is also likely to garner more political support. It should be mentioned, how- ever, that even under this alternative a majority of states are still worse off under the new allocation percentages, and hence, even the passage of compromise legislation might fail to generate sufficient support. Legislation that sponsors a trigger-level reduction must be crafted to ensure the reduction is significant or further drift between D and F may result in the future. Re-adjusting F downward may yield the desired outcome, but from a programmatic perspective, also perpetrates the applica- tion/justification of the 1981 formula, which is con- sidered a major drawback.
In the third alternative, a simple allocation mechan- ism was examined that set-aside a given percentage of
ARTICLE IN PRESS M.J. Kaiser, A.G. Pulsipher / Energy Policy 31 (2003) 1441–1458 1455
the regular appropriation to be used exclusively with the 1981 formula with the remaining funds allocated under the 1984 formula. The advantage of this mechanism is that the 1984 formula will always be incorporated within the allotment procedure regardless of the level of regular appropriation, and hence the downside risk (i.e., when DpF ) is always covered. On the other hand, it may be difficult to negotiate the value of r; and once selected, r fixes the allotment percentages and does not allow the upside potential of the 1984 formula (i.e., when D > F ) to be realized. Since the upside potential of the 1984 formula has only a low probability of occurrence, however, this does not seem to be an extenuating factor. ‘‘Mixing’’ the formula allocations is also not a preferred procedure from a legislative or methodological perspec- tive, although as shown in Theorem 4, a correspondence does exists between the mixed formula and the triggered hold-harmless provision. Hence, the user can always view the allotment block procedure in terms of a trigger- level enabled mechanism.
Acknowledgements
The authors would like to acknowledge the role Carolyn Drake had in introducing the LIHEAP problem to the authors and suggesting its initial formulation. Her encouragement, guidance, and pa- tience is kindly recognized. The authors would also like to thank Kathy Baskin, Elizabeth Jones, Beth Osborne, Ed Rissing, and Fred Zeytoonjian for their support, insight, and useful criticisms of this work. Funding for this research was provided in part by the Southern States Energy Board, but all opinions, findings, and results remain the responsibility of the authors.
Appendix A. Derivation of the LIHEAP structure
equations
Theorem 1. The LIHEAP allocation mechanism under
the primary HH-GB provision yields state allotments
according to the following recipe:
AiðF; DÞ ¼
Dfi; iAfSg; DpF; Ffi; iAfHg; D > F;
Ffi þ Ei; iAfNg; D > F;
8>< >:
where fSg ¼ fHg,fNg; fHg ¼ fi j Ffi=Dgi > 1g; fNg ¼ fi j Ffi=Dgip1g; Ei ¼ gðDgi � FfiÞ; g ¼ X =Q; X ¼ D � F; and Q ¼
P iAfNg ðDgi � FfiÞ:
Proof. If DpF; then following the current LIHEAP statute, AiðF; DÞ ¼ Dfi for all iAfSg: If D > F; then partition fSg into two sets according to whether state i satisfies Ffi > Dgi or FfipDgi: The class of states labeled
fHg ¼ fi j Ffi=Dgi > 1g are the HH states which are held at their statutory floor. The class of states labeled fNg ¼ fi j Ffi=Dgip1g will have an allotment falling above their floor. All states in fHg are assigned their statutory floor, while states in fNg are assigned their statutory floor plus an amount Ei that needs to be determined
AiðF; DÞ ¼ Ffi; iAfHg;
Ffi þ Ei; iAfNg:
(
Since statutory floors are maintained across all states, the HH-GB provision in essence captures F of the regular appropriated dollars D:
D ¼ X
iAfSg
AiðF; DÞ ¼ X
iAfHg
Ffi þ X
iAfNg
ðFfi þ EiÞ
¼ X
iAfHg
Ffi þ X
iAfNg
Ffi þ X
iAfNg
Ei
¼ F X
iAfSg
fi þ X
iAfNg
Ei ¼ F þ X
iAfNg
Ei:
The value
X ¼ X
iAfNg
Ei ¼ D � F
represents the dollars available to the states in fNg after the statutory floor money is assigned to all the states in fSg:
To determine the extra allotment that each state in fNg receives, state i’s allotment ‘‘above the floor’’
Dgi � Ffi; iAfNg
is computed, and since
Q ¼ X
iAfNg
ðDgi � FfiÞ > X ;
the value of Q will need to be reduced so that it matches the available funds X : The value of X is thus a binding constraint. The reduction factor g ¼ X =Q is applied across each state’s money above the floor, and the value of Ei is calculated as
Ei ¼ gðDgi � FfiÞ:
Observe that the sum of the ‘‘extra’’ state allotments is by construction equal to the available funds X as required:X iAfNg
Ei ¼ g X
iAfNg
ðDgi � FfiÞ
¼ X
Q
X iAfNg
ðDgi � FfiÞ ¼ X : &
Theorem 2. The LIHEAP allocation mechanism under
the primary HH-GB yields the following limiting relations:
lim D=F-1
aiðF; DÞ ¼ fi;
lim D=F-N
aiðF; DÞ ¼ gi:
ARTICLE IN PRESS M.J. Kaiser, A.G. Pulsipher / Energy Policy 31 (2003) 1441–14581456
Proof. By the definition of the allocation mechanism, when D=F-1; it is clear that
lim D=F-1
aiðF; DÞ ¼ fi; iAfSg:
On the other hand, as the ratio D=F increases it is clear that #fHg-0 and fNg-fSg: Then since Q ¼P
iAfNg ðDgi � FfiÞ;
lim D=F-N
X iAfNg
ðDgi � FfiÞ ¼ X
iAfSg
ðDgi � FfiÞ
¼ D X
iAfSg
gi � F X
iAfSg
fi ¼ D � F;
i.e.,
lim D=F-N
g ¼ lim D=F-N
X
Q ¼
D � F D � F
¼ 1:
Then since AiðF; DÞ ¼ Ffi þ Ei ¼ Ffi þ gðDgi � FfiÞ ¼ ð1 � gÞFfi þ gDgi;
lim D=F-N
AiðF; DÞ ¼ lim g-1
½ð1 � gÞFfi þ gDgi ¼ Dgi
or
lim D=F-N
aiðF; DÞ ¼ AiðF; DÞ
D ¼ gi: &
Appendix B. One-dimensional LIHEAP system measures
Theorem 3. For D > F; the LIHEAP allocation mechan- ism under the primary HH-GB provision yield state
allotment percentages
aiðF; DÞ ¼ F D
fi; iAfHg; F D
fi þ Ei D ; iAfNg
(
that are functions of the ratio F/D.
Proof. Let k ¼ F=Da0: It is sufficient to show that aiðF; DÞ is a constant:
(i) If iAfHg; then aiðF; DÞ ¼ ðF=DÞfi ¼ kfi; which is a constant since fi is constant.
(ii) If iAfNg; then aiðF; DÞ ¼ ðF=DÞfi þ ðEi=DÞ ¼ kfi þ ðEi=DÞ:
It is sufficient to show Ei=D is constant.
Ei
D ¼
X
DQ ðDgi � FfiÞ ¼
X
Q gi �
F
D fi
�
¼ X
Q ðgi � kfiÞ ¼ k
0 X
Q ;
where k0 ¼ gi � kfi is constant since gi is constant. Now X =Q ¼ X =D=Q=D and
X
D ¼
D � F D
¼ 1 � F
D ¼ 1 � k
a constant, and
Q
D ¼
D P
iAfNg gi � F P
iAfNg fi
D
¼ X
iAfNg
gi � F
D
X iAfNg
fi ¼ X
iAfNg
gi � k X
iAfNg
fi:
The summation terms P
iAfNg fi; P
iAfHg fi; P
iAfNg gi; and
P iAfHg gi are all constant since the elements of
fHg ¼ i j Ffi
Dgi > 1
� ¼ i j k
fi
gi > 1
� ;
fNg ¼ i j Ffi
Dgi p1
� ¼ i j k
fi
gi p1
�
are constant, and so Q=D is constant, and thus, X =Q is constant. Hence, aiðF; DÞ is a constant and can be expressed as aiðF=DÞ; or equivalently, since k ¼ F=Da0; as aiðD=FÞ: &
Theorem 4. The LIHEAP system measures
sðF; DÞ; iðF; DÞ; and rðF; DÞ can be expressed in terms of the ratio D=F as sðD=FÞ; iðD=FÞ; and rðD=FÞ:
Proof. Let k ¼ F=Da0: It suffices to show that sðF; DÞ; iðF; DÞ; and rðF; DÞ are constants. By definition of the functionals sðF; DÞ; iðF; DÞ; and rðF; DÞ;
sðF; DÞ ¼ H � G
D ¼
H
D �
G
D
¼ F
P iAfHg fi
D �
D P
iAfHg gi
D
¼ k X
iAfHg
fi � X
iAfHg
gi;
is a constant, and so sðF; DÞ can be expressed in terms of the ratio sðD=FÞ: Similarly,
iðF; DÞ ¼ H � G
G ¼
H
G � 1
¼ F
P iAfHg fi
D P
iAfHg gi � 1 ¼ k
P iAfHg fiP iAfHg gi
� 1
a constant, and
rðF; DÞ ¼ H � G
M ¼
H
M �
G
M
¼ F
P iAfHg fi
D P
iAfNg gi �
D P
iAfHg gi
D P
iAfNg gi
¼ k
P iAfHg fiP iAfNg gi
�
P iAfHg giP iAfNg gi
a constant, and so iðF; DÞ and rðF; DÞ can be expressed as iðD=FÞ and rðD=FÞ: &
Appendix C. Derivation of the least-squares estimator
Theorem 5. For a given value of D/F, the final allocation percentages specified by aðD=FÞ correspond to aðrÞ ¼
ARTICLE IN PRESS M.J. Kaiser, A.G. Pulsipher / Energy Policy 31 (2003) 1441–1458 1457
rf þ ð1 � rÞg for the least-squares estimator
r ¼ rðD=FÞ ¼ P
aiðD=FÞðfi � giÞ � P
giðfi � giÞP ðfi � giÞ
2 :
Proof. For a given value of the ratio D=F; the value of r that minimizes the sum of the squared differences between aðD=FÞ and aðrÞ is determined as follows. Let
f ðrÞ ¼ X
ðaiðD=FÞ � aiðrÞÞ 2:
Then the solution to minr f ðrÞ is determined by solving dðf ðrÞÞ=dr ¼ 0 orX
ðaiðD=FÞ � aiðrÞÞ daiðrÞ
dr ¼ 0:
Since aiðrÞ ¼ rfi þ ð1 � rÞgi; dðaiðrÞÞ=dr ¼ fi � gi; and solving for r yields
r ¼ P
aiðD=FÞðfi � giÞ � P
giðfi � giÞP ðfi � giÞ
2 :
The second-derivative test yields
d 2 f ðrÞ dr
¼ � daiðrÞ
dr
� 2 o0
ensuring that r is a minimum value of f ðrÞ: &
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