Statistical Issues in Allocating Funds by Formula (2003)

Each year, formulas are used to allocate large amounts of federal funds to state and local governments via federal programs designed to meet a wide spectrum of economic and social objectives. These programs address societal goals, such as improving educational outcomes and accessibility of medical care, and some of them are designed to equalize fiscal capacity to address identified needs.

Funds distributed by federal formula allocation grants have more than doubled in real terms over the past 25 years. A 1975 study estimated that $35.6 billion was allocated under grants using population or per capita income as formula components. By fiscal year (FY) 2000, Medicaid was by far the largest formula allocation program, with $111.1 billion disbursed. Federal-Aid Highway Program grants were the next largest, with $25.9 billion, followed by allocations under the Temporary Assistance for Needy Families program (TANF), with $19.1 billion. The December 2001 update of the Catalog of Federal Domestic Assistance showed 180 federal formula allocation programs with FY 2000 obligations totaling approximately $262.3 billion. Table 1-1 lists the 11 largest programs for fiscal years 1999 and 2000 (see Appendix B for more details).

Formula allocation programs are characterized by the allocation of money to states or their subdivisions in accordance with a distribution formula prescribed by law or administrative regulation, for activities of a

continuing nature not confined to a specific project. For some programs, the distribution formula used is a closed mathematical expression; for others, iterative processes are used to arrive at the final allocations. Block grant programs are a subset of formula allocation programs in which the recipient jurisdiction has broad discretion for the application of funds received in support of such programs as community development or the prevention and treatment of substance abuse, which are specified in the authorizing legislation. Matching grant programs, such as Medicaid and certain transportation programs, require that the recipient state provide a matching percentage of funds from state sources.

Allocation formulas are usually designed with one or more objectives: to distribute funds to recipient governments in proportion to some measure of program need, to equalize their fiscal capacities to meet program needs, or to influence their spending decisions. They are developed in the context of a complex political process. Use of a formula (rather than a specification

of the amount to be given to each recipient jurisdiction) facilitates informed debate and a degree of transparency about the allocation process by providing documentation of assumptions and computations. Furthermore, a formula offers legislators an effective way of explaining the allocation process to their constituents. However, when funds are allocated according to a formula, there is no guarantee that objectives will be fully met. In particular, properties of data sources and statistical procedures used to produce formula inputs can interact in complex ways with formula features to produce consequences that may not have been anticipated or intended. Sometimes these problems arise when a formula developed in authorizing legislation is modified by provisions introduced in the appropriations process.

The use of formulas to allocate federal funds to subordinate jurisdictions is part of a broader process of government-to-government transfer of funds. Uses of such funds by the recipients may be unrestricted (as in the General Revenue Sharing Program of the 1970s and 1980s and the Canadian Equalization Program), or they may be limited to specific purposes, for example, to provide medical care and improve the education of children in poor families, assist persons to end their dependence on welfare, revitalize economically depressed cities and neighborhoods, or provide assistance to localities disproportionately affected by HIV infection. At the federal level, the U.S. Congress determines how much money will be distributed and for what purposes. For some programs, Congress appropriates a fixed total amount each year to be allocated among states or other recipients; for others, such as Medicaid, amounts may be specified as a certain proportion of all qualified expenditures by a state or other jurisdiction. In the former case, a formula dictates how much of the total goes to each recipient; in the latter case, a formula determines what proportion of each jurisdiction’s amount will be matched by the federal government.

In a few programs, federal funds are allocated directly to jurisdictions below the state level, such as school districts. In most federal formula allocation programs, however, state agencies take responsibility for within-state distribution of the funds allocated to them, subject to program regulations and various kinds of audits and reviews. Importantly, states use formulas to distribute a substantial amount of their own funds. For example, state funds provided for public education substantially exceed those provided by the federal government.

Governments and international organizations use mathematical and statistical formulas for many purposes besides the allocation of funds to subordinate jurisdictions. Examples include the apportionment of seats in

the U.S. House of Representatives, judicial determination of amounts awarded for child support (National Research Council, 2001:34-35), payments to health plans in the Federal Employee Health Benefits Program, and “taxes” on federal agency appropriations to cover overhead expenses. However, to make its task more manageable and to intensify its focus, the panel decided to restrict attention to formulas used to allocate funds to government units, with primary, but not unique, focus on allocations by the U.S. federal government.

The use of formulas to allocate federal and state funds to subordinate jurisdictions can be traced back to the 19th century. The Morrill Act of 1862 allotted to each state 30,000 acres of public land for each of its senators and representatives in Congress. The land was to be sold and the proceeds used to establish the institutions of higher learning that came to be known as the land grant colleges. Although originally started as agricultural and technical schools, many land grant colleges grew, with additional state aid, into large public universities, which over the years have educated millions of Americans.

Extensive use of formula grants to states did not begin until the 20th century. The Federal-Aid Road Act of 1916, the forerunner of today’s Federal-Aid Highway Program, established important precedents for provision of federal assistance through state and local governments, requirements for recipient governments to expend matching funds, and the use of formulas to determine the distribution of program funds (Weingroff, 2001). The broad category of intergovernmental expenditures to state and local governments grew from $2 million in 1902, to $118 million in 1922, to $2.371 billion in 1950 (U.S. Bureau of the Census, 1975).

Growth continued in the second half of the century. A broad-based grant program to provide school aid to federally impacted areas was enacted in 1950, and the first block grant program, comprehensive health grants, was established in 1962 (Break, 1980). In 1965, Title I of the Elementary and Secondary Education Act established a formula-based allocation of federal funds designed to improve educational opportunities for school-age children from poor families.

The General Revenue Sharing Program, enacted in 1972, represented an attempt to equalize fiscal capacity across jurisdictions by providing federal funds for unrestricted use by the recipients. Similar programs of

unrestricted grants, which continue to the present, already existed in Canada and Australia. Under general revenue sharing, federal funds were allocated to approximately 39,000 local jurisdictions using a formula based on tax effort, population, and per capita income. The program continued through 1986, but during a period of budgetary stringency was not reauthorized by Congress. However, categorical and block grant programs continued to grow. Today, formulas are used to allocate well over $250 billion of federal funds annually to state and local governments via approximately 180 federal programs designed to meet a wide spectrum of economic and social objectives.¹ Large amounts of state revenues are distributed to cities, counties, and other local governments by way of formula allocation programs.

The essence of a formula allocation program is that the amounts of money to be allocated through the program are calculated by a specified formula that requires various statistical inputs and other information. The formula may be expressed as a mathematical equation that directly provides the amounts, given the appropriate input quantities, or as an algorithm for calculating the amounts. Frequently the formula consists of a basic calculation (an equation or an algorithm) that provides initial amounts, followed by a series of additional rules or adjustments that amend these amounts through constraints on their level or change over time. In some cases, the formula produces the actual dollar amounts to be allocated; in other cases, it produces the shares of an available total amount to go to each jurisdiction; in yet other cases (e.g., Medicaid), it provides a matching percentage for supplementing expenditures being made by the recipient jurisdiction from its own funds.

Formula allocation programs also vary in terms of the uses to which funds can be put. Many are directed at alleviating particular socioeconomic problems, with the use of funds restricted to that purpose. Of course, a recipient jurisdiction might, on receiving directed federal funding, be tempted to reduce its own expenditure in the problem area, thus freeing up funds for other purposes.² Some programs, such as Medicaid, try to

discourage such substitution by making allocations proportional to the recipients’ expenditures for the program. Others, such as TANF, require the recipient to spend at least a specified amount—for example, some proportion of its federal grant—from its own funds for the program.

The potential benefits of a formula-driven approach can be understood by considering the alternatives. In the absence of a formula, one of two scenarios may arise. If responsibility for the distribution of funds remains with Congress, members would have to negotiate an allocation afresh for each fiscal period. If the allocation of funding has been delegated to program administrators, they would have to deal with the pressures from recipient jurisdictions in making their allocation decisions each year. In either case it makes good sense that, if an initial round of allocations has been negotiated on the basis of certain principles or rationale, those principles or rationale be embedded in a formula so that equivalent results may be reproduced (with necessary changes being made) in subsequent years. Of course, the formula-based approach is not perfect, and much of this report is concerned with identifying and dealing with problems that arise in the design and implementation of formula-based programs.

The crucial concept in the previous paragraph—that allocations are based on some principles or rationale—is critical to one’s view of formula allocation programs. Two extreme views of formula allocation programs can be contrasted. At one extreme is the view that underlying any formula allocation program there exists a clear set of principles that define how funds should be allocated. For example, the funds a jurisdiction receives should rise with the size of its program caseload, fall with its capacity to raise revenue itself, and perhaps rise with the relative cost of providing the program in that jurisdiction. Even with such principles defined, there is still plenty of room to negotiate formulas in terms of, for example, how exactly caseload, capacity, and cost are to be measured, what weights they are to be given, and how steeply funds should rise or fall with each of these measures. But the existence of such principles or objectives at least provides criteria for assessing how well a given formula program performs, as well as for judging the value of proposed improvements to a formula.

At the other extreme is the view that program allocations are purely the result of political negotiations, that political trade-offs may be made across unrelated programs and issues, and that, while principles may have under-

lined initial negotiating positions, these will not be recognizable in final allocations. Under this scenario, the use of a formula that yields the negotiated results becomes little more than cover to provide pseudorationality for the results of such a process. No basis for judging improvements to the formula exists under this scenario.

Each of these views is extreme, and reality undoubtedly occupies a middle ground. But, when designing, assessing, or revising formula allocation programs, it is important to recognize the conflicting pressures produced by these alternative views of the role of formulas in the allocation of funds.

Paralleling this range of views on how formula allocations come about are various approaches to it. In some cases, the dollar amounts (or shares) to be provided to each jurisdiction are specified in the legislation itself with no pretense of a formula. For example, several years ago in the State Capitalization Grants Program of the Environmental Protection Agency, a formula based on a periodic survey of clean water needs was replaced by legislated shares. In the majority of formula allocation programs, the formula is specified in legislation and leaves little or no discretion to the program administrators to influence the method of calculation. For some of these programs, for example, the State Children’s Health Insurance Program (SCHIP), the data sources for each formula element are specified in precise detail in the legislation, leaving the program agency with even less discretion to decide how the formula inputs should be derived. For others, such as Title I education, the program agency has been given substantial leeway in developing estimates of formula components. In a few cases, only the program goals are included in legislation and the formula is developed by the administering program agency. For example, for the food grant portion of the Special Supplemental Nutrition Program for Women, Infants, and Children (the WIC program), Congress established the general objectives in legislation and left the program agency, the U.S. Department of Agriculture’s Food and Nutrition Service, to develop the allocation formula, which is set out in regulations published by the agency.

Several key players have important roles in the development and administration of formula allocation programs. Congress, as the legislative

authority, plays a determining role. Funding through formula allocations represents just one of several methods of funding programs. The funding decisions of Congress require the agreement of its members, who represent a variety of different constituents and interests. They may also require negotiations with the presidential administration. Therefore, the funding decisions that emerge from Congress, including any specific formulas and allocation procedures, generally represent compromises between divergent interests within a general agreement to act in a particular problem area.

Program agencies are required to administer programs within the prescribed legislation. The discretion they are given varies from program to program. Typically, many of the decisions on which estimates or quantities to use in formulas will be theirs, but the formulas themselves will not. Agencies often play an important role in formulating the rules and regulations that govern how funds are to be used by recipient jurisdictions. They have a responsibility to monitor programs and identify whether they are achieving the objectives implied in legislation.

The first-level recipients of most programs are the states and territories. They can be expected to exert pressure on program agencies to maximize their gain from formula programs. In many cases, the state government is not the ultimate spender of funds but an intermediary between the federal government and the local government. In that role, the state will sometimes have its own formula allocation program for distributing state funds to localities—counties, cities, school districts, etc. (see Chapter 7).

Two other types of organization also influence this process. National, state, and local advocacy groups with an interest in the problem being addressed by a formula allocation program will take a close interest in the operation and success of a program and may be a significant voice in promoting change or expansion of programs. Statistical agencies play a key role in providing the statistical estimates required by formulas, and in some cases in developing new data collection programs to meet the legislated needs of programs.

Table 1-1 provides a list of the 11 largest formula allocation programs operating in FY 1999 and 2000, with the amounts of funds obligated in each of those years. The dominant role of Medicaid in financial terms is evident. A further elaboration of a selection of key programs, with a discussion of some of their distinguishing features, is provided in Appendix B.

At the state level, large amounts of state revenues are transferred to counties, cities, school districts, and other jurisdictions through a variety of formula allocation programs. Some states have general revenue sharing programs, in which uses by the recipients are unrestricted. State programs are especially significant in the field of public education, in which state funds cover nearly half of all expenditures, compared with a less than 10 percent share provided by Title I education and other federal programs (National Research Council, 1999a:Table 2).

The use of formulas to allocate funds is not restricted to the United States, although to our knowledge no country has quite the variety of separate formula allocation programs of the United States. The federal governments of both Canada and Australia operate equalization programs for their respective provinces or states that aim to ensure that residents of poorer jurisdictions can receive common basic services at comparable rates of taxation. These programs are similar in intent to the General Revenue Sharing Program that operated in the United States between 1972 and 1986. In the United Kingdom, large amounts of money (84 billion pounds, or $126 billion, by one estimate) are distributed annually by the national government to local areas (Derbyshire, 2001). The favored approach to distributing public funds is increasingly to place more weight on objective means of allocation through mathematical formulas that model expenditure needs (Smith et al., 2001). Although some formulas are rudimentary, the general tendency has been toward ever-greater intricacy. Attempts to simplify formulas have not met with much success, as the demands for sensitivity to local conditions (and therefore complexity) seem to dominate.

International organizations face a similar allocation problem but in reverse—how to calculate the contributions that individual member countries should provide toward the organization’s financing. The United Nations, for example, operates a formula-based assessment system based primarily on gross national product, but taking into account debt burden, per capita income, a ceiling for the least developed countries, and other adjustments.

The use of formulas for allocating funds is pervasive and increasing worldwide. On one hand, the appeal of a formula approach is its apparent objectivity and transparency and some separation of funding decisions from political influence once the formula has been agreed on. On the other hand, the inevitable political negotiations that must precede agreement to a formula can result in one that deviates significantly in its impact from a formula that might appear fair and rational to a disinterested bystander.

Building on a workshop held in Spring 2000, the Committee on National Statistics (CNSTAT) formed the panel to study formulas used to allocate federal funds to states and localities. The purpose of this panel study was not to recommend changes in existing formulas, new formulas, or the use of particular data sets. Rather, through its deliberations and commissioned papers, the panel was to assess and illustrate the interplay among statistical estimates used as inputs, formula features, and the legislative process, with specific attention to attainment of program goals and the objectives of the allocation process. The charge to the panel stated:

The panel started with a primarily technical focus but soon became aware that meaningful recommendations for improvements in the structure of formulas and inputs to formula allocation processes would have to take full account of the policy and legislative contexts in which these processes are developed. The panel’s thinking evolved in other ways as well. The charge to the panel made no mention of various special features of the allocation process (such as hold-harmless provisions or eligibility thresholds) and how they interact over time with basic formulas and their data inputs. However, it became evident that failure to consider such interactions can

lead to unintended consequences and that this was a topic that deserved careful attention.

At the end of its first year of study, the panel issued a preliminary report of its findings and analyses, which included case studies of education programs. This final report explores the broad statistical issues, embedded in a political and policy framework, that arise in the development and use of formulas for allocating federal funds to state and local governments. The report identifies specific statistical and methodological problems associated with the use of formulas for fund allocation, such as those involving estimation, sampling and measurement error, and sensitivity analysis, but it does not seek to characterize them in detail, as most such description would necessarily be highly program specific. This type of description can be found in the literature; for example, see National Research Council (1999b and 2000) for a discussion of the statistical issues relating to Title I formula allocations.

As part of its work, the panel commissioned a set of papers that provide further support for its conclusions and recommendations by dissecting the technical underpinnings of formulas, as well as the practical application issues. The papers appear in the September 2002 issue of the Journal of Official Statistics, which should be viewed as a companion volume to this report. The first group of papers lays out how the formula allocation process works—the authors examine the goals and underlying structures of fund allocation formulas, describe the legislative development process, and explore how formula features, data, and estimation procedures interact in producing formula outputs. These papers are followed by several case studies that serve to illustrate many of the issues raised in the first three papers. The case studies are drawn from U.S. programs in the areas of children’s health, women’s and children’s nutrition, and education; from a Canadian program designed to reduce discrepancies in the fiscal capacities of the provinces; and from the United Nations’ dues assessment procedures.

In this final report, Chapter 2 reviews objectives of funding aid to subordinate jurisdictions, such as closing the gap between need and effort, treating economic equals equally, redistributing economic well-being, and encouraging spending on certain services. Implications of these goals for the structure of aid formulas are discussed and illustrated with some examples.

Chapters 3 through 6 provide a more detailed treatment of features that are common to most formula allocation programs, using numerous examples from U.S. federal and other aid programs. Chapter 3 outlines the

broad context in which the programs operate. Who are the initial recipients and what are the program rules that govern their use and further distribution of funds? What is the timing and frequency of disbursements? What provisions are made to cover the administrative costs incurred by federal and state program agencies?

Chapter 4 identifies and provides examples of components frequently included in allocation formulas: measures of need (including geographic cost differentials), measures of the fiscal capacity of recipient governments, and measures of effort by those governments. The final section provides a model for combining those elements and discusses how errors in the input data can affect the allocations.

Chapter 5 identifies and gives examples of special features that are frequently part of the allocation process and interact with the basic formula and data inputs in determining the final results. The chapter discusses eligibility thresholds, limits, hold-harmless provisions, caps, moving averages, step functions, bonuses, and penalties.

Chapter 6 looks at sources of data for estimating formula elements. Sources are sometimes specified in legislation and sometimes determined by program agencies. The chapter discusses the advantages and disadvantages of each of the major sources used: the decennial census, current population estimates, household surveys, other statistical programs, and administrative records from sources like the Internal Revenue Service and the Food Stamp Program. Choices among these sources are influenced by both data quality and cost considerations.

In Chapters 7 and 8 the focus shifts from U.S. federal aid programs to those operated by the State of California, several foreign countries, and the United Nations. Chapter 7 begins with a brief discussion of how funds received under a number of federal formula allocation programs are managed and used by various agencies of the California state government. It then describes several programs that distribute state revenues to Californian cities, counties, and other local governments. Chapter 8 describes revenue-sharing programs in Canada, Australia, and the United Kingdom. It also describes the formula-based dues assessment system used by the United Nations to finance its operations. Although the UN system collects rather than distributes money, it is found to have many of the same features as the aid programs that are the main focus of this study.

Chapter 9 presents the panel’s conclusions and recommendations.

This report has six appendices. Appendix A briefly describes the papers that were commissioned by the panel. Appendix B summarizes findings

from a review of 12 federal formula allocation programs, including the 10 programs with the largest obligations for FY 1999 and two others with features of special interest. Appendix C describes the main sources of data about formula allocation programs that were used by the panel to obtain background information for this study. Appendix D outlines the contents of a proposed handbook on fund allocation formulas and processes. Appendix E lists participants in the panel’s meetings and the April 2000 Workshop on Formulas for Allocating Program Funds. Appendix F provides biographical sketches of members of the panel and staff.