Human Capital Development
HCD
Working Papers
Equitable Allocation of Ceilings on Public
Investment:
A General Formula and a Brazilian Example in the
Health Sector
Philip Musgrove
August 1996
HCDWP 69



Papers in this series are not formal publications of the World Bank. They present preliminary and unpolished results of
analysis that are circulated to encourage discussion and comment; citation and the use of such a paper should take account of its
provisional character. The findings, interpretations, and conclusions expressed in this paper are entirelv those of the author(s)
and should not be attributed in any manner to the World Bank, to its affiliated organizations, or to members of its Board of
Executive Directors or the countries they represent.



Equitable Allocation of Ceilings on Public Investment: a General
Formula and a Brazilian Example in the Health Sector
by
Philip Musgrove






Abstract
Funds for investment (or recurrent expenditure) should often be pre-allocated
among competing entities-territorial units, sub-national governments or others-to give
substantial weight to population but also to recognize greater per capita needs in some
entities than in others. Such ceilings prevent better-prepared entities from acquiring all the
resources, and thereby protect weaker claimants. A general class of simple linear formulas
can be used for this purpose: it minimizes arbitrary adjustments, allows for transparent and
politically acceptable calculations, and guards against unreasonable extreme values. The
procedure is illustrated with allocation of health investment funds among Brazilian states.






Contents
Introduction         ..................,..,,,,,,.....,,,,,,,......,,,,,,,.....,,,..,.,..,,,,.,,.,,.....,.,                                     1
The Formula .,..,,,,..,,,,,..,,...2
The Brazilian Case                                                                   .5
Complications .                                                                                                                             10
Concluding Comment .13






Introduction
Not only are resources always limited, but often there are a number of entities-
territorial units, sub-national governments, ministries, or other potential recipients-
competing for a more-or-less fixed amount to be spent, invested or otherwise applied.
When the market cannot be relied on to allocate the funds; and when it is desired to keep
the distribution equitable in some sense, which often means keeping it partly proportional
to the beneficiary population, it is desirable to 'pre-allocate" the total fund among
entities-that is, to set ceilings on how much each of them can receive, subject perhaps to
subsequent revision in the light of how funds are actually invested or spent. This note sets
out a simple general formula for such pre-allocation and discusses some of the
requirements for its application.
A specific example of a situation calling for pre-allocation of a fixed fund is
provided by the REFORSUS health sector reform and investment project in Brazil,
approved in June 1996. It will finance investments under a decentralized, competitive
arrangement in which states, municipalities and philanthropic providers submit proposals
for sub-projects. This procedure runs the risk that some states, with a greater capacity to
generate proposals, would end up getting too large a share of the funds, leaving too little
for states which were slower to submit proposals. To avoid this, the total fund available
for investment will be pre-allocated among the states, establishing an initial ceiling for each
state. A state will not automatically receive that much money; instead the ceiling will



2
reserve those funds to be invested if there are enough acceptable sub-project proposals to
absorb them. If the ceiling for a particular state is not reached during the life of the
project, despite technical assistance from the Ministry of Health in preparing sub-projects,
the unused funds can be reallocated to other states which still have acceptable proposals
pending.
The Formula
The formula for the state-by-state pre-allocation of investment funds has to meet
a number of conditions which are likely to recur, perhaps in slightly modified form, in
similar situations. The formula must:
.  Allocate not less than half the total investment fund in proportion to
population, in order to be consistent with the rule that at least 50 percent of
recurrent expenditures by the Ministry for the purchase of health care be
allocated by population. More generally, the share to be assigned according to
population-the weight of population in the decision on how to distribute
funds-is one of the major choices to be made in these circumstances.
Allocating part of a fund by head-count is often regarded as a first, crude
approximation to an equitable distribution, to be '<djusted" or 'zorrected" by
some mechanism which accomplishes the second condition.



3
* Take into account, in order to allocate the rest of the funds not strictly
proportional to population, one or more variables associated with greater
needs or justifications for investment, such as health status, poverty, or the
current pattern of expenditure.
*  Be transparent and easy to explain and justify to state and municipal officials,
directors of health facilities, and the public. If only the people who made up
the formula can understand it, it is politically worthless.
The first two of these requirements are satisfied by the general class of formulas
Fe/F = a(Pe/P) + (1-a)(Xe/X), where
e refers to a particular entity. In the Brazilian example, this is a state
(estado).
F is the investment (or other)fund to be pre-allocated
P is population



4
a is a parameter to be chosen, subject to the general restriction that it be
less than 1.0 and greater than zero. In the Brazilian REFORSUS example,
a is further restricted to be greater than or equal to 0.5 (1 > a > 0.5); and
X is an indicator of need for additional or excess investment, beyond that
associated with population alone. This may be a single variable or a
function of several variables. Obviously the construction of X involves all
the allocation choices other than the relative weight of population versus all
other factors.
In order for the formula never to take away from a state or entity, any part of the
ceiling distributed according to population, Xe must be non-negative. A zero value
implies that the state gets no more funds than it is entitled to on the basis of population
alone-that is, it is already a favored state. In addition, in order for the formula to
allocate all of F exactly, it is necessary for the national value of X to be the sum of the Xe
across all the states, just as P is the sum of the state populations, Pe. With this condition,
summing the formula across entities makes it clear that a is exactly the share of the total
fund allocated according to population, and (1-a) is the share allocated according to X.
This restriction has the great advantage that no other arbitrary parameters are
needed in order to '<djust" the shares Fe so that they will add up to the total F which has
to be allocated-any such adjustment would violate the condition that the calculation be



5
simple and transparent. The restriction that the Xe add up to total X also means that X
cannot be a ratio or other non-linear function of whatever variables are used to compose
it. (Ratios can nonetheless be used in the construction of X, as will be shown below.)
The Brazilian Case
Excluding ratios and similar non-linear formulations of X incidentally protects the
calculation against extreme values that could result from seemingly reasonable and
innocuous measures of 'heed". In the Brazilian example, the first attempt to construct an
indicator used the inverse of recurrent expenditure in each state as the value of X, on the
presumption that states where little was spent needed more investment so as to be able to
provide more services and therefore absorb more recurrent expenditure. Total spending
rather than per-capita expenditure was used because it was believed, incorrectly, that the
first term, based only on population, already took the size of the state adequately into
account. The result was to assign to Amapa, an almost empty state with only 0.21 percent
of the country's population, fully 1.47 percent of the investment funds. The ceiling
allocated according to the inverse of expenditure was nearly eight times the amount to
which the state would be '%ntitled" on the basis of population alone, and certainly much
more than was needed or could be absorbed. An inverse formula makes little distinction
among large values of the variable, but explodes as the value approaches zero. And of
course a series of inverses does not sum up to anything meaningful, so the formula
required a constant multiplier to bring the state-by-state ceilings into line with the total



6
allocable funds. Other non-linear functions could also yield unacceptable results at
extreme values.
To satisfy the third requirement, that the formula be comprehensible and
politically acceptable, imposes other, less exact, restrictions on the indicator X. It must be
based on variables that can be measured without ambiguity, and it must not involve any
calculation so complicated that it will appear designed to mystify people or manipulate the
outcome.
The Bank initially proposed that X should be defined by economic, demographic
and epidemiologic variables, particularly the share of people in poverty, the age and sex
composition of the population and the standardized mortality rate. Poor people generally
need more care, and because they cannot afford much privately-financed care, depend
particularly on the Ministry for health services. The demographic variables were expected
to account for the predictable differences, not related to income, in the use of health care,
so tha', for example, a state with more children, or more elderly, than the average for the
Brazilian population would need to provide more care, and therefore should be eligible for
a larger investment ceiling. The standard mortality rate was contemplated as a measure of
health status; the higher the rate, again, the more need for care and therefore for
investment. Ideally, X might be based on a more exact estimate of the burden of disease
and on the cost-effectiveness of medical services, so that the 'heed" in each state would be
a need for investment in services above some threshold of cost-effectiveness, and in



7
proportion to the size of the disease burden or the rate at which it could be reduced by
expanding care. However, the studies to estimate the burden of disease in Brazil and the
costs of interventions (with or without estimating their effectiveness) will only be
undertaken as part of the project; no such information is available at the outset.
In these circumstances, the Ministry of Health of Brazil decided that X should be
based only on recurrent expenditures (labeled G for gasto), on the grounds that:
*    Expenditure is under the Ministry's control: spending money is what the
Ministry does;
*    Expenditure already bears some relation to other variables which might be
used to construct X, such as the prevalence of poverty in the state or the
size and nature of the burden of disease;
*    Tying X too closely to estimates of the care the population needs or should
receive takes no account either of actual demand for services nor of the
capacity to absorb investment funds, and so could yield unrealistic ceilings;
*    Data on expenditure by state are unambiguous and publicly available; and



8
*    Since state officials are always likely to claim that their state does not get
enough funds, what is needed is simply a measure of what is "enough', and
that can better be derived from actual spending than from an estimate of
needs unrelated to expenditure.
Of course, present expenditure will not always be reasonably related to needs,
when allocating funds across entities. However, the reasons for choosing expenditure in
the Brazilian case are likely to be generally valid: whatever variable(s) is used needs to be
unambiguous, easy to measure, and-perhaps--under the control of the institution which
is allocating, or will disburse, the funds. Discussion among the Ministry of Health, the
Inter-American Development Bank and the World Bank ended with agreement on the
following construction of the indicator X:
e    First, per capita expenditure Ge/Pe is calculated in each state. The highest
value is denoted (G/P)m. (m indicates the state where the ratio is
maximal-it does not necessarily mean the state with the highest
expenditure or the highest population. m happens at present to refer to the
state of Parana; per capita expenditure is only slightly lower in Sao Paulo.
Parana and Sao Paulo are also two of the richest states in the country, so it
is equitable for them to receive little or none of the investment funds not
allocated according to population.)



9
*    Next, the ratio (G/P)m is multiplied by Pe, the population of state e: this
creates the variable
*    Gem = (GIP)m(Pe)
which is what expenditure in state e would be if per capita spending there
were as high as in the state with the most expenditure per person-for
example, how much the Ministry would spend in Para if the expenditure
per head in that state were the same as in Parana. The sum of the Gem
over all the states is denoted Gm: it is the amount that would be spent in
all of Brazil if the per capita expenditure were everywhere as high as in
Parana. (It is not assumed, nor does it need to be supposed, that the
country could actually afford to spend that much. The values of Gem
determine the investment ceilings, but are not equal to them.)
*    Then, actual expenditure Ge is subtracted from this estimate of maximal
expenditure to yield
Xe = Gem - Ge
which means that Xe can be interpreted as a kind of 'gap" or shortfall of
expenditure in state e compared to state m. Clearly,



10
Xm = 0 (State m, being the most favored already, has no shortfall to make
up and therefore is not entitled to any more investment funds than are
distributed according to population alone);
Xe > 0 for every other state e (So no state has any money taken away, that
was already assigned according to population); and
The sum of the Xe is equal to X, which is the total shortfall in the whole
country, or the amount of additional expenditure that would be needed in
order to raise per capita spending everywhere to the level in the state
(Parana) where it is highest now.
The final formula is then
Fe/F = a(Pe/P) + (1-a)(Gem - Ge)/(Gm - G)
which is linear in Ge and therefore avoids the risks of non-linearity referred to earlier.
Complications
Several possible and reasonable-sounding complications to this formula were
debated and discarded. For example, the need for medical care is not simply proportional



11
to population but depends on the age (and sex) distribution, so it is arguable that Pe
should somehow be adjusted for these factors. However, this would make the formula for
investment allocation differ from the pure population basis on which part of recurrent
spending is assigned. And since there is no one way to '"djust" for age and sex
distribution, this would cause more argument over the formula and make it harder to
accept politically. Similarly, Ge might be complicated by including state or municipal as
well as federal expenditures. But that would take the variable out of the control of the
Ministry, would make it less transparent, and might even create a perverse incentive for a
state to spend less of its own resources in order to claim a larger share of the investment
funds. None of the possible subtleties or complications to the formula appeared to
improve it sufficiently in technical terms, to justify the consequent ambiguities and political
difficulties. In general, apparent refinements can easily make things worse.
The parameter a is necessarily arbitrary, since the only requirement is that it lie
between 0.5 and 1.0 The Banks and the Ministry of Health agreed that it should fall
somewhere in the middle of that range, so that Xe, the expenditure-related indicator, can
actually have some impact on the allocation of funds, but not so much as to make the
allocation depart greatly from proportionality to population. In the circumstances of
Brazil, a value of a = 0.7 appears satisfactory: 70 percent of the ceiling allocated
according to population and 30 percent according to expenditure shortfall. In general, the
arbitrary nature of the parameter a is not entirely a bad thing, because it focuses the
discussion on the relative importance of population versus everything else, rather than



12
trying to decide on the specific weight of each of a number of variables. Having only one
arbitrary number in the decision rule may be the best situation possible: no arbitrariness at
all is probably infeasible in many situations, whereas having several unrestrained
parameters could make it impossible to reach consensus on the formula or to understand
how it would work.
Table 1 shows the share of population Pe/P which determines 70 percent of the
allocation, the per capita recurrent spending Ge/Pe which determines the other 30 percent,
and the resulting investment ceilings, as shares of the total, Fe/F and in per capita terms,
Fe/Pe. States are in alphabetical order by two-letter abbreviations; shares are shown as
percentages.
The formula does reward the states where per capita federal health expenditure is
low, particularly those in the North (Acre, Amapa, Amazonas, Para) and some in the
Northeast (Bahia, Sergipe) at the expense of the South and Southeast (Rio de Janeiro, Sao
Paulo, Parana, Rio Grande do Sul). But since population weighs heavily in the allocation,
the range of per capita ceilings is only 2.2:1 (from 2.4 to 5.3), and the need-based
allocation never adds more than about half as much as a state would receive on the basis
of population alone. (The share of investment is about 1.5 times the share of population,
for Acre, Amapa, Para, Rondonia and Roraima, all very small states whose populations
have grown by migration faster than health facilities have expanded, so that per capita
spending is low and there is a need for public investment to keep up with demographic



13
growth.) The largest absolute gain from basing part of the allocation on current per capita
spending occurs in Bahia, because it includes a large share of the population (8.12
percent) with much lower expenditure per head than the maximum in Parana (32.46 versus
58.09). In the other large states (Minas Gerais, Rio de Janeiro and Sao Paulo) the per
capita expenditure gap is much smaller and so the impact on investment ceilings is less.
Concluding Comment
It remains to be seen how these proposed ceilings will be accepted politically, and
then whether the states will all be able to propose enough good investment sub-projects to
use up all the funds reserved for them. The final distribution of investments may differ
from this proposal; but the changes will probably not be very large. The formula provides
a simple and transparent basis not only for the calculations shown in the table, but for the
discussion and negotiation that will be integral to the project. The same logic, using
different variables to construct the second term of the formula, can be applied to many
similar situations in which resources need to be assigned in advance on a mixed basis of
population size and some measure of additional needs.



14
Table 1: Distribution of Population, Per Capita Recurrent Federal Health
Expenditure and Investment Ceilings by State in Brazil
Population     Per Capita      Investment       Per Capita
Share         Spending          Share           Ceiling
State          (Pe/P)          (Ge/Pe)          (Fe/F)          (Fe/Pe)
Acre                   0.29           24.46            0.44            5.3
Alagoas                1.72           35.93            2.17            4.3
Amazonas               1.49           29.37            2.08            4.9
Amapa                  0.21           20.94            0.33            5.5
Bahia                  8.12           32.46            11.12           4.8
Ceara                  4.31           43.31            4.60            3.7
Distrito Federal       1.12           52.66            0.93            2.9
Espirito Santo         1.79           38.39            2.28            4.4
Goias                  2.76           48.16            2.71            3.4
Maranhao               3.36           39.29            3.91            4.0
Minas Gerais          10.59           43.92           10.12            3.3
Mato Grosso do         1.23           39.64            1.39            3.9
Sul
Mato Grosso            1.48           38.68            1.72            4.0
Para                   3.50           23.16            5.37            5.3
Paraiba                2.14           43.06            2.28            3.7
Pernambuco             4.78           42.64            5.13            3.7
Piaui                  1.75           42.49            1.87            3.7
Parana                 5.59           58.09            3.91            2.4
Rio de Janeiro         8.53           52.58            7.23            2.9
Rio Grande do          1.66           36.34            2.06            4.3
Norte
Rondonia               0.86           31.51            1.22            4.9
Roraima                0.17           28.94            0.24            5.0
Rio Grande do Sul      6.15           53.08            5.43            3.1
Santa Catarina         3.10          43.14             3.34            3.7
Sergipe                1.03           32.42            1.39            4.7
Sao Paulo             21.63           57.97           15.93            2.6
Tocantins              0.66           33.57            0.82            4.4
Total or Average     100.00           46.15          100.00            3.5



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General Formula and a
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Sector