Full text of DOI:10.1177/109114210203000505

PUBLICUsFhIeNrA/ NPRCOEVRIESVIOIENWOF INDIVISIBLE PRIVATE GOODS
Public provision of private goods is usually modeled as the displacement of a market
where people can buy as much or as little of a good as they please, with tax-financed pro-
vision by the government of equal and identical amounts of the good per person. But pub-
lic provision is not always equal per person because some goods cannot be supplied
equally or because the government does not choose to supply goods equally. There may
be fewer organs available for transplant than there are people who need them. The best
education and medical care are unavoidably rationed because teachers and doctors dif-
fer in skill or dedication. Total public expenditure may be insufficient to provide for ev-
erybody when there is a lower limit to useful expenditure per person.
PUBLIC PROVISION OF INDIVISIBLE
PRIVATE GOODS IN SHORT SUPPLY
DAN USHER
Queen’s University
It takes months to get an appointment with a specialist and months to
get badly needed surgery for hip replacement or cataracts—unless one
has enough personal contacts to climb up the waiting list. (The exis-
tence of a “nomenclatura” that gets preferential treatment is the best-
kept secret of our celebrated medical care.)
—Lysiane Gagnon (1998, D3)
Two people are traveling in the desert with enough water for one of
them to survive the journey but not both. Who gets the water?
This ancient conundrum has its counterparts in contemporary pub-
lic finance:
Long ago at Manchester University, a cleaning lady who was one of Man-
chester’s little people (for you could guess a person’s social class by his
height in those days) complained to me that her family had been wait-
AUTHOR’S NOTE: With thanks for helpful suggestions to my colleagues Frank Lewis and Filip
Palda, to Larry Solomon at the Energy Probe Foundation, and to Carolyn DeCosta at the Mani-
toba Centre for Health Policy and Evaluation.
PUBLIC FINANCE REVIEW, Vol. 30 No. 5, September 2002 385-415
© 2002 Sage Publications
385

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ing for years to be assigned public housing, though other people with
no better claim than they had been assigned public housing already.
Today in Ontario, a child in elementary school has a significant chance of
sharing the class with children who are mentally ill and who take up a
disproportionate amount of the teacher’s time, lowering the quality of
education. A parent can avoid this risk by enrolling his or her child in
Frenchemersion or in an alternative schoolwith restricted enrollment.
Women about to give birth are sometimes flown from Toronto to obstetric
wards in Kingston when Toronto hospitals are especially overcrowded.
This is not a pleasant experience for the women involved, especially as
they are often put on the bus back home a fewhours after their children
are born. It has been alleged that a disproportionate number of such
women are Black.
A hospital in Ottawa treated senior politicians, senior bureaucrats, top sol-
diers, and nobody else.
Some time ago, an aged and distinguished politician received a heart
transplant, although similar transplants are usually denied to people of
his age and state of health.
Recently, a woman died after waiting for 20 hours in the emergency ward
of a hospital. Not everybody waits that long. By what process was it de-
termined that she should do so, and could her life have been saved if
some influential person had intervened on her behalf? It is a fair bet that
the woman was not a doctor, a dependent of a doctor, or a premier of the
province.
Many Queen’s students believe that the allocation of loans to students by
the Ontario government is arbitrary and unfair.
These examples have a common theme: It is exceedingly difficult
for the government to allocate indivisible private goods when there is
not enough for everybody. Goods are said to be indivisible when they
are only available or useful in lumps of a certain size. Kidneys avail-
able for transplant are indivisible in this sense of the term. If there are
only three kidneys available for transplant and there are six people
who need them, then three people must do without, for half a kidney is
no use at all. There may be fewer places in university than qualified
students to fill them. Almost by definition, there are fewer highly qual-
ified surgeons than patients wanting their services. Indivisibilities are
not especially problematic in the private sector, where the price mech-

Usher / PROVISION OF INDIVISIBLE PRIVATE GOODS
387
anism automatically allocates the available supply to whomever is
prepared to pay the most for access. Indivisibilities become problem-
atic in the public sector.
The crux of the problem is that the preferred principle of public-
sector allocation—that everybody be treated alike—may be inopera-
tive because it is in the very nature of certain goods that they cannot be
allocated equally. The preceding examples suggest not just that we
sometimes deviate from equality—falling short of our ideal as people
always do—but that there are times when our ideal of equality cannot
be realized for technical rather than moral reasons. The problem arises
in the allocation of places in old people’s homes, medical care, access
to education, public housing, and a variety of other goods supplied by
the government.
We think of people in a capitalist democracy as having two types of
rights: property rights, which are intrinsically unequal, and political
rights, such as the rights to vote and protection under the law, which
are intrinsically equal. When a good or service is supplied in the pri-
vate sector, it is expected to be allocated unequally according to what
people are willing or can afford to buy. When a good or service is sup-
plied in the public sector, it is expected that everybody’s right to the
good is the same. When governments deliberately depart from equal
allocation, we say they are unjust. But governments may sometimes
allocate unequally because they cannot do otherwise. Some goods
must, by their very nature, be allocated on different principles. The
preceding examples are of instances when the allocation goes sour.
This article is an attempt to formalize the problem with particular ref-
erence to medical care.
Equality of public provision is less straightforward than one might
suppose:Equality may be of inputor output. In allocating funds for the
police among the different localities, a central government might sup-
ply equal expenditure per person or attempt to equalize crime rates by
providing extra funds to relatively crime-ridden localities. Equality
may be ex post or ex ante. Education would be equal ex post if expen-
diture on every child—clever or not, disabled or in good health—were
the same. Education would be the equal ex ante if entry to limited
places in university were allocated by competitive examination in cir-

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cumstances where the wealth or social class of the parents played no
role whatsoever. This is a slippery distinction because even ex ante
equality is at most the disregarding of some circumstances of candi-
dates for limited places and because wealth can influence a candi-
date’s preparation. Equality may be of offer or supply. We would say
that public housing is offered equally if accommodation of a given
standard is made available to everybody, even though the wealthier
part of the population prefers to acquire better accommodation on the
open market. Nor is it obvious what the domain of equality should be.
Public funds devoted to the treatment of prostate cancer would be allo-
cated equally if the same rules applied to everybody with the disease,
but women would not be treated equally with men if a disproportion-
ate amount of medical funds were devoted to prostate cancer rather
than breast cancer. The distinction between equality of input and
equality of output may be important in this context because research in
one disease may be much more promising in one disease than in
another.1
Reasons for public provision—some good, some bad—are not the
subject of this article. For instance, medical care may be socialized to
avoid the “Samaritan’s dilemma,” in which a person neglects to care
for himself or herself or to provide for the risk of illness, knowing that
others will not allow him or her to die for want of medical care, no mat-
ter howimprovident the ill person may have been; or to avoid the con-
siderable overhead cost of private medical insurance; or in fear that
insurance companies and health maintenance organizations may let
sick people die when medical care becomes especially expensive; or
because universal insurance is blocked by too much readily available
information on people’s health status when they apply for insurance.
Education might be in the public sector because each of the present
generation of adults is concerned not just about their own children but
about the entire future generation. Regardless, the starting point of this
article is that, for whatever reasons, indivisible goods are provided by
thegovernment. The question ishowand to whomthey are supplied.

Usher / PROVISION OF INDIVISIBLE PRIVATE GOODS
389
INDIVISIBLE GOODS DEFINED
An indivisible good is available altogether in a lump or not at all. It
does not admit more or less. You cannot have half a kidney transplant
as you can have half a loaf of bread or half a glass of water. People may
differ in their valuations of indivisible goods. Some would give up a
great deal for a kidney transplant; others would give up a great deal to
avoid one. The essential feature of indivisible goods is that the avail-
able supply cannot be divided up equally among would-be recipients
unless there is enough to go around. Otherwise, some people acquire
the good, and others have to do without.
Indivisible goods can be seen as one of many types of goods that
economists have come to recognize once Samuelson opened the field
with his articles on public goods. At first, we recognized only private
goods and public goods, as though all goods in the world could be
classified as one or the other. Deluded, perhaps, by the connotations of
words, we were inclined to suppose that public goods were what the
government supplies. In introducing club goods, Buchanan (1965)
was one of the first to recognize the existence of other types of goods
in between, with other implications about the role of government in
the economy. Since then, there has been a profusion of newtypes of
goods: congested public goods, shared goods (such as the police force,
with total cost dependent on the number of beneficiaries but with iden-
tical benefits to everybody in the relevant catchment area), personal
goods, privately provided public goods, publicly provided private goods,
preclusive goods, and so on. Indivisible goods are one more type.
Indivisible goods may be uniform or variegated. Uniform indivisi-
ble goods, exemplified by kidney transplants, appear in the utility
function as either 1 (if you have it) or 0 (if you do not).
ui = ui(xi; δ),
(1)
where ui is theutility of person i, xi isa vector of his or her consumption
of all other goods, and δ is either 1 or 0 depending on whether person i
does or does not acquire the indivisible good. People vary in their eval-
uation of the indivisible good, and some people may not want it at all.
Person i wants the indivisible good if and only if

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ui(xi; 1) > ui(xi; 0).
(2)
Variegated indivisible goods, exemplified by doctors who can be or-
dered from best to worst, are represented in the utility function as
numbers. The utility function of person i becomes
ui = ui(xi; q),
(3)
where q is the quality of medical care, graduated so that q = 0 signifies
either no medical care or a quality of medical care so poor that one
would be just as well off without it. Medical care becomes a variegated
indivisible good when, for example, there are a total of D doctors, each
doctor can service r people, and doctors differ in skill. With s distinct
categories of medical skill, there are ds doctors supplying medical care
of top quality, qs; ds – 1 doctors supplying medical care of somewhat
lower quality, qs – 1; and so on. There may or may not be enough doctors
to go around; that is, rD may or may not be greater than P (total popu-
lation). Regardless, the superior-quality doctors must somehowbe as-
signed to patients through the market or in some other way.
In a competitive market, indivisible goods are allocated by prices.
The price of a uniform indivisible good is the lowest of the reservation
prices of the people who acquire the good, where each person’s reser-
vation price is the most he or she would be prepared to pay to acquire
the good rather than to do without it. Suppose K kidney transplants are
available for a population of P people who need them. Suppose P > K,
so that some people must be denied the indivisible good. Let Ri be the
reservation price of the ith person ordered by reservation prices so
that, forall i, Ri > Ri + 1. The first K people gettransplants. The restof the
population does not. The market price of a kidney transplant would be
RK. Everybody except the Kth purchaser enjoys a surplus, the differ-
ence between his or her reservation price and the market price, RK.
The allocation of variegated indivisible goods is similar but more
complex. With D doctors to serve P would-be patients, and when each
doctor can service r patients, the market must churn up a fee schedule
f(qj), where j refers to the jth level of skill, so that the earnings of a doc-
tor at the jth level of skill becomes rf(qj). The fees—f(qj) per patient for
a doctor with skill j—must have the property that each patient is satis-
fied with his or her doctor. The patient of a doctor with skill j would be

Usher / PROVISION OF INDIVISIBLE PRIVATE GOODS 391
unwilling to pay the extra f(j + 1) – f(j) for the services of a doctor of
skill j + 1 or to become a patient of a doctor with skill j – 1 for a saving
of f(j) – f(j – 1). In equilibrium, some would-be patients may have no
doctor, and there may be some doctors with no patients—incompetent
doctors for whom f is negative. The skill bands can be wide or narrow.
At one extreme, all doctors are equally skilled, but there are not
enough to go around. At the other extreme, each and every doctor can
be ranked according to his or her skill. In the examination of public
provision to follow, we flip back and forth between specifications of
indivisible goods, assuming them to be uniform or variegated depend-
ing on which is the more appropriate for the purpose at hand.2
METHODS OF ALLOCATING INDIVISIBLE GOODS
Nine methods are to be discussed and compared: sale, merit, cost-
benefit rules, random allocation, extra billing, a private fringe, queu-
ing, rent seeking, and influence.
SALE
The public sector could replicate the private sector. It does pre-
cisely that in the allocation of tickets for trips on the state-owned rail-
way or access to toll roads. Publicly administered medical care could
also be allocated by sale, but with that allocation, it would make little
sense to supply medical care in the public sector at all. We shall never-
theless use medical care as an example to trace out the implications of
this form of public allocation. Assume that medical services are uni-
form rather than variegated but that there are simply not enough ser-
vices to go around, so that some people must do without. Imagine a so-
ciety of identical people whose tastes are represented by the common
utility function
(4)
u(x; δ) = ln(x) + δ,
which is the same as Equation (1) above except that there is only one
private good, x, and the utility function is separable in x and δ. As
above, the value of δ is 1 if one is allocated medical services, and it is 0

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if one is not. Less than one unit is useless, and any excess over and
above one unit is useless as well. Each person is endowed with an in-
come y, varying uniformly from a lowof $10,000 to a high of
$190,000, so that the median income is $100,000. The key assumption
here is that there is only enough of the indivisible good for half the
population.
Suppose initially that medical services are privately supplied in a
market, where the price of the ordinary private good is 1 and the fee for
medical services, f, must be just high enough to deter half the popula-
tion from purchasing medical services. A person’s consumption, x, of
the ordinary private good is either y or y – f, depending on whether one
chooses to purchase a unit of the indivisible good.
If the indivisible good were provided in a competitive market, its
market-clearing price would allocate the available supply so that ev-
erybody who wants to buy access at that price may do so, with nothing
left over. As an intermediate step, define fR(y) as the reservation price
of the indivisible good for a person with an income of y, the price at
which he or she is indifferent between acquiring and not acquiring a
unit of the indivisible good. Indifference implies that
ln(y – f R(y)) + 1 = ln(y)
(5)
or, equivalently,
f R(y) = y(1 – e–1),
(6)
which is a simple multiple of y.
A person acquires medical services if and only if his or her reserva-
tion price exceeds the market price. Because the reservation price in-
creases with income, the available medical services are purchased by
the richer half of the population. To clear the market, the price of medi-
cal services must equal the reservation price of the person with the me-
dian income. With incomes varying uniformly from $10,000 to
$190,000, people’s reservation prices vary from a lowof $6,321
[f R ($10,000)] in accordance with Equation [6] to a high of $120,103
[fR ($190,000)]. The market-clearing price is the median reservation
price, which turns out to be $63,212 [fR ($100,000)].

Usher / PROVISION OF INDIVISIBLE PRIVATE GOODS
393
Public provision by sale would be identical to private provision in
an ordinary competitive market if the government purchased medical
services at what would have been the market-clearing price. The out-
come differs if the good is acquired by the government—like blood
donations—free or at less than the market-clearing price under private
provision. Consider a government that has already acquired enough
medical services to supply half the population and that cannot acquire
any more. With a fee, f, paid to the government and with only half
enough medical care to go around (so that revenue per head from the
sale of medical care becomes f/2), each person’s gross income before
the purchase of medical care is augmented by whatever portion he or
she is assigned of the total revenue from the sale of medical care. Sup-
pose, for simplicity, that revenue from medical care is allocated
among all citizens equally, raising everybody’s income from its initial
level, y, to y + f/2. One’s consumption of other goods becomes y – f/2 if
one purchases medical care or y + f/2 if one does not. Once again, the
market-clearing price of medical care must be the reservation price of
the person with the median income, where the median income and the
average income are the same because the distribution of income has
been assumed to be uniform. The reservation price, fR, of a patient
with income y is identified by
ln(y – f R/2) + 1 = ln(y + f R/2).
(7)
A simple manipulation of Equation (7) reveals that with a median in-
come of $100,000, the market-clearing price of medical care has to be
$92,423, somewhat higher than with out-and-out private provision be-
cause everybody’s income is augmented by the revenue from the sale
of medical care.
Note particularly that the poorest of the would-be patients are actu-
ally better off when medical care is sold for whatever the market will
bear than they would be if they had received medical care “free” from
the government. The revenue from the sale of medical care finances a
transfer per person of $46,212. A person is better off with the transfer
than he or she would be with “free” medical care when his or her in-
come is lowenough that
ln(y + 46,212) > ln(y) + 1.
(8)

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The critical income is $26,894. Anybody whose income is less than
$26,894 is better off with his or her share of the proceeds of the sale of
medical care than he or she would be if supplied with medical care
instead.
Of course, this result is entirely contingent on arbitrarily chosen
numbers. The point of the example is that public provision of private
goods, especially when there is not enough to go around, may be a
poor instrument for the promotion of equality among people. The nat-
ural vehicle for redistribution is the simple transfer of money, ideally
through some variant of the negative income tax. The principal justifi-
cation for public provision of private goods must lie elsewhere, in ex-
ternalities or in some impediment to redistribution through the inter-
mediary of money. Otherwise, when goods are allocated equally, the
discrepancy between the values to the rich and to the poor creates an
unnecessary inefficiency that is beneficial neither to the rich nor to the
poor.3
MERIT
The limited supply of publicly provided private goods may be as-
signed to those who “deserve” them, as when scarce places in univer-
sity are assigned to the best students. Merit has at least three distinct
interpretations. It may be a throwback to an aristocratic view of the
world in which some people are simply better than others—people of
good breeding, people of quality, a Nomenklatura of the politically or
the socially well connected. Especially in universities, it would be
rash to assume that such a viewof the world, once commonplace, has
entirely disappeared or has no residual bearing on public decisions.
Alternatively, merit may refer to the effect of allocating the scarce
good on the marginal contribution of the recipient to the welfare,
somehowdefined, of the community as a whole. Everyone may agree
to allocate places in the physics department to students expected to be-
come the best physicists. Fewwould gain from any other principle of
allocation, unfair as it may be to the less competent among us who
bear the cost of education but get little or none themselves. The attrac-
tiveness of this interpretation of merit depends very much on the antic-
ipated relation between the wage and the marginal product of labor. If

Usher / PROVISION OF INDIVISIBLE PRIVATE GOODS
395
full contribution to society is reflected in one’s wage, it might be best
for the government to allocate scarce benefits by sale rather than by
gift. Only when one’s prospective contribution to society is not re-
flected in prospective income—when significant externalities are
anticipated—might allocation by merit as anticipated contribution be
warranted.
A third interpretation of merit is that scarce privileges should ac-
crue to those who will gain the most from them, when gain refers not
to monetary valuation but to some absolute measure of utility: food for
the hungry, housing for the homeless, education in the arts to those
with the keenest sense of beauty. As a criterion for the public sector,
this interpretation of merit has much to be said for it, but it is not within
the subject of this article. This article is about indivisible goods in
short supply. As a criterion for allocation, “food for the hungry” refers
to a world where there is already plenty of food for the rich over and
above what the government has at its disposal to supply.
COST-BENEFIT RULES
A cost-benefit rule is somewhat like merit, but with an emphasis on
evaluation in dollars’worth to the recipients of public projects or pro-
grams. The fundamental principles of cost-benefit analysis—that
benefits of projects be evaluated by the government according the val-
uations of the beneficiaries and that all dollar values of benefits be
weighed equally “to whomsoever they may accrue”—might be ex-
tended from the choice of projects to the choice of beneficiaries of
public provision of indivisible and heterogeneous goods. Indispens-
able for the public provision of roads, bridges, and airports, cost-bene-
fit analysis has its place in public provision of medical care too, but it
is a tool that must be employed with care and subject to important
qualifications.
To see what is at stake, consider once again the simple model of
medical care in which all doctors are equally skilled, but there are not
enough doctors to go around. Suppose medical care is socialized with
a viewto providing the same standard of care for everyone, insofar as
possible. Withthat objective of socialization, the allocation of medical
care by strict adherence to the principles of ordinary cost-benefit anal-

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PUBLIC FINANCE REVIEW
ysis can be dismissed out of hand because the recipients of medical
care would be precisely those rich folks who would purchase medical
care if it were for sale. Public provision would then provide “for free’
what the recipients would otherwise have to buy. The reason is simple
enough. With only half enough doctors to go around and on the as-
sumptions we have made about taste and the distribution of income,
patients’ reservation prices for medical care are proportional to their
incomes: f R(y) = y(1 – e–1) as in Equation (6). A cost-benefit rule
would automatically allocate medical care to people for whom the
benefit, as reflected in their reservation prices, fR(y), is largest. By
comparison with private provision of medical care, the poor have ev-
erything to lose because they pay part of the cost of medical care but
receive none of the benefits. It is difficult to say what the practical im-
plications of this extreme case may be, except perhaps that govern-
ments do not and ought not to carry cost-benefit principles to their log-
ical extreme.
If cost-benefit analysis is to be adopted for medical care at all, it can
only be with reference to a special calculus of pain, suffering, and
death, where costs are assigned to aspects of ill health independently
of wealth-induced variations in people’s willingness to pay. Suppose
reservation prices for hip replacements are R(yi, xi) = yi A(xi), where xi
is a vector representing aspects of a person i’s medical condition and yi
is his or her income. For medical cost-benefit analysis, the assigned
value of person i’s hip replacement for cost-benefit analysis would
have to be something such as YA(xi), where Y is the average income in
the community as a whole. For medical cost-benefit analysis, a life is a
life is a life, regardless of whether one is rich or poor, although other
circumstances, such as whether one is old or young, would have to be
considered.
RANDOM ALLOCATION
With only enough medical care for half the population, equality
ex post is unattainable, but equality ex ante can be attained by random-
ization. Would-be patients could be treated alike by assigning access
to medical care by chance. Names of all patients are put in a hat, half
the names are chosen at random, and only the winners in this lottery

Usher / PROVISION OF INDIVISIBLE PRIVATE GOODS
397
are provided with medical care. Rich and poor are placed on exactly
the same footing.
To assess the consequences of random allocation, consider once
again a society where the government controls a supply of medical
care, where there is exactly half enough medical care to go around,
where initially the available medical care is sold for what the market
will bear, and where the proceeds of the sale are redistributed equally
to everybody, those who buy medical care and those who do not. The
question at hand is, Who gains and who loses from the transition to
random allocation? This broad question breaks down into three more
specific questions: (1) Who gains and who loses ex post from the tran-
sition? (2) Who gains and who loses ex ante from the transition? (3)
Could the rich bribe the poor to give up their entitlement to a chance of
access to medical care under random allocation? These questions will
be discussed in turn.
To deal with the first question, think of random allocation as creat-
ing four groups of people ex post: the lucky rich, the lucky poor, the
unlucky rich, and the unlucky poor. The lucky rich are unambiguously
better off than they would be if medical care were sold because they
are getting free what they would otherwise have to buy. The unlucky
poor are unambiguously worse off because they lose their share of the
proceeds from the sale of medical care. The unlucky rich are also
worse off because their valuation of medical care exceeds their share
of the proceeds from the sale. One might suppose by symmetry that
the lucky poor would be better off too, but that is not always so. The
richer among them are better off, but the poor among them are worse
off because access to medical care is worth less to them than the trans-
fer forgone.
If medical care were sold by the government for what the market
would bear and if the proceeds of the sale were divided equally among
all would-be patients, the market price would have to be the reserva-
tion price of the person with the median income. As shown above, the
market price turns out to be $92,423—providing a transfer from the
government of $46,212 per person—when the incomes of the would-
be patients vary uniformly from $10,000 to $190,000. The transfer is
automatically forgone when the available medical care is allocated at
random.

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To see who among the lucky poor are made worse off by the switch
from sale to random allocation, note that a lucky person is made worse
off under random allocation when his or her income is small enough
that
(9)
ln(y) + 1 ≤ ln(y + f*/2),
where ln(y) + 1 is his or her utility with medical care but no transfer,
ln(y + f*/2) is his or her utility with a transfer but no medical care, and
f*/2 = $46,212. Rearranging Equation (9), we see that a lucky person
is worse off under random allocation when
(10)
y ≤ [f*/2]/[e – 1] = $26,894.
Among the poorest of the poor (those with incomes less than
$26,894), even the lucky patients to whom medical care is provided
free are worse off when the medical care is allocated randomly than
when medical care is sold. Thus, more than half of the population is
better off ex post when medical care is sold in the market: the unlucky
rich, the unlucky poor, and the poorest of the lucky poor with incomes
between $10,000 and $26,894.
The comparison ex ante is somewhat different. Under random allo-
cation, the expected utility of a patient with income y is
½ [ln(y) + 1] + ½ [ln(y)] = ln(y) + ½.
(11)
By contrast, when medical care is sold, a patient’s utility depends on
whether he or she buys medical care, and that, in turn, depends on his
or her income. Let f* be the price of medical care as it would be if med-
ical care were sold and the proceeds redistributed equally, so that f*/2
is at once the net financial gain of someone who does not purchase
medical care and the net financial loss of one who does. If y >
$100,000, the patient buys medical care, and his or her utility becomes
[ln(y – f*/2) + 1]. If y < $100,000, the patient buys no medical care, and
his or her utility becomes [ln(y + f*/2)]. Someone whose income is ex-
actly $100,000 is indifferent between buying medical care and not
buying medical care. For such a person,
[ln(y + f*/2)] = [ln(y – f*/2) + 1]

Usher / PROVISION OF INDIVISIBLE PRIVATE GOODS
399
= ½ [ln(y + f*/2) + ln(y – f*/2)] + ½
= ln[y2 – (f*/2)2]1/2 + ½ < ln(y) + ½,
(12)
indicating that, ex ante, a person whose income is exactly $100,000 is
actually better off under random allocation than he or she would be if
medical care were sold. This is an unusual outcome in which risk
would seem to be preferred to certainty. Because, as assumed in Equa-
tion (4), utility is additive in δ and concave in x, a random provision of
medical care is evaluated at its expected value, but the corresponding
certainty of income under random allocation of medical care is prefer-
able to the choice between y – f*/2 and y + f*/2, depending on whether
medical care is purchased.4
There is a band of incomes around $100,000 within which random
allocation is preferable to sale and outside of which sale is preferable
to random allocation. Upper and lower limits of the band are easily
computed. To determine the lower limit, yL, note that a patient is indif-
ferent between random allocation and sale (together with the transfer
of income when medical care is for sale) if
ln(yL) + ½ = ln(yL + f*/2),
(13)
(14)
(15)
(16)
so that
yL = f*/[2(e1/2 – 1)] = $71,234.
Similarly, the upper limit, yH, is an income such that
ln(yH) + ½ = ln(yH – f*/2) + 1,
so that
yH = (e1/2)f*/[2(e1/2 – 1)] = $117,446.
Thus, ex ante, one is better off with random allocation than if health
care is sold when one’s income is between $71,234 and $117,446. A
patient whose income lies outside these limits would prefer health
care to be sold for what the market will bear. If the patient’s income is
belowthe lower limit, he or she places a higher value on the transfer

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PUBLIC FINANCE REVIEW
than on the chance of “free” medical care. If the patient’s income is
above the upper limit, he or she places a high value on the assurance
that medical care will be provided.
Whether the rich could bribe the poor to relinquish their entitlement
to a chance of acquiring medical care depends critically on howand
about whom the question is posed. Consider a pair of patients with in-
comes equally above and equally belowthe median income, y*, of
$100,000. The assumed uniformity of the income distribution guaran-
tees that the entire population can be divided into such pairs with no-
body left over. The incomes of any such pair are y* + x and y* – x, for
all x greater than zero and less than $90,000. Nowimagine a random
allocation slightly different from what we have postulated so far. Sup-
pose health care is provided to one randomly chosen person in each
pair. The question at hand is whether, before it is known who will re-
ceive medical care, a mutually advantageous deal can be struck in
which the richer party pays an amount P(x) to the poor party to give up
his or her right to a chance at medical care.
Their expected utilities under random allocation are ln(y* + x) + ½
and ln(y* – x) + ½. After payment of P(x) from the richer to the poorer
party in return for giving up the entitlement to a chance at health care,
their utilities become ln(y* + x – P(x)) + 1 and ln(y* – x + P(x)). The
question becomes whether there is any range of P(x) for which both
parties are better off. The poor person is better off if
ln(y* – x + P(x)) > ln(y* – x) + ½.
(17)
(18)
(19)
(20)
The rich person is better off if
ln(y* + x – P(x)) + 1 > ln(y* + x) + ½.
From these inequalities, it follows that
[P(x)/(y* – x)] > (e1/2 – 1) = .649
and
[P(x)/(y* + x)] < (1 – 1/e1/2) = .393.

Usher / PROVISION OF INDIVISIBLE PRIVATE GOODS
401
These conditions will hold simultaneously when x is large but not
when x is small. Equations (19) and (20) are inconsistent when x = 0;
no deal is possible in that case. But if x = $50,000, then y* – x =
$50,000 and y* + x = $150, so that any P(50,000) between $32,450
and $58,950 would make both parties better off. Whether a more com-
plex deal makes every single person better off than he or she would be
under random allocation depends on the range of disparities of in-
come. The wider the initial disparity, the more likely is such a deal to
be feasible.
EXTRA BILLING
Little has been said so far about the remuneration of doctors. The
government was assumed to have acquired a supply of doctors suffi-
cient to provide medical care to half the population. The government’s
pay scale might reflect skill, or it might not. If not, doctors’skill rents
would, in effect, be shared among all doctors. But skill rents might re-
vert totheskilled doctors themselves undera systemof extra billing.
In practice, the term extra billing is a catchall term covering a vari-
ety of methods of compensation. The method examined here is simple
and uncompromising to emphasize a particular aspect of extra billing.
Suppose that doctors’ skills vary uniformly over some range, that the
patient-doctor ratio is invariant but there are not enough doctors to go
around. Suppose also that all doctors are paid a stipend by the govern-
ment, but that extra billing is permitted.
It is obvious what happens. In the short run, with a given supply of
doctors of different skills, extra billing replicates market-determined
fees in the allocation of medical care. Recipients of medical care are
precisely those people who would receive medical care under private
provision, and the assignment of skilled doctors remains exactly the
same. Poor patientsstillreceive no medical care. The bestdoctors con-
tinue to treat the wealthier patients. Having no role to play in the allo-
cation of scarce resources in medical care, a publicly supplied stipend
to doctors becomes like a transfer of income from all patients to all
doctors. As a device for equalizing the provision of medical care or for
shifting skill rentsfromdoctors to patients, public provision with extra
billing is a complete failure.

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PUBLIC FINANCE REVIEW
Governments are unlikely to be as shortsighted in designing a pro-
gram of public provision of medical care as this analysis would sug-
gest. Doctors may be paid per patient or per unit of medical service.
Limits may be imposed on the amount of extra billing. Nevertheless, a
residue of our extreme example may persist under more sophisticated
methods of medical care. Extra billing may not be a good idea.
PUBLIC PROVISION WITH A PRIVATE FRINGE
Normally, public provision drives out private provision because no-
body would willingly pay for what the government provides free of
charge. A private sector where patients pay for medical care can only
coexist side by side with a public sector where medical care is pro-
vided free when private care provides something over and above what
is provided in the public sector. Medical care may be a mix of services,
only some of which are supplied free in the public sector. Here the fea-
ture differentiating private from public medical care is the quality of
medical care as a reflection of doctors’skill. Medical care is looked on
as a variegated indivisible good. Suppose that every doctor serves a
fixed number of patients, that there are just enough doctors to go
around, and that doctors’ skill, represented by the quality of medical
service q, varies uniformly from a minimum of qmin to a maximum of
qmax. Coexistence of public and private sectors is only possible when
better doctors work in the private sector and worse doctors work in the
public sector. That, in turn, requires all doctors in the private fringe to
earn more than or at least as much as any doctor in the public sector
when doctors are free to work in one sector or the other.
Suppose the government sets these rules for the provision of medi-
cal care: (a) Doctors may choose to work in the public sector for a
fixed stipend regardless of skill, or they may choose to work in the pri-
vate sector for whatever fees their patients are prepared to pay. (b) Pa-
tients may receive “free” medical care in the public sector, or they may
buy medical care in the private sector at whatever the equilibrium
skill-determined fee schedule happens to be. (c) Publicly provided
medical care is financed by proportional income taxation, with no ex-
emption for people acquiring medical care privately. (d) Matching of
patients and doctors in the public sector is entirely random. A patient’s
expected quality of care in the public sector is the average quality of all

Usher / PROVISION OF INDIVISIBLE PRIVATE GOODS
403
doctors in the public sector. Assume for convenience that each doctor
treats one, and only one, patient so that the doctor’s wage and the pa-
tient’s fee in the private sector are the same.
In entirely private medical care, the market would churn up a fee
schedule with higher fees for better doctors. Such a schedule is illus-
trated as a curve on the left-hand side of Figure 1. Doctors’ skill—
represented by the quality of care they provide from a minimum of qmin
to a maximum of qmax—is shown on the horizontal axis, and doctors’
fees is shown on the vertical axis. The market determined “fee-skill”
schedule, f(q), must be upward sloping and might be convex as well.
Equilibrium in the market for medical care requires that very patient
must prefer the fee-quality combination he or she acquires to every
other combination along the fee-quality curve.
In public medical care with a private fringe, the government sets a
standard fee, called f*, for all doctors in the public sector, and all pa-
tients must decide whether to accept “free” medical care or to buy
medical care in the private sector. In response to the government’s
choice of f*, there must emerge a cutoff quality of care, q*, such that
all doctors with skill less than q* work in the public sector, and all doc-
tors with skill greater than q* work in the private sector. The resulting
fee schedule is illustrated on the right-hand side of Figure 1, which is
identical to the left-hand side except for the shape of the fee-skill
schedule, designated by g(q) rather than by f(q) to signify that even
over the range of skill in the private fringe, the two schedules need not
be quite the same. From qmin to q*, the schedule g(q) is flat because all
doctors in the public sector are paid the same. From q* to qmax, the
schedule g(q) rises because patients in the private sector are prepared
to pay higher fees to better doctors.
This arrangement is characterized by two equilibrium conditions,
one pertaining to doctors and the other to patients. First, the fee of the
least skilled doctor in the private sector must equal the fee that the gov-
ernment provides for doctors in the public sector:
g(q*) = f*.
(21)
These must be equal because if g(q*) < f*, the least skilled among the
doctors in the public sector would offer his or her services to the public
sector instead, and if g(q*) > f*, the most skilled doctor in the public

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PUBLIC FINANCE REVIEW
Quality-adjusted
fee for medical
care, g
Quality-adjusted
fee for medical
care, f
Fee-skill
schedule,
g(q)
Fee-skill
schedule f(q)
f*
Quality
of medical
care(q)
qmin
qmax
qmin
q*
qmax
Private provision
Public provision with a
private fringe
Figure 1: Private Provision Compared to Public Provision With a Private Fringe
sector would undercut the fee of the least skilled doctor in the private
sector. Second, the patient using the services of the least skilled doctor
in the private sector must be indifferent between public and private
provision. If a patient opts for the public sector, his or her expected
quality of care is (qmin + q*)/2, and the patient pays nothing for medical
care over and above the tax that must be paid regardless of the choice
between the public system and the private system. If he or she opts for
the private sector, the quality of care is q, and the patient must pay a fee
of g(q), where q is necessarily greater than q*. A patient whose in-
come is y and whose utility function is as shown in Equation (4) is in-
different between public and private care if and only if
ln(y – g(q*)) + q* = ln(y) + (qmin + q*)/2.
(22)
It is immediately evident from Equation (22) that there is some critical
income, y*, for which the equation holds and that a person with an in-
come larger than y* opts for private medical care and a person with an
income smaller than y* opts for the public system.
Although less than fully rigorous, this analysis of public provision
with a private fringe suggests three reasons why it might be inadvis-

Usher / PROVISION OF INDIVISIBLE PRIVATE GOODS
405
able. First, with a given skill distribution of doctors, the private fringe
necessarily attracts the more skillful doctors, automatically lowering
the quality of medical care in public sector. Second, although the gov-
ernment is relieved of the requirement to pay doctors in the private
fringe, the prospect of employment in the private fringe may drive up
the required payment to doctors in the public sector, lest too many
skilled doctors choose to work in the private fringe instead. Third,
there is a political danger in this method of provision. The possibility
of escape from public provision to a private fringe creates an incentive
among the rich to favor cost cutting in public provision, no matter
what the consequences for the quality of care. This could be a serious
concern if the rich are disproportionately influential in public decision
making. A case can be made for forbidding a private fringe to ensure
that the rich and influential must partake of whatever services they
provide for the peasants.5
QUEUING
Strictly speaking, there is no place for queuing in the models in this
article because queuing takes time, and the models, as developed so
far, are atemporal. Queuing can, nevertheless, be contrasted with ran-
dom allocation in a simple example. Suppose 1,000 people require
heart surgery each year, and there is only capacity for 500 operations.
Inevitably, 500 people each year must do without. Assume for the sake
of the argument that all cases are equally grave so that the medical es-
tablishment does not have the option of directing scarce resources to
the worst or most needy cases. With random allocation, names of 500
out of the 1,000 newcases each year are picked out of a hat, operations
are provided for the lucky people whose names have been chosen, and
the rest are left to their fate. With queuing, everybody in need of heart
surgery is placed on a list that must growlong enough for 500 people
to die each year waiting for heart surgery, so that the 1,000 new cases
each year are balanced by 500 surgeries and 500 deaths. For example,
if the common mortality rate of people in need of heart surgery is 20%
per year, there must in equilibrium be 2,500 people on the list (so that
500 people die each year), and the waiting time must be 3½ years.
(The waiting time is t, where 500 = 1,000e–(.2)t.) If waiting is unpleas-
ant, then the misery of waiting must be counted as a cost of queuing

406
PUBLIC FINANCE REVIEW
that could be avoided, for those who ultimately receive operations, by
choosing recipients of the operation by lottery instead. If life expec-
tancyonreceiving the operation aftera delay of 3½ years isreduced by
more than 3½ years, then that loss of life expectancy constitutes an ad-
ditional cost of queuing that could be avoided by immediate random
allocation. Human nature being what it is, discretion on the part of the
medical establishment about who belongs where in the queue is likely
to be employed unfairly, as will be discussed below under the heading
of “Influence.”
RENT SEEKING
Rent seeking is like queuing in some respects, but it differs from
queuing in that resources of patients are directed to influencing the al-
location of public largesse. Think of the government as intending to
allocate medical care by merit, although there really is no solid crite-
rion of merit, and patients are seen by the government as meritorious
according to their efforts or resources deployed in puffing up their
claims. Such rent seeking is pure waste rather than disguised bribery.
Nevertheless, the present context suggests a distinction I have not seen
elsewhere in the rent-seeking literature. Patients may compete for
medical care by expenditure of goods or by expenditure of labor. Both
are wasteful, but they are wasteful in different ways, and the form of
competitionmay determine whether medical care accrues in the end to
the rich or to the poor.
Suppose once again that all doctors are equally skilled but that there
are not enough doctors to go around. Specifically, there are only
enough doctors to supply half of the would-be patients with medical
care. To distinguish rent seeking in labor from rent seeking in goods,
assume that all would-be patients have equal supplies of labor time but
that some patients are more productive than others. Everybody has the
same initial supply of labor, L, which may be used either for produc-
tion of ordinary nonmedical goods and services or for rent seeking to
obtain medical services. Leisure plays no role in this analysis because
everybody’s leisure time is assumed to be the same. Productivity of la-
bor, p, varies uniformly from a minimum of pmin to a maximum of pmax.
A person’s income, y, and his or her consumption, x, of nonmedical

Usher / PROVISION OF INDIVISIBLE PRIVATE GOODS
407
goods depend on his or her productivity of labor, p; rent seeking in la-
bor, R; and rent seeking in goods, r. Specifically,
y = p(L – R) and y = x + r.
(23)
The mode of rent seeking—in labor, goods, or both—is chosen to
maximize utility, which is dependent on the amount of nonmedical
goods and on the quality of medical care.
Two possibilities will be considered in turn:
1. Would-be patients may compete for medical care with goods, in which
case there is established a price in goods.
2. Would-be patients may compete for medical care with labor, in which
case there is established a price in labor.
Competing withgoods . Rent seeking is usually modeled as expen-
diture to influence the allocation by the government of a benefit or
prize that is what it is regardless of anybody’s efforts to attain it (see,
e.g., Tullock 1980). Suppose access to medical care requires a contri-
bution of goods to be burned at the altar of Hippocrates. This method
of allocation is like the purchase of medical care except that every-
body’s net income is lower because the sacrifice makes no contribu-
tion to public revenue, and doctors have to be paid regardless. Each
person’s reservation price for medical care is determined in accor-
dance with Equation (5) above rather than in accordance with Equa-
tion (9) because income devoted to rent seeking is, by definition,
wasted rather than returned in a lump sum to taxpayers. (Remember,
we are assuming that the government has already acquired a stock of
the scarce private good. Had we allowed for the cost of the govern-
ment’s acquisition of the scarce private good, the detail of the example
would have been more complex, but the story would be essentially the
same.) The cost of successful rent seeking rises to the point where half
the population of would-be patients drops out of the race. Access to
medical care is supplied to people who are prepared to pay as much in
rent seeking as they would have to pay if medical care were sold.
Allocation by rent seeking in goods leaves everybody worse off
than if medical care were sold outright: The rich still acquire the avail-
able medical care, the poor still get none, and everybody, rich and poor

408
PUBLIC FINANCE REVIEW
alike, loses a share of the revenue from the sale of medical care. Alter-
natively, if the cost to the government of acquiring medical care is
taken into account, the rich lose because they pay twice for medical
care—once involuntarily through taxation and again voluntarily through
their offerings to the statue of Hippocrates—and the poor lose the tax
they pay to cover their share of the government’s cost of medical care
that accrues in the end to others.
Competing withlabor. Rent seeking nowconsists not of goods sup -
plied to the rent setter but to time devoted to meditation in the presence
of the statue of Hippocrates. When scarce medical care is provided to
those who devote the most time to meditation, there must emerge a
critical meditation time, R*, such that everybody who meditates more
than R* hours gets medical care and everybody who meditates less
gets none. The population must then divide itself into two groups,
those who meditate for R* hours and those who do not meditate at all.
The utility of a person with productivity p who does not choose to
meditate—and forgoes medical care—becomes ln(pL). The utility of
a person with productivity p who chooses to meditate becomes
ln(p(L – R*)) + 1. To determine what R* must be, let R(p) be the reser-
vation price in labor for medical care of a person whose productivity is
p. This reservation price is the maximum amount of time one would
willingly devote to obtaining medical care. For every value of p, the
reservation price is determined implicitly by the equality
ln(pL) = ln(p(L – R(p))) + 1,
(24)
where the left-hand side is one’s utility without medical care, and the
right-hand side is one’s utility with medical care acquired at a rent-
seeking cost of R(p). A little manipulation of Equation (24) reveals
that
R(p) = L(1 – e–1),
(25)
which means that everybody’s reservation price, regardless of produc-
tivity, is the same.
That being so, the equilibrium price, R*, must be the common value
of everybody’s reservation price, and the allocation among people of
medical care must be indeterminate. At a rent-seeking cost of R* units

Usher / PROVISION OF INDIVISIBLE PRIVATE GOODS
409
of labor, everybody is indifferent between acquiring medical care and
not acquiring medical care, and one cannot predict who purchasers of
medical care will turn out to be. The rich place the higher monetary
value on access to medical care as well as on time; there is no predict-
ing which will predominate. The implication in practice is that scarce
access to medical care might be acquired by the rich or by the poor de-
pending on the specifics of the tax system and the shapes of the utility
functions.
Suppose, for example, that the utility function is u = .x + q rather
than u = ln(x) + q. It can then be shown that the reservation price, in la-
bor rather than in money, decreases with productivity, p, so that the
poor, rather than the rich, become the “purchasers” through rent seek-
ing of medical care.6 There is nothing queer or anomalous about this
result. Some public services are occupied by the poor rather than by
the rich when access is by queuing rather than by willingness to pay
cash.
INFLUENCE
In the old Soviet Union, bureaucrats were not especially well paid,
but they were compensated by preferential treatment in the allocation
of goods, including medical care, and by the right to shop at stores
where goods unavailable to ordinary folk could be purchased at mod-
erate prices. The delightful term Nomenklatura was invented to iden-
tify members of the ruling class not by their contribution to society or
even by their authority but by their privileges in an officially egalitar-
ian society. Nothing so blatant can be found in Canada or the United
States, but “everybody knows” that there are privileged people among
us who, by virtue of their influence, get first pick at government lar-
gesse. Admittedly, much that everybody knows is false, and it is diffi-
cult to produce hard evidence on the matter.
***
There is considerable evidence that the poor have worse health and
shorter lives than the rich, but it is hard to sort out cause and effect (see

410
PUBLIC FINANCE REVIEW
Daniels, Kennedy, and Kawachi 2000). Do the poor have shorter lives
because they have worse medical care, because income buys good
health through better food and a healthier style of life, or because cer-
tain characteristics render people less likely to be wealthy and more
likely to be ill? I cannot get to the bottom of this question. The most I
can do is to present a fewscraps of data from two studies, one from
Great Britain and the other from Canada, on variations in life expec-
tancy by social class.
The British data presented in Table 1 are remarkable for the size of
the gaps between occupational classes and because, despite the intro-
duction of the socialized medicine, these gaps appear to have widened
over the period from 1930 to 1970. As shown in Table 1, people are di-
vided into five occupational classes:
I: professional,
II: managerial,
III: skilled manual and nonmanual,
IV: partly skilled,
V: unskilled.
For each class, the table shows the “percentage of the group reporting
illness” and “standardized mortality ratios,” defined as the groups’
percentages of the average mortality rate for the entire population with
corrections for differences among classes in age and gender. For in-
stance, the top right-hand corner of Table 1 shows that, among males
from ages 15 to 65, the mortality rate of professionals is 77% of the av-
erage mortality rate for all occupations. The table shows that class dif-
ferences are large and widening, although it should be borne in mind
that a widening of a percentage difference may coincide with a nar-
rowing of an absolute difference when, as is the case, all mortality is
falling over time.
The Canadian data in Table 2 are for the city of Winnipeg in 1992,
with people classified by income rather than by occupation. The five
social classes are based on average income by neighborhood, a princi-
ple of classification that was, presumably, made necessary by the ab-
sence of direct evidence of health status by income. Among men, there
is a full 10-year gap between the life expectancies of the wealthiest
and the poorest quintiles of the population; among women, the gap is

Usher / PROVISION OF INDIVISIBLE PRIVATE GOODS 411
TABLE 1: Illness and Mortality by Social Class in Great Britain
Percentage
of Group
Reporting
Mortality Rates (national average = 100)
Social Class
Illness, 1976 1930-1932 1949-1953 1959-1963 1970-1972
I: Professional
II: Managerial
III: Skilled manual
IV: Partly skilled
V: Unskilled
17.0
18.7
21.9
25.5
32.1
90
94
97
102
111
86
92
101
104
118
76
81
100
103
143
77
81
104
114
137
SOURCE: Le Grand (1982, Tables 3.2 and 3.3).
NOTE: Some of Le Grand’s categories have been combined or ignored.
only slightly less, 8 years rather than 10. Note also the unfortunate de-
parture from gender equity among the poor, although the mortality ex-
perience of men and women is similar among the rich.
Two questions arise at this point: Who gets to allocate indivisible
goods in the gift of the government? For whose benefit is the alloca-
tion undertaken? Whoever the allocator of public services may be, one
would expect the allocation to be biased in his or her favor to a greater
or lesser extent. Bias could range from relatively innocuous privi-
leges, such as queue jumping for his or her family, to the out-and-out
sale of access to medical care and other publicly provided private
goods. With public provision of medical care, allocation would have
to be undertaken by public officials or by the doctors themselves.
When allocation is the responsibility of the medical profession, one
would expect a doctor who can only care for so many patients and
whose list is closed to the clerk at the local supermarket to extend
“professional courtesy” to other doctors and their families and to ac-
commodate a range of people who he or she finds especially sympa-
thetic: golf buddies, a prominent lawyer in town, or an official of the
Ministry of Health who can influence the remuneration of the medical
profession. When allocation is the responsibility of public servants,
medical services may be allocated in accordance with the interests of
their political masters. First-class medical facilities may appear in
constituencies that support the party in office: “Vote for me or expect
your illnesses to remain untreated.” Alternatively, influence may be
exerted politically by pressure groups that may or may not be wealthy.

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TABLE 2: Life Expectancy in Winnipeg, 1992
Second
Second
Poorest
Quintile
Poorest
Quintile
Third
Quintile
Wealthiest Wealthiest
Quintile
Quintile
Males
Females
65.3
74.4
70.5
77.8
72.8
79.5
74.3
80.0
76.6
82.1
SOURCE: Roos and Mustard (1997, 89-111, Table 2).
Such influence intensifies conflict among social groups and regions of
a country. Join a union and you can expect better medical care not be-
cause you pay for it through your dues but because the union is influ-
ential in the Ministry of Health. Vote for the party in office and you can
expect better doctors and better medical facilities in your constitu-
ency. Call attention to yourself or convey the impression that you have
favors to grant or can make life unpleasant if you are crossed, and you
can expect better medical care at no extra cost. Politics becomes that
much nastier as a win for one’s party in an election becomes entitle-
ment to medical care and for whatever else is being allocated by influ-
ence. A vote for the socialist party becomes a vote to supply medical
care for union members but not for farmers. A vote for the conserva-
tive party becomes a vote to supply medical care for farmers but not
for union members. Political leaders would be well looked after in ei-
thercase. Politics becomes, quite literally, a matter of life and death.
CONCLUSION
One might drawtwo morals from the analysis. The first is that pub -
lic provision of private goods may be a poor vehicle for the promotion
of equality, especially when goods are indivisible. The reason is en-
capsulated in the old saying that “it’s only money,” that money is un-
important by comparison with goods. Take away a dollar, and one au-
tomatically loses a dollar’s worth of the most dispensable bits of the
bundle of goods that one consumes. Take away a dollar’s worth of salt
(on the understanding that the loss cannot be made up by purchase),
and one may be seriously harmed. Public disposition of scarce goods
and services can wreak enormous harm on those left out. Financially

Usher / PROVISION OF INDIVISIBLE PRIVATE GOODS
413
equivalent redistribution from rich to poor may be borne with little in-
convenience by the ultimate benefactors. Redistribution should be un-
dertaken, wherever possible, with money. Public provision of private
goods requires justification in market failure or externalities.
The other moral is that if the government is to provide indivisible
goods, it should strive to ensure that there is enough to go around. That
is not always possible. There may be an unavoidable shortage of kid-
neys available for transplant. Nevertheless, when the government has
some choice in the matter, ordinary market allocation of scarce goods
at one extreme and universal public provision at the other extreme
may both be preferable to poorly funded public provision, with many
omissions and exceptions. Poorly funded socialization gives rise to a
great scramble for privileges in the gift of the government of the day.
Strict uniform provision is the counterpart in expenditure to horizontal
equity in taxation. Divergence from either has similar effects on the
economy and on society in general. Travelers in the desert should
carry enough water for everybody.
In the words of Carl Shoup (1964, 383),
Little is known about the distribution of government services by loca-
tion, race, race income class, or other category. Usually, no record is
made, no estimate attempted. The laws providing the services are silent
in this respect; the authorizing or appropriating committees of legisla-
tures do not discuss it; budgets submitted by the executive say nothing
about howa given service is to be distributed among the users. The si -
lence reflects in part a social propensity to discriminate covertly in
ways that are not tolerable in taxation, where the pattern of impact is
more obvious.
NOTES
1. The natural first assumption about the public provision of private goods is that they would
be allocated equally among all eligible citizens: equal schooling for all children, equal medical
care for all sick people, equal public housing for all who would otherwise be homeless, and so
on. I was typical of the literature in employing that assumption in my earlier piece on public pro-
vision more than 20 years ago (Usher 1977). Epple and Romero (1996a, 1996b) employed that
assumption with some very interesting wrinkles. But even when goods are divisible, the meaning
of equality is not as clear-cut and unambiguous as one might at first suppose, for equality may be

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defined as input or as output, in total or at the margin. The problem was identified by Shoup
(1964) and then developed by Kennett (1982).
2. On markets where goods vary in quality, see Rosen (1986). The market for medical care
of varying quality has been discussed in an earlier version of this article (see Usher 1998).
3. In The Strategy of Equality, Julian Le Grand (1982) reviewed the principal items of public
expenditure on private goods in the United Kingdom, found their redistributive impact to be dis-
appointingly small, and concluded that public provision of private goods is no substitute for the
simple redistribution of money.
4. The argument here is similar to an argument for random allocation of access to university
education by Garratt and Marshall (1994). There is some question about the validity of the argu-
ment under different specifications of utility.
5. What is here called “public provision with a private fringe” is institutionally equivalent to
what Epple and Romano (1996a, 1996b) called a GM (mnemonic for government-market) re-
gime. Their model differs from the model presented here in that their publicly provided good
(also called medical care) is not assumed to be indivisible. It is an ordinary good available in any
amount at a constant rate of trade-off in production with the other good in the economy.
6. If u = x + q, where q equals 0 or 1, then each person’s reservation rent-seeking price for
medical care becomes the solution to
[ p(L − R( p))] +1 = [ pL].
Differentiating this totally with respect to p, we see that
dR
=
[L − R]
[L − R] − [L] / p < 0.
{
}{
}
dp
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