The Efficiency of Rationing and Resale∗

Yeon-Koo Che† Ian Gale‡

September 12, 2005

Abstract: Housing, health care, and education are often rationed in the

sense that the allocation process involves non-market mechanisms. We study

rationing in a market in which some buyers face binding wealth constraints and

resale is permitted. Buyers with a high willingness-to-pay for a good but a low

ability-to-pay may not acquire it in a free market. Rationing gives these buyers

a chance to acquire the good, possibly at the expense of buyers with a high

willingness-to-pay and high ability-to-pay. Rationing also lets some buyers with

a low willingness-to-pay acquire the good, but the latter buyers will resell the

good to the former. The ultimate allocation with rationing and resale is more

efficient than if the free market had allocated the good. We also show that need-

based rationing rules that favor the poor produce a more efficient allocation

than need-blind rules; need-based rules may even achieve full efficiency. The

crucial role played by resale suggests that its introduction may improve existing

schemes that utilize rationing. Moreover, the introduction of rationing and

resale together may be beneficial for allocating many goods and services.

Keywords: Efficiency, Rationing, Resale, Merit-Based Rules, Need-Based

Rules.

∗The authors thank Susan Athey, Faruk Gul, Billy Jack, Jon Levin, Albert Ma, Paul Milgrom, Bernard

Salanie, Ananth Seshadri, Joel Shapiro, and Andy Skrzypacz for helpful discussions.†Department of Economics, Columbia University and University of Wisconsin-Madison.‡Department of Economics, Georgetown University.

1

1 Introduction

Competitive markets are praised for allocating goods to the individuals who value them

most highly. Despite this, many goods and services are rationed in the sense that the

allocation process involves non-market mechanisms.1 Health care, housing, and education

are among numerous markets in which price is capped and a rationing scheme is used to

allocate supply. For instance, many countries prohibit the sale of human organs, meaning

that the price is capped at zero. Patients in need of a transplant are put on a waiting list

and organs are allocated based on factors such as the potential recipient’s age, the severity

of the condition, and the distance between the donor and recipient. Likewise, access to a

physician may depend on a patient’s arrival time, a nurse’s assessment of the condition, or

the insurance company’s reimbursement criteria.

In many housing markets, prices of some units are held below market-clearing levels,

and a lottery or priority scheme is used to allocate them. A common example has a local

government requiring that a certain percentage of new housing units be sold at a discount

to low-income buyers. Another notable example comes from Singapore, where most citizens

live in units sold by the government at prices that are well below the market.2

Rationing occurs when a public school enrols students according to a priority scheme,

possibly a lottery. There may be a pure lottery in which all applicants have the same

probability of admission, or there may be different probabilities for different applicants.

An example of the latter arises when the students living near a school are guaranteed

admission whereas all others face a common lottery.

Rationing has also been used to allocate unclaimed property (known as “fugitive prop-

erty”) as well as government-owned resources. For instance, land was allocated on a first-

come-first-served basis during the 1889 Oklahoma Land Rush, and subsequently by lottery

in 1901. Lotteries have been used to allocate cellular telephone licenses as well as airport

takeoff and landing slots.3 Rationing also occurs when the U.S. allocates immigrant visas

1In other words, instruments other than price are employed to allocate supply.2Some 86 per cent of Singapore’s citizens live in such units and 92 per cent of those residents own their

units. (See “Building Homes, Shaping Communities,” at http://www.mnd.gov.sg/, accessed on May 28,

2005.) The price of a new housing unit in Singapore is as low as half of the price on the resale market (Tu

and Wong (2002)).3For the former, see Hazlett and Michaels (1993). For the latter, see “Notice of Lottery for Takeoff and

2

through a lottery. Even a military draft can be seen as rationing in which conscripts are

selected to serve by lottery.4 These examples suggest that rationing is common, even in

market economies.

The existing literature offers different possible justifications for rationing. Wijkander

(1988) showed that rationing can be used to redistribute a good in favor of a certain

income group when direct redistribution is infeasible. Given a welfare function that puts

different weights on different consumers’ utility, rationing may raise welfare. Weitzman

(1977) justifies rationing on similar redistributive grounds, but with the flavor of efficiency.

He considered consumers who differed in their marginal utility of wealth and in a taste

parameter. The main finding was that rationing may produce an allocation closer to the

one that would arise if consumers had identical incomes.5

There are problems with explanations for rationing that are not efficiency-based. First,

if redistribution is the goal, there may be more effective methods than capping prices and

rationing. For instance, one could subsidize the “target” individuals directly, financing the

subsidy through general taxation. Second, if the goal is to raise one group’s consumption of

a good, that goal may be undermined by resale since some recipients will have an incentive

to resell the good rather than consuming it.

Guesnerie and Roberts (1984) offer a justification for rationing that is efficiency-based.

They considered a market in which the equilibrium allocation was not Pareto optimal

because of distortionary excise taxes, for example. In that context, rationing of a divisible

good may improve allocative efficiency.6 In particular, they show that an upper bound on

consumption of a particular good may produce a Pareto improvement.

All of the aforementioned papers assume that resale cannot occur; in fact, it is common

for resale to occur following rationing. It is well documented that there was substantial

Landing Times at LaGuardia Airport,” at http://www.faa.gov/apa/pr/pr.cfm?id=1192, accessed on June

1, 2005.4Demand for the right not to serve exceeds supply.5Weitzman first determined the equilibrium market allocation. Next, he considered quantity rationing

(i.e., allocating the good evenly). Finally, he determined a benchmark allocation—the market allocation

when all consumers had the mean income. The distance of the former two allocations to the benchmark

was determined using a mean-square-error criterion. The latter criterion does not give an exact measure

of the welfare cost of misallocation, however.6The authors also consider in-kind redistribution and taxation.

3

resale of rationed goods during World War II, for instance.7 More recently, 643 cellular

telephone licenses were allocated by lottery in the U.S. between 1986 and 1989 (of which

428 were in rural areas); 238 of the licenses in rural areas were resold in 1991 (see Hazlett

and Michaels (1993)). While resale per se is beneficial, the possibility of resale attracts

speculators hoping to acquire and resell the good. That was certainly true for cellular

telephone licenses. A recent case involved orders for tickets to the 2006 World Cup. Two

million orders were deemed to be fraudulent attempts to circumvent quantity limits; the

aim was to acquire tickets in order to resell them (see “World Cup Ticket Scam Uncovered,”

The Washington Post, April 16, 2005, page D2).

The current paper considers the allocation of an indivisible good when some buyers face

binding wealth constraints. These constraints could arise from capital market imperfec-

tions, which limit a buyer’s ability to borrow against future income, or simply from a low

lifetime income. Some buyers’ wealth (or ability-to-pay) falls short of their valuation (or

willingness-to-pay). This means that, although the competitive market allocation is Pareto

optimal, it is inefficient in the sense that some high-wealth buyers get the good instead of

some low-wealth buyers with higher valuations.8

Rationing and resale acting together may enhance efficiency here. To see why, consider

purely random rationing. Capping the price below the market level and rationing randomly

gives buyers with lower wealth access to the good (albeit with probability less than one),

but at the cost of allowing some lower-valuation buyers to consume the good. Conversely,

some high-wealth-high-valuation buyers who would have consumed for sure in a free market

may not consume now. Random rationing is thus less efficient than the market. In other

words, rationing alone does not perform well.

When resale is permitted, the ultimate allocation dominates the free market allocation

(that is, the aggregate realized value is higher). Buyers with low valuations who get the

good initially resell it to those with high wealth and high valuations who did not get it.

Buyers with high valuations and low wealth who get the good initially will keep it. Hence,

7Many commodities were subject to coupon rationing, and there was a great deal of (black-market)

trade in coupons. See Mills and Rockoff (1987) for a comparison of the U.S. and U.K. experiences.8Gali and Fernandez (1999) considered the assignment of students to schools when output was super-

modular in student ability and school quality. Given wealth constraints, testing dominates the market in

terms of matching efficiency.

4

rationing followed by resale improves access for those with high valuations and low wealth,

raising the total realized value.

The merits of rationing and resale are robust in the sense that they hold for a broad class

of rationing technologies. At the same time, changing the technology will alter allocative

efficiency. For instance, if resale is not permitted, a merit-based rationing scheme that

favors buyers with high valuations yields a more efficient allocation than merit-blind ones,

which do not screen based on valuations. In particular, need-based rationing, which favors

buyers based on wealth, has no impact on allocative efficiency.

Both merit-based and need-based rationing rules improve allocative efficiency when

resale is permitted. The benefit of merit-based rationing is obvious, but for need-based

rationing it is somewhat surprising. The latter effect can be traced to differences in ability

to pay: While the wealthy can purchase the good on the resale market, the poor are unable

to do so. Favoring the poor in the rationing process allows a more efficient allocation to

arise from the resale market equilibrium. In fact, allocating the entire supply to the poor

may eliminate the effects of wealth constraints, leading to the first-best allocation. This

result provides an efficiency rationale for rationing schemes and social programs that favor

the poor.

The gains from allowing resale shed light on how different social programs perform

efficiency-wise and, more importantly, how they can be improved. Consider U.S. immigra-

tion policy. Visas are currently rationed using methods that include a lottery, but with no

possibility of resale. Becker (1987) proposed selling visas to a qualified pool of applicants

instead. A simple alternative is to retain the lottery system but permit resale to qualified

applicants. Our results suggest that this change would yield greater efficiency than both

the current mechanism and the Becker proposal.9 Another possible application concerns

the provision of health care. Doctors’ appointments are allocated primarily on a first-come-

first-served basis, with no possibility of transfer to other patients. One could use the same

first-come-first-served scheme but allow patients to trade their appointment slots (i.e., hold

a resale market). The list of such examples is long indeed.

9An implicit assumption is that immigrants share a certain fraction of the rents they create with society.

Hence, those who create the most rents are the ones who have the most to gain. Other factors may enter a

policymaker’s welfare function. For example, visa sales generate revenue for the Treasury whereas a lottery,

with or without resale, does not.

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Resale can also be used in conjunction with sophisticated rationing schemes. Some re-

cent studies have proposed ingenious algorithms for improving allocative efficiency without

using transfers, in settings where wealth constraints may be important. (See Abdulka-

diroglu and Sonmez (1999) and (2003), or Roth, Sonmez, and Unver (2004).) The best

algorithm may still not deliver efficiency, however, and small transfers may facilitate a big

improvement. The results here imply that introducing transfers in the form of resale could

improve efficiency of these rationing schemes. Our approach is therefore complementary to

this literature.

Section 2 lays out the basic model and describes the benchmark allocation and the

competitive market allocation. Section 3 characterizes the outcome under random rationing

(with or without resale), and makes comparisons to the benchmarks. Section 4 analyzes

general rationing technologies and derives conditions under which rationing with resale

outperforms the market. Extensions are in Section 5, with concluding remarks in Section

6.

2 The Model

2.1 Primitives

A good is available in fixed supply, S ∈ (0, 1). There exists a unit mass of buyers who

each demand one unit. Each buyer is characterized by two attributes: her wealth, w,

and her valuation, v. A buyer’s type, (w, v), is distributed continuously over [0, 1]2. The

two components are independent, with w having the cumulative distribution function (cdf)

G(w) and v having the cdf F (v). The corresponding densities are non-zero for almost every

v and w. Independence makes the impact of the two attributes transparent. This helps

to isolate the role each attribute plays and the effect of policy treatments that focus on a

single attribute.10

Buyers are risk neutral, with quasilinear utility. Specifically, a type-(w, v) buyer receives

10If the attributes were negatively correlated, say, favoring buyers with low wealth would indirectly favor

buyers with high valuations. So, it would be difficult to say whether any change in the final outcome

is attributable to favoring buyers based on wealth or based on valuations. Independence enables us to

separate the effects of the two attributes.

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utility v +w−p if she consumes the good and pays p; her utility is w if she never consumes

it. A buyer with v > w is wealth constrained in the sense that she is not able to pay as much

as she is willing to pay. The presence of wealth-constrained buyers plays an important role

in our analysis, as noted in the Introduction.

The good is indivisible and supplied competitively at a constant marginal cost, c ≥ 0,

up to the capacity limit, S. Marginal cost is sufficiently low that there may be a shortage

of supply. Inelastic supply and shortages are characteristic of many many markets in which

rationing is considered. We will focus on methods of allocating the fixed supply to a

population of buyers. We will subsequently consider elastic supply.

We evaluate the welfare consequences of using several allocation mechanisms. Before

describing the mechanisms, it is necessary to discuss our welfare criterion.

2.2 Welfare Criterion and Efficient Allocation

Given quasilinear preferences, the aggregate valuation (the sum of valuations for those who

consume) reflects the sum of buyers’ utilities, which is the sensible criterion when utility

is fully transferable. Aggregate valuation is also a compelling criterion when utility is not

transferable, since it reflects buyers’ (ex ante) expected payoffs. If the buyers were to select

an allocation mechanism prior to realizing their types, they would vote for the one with

the higher aggregate valuation.11 Thus, the natural welfare measure here is the realized

aggregate valuation.

Given this measure, the first-best allocation would award the good to the buyers with

the highest valuations. Let v∗ denote the critical valuation such that 1− F (v∗) = S. If all

buyers with valuations of v∗ and above acquire the good, the supply is allocated efficiently.

The total realized value in this case is

V ∗ :=

∫ 1

0

∫ 1

v∗vdF (v)dG(w) =

∫ 1

v∗vdF (v) = Sφ(v∗),

where

φ(z) :=

∫ 1

zvdF (v)

1− F (z)

11A similar argument is given in Vickrey (1945).

7

is the expectation of a buyer’s valuation, conditional on exceeding z. The first equality

follows by integrating over w, while the second holds since 1− F (v∗) = S. Intuitively, the

aggregate quantity is S, and the average valuation among those who acquire the good is

φ(v∗), so the total realized value in the efficient allocation is Sφ(v∗).

The first-best allocation and the associated realized value, V ∗, will serve as benchmarks

for the mechanisms that we consider.

2.3 Allocation Mechanisms

Within the framework introduced above, we will examine three allocation mechanisms:

(1) a competitive market, (2) rationing without resale, and (3) rationing with resale. We

will also compare them with the first-best benchmark. The competitive market is simply

characterized by the price at which the effective demand — the measure of buyers willing

and able to pay the price — equals the supply. Rationing without resale entails a binding

price cap, and it allocates the good according to some non-market criterion or priority rule.

It is described by the probability of allocation for those buyers who participate. Formally,

a rationing rule is a (measurable) function, x : [0, 1]2 7→ [0, 1], which maps each type

(w, v) ∈ [0, 1]2 to a probability of allocation. For instance, a completely random rule will

give the same probability of allocation to all participants. In the case of rationing with

resale, there will be a competitive market after the initial rationing in which those who

obtained the good may resell to those who did not.

These three mechanisms are used prominently, as illustrated by the following examples:

• Fugitive Property and Government-Owned Resources: Fugitive property,

which is an object or resource whose ownership is not yet established, is allocated

according to one of three rules. It can be allocated to whoever claims it first (the rule

of first possession). It can be allocated to the individual who owns property tied to

that object (tied ownership).12 Lastly, ownership can be established by an auction.

The first two methods correspond to rationing with resale, since the allocation is

via a nonmarket mechanism and the property can be transferred to another party.

Auctions correspond to the market. The 1889 Oklahoma Land Rush and the 1901 Ok-

12For instance, a landowner has the subsurface right to natural gas deposits underneath the land.

8

lahoma Land Opening are examples of rationing with resale. Government resources

such as land, radio spectrum, timber rights, and oil rights have been allocated using a

number of methods including hearings, lotteries, and auctions. Sometimes, auctions

have preferences for “designated entities” such as minorities and small businesses,

which gives the flavor of need-based rationing.

• Job recruiting, internal labor markets and promotion: Many jobs are al-

located through the labor market, but with rigid wages set at levels that leave a

surplus of candidates. Criteria such as performance history, educational attainment,

and written tests are used to “ration” the slots. This is even more true for internal

labor market allocations such as task assignment and promotion decisions, which rely

primarily on nonmarket evaluations. Hence, job allocation (external or internal) is

often determined via rationing without resale.

• Education: Public school enrollment is typically tied to one’s residence. This tie

means that the housing market serves indirectly as a market for school enrolment.

Suppose that there are two public schools, A and B. The schools draw students from

local attendance areas that are mutually exclusive and exhaustive. More than half of

the students prefer A, resulting in excess demand for that school. The valuation, v,

now represents the premium that a student is willing to pay for the right to attend A.

Since the price of attending the school is zero, the preference for A will be capitalized

in housing prices. This scenario corresponds to the market regime. Now suppose

that slots in school A are awarded by lottery. (Once again, the price of attending is

capped at zero.) This corresponds to rationing without resale. Finally, if a lottery

awards transferable vouchers that confer the right to attend A, there is rationing with

resale.

• Housing: The three regimes arise naturally in housing markets. Consider a rental

market that is subject to rent control. That is, there is a binding price cap. If leases

are not transferable, we have the regime of rationing without resale. If leases are

transferable, we have rationing with resale. If rent control is abolished, we have a

competitive market.

• Health care: Health care is often provided via rationing without resale. A specific

9

example involves organ transplants. Patients are placed in a queue and cannot ex-

change their places in the queue.13 In principle, there could also be a competitive

market, which allocates organs to those willing and able to pay the market price.14

Finally, there could be rationing with resale in which patients waiting for a transplant

may sell their places in the queue.

• Military recruitment: An all-volunteer army corresponds to the market regime.

A universal draft in which conscripts are called to duty based on a lottery is akin to

rationing without resale. (Think of using a lottery to give deferments rather than to

call people to serve. There is rationing since the price of a deferment is capped at

zero, and demand exceeds supply.) Finally, the regime of rationing with resale can be

realized with a draft and tradable deferments. An early example occurred in ancient

Korean kingdoms when wealthy families were allowed to pay sharecroppers to enlist

on their behalf.

Section 5 discusses modifications of these mechanisms that are used in some circum-

stances. More detailed descriptions of rationing rules will also be given in the subsequent

sections, where we analyze the rationing mechanisms. In the remainder of this section, we

characterize the competitive market.

2.4 A Competitive Market

A competitive market is characterized by the price at which the supply is exhausted by

effective demand — the measure of buyers who are willing and able to pay the prevailing

price. When the market price is p, effective demand is given by

D(p) := [1−G(p)][1− F (p)].

Hence, the competitive equilibrium, if it exists, will be achieved at a price, pe, such that

D(pe) = [1−G(pe)][1− F (pe)] = S. (1)

13In the U.S., waiting lists for organ transplants are maintained by the United Network for Organ Sharing,

which matches donated organs to recipients. Additional issues arise with kidney transplants because of

the possibility of live donors. See Roth, Sonmez, and Unver for a discussion of kidney exchanges wherein

patients with incompatible live donors essentially trade donor kidneys.14Organs are available in certain countries at market prices.

10

Assuming that pe > c, this is a competitive equilibrium.

A couple of remarks are in order. First, 1− F (v∗) = S, so [1−G(v∗)][1− F (v∗)] < S,

which implies pe < v∗. This means that the market equilibrium does not replicate the

efficient allocation. As can be seen in Figure 1, the first-best allocates to all buyers in

the region A + B while the market allocates to those in B + C. Relative to the first-best

allocation, the market favors high-wealth-low-valuation buyers (region C) over low-wealth-

high-valuation buyers (region A).

[PLACE FIGURE 1 ABOUT HERE.]

The aggregate realized value under the market is:

V e :=

∫ 1

pe

∫ 1

pe

vdF (v)dG(w) = [1−G(pe)]

∫ 1

pe

vdF (v) = Sφ(pe) < Sφ(v∗) = V ∗.

The second equality holds by integrating over w; the third holds because [1 − G(pe)][1 −F (pe)] = S, by (1); and the inequality holds since pe < v∗ and φ is an increasing function.

The inefficiency is entirely attributable to the binding wealth constraints. If no buyers were

constrained, the effective demand would simply be 1 − F (p), so the market-clearing price

would satisfy 1− F (p) = S, or p = v∗, implying that the allocation would be efficient.

Second, even though the allocation is inefficient under the market regime, it would not

trigger any resale. (In other words, it is Pareto optimal.) Those who have the good would

only be willing to resell at prices exceeding pe, but there are no new buyers who are willing

and able to pay that much.15 Simply put, the inefficiency resulting from the market will not

be resolved by opening another market. With rationing, by contrast, inefficiencies may be

resolved through resale. Moreover, the resale process may lead to a more efficient allocation

than if the market regime were used in the beginning.

3 Analysis with Random Rationing

In this section, we consider completely random rationing, with and without resale. To

begin, we suppose that the price is capped at a below-market level, p ∈ [c, pe), and excess

15By the same token, if there were an active resale market, the resale market price would be the same as

the price in the original equilibrium; otherwise, buyers would switch from one market to the other. Hence,

the equilibrium price must be pe, so the allocation is the same.

11

demand is rationed randomly. There is a constant allocation probability, x(w, v) = ρ, for

all who participate. Random rationing is appealing for various reasons. It is simple to

implement as it does not require knowledge of buyers’ preferences or budgets. While a

government may try to screen buyers to favor one group, such screening may be imperfect

because buyers’ types are private information and any screening method may be subject

to manipulation. (For instance, if rationing is based on wealth, buyers may attempt to

hide or divest their wealth.) The analysis is also more accessible with random rationing,

and the results are unambiguous. We present the case of random rationing first. We will

analyze general rationing schemes in Section 4, so details of the equilibrium analysis will

be presented then.

3.1 Rationing without Resale

Consider rationing without resale first. The assumption that resale is not possible is rea-

sonable for many of the goods we have discussed. Health care and education are made

nontransferable by requiring identification of the patient or student. Cellular telephone

licenses or takeoff-and-landing slots are easily made nontransferable since ownership or use

can be monitored.

Buyers whose ability-to-pay and willingness-to-pay both exceed p will want to buy the

good now.16 The expected valuation for such buyers is φ(p), and the aggregate quantity

is S, so random rationing gives a total realized value of Sφ(p). Since p < pe, we have

Sφ(p) < Sφ(pe), so efficiency is lower with rationing than without. The result is intuitive:

When the cap binds, it expands the set of buyers who get the good. In particular, some

buyers with lower valuations now acquire the good. The effect on efficiency is negative

because these buyers crowd out some buyers with higher valuations. Although some buyers

with lower wealth now acquire the good, this has no effect since low-wealth buyers are as

likely to have low valuations as high-wealth buyers are since v and w are independent.

16We can assume that if a buyer participates in rationing, she must take delivery and pay for the good

if successful. If the buyer does not fulfill these obligations, she will not get the good and she must pay

a small penalty. This will mean that buyers only participate in the rationing if they are able to pay the

price.

12

3.2 Rationing with Resale

Under random rationing, some high-valuation buyers did not get the good whereas some

low-valuation buyers did. The latter buyers have an incentive to resell the good so we now

allow resale. All else equal, allowing resale raises allocative efficiency; however, it induces

speculation by some buyers seeking to resell the good. We now show that the former effect

dominates to such an extent that the ultimate allocation is more efficient than even the

market allocation. For simplicity, we assume that each individual may only attempt to

purchase one unit of the good. This simply avoids having to deal with the possibility that

some speculators obtain multiple units. It is also easily enforceable in many circumstances.

Following rationing, a competitive resale market opens and the price clears the market.

The equilibrium resale price, r(p), must exceed p. (If not, we have r(p) < pe, so there

would be excess demand on the resale market.) A buyer who is successful in the rationing

can then pocket a strictly positive gain, r(p) − p, by reselling. Thus, all buyers who can

afford to pay p, including those with very low valuations, will participate in the rationing.

In particular, buyers with v < r(p) enter solely to profit from speculation.

We now characterize the equilibrium resale price. Given any resale market price, r > p,

resale demand comprises those buyers who were unsuccessful in rationing but are willing

and able to pay r. Hence, the resale demand is

RD(r) :=

(1− S

1−G(p)

)D(r),

since 1− S1−G(p)

is the probability of being unsuccessful under random rationing.

Resale supply comprises those buyers successful in the rationing who would rather resell

the good than consume it. If a buyer with valuation v gets the good and keeps it, she will

receive a surplus of v− p. If she resells the good, she will net r− p, so the buyer will resell

if and only if v − p < r − p, or v < r. Thus, resale supply is the fraction F (r) of those

buyers with v < r and w ≥ p who are successful in the rationing, or

RS(r) :=

(S

1−G(p)

)F (r)[1−G(p)] = F (r)S.

The resale equilibrium arises at the price r∗ such that RD(r∗) = RS(r∗).

13

To characterize the price, write excess demand as

RD(r)−RS(r) = D(r)− S +

(S

1−G(p)

)[G(r)−G(p)][1− F (r)]. (2)

The last term is the measure of buyers who receive the good and retain it, but would be

unable to purchase on the resale market if unsuccessful initially. Clearly, for any r ≤ (<) pe,

D(r) ≥ (>)S, so RD(r) > RS(r). Hence, the resale equilibrium must have r(p) > pe. This

can be seen intuitively through Figure 2.

[PLACE FIGURE 2 ABOUT HERE.]

Suppose that the resale price were r = pe. Then, the buyers who are both willing and

able to pay r = pe (area B) would end up with the good, just as in the competitive market

equilibrium. (They either acquire the good in the resale market if initially unsuccessful or

they keep the good if initially successful.) There are also buyers with (w, v) ∈ [p, r]× [r, 1]

who would not obtain the good in the market regime but may obtain it and keep it with

rationing. This group accounts for the third term in (2) and area A, so the resale price

must be higher than pe.

That the equilibrium resale price exceeds pe has an immediate welfare implication. The

total realized value from rationing with resale is Sφ(r(p)) > Sφ(pe) = V e.17 In other

words, rationing with resale yields a strictly more efficient allocation than the competitive

market.18

The welfare ranking rests on the following logic: The market produces a more efficient

allocation when the initial endowment is shifted away from those with the purchasing power

to acquire the good (i.e., the wealthy) and toward those without (i.e., the poor). Compared

with the market allocation, random rationing at a below-market price reallocates the good

from the wealthy to the poor, which makes the ensuing resale market allocation more

efficient. The same logic applies to a change in p. As the cap drops, random rationing

17Independence of v and w implies that, for any given w ≥ p, the expected value of v, conditional on

exceeding r(p), is φ(r(p)). When buyers with wealth w and valuation v ∈ [r(p), 1] all get the good with

probability ρw, say, the expected value for such bidders is φ(r(p)). Since this holds for all w ≥ p, and since

quantity equals S, the total realized value is Sφ(r(p)).18At the same time, r(p) < v∗ since high-valuation buyers with wealth w < p cannot get the good. Since

Sφ(r(p)) < Sφ(v∗) = V ∗, rationing with resale does not deliver full efficiency.

14

reallocates more of the good from the wealthy to the poor, so the ultimate allocation

arising from the resale market becomes more efficient, as demonstrated by the fact that

r(p) is increasing as p falls. Consequently, allocative efficiency is highest when the cap is

at the lowest level at which supply is available, c.

We close the section by summarizing our findings.

Proposition 1. Random rationing without resale at a below-market price is less efficient

than the competitive market, and becomes progressively less so as the price cap falls. Ran-

dom rationing with resale is strictly more efficient than the competitive market, however,

and it becomes more so as the price cap falls.

4 Analysis with a General Rationing Technology

The previous section assumed that rationing was completely random. In practice, the

government or the suppliers may have information about buyers’ wealths or valuations,

allowing them to implement schemes besides random rationing. When one patient is given

priority for an organ transplant based on age and medical urgency, that is an implicit

assessment that the valuation is higher for the one patient than the other. A government

agency may rank applicants for subsidized housing based on family size, which is one

indication of need. These observations suggest that the nature of the rationing rule may

be affected by the available information.

Rationing rules may also depend on the government’s or suppliers’ objectives. Some

schemes are geared to help the poor and some are geared to select those deemed to have the

greatest merit. For example, need-based scholarships target those with low wealth while

merit-based scholarships could be seen as targeting those with high valuations. In this

section we will consider a family of general rationing rules. We do not model the objectives

or the information available when selecting the rationing rule; instead, we simply take as

given the resulting technology.

A rationing rule, x, is feasible if∫ 1

0

∫ 1

0x(w, v)dF (v)dG(w) = S. A feasible rationing rule

is separable if x(w, v) = αx(w)βx(v) for functions αx : [0, 1] 7→ <+ and βx : [0, 1] 7→ <+.

Separable rationing rules are attractive because they allow us to examine the effect of

policies targeting a single attribute in the most transparent way. Let X denote the set of

15

feasible, separable rationing rules. We henceforth consider rationing rules in this family.

4.1 Characterization of Rationing Rules

Let

ax(w) :=αx(w)∫ 1

0αx(w)dG(w)

and

bx(v) :=βx(v)∫ 1

0βx(v)dF (v)

.

The probability of allocation satisfies

x(w, v) = S · ax(w)bx(v),

since feasibility implies

∫ 1

0

∫ 1

0

αx(w)βx(v)dF (v)dG(w) =

(∫ 1

0

αx(w)dG(w)

)(∫ 1

0

βx(v)dF (v)

)= S.

Then, Ax(w) :=∫ w

0ax(w)dG(w) and Bx(v) :=

∫ v

0bx(v)dF (v) can be seen as cumulative

distribution functions of the allocations across different wealths and valuations, respectively.

It will prove useful for comparing allocative efficiency to characterize rationing rules in terms

of these functions.

We begin with rationing rules that favor those with higher valuations. We say that

a rationing rule, x ∈ X merit-dominates another rationing rule, y ∈ X , if Bx first-order

stochastically dominates (FOSD) By: ∀v, Bx(v) ≤ By(v).19 In words, x assigns a higher

probability to high-valuation buyers than y does. Rules x and y are merit-equivalent if

bx(·) = by(·).A rationing rule, x ∈ X , is merit-blind if bx(v) is constant for all v ≥ p. Random

rationing is an obvious example of a merit-blind rule as it awards the good with the same

probability to all participants whose valuations exceed the price cap.

There are analogous conditions for cases in which lower wealth is favored. A rationing

rule, x ∈ X , need-dominates another rule, y ∈ X , if Ay FOSD Ax: ∀w, Ax(w) ≥ Ay(w). In

19The merit-dominance is strict if the inequality is strict for a positive measure of valuations. The

analogous condition makes subsequent dominance definitions strict as well.

16

words, x is more likely to allocate the good to low-wealth buyers than y is. Rules x and

y are said to be need-equivalent if ax(·) = ay(·). Finally, x ∈ X is need-blind if ax(w) is

constant for all w ≥ p. Fixing the valuation, the probability of receipt does not depend on

wealth now.

The outcomes of rationing with and without resale can now be characterized.

4.2 Rationing without Resale

Given a price cap, p < pe, only buyers with (w, v) ≥ (p, p) will participate in the rationing

if resale is not permitted. The rationing rule will then have x(w, v) = 0 if w < p or v < p.

The total realized value is

Vx :=

∫ 1

0

∫ 1

0

vx(w, v)dF (v)dG(w).

The following proposition provides a ranking based on merit-dominance.

Proposition 2. If x ∈ X [strictly] merit-dominates y ∈ X , then Vx ≥ [>]Vy.

Proof: Rewrite the total realized value as

Vx = S

∫ 1

0

vax(w)bx(v)dF (v)dG(w)

= S

∫ 1

0

vbx(v)dF (v)

= S

∫ 1

0

vdBx(v).

If x ∈ X [strictly] merit-dominates y ∈ X , then Bx [strictly] FOSD By, so the result

follows.

This result says that a rationing rule that puts relatively more weight on high valuations

yields greater efficiency. A couple of implications can be drawn immediately. We first note

that it does not matter how a rationing rule treats buyers with different wealth levels: All

that matters for efficiency is how it screens based on valuations.

Corollary 1. (Irrelevance of need-based screening) If x ∈ X and y ∈ X are

merit-equivalent, then Vx = Vy.

17

Any rationing rule that is merit-dominated by a merit-blind rule is less efficient than the

merit-blind rule, which is welfare-equivalent to random rationing, by Corollary 1. The pre-

vious section established that random rationing is strictly less efficient than the competitive

market, so the following result is also immediate.

Corollary 2. (Drawback of rationing without resale) Any rationing rule in Xthat is merit-dominated by the random rationing rule (given the same price cap, p < pe)

yields a strictly less efficient allocation than the market does.

It follows that a merit-blind rationing rule, which is merit-equivalent to random ra-

tioning, is strictly less efficient than the competitive market. That is, any rationing rule

associated with purely need-based screening cannot be justified from an efficiency perspec-

tive, when resale is not permitted.

The strict dominance by the market over all merit-blind rules implies that any rule

that is modestly merit-superior to random rationing cannot be justified from an efficiency

perspective either. In order for rationing without resale to improve efficiency, substantial

merit-based screening must be feasible (i.e., buyers’ valuations must be identifiable with

sufficient accuracy.) In the extreme case in which buyers’ valuations are observable, the

first-best allocation can be achieved with a rationing rule having x(w, v) = 1 for v ≥ v∗

and x(w, v) = 0 for v < v∗. In practice, however, such precise information will typically

not be available.

4.3 Rationing with Resale

The final regime has rationing with resale permitted. Suppose that the price cap is p < pe

and the resale market price is r > p. (The latter inequality, which makes speculation

profitable, will be shown to hold.) Then, a buyer would participate in the rationing if and

only if (w, v) ≥ (p, 0). Hence, any feasible rationing rule must have x(w, v) = 0 for any

w < p.

A buyer who fails to get the good initially will demand a unit on the resale market if

she is willing and able to pay r. Such buyers represent the resale demand:

RD(r) :=

∫ 1

r

∫ 1

r

(1− x(w, v))dF (v)dG(w).

18

Meanwhile, buyers who get the good will optimally keep it if v ≥ r. The measure of these

buyers is∫ 1

p

∫ 1

rx(w, v)dF (v)dG(w) since all buyers with w ≥ p participate. The quantity

supplied in the resale market is therefore

RS(r) := S −∫ 1

p

∫ 1

r

x(w, v)dF (v)dG(w).

Observe that RD(·) is nonincreasing and RS(·) is nondecreasing, and both are contin-

uous functions. Further, RD(1) = 0 < S = RS(1), and RD(0) = 1 − S > 0 = RS(0).

Hence, there exists a resale price, rx, that clears the market: RD(rx) = RS(rx).

Rewrite the market-clearing condition as:

RD(r)−RS(r) = D(r)− S + Kx(r) = 0, (3)

where

Kx(r) :=

∫ r

p

∫ 1

r

x(w, v)dF (v)dG(w) = S · Ax(r)[1−Bx(r)]

is the measure of buyers with wealth w ∈ [p, r] who get the good and keep it. Note that if

Kx(pe) > 0, then RD(pe)− RS(pe) = Kx(p

e) > 0. It follows that RD(r) > RS(r) for any

r ≤ pe, implying that rx > pe, which confirms that the resale price exceeds pe.

The total realized value is now

Vx :=

∫ rx

p

∫ 1

rx

vx(w, v)dF (v)dG(w) +

∫ 1

rx

∫ 1

rx

vdF (v)dG(w).

The first term pertains to high-valuation-low-wealth buyers who keep the good, while the

second pertains to those with high valuations and high wealth; some of them get the good

through rationing, while the others purchase it on the resale market.

The subsequent characterization refers to a new property. We say that the rationing

rule x relatively merit-dominates the rule y if, for any v′ > v,

bx(v′)

bx(v)≥ by(v

′)by(v)

.

This notion implies merit-dominance as it requires the latter to hold for all subsets of

valuations. We say that x is meritorious if it relatively merit-dominates a merit-blind rule;

i.e., bx(v′) ≥ bx(v) for all v′ > v. Likewise, x is demeritorious if bx(v

′′) ≤ bx(v) for all

19

v′′ < v. Roughly speaking, (de)meritorious rationing rules dominate (are dominated by)

merit-blind rules, in a stronger sense than merit-based dominance.

Let X+ and X− denote, respectively, the set of all meritorious and demeritorious ra-

tioning rules in X . Since these are weak notions, both sets include merit-blind rules.

Proposition 3. If a meritorious rule, x ∈ X+, relatively merit-dominates y ∈ X , and if

rx ≥ [>]ry, then Vx ≥ [>]Vy.

Proof: See the Appendix.

As was the case with random rationing, Proposition 3 shows that a higher resale price

is indicative of a more efficient allocation. Several important implications can be drawn

from this proposition. First of all, we can provide a sufficient condition for rationing with

resale to improve upon the competitive market equilibrium.

Corollary 3. (Benefit of rationing and resale over the market) Any merito-

rious rationing rule, x ∈ X+, with p < pe and Ax(pe) > 0, produces a strictly more efficient

allocation than the competitive market.

Proof: Fix x ∈ X+ with px < pe. As noted, the total realized value in the competitive

market is equal to Vy for any y ∈ X satisfying py = ry = pe. In particular, we can choose

y to be relatively merit-dominated by x. Since x is meritorious, we have Bx(pe) ≤ pe < 1;

together with Ax(pe) > 0, this means Kx(p

e). It follows that rx > pe. In sum, x ∈ X+

relatively merit-dominates y, and rx > pe = ry. Proposition 3 then implies Vx > Vy.

The condition Ax(pe) > 0 simply means that x awards the good to some buyers who,

despite their willingness-to-pay the market price, would not get the good in the competitive

market due to their limited wealth. Since these buyers would not resell the good if r = pe,

the resale price must exceed the competitive market price. Except for this condition,

the result does not require much in terms of how rationing depends on wealth levels. In

other words, a weakly meritorious rationing rule (with resale) does strictly better than the

competitive market, largely independent of how it treats the poor relative to the wealthy.

Also worth noting is the strict dominance of rationing over the market even when the

rationing rule is merit-blind. This means that even demeritorious rationing rules could do

better than the market when resale is permitted.

20

We next examine the effect of a change in the price cap. This analysis involves a delicate

issue: When the cap changes, it alters the set of buyers who participate, so the rationing

rule itself changes. One must therefore specify how the rationing rule changes when the

price cap changes. We make the following assumptions.

Condition (RC): Suppose that x and x′ are rationing rules induced by a given rationing

technology, with price caps p < pe and p′ < p, respectively. Then, (i) x and x′ are merit-

equivalent (i.e., bx′(·) = bx(·)), and (ii) ax′(w) < ax(w) for w ∈ [p, 1].

Property (i) is appealing since a change in the price cap affects the set of participants

only along the wealth dimension. It is then reasonable that the merit aspect of the assign-

ment does not change. Property (ii) reflects the equally plausible assertion that introducing

buyers with lower wealth into the pool reduces the allocation probability for all of the orig-

inal participants. A special case of Condition (RC) would arise if the relative probabilities

among the original participants remain unchanged; i.e., x′(w, v) = λx(w, v) for w ∈ [p, 1],

for some λ < 1.

Corollary 4. (Benefit of lowering price caps) Lowering the price cap increases

allocative efficiency, given a meritorious rationing technology satisfying Condition (RC).

Proof: Let x and x′ be meritorious rationing rules induced by a given rationing

technology, with price caps p < pe and p′ < p, respectively. Then, Condition (RC) means

that x and x′ are also merit-equivalent. Hence, x′ relatively merit-dominates x, and Bx′(·) =

Bx(·).We next prove Ax′(w) > Ax(w) for all w ∈ [p, 1). Observe that, ∀w ≥ p, we have

Ax(w) =

∫ w

p

ax(w)dG(w),

and

Ax′(w) =

∫ w

p′ax′(w)dG(w).

Hence, for w ≥ p,

dAx′(w)

dw= ax′(w)g(w) < ax(w)g(w) =

dAx(w)

dw,

where the inequality follows from Condition (RC). This, together with Ax′(1) = Ax(1),

implies that Ax′(w) > Ax(w) for all w ∈ [p, 1).

21

Combining these facts, we have

Kx′(r) = Ax′(r)[1−Bx′(r)] > Ax(r)[1−Bx(r)] = Kx(r)

for any r > p, from which it follows that rx′ > rx. Consequently, Proposition 3 implies that

Vx′ > Vx.

This result shows that the lowest feasible cap is optimal. (Once again, this result covers

the case of random rationing.) The proposition also allows us to investigate which rationing

schemes promote efficiency.

Corollary 5. (Benefit of need-based screening) If x ∈ X relatively merit-dominates

and need-dominates y ∈ X , then Vx ≥ Vy. If either dominance is strict, then Vx > Vy.

Proof: Given Proposition 3, it suffices to show that rx ≥ ry, with a strict inequality

for a strict ranking. Relative merit-dominance by x over y implies 1 − Bx(r) ≥ 1 − By(r)

for all r, whereas need-dominance implies Ax(r) ≥ Ay(r). Hence, Kx(r) ≥ Ky(r), for all r,

which implies rx ≥ ry. If either dominance is strict, then Kx(r) > Ky(r) when r = ry, so

rx > ry.

Need-based rationing can yield greater efficiency than the market does. While it may

be unsurprising that merit-based rationing can improve efficiency, it is striking that need-

based rationing can have the same effect. If x and y are merit-equivalent, but the former

need-dominates the latter, x produces a more efficient allocation. This result stands in

contrast to Corollary 1, which reported that need-based rationing has no effect when resale

is not permitted. The benefit from need-based rationing here arises because of differences

in buyers’ ability to trade on the resale market. Whereas wealthy buyers can buy the

good from a reseller if they do not get it initially, the poor lack the means to do so. As a

consequence, an initial allocation that gives the good disproportionately to the poor allows

the resale market to produce a more efficient allocation.

Corollary 5 also provides a strong result if the rationing rule favors low-wealth buyers

absolutely. Consider a merit-blind rule, x∗ ∈ X , with x∗(w, v) = 1 if w ≤ w∗ and x∗(w, v) =

0 if w > w∗, where G(w∗) = S. (It is implicit that p = 0.) That is, x∗ assigns the good to

a buyer if and only if his wealth is w∗ or below. This group comprises the region A + B

22

in Figure 3. This rule will achieve full efficiency if v∗ ≤ w∗, where [1 − F (v∗)] = S.20 To

see this, suppose that v∗ ≤ w∗ and that the resale price is r. Then, the resale supply will

be RS(r) = SF (r). Meanwhile, those not assigned the good will demand it if v ≥ r, so

resale demand is RD(r) = [1 − F (r)][1 − G(max{r, w∗})]. Since RD(r)>=<

RS(r) if r<=>

v∗,

the unique resale equilibrium price is v∗. Consequently, the buyers with v ≥ v∗ will end

up with the good, which is the full-efficiency outcome. This group comprises the region

A + C in Figure 3. Note that this benefit from need-based screening does not depend on

independence of v and w.

[PLACE FIGURE 3 ABOUT HERE.]

When the supply, S, is sufficiently large, the infimum wealth among buyers who do

not get the good initially is high. These buyers are able to purchase on the resale market.

Consequently, any buyer who did not get the good initially, and who is willing to pay the

resale price, gets the good.

5 Discussion

In this section we explore the impact of several natural extensions to our analysis of ra-

tioning and resale. We will see that the basic results continue to hold in a range of circum-

stances.

5.1 A Dual System with Rationing and a Market

When we considered rationing we assumed that the entire supply was offered for sale at

the price cap. Suppose that only part of the supply is sold on a market with rationing;

the remainder is sold on a market without rationing. The most common example of this

phenomenon is a black market. In the presence of rationing, incentives exist to operate an

illegal market where the goods can be sold at a higher price.21 There were black markets

20This condition is satisfied if S is sufficiently large. Letting t(z) := [1 − F (z)] − G(z), we just need to

find a root of the equation t(z) = 0. Since t(0) = 1, t(1) = 0, and t(·) is continuous and strictly decreasing,

there is a unique root, z∗. Full efficiency requires S ≥ G(z∗) = [1− F (z∗)].21There are other means of circumventing price caps. One is a tied sale in which the purchase of a good

subject to a binding price cap is conditioned on the purchase of another good that is not.

23

for beef, chicken, gasoline, and tires in the U.S. during World War II. For instance, some

farmers sold part of their crop on the market at the price cap, and the remainder at the

farm gate at higher prices. (For a general discussion see Rockoff (1998) and references

therein.) Underground markets for a multitude of goods thrived for decades in Soviet-style

planned economies. More recently, news reports from Baghdad indicated that gasoline was

selling for 2,000 dinars on the black market whereas the price cap was only 80 dinars.22

Finally, despite restrictions on the sale of human organs, there have been sales of organs

on black markets. These anecdotes illustrate that dual markets exist and have existed for

a long time.

Dual market systems may also arise by design. Consider school choice. A familiar

scheme guarantees admission to students living in the vicinity of a public school and uses

a lottery to allocate the remaining slots (Abdulkadiroglu and Sonmez (2003)). Thus, one

can “buy” a slot by purchasing a house in the immediate vicinity of the school. This means

that there is a high price to get a slot with probability one, and a low (zero) price to get

it with probability below one through the lottery.

An analogous situation occurs when a high school senior applies to a college or university

under the “early decision” program. All else equal, the probability of being accepted is

higher when one applies for an early decision (Avery, Fairbanks, and Zeckhauser (2003)).

The “price” is also higher, however, since financial aid offers may be significantly lower

when one applies for an early decision. (See “The Benefits and Drawbacks of Early Decision

and Early Action Plans,” at http://www.collegeboard.com/prof/counselors/apply/5.html,

accessed on June, 8, 2005.) In other words, students who are accepted under the early

decision program tend to pay more, all else equal.

Our model can be altered to accommodate dual systems. Suppose that a fraction

γ ∈ (0, 1) of the supply is sold in a competitive market, while the remaining 1−γ is subject

to a price cap, p < pe, and random rationing. Assume that resale is not permitted.

In equilibrium, buyers with (w, v) ≥ (p, p) enter the rationing pool. There is also a

market price, p∗ > pe, such that unsuccessful buyers with (w, v) ≥ (p∗, p∗) will purchase

on the market.23 Let L(z) := {(w, v)|(w, v) ≥ (p, z), min{w, v} < p∗} be the L-shaped set

22See “Iraqis’ Dismay Surges as Lights Flicker and Gas Lines Grow,” The Washington Post, Dec. 24,

2004, A1.23The organizer may be able to monitor participation in the dual system so that one individual may get

24

containing types with valuations exceeding z who participate in the rationing but who will

not purchase at p∗; and let `(z) := Pr{(w, v) ∈ L(z)} and ξ(z) := E[v|(w, v) ∈ L(z)]. The

total realized value from the dual system is

VD := σ`(p)ξ(p) + D(p∗)φ(p∗),

where σ is the probability of allocation in the lottery, which satisfies the market-clearing

condition, σ`(p) + D(p∗) = S.

The efficiency comparison between the dual system and the unregulated market is ac-

tually ambiguous.24 At the same time, incorporating the rationing-resale mechanism above

would enhance the dual system. To be concrete, suppose that the quantity γS is allocated

via “transferable” vouchers and the remaining (1− γ)S is allocated via “nontransferable”

vouchers, both at the price of p. Consistent with the earlier approach, we assume that

each individual may only enter the rationing pool for a single type of voucher. Those who

are unsuccessful in the rationing can purchase a transferable voucher in the resale market,

which is again assumed to be competitive.

We look for an equilibrium with two properties: (1) There exists v ∈ [0, p∗] such that

buyers with v ≥ v enter the rationing pool for nonrefundable vouchers, and those with

v < v enter the pool for refundable ones; and (2) the resale price is p∗. Sufficient conditions

for such an equilibrium to exist are derived as follows. First, the critical type, v, must be

indifferent between the two types of vouchers:

(1− γ)S

[1− F (v)][1−G(p)][v − p] =

γS

F (v)[1−G(p)][p∗ − p]. (4)

Given (4), any buyer with v > v [resp., v < v] will strictly prefer the nontransferable [resp.,

transferable] vouchers. Next, the resale market will clear at p∗ if(

1− (1− γ)S

[1− F (v)][1−G(p)]

)D(p∗) = γS. (5)

If there exists a pair, (v, γ), satisfying (4) and (5), then such an equilibrium exists. The

aggregate value realized in that equilibrium is

VD = σ`(v)ξ(v) + D(p∗)φ(p∗),

access to either rationing or market. The result is qualitatively the same.24When γ = 1, this is the same as a competitive market. Lowering γ slightly raises p∗. Moreover, buyers

in L(p) now get the good with some small probability. These changes may raise or lower the total realized

value, depending on the density in this region.

25

where σ`(v) + D(p∗) = S. Since v > p, we conclude that VD > VD.

Proposition 4. For any dual system, there exists a rationing scheme with vouchers (some

transferable) that is more efficient.

Proof: The proof is in the Appendix.

5.2 Regulation of Resale

When resale is permitted, speculation by low-valuation buyers reduces the probability that

any given buyer gets the good initially, which ultimately lowers efficiency. One possible

response is to tax resales. Suppose that the government caps the price of the good at

p < pe, rations randomly, and imposes a lump-sum tax of τ ≥ 0 on resellers. All buyers

with w ≥ p will participate in the rationing if the resale price, r(τ), satisfies r(τ) ≥ p + τ ,

which we will assume. This condition means that it is profitable to engage in resale.25 The

equilibrium resale price increases with the tax, and the net proceeds from resale fall, in this

region.

To see the effect of the higher tax, it is useful to distinguish the two groups that get the

good ultimately. First, some buyers with (w, v) ≥ (p, r(τ) − τ) get the good initially and

find it optimal not to resell. Second, some buyers with (w, v) ≥ (r(τ), r(τ)) purchase on

the resale market. When the tax rate rises from τ to τ ′, the former group expands and the

latter group contracts. In particular, some buyers with v ≥ r(τ) now get the good with a

probability below one. Conversely, buyers with w ≥ p and r(τ ′) − τ ′ ≤ v ≤ r(τ) − τ now

get the good with strictly positive probability, rather than zero. Since the probability of

receipt is unchanged in the other regions, the total realized value is lower with the higher

tax, τ ′. It follows that a tax on resales is counterproductive.26

25If the inequality does not hold, speculation is unprofitable. Only those with (w, v) ≥ (p, p) will

participate in the rationing, and the resale market will be inactive.26One can also interpret τ as a measure of resale transaction costs, unrelated to government regulation.

Then, τ is a social cost as well.

26

5.3 Pre-payment Resale

Since restricting resale is not beneficial, another possibility to consider is encouraging even

more activity on the resale market. In certain situations a buyer may be able to resell the

good before taking delivery. This happens when transferable vouchers are used to ration

goods. We consider such cases by allowing a buyer who acquires the good to resell it after

paying a deposit δ ≤ p.27 This allows the buyer to engage in speculation even if his wealth

is insufficient to pay the price.

Fixing p, speculation declines as the required deposit rises, so more high-valuation

buyers with lower wealth get the good through rationing. The resale price must therefore

rise, so a higher deposit means improved allocative efficiency. Setting the deposit equal to

the price cap eliminates speculation by buyers with w < p. These buyers would not be able

to keep the good anyway, so there is no efficiency loss from excluding them. We conclude

that a deposit equal to the cap gives the highest efficiency.

5.4 Direct Subsidy vs. Rationing with Resale

An alternative method of improving the allocation is to alleviate wealth constraints directly.

It is possible to achieve the efficient allocation by subsidizing individuals directly; however,

there are costs associated with the direct subsidy approach. First, a direct subsidy may

require financing a large sum of money through a distortionary tax. Hence, there is an

efficiency loss associated with a subsidy. Second, the efficiency loss may be magnified by

the difficulty of targeting the subsidy. If the government offers a subsidy based on wealth,

individuals may conceal their wealth. Targeting a subsidy based on individuals’ behavioral

choices is not foolproof either. Suppose that the government offers a subsidy only to those

who purchase the good. This will simply raise all buyers’ willingness-to-pay by exactly the

amount of the subsidy, so the wealth constraint problem is not alleviated at all.

Our rationing-resale mechanism can be seen as a method of subsidizing the poor, but

unlike a direct subsidy it has the advantage of balancing the budget. It is also targeted

in the sense that not all of the wealthy people are subsidized. (In fact, the wealthy can

27Requiring a deposit δ > p has essentially the same effect as raising the price cap to δ, so there is no

loss of generality in assuming that δ ≤ p.

27

be worse-off, on average, since there is probability (1 − S) that they pay more with the

price cap and resale than without the cap.) The only unintended beneficiaries are the

speculators, who have low valuations. But, enriching this group appears to be a necessary

consequence of subsidizing the buyers in a budget-balanced fashion.

5.5 Social Cost of Speculation

In our model, speculators buy and resell the good without generating any direct welfare

cost. The indirect welfare cost is that they reduce the initial probability of allocation for

those with low wealth but a high valuation. In reality, speculation may entail a direct

welfare cost, however, when it involves real transaction costs or opportunity costs. Indeed,

the price caps on new housing in Korea have been criticized for encouraging speculation

and diverting resources away from other productive investment activities. This cost need

not overturn the desirability of the rationing-resale mechanism, however.

Suppose that there exists a project requiring a capital investment of k ∈ [0, 1) that

yields a net return of R ∈ (0, 1). An individual who participates in the rationing process

must forego that project. (For simplicity, assume that k + c > 1, so no individual can both

purchase the good and pursue the project.) Clearly, it would be socially desirable for those

individuals with w ≥ k and v < R + c to invest in the project. Indeed, this will occur in

the competitive equilibrium since pe ≥ c, which implies v − pe < R if v < R + c.

Now suppose that the price is capped at p < pe such that

ρ(p)[r(p)− p] ≥ R,

where

ρ(p) :=S(p)

1−G(p)

is the probability of receipt for those participating in the rationing pool. Speculation then

offers a higher expected profit than the project return, so no agent will invest in the project.

In this case, our rationing-resale mechanism has a direct welfare cost — the unrealized

project return for agents with w ≥ k and v < R + c. This cost need not overturn the

desirability of the rationing-resale mechanism, although moderation in capping the price

may be warranted. The improvement in allocative efficiency that the cap brings may still

outweigh the cost. But if not, there exists p sufficiently close to pe such that r(p)−p < R. It

28

follows that such a cap would not create the kind of speculation that would entail inefficient

project choices.28

5.6 Elastic Supply

Our analysis has assumed that supply is perfectly inelastic, which may be a reasonable

assumption for health care, school choice, and even housing, where rationing is prompted

by limited capacity. Supply may be responsive to price in situations where capacity is not

an issue, however. Moreover, even if capacity is limited in the short run, the ability to

invest means that supply may be elastic in the long run.29 It is therefore important to

establish the robustness of our result — the desirability of rationing and resale — to elastic

supply.

Suppose that the good is supplied competitively according to a twice-differentiable,

increasing aggregate cost function, c(·). The supply at price p is then given by S(p) ∈arg maxq≥0 pq − c(q), or implicitly by c′(S(p)) = p; S(·) is increasing and differentiable.

The competitive equilibrium is characterized by a price, pe, satisfying D(pe) = S(pe).

When the price is capped at p < pe, supply is S(p) < S(pe). Since D(p) > D(pe),

there is excess demand, which will be resolved by random rationing. Resale is permitted,

so all individuals with (w, v) ≥ (p, 0) will participate in the rationing pool. Following the

same argument as before, a resale equilibrium exists and is characterized by the price r(p)

satisfying

D(r(p)) = S(p)− ρ(p)[1− F (r(p))][G(r(p))−G(p)]. (6)

The total realized value is now

V (p) = S(p)φ(r(p))− c(S(p)). (7)

28Essentially the same conclusion would arise with a continuum of projects with returns ranging from

0 to R > 0. In this case, our rationing-resale mechanism will crowd out some low-return projects. Yet,

lowering the cap slightly below pe will entail only a negligible welfare cost, since the projects that are

foregone in pursuit of speculation would have generated only small social returns, whereas the gain from

improved allocative efficiency would be nonnegligible.29For example, new apartments can be built in the long run. Conversely, apartments can be converted

to offices while rent control is in place.

29

Capping the price and rationing has a tradeoff. On the positive side, it yields the now-

familiar benefit of reallocating the good from some low-valuation-high-wealth buyers to

some with high valuations and low wealth. On the negative side, supply falls, so some

current buyers lose access to the good. Moreover, this latter effect is nonnegligible, even

if the cap is just below the market price, since the buyers losing access may have high

valuations. (Some current buyers are wealth constrained, with wealths just above pe but

valuations well above pe.) The net effect is positive if supply is not too elastic.

Proposition 5. If S′(pe)S(pe)

< f(pe)F (pe)

, there exists a price cap, p < pe, such that random

rationing followed by resale yields higher total realized value than the competitive market

does.

Proof: The proof is in the Appendix.

The applicability of Proposition 5 is clear in the two polar cases. If supply is almost

perfectly inelastic, the proposition applies. The previous results continue to hold and

rationing with resale outperforms the competitive market. Conversely, if supply is almost

perfectly elastic, the proposition does not apply. The drop in quantity is so great that this

effect swamps the improved mix of buyers.

It is also worth pointing out that, while the condition in Proposition 5 is sufficient given

random rationing, it may not be necessary in that case or in other cases. If the speculation

problem could be addressed directly, the rationing and resale mechanism would be desirable

regardless of the supply elasticity. That is, if x(w, v) = 0 for all v < p (instead of random

rationing), the result of Proposition 5 always holds. In addition, the same result would hold

if the buyers’ price could be lowered without altering the price that the sellers command,

although such a policy would violate the budget-balancing property, one of the attractive

features of the rationing mechanism.

6 Concluding Remarks

This paper has shown that rationing followed by resale may increase allocative efficiency in

a market in which wealth constraints are important. In fact, a range of rationing schemes

may do better than the unregulated market outcome. Schemes that place goods in the

30

hands of high-valuation buyers obviously work well. It is striking, however, that schemes

placing goods in the hands of low-wealth buyers also do well if resale is permitted. Even

more striking is the possibility that full efficiency can be realized by allocating the entire

stock of the good to the poor and then allowing resale.

The benefits of rationing and resale here suggest a paradigm shift in how particular

goods are allocated. Certain goods tend always to be allocated by a pure market mechanism

whereas others tend to be subject to rationing. The current paper suggests a blurring of

this line when wealth constraints are important. Concretely, the results suggest that one

should consider removing prohibitions on resale in certain markets with price caps, and one

should consider rationing and resale in certain unregulated markets.

There are numerous possible applications of the results. Return to the examples in

Section 2. The benefits of rationing with resale suggest that using a lottery to allocate

transferable educational vouchers may be preferable to a system of local attendance zones

or to the use of a lottery to allocate nontransferable vouchers. Likewise, a case has been

made for rationing and resale in the housing market. The analysis also suggests that keeping

the price of health care below competitive market levels, but allowing patients to sell their

spot in the queue, could improve efficiency. Finally, the analysis provides an efficiency

rationale for employing a draft with tradable deferments for military recruitment. In each

of these cases there may be other objectives or other institutional details that loom large,

but the results here argue for consideration of rationing and resale.

References

[1] Abdulkadiroglu, A., and Sonmez, T. (2003), “School Choice: A Mechanism Design

Approach,” American Economic Review, 93, 729-747.

[2] Abdulkadiroglu, A., and Sonmez, T. (1999), “House Allocation with Existing Ten-

ants,” Journal of Economic Theory, 88, 233-260.

[3] Avery, C., Fairbanks, A. and Zeckhauser, R. (2003), The Early Admissions Game:

Joining the Elite, Cambridge, MA: Harvard University Press.

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[4] Becker, G. (1987), “Why Not Let Immigrants Pay for Speedy Entry?” Business Week,

Mar. 2, 1987, p. 20.

[5] Gali, J. and Fernandez, R. (1999), “To each according to...? Markets, Tournaments,

and the Matching Problem with Borrowing Constraints,” Review of Economic Studies,

66, 799-824.

[6] Guesnerie, R. and Roberts, K. (1984), “Effective Policy Tools and Quantity Controls,”

Econometrica, 52, 59-86.

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[9] Mills, G. and Rockoff, H., “Compliance with Price Controls in the United States and

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rison, ed., The Economics of World War II: Six Great Powers in International Com-

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[11] Roth, A., Sonmez, T. and Unver, M. (2004), “Kidney Exchange,” Quarterly Journal

of Economics, forthcoming.

[12] Tu, Y., and G. Wong, (2002), “Public Policies and Public Resale Housing Prices in

Singapore,” International Real Estate Review, 5, 115-132.

[13] Vickrey, W. (1945), “Measuring Marginal Utility by Reactions to Risks,”Econometrica,

13, 319-333.

[14] Weitzman, M. (1977), “Is the Price System or Rationing More Effective in Getting a

Commodity to Those Who Need it Most?” The Bell Journal of Economics, 8, 517-524.

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Stochastic Rationing,” Econometrica, 56, 1455-1465.

32

Appendix: Proofs

Proof of Proposition 3: Let ψx(v) :=R 1

v vdBx(v)

1−Bx(v)denote the expected value conditional

on exceeding v, given the ex post (i.e., after rationing) distribution. The total realized

value can be expressed as:

Vx =

∫ rx

p

∫ 1

rx

vx(w, v)dF (v)dG(w) +

∫ 1

rx

∫ 1

rx

vdF (v)dG(w)

= S ·∫ rx

p

∫ 1

rx

vax(w)bx(v)dF (v)dG(w) + (1−G(rx))

∫ 1

rx

vdF (v)

= S · Ax(rx)[1−Bx(rx)]

(∫ 1

rxvdBx(v)

1−Bx(rx)

)+ D(rx)

(∫ 1

rxvdF (v)

1− F (rx)

)

= Kx(rx)ψx(rx) + D(rx)φ(rx)

= S

[(1− D(rx)

S

)ψx(rx) +

(D(rx)

S

)φ(rx)

];

where the first equality follows by definition; the second and third follow by substituting

for x(w, v) and integrating; and the last one follows from (3). If x is meritorious, then

ψx(rx) ≥ φ(rx).

Suppose, further, that x merit-dominates y ∈ X and rx ≥ [>]ry. Then,

Vx = S

[(1− D(rx)

S

)ψx(rx) +

(D(rx)

S

)φ(rx)

]

≥ S

[(1− D(ry)

S

)ψx(rx) +

(D(ry)

S

)φ(rx)

]

≥ S

[(1− D(ry)

S

)ψy(rx) +

(D(ry)

S

)φ(rx)

]

≥ [>] S

[(1− D(ry)

S

)ψy(ry) +

(D(ry)

S

)φ(ry)

]

= Vy;

where the first inequality follows from rx ≥ ry (which implies D(rx) ≤ D(ry)) and from

ψx(rx) ≥ φ(rx); the second follows from the relative merit-dominance of x over y; and the

third one follows from the fact that the conditional expectations, ψy(·) and φ(·), are strictly

increasing in the relevant region.

Proof of Proposition 4: It suffices to show that there exists a pair (v, γ) satisfying

the system of nonlinear equations (4) and (5). Observe first that for each v ∈ [p, p∗],

33

there exists γ = t1(v) satisfying (4). Further, t1(·) is increasing and satisfies t1(p) = 0.

Next, let p > pe be such that [1 − F (p)][1 − G(p)] = S. Then, for each v ∈ [p, p], there

exists γ = t2(v) satisfying (5). Further, t2(·) is decreasing and satisfies t2(p) > 0 and

t2(p) = 0. Since both t1(v) and t2(v) are continuous, there exists a unique v∗ ∈ (p, p) such

that t1(v∗) = t2(v

∗) =: γ∗. Obviously, (v∗, γ∗) satisfies (4) and (5). Since v∗ > p and ξ(·) is

increasing, we have VD > VD.

Proof of Proposition 5: It suffices to show that, given the condition, V ′(pe) < 0,

which would mean that lowering the cap increases the total realized value. To that end,

for p ≤ pe, rewrite

V (p) = {ρ(p)[G(r(p))−G(p)] + [1−G(r(p))]}∫ 1

r(p)

vdF (v)− c(S(p)).

Since r(pe) = pe, we have

V ′(pe) = −(1−G(pe))f(pe)r′(pe)pe − (1− ρ(pe))g(pe)r′(pe)

∫ 1

pe

vdF (v)

−ρ(pe)g(pe)

∫ 1

pe

vdF (v)− c′(S(pe))S ′(pe)

= −(1−G(pe))f(pe)r′(pe)pe − F (pe)[1− F (pe)]g(pe)r′(pe)φ(pe)

−[1− F (pe)]2g(pe)φ(pe)− S ′(pe)pe, (8)

where the second equality holds since ρ(pe) = S(pe)/[1 − G(pe)] = D(pe)/[1 − G(pe)] =

1− F (pe), φ(z) =∫ 1

zvdF (v)/[1− F (z)], and c′(S(pe)) = pe.

Meanwhile, totally differentiating both sides of (6) and using ρ(pe) = 1− F (pe) yields

r′(pe) = − S ′(pe) + g(pe)[1− F (pe))]2

g(pe)(1− F (pe))F (pe) + f(pe)(1−G(pe)).

Substituting this into (8) and collecting terms, we get

V ′(pe) = − [φ(pe)− pe](1− F (pe))g(pe)[D(pe)f(pe)− S ′(pe)F (pe)]

g(pe)(1− F (pe))F (pe) + f(pe)(1−G(pe))

= − [φ(pe)− pe](1− F (pe))g(pe)[S(pe)f(pe)− S ′(pe)F (pe)]

g(pe)(1− F (pe))F (pe) + f(pe)(1−G(pe)).

Hence, V ′(pe) < 0 if and only if S′(pe)S(pe)

< f(pe)F (pe)

.

34

v*

pe

pe

MarketEfficient allocation

w

v

Figure 1: Benchmark allocations

pe

p

r(p)

r(p)

Figure 2: Rationing with resale

Probability = 1Probability = ρ < 1

w

v

v*

Need-based allocationEfficient allocation

w

v

Figure 3: Full efficiency of need-based screening

v* w*