# Who Cares More? Allocation with Diverse Preference Intensities††thanks: We
thank Mariagiovanna Baccara, Kfir Eliaz, Faruk Gul, SangMok Lee, Alessandro
Lizzeri, Stephen Morris, and Wolfgang Pesendorfer for very helpful discussions
and feedback. Yariv gratefully acknowledges the support of NSF grant
SES-1629613.

Pietro Ortoleva Evgenii Safonov Leeat Yariv Department of Economics and SPIA,
Princeton University.Department of Economics, Princeton University.Department
of Economics, Princeton University, CEPR, and NBER.

(_This version_ : )

###### Abstract

Goods and services—public housing, medical appointments, schools—are often
allocated to individuals who rank them similarly but differ in their
preference intensities. We characterize optimal allocation rules when
individual preferences are known and when they are not. Several insights
emerge. First-best allocations may involve assigning some agents “lotteries”
between high- and low-ranked goods. When preference intensities are private
information, second-best allocations always involve such lotteries and,
crucially, may coincide with first-best allocations. Furthermore, second-best
allocations may entail disposal of services. We discuss a market-based
alternative and show how it differs.

Keywords: Market Design, Mechanism Design, Allocation Problems.

JEL: C78, D02, D47.

##  1 Introduction

In many allocation problems, from public housing to appointment scheduling to
some school assignments, individuals largely agree on ordinal rankings of
goods: prospective public-housing residents usually prefer earlier
availability, appointment schedulers tend to prefer sooner service, and
parents desire schools in which students perform well on standardized tests.
In such settings, heterogeneity appears in preference intensities, the
cardinal rankings. Public housing recipients or appointment schedulers can
vary in their urgency, parents may differ in their sensitivity to the ranking
of schools their kids attend. Traditional market pricing tools can be used to
account for such preference intensities, and lead to efficient allocations.
However, in many environments jurisprudence and ethical norms prohibit
pricing, making transfers unavailable. How should a social planner allocate
services when prices cannot be used? If preferences are not transparent, how
can a social planner screen those who care more?

We consider a social planner maximizing utilitarian efficiency and
characterize the optimal allocation rules both when preferences are observable
and when they are not. When preferences are observable, the unique first-best
allocation may probabilistically assign either very high-ranked or very low-
ranked services; it may involve a “lottery.” When preferences are not
observable (private information), the planner faces a screening problem. We
show that the unique optimal allocation is fully separating and always
involves such lotteries. Crucially, it coincides with the first-best
allocation for a non-trivial set of environments. In other cases, first- and
second-best allocations differ qualitatively and the second-best allocation
may exhibit disposal of services.

Specifically, we consider an environment in which a continuous supply of goods
differing in quality is allocated to a population of agents. Each agent
requires only one good. There is a natural ranking of the goods’ quality, on
which all agents agree. However, agents’ preference intensities vary: their
valuation of marginal quality changes is different. We capture this
disagreement through a natural comparison of marginal utilities, translating
into an ordering of utility curvature that can be expressed as an ordering of
agents’ absolute risk aversion parameters.

This formulation captures many environments. One example is that of
homogeneous services supplied over time—identical public-housing units that
vary in availability, academic or medical appointments that vary in timing.
When agents are exponential discounters, some may be more patient than others,
corresponding to well-ranked absolute risk aversion comparisons over timed
services. Another example is school choice, where some parents discern small
but compelling differences across highly-ranked schools but view lower-ranked
schools mostly as an undifferentiated mass; other parents place more equal
values on increased school rank. In our baseline setting, agents are of two
types: P for prudent or patient, and I for imprudent or impatient.

Suppose first that the social planner observes agents’ preferences. This is a
convenient technical benchmark that is also relevant for some applications:
public housing officials may be informed of home-seekers’ circumstances and
resulting urgency, academic advisors may be cognizant of students’ deadlines
when scheduling meetings. We show that the optimal, first-best allocation
takes two possible forms. The first has all agents of one type served with
goods of the highest quality and all agents of the other type served with
goods of lower quality: either all $I$-agents are served with higher-quality
goods than $P$-agents, or vice versa. The second possible structure has all
$P$-agents, who are more risk averse, served with goods of middling quality,
while $I$-agents are served with a “lottery” and get, with some probability,
either goods of the highest quality or goods of substantially lower quality.

When are such distributed allocations optimal? Intuitively, the highest-
quality goods are more valuable to $I$-agents. When there are many $I$-agents,
limited supply of the highest-quality goods implies that some $I$-agents must
receive goods that are not of the highest quality. Those $I$-agents who
receive lower-quality goods experience a substantially lower utility. For such
$I$-agents, a further reduction in quality does not come at a substantial
loss. Instead, an equivalent quality reduction for $P$-agents is costlier. It
is therefore optimal to serve the highest-quality goods to some of the
$I$-agents, then serve the $P$-agents with intermediate-quality goods, and,
finally, serve remaining $I$-agents with the lowest-quality goods.

Next, we characterize the optimal allocation when types are unobservable.
Extant literature on allocation problems absent transfers commonly assumes
complete information of preferences (see our literature review below for a few
exceptions). In many cases, however, preference intensities, e.g. urgency of
public-housing seekers or appointment schedulers, cannot be observed or
confirmed. As is standard in screening problems, the social planner then
offers a menu of allocations, tailoring the allocations to agent types.

If the first-best allocation serves all agents of one type with goods of
uniformly higher quality than those other agents receive, there is no hope for
its implementation when types are unobservable: some agent type would have an
incentive to mimic the other. However, the first-best allocation can be
incentive compatible if $I$-agents are served probabilistically with either
the highest- or the lowest-quality goods. $I$-agents may prefer their
allocation to that of $P$-agents as it guarantees them a chance of higher-
quality service, which they value greatly. $P$-agents may prefer their
assigned allocation since it shields them from the lowest-quality goods.
Indeed, we show that, in some cases, the first-best allocation is incentive
compatible.

What happens when the first-best allocation is not incentive compatible? We
show that the second-best allocation is unique and fully separating. It again
takes the form of a distribution over high- and low-quality goods for
$I$-agents, and the allocation of goods of contiguous and intermediate quality
for $P$-agents. In particular, the pooling allocation, which offers all agents
an identical share of the goods’ supply and is inherently fair, is never a
second-best solution.

Another feature unique to the second-best allocation is that it may exhibit
disposal of goods: some agents may receive nothing, even when there is
sufficient supply. This occurs when $P$-agents place little marginal value on
improving the quality of the goods they receive. In this case, the first-best
allocation provides them with the lowest-quality goods. To have an incentive-
compatible allocation, the planner needs to eliminate the appeal of the
$I$-agents’ allocation to the $P$-agents. Certainly, the planner can improve
$P$-agents’ allocation at the expense of $I$-agents. Alternatively, the
planner can worsen $I$-agents’ allocation: instead of providing some
$I$-agents lower-quality goods, the planner can deny them service altogether.
$I$-agents welfare would then be diminished only slightly—their value for the
lower-quality goods is relatively low. However, for $P$-agents, such a change
can make $I$-agents’ allocation substantially less appealing. Denial of
service for $I$-agents can therefore be optimal.

Overall, we show that the infinite-dimensional screening problem can be easily
reformulated as a simple two-dimensional constrained maximization. Second-best
allocations are determined through two levers the planner controls: the set of
high-quality goods $I$-agents receive, and the fraction of $I$-agents that are
served.

While we describe most of our results for the two-type environment, we show
that our qualitative results extend to a setting with an arbitrary number of
types. In particular, the first-best solution can be incentive compatible.
Furthermore, screening agents for their cardinal preferences is always
beneficial: the pooling allocation is never optimal.

We also explore a natural alternative to the second-best solution, in the
spirit of Hylland and Zeckhauser (1979), where agents receive equal shares of
the available goods and can trade through a market. The first welfare theorem
ensures that induced allocations are Pareto efficient. Nonetheless, we show
that resulting allocations may still entail significant efficiency losses
relative to second-best allocations.

#### Related Literature.

A large literature considers screening agents with diverse risk
attitudes—through markets, starting from Rothschild and Stiglitz (1976), or
through auctions, starting from Maskin and Riley (1984). While addressing
related questions to ours, this literature relies heavily on pricing
mechanisms. Therefore, our analysis and the relevant applications are
different.

In the context of time preferences, DellaVigna and Malmendier (2004) and Eliaz
and Spiegler (2006) study screening of time-inconsistent agents. We are not
aware of work on screening of time-consistent agents who vary in patience,
which our analysis encompasses.

Most work on matching and assignment problems assumes complete information of
preferences. Roth (1989) and, more recently, Fernandez et al. (2021)
illustrate some of the new phenomena that emerge in centralized one-to-one
matching markets with incomplete information. Our paper provides insights on
the optimal design of allocation protocols in the presence of a particular
form of incomplete information.

A recent and growing literature studies dynamic matching and allocations: see
Akbarpour et al. (2020), Baccara et al. (2020), Bloch and Cantala (2017), and
the survey by Baccara and Yariv (2021). Leshno (2019) considers the
implications of a desire to speed up assignments on the design of a dynamic
allocation procedure. Dimakopoulos and Heller (2019) show that using wait time
as a contractual term can be beneficial in the German market for entry-level
lawyers.111In a somewhat different setting, Ely and Szydlowski (2020)
illustrate how goalposts can be efficiently modified over time in a moral-
hazard environment in which tasks take different amounts of time depending on
their (uncertain) difficulty. Schummer (2021) studies the impacts of risk
aversion and impatience on the performance of deferral rights in waiting
lists.

The link between exponential discounting and risk attitudes over lotteries
involving timed services has been illustrated by Dejarnette et al. (2020) and
is utilized in some of our examples.

The idea that disposal can be a useful instrument for relaxing incentive
constraints is present in other environments, as seen, for example, in Alatas
et al. (2016) in the context of applications for aid programs and Austen-Smith
and Banks (2000) in the context of cheap talk. Technically, our observation
regarding the optimal use of probabilistic allocations relates to Gauthier and
Laroque (2014), who provide general necessary and sufficient conditions for
stochastic optimization solutions.222When a buyer and seller have correlated
valuations of a good, Kattwinkel (2020) shows that the seller’s optimal
mechanism may involve randomization. Reminiscent of some of our results, with
positive correlation, the good may not be allocated to a higher-value buyer
with higher probability.

Finally, we discuss a market-based implementation that is inspired by Hylland
and Zeckhauser (1979) and work that followed.

##  2 The Allocation Problem

###  2.1 Setup

We study the allocation of a continuum of goods to agents of heterogeneous
preferences.

#### Goods.

Goods are characterized by a one-dimensional attribute $x\in[0,X]$. They can
stand for public housing units available at different times, doctor
appointments that vary in physician’s expertise or date of service, schools
that vary in quality, etc. The available supply of different goods is captured
by a strictly positive continuous density $f$ over $[0,X]$, with cumulative
distribution $F$.

#### Agents.

We start by considering two types of agents. We extend our analysis to $N$
types in Section 5. Agents are of type $P$ and $I$, with strictly positive
masses $\mu_{P}$ and $\mu_{I}$, respectively. Each agent demands one unit of
the good. For presentation ease, we assume there is sufficient supply; that
is, $F(X)\geq\mu_{P}+\mu_{I}$.333When supply in insufficient, the analysis is
broadly similar but requires consideration of multiple cases depending on the
severity of the goods’ scarcity.

$P$\- and $I$-agents have utilities $u_{P}$ and $u_{I}$ over
$\mathbb{R}_{+}\cup{\diamond}$, respectively, where $\diamond$ denotes
receiving none of the goods. Agents agree on the ordinal ranking of goods:
both $u_{P}$ and $u_{I}$ are strictly decreasing and twice continuously
differentiable over $\mathbb{R}_{+}$.444Goods’ labels can therefore be thought
of as a continuous rank, with $x=0$ representing the most-preferred good, and
$x=X$ representing the least-preferred good. They also rank any of the goods
strictly higher than $\diamond$. Without loss of generality, we posit
$u_{P}(\diamond)=u_{I}(\diamond)=0$. We also assume that as goods’ quality
deteriorates to $\infty$, their value approaches that of not receiving any
good: $\lim_{x\to\infty}u_{P}(x)=\lim_{x\to\infty}u_{I}(x)=0$.

While agents have the same ordinal ranking of goods, they disagree on cardinal
assessments. Specifically, we assume that for all $x\in\mathbb{R}_{+}$,

$\quad\quad\frac{u_{P}^{\prime\prime}(x)}{u_{P}^{\prime}(x)}>\frac{u_{I}^{\prime\prime}(x)}{u_{I}^{\prime}(x)}.$
(1)

That is, since agents’ utilities are decreasing, $P$-agents have greater
absolute risk aversion than $I$-agents. Put another way, $I$-agents have “more
convex” or “less concave” utility functions. We emphasize that this comparison
is relative—both utilities can be convex, concave, or change curvature within
the domain. Below we discuss a few examples that fit our framework.

#### Allocations.

A lottery is a probability distribution on $[0,X]\cup\\{\diamond\\}$. Since
the supply $f$ has continuous density, we focus on lotteries that have no mass
points on $[0,X]$. A lottery $q$ then specifies a density over $[0,X]$.
Abusing notation, we denote by $q(\diamond)$ the remaining probability:
$q(\diamond):=1-\int_{[0,X]}q(x)\mathrm{d}x$. For any measurable
$A\subseteq[0,X]\cup\\{\diamond\\}$, we denote
$q(A):=\int_{A\backslash\\{\diamond\\}}q(x)\mathrm{d}x+\mathbbm{1}_{\diamond\in
A}q(\diamond)$.

An allocation is a pair $(q_{P},q_{I})$, where $q_{P}$ and $q_{I}$ are
$P$-agents’ and $I$-agents’ allocation, or lottery, respectively. An
allocation is feasible if goods assigned are available: for (almost) all
$x\in[0,X]$

$\mu_{P}q_{P}(x)+\mu_{I}q_{I}(x)\leq f(x).$

While feasibility is a natural requirement, in some applications the planner
may be able to weaken it by lowering the quality of some good: for example,
for allocations of goods over time, the planner may be able to store some of
the unassigned goods, increasing availability in future periods. This
naturally expands the set of feasible allocations. As it turns, allowing for
such expansion does not alter our results. We discuss details in our
Conclusions and Online Appendix.

Finally, the following notation will be useful: for any measurable
$A,B\subseteq[0,X]\cup\\{\diamond\\}$, $A\triangleleft B$ denotes the case in
which any element in $A$ has strictly higher quality than any element of $B$;
that is, $x<x^{\prime}$ for any $x\in A$, $x^{\prime}\in B\backslash\diamond$,
and $\diamond\notin A$.

#### Expected Payoffs.

Agents evaluate their allocation using expected utility. That is, for
$k\in\\{P,I\\}$, $k$-agents’ utility from allocation $q$ is given by:555Recall
that $u(\diamond)$ has been normalized to 0, and therefore does not appear in
the expression.

$V_{k}(q)=\overset{X}{\underset{0}{\int}}u_{k}(x)q_{k}(x)\mathrm{d}x.$

###  2.2 Examples

#### Assignment Problems and Risk Attitudes.

The set $[0,X]$ may stand for different qualities of goods assigned at the
same time: the quality of schools in a school-choice problem, of houses in a
real-estate market, of doctors in healthcare allocation problems, and so
forth. In many applications, ordinal preferences of agents are highly
correlated—schools may have publicly available rankings and houses may have
features desired by most. Our model focuses on the case in which ordinal
preferences are identical, but cardinal preferences are different. Some
individuals have large marginal returns for improved quality, others less so.
In this context, our condition (1) on utilities fits many common functional
forms: for example, CRRA or CARA utilities of varying parameters are ranked
via our condition.

Our condition on utilities can also be read directly in terms of heterogeneity
in risk attitudes. The planner’s problem can thus be seen as one of screening
different risk attitudes over quality. As discussed above, most extant work
considers screening of agents over risk attitudes using pricing mechanisms,
which may not be germane to many of the applications we consider.

#### Timing of Goods and Services.

A natural application is to homogeneous goods available at different dates,
where $x\in[0,X]$ denotes a delivery time. Agents have instantaneous utility
for the good normalized to $1$, but discount at different rates: $r_{P}$ for
the more patient $P$-agents and $r_{I}$ for the more impatient $I$-agents,
where $0<r_{P}<r_{I}$.

Many examples fit this application: scheduling appointments or services,
public housing available over time, etc. The planner’s problem is then one of
general scheduling, determining how a given supply of timed appointments,
houses, and the like should be allocated when agents have different discount
rates, or “urgency”.

Dejarnette et al. (2020) illustrate that exponential discounters are strictly
risk seeking over the date at which they receive the good. Indeed, impatient
$I$-agents are strictly more risk seeking than patient $P$-agents: since they
care more about immediate service, they are more willing to take lotteries
with larger spread that provide either highly-desirable quick service, or
greatly delayed service. Formally, we have
$e^{-r_{P}x}=\left(e^{-r_{I}x}\right)^{\frac{r_{P}}{r_{I}}}$, and utilities
satisfy our assumption (1).

While exponential discounting is a natural case, our analysis applies also to
other forms of utility loss from waiting, as long as (1) holds. For example,
agents could be present-biased, discounting can be hyperbolic or quasi-
hyperbolic, and so on.

##  3 First-best Allocations

We begin with the case in which types are observable. This is not only a
natural theoretical benchmark, it also speaks to various applications:
healthcare systems may be able to assess patients’ urgency, social services
may be able to gauge individuals’ immediate needs for public housing.

The planner’s problem is to find a feasible allocation $(q_{P},q_{I})$ that
maximizes the weighted utilitarian welfare function $W$:

$W(q_{P},q_{I}):=\alpha\mu_{P}V_{P}(q_{P})+(1-\alpha)\mu_{I}V_{I}(q_{I}),$ (2)

where $\alpha\in(0,1)$ denotes the weight placed on $P$-agents’ expected
utility. While we consider a general model, $\alpha=\frac{1}{2}$ is a natural
special case in which agents are valued equally. We call a solution to this
problem the solution to the planner’s problem or the first-best.

Denote by $\overline{X}:=F^{-1}(\mu_{I}+\mu_{P})$ the lowest quality needed to
exhaust demand if only the best qualities are used. Since we assume sufficient
supply, in the first-best allocation all agents receive a good and only
highest-quality goods are used. Thus, the trade-off that the planner needs to
resolve pertains only to the allocation of goods within $[0,\overline{X}]$.
Which goods should go to $I$-agents and which to $P$-agents? The trade-off is
captured by the difference between agents’ utilities and summarized by the
function $g:\mathbb{R}_{+}\cup\\{\diamond\\}\to\mathbb{R}$ defined as

$g(x):=\alpha u_{P}(x)-(1-\alpha)u_{I}(x).$ (3)

The planner would like to assign a good of quality $x$ to $P$-agents when
$g(x)$ is high, and to $I$-agents when $g(x)$ is low. The characterization of
the first-best solution thus tracks the shape of $g$.

######  Lemma 1.

The function $g$ is single-peaked and strictly quasi-concave.

For intuition, take our example of homogeneous goods over time, where utility
is of the form $e^{-r_{j}x}$. If equal weights are placed on both types, then
$g(0)=0$—there is no valuation difference between types at time zero. The
difference $g(x)$ becomes arbitrarily small for very late delivery time.
However, $g$ remains strictly positive in between and there is a unique
maximum of $g(x)$ for some strictly positive $x\in[0,X]$.

The first-best solution can be characterized using the function $g$. Recall
that $\overline{X}$ is the lowest quality of goods assigned. For any $j=I,P$,
denote by $\overline{X}_{j}:=F^{-1}(\mu_{j})$ the lowest quality of goods
assigned to $j$-agents if $j$-agents were to receive the best-quality goods.
Naturally, $\overline{X}_{j}<\overline{X}$ for $j=I,P$.

Now consider $g(\overline{X}_{I}),g(\overline{X}_{P})$, and $g(\overline{X})$.
Suppose first that $g(\overline{X}_{I})<g(\overline{X})$. This is illustrated
in panel (a) of Figure 1. The planner assigns goods to $I$-agents when $g$ is
low and to $P$-agents when $g$ is high. This is achieved by exhausting
$I$-agents’ demand with the highest-quality goods, namely allocating them
goods in $[0,\overline{X}_{I}]$. Lower-quality goods in
$[\overline{X}_{I},\overline{X}]$ are then allocated to $P$-agents. We term
the resulting allocation structure IP.

Figure 1: Computing the first-best allocation.

Suppose $g(\overline{X}_{P})\leq g(0)$, as depicted in panel (b) of Figure 1.
Because $g$ is single peaked, $g$ is decreasing to the right of
$\overline{X}_{P}$. Thus, the highest values of $g$ are achieved to the left
of $\bar{X}_{p}$, and it is optimal to exhaust $P$-agents’ demand with the
highest-quality goods, those with quality in $[0,\overline{X}_{P}]$. Lower-
quality goods, in $[\overline{X}_{P},\overline{X}]$ are assigned $I$-agents.
We term this allocation structure PI.

In the remaining case, $g(\overline{X}_{I})>g(\overline{X})$ and
$g(\overline{X}_{P})>g(0)$, as in panel (c) of Figure 1. The highest levels of
the $g$ occur between $\overline{X}_{I}$ and $\overline{X}$. The optimal
allocation then has $P$-agents served with goods in
$(x_{1},x_{2})\subsetneq(0,\overline{X})$, with $g(x_{1})=g(x_{2})$ and
$F(x_{2})-F(x_{1})=\mu_{P}$.666If $g(x_{1})<g(x_{2})$, since $g(x)$ is
continuous, the planner would benefit from serving a small mass of $P$-agents
with goods of qualities just below $x_{2}$ instead of qualities just below
$x_{1}$. A similar argument follows if $g(x_{1})>g(x_{2})$. The interval
$(x_{1},x_{2})$ is uniquely determined by these two constraints. The resulting
allocation has $I$-agents served with relatively high- and low-quality goods,
while $P$-agents are served with intermediate-quality goods. We term this
allocation structure IPI.

The following proposition summarizes our discussion. For any $A\subset[0,X]$,
denote by $f~{}\big{|}~{}A$ the allocation that assigns the full supply
available in $A$.777Formally,
$\left(f~{}\big{|}~{}A\right)~{}(x)=f(x)\cdot\mathbbm{1}\\{x\in
A\\}\big{/}\int_{y\in A}f(y)dy$.

######  Proposition 1 (First-Best).

There exists a unique first-best allocation $(q_{P},q_{I})$. Moreover:

  1. 1.

If $g(\overline{X}_{I})\leq g(\overline{X})$, then $I$-agents are assigned
higher-quality goods than $P$-agents:

$q_{P}~{}=~{}f~{}\big{|}~{}[\overline{X}_{I},\overline{X}]\quad\quad
q_{I}~{}=~{}f~{}\big{|}~{}[0,\overline{X}_{I}]\quad\quad\quad\quad(IP~{}~{}structure)$

  2. 2.

If $g(\overline{X}_{P})\leq g(0)$, then $P$-agents are assigned higher-quality
goods than $I$-agents:

$q_{P}~{}=~{}f~{}\big{|}~{}[0,\overline{X}_{P}]\quad\quad
q_{I}~{}=~{}f~{}\big{|}~{}[\overline{X}_{P},\overline{X}]\quad\quad\quad\quad(PI~{}~{}structure)$

  3. 3.

Otherwise, $P$-agents are assigned goods with quality in between that of those
assigned to $I$-agents:

$q_{P}~{}=f~{}\big{|}~{}[x_{1},x_{2}],\quad\quad
q_{I}~{}=f~{}\big{|}~{}[0,x_{1}]\cup[x_{2},\overline{X}]\quad\quad(IPI~{}~{}structure)$

where $0<x_{1}<x_{2}<\overline{X}$ is the unique solution of
$g(x_{1})=g(x_{2})$ and $F(x_{2})-F(x_{1})=\mu_{P}$.

Our discussion also suggests how, in general, neither of the three cases—IP,
PI, or IPI—is knife-edge. This is immediate to see in our example of
homogenous goods over time. When $\alpha\leq\frac{1}{2}$, we have
$g(x)>g(0)=0$ for all $x>0$, and the PI structure is never optimal. However,
there are positive measures of discount factors, welfare weights, and masses
of $I$\- and $P$-agents, for which the IP structure or the IPI structures are
optimal. Similarly, if $\alpha>\frac{1}{2}$, then for any $r_{P}<r_{I}$, we
have $g(x)<g(0)$ for large enough $x$. Thus, the PI structure is optimal when
a large enough mass of $P$-agents is present.888Arguments of robustness can be
made stronger. We could endow both the set of viable utilities and the set of
supply functions with a specific metric on each dimension of the problem
(e.g., the sup-norm for utilities or supply functions) and show that, for any
of the three plausible structures, there exists an open set of
parameters—masses, utilities, and welfare weights for either agent type, as
well as supply functions—so that the specific structure is optimal within that
set.

#### Comparative Statics.

How does the solution structure change with the underlying parameters of the
environment? Fix the overall mass of agents and assume first that the
maximizer of $g$, call it $x^{*}$, is in $(0,\overline{X})$.

If $g(0)<g(\overline{X})$, then $g(x)>g(0)$ for all $x\in(0,\overline{X}]$ and
only cases (1) and (3) of Proposition 1 can occur. When $\mu_{I}$ is small,
$\overline{X}_{I}$ is close to $0$ and an IP structure is optimal. As
$\mu_{I}$ increases, $\overline{X}_{I}$ increases and, for some level of
$\mu_{I}$, $g(\overline{X}_{I})=g(\overline{X})$. Any further increase of
$\mu_{I}$ yields an IPI structure as the solution.

Similarly, if $g(0)>g(\overline{X})$, then $g(x)>g(\overline{X})$ for all
$x\in[0,\overline{X})$ and only cases (2) and (3) of Proposition 1 can occur.
When $\mu_{I}$ is small, $\overline{X}_{P}$ is close to $\overline{X}$ and a
PI structure is optimal. As $\mu_{I}$ increases, or $\mu_{P}$ decreases,
$\overline{X}_{P}$ decreases. As before, for some level of $\mu_{I}$,
$g(\overline{X}_{P})=g(0)$. Any further increase of the mass of $I$-agents
yields an IPI structure as optimal.

If $x^{*}$ is either $0$ or $\overline{X}$, then $g(x)$ is monotone over
$[0,\overline{X}]$, and changes in the relative masses of agent types do not
affect the structure of the first-best solution: if $x^{*}=0$, we have a PI
structure; if $x^{*}=\overline{X}$, an IP one.

######  Corollary 1.

Fix the overall mass of agents $\mu_{I}+\mu_{P}$. Let
$x^{*}:=\underset{x\in[0,\overline{X}]}{\textrm{arg~{}max}}~{}g(x)$, then:

  1. 1.

If $x^{*}\in(0,\overline{X})$ and $g(0)<g(\overline{X})$, then there exists
$\tilde{\mu}_{I}$ such that, for all $\mu_{I}<\tilde{\mu}_{I}$, the first-best
solution exhibits the IP structure and, for all $\mu_{I}>\tilde{\mu}_{I}$, the
first-best solution exhibits the IPI structure.

  2. 2.

If $x^{*}\in(0,\overline{X})$ and $g(0)>g(\overline{X})$, then there exists
$\overline{\mu}_{I}$ such that, for all $\mu_{I}<\overline{\mu}_{I}$, the
first-best solution exhibits the PI structure and, for all
$\mu_{I}>\overline{\mu}_{I}$, the first-best solution exhibits the IPI
structure.

  3. 3.

If $x^{*}\in\\{0,\overline{X}\\}$, the first-best solution exhibits the same
structure, either PI or IP, regardless of the relative masses of agents.

We can also analyze the impact of the relative curvature of utilities.
Considering again our example of homogeneous goods over time, we have
$g(x)=\alpha e^{-r_{P}x}-(1-\alpha)e^{-r_{I}x}$. If agents are weighted
equally ($\alpha=\frac{1}{2}$), then $g(0)<g(x)$ and $g$ is maximized at
$x^{*}=\frac{\ln{r_{I}}-\ln{r_{P}}}{r_{I}-r_{P}}$. When $r_{P}$ is low, so
that $P$-agents are patient and their utilities are fairly flat over delivery
times, serving the impatient $I$-agents first is optimal and the first-best
solution exhibits an IP structure. As $r_{P}$ grows, the maximizing $x^{*}$
becomes small, and the first-best solution exhibits an IPI structure.
Intuitively, $I$-agents are impatient enough so that, beyond a certain early
time, further delays do not entail significant welfare costs. However, for the
more patient $P$-agents, minor delays are not as costly, but substantial
delays are. Figure 2 depicts the regions of discount factors corresponding to
each first-best structure, IP or IPI, assuming equal mass of agents and
uniform supply.999In particular, $\alpha=\mu_{I}=\mu_{P}=\frac{1}{2}$, and
supply with total mass $3/2$ is distributed uniformly over $[0,5]$.

Figure 2: First-best allocations in the timed homogeneous-good example as a
function of agents’ discounts.

##  4 Incentive-Compatible Mechanisms

We now turn to the case in which types are not observable: in some cases,
urgency for or preferences over public-housing units are difficult to
ascertain; for mundane treatments, medical offices may be unable to assess the
necessity for quick attention; universities may be unable to assess preference
intensities over dorm rooms; school systems may not be privy to parents’ or
students’ strength of preferences for one school over the other; etc. As
before, the welfare-maximizing mechanism designer would like to associate a
different lottery to each agent type. However, now the choice of who receives
which lottery is effectively done by the agents—they report their type, which
yields an allocation. The designer then needs to take care of additional
constraints, corresponding to agents picking lotteries tailored for them. This
boils down to a standard screening problem. We call it the mechanism-designer
problem:

$\begin{array}[]{lrlll}&\lx@intercol\hfil\underset{(q_{P},q_{I})}{\text{max}}\
\ \alpha\mu_{P}V_{P}(q_{P})+(1-\alpha)\mu_{I}V_{I}(q_{I})\text{ \ \ \ such
that }\hfil\lx@intercol\\\ (IC_{kj})&\ \ \ \ \ \ \ \ \ V_{k}(q_{k})&\geq\
V_{k}(q_{j})&\ \ \ \ \ \ \forall k,j\in\\{P,I\\}\\\ (\text{Feasibility})&\ \ \
\ \ \ \ \ \ \mu_{P}q_{P}(x)+\mu_{I}q_{I}(x)&\leq\ f(x)&\ \ \ \ \ \ \forall
x\in[0,X].\end{array}$

Like the planner, the mechanism designer chooses a menu $(q_{P},q_{I})$, where
$q_{k}$ is the lottery designed for $k$-types, $k=P,I$. However, the mechanism
designer needs to respect additional incentive-compatibility constraints,
$IC_{kj}$, with $k,j\in\\{P,I\\}$, ensuring that $k$-agents do not want to
emulate $j$-agents.

Standard arguments guarantee that a solution to the mechanism designer’s
problem always exists (see Appendix). We refer to this solution as the second-
best.

###  4.1 Can the First-Best be Achieved?

We begin our analysis by asking: Can first-best allocations be achieved when
types are unobservable? Solutions with an IP or a PI structure are naturally
not incentive compatible: agents receiving lower-quality goods would benefit
from misreporting their type—no matter the curvature of the utility function,
their allocation is first-order stochastically dominated.

When the first-best is of the IPI form, however, allocations are no longer
ranked in terms of first-order stochastic dominance; which one is preferred
depends on agents’ utility functions. We assumed that $u_{I}$ decreases more
sharply at higher qualities than $u_{P}$. Thus, $I$-agents may be willing to
accept the risk of getting lower-quality goods in exchange for the chance of
getting higher-quality ones; $P$-agents, instead, may prefer a more “balanced”
allocation. Put differently, in an IPI allocation, the lottery for $I$-agents
is more risky than the one for $P$-agents. Because $I$-agents are more risk
seeking, some IPI allocations can be incentive compatible.

######  Proposition 2.

There exists a non-degenerate closed interval of positive weights $\alpha$ for
which the first-best allocation is incentive compatible.

The proposition shows that there are non-trivial cases in which the first-best
is achievable via an incentive compatible mechanism.101010In fact, the proof
shows that there exists a closed interval within $(0,1)$ such that the first-
best allocation is incentive compatible if and only if the welfare weight
$\alpha$ is in that interval. For the time-discounting case, for example, for
any supply function, there is an open set of the parameters—planner’s weights,
discount factors, masses of types—for which the first-best allocation
coincides with the second-best allocation.

###  4.2 Features of Incentive-Compatible Mechanisms

To characterize the solution of the mechanism designer’s problem, we identify
several necessary features it must exhibit.

#### No inverted spread and its implications.

We already saw that allocations of the form IPI may be incentive compatible
since they exhibit a larger “spread” in the allocation tailored to the more
risk-seeking $I$-agents. We now define the mirror image, what we call
an“inverted spread,” where $I$-agents receive goods of quality in-between that
provided to $P$-agents.

######  Definition 1.

An allocation $(q_{P},q_{I})$ exhibits an _inverted spread_ if there exist
$A,B,C\subseteq[0,X)\cup\\{\diamond\\}$ such that $A\triangleleft
B\triangleleft C$ and $q_{P}(A),q_{I}(B),q_{P}(C)>0$.

Our first step illustrates inverted spreads never occur in the second-best
solution.

######  Lemma 2.

Solutions of the mechanism designer’s problem never exhibit an inverted
spread.

The intuition is illustrated in Figure 3. Suppose an allocation exhibits an
inverted spread. Fix one lottery corresponding to intermediate-quality goods
within the support of $I$-agents’ allocation—the small yellow rectangle. There
exist a lottery that provides, with some probability, each of two qualities
within the support of $P$-agents’ allocation—the small red rectangles in the
figure—such that $P$ agents are indifferent between the two lotteries. Since
$I$-agents are more risk loving, they strictly prefer the lottery that has
more extreme qualities in its support. Thus, the original allocation cannot be
optimal: the designer can “swap” some of the goods, assigning a small amount
of intermediate-quality goods to $P$-agents, and more spread-out quality goods
to $I$-agents. This increases welfare preserving incentive compatibility.

Figure 3: Intuition for Lemma 2, ‘No inverted spread.’

Three implications follow. First, all solutions must be “fully” separating: a
given quality level is never assigned to both types (except for measure-zero
sets). If it did, the allocation would exhibit an inverted spread. This
immediately rules out the pooling allocation—placing uniform probability over
qualities from 0 to $\overline{X}$—as a possible solution.111111Formally, the
pooling allocation $\big{(}q^{\mathrm{pool}}_{P},q^{\mathrm{pool}}_{I}\big{)}$
is defined by
$q_{I}^{\mathrm{pool}}=q_{P}^{\mathrm{pool}}=f~{}|~{}[0,\overline{X}]$.

A second implication is that, in any solution, the two IC constraints cannot
bind at the same time. Indeed, suppose both $IC$ constraints bind. Then, both
types of agents must be indifferent between the allocations. As Expected
Utility is linear in probabilities, agents must also be indifferent between
any convex combination of the allocations. Thus, any such convex combination,
and in particular the pooling allocation, is also a solution, in
contradiction.

The third implication is that all $P$-agents receive a good:
$q_{P}(\diamond)=0$. Certainly, if all $P$-agents receive lower-quality goods
than $I$-agents or no goods at all, incentive compatibility is violated.
Suppose some $I$-agents receive lower-quality goods than some $P$-agents and
that other $P$-agents are denied service. That would yield an inverted spread,
which cannot occur.

######  Corollary 2.

If $(q_{P},q_{I})$ is a solution of the mechanism designer’s problem, then
$\mathrm{supp}(q_{P})\cap$ $\mathrm{supp}(q_{I})$ $\cap$ $[0,X]$ has measure
zero. Moreover, $IC_{IP}$ and $IC_{PI}$ are not both binding.

####  $P$-agents served in “one-block.”

The next step illustrates that $P$-agents must be served in one continuous
block. That is, there exists an interval of qualities such that all supply in
that interval is given to $P$-agents, and this exhausts their demand.

######  Lemma 3.

If $(q_{P},q_{I})$ is a solution of the mechanism designer’s problem, then
there exists $x_{1},x_{2}\in[0,X]$ such that $F(x_{2})-F(x_{1})=\mu_{P}$, and
$q_{P}=f~{}|~{}[x_{1},x_{2}]$.

Intuitively, $P$-agents are more risk averse, and are therefore optimally
served contiguously. Because there is no inverted spread, no $I$-agents are
served in between. The lemma shows that there are no “gaps” with goods left
unassigned. To see why, suppose that a solution with such a gap exists and
recall that IC-constraints cannot bind simultaneously.

If $IC_{IP}$ does not bind, consider a small quantity of the lower-quality
goods served to the $P$-agents and swap it for a small quantity of equal mass
of the higher, unassigned quality goods in the “gap”—see the left panel of
Figure 4. This would improve $P$-agents’ allocation and welfare. Since
$IC_{IP}$ does not bind, for a small enough mass of goods swapped, incentive
compatibility for $I$-agents is preserved, in contradiction.

Suppose $IC_{IP}$ binds, so that $IC_{PI}$ does not. Consider a small quantity
of lower-quality goods served to $I$-agents and swap it for an equal quantity
of higher, unassigned quality goods in the “gap”—see the right panel of Figure
4. Such a swap improves $I$-agents’ utility and, if it entails a small enough
mass, does not violate incentive compatibility for $P$-agents, again in
contradiction.

Figure 4: Intuition of Lemma 3, $P$-agents served in one block.

#### Disposal of goods for $I$-agents.

We established that all $P$-agents are served in one continuous block. We now
discuss the use of disposal, when some agents—namely, the $I$-agents—are
denied service despite the availability of goods.

######  Definition 2.

An allocation $(q_{P},q_{I})$ exhibits _disposal_ for agent of type
$k\in\\{P,I\\}$ if there exist sets $A,B\subset[0,X]\cup\\{\diamond\\}$ such
that $A\triangleleft B$, $f(A)-\mu_{P}q_{P}(A)-\mu_{I}q_{I}(A)>0$, and
$q_{k}(B)>0$.

We show that the only possible form of disposal is one that precludes some
$I$-agents from receiving goods.

######  Lemma 4.

If $(q_{P},q_{I})$ is a solution of the mechanism designer’s problem, then:

  1. 1.

$(q_{P},q_{I})$ does not exhibit disposal for $P$-agents;

  2. 2.

If $(q_{P},q_{I})$ exhibits disposal for $I$-agents, then $q_{I}(\diamond)>0$.
Moreover, there are no $A,B\subset[0,X]$ such that $A\triangleleft B$,
$\mu_{I}q_{I}(A)+\mu_{P}q_{P}(A)<f(A)$, and $q_{I}(B)>0$.

Suppose a second-best solution exhibits disposal such that some $I$-agents are
served with a lower-quality good when higher qualities are available, as
depicted in Figure 5. We can use similar arguments to those already used to
show there are no inverted spreads. Consider two lotteries. One lottery is
supported by unassigned higher-quality goods and available lower-quality
goods—the small red rectangles in the figure. A second lottery is supported by
intermediate-quality goods within the support of $I$-agents—the yellow
rectangle in the figure. We can set the probabilities so that $P$-agents are
indifferent. Since $I$-agents are more risk loving, they strictly prefer the
first, more diverse lottery. As before, a swap of these lotteries increases
welfare while preserving incentive compatibility, contradicting optimality.

Figure 5: Intuition for Lemma 4, only disposal can be denial of goods for
$I$-agents.

This line of arguments implies that if the mechanism designer utilizes
disposal, she would do so even if an additional supply of goods of quality
lower than $X$ were added. We show below that disposal may indeed occur in the
second-best allocation.

###  4.3 The Optimal Mechanism

We are now ready to state the characterization of the second-best allocation.

######  Proposition 3.

There exists a unique solution of the mechanism designer’s problem, given by

$q_{P}~{}=~{}f~{}\big{|}~{}[x_{1},x_{2}]$ $q_{I}~{}=~{}(1-\beta)\cdot
f~{}\big{|}~{}[0,x_{1}]\cup[x_{2},x_{3}]~{}+~{}\beta\cdot\delta_{\diamond}$

where $\beta\in[0,1)$, $0<x_{1}<x_{2}\leq x_{3}\leq X$,
$F(x_{2})-F(x_{1})=\mu_{P}$, and
$F(x_{1})+\left(F(x_{3})-F(x_{2})\right)=(1-\beta)\mu_{I}$.

The proposition shows that the second-best allocation is unique, and always
takes the IPI structure, with the caveat that some $I$-agents may not be
served at all. $P$-agents’ demands are exhausted over an interval
$[x_{1},x_{2}]$, where $x_{1}>0$. $I$-agents are served with high-quality
goods in $[0,x_{1})$ and lower-quality goods in $[x_{2},x_{3}]$. These may not
exhaust their demands and with probability $\beta$ they receive no good.

The proposition also illustrates that the original, infinite-dimensional
problem is effectively reduced to a two-dimensional problem. There are only
two levers the mechanism designer can use. The first is $x_{1}$, which
specifies the quantity of the highest-quality goods $[0,x_{1}]$ distributed to
$I$-agents. Once $x_{1}$ is set, the length of the interval $[x_{1},x_{2}]$
follows: the supply of goods over that interval must coincide with the
$P$-agents’ mass. Any choice of $x_{1}$ therefore uniquely pins down $x_{2}$.
The second lever is then $\beta$, or equivalently $x_{3}$, which governs the
probability that $I$-agents are served with a good and, in turn, the interval
of lower-quality goods $[x_{2},x_{3}]$ they are served with.

Why might it be optimal to dispose of some goods and not serve $I$-agents?
Consider a candidate allocation in which all $I$-agents are served and receive
goods in $[0,x_{1}]$ and $[x_{2},x_{3}]$, while $P$-agents receive goods in
$[x_{1},x_{2}]$. Suppose it violates $IC_{PI}$: $P$-agents prefer $I$-agents’
allocation. There are two adjustments the mechanism designer can contemplate.
She can make $P$-agents’ allocation more attractive, reducing $x_{1}$ and
$x_{2}$ to improve the quality of goods they receive. Alternatively, the
designer can reduce the desirability of $I$-agents’ allocation by decreasing
the quality of goods they receive, potentially precluding some $I$-agents from
goods altogether. Taking away goods of very low quality from $I$-agents has
little impact on their welfare. But it may substantially reduce the appeal of
the $I$-agents’ allocation for $P$-agents. Due to the asymmetry in how agents
evaluate lower-quality goods, it may be efficient for the mechanism designer
to use disposal.

We illustrate the different regions corresponding to second-best allocations
for our example of homogeneous goods that vary in their delivery times. Figure
6 displays the structure of the second-best solution for different discounts
(Full disposal refers to solutions in which $I$-agents are either served with
higher-quality goods than $P$-agents, or not served at all).121212As before,
the figure corresponds to an environment with an equal mass of each agent
type: $\mu_{I}=\mu_{P}=1/2$, weighted equally by the mechanism designer. The
supply has total mass of $3/2$ and is distributed uniformly over $[0,5]$.
Clearly, all allocation forms occur for a substantial set of parameters.
Second-best allocations are also first-best allocations for intermediate
values of discount factors. Finally, disposal occurs when $P$-agents’ discount
factor is low and there is a sufficient wedge between the discount factors.

Figure 6: Second-best allocations in the timed homogeneous-good example.

###  4.4 Welfare Implications

We now discuss welfare properties of our second-best allocation. As a
benchmark, we consider the pooling allocation, which is incentive compatible
and inherently “fair,” but never optimal. Who gains and who loses as we move
from the pooling allocation to the second-best allocation?

Welfare comparisons can be determined via which $IC$ constraint binds, using
two observations. First, when $IC_{kj}$ binds, $k$-agents are indifferent
between their allocation and $j$-agents’ allocation. They are therefore
indifferent between the allocation they receive and any mixture of the two.
When there is no disposal, they are then also indifferent between the
allocation they receive and the pooling allocation. It follows that, when
there is no disposal, if $IC_{kj}$ binds, $k$-agents are as well off as in the
pooling allocation. With disposal, mixtures of the second-best allocations
each type receives generate strictly lower utility than the pooling
allocation.

Second, since the pooling allocation is never optimal, at least one of the
agents must prefer the second-best. It follows that $k$-agents have strictly
higher welfare in the second-best than in the pooling allocation if and only
if $IC_{kj}$ does not bind.

######  Corollary 3.

Suppose $(q_{P},q_{I})$ is a second-best solution. Then, one of the following
must hold:

  1. 1.

Neither $IC_{IP}$ nor $IC_{PI}$ binds. Then, the solution coincides with the
first-best solution. Both $I$\- and $P$-agents strictly prefer it to the
pooling allocation.

  2. 2.

Only $IC_{IP}$ binds. Then, $P$-agents strictly prefer the second-best to the
pooling allocation, while $I$-agents are indifferent.

  3. 3.

Only $IC_{PI}$ binds and there is no disposal. $I$-agents strictly prefer the
second-best to the pooling allocation, while $P$-agents are indifferent.

  4. 4.

Only $IC_{PI}$ binds and there is disposal. $I$-agents strictly prefer the
second-best to the pooling allocation, while $P$-agents strictly prefer the
pooling to the second-best allocation.

##  5 Beyond Two Types

We now turn to discuss how our results extend to the case of any finite number
$N$ of types. We maintain the same assumptions for the utility $u_{i}$ of each
type $i$. As before, we posit that all types have the same ordinal
preferences, but differ in their cardinal valuations, with utilities ordered
via absolute risk aversion: $\forall x\in\mathbb{R}_{+}$,

$\frac{u_{i}^{\prime\prime}(x)}{u_{i}^{\prime}(x)}>\frac{u_{i+1}^{\prime\prime}(x)}{u_{i+1}^{\prime}(x)}\
\ \ \ \forall i\in\\{1,\dots,N-1\\}\text{.}$ (4)

Each type $i$ has mass of $\mu_{i}>0$ and we continue to focus on a setting
with sufficient supply: $\sum_{i=1}^{N}\mu_{i}\leq F(X)$.

###  5.1 First-best with Many Types

As in the two-type case, the social planner selects an allocation
$(q_{1},...,q_{N})$, where each $q_{i}$ is a lottery and the allocation is
feasible:

$\sum_{i=1}^{N}\mu_{i}q_{i}(x)\leq f(x)\text{ \ \ }\forall x\in[0,X].$

The goal is to maximize welfare

$W(q_{1},...,q_{n})=\sum_{i=1}^{N}\alpha_{i}\mu_{i}\int_{0}^{X}u_{i}(x)q_{i}(x)dx\text{,}$

where $\\{\alpha_{i}\\}_{i=1}^{n}$ are arbitrary weights with $\alpha_{i}>0$
for all $i$.

Following our results for the two-type case, we can expect any first-best
solution to satisfy two properties. First, it is never optimal for the planner
to dispose of goods. Second, there are no inverted spreads: for any $i$ and
$j$ with $i<j$, if any part of $i$-agents’ allocation is a spread of any part
of $j$-agents’ allocation, there is a beneficial swap that violates
optimality.

As it turns out, these two restrictions fully characterize the first-best: the
set of allocations that exhibit no disposal and no inverted spread coincides
with the set of first-best allocations for some welfare weights. Moreover,
this set also coincides with the set of Pareto efficient allocations.

######  Proposition 4.

The following sets coincide:

  1. 1.

The set of feasible allocations that do not exhibit disposal or inverted
spread;

  2. 2.

The set of feasible allocations that are first-best for some strictly positive
welfare weights $\\{\alpha_{i}\\}_{i=1}^{N}$;

  3. 3.

The set of feasible Pareto efficient allocations.

To glean intuition, consider a market for allocations in which agents’ initial
allocations serve as their endowments and trade can take place freely—we
further elaborate on such markets in the following section. From the second
welfare theorem, any Pareto efficient allocation can be mapped to a
competitive equilibrium for some endowments. An inverted spread would leave
room for beneficial trades and cannot occur in a competitive equilibrium.
Conversely, without an inverted spread, agents have no opportunities for
profitable bilateral trade. As it turns out, there is also no profitable
multilateral trade.131313The proof itself uses alternative arguments. An
allocation without an inverted spread is then a competitive equilibrium for
some endowments and thus Pareto efficient. Finally, standard arguments show
the equivalence between Pareto efficiency and utilitarian efficiency with some
welfare weights.

#### Shape of the first-best allocation

What do allocations that exhibit no inverted spread and no disposal look like?
As in the $N=2$ case, we denote by $\overline{X}:=F^{-1}(\mu_{1}+...+\mu_{N})$
the lowest quality needed to exhaust demand when only the best-quality goods
are used. Any allocation satisfying no disposal fully utilizes goods in
$[0,\overline{X}]$.

The most risk-averse agents must be served in one contiguous block
$[x_{1},x^{1}]\subseteq[0,\overline{X}]$, exhausting the supply available
there; if another type were served within the block, it would yield an
inverted spread. Thus, we have $q_{1}=f~{}|~{}[x_{1},x^{1}]$ with
$f([x_{1},x^{1}])=\mu_{1}$. Once we determine type-$1$ agents’ allocation, we
can consider a reduced problem. We adjust the supply by eliminating the block
of goods already promised to type-$1$ agents. Namely, we let
$f^{1}:=f-\mu_{1}q_{1}$ and focus on agents of types $2,...,N$. Of those,
type-$2$ agents are the most risk averse and, as before, must be served in one
block, exhausting the supply $f^{1}$ there. Thus,
$q_{2}=f^{1}~{}|~{}[x_{2},x^{2}]$ for some $0\leq
x_{2}<x^{2}\leq\overline{X}$. While $q_{2}$ exhausts the supply $f^{1}$ within
$[x_{2},x^{2}]$, it need not be a continuous block within
$[0,\overline{X}]$—we may have $[x_{1},x^{1}]\subset[x_{2},x^{2}]$. We can
continue to generate assignments for all types recursively.141414Namely, at
step $k$, define the remaining supply
$f^{k-1}=f-\sum_{i=1}^{k-1}\mu_{i}q_{i}$. Since there is no inverted spread,
$q_{k}=f^{k-1}~{}|~{}[x_{k},x^{k}]$ for some $0\leq
x_{k}<x^{k}\leq\overline{X}$. Without loss of generality, we assume that
$x_{k}=\textrm{inf}(\textrm{supp}(q_{k}))$ and
$x^{k}=\textrm{sup}(\textrm{supp}(q_{k}))$. At each step $k$, the pair
$x_{k},x^{k}$ respects a feasibility constraint
$f^{k-1}([x_{k},x^{k}])=\mu_{k}$. Denote by $\mathcal{A}$ the set of all
allocations that can be constructed using this procedure. It turns out that
all first-best allocations can be constructed in this way and that any such
construction leads to a first-best allocation.

######  Corollary 4.

The set $\mathcal{A}$ of allocations coincides with the set of allocations
that do not exhibit disposal or inverted spread. In particular, any first-best
allocation consists of no more than $2N-1$ blocks and type-$k$’s allocation
consists of no more than $k$ disjoint blocks.

###  5.2 Many Unobservable Types

The mechanism designer’s problem is defined analogously to that for $N=2$: the
designer chooses feasible lotteries to maximize the weighted sum of utilities
subject to incentive-compatibility constraints $IC_{kj}$ with $k\neq j$ and
$k,j=1,...,N$, ensuring that $k$-agents do not want to emulate $j$-agents.

To derive properties of the solution, we further assume that utility functions
of different types are linearly independent. Formally, for any
$\lambda_{0},...,\lambda_{N}\in\mathbb{R}$, the set
$\left\\{~{}x~{}\big{|}~{}\sum_{j=1}^{N}\lambda_{j}u_{j}(x)=\lambda_{0}~{}\right\\}$
has measure zero. This assumption, which holds automatically with only two
types, is valid for many classes of utilities, including CRRA, CARA,
exponentially discounted utilities, or present-biased ones. This regularity
assumption guarantees that a social planner is never indifferent between
randomly supplying an interval of goods to several types, or assigning those
goods to another, different type.

######  Proposition 5.

A solution of the mechanism designer’s problem with $N$ types exists and is
unique. If $(q_{i})_{i\in\\{1,\dots,N\\}}$ is a solution,

  1. 1.

For almost every $x\in[0,X]$, either one type of agent gets the entire supply
of the good or the entire supply remains unused. That is, for almost every
$x\in[0,X]$, either $q_{i}(x)=f(x)$ or $q_{i}(x)=0$ for all
$i\in\\{1,\dots,N\\}$. Thus, the solution is fully separating.

  2. 2.

The graph of binding IC constraints for the optimal allocation has no directed
cycles. That is, there is no subset of types $(k_{1},\dots,k_{m})\subset
N^{m}$, with $k_{i}\neq k_{j}$ for all $i,j\in\\{1,\dots,m\\}$ such that
$IC_{k_{i},k_{i+1}}$ for all $i\in\\{1,\dots,m-1\\}$ and $IC_{k_{m},k_{1}}$
all bind.

The proposition indicates that some of the main properties of our solution
with two types continue to hold. A solution exists and is unique. Part 1 of
the proposition illustrates that it is “fully” separating—not only do
different types get different allocations, but their allocations’ supports do
not meaningfully overlap. This implies, once again, that the pooling
allocation is never a solution and yields strictly lower welfare.

Part 2 of the proposition asserts that the $IC$ constraints do not form a
cycle, generalizing the observation from our two-type setting that both $IC$
constraints cannot bind simultaneously. It implies that the $IC$ constraints
cannot all be binding even in the $N$-type case.

Recall our Corollary 4 that suggested that any allocation in the set
$\mathcal{A}$ is a first-best allocation for some welfare weights. In general,
to establish incentive compatibility, there are $N\times(N-1)$ constraints
that need to be satisfied. However, by construction, for any allocation in
$\mathcal{A}$, it suffices to check the constraints corresponding to adjacent
types—$k$-agents’ allocation contains more extreme-quality goods than
$k-1$-agents’. This insight allows us to illustrate the possible coincidence
of the first-best and second-best allocations for a non-trivial set of welfare
weights.

Intuitively, construct an allocation in $\mathcal{A}$ as follows. Pick some
$[x_{1},x^{1}]\subset(0,\overline{x})$. We can find an allocation for type-$2$
agents, defined by $y_{2}$ and $y^{2}$ so that type-$1$ agents are indifferent
between their allocation and the resulting type-$2$ agents’
allocation.151515As in the construction of the set $\mathcal{A}$, type-$2$
agents are provided goods of quality $[y_{2},x_{1}]\cup[x^{1},y^{2}]$. Since
type-$2$ agents are less risk averse, they strictly prefer their allocation.
We then find an allocation for type-$2$ agents, defined similarly by $z_{2}$
and $z^{2}$, such that type-$2$ agents are indifferent between their
allocation and type-$1$ agents’ allocation. It follows that type-$1$ agents
would strictly prefer their allocation. Consider now the allocation defined by
$x_{2}=\frac{y_{2}+z_{2}}{2}$ and $x^{2}=\frac{y^{2}+z^{2}}{2}$, assuming it
is feasible. Our construction guarantees that neither $IC_{12}$ nor $IC_{21}$
binds. We can complete the construction of such an allocation for higher
types, each step ensuring that incentive compatibility constraints of adjacent
types do not bind (in the proof, we make appropriate adjustments to guarantee
feasibility). By Corollary 4 and Proposition 4, this allocation is a first-
best solution for some welfare weights. Since it is incentive compatible,
under those same welfare weights, it is also a second-best solution.
Furthermore, any small enough perturbation of the welfare weights corresponds
to a small perturbation of the first-best solution—namely, a small change in
the end-points defining the allocations of each agent type—and remains
incentive compatible.

######  Corollary 5.

For all $N>2$, there is an open set of welfare weights for which the first-
best allocation is incentive compatible.

To illustrate graphically the welfare implications of the first- and second-
best allocations for different type volumes, we focus on the special case of
time discounting. We consider random profiles of $N$ discount factors, where
$N=2,...,10$ and each discount factor is randomly and drawn uniformly from
$[0,2]$. We assume all types have equal masses, supply is uniform, and all
types are weighted equally in the expected welfare.161616For any $N$,
$\alpha_{i}=1$, $\mu_{i}=\frac{1}{N}$ for all $i\in\\{1,...,N\\}$. We simulate
$100,000$ discount rates within $[0,2]$ and partition those randomly to sets
of $N$. The supply has total mass $3/2$ and is distributed uniformly over
$[0,5]$.

Figure 7: Welfare levels for $N$ types.

Figure 7 displays the expected welfare from the first-best, second-best, and
pooling allocations. Naturally, the welfare from the first-best allocation
exceeds that from the second-best allocation, and is lowest for the pooling
allocation. As the figure illustrates, the expected welfare values plateau as
the number of types increase. Furthermore, the second-best allocation
establishes substantially higher welfare than the pooling allocation.

##  6 A Market for Allocations

We now consider a simple method for generating incentive-compatible
allocations that Pareto dominate the pooling allocation for an arbitrary
number of types: a market for allocations, in the spirit of Hylland and
Zeckhauser (1979). Agents receive an endowment and can trade freely through
market interactions, which determine the prices of various allocations. As
usual, such market interactions need not take place literally—they can be
emulated after types are reported. We maintain the previous section’s
assumptions on agents’ utilities.

Without loss of generality, we consider a symmetric notion of competitive
equilibrium, where all agents of a certain type have the same demand for
lotteries.171717This is without loss of generality since, for any asymmetric
equilibrium, we can construct a corresponding symmetric equilibrium with the
same aggregate demand per type and the same price schedule. A price schedule
is a measurable function $p:~{}[0,\overline{X}]\rightarrow R_{+}$. A demand of
type-$i$ agent is a measure $q_{i}$ over $[0,\overline{X}]$.181818In
principle, unlike lotteries described in prior sections, a demand function can
exhibit mass points.

An agent of type $i$ optimizes the allocation subject to a budget constraint,
determined by the endowment $\omega_{i}$ and the price schedule $p(x)$: the
agent’s problem is

$\underset{q_{i}}{\textrm{max}}~{}\int_{0}^{\overline{X}}u_{i}(x)q_{i}(x)dx$
(5)

such that

$~{}~{}~{}~{}~{}~{}1-q_{i}\big{(}[0,\overline{X}]\big{)}~{}\geq~{}0~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathrm{and}~{}~{}~{}~{}~{}~{}~{}~{}\omega_{i}-\int_{0}^{\overline{X}}p(x)dq_{i}(x)~{}\geq~{}0.$

A natural, seemingly equitable case to consider entails all agents receiving
equal endowments. For any given profile of prices, it is tantamount to giving
all agents an equal share of the supply, essentially their pooling
allocation.191919Targeted endowments would introduce incentive problems at the
endowment-distribution stage. One could also consider random, but unequal
endowments. Our main insights carry over. We call the resulting allocation a
fair competitive equilibrium.

Definition. A fair competitive equilibrium consists of a price schedule $p(x)$
and demand functions $\\{q^{i}\\}_{i=1,...,N}$ such that

  1. 1.

$q_{i}$ solves (5) for each type of agent $i$.

  2. 2.

All agents receive the same strictly-positive endowment,
$\omega_{i}~{}=~{}\omega$.

  3. 3.

Market clearing holds,
$\sum_{i=1}^{N}\mu_{i}q_{i}(x)~{}=~{}f(x)\cdot\mathbbm{1}\big{\\{}x\in[0,\overline{X}]\big{\\}}$.

The following proposition characterizes the structure of fair equilibrium
outcomes.

######  Proposition 6.

A fair competitive equilibrium exists. If
$\big{(}p,\\{q_{i}\\}_{i=1,...,N}\big{)}$ is a fair competitive equilibrium,
there are threshold qualities
$0=\underline{x}_{N}<\underline{x}_{N-1}<...<\underline{x}_{2}<\underline{x}_{1}<\overline{x}_{1}<\overline{x}_{2}<....<\overline{x}_{N-1}<\overline{x}_{N}=\overline{X}$
such that

$q_{k}~{}~{}=~{}~{}f~{}\big{|}~{}[\underline{x}_{k},\underline{x}_{k-1}]\cup[\overline{x}_{k-1},\overline{x}_{k}]~{}~{}~{}\textrm{for}~{}k>1,~{}~{}and~{}~{}q_{1}~{}~{}=~{}~{}f~{}\big{|}~{}[\underline{x}_{1},\overline{x}_{1}].$

The proposition demonstrates that the structure of a fair competitive
equilibrium extends the IPI structure we obtained as a second-best solution
for two types. The most risk-averse agents receive goods in one contiguous
interval; the second most risk-averse agents receive goods in two surrounding
intervals; and so on, with the least risk-averse agents receiving either the
best or the worst goods. The resulting allocation belongs to $\mathcal{A}$
and, as Corollary 4 indicates, does not exhibit disposal or inverted spread.

From the first welfare theorem, we know the fair competitive equilibrium
allocation is Pareto efficient. In particular, it dominates the pooling
allocation for all agents. Furthermore, by construction, it satisfies all
agents’ incentive compatibility constraints. Nevertheless, it need not
coincide with the second-best solution. Furthermore, while the second-best
solution may entail disposal of goods, the fair competitive equilibrium never
does.

To illustrate the wedge between the market and second-best solutions, consider
again the discounting case with two types. Standard techniques can be used to
show that the fair competitive equilibrium is unique.

Panel (a) of Figure 8 displays the ratio of welfare generated by the fair
competitive equilibrium and the second-best allocation, adjusted by welfare
generated by the pooling allocation. There is a zero-measure set of discount
rates—corresponding to the line with a ratio of $1$—for which there is no
welfare loss produced by the market solution. For all other parameters, the
second-best solution yields greater welfare levels. Welfare losses are more
pronounced for more patient $P$-agents. Certainly, these discount-rate regions
may entail disposal in the second-best allocation, which is never featured in
the fair competitive equilibrium. Nonetheless, the wedge in welfare is not
solely due to disposal: panel (b) of Figure 8 considers restricted second-best
solutions in which disposal is banned (see the Online Appendix for a formal
analysis). Welfare losses generated by the market solution are still
pronounced, particularly when $P$-agents are very patient.

Figure 8: Ratio of welfare generated by the fair competitive equilibrium
relative to the second-best allocation, adjusted by welfare generated by the
pooling allocation, with disposal (in left panel) and without (in right
panel).

##  7 Conclusions

Goods and services—public housing, medical appointments, schools—are often
allocated to individuals who rank them similarly but differ in their
preference intensities. We characterize optimal allocation rules in such
settings, considering both the case in which individual preferences are known
and ones in which they need to be elicited. We show that first-best
allocations may involve assigning some agents lotteries between high- and low-
ranked goods. When preference intensities are private information, second-best
allocations always involve such lotteries and may coincide with first-best
allocations. Furthermore, second-best allocations may entail disposal of
services. We also illustrate the potential drawbacks of utilizing a simple
market solution in lieu of the optimal mechanism.

Our analysis assumes a fixed supply that cannot be altered, but complete
freedom in selecting which agents are offered a good. In many applications,
however, goods’ quality can be reduced and denial of service is not viable. In
the Online Appendix, we show that the possibility to reduce goods’ qualities
would never be utilized in the first- and second-best solutions. Furthermore,
the qualitative features of the second-best solution are retained when all
agents need to be catered to with certainty.

## Appendix

Proposition 0 (Existence)

For any number $N$ of types, a first-best and a second-best allocation exists.

Proof of Proposition 0. Since $Y=[0,X]$ is a compact metric space, the space
of distributions $\mathcal{P}(Y)$ is a metric space. Moreover, it is a closed
subset of the unit ball with respect to the weak∗ topology. The latter is a
compact set by the Banach-Alaogly Theorem, hence $\mathcal{P}(Y)$ is a compact
metric space. The space of allocations is a direct product
$\left(\mathcal{P}(Y)\right)^{N}$, where $N$ is the number of agent types.
Therefore, the space of allocations is a compact metric space with the $sup$
metric induced by the metric on $\mathcal{P}(Y)$. The $IC$ and feasibility
constraints are not strict and are linear in the allocation. Hence, the subset
of feasible and incentive-compatible allocations is a closed subset of
$\left(\mathcal{P}(Y)\right)^{N}$, and in itself a compact set. Finally, the
objective function (of both the social planner and the mechanism designer) is
linear in the allocation, and, therefore, continuous. A first-best and second-
best allocation exists by the Weierstrass Theorem. $\blacksquare$

Lemma 0

Agents of type $P$ are strictly more risk averse than agents of type $I$.

Proof of Lemma 0. The result is well known for lotteries with support on
$[0,\infty)$. For a lottery with
$\textrm{supp}(q)\subseteq[0,X]\cup\\{\diamond\\}$ and $q_{\diamond}>0$, there
is a sequence of lotteries $q^{(k)}=\big{(}1-q(\diamond)\big{)}\cdot
q~{}|~{}[0,X]+q(\diamond)\cdot\delta_{k}$, $k=1,2,...$, defined on the Borel
subsets of $[0,\infty)$, such that
$V_{j}(q^{(k)})\overset{k\rightarrow\infty}{\longrightarrow}V_{j}(q)$. Thus,
we can use known results for lotteries with support on $[0,\infty)$ to
conclude that $P$-agents are weakly more risk averse than $I$-agents. The
proof that this ordering is, in fact, strict for lotteries with
$q(\diamond)>0$ is also standard, but somewhat more involved: it appears as
Lemma A1 in the Online Appendix. $\blacksquare$

Proof of Lemma 1. We are interested in the sign of the derivative of
$g(\cdot)$:

$g^{\prime}(x)~{}=~{}\alpha u_{P}^{\prime}(x)-(1-\alpha)u_{I}^{\prime}(x)$

Since utilities are decreasing,

$\textrm{sign}(g^{\prime}(x))~{}=~{}-\textrm{sign}\left(\frac{u_{P}^{\prime}(x)}{u_{I}^{\prime}(x)}~{}-~{}\frac{1-\alpha}{\alpha}\right)~{}=~{}-\textrm{sign}(\theta(x)),$

where
$\theta(x)=\frac{u_{P}^{\prime}(x)}{u_{I}^{\prime}(x)}-\frac{1-\alpha}{\alpha}$.
We have

$\theta^{\prime}(x)~{}=~{}\frac{u_{P}^{\prime}(x)}{u_{I}^{\prime}(x)}\cdot\left[\frac{u_{P}^{\prime\prime}(x)}{u_{P}^{\prime}(x)}~{}-~{}\frac{u_{I}^{\prime\prime}(x)}{u_{I}^{\prime}(x)}\right]~{}>~{}0.$

Thus, $\theta$ is a strictly increasing function. There are three possible
cases. If $\textrm{sign}(\theta(x))>0$ always, then $g(\cdot)$ is strictly
decreasing and, hence, strictly quasi-concave. If $\textrm{sign}(\theta(x))<0$
always, then $g(\cdot)$ is strictly increasing and, hence, strictly quasi-
concave. Last, suppose there is $x_{PI}$ such that
$\textrm{sign}(\theta(x))<0$ for $x<x_{PI}$, $\textrm{sign}(\theta(x))>0$ for
$x>x_{PI}$, and $\textrm{sign}(\theta(x_{PI}))=0$. Consider
$x^{\prime}<x^{\prime\prime}$ and $\lambda\in(0,1)$. If $x^{\prime}<\lambda
x^{\prime}+(1-\lambda)x^{\prime\prime}\leq x_{PI}$, then $g(\lambda
x^{\prime}+(1-\lambda)x^{\prime\prime})>g(x^{\prime})$. Otherwise,
$x_{PI}\leq\lambda x^{\prime}+(1-\lambda)x^{\prime\prime}<x^{\prime\prime}$
and $g(\lambda x^{\prime}+(1-\lambda)x^{\prime\prime})>g(x^{\prime\prime})$.
It follows that $g$ is a strictly quasi-concave function, since $g(\lambda
x^{\prime}+(1-\lambda)x^{\prime\prime})>\textrm{min}\\{g(x^{\prime}),g(x^{\prime\prime})\\}$
for all $\lambda\in(0,1)$. $\blacksquare$

Proof of Proposition 1 An optimal allocation exists by Proposition 0. If $q$
is an optimal allocation, then
$(\mu_{I}q_{I}+\mu_{P}q_{P})([0,\overline{X}])=\mu_{I}+\mu_{P}=F(\overline{X})$.
Otherwise, an allocation for at least one of the agents’ types could be
improved through the provision of superior-quality goods instead of goods on
$(\overline{X},\infty)\cup\\{\diamond\\}$, without altering the other type
agents’ allocation. Therefore, if $q_{P}$ is chosen optimally, we can write
$q_{I}=q_{I}(q_{P})$, where
$q_{I}(x)=\mu_{I}^{-1}\cdot(f(x)-\mu_{P}q_{P}(x))\cdot\mathbbm{1}\\{x\in[0,\overline{X}]\\}$.
The social planner’s optimization problem is then:

$\max_{q_{P}}~{}~{}(1-\alpha)\int_{0}^{\overline{X}}f(x)u_{I}(x)dx+\mu_{p}\int_{0}^{\overline{X}}g(x)q_{P}(x)dx$
$\hbox{s.t.:}~{}~{}~{}~{}0\leq
q_{P}(x)\leq\mu_{P}^{-1}f(x)~{}~{}~{}~{}~{}~{},~{}~{}~{}~{}~{}~{}\int_{0}^{\overline{X}}q_{P}(x)dx=1$

This problem has a unique solution given by
$q_{p}(x)=\mu_{P}^{-1}f(x)\mathbbm{1}\\{g(x)\geq c\\}$, where
$c=\inf\big{\\{}t\in
R~{}\big{|}~{}\mu_{P}^{-1}f(\\{x\in[0,\overline{X}]~{}|~{}g(x)>t\\})\leq
1\big{\\}}$ by the so-called “Bathtub” principle (Theorem 1.14 in H.Lieb and
Loss (2001)). The arguments provided in Section 3 show that the solution has
the desired form in all three cases considered in the proposition. To see that
the 3 cases are not overlapping, assume that $g(\overline{X}_{P})\leq g(0)$.
The strict quasi-concavity of $g(\cdot)$ implies that
$\min\\{g(0),g(\overline{X})\\}<g(\overline{X}_{P})\leq g(\overline{X}_{P})$.
Therefore,
$g(\overline{X})=\min\\{g(0),g(\overline{X})\\}<g(\overline{X}_{I})$.
$\blacksquare$

The following lemma generalizes Lemma 2 in the text and will be used in our
$N$-type analysis.

Lemma 2* (No Inverted Spread). The first-best allocation never exhibits an
inverted spread for any number of types. The second-best allocation exhibits
no inverted spread for N=2 types.

Proof of Lemma 2*. Consider first the case of $N=2$ types. Suppose the
allocation $(q_{P},q_{I})$ exhibits an inverted spread; that is, there are
$A\triangleleft B\triangleleft C$ such that $q_{P}(A),q_{I}(B),q_{P}(C)>0$.
For an arbitrary $x\in B$, let $\gamma(x)\in(0,1)$ be such that
$V_{P}\big{(}\gamma(x)\cdot q_{P}~{}|~{}A+(1-\gamma(x))\cdot
q_{P}~{}|~{}C\big{)}=V_{P}\big{(}\delta_{x}\big{)}$, then
$V_{I}\big{(}\gamma(x)\cdot q_{P}~{}|~{}A+(1-\gamma(x))\cdot
q_{P}~{}|~{}C\big{)}>V_{I}\big{(}\delta_{x}\big{)}$ by Lemma 0. Integrating
these inequalities with respect to $q_{I}~{}|~{}B$ and using the linearity of
$V_{P}(\cdot),V_{I}(\cdot)$, we get

$V_{P}\big{(}\gamma\cdot q_{P}~{}|~{}A+(1-\gamma)\cdot
q_{P}~{}|~{}C\big{)}=V_{P}\big{(}q_{I}~{}|~{}B\big{)}~{}~{}~{}~{}~{}~{}~{},~{}~{}~{}~{}~{}~{}~{}V_{I}\big{(}\gamma\cdot
q_{P}~{}|~{}A+(1-\gamma)\cdot
q_{P}~{}|~{}C\big{)}>V_{I}\big{(}q_{I}~{}|~{}B\big{)}$

where $\gamma=\int\gamma(x)dq|B(x)\in(0,1)$. Let
$\epsilon=\textrm{min}\\{q_{P}(A),q_{I}(B),q_{P}(C)\\}$, then $\epsilon>0$.
Consider

$q^{\prime}_{I}~{}~{}=~{}~{}q_{I}~{}-~{}\epsilon\mu_{P}(\mu_{P}+\mu_{I})^{-1}\cdot
q_{I}~{}|~{}B~{}+~{}\epsilon\mu_{P}(\mu_{P}+\mu_{I})^{-1}\cdot\big{(}\gamma\cdot
q_{P}~{}|~{}A+(1-\gamma)\cdot q_{P}~{}|~{}C\big{)}$
$q^{\prime}_{P}~{}~{}=~{}~{}q_{P}~{}+~{}\epsilon\mu_{I}(\mu_{P}+\mu_{I})^{-1}\cdot
q_{I}~{}|~{}B~{}-~{}\epsilon\mu_{I}(\mu_{P}+\mu_{I})^{-1}\cdot\big{(}\gamma\cdot
q_{P}~{}|~{}A+(1-\gamma)\cdot q_{P}~{}|~{}C\big{)}$

It is easy to see that $q^{\prime}$ is feasible. Since
$V_{P}(q^{\prime}_{P})=V_{P}(q^{\prime}_{P})$,
$V_{I}(q^{\prime}_{I})>V_{I}(q_{I})$, then $W(q^{\prime})>W(q)$. Thus, $q$
cannot be a first-best allocation. Notice also that
$V_{P}(q^{\prime}_{I})=V_{P}(q_{I})$, and
$V_{I}(q^{\prime}_{P})<V_{I}(q_{I})$. Therefore, if $q$ is incentive
compatible, then $q^{\prime}$ is also incentive compatible. We conclude that
$q$ cannot be a second-best allocation as well.

Suppose that $N>2$ and types $j,k$ exhibit an inverted spread, where type $j$
is more patient than type $k$. Fix an allocation for all types $i\neq j,k$ and
repeat the argument used for $N=2$ types for $j=P$ and $k=I$ to conclude that
the first-best allocation cannot exhibit an inverted spread.202020Our argument
for the second-best allocation does not extend beyond two types. Indeed, using
the notation above, some of the incentive constraints $IC_{ij},IC_{ik}$ for
$i\neq j,k$ may be violated for $q^{\prime}$. $\blacksquare$

Proof of Corollary 2. Let $q$ be an optimal allocation. First, let
$\nu(\cdot)$ be a $\mathcal{L}$ebesque measure, and assume that
$\nu(\textrm{supp}(q_{I})\cap\textrm{supp}(q_{P})\cap[0,X])>0.$ Since
$q_{P},q_{I}$ are non-atomic, there are $0<x^{\prime}<x^{\prime\prime}<X$ such
that $\nu(\textrm{supp}(q_{I})\cap\textrm{supp}(q_{P})\cup[0,x^{\prime}])>0$,
$\nu(\textrm{supp}(q_{I})\cap\textrm{supp}(q_{P})\cup(x^{\prime},x^{\prime\prime}))>0$,
and
$\nu(\textrm{supp}(q_{I})\cap\textrm{supp}(q_{P})\cup[x^{\prime\prime},X])>0$.
Then
$[0,x^{\prime}]\triangleleft(x^{\prime},x^{\prime\prime})\triangleleft[x^{\prime\prime},X]$
constitutes an inverted spread, contradicting Lemma 2.

Furthermore, if $q_{j}(\diamond)=1$ for some type, then by incentive
compatibility $q_{P}=q_{I}=\delta_{\diamond}$, which clearly cannot be
optimal. Hence, $q_{I}(\diamond),q_{P}(\diamond)<1$. Assume, towards a
contradiction, that both $IC$ constraints are binding. Then the allocation
$\widetilde{q}_{P}=\widetilde{q}_{P}=\dfrac{\mu_{P}q_{P}+\mu_{I}q_{I}}{\mu_{P}+\mu_{I}}$
is also feasible, incentive compatible, and provides the same welfare as $q$.
It follows that this allocation is also optimal. The supports of
$\widetilde{q}_{I}$ and $\widetilde{q}_{P}$ coincide, and
$\widetilde{q}_{I}(\diamond),\widetilde{q}_{P}(\diamond)<1$. These
observations imply that
$\nu\big{(}\textrm{supp}(\widetilde{q}_{I})\cap\textrm{supp}(\widetilde{q}_{P})\cap[0,X]\big{)}>0$,
in contradiction. $\blacksquare$

Proof of Proposition 2. Define
$x_{2}:[0,F^{-1}(\mu_{I})]\rightarrow[F^{-1}(\mu_{P}),\overline{X}]$ as
$x_{2}(x_{1})=F^{-1}(F(x_{1})+\mu_{P})$,

$\underline{x}_{1}~{}~{}=~{}~{}\min\Big{\\{}x_{1}\in[0,F^{-1}(\mu_{I})]~{}~{}\Big{|}~{}~{}V_{I}\big{(}f~{}\big{|}~{}[0,x_{1}]\cup[x_{2}(x_{1}),\overline{X}]\big{)}\geq
V_{I}\big{(}f~{}\big{|}~{}[x_{1},x_{2}(x_{1})]\big{)}\Big{\\}},$
$\overline{x}_{1}~{}~{}=~{}~{}\max\Big{\\{}x_{1}\in[0,F^{-1}(\mu_{I})]~{}~{}\Big{|}~{}~{}V_{P}\big{(}f~{}\big{|}~{}[0,x_{1}]\cup[x_{2}(x_{1}),\overline{X}]\big{)}\leq
V_{P}\big{(}f~{}\big{|}~{}[x_{1},x_{2}(x_{1})]\big{)}\Big{\\}}.$

Thus, if $x_{1}=\underline{x}$, then $I$-agents are indifferent between
lottery $f~{}\big{|}~{}[0,x_{1}]\cup[x_{2}(x_{1}),\overline{X}]$ and lottery
$f~{}\big{|}~{}[x_{1},x_{2}(x_{1})]$. It follows that the more risk averse
$P$-agents strictly prefer the second lottery. Therefore,
$\overline{x}_{1}>\underline{x}_{1}$. Similarly, if $x_{1}=\overline{x}_{1}$,
$P$-agents are indifferent between the two lotteries above. In this case,
$I$-agents strictly prefer the first lottery. By construction, for any
$x_{1}\in[\underline{x}_{1},\overline{x}_{1}]$, the allocation $q^{x_{1}}$
given by $q_{P}^{x_{1}}=f~{}\big{|}~{}[x_{1},x_{2}(x_{1})]$,
$q_{I}^{x_{1}}=f~{}\big{|}~{}[0,x_{1}]\cup[x_{2}(x_{1}),\overline{X}]$ is
feasible and incentive compatible. It follows that
$0<\underline{x}_{1}<\overline{x}_{1}<F^{-1}(\mu_{I})$.

By Proposition 1, for any $\alpha\in(0,1)$, the first-best allocation takes
the form $q^{x_{1}}$, as defined above, for some
$x_{1}\in[0,F^{-1}(\mu_{I})]$. Therefore, we can define a function
$x_{1}:(0,1)\rightarrow[0,F^{-1}(\mu_{I})]$ by identifying $x_{1}(\alpha)$
such that $q^{x_{1}(\alpha)}$ is the unique first-best allocation for welfare
weight $\alpha$. For brevity, in what follows, we drop the arguments of
$x_{1}$ and $x_{2}$ whenever there is little risk of confusion. Suppose
$x_{1}\in(0,F^{-1}(\mu_{I}))$. By Proposition 1, $g(x_{1})=g(x_{2})$. There is
therefore an inverse function $\alpha:(0,F^{-1}(\mu_{I}))\rightarrow(0,1)$
given by

$\alpha(x_{1})=\dfrac{u_{I}(x_{1})-u_{I}(x_{2})}{u_{P}(x_{1})-u_{P}(x_{2})+u_{I}(x_{1})-u_{I}(x_{2})}=\dfrac{1}{1+\gamma(x_{1})},$

where

$\gamma(x_{1})=\dfrac{u_{P}(x_{1})-u_{P}(x_{2})}{u_{I}(x_{1})-u_{I}(x_{2})}.$

Our assumptions on utilities guarantee that $\gamma$ is strictly increasing:
see Lemma A2 in the Online Appendix for a complete argument. Therefore,
$\alpha(\cdot)$ is strictly decreasing. It follows that the first-best
allocation is incentive compatible if and only if
$\alpha\in[\alpha(\underline{x}_{1}),\alpha(\overline{x}_{1})]$.
$\blacksquare$

Proof of Lemma 3. Assume that $IC_{ij}$ does not bind for $i\in\\{I,P\\}$,
$j\in\\{I,P\\}\backslash i$. Then, $q_{j}(\diamond)=0$. To see this, suppose
$q_{j}(\diamond)>0$. Since the supply is sufficient,
$(f-\mu_{P}q_{P}-\mu_{I}q_{I})([0,X])>0$. Then for sufficiently small
$\epsilon>0$ the allocation $q^{\prime}$ with $q^{\prime}_{i}=q_{i}$ and

$q^{\prime}_{j}(x)~{}~{}~{}~{}~{}=~{}~{}~{}\Big{(}q_{j}(x)+\epsilon\cdot
q_{j}(\diamond)\cdot\big{(}f(x)-\mu_{P}q_{P}(x)-\mu_{I}q_{I}(x)\big{)}\Big{)}\cdot\mathbbm{1}\big{\\{}x\in[0,X]\big{\\}}~{}+~{}$
$~{}+~{}\Big{(}1-\epsilon\cdot(f-\mu_{P}q_{P}-\mu_{I}q_{I})\big{(}[0,X]\big{)}\Big{)}\cdot
q_{j}(\diamond)\cdot\delta_{\diamond}(x)$

is feasible, incentive compatible, and provides a strict welfare improvement
with respect to $q$, thereby producing a contradiction.

We now show that $q_{P}=f~{}|~{}[x_{1},x_{2}]$ with
$F(x_{2})-F(x_{1})=\mu_{P}$. By Corollary 2, both $IC$ constraints cannot bind
for $q$.

There are two cases to consider. First, suppose $IC_{IP}$ is not binding. Our
argument above shows that $q_{P}(\diamond)=0$. Therefore, we can define
$x_{1}=\inf(\textrm{supp}(q_{P}))$, $x_{2}=\sup(\textrm{supp}(q_{P}))$, with
$x_{1},x_{2}\in[0,X]$. By Corollory 2,
$\nu(\textrm{supp}(q_{I})\cap(x_{1},x_{2}))=0$, and hence
$q_{I}([x_{1},x_{2}])=0$. Assume, towards a contradiction, that
$(f-\mu_{p}q_{p})([x_{1},x_{2}])>0$. Let
$x^{\prime}_{2}=F^{-1}\big{(}F(x_{1})+\mu_{P}\big{)}$. Then
$x^{\prime}_{2}<x_{2}$ and $q_{P}\big{(}(x^{\prime}_{2},x_{2}]\big{)}>0$.
Consider allocation $q^{\prime}$ with $q^{\prime}_{I}=q_{I}$ and

$q^{\prime}_{P}(x)~{}~{}~{}~{}~{}=~{}~{}~{}\Big{(}q_{P}(x)+\epsilon\cdot
q_{P}\big{(}[x_{2}^{\prime},x_{2}]\big{)}\cdot\big{(}f(x)-\mu_{P}q_{P}(x)\big{)}\Big{)}\cdot\mathbbm{1}\big{\\{}x\in[x_{1},x^{\prime}_{2}]\big{\\}}~{}+~{}$
$~{}+~{}\Big{(}1-\epsilon\cdot(f-\mu_{P}q_{P})\big{(}[x_{1},x^{\prime}_{2}]\big{)}\Big{)}\cdot
q_{P}(x)\cdot\mathbbm{1}\big{\\{}x\in(x^{\prime}_{2},x_{2}]\big{\\}}$

For small enough $\epsilon>0$, the allocation $q^{\prime}$ is feasible,
incentive compatible, and provides a strict welfare improvement with respect
to $q$, in contradiction.

The second case to consider corresponds to $IC_{IP}$ binding, in which case
$IC_{PI}$ does not bind. As above, $q_{I}(\diamond)=0$. Using our definitions
of $x_{1}$, $x_{2}$, and $x^{\prime}_{2}$, Corollary 2 implies that
$q_{I}([x_{1},x_{2}])$=0. If $q_{I}\big{(}[x_{2},X]\big{)}=0$, then
$V_{P}(q_{I})\geq u_{P}(x_{1})>V_{P}(q_{P})$, contradicting incentive
compatibility of $q$. Thus, $q_{I}\big{(}[x_{2},X]\big{)}>0$. Assume, towards
a contradiction, that $x^{\prime}_{2}<x_{2}$, so that
$(f-\mu_{p}q_{p})([x_{1},x_{2}])>0$. Consider allocation $q^{\prime}$ with
$q^{\prime}_{P}=q_{P}$ and

$q^{\prime}_{I}~{}~{}~{}~{}~{}=~{}~{}~{}q_{I}\cdot\mathbbm{1}\big{\\{}1\in[0,x_{1}]\big{\\}}~{}+~{}\epsilon\cdot
q_{I}\big{(}(x_{2},X]\big{)}\cdot\big{(}f(x)-\mu_{P}q_{P}(x)\big{)}\cdot\mathbbm{1}\big{\\{}x\in(x_{1},x^{\prime}_{2})\big{\\}}~{}+~{}$
$~{}+~{}\Big{(}1-\epsilon\cdot(f-\mu_{P}q_{P})\big{(}[x_{1},x^{\prime}_{2}]\big{)}\Big{)}\cdot
q_{I}(x)\cdot\mathbbm{1}\big{\\{}x\in[x_{2},X]\big{\\}}.$

For small enough $\epsilon>0$, allocation $q^{\prime}$ is feasible, incentive
compatible, and a provides strict welfare improvement with respect to $q$, in
contradiction. The claim follows. $\blacksquare$

Proof of Lemma 4. Let $q$ be an optimal allocation. Recall that an allocation
$q$ exhibits disposal for type $k$ if there are sets $A,B$ such that
$A\triangleleft B$, $(f-\mu_{P}q_{P}-\mu_{I}q_{I})(A)>0$, and $q_{k}(B)>0$.
Although Lemma 3 shows that $P$-agents are served in one continuous block
$[x_{1},x_{2}]$, this still leaves room for disposal involving $P$-agents, if
there is some unused suppply on $[0,x_{1}]$. Thus, we formulate the next claim
for both types:

Claim A1. If $IC_{ik}$ does not bind for $q$, then $q$ does not exhibit
disposal for $k$-agents.

Proof. If $q$ exhibits disposal for $k$-agents, then for sufficiently small
$\epsilon>0$, the allocation $q^{\prime}$ with $q^{\prime}_{i}=q_{i}$, and

$q^{\prime}_{k}(x)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=~{}~{}~{}~{}~{}~{}~{}\Big{(}q_{k}(x)+\epsilon\cdot
q_{k}(B)\cdot\big{(}f(x)-\mu_{P}q_{P}(x)-\mu_{I}q_{I}(x)\big{)}\Big{)}\cdot\mathbbm{1}\big{\\{}x\in
A\big{\\}}~{}+~{}$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+~{}\Big{(}1-\epsilon\cdot(f-\mu_{P}q_{P}-\mu_{I}q_{I})(A)\Big{)}\cdot
q_{k}(x)\cdot\mathbbm{1}\big{\\{}x\in
B\big{\\}}~{}+~{}q_{k}(x)\cdot\mathbbm{1}\big{\\{}x\not\in(A\cup B)\big{\\}}$

is feasible, incentive compatible, and provides a strict welfare improvement,
in contradiction. $\square$

Assume $IC_{PI}$ does not bind. Then $q_{I}(\diamond)=0$, and
$x_{3}=\sup(\textrm{supp}(q))\in[0,X]$. If $x_{3}\leq x_{2}$, then by Lemma 3,
$x_{3}\leq x_{1}$, where $q_{P}=f~{}|~{}[x_{1},x_{2}]$. In this case,
$V_{P}(q_{I})>u_{P}(x_{1})>V_{P}(q_{P})$, contradicting incentive
compatibility. If $(f-\mu_{P}q_{P}-\mu_{I}q_{I})\big{(}[0,x_{3}]\big{)}>0$,
then for small enough $\delta>0$, we have
$(f-\mu_{P}q_{P}-\mu_{I}q_{I})\big{(}[0,x_{3}-\delta]\big{)}>0$. Notice also
that $q_{I}\big{(}(x_{3}-\delta,x_{3}]\big{)}>0$; otherwise,
$x_{3}=\sup(\textrm{supp}(q))\leq x_{3}-\delta<x_{3}$, in contradiction. Thus,
the allocation $q$ exhibits disposal for type $I$ with $A=[0,x_{3}-\delta]$
and $B=(x_{3}-\delta,x_{3}]$, leading to a contradiction. Therefore, an
optimal allocation in this case takes the form $q_{P}=f~{}|~{}[x_{1},x_{2}]$,
$q_{I}=f~{}|~{}[0,x_{1}]\cup[x_{2},x_{3}]$, where $F(x_{3})=\mu_{P}+\mu_{I}$,
consistent with the lemma’s statement.

Assume $IC_{PI}$ binds. Then, $IC_{IP}$ does not bind by Corollary 2. By Lemma
3, $q_{P}=f~{}|~{}[x_{1},x_{2}]$. By our arguments above, $q$ does not exhibit
disposal for $P$-agents. Thus, $q_{I}([0,x_{1}])=\mu_{I}^{-1}F(x_{1})$. Denote
$x_{3}=\sup\big{(}\textrm{supp}(q_{I})\cap[0,X]\big{)}$.

If $x_{3}=x_{1}$, then $q=(1-\beta)\cdot
f~{}|~{}[0,x_{1}]+\beta\cdot\delta_{\diamond}$, which satisfies the lemma’s
statement.

Otherwise, $x_{3}>x_{2}$. Towards a contradiction, assume that
$(f-\mu_{I}q_{I})\big{(}[x_{2},x_{3}]\big{)}>0$. Then, for small enough
$\delta>0$, we have $x_{2}<x_{3}-\delta<x_{3}$, where $(f-\mu_{I}q_{I})(A)>0$
and $q_{I}(B)>0$ with $A=[x_{2},x_{3}-\delta]$. and $B=(x_{3}-\delta,x_{3}]$.
By definition, $A\triangleleft B$. Let $\gamma(x)\in(0,1)$ for $x\in B$ be
such that

$V_{P}\Big{(}\gamma(x)\cdot(f-q_{I}\mu_{I})~{}|~{}A+(1-\gamma(x))\cdot\delta_{\diamond}\Big{)}=V_{P}(\delta_{x}).$

By Lemma 0,

$V_{I}\Big{(}\gamma(x)\cdot(f-q_{I}\mu_{I})~{}|~{}A+(1-\gamma(x))\cdot\delta_{\diamond}\Big{)}=V_{I}(\delta_{x}).$

Integrating these equations with respect to $q_{I}~{}|~{}B$ yields

$V_{P}\Big{(}\overline{\gamma}\cdot(f-q_{I}\mu_{I})~{}|~{}A+(1-\overline{\gamma})\cdot\delta_{\diamond}\Big{)}~{}=~{}V_{P}(q_{I}~{}|~{}B)~{}~{}~{},~{}~{}~{}V_{I}\Big{(}\overline{\gamma}\cdot(f-q_{I}\mu_{I})~{}|~{}A+(1-\overline{\gamma})\cdot\delta_{\diamond}\Big{)}~{}>~{}V_{I}(q_{I}~{}|~{}B)$

for some $\overline{\gamma}\in(0,1)$. Consider allocation $q^{\prime}$ given
by $q^{\prime}_{P}=q_{P}$ and

$q^{\prime}_{I}~{}~{}~{}~{}=~{}~{}~{}~{}q_{I}~{}+~{}\epsilon\cdot\big{(}\overline{\gamma}\cdot(f-q_{I}\mu_{I})~{}|~{}A+(1-\overline{\gamma})\cdot\delta_{\diamond}\big{)}~{}-~{}\epsilon\cdot
q_{I}~{}|~{}B.$

For $\epsilon>0$ small enough, $q^{\prime}$ is feasible, incentive compatible,
and provides strict welfare improvement, in contradiction. Therefore,
$q_{I}~{}|~{}[x_{2},x_{3}]=f~{}|~{}[x_{2},x_{3}]$, and $q_{I}=(1-\beta)\cdot
f~{}|~{}[0,x_{1}]\cup[x_{2},x_{3}]+\beta\delta_{\diamond}$ and
$q_{P}=f~{}|~{}[x_{1},x_{2}]$, satisfying the lemma’s statement.
$\blacksquare$

Proof of Proposition 3. The optimal allocation exists by Proposition 0. Let
$q$ be an optimal allocation. From the proof of Lemma 4,
$q_{P}=f~{}|~{}[x_{1},x_{2}]$ and $q_{I}=(1-\beta)\cdot
f~{}|~{}[0,x_{1}]\cup[x_{2},x_{3}]+\beta\cdot\delta_{\diamond}$. It remains to
show that $q$ is a unique optimal allocation. Assume, towards a contradiction,
that there is another optimal allocation
$q^{\prime}_{P}=f~{}|~{}[x^{\prime}_{1},x^{\prime}_{2}]$,
$q^{\prime}_{I}=(1-\beta^{\prime})\cdot
f~{}|~{}[0,x^{\prime}_{1}]\cup[x^{\prime}_{2},x^{\prime}_{3}]+\beta^{\prime}\cdot\delta_{\diamond}$.
Consider the allocation $q^{\prime\prime}=0.5\cdot q+0.5q^{\prime}$. By
linearity of the welfare function and the constraints, $q^{\prime\prime}$ is
an optimal allocation as well. Hence,
$q^{\prime\prime}_{P}=f~{}|~{}[x^{\prime\prime}_{1},x^{\prime\prime}_{2}]$,
$q^{\prime\prime}_{I}=(1-\beta^{\prime\prime})\cdot
f~{}|~{}[0,x^{\prime\prime}_{1}]\cup[x^{\prime\prime}_{2},x^{\prime\prime}_{3}]+\beta^{\prime\prime}\cdot\delta_{\diamond}$,
which is impossible unless $x_{i}=x^{\prime}_{i}=x^{\prime\prime}_{i}$ for
$i=1,2,3$ and $\beta=\beta^{\prime}=\beta^{\prime\prime}$, in which case
$q^{\prime\prime}=q^{\prime}=q$. To conclude the proof, feasibility requires
$F(x_{2})-F(x_{1})=\mu_{P}$, and
$F(x_{3})-F(x_{2})+F(x_{1})=(1-\beta)\mu_{I}$. Incentive compatibility for
$I$-agents implies that $\beta<1$. $\blacksquare$

Proof of Proposition 4 and Corollary 4. First, we show that if an allocation
$q$ is feasible and does not exhibit disposal or an inverted spread, then
$q\in\mathcal{A}$, where $\mathcal{A}$ is the set of allocations that are
constructed recursively as described in Section 5.1. Absence of disposal means
that
$\sum_{i}\mu_{i}q_{i}(x)=f(x)\cdot\mathbbm{1}\big{\\{}x\in[0,\overline{X}]\big{\\}}$
and $q_{i}\big{(}[0,\overline{X}]\big{)}=1$ for all $i$.

Claim B1. Suppose allocation $q$ does not exhibit an inverted spread or
disposal, and let $j<k$. Then
$\nu\Big{(}\textrm{conv}\big{(}\textrm{supp}(q_{j})\big{)}\cap\textrm{supp}(q_{k})\Big{)}=0$.

Proof. Since $q$ does not exhibit disposal,
$\diamond\not\in\textrm{supp}(q_{j})$ and
$\textrm{conv}\big{(}\textrm{supp}(q_{j})\big{)}$ is well-defined. Let
$x_{j}\equiv\inf(\textrm{supp}(q_{j}))$, and
$x^{j}\equiv\sup(\textrm{supp}(q_{j}))$. Then
$D\equiv\textrm{conv}(\textrm{supp}(q_{j}))\cap\textrm{supp}(q_{k})$ $=$
$[x_{j},x^{j}]\cap\textrm{supp}(q_{k})$. Towards a contradiction, assume that
$\nu(D)>0$. Since $q_{k}(x)\leq\mu_{k}^{-1}\cdot f(x)$ is bounded above on
$[x_{j},x^{j}]$, there exists $\epsilon>0$ such that
$x_{j}+\epsilon<x^{j}-\epsilon$ and
$q_{k}\big{(}(x_{j}+\epsilon,x^{j}-\epsilon)\big{)}>0$. The definition of
$x_{j},x^{j}$ implies that $q_{j}\big{(}[x_{j},x_{j}+\epsilon]\big{)}>0$ and
$q_{j}\big{(}[x^{j}-\epsilon,x^{j}]\big{)}>0$. Therefore,
$[x_{j},x_{j}+\epsilon]\triangleleft(x_{j}+\epsilon,x^{j}-\epsilon)\triangleleft[x^{j}-\epsilon,x^{j}]$
constitutes an inverted spread, in contradiction. $\square$

Let $x_{i}=\textrm{inf}(\textrm{supp}$ $(q_{i}))$ and
$x^{i}=\textrm{sup}(\textrm{supp}$ $(q_{i}))$ for $i=1,...,N$. Define
recursively $f^{0}=f$ and
$f^{k}=f^{k-1}-\mu_{k}q_{k}=f-\sum_{i=1}^{k}\mu_{i}q_{i}$ for $k=1,...,N-1$.
It is easy to see that Claim B1 and $\sum_{i}\mu_{i}q_{i}(x)=f(x)$ for
$x\in[0,\overline{X}]$ imply that $q_{k}=f^{k-1}~{}|~{}[x_{k},x^{k}]$ and
$f^{k-1}\big{(}[x_{k},x^{k}]\big{)}=\mu_{k}$. Thus, $q\in\mathcal{A}$.

Assume now that $q\in\mathcal{A}$. We have that $q_{1}=f~{}|~{}[x_{1},x^{1}]$
and $q_{k}=f^{k-1}~{}|~{}[x_{k},x^{k}]$, where
$f^{k-1}=f-\sum_{i=1}^{k-1}\mu_{i}q_{i}$ and
$f^{k-1}\big{(}[x_{k},x^{k}]\big{)}=\mu_{k}$. This implies that
$\big{(}f-\sum_{i=1}^{k}\mu_{i}q_{i}\big{)}\big{(}[x_{k},x^{k}]\big{)}=0$. We
now show that $q$ is a first-best allocation for some strictly positive
welfare weights $(\alpha_{1},...,\alpha_{N})$. Define
$w_{k}=\textrm{min}\\{x_{k},...,x_{N}\\}$ and
$w^{k}=\textrm{max}\\{x^{k},...,x^{N}\\}$.

Consider the following procedure that defines weights
$\alpha_{1},...,\alpha_{N}$, auxiliary real numbers $b_{1},...,b_{N}$, and
functions $v_{1},...,v_{N}:~{}[0,X]\rightarrow R$ recursively for
$k=N,N-1,...,2,1$.

For $k=N$, define $\alpha_{N}=1$, $b_{N}=0$, and $v_{N}(x)=u_{N}(x)$.

For $k<N$, suppose we constructed our desired objects up to $k+1$. Consider
the following three cases that we soon show are exhaustive:

Case 1: If $w_{k+1}<x_{k}<x^{k}<w^{k+1}$, define
$\alpha_{k}=\dfrac{v_{k+1}(x_{k})-v_{k+1}(x^{k})}{u_{k}(x_{k})-u_{k}(x^{k})}$
and $b_{k}=v_{k+1}(x_{k})-\alpha_{k}u_{k}(x_{k})$. Then,
$\alpha_{k}u_{k}(x_{k})+b_{k}=v_{k+1}(x_{k})$ and
$\alpha_{k}u_{k}(x^{k})+b_{k}=v_{k+1}(x^{k})$.

Case 2: If $x_{k}<x^{k}\leq w_{k+1}<w^{k+1}$, define
$\alpha_{k}~{}=~{}\dfrac{v_{k+1}(0)-v_{k+1}(x^{k})}{u_{k}(0)-u_{k}(x^{k})}$
and $b_{k}=v_{k+1}(0)-\alpha_{k}u_{k}(0)$. Then,
$\alpha_{k}u_{k}(0)+b_{k}~{}=~{}v_{k+1}(0)$ and
$\alpha_{k}u_{k}(x^{k})+b_{k}~{}=~{}v_{k+1}(x^{k})$.

Case 3: If $w_{k+1}<w^{k+1}\leq x_{k}<x^{k}$, define
$\alpha_{k}~{}=~{}\dfrac{v_{k+1}(x_{k})-v_{k+1}(\overline{X})}{u_{k}(x_{k})-u_{k}(\overline{X})}$
and $b_{k}=v_{k+1}(x_{k})-\alpha_{k}u_{k}(x_{k})$. Then,
$\alpha_{k}u_{k}(x_{k})+b_{k}~{}=~{}v_{k+1}(x_{k})$ and
$\alpha_{k}u_{k}(\overline{X})+b_{k}~{}=~{}v_{k+1}(\overline{X})$.

In all three cases, define

$v_{k}(x)\equiv\max\\{v_{k+1}(x),\alpha_{k}u_{k}(x)+b_{k}\\}=\textrm{max}\\{\alpha_{N}u_{N}(x)+b_{N}~{},~{}\alpha_{N-1}u_{N-1}(x)+b_{N-1}~{},~{}...~{},~{}\alpha_{k}u_{k}(x)+b_{k}\\}.$

Since $u_{k}(\cdot)$ is strictly decreasing, $\alpha_{k}$ is well-defined. The
function $v_{k}(x)$ is also strictly decreasing, hence $\alpha_{k}>0$. Thus,
$\alpha_{1},...,\alpha_{N}$ are conceivable welfare weights.

Claim B2. The three cases considered above are exhaustive.

Proof. Towards a contradiction, assume that $w^{k+1}\in(x_{k},x^{k}]$. Then
there is some $j>k$ such that $\sup(\textrm{supp}(q_{j}))=x^{j}=w^{k+1}$,
hence $q_{j}\big{(}[x_{k},x^{k}]\big{)}=q_{j}\big{(}[x_{k},x^{j})\big{)}>0$,
contradicting
$\big{(}f-\sum_{i=1}^{k}\mu_{i}q_{i}\big{)}\big{(}[x_{k},x^{k}]\big{)}=0$ for
$q\in\mathcal{A}$. Thus, $w^{k+1}\not\in(x_{k},x^{k}]$. A symmetric argument
shows that $w_{k+1}\not\in[x_{k},x^{k})$. Since $w_{k}<w^{k}$, the statement
of the Claim follows. $\square$

Claim B3. For all $k=N,N-1,...,2,1$, for all $j=k,k+1,...,N$:

$\Big{\\{}~{}x\in[w_{k},w^{k}]~{}\Big{|}~{}\alpha_{j}u_{j}(x)+b_{j}=v_{k}(x)~{}\Big{\\}}~{}~{}~{}=~{}~{}~{}[x_{j},x^{j}]~{}\Big{\backslash}~{}\left(\overset{j-1}{\underset{i=k}{\bigcup}}(x_{i},x^{i})\right)$

and $\alpha_{j}u_{j}(x)+b_{j}<v_{k}(x)$ for all
$x\in[w_{k},w^{k}]\big{\backslash}\left([x_{j},x^{j}]~{}\Big{\backslash}~{}\left(\overset{j-1}{\underset{i=k}{\bigcup}}(x_{i},x^{i})\right)\right)$.

Proof. We use induction on $k=N,N-1,...,2,1$.

For $k=N$, the claim is true since
$v_{N}(x)=u_{N}(x)=\alpha_{N}u_{N}(x)+b_{N}$.

Assume the claim holds up to $k+1$. Consider
$\eta_{k}(x)~{}=~{}\alpha_{k}u_{k}(x)+b_{k}-v_{k+1}(x)$. Let
$x^{\prime}<x^{\prime\prime}$ and $\lambda\in(0,1)$. Denote by
$g_{kj}(x)=\alpha_{k}u_{k}(x)+b_{k}-\alpha_{j}u_{j}(x)-b_{j}$ for $j>k$.
Type-$k$ agents are more risk averse than type-$j$ agents for any $j>k$. Since
strict quasi-concavity is invariant with respect to positive affine
transformations, Lemma 1 implies that $g_{jk}(\lambda
x^{\prime}+(1-\lambda)x^{\prime\prime})>\textrm{min}\\{g_{jk}(x^{\prime}),g_{jk}(x^{\prime\prime})\\}$
and

$\eta_{k}(\lambda
x^{\prime}+(1-\lambda)x^{\prime\prime})~{}=~{}\alpha_{k}u_{k}(\lambda
x^{\prime}+(1-\lambda)x^{\prime\prime})+b_{k}~{}-\underset{j=k+1,...,N}{\textrm{max}}\left\\{\alpha_{j}u_{j}(\lambda
x^{\prime}+(1-\lambda)x^{\prime\prime})+b_{j}\right\\}~{}=~{}$
$~{}=~{}\underset{j=k+1,...,N}{\textrm{min}}\left(g_{kj}(\lambda
x^{\prime}+(1-\lambda)x^{\prime\prime})\right)~{}>~{}\underset{j=k+1,...,N}{\textrm{min}}\left(\textrm{min}\left\\{g_{kj}(x^{\prime}),g_{kj}(x^{\prime\prime})\right\\}\right)~{}=~{}$
$~{}=~{}\textrm{min}\left\\{\underset{j=k+1,...,N}{\textrm{min}}\left(g_{kj}(x^{\prime})\right)~{},~{}\underset{j=k+1,...,N}{\textrm{min}}\left(g_{kj}(x^{\prime\prime})\right)\right\\}~{}=~{}\textrm{min}\left\\{\eta_{k}(x^{\prime}),\eta_{k}(x^{\prime\prime})\right\\}$

We conclude that the function $\eta_{k}(x)$ is strictly quasi-concave.

Consider case 1 above, corresponding to $w_{k+1}<x_{k}<x^{k}<w^{k+1}$. Then
$w_{k}=w_{k+1}$ and $w^{k}=w^{k+1}$. We deal with the case of $j=k$ first. By
construction, $\eta_{k}(x_{k})=\eta_{k}(x^{k})=0$. Hence, $\eta_{k}(\lambda
x_{k}+(1-\lambda)x^{k})>0$ for $\lambda\in(0,1)$. We conclude that
$v_{k}(x)=\alpha_{k}u_{k}(x)+b_{k}$ for $x\in[x_{k},x^{k}]$, and
$v_{k}(x)>v_{k+1}(x)\geq\alpha_{j}u_{j}(x)+b_{j}$ for $j=k+1,...,N$ and
$x\in(x_{k},x^{k})$.

Assume, towards a contradiction, that $\eta_{k}(x)\geq 0$ for some
$x\not\in[x_{k},x^{k}]$. If $x<x_{k}$ then, by strict quasi-concavity of
$\eta_{k}$, we have that $\eta_{k}(x_{k})>0$, contradiction; similarly if
$x>x^{k}$, then $\eta_{k}(x^{k})>0$, contradiction. The claim is true for
$j=k$.

Consider now $j>k$. From continuity of $v_{k}(\cdot)$ and our arguments above,
$v_{k}(x)=v_{k+1}(x)$ for $x\not\in(x_{k},x^{k})$. Then,

$\Big{\\{}~{}x\in[w_{k},w^{k}]~{}\Big{|}~{}\alpha_{j}u_{j}(x)+b_{j}=v_{k}(x)~{}\Big{\\}}~{}=~{}\Big{\\{}~{}x\in[w_{k+1},w^{k+1}]~{}\Big{|}~{}\alpha_{j}u_{j}(x)+b_{j}=v_{k}(x)~{}\Big{\\}}~{}=~{}$
$~{}=~{}\Big{\\{}~{}x\in[w_{k+1},w^{k+1}]~{}\Big{|}~{}\alpha_{j}u_{j}(x)+b_{j}=v_{k+1}(x)~{}\Big{\\}}\Big{\backslash}(x_{k},x^{k})~{}=~{}[x_{j},x^{j}]~{}\Big{\backslash}~{}\left(\left(\overset{j-1}{\underset{i=k+1}{\bigcup}}(x_{i},x^{i})\right)\cup(x_{k},x^{k})\right),$

where the last equality follows from our induction hypothesis. The claim then
follows.

Consider case 2 above, so that $x_{k}<x^{k}\leq w_{k+1}<w^{k+1}$. Then
$w_{k}=x_{k}$ and $w^{k}=w^{k+1}$. By construction,
$\eta_{k}(0)=\eta_{k}(x^{k})=0$. Repeating similar arguments to those used for
case 1, we conclude that $v_{k}(x)=\alpha_{k}u_{k}(x)+b_{k}>v_{k+1}(x)$ for
all $x\in(0,x^{k})\supseteq(x_{k},x^{k})$ and
$v_{k}(x)=v_{k+1}(x)>\alpha_{k}u_{k}(x)+b_{k}$ for all
$x\in(x^{k},w^{k}]=[w_{k},w^{k}]\backslash[x_{k},x^{k}]$. Therefore, we can
follow the same arguments used for case 1 to show the claim for
$j=k,k+1,...,N$.

Finally, consider case 3, where $w_{k+1}<w^{k+1}\leq x_{k}<x^{k}$. Then
$w_{k}=w_{k+1}$ and $w^{k}=x^{k}$. By construction,
$\eta_{k}(x_{k})=\eta_{k}(\overline{X})=0$. Repeating the same arguments
again, we conclude that $v_{k}(x)=\alpha_{k}u_{k}(x)+b_{k}>v_{k+1}(x)$ for all
$x\in(x_{k},\overline{X})\supseteq(x_{k},x^{k})$ and
$v_{k}(x)=v_{k+1}(x)>\alpha_{k}u_{k}(x)+b_{k}$ for all
$x\in[w_{k},x_{k})=[w_{k},w^{k}]\backslash[x_{k},x^{k}]$. Therefore, we can
repeat the arguments pertaining to case 1 above and the claim follows.
$\square$

By Clam B3 for $k=1$, and the fact that $q\in\mathcal{A}$, it follows that for
all $x\in[0,\overline{X}]$,

$\begin{array}[]{lllllllll}a_{j}u_{j}(x)+b_{j}&=&v_{1}(x)&\Longleftrightarrow&x\in[x_{j},x^{j}]\Big{\backslash}\left(\overset{j-1}{\underset{i=1}{\bigcup}}(x_{i},x^{i})\right)&\Longleftrightarrow&x\in\textrm{supp}(q_{j}),\\\
a_{j}u_{j}(x)+b_{j}&<&v_{1}(x)&\Longleftrightarrow&x\not\in[x_{j},x^{j}]\Big{\backslash}\left(\overset{j-1}{\underset{i=1}{\bigcup}}(x_{i},x^{i})\right)&\Longleftrightarrow&x\not\in\textrm{supp}(q_{j}),\end{array}$

where, for any $q\in\mathcal{A}$, we use
$(x_{j},x^{j})\cap\textrm{supp}(q_{m})=\varnothing$ for $m>j$ since
$q_{m}=\big{(}f-\sum_{i<m}\mu_{i}q_{i}\big{)}$ $~{}|~{}[x_{m},x^{m}]$.

Consider welfare weights $\alpha_{1},...,\alpha_{N}$ and any allocation
$q^{\prime}$ that does not exhibit disposal.212121As already noted, an
allocation that exhibits disposal cannot be first-best—there is an obvious way
to improve the welfare of such an allocation by assigning the unused supply of
a better quality to an agent who exhibits disposal. Then,

$W(q^{\prime})~{}+~{}\sum_{j=1}^{N}\mu_{j}b_{j}~{}=~{}\overset{\overline{X}}{\underset{0}{\int}}\sum_{j=1}^{N}\mu_{j}(\alpha_{j}u_{j}(x)+b_{j})q^{\prime}_{j}(x)dx~{}\leq~{}\overset{\overline{X}}{\underset{0}{\int}}\sum_{j=1}^{N}\mu_{j}v_{1}(x)q^{\prime}_{j}(x)dx~{}\leq~{}\overset{\overline{X}}{\underset{0}{\int}}v_{1}(x)f(x)dx~{}=~{}$
$~{}=~{}\sum_{j=1}^{N}\underset{\textrm{supp}(q_{j})}{\int}v_{1}(x)f(x)dx~{}=~{}\sum_{j=1}^{N}\underset{\textrm{supp}(q_{j})}{\int}(\alpha_{j}u_{j}(x)+b_{j})\mu_{j}q_{j}(x)dx~{}=~{}W(q)~{}+~{}\sum_{j=1}^{N}\mu_{j}b_{j}$

We conclude that $q$ is a first-best allocation for strictly positive welfare
weights $\alpha_{1},...,\alpha_{N}$.

The proof that any first-best allocation is Pareto efficient is standard and
omitted. Suppose now that $q$ is a Pareto efficient allocation. As already
argued, $q$ does not exhibit disposal. Suppose $q$ exhibits an inverted spread
between types $j,k$ with $j<k$. We can fix the allocation of all other types
$i\neq j,k$, and use an allocation $q^{\prime}$ following the construction in
the proof of Lemma 2 (with $P=j$ and $I=k$) to get a strict improvement for
$k$-agents without altering other agents’ payoffs. It follows that any Pareto
efficient allocation does not exhibit disposal or an inverted spread. This
completes the proof of Proposition 4.

To complete the proof of Corollary 4, two claims remain. First, we need to
show that the number of different blocks is no more than $2N-1$. This follows
from the fact that the boundaries of the blocks defining the allocation are
$x_{N},...,x_{1},x^{1},...,x^{N}$, with some of these points possibly
coinciding. Second, we need to show that the allocation given to $k$-agents
consists of no more than $k$ disjoint blocks. This follows from
$\textrm{supp}(q_{k})=[x_{k},x^{k}]\Big{\backslash}\left(\overset{k-1}{\underset{i=1}{\bigcup}}(x_{i},x^{i})\right)$.
$\blacksquare$

Proof of Proposition 5.

By Proposition 0, a solution to the mechanism designer’s problem exists. We
show uniqueness after proving the first part of the proposition.

Proof of Part 1. Suppose $q=(q_{1},...,q_{N})$ is a solution to the mechanism
designer’s problem. Consider two different agent types, $j$ and $k$. For
$\epsilon>0$, consider the set

$A_{jk}^{\epsilon}~{}\equiv~{}\left\\{~{}x\in[0,X]~{}\Big{|}~{}\text{min}\big{\\{}q_{j}(x),q_{k}(x)\big{\\}}>\epsilon~{}\right\\}.$

Since $q^{j}$ and $q^{k}$ are measurable, the set $A_{jk}^{\epsilon}$ is
measurable as well. Let $\nu(\cdot)$ be a $\mathcal{L}$ebesque measure.
Towards a contradiction, assume that $\nu(A_{jk}^{\epsilon})>0$. Since $\nu$
is non-atomic, we can partition $A_{jk}^{\epsilon}$ into $N+1$ disjoint
subsets $\\{A_{i}\\}_{i=1}^{N+1}$ such that $\nu(A_{i})>0$ for all
$i=1,...,N+1$. Consider any arbitrary family of disjoint measurable sets
$\\{B_{i}\\}_{i=1}^{N+1}$, with $B_{i}\subseteq A_{i}$ for all $i=1,...,N+1$.
Denote by $B=\underset{i}{\bigcup}B_{i}$. Let
$\omega~{}=~{}(\omega_{1},...,\omega_{N+1})~{}~{}\in~{}~{}\left[0,1\right]^{N+1}$
and define the following allocation:

$\begin{array}[]{llll}\widetilde{q}_{j}(\omega)(x)&=&q_{j}(x)\cdot\mathbbm{1}\\{t\not\in
B\\}~{}+~{}\left[q_{j}(x)-\epsilon+\epsilon\cdot\dfrac{\mu_{j}+\mu_{k}}{\mu_{j}}\cdot\displaystyle\sum_{i=1}^{N+1}(1-\omega_{i})\cdot\mathbbm{1}\\{x\in
B_{i}\\}\right]\cdot\mathbbm{1}\\{x\in B\\},\\\
\widetilde{q}_{k}(\omega)(x)&=&q_{k}(x)\cdot\mathbbm{1}\\{t\not\in
B\\}~{}+~{}\left[q_{k}(x)-\epsilon+\epsilon\cdot\dfrac{\mu_{j}+\mu_{k}}{\mu_{k}}\cdot\displaystyle\sum_{i=1}^{N+1}\omega_{i}\cdot\mathbbm{1}\\{x\in
B_{i}\\}\right]\cdot\mathbbm{1}\\{x\in B\\},\\\
\widetilde{q}_{l}(\omega)(x)&=&q_{l}(x)~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{for}~{}l\neq
j,k.\end{array}$

Define also

$\begin{array}[]{llllll}h^{0}(\omega)&\equiv&\displaystyle\sum_{i=1}^{N+1}\omega_{i}\cdot\nu(B_{i})~{}~{}-~{}\dfrac{\mu_{k}}{\mu_{j}+\mu_{k}}\cdot\nu(B)&=&\epsilon^{-1}\cdot\dfrac{\mu_{k}}{\mu_{j}+\mu_{k}}\cdot\big{(}\widetilde{q}_{k}([0,X]-q_{k}([0,X])\big{)},\\\
h^{l}(\omega)&\equiv&\displaystyle\sum_{i=1}^{N+1}\left(\omega_{i}~{}-~{}\dfrac{\mu_{k}}{\mu_{j}+\mu_{k}}\right)\underset{B_{i}}{\displaystyle\int}u_{l}(x)dx&=&\epsilon^{-1}\cdot\dfrac{\mu_{k}}{\mu_{j}+\mu_{k}}\cdot\big{(}V_{l}(\widetilde{q}_{k}(\omega))-V_{l}(q_{k})\big{)},\end{array}$
(6)

for all $l\in\\{1,...,N\\}$.

Claim C1. If $h^{0}(\omega)=0$, then the allocation $\widetilde{q}(\omega)$ is
feasible.

Proof. By construction, $q_{j}(x),q_{k}(x)>\epsilon$ for $x\in B$. Thus,
$\widetilde{q}_{i}(x)\geq 0$ for all $x\in[0,X]$ and $i\in\\{1,...,n\\}$.
Additionally,
$\mu_{j}\widetilde{q}_{j}+\mu_{k}\widetilde{q}_{k}=\mu_{j}q_{j}+\mu_{k}q_{k}$,
implying that
$f(x)-\sum_{i}\mu_{i}\widetilde{q}_{i}(x)=f(x)-\sum_{i}\mu_{i}q_{i}(x)\geq 0$
from $q$’s feasibility. It also implies that
$\widetilde{q}_{j}([0,X])=q_{j}([0,X])$ since
$\widetilde{q}_{k}([0,X])=q_{k}([0,X])$. Thus, $\widetilde{q}_{i}([0,X])\leq
1$ for all $i\in\\{1,...,N\\}$, again using $q$’s feasibility. $\square$
The functions $h^{l}(\omega)$ are linear in $\omega$, and their gradients are
given by

$\begin{array}[]{lllllllll}\triangledown_{\omega}h^{0}&=&\big{(}\nu(B_{1})&,&...&,&\nu(B_{N+1})\big{)},&\\\
\triangledown_{w}h^{l}&=&\left(\underset{B_{1}}{\displaystyle\int}u_{l}(x)dx\right.&,&...&,&\left.\underset{B_{N+1}}{\displaystyle\int}u_{l}(x)dx\right),&~{}~{}~{}l=1,...,N.\end{array}$

Let $\zeta^{l}_{m}=\left(\dfrac{\partial
h^{l}}{\partial\omega_{1}}~{},~{}\right.$ $...~{},~{}$ $\left.\dfrac{\partial
h^{l}}{\partial\omega_{m}}\right)$ be a truncated gradient of $h^{l}$.

Claim C2. There exist $\\{B_{i}\\}_{i=1}^{N+1}$ as above such that the vectors
$\zeta_{m}^{0},...,\zeta_{m}^{m-1}$ are linearly independent for
$m\in\\{1,...,N+1\\}$.

Proof. We prove the claim by induction on $m=1,...,N+1$.

For $m=1$, take $B_{1}=A_{1}$. Then, $\zeta_{1}^{0}=\nu(B_{1})>0$.

Suppose the statement is true for $m$. Fix $B_{1},...,B_{m}$. Since
$\zeta_{m}^{0},...,\zeta_{m}^{m-1}$ are linearly independent, these vectors
constitute a basis in $\mathbb{R}^{m}$. Therefore,

$\left(\dfrac{\partial
h^{m}}{\partial\omega_{1}}~{},\right....~{},\left.~{}\dfrac{\partial
h^{m}}{\partial\omega_{m}}\right)=\overset{m-1}{\underset{i=0}{\sum}}\lambda_{i}\zeta_{m}^{i},$

where $\lambda_{i}$, $i=0,...,m-1$, are determined uniquely.

Towards a contradiction, assume that for all $B_{m+1}\subseteq A_{m+1}$ such
that $\nu(B_{m+1})>0$, the vectors $\zeta_{m+1}^{0},...,\zeta_{m+1}^{m}$ are
not linearly independent. Then,

$\zeta^{m}_{m+1}\equiv\left(\dfrac{\partial
h^{m}}{\partial\omega_{1}}~{},\right....~{},\left.~{}\dfrac{\partial
h^{m}}{\partial\omega_{m+1}}\right)=\overset{m-1}{\underset{i=0}{\sum}}\lambda_{i}\zeta_{m+1}^{i},$

where $\lambda_{i}$, $i=0,...,m-1$, do not depend on $B_{m+1}$. Hence,

$\underset{B_{m+1}}{\int}u_{m}(x)dx~{}=~{}\dfrac{\partial
h^{m}}{\partial\omega_{m+1}}~{}=~{}\sum_{i=0}^{m-1}\lambda_{i}\cdot\dfrac{\partial
h^{i}}{\partial\omega_{m+1}}~{}=~{}\overset{m-1}{\underset{i=1}{\sum}}\lambda_{i}\underset{B_{m+1}}{\int}u_{i}(x)dx+\lambda_{0}\nu(B_{m+1})$
(7)

For any $\delta>0$, denote by

$D^{+}_{\delta}\equiv\Big{\\{}x\in
A_{m+1}~{}\Big{|}~{}u_{m}(x)>\overset{m-1}{\underset{i=1}{\sum}}\lambda_{i}u_{i}(x)+\lambda_{0}+\delta\Big{\\}},~{}D^{-}_{\delta}\equiv\Big{\\{}x\in
A_{m+1}~{}\Big{|}~{}u_{m}(x)<\overset{m-1}{\underset{i=1}{\sum}}\lambda_{i}u_{i}(x)+\lambda_{0}-\delta\Big{\\}}.$

Then,

$\underset{D_{\delta}^{+}}{\int}u_{m}(x)dx>\overset{m-1}{\underset{i=1}{\sum}}\lambda_{i}\underset{D_{\delta}^{+}}{\int}u_{i}(x)dx+\lambda_{0}\nu(D_{\delta}^{+})+\delta\nu(D_{\delta}^{+}),~{}~{}~{}\textrm{and}$
$\underset{D_{\delta}^{-}}{\int}u_{m}(x)dx<\overset{m-1}{\underset{i=1}{\sum}}\lambda_{i}\underset{D_{\delta}^{-}}{\int}u_{i}(x)dx+\lambda_{0}\nu(D_{\delta}^{-})-\delta\nu(D_{\delta}^{-}).$

If $\nu(D_{\delta}^{+})>0$ or $\nu(D_{\delta}^{-})>0$, we can pick
$B_{m+1}=D_{\delta}^{+}$ or $B_{m+1}=D_{\delta}^{-}$, respectively, to achieve
a contradiction through equation (7). It follows that
$\nu(D_{\delta}^{+})=\nu(D_{\delta}^{-})=0$.

It follows that

$\nu\left(\left\\{x\in
A_{m+1}~{}\Big{|}~{}u_{m}(x)=\overset{m-1}{\underset{i=1}{\sum}}\lambda_{i}u_{i}(x)+\lambda_{0}\right\\}\right)~{}=~{}\nu\left(A_{m+1}\Big{\backslash}\left(\overset{\infty}{\underset{r=1}{\bigcup}}\left(D_{1/r}^{+}\cup
D_{1/r}^{-}\right)\right)\right)~{}=~{}\nu(A_{m+1})>0,$

in contradiction to the independence of different agent types’ utility
functions. $\square$

Claim C3. There exists a unit vector $e\in R^{N+1}$ such that
$(\triangledown_{\omega}h^{k}\cdot e)$$>0$ and
$(\triangledown_{\omega}h^{i}\cdot e)=0$ for
$i\in\\{0,1,...,N\\}\backslash\\{k\\}$.
Proof. If $m=N+1$, then $\zeta_{m}^{l}=\triangledown_{\omega}h^{l}$. Thus, by
Claim C2, the vector $\triangledown_{\omega}h^{k}$ is linearly independent of
$\\{\triangledown_{\omega}h^{i}\\}_{i\in\\{0,1,...,N\\}\backslash\\{k\\}}$.
The claim follows. $\square$

Denote by
$\omega^{*}=\frac{\mu_{k}}{\mu_{j}+\mu_{k}}\cdot(1,...,1)\in[0,1]^{N+1}$ and
define $\omega=\omega^{*}+c\cdot e\in[0,1]^{N+1}$ for $c>0$ small enough, say,
$c=(1/2)\cdot\min\big{\\{}\frac{\mu_{k}}{\mu_{j}+\mu_{k}},\frac{\mu_{j}}{\mu_{j}+\mu_{k}}\big{\\}}$.

Claim C4. Let $\widetilde{q}=\widetilde{q}(\omega)$ be an allocation defined
as above. Then, (1) $V_{k}(\widetilde{q}_{k})>V_{k}(q_{k})$; (2)
$V_{k}(\widetilde{q}_{j})<V_{k}(q_{j})$; (3)
$V_{k}(\widetilde{q}_{l})=V_{k}(q_{l})$ for all
$l\in\\{1,...,n\\}\backslash\\{j,k\\}$; and (4)
$V_{r}(\widetilde{q}_{l})=V_{r}(q_{l})$ for all
$r\in\\{1,...,n\\}\backslash\\{k\\}$, $l\in\\{1,...,n\\}$.
Proof. Recall that $\widetilde{q}(\omega^{*})=q$ and $h^{l}(\omega^{*})=0$,
for all $l\in\\{0,1,...,N\\}$. Thus,

$V_{k}(\widetilde{q}_{k})-V_{k}(q_{k})=\epsilon\cdot\frac{\mu_{j}+\mu_{k}}{\mu_{k}}\cdot
c\cdot(\triangledown_{\omega}h^{k}\cdot e)>0,$

so statement (1) holds. Next,
$\mu_{j}\widetilde{q}_{j}+\mu_{k}\widetilde{q}_{k}=\mu_{j}q_{j}+\mu_{k}q_{k}$.
Therefore,

$V_{k}(\widetilde{q}_{j})=V_{k}\Big{(}\frac{\mu_{j}q_{j}+\mu_{k}q_{k}-\mu_{k}\widetilde{q}_{k}}{\mu_{j}}\Big{)}=V_{k}(q_{j})+\frac{\mu_{k}}{\mu_{j}}\cdot\Big{(}V_{k}(q_{k})-V_{k}(\widetilde{q}_{k})\Big{)}<V_{k}(q_{j}),$

proving statement (2). Statement (3) follows from
$\widetilde{q}_{l}(\omega)=q_{l}$ for $l\neq j,k$. We now show statement (4).
If $l\in\\{1,...,n\\}\backslash\\{j,k\\}$, then statement (4) follows from
$\widetilde{q}_{l}(\omega)=q_{l}$. From Claim C3,
$(\triangledown_{\omega}h^{r}\cdot e)=0$ for all $r\neq k$. Therefore,
$h^{r}(\omega)=h^{r}(\omega^{*})+\epsilon\cdot\frac{\mu_{j}+\mu_{k}}{\mu_{k}}\cdot
c\cdot(\triangledown_{\omega}h^{r}\cdot e)=0$ for all $r\neq k$. Thus, if
$l=k$, then
$V_{r}(\widetilde{q}_{k})=V_{r}(q_{k})+h^{r}(\omega)=V_{r}(q_{k})$. Finally,
for $l=j$,

$V_{r}(\widetilde{q}_{j})=V_{r}\big{(}\frac{\mu_{j}q_{j}+\mu_{k}q_{k}-\mu_{k}\widetilde{q}_{k}}{\mu_{j}}\big{)}=V_{r}(q_{j})+\frac{\mu_{k}}{\mu_{j}}\cdot\big{(}V_{r}(q_{k})-V_{r}(\widetilde{q}_{k})\big{)}=V_{r}(q_{j}).$

$\square$

Claim C5. Allocation $\widetilde{q}(\omega)$, constructed above, is feasible
and incentive compatible.
Proof. Feasibility follows from Claim C1 since, from Claim C3,
$h^{0}(\omega)=h^{0}(\omega^{*})+\epsilon\cdot\frac{\mu_{j}+\mu_{k}}{\mu_{k}}\cdot
c\cdot(\triangledown_{\omega}h^{0}\cdot e)=0$. Incentive compatibility follows
from Claim C4 and incentive compatibility of $q$. $\square$

By Claim C4,

$W(\widetilde{q})=\sum_{i=1}^{n}\mu_{i}V_{i}(\widetilde{q}_{i})=\sum_{i\neq
k}\mu_{i}V_{i}(q_{i})+\mu_{k}V_{k}(\widetilde{q}_{k})>\sum_{i=1}^{n}\mu_{i}V_{i}(q_{i})=W(q).$

Using Claim C5, the allocation $\widetilde{q}$ is a viable improvement and $q$
cannot be optimal. Therefore, $\nu(A^{\epsilon}_{jk})=0$ for all $\epsilon>0$
for all $j,k\in\\{1,...,n\\}$ such that $j\neq k$. It follows that, at the
optimal allocation, supply can be used by at most one type of agent almost
everywhere.

We now show that, whenever a supply of some quality good is used non-
trivially, its supply is exhausted. The proof mirrors our construction above.
Namely, for any $\epsilon>0$, we now consider

$A_{0k}^{\epsilon}~{}\equiv~{}\left\\{~{}x\in[0,X]~{}\Big{|}~{}\text{min}\big{\\{}q_{k}(x),f(x)-\sum_{i}\mu_{i}q_{i}(x)\big{\\}}>\epsilon~{}\right\\}.$

Towards a contradiction, assume that $\nu(A_{0k}^{\epsilon})>0$. Introduce a
partition $\\{A_{i}\\}_{i=1}^{N+1}$ of $A_{0k}^{\epsilon}$, and then
corresponding subsets $\\{B_{i}\\}_{i=1}^{N+1}$ and $B$ as in our analysis of
$A^{\epsilon}_{jk}$. Let $\omega\in[0,1]^{N+1}$, and let $q^{\prime}(\omega)$
be given by

$q^{\prime}_{k}(\omega)(x)~{}=~{}q_{k}(x)\cdot\mathbbm{1}\\{t\not\in
B\\}~{}+~{}\left[q_{k}(x)-\epsilon+\epsilon\cdot\frac{1+\mu_{k}}{\mu_{k}}\overset{N+1}{\underset{i=1}{\sum}}\omega_{i}\cdot\mathbbm{1}\\{x\in
B_{i}\\}\right]\cdot\mathbbm{1}\\{x\in B\\},$

with $q^{\prime}_{l}=q_{l}$ for $l\neq k$. We then have

$\epsilon^{-1}\cdot\frac{\mu_{k}}{1+\mu_{k}}\big{(}q^{\prime}_{k}([0,X]-q_{k}([0,X])\big{)}~{}=~{}h^{0}(\omega)~{}~{}~{}~{}~{}~{},~{}~{}~{}~{}~{}~{}\epsilon^{-1}\cdot\frac{\mu_{k}}{1+\mu_{k}}\big{(}V_{l}(q^{\prime}(\omega)_{k})-V_{l}(q_{k})\big{)}~{}=~{}h^{l}(\omega)$

for $l=1,...,N$, where $h^{i}(\omega)$, $i=0,1,...,N$ are defined in eq. (6).

Claim C1’. If $h^{0}(\omega)=0$, the allocation $q^{\prime}(\omega)$ is
feasible.
Proof. Since $q_{k}(x)>\epsilon$ for $x\in B$, and $q^{\prime}_{l}=q_{l}$ for
$l\neq k$, then $q^{\prime}_{i}(x)\geq 0$ for all $x\in[0,X]$ and
$i\in\\{1,...,n\\}$. Next, since $f(x)-\sum_{i}\mu_{i}q_{i}(x)>\epsilon$ for
$x\in B$, then $f(x)-\sum_{i}\mu_{i}q^{\prime}_{i}(x)\geq
f(x)-\sum_{i}\mu_{i}q_{i}(x)-\epsilon\geq 0$ for $x\in B$, and
$f(x)-\sum_{i}\mu_{i}q^{\prime}_{i}(x)=f(x)-\sum_{i}\mu_{i}q_{i}(x)\geq 0$ for
$x\not\in B$. Finally, $q^{\prime}_{k}([0,X])=q_{k}([0,X])+\epsilon\cdot
h^{0}(\omega)=q_{k}([0,X])$, so that $q^{\prime}_{i}([0,X])=q_{i}([0,X])\leq
1$ for all $i$. $\square$

By Claim C3, there exists a unit vector $e\in R^{N+1}$ such that
$(\triangledown_{\omega}h^{k}\cdot e)$ $>0$ and
$(\triangledown_{\omega}h^{i}\cdot e)=0$ for
$i\in\\{0,1,...,N\\}\backslash\\{k\\}$. Denote by
$\omega^{*}=\frac{\mu_{k}}{1+\mu_{k}}\cdot(1,...,1)\in[0,1]^{N+1}$, and
consider $\omega=\omega^{*}+c\cdot e\in[0,1]^{N+1}$, where $c>0$ is small
enough, say,
$c=(1/2)\cdot\min\big{\\{}\frac{\mu_{k}}{1+\mu_{k}},\frac{1}{1+\mu_{k}}\big{\\}}$.

Claim C4’. Let $q^{\prime}=q^{\prime}(\omega)$ be defined as above. Then, (1)
$V_{k}(q^{\prime}_{k})>V_{k}(q_{k})$; (2) $V_{k}(q^{\prime}_{l})=V_{k}(q_{l})$
for all $l\in\\{1,...,n\\}\backslash\\{k\\}$; and (3)
$V_{r}(q^{\prime}_{l})=V_{r}(q_{l})$ for all
$r\in\\{1,...,n\\}\backslash\\{k\\}$, $l\in\\{1,...,n\\}$
Proof. Recall that $q^{\prime}(\omega^{*})=q$ and $h^{l}(\omega^{*})=0$ for
all $l\in\\{0,1,...,N\\}$. Thus,

$V_{k}(q^{\prime}_{k})-V_{k}(q_{k})=\epsilon\cdot\frac{1+\mu_{k}}{\mu_{k}}\cdot
c\cdot(\triangledown_{\omega}h^{k}\cdot e)>0,$

so statement (1) holds. For $l\neq k$, statements (2) and (3) follow from
$q^{\prime}_{l}(\omega)=q_{l}$. Finally, since
$(\triangledown_{\omega}h^{r}\cdot e)=0$ for all $r\neq k$, we have
$V_{r}(q^{\prime}_{k})=V_{r}(q_{k})+h^{r}(\omega)=V_{r}(q_{k})$. $\square$

Claim C5’. Allocation $q^{\prime}(\omega)$, constructed above, is feasible and
incentive compatible.
Proof. Feasibility follows from
$h^{0}(\omega)=h^{0}(\omega^{*})+\epsilon\cdot\frac{1+\mu_{k}}{\mu_{k}}\cdot
c\cdot(\triangledown_{\omega}h^{0}\cdot e)=0$ and Claim C1’. Incentive
compatibility follows from Claim C4’. $\square$

By Claim C4’,

$W(q^{\prime})=\sum_{i=1}^{n}\mu_{i}V_{i}(q^{\prime}_{i})=\sum_{i\neq
k}\mu_{i}V_{i}(q_{i})+\mu_{k}V_{k}(q^{\prime}_{k})>\sum_{i=1}^{n}\mu_{i}V_{i}(q_{i})=W(q).$

Using Claim C5’, we conclude that allocation $q^{\prime}$ is feasible,
incentive compatible, and provides a strict welfare improvement over $q$,
contradicting $q$’s optimality. Therefore, $\nu(A^{\epsilon}_{0k})=0$ for all
$\epsilon>0$ and all $k$. Hence, at the optimal allocation, for almost all
$x$, either the entire supply is utilized or none of it is.

Proof of uniqueness. Assume, towards a contradiction, that there are two
different optimal allocations $q$ and $q^{\prime}$. Since neither $q$ nor
$q^{\prime}$ have mass points, they differ on a set $A\subseteq[0,X]$ of
positive $\mathcal{L}$ebesque measure. Since the mechanism design problem
entails a linear objective subject to linear constraints, the set of
optimizers is convex. In particular, $0.5q+0.5q^{\prime}$ is also an optimal
allocation. However, $0.5q+0.5q^{\prime}$ violates our conclusions above.
Indeed, for any $x\in A$, we have $q(x)\neq q^{\prime}(x)$. It follows that
there is $k\in\\{1,...,N\\}$ such that $q_{k}(x)\neq q^{\prime}_{k}(x)$. Since
$0\leq q_{k}(x)\leq\frac{f(x)}{\mu_{k}}$, and $0\leq
q^{\prime}_{k}(x)\leq\frac{f(x)}{\mu_{k}}$, we have
$0<0.5q_{k}(x)+0.5q^{\prime}_{k}(x)<\frac{f(x)}{\mu_{k}}$. Because the number
of types is finite, this happens for a positive measure of $x\in A$ for at
least one agent type, contradicting what we already showed.

Proof of Part 2. Assume, towards a contradiction, that an optimal allocation
$q$ exhibits a directed cycle in the graph of binding IC constraints. That is,
there exist a sequence of types, $k_{1},...,k_{m}$, such that
$V_{k_{1}}(q_{k_{1}})=V_{k_{1}}(q_{k_{2}})~{},~{}V_{k_{2}}(q_{k_{2}})=V_{k_{2}}(q_{k_{3}})~{},~{}...~{},~{}V_{k_{m}}(q_{k_{m}})=V_{k_{m}}(q_{k_{1}})$.
Consider the following allocation $q^{\prime}$:

$q^{\prime}_{k_{i}}~{}=~{}\left(1-\epsilon/\mu_{k_{i}}\right)q_{k_{i}}+\left(\epsilon/\mu_{k_{i}}\right)q_{k_{i+1}}~{}~{}~{}~{},~{}~{}~{}~{}\textrm{where}~{}~{}~{}~{}\epsilon=\left(1/2\right)\cdot\underset{j}{\text{min}}(\mu_{j})~{}>0,$
$q^{\prime}_{j}~{}=~{}q_{j}~{}~{}~{}~{}\text{for}~{}j\neq k_{1},...,k_{m},$

where we define $k_{m+1}\equiv k_{1}$. The allocation $q^{\prime}_{l}$ is a
convex combination of $q_{j}$ for all $l$ because of our choice of $\epsilon$.
It is straightforward to see that the allocation $q^{\prime}$ is feasible.
Next,

$V_{j}(q^{\prime}_{j})~{}=~{}V_{j}(q_{j})~{}~{}~{}~{}\text{for}~{}j\neq
k_{1},...,k_{m}$

and

$V_{k_{i}}(q^{\prime}_{k_{i}})~{}=~{}V_{k_{i}}\left(\left(1-\epsilon/\mu_{k_{i}}\right)q_{k_{i}}+\left(\epsilon/\mu_{k_{i}}\right)q_{k_{i+1}}\right)=\left(1-\epsilon/\mu_{k_{i}}\right)V_{k_{i}}(q_{k_{i}})+\left(\epsilon/\mu_{k_{i}}\right)V_{k_{i}}(q_{k_{i+1}})=V_{k_{i}}(q_{k_{i}}).$

Therefore, the payoffs of all agent types under the allocation $q^{\prime}$
are the same as under the allocation $q$. The resulting welfare is then
identical.

Now, for an arbitrary type $l$ we have:

$V_{l}(q^{\prime}_{l})~{}=~{}V_{l}(q_{l})~{}\geq~{}V_{l}(q_{j})~{}=~{}V_{l}(q^{\prime}_{j})~{}~{}~{}~{}\text{for}~{}j\neq
k_{1},...,k_{m},$
$V_{l}(q^{\prime}_{l})~{}=~{}V_{l}(q_{l})~{}\geq~{}\left(1-\epsilon/\mu_{k_{i}}\right)V_{l}(q_{k_{i}})+\left(\epsilon/\mu_{k_{i}}\right)V_{l}(q_{k_{i+1}})~{}=~{}V_{l}(q^{\prime}_{k_{i}}).$

We conclude that $q^{\prime}$ is also incentive compatible and, therefore,
optimal. However, since $q_{k_{i}}\neq q_{k_{i+1}}$ as a consequence of the
proposition’s first part proven earlier, $q^{\prime}\neq q$. This contradicts
the uniqueness of an optimal allocation, already shown. $\blacksquare$

Proof of Corollary 5. Proposition 2 proven above is a special case of
Corollary 5 when $N=2$. Thus, we assume $N>2$. We identify an incentive-
compatible allocation $q^{*}$ for which none of the $IC$ constraints is
binding, and $q^{*}$ is a first-best solution for some welfare weights
$\alpha^{*}$ (now, a vector). The statement of the proposition then follows
directly from Berge’s theorem—the first-best allocation is continuous with
respect to the welfare weights and therefore so are the functions
$V_{j}(q_{j})-V_{j}(q_{k})$ for all $j,k$. Therefore, for any vector of
weights $\alpha$ in a small enough neighborhood of $\alpha^{*}$, we have
$V_{j}(q_{j})-V_{j}(q_{k})>0$ for all $j\neq k$ for the corresponding first-
best allocation $q$.

We call any pair of types $i,i+1$ for $i=1,..,N-1$ _adjacent_. We say that a
feasible allocation $q$ exhibits an _accordion structure for adjacent types
$i,i+1$_, if $q_{i+1}=f~{}|~{}[x_{i+1},x_{i}]\cup[x^{i},x^{i+1}]$, and
$q_{i}=f~{}|~{}[x_{i},x_{i-1}]\cup[x^{i-1},x^{i}]$ for some $0\leq x_{i+1}\leq
x_{i}\leq x_{i-1}\leq x^{i-1}\leq x^{i}\leq x^{i+1}$. A feasible allocation
$q$ exhibits an _accordion structure_ if it exhibits an accordion structure
for any adjacent pairs of types. In this case, there are $0=x_{N}\leq
x_{N-1}\leq...\leq x_{1}<x^{1}\leq...\leq x^{N-1}\leq x^{N}=\overline{X}$ such
that $q_{i}=f~{}|~{}[x_{i},x_{i-1}]\cup[x^{i-1},x^{i}]$ and
$q_{1}=f~{}|~{}[x_{1},x^{1}]$, where $x_{0}=x^{0}=\frac{x_{1}+x^{1}}{2}$.

Claim F1. Let $A\triangleleft B\triangleleft C$ for non-empty measurable
subsets $A,B,C\subseteq[0,\overline{X}]$. Then, for each type of agent $i$,
there is a unique number $\gamma=\gamma_{i}(A,B,C)\in(0,1)$ such that an
$i$-type agent is indifferent between $\gamma\cdot f~{}|~{}A+(1-\gamma)\cdot
f~{}|~{}C$ and $f~{}|~{}B$. Moreover, if $j>i$, then
$\gamma_{j}(A,B,C)<\gamma_{i}(A,B,C)$.
Proof. Existence and uniqueness follow from the continuity and strict
monotonicity of $V_{i}\big{(}\gamma\cdot f~{}|~{}A+(1-\gamma)\cdot
f~{}|~{}C\big{)}$ with respect to $\gamma\in[0,1]$ and the fact that
$V_{i}(f~{}|~{}A)>V_{i}(f~{}|~{}B)>V_{i}(f~{}|~{}C)$. In fact, we can directly
identify

$\gamma_{i}(A,B,C)~{}=~{}\int\gamma_{i}(A,\delta_{x},C)d(f~{}|B).$

By Lemma 0, $\gamma_{j}(A,\delta_{x},C)<\gamma_{i}(A,\delta_{x},C)$ for all
$x\in B$. It follows that $\gamma_{j}(A,B,C)<\gamma_{i}(A,B,C)$. $\square$

Claim F2. Suppose $q$ has an accordion structure for adjacent types $i,i+1$.
Then at least one of the constraints $IC_{i(i+1)}$ and $IC_{(i+1)i}$ does not
bind.
Proof. Let $A=(x_{i+1},x_{i})$, $B=(x_{i},x_{i-1})\cup(x^{i-1},x^{i})$, and
$C=(x^{i},x^{i+1})$. If $A=\varnothing$, then $V_{i}(q_{i})>V_{i}(q_{i-1})$,
and if $C=\varnothing$, then $V_{i+1}(q_{i+1})>V_{i+1}(q_{i})$. In these
cases, the statement of the claim holds.

Suppose $A,C\neq\varnothing$. From $q$’s feasibility, $B\neq\varnothing$.
Thus, $A\triangleleft B\triangleleft C$. Define $\beta$ by

$\beta\cdot f~{}|~{}[x_{i+1},x_{i}]+(1-\beta)\cdot
f~{}|~{}[x^{i},x^{i+1}]=f~{}|~{}[x_{i+1},x_{i}]\cup[x^{i},x^{i+1}].$

Thus,
$\beta=\dfrac{f\big{(}[x_{i+1},x_{i}]\big{)}}{f\big{(}[x_{i+1},x_{i}]\cup[x^{i},x^{i+1}]\big{)}}$.
If $V_{i}(q_{i})\leq V_{i}(q_{i+1})$, then
$\beta\leq\gamma_{i}(A,B,C)<\gamma_{i+1}(A,B,C)$, where we use Claim F1 and
the fact that $f~{}|~{}A=f~{}|~{}[x_{i+1},x_{i}]$,
$f~{}|~{}B=f~{}|~{}[x_{i},x_{i-1}]\cup[x^{i-1},x^{i}]$, and
$f~{}|~{}C=f~{}|~{}[x^{i},x^{i+1}]$. Therefore,
$V_{i+1}(q_{i})<V_{i+1}(q_{i+1})$. Otherwise, $V_{i}(q_{i})>V_{i}(q_{i+1})$.
In both cases the claim’s statement holds. $\square$

Claim F3. If $q$ has an accordion structure and all constraints $IC_{i(i+1)}$,
$IC_{(i+1)i}$ for $i=1,...,N-1$ do not bind, then $q$ is incentive compatible,
and no incentive constraint binds.
Proof. Define $\beta_{i}$ as above:

$\beta_{i}\cdot f~{}|~{}[x_{i+1},x_{i}]+(1-\beta_{i})\cdot
f~{}|~{}[x^{i},x^{i+1}]=f~{}|~{}[x_{i+1},x_{i}]\cup[x^{i},x^{i+1}].$

Let $A_{i}\equiv(x_{i+1},x_{i})$, $C_{i}\equiv(x^{i},x^{i+1})$, and
$B_{i}\equiv(x_{i},x_{i-1})\cup(x^{i-1},x^{i})=A_{i-1}\cup C_{i-1}$. Since
$IC_{i(i+1))}$ and $IC_{i(i+1)}$ do not bind,
$\gamma_{i+1}(A_{i},B_{i},C_{i})<\beta_{i}<\gamma_{i}(A_{i},B_{i},C_{i})$ for
all $i=1,...,N-1$. Consider any types $j<k$. By Claim F1,
$\gamma_{k}(A_{m},B_{m},C_{m})<\beta_{m}$ for all $m<k$. Therefore,
$V_{k}(q_{k})>V_{k}(q_{k-1})>...>V_{k}(q_{j})$. Similarly,
$\gamma_{j}(A_{m},B_{m},C_{m})>\beta_{m}$ for all $m\geq j$. Thus,
$V_{j}(q_{j})>V_{j}(q_{j+1})>...>V_{j}(q_{k})$, as needed. $\square$

We use a recursive procedure to define a parametric family of allocations
$\\{q^{x_{1}}\\}_{x_{1}\in Y}$, where $Y=[0,F^{-1}(F(\overline{X})-\mu_{1})]$,
that have an accordion structure. Let $k\in\\{1,...,N\\}$ denote the state of
the procedure. The procedure starts at state $k=1$ and ends at state $k=N$.

If $k=1$, define $x^{1}=F^{-1}(F(x_{1})+\mu_{1})$ and proceed to state $2$.

If $k\in\\{2,...,N-1\\}$, assume that $x_{k-1},...,x_{1},x^{1},...,x^{k-1}$
have already been defined. There are then three cases.

First, if $F(x_{k})<\mu_{k+1}$ and
$V\big{(}f~{}|~{}[0,x_{k}]\cup[x^{k},x^{\prime}]\big{)}\leq V_{k+1}(q_{k})$,
where $x^{\prime}=F^{-1}\big{(}\sum_{i=1}^{k}\mu_{i}\big{)}$. In this case,
define $x_{k}=x_{k+1}=...=x_{N-1}=0$,
$x^{j}=F^{-1}\big{(}\sum_{i=1}^{j}\mu_{i}\big{)}$ for $j=k,...,N-1$ and
proceed to state $N$.

Second, $F(\overline{X})-F(x^{k-1})\leq\mu_{k}$ and
$V_{k-1}\big{(}f~{}|~{}[x^{\prime\prime},x_{k-1}]\cup[x^{k-1},\overline{X}]\big{)}\geq
V_{k-1}(q_{k-1})$, where
$x^{\prime\prime}=F^{-1}\big{(}\sum_{i=k+1}^{N}\mu_{i}\big{)}$. In this case,
define $x^{k}=x^{k+1}=...=x^{N-1}=\overline{X}$,
$x_{j}=F^{-1}\big{(}\sum_{i=j+1}^{N}\mu_{i}\big{)}$ for $j=k,k+1,...,N-1$ and
proceed to state $N$.

Otherwise, let $y_{k}$ and $y^{k}$, with $0<y_{k}<y^{k}<\overline{X}$, be the
unique quality levels satisfying

$V_{k-1}\big{(}f~{}|~{}[y_{k},x_{k-1}]\cup[x^{k-1},y^{k}]\big{)}~{}=~{}V_{k-1}(q_{k-1})~{}~{}~{}~{}~{}~{},~{}~{}~{}~{}~{}f\big{(}[y_{k},x_{k-1}]\cup[x^{k-1},y^{k}]\big{)}~{}=~{}\mu_{k}.$

Similarly, let $z_{k}$ and $z^{k}$ be the unique quality levels satisfying

$V_{k}\big{(}f~{}|~{}[z_{k},x_{k-1}]\cup[x^{k-1},z^{k}]\big{)}~{}=~{}V_{k}(q_{k-1})~{}~{}~{}~{}~{}~{},~{}~{}~{}~{}~{}f\big{(}[z_{k},x_{k-1}]\cup[x^{k-1},z^{k}]\big{)}~{}=~{}\mu_{k}.$

Define $x_{k}=\frac{y_{k}+z_{k}}{2}$,
$x^{k}=F^{-1}(F(x^{k-1})+\mu_{k}-F(x_{k-1})+F(x_{k}))$, and proceed to state
$k+1$.

If $k=N$, define $x_{N}=0$, $x^{N}=\overline{X}$ and end the procedure.

It follows directly that $x_{N},x_{N-1},$ $...,$ $x_{2},$ $x^{1},...,x^{N}$
and the corresponding accordion allocations are continuous in $x_{i}\in Y$.
Let $A=\big{\\{}x_{1}\in
Y~{}\big{|}~{}V_{N-1}(q^{x_{1}}_{N-1})>V_{N-1}(q^{x_{1}}_{N})\big{\\}}$ and
$B=\big{\\{}x_{1}\in
Y~{}\big{|}~{}V_{N}(q^{x_{1}}_{N})>V_{N}(q^{x_{1}}_{N-1})\big{\\}}$. If
$x_{1}=0$, then $q_{N-1}=f~{}|~{}[x^{N-2},x^{N-1}]$ and
$q_{N}=f~{}|~{}[x^{N-1},\overline{X}]$. Therefore, $0\in A\neq\varnothing$.
Similarly, if $x_{1}=F^{-1}(F(\overline{X})-\mu_{1})$, then
$q_{N-1}=f~{}|~{}[x_{N-1},x_{N-2}]$ and $q_{N}=f~{}|~{}[0,x_{N-1}]$.
Therefore, $F^{-1}(F(\overline{X})-\mu_{1})\in B\neq\varnothing$. Since
$V_{i}(q^{x_{1}}_{j})$ for all $i,j$ are continuous in $x_{1}$, it follows
that $A$ and $B$ are open sets. By Claim F2, $A\cup B=Y$. Since $Y$ is a
connected set, there is $x^{*}_{1}\in A\cap B\neq\varnothing$.

Finally, we show that $q^{x^{*}_{1}}$ is an allocation as desired. If the
procedure with parameter $x^{*}_{1}$ encounters the first case in its
specification for some $k\in\\{2,...,N-1\\}$, then
$q_{N-1}=f~{}|~{}[x^{N-2},x^{N-1}]$ and
$q_{N}=f~{}|~{}[x^{N-1},\overline{X}]$, contradicting $x^{*}_{1}\in B$. If the
procedure encounters the second case in its specification for some
$k\in\\{2,...,N-1\\}$, then $q_{N-1}=f~{}|~{}[x_{N-1},x_{N-2}]$ and
$q_{N}=f~{}|~{}[0,x_{N-1}]$, contradicting $x^{*}_{1}\in A$. Consider then an
arbitrary state of the procedure, $k\in\\{2,...,N-1\\}$. For arbitrary
$w_{k}<x_{k}$, denote by
$w^{k}=F^{-1}(F(x^{k-1})+\mu_{k}+F(x_{k-1})-F(w_{k}))$. Then
$V_{j}\big{(}f~{}|~{}[w_{k},x_{k-1}]\cup[x^{k-1},w^{k}]\big{)}$ is strictly
decreasing in $w_{k}$ for any $j$. By Claim F2, $y_{k}<z_{k}$. Therefore,
$y_{k}<x_{k}<z_{k}$, and we conclude that both $IC_{k(k-1)}$ and $IC_{(k-1)k}$
are satisfied and not binding. By our choice of $x^{*}_{1}$, we also know that
both $IC_{N(N-1)}$ and $IC_{(N-1)N}$ do not bind. By Claim F3, the allocation
$q^{x^{*}_{1}}$ is incentive compatible. Since $q^{x^{*}_{1}}$ has an
accordion structure, $q^{x^{*}_{1}}\in\mathcal{A}$. By Proposition 4 and
Corollary 4, there are positive welfare weights $\alpha$ such that
$q^{x^{*}_{1}}$ is a first-best solution, concluding the proof. $\blacksquare$

Proof of Proposition 6. For brevity, we use the shorthand of “equilibrium” to
represent “fair competitive equilibrium.” The proof that an equilibrium exists
follows standard arguments and, for completeness, provided in the Online
Appendix (Lemma A3). We now show the asserted resulting structure of any
equilibrium allocation.

Consider an $i$-agent’s problem. Let $\eta_{i}$ be the Lagrange multiplier of
the feasibility constraint $1-q_{i}\big{(}[0,\overline{X}]\big{)}\geq 0$, and
$\xi_{i}$ be the Lagrange multiplier of the budget constraint. The Lagrange
function for agent $i$’s problem is

$\mathcal{L}(q_{i})~{}=~{}\int_{0}^{\overline{X}}\big{[}u_{i}(x)-\eta_{i}-\xi_{i}p(x)\big{]}dq_{i}(x)+\xi_{i}\omega_{i}+\eta_{i},$

where the optimization of $\mathcal{L}$ is over all distributions $q_{i}$ on
Borel subsets of $[0,\overline{X}]$. Denote

$p^{i}(x)~{}~{}=~{}~{}\frac{1}{\xi_{i}}\cdot
u_{i}(x)-\frac{\eta_{i}}{\xi_{i}}.$

If $(p,q)$ is an equilibrium, then $\xi_{i}>0$. Indeed, if $\xi_{i}=0$, then
either $u_{i}(0)-\eta_{i}>0$, in which case the problem has no solution, or
$u_{i}(x)-\eta_{i}<0$ for all $x\in(0,\overline{X}]$, in which case
$q_{i}\big{(}(0,\overline{X}]\big{)}=0$. In both cases market-clearing fails.
Hence, $p^{i}(x)$ is well-defined. It follows that $1/\xi_{i}>0$ and
$\eta_{i}/\xi_{i}\geq 0$. We can write $i$-agents’ Lagrange function as
follows:

$\mathcal{L}(q_{i})~{}=~{}\xi_{i}\cdot\int_{0}^{\overline{X}}\left[p^{i}(x)-p(x)\right]dq_{i}(x)+\xi_{i}\omega_{i}+\eta_{i}.$

Claim D1. Suppose $(p,q)$ is an equilibrium, then: (1) $p(x)\geq p^{i}(x)$ for
all $x\in[0,\overline{X}]$; (2)
$B_{i}\equiv\big{\\{}x\in[0,\overline{X}]~{}\big{|}~{}p(x)=p_{i}(x)\big{\\}}\neq\varnothing$;
and (3) $q_{i}(B_{i})=1$.

Proof. Assume that $p(x)<p^{i}(x)$ for some $x\in[0,\overline{X}]$, then the
problem of the Lagrangian maximization has no solutions, since substituting
$q_{i}=l\cdot\delta_{x}$ with $l=1,2,..$ yields an unbounded sequence of