Subsidy Design and Asymmetric Information:Wealth versus Bene�ts

Simona Grassi and Ching-to Albert Ma

Department of EconomicsBoston University270 Bay State Road

Boston, MA 02215, USA

emails: sgrassi@bu.edu and ma@bu.edu

February 15, 2007

Preliminary and incomplete. Not for citation or circulation.

Abstract

A government or public organization would like to subsidize the provision of an indivisible good. Con-sumers�valuations of the good vary according to their wealth and potential bene�ts from the good. Edu-cation, medical care, and housing are common examples. A regulator has access to either wealth or bene�tinformation, but not both. We present a method to translate a wealth-based policy to a bene�t-based policy,and vice versa. We characterize environments in which the wealth-based policy and translated bene�t-basedpolicy implement the same assignment: consumers choose to purchase the good under the wealth-basedpolicy if and only if they choose to do so under the translated bene�t-based policy. Examples of these envi-ronments are common in monopoly pro�t maximization outcomes in a private market and optimal subsidydesign. General taxation allows equivalent wealth-based and bene�t-based policies to generate the samerevenue from consumers. Our results provide a foundation for the optimal choice of information on whichsubsidy schemes may be based.

Acknowledgement: We thank seminar participants at Boston College, Boston University, University ofLausanne, Univesity of Liège, and Rutgers University for their comments. We also thank Ingela Alger, SteveCoate, Preston McAfee, Dilip Mookherjee, Larry Samuelson, and Thomas Sjostrom for their suggestions.The �rst author is grateful to the Fulbright Foundation of Italy for �nancial support.

1 Introduction

A government or public organization often subsidize goods and services such as education, health care and

housing. It is widely recognized that asymmetric information is an important considerationin for subsidy

design. Subsidies are always implemented under limited budgets, and soliciting many pieces of information

may be too costly. In this paper, we investigate the relationship between subsidy schemes that are based on

di¤erent kinds of information. Our research here provides the foundation for the choice of information for

subsidy schemes.

In our model, a regulator provides subsidies to consumers for an indivisible good. The subsidy may

be based either on a consumer�s wealth or potential bene�t from consuming the good, but not both. A

consumer�s willingness to pay for the good, however, depends on both pieces of information, with this

willingness-to-pay increasing in both wealth and bene�ts.

As an example, suppose that a government o¤ers subsidized college education. It may solicit information

about family income and wealth, or hire education consultants to analyze students� examination scores,

which determine students�potential bene�ts from the education program. Scholarships that are based on

merit or �nancial needs are common. Suppose that the regulator �nds it too expensive to solicit or process

both family wealth and test score information.

Obviously, the lack of all relevant information is a drawback. With wealth information, a regulator can

help students with low family resources; with test score or bene�t information, a regulator can encourage

more able students to attend college. Without both pieces of information, a regulator cannot subsidize

students who are simultaneously low-income and capable.

For another example, suppose that a government subsidizes a course of medical treatment. The subsidies

may be based on the patient�s wealth or health status (with the latter determining potential bene�ts). Again,

the ideal policy would set subsidies based on both wealth and health conditions, but it may be too costly to

implement such a policy.1 In the above examples, a consumer�s willingness-to-pay for education or medical

1 In the United States, the Medicaid program provides insurance (and hence subsidized medicines) for those withlittle wealth, while the Canadian single-payer national health insurance system rations health care according to disease

1

treatment depends on both his wealth and bene�t.

A wealth-based subsidy induces a set of consumers to decide to purchase the good; likewise, a bene�t-

based subsidy induces another set of consumers to do the same. Each of these consumers may be purchasing

the good at quite di¤erent subsidized prices. In this paper, we present a way to translate a wealth-based

subsidy policy to a bene�t-based policy, and vice versa. In other words, we �nd conditions on wealth-based

and bene�t-based policies where the same set of consumers will purchase the good given either policy. We

say that such wealth-based and bene�t-based policies are equivalent, and that they implement the same

assignment.

The translation method can be described as follows. Let the wealth of a consumer be observed by the

regulator, and suppose that it is $1000. Let the wealth-based subsidized price for the good at this wealth

level be $200. The regulator does not have the bene�t information but can compute the smallest bene�t

level2 at which this consumer with wealth $1000 is willing to buy the good at $200. Suppose that this bene�t

level is 300. Then the bene�t-based subsidized price for the good at bene�t level 300 will be $200, the price

at which a consumer with wealth $1000 and bene�t 300 is indi¤erent between purchasing and not. For a

given policy, say wealth-based policy, the translation method works through this �indi¤erent boundary�at

which a consumer with a combination of wealth and bene�t is the marginal consumer.

What motivates the interest in equivalent wealth-based and bene�t-based policies? It simply is a practical

issue. Suppose that a government is interested in subsidizing college education for poor students, but suppose

that the wealth and income information may be di¢ cult or impossible to collect. In many economies,

widespread tax evasion and wealth hiding are common. Suppose that the government has in mind a wealth-

based subsidy scheme, and if there was wealth information, this scheme could be implemented and generate

an assignment of students into college education. Now through our method, the government would be able

to use test score information instead of wealth, design a bene�t-based subsidy policy, and implement the

same assignment as if wealth information was available. That is, the government can overcome some of the

severities.

2We measure bene�t in utility units.

2

di¢ culties arising from unobserved wealth. In less extremely cases, collecting wealth or bene�t information

may be possible, but their collection and processing costs may be di¤erent. Our theory then gives a ranking

of the cost of information, given a policy scheme.

Our use of a discrete, indivisible good is important for the analysis, although it is a natural assumption

to make in the markets we have mentioned.3 Under equivalent wealth-based and bene�t-based policies, a

consumer will receive di¤erent subsidies and is almost never indi¤erent between whether he is subsidized

according to wealth or bene�ts. Because of the discreteness of the purchase decision, consumers may still

decide to purchase the good under these schemes even when they receive di¤erent subsidies. The focus on

assignment rather than the exact utility (which takes into account consumers�payments) is perhaps a limi-

tation. It is clear, however, that schemes that are based on di¤erent information cannot exactly implement

the same allocation (the assignment and payment for each consumer). Focusing on assignment rather than

the allocation is natural for an indivisible good, and can be regarded as a second-best consideration.

Our analysis takes as given subsidy policies, and we impose conditions on them for the translation

between wealth-based and bene�t-based policies. These conditions concerns the monotonicity properties of

indi¤erence boundaries. We have identi�ed examples in which such indi¤erence boundaries arise naturally. In

the �rst example, the government considers nationalizing a market that is served by an incumbent monopolist.

We characterize the monopolist�s optimal pricing strategy when the bene�t information is available to the

�rm, but wealth remains consumers� private information. Because the bene�t information is available,

the monopolist practices price discrimination, o¤ering consumers a price contingent to their bene�t level.

If the monopolist sells to more consumers when the consumer bene�t becomes higher, our monotonocity

requirement for equivalent bene�t-based and wealth-based translations is satis�ed.

A government regulator may not have the same sort of information as the monopolist. In particular, it

may have access to consumers�wealth information, but not their bene�ts. Then the regulator can nationalize

the market, translate the monopolist�s pricing scheme to a wealth-based policy, and implement the same

assignment.

3Ma and Riordan (2002) uses a discrete treatment for the analysis of optimal insurance and moral hazard.

3

In the second example, the government sets a budget policy based on consumer bene�ts. Those consumers

with the same bene�t level receive a subsidy when they purchase the good. For each bene�t level, the

budget allocated determines the subsidy and the corresponding set of buyers. We show that when the

budget amount is increasing in bene�ts, the indi¤erence boundary satis�es the monotonicity requirement for

equivalent translation.

We further prove that with general taxation, the regulator can implement equivalent policies at the same

budget. Now consumers are required to pay an amount based on the available information (either wealth or

bene�t), and an extra amount if they decide to purchase the good. The general taxation acts as a lump-sum

transfer. Wealth-based and bene�t-based schemes that implement the same assignment as well as generate

the same amount of revenue can be found.

Our work is related to the literatures on the public provision of private goods and the optimal design

of tax and transfer schemes. In both literatures, the research focus is on the public sector�s policies in the

presence of heterogenous individuals and asymmetric information. Nevertheless, most papers use a di¤erent

approach: the goal is to de�ne the optimal provision or the optimal tax or transfers given some constraints.

The missing information is often the component generating relevant constraints. The equivalence between

subsidy schemes that are based on di¤erent information has not been studied before.

Arrow (1971) gives a benchmark for the optimal provision of private goods, under perfect information. He

considers individuals di¤ering in abilities; under a utilitarian social welfare functiona he derives conditions

for the optimal expenditure policy. The subsequent literature focuses on asymmetric information between

the social planner and individuals who have private information about their own incomes or abilities but

no incentive to reveal the information to the social planner. Blackorby and Donaldson (1988) show that

when the social planner cannot distinguish between able and less able individuals, in-kind transfers may be

preferred over monetary transfers. Assuming that income information cannot be used by the social planner,

Besley and Coate (1991) justify the public provision of private goods as a way to redistribute income from

the rich to the poor.

Following Mirrlees (1971), Boadway and Marchand (1995) model the public provision of a private good

4

in the context of optimal income taxation, where individuals di¤er in labor productivities, and where labor

productivities are private information. More recently, De Fraja (2002) investigates the design of optimal

education policies, when children in households have di¤erent abilities, and when household incomes di¤er.

In De Fraja�s model, income is observable, but ability is private information.

Some of the literature on the optimal tax and subsidy design deals with inequality under asymmetric

information. It is recognized that inequality depends on income or wealth, as well as characteristics such as

age, health status, gender, etc. As a consequence, transfers should take into account these characteristics.

Atkinson (1992) concludes that �the issue of policy design is not therefore a confrontation between fully

universal bene�ts and pure income testing; rather the question is that of the appropriate balance of categorical

and income tests.� Blackorby and Donaldson (1994) consider information other than income as a way to

de�ne di¤erent groups (�people with serious illness, the disabled, racial and ethnic goups,�etc., p.440). They

study the conditions for the optimality of transfers between groups when the planner does not know the

distribution of income within groups.

The usual method for deriving optimal schemes involves recognizing missing information, and the corre-

sponding constrained maximization of some welfare index. The relevance of optimal schemes is limited by

the perspective of missing information that a particular model focuses on. Robustness is therefore an issue

for many models. Furthermore, optimal schemes are contingent a choice of a welfare index. In practice, it

may be questionable whether a given policy in practice may be construed as an optimal choice.Our approach

is more practical. We take as given a policy scheme that is based on some information, and relate it to an

equivalent scheme that is based on some other information. So our analysis does not rely on a choice of a

welfare index. Nor do we rely on the optimality properties of policy schemes.

We introduce the model in the next section. Conditions for equivalent policies are presented in Section

3. We also demonstrate that equivalent policies collect di¤erent revenues from consumers. In Section 4, we

present two examples in which policies that can be successfully translated arise naturally. Section 5 expands

the policies to allow for general taxation. Equivalent policies that collect the same revenue from consumers

can be constructed. The last section draws some conclusions.

5

2 The Model

We consider a regulator allocating a private good to a set of consumers. The good is indivisible, but each unit

of the good may give di¤erent bene�ts to di¤erent consumers. In both the education and health markets,

there are many such examples. A course of study confers di¤erent bene�ts depending on students�abilities; a

course of treatment or surgery may heal an illness, but consumers may experience di¤erent utility recoveries.

Nevertheless, the cost of a study or treatment program may not vary according to consumer characteristics.4

We normalize the total mass of consumers to 1. Each consumer may get at most one unit of the good.

Each unit of the good costs c > 0. The regulator has available a budget B to pay for these goods. We

assume that 0 � B < c; that is, the regulator�s budget is insu¢ cient to supply the good to all consumers

at a zero price. It is unimportant for the analysis whether the government actually produces the good or

contracts with a �rm to do so.

A consumer has wealth or income, w. A consumer obtains some bene�t ` when he receives the good. We

let w and ` be random variables. Respectively, the variables w and ` have supports on the intervals [w;w]

and�`; `�. Let F and f be the distribution and density functions of w; let G and g be the distribution and

density functions of `. Both f and g are assumed to be strictly positive and continuous. We say that a

consumer is type (w; `) if he has wealth w and derives bene�t ` from the good.

If a type (w; `) consumer pays p and obtains the good, his utility is U(w � p) + `, where U is a strictly

increasing and strictly concave function. If a consumer does not obtain the good (and pays nothing), his

utility is U(w). In the education example, the variable ` measures his (expected) bene�t from a course of

study. In the health care example, ` represents the (expected) loss of illness. If a sick consumer goes without

a course of treatment (the good), his utility is U(w)� `; if he pays p to obtain treatment his utility becomes

U(w � p). The bene�t ` is assumed to be separable from the utility of wealth. The utility from bene�t ` is

measured linearly, but this is without any loss of generality.5

4We do not consider cost selection issues here. For some services, the provision cost may well depend on consumercharacteristics. This is a possibility that we leave for future research.

5The utility from bene�t ` can be written generally as V (` ), where V is strictly increasing. We de�ne a newbene�t variable `0 � V (`) and adjust the distribution and density functions G and g accordingly.

6

The variables w and ` determine how much a consumer is willing to pay for the good. A type (w; `)

consumer is willing to pay for the good at a price p if

U(w � p) + ` � U(w): (1)

The consumer�s willingness to pay exhibits monotonicity with respect to both wealth and bene�t. If a type

(w; `) consumer is willing to pay for the good at price p, so are those who have higher incomes and those who

derive higher bene�ts. Consider a type (w0; `0) consumer, where w0 > w and `0 > `, the following inequalities

follow from (1):

U(w0 � p) + ` � U(w0) U(w � p) + `0 � U(w): (2)

Our basic hypothesis is that a consumer�s willingness to pay is his private information, because either w

or ` is assumed to be private information. We consider each of these two possibilities, and call these cases

�unknown bene�t�and �unknown wealth.�

The regulator o¤er subsidy schemes to consumers. We consider two cases separately. In the �rst, subsidies

are speci�c to the good; in the second, general subsidies (or taxes) may also be used by the regulator. We

consider speci�c subsidies in the following section. If w is public information while ` unknown, a wealth-

based policy is a function t(w); a consumer pays t(w) if he purchases the good. If ` is public information

while w unknown, a policy is a function s(`); a consumer pays s(`) if he purchases the good. If the regulator

intends to give subsidies, s(`) and t(w) will be less than c, the cost of the good.

In a later section, we consider both speci�c and general subsidies. If w is public information while `

unknown, the policy is a pair of functions t1(w) and t2(w), where t1(w) is the payment when the individual

does not buy the good and t2(w) is the payment when the individual does. Similarly, if ` is public information

while w unknown, the policy is a pair s1(`) and s2(`), where s1(`) is the payment when the individual does

not get the good and s2(`) is the payment when the individual does get the good. The payments are allowed

to be positive or negative.

In each regime, the game proceeds as follows. The regulator sets up the policy, which is either s(`), t(w),

[s1(`),s2(`)], or [t1(w); t2(w)]. Each consumer then decides whether to purchase the good at these subsidized

prices. The regulator pays c for each unit purchased by the consumers.

7

If consumers have access to a private market, the regulator�s policies will be constrained by the price

that is available in the market. For example, if a consumer can purchase the good in the private market at

d (which may be higher than c), then s(`) and t(w) must not be higher than d. The results in the following

section do not depend on this restriction.

3 Equivalent assignments

In this section we consider speci�c subsidies; consumers pay the regulator if and only if they actually choose

to obtain the good. We begin with the case of consumers�wealth w being public information, while their

bene�ts ` are their private information, unknown to the regulator. A policy is a function t(w) : [w;w]! R+;

a consumer with wealth w pays the regulator t(w) if he buys the good.

We de�ne the assignment set �(t) due to the payment scheme t(w) by

�(t) � f(w; `) : U(w � t(w)) + ` � U(w)g : (3)

The inequality in (3) says that a type (w; `) consumer weakly prefers to buy the good at price t(w). We will

only consider those wealth-based policies that implement nontrivial assignments; the set �(t) is a nonempty,

proper subset of all consumers.

Next, suppose that consumers�wealth ` is public information, but w unknown to the regulator. A policy

is a function: s(`) :�`; `�! R+; a consumer with bene�t ` pays the regulator s(`) if he chooses to get the

good.

We de�ne the assignment set �(s) due to the payment scheme s(`) by

�(s) � f(w; `) : U(w � s(`)) + ` � U(w)g : (4)

The inequality in (4) says that a type (w; `) consumer weakly prefers to buy the good at price s(`). Again,

we only consider those bene�t-based policies that implement nontrivial assignments; that is, the set �(s) is

a nonempty, proper subset of all consumers.

When the wealth-based policy t(w) implements the assignment �(t), we can evaluate the total required

subsidy. Because of the subsidy, the regulator is responsible for the balance c� t(w). Hence the total subsidy

8

under policy t(w) is Z�(t)

[c� t(w)]dF (w)dG(`): (5)

The total subsidy under policy t(w) coincides with the required budget.

Similarly, when the bene�t-based policy s(`) implements the assignment �(s), we can evaluate the total

required subsidy: Z�(s)

[c� s(`)]dF (w)dG(`): (6)

Again, the total subsidy under policy s(`) coincides with the required budget.

Our �rst set of results concerns the relationship between the policy schedules and their assignment sets.

Then we look at the corresponding total subsidies.

3.1 Unknown Bene�t

First for a given t, and for any w 2 [w;w], de�ne ^ by the equation:

U(w � t(w)) + ^= U(w): (7)

For each w, a type (w; ^) consumer is indi¤erent between purchasing the good at t(w) and the status quo.

The equation (7) de�nes a functional relationship between ^ and w; we denote this function by �:

^= �(w; t) � U(w)� U(w � t(w)): (8)

We suppress the policy t in the argument of �. From (2), at each w, consumers with bene�ts ` > ^ strictly

prefer to purchase the good; at each w, �(w) is the minimal level of bene�t at which consumers prefer to

purchase at t(w). We call � the �indi¤erence boundary�with respect to t(w).

Figures 1, 2 and 3, show possible indi¤erence boundaries for various policies. In Figure 1, the boundary

is increasing. Under this t(w) if a consumer with wealth w and bene�t ` is indi¤erent between paying

t(w) to obtain the good and not, a consumer with wealth w0 > w actually declines to pay t(w0) to get

the same bene�t. The wealth-based policy is progressive and increases so rapidly that the consumer with

wealth w0 > w must receive more bene�t than ` to be willing to pay t(w0). In Figure 2, the boundary

is decreasing, and the comparison between decisions made by consumers with wealth levels w and w0 goes

9

exactly the opposite way. In Figure 3, the boundary is due to a discontinuous policy: t(w) =1

4c for c < ew

and t(w) =3

4c otherwise. It is of some interest to note that if t(w) = k, a constant, the boundary is strictly

decreasing due to the strict concavity of U .6

l

w

Figure 1: Increasing Indi¤erence Boundary

l

w

Figure 2: Decreasing Indi¤erence Boundary

For a given utility function U and a policy t, the indi¤erence boundary � de�ned above is a function

� : [w;w]! R. The following refers to conditions of the policy t and its associated indi¤erence boundary.

Condition 1 (Decreasing Indi¤erence Boundary) The wealth-based policy t(w) is continuous (equiva-

lently the function �(w) is continuous). The indi¤erence boundary �(w) is strictly decreasing.

The two indi¤erence boundaries in Figures 1 and 2 satisfy Condition 1, but the one in Figure 3 is neither

6 If t(w) is di¤erentiable, the slope of the indi¤erence boundary isd`

dw= U 0(w)�U 0(w� t(w))[1� t0(w)] from total

di¤erentiation of (7). Note thatd`

dw< 0 if t(w) is a constant.

10

w

l

w~

cp43

=

cp41

=

Figure 3: Discontinuous Indi¤erence Boundary

continuous nor monotone. Now we show that under Condition 1, we can translate a wealth-based policy t(w)

to a bene�t-based policy s(`) in such a way that the assignment sets under the two policies are identical.

Condition 1 implies that the inverse of � exists for the set of bene�ts [`0; `0] � �([w;w]), the range of the

function �. Let this inverse be � : [`0; `0] ! [w;w]. That is, for any bene�t level in ` in [`0; `

0], the function

� gives the wealth level at which the consumer will be just willing to pay t(w) to purchase the good. Note

that under Condition 1, �(w) = `0and �(w) = `0.

The range of �, [`0; `0], need not be exactly

�`; `�, but because the assignment set �(t) is nonempty and

a proper subset of all consumers, it must intersect�`; `�. The next two diagrams illustrate two possibilities.

In Figure 4, the range of � contains�`; `�, while in Figure 5, the range of � is a proper subset of

�`; `�.

Now we construct a bene�t-based policy, s(`), which implements the same indi¤erence boundary as t(w).

For each ` 2�`; `�\ [`0; `0], we de�ne a payment s(`) by

U(�(`)� s) + ` = U(�(`)): (9)

In words, we replace the wealth variable in the de�nition of the indi¤erence boundary (7) by �(`). The

equation in (9) yields an implicit function s(`), a bene�t-based policy. The construction of such an s(`)

yields an identical boundary: U(w� s(`)) + ` = U(w), but now the policy de�ned by (9) is written in terms

of bene�ts instead of wealth.

There remain possibles values of bene�ts which are not in the range [`0; `0]. These cases, when they exist,

11

l

w

l

ww

'l

'l

l

Figure 4: [`0; `0] contains

�`; `�

l

w

l

ww

'l

'l

l

Figure 5:�`; `�contains [`0; `

0]

12

correspond to either ` < `0, `0< `, or both (see Figure 5). We complete the de�nition of s by the following.

For ` 2�`; `0

�, let s(`) = s(`0): For ` 2 [`0; `], let s(`) = s(`0): (10)

The two sets,�`; `0

�and [`

0; `]; contain consumers with very low or very high bene�ts. Under t(w), those

consumers with very low bene�ts will never purchase the good at s(`0) no matter how high their wealth w;

those with very high bene�ts will always purchase at s(`0). This completes the translation of a wealth-based

policy t(w) to a bene�t-based policy s(`).

Proposition 1 Suppose that a wealth-based payment schedule t(w) satis�es Condition 1 (Decreasing Indif-

ference Boundary). The bene�t-based payment policy s(`) de�ned in (9) and (10) implements the assignment

as the wealth-based policy t(w). That is, assignment sets �(t) and �(s) are identical.

Proposition 1 (whose proof is in the appendix) makes use of the strictly decreasing monotonicity of the

indi¤erence boundary. Given a wealth-based policy, to each wealth level, we associate a bene�t threshold

at which the consumer is indi¤erent between purchasing and not. The strict monotonicity of the boundary

allows us to invert this relationship. So for each bene�t level, we are able to associate a wealth threshold.

This accounts for the construction of the bene�t-based policy.

Monotonicity alone does not guarantee that the assignments are identical when a wealth-based policy is

translated to a bene�t-based policy according to the method just described. The preferences of consumers

determine which side of the indi¤erence boundary the assignment set lies. By (2), the assignment set is

always the half space above the indi¤erence boundary. Given a boundary �(w) on w-` space, then at a point

(w; `) on the boundary, those points above it are those consumers with higher bene�t, and these belong to

the set �(t). Conversely, given the equivalent boundary �(`) (the inverse of �) on the same w-` space, then

at a point (w; `) on the boundary, those points to the right of it are those consumers with higher bene�t, and

they belong to the set �(s). When an indi¤erence boundary is strictly decreasing, the assignment sets of the

wealth-based and translated bene�t-based policies coincide. The following diagram in Figure 6 illustrates

this.

Our next condition says that the indi¤erence boundary is strictly increasing. Then the union of the

13

l

w

↑)(tα

→)(sβ

Figure 6: Downward Sloping Boundary: Direction of Preferences

assignment sets �(t) and �(s) of equivalent boundaries is the sets of all consumers and the intersection

contains only the indi¤erence boundary. The following Figure 7 illustrates the direction of the preferences

when the boundary is upward sloping.

Condition 2 (Increasing Indi¤erence Boundary) The wealth-based policy t(w) is continuous (equiva-

lently the function �(w) is continuous). The indi¤erence boundary �(w) is strictly increasing.

Corollary 1 Suppose that a wealth-based payment schedule t(w) satis�es Condition 2 (Increasing Indi¤er-

ence Boundary). The bene�t-based payment policy s(`) de�ned in (9) and (10) implements an assignment

�(s) that intersects the assignment �(t) implemented by the wealth-based policy t(w) only for the indi¤erent

consumers. That is, the union of �(t)and �(s) is the set of all consumers [(w;w) � (`; `)]; the intersection

of �(t)and �(s) is the set of the indi¤erent consumers.

We omit the proof; it is symmetric to the proof of Proposition 1. We present an example of the translation

of a wealth-based policy to a bene�t-based policy.

Example 1 Let U be the logarithmic function. Let a wealth-based policy be quasi-linear, t(w) = a+bw. The

indi¤erence boundary is given by ln(w� a� bw)+ ` = lnw, or ` = �(w) � ln w

(w � a� bw) . We assume that

14

l

w

→)(sβ

↑)(tα

Figure 7: Upward Sloping Boundary: Direction of Preferences

the denominator is strictly positive; that is, w � a� bw > 0 . The derivative of � is

d�dw

= � a

w(w � a� bw) :

The boundary is strictly decreasing if and only if a > 0 (conditional on w� a� bw > 0). The inverse of � is

w = �(`) � ae`

(1� b)e` � 1 . By substituting w in (7) by �(`) and then solving for s, we obtain the bene�t-based

policy

s(`) =a�e` � 1

�(1� b)e` � 1 : (11)

3.2 Unknown Wealth

In this subsection, we consider the translation of a bene�t-based policy to a wealth-based policy; the analysis

is similar to the previous subsection, and we will prove the converse of Proposition 1. A policy based on

bene�t is a function s(`) :�`; `�! R+. Here, a consumer�s bene�t ` is observable, but his wealth w is

unknown to the regulator.

For a given s, and for any ` 2�`; `�, de�ne w by the equation:

U(w � s(`)) + ` = U(w): (12)

Equation (12) de�nes a functional relationship between w and `; we denote this function by '. From (12), at

15

each `, consumers with wealth w > w strictly prefer to purchase the good; at each `, '(`) is the minimal level

of wealth at which consumers prefer to purchase at s(`). We call ' :�`; `�! R the �indi¤erence boundary�

with respect to s(`).

Condition 3 (Decreasing Indi¤erence Boundary) The bene�t-based payment schedule s(`) is continu-

ous (equivalently the function '(`) is continuous). The indi¤erence boundary '(`) is strictly decreasing.

Condition 3 implies that the inverse of ' exists for the set of bene�ts [w0; w0] � '(�`; `�), the range of the

function '. Let this inverse be # : [w0; w0] !�`; `�. That is, for any wealth level in [w0; w0], the function #

gives the bene�t level at which the consumer will be just willing to pay s(`) to purchase the good.7 Under

Condition 3, '(`) = w0 and '(`) = w0. As in the case of unknown bene�t, the range [w0; w0] need not be

identical to [w;w], but because the assignment �(s) is nonempty and a proper subset of all consumers, the

intersection between [w0; w0] and [w;w] is nonempty.

Now we construct a wealth-based policy t(w) that implements the same indi¤erence boundary as s(`).

For each w 2 [w;w] \ [w0; w0], we construct a payment t(w) that satis�es

U(w � t) + #(w) = U(w): (13)

That is, we replace the variable ` in equation (29) by #(w) The above equation (13) de�nes a functional

relation between t and w. For w =2 [w;w] \ [w0; w0], if it exists, we de�ne the following payments:

For w 2 [w;w0] , let t(w) = t(w0): For w 2 [w0; w], let t(w) = t(w0): (14)

Note that t(w0) = s(`) and t(w0) = s(`).

We now state the following proposition (whose proof is omitted since it is identical to the one for Propo-

sition 1).

Proposition 2 Suppose that a bene�t-based payment schedule s(`) satis�es Condition 3 (Decreasing Indif-

ference Boundary). The wealth-based payment schedule t(w) de�ned in (13) and (14) implements the same

7We assume that the domain of the function U can be extended beyond [w;w], say to [w0; w0], and that on theextended domain, U remains strictly increasing and strictly concave.

16

assignment as the bene�t-based schedule s(`). That is, the two sets �(t) and �(s) are identical.

The following Condition 4 and Corollary 2 describe the relation between the sets �(t) and �(`) when the

indi¤erence boundary is strictly increasing.

Condition 4 (Increasing Indi¤erence Boundary) The bene�t-based policy s(`) is continuous (equiva-

lently the function '(`) is continuous). The indi¤erence boundary '(`) is strictly increasing.

Corollary 2 Suppose that a bene�t-based payment schedule s(`) satis�es Condition 4 (Increasing Indi¤er-

ence Boundary). The wealth-based payment policy t(w) de�ned in (13) and (14) implements an assignment

�(t) that intersects the assignment �(s) implemented by the bene�t-based policy s(`) only for the indi¤erent

consumers. That is, the union of �(t)and �(s) is the set of all consumers [(w;w) � (`; `)]; the intersection

of �(t)and �(s) is the set of the indi¤erent consumers.

We again present an example to illustrate the translation from s(`) to t(w).

Example 2 Again let U be the logarithmic function. Let a bene�t-based payment schedule be quasi-linear

in e�`, s(`) = a+ be�`. The indi¤erence boundary is given by ln�w � a� be�`

�+ ` = lnw, or w = '(`) �

e`a+ be�`

e` � 1 . We assume that w � a� be�` > 0, and that ` > 0. The derivative of ' is

d'd`= � (a+ b)e

�`

(e�` � 1)2 :

The boundary is strictly decreasing if and only if (a + b) > 0. The inverse of ' is ` = #(w) � ln w + b

(w � a) .

By substituting ` in equation (29) by #(w) and solving for t, we obtain the wealth-based policy

t(w) =w

w + b(a+ b) :

3.3 Equivalent Policies and Revenues

Propositions 1 and 2 relate the wealth-based and bene�t-based policies that implement the same assignment.

All consumers get the good under these two assignment-equivalent policies based on di¤erent information.

Consumers pay di¤erent costs according to whether payment is based on wealth or bene�t, and are generally

17

not indi¤erent across these regimes. For example, suppose that a type (w; `) consumer is charged t(w) to

obtain bene�t `, and suppose that ` is very large and the consumer obtains a large surplus. Now if he is

faced with the equivalent schedule s(`), and the equivalent schedule turns out to be sharply increasing in `,

he may have to pay much more and his surplus is reduced.

The equivalence of s and t only says that the assignment of the good for type (w; `) consumer must

be identical across the two information regimes. The required budgets (see 5 and 6) for each of the two

equivalent may also be di¤erent because each of the equivalent policy generates a di¤erent level of revenue.

Consider three types of the consumer: (w1; `2), (w2; `1), and (w2; `2), with w1 < w2 and `1 < `2.

Suppose that there is a wealth-based policy, t(w), and it generates a strictly monotone decreasing indi¤erence

boundary. Let types (w1; `2), (w2; `1) be on the indi¤erence boundary, and therefore, type (w2; `2) is in the

interior of the assignment set. See Figure 8. Let s(`) be the bene�t-based policy that implements the same

assignment set. By construction, we have t(w1) = s(`2), and t(w2) = s(`1). Under the wealth-based policy,

consumer type (w2; `2) pays t(w2); under the equivalent bene�t-based policy, the same consumer (type

(w2; `2)) pays s(`2). Unless t and s are constant functions, t(w2) 6= t(w1) = s(`2). So any consumer (w2; `2)

in the interior of the assignment set pays di¤erent amount for getting the good. The regulator collects

di¤erent revenues from any type of consumer who is in the interior of an assignment set under equivalent

wealth-based and bene�t-based policies.

Proposition 3 (Revenue Nonequivalence) If a wealth-based payment schedule t(w) and a bene�t-based

payment schedule s(`) implement the same assignment, they generate di¤erent revenues (and therefore require

di¤erent budgets) for generic distributions of wealth (F (w)) and bene�ts (G(`)) except when t(w) = s(`) = k,

a constant.

An example below illustrates that the di¤erence in collected revenue can be signi�cant.

Example 3 (Revenue Nonequivalence) Let w and ` be uniformly and independently distributed over the

interval [0:5; 1:5]. Consider the wealth-based bolicy t(w) = 0:1+0:3w. We obtain the equivalent bene�t-based

policy s(`) =0:1�e` � 1

�0:7e` � 1 . >From these, we compute the revenue that will be collected under each of these

18

l

w

1w

2w

1l 2l

( ) ( )1212 )(:, ll swtw =

( ) ( )2222 )(:, ll swtw ≠

( ) ( )2121 )(:, ll swtw =

Figure 8: Nonequivalent Revenue

schemes:

R(t) �Z�(t)

t(w)dF (w)dG(`) = :393 and R(s) �Z�(t)

s(`)dF (w)dG(`) = :229:

In percentage terms we have:

R(t)�R(s)R(s)

= 70% andR(t)�R(s)

R(t)= 42%:

Example 3 shows a large di¤erence (in percentage) of revenues collected by the regulator under wealth-

based and bene�t-based policies. In turn these policies imply a large di¤erence of required budgets to

implement the same assignment set.

Proposition 3 has a straightforward policy implication. The regulator has to decide which information,

wealth or bene�ts, need to be collected. If there is any �xed cost in information collection in order to

implement a policy, the total cost of a policy under an information regime can be computed. This cost is the

sum of the required budget and the information collection cost. Cost savings can be achieved by selecting

the policy that requires a smaller total cost.

19

4 Endogeneous Payment Policies and Indi¤erence Boundaries

In the previous sections, the analysis builds upon a given wealth based or bene�t based policy (t(w) or s(`))

taken as exogenous. In this section, we expand the analysis to consider two cases where payment policies

are endogeneous. We show that these policies often lead to strictly decreasing indi¤erent boundaries.

First we look at a monopolist selling the good to consumers and at a regulator replacing the monopoly by

a public subsidy program. Next, we consider a social planner deciding on the distribution of a given budget

across consumers. In both cases, the endogenous selection of the policy implies a relation between w and `.

When this relation is negative, the translation between policies based on di¤erent information is feasible.

4.1 Monopoly prices and indi¤erence boundary

Suppose that a monopolist now sets pro�t-maximizing prices to sell the good to consumers. There is no public

subsidy available to consumers at this point. We let the monopolist observe a consumer�s potential bene�t,

`, but a consumer�s wealth, w, remains his private information. The monopolist is able to discriminate

consumers according to their bene�ts, but not to their wealth. The pro�t-maximizing price is a schedule

s(`); a consumer with (observable) bene�t ` will be o¤ered a price s(`) for the purchase of the good from

the monopolist.

A type (w; `) consumer �nds it optimal to purchase the good from the monopolist if U(w�s(`))+` � U(w).

For a price schedule s(`), let bw be the wealth level of the consumer who is just indi¤erent between purchasingfrom the monopolist and rejecting the o¤er: U( bw � s(`)) + ` = U( bw). From (1) and (2), those consumers

with w > bw will striclty prefer to purchase. So at price s(`), the monopolist�s demand is 1 � F ( bw). Fora given `, the pro�t is [s(`) � c][1 � F ( bw)]. We assume that the pro�t function is quasi-concave (so thatthe isopro�t lines in bw-s space are convex to the origin). For a set of consumers each with bene�t `, themonopolist chooses the price s and bw to maximize

�(s; `) � [s� c][1� F ( bw)] (15)

subject to

U( bw � s) + ` = U( bw): (16)

20

We can use standard techniques to characterize the solution of the constrained maximization problem. It

is su¢ cient for our purpose to illustrate the solution with a simple graph. See Figure 9. The upward sloping

curve is the �demand�(equation (16)),8 while the curve that is convex to the origin is the iso-pro�t line.9

w

s

Iso­profit

Demand

Figure 9: Monopolist�s Pro�t Maximization

The tangency point is the pro�t-maximizing choice of the price and the wealth level of the marginal

consumer. We may repeat the above for each bene�t ` to obtain the optimal pricing policy.

To save on notation, we let s(`) denote the solution of the above constrained maximization problems.

How is the demand� the wealth level of the marginal consumer� related to the bene�t? Figure 10 shows

two such possibilities.The two upward sloping lines are two demands (equation (16)), at ` = `1 and ` = `2

with `1 < `2. At ` = `1, the solution yields a marginal wealth level bw1. From (16) a consumer with bene�t

`1 purchases from the monopolist (at the pro�t-maximizing price s(`1)) if and only if his wealth is above bw1.8From (16), total di¤erentiation yields

dsd bw =

U 0( bw � s)� U 0( bw)U 0( bw � s) > 0

9The slope of the iso-pro�t line isdsd bw =

f( bw)(s� c)1� F ( bw) :

21

w

s

B

A

1ll =

2ll =

1w2w 2'w

Figure 10: Monopolist pro�t maximization: ` and w

Consider a consumer with bene�t `2 > `1. The new demand (16) is now the higher curve. We have drawn

two potential tangency, pro�t-maximizing points, A and B on the diagram. If the isopro�t lines result in a

new pro�t-maximizing point such as A, then the monopolist sells to consumers with wealth above bw2 < bw1;that is, the monopolist sells to more consumers as the bene�t increases. At point B, the monopolist sells to

consumers with wealth bw02 higher than bw1.Our general assumptions on the utility function U and the distribution function F do not allow us

to exclude either A or B in the comparative statics; in the appendix, we derive this comparative static

expression. If the monopolist always sells to consumers with lower wealth thresholds as the bene�t parameter

increases, the pro�t-maximizing policy s(`) results in a strictly decreasing indi¤erence boundary. This seems

like a natural outcome of pro�t-maximization but the precise details of the optimization do not predict

this.10

Suppose that a monopolist does use a pro�t-maximizing policy s(`) that gives rise to a comparative

static of the sort for a strictly decreasing indi¤erence boundary. That is, if bw(`) is the solution, then it10The observation is reminisicient of the lack of strict prediction of the monotonicity of demand function in standard

consumer choice theory.

22

is strictly decreasing. Consider now a regulator taking over the market. Suppose that the regulator does

not have access to the bene�t information as the monopolist, but does have access to consumers�wealth

information. The regulator can make use of the monopolist�s optimal pricing schedule s(`), translates it

to a wealth-based payment policy t(w) by the method in the previous section, and implements the same

assignment. By nationalizing the monopolist, the regulator may still let the same set of consumers obtain

the good.

Proposition 4 Let s(`) be the monopoly pro�t-maximizing bene�t-based price schedule. Suppose that the

corresponding wealth threshold bw(`) gives rise to a strictly decreasing indi¤erence boundary. A regulator na-tionalizing the private market can �nd a wealth-based payment policy t(w) to implement the same assignment

as in the private market.

A regulator may desire to subsidize the consumers. It may so happen that the wealth-based policy t(w)

does collect less revenue from consumers, and subsidization results as a consequence. In the next section, we

will show that with general taxation, the regulator can indeed guarantee that less revenues will be collected

from consumers for the implementation of the same assignment. It may be plausible for the regulator to have

access to consumer wealth information but not the bene�t information, and conversely for the monopolist.

The policy implication of a strictly decreasing indi¤erence boundary due to pro�t maximization is simply

that at a minimum the regulator can duplicate the assignment in the private market; with more taxation

instruments, the regulator can implement subsidization.

4.2 Budget allocation and indi¤erence boundary

In this subsection, we consider a regulator who intend to change information regimes. We consider a resource

or budget allocation mechanism. Suppose initially that consumer bene�t ` is observed. The regulator may

have a given budget and want to allocate it on the basis of the available information `. Let this allocation

be B(`). We assume that B(`) is di¤erentiable.

An allocation policy is s(`) where a consumer with bene�t ` pays s(`) to buy the good. For a given `

and the correspondent s(`), we can �nd the level of wealth bw(`) that makes a consumer with bene�t ` and23

paying s(`) indi¤erent between purchasing the good or not. By (1) and (2), if a consumer with wealth level

bw(`) is indi¤erent between getting the good at s(`) and going without, all consumers with w > bw(`) muststrictly prefer to purchase the good. To exhaust the budget allocated for consumers with bene�t `, for each

` the following two equations must hold:

[1� F ( bw)][c� s] = B(`) (17)

U( bw � s) + ` = U( bw): (18)

Equation (17) says that if consumers with wealth above bw purchase the good from the regulator at s, the

total subsidy (equal to per consumer subsidy c � s multiplied by the total demand) exhausts the available

budget for consumers with bene�t `. Equation (18) is the de�nition of the marginal or indi¤erent consumer,

just as in the previous subsection. These two equations de�ne s and bw as functions of `, and the bene�t-based payment policy s(`) is consistent with equations (17) and (18). Viewed as a function of `, bw(`) isthe indi¤erence boundary; it describes the relationship between the indi¤erent consumer�s wealth and the

bene�t `, taking into account both the direct e¤ect of ` in (18) and the indirect e¤ect through the budget in

(17).

Computation from total di¤erentiation of the two equations yields the following:

d bwd`

=�[1� F ( bw)]� U 0( bw � s)B0(`)

[1� F ( bw)][U 0( bw � s)� U 0( bw)] + f( bw)U 0( bw � s)(c� s) (19)

which is negative for all w if and only if B0(`) � 0. Figure 11 illustrates this.

The upward sloping curves are two examples of equation (18), with `1 < `2. The downward sloping

curves are two examples of equation (17) under the assumption that B(`) is increasing; when ` increases,

the locus of equation (17) shifts to the left.

The two intersection points yield two wealth levels, bw1 and bw2; at ` = `i, type ( bwi; `i) consumer (i = 1; 2)is indi¤erent between purchasing the good and not. In Figure 11 , bw1 > bw2. So in this case, the indi¤erenceboundary bw(`) is strictly monotone decreasing. Nevertheless, if B(`) is decreasing and the magnitude ofB0(`) is large, it is quite possible that as ` increases, the shift of the locus due to equation (17) shifts to the

right so much that bw1 < bw2, which results in an upward sloping indi¤erence boundary. Figure 12 shows how24

w

s

1ll =

2ll =

2w 1w

1( )B l

2( )B l

Demand

Budget

Figure 11: Budget allocation: ` and w

a decreasing B(`) can give rise to an increasing bw(`).Proposition 5 Suppose that a regulator sets a budget B(`) for consumers with bene�t `, and suppose that

the budget B(`) is increasing. This gives rise to a strictly decreasing indi¤erence boundary. The regulator

can translate the bene�t-based policy s(`) satisfying (17) and (18) to a wealth-based policy to implement the

same assignment.

The choice of the budget allocation rule B(`) may be a result of an optimization of a social welfare

function. At each `, the solutions of equations (17) and (18), yield the set of consumers who will obtain the

good (those with wealth above bw(`)) and the price they pay (s(`)). Given this allocation, we may obtain avaluation according to some welfare function.

5 Equivalence Revenue and General Subsidy

In this section, we expand the regulator�s subsidy policies. We have assumed that a consumer makes a

payment to the regulator only when he purchases the good. We now allow the regulator to impose another

25

w

s

1ll =

2ll =

2w1w

1( )B l

2( )B l

Deman

d

BudgetFigure 12: Budget allocation: B(`) increasing

tax or subsidy when the consumer decides not to purchase. This can be regarded as a general taxation

scheme. We have seen that equivalent bene�t-based and wealth-based policies (those that implement the

same assignment set) may generate di¤erent revenues. We show that with general taxation, equivalent

bene�t-based and wealth-based policies may be so chosen that they generate the same revenue. General

taxation is assumed to be feasible and consumers cannot opt out of the system altogether.11

Consider �rst the case that wealth is known while bene�t remains consumers�private information. A

wealth-based policy is now a pair of payment function [t1(w); t2(w)]; a consumer with wealth w pays t1(w)

when he does not purchase the good, and t2(w) when he does. In any case, a type (w; `) consumer decides

to purchase the good if

U(w � t2(w)) + ` � U(w � t1(w)): (20)

For a given policy, we de�ne the assignment set analogously:

�(t1; t2) � f(w; `) : U(w � t2(w)) + ` � U(w � t1(w))g : (21)

Again, consider the equation U(w � t2(w)) + ` = U(w � t1(w)). This de�nes a relationship between bene�t

` and wealth w, and generates the indi¤erence boundary ` = �(w; t1; t2).

11We therefore can ignore any individual rationality constraint.

26

When [t1(w); t2(w)] satis�es Condition 1 ([t1(w); t2(w)] continuous and �(w; t1; t2) strictly decreasing in

w), Proposition 1 applies. There is a bene�t-based policy [s1(`); s2(`)] to implement the same assignment

set:

�(s1; s2) � f(w; `) : U(w � s2(`)) + ` � U(w � s1(`))g (22)

where s1(`) is the payment by a consumer with bene�t ` when he does not purchase, and s2(`) is the

payment when he does. The construction of the bene�t-based policy uses the same procedure: we replace w

in U(w � t2(w)) + ` = U(w � t1(w)) by the inverse of the (strictly decreasing) indi¤erence boundary �, say

�. So for each ` we choose s1(`) and s2(`) to satisfy

U(�(`)� s2(`)) + ` = U(�(`)� s1(`)): (23)

Clearly many bene�t-based policies satisfy (23).

The revenue that is being generated by [t1(w); t2(w)] is

R(t1; t2) �Z�(t1;t2)c

t1(w)dF (w)dG(`) +Z�(t1;t2)

t2(w)dF (w)dG(`): (24)

The revenue that is being generated by [s1(`); s2(`)] is

R(s1; s2) �Z�(s1;s2)c

s1(`)dF (w)dG(`) +Z�(s1;s2)

s2(`)dF (w)dG(`): (25)

(Here, the superscript c over the sets �(t1; t2) and �(s1; s2) denotes their complements.)

Proposition 6 ( Revenue equivalence) Suppose that a wealth-based policy [t1(w); t2(w)] satis�es Condi-

tion 1 (Decreasing Indi¤erence Boundary). There exists a bene�t-based policy [s1(`),s2(`)] such that they im-

plement the same assignment and generate the same revenue: �(t1; t2) = �(s1; s2) and R(t1; t2) = R(s1; s2):

The intuition for Proposition 6 (whose proof is in the appendix) is best illustrated by a two-part tari¤

policy. Consider a wealth-based policy [t1(w); t2(w)] and the assignment that it implements. Let [s1(`),s2(`)]

be the policy that implements the same assignment. We know that there are many such policies. Let

s1(`) =M , a constant, and s2(`) =M + s(`), where M is now interpreted as a lump sum tax or subsidy for

each consumer, and s(`) is the incremental payment for purchasing the good. The revenue that is generated

27

by [s1(`),s2(`)] = [M;M + s(`)] is now

M +

Z�(M;M+s)

s(`)dF (w)dG(`): (26)

The level of M uses up one degree of freedom in the choice of the bene�t-based policy [s1(`),s2(`)]. For each

given value of M , we can set s(`) to satisfy

U(�(`)�M � s(`)) + ` = U(�(`)�M) (27)

maintaining the same assignment set . So now we can adjust the level of M to obtain the same revenue:

R(M;M + s) = R(t1; t2).

The result in Proposition 6 assumes implicitly that the lump sum payment M is feasible. If consumers

have limited wealth so that some of them cannot pay the standard taxes then the result need not hold.

6 Conclusions

The analysis in the paper sheds a new perspective on subsidy policies. Whereas previous reseach tends to

focus on the characterization of optimal schemes, we have looked at properties of schemes that are based on

di¤erent information. The focus on optimal schemes can be misleading. Often a research focuses on one set

of problems when deriving an optimal scheme; this method lacks robustness. Our approach does not impose

optimality properties on subsidy schemes; they might have come from historical, political, or sociological

considerations. Instead, we look at how schemes that are based on one type of information can be related

to schemes that are based on another type of information.

Our point is that these schemes may perform quite similarly even when they are based on di¤erent infor-

mation. Our analysis provides a perspective to think about the cost of collecting and processing information.

28

Appendix

Proof of Proposition 1: Consider those ` 2�`; `�\ [`0; `0]. By construction, the payment schedule s(`)

satis�es U(w � s(`)) + ` = U(w). Any type (w; `) consumer is indi¤erent between purchasing the good at

price t(w) and not if and only if he is indi¤erent at price s(`). For each such `, we show that (w; `) belongs to

�(t) if and only if it belongs to �(s). Suppose (w; `) 2 �(t) where �(t) � f(w; `) : U(w � t(w)) + ` � U(w)g.

We have

U(w � t(w)) + ` � U(w � t(w)) + �(w) = U(w); (28)

with a strict inequality if ` > �(w). Let ^= �(w) � `. By the de�nition of s, we have U(w�s(^))+ ^= U(w).

Because � is strictly decreasing, for ` � ^, there exists w � w such that

U(w � s(`)) + ` = U(w): (29)

But w � w. Hence equation (29) implies

U(w � s(`)) + ` � U(w);

which says that (w; `) 2 �(s). We can use a symmetric argument to show that if (w; `) does not belong to

�(t), then it does not belong to �(s).

Next consider ` 2�`; `0

�(if it exists). According to �(t), a type (w; `) consumer does not purchase the

good when ` < `0, for all w. By construction, U(w � s(`0)) + `0 = U(w). Hence, U(w � s(`0)) + ` < U(w),

and therefore, U(w � s(`0)) + ` < U(w), all w. A type (w; `) consumer does not purchase the good under

s(`) either.

Finally, consider ` 2 [`0; `] (if it exists). According to �(t), a type (w; `) consumer purchases the good

when ` > `0, for all w. By construction, U(w � s(`0)) + `0 = U(w). Hence U(w � s(`0)) + ` > U(w), and

therefore, U(w � s(`0)) + ` > U(w), all w. A type (w; `) consumer always purchases the good under s(`).

This concludes the proof that �(t) = �(s).

Proof of Proposition 6: Let s1(`) and s2(`) satisfy equation (23). Choose (continuous) functions �

and � such that

U(�(`)� s2(`) + �(`)) + ` = U(�(`)� s1(`) + �(`)): (30)

29

By the continuity of U , such � and � functions exist. The policy [s1(`) + �(`); s2(`) + �(`)] also satisifes (23).

From (30), for each ` the values of �(`) and �(`) must follow:

d�d�=U 0(�(`)� s2(`) + �(`))U 0(�(`)� s1(`) + �(`))

> 0: (31)

The collected revenue under [s1(`) + �(`); s2(`) + �(`)] is

R(s1(`) + �(`); s2(`) + �(`)) �Z�(s1;s2)c

[s1(`) + �(`)]dF (w)dG(`) +Z�(s1;s2)

[s2(`) + �(`)]dF (w)dG(`)

= R(s1; s2) +

Z�(s1;s2)c

�(`)dF (w)dG(`) +Z�(s1;s2)

�(`)dF (w)dG(`):

For a given policy [s1(`); s2(`)], R(s1(`) + �(`); s2(`) + �(`)) is monotone in � due to (31). So there exist �(`)

and �(`) such that R(s1(`) + �(`); s2(`) + �(`)) = R(t1; t2).

Comparative static on the monopolist�s pro�t maximization problem (section 4.1):

The monopolist solves the following problem:

Maxs; bw �(s; `) � [s� c][1� F ( bw)]s.t. U( bw � s) + ` = U( bw):

The Lagrangean function is

L � [s� c][1� F ( bw)] + � [U( bw � s) + `� U( bw)]The system of �rst-order conditions is

8>>>><>>>>:�L

� bw = �f( bw) + �[U 0( bw � s)� U 0( bw)] = 0�L

�s= [1� F ( bw)] + �[U( bw � s) + `� U( bw)] = 0

�L

��= U( bw � s) + `� U( bw) = 0

We totally di¤erentiate the �rst-order conditions and solve the system of equations ford bwd`. After sim-

plifcation, we obtain

d bwd`

=��U 0( bw)U 00( bw � s)� U 0( bw � s)f( bw)

jHj

30

The determinant of the bordered Hessian matrix, jHj, is positive, where as � is positive. It follows that

the sign ofd bwd`

is ambiguous.

31

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