Monopolistic Provision of Excludable Public goods with an application to Education∗ Jishnu Das
Asim Ijaz Khwaja
The World Bank
Harvard University
[email protected]
[email protected] July 5, 2002
Abstract We examine the monopolistic provision of an excludable public good under imperfect information, and characterize the form of the optimal contract. We show that the optimal contract consists of a single price posted by the monopolist as long as the ratio of marginal willingness to pay between any two types is constant for all levels of the good. Thus, the optimal contract is characterized by a ‘threshold’ value, whereby all individuals whose willingness to pay is above the threshold receive the same positive quantity of the good and those below are fully excluded. This result is then used to examine the impact of changes in the distribution of willingness to pay on the provision of the public good. Finally, we develop an application of this result to the educational sector that derives the optimal mix of public and private provision of education under different assumptions regarding the information available to the monopolistic provider.
1
Introduction
Recently, there has been considerable interest in the private provision of public goods with the on-going privatization of government utilities in various countries. Several important questions have arisen in this regard: What are the costs of allowing the private sector to provide such goods? What will optimal contracts look like? How does this depend on the underlying characteristics of the economy such as the wealth distribution and the presence and verifiability of information? In this paper we address these questions by examining the provision of excludable public goods by a monopolist to a agents indexed by willingness to pay. Our interest in examining this class of goods follows from the observation that while the commodities that are being privatized often display non-rivalness in consumption, they are seldom non-excludable. Examples of such goods would include roads, television/cable, radio frequencies, security, and education. To map these goods into our framework, consider for instance, the case of cable television. The commodity is non-rival since the consumption of a number of channels by one ∗ We would like to thank Abhijit Banerjee, Sehr Jalal, Michael Kremer, Dilip Mokherjee, Debraj Ray, Carolina Sánchez and Jeffrey Williamson for helpful comments. We would also like to thank Dr.Iota for her constant devotion.
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viewer does not decrease the amount available to another. Simultaneously, the commodity is excludable since the company providing the cable channels can restrict the number of channels available to any one viewer. Similarly, in the case of private security the oversight a guard provides is independent of the number of houses under her detail. However, such houses can be excluded from such coverage for any arbitrary number of hours during the day.1 Finally, since such goods also typically exhibit scale and location economies we examine their provision by a monopolist. We first analyze the benchmark perfect information case, where the monopolist can write contracts contingent on willingness to pay. As expected, the optimal contract is characterized by perfect price discrimination with all agents receiving identical amounts of the good. We then analyze the case where the agent’s willingness to pay for the public good is non-verifiable but the monopolist can offer a menu of potentially screening contracts as in the standard adverse selection problem. The main theoretical contribution of our paper is that our focus on excludable public goods allows us to provide a much simpler characterization of the solution in this problem. We prove a somewhat surprising ‘threshold’ result in this context: In equilibrium the monopolist only offers a single contract. All individuals whose willingness to pay exceeds a certain threshold choose this contract, and those below the threshold choose not to consume the public good. We show that this general result is robust to the functional form assumed. This simple characterization of the optimal contract is particularly amenable to comparative static analysis of changes in the underlying environment on the provision of the public good. Section 3 focuses on one such characteristic, the distribution of willingness to pay for the good in the environment, and looks at comparative static results for changes in this distribution. For the case of the monopolist with perfect information we prove a neutrality result, analogous to Schmitz (1997), whereby the distribution of willingness to pay has no effect on the allocation or overall production of the commodity. This result differs from the standard neutrality results concerning the provision of a public good in the literature (see for example, Warr (1983) and Bergstrom (1986)) in that we allow consumers to be excluded, and the equilibrium is not the outcome of a contribution game, but the outcome of a series of take-it-or-leave-it offers from the monopolist. In the case of imperfect information we show that despite the simple characterization of the optimal contract, general results with changes in the distribution of willingness to pay do not exist. In particular, we show that increases in heterogeneity as represented by a mean preserving spread in the distribution of willingness to pay among consumers has two effects: By moving some part of the distribution from the middle to the left tail, such a spread may allow the monopolist to increase her profits through a decrease in the threshold (to take advantage of the mass in the lower tail of the distribution). On the other hand, a mean preserving spread also increases mass in the right tail of the distribution, and this may cause the monopolist to increase her threshold, leading to a lower mass but a higher price. For two important distributions the uniform and the normal - we show that from perfect equality, the threshold value first decreases and then increases with increases in heterogeneity. However, the mass of individuals who receive the public good always weakly decreases with heterogeneity. 1 There is an issue of congestion costs in most of these commodities. While we acknowledge the importance of such costs, for the purposes of tractability, we abtsract from them. However, as we mention later, we believe that the main results of this paper continue to hold under some reasonable restrictions on the structure of these costs.
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We then present an application of our results to the case of educational provision in low income countries that addresses an ongoing debate over the private provision of education, both in the form of private schooling but also as private tutoring by public school teachers. Several countries have argued for and implemented bans on such private provision by public sector employees and remain suspicious of private provision in general. In this paper we argue that the debate can be enriched by examining the excludable public good aspect of education. Our contribution to this debate is based on the observation that an important input into learning is the educational ‘investment’ or ‘knowledge’ of the teacher. While this investment is costly, the consumption of ‘knowledge’ by one student does not decrease the amount available for another. At the same time however, a teacher can exclude students to any extent by screening on the number of hours of education that the student receives. These two characeristics together allow us to examine the case of private/public education within the framework of our model. Further, our assumptions on monopolistic provision and informational constraints are also particularly pertinent to low income countries where the provision of education is often characterized by a scarcity of providers and weak contract enforcement. We show that the presence of private tuition has two opposing effects. First, as the popular press argues, private tuition does indeed decrease the incentive to provide education in the public sector. On the other hand, if there are any externalities between private and public provision (perhaps through increased investment in learning), decreasing the amount of private tuition that the teacher provides may worsen educational outcomes. We show that the optimal government policy is sensitive to informational assumptions regarding the private market, and use our threshold result to derive such a policy. There has been considerable emphasis in the literature on the provision of public goods and it’s relationship to the distribution of willingness to pay in the population (see for instance, Bergstrom 1986 or deMeza and Myles, 1997). Two assumptions characterize this body of work: First, it is assumed that the good provided is a ‘pure’ public good, characterized by non-rivalry and non-excludability in consumption. Second, the provision of a public good is through private individual contributions, and the amount of the good provided is the Nash equilibrium of this contribution game. We depart from this literature by focusing on excludable public goods with monopolistic provision. The monopolist provision of such goods has been analyzed by Brito and Oakland (1980) and Schmitz (1997) in two related papers. While our paper builds on the analysis presented in these studies, there are important differences. Specifically, Brito and Oakland examine only posted-price schedules, and therefore solve for the optimal contract within this restricted space of contracts. In this study, we allow the menu of contracts offered by the monopolist to depend on the type of the individual, and study truthful revelation mechanisms that maximize the profit of the monopolist. The setup described by Schmitz corresponds more closely to our study with the monopolist determining a set of allocations and prices as a functions of the type of the individual. However, Schmitz examines the allocation rules only for an indivisible excludable public good - the monopolist is restricted in the choice of quantity to a binary set, Q ∈ {0, 1} - and hence, only for goods for which the consumers have constant marginal valuations and constant marginal cost of provision. We extend Schmitz’s analysis to the case of divisible public goods and to a more general class of valuation and cost of provision functions, and derive the optimal pricing scheme under imperfect information.
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2
The Model:
2.1
Assumptions
We imagine a monopolist providing the excludable public good, E, to a set of consumers (or a continuum of consumers), indexed by each individual consumers willingness to pay. Formally, the general problem considers the determination of a vector of allocations (E1 , E2 , ...., EN ) and a vector of prices (T1 , T2 , .....TN ) where the index refers to the ‘type’ of the individual in an economy consisting of: 1. N consumers endowed with the weak preference relation
i
.Following Maskin-Riley (’84), we take
a consumer surplus approach to model the utility of individuals in our population. Specifically, let xi (p, vi ) be the derived demand curve of individual i. Then, p(x, vi ) represents the maximum that the individual is willing to pay for the xth marginal unit of the good and
E 0
p(x, vi )dx is the maximum
that she is willing to pay for E units of the commodity. Thus, the consumer surplus or ‘rent’ that the individual gets from paying Ti for E units of the commodity is simply label this rent as the utility of type i, so that Ui (E, Ti ) = assumptions regarding p(x, vi ):
E 0
E 0
p(x, vi )dx − Ti . We
p(x, vi )dx − Ti . We make two important
[A1]: The demand price function p(x; v) is non-increasing in x and non-negative. Moreover, p(x; v) is twice continuously differentiable, strictly increasing in v, and decreasing in x. [A2]: p(x, vi ) can be written as a separable function p1 (x)p2 (vi )
2. A commodity E produced by a monopolist with a concave production function E = g(w) and an associated cost function C(E) where w is the vector of inputs.2 Therefore the solution must be such that: 1. (Ei , Ti )
i
(0, 0) : The vector of allocation and prices satisfy individual rationality constraints for all
individuals. 2. (Ei , Ti )
i
(Ej , Tj ) ∀j : The vectors satisfy incentive compatibility constraints implying that individuals
prefer their own allocation to that of any other consumer. In this formulation, the excludable nature of the public good is implicitly built in, since the agent receives no (dis)utility from the consumption of others. 3. (E1 , E2 , ...., EN ), (T1 , T2 , .....TN ) ∈ arg max
i
Ti − C(max Ei ): The vector of allocations and prices is
such that the monopolist maximizes her profits. Note that the second term in the profit expression
arises due to the non-rival nature of the commodity, whereby the monopolist produces max(Ei ) for any allocation bundle (E1 , E2 , ...., EN ). This completes the description of our environment. In the next section we first consider the nature of the optimal contracts offered when there is no private information and the monopolist can write contracts 2 Formally, C(E) is the solution to the cost minimization problem, min p .w s.t. g(w) ≥ E. From a standard result in w w production theory (see for instance Mas-Collel, Whinston and Green, 1995), the single output concave production function guarantees the convexity of C(E).
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contingent on the type of the agent. Next we look at the more realistic case of imperfect information. We then examine the comparative statics of changes in the willingness to pay distribution in the economy under both informational regimes. Finally, we end with an application to education.
2.2
Solution: Perfect Information
Under perfect information, the ‘types’ of individuals as well as the amount of public good received is verifiable. Hence, it is possible for the monopolist to write contracts that specify the amount of public good received as well as a type contingent payment. Under this environment we have the standard result for a perfectly discriminating monopolist. Claim 1 Under perfect information the monopolist offers different contracts to each consumer type such that she fully extracts their surplus i.e. All consumers are offered the same amount E ∗ of the public good, and consumer’s of type v are charged an amount T (v) =
E∗ 0
p(x, vi )dx.
Proof. Under perfect information, the monopolist can extract full rent from all consumers and therefore sets the fees of each consumer to satisfy each individual rationality constraints i.e, T (v) =
E(v) p(x, vi )dx. 0
Thus the monopolist’s problem is to, v
max
E(v)
[
E(v)
0
p(x, vi )dx]f (v)dv − C(max E(v))
0
From the form of the maximand, we have E(v) = E ∀ v. To see this , if ∃Ej < max(E(v)), providing Ej + ε
to type j is always strictly better, since the cost remains unchanged and revenue is strictly increasing in E(v). Using this property and assumption [A2] above we can rewrite the objective function as v
max E
E
[ 0
0
p1 (x)p2 (v)dx]f(v)dv − C(E)
with associated first order condition used to obtain E ∗ v
p1 (E ∗ )p2 (v)f(v)dv
= C (E ∗ )
0 v
or, p1 (E ∗ )
p2 (v)f(v)dv
= C (E ∗ )
0
2.3
Solution: Imperfect Information
Under imperfect information, the monopolist does not observe the type of each individual, vi , but knows the distribution of types, f (v) in the population. Thus, the problem for the monopolist is to choose a set of allocations and fees that satisfy the individual rationality and incentive compatibility constraints for each consumer type. We show that despite the general setting, as long as the ratio of consumer of different types’ marginal willingness to pay at changing levels of the public good remains constant, the optimal contract 5
Up=0
Ur = U>0
Loss from dE
E*r
Pub Good
Ur=0
A
Er
(dT, dE)
Tnewr
Gain from dE
Fees
Tr
Figure 1: Optimal Contracts with Decreasing Marginal Willingness to Pay offered by the monopolist takes a particularly simple form, which consists of choosing a ‘threshold’ type v∗ s.t. for all v > v∗ , E = E(v ∗ ) and T = T (v ∗ ) and for all v < v∗ , E = T = 0. To develop intuition for this result, we start by analyzing an economy consisting of only two types of agents and then extend the argument to an arbitrarily large number of consumer types. 2.3.1
Two Consumer Types
To examine the threshold result in a simple setting, consider an economy with only two consumer types, ‘rich’ and ‘poor’ with demand price functions given by p(x, rich) = p1 (x)(1+α) and p(x, poor) = p1 (x)(1−α) with p1 (x) decreasing monotonically in x and p1 (x) → ∞ as x → 0 and p1 (x) → 0 as x → ∞, so that Inada conditions are satisfied. Therefore the utility each type for public good level E is: E
Ur
= 0
Up where g (E) =
∂(
E 0
p1 (x)dx) ∂E
(1 + α)p1 (x)dx − Tr = (1 + α)g(E) − Tr
= (1 − α)g(E) − Tp
= p1 (E). Thus, g(E) → ∞ as E → 0 and g(E) → 0 as E → ∞.
Proposition 1 The optimal contract offered by the monopolist is characterized by Er Er
1 ( low heterogeneity: pooling) 3 1 = Er∗ , Ep = 0∀α > (high heterogeneity:separation) 3 ∗ = Ep = Er,p ∀α
13 . Consider a marginal increase in Ep . We have to show that this leads to lower profits for the monopolist if α >
1 3
but to higher profits if α < 13 . For any increase dEp = dE, the
3 Note that in this case, there is a clear relationship between heterogeneity (α) and public good provision: the public good provided to the poor is (weakly) decreasing with heterogeneity, while the public good provided to the rich as well as the monopolist’s total public good provision follows a U-shaped pattern .
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(maximal) gain in fees from the poor is: dTp = (1 − α)g (0)dE illustrated in Figure 1 (i.e. move higher along the 0 rent indifference curve for the poor). To satisfy the rich consumer’s incentive compatibility constraint, it must be that any contract offered to the rich satisfies Ur (Tr , Er ) ≥ U where U is the utility received by the rich consumer if he accepts the new contract (dE, dT ) intended for the poor consumer. Note that U > 0 as indicated in the expression below. U = (1 + α)g(dE) − (1 − α)g (0)dE Since, the rich consumer was previously earning 0 rent, the above expression also represents the (minimum) increase in the rich consumers utility. Thus, the space of contracts that the monopolist can potentially offer to the rich, (Tr , Er ) is characterized by (1 + α)g(Er ) − Tr ≥ (1 + α)g(dE) − (1 − α)g (0)dE This corresponds to the indifference curve labelled Ur = U in the figure. Assume that the monopolist picks that same maximal public good level Er∗ to offer to the rich consumer on this new indifference curve (later on we relax this assumption and show that the proof still holds).Therefore, the maximum amount of fees that can be now raised from the rich are given by: Trnew = (1 + α)(g(Er∗ ) − g(dE)) + (1 − α)g (0)dE These fees are clearly lower than the fees being raised from the rich before (this has to be the case since, the rich must now be offered a strictly greater utility for the same level of public good). The loss in fees raised from the rich (indicated in the graph) is given by: dTr
= (1 + α)(g(Er∗ )) − [(1 + α)(g(Er∗ ) − g(dE)) + (1 − α)g (0)dE] = (1 + α)g(dE) − (1 − α)g (0)dE
dTtotal
= dTp − dTr = (1 − α)g (0)dE − (1 + α)g(dE) + (1 − α)g (0)dE = 2(1 − α)g (0)dE − (1 + α)g(dE)
Now, from a Taylor expansion at 0, g(dE) = g(0) + g (0)dE + higher order terms
7
Disregarding the higher order terms and recognizing that f(0) = 0, we get dTtotal = (1 − 3α)g (0)dE and hence, dTtotal < 0 ∀α >
1 3,
and dTtotal > 0 ∀α
Ep by offering slightly more public good to the poor. Thus for
α < 13 , the optimal contract will be a pooling contract. The above proves the proposition assuming that the optimal Er∗ remains at the same level. To complete the proof, we also have to show that the proposition continues to hold once the optimal E ∗ changes. Since the proof is similar to the above it is provided in the appendix. Our proof relies on two assumptions: the non-rival nature of the commodity and the form of the utility function, whereby the indifference curves represent horizontal shifts along the (E, T ) axis and the ratio of slopes of the two types’ indifference curves remains the same for all value of E. To recognize the role of the first assumption, note that the fact that E is a public good means that we can restrict ourselves to the comparison only in terms of revenue as long as E ∗ is fixed - providing an extra (dE) to the poor consumer has no impact on costs. Thus simply considering equation 1, which gives us the total change in revenue, ensures that the conclusion is the same for the change in total profits as well. The role of the second assumption is made clear by rewriting equation 1 as: dTtotal = (1 − α)g (0)dE − {(1 + α)g (0)dE − (1 − α)g (0)dE} This equation considers the relative gain (loss) from moving to a new contract assuming that the total amount of public good provided remains unchanged. The fact that indifference curves are horizontal shifts in (E, T ) space implies that we can decompose the change in profits from moving to a new level of E into two components. The first component is the change in profits from moving along the indifference curve of the poor, and the second is the change in profits from the movement of the indifference curve of the rich. The second term represents the additional utility (and hence decrease in fees raised) that must be provided to the rich consumer and, given our second assumption, this change is equal to the change evaluated at the initial public good level of the poor. With the Inada conditions, while it is true that the poor consumer is willing to pay on the margin a very large amount for the first ε units of public good, it is also true that the rich consumer is always willing to pay more for this first marginal unit. Thus the rich consumer will strictly prefer this new contract to the one he was receiving before (which gave him no rents). By accepting this new ε public good contract he earns a strictly positive rent and this savings in fees is a function, not only of the poor consumers marginal willingness to pay at zero, but also that of the rich consumer’s - both the rich and poor consumers’ initial 8
marginal conditions are relevant and so the Inada conditions affect both. Thus to be able to offer this new ε public good contract, the monopolist must provide at least as much of an utility increase as the rich gets from accepting the ε contract. In other words, she has to lower fees by the same amount as the savings in fees by the rich on that first marginal unit of public good offered to the poor. With high heterogeneity, this fee reduction from the rich is substantial and, in particular, larger than the fees gained from the poor thus giving the complete separation result. 2.3.2
Multiple Consumer Types
We now extend our model to the N consumer types. For purposes of tractability we first consider the case where a consumer of type i s utility is: Ui = vi E − T where vi ∈ Λ = set of all consumer types. and vi ≥ 0 ∀i. Note that a higher value of vi indicates a
richer or higher type consumer i.e. one who is willing to pay more for the same amount of public good. This
simpler formulation of the consumer’s utility is derived from our general setup by considering the special case when p(x; v) = v. (and therefore we get that
E 0
p(x, vi )dx = vi E). As before we continue to assume
that E = g(w). Henceforth, i > j will be used to indicate vi > vj . Thus the consumer’s utility will be a function of the total public good he receives, E, and the total amount of money he has to pay for this public good, T. The monopolist therefore offers contracts of the form E, T i.e. a prescribed fee for a given level of public good. The primary purpose of this section is to characterize the optimal solution of the monopolist’s problem when she faces a large number of consumer types . In order to do so we shall state some claims leading to the main ’threshold result’ proposition. The proofs of the claims are fairly standard and are presented as an appendix. Claim 2 If a consumer of a given type weakly prefers a contract with higher public good to another, any consumer of a higher type will strictly prefer the former to the latter contract. ∀E2 ≥ E1 &
E1 , T1 = E2 , T2 ,
E1 , T1 0i E2 , T2 ⇒ E1 , T1 ≺j E2 , T2 ∀j > i
Claim 3 Any optimal menu of contracts offered by the monopolist,
Ei∗ , Ti∗
i∈Λ ,
has to be such that the
contract chosen by the higher consumer type offers an equal or higher level of public good. i.e. Ej∗ ≥ Ei∗ ∀j > i, where Ei∗ , Ti∗ denotes i s preferred contract choice from the menu of contracts offered.
Claim 4 If Ei∗ , Ti∗
i∈Λ
represents the optimal (for the monopolist) implementable menu of contracts, then
it must be that for any given consumer type, his downward incentive constraint will always be binding i.e. ∗ ∗ Ei∗ , Ti∗ ∼i Ei−1 ∀i , Ti−1
Now we turn to the monopolist’s full problem and a characterization of the solution to this problem. The monopolist solves: max
Ei ,Ti
i∈Λ
ρi Ti − C(max{Ei |i ∈ Λ}) 9
where ρi = proportion of consumers in the population of type i. (i.e. ρi ≥ 0 and
i∈Λ ρi
= 1) subject to
each consumer type’s participation and incentive constraints being satisfied. The above claims allow us to simplify the monopolist’s problem considerably: Proposition 2 .Threshold Result: The monopolist’s optimal menu of contracts is characterized by a simple threshold rule. All consumers of type j ≥ I ∗ will be offered the same contract EI∗ , TI∗ , whereas all types below I ∗ will not be offered the good i.e. given a contract where E, T = 0, 0 .
∗ ∗ Proof. From the previous claim we know that Ei∗ , Ti∗ ∼i Ei−1 ∀i. An implication is that for , Ti−1
any solution to the monopolist’s problem the lowest consumer type must be earning no rent i.e. E1∗ , T1∗ ∼1 0, 0 ⇒ T1∗ = a1 E1∗ . Moreover, this claim also provides a simpler representation of Tj∗ i.e. Tj∗ =
∗ ∗ Ej−1 ] ∀j > 1. where Ej∗ ≥ Ej−1 . This allows us to rewrite the monopolist’s problem as
N
max
Ei ,Ti
i=1
Ei ρi vi − (vi+1 − vi )
N
j=i+1
ρj − C(max{Ei |i s.t.Ej∗
∈
∗ j∈Λ aj [Ej −
Λ})
≥ Ei∗ ∀j > i
where N is the highest type in set Λ. Note that the first term multiplying Ei inside the summation, ρi vi , is always positive and the second term, −(vi+1 − vi )
N j=i+1 ρj ,
always negative. Thus the sign of the ‘coefficient’ of Ei depends on the
values of vi and the probability weights on these consumer types. Note also that the cost of public good, C(max{Ei |i ∈ Λ}), depends only on the highest level of E provided. This simplifies the principal’s problem
to maximizing revenue for a given value of max{Ei |i ∈ Λ}. Examining the revenue term above leads to an
immediate though slightly incorrect solution that the principal should choose Ei = 0 whenever its coefficient
is negative and Ei = E ∗ when it’s positive. However, this solution may violate the constraint that Ej∗ ≥ Ei∗
∀j > i. Introducing this constraint provides the threshold result: It is still true that the monopolist offers
contracts 0, 0 and E ∗ , T ∗ (since the revenue function is linear in Ei this must always be the case). However, the constraint now implies that ∃ I ∗ such that the contract offered to I ∗ is E ∗ , T ∗ , and all types
higher than I ∗ are also offered and choose this contract. This also implies that all types below I ∗ are not provided any public good in equilibrium. The monopolist’s problem is now simply reduced to choosing the optimal type-threshold I ∗ . T ∗ will then be set such that I ∗ earns no rent i.e. T ∗ = vI ∗ E ∗ . The monopolist’s problem is reduced to the following two steps. First choose the optimal threshold: N
max I∈Λ
i=I
ρi vi − (vi+1 − vi )
N j=i+1
and then choose the optimal level of E given this threshold I ∗ : N
max E
i=I
E ρi vi − (vi+1 − vi ) ∗ 10
N
j=i+1
ρj
ρj − C(E)
The monopolist will then offer two contracts: 0, 0 and E ∗ , aI ∗ E ∗ and all types i ≥ I ∗ will prefer the
latter and all types below I ∗ , the former contract. A Continuum of consumer Types
In order to examine the effects of heterogeneity, which we will do in the next section, it is helpful to extend our previous proposition to a continuum of consumer types and a general consumer utility function. E 0
A consumer is now indexed by type v ∈ [0, v] and his utility is given by Uv =
p(x; v)dx −T where T is the
price a consumer has to pay if he consumes E units of the public good and the return to the consumer from
this public good is captured in the standard consumer surplus formulation by aggregating his willingness to pay, p(x; v), for each unit of public good. Moreover, p(x; v) is strictly increasing in v and non-increasing in x. Thus “richer” consumers are captured by a higher v and each consumer experiences diminishing utility from each additional unit of public good. Our previous claims continue to hold. In particular, for the same reasons as above, in equilibrium the downward incentive compatibility constraint of any type always binds.
Using this we can derive
an expression for each type’s utility by considering the discrete case and then taking limits. We know Ev∗ 0
∗ ∗ from above that Ev∗ , Tv∗ ∼v Ev−1 ⇒ Uv∗ = , Tv−1
gives us that get v 0
Uv∗
E∗ ∗ − Uv−1 = 0 v−1 [p(x; v) − p(x; v − 1)] dx ∗ i Ei−1 ∂p(x;y) ∂v dxdy . Taking the limit i−1 0
v Uv∗ = i=1 Ey∗ ∂p(x;y) ∂v dxdy 0
∗ Ev−1 ∗ p(x; v)dx − Tv−1 . 0 ∂p(x;y) ∗ ∂v dxdy. Since U0 =
p(x; v)dx − Tv∗ = =
∗ v Ev−1 v−1 0
This 0 we
as types becomes continuous gives us Uv∗ =
giving: Tv∗
Ev∗
= 0
Ey∗
v
p(x; v)dx −
0
0
∂p(x; y) dxdy ∂v
(2)
Now let the cumulative distribution of v be T (v) (and let f (v) denote the density function.). Therefore the total return of the monopolist is given by: Ev∗
v 0
0
Ey∗
v
p(x; v)dx −
0
0
∂p(x; y) dxdy dF (v) − C(Ev ) ∂v
(3)
After integration by parts and further simplification, we get: Ev∗
v 0
0
f(v)p(x; v) − (1 − F (v))
∂p(x; v) dxdv − C(Ev ) ∂v
(4)
The intergrand has a natural interpretation. The first term represents the increase in fees the monopolist gets by offering the consumer of type v a marginal unit of public good while the second term represents the loss in fees from all consumer types higher than v i.e. now that type v is being offered marginally more of the public good, to satisfy the incentive compatibility of all types higher than v, each of their fees must be reduced. Thus the monopolist’s problem is to choose functions E ∗ (v) and T ∗ (v) that maximize the above expression subject to the constraint that E ∗ (v) is non-decreasing (recall from above that this constraint has to be satisfied in any equilibrium). We now check for sufficient conditions on p(x; v) to obtain the threshold result. Note that as long as the integrand is always positive (or negative) for E ∈ [0, Ev∗ ] then it appears the monopolist will maximize by 11
setting the maximum non-zero Ev∗ (and Ev∗ = 0 when its negative). However, this ignores the non-decreasing constraint on E ∗ (v). The following proposition takes proves this “threshold result”. Proposition 3 If the ratio of marginal willingness to pay is constant across consumer types for a given level of consumption (i.e. assumption A2 holds; p(x, vi ) can be written as a separable function p1 (x)p2 (vi )), then the monopolist’s optimum is characterized by a simple threshold rule: All consumers of type v ≥ v are offered the same contract E ∗ (v), T ∗ (v), whereas all types v < v are not provided the good. i.e. offered a contract
where E ∗ (v) = 0 and therefore T ∗ (v) = 0. Proof. Substituting p(x, vi ) = p1 (x)p2 (v) the monopolist’s problem can be re-written as: v
max E(v)
0
f (v)p2 (v) − (1 − F (v))
E(v)
∂p2 (v) ∂v
Note that since p1 (x) ≥ 0 this implies that
0
p1 (x)dx dv − C(Ev ) s.t.
E(v) p1 (x)dx 0
∂E ∗ (v) ≥0 ∂v
(5)
≥ 0 for E(v) ≥ 0. However, the ‘coefficient’
of this integrand, may be positive or negative for a given type v. Ignoring the non-decreasing constraint on E ∗ (v) this implies that E ∗ (v) should be equal to 0 when this coefficient is negative and E ∗ (v) should be set to it’s maximum optimal value, E(v), when this term is positive (since the cost of E only depends on the maximum E provided, it is never optimal to provide different positive amounts of public good). However, such a solution would violate the non-decreasing constraint on E ∗ (v). Adding this constraint gives our desired threshold result. Given this, the monopolist first solves for this threshold type, v: v
v ∈ arg max v
v
f(s)p2 (v) − (1 − F (s))
∂p2 (v) ∂v
ds
(6)
and then chooses the amount of public good to offer to all types v ≥ v: v
max E
v
f (s)p2 (v) − (1 − F (s))
∂p2 (v) ∂v
E 0
p1 (x)dx ds − C(E)
(7)
All types v < v are offered none of the public good.
3
Changes in Heterogeneity
We now consider what happens to the provision of the good when there is a change in the distribution of willingness to pay. In general this will depend on the underlying informational environment. In this section we first prove a “neutrality result" for the case of perfect information: We show the solution is invariant to mean-preserving spreads (increasing heterogeneity) of the willingness-to-pay distribution. We then consider the case of imperfect information. We show that despite the simple form of the contract, general results for distributional changes do not exist: Even if we impose a fairly restrictive structure on the preference relation, results can only be obtained for specific distributions and classes of distributional changes, but cannot be extended without further constraints on the class of distributional changes considered. Before proceeding though, we need to be careful in defining changes in heterogeneity. Specifically, we think of an increase in heterogeneity as a mean preserving spread in the distribution of the willingness to pay 12
at all possible levels of E in the population. Thus, an increase in heterogeneity must satisfy the following condition. Definition 1 Let Z =
E 0
p(x, vi )dx be a random variable. An increase in heterogeneity is represented by a
mean preserving spread of the random variable Z at every possible value of E. From assumption [A2] in section 2.1, we can rewrite Z =
E 0
p1 (x)p2 (vi )dx = p2 (vi )
E 0
p1 (x)dx. Noting
that the term in the integral is a positive constant, an increase in heterogeneity is a mean preserving spread of the transformation, p2 (vi ) of individual income for any level of E (if the condition holds for one E, it must hold for all E).4
3.1
Perfect Information
Claim 5 (Neutrality Result): Under perfect information the amount of public good provided to each type depends only on the total willingness to pay and is independent of the distribution of willingness to pay. Proof. In characterizing the solution under perfect information we had previously proven that the monopolist would extract full rent from all consumers and offer each the same amount E ∗ of the public good such that: v
p1 (E ∗ )p2 (v)f(v)dv
= C (E ∗ )
0 v
or, p1 (E ∗ )
= C (E ∗ )
p2 (v)f(v)dv 0
The neutrality result follows directly from this expression, since we are considering mean preserving spreads in p2 (v), implying that the term in the integral depends only on the mean of the distribution. To see this formally, define as above the random variable Z = p2 (v) Z, f (z) is then given by f (z) =
f (v) dv dz
E 0
p1 (x)dx = p2 (v)K(E). The density function of
and the mean of the random variable Z is p2 (v)k(E)
zf(z)dz p2 (0)K(E) p2 (v)k(E)
=
p2 (v)K(E)f (v) p2 (0)K(E) v
=
dv dz dz
p2 (v)K(E)f (v)dv 0 v
= K(E)
p2 (v)f (v)dv 0
Thus, any mean preserving spread in the willingness to pay keeps constant
v 0
p2 (v)f(v)dv. Hence for any
4 It is worth noting that heterogeneity in comsumer types can map into income/wealth inequality provided there are no/negligible indirect income effects (i.e changes in expenditure on the public good do not lead to significant income effects) and income/wealth acts as a ‘taste shifter’. In other words, we would require that, on the margin, a wealthier individual is willing to pay more for the same level of the public good than a poorer one.
13
two distributions with the same mean, µ, the first order condition is p1 (E ∗ )µ = C (E ∗ ). Since this depends only on the mean and not the distribution of Z, the proof is complete. Hence, the optimal E ∗ that the monopolist provides as well as the public good level achieved for both the rich and the poor consumers is independent of the distribution of willingness to pay in the population. The intuition for this neutrality result is straight forward - since there is non-rivalry in the consumption of public good and the monopolist can identify consumer types, she will always offer contracts that fully extract the rent from each consumer type i.e. she acts as a perfectly price discriminating monopolist. Thus, the total wealth that she can extract is independent of the distribution of wealth as long as mean wealth remains unchanged. Note that this result should be interpreted as an extension to the neutrality results of Bergstrom and Ware to the case of excludable public goods provided by a monopolist. As in the literature on the relationship between the wealth distribution and the provision of a public good, the central intuition that only the total willingness to pay in the economy and not it’s distribution matters, remains unchanged.
3.2
Imperfect Information
From the previous result, the monopolist’s problem in the first stage can be reduced to the choice of a threshold value, with agents whose willingness-to-pay (WTP) is less than the threshold not receiving any amount of the excludable public good, and above which all agents receive the same amount of the good. Moreover, the ‘investment’ effort and hence the level of public good provided to those above the threshold value depends directly on the maximum profits that the monopolist can raise. To simplify our problem, we consider a continuum of agents endowed with preferences such that p1 (x) = 1 and p2 (vi ) = vi where vi is the income of agent i. Thus, the willingness to pay for an amount of public good E =
E 0
vi dx = Evi , and Ui = Evi − Ti . We consider the general class of increases in heterogeneity as
represented by a mean preserving spread5 in the distribution of vi , f (vi ). Thus, the monopolists maximization problem can be stated as B
max θ θ
f (α, r)dα θ
where θ is the threshold value, and increases in r represent a mean preserving spread if and
B 0
Fr (s, r)ds = 0.
y 0
Fr (s, r)ds ≥ 0
Since a mean preserving spread implies that there is greater mass on the tails of the distribution, intuitively, it may appear that such a spread will increase the threshold value that the monopolist sets, and decrease the mass of consumers receiving the good. However, this is not necessarily true since there are two opposing effects of such a spread. First, there is an increase in the mass of individuals on the right tail of the distribution, which leads the monopolist to increase her threshold value, and decrease the mass of individuals receiving the good. Correspondingly, there is an increase in the mass of individuals on the left tail of the distribution, and to take advantage of this increase in mass, the monopolist may decrease 5 Since
mean preserving spreads imply Lorenz dominance, such a sequence would appear to be useful in characterizing changes in inequality in the context of this paper.
14
the threshold value, leading to an increase in the mass of individuals receiving the good. Hence the optimal threshold may be a non-monotonic function of heterogeneity. The example below (in the case of the uniform distribution) and the example in the appendix (in the case of the normal distribution) illustrate this point 3.2.1
Example: Willingness to Pay is Uniformly Distributed
Thus, let f (α) ∼ u[m, n] and consider ‘stretching’ the distribution s.t. g(α) ∼ u[m − y, n + y]. Remark 1 g(α) is a mean preserving spread of f (α) Claim 6 The threshold value decreases and then increases with mean preserving spreads, while the mass of individuals receiving the good is weakly decreasing. Finally, the maximum fees that the monopolist can raise is U-shaped w.r.t. mean preserving spreads in WTP. Proof. For an optimal contract, the monopolist chooses n
θ s.t. θ
:
θ ∈ arg max{θ
≥ m
θ
1 dx} n−m
Solving this, there are two regions λ1 λ1
n − 2m ∗ n , θ =m∀ 1/3 however, this result breaks down. From Equation 8 we have that the maximand is strictly decreasing in ppp , and thus the monopolist sets ppp = 0. Conditional on pG , the monopolist now solves max(1 + α) ln(1 + e) + (1 − 3α) ln(1 + epG ) − 2(1 − α) ln(1 + epG ) e
Solving for e, we have e∗sp,sep |pG =
−(1 + pG ) +
(1 + pG )2 − 4pG [pG (1 + α) − α] 2pG
In the first stage, the government now chooses pG to maximize pG e∗sp,sep |pG with the corresponding solution pG (α) =
√ 1 + 2α − 1 + α 3 + 4α
Figure 5a below show the policy function pG (α), the education received by the poor epG (α) + eppp , and the investment induced by pG , e∗ |pG . As is clear, the positive externality from the private to the public sector
implies that even at very high levels of heterogeneity, government intervention is fairly limited.
4.4
Pure Monopolist
Finally, consider the case of the pure monopolist who offers a single price, f in the private market. Note that in such a market, each consumer will choose a quantity of education, pi , that maximizes her utility and these quantities may differ across individuals. The monopolist thus solves: max f (ppr + ppp ) − c(e)
p e,f,pp p ,pr
s.t.M RSr = f = M RSp (interior solution) or, M RSr = f & M RSp < f (corner solution for poor)
22
Ur
Education
Up
Direction of Increasing Fees
Er
Ep
fEp
fEr
Fees
Figure 4: Pure Monopolist’s Provision of Education (1+α)e and G (1+epp r +ep ) demands ppr = 1+α f
(1+α)e p p G , represented by Figure 4. Solving for pp and pr gives (1+epp r +ep ) G ) 1+epG and ppp = 1−α and total demand, ppp + ppr = f2 − 2(1+ep : f − e e
where MRSr =
MRSp =
us individual
−
1+epG e
Any increase in the price of education f decreases total demand, while any increase in the learning investment by the teacher, e, has the reverse effect. The monopolists problem for an interior solution is then reduced to: 2f − 2fpG − c(e) e = f = M RSp (interior solution) max 2 −
p e,f,pp p ,pr
s.t.M RSr
Solving for e by noting that since p is a costless input, the monopolist will always provide ppr = (1 − pG ), we
have
e∗pm,interior =
2(1 − pG )(1 + α) − 1
The government’s problem is: (1 − α) (e∗pm,interior pG + e∗pm,interior ppp ) = √ max G p 1+α
2(1 − pG ) − 1
Note that in this case, as with the separating monopolist above, we have that pG = 0; the government leaves all educational provision to the private market. However, an important difference with the previous case is that the range of α over which the monopolist provides education to both the rich and the poor is smaller. For a separating monopolist education is provided to both consumers as long as α < 1/3; in this case we have educational provision to both consumers only for α < .22. For α values higher than this the monopolist provides education in the private market only to the rich student, and her maximization problem is thus to max f ppr − c(e) s.t.M RSr = f p f,pr
Proceeding in an analogous manner, we have e∗pm,corner =
(1 + α)(1 − pG ) − 1 23
and the government chooses pG to maximize pG e∗pm,corner to give √ 2(2 + 3α − 4 + 3α) p = 9(1 + α) G
4.5
Discussion
Figures 5a-b below plot the optimal policy function, pG , and the education received by the poor under the three private market regimes. Figure 5c also shows the loss from the government mistakenly using the optimal pG for a pure monopolist when the underlying private market is instead characterized by a separating monopolist. Two observations follow immediately: From the derivation of the optimal time that a teacher should spend in the public school, it is not at all clear that ‘banning’ the use of private tuition is optimal, even at very high levels of inequality (in our example for instance, the teacher never spends more than a quarter of her time in the public school). While there are clearly other functional specifications that may lead to different results (for instance, decreasing the size of the externality), the example shows that the extent to which a government should disallow the use of private tuition is an open empirical question and care should be taken in evaluating the effects of any policy prior to its use. g
Optimal pG(SM)
Optimal pG(PM)
Percentage drop (PD-PDwrong)
Poor Education
Governments Optimal P 0.3
60
Optimal pG(PDM)
Percentage Difference
1
0.25
50
Separating Monopolist
0.9 0.8
Epoor
Pg 0.1
40
Pure Monopolist
0.7
% diff
0.2 0.15
0.6
Separating Monopolist (wrong policy)
0.5 0.4
20 10
0.3
0.05
30
0.2
0
0
0.1 0
-0.05
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.2
0.4
0.6
0.8
1
-10 0
alpha
0.2
alpha
F ig 5 a : O p t im a l T im e in P u b lic S ch o o l
0.4
0.6
0.8
1
alpha
F ig 5 b : E d u c a t io n R e c ie v e d b y t h e P o o r
F ig 5 c : P e r c e n t a g e L o ss U n d e r M is t a k e n P o lic y
The second and third graph then show that any such evaluation should carefully account for the contractual and informational regime present in the private market. Essentially, the degree of government involvement in the provision of education (at least in terms of pure educational attainment) depends on the efficiency of the private market. Typically, the private market has either been modelled as fully efficient (the price-discriminating monopolist) or extremely inefficient (the pure-monopolist). Our example shows that policy regimes derived under these two environments may be harmful if the true world is different- since the information available to a separating monopolist is different, policy regimes that assume a more inefficient (efficient) state of private provision (pure-monopoly or price-discriminating monopolists) will argue for greater (lesser) government intervention and subsequent decreases in educational outcomes. Our model thus provides a middle ground between the price discriminating and the pure monopolist, in which the monopolist has some information and some contractual flexibility- what is particularly useful about our characterization is the simplicity of the optimal contract, and hence it’s use in such policy exercises.
5
Conclusion
In this paper we have considered the problem of a monopolist supplying an excludable, non-rival good to a population characterized by agents who have differing willingness to pay for the commodity. We have shown 24
that when the monopolist can observe the types of the individual, she perfectly price discriminates, and all agents receive exactly the same amount of the good in equilibrium. The main theoretical contribution of this paper is to study optimal contracts when the types of individuals are unobservable. In this case, we prove a ‘threshold’ result that greatly simplifies the characterization of the optimal contract. Under this result, the optimal contract offered by the monopolist is characterized by a threshold - everyone above the threshold receives the same positive amount of the good, and everyone below the threshold is not provided with the commodity. This threshold result follows directly from the non-rival nature of the good in combination with the assumption regarding the constant ratio of marginal willingness to pay of any two individuals. We then examine the provision of the public good with changes in the distribution of willingness to pay in the population. Under perfect information, the amount of the good provided depends only on the mean of the distribution. However, when the types of the individuals are unobservable, this neutrality result breaks down - increases in heterogeneity may hurt the poor, and may or may not decrease provision for the rich. Finally, we apply the threshold result to a model of public and private provision in the educational sector. In this application, we provide characterizations of the private market under different contractual regimes and derive optimal government policy. We show that the extent of government intervention in private provision depends both on the informational environment in the private sector, as well as the strength of the externality effect from private to public provision. There are several avenues that we hope to pursue in future work: First, the heterogeneity results here show that even with single-crossing mean preserving spreads, it is not possible to obtain unambiguous results. It is therefore worth examining further restrictions on distributional change that would allow for tighter results (analogous to that for the uniform and normal distributions) on a general class of distributions. Second, throughout the paper we have examined the private sector as monopolistic. Our future work examines how these results are affected if we allow for greater competition in this sector. Moreover, our model may be extend by explicitly introducing congestion costs and by considering other changes to our basic setup to include industries such as the pharmaceutical sector where there is an initial large (excludable) public investment of developing a new drug followed by smaller fixed costs of producing it. Finally, another potential avenue to explore involves introducing dynamics explicitly in our setup; not just in terms of the dynamic implications of the contracts but also allowing greater flexibility in these contracts over multiple periods. Doing so would allow us to explore far richer contracts such as a “triggered ban" (where the amount of time a teacher is allowed to spend in private provision in the second period is contingent of first period outcomes) instead of the simpler private provision ban we have considered here.
References [1] Baland, J.M, and J.P.Platteau (1996) : ”Heterogeneity and Collective Action in the Commons”, Mimeo, University of Namur. [2] Bardhan, P., M. Ghatak and A. Karainov (2000) : ”heterogeneity and Collective Action Problems”, Mimeo, University of Berkeley.
25
[3] Baron, D. and Myerson, R.B. (1982) : ”Regulating a Monopolist with Unknown Costs”, Econometrica, 50. pp. 911-930. [4] Bergstrom, T., L.Blume and H.Varian (1986) : ”On the Private Provision of Public Goods”, Journal of Public Economics, 29, pp. 25-49. [5] Bernheim, B.D. (1986): On the Voluntary and Involuntary Provision of Public Goods”, American Economic Review, September. [6] Biswal, B.P. (1999) : ”Private Tutoring and Public Corruption: A Cost-Effective Education System for Developing Countries”, The Developing Economies, 37-2, pp. 222-40. [7] Brito, D.L. and W.H. Oakland (1980), ”On the Monopolisitc Provision of Escludable Public Goods”, American Economic Review, 70, pp. 691-704. [8] Cornes, R. and T.Sandler (1984): ”The Comparative Static Properties of the Impure Public Good Model” Journal of Public Economics, 54, pp.403-21. [9] Itaya, J., D. de Meza and G.D.Myles (1997): ”In Praise of heterogeneity: Public Good Provison and Income Distribution”, Economic Letters, 57, pp. 289-296. [10] Gradstein, M., S. Nitzan, and S.Slutsky (1994): ”Neutrality and the Private Provision of Public Goods with Incomplete Information”, Economics Letters, pp.69-75. [11] Mas-Colell, A., M.D. Whinston and J.R. Green (1995) Microeconomic Theory, Oxford University Press, New York. [12] Maskin, E. and J.Riley (1984) : ”Monopoly with Incomplete Information”, Rand Journal of Economics, 15,2. pp. 171-195. [13] PROBE Team in association with the Center for Development Economics (1999) : ‘Public Report on Basic Education in India’, Oxford University Press, New Delhi [14] Rothschild, M. and J.E. Stiglitz (1970) : ”Increasing Risk I: A Definition”, Journal of Economic Theory, 2, pp.225-243. [15] Rothschild, M. and J.E. Stiglitz (1971) : ”Increasing Risk II: It’s Economic Consequences”, Journal of Economic Theory, 3, pp.66-84. [16] Rothschild, M. and J.E. Stiglitz (1973) : ”Some Further Results on the Measurement of heterogeneity”, Journal of Economic Theory, 6, pp.188-204. [17] Sen, A. and J.E. Foster (1997) : On Economic heterogeneity, Clarendon Press, Oxford. [18] Schmitz, P.W. (1997) : ”Monopolistic Provision of Excludable Public Goods Under Private Information”, Public Finance/Finances Publiques, 52(1), pp.89-101. [19] Warr, P.G. (1983) ” ”The Private Provision of a Public Good is Indpenedent of the Distribution of Income”, Economic Letters, 13, Pp.207-11. 26
Appendix: Proofs and Examples Proof of Proposition 1 (cont.). In the paper we had proven porposition 1 assuming that the optimal choice Er∗ remains at the same level under the perturbation to the separating contract. To complete the proof, we now show that the proposition continues to hold once the optimal E ∗ changes. Under the initial menu of contracts, the monopolist earns πold = Trold − C(Er∗ ) and other the new menu she earns πnew = Trnew (E r ) + Tpnew (E r ) − C(E r ) where E r is the optimal public good chosen and may not be equal to Er∗ . Trnew (E r ), Tpnew (E r ) are the fees corresponding to this public good level. When α < 13 , from the proof in the paper we know that Trnew (Er∗ ) + Tpnew (dE) − C(Er∗ ) > Trold (Er∗ ) − C(Er∗ ) But since E r is the new optimal effort chosen by the monopolist, by revealed preference it must be that π new ≥ Trnew (Er∗ ) + Tpnew (dE) − C(Er∗ ) and therefore πnew > Trold (Er∗ ) − C(Er∗ ) completing the proof. For α > 13 , towards a contradiction, assume that πnew
=
sin ce Trold (Er∗ ) − C(Er∗ )
≥
but, Trold (E r ) − Trnew (E r )
Trnew (E r ) + Tpnew (dE) − C(E r ) > π old = Trold (Er∗ ) − C(Er∗ )
Trold (E r ) − C(E r )) by revealed preference
⇒ Trnew (E r ) + Tpnew (dE) − C(E r ) ≥ Trold (E r ) − C(E r ) ⇒ Tpnew (dE) ≥ Trold (E r ) − Trnew (E r )
= Trold (Er∗ ) − Trnew (Er∗ ) by the form of the utility function ⇒ Tpnew (dE) ≥ Tpold (Er∗ ) − Trnew (Er∗ )
(9)
establishing a contradiction with our previous proof. Thus it must be that π new < πold establishing that the monopolist (weakly) does better by continuing to offer a separating contract. Claim 2: If a consumer of a given type weakly prefers a contract with higher public good to another, any consumer of a higher type will strictly prefer the former to the latter contract. ∀E2 ≥ E1 &
E1 , T1 = E2 , T2 ,
E1 , T1 0i E2 , T2 ⇒ E1 , T1 ≺j E2 , T2 ∀j > i
Proof. This claims follows from the fact that for any given public good level, a consumer of higher type is always willing to pay more than a consumer of lower type. More specifically, the slopes of the consumer’s indifference curves can always be type-ordered and the ordering remains that same regardless of the public dT |dUi =0 = vi g´(.). This implies that, for any given level of public good level. In the above setup, we get: dE dT dT good received dE |dUj =0 > dE |dUi =0 ∀j > i. Therefore it must be that if a consumer of type i weakly prefers contract 2 with higher E to contract 1, then the indifference curve of any higher type j through contract 1 must lie below contract 2 and hence type j strictly prefers contract 2 to contract 1. Claim 3: Any optimal menu of contracts offered by the monopolist, Ei∗ , Ti∗ i∈Λ , has to be such that the contract chosen by the higher consumer type offers an equal or higher level of public good. i.e. Ej∗ ≥ Ei∗ ∀j > i, where Ei∗ , Ti∗ denotes i s preferred contract choice from the menu of contracts offered. Proof. Let E1∗ , T1∗ be consumer type 1’s optimal contract choice. This implies that E1∗ , T1∗ (1 ∗ Ei , Ti∗ ∀i i.e. all other contracts offered in the menu must lie either on or below Type 1’s indifference 27
U1
E
U2 A ( E1*, T1?*)
T
Figure 5: Higher Types Receive Higher E curve (IC) through E1∗ , T1∗ . Now consider the Type 2 consumer. From above we know that Type 2’s IC through point E1∗ , T1∗ will be as drawn in Figure 2 below i.e. the slope of the IC is lower for type 2 at point E1∗ , T1∗ . Now consider Type 2’s optimal contract choice E2∗ , T2∗ . Since this is his optimal choice it must be that E2∗ , T2∗ (2 E1∗ , T1∗ ∀i i.e. E2∗ , T2∗ must lie on the same or higher IC for Type 2. Together these two conditions imply that E2∗ , T2∗ must lie in the shaded area shown in Figure 3 and hence E2∗ ≥ E1∗ . More generally, the same argument shows that Ej∗ ≥ Ei∗ ∀j > i Claim 4: If Ei∗ , Ti∗ i∈Λ represents the optimal (for the monopolist) implementable menu of contracts, then it must be that for any given consumer type, his downward incentive constraint will always be binding ∗ ∗ ∀i , Ti−1 i.e. Ei∗ , Ti∗ ∼i Ei−1 ∗ ∗ ∗ ∗ Proof. Suppose that the above statement is not true i.e. Ei∗ , Ti∗ i Ei−1 , Ti−1 . Since Ei−1 , Ti−1 ∗ ∗ ∗ ∗ is by definition the optimal contract for type i − 1, it must be that Ei−1 , Ti−1 (i−1 Ek , Tk ∀k and in ∗ ∗ ∗ ∗ particular for k = 1, .., i − 1. Using the above two claims this then implies that Ei−1 , Ti−1 i Ek , Tk ∀k = 1, .., i − 1. Now consider an alternate menu of contracts such that: Ei
= Ei∗
Ti
= {
Ti∗ f or k < i + δ for k ≥ i
Ti∗
where δ > 0. Note that given our assumption that Ei∗ , Ti∗ enough δ s.t.
Ei , Ti
i
i
∗ ∗ Ei−1 , we can always choose a small , Ti−1
Ei−1 , Ti−1 and hence we know that Ei−1 , Ti−1
i
Ek , Tk ∀k = 1, .., i − 1.
Thus under the new menu of contracts offered, consumer Type i continues to prefer contract Ei , Ti i.e. he does not want to move to a contract with a lower E value despite having to pay more under this new menu of contracts. Similarly Type i will not want to move to any contract with a higher E value as that would violate Ei∗ , Ti∗ having been optimal (in comparison to higher types there has been no effective change under the new menu of contracts as the incentive constraints remain the same i.e. −δ is added to both sides of any incentive constraint between type I and a higher type). Thus i continues to choose Ei , Ti . Similarly, we can show that all other types l continue to prefer contract El , Tl . For any type j > i, using the above two claims and that Ei , Ti i Ek , Tk ∀k < i we know that Ei , Ti j Ek , Tk ∀k < i and ∀j > i. Thus type j > i will not want to switch to any contract with a lower E. For contracts with higher E, since the relevant ICs have not changed they will not want to switch to such contracts either. Finally no type k < i will want to switch. For a contracts with higher E, they now have to pay even greater fees and so if they didn’t prefer such contracts under the old menu, they will definitely not prefer them now. For contracts with lower E, the new menu of contracts is exactly the same and so type k < i will optimally choose Ek , Tk . Thus we have shown that under the new menu of contracts each type i chooses contract Ei , Ti and therefore there is no change in the amount of public good provided in comparison will the old menu of contracts. However it is clear that the monopolist’s total fees collected have increased which contradicts our initial assumption 28
Optimal Thresholds and Mass above Threshold 350
1.2
1
250 0.8 200 0.6 150 0.4 100
M as s of Stud e nts ab o ve T h r e s h old
Op tim al T h r e s h old
300
0.2
50 0
0 0.5
25.5 50.5 75.5 100.5 125.5 150.5 175.5 200.5 225.5 250.5 275.5 300.5 325.5
Optimal Thres hold Mass of Students above Threshold
Sigma
Figure 6: Willingness to Pay is Normally Distributed that Ei∗ , Ti∗
i∈Λ
∗ ∗ ∀i. was optimal for the monopolist. Therefore it must be that Ei∗ , Ti∗ ∼i Ei−1 , Ti−1
Example: Willingness to Pay is Normally Distributed Let f (α) be distributed N (µ, σ 2 ) and consider a sequence of distributions with constant µ, indexed by σ. Note that for any σ and σ (> σ) N (µ, σ 2 ) constitutes a mean preserving spread of N (µ, σ 2 ). ∂θ∗ To sign ∂σ , the change in the threshold with a mean preserving spread, note that −θ∗ fr (θ ∗ , r) − Fr (θ∗ , r) = −Fr (θ∗ , r) + {
1 σ2 [2Π
1
e− 2 (
θ ∗ −µ 2 ) σ
}[1 − (
θ∗ − µ 2 ) ] σ
From the single crossing property, we know that if θ∗ < µ, Fr (θ∗ , r) > 0 and if θ∗ > µ, Fr (θ∗ , r) < 0. Thus, ∗ for all θ∗ > µ, the sign of this expression depends only on the sign of 1−( θ σ−µ )2 . However, for θ ∗ < µ, the sign 1
θ ∗ −µ 2
∗
1 of the expression depends on the magnitudes of both Fr (θ∗ , r) as well as { σ2 [2Π e− 2 ( σ ) }[1 − ( θ σ−µ )2 ]. In θ∗ −µ 2 particular, for low values of σ, we might expect 1 − ( σ ) < 0, but as σ increases, and the threshold decreases, at some stage, we might expect this heterogeneity to reverse and for the threshold to start increasing. The graph below, which shows the optimal threshold for increasing values of σ (with a constant mean of µ = 100) exhibit exactly this property. Finally, note that the profits of the monopolist can be simply derived from the envelope theorem in addition to the single crossing property of the sequence of mean preserving spreads considered here. In ∞ particular, let the profits of the monopolist, V (r) = θ∗ θ∗ f (α, r)dα. Then, from the envelope theorem, ∗ ∗ ∂V ∂r = Fr (θ , r). Thus, for all θ < µ, profits decrease with an increase in heterogeneity, while the reverse is ∗ µ 2µ true when θ > µ. Moreover, from the first order conditions, θ∗ = µ when 1 − F (µ) − [2πσ 2 = 0, or σ = [2π , ∗ ∗ θ > µ for all σ greater than this value and θ < µ for all σ below this value. Hence, as σ increases, profits first decline and then increase, exhibiting a U-shape in heterogeneity.
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