Mechanism Design with Financially Constrained Agents and Costly Verification

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1 Mechanism Design with Financially Constrained Agents and Costly Verification Yunan Li City Uniersity of Hong Kong Noember, 07 Abstract A principal wishes to distribute an indiisible good to a population of budget-constrained agents. Both aluation and budget are an agent s priate information. The principal can inspect an agent s budget through a costly erification process and punish an agent who makes a false statement. I characterize the direct surplus-maximizing mechanism. This direct mechanism can be implemented by a two-stage mechanism in which agents only report their budgets. Specifically, all agents report their budgets in the first stage. The principal then proides budgetdependent cash subsidies to agents and assigns the goods randomly (with uniform probability) at budget-dependent prices. In the second stage, a resale market opens, but is regulated with budget-dependent sales taxes. Agents who report low budgets receie more subsidies in their initial purchases (the first stage), face higher taxes in the resale market (the second stage) and are inspected randomly. This implementation exhibits some of the features of some welfare programs, such as Singapore s housing and deelopment board. Keywords: Mechanism Design, Budget Constraints, Efficiency, Costly Verification JEL Classification: D45, D6, D8, H4 This paper is based on a chapter of my dissertation at Uniersity of Pennsylania. I would like to express my gratitude to Rakesh Vohra for his encouraging support from the ery beginning of the project. I am also grateful to Arjada Bardhi, Aislinn Bohren, Hanming Fang, Qing Gong, Daniel Hauser, Ilwoo Hwang, Nick Janetos, George Mailath, Steen Matthews, Mariann Ollar, Mallesh Pai, Andrew Postlewaite, Debraj Ray, Nishant Rai, Michael Richter, Phillip Strack and the participants in seminars at Uniersity of Pennsylania, City Uniersity of Hong Kong, Tsinghua Uniersity, Hong Kong Baptist Uniersity, The Uniersity of Hong Kong, 8th ACM Conference on Economics and Computation, 06 NASM of the Econometric Society, the 7th International Conference on Game Theory and EconCon 06 for aluable discussions. All remaining errors are my sole responsibility. yunanli@cityu.edu.hk.

2 Introduction Goernments around the world allocate a ariety of aluable resources to agents who are financially constrained. In Singapore, for example, 80% of the population s housing needs are met by the Housing and Deelopment Board (HDB), a goernment agency founded in 960 to proide affordable housing. In the United States, Medicaid has proided health care to indiiduals and families with low income and limited resources since 965. Medicaid currently accounts for 6.% of the state general funds and proides health coerage to 80 million low-income people. 3 Similar public housing and social health care programs preail in many other countries. 45 In China, seeral cities limit the supply of ehicle licenses to curb the growth in priate ehicles, and different cities hae implemented different mechanisms. For example, Shanghai allocates ehicle licenses through an auction-like mechanism, while Beijing uses a ehicle license lottery (see Rong et al. 05). The ealuation of existing mechanisms has attracted attention from researchers and policymakers. In comparison to auction-like mechanisms, lotteries are considered less efficient but benefits low-income families. One justification for this role of goernments is that a competitie market outcome will not maximize social surplus if households are financially constrained. Financial constraints mean that in a competitie market some households with high aluations will not obtain resources, while households with low aluations but access to cash will. The natural question arises as to what the surplus-maximizing (or optimal) mechanism is in these circumstances when both aluations and financial constraints are households priate information. The mechanism design literature concerning this question has focused on mechanisms with only monetary transfers and has ignored the possibility of goernments (or the principal) erifying the priate information supplied by households (or agents). Indeed, in many instances, goernments id=

3 rely on households reports of their abilities to pay when deciding allocations, and goernments can erify this information and punish an household who makes a false statement. For example, applicants for HDB flats in Singapore and Medicaid in the United States are subject to a set of eligibility conditions on age, family nucleus, monthly income, and so on. Verification presumably allows goernments to more accurately target the indiiduals in need for help. Howeer, erification process can be costly. First, in some deeloping countries, erifiable records on household income or wealth are rarely aailable, and goernments lack the administratie capacity to process this information. In such cases, alternatie erification methods such as a isit to a household to inspect the isible liing conditions are not uncommon but are often costly (see Coady et al. 004). Second, certain types of income such as tips, side-jobs and cash receipts are costly to erify. Similarly, goernments hae few ways to erify the income reports by indiiduals who are self-employed or run small business without performing a costly inestigation. Third, households may be financially constrained due to limited access to financial markets or high expenditures, such as medical expenses or education costs. This information is often costly for goernments to erify. Last but not least, een if the erification cost for one indiidual is low, the total cost can be substantial for a large population. Hence, it is important to explore how the option of costly erification affects the optimal mechanism. On the one hand, erification allows the principal to better target financially constrained agents and potentially improe their welfare. On the other hand, erification is costly and reduces the amount of money aailable for subsidies. The principal must now trade allocatie efficiency for erification cost. The cost of erification also influences whether the principal chooses to use cash subsidies or in-kind subsidies (the proision of goods at discounted prices). The latter is less efficient because it often inoles rationing, but saes erification cost because it only benefits lowbudget agents with high aluations. To study these questions, I consider a mechanism design problem in which there is a unit mass of a continuum of agents and a limited supply of indiisible goods. Each agent has two-dimensional priate information his aluation of the good [, ] and his exogenous budget constraint b. 3

4 The budget constraint is a hard one in the sense that agents cannot be compelled to pay more than their budgets. For simplicity, I assume there are only two possible types of budgets, b > b. The principal can inspect an agent at a cost, perfectly reealing his budget, and impose a penalty on detected misreporting. The principal is also subject to a budget balance constraint which requires that the reenue from selling the goods must exceed the total inspection cost. This constraint rules out the possibility that the principal can simply inject money and reliee all budget constraints. I first focus on direct mechanisms in which each agent reports his priate information directly and is punished if and only if he is found to hae lied about his budget. Gien their reports, a direct mechanism specifies for each agent his probability of receiing one unit good, his payment and his probability of being inspected. I characterize the optimal direct mechanism which maximizes utilitarian efficiency among all mechanisms that are incentie compatible and indiidually rational, and that satisfy the resource constraint, agents budget constraints and the principal s budget balance constraint. The optimal direct mechanism can be described as follows. Let u(, b) denote the utility of an agent with the lowest aluation and budget b, which is also the amount of cash subsidies receied by agents with budget b. There exist three cutoffs. Firstly, low-budget agents whose aluations are below and high-budget agents whose aluations are below receie only cash subsidies. Not surprisingly, these low-budget agents receie higher cash subsidies (u(, b ) u(, b )) and are inspected with a probability proportional to the difference in cash subsidies between the two budget types, u(, b ) u(, b ). Secondly, low-budget agents whose aluations exceed receie the good with probability a and make a payment of a u(, b ). High-budget agents whose aluations lie in [, ] are pooled with low-budget agents whose aluations are aboe. They also receie the good with probability a, but they make a payment of a u(, b ). These low-budget agents are inspected with a probability proportional to the difference in payments between the two budget types, u(, b ) u(, b ) + a ( ). Finally, high-budget agents receie the good for sure and make a payment of u(, b ) if their aluations exceed. In the absence of erification, high-budget agents whose aluations are below hae incenties 4

5 to misreport as low-budget types to receie more cash subsidies; and high-budget agents whose aluations are slightly aboe hae incenties to misreport as low-budget types to receie the good at a lower payment. As a result, in this case, agents must make the same payment if they receie the good with the same probability regardless of their budgets: u(, b ) = u(, b ) and =. In a direct mechanism, agents report their budgets as well as aluations. But in practice households are rarely asked to report their aluations. As the second main result of the paper, I show that the optimal direct mechanism can be implemented by a simple two-stage mechanism in which agents are asked to report only their budgets. Specifically, all agents report their budgets in the first stage. The principal then proides budget-dependent cash subsidies to agents and the opportunities to participate in a lottery at budget-dependent prices. the principal then assigns the goods randomly (with uniform probability) among all lottery participants. Agents who report low budgets receie higher cash subsidies and face lower prices. In the second stage, a resale market opens, but is regulated with budget-dependent sales taxes. Agents who report low budgets in the first stage are subject to higher sales taxes. Only agents who report low budgets are inspected randomly. Unlike the case without inspection, in which all agents are subsidized and regulated equally regardless of their budgets, the two-stage mechanism proides more subsidies to low-budget agents in their initial purchases (the first stage) and imposes more restrictions on them in the resale market (the second stage). Although in my analysis the principal s objectie is to maximize social surplus, I conjecture that these features would continue to apply when the principal wants to benefit only low-budget agents. This implementation exhibits some features of the public housing program in Singapore, as shown in Table. In Singapore, buyers of resale HDB flats can apply for additional housing grants. If these flats are purchased with housing grants, these buyers are required to reside in their flats for at least 5 years before they could resell or sublet. In contrast, flats purchased without housing grants are subject to no requirement or a shorter one. It is interesting to see how erification cost, the supply of goods and other parameters affect 5

6 Table : Minimum occupation periods (MOP) of housing and deelopment board (HDB) flats Types of HDB flats MOP Sell Sublet Resale flats w/ Grants 5 7 years 5 7 years Resale flats w/o Grants 0 5 years 3 years Sources. Sell: and Sublet: hdb.go.sg/cs/infoweb/residential/renting-out-a-flat-bedroom/renting-out-your-flat/eligibility. the optimal mechanism and agents welfare. I proide analytic results of comparatie statics for some extreme cases, such as when erification cost is sufficiently large and the supply of goods is sufficiently large or small, and I explore the intermediate cases numerically. Verification allows the principal to better target low-budget agents and improes their welfare. Intuitiely, as erification becomes costly, the principal tends to proide relatiely smaller subsidies to low-budget agents and inspect them less frequently. More interestingly, the optimal mechanism makes use of both cash and in-kind subsidies, and the change in erification cost affects that mechanism s reliance on each of them. If erification is cheap, then the principal achiees efficiency mainly by offering more cash subsidies to low-budget agents. As erification becomes costly, the difference in cash subsidies declines but the difference in in-kind subsidies increases. This is because in-kind subsidies are attractie only to high-aluation agents, which is cheaper in terms of erification cost. Eentually, the difference in in-kind subsidies also declines as erification becomes sufficiently costly. Though reducing erification cost improes the welfare of low-budget agents, it may hurt high-budget agents as more subsidies are dierted to low-budget agents. Another interesting obseration is that although an increase in the supply of goods improes the total welfare, its impact on the welfare of each budget type is not monotonic. This is because an increase in the supply has two opposite effects. On the one hand, the principal becomes less budget constrained, and can direct more subsidies to low-budget agents and inspect them more frequently. On the other hand, low-budget agents also become less budget constrained, which reduces the needs to subsidize and inspect them. As a result, the differences in cash and in-kind subsidies and the inspection probability are hump-shaped. Initially, the welfare of both budget types increases as the 6

7 supply increases. When the supply is large enough that the principal can afford to proide more subsidies to low-budget agents, the welfare of high-budget agents begins to decrease. Eentually, the need to subsidize low-budget agents decreases as the supply increases. As a result, the welfare of low-budget agents begins to decrease and that of high-budget agents begins to increase, until they coincide. Introducing costly erification is technically challenging because it is no longer sufficient to consider local incentie compatibility (IC) constraints. Because the IC constraints between distant types can also bind, one cannot anticipate a priori the set of binding IC constraints. More specifically, if each agent has only one-dimensional priate information, i.e., aluation, then it is sufficient to consider adjacent IC constraints; if each agent has two-dimensional priate information but the principal cannot inspect budgets, then it is sufficient to consider two one-dimensional deiations. These, howeer, no longer apply in the case that each agent has two dimensional priate information and the principal can inspect budget at a cost. In this case, in addition to downward adjacent IC constraints of misreporting alues, one must consider deiations in which an agent can misreport both dimensions of his priate information. As a result, the local approach commonly used does not work here. In this paper, I deelop a noel method that can potentially be used in soling other mechanism design problems with multidimensional types. First, I restrict attention to a class of allocation rules that hae enough structures to help me keep track of binding IC constraints, and that are also rich enough to approximate any general allocation rule well. Specifically, I approximate the allocation rule of each budget type using step functions. When restricting attention to step functions, binding IC constraints corresponding to the under-reporting of budgets are between different budget types whose aluations are the jump discontinuity points of their allocation rules. This structure allows me to write the optimal inspection rule as a function of the possible alues and jump discontinuity points of the allocation rule. I then sole a modified problem of the principal in which the allocation rule of low-budget types are restricted to take at most M distinct alues. Finally, b ecause for M sufficiently large step-functions can approximate the optimal allocation rule arbitrarily well, I can 7

8 obtain a characterization of the optimal mechanism in the limit. The rest of the paper is organized as follows. Section. discusses related work. Section presents the model. Section 3 characterizes the direct optimal mechanism when all agents budget constraints are common knowledge. Section 4 characterizes the direct optimal mechanism when an agent s budget is his priate information. Section 5 proides a simple implementation. Section 6 studies the properties of the optimal mechanism. Section 7 considers arious extensions of the model. Section 8 concludes. All the proofs are relegated to the appendix.. Related Literature This paper is related to two branches of literature. First, it contributes to the literature studying mechanism design problems when agents are financially constrained by incorporating costly erification. Prior work analyzes the reenue or efficiency of a gien mechanism or the design of an optimal mechanism when either budgets are common knowledge, or budgets are agents priate information but cannot be erified. See Che and Gale (998, 006, 000), Laffont and Robert (996), Maskin (000), Benoit and Krishna (00), Brusco and Lopomo (008), Malakho and Vohra (008) and Pai and Vohra (04). In this first branch of literature, the two closest papers to the current paper are Che et al. (03) and Richter (05). In Che et al. (03) and Richter (05), like in this paper, there is a unit mass of a continuum of agents and a limited supply of goods. In Richter (05) agents hae linear preferences for an unlimited supply of the goods. He finds that both the reenue-maximizing mechanism and surplus-maximizing mechanism feature a linear price for the good. In addition, the surplusmaximizing mechanism has a uniform cash subsidy. In both Che et al. (03) and this paper, each agent has a unit demand for an indiisible good, and the surplus-maximizing mechanism can be implemented ia a random assignment with a regulated resale and cash subsidy scheme. Howeer, Che et al. (03) does not consider the possibility that the principal can erify an agent s budget at a cost. This feature also distinguishes the current paper from all the other papers on mechanism design with financially constrained agents. Che et al. (03) first compare three different methods 8

9 of assigning the goods when agents hae a continuum of possible aluations and a continuum of possible budgets, and then characterize the optimal mechanism in a simple model, in which each agent has two possible aluations of the good and two possible budgets. In the presence of costly erification, unlike Che et al. (03), in which all agents are subsidized and regulated equally regardless of their budgets in an optimal mechanism, I show that an optimal mechanism proides more subsidies to low-budget agents in their initial purchases and imposes more restrictions on them in the resale market. Second, this paper is related to the costly state erification literature. The first significant contribution to this series is from Townsend (979), who studies a model of a principal and a single agent. In Townsend (979) erification is deterministic. Border and Sobel (987) and Mookherjee and Png (989) generalize it by allowing random inspection. Gale and Hellwig (985) consider the effects of costly erification in the context of credit markets. Recently, Ben-Porath et al. (04) study the allocation problem in the costly state erification framework when there are multiple agents and monetary transfer is not possible. Li (06) extends Ben-Porath et al. (04) to enironments in which the principal s ability to punish an agent is sufficiently limited. These models differ from what I consider here in that in their models each agent has only one-dimensional priate information. This paper is also somewhat related to the literature on costless or ex-post erification. Glazer and Rubinstein (004) can be interpreted as a model of a principal and one agent with limited but costless erification and no monetary transfers. Myloano and Zapechelnyuk (04) study a model of multiple agents with costless erification but sufficiently limited punishments. This paper differs from these earlier studies in that erification is costly and there are monetary transfers. In the literature discussed aboe, one can anticipate a priori the set of binding IC constraints, which is no longer true here. Instead, I deelop new techniques for keeping track of binding IC constraints. 9

10 Model There is a unit mass of a continuum of agents. There is a mass S (0, ) of indiisible goods. 6 Each agent has a priate aluation of the good V = [, ] R +, and a priately known budget b B = {b, b }. I assume that b > and b >. 7 Thus, a high-budget agent is neer budget constrained in an indiidually rational mechanism. The type of an agent is a pair consisting of his aluation and his budget: t = (, b); and the type space is T = V B. I assume and b are independent. Each agent has a high budget with probability π and a low budget with probability π. The aluation is distributed with cumulatie distribution function F and strictly positie density f. The principal can inspect an agent s budget at a cost k 0, and can impose a penalty c > 0. Inspection perfectly reeals an agent s budget. 8 I assume that the penalty c is large enough that an agent neer find it optimal to misreport his budget if he is certain that he will be inspected. For the main body of the paper, I assume that the penalty is not transferable. In Section 7., I study the case in which penalty is transferable and show that all results hold in that case. For later use, let ρ = k c. As it will become clear, ρ measures the effectie inspection cost to the principal. The cost to an agent to hae his report erified is zero. This assumption is reasonable if the goods are aluable to agents and disclosure costs are negligible. In Section 7.3, I discuss what happens if it is also costly for an agent to hae his report erified. The usual ersion of the reelation principle (see, e.g., Myerson 979 and Harris and Townsend 98) does not apply to models with erification. Howeer, it is not hard to extend the argument to this type of enironment. 9 Specifically, I show in Appendix A that it is without loss of generality to restrict attention to direct mechanisms. Furthermore, I assume that the principal can only punish an agent who is inspected and found to hae lied about his budget. This assumption, howeer, is not 6 The model is also applicable to diisible goods when an agent s per-unit alue for the good is constant up to an upper bound. 7 All the results can be easily extended to any b 0. In the paper, I assume b > to make the statement more concise. 8 The paper s results will not change if the principal cannot detect a lie with some probability. 9 See Townsend (988) and Ben-Porath et al. (04) for more discussion and extension of the reelation principle to arious erification models, not including the enironment considered in this paper. 0

11 without loss of generality. Roughly speaking, if we relax this assumption, in an optimal mechanism the principal will sometimes punish an low-budget agent without erifying his budget. In this case, punishment plays a similar role as red tape" in Banerjee (997) and is used to screen agents with different aluations when their aluations exceed their abilities to pay. 0 In this paper I want to first abstract way from this role of punishment. I will relax this assumption in Section 7.4, where the principal is allowed to punish an agent without erifying his budget or to punish an agent who is found to hae reported his budget truthfully. A direct mechanism is a triple (a, p, q), where a T [0, ] is the allocation rule, p T R is the payment rule and q T [0, ] is the inspection rule. Specifically, for each reported type t T, a(t) denotes the probability an agent obtains the good, p(t) denotes the payment an agent must make and q(t) denotes the probability of inspection. In this definition, I implicitly assume that payment rules are deterministic. I discuss random payment rules at the end of this section and show that it is without loss of generality to focus on deterministic payment rules. The utility of an agent who has type t = (, b) and reports t is u( t, t) = a( t) p( t) a( t) q( t)c p( t) if b = b and p( t) b, if b b and p( t) b, if p( t) > b. That is, an agent has a standard quasi-linear utility up to his budget constraint, and cannot pay more than his budget. With transferable utilities, the welfare criterion I use is simply utilitarian efficiency. For why utilitarian efficiency is a reasonable welfare criterion, see Vickrey (945) and Harsanyi (955). Gien quasi-linear preferences, the total alue realized minus total inspection cost is an equialent 0 Note that this argument is alid only if penalty is not transferable. Indeed, if penalty is transferable, this assumption is without loss of generality as demonstrated in Section 7..

12 criterion. The principal s problem is max a,p,q E t [a(t) q(t)k], ( ) subject to u(t) u(t, t) 0, t T, u(t) u( t, t), t T, t { t T p( t) b }, p(t) b, t T, E t [p(t) q(t)k] 0, E t [a(t)] S. (IR) (IC) (BC) (BB) (S) The indiidual rationality (IR) constraint requires that each agent gets a non-negatie expected payoff from participating in the mechanism. The incentie compatibility (IC) constraint requires that it is weakly better for an agent to report his true type than any other type whose transfers he can afford. The budget constraint (BC) states that an agent cannot be ask to make a payment larger than his budget b. To be clear, note that (BC) follows from (IR). This budget constraint is the same as that found in Che and Gale (000) and Pai and Vohra (04), but different from Che et al. (03), who use a per unit price constraint. 3 I discuss the differences of the two frameworks in Section 7.. The principal s budget balance (BB) constraint requires that the reenue raised from selling the goods must exceed the inspection cost. (BB) rules out the possibility that the principal can inject money and reliee all budget constraints. Finally, the limited supply (S) constraint requires that the To see this, consider a feasible mechanism (a, p, q). Note that if (a, p, q) maximizes welfare, then (BB) must hold with equality. Otherwise the principal can improe welfare through lump-sum transfers. Then the principal s objectie function becomes E[u(t)] = E[a(t) p(t)] = E[a(t) q(t)k], where the last equality holds since (BB) holds with equality. There are some subtle issues regarding a continuum of random ariables (see Judd (985)). Howeer, if we interpret the continuum model as an approximation of a large economy, then Al-Najjar (004) makes the limiting argument rigorous. 3 This constraint is called ex-post budget constraint in Che et al. (03).

13 amount of good assigned cannot exceed the supply. We say a mechanism (a, p, q) is feasible if it satisfies constraints (IR), (IC), (BC), (BB) and (S). Throughout the paper, I assume that S < F (b ) since otherwise the first-best can be achieed ia a competitie market. I also impose the following two assumptions throughout the paper. Assumption F f is non-increasing. Assumption f is non-increasing. Assumption is the standard monotone hazard rate condition, which is often adopted in the mechanism design literature. This assumption ensures that allocating more goods to agents with higher aluations rather than to those with lower aluations yeilds higher reenues for the principal. Assumption says that agents are less likely to hae higher aluations than to hae lower aluations. These two assumptions are also imposed in Richter (05) and Pai and Vohra (04). These two assumptions are satisfied by some commonly used distributions such as uniform distributions, exponential distributions and left truncation of a normal distribution. I conclude this section with a discussion of random payment rules.. Random Payment Rules When defining a direct mechanism, I implicitly assume that the payment rule is deterministic. I argue that this is without loss of generality. Consider a random payment rule p T Δ(R). Let supp( p(t)) denote the supremum of payments in the support of p(t). The utility of an agent who has type t and report t is u( t, t) = a( t) E[ p( t)] a( t) q( t)c E[ p( t)] if b = b and supp( p( t)) b, if b b and supp( p( t)) b, if supp( p( t)) > b. 3

14 In other words, an agent suffers an unbounded dis-utility if his budget constraint is iolated with a positie probability. Then IC constraints become u(t) u( t, t), t T, t { t T supp( p( t)) b }. The principal s objectie function and all the other constraints remain intact. By a similar argument to that used in Pai and Vohra (04), for any feasible mechanism (a, p, q), one can construct another feasible mechanism (a, p, q) by setting p(t) = E[ p(t)] ε b with propability with propability b E[ p(t)] b E[ p(t)]+ε, ε b E[ p(t)]+ε, for some ε > 0 sufficiently small. Furthermore, both mechanisms hae the same welfare. Obsere that, under this construction, IC constraints corresponding to oer-reporting of budgets are satisfied for free. Gien these obserations, it is not hard to see that one can sole the principal s problem (allowing for random payment rules) by restricting attention to deterministic payment rules but relaxing IC constraints corresponding to the oer-reporting of budgets. As I will show later, in the optimal mechanism of no low-budget agent has any incentie to oer report his budget. Hence, it is without loss of generality to focus on deterministic payment rules. 3 Common Knowledge Budgets As a benchmark, I first analyze the case in which all agents budgets are common knowledge. This case can be iewed as situations in which the principal can inspect agents budgets for free or the penalty is infinitely large so that agents neer lie about their budgets (i.e., ρ = k c = 0). Since budgets are common knowledge, IC constraints hold as long as no agent has incentie to misreport his alue: a(, b) p(, b) a(, b) p(, b),,, b. (IC-) 4

15 The principal s problem becomes max a,p,q E t [a(t)], ( CB ) subject to (IR), (IC-), (BC), (S) and E t [p(t)] 0, t T. (BB CB ) By the standard argument, (IC-) holds if and only if for all b B, a(, b) is non-decreasing in and p(, b) = a(, b) a(ν, b)dν u(, b) for all. Since a(, b) is non-decreasing in, the payment p(, b) is also non-decreasing in. Hence, (BC) holds if and only if p(, b) b for all b. Let χ denote the characteristic function. The following theorem characterizes the optimal mechanism. Theorem Suppose Assumption holds, and budgets are common knowledge. There exist (0), (0), u (0) and u (0) such that an optimal mechanism of CB is gien by a(, b ) = χ { (0) } a (0), p(, b ) = χ { (0) } (u (0) + b ) u (0), a(, b ) = χ { (0) }, p(, b ) = χ { } (0), where a (0) = [ u (0) + b ] (0), b < (0) (0) < and 0 = u (0) < u (0) (0) b. In notations a (0), i (0) and u i (0) (i =, ), subscript i indicates the corresponding budget b i and argument 0 indicates that this can be iewed as an optimal mechanism when ρ = 0. As expected, when budgets are common knowledge, the two budget groups can be treated separately. Only low-budget agents receie positie cash subsidies aiming to relax their budget constraints: u(, b ) = u (0) > 0 = u (0) = u(, b ). There are two cutoffs: (0) (0). All high-budget agents whose aluations are aboe (0) receie the good with probability one. This allocation can be implemented by posting a price (0) for high-budget agents. All low-budget agents whose aluations are aboe (0) receie the good with positie probability but are possibly rationed. The intuition for rationing is familiar from the literature. Increasing allocations to low 5

16 alue agents reduces the payment of high alue agents by increasing their information rents and therefore relaxes their budget constraints. Clearly, a high-budget agent whose alue is below (0) has a strict incentie to misreport as a low-budget agent to receie higher cash transfers since u(, b ) > 0 = u(, b ). A high-budget agent whose alue is slightly aboe (0) also has strict incenties to misreport as a low-budget agent: ( u(, b ) + b ) (0) b > ( (0)) b (0) max { (0), 0}. The last inequality holds for > (0) sufficiently close to (0). As it will become clear in Section 4., when budgets are agents priate information and cannot be inspected by the principal, to discourage agents from under reporting their budgets, it must be that u(, b ) = u(, b ) and a highbudget agent must receie the good with a probability no less than that of a low-budget agent who has the same aluation. 4 Priately Known Budgets In this section, I analyze the case in which an agent s budget is his priate information. In this case, IC constraints can be separated into two categories: Misreport alue: a(, b) p(, b) a(, b) p(, b),,, b, (IC-) Misreport both: a(, b) p(, b) χ { p(, b) b} ( a(, b) q(, b)c p(, b) ),,, b, b. () As I stated in the preious section, (IC-) holds if and only if for all b B, a(, b) is non-decreasing in and p(, b) = a(, b) a(ν, b)dν u(, b) for all. The difficulty arises from (). In what follows, I first consider a relaxed problem by replacing () with the following constraint: a(, b ) p(, b ) a(, b ) q(, b )c p(, b ),,. (IC-b) 6

17 This relaxation formalizes the intuition that the principal s main concern is to preent high-budget agents from falsely claiming to be low-budget agents. Later, I erify that an optimal mechanism of the relaxed problem automatically satisfies IC constraints corresponding to oer-reporting of budgets. In other words, a solution to the relaxed problem also soles the original problem. To summarize, the principal s relaxed problem is max a,p,a E t [a(t) q(t)k], ( ) subject to (IR), (IC-), (IC-b), (BC), (BB) and (S). 4. No Verification Before soling the general model, I first consider the special case in which the principal does not inspect agents, i.e., q 0. In this case, as will become clear in the discussion below, it is sufficient to consider two one-dimensional deiations, which greatly simplifies the analysis. Although some of the results may be familiar, it highlights the differences in my approach. Denote the principal s problem in this case by NI and the corresponding relaxed problem by. As will become clear NI in Section 6, if the inspection cost, k, is sufficiently high relatie to the punishment, c, it is optimal for the principal not to use inspection. In particular, this is the case when the principal s effectie inspection cost is infinity (i.e., ρ = k c = ). Obsere first that in this case (IC-b) holds if and only if (IC-) holds and a high-budget agent does not hae any incenties to misreport only his budget: a(, b ) p(, b ) a(, b ) p(, b ),. () 7

18 To see this, note that if both (IC-) and () hold, then a(, b ) p(, b ) a(, b ) p(, b ) a(, b ) p(, b ). Thus, it is sufficient to consider the two one-dimensional deiations: misreport only alue and misreport only budget. The aboe inequality says that if a type (, b ) agent has no incentie to misreport (, b ), then he has no incentie to misreport (, b ). This argument is not true when there is erification because it is possible that types (, b ) and (, b ) are inspected with different probabilities. Instead, one must identify for each high-budget type (, b ) the high-budget type who benefits most from misreporting (, b ) in the absence of inspection, which determines the set of binding (IC-b) constraints. Using the enelope condition, () can be rewritten as u(, b ) + a(ν, b )dν u(, b ) + a(ν, b )dν,. (3) If =, then (3) implies that u(, b ) u(, b ). If u(, b ) > u(, b ), then one can construct another feasible mechanism by reducing cash subsidies to high-budget agents while increasing their probabilities of receiing the goods, which generates the same welfare. Hence, it is without loss of generality to assume that u(, b ) = u(, b ). This result is summarized in Lemma, and a complete proof can be found in the appendix. 4 Lemma Suppose Assumption holds, and the principal does not inspect agents. In an optimal mechanism of NI, it is without loss of generality to assume that u(, b ) = u(, b ). One implication of Lemma is that in an optimal mechanism agents receie positie cash subsidies regardless of their budgets. This result contrasts the case of common knowledge budgets in which only low-budget agents receie positie cash subsidies. 4 It is immediate that u(, b ) = u(, b ) if one also requires that a low-budget agent has no incentie to misreport as a high-budget agent. 8

19 Next, I show that, for any gien, an optimal mechanism on aerage allocates weakly more resources to high-budget agents whose aluations are below than to low-budget agents whose aluations are below. Lemma Suppose Assumptions and hold, and the principal does not inspect agents. In an optimal mechanism of, the allocation rule satisfies NI a(ν, b )f(ν)dν a(ν, b )f(ν)dν,. (4) Gien Lemma, (4) follows immediately from (3) if is uniformly distributed. Lemma shows that the result holds more generally for any distribution with non-increasing density. Using Lemmas and, one can proe the following theorem, which characterizes the optimal direct mechanism. Theorem Suppose Assumptions and hold, and the principal does not inspect agents. There exist ( ), ( ), ( ), u ( ) and u ( ) such that an optimal mechanism of NI satisfies a(, b ) = χ { ( )} a ( ), p(, b ) = χ { ( ) } (u ( ) + b ) u ( ), a(, b ) = χ { ( ) } a ( ) + χ { ( ) } ( a ( )), p(, b ) = χ { ( ) } (u ( ) + b ) + χ { ( ) } ( a ( )) ( ) u ( ), where a ( ) = u ( ) + b ( ), b < ( ) = ( ) ( ) and 0 < u ( ) = u ( ) ( ) b. In notations a ( ), ( ), ( ) and i u ( ) (i =, ), subscript i indicates the corresponding i budget b i and argument indicates that this can be iewed as an optimal mechanism when ρ =. Not surprisingly the optimal allocation rule obtained here shares similar features with the one found in Pai and Vohra (04). There are three cutoffs: ( ) = ( ) < ( ). All high-budget agents whose aluations are aboe ( ) receie the good with probability one. All low-budget 9

20 agents whose aluations are aboe ( ) receie the good with positie probability but may be rationed. In addition, high-budget agents whose aluations are in [ ( ), ( )] are pooled with low-budget agents whose aluations are at least ( )(= ( )). To understand this pooling result, consider two agents with the same aluation, but different budgets b > b. Then (IC-b) implies that as long as agent (, b ) s payment is less than b, he must receie the good with the same probability as (, b ) does. The proof of Theorem follows a weight-shifting argument similar to that of Lemma in Richter (05). Consider a feasible mechanism (a, p, 0) in which a high-budget agent s allocation rule is indicated by the dotted blue cure and a low-budget agent s allocation rule is indicated by the dash-dotted red cure in Figure. One can construct another feasible mechanism (a, p, 0), in which a high-budget agent s allocation rule is indicated by the solid blue line and a low-budget agent s allocation rule is indicated by the dash-two-dotted red line, in the following way. Find a and shift the allocation mass of low-budget agents from the region to the left of to the region to the right of. The choice of is uniquely determined so that the supply to low-budget agents remains unchanged. Let denote the minimum aluation of high-budget agents who receie the good with a probability of at least a(, b ) ( = a (, b ) ). Find and such that. Shift the allocation mass of high-budget agents from the region to the left of to [, ] and from [, ] to the region to the right of. The choice of (and, respectiely) is uniquely determined so that the supply to high-budget agents whose aluations are in [, ] (and [, ], respectiely) remains unchanged. Finally, define the new payment rule using the enelope condition. If f is regular, i.e., satisfies Assumptions and, then the new mechanism improes welfare while remaining affordable. Lemma guarantees that. Thus, no high-budget agent has incentie to misreport his budget. It is easy to see that one can further improe welfare by increasing and reducing. Hence, in an optimal mechanism ( ) = ( ). 0

21 a a (, b ) a(, b ) a(, b ) a (, b ) Figure : Proof sketch of Theorem 4. The General Case I now turn to the general problem of the principal. Using the enelope condition, (IC-b) becomes the following: For all and, u(, b ) + a(ν, b )dν u(, b ) + a(, b )( ) q(, b )c + a(ν, b )dν. (IC-b) First, for each, I identify the type of high-budget agents whose gains from falsely claiming to be a type (, b ) agent are the largest. (IC-b) holds if and only if for each V, q(, b )c sup Δ(, ), where Δ(, ) u(, b ) u(, b ) a(ν, b )dν + a(, b )( ) + a(ν, b )dν. Since Δ(, ) = a(, b )+a(, b ) is non-increasing in, Δ(, ) is concae in and achiees its maximum at = d ( ), where d ( ) inf { a(, b ) a(, b ) }. (5)

22 a a(, b ) a(, b ) d ( ) Figure : The set of binding (IC-b) constraints Suppose the allocation rules for both budget types are continuous in alue. Then the high-budget agents who benefit most from falsely claiming to be (, b ) are those who get the goods with the same probability as type (, b ) agents do. This point is illustrated by Figure, which plots an allocation rule for high-budget agents, a(, b ), and an allocation rule for low-budget agents, a(, b ), as a function of their aluations. Since the principal s objectie function is strictly decreasing in q, the optimal inspection rule satisfies q(, b ) = c max { 0, Δ( d ( )) }. (6) Note that d ( ) is defined using the allocation rule. As a result, one cannot anticipate, a priori, which (IC-b) constraint binds. Furthermore, (IC-b) constraints are frequently binding not only among local types. These difficulties are inherent in all multidimensional problems, and as a result the existing approaches in the mechanism literature do not apply to this problem. 5 In order to keep track of the binding (IC-b) constraints, we sole the principal s problem by approximating the allocation rule using step functions. Fix M. Let = 0 < < < M = and 0 = a 0 a < a < < a M a M+ =. Suppose the allocation rule for type b agents takes M distinct alues: a(, b ) = a m if ( m, m ) for m =,, M. The next lemma 5 See Rochet and Stole (003) for a surey on multidimensional mechanism design problem.

23 a a(, b ) a(, b ) Figure 3: Proof Sketch of Lemma 3 shows that the optimal allocation rule for type b agents can take at most M + distinct alues: a 0, a,, a M+. Lemma 3 Suppose Assumptions and hold. Suppose a(, b ) = a m if ( m, m ) for m =,, M. Then there exists 0 M such that an optimal allocation rule for b satisfies a(, b ) = a m if ( m, m ) for m =,, M, a(, b ) = 0 if < 0 and a(, b ) = if > M. The proof of Lemma 3 is similar to that of Theorem and illustrated by Figure 3, where the allocation rule for low-budget agents (the solid red line) takes three distinctie alues: a < a < a 3. Consider a feasible allocation rule for high-budget agents indicated by the dotted blue cure. Suppose there exist a payment rule and an inspection to be used in conjunction with the allocation rule so that the resulting mechanism is feasible. For ease of exposition, suppose a(, b ) is continuous and let m be such that a( m, b ) = am for m =,, 3. For each m =,, 3, find m and moe the allocation mass of high-budget agents from [ m, m ] to [m, m+ ], where 4 =. The choice of m is uniquely determined so that the supply to high-budget agents whose alue is in [ m, m+ ] remains unchanged. Redefine the payment rule using the enelope condition and let the inspection rule remain the same. One can erify that the new mechanism is feasible and clearly improes welfare. We say an allocation rule a is an M-step allocation rule if there exist = 0 < < < 3

24 M =, 0 M and 0 = a 0 a < a < < a M a M+ = for some M such that a(, b ) = a m if ( m, m ) for m =,, M and a(, b ) = am if ( m, m ) for m = 0,,, M +. Lemma 3 shows that it is without loss of generality to focus on M-step-allocation rules among all step allocation rules. Consider a mechanism using a M-step allocation rule. It is easy to see that for ( m, m ), the type b agents who benefit most from falsely claiming to be type (, b ) hae aluations d () = m. Hence, we can keep track of the binding (IC-b) constraints by keeping track of the jump points of the allocation rule. In this case, the optimal inspection rule satisfies q(, b ) = q m for all ( m, m ) and q m = c max {0, u(, b ) u(, b ) + } m (a j a j )( j j ) j= (7) for m =,, M. Consider the principal s problem ( ) with two modifications: max E t [a(t) q(t)k], ( a,p,q (M, d)) subject to (IR), (IC-), (IC-b), (BC), (S), a is a M -step allocation rule for some M M, E[p(t) q(t)k] d. (BB-d) The second modification is to relax the goernment s budget balance constraint by d 0. As it will become clear later, any feasible mechanism of can be approximated arbitrarily well by a feasible mechanism of (M, d) for M sufficiently large and d sufficiently small. Next, I show that in an optimal mechanism of (M, d), in the absence of erification, either no high-budget agent has incenties to misreport as low budget, or all high-budget agents weakly prefer to misreport as low budget. 4

25 Lemma 4 Suppose Assumptions and hold. An optimal mechanism of (M, d) satisfies one of the following two conditions: (C) For all m =,, M, u(, b ) u(, b ) + m (a j a j )( j j ) 0. (8) j= (C) For all m =,, M, u(, b ) u(, b ) + m (a j a j )( j j ) 0. (9) j= The basic intuition underlying Lemma 4 is as follows: As long as a mechanism satisfies neither (C) nor (C), one can strictly improe welfare by adjusting the allocation rule in regions in which high-budget agents find it strictly optimal to report their budgets truthfully. I proide only a proof sketch of Lemma 4 here. The full proof can be found in the appendix. Proof Sketch. The proof is by contradiction. Let (a, p, q) be a feasible mechanism, where a is a Mstep allocation rule. Suppose (a, p, q) satisfies neither (C) nor (C). I show that one can construct another feasible mechanism (a, p, q ), which strictly improes welfare and satisfies one of the two conditions. Furthermore, a is a M -step function for some M M. I break the proof into two steps. Step. I show that it is without loss of generality to assume that (8) holds for m =. Suppose, on the contrary, that u(, b ) u(, b ) + a 0 < 0. Then there exists m > such that m m 0 for all m < m and m m > 0. One can construct another feasible mechanism by redirecting cash subsidies from high-budget agents to low-budget agents, and shifting the allocation mass from low-budget agents in [ m, m ] to high-budget agents in [ m, m ] for some m m m m. Step. Suppose u(, b ) u(, b )+a 0 0. There exists m > such that (8) holds for all m < m 5

26 and u(, b ) u(, b ) + m (a j a j )( j j ) < 0. j= It must be the case that m < m. For ease of exposition, assume that m > m. 6 One can construct another feasible mechanism by either shifting the allocation mass from high-budget agents in [ m, ] to high-budget agents in [, m ] for some m < < m, or shifting the allocation mass from high-budget agents in [ m, m ] to low-budget agents in [ m, m ] for some m m m m. If (C) holds, then the optimal inspection rule is q 0. The optimal mechanism of in this case, which is characterized in Section 4., is a feasible mechanism of (M, d) and satisfies (C) with equality. Thus, I can conclude that an optimal mechanism of (M, d) satisfies (C). Corollary Suppose Assumptions and hold. An optimal mechanism of (M, d) satisfies (C). Hence, an optimal inspection rule satisfies q(, b ) = q m for all ( m, m ), where q m = c [ u(, b ) u(, b ) + ] m (a j a j )( j j ) j= (0) for m =,, M. Now the principal s problem (M, d) can be written as follows, where the Greek letters in parentheses denote the corresponding Lagrangian multipliers. max u(,b ),u(,b ), {a m } M m=,{m }M m=,{m }M m=0 ( π) k c M m m= m [ π M+ m m= m u(, b ) u(, b ) + a m f()d + ( π) M m m= m a m f()d ] m (a j a j )( j j ) f()d, j= m 6 In the appendix, I break the proof in three steps. I consider the case in which m < M = M in Step 3. in Step and the case 6

27 subject to π M+ m= a m [F ( m ) F (m )] + ( π) M m= a m [F ( m ) F (m )] S, (β) M a M M a j ( j j ) u(, b ) b, (η) j= ( π)u(, b ) + ( π) ( π) k c M πu(, b ) + π m m= m M+ [ m= m M m m= m a m [ u(, b ) u(, b ) + m a m [ F () f() ] F () f()d f() ] m (a j a j )( j j ) f()d j= ] f()d d, u(, b ) 0, u(, b ) 0, (ξ, ξ ) m u(, b ) u(, b ) + (a j a j )( j j ) 0, m =,, M, (μ m ) j= 0 = a 0 a a a M a M+ =, (α,, α M+ ) = 0 M =, (γ,, γm ) 0 M. (γ 0,, γm+ ) (λ) To sole this problem, I first show that in an optimal mechanism of (M, d), the inspection probability is non-decreasing in a low-budget agent s reported alue: Lemma 5 Suppose Assumptions and hold. In an optimal mechanism of (M, d), 0. Suppose in addition that V (M, d) > V (M, d) for M 3, then M M > > 0. As a result, the inspection probability in an optimal mechanism of (M, d) is non-decreasing in reported alue, i.e., q M q 0. To understand the intuition behind the monotonicity of inspection probability, consider a low- 7

28 budget agent and a high-budget agent both receiing the good with probability a m. Let p m and p m denote their payments respectiely. The difference in their payments, to which the inspection probability is proportional, is m p m pm = u(, b ) u(, b ) + (a j a j )( j j ). j= Clearly, this difference is non-decreasing in m since m m 0. Suppose, on the contrary, that q m > q m. Then the principal can shift allocation from low-budget agents in [ m, m ] to lowbudget agents in [ m, m ], which clearly improes allocation efficiency and reenue. This shift also strictly reduces inspection cost because more low-budget agents are inspected with probability q m rather than q m and q m > q m. The inequality constraints corresponding to μ m s in (M, d) are non-negatiity constraints on inspection probabilities. As shown in Lemma 5, in an optimal mechanism of (M, d), the inspection probability is non-decreasing in a low-budget agent s reported alue. As a result, it is sufficient to consider the inequality constraint corresponding to μ : u(, b ) u(, b ) + a ( 0 0 ) 0. Note that for fixed jump discontinuity points m s, the principal s problem (M, d) is linear in i u(, b ), u(, b ) and a m s. Hence, an optimal solution can be obtained at an extreme point of the feasible region. The monotonicity of inspection probability implies that in addition to the monotonicity constraints on a m s there are only finitely many other constraints binding. As a result, for an M sufficiently large, there are finitely many distinct a m s in an optimal mechanism. More formally, let V (M, d) denote the alue of (M, d). Then V (M, d) = V (M, d) for M sufficiently large. This result still holds if I replace (BC) with a per-unit price constraint, as shown in Section 7.. If I impose only (BC), then I can further proe that in an optimal mechanism of (M, d) the allocation rule is a -step allocation rule, i.e., V (M, d) = V (M, d) for M 3. 8