Optimally Constraining a Bidder Using a Simple Budget

Transcription

1 Optimally Constraining a Bidder Using a Simple Budget Justin Burkett December 18, 213 Abstract Motivated by the observation that bidders in actual auctions are budget constrained, I study a principal s optimal choice of constraint for an agent participating in an auction (or auction-like allocation mechanism). I give necessary and sufficient conditions on the principal s beliefs about the value of the item for a simple budget constraint to be the optimal contract. The results link the observed use of budget constraints to their use in models incorporating budget-constrained bidders. Other implications of the model are that a general revenue equivalence result applies and that the optimal auction with budgetconstrained bidders has a standard solution analogous to the one for classic models. 1 Introduction A firm (principal) employs a manager (agent) to bid on an asset available to the firm via auction, because the manager is more capable of determining the asset s value than the board of directors. If the firm wants to constrain the bidding behavior, perhaps because they expect the manager to exaggerate the value of the asset, 1 how should it do so? Evidence from actual auctions suggests that a common method used to constrain bidders is a simple budget constraint or a limit on the highest bid that may be placed (Cramton, 1995; Bulow et al., 29), and models incorporating Department of Economics, Wake Forest University, Address: 1834 Wake Forest Rd., Winston- Salem, NC 2716, USA, Telephone: , burketje@wfu.edu. The author is grateful to Thayer Morrill for advice and suggestions. 1 Controlling the asset may privately benefit the manager beyond its contribution to firm profits (Jensen, 1986). 1

2 financially constrained bidders have generally assumed that simple budget constraints are used, whether or not an explicit principal-agent relationship is modeled. 2 One might argue that the prevalence of budget constraints is the result of practical considerations, such as enforceability and ease of implementation. In this paper, I show that budget constraints are also optimal in a general environment. Specifically, I provide sufficient conditions on the distribution of the principal s beliefs about the value of the asset to the firm that guarantee that a budget constraint is optimal for the principal under two different regimes, which specify the feasible set of contracts. When the conditions fail, I show how to construct profitable deviations, proving necessity. Importantly, the conditions do not depend on the details of the mechanism used to allocate the good (the auction is modeled as a direct revelation mechanism), and hence the same conditions apply across a variety of auctions and auction-like situations in which a principal constrains an agent charged with obtaining a good that benefits the two of them. Furthermore, the conditions hold for any distribution of beliefs when the bias in the agent s preferences is not too large. The regimes specify the set of alternative contracts that the principal may choose, and both require that the contract be incentive compatible and individually rational for the agent. A contract specifies the reports the agent makes to the mechanism and side payments for each of the agent s signals. In the first regime, no conditional transfers between the principal and agent are allowed but the principal may conditionally require the agent to take a costly action that hurts both parties (i.e., only negative side payments are allowed). Although somewhat restrictive, this first regime includes any contract that does not specify side payments, which have been studied in the literature on optimal delegation (e.g., Alonso and Matouschek, 28). In the second regime, the principal is allowed to use conditional transfers subject to a limited-liability condition. These regimes allow for a variety of contractual arrangements, but budget constraints, which do not utilize either type of conditional payment, are shown to be optimal given simple conditions are satisfied. In fact, the budget chosen is the same in both regimes. Budget constraints would be classified as constrained delegation in 2 Early contributions to this literature were made by Che and Gale (1996, 1998). Che and Gale (1998) do consider an additional case where bidders may face an increasing cost of financing their bid. However, the literature since has focused on the simple budget constraint case (see, for example, Che and Gale (2), Pai and Vohra (213), and the citations in Burkett (213)). 2

3 the optimal delegation literature, so another interpretation of this result is that it provides conditions under which delegation is optimal despite the feasibility of a wide range of contracts that allow for conditional transfers. These results provide a link between the observed use of budget constraints in auctions and their use in models that assume optimizing behavior. The fact that the optimality of the budget constraint does not depend on specifics of the mechanism used to allocate the good also implies several important and useful results. One consequence is a more general revenue equivalence result between auctions with budget-constrained bidders than the one given in Burkett (213), from which the basic model of the principal-agent interaction is taken. Another is that the the structure of the model here allows for budget constraints to be easily incorporated into more complex ones. It is straightforward, for example, to use the results in this paper to incorporate budget-constrained bidders into a seller s revenue-maximization problem. In Section 5.2, I show that the revenue-maximizing auction with budget-constrained bidders is nearly identical to the one developed in Myerson (1981), after appropriately redefining some of the key terms. This result stands in contrast to the existing literature on optimal auctions with budget-constrained bidders, which among other results finds that the optimal auction is a modified all-pay auction (Pai and Vohra, 213). The primary distinction between the present model and the most common one in this literature is that I allow for an endogenous choice of budget constraint. The structure of the paper is as follows. Section 2 discusses related literature. Section 3 introduces the model and states the principal s optimization problem. Section 4 reports the main results of the paper. After a brief discussion of incentive compatibility of the mechanism (Section 4.1), the sufficiency results for each regime are presented (Section 4.2), followed by results on the necessity of the sufficient conditions (Section 4.3). Section 5 discusses applications, including to the optimal auction problem, and Section 6 concludes. 2 Related Work The effect of budget constraints on bidder behavior and auction outcomes was initiated by Che and Gale (1996, 1998), who show that when budget constraints are exogenously determined the expected revenue is highest in the all-pay auction out 3

4 of the all-pay, first-price, and second-price formats. Recent work by Pai and Vohra (213) shows that a modified all-pay auction is the seller s revenue maximizing choice. Burkett (213) shows that by allowing for the budget constraint to be determined endogenously in the model, varying according to the auction rules, revenue and efficiency equivalence between the first- and second-price auctions with independently distributed valuations can be restored. In the model, there are many principal-agent pairs, where each principal decides on a budget for their respective agent (bidder) and each bidder places a bid that respects the budget. In this paper, I use the payoff structure between the principals and the agents used in Burkett (213), but allow the principal more freedom in how the agent is constrained. With respect to Burkett (213), this paper serves two purposes. One is to provide a theoretical justification for the use of a standard budget constraint in that paper, and the other is to extend the analysis beyond the first- and second-price auction rules. In addition to the auction literature, this paper is a contribution to the very large literature on agency problems and organizational decision making. The two regimes in this paper can be considered as instances of well-studied problems in this literature. In the delegation problem (introduced by Holmstrom (1977)), a principal delegates decision-making authority, without the use of transfers, to an agent who is better informed about the state of nature but is potentially biased towards picking sub-optimal outcomes. 3 Alonso and Matouschek (28) give conditions under which interval delegation is optimal in a setting with a biased agent with symmetric and single-peaked preferences (see this paper for citations to the delegation literature). Under interval delegation, the agent is allowed to choose her most preferred choice in an intermediate range of states, but is restricted otherwise. 4 Interval delegation is analogous to a budget constraint with the additional constraint that low types are required to place higher bids than they would like. Amador and Bagwell (213) analyzes a model with fewer restrictions on the pay- 3 Several types of delegation problems have been studied that differ conceptually from the one considered here, including for example, Armstrong and Vickers (21) who study a delegation problem where the agent selects from a set of two-dimensional projects, but the principal does not know which projects are available. 4 For a precise definition, consider an agent who chooses an action from [, 1] based on a private observed signal, s [, 1]. An example of interval delegation is a contract where the agent must select a l > when < s < a l, can choose his most preferred action when a l < s < a h, and must select a h when a h < s < 1. 4

5 offs, providing conditions for the optimality of interval delegation. 5 They also consider an extension of the delegation problem where the agent may be required to take a costly action that is contingent on the state of nature and entails a loss of surplus by both parties. They refer to this case as money burning, and provide several interpretations and motivations for considering it. 6 Their description of money burning corresponds to the first regime in the present paper, and it is not surprising that the results are similar given the similarity in the models. However, I cannot simply apply their results in the first regime, primarily because their results depend on assumptions of concavity that I do not make. 7 The second regime I consider allows for the use of transfers between the principal and agent. The specific problem I analyze allows for transfers but imposes a limitedliability condition, requiring that side payments to the agent be non-negative. This corresponds to an extension of the Crawford and Sobel (1982) cheap talk model analyzed in Krishna and Morgan (28). Krishna and Morgan (28) consider a scenario where the principal can fully commit to a contract specifying the choice of project (the project in their model corresponds to the value of the good in mine) and transfer, conditional on the agent s report, and one where the principal can only commit to the choice of transfer. I only consider an environment where the principal can fully commit. Krishna and Morgan (28) describe some features of the optimal contract for the principal in this case, including that the principal never chooses to implement his most preferred contract (it is feasible to do so) and that the principal always sets the transfer to zero in the highest states. A budget constraint shares these features, because it never implements the principal s most preferred action with positive probability and never specifies a transfer. Krishna and Morgan (28) also provide a complete solution in the uniformquadratic version of their model. 8 Although it shares the features mentioned, the 5 In Proposition 3 of that paper, they consider a special case of the model where the optimal contract is actually a budget constraint as I use the term here. 6 For example, they suggest that the principal may require the agent to undertake wasteful administrative tasks if the agent wants to take an exceptional action. See Amador and Bagwell (213) for more examples. 7 Essentially, applying their results would require making concavity assumptions about the mechanism that the agent reports to. As I show, these assumptions are unnecessary for the purposes of this paper, and ultimately this means that the method of proof differs (see Section 4 for a brief discussion of these differences). 8 The uniform-quadratic model specifies that the principal s and agent s preferences are each 5

6 solution they find in this case cannot be translated into a budget constraint, because it does specify non-zero transfers in some states. One important distinction between their model and the one I analyze is that the bias between principal and agent can remain relatively large in their case when the realized state is small, whereas in the model I analyze the bias is proportional to the value of the asset. It seems that this is the important difference driving the divergent results. Given the interpretation of the state as the value of an asset, the assumption that the bias is smaller when the value is smaller seems natural here. 3 Model Burkett (213) describes a model where principal-agent pairs compete in a sealed-bid auction for a single good. Each agent observes the value of the good to the pair and submits a bid for the good that is constrained by the budget the agent receives from the principal. The principal only has a noisy signal of the value and constrains the behavior of the agent because the agent derives more value from the good than the principal does. The design of the model employed in this paper is motivated by this situation, but focuses on a single principal-agent pair, and abstracts away from the details of an auction. The basic idea is that the principal-agent pair will jointly decide on a report of a type to an incentive-compatible direct-revelation mechanism (the mechanism), representing a game like an auction. The report determines a probability of receiving a good and an expected transfer. Models where multiple agents agree on how to manipulate their report to a central direct revelation mechanism have been used to study mechanism design in the presence of collusive agents (Laffont and Martimort, 1997). To decide on the report to the mechanism, the principal designs a second directrevelation mechanism (the contract), to which the agent makes a report after learning the value of the good to the pair (the agent has the option of rejecting the contract and not participating in the mechanism). I consider two regimes governing the set of feasible contracts. I retain the assumptions that the principal and agent are risk neutral, the agent has better information than the principal, and that the agent values quadratic functions of the state and a bias term, and that the principal s beliefs about the state are uniform. This is a common specification in this literature. 6

7 the good more than the principal. 3.1 Mechanism and Contract The mechanism is proposed by a third party before any actions are taken and is composed of two functions, P : [, 1] [, 1] and T : [, 1] R, specifying the probability that the good is awarded as a function of the report t and the expected transfer made to the third party. The mechanism is assumed to be incentive compatible, which via standard arguments implies that P (t) is nondecreasing and that T (t) can be expressed as a function of P (t) and T (). I also assume that the mechanism is individually rational, which requires that the payment made by the lowest report be non-positive, T (). Incentive compatibility will be discussed further below. After the mechanism is announced, the principal proposes a second mechanism to the agent, which I refer to as the contract. The contract specifies as many as three functions (all functions of the agent s report), θ : [, 1] {O [, 1], and τ i : [, 1] R +, i = p, a. θ(t) is the report ultimately submitted to the mechanism and may specify that the agent and principal not participate (action O), while τ a (t) (τ p (t)) is an additional transfer payment made by the agent (principal). I will assume throughout that τ a (t) + τ p (t), or that the contract is not subsidized by a third party. The principal is assumed to be able to commit to the offered contract, and the agent cannot participate in the mechanism without a contract with the principal. 3.2 Payoffs and Information Ignoring any transfers specified by the contract, the principal and the agent earn a constant fraction of the difference between their perceived value of the good and the transfer made to the third party. For example, they may initially own equity shares in the firm or have some prior contractual arrangement equivalent to equity for this decision (e.g., a bonus tied to the firm s profit at the auction). Let σ a and σ p be the shares for the agent and principal respectively. The value of the good to the principal (and to the firm if the principal is taken to be the owner) is given by δt where < δ < 1. The value to the agent is t or δt + (1 δ)t where the second term represents the difference between the agent s and the principal s values. This difference is the private gain the agent receives from 7

8 winning the good. If the good is assigned to the firm for payment of p, the principal receives σ p (δt p) and the agent receives σ a (t p). Therefore, if the report to the mechanism is t and the agent s true value is t, the respective expected payoffs (conditional on t) are σ p (P (t )δt T (t )) and σ a (P (t )t T (t )). When the contract is proposed by the principal, the principal only knows that t is distributed on [, 1] according to an absolutely continuous distribution, F (t). The density is denoted by f(t) and exists (a.e.). The agent learns the value of t after the contract is proposed but before she is required to make any decisions in the model. To incorporate transfers required by the contract (τ a (t) and τ p (t)), it will be convenient to assume that they are specified so that the principal (agent) receives an amount equal to σ p τ p (t) (σ a τ a (t)) when the agent reports t. Incorporating the contract, the expected payoffs to the principal and the agent when the agent has a type t but reports t are π p = σ p {P (θ(t )) δt T (θ(t )) + τ p (t ) π a = σ a {P (θ(t )) t T (θ(t )) + τ a (t ) Since the shares (σ i ) are assumed constant, I will omit them from the remainder of the paper. 3.3 Timing Putting everything together, the timing of the game is as follows: 1. The mechanism (P, T ) is announced by a third party. 2. The principal proposes a contract (θ, τ p, τ a ) to the agent. 3. The agent observes t and accepts or rejects the contract. If the agent rejects the contract, the game ends and the principal and agent each receive a zero payoff. If the agent accepts, the agent reports a type t {O [, 1], which is translated into the report θ(t ) to the mechanism. 4. The uncertainty is resolved and payoffs are realized. 8

9 3.4 Statement of the Principal s Problem Given the mechanism (P, T ) proposed by the third party, the principal s problem is to design a contract (θ, τ p, τ a ) to maximize his expected payoff, subject to the contract being incentive compatible (IC) and individually rational (IR) for the agent. In addition, I will separately consider two types of constraints on (τ p, τ a ), labeled (R1) and (R2) below. Formally, the set of problems considered can be described as follows: maximize θ,τ p,τ a subject to E t [P (θ(t))δt T (θ(t)) + τ p (t)] t arg max P (θ(t ))t T (θ(t )) + τ a (t ) t [,1] P (θ(t))t T (θ(t)) + τ a (t) (IC) (IR) τ a (t), τ p (t) = τ a (t), t [, 1] (R1) τ a (t), τ p (t) = τ a (t), t [, 1] (R2) The (IR) constraint is imposed after t is revealed to the agent. This assumption can be justified by supposing that the agent knows t when she decides whether or not to accept the contract. It may also be that the agent is able to irrevocably cancel the contract (by quitting the firm perhaps), so that acceptance of the contract is not complete until the agent learns t. Regime 1 (R1) allows for the principal to impose a state-contingent cost on the agent that affects both parties equally, such as a wasteful administrative process. This is the money burning referred to in the Introduction. Note that optimality under (R1) implies optimality under the stricter condition τ a (t) = τ p (t) =. This would be the restriction if I were separately considering the delegation problem (see the discussion in Section 2). Regime 2 (R2) allows for transfers between the principal and agent in a proper sense but imposes a limited-liability constraint on the principal, restricting the principal to providing positive transfers. Relaxing the limited-liability constraint alters the character of the problem for the principal. Under (R2), transfers are costly for the principal and are purely an instrument for manipulating the preferences of the 9

10 agent. Without limited liability, the principal may in addition use the transfers to extra profit from the agent, and the problem takes on some of the characteristics of a revenue maximization problem. 4 Results The results are primarily concerned with the optimality of budget constraints in this environment. By budget constraint I mean a contract between the principal and the agent that sets a cap on the highest type the agent may report to the mechanism, leaves the agent unconstrained otherwise, and involves no conditional transfers. Formally, I define a budget constraint as a contract that takes the following form for a given. θ BC (t) = min ( t, ) τ BC i (t) =, i = a, p When the mechanism is an auction and equilibrium bid functions are increasing (and continuous) functions of the agent s valuation, the budget constraint contract can be implemented as a upper limit on the amount that is bid. It turns out that the optimal choice of for a given F (t) and δ is the same in both the money burning case (Section 4.2.1) and the transfer case (Section 4.2.2) despite important differences between the two. I leave the discussion of this point for later in the paper, and only provide a definition for here. 9 inf { t 1 F (x) (1 δ)xf(x) dx, t > t t In words, is the lowest type for which the integral is negative for all t >. When (, 1), it must be that 1 F (x) (1 δ)xf(x) dx =, by continuity of the integral. Integrating by parts and rearranging this expression yields 9 This definition is an adaptation of the one provided by Amador et al. (26). (1) 1

11 1 F (x) (1 δ)xf(x) dx = (1 F ()) F () δxf(x) dx = δxf(x) dx =, which is the same value for in Burkett (213). The left-hand side is the expected value of δt given that t or the expected value of the principal given that the budget binds. This all assumes that is an interior solution, which turns out to be the only relevant case. 1 To prove that the budget constraint contract is optimal, I employ the following modified version of Luenberger s Sufficiency Theorem (Luenberger, 1969, Theorem 1, p. 22) given by Amador and Bagwell (213). The theorem provides sufficient conditions for the solution of a general constrained optimization problem on vector spaces. Theorem 1 (Amador and Bagwell (213) Theorem 1). Let f be a real-valued functional defined on a subset Ω of a linear space X. Let G be a mapping from Ω into the normed space Z having nonempty positive cone Z +. Suppose that (i) there exists a linear functional S : Z R such that S(z) for all z Z +, (ii) there is an element x Ω such that for all x Ω, f (x ) + S(G(x )) f (x) + S(G(x)), 1 >, because = would imply that δxf(x) dx, a contradiction. Suppose instead that = and δxf(x) dx >, then continuity would imply that (1 F (ε))ε + δxf(x) dx > for ε an arbitrarily small ε >, contradicting that =. < 1 follows from the following observation (1 F (t))t + The last term is less than when δ < t < 1. t δxf(x) dx (1 F (t))t + δ(1 F (t)) = (δ t)(1 F (t)) 11

12 (iii) G(x ) Z +, and (iv) S(G(x )) =. Then x solves: minimize f (x) subject to G(x) Z +, x Ω. The functional L(x) = f (x) + S(G(x)) plays the role of a Lagrangian function with S(x) being constructed from Lagrange multipliers in a manner to be described shortly. As explained in Amador et al. (26) and Amador and Bagwell (213), one advantage to using this theorem is that monotonicity constraints on the choice of function, x, can be embedded in the description of Ω, instead of being described by the functional G(x). The basic strategy used for proving optimality of budget constraints is as follows. First, standard arguments about incentive compatibility of the contract imply that an integral equation and monotonicity constraint hold. The integral equation can be moved into the objective function using the relation between τ a and τ p, forming f in the theorem above. In combination with defining Ω to be the subset of increasing functions on [, 1], this takes care of the (IC) constraint. G(x) can then be used to incorporate the restrictions imposed by (R1) or (R2). This leaves (IR) which can safely be ignored and checked at the end. I then apply the theorem by first constructing the functional S(x) and then proving directly that L(x) is minimized by the budget constraint contract. This is a different approach to the one used in Amador et al. (26) and Amador and Bagwell (213), who after constructing S(x), show that L(x) is concave in x making certain differential conditions sufficient for the minimization of L(x). Despite the mathematical similarity between those papers and Regime 1 (R1), I cannot directly apply their techniques here, because without imposing significant restrictions on the mechanism (P, T ), L(x) will fail to be everywhere concave for this problem. The S(x) functional I use is determined by the selection of a nondecreasing function Λ(t) as follows. S(x) = x(t) dλ(t) The requirement that Λ(t) be nondecreasing comes from the requirement that S(z) for all z Z + (see the proof of Proposition 1). 12

13 4.1 Incentive Compatibility By assumption the mechanism offered to the principal-agent pair, (P, T ), is incentive compatible (i.e., given a value t, a report of t maximizes the payoff from the mechanism). Following standard arguments this requires that P (t) be nondecreasing and that the payoff from a report of t with a value of t is U(t, t) = P (t)t T (t) = t P (x) dx T (). The payoff from a report of θ(t) is therefore U(θ(t), t) = P (θ(t))(t θ(t)) + θ(t) P (x) dx T () It will be useful to observe that ignoring τ a and τ p the principal s payoff from a report of θ(t) is U(θ(t), δt) = U(θ(t), t) (1 δ)p (θ(t))t. The incentive compatibility of the composition of the contract and the mechanism implies (using the same standard envelope theorem argument) that the following holds for the agent s payoff t P (θ(x)) dx = U(θ(t), t) + τ a (t) τ a () U (θ ) (2) where U (θ ) = P (θ())θ() + θ() P (x) dx T (). It also must be the case that P (θ(t)) is a nondecreasing function of t, which given the assumption on P implies that θ(t) is nondecreasing. Together, these two conditions completely characterize the incentive compatibility constraint on the principal. 4.2 Sufficiency Money Burning Case When money burning is allowed, the constraints on the principal are (IC), (IR) and (R1). The budget constraint contract is feasible for the problem, because θ BC (t) is weakly increasing, it involves no transfers and satisfies (IR) given the assumption that (P, T ) is individually rational. In this section, I show that it is also optimal under the following assumption. Assumption 1. F Λ (t) F (t) + (1 δ)f(t)t is nondecreasing for t [, ]. This assumption essentially puts a lower limit on how quickly f(t) can decrease. It obviously holds if f(t) is nondecreasing. As mentioned in Amador et al. (26), 13

14 who introduce the same assumption in their paper, when f(t) is differentiable the assumption is equivalent to a lower bound on the density s elasticity, which is satisfied for many commonly used distributions with decreasing densities. 11 f (t)t f(t) 2 δ 1 δ As the difference in valuations between the principal and agent becomes less severe (i.e., as δ 1), so does the restriction on F (t). In other words, given an arbitrary distribution F (t), there exists a δ sufficiently close to 1 such that Assumption 1 is satisfied. The function F Λ (t) will be the basis for the construction of the multiplier function (Λ(t)), and the requirements put on F Λ (t) are directly derived from requirements put on Λ(t) by Theorem 1 (see the proof of Proposition 1). That the two should be related becomes clear after forming the Lagrangian for this problem. Using (2) to replace τ a (t) in the principal s objective, I get: L(θ) = = + { U(θ(t), δt) + { U(θ(t), t) t t P (θ(x)) dx U(θ(t), t) df (t) P (θ(x)) dx dλ(t) (Λ(1) Λ() 1)(τ a () + U (θ )) P (θ(t)) (1 F Λ (t) Λ(1) + Λ(t)) dt + U(θ(t), t) dλ(t) (Λ(1) Λ() 1)(τ a () + U (θ )) (3) The second equation follows from integration by parts and the definitions of U(θ(t), t) and F Λ (t). Note that I form the negative of the Lagrangian, because Theorem 1 refers to a minimization problem. Proposition 1. Under Assumption 1, a budget constraint contract is optimal when the constraints on the principal are (IC), (IR) and (R1). 11 Amador et al. (26) mention the exponential, log-normal, and the Pareto and Gamma distributions for some parameter values. 14

15 Proof. Consider the multiplier, F Λ (t) Λ(t) = sup t [,t] { 1 t t F Λ (x) dx if t if t > Assumption 1 guarantees that Λ(t) is nondecreasing for t and the definition guarantees that it is nondecreasing for t >. Clearly, Λ() =. It is continuous at, and it must be that Λ(t) 1, 12 and that Λ(1) = Combined these properties make Λ(t) a distribution function and imply that it is differentiable (a.e.). I next show that a budget constraint contract is optimal. Incorporating the multiplier, write the Lagrangian as L(θ) = = + U(θ(t), t) dλ(t) + U(θ(t), t) dλ(t) + P (θ(t)) ( t ) dλ(t) + U(θ(t), t) dλ(t) U(θ(t), ) dλ(t) P (θ(t)) (Λ(t) F Λ (t)) dt (F Λ (t) Λ(t))P (θ(t)) dt The first two terms are maximized by choosing θ(t) = θ BC (t), so if the last term is also maximized by this choice it is optimal. Using the identities, P (θ(t)) = P () + t dp (θ(x)) and b (t a) dλ(t) = b (Λ(b) Λ(t)) dt, the third term becomes a a 12 To see that this is true, suppose that Λ(t) > 1 for some t [, 1]. Then for some t [, t], 1 1 t t F t Λ (x) dx < (1 F Λ (x)) dx < (1 F Λ (x)) dx > t where the last implication follows from (1 F Λ(x)) dx =. This contradicts the definition of, so Λ(t) This follows from the definition of and that it must be interior (see footnote 1). From (1), F Λ(x) dx = 1 15

16 P () = = = { ( (1 F Λ (t)) dt + t ) dλ(t) + x x { (1 F Λ (t)) dt + (x )(1 Λ(x)) dp (θ(x)) x { 1 x (1 F Λ (t)) dt + 1 Λ(x) (x ) dp (θ(x)) x { x F Λ (t) dt Λ(x) (x ) dp (θ(x)) x (Λ(t) F Λ (t)) dt dp (θ(x)) The definition of Λ(t) implies that the term inside the brackets is non-positive, so this term is maximized by setting dp (θ(x)) = which is the case with the budget constraint contract. To formally apply Theorem 1, I follow Amador and Bagwell (213) in letting X {θ θ : [, 1] [, 1], Z {z z : [, 1] R and z integrable, Ω be the set of nondecreasing functions in X, and Z + be the positive functions in Z (i.e., Z + = {z z Z and z(t), t). Condition (i) is clearly satisfied, and condition (ii) follows from the previous discussion. Finally, since Λ(t) is nondecreasing and the constraint (R1) binds everywhere (τ a (t) =, t), conditions (iii) and (iv) are also satisfied. As mentioned above, optimality of budget constraints in the delegation problem, which maybe defined in this context as the stronger constraint that τ a (t) = τ p (t) =, is implied by the optimality of the budget constraint under the weaker restriction (R1). That is, under Assumption 1, budget constraint contracts are necessarily optimal in the delegation problem. This follows from the observation that budget constraint contracts remain feasible in the problem with tighter constraints Transfer Case This case allows the principal to pay non-negative transfers to the agent condition on the realized signal. The constraints in this case are (IC), (IR) and (R2). Again, the budget constraint contract is feasible for the problem, because θ BC (t) is weakly 16

17 increasing, it involves no transfers and satisfies (IR). The conditions under which a budget constraint contract is optimal in the transfer case will be more restrictive than they are in the money burning case. I will require the following additional assumptions. Assumption 2. F Γ (t) F (t) (1 δ)f(t)t is nondecreasing for t [, ]. 1 t Assumption 3. t F Γ (x) dx F Γ ( ) for t [, 1 ]. Assumption 2 functions as a upper bound on the rate at which f(t) can increase. Together, Assumptions 1 and 2 put restrictions on how quickly f(t) can vary as t increases. To see this, observe that together they imply upper and lower bounds on the elasticity of the density function when it is differentiable, 2 δ 1 δ f (t)t f(t) δ 1 δ. Assumption 3 requires that the expected value (with respect to the uniform distribution) of F Γ (t) not fall below F Γ ( ). If F Γ (t) F Γ ( ) for t > this assumption is satisfied, but it allows for F Γ (t) to decrease to a limited extent on [, 1]. As with Assumption 1, Assumptions 2 and 3 become less severe as δ 1, and all three are satisfied with an arbitrary distribution for a δ sufficiently close to 1. For arbitrary δ, the assumptions are satisfied by common families of distribution functions on subsets of their parameter spaces. 14 Again, by using (2) to replace τ a (t) in the principal s objective, I get the following for the Lagrangian for this case, where Γ(t) is the multiplier function. 14 The following assumptions guarantee that F Λ (t) and F Γ (t) are nondecreasing on [, 1]. The uniform distribution clearly satisfies the assumptions, but so do all Gamma distributions where the shape (k) and scale (θ) parameters satisfy k < 1 1 δ + 1 θ. Special cases of the Gamma distributions are the exponential distribution (k = 1, θ = 1/λ), which always satisfies this inequality, and the χ 2 distribution (k = v/2, θ = 2), which satisfies the inequality for any δ if v 3 and for all degrees of freedom (v) up to a δ-dependent upper bound. For more examples, there are subsets of parameters for the beta, normal, Pareto densities that satisfy both assumptions with arbitrary δ. 17

18 L(θ) = = = + + { U(θ(t), δt) + U(θ(t), t) t { t P (θ(x)) dx U(θ(t), t) {2U(θ(t), t) (1 δ)p (θ(t))t f(t) dt (Γ(1) Γ(t) 1 + F (t))p (θ(t)) dt + (Γ(1) Γ() 1)(τ a () + U (θ )) U(θ(t), t)d[2f (t) Γ(t)] + (Γ(1) Γ() 1)(τ a () + U (θ )) P (θ(x)) dx df (t) dγ(t) + (Γ(1) Γ() 1)(τ a () + U (θ )) U(θ(t), t)dγ(t) P (θ(t))(1 F Γ (t) Γ(1) + Γ(t)) dt+ The first equation follows from integration by parts and the definition of U(θ(t), t). The second uses the definition of F Γ (t). Proposition 2. Under Assumptions 1, 2 and 3, a budget constraint contract is optimal when the constraints on the principal are (IC), (IR) and (R2). Proof. The proposed multiplier is 2F (t) Λ(t) Γ(t) = inf t [t,1] {2F (t ) Λ(t ) if t if t >. Note that 2F (t) Λ(t) = F Γ (t) for t. Assumption 2 and the definition of Γ(t) for t > guarantee that Γ(t) is nondecreasing. As with Λ(t), Γ() = and Γ(1) = Γ(1) = 1 implies that Γ(t) 1. Assumption 3 guarantees that Γ(t) Γ() for t >. 16 Γ(t) is therefore a distribution function and is differentiable (a.e.). Following the idea of the proof of Proposition 1, and incorporating the new multiplier, the Lagrangian for the problem becomes 15 Since Λ(1) = 1, Γ(1) = Suppose that Γ(t) < F Γ (). Then for some t t, F Γ () > 2F (t) 1 1 t F t Λ (x)dx = 1 t F t Γ (x)dx + 1 t t the second term must be nonnegative. t t F Λ (x)dx 2F (t ) 2(F (t ) F (x))dx. This is a contradiction, because 18

19 L(θ) = = + + U(θ(t), t)d[2f (t) Γ(t)] + P (θ(t)) (F Γ (t) Γ(t)) dt + U(θ(t), t)dλ(t) + P (θ(t)) (F Γ (t) Γ(t)) dt + U(θ(t), )d[2f (t) Γ(t)] P (θ(t))(t )d[2f (t) Γ(t)] U(θ(t), )d[2f (t) Γ(t)] P (θ(t))(t )d[2f (t) Γ(t)] The choice θ(t) = θ BC (t) maximizes the first term pointwise, because Λ(t) is nondecreasing. The definition of Γ(t) for t > implies that either Γ(t) is constant or that Γ(t) = 2F (t) Λ(t). In either case, it is nondecreasing, so the second term is also maximized pointwise by the budget constraint. This leaves the third term, which can be rewritten as 17 { P () (F Γ (t) Γ(t)) dt + { + (F Γ (t) Γ(t)) dt + x (t )d[2f (t) Γ(t)] x (t )d[2f (t) Γ(t)] dp (θ(x)) = P () {1 F Λ (t) dt { + (1 F Λ (t)) dt + (x )(1 2F (x) + Γ(x)) dp (θ(x)) x { 1 x = (1 F Λ (t)) dt + 1 2F (x) + Γ(x) (x )dp (θ(x)) x { x = F Λ (t) dt 2F (x) + Γ(x) (x )dp (θ(x)) x The term in brackets is non-positive because Γ(x) 2F (x) Λ(x) 2F (x) 17 I use the same idea as in the proof of Proposition 1. The key observation is that x (t )d[2f (t) Γ(t)] = x (t x)d[2f (t) Γ(t)] + (x )(1 2F (x) + Γ(x)) = x (1 2F (t) + Γ(t))dt + (x )(1 2F (x) + Γ(x)) 19

20 1 x F x Λ (t) dt. This implies that a budget constraint also maximizes this third term, so it is optimal for the problem. The remainder of the proof is exactly as it is in the proof to Proposition 1. The spaces are defined the same way, and the conditions hold for analogous reasons. 4.3 Necessity This section shows that Assumptions 1 and 2 are also necessary for budget constraints to be optimal. 18 To prove necessity of these conditions I construct profitable deviations from the budget constraint contract, which suggests some intuition for the main results of the paper. To find profitable deviations when the assumptions are violated it is useful to focus on the term (1 δ)tf(t). This term is the weighted difference between the principal and the agent s payoffs, and can be thought of as capturing the importance of the bias of a type t agent. Assumption 1 requires that this term not decrease too quickly on [, ], while Assumption 2 requires that it not increase too quickly over the same interval. Assumption 1 is necessary in both regimes, because without this assumption it is possible to construct a profitable deviation that does not require the use of transfers. If the bias term decreases too quickly on some interval [a, b], the principal can improve on the budget constraint contract by requiring the agent to make a low report on the first part of this interval and a high report on the remainder of this interval. The high report is necessary to preserve incentive compatibility and is not too costly because the bias term is relatively low when the agent is required to make this high report. When the principal can make positive transfers to the agent, the principal is able to implement contracts that are not feasible in the first regime. This leads to the necessity of Assumption 2 in the second case. When this assumption is violated, the bias term increases quickly over a sub-interval. The principal in this case can benefit from requiring the agent to report higher types in the first part of the interval, while reporting lower types in the remainder of the interval. This deviation is feasible with transfers, because the principal can pay the agent to over-report, then gradually adjust the payment over the interval. 18 It is clear from footnote 16 that Assumption 3 can be relaxed somewhat. 2

21 θ(t) t 1 t 2 t 3 t 4 t Figure 1: Profitable Deviations The two types of deviations are illustrated in Figure 1. The first type of deviation occurs between [t 1, t 2 ]. The second type occurs between [t 3, t 4 ]. These are exactly the types of deviations that I use in the proofs that follow. The remainder of the section formalizes these ideas. First, I show that a failure of Assumption 1 allows for a profitable deviation. Proposition 3. If F Λ (t) decreases on [t 1, t 2 ] with t 2 (i.e. Assumption 1 does not hold), then there is a profitable deviation from the budget constraint that requires no transfers. Consequently, this assumption is necessary in both the Money Burning and Transfer Cases. Proof. Modify the budget constraint contract to have the agent report θ(t) = t 1 t for t 1 t t m and θ(t) = t 2 t for t m < t t 2. The contract is unchanged otherwise. θ(t) is clearly still nondecreasing. Incentive compatibility also requires that the agent be indifferent between reporting t 1 and t 2 when t = t m. 19 This means that t m must satisfy P (t 1 )(t m t 1 ) + tm t1 t 1 (P (x) P (t 1 )) dx = 19 This follows from the continuity of t P (θ(x)) dx P (x) dx = P (t 2 )(t m t 2 ) + 21 t2 t2 t m (P (t 2 ) P (x)) dx P (x) dx

22 . Next consider π D π BC, the difference between the principal s payoff with the deviation and that with the budget constraint. The principal s payoff in the money burning case is given by (3) ignoring the multipliers. Since the contracts are identical outside [t 1, t 2 ], this difference is tm t2 (1 F Λ (x))(p (t 1 ) P (x)) dx + (1 F Λ (x))(p (t 2 ) P (x)) dx t 1 t ( m tm t2 ) > (1 F Λ (t m )) (P (t 1 ) P (x)) dx + (P (t 2 ) P (x)) dx = t 1 t m. The inequality follows because 1 F Λ (t) increases by the assumption of the proposition. This proves that the deviation is profitable in the money burning case. It is easy to check that with this deviation t P (θ(x)) dx = U(θ(t), t), so the deviation involves no transfers. This implies that the principal s payoff in the trans- fer case is U(θ(t), δt)f(t) dt = {U(θ(t), t) P (θ(t))(1 δ)t f(t) dt = (1 F Λ (t))p (θ(t)) dt, so the result holds in this case, too. The next proposition shows the necessity of Assumption 2 in the transfer case. Proposition 4. If F Γ (t) decreases on [t 3, t 4 ] with t 4 (i.e. Assumption 2 does not hold), then a profitable deviation from the budget constraint contract exists in the Transfer Case. Proof. The idea is to construct a deviation where the agent reports a type greater than his true type for low values in [t 3, t 4 ] and one lower than his type for high values (see Figure 1). It will be sufficient to consider a linear deviation on [t 3, t 4 ], given by θ(t, α) = αt + (1 α)t m (α) with < α < 1 and t m (α) chosen so that τ(t 3, α) = τ(t 4, α). This deviation will require that the principal pay some transfer to get the agent to over-report his type at t 3, gradually increase the transfer across [t 3, t m ], then gradually decrease until t 4. The adjustment in the transfer must satisfy τ t (t, α) = P (θ(t, α))θ t (t, α)(t θ(t, α)) = P (θ(t, α))α(1 α)(t t m (α)). Define t m (α) implicitly as the solution to 22

23 . t4 τ(t 3, α) τ(t 4, α) = α(1 α) P (θ(t, α))(t t m (α)) dt = t 3 The middle term is continuous in t m (α), less than zero at t m (α) = t 3, and greater than zero at t m (α) = t 4. So it has a solution. This contract is feasible, because the transfers are smallest at τ(t 3, α) and τ(t 4, α), which are both positive (τ(t 3, α), for example, is chosen so that the agent is indifferent between reporting θ(t 3, α) and receiving τ(t 3, α) and reporting t 3 with no transfer.) Evaluating the principal s payoff (over the interval [t 3, t 4 ]) from this deviation, then differentiating with respect to α and evaluating at α = 1, I get π D (α) = π D(α) = π D(1) = t4 t 3 t4 t 3 t4 t 3 2U(θ(t, α))f(t) P (θ(t, α))(1 F Γ (t)) dt 2U θ (θ(t, α))θ α (t, α)f(t) P (θ(t, α))θ α (t, α)(1 F Γ (t)) dt < (1 F Γ (t m )) P (t)(t t m (1))(1 F Γ (t)) dt t4 t 3 P (t)(t t m (1)) dt =. The inequality follows from the assumption of the proposition that 1 F Γ (t) is increasing, so that t m t 3 P (t)(t t m (1))(1 F Γ (t)) dt > (1 F Γ (t m )) t m t 3 P (t)(t t m (1)) dt and t 4 t m P (t)(t t m (1))(1 F Γ (t)) dt > (1 F Γ (t m )) t 4 t m P (t)(t t m (1)) dt. Therefore, for some α < 1 the principal s payoff can be made greater, and the proposition follows. 5 Applications To discuss applications, I extend the above model to allow for many, possibly asymmetric, principal-agent pairs. This extension is based on the model in Burkett (213). 23

24 Suppose that principal i (i will index the principal-agent pairs) privately observes a signal, s i, before deciding on the agent s constraint that determines the distribution F i. That is, let s i be distributed on [, 1] according to some absolutely continuous distribution function G i (s i ) with density g i (s i ), and replace the distribution F (t) used above with F i (t i s i ). If F i (t i s i ) satisfies Assumption 1 in the money burning case or Assumptions 1, 2 and 3 in the transfer case, for all s i and all pairs i, then a budget constraint contract is optimal. With a budget constraint the agent reports min(t i, i (s i )) to any incentive compatible and individually rational mechanism, where i (s i ) is defined as above. 2 I assume for the remainder of this section that the assumptions guaranteeing the optimality of budget constraints hold. 5.1 Revenue Equivalence A general statement of the revenue equivalence principle is that the expected payment of a bidder with a given valuation is the same for two mechanisms that have the same allocation rule (and require the same payment by the lowest type). When the optimal constraint is a budget constraint for the principal, the agent s report to the mechanism is min(t i, i (s i )) and is independent of the choice of mechanism, (P, T ). Therefore, the agent s expected payment is completely determined by the value of min(t i, i (s i )), the allocation rule P, and the payment T (). This is the sense in which the revenue equivalence principle extends to this case. It is not true, for example, that any two bidders with the same value make the same expected payment, because one may be constrained and the other not (depending on s i ). However, from the mechanism designer s perspective, the bidders report min(t i, i (s i )) to any incentive compatible mechanism, so the designer cannot hope to raise more revenue without changing the allocation rule (e.g., the switch from a second-price to first-price auction will not yield a higher expected revenue). 5.2 Optimal Auctions That the optimal auction problem should be standard (Myerson, 1981) follows directly from the logic of the last section. The key observation to make is that from the mechanism designer s perspective, one can think of a budget-constrained bidder 2 It is now a function of s i because the distribution F i (t i s i ) is determined by the realization of s i. 24

25 reporting min(t i, i (s i )) as being equivalent to an unconstrained bidder with valuation distributed according to the distribution of the random variable min(t i, i (s i )), which can be written as. P{min(t i, i (s i )) x = 1 { min(t i, i (s i )) x f(t i s i )g(s i ) dt i ds i By considering equivalent bidders in this way, the optimal auction problem with budget-constrained bidders can be translated into a model that fits the assumptions of the standard model by Myerson (1981). In other words, the qualitative results of the optimal auction carry over to this model with the proviso that instead of bidder valuations we consider min(t i, i (s i )) as each bidder s valuation. For example, if one defines v i = min(t i, i (s i )) and Fi v (x) = P{min(t i, i (s i )) x, then the revenue maximizing auction allocates to the bidder with the largest virtual valuation, v i 1 Fi v(v i), when this virtual valuation is increasing. As with the classic results, when fi v(v i) the bidders are symmetric (now the principal-agent pairs need to be symmetric) the revenue maximizing auction can be implemented as a first- or second-price auction with a reserve price. 21 This solution to the optimal auction problem with budget-constrained bidders contrasts starkly with the existing literature on the optimal choice of auction with budget constrained bidders, which concludes, for example, that the optimal auction is close to an all-pay auction (Pai and Vohra, 213) Conclusion The simplicity of a budget constraint is highlighted by the fact that the choice of a budget constraint involves deciding on the value of a single (scalar) parameter. This paper gives conditions, which are satisfied by common distribution functions, that guarantee this contract is optimal in an infinite-dimensional choice set. Intuitively, the conditions require that there be no abrupt changes in the distribution of valua- 21 The optimal reserve price in the symmetric case is the solution to v i 1 F v i (vi) fi v(vi) = when the virtual valuation is increasing (Myerson, 1981). 22 Of course, if there is reason to believe that budget constraints are exogenous (not affected by a change in auction rules), then the conclusions of the prior literature would hold. 25

26 tions (i.e., that f(t)t not change too quickly ), and the conditions are satisfied for commonly used distributions, including the uniform distribution. The results of Che and Gale (1996, 1998) and subsequent papers suggest that budget constraints cannot be incorporated into auction models by simply redefining the valuation of a bidder as the minimum of the value and a budget. If this were the case, one could appeal to the classic results of auction theory. However, that conclusion is to some extent based on the model of budget constraints that those papers were based on, specifically, the assumption that budget constraints are exogenously specified. The results in Burkett (213) and this paper show that when budget constraints are set endogenously in the model there is a way to incorporate them that does not upset the classic auction theory results. Sections 5.1 and 5.2 of this paper illustrate this point. References Alonso, R. and N. Matouschek (28). Optimal delegation. The Review of Economic Studies 75 (1), Amador, M. and K. Bagwell (213). The theory of optimal delegation with an application to tariff caps. Econometrica 81 (4), Amador, M., I. Werning, and G.-M. Angeletos (26, March). flexibility. Econometrica 74 (2), Commitment vs. Armstrong, M. and J. Vickers (21). A model of delegated project choice. Econometrica 78 (1), Bulow, J., J. D. Levin, and P. R. Milgrom (29, March). Winning play in spectrum auctions. Working Paper 14756, National Bureau of Economic Research. Burkett, J. (213, March). Endogenous budget constraints in auctions. Unpublished. Che, Y.-K. and I. Gale (1996). Expected revenue of all-pay auctions and first price sealed-bid auctions with budget constraints. Economics Letters 5 (3), Che, Y.-K. and I. Gale (1998). Standard auctions with financially constrained bidders. Review of Economic Studies 65 (1),