Dynamic screening with liquidity constraints

Krähmer, Daniel; Strausz, Roland Article — Published Version Dynamic screening with liquidity constraints Economic Theory Provided in Cooperation with: Springer Nature Suggested Citation: Krähmer, Daniel; Strausz, Roland (2024) : Dynamic screening with liquidity constraints, Economic Theory, ISSN 1432-0479, Springer, Berlin, Heidelberg, Vol. 79, Iss. 4, pp. 1421-1453, https://doi.org/10.1007/s00199-024-01616-2 This Version is available at: https://hdl.handle.net/10419/323258 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ Economic Theory (2025) 79:1421–1453 https://doi.org/10.1007/s00199-024-01616-2 RESEARCH ARTICLE Dynamic screening with liquidity constraints Daniel Krähmer1 ·Roland Strausz2 Received: 22 March 2024 / Accepted: 12 October 2024 / Published online: 16 November 2024 © The Author(s) 2024 Abstract We consider a dynamic screening model with serially independent types where the agent is short-term liquidity constrained. We model a liquidity constraint as a hard constraint that forces the agent to renege whenever he would suffer a loss from fulfilling the contract terms in a given period. In particular, the violation of a liquidity constraint is a verifiable event that future contract terms can condition on. This verifiability leads to less stringent truth-telling constraints than those considered in the existing literature. We show that the weaker constraints do not affect optimal contracting, however. Moreover, we develop a novel method to study private values settings with continuous types and show that a regularity condition that has analogues in the literature on multi-dimensional screening ensures that the optimal contract is deterministic. Keywords Dynamic screening ·Liquidity constraints ·Verifiability ·Mean preserving spread JEL Classification D82 ·H57 1 Introduction A recent literature studies the role of short-term liquidity constraints in dynamic screening models where a procurer (the principal) procures goods or services over multiple periods from a supplier (the agent) whose costs evolve dynamically over time and are the supplier’s private information (e.g. Krishna et al. 2013; Mirrokni et al. 2020; Krasikov and Lamba 2021; Ashlagi et al. 2023). A liquidity constraint is a hard (phys- We thank Nima Haghpanah, Mathijs Janssen, and Rohit Lamba for useful comments as well as seminar participants in Berlin, Bonn, Duisburg, Jerusalem/Tel-Aviv, Montreal, Peking, and Singapore. In addition we thank the editor and three anonymous referees. BRoland Strausz [email protected] Daniel Krähmer [email protected] 1Department of Economics, Universität of Bonn, Bonn, Germany 2School of Business and Economics, Humboldt Universität zu Berlin, Berlin, Germany 123 1422 D. Krähmer, R. Strausz ical) constraint that captures the fact that, in practice, suppliers are often forced to renege on the contract because they are unable to raise cash for paying short-term bills.1By contrast, the classical literature on dynamic screening/mechanism design (e.g., Baron and Besanko 1984; Battaglini 2005, Pavan et al. 2014, Esö and Szentes 2017) neglects such concerns, effectively assuming that the supplier has sufficiently deep pockets to overcome short-term liquidity needs. In this case, optimal contracts exploit this feature and, in fact, impose short-term losses on the supplier. These contracts are thus no longer feasible in situations when the supplier is unable to cope with short-term liquidity needs. We make two contributions to the literature. First, existing literature focusses on direct revelation mechanisms and imposes liquidity constraints by requiring the mechanism to ensure that the agent gets non-negative utility on the path, that is, when the agent reveals his private information truthfully. This approach is, however, difficult to interpret because it does not ensure that the agent obtains a non-negative periodic utility off the path, that is, when the agent misreports.2In fact, the mechanisms that this literature identifies as optimal typically exhibit a binding liquidity constraint for some cost type θso that the liquidity constraint is violated for any cost type θ>θ that misreports to be type θ.3 To clarify these issues, we take serious the idea of the literature that a liquidity constraint is a hard constraint that kicks in whenever the agent would suffer a financial loss when fulfilling the contract terms in the current period. In other words, when illiquid, the agent has no choice but to renege on the contract. Taking this idea to its logical conclusion implies that a violation of liquidity constraints is inherently verifiable. Moreover, since the agent is forced to renege only because his pockets are empty, we assume that he obtains a payoff equal to (his outside option of) zero.4This assumption also captures that the agent as a firm is protected by limited liability. However, as the agent’s liquidity is verifiable, the future terms of the contract can be conditioned on whether the current liquidity constraint is violated or not. Micro-founding liquidity constraints in this way allows us to deduce the contractual feasibility constraints from the underlying physical environment. In particular, the 1According to one study, about 80% percent of failing small business in the US attribute their bankruptcy to cash flow problems. https://www.visualcapitalist.com/why-do-businesses-fail/ 2Krishna et al. (2013) call these liquidity constraints “non-negative cash flows” and claim in footnote 10 that incentive compatibility and on-path liquidity constraints imply that liquidity constraints are never violated off-path and can therefore be neglected. Krasikov and Lamba (2021) refer to the agent’s liquidity constraints as “cash-strapped” and while they state in footnote 22 that “Even if the agent may have misreported in the past, the principal delivers a non-negative stage utility to him if he is truthful today” (emphasis added), they however leave unspecified the payoff of an agent who is not truthful today. Ashlagi et al. (2023) impose ex post individual rationality, requiring that the agent gets non-negative life-time utility along the (truth-telling) equilibrium path. They note that, in their context, ex post individual rationality is equivalent to requiring that the agent obtains a non-negative per-period utility (along the equilibrium path). However, no restrictions are imposed on life-time or periodic utility off the path. 3Hence, this observation is in conflict with footnote 10 in Krishna et al. (2013) claiming that incentive compatibility and on-path liquidity constraints imply that liquidity constraints are satisfied off-path. 4Thus, we abstract from potential non-monetary costs of reneging on the contractual terms, for example reputational costs. As we point out in more detail in Footnote 8, the presence of such costs would possibly allow the principal to induce violations of the liquidity constraint as a screening instrument. 123 Dynamic screening with liquidity constraints 1423 resulting dynamic incentive compatibility constraints account for the possibility that the agent violates the liquidity constraint after a deviation from truth-telling. Moreover, since violations of liquidity constraints are verifiable, certain deviations are detectable, and this enlarges the contractual design choices to dissuade them. Specifically, we show that an optimal contract has to satisfy only uni-directional incentive constraints that only prevent an agent from overstating her costs. The reason is that understating one’s cost results in a verifiable violation of the liquidity constraint, thus revealing a lie. Even though our uni-directional incentive constraints are weaker than the feasibility constraints posited by the existing literature, a key insight of our paper is that optimal contracts do not, however, exploit the additional slack thus gained. In this sense, our approach validates the literature’s approach to impose cash constraints only on but not off the equilibrium path.5 Our observation that a liquidity constraint affects the agent’s incentive constraint has a counterpart in the literature on mechanism design with quitting or withdrawal rights. In particular, that literature points out that incentive constraints are affected by such rights in that they have to account for the “double deviation” that an agent misreports and subsequently quits (e.g. Matthews 1989; Forges 1999; Compte 2007; Compte and Jehiel 2009; Krähmer and Strausz 2015, Bergemann et al. 2020). We stress, however, that despite this similarity, liquidity constraints conceptually differ from quitting or withdrawal rights. This is so because with a quitting or withdrawal right, the agent can strategically decide whether to sustain a loss ex post or not. By contrast, and as mentioned above, the agent cannot do so in case of a liquidity constraint. Hence, the consideration of quitting or withdrawal rights introduces a moral hazard problem, which does not arise in the case of liquidity constraints. We refer to Krishna et al. (2013, p106) for a more extensive discussion of this distinction. Our second contribution is to extend the existing literature’s analysis of liquidity constraints with two agent types to settings with continuous types. This extension is not straightforward, because with liquidity constraints, the principal’s ex ante payoff is a non-linear and non-monotone function of the agent’s (future) information rents. Hence, contrary to dynamic screening without liquidity constraints, the problem cannot be reduced to maximizing a virtual surplus representation where allocations are additively separable by type. Consequently, the problem becomes difficult to solve with standard techniques when there are more than two types. For this reason, we develop a novel solution method. The basic idea behind this method is based on the observation that every dynamic contract induces a continuation value for the agent which, from the principal’s perspective, is a random variable, as it depends on the agent’s privately known type. A standard argument from dynamic programming implies that the principal’s continuation profit is concave in the agent’s continuation value. This observation allows us to rank contracts in terms of second order stochastic dominance of the induced con- 5Because our micro foundation implies that an optimal contract has to satisfy only uni-directional incentive constraints, our study of dynamic setting with bankruptcy constraints links to the literature that considers static settings in which such uni-directional incentive constraints exist for exogenous reasons (e.g., Moore 1984, Celik 2006, Krähmer and Strausz 2024). In line with our finding, this literature shows that, in static settings, these weaker incentive constraints do not give rise to different predictions in settings with private values or, more generally, when the aggregate surplus is monotone in the allocation. 123 1424 D. Krähmer, R. Strausz tinuation value. As a result, we can identify an optimal contract as a contract that, among the set of feasible contracts, displays minimal dispersion in the second order sense. We show that, under a regularity condition, an optimal contract has a simple, deterministic cutoff structure where cost types below a cutoff produce the good and types above the cutoff do not. The regularity condition differs from the more familiar monotone virtual surplus kind of conditions, and also appears in the (static) multidimensional screening literature (e.g. Manelli and Vincent 2006). The connection is that, as in this literature, we write the principal’s optimization problem in terms of the agent’s (continuation) value rather than the allocation rule. 2Themodel A principal (the buyer, she) and an agent (the seller, he) interact over two periods τ=1,2.6In each period, the principal seeks to procure one good from the agent. In period τ, the terms of trade are the probability of trade xτand a transfer tτfrom the principal to the agent.7The principal’s valuation for the good is vτ, and the agent’s cost to produce the good is θτ. While vτis commonly known, θτ, the agent’s cost type in period τ, is privately known to the agent in period τ, and it is commonly known that θτis distributed with cdf Fτwith support τ≡[θτ,¯ θτ]and differentiable pdf fτ. We assume that θ1and θ2are stochastically independent. Moreover, we assume that production is only efficient when costs are low enough, i.e. vτ∈(θτ,¯ θτ]. The parties have time-separable quasi-linear utilities. That is, under the terms of trade xτ,tτthe principal’s utility in period τis vτxτ−tτ, and the agent’s utility is tτ−θτxτ. A party’s overall utility is the sum over the per-period utilities. The key feature of our paper is that the agent is short-term liquidity constrained. This means that the agent cannot honor the contract in a given period if this would require him to make a loss in this period. We say that the agent is “illiquid” in this case and, as explained in the introduction, the fact that the liquidity constraint is hard implies that being illiquid is a verifiable event. In particular, future contract terms can condition on past liquidity states. Moreover, we assume that when the agent is illiquid, both the agent and the principal receive their reservation utility of zero.8 The timing is as follows: 1. At the outset, the principal commits to a long-term contract which specifies the terms of trade over the two periods. If the agent rejects the contract, both parties receive their reservation utility of 0 and the game ends. 6At the end of Sect.4, we show that our analysis and results extend to a setting with infinitely many periods. 7As is standard, we interpret tτas the expected payment t(0) τ(1−xτ)+t(1) τxτ,wheret(0)(resp. t(1))is the payment when trade does not (resp. does) occur. Alternatively, for a divisible good, we may interpret xτas the share of the good traded. 8Thus, we assume that the agent is protected by limited liability and cannot be penalized to a level below her outside option when becoming illiquid. This simply reflects that being illiquid means that the agent has empty pockets. In practice, reneging on a contract often comes with additional costs such as reputational costs or the opportunity costs entailed by legal proceedings. To keep the analysis simple, we abstract from the presence of such costs because they would introduce the possibility of screening the agent by sometimes deliberately inducing illiquidity (a form of “money burning”). 123 Dynamic screening with liquidity constraints 1425 2. If the agent accepts, then in period 1, he privately learns θ1.Ift1−θ1x1≥0, the agent is liquid and the terms of trade x1,t1are implemented. If t1−θ1x1<0, the agent is illiquid and both parties receive 0.9 3. In period 2, the agent privately learns θ2.Ift2−θ2x2≥0, the agent is liquid and the terms of trade x2,t2are implemented. If t2−θ2x2<0, the agent is illiquid and both parties receive 0. Benchmarks: It is useful to contrast our setting to various benchmarks. In the absence of a liquidity constraint, the model corresponds to a traditional dynamic screening model where the agent has an outside option of zero at the contracting stage. If contracting takes place under symmetric information, then the principal can implement the firstbest by “selling the firm” to the agent at a price equal to the expected first-best surplus (see, e.g., Harris and Raviv 1979). The agent will therefore make a loss for high cost realizations. This outcome is therefore not feasible with liquidity constraints. If contracting takes place after the agent observes θ1, then an optimal mechanism features a distorted allocation in the first period but implements the first-best in the second period because cost types are independent (see, e.g., Baron and Besanko 1984). Again, for high cost realizations, the liquidity constraint will be violated in some period. By backloading payments to the agent from the first to the second period, however, the agent’s second period payoff can be made non-negative for all θ2so that second period liquidity constraints are automatically satisfied. Since these constraints effectively correspond to second period participation constraints, the same outcome can be implemented even if the agent can walk away from the contract after observing θ2. Note that in our setting, it is immaterial whether contracting takes place after or before the agent observes θ1because the liquidity constraint ensures that the participation constraint is satisfied for all types θ1even if contracting takes place under symmetric information (see Sappington 1983). Finally, if the principal can offer only one-period spot contracts, then he will offer in each period the static one-period second best mechanism, which is a posted price mechanism (see Riley and Zeckhauser 1983). Example: To illustrate our analysis, we use the uniform example, where θ1and θ2 are both uniformly distributed over the interval [0,1], and v1=v2=¯ θ=1. For this example, trade is efficient for all types and the per-period first-best surplus equals SFB =SFB 1=SFB 2=1 01−θdθ=1/2, yielding an aggregate surplus of SFB 1+SFB 2=1. In the static second best, the optimal mechanism is a posted price of 1/2, yielding the principal a per-period profit of SB ≡(1−1/2)∗1/2=1/4 and the agent a per-period second best utility of USB ≡1/2 01/2−θdθ=1/8. Implementing a posted price of 1/2 for each of the two periods, yields an overall profit of 2SB =1/2 to the principal and an overall utility of 2USB =1/4 to the agent, resulting in aggregate surplus of 3/4. In the benchmark case in which there is an interim participation constraint in period 1 but no liquidity constraint, the optimal mechanism implements a posted price of p=1/2 for the first period, and extracts the whole first-best surplus in the second 9Related to footnote 7,ifx1∈(0,1), the agent is illiquid if t(0)<0 and the mechanism does not prescribe trade, as well as if t(1)−θ1<0 and the mechanism does prescribe trade. 123 1426 D. Krähmer, R. Strausz period. This yields an overall profit of SB +SFB =3/4 to the principal, an overall utility of USB =1/8 to the agent, resulting in aggregate surplus of 7/8.  3 The principal’s problem The principal’s objective is to design a contract to maximize her profits. Because the principal has full commitment, the revelation principle applies, implying that an optimal contract is in the class of direct mechanisms where, on the equilibrium path, the agent reports his type truthfully in every period (Myerson 1986). Moreover, because the agent’s liquidity in period 1 is verifiable, a mechanism can condition the terms of trade in period 2 on whether the agent was illiquid in period 1 or not. Without loss, we can therefore restrict attention to contracts of the form (x1,t1,xL 2,tL 2,xI 2,tI 2), where (x1,t1):[θ1,¯ θ1]→[0,1]×R,(x 2,t 2):[θ1,¯ θ1]×[θ2,¯ θ2]→[0,1]×R,(1) where ∈{I,L}indicates whether the agent was illiquid (=I)orliquid(=L) in period 1. To state the incentive compatibility constraints, we denote for ∈{I,L}the agent’s expected period 2 utility from a report ˆ θ1, conditional on truthfully reporting in period 2, by U(ˆ θ1)=¯ θ2 θ2 max{0,t 2(ˆ θ1,θ 2)−θ2x 2(ˆ θ1,θ 2)}dF 2(θ2). (2) Moreover, let L 1={θ1|t1(θ1)−θ1x1(θ1)≥0}(3) be the set of period 1 types who are liquid in period 1 under a given mechanism. Definition 1 A contract (x1,t1,xL 2,tL 2,xI 2,tI 2)is feasible if: (i) It is incentive compatible in period 2, that is, for ∈{I,L}10: max{0,t 2(θ1,θ 2)−θ2x 2(θ1,θ 2)}≥max{0,t 2(θ1,ˆ θ2)−θ2x 2(θ1,ˆ θ2)}∀θ1,θ 2,ˆ θ2. (4) (ii) It is incentive compatible in period 1, that is: For all θ1∈L 1,wehave: t1(θ1)−θ1x1(θ1)+UL(θ1)≥t1(ˆ θ1)−θ1x1(ˆ θ1)+UL(ˆ θ1) 10 The revelation principle for dynamic games requires truthful reporting in period 2 only after a truthful report in period 1 (see Myerson 1986). In our context, where types are independent, the support of period 2 types is “non-shifting”, that is, is independent of the period 1 type. It then follows with standard arguments that if truth-telling in period 2 is optimal for the agent after telling the truth in period 1, then it is so after any report in period 1. 123 Dynamic screening with liquidity constraints 1427 ∀ˆ θ1:t1(ˆ θ1)−θ1x1(ˆ θ1)≥0,(5) t1(θ1)−θ1x1(θ1)+UL(θ1)≥UI(ˆ θ1)∀ˆ θ1:t1(ˆ θ1)−θ1x1(ˆ θ1)<0,(6) For all θ1/∈L 1,wehave: UI(θ1)≥t1(ˆ θ1)−θ1x1(ˆ θ1)+UL(ˆ θ1)∀ˆ θ1:t1(ˆ θ1)−θ1x1(ˆ θ1)≥0,(7) UI(θ1)≥UI(ˆ θ1)∀ˆ θ1:t1(ˆ θ1)−θ1x1(ˆ θ1)<0.(8) Part (i) of the definition captures the truth-telling constraints for the agent in period 2, explicitly taking into that bankruptcy in period 2 occurs whenever the terms of trade would impose a loss on the agent. Similarly, part (ii) describes the truth-telling constraints for the agent in period 1. This requires a distinction between four cases, depending on how both truth-telling and lying affects the agent’s liquidity in period 1. The principal’s problem is thus to select a feasible contract that maximizes her profits L 1v1x1(θ1)−t1(θ1)+L,L 2(θ1) v2xL 2(θ1,θ 2)−tL 2(θ1,θ 2)dF 2(θ2)dF 1(θ1) (9) +1\L 10+I,L 2(θ1) v2xI 2(θ1,θ 2)−tI 2(θ1,θ 2)dF 2(θ2)dF 1(θ1), (10) where ,L 2(θ1)≡{θ2∈2|t 2(θ1,θ 2)−θ2x 2(θ1,θ 2)≥0}(11) denotes the set of types θ2who are liquid in period 2 given their liquidity state ∈ {L,I}in period 1. To solve the principal’s problem, we first show that it is without loss to focus on contracts with the property that the agent is liquid on the equilibrium path where the agent tells the truth. The intuition is simply that the outcome when the agent is illiquid is equivalent to not trading the good (x=0) and making no payments (t=0), which keeps the agent just liquid. Thus, the outcome of a mechanism γwhere the agent becomes illiquid on the equilibrium path can be replicated by the mechanism which differs from γonly in that it specifies no trade and zero payments for any contingency where the agent becomes illiquid on path under γ. Second, it is without loss to focus on mechanisms in which the agent exactly breaks even in the first period, and backloads any potential profit for the agent in that it accrues only in the second period.11 The reason is that if the agent were to make a profit in the first period, the principal could deduct it from the agent’s first period 1 payments and pay it out in period 2 instead. This would not affect the principal’s profit and would maintain truth-telling incentives for which only total payments matter. 11 This argument also appears in Ashlagi et al. (2023). 123 1428 D. Krähmer, R. Strausz We summarize these considerations in the next lemma. Lemma 1 For any feasible contract there is a payoff-equivalent feasible contract (x1,t1,xL 2,tL 2,xI 2,tI 2)with the following properties: •The agent exactly breaks even, and is never illiquid in period 1 (on path): t1(θ1)−θ1x1(θ1)=0for all θ1.(12) •The agent is never illiquid in period 2 (on path): tL 2(θ1,θ 2)−θ2xL 2(θ1,θ 2)≥0for all θ1,θ 2.(13) •After the off-path event that the agent becomes illiquid in period 1, the relationship is terminated: xI 2(θ1,θ 2)=tI 2(θ1,θ 2)=0for all θ1,θ 2.(14) Lemma 1implies that we can find an optimal contract in the class of feasible contracts that satisfy (12)-(14). Since properties (12) and (14)pindownt1,xI 2, and tI 2, we are actually left to determine only the triple (x1,xL 2,tL 2). We therefore introduce the following definition. Definition 2 Atriple(x1,x2,t2)with x1:1→[0,1],x2:1×2→[0,1], and t1:1×2→Ris called a backloaded contract if L2:t2(θ1,θ 2)−θ2x2(θ1,θ 2)≥0∀θ1,θ 2.(15) A backloaded contract uniquely induces a contract (x1,t1,xL 2,tL 2,xI 2,tI 2)with the properties (12)-(14) by setting t1=θ1x1,xL 2=x2,tL 2=t2, and xI 2=tI 2=0. For a backloaded contract, we write U(θ1)=¯ θ2 θ2 t2(θ1,θ 2)−θ2x2(θ1,θ 2)dF 2(θ2)(16) for the agent’s expected period 2 utility. The next lemma characterizes when a backloaded contract is feasible. Lemma 2 A backloaded contract (x1,x2,t2)induces a feasible contract (x1,t1, xL 2,tL 2,xI 2,tI 2)if and only if IC2:t2(θ1,θ 2)−θ2x2(θ1,θ 2)≥t2(θ1,ˆ θ2)−θ2x2(θ1,ˆ θ2)∀θ1,θ 2,ˆ θ2(17) IC1:U(θ1)≥(ˆ θ1−θ1)x1(ˆ θ1)+U(ˆ θ1)∀θ1<ˆ θ1(18) IC0 1:U(θ1)≥U(ˆ θ1)∀ˆ θ1∈0 1,∀θ1∈1,(19) where 0 1={θ∈1|x1(θ) =0}is the set of types who do not trade in period 1. 123 Dynamic screening with liquidity constraints 1435 We now solve R1and then show that its solution also solves P 1. We proceed in two steps. We first show that at a solution to R1, trade never happens if it is inefficient, and the constraint ICLis binding. In the second step, we use these properties to establish asolutiontoR1. To establish the first step, let be the (non-empty) feasible set for problem R1. We then obtain the following result. Lemma 5 Let (˜x,˜ U)∈. Then there is (x,U)∈which delivers the principal a (weakly) higher profit than (˜x,˜ U)and has the following properties: (i) If v<¯ θ, then x(θ) =0for all θ>v. (ii) U(θ) =−x(θ) for all θ. The first part makes the familiar point that an optimal contract induces a downward distortion. To understand the second part, recall that is concave with a maximum at USB. For a given trading probability x, the principal therefore seeks to choose Uas closely as possible to USB while maintaining the incentive compatibility requirement that U(θ) ≤−x(θ). Thus, an optimal choice of Uis maximally flat, implying that U(θ) =−x(θ). We emphasize that although property (ii) corresponds to the revenue equivalence property from standard screening models where IC is required for all reports ˆ θ,in our setting, property (ii) expresses an optimality rather than a feasibility condition. In standard screening models, property (ii) is useful, because it pins down the agent’s utility Uas an integral over the trading probability x. If, in addition, is linear, an integration by parts argument can be used to replace Uin the objective function of (35), and the problem can then be solved by point-wise maximization over x(θ).In our case, because is concave, this approach does not work. Our alternative approach is to instead use property (ii) to replace the trading probability xby the agent’s utility function Uin the objective function of (35) and then maximize over U. This allows us to show that an optimal contract is in the class of cutoff-contracts where the good is traded if and only if that agent’s cost is below a cutoff θ0∈[θ. ¯ θ]. Definition 3 A cutoff-contract (x,U)is characterized by two parameters: a cutoff θ0∈[θ, ¯ θ]and an intercept U0≥θ0−θsuch that x(θ) =1if θ≤θ0 0else ,U(θ) =U0−(θ −θ) if θ≤θ0 U0−(θ0−θ) else.(36) We denote by the set of cutoff contracts. Clearly, ⊂. We now state the main result of this section that, under a regularity condition, a cutoff-contract is a solution to the relaxed problem R1. Proposition 1 Let (v −θ) f(θ) f(θ) be increasing on the range [θ, min{v, ¯ θ}]. Consider (˜x,˜ U)∈. Then there is a cutoff-contract (x,U)∈which delivers a (weakly) higher profit than (˜x,˜ U). While we prove the proposition in the appendix, the underlying logic is best understood in the context of our uniform example. Note that the uniform example satisfies the regularity condition trivially, as f(θ) =0. 123 1436 D. Krähmer, R. Strausz Fig. 1 The left panel illustrates, given ˜ Uand that θis uniformly distributed over [0,1], the construction of the cutoff contract U(.) such that U0=˜ U(0)and 1 0U(θ) dθ=1 0˜ U(θ) dθ. The right panel shows the associated probability distributions FUand F˜ Uof Uand ˜ Uin utility space. F˜ Uis a mean preserving spread of FU Example: Consider some (˜x,˜ U)∈. As indicated earlier, we can use part (ii) of Lemma 5to replace ˜xby ˜ Uin the objective of (35). Using integration by parts and v=1, the objective then rewrites as ¯ θ θ[v−θ]˜x(θ) +( ˜ U(θ)) dF(θ) =1 0−[v−θ]˜ U(θ) +( ˜ U(θ)) dθ(37) =˜ U(0)+1 0˜ U(θ) dθ+1 0 ( ˜ U(θ)) dθ. (38) We now construct a function Ubelonging to a cutoff-contract for which expression (38) is at least as large as for ˜ U. To do so, note that Lemma 5implies that ˜ Uis a decreasing continuous function with a slope between −1 and 0. Therefore, because under a cutoff-contract, Uhas slope −1 up to the cutoff θ0and then slope 0, an intermediate value argument implies that we can find Uso that U0=˜ U(0), 1 0 U(θ) dθ=1 0˜ U(θ) dθ. (39) In particular, there is a ˜ θ∈[0,1]so that U(θ) ≤˜ U(θ) for θ≤˜ θand U(θ) ≥˜ U(θ) for θ≥˜ θ. (40) The first panel of Fig.1illustrates the construction graphically. By (39), the first two terms in (38)arethesameforUand ˜ U. The key idea to analyze the third term in (38) is to interpret the agent’s utility as a random variable which induces a probability distribution in utility space (the pushforward). Formally, and as illustrated in the second panel of Fig. 1, the distributions induced by ˜ Uand U 123 Dynamic screening with liquidity constraints 1437 correspond to the cumulative distribution functions F˜ U(u)=Pr(θ ∈[0,1]: ˜ U(θ) ≤u)and FU(u)=Pr(θ ∈[0,1]:U(θ) ≤u). (41) The key observation is now that the second part of (39) and (40) imply that F˜ Uis a mean preserving spread of FU. Therefore, because is concave, the third term in (38) is larger for Uthan for ˜ U. For the general case without uniform distribution, the construction is analogous. The mean preserving spread argument carries over unchanged. The role of the regularity condition is to sign what corresponds to the first and second terms in (38), since these terms depend in general on the density f. The regularity condition in Proposition 1is not entirely new to the literature. In a context where the principal is a seller and the agent is a buyer, Manelli and Vincent (2006, Theorem 4) impose an equivalent regularity condition when characterizing the profit maximizing solution in a multi-dimensional screening problem. A sufficient condition for the regularity condition is that jointly f≤0 and fis log-convex.19 Examples include the uniform distribution of our example, and, more generally the family of power distributions F(θ) =θα,θ∈[0,1],forα≤1, or the family of exponential distributions F(θ) =1−e−λθ ,θ≥0, λ≥0. While our regularity condition is more restrictive than other regularity conditions often found in mechanism design (such as monotone hazard rates), we stress that the condition is not a tight, but only a sufficient condition that ensures the optimality of a cutoff-contract and thus a deterministic allocation in period 1. In general, if our regularity condition is violated, solving R1becomes complicated for two reasons: first, the incentive constraints do not rule out non-monotone allocations. Second, even if one could show that a monotone allocation is optimal, the principal’s problem is not a linear problem, and the trade-off between period 1 profits (which are linear in U) and period 2 profits (which are concave in U) does generally lead to “interior” solutions that do not correspond to allocations with possibly multiple cutoffs. As mentioned above, the regularity condition is needed to control the sign of the principal’s first period profit. Therefore, our results go through without the regularity condition whenever second period profits are sufficiently higher than first period profits, for example, when the period 2 project has a much larger scale than the period 1 project.20 19 To see this, note d dθ(v −θ) f(θ) f(θ) =−f(θ) f(θ) +(v −θ) d dθ f(θ) f(θ) =−f(θ) f(θ) +(v −θ) d dθlog(f(θ)). (42) Because v−θis positive on the range [θ, min{v, ¯ θ}], this expression is postive if f≤0andlogfis increasing, that is, fis log-convex. 20 More precisely, scale up the second period by multiplying the valuation v2and costs θ2by a factor λ≥1 so that the principal’s continuation profit is increasing in λ.Asshownin(99) in the appendix, the difference between the principal’s profit from an arbitrary and a cutoff contract becomes W=W1+W2 with W1=v θ[(v −θ) f(θ) f(θ) −1][ ˜ U(θ) −ˆ U(θ)]dF(θ) and W2=¯ θ θ( ˆ U(θ)) −( ˜ U(θ)) dF(θ). 123 1438 D. Krähmer, R. Strausz Proposition 1shows that a cutoff-contract is a solution to the relaxed problem R1. It is straightforward to verify that any cutoff-contract satisfies the constraints IC of the original problem. Therefore, we have: Proposition 2 Let (v −θ) f(θ) f(θ) be increasing on the range [θ, min{v, ¯ θ}], then there is a cutoff-contract (x,U)∈which solves problem P 1. Moreover, a cutoff contract also satisfies constraint IC0. Thus, it is a solution to the original problem P. Since a cutoff-contract consists only of the two parameters θ0,U0, finding the optimal cutoff-contract comes down to solving an optimization problem in two variables. We illustrate this exercise in our running example. Example: For our uniform example, the principal’s objective is W(θ0,U0)=θ0 0 1−θdθ+θ0 0 (U0−θ) dθ+1 θ0 (U0−θ0)dθ(43) with U0≥θ0and where is given by (28). To determine the maximizer, we first determine an optimal θ∗ 0(U0)for a given U0. A tedious but otherwise straightforward analysis of the first and second order condition with respect to θ0yields21 θ∗ 0(U0)=U0−1/18.(46) Next, we maximize W(θ∗ 0(U0), U0)=W(U0−1/18,U0)with respect to U0.For U0≤1/2, this expression reduces to 109/648 +(5+4√2U0−9U0)U0/6 which is strictly increasing for U0≤1/2 so that a maximum exhibits U0≥1/2. For U0>1/2, the expression W(U0−1/18,U0)reduces to the quadratic expression 41/324 +4/3·U0−U2 0which attains a maximum at U0=2/3. We therefore conclude that (θ∗ 0,U∗ 0)=(11/18,2/3)maximizes W(θ0,U0)with a payoff of 185/324 ≈0.571, exceeding by 14% the principal’s payoff of 2SB =1/2, from charging twice the static optimal price p=1/2. Recall from above that in the uniform example, the period 2 terms of trade can be implemented by offering the agent a period 2 price p2=√2Ufor U≤1/2, and Our stochastic dominance argument implies that for any distribution, we have W2>0. The regularity condition implies that W1>0 so that in this case, Wis always positive for any λ. If the regularity conditions fails, then we have W2>0 but may have W1<0. In this case, Wis positive when W2>0 is sufficiently large, which is the case if λis sufficiently large. 21 The first order condition with respect to θ0is: ∂W ∂θ0=(1−θ0)(1−(U0−θ0)) =0⇔θ0=1or(U0−θ0)=1.(44) It is easy to check that θ0=1 is not a maximizer of W.By(28), the unique solution to (U0−θ0)=1 is θ0=U0−1/18. This is indeed a maximizer of W(θ0,U0), because the second order condition is ∂2W ∂θ2 0=−1+(U0−θ0)+(U0−θ0)(1−θ0)<0,(45) is satisfied for θ0=U0−1/18, since the first two terms cancel, while (U)=− √2U−3/2/4<0for U≤1/2andU0−θ0=1/18 <1/2. 123 Dynamic screening with liquidity constraints 1439 Fig. 2 The table in the left panel presents the payoff comparisons of the four cases i) first best (FB); ii) static second best (sSB); iii) no liquidity constraints (no LC); and iv) liquidity constraints (LC). The graph in the right panel displays how optimal production (x1,x2)over the two periods depend on type combinations (θ1,θ 2), splitting the type space [0,1]×[0,1]into four regions, where the curve p2(θ1)is the price that is paid to the agent for production in period 2 after having reported θ1in period 1 p2=U+1/2forU>1/2. With this in mind, period 1 cost types θabove the cutoff θ∗ 0=11/18 do not produce in period 1 and obtain expected period 2 utility of U(θ) =U∗ 0−θ∗ 0=2/3−11/18 =1/18 corresponding to a period 2 price offer p2=1/3. All period 1 cost types θbelow the cutoff θ∗ 0=11/18 produce in the first period and obtain expected period 2 utilityU(θ) =U∗ 0−θ=2/3−θ. This corresponds to a period 2 price offer p2(θ1)=U(θ1)+1/2=7/6−θ1for θ1∈[0,1/6), and p2(θ1)=√2U(θ1)=√4/3−2θ1for θ1∈[1/6,11/18). Interestingly, period 1 cost types θ<1/6 obtain a continuation utility Ularger than 1/2. These types always produce in period 2, since they receive the offer to produce at a price larger than 1 in period 2. In particular, the principal suffers a period 2 loss in this case. The ex ante expected utility of the agent is 157/648 so that expected aggregate surplus is 185/324+157/648 =527/648 ≈0.813, compared to the first best surplus of 1. Without liquidity constraints, aggregate surplus is 7/8=.875, while the twicely repeated static second best contract yields aggregate surplus of.75. The table in the left panel of Fig.2summarizes payoffs for the different types of models. The graph in the right panel displays the dependence of optimal production over the two periods on the first and second period type combinations (θ1,θ 2). At our optimum, period 1 types produce if and only if θ1≤11/18, whereas period 2 types produce if and only if θ2≤p2(θ1), i.e., when their costs lies below the price p2(θ1). As the graph illustrates, this yields four different regions. While the aggregate trade volume slightly increases from 11/18 in period 1 to 50/81 in period 222, we point out that conditional on cost type, trade does not become more efficient: for example, the first period type θ1=11/18 −always trades, but the same second period type θ2=11/18 −does not. In this sense, it is not clear-cut in which of the periods the allocation is more efficient.  Remark 2 (Implementation) We now briefly discuss how an optimal cutoff contract can be indirectly implemented by a menu of prices. For simplicity, suppose that the 22 The trade volume in period 2 is 1/6+11/18 1/6√4/3−2θ1dθ1+7/18 ×1/3. 123 1440 D. Krähmer, R. Strausz optimal period 2 terms of trade can be implemented by a posted price. Recall from Remark 1 that this is the case if, for example, F2/f2is increasing. An optimal contract can then be implemented by a menu {(r,p2(r)) |r∈[θ,θ 0]} where the agent can choose to produce the good in period 1 for a price rand conditional on not going bankrupt in period 1, obtains the option to produce the good in period 2 for the price p2(r)where p2is decreasing in r. Moreover, if the agent goes bankrupt in period 1, the relationship is terminated. To see this, recall that under a backloaded contract, the agent breaks even in period 1. Under a cutoff contract, the agent therefore receives in period 1 the transfer ˆ θ1 and produces the good if he announces ˆ θ1∈[θ,θ 0]and stays liquid. If he announces ˆ θ1∈(θ0,¯ θ], he receives the transfer 0 and does not produce the good. This corresponds to choosing a price r=ˆ θ1∈[θ,θ 0]at which to deliver the good in period 1. Moreover, after announcing ˆ θ1, the agent obtains expected utility U(ˆ θ1)in period 2 which can be implemented by a posted price p2(ˆ θ1)which is decreasing in ˆ θ1because U(ˆ θ1) is decreasing in ˆ θ1. This corresponds to obtaining the option to produce the good at p2(r)=p2(ˆ θ1)in period 2 after choosing the price rin period 1. Remark 3 (Finitely many types) A novelty of our paper is that we study continuous types. Ashlagi et al. (2023), like us, consider a linear, unit-good framework but with finitely many types and where trade is always efficient. Their analysis shows that the case with more than two finitely many types is analytically intractable. When there are two types, any contract that is feasible (with their liquidity constraint) is a cut-off contract by definition, and their Proposition 4 shows that a variety of deterministic contracts can be optimal, among them non-dynamic posted-price contracts where the second period posted-price is independent of the first period report. Our analysis suggests that this is a special feature of the two types case since with continuous types the second period posted price always depends on the first period report (see the previous remark). Remark 4 (Off-path liquidity constraints) As explained before Sect.4, the approach of the existing literature differs from our approach in that it considers the bi-directional version of IC1and not having IC0 1in problem P. Therefore, in the recursive formulation, the period 2 problem is the same under both approaches. Moreover, since we consider a relaxed version without IC0 1, the period 1 problem the literature considers is precisely problem P1with the difference that IC1is replaced by its bi-directional counterpart. Now, the solution to P1exhibits the revenue equivalence property U(θ) =−x(θ) (by Lemma 5) and, as a cutoff contract, displays a monotone allocation. Therefore, the solution indeed satisfies the bi-directional counterpart of IC1and is thus also a solution to the problem studied in the literature. When considering the bi-directional version of IC1, however, the following inconsistency arises: consider a high cost type θ>θ 0(who does not produce the good) and a low cost type ˆ θ<θ 0(who does produce the good). Type θ’s continuation value is lower than that of type ˆ θ:U(θ) < U(ˆ θ). Imposing the bi-directional version of IC1 implies that type θcould obtain this higher continuation value by reporting ˆ θ. Therefore, what ensures bi-directional incentive compatibility is that one needs to assume that type θwould suffer a period 1 loss of t(ˆ θ) −θ<0 from reporting to be type ˆ θ. Clearly, this is inconsistent with the agent being periodically liquidity constraint. In our approach, this inconsistency does not arise, because what ensures incentive 123 Dynamic screening with liquidity constraints 1441 compatibility is that type θwould become illiquid when reporting to be type ˆ θ, and then, as this is verifiable, obtain a continuation value of zero. Remark 5 (More than two periods) While we performed our analysis only for two periods, the extension to multiple periods is straightforward. To illustrate, suppose that there are infinitely many periods and that cost types θτare i.i.d. with time-independent cdf Fon the support [θ,¯ θ].23 For the problem to be well-defined, assume that both parties discount future payoffs with a discount factor δ∈[0,1). Under the dynamic programming formulation, the principal’s choice variables are a probability of trade x(θ) for the current period and the expected continuation utility for the agent U(θ) that both depend on a report θby the agent about his current type (as well as on the history of past reports which we suppress). The principal’s value function (V)is now defined recursively as a function of the agent’s expected utility V(starting as of now) according to the dynamic program24: P∞:(V)=max x,U¯ θ θ (v −θ)x(θ) +δ(U(θ)) dF(θ) s.t.(47) IR:U(θ) ≥0∀θ(48) IC :U(θ) ≥(ˆ θ−θ)x(ˆ θ)+U(ˆ θ) ∀θ≤ˆ θ(49) UG :x(θ) ∈[0,1]∀θ(50) PK :¯ θ θ δU(θ)dF(θ) =V.(51) While problem P∞yields the principal’s value function, the solution to the principal’s overall problem starting in the initial period is obtained by maximizing with respect to V. The essential difference between P∞and P 1is the presence of the promise keeping constraint PK which ensures that the agent’s expected utility from the contract is V. As above, we consider a relaxed problem where we localize IC and replace it with the constraints Mand ICLas stated in Lemma 4: R∞:˜ (V)=max x,U¯ θ θ (v −θ)x(θ) +δ˜ (U(θ)) dF(θ) s.tIR,M,ICL,UG,PK.(52) It follows from standard arguments (see Stokey et al. 1989 or Krishna et al. 2013) that ˜ exists. Crucially, as in the two-period case, ˜ is concave. Recall that to establish the optimality of a cutoff contract for the two-period problem R1, we exploited the concavity of ˜ to construct for a every feasible contract (˜x,˜ U)a feasible cutoffcontract (x,U)that is an improvement. Note that, in contrast to problem R1, feasibility 23 The extension to an arbitrary finite time horizon is analogous but all expressions are time-dependent. 24 The formulation implicitly assumes that contracts are backloaded and excess payments are paid out “at infinity”. This assumption is standard in the literature, and one motivation of it is that the discount factor corresponds to the probability that the relationship does not terminate in the next round, and excess payments are made after termination (which happens in finite time with probability 1). 123 1442 D. Krähmer, R. Strausz in problem R∞requires that a contract, in addition, satisfies PK. Therefore, to extend the argument from R1to R∞, we have to ensure that the cutoff contract (x,U)that improves a given feasible contract does satisfy PK. However, note that the cutoff contract (x,U)constructed in the two-period problem to improve upon (˜x,˜ U)has the property that25 ¯ θ θ U(θ)dF(θ) =¯ θ θ˜ U(θ)dF(θ). (53) Therefore, as (˜x,˜ U)is an arbitrary feasible contract and thus satisfies PK by definition, so does (x,U). This shows that a cutoff contract is optimal also when there are more than two periods. Remark 6 (Portability) We have solved for an optimal contract using a new method that ranks contracts in terms of the spread of the distribution of the induced continuation values for the agent. An open question is to what extent this method can be employed in a model with correlated cost types (as in Krasikov and Lamba 2021). Such an extension is beyond the scope of the current paper because it implies that the agent’s continuation value becomes type dependent, thus constraining the principal’s choice of continuation values. Outside of the context of this paper, our method is applicable to static mechanism design problems with standard bilateral incentive constraints where agents have linear utility functions and the principal’s payoff is concave in the agent’s information rent. An example is optimal redistribution by a social planner who assigns (after-tax) payments tand (pre-tax) “labour income” xto each of a continuum of agents’ who each privately know their labour cost θ. The social planner seeks to maximize a social welfare function (U(θ)) dF(θ) subject to the budget constrained that after-tax payments are lower than pre-tax income in the aggregate. The concavity of captures the planner’s redistribution concerns. Our solution method is applicable when agents’ preferences are linear in type and transfer. In this case, the (slope of) the indirect utility Uis a function of the allocation, and analogous steps as above can be used to write the planner’s problem as a constrained maximization problem over the indirect utility function where the (Langrangian) objective ranks indirect utility functions depending on how spread out they are. We leave a detailed analysis for future research. 5 Conclusion We study short-term liquidity constraints in an otherwise standard dynamic screening model. We argue that modelling a liquidity constraint as a hard, physical constraint whose violation forces the agent to renege on the current contract terms implies that such a violation must be seen as a verifiable event on which a long-term contract can condition. We show how this yields a consistent framework in which liquidity concerns affect contractual feasibility constraints by giving rise to one-sided incentive compatibility constraints. 25 This corresponds to the right part of (39) where we defined (x,U)in the uniform example. 123 Dynamic screening with liquidity constraints 1443 While our assumption that the agent runs into liquidity problems results whenever the agent makes short term losses is in line with standard approaches, in practice the occurrence and consequences of liquidity problems may be more complicated than that, since they may, for example, be partially discretionary or involve restructuring processes which would affect the extent to which liquidity problems are verifiable. It is an interesting avenue for future research to capture such richer forms of liquidity concerns. Appendix Proof of Lemma 1Let ˜γ=(˜x1,˜ t1,˜xL 2,˜ tL 2,˜xI 2,˜ tI 2)be a feasible contract. Our proof strategy is to first define an auxiliary contract ˆγthat is feasible and payoff-equivalent to ˜γbut under which the agent never becomes illiquid. In a second step, we modify ˆγto obtain the desired contract γthat has the properties stated in the lemma. In what follows, we indicate all variables pertaining to ˜γand ˆγwith a tilde and a hat. Step 1: Define the auxiliary contract ˆγ=(ˆx1,ˆ t1,ˆxL 2,ˆ tL 2,ˆxI 2,ˆ tI 2)by (ˆx1(θ1), ˆ t1(θ1)) =(˜x1(θ1), ˜ t1(θ1)) if θ1∈˜ L 1, (0,0)otherwise (54) (ˆxL 2(θ1,θ 2), ˆ tL 2(θ1,θ 2)) =⎧ ⎨ ⎩ (˜xL 2(θ1,θ 2), ˜ tL 2(θ1,θ 2)) if θ1∈˜ L 1,θ 2∈˜ L,L 2(θ1) (˜xI 2(θ1,θ 2), ˜ tI 2(θ1,θ 2)) if θ1/∈˜ L 1,θ 2∈˜ I,L 2(θ1) (0,0)otherwise, (55) (ˆxI 2(θ1,θ 2), ˆ tI 2(θ1,θ 2)) =(0,0)∀θ1,θ 2.(56) We show that ˆγis feasible and payoff-equivalent to ˜γ. To see this, note first that, by construction, we have ˆ L 1=1and ˆ L,L 2(θ1)=2for all θ1. Furthermore, ˆ UL(θ1)=˜ UL(θ1)for θ1∈˜ L 1and ˆ UL(θ1)=˜ UI(θ1)for θ1/∈˜ L 1.(57) To see feasibility, observe that ˆγtrivially satisfies (4)for=I, and inherits (4)for =Lby construction. To see (5), let ˆ t1(ˆ θ1)−θ1ˆx1(ˆ θ1)≥0. Consider first the case that θ1∈˜ L 1and ˆ θ1∈˜ L 1. Then, we have: ˆ t1(θ1)−θ1ˆx1(θ1)+ˆ UL(θ1)=˜ t1(θ1)−θ1˜x1(θ1)+˜ UL(θ1)(58) ≥˜ t1(ˆ θ1)−θ1˜x1(ˆ θ1)+˜ UL(ˆ θ1)(59) =ˆ t1(ˆ θ1)−θ1ˆx1(ˆ θ1)+ˆ UL(ˆ θ1), (60) where the inequality follows, because ˜γsatisfies (5) and the two equalities follow from (57). The other cases can be shown analogously. To see (6), note that the left hand side of (6) is non-negative by definition of ˆγ. Moreover, because ˆxI 2=ˆ tI 2=0, we have ˆ UI(ˆ θ1)=0 for all ˆ θ1so that the right hand 123 1444 D. Krähmer, R. Strausz side is zero. Therefore, (6) follows. To complete the proof of feasibility, note that (7) and (8) are void for ˆγ, because ˆ L 1=1. Finally, ˆγand ˜γare payoff-equivalent, because by construction, if the agent is liquid under ˜γ, then ˆγimplements the same terms of trade as ˜γ, and when the agent becomes illiquid under ˜γ, no trade occurs under ˆγso that under either contract both the principal and the agent get zero. Step 2: We now construct a feasible contract γ=(x1,t1,xL 2,tL 2,xI 2,tI 2)which is payoff-equivalent to ˆγand satisfies (12)-(14). To do so, note first that ˆγsatisfies (13) and (14), but may violate (12) and display ˆ t1(θ1)−θ1ˆx1(θ1)>0forsomeθ1. Define γas the contract that differs from ˆγonly in that the period 1 profits for the agent are backloaded to period 2. Formally, γdisplays x1=ˆx1,xL 2=ˆxL 2,xI 2= ˆxI 2,tI 2=ˆ tI 2and payments t1(θ) =θ1x1(θ1), tL 2(θ1,θ 2)=ˆ tL 2(θ1,θ 2)+ˆ t1(θ1)−t1(θ1). (61) Note first that γsatisfies (12) by construction. Moreover, it inherits (14)from ˆγand also property (13) because tL 2(θ1,θ 2)−θ2xL 2(θ1,θ 2)=ˆ tL 2(θ1,θ 2)+ˆ t1(θ1)−t1(θ1)−θ2ˆxL 2(θ1,θ 2)(62) =ˆ tL 2(θ1,θ 2)−θ2ˆxL 2(θ1,θ 2)+ˆ t1(θ1)−θ1ˆx1(θ1)≥0, (63) where the inequality follows since under ˆγ We next show that γis feasible. Indeed, γtrivially satisfies (4)for=Ibecause xI 2=tI 2=0. For =L, we have for all θ1,θ 2,ˆ θ2: tL 2(θ1,θ 2)−θ2xL 2(θ1,θ 2)=ˆ tL 2(θ1,θ 2)+ˆ t1(θ1)−t1(θ1)−θ2xL 2(θ1,θ 2)(64) ≥ˆ tL 2(θ1,ˆ θ2)+ˆ t1(θ1)−t1(θ1)−θ2xL 2(θ1,ˆ θ2)(65) =tL 2(θ1,ˆ θ2)−θ2xL 2(θ1,ˆ θ2), (66) where the first and the third lines use the definition of tL 2, and the second line follows because ˆγsatisfies (4)for=Land since xL 2=ˆxL 2. To see (5), consider θ1,ˆ θ1so that t1(ˆ θ1)−θ1x1(ˆ θ1)≥0. Because ˆ t1(ˆ θ1)≥t1(ˆ θ1) and ˆx1(ˆ θ) =x1(ˆ θ), this implies that also ˆ t1(ˆ θ1)−θ1ˆx1(ˆ θ1)≥0. Therefore, since ˆγ satisfies (5), we have ˆ t1(θ) −θ1ˆx1(θ1)+ˆ UL(θ1)≥ˆ t1(ˆ θ)−θ1ˆx1(ˆ θ1)+ˆ UL(ˆ θ1). (67) Moreover, by construction, we have that t1(θ1)+UL(θ1)=ˆ t1(θ1)+ˆ UL(θ1). These two observations imply that t1(θ1)−θ1x1(θ1)+UL(θ1)=ˆ t1(θ) −θ1ˆx1(θ1)+ˆ UL(θ1)(68) ≥ˆ t1(ˆ θ)−θ1ˆx1(ˆ θ1)+ˆ UL(ˆ θ1)(69) =t1(ˆ θ)−θ1x1(ˆ θ1)+UL(ˆ θ1). (70) 123 Dynamic screening with liquidity constraints 1451 contract with cutoff θ0=ˆ θand an intercept U0∈[ˆ U(v) +ˆ θ−θ, ˆ U(θ)]such that26 ¯ θ θ U(θ) dF(θ) =¯ θ θˆ U(θ) dF(θ). (108) This also implies that U(θ) −ˆ U(θ) ≤0∀θ≤vand U(θ) −ˆ U(θ) ≥0∀θ≥v. (109) Because θ0=ˆ θimplies x=ˆx, the difference in the principal’s profit from (x,U)and (ˆx,ˆ U)can be written as W(x,U)−W(ˆx,ˆ U)=¯ θ θ (v −θ)[x(θ) −ˆx(θ)]+(U(θ)) −( ˆ U(θ)) dF(θ) =¯ θ θ (U(θ)) −( ˆ U(θ)) dF(θ). (110) Similarly to the argument at the end of the first step, (108) and (109) imply that Fˆ Uis a mean preserving spread of FU, and hence (110) is positive, and this completes the proof.  Proof of Proposition 2Note first if there is a solution (x,U)∈to the relaxed problem R1, then because (x,U)∈is obviously feasible for the problem P 1,itisalsoa solution to P 1. Moreover, any contract (x,U)∈satisfies the constraint IC0and thus a solution (x,U)∈to P 1is also a solution to P. To see this, observe that for (x,U)∈we have that 0 1=(θ0,¯ θ]by (36). To show IC0, we thus have to show that U(θ) ≥U(ˆ θ) for all ˆ θ∈(θ0,¯ θ]and θ∈. But this is immediate from the definition of Uin (36). It remains to show existence of a solution (x,U)∈to R1. For this recall that a cutoff contract is characterized by cutoffs θ0∈[θ, ¯ θ]and U0≥θ0−θ. We first show the auxiliary claim that for any (˜x,˜ U)∈there is a (x,U)∈which yields a (weakly) higher profit than (˜x,˜ U)and has the property that U0≤USB +(¯ θ−θ). (111) Indeed, consider a (˜x,˜ U)with cutoffs (˜ θ0,˜ U0)that violates (111). Since (˜x,˜ U)is a cutoff contract, this implies that ˜ U(θ) > USB for all θ. Define (x,U)∈with cutoffs θ0=˜ θ0,U0=˜ U0−(˜ U(¯ θ)−USB). (112) 26 Given θ0=ˆ θ, the cutoff U0exists by the intermediate value theorem, because the integral on the left hand side of (108) is strictly larger than the right hand side for U0=ˆ U(θ), strictly lower for U0=ˆ U(v) +ˆ θ−θ, and changes continuously in U0. 123 1452 D. Krähmer, R. Strausz By construction, we have that USB ≤U(θ) ≤˜ U(θ) for all θ. Thus, because is concave and uniquely maximized at USB by Lemma 3, this implies that (U(θ)) ≥ ( ˜ U(θ)) for all θ. Therefore, and since x=˜x, we obtain the profit W(x,U)=(v −θ)˜x(θ) +(U(θ)) dF(θ) (113) ≥(v −θ)˜x(θ) +( ˜ U(θ)) dF(θ) =W(˜x,˜ U), (114) and this proves the auxiliary claim. Now, let ¯ be the set of cutoff contracts that satisfy (111). That is, (x,U)∈¯  if we can express (x,U)as a cutoff contract with cutoff θ0∈[θ, ¯ θ]and intercept U0∈[θ0−θ, USB +(¯ θ−θ)]. The auxiliary claim and Proposition 1then imply that there is a solution (x,U)∈ to R1if there is a solution to the problem Q:max (x,U)W(x,U)s.t.(x,U)∈¯ . (115) Because the profit W(x,U)of a cutoff contract is pinned down by (θ0,U0), problem Qboils down to the problem of choosing a two-dimensional variable (θ0,U0)from the compact set [θ,¯ θ]×[θ0−θ, USB +(¯ θ−θ)]. Because profit is continuous in (θ0,U0), there is a solution to Q. Therefore, there is a solution (x,U)∈to R1, and this completes the proof.  Funding Open Access funding enabled and organized by Projekt DEAL. Daniel Kähmer acknowledges financial support from the German Research Foundation (DFG) through Germany’s Excellence Strategy EXC 2126/1–390838866 and CRC TR 224. Roland Strausz acknowledges funding by the European Union (ERC, PRIVDIMA, 101096682) and from the German Research Foundation (DFG) through CRC TRR 190 (No. 280092119). Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. 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